Stability and Time-Optimal Control of Hereditary Systems
This isVolume 188 in MATHEMATICS IN SCIENCE AND ENGINEERING Edited by William E Ames, Georgia Institute of Technology A list of recent titles in this series appears at the end of this volume.
STABILITY AND TIME-OPTIMAL CONTROL OF HEREDITARY SYSTEMS E.N Chukwu DEPARTMENT OF MATHEMATICS NORTH CAROLINA STATE UNIVERSITY RALEIGH, NORTH CAROLINA
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Library of Congress Cataloging-in-Publication Data Chukwu, Ethelbert N. Stability and time-optimal control of hereditary systems / E.N. Chukwu. cm. - (Mathematics in science and engineering; v. 188) p. Includes bibliographical references and index. ISBN 0-12-174560-0 1. Control theory. 2. Mathematical optimization. 3. Stability. I. Title. 11. Series. QA402.3.C5567 1992 91-37886 629.8 312-dc20 CIP
PRINTED IN THE UNITED STATES OF AMERICA 92939495
98765432 1
DEDICATION It is with gratitude and great joy that I dedicate this book to the following good and courageous men and women whose faith in my family sustained us through our recent difficulties. My entire family, including my wife Regina, and our children Ezeigwe, Emeka, Uchenna, Obioma, Ndubusi, and Chika are most humbled by the friendship from so many. Our trials were very harrowing, but with the support and belief of all these fine women and men, our faith in God and belief in the principles of truth and justice were reaffirmed. Dr. A. Schafer (Chair, Committee of American Mathematical Society, Human Rights of Mathematicians, American Mathematical Society), Dr. William Browder (Princeton University) President , American Mathematical Society), Dr. M. Brin, Professor 0. Hzijek, Professor G. Leitmann, Professor R. Martin, Professor A. Fauntleroy, Professor E. Burniston, Dr. C. Christensen, Mr. R. O’Connor, Bishop F. Joseph Gossman, Father 0. Howard, Father G. Wilson, Mr. H. Ray Daley, Jr., Dr. J . Nathan, Mr. J. Downs, Mr. G. Berens and Mrs. Marie Berens, Professor T. Hallam, Dr. H. Liston, Dr. S. Lenhart, Dr. S. Middleton, Professor H. Petrea, Professor and Mrs. C. Wolf, Dr. G. Briggs, Dr. C. Jefferson, Reverend Fr. Doherty, Reverend Fr. J . Scannel, Mr. A. Macnair, Mr. J. Barry, Reverend M. P. Shugrue, Dr. B. Westbrook, Dr. D. M. Lotz, Ms. S. Mckenna, Ms. J. Hunt, Rev. G. Lewis, Rev. M. J. Carter, Mr. and Mrs. B. E. Spencer. My family and I are eternally grateful t o the following political leaders who provided support and assistance during a very difficult time that occurred while I was in the process of developing and writing this book. Without their support and assistance I could not have completed this work, and our difficulties would have overwhelmed us. The Honorable Terry Sanford, U.S. Senator for North Carolina. The Honorable Jesse Helms, U.S. Senator for North Carolina. The Honorable Albert Gore, U.S. Senator for Tennessee. The Honorable Clairborne Pell, U.S. Senator for Rhode Island. The Honorable David Price, U.S. Congressman for the 4th District, for North Carolina. The Honorable Rufus L. Edmisten, North Carolina Secretary of State. My family and I rejoice in your generosity, and thank you with all our hearts and souls. It is with humility that I pay tribute to my undergraduate teacher Professor Harold Ward and my Ph.D. supervisor and friend Professor 0. HAjek. Ethelbert Nwakuche Chukwu January, 1991 Raleigh, NC, USA
V
ACKNOWLEDGEMENTS The author is indebted t o Professors T. Angel1 and A. Kirsch for their kind permission to use their recent published paper. The author has consulted the work of Professors J. Hale, Haddock and Terjkki, T . Banks, M. Jacobs and Langenhop, A. Manitius, Drs. G . Kent and D. Salamon, to all of whom he is greatly indebted. He acknowledges the enduring inspiration of Professors 0. Hhjek, L. Cesari, H. Hermes and J. P. LaSalle, and R. Bellman. It was Bellman who invited the author to write this book. He is particularly appreciative of the joyous efforts of Mrs. Vicki Grantham in typing the manuscript. The author acknowledges with immense gratitude the support of NSF for the summer of 1991, which enabled him to complete some of the original work reported here. It was a pleasure to work with the professional staff of Academic Press, especially Charles B. Glaser and Camille Pecoul, and the author offers his gratitude to them.
Ethelbert Nwakuche Chukwu January, 1991 Raleigh, NC, USA
vi
CONTENTS xi
Preface
Chapter 1 Examples of Control Systems Described by Functional Differential Equations 1.1 1.2 1.3 1.4 1.5 1.6 1.7
Ship Stabilization and Automatic Steering Predator-Prey Interactions Fluctuations of Current Control of Epidemics Wind 'Alnnel Model; Mach Number Control The Flip-Flop Circuit Hyperbolic Partial Differential Equations with Boundary Controls 1.8 Control of Global Economic Growth 1.9 The General Time-Optimal Control Problem and the Stability Problem
Chapter 2 General Linear Equations
2.1 The Fundamental Matrix of Retarded Equations 2.2 The Variation of Constant Formula of Retarded Equations 2.3 The Fundamental Solution of Linear Neutral Functional Differential Equations 2.4 Linear Systems Stability 2.5 Perturbed Linear Systems Stability Chapter 3
Lyapunov-Razumikhin Methods of Stability of Delay Equations
3.1 Lyapunov Stability Theory 3.2 Razumikhin-type Theorems 3.3 Lyapunov-Razumikhin Methods of Stability in Delay Equations
vii
1 1 3 6 9 17 18 22 25 30
35
35 44 46 52 58
67
67 73 75
...
Con t e n t s
Vlll
Chapter 4 4.1 4.2 4.3 4.4
Global Stability of Functional Differential Equations of Neutral Type
Definitions Lyapunov Stability Theorem Perturbed System Perturbations of Nonlinear Neutral Systems
Chapter 5 Synthesis of Time-Optimal and Minimum-Effort Control of Linear Ordinary Systems 5.0 Control of Ordinary Linear Systems 5.1 Synthesis of Time-Optimal Control of Linear Ordinary Systems 5.2 Construction of Optimal Feedback Controls 5.3 Time-Optimal Feedback Control of Nonautonomous Systems 5.4 Synthesis of Minimum-Effort Feedback Control Systems 5.5 General Method for the Proof of Existence and Form of Time-Optimal, Minimum-Effort Control of Ordinary Linear Systems Chapter 6
Control of Linear Delay Systems
6.1 Euclidean Controllability of Linear Delay Systems 6.2 Linear Function Space Controllability 6.3 Constrained Controllability of Linear Delay Systems Chapter 7 Synthesis of Time-Optimal and Minimum-Effort Control of Delay Systems 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8
7.9
Linear Systems in Euclidean Space Geometric Theory and Continuity of Minimal Time Function The Index of a Control System Time-Optimal Feedback Control of Autonomous Delay Systems Minimum-Effort Control of Delay Systems Proof of Minimum-Effort Theorems Optimal Absolute Fuel Function Control of Nonlinear Functional Differential Systems with Finite Delay-Existence, Uniqueness, and Continuity of Solutions Sufficient Conditions for the Existence of a Time-Optimal Control
87 87 89 94 101 105 105 106 111 120 156
164 193 193 197 202 207 207 217 223 237 247 259 268
277 28 1
ix
Contents
7.10 Optimal Control of Nonlinear Delay Systems with Target in En 7.11 Optimal Control of Delay Systems in Function Space 7.12 The Time-Optimal Problem in Function Space Chapter 8 Controllable Nonlinear Delay Systems 8.1 Controllability of Ordinary Nonlinear Systems 8.2 Controllability of Nonlinear Delay Systems 8.3 Controllability of Nonlinear Systems with Controls Appearing Linearly Control of Interconnected-Delay Differential Equations in Introduction Nonlinear Systems General Nonlinear Systems Examples Control of Global Economic Growth Effective Solidarity Functions
Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6
w!)
Chapter 10 The Time-Optimal Control of Linear Differential Equations of Neutral Type 10.1 The Time-Optimal Control Problem of Linear Neutral F’unctional Systems in En:Introduction 10.2 Forcing to Zero 10.3 Normal and Proper Autonomous Systems 10.4 Pursuit Games and Time-Optimal Control Theory 10.5 Applications and Economic Growth 10.6 Optimal Control Theory of Linear Neutral Systems 10.7 The Theory of Time-Optimal Control of Linear Neutral Systems 10.8 Existence Results 10.9 Necessary Conditions €or Optimal Control 10.10 Normal Systems 10.11 The Geometric Theory of Time-Optimal Control of Linear Neutral Systems 10.12 Continuity of the Minimal-Time Functions 10.13 The Index of the Control System 10.14 Examples
288 29 1 299 303 303 308 32 1 337 337 34 1 345 349 355 367 37 3 373 382 385 388 397 400 408 412 419 422 423 426 432 433
X
Chapter 11 The Time-Optimal Control Theory of Nonlinear Systems of Neutral Type
11.1 Introduction 11.2 Existence, Uniqueness, and Continuity of Solutions of Neutral Systems 11.3 Existence of Optimal Controls of Neutral Systems 11.4 Optimal Control of Neutral Systems in Function Space Chapter 12 Controllable Nonlinear Neutral Systems
12.1 General Nonlinear Systems 12.2 Nonlinear Interconnected Systems 12.3 An Example: A Network of Flip-Flop Circuits
Contents 44 1 44 1 443 452 458 473 473 485 489
Chapter 13 Stability Theory of Large-Scale Hereditary Systems
495
13.1 Delay Systems 13.2 Uniform Asymptotic Stability of Large-Scale Systems of Neutral Type 13.3 General Comments
495
Index
505
500 504
PREFACE This monograph introduces the theory of stability anJ of time-optimal control of hereditary systems. It is well known that the future state of realistic models in the natural sciences, economics and engineering depends not only on the present but on the past state and the derivative of the past state. Such models that contain past information are called hereditary systems. There are simple examples from biology (predator-prey, Lotka-Volterra, spread of epidemics, e.g., AIDS epidemic), from economics (dynamics of capital growth of global economy), and from engineering (mechanical and aerospace: aircraft stabilization and automatic steering using minimum fuel and effort, control of a high-speed closed air circuit wind tunnel; computer and electrical engineering: fluctuations of current in linear and nonlinear circuits, flip-flop circuits, lossless transmission line). Such examples are used t o study the stability and the time-optimal control of hereditary systems. Within this theory the problem of minimum energy and effort control of systems are also explored. Conditions for stability for both linear and nonlinear systems are given. Questions of controllability are posed and answered. Their application to the growth of the global economy yield broad, obvious but startling policy prescription. For the problem of finding some control to use in order to reach, in minimum time, an equilibrium point from any initial point and any initial state, existence theorems are given. Some effort is made to construct an optimal control strategy for the simple linear systems cited in the examples. For finite dimensional linear systems, time-optimal, and minimum effort feedback controls are constructed. It is assumed that the reader is familiar with advanced calculus and has some knowledge of ordinary differential equations. Some familiarity with real and functional analysis is helpful. The contents of this book were offered as a year-long course a t North Carolina State University in the 1989-1991 sessions. The students were mainly from engineering, economics, mathematical biology, and Applied Mathematics. Though in many aspects the book is self-contained, theorems often are stated and the appropriate references cited. The references should help a further exploration of the subject.
xi
xii
Preface
There are thirteen chapters in this book. The first chapter is devoted to examples from applications in which delays are important in time-optimal, minimum effort, and stability problems. The second chapter studies linear systems. The fundamental matrices of linear delay and linear neutral matrices are calculated and used t o obtain the variation of parameter formulae for nonhomogeneous systems. The stability theory of linear and perturbed linear systems is considered. Chapter 3 presents the basic Lyapunov and Razumikhin theories and their extensions by Hale and LaSalle and recently by Haddock and TerjCki. On the basis of these, a simple quadratic form is used to deduce a necessary and sufficient condition for stability of linear delay equations. The fourth chapter treats the global stability of functional differential equals of neutral type. The second part of the book begins in Chapter 5. It deals with the construction of minimum time and minimum effort optimal feedback control of linear ordinary differential systems. It is both an introduction and a model for such studies of linear hereditary systems. Most of the theory presented here is only available in research articles and theses. It will be most interesting to present parallel results for hereditary systems. Chapter 6 presents the theory of the underlying controllability assumptions required for the existence of optimal control of delay equations. The geometric theory of time-optimal control is then presented in Chapter 7, where numerous examples are considered. Here also the maximum principle both in Euclidean and function space is formulated. The fundamental work of Angel1 and Kirsch is stated: It gives a maximum principle in a setting that guarantees the existence of non-trivial regular multipliers for nonlinear problems involving point-wise control constraints. It presents an introduction t o the synthesis of optimal feedback control of linear hereditary systems. The synthesis of optimal control of simple examples is attempted. In Chapter 8 controllability of nonlinear systems is treated. Interconnected systems are studied in Chapter 9. There universal principles for the control of interconnected organizations are formulated. They have fundamental economic policy implications of immense practical interest. Chapters 10 through 12 consider the corresponding controllability and time-optimal control of linear and nonlinear neutral equations. Chapter 13 presents the stability theory of large-scale hereditary systems in the spirit of Michel and Miller.
Chapter 1 Examples of Control System. Described b Functional Differential l3quations 1.1 Ship Stabilization and Automatic Steering A ship is rolling in the waves. The differential equation satisfied by the angle of tilt z from the normal upright position is given by
+ b-d zdt( t ) + kx(t) = 0 ,
d%(t)
m$t2
(1.1.l)
where m > 0 is the mass, b > 0 is the damping constant, and k > 0 is a constant. The damping constant b determines how fast the ship returns to its upright position. It is known that if b2 < 4mk, the ship is “underdamped” and the ship oscillates as it returns to its upright position. If b2 > 4mR, it is “overdamped,” there is no oscillation, but it returns to its upright position more slowly the larger b becomes. When b2 = 4mk, it is “critically damped” and upright position is achieved more rapidly. This “roll-quenching” was a very important problem tackled by engineers for ships and destroyers of the second world war. In one such research by Minorsky [5,6],ballast tanks, partially filled with water, are introduced in each side of the ship in order to obtain the best value of b. A servomechanism is introduced and is designed dX to reduce the angle of tilt z, and its velocity, -, to 0 as fast as possible. dt What the contrivance does is t o introduce an input to the natural damping of the rolling ship, a term proportional to the velocity at an earlier instant t - h : q i ( t - h ) . The delay h is present because the servomechanism cannot respond instantaneously. It takes this time delay h to respond. Also introduced by the servomechanism is a control with components (u1, u2), which yields the following equations:
b m
i ( t ) = --y(t)
Q k - -y(t - h ) - - z ( t )
mZ(t)
m
m
+ b i ( t ) + c z ( t ) = 0.
+ uz(t).
(1.1.2)
Stability and Time-Optimal Control of Hereditary Systems
2
x =Angle of tiit
FIGURE1.1.1. b2 < 4 m c Decreasing oscillations b2 > 4mc No oscillations b2 = 4mc Critically damped We note that (1.1.1) is equivalent to
X(t) = Y(t) b Y(t)= --y(t) m
(1.1.3)
k
- -m- z ( d ) .
Equation (1.12) can be written in matrix form
where A0
[
-X(t) = Aog(t) + A ~ g (-t h) + Bu(t)i 0
= -- ,
1
]
--b , A1
=
[: ]:-
,B=
m m Using matrix notation, (1.1.3) assumes the form
[
(1.1.4) 1 0 1], E =
[;].
3
Examples of Control Systems
The point (0,O)can be regarded as an equilibrium position and the aim of the gadget is t o steer the angle of tilt and its velocity to the equilibrium position. There are three problems. The first is the problem of stability: Determine necessary and sufficient conditions on A,,, A1 such that the solution of (1.1.5) satisfies . . ( t ) - + O as t -+m.
(1.1.6)
But it is undesirable to wait forever for the system to attain the upright position. The second problem is: Is it possible that a control u, introduced by the servomechanism, can drive the system to the equilibrium in finite time? Is the system controllable? The third problem is: Find a control strategy u that will drive the angle of tilt and its velocity to zero in minimum time. Such a control is called a time-optimal control. Our ultimate goal in optimal control theory is to get an optimal control as a function of the appropriate state space, i.e., t o obtain the “feedback” or “closed loop” control. The major advantage of such a control, as opposed to an “open loop” one with u as a function o f t , is that the system in question becomes self-correcting and automatic.
1.2
Predator-Prey Interactions
Let z ( t ) be the population a t time t of a species of fish called prey, and let y(t) be the population of another species called the predator, which lives off the prey. Under the assumption that without the predator present the prey will increase at a rate proportional t o z ( t ) , and that the feeding action of the predator reduces the growth rate of the prey by an amount proportional t o the product z(t)y(t), we have .(t)
= a12(t) - b l Z ( t ) Y ( t ) .
If the predator eats the prey and breeds at a rate proportional to its number and the amount of food available, then
where
a l , ~ 2 ~ 6 b2 1 ,are
positive constants. The system of two equations
4
Stability and Time-Optimal Control of Hereditary Systems
is rather naive. A more realistic model assumes that the birthrate of the prey will diminish as z ( t ) grows because of overcrowding and shortage of available food. In this model it is assumed that there is a time delay of period h for the predator to respond to changes in the sizes of x and y. Thus
& ( t )= a1 [l
-
Qi(t) = -azy(t)
Y]
+ ( t )- b 1 z ( t ) y ( t ) ,
+ h ~ (-t h)y(t - h ) ,
(1.2.2)
where p is a positive constant. Voltetra 191 studied (1.2.1) and (1.2.2) and various generalizations of (1.2.2). One such system is given by
where 91, g 2 are continuous, nonnegative functions, and c1, c2 are constants. In the interaction between the prey and the predator it is important to ask whether there are equilibrium states that may be reached for the systems (1.2.3). If these states are not zero, then neither the predator nor the prey is extinct and the following is true: a1 - c l + ( t )- blY(t) (12
-c 4 t )
J_o, 0
+ b22(t) +J
-h
+
g l ( s ) y ( t s)ds = 0, ga(s)t(t
+ s)ds = 0.
The function z* = (+*, y*), which solves the above functional equation, is the equilibrium; it may well be the saturation levels of the given species. In this case it may be desirable that every solution ( ~ ( t y) (,t ) ) = ~ ( tof) (1.2.3) satisfies z ( t ) + z* as t + 00. This is an asymptotic stability property. The state z* is not attained in finite time. A more desirable objective is to “manage” the interaction and to have the population as near as possible to its equilibrium position, and to prevent near periodic outbreaks of predator population y beyond its equilibrium. We aim at coexistence of the two species at their equilibrium states, which are also the saturation levels of the two populations.
Examples of Control Systems
5
One management strategy is fishing. Man is interested in harvesting one e i , i = 1,2. For this situation the dynamics of the interaction are
or both species at some rates
If u l ( t ) = e l z ( t ) , uz(t) = e g y ( t ) , then u i ( t ) is the harvest rate that is proportional to the population density. The effort level e i ( t ) is a positive function with dimension l/time, which measures the total effort at time t made to harvest a given species, and uj is a piecewise continuous function [O,T] + [ O , b ] , T > 0, b > 0. Thus e i ( t ) can be considered a control ) the rate the fisherman function. The function u = ( 2 1 1 , u ~ represents (i) selectively kills the prey z, (ii) selectively kills the predator y , or (iii) kills both z and y . There are two problems. By harvesting, what conditions on the systems’ parameters ensure that the two species can be driven to the equilibrium z* in finite time. This is a problem of controllability. If e i ( t ) is negative in (1.2.4), u can be said to represent the rate at which laboratory-reared (iv) fishes z or (v) predators y are released into the system. One can use a combination of release and harvesting strategies to drive z , y to the saturation level t*in finite time. The second question is optimality: What is the best harvesting strategy that will, as quickly as possible, drive the system from any initial population to this equilibrium? Good fish management is interested in the time-optimal control problem for the system (1.2.4). A more general version of this system is given by
where L ( t , g5) is a 2 x 2 matrix function, which is linear in z ( t ) ;and B ( z ( t ) ,t ) is the matrix
6
Stability and Tirne-optimal Control of Hereditary Systems
A more general n-dimensional version of this system is given by
where B is an n x m matrix function, u is in Em,and
Here Ak is an n x n continuous matrix function, A ( t , s ) is integrable in s for each t , and there is a function a ( t ) that is integrable such that
The three questions that we posed for (1.2.4) can also be asked for (1.2.5) or for the general nonlinear equation
where f : E x C -+ En,B : E x C -+ Enxmis an n x m matrix valued function, and u is a rn-vector valued function. Here E = (--oo,oo), E" is the n-dimensional Euclidean space, and C = C([-h,01, En)is the space of continuous functions from [-h, 01 + En.The symbol zt E C is a function defined by q ( s ) = z ( t s) s E [-h, 01. In the discussion above we required all populations t o be driven to the equilibrium, which can be taken as a target. A more general target can be assumed t o be (1.2.7) T = {$ E C : HdJ = p ) .
+
Here H is a linear operator and p E C. The target represents the final configuration at which it is desirable for the species to lie after harvesting.
1.3
Fluctuations of Current
Let us consider an electric circuit in which two resistances, a capacitance and inductance, are connected in series. Assume that current is Rowing through the loop, and its value at time t is z ( t ) amperes. We use also the following units: volts for the voltage, ohms for the resistance R, henry for the inductance L , farads for the capacitance c , coulombs for the charge on the capacitance, and seconds for the time t . It is well known that with
7
Examples of Control Systems
these systems of units, the voltage drop across the inductance is L-d W dt ’ and that across the resistances it is ( R R l ) x ( t ) . The voltage drop across the capacitance is q / c where q is the charge on the capacitance. It is also dq A fundamental law of Kirchhoff states that the sum known that x ( t ) = -. dt of the voltage drops around the loop must be equal to the applied voltage:
+
dx Ldt
+ ( R + R I ) + ( t ) + -q1 C
= 0.
On differentiating with respect to t we deduce
d2x(t) Ldt2
+ ( R + R1)-ddtx + -c1x ( t ) = 0.
(1.3.1)
In Figure 1.3.2, the voltage across R1 is applied to a nonlinear amplifier A. The output is provided with a special phase-shifting network P. This introduces a constant time lag between the input and output P. The voltage drop across R in series with the output P is
q is the gain of the amplifier to R measured through the network. The equation becomes
L
2 t a dt2
+ R i ( t ) + q g ( i ( t - h ) ) + -1Cx ( t ) = 0 .
Finally, a control device is introduced to help stabilize the fluctuations of the current. If i ( t ) = y ( t ) , the “controlled” system may be described by
Y(t)= --R y(t) L
- -Qg ( y ( t - h ) ) - -4) 1 +u 2 ( 9 L CL
(1.3.2)
The system (1.3.2) can be put in the matrix form
a ( t ) =Ag(t)
+ G ( g ( t - h ) ) + Bu
(1.3.3)
8
Stability and Time-Optima[ Control of Hereditary Systems
where
-x =
(a)
u= 0
A=
[-&
[:;I
-4 1
1 0
B = [0 1
1
FIGURE 1.3.1 The control u = (ul,u2) is “created” and introduced by the stabilizer. The three basic questions are now natural: With ti = 0, what are the properties of the systems parameter which will ensure that t 2 ( t ) + y 2 ( t ) ---t 0 as t -+ m? Will “admissible controls” ti (say luj I 5 1, j = 1,2) bring any wild fluctuations of current (any initial position) to a normal equilibrium in finite time? Can this be done in minimum time?
Examples of Control Systems
9
FIGURE 1.3.2. [5, P . 361 1.4
Control of Epidemics
A. First Model [12, p. 1131 In this section we formulate a theory of epidemics, a problem of timeoptimal control theory. It is a slight modification of Bank’s presentation of the work of Gupta and Rink [2]. We assume as basic that there are four types of individuals in our population: (i) susceptibles: X , ( t ) ,
10
Stability and Time-Optimal Control of Hereditary Systems
(ii) exposed and infected but not yet infectious (infective) individuals: Xdt), (iii) infectives: X 3 ( t ) , and (iv) removed individuals: Xq. We include in (iv) those who have recovered and gained immunity and those who have been removed from the population because of observable symptoms. We assume a constant rate A of inflow of new susceptible members. The latent period hl is the period from the time an individual is exposed and becomes infected to the time he becomes infective. The infectious period hz is the period the individual is in the infectious (infective) class. The incubation period will denote the time from exposure and infectedness to hz = T). After normalization by setting the time of removal (i.e., hl x i = & / N , where N is the population, the dynamics of interaction is
+
The constant p is the average number of individuals per unit time that any individual will encounter, sufficient to cause infection. To control the epidemic we allow two strategies: (i) The removal at some rates ui i = 2 , 3 of both the infectives and exposed and infected individuals. The removal rate ui(t) = ei(t)bi(zi(t)) is proportional to a function bi of the population density and the effort level e i ( t ) . (ii) active immunization, the injection of dead or live but attenuated d i 5 ease microorganisms into members of the population, resulting in antibodies in the population of vaccinated individuals. If El ( t ) represents the reliable rate per day at which members of the population are being actively vaccinated at time t , the normalized rate is
e1(t) = E l ( t ) / N . We assume that the effective immunization rate is b l ( z l ( t ) ) e l ( l )= u l ( t ) . If we assume that there is a delay h (hi 5 h , i = 1,2) days before the antibodies become effective, then the rate is b l ( t l ( l ) ) e l ( t- h ) = u t ( t ) .
Examples of Control Systems
11
Thus the dynamical system is
+
& l ( t )= - P z l ( t ) z 3 ( t ) - h ( z i ( t ) ) e l ( t- h ) A , 52(t) = Pzl(t)23(t)- p z l ( t - h1)23(t - hl) - uZ(t)i i 3 ( t ) = pzl(t - hl)Z3(t h l ) - pzl(t - hl - h2)23(t - hl
-
- h2)
(1.4.2)
- u3(t). We use control strategies (i) and (ii) to reduce the total number of infected to an “acceptable” size in finite time. This health policy may be used to “prevent an epidemic” in minimum time, where preventing an epidemic means that the solution of (1.4.2) satisfies (i) I z z ( t ) l + Iz:s(t)l 5 A for all t 2 0. (ii) max 1z3(8)1 5 B, where A, B are prescribed constants. JE[O,TI
The solution of (1.4.2) may be viewed as lying in the space C([-h,O],E 3 ) the space of continuous functions mapping [-h,O] into E3 with the sup norm. Or it may be viewed as lying in E3. We define a subset T c E3 by
The best health policy may be that solutions of (1.4.2) hit the target, TIin minimum time. Also desirable is to have solutions of (1.4.2) to reach and remain in T, and to do so as fast as possible. The time-optimal control problem formulated in relation to the epidemic models above can also be formulated for the more general nonlinear system
i ( t ) = f(t, ~ ( t )~, (-th l ) * . * z(t - h a ) , u ( t ) , ~ (-t h ) ) , with target set
T = {z E En : Hx = T ,
T
(1.4.3)
E En},
and H an n x n matrix. Equation (1.4.3) is a special case of the delay system (1.4.4) which includes the ordinary differential equation (1.4.5)
i ( t ) = q t , z ( t ) ,U ( t ) ) . In (1.4.4) xt is a function defined by q ( s ) = z(t The symbol ut is defined similarly.
+
S) s
E [-h, 01.
12
Stability and Time-Optimal Control of Hereditary Systems
B. Control of Epidemics: AIDS Just as in A, we formulate a theory of acquired immunodeficiency syndrome (AIDS) epidemic as a problem of time-optimal control theory. It is a modification of the recent works of Castillc-Chavez, Cooke, Huang, and Levin [14].We assume as basic that there are five classes (of sexually active male homosexuals with multiple partners): 21 : susceptible individuals, 22: those infectious individuals who will go on to develop full-blown AIDS, 2 3 : infectious individuals who will not develop full-blown AIDS, 2 5 : those former 23 who are no longer sexually active, and 2 4 : those former x 2 who have developed full-blown AIDS. Note that if an individual enters 2 5 or 2 4 , he no longer enters into the dynamics of the disease. In contrast to our earlier treatment of the theory of epidemics, a latent class is excluded, i.e., those exposed individuals who are not yet infectious, since only a very short time is spent in that class. It is our assumption that an individual with full-blown AIDS has no sexual contacts and is therefore not infectious. Also, once infected, an individual is immediately infectious, and sexual inactivity or acquisition of AIDS are at the constant rates of a3 and a2,respectively, per unit time. We let A denote the total recruitment rate into the susceptible class, defined t o be those individuals who are sexually active. Let p be the natural mortality rate, and d the disease-induced mortality due t o AIDS. Suppose p is the fraction of the susceptible individuals that after becoming infectious will enter the AIDS class, and (1 - p ) the fraction that become infectious but will not develop full-blown AIDS. We use the following diagram:
FIGURE 1.4.1
13
Examples of Control Systems to determine the epidemiological dynamics:
(1.4.6)
+
-d x z ( t ) - X p C ( T ) ( t ) x l ( t )W - ( t )- ( a ~p ) x z ( t ) , dt T(t)
(1.4.7)
-dx4(t) - a z x z ( t )- (d + p ) x 4 ( t ) ,
(1.4.9)
dt
( 1.4.10)
d 2 5 ( t ) = a3x3(t) - p25(t),
dt
where
W=xz+x3
and
T = W+z1.
( 1.4.11 )
In the above formulation, C ( T )represents the mean number of sexual partners an average individual has per unit time when the total population is T , and the constant X denotes the average sexual risk per partner. Thus the expression X C ( T ) x l ( t ) W / T denotes the number of newly infected individuals per unit time. Very often the Michaelis-Menten type contact law is accepted for C(T):
C ( T )=
~
1
PT
+ kT’
( 1.4.12)
where P,k is constant. Systems (1.4.6 - 1.4.10) are ordinary differential equations. Further to the assumption above where individuals become immediately infectious and the infectious period is equal to the incubation period, we designate h z to be the fixed period of infection for 2 2 , and h3 to be that of c3. We set 2 2 0 ( t ) ] 2 3 0 ( t ) , to be infectious x z ( t ) and x 3 ( t ) at time t = 0, and 2 4 0 ( t ) , ~ 5 0 ( t living ) x 4 ( t ) ,x5(t) at time 2 = 0. Clearly 2 4 0 , 2 5 0 have compact support, i.e., ~ 4 O ( t ) , ~ 5 O vanish (t) as t +. 00. Note that x z o ( t ) = ~ ~ ~= (0 for 0 )t > max(hl,hz). If H ( t ) is the unit step or 0, t < O Heaviside function, i.e., H ( t ) = , then the system is modified 1, t > 1
14
Stability and Tame-Optimal Control of Hereditary Systems
with initial conditions,
Z l (t ) = ZlO(t), (1.4.16)
Z3(t) = 230(t). This is a delay differential equation whose existence and uniqueness of solutions are easily proved. We now restrict our analysis to the case p = l (i.e., we model AIDS as a progressive disease). The study of the steady states reduces t o the following set of equations:
i l ( t )= A
x2(t) - p x ~ ( t ) , T(t)
- XCT’(t)t1(t)-
with infection free state
(:-, )
0 as equilibrium. It is possible t o allow h2, h3
t o be distributed. For this we follow the generalizations of 1141 and define the survivorship functions P~(s), P3(s), which are the proportions of those individuals who become t 2 or t g infective a t time t and if alive are still infectious at time t s. These are nonnegative nonincreasing with
+
P2(0) = p3(0)= 1, (1.4.17)
Examples of Control Systems
15
are the removal rates of indiUnder these assumptions, -Pz(s)and -&(s) viduals from 2 2 and 23 into 2 4 , and x 5 classes s time units after infection. We therefore have
(1.4.18) (1.4.19)
+
t
W(s) x g ( t ) = z ~ o ( t ) (1 - p~AC(T)(s)zl(s)-e-'('-')Ps(t T(s)
0
- s)ds,
(1.4.20)
(1.4.22) Now denote by B the rate of infection-the number of new cases of infection per unit of time: B ( t ) = C ( T ( t ) ) z l ( tW ) - .( t ) (1.4.23)
T(t)
Since p is the mortality rate, we denote the rate of attrition by A ( t ) : In (1.4.6) this is given by A ( t ) = - p l ( t ) . Thus, (1.4.6) becomes i1(t)
= A - B ( t )- A ( t ) .
(1.4.24a)
This can be generalized. If A is time varying, then A ( t ) is defined as the total rate of recruitment into the susceptible class at time t . Assuming a certain fraction of the total dies in each future time period after recruitment, then n ( r ) d r is the natural mortality density, i.e., the fraction that dies in any small interval of length d r around the time point r. Then A ( t ) n ( t ) d . r will die in a small interval about t T , and if we replace t by t - 7 then A(t - r)na(r)drof the total recruitment made in a small interval about t - T will die. The total natural death at time t of all previous total recruitments A(t - r)m(r)dr.Equation (1.4.24a) becomes is
+
(1.4.24b)
Stability and Time-Optimal Control of Heredatay Systems
16
This can be considered an integral equation of renewal theory, where i l ( t ) and B ( t ) are known and A ( t ) is unknown. If we define the nth convolution, rnn(r)of m ( ~recursively ) by the relations rn(')(T)
= rn(T),
?7l"+'(T)
and the replacement density,
=
r(T)
l7
rnn(U)rn(T
=
of (1.4.24b) is
- U)dU,
(1.4.25a)
00
C rn(")(~), then the unique solution
n=l
roo
~ ( t=)i l ( t )+ ~ ( t+)J, i l ( t - T)r(T)dr. If the mortality density is exponential,
m ( ~= ) pe-p7,
(1.4.25b)
then (1.4.26)
so that
r(T) =
p . With this (1.4.25b) becomes A(t)
= i l ( t )+
or A(2)
Jd
00
i l ( t - r ) r ( T ) d T+ ~ ( t )
+
=ii(t) pi(t)
+ B(t).
From this analysis one has (1.4.27) Since r ( ~2) 0 for all
T , there
is a number 0
lW
5 h(t) = h < 00 such that
i l ( t - T)r(T)dT = i l ( t - h)
W
r(T)dT,
and we can then postulate that for some finite number
a-1,
(1.4.28) We have derived the neutral equation (1.4.29)
17
Examples of Control Systems
which now replaces (1.4.13). Because of the delays in (1.4.14) and (1.4.15) there is a possibility of oscillation and endemic equilibrium. One can now study the stability of the disease as well as its controllability by using any of the control strategies of Example A, i.e., allowing removal and active immunization. The results are available in Chukwu [15]. The interventions studied in [15]include the use of “positive controller,” (e.g., putting (known) infected individuals in quarantine, passive immunization-the direct injection of antibodies). The model (1.4.29) and the intervention of controllers suggest possible consequences for the spread of AIDS epidemics. The study of such control systems will help think through and focus on the moral and ethical questions raised by the controls used in the study. 1.5 Wind Tunnel Model; Mach Number Control
Delay equations of type (1.4.4) are also encountered in the design of an aircraft control facility. For example, in the control of a high-speed closed-air unit wind tunnel known as the National Transonic Facility (NTF), which is at present under development at NASA Langley Research Center, a linearized model of the Mach number dynamics is the system of three state equations with a delay, given by
<
1 with Q = - k. = 0.117, w = 6, = 1.6, and h = 0.33s. luil 5 K 1.964’ with K a positive constant. The state variables x 1 , x 2 , x 3 represent deviations from a chosen operating point (equilibrium point) of the following quantities: x l = Mach number, 2 2 = actuator position guide vane angle in a driving fan), and 23 = actuator rate. The delay represents the time of the transport between the fan and the test section. Though in many cases only k.1 = k.2 = 0 are considered ([1,4]), we assume here that there is a control device that creates the control variable with three components that constitute an input t o the rate of the Mach number, the actuator velocity, and the actuator servomechanism. The parameter values correspond to one particular operating point. Equation (1.5.1) can be written as
i ( t ) = A o ~ ( t+) A
- h) + Bu(t),
l ~ ( t
(1.5.2)
18
Stability and Time-Optimal Control of Hereditary Systems
where
Ao=[O -a 0
0 -w2
-yw
],A1=[:
7
: ] , B = [ D kl 0
k2 0
0
q.
w2
The following problem can be proposed (Manitius and Tran [4]): Find an optimal control subject t o its constraints such that the solution of (1.5.2), with this control ( k l = k2 = 0) and with an initial configuration
will hit a target
in minimum time T and remain there forever after.
Remark 1.5.1: In Equations (1.3.2) and (1.5.1) the controls ( ~ 1 , 2 1 2 )and (u1, u2, u3) introduced are such that additional variables have been added in the equations describing the relations between velocity and position. One can question the physical validity of such an effort since velocity is the derivative of position, and no extra control may be allowed in the relation. Such an objection is shallow. Just as a bird flying in the wind with z1 as its position may have its resultant velocity il = y + u l , where u1 is the velocity of the wind (control), so may the control device bring in an additional u1 to bear in (1.3.2) and an additional ( u l , u 2 ) to bear in (1.5.1). These are possibilities. If these components are zero, the Euclidean controllability of these equations can still be studied.
1.6
The Flip-Flop Circuit
In the general system (1.4.4) whose special cases are given in the previous examples, we deduced the equilibrium position by setting f(t, z t , u t ) = 0. The state z; for which this is true is the equilibrium state. The problem of stability is to deduce conditions on f for which every solution z of (1.4.4) satisfies zt + z; as t + 00. It is possible that some dynamical systems possess multiple equilibria and are therefore suited to be used as a memory storage device in the design of a digital computer. The flip-flop circuit has such dynamics [8]. It is the basic element in a digital computer, and a standard model is given in Figure 1.6.1 [S].
Examples of Control Systems
19
In this model the portion between 0 and 1 is a lossless transmission line with inductance L and capacitance C. The current i flowing through the line and the voltage v across it are both functions of z and t . In Figure 1.6.2 [8] the function g ( v ) is a nonlinear function of v and gives the current in the indicated box in the direction shown. The lossless transmission line can be described by the hyperbolic partial differential equations aV di = -c-, dV -- -L- ai -
ax
d t ' ax
dt
0 <2
< 1 , t > 0,
(1.6.1)
with boundary conditions
E
- v(0,t) - Ri(0,t ) = 0,
(1.6.2)
+
If we let s = (LC)-a, t = ( L / C ) i , k = (2 - R ) / ( z R), and a = 2 E / ( z R), we can convert Equation (1.6.1) with boundary conditions (1.6.2) into a neutral functional differential equation, i.e., a system with delay in the derivative of the state as well as in the state itself. Indeed, the general solution of (f.6.1J is
+
+
+
) = 4(z - St) 4(2 s t ) , 1 i ( z ,t ) = -[4(z - s t ) - $J(Z s t ) ] ,
V(Z, t
t
(1.6.3)
+
which is equivalent to
This implies that
2 4 ( - s t ) = 2, ( 1 , t
+ :) + t i ( 1 , t + :)
,
We use these expressions in the general solution (1.6.3) and in the first expression of (1.5.2) a t the moment (t to deduce
(5))
t
Stability and Time-Optimal Control of Hereditary Systems
20
On using the second boundary condition of (1.6.2) and setting u ( t ) = v(1, t ) , we obtain the equation
?qt) -hi (t
-
a)
=f
(t-
?)),
(1.6.4)
where
Clf(U(t),U(t
- h ) ) = cr - $ u ( t ) - $u(t - h) - g (u (t)) + kgu(t - h ) ) .
i = g(v)
x =o
x=I
FIGURE 1.6.1. [8]
x =o
x=i
FIGURE 1.6.2. [8] For the flip-flop circuit to operate as a memory storage device, one equilibria it may possess is assumed to be globally asymptotically stable: Criteria on the circuit parameters for a single equilibria point to be globally, asymptotically stable are deduced by a Lyapunov stability theory for a more general system than (1.6.4), namely
d dt
-[s(t)
- A - i z ( t - h)] = f ( t ,z t ) ,
(1.6.5)
where A-1 is an n x n matrix and f : E x C 3 En is a continuous function.
Examples of Control Systems
21
Equation (1.6.5) can be obtained from a lossless transmission line in which conditions (1.6.2) are more complicated. See bference [7]. In Figure 1.6.3, for example, (1.6.1) is satisfied and the following boundary conditions hold: v ( t , 0) = uo(t) - Roi(t, 0) Eo(t), (1.6.6) v ( t , 1) = U l ( t ) Rli(t, I),
+
+
If we integrate (1.6.1) along its characteristics, then we have z1(t)
= &(t,
0)
+ Qt,
0) = &v(t
Q ( t ) = Jcv(t, 1) - d z i ( t ,1)
= &(t
+ h, 1) + a i ( t + h , l ) ,
+ h,O) - &i(t + hO),
m.
where h = Setting z S ( t ) = 2t/Zio(t), z*(t) = 2&il(t), and u ( t ) = 2&Eo(t), and using the boundary conditions (1.6.6) - (1.6.8), we deduce an equation of the form d -[z(t) -A-lt(t dt
0
- h ) - B - o ~ ( t )=] A o ~ ( t+) A l z ( t - h ) + Bou(t), (1.6.9)
1
FIGURE 1.6.3 [7]
X
22
Stability and Time-Optimal Control of Hereditary Systems
where
0
[i
0
0 0 0 0]1
0
0 0
a 4
[a{pO]’
A1=
B-o=
[i],
and where
a 4
=
RoJE-A R o f i i-
a’
ag
=
R1fi-JZ RI@+fl‘
Equation (1.6.9) is a neutral functional differential equation with control u that is related t o the initial power source E o ( t ) . Equations (1.6.4), (1.6.5), and (1.6.9) are special cases of the system (1.6.10) where D : E x C with
+
En, f : E x C x Em -+En are continuous functions D ( t , .*) = .(t)
and g : E x C
1.7
+ En
- g ( t , a),
(1.6.11)
is continuous.
Hyperbolic Partial Differential Equations with Boundary Controls [3]
A final example involves the derivation of Equation (1.6.10) from linear hyperbolic partial differential equations with boundary control. The derivation is treated in [3].Let the wave equation for w ( t , z )be given by wtt
- c2 wxr = 0,
t E [O’T],
2
E [O, 11,
(1.7.1)
23
Examples of Control Systems
and initial conditions
where the initial functions satisfy the boundary conditions at t = 0. Here the subscript t or z denotes partial derivative. The prime in the sequel denotes total derivative. The terms Go, GI contain the controls if
and
(:)
is the control to be chosen from a prescribed class. We assume
that Ai, Bi are continuously differentiable, Gi are absolutely continuous in t , continuously differentiable in other arguments with Git dominated by a square integrable function. We assume further that
fort E
[O,T].
We assume a D’Alamberto solution of the form
w(t, z) = +(t
+ z / c ) + $(t - z / c ) ,
(1.7.5)
24
Stability and Time-Optimal Control of Heredita y Systems
and
(t - f) @ ( t ) + p1 (t - f) $" (1- t) = GI (t + l-! 4'(t) + (t - %) (t - a,)
a1
$l'(s)ds,
$I
C
I
It follows from condition (1.7.4) that we can multiply the last pair of equations by the matrix 1
to deduce the equations
Denoting
$I)
by (yl z ) , this equation is a neutral equation of the form
The data of (1.7.3) can now be used to produce initial and terminal functions of (y, z ) for (1.7.6). Since there are controls on Gi and therefore on
Hi,
the system (1.7.6) is a control system for t E
I : [-t,:] [. [
[: 1 -T
with initial and
terminal values for y given on 0,
and at t = T , and the corresponding
values of z given on
- :,TI. Because of the smoothness
and
25
Examples of Control Systems
conditions on G;and therefore on H i , (1.7.6) may be argued from the fundamental existence theorem beIow t o have a solution (y, z ) = (@,$’), which is absolutely continuous with square integrable derivatives. One argues that this (4, q!~)utilized in (1.7.5) yields a solution of (1.7.1) in the generalized X X sense or in the sense of almost everywhere, i.e., w(t, x) = 6(t+); +$(t -); is continuously differentiable with wt,w, being absolutely continuous and possessing square integrable partials that satisfy (1.7.1) a.e. It is interesting t o note that the boundary conditions of (1.7.2) cover the usual ones for transverse vibrations of a string or longitudinal vibrations in an elastic rod with elastically supported ends. Just as in earlier examples, it is there at the boundary that controls are introduced. It sometimes requires some ingenuity to introduce in practice the type of control that is required d for the system -[D(t)xt] = f(t,xt) B(t)u(t)(or of (1.6.10)) to have a dt desired effect on B ( t ) . One can then relate this t o the control device. An appropriate time-optimal control problem can now be formulated for such systems.
+
1.8
Control of Global Economic Growth
Let x ( t ) denote the value of capital stock at time t . Let u ( t ) , 0 5 u ( t ) 5 1 be the fraction of the stock that is allocated to investment. This investment is used to increase productive capacity. We can assume the value of this investment I ( t ) is given by
I(t) = ku(t)z(t). where k is the constant of proportionality. There is depreciation, D ( t ) , of capital stock, and it is proportional to capital stock
D ( t ) = -Sz(t). The net investment is
N ( t ) = I ( t ) - Sz(t),
and it is used to produce new capital. Thus net capital formation i ( t ) is given by k(2) = k u ( t ) x ( t )- 6 + ( t ) . (1.8.1) If -1 5 u(t) 5 0, we can ifiterpret u(t) to be the fraction of the value of stock that is used for payment of taxes, etc. Thus, in general, u ( t ) satisfies -1 _< u ( t ) 5 1. Such models are very naive and unrealistic. As pointed out
Stability and Time-Optimal Control of Hereditary Systems
26
by Takayama [lo, p. 7051, this implies that adjustment of desired stock of capital is instantaneous and frictionless, an implication that has no valid empirical foundation. There is a time lag that needs to be incorporated to the control procedure of the firm. Investment in new capital equipment does not yield new productive capacity until the equipment is delivered, installed, and tested. There is a time delay h > 0. It is more realistic to express I ( t ) as a function of the present and past values of u ( t ) and z ( t ) and of time: q t ) = g ( t , z ( t ) U(t), , 4 t - h ) ,4 t - h ) ) . In general,
I ( t ) = B ( t , z t ,~
t
)
G ~ t E h
d,H(t,z(t
+ s ) , u ( t + s ))u (t+ s ) .
Here the past history of 2 and u over an entire interval of length h enters equations through the Riemann-Stieltjes integral. Realism dictates that depreciation be incorporated into the model. We assume that the value of the capital stock that has depreciated is not just proportional to z ( t ) as in (1.8.1), but is a function g of ( z ( t ) u , ( t ) ) and is given by g ( z ( t ) u, ( t ) ) z ( t )at time t . Suppose this value decreases by a factor P ( a ) at time a ( P ( 0 ) =
1, P ( L ) = 0 where L is the lifetime of the asset). If p ( a ) = d P ( a ) , the da depreciation density or mortality density, p ( ~ ) d r represents , the fraction lost to productive use in any small time interval of length d r around the time T . Therefore, g(z(T),u ( ~ ) ) z ( r ) p ( ~will ) d rdisappear in a small interval about t T. If we replace t by t - T , we can say that at time t the amount
+
g ( z ( r - h ) ,u(7 - h ) ) X ( T - h)P(T)dT in a small time interval about t - T will disappear. The total evaporation at time t is
or -
lL
g(z(T - h ) , U ( T - h ) ) z (~ h)dP(T)
if we use Riemann-Stieltjes integral. A general representative expression is
L ( t , z t ,ut>.t
=
Ih
d,V(t, s,z ( t
+ s), u(t + s ) ) z ( t+ .)
27
Examples of Control Systems
where we use Riemann-Stieltjes integral. We can therefore state that net capital formation i ( t ) is given by
w = L(4zt,ut)zt + B(t,zt,ut)ut.
(1.8.2)
We postulate initial capital endowment function as 2, = 4, and initial fraction of stock allocated to investment as u, = v . The key elements in (1.8.2) are the time patterns of capital z ( t ) and the capital policy described by the function ~ ( 2 ) . The problem of optimal capital policy is to describe u ( t ) subject t o its constraints -1 5 u ( t ) 5 1, such that the system with initial endowment function 4 will hit a prescribed target z1 while minimizing the firms' power or effort defined by some function E ( u ) , and perhaps to do it in minimum time. Possible definitions of effort or power are given by
(1.8.3a) (1.8.3b) where u E U = { u E E, u piecewise continuous lu(t)l 5 1 0 5 t
5 tl}.
This represents maximum investment thrust available to the firms.
(1.8.3~) where 2t
E U = { u E E,
2t
measurable liullp = E(u(t))5 1).
If p = 2, E ( u ( t ) ) represents the investment energy or power that must be minimized. JD
(1.8.3d)
where u E U = { u : IIuIIL, 5 1). The expression E4(u(t)) can be called investment; the problem is to minimize it when we achieve our growth target from the initial endowment q5 to our prescribed target 21. We may wish to maximize some attainable value of the social welfare criterion
J=
1'
E ( u ) exp( - y t ) d t ,
28
Stability and Time-Optimal Control of Hereditary Systems
where E is a specified “well-behaved” utility function and 7 is a fixed discount rate. E is usually a strictly concave monotone increasing in u with second derivative defined everywhere and such that lim E ( u ) = 00.
u-0
The fixed time T is the term of years in which the above objectives will be fulfilled. To solve this problem one must first solve the problem of controllability. The controllability question is investigated in Chapters 8 and 12. Problems of optimality are reported in Chapter 7. Our research here deals with -1 5 u ( t ) 5 1. Currently in press, [15],the situation 0 5 u ( t ) 5 1 will be explored. In the previous discussion z ( t ) is a real number, the value of one stock. We can denote z ( t ) = ( z l ( t ) .. . zn(t)) to be the value of n capital stocks at time t , with investment and tax strategy u = (u1 -”U,,), where -1 5 u j ( t ) 5 1. We therefore consider (1.8.2) as the equation of the net capital function for n stocks in a region that is isolated. These stocks are linked to I other such systems in the globe, and the “interconnection” or “solidarity function” is given by
This function describes the effects of other subsystems on the ith subsystem as measured locally at the ith location. Thus, i i ( t ) = &(t, zit ,U i t ) z i t
+ Bi(t,
+ gi(t,zit
zit, u i t ) ~
i = 1, ... ,e
7.
9
Z t t t u1t
. ..~
(1.8.4)
is the decomposed interconnected system whose free subsystem is
We now introduce the following notation: Let
t
L i=l
n; = n,
L i=l
m;= m.
)
Examples of Control Systems
29
Then (1.8.3) is given as
i ( t ) = L ( t , z t , u t ) + B ( t , z t , u t ) u t+ g ( t , . t , u t ) .
(S)
To clarify our terminology we introduce the following notations and definitions. Let E = (--00, oo),and E‘ be the r-dimensional Euclidean space with norm I I. The symbol C denotes the space of continuous functions mapping the interval [-h, 01, h > 0 , h E E into Enwith the sup norm 11 defined by 11411 = sup l4(s)l, 4 E C. If t E [ u l t l ] ,we let zt E C be de-h<s
+
fined by z t ( s ) = z ( t s), s E [-h, 01. The symbol L , ( [ u , t l ] , Em)= L , denotes the space of essentially bounded measurable functions on [u, tl]
Definition 1.8.1: The system (S) is Euclidean controllable on [ u , t l ] if for any function 4 E C and any vector z1 E En,there is a control u E L , ( [ u , t l ] , Em) such that the solution z ( t ) = z ( t , u , + , u )of (S) satisfies z,,(-, u,4, u) = 4, z(t1,u,4, u) = 21. It is Euclidean null controllable on [ u , t l ]if z1 = 0 in the above definition.
+
The system (S) is controllable on [a,t13, t l > u h , if for each 6,$ E C there is a control function u E L , ( [ u , t l ] , E m ) such that the solution z ( t ) = z ( t , u , 4 , u ) satisfies zu(.)= 4, ztl = $. It is null controllable on 0 in the preceding definition. In C we drop the qualifying [a,tl]if $ phrase “on the interval [u,tl],” if controllability obtains on every interval [u,2 1 1 , with tl > u + h. In En we drop “on the interval [u,tl]”if Euclidean controllability holds on every interval [u,t1],tl 2 u. The problem of controllability of (S) will be explored in Chapters 8 and 9. Conditions are stated for the controllability of the isolated system (S,). Assuming that the subsystem (Si) is controllable we shall deduce conditions for (S) t o be controllable when the solidarity function is “nice”. In (1.8.2) we postulated that capital growth is described by a functional differential equation with delay. The crucial assumption is that the net capital formation i ( t ) is given by the nonlinear function, I ( t ) , the gross investment, on the right. A more general situation can be obtained. Arrow [ll,p. 1841 showed that indeed it is realistic to have
~ ( t=) i ( t >+
J i ( t - s)r(s)ds,
where r ( t ) 2 0 is the replacement density and i ( t )the net capital formation. Since I ( t ) is finite valued and r ( t ) 2 0, the mean value theorem for integrals
Stability and Time-Optimal Control of Hereditary Systems
30
allows one to write
lo where 0 5 h
1
i ( t - s ) r ( s ) d s= i ( t - h ( t ) )
5 00. It
r(s)ds,
0
is therefore reasonable to postulate that
I ( t ) = Z(t)
+ i!(t - h(t))a,
where 0 5 h < 00 and a is determined from the replacement density. As a consequence of this hypothesis we can reasonably postulate that
in place of (1.8.2). This is a functional differential equation of neutral type. Analogous systems (1.8.4) (Sj) and (S) can be formulated.
1.9
The General Time-Optimal Control Problem and the Stability Problem
The general form of equation studied is (1.9.1)
g ( t ,p) is linear in p, and f ( t ,p, u) may be nonlinear. It includes the delay system
(1.9.2) and the ordinary differential system (1.9.3) The initial state p can appropriately be given as the space C of continuous functions from [-h,O] into En with the uniform norm. In this case,
31
Examples of Control Systems
finding a solution of (1.9.2) is equivalent to finding a solution of the integral equation
Other spaces with this property of equivalence of solution may be taken, and will be considered in subsequent discussions. If spaces other than C are used, the final point and the solution lie in C for t 2 6 h. Often the final point we reach may be taken to be in En,or in C. The space W i , consisting of all absolutely continuous functions from [-h,O] 4 En that have pintegrable derivatives where 1 5 p 5 00, will play a significant role in our investigations. Another space of initial functions that is found to be very useful in application is the space Lp([-a, 01, En),i.e., the space of pintegrable functions with the usual norm. If we appropriate this as the state of initial conditions, the solutions of (1.9.1) can be considered in the product space
+
MP= E” x Lp([-h,O], E”) x LP([O,Tj,Em),
15p
5 00, endowed with the norm
With the state space selected, the target is either a point or a subset of the appropriately selected state space. The admissible controls are measurable vector valued functions with values constrained to lie in a compact convex set U of Em. Often U is specialized t o be the unit cube,
C” = {u € Em : lujl 5 1, j = 1 , 2 , . . . ,m}. The time-optimal control problem is now formulated as follows: Determine an admissible control u* such that the solution z(’p,u,u*) of (1.9.1) hits a continuously moving target point or set in the appropriate space, in minimum time, t* 2 0. Such controls u* are called time-optimal controls. Our ultimate goal is to get an optimal control ti* as a function of the appropriate state space, i.e., to obtain the “feedback” or “closed loop” control. The major advantage of such a feedback-optimal control as against
32
Stability and Tame-Optimal Control of Hereditary Systems
an “open-loop” one with u as a function o f t is that the system in question becomes self-correcting and automatic. Thus if C is the state space, we hunt for a measurable function m : C -+ Emsuch that (i) rn(+t) E u ( t ) , u(t) E U. (ii) rn is optimal feedback control for (1.9.1) in the following sense. In addition t o (1.9.1), consider the differential equation
(1.9.4) Then each optimal control of (1.9.1) is a solution of (1.9.4), and conversely each solution of (1.9.4) is an optimal control solution of (1.9.1). Once found, the time-optimal control problem for the system (1.9.1) is completely solved. In Example 1.2.4 for instance, the optimal feedback control is the fishing strategy that would drive the system to the target (equilibrium) as quickly as possible. Sometimes the theory and techniques of the solution of the time-optimal problem can be appropriated to tackle the problem of minimizing a cost function which, for example, is given in (1.8.3). The general situation will be treated in Chapters 7 and 9.
The Stability Problem The complete solution of the time-optimal problem is dependent on the solution of the corresponding stability problem, which is formulated as follows: Let I* be a solution of
Find necessary and sufficient conditions on D and g such that every solution 4 4 ) of
28
(1.9.6)
=4
is uniformly, globally, asymptotically stable in the following sense: For each 4 , the solution I($) of (1.9.6) satisfies
q ( 4 )+ Z f
as t + w .
(1.9.7)
33
Examples of Control Systems
These two problems of stability and optimal control are the primary questions we attempt to address in this monograph.
Remark 1.9.1: In the time-optimal or minimum-effort problem treated here, the controls of essential interest are small and bounded in components: Iuj(t)l 5 1, j = 1 , . . . ,m. Because of this constraint, the global asymptotic stability of the system without control are needed and will be studied to ensure global constrained controllability of our dynamics, on which rest the existence of an optimal control. This contrasts with the situation in which the constraint I u j ( t ) l 5 1, j = 1 , . . . ,rn is removed, and we require admissible controls to be square integrable. For such “big” controls one could study feedback stabilization and optimal control with quadratic cast functional. Though such a study is important, we shall not pursue it in depth because the main applications we have in mind for this introductory text have essentially bounded controls.
REFERENCES 1. E. S. Armstrong and J. S. Tripp, “An Application of Multivariate Design Techniques to the Control of the National Transonic Facility,” NASA Technical paper 1887,1981. 2. H. T. Banks, “Modelling and Control in Biomedical Sciences,” Lecture Notes in Biomathetics, 6 , Springer-Verlag, 1975. 3. G. A. Kent, “Optimal Control of Functional Differential Equations of Neutral Type,” Ph.D. Thesis, Brown University, 1971.
H. Tran, “Numerical Simulation of a Nonlinear Feedback Controller for a Wind Tunnel Model Involving a Time Delay,” Optimal Control Application and Methods, 7 (1986) 19-39.
4. A. Manitius and
5. N. Minorsky, Nonlinear Oscillations, D. Van Nostrand Co., Inc., Princeton, 1962.
6. N. Minorsky, “Self-Existed Oscillations in Dynamical Systems Possessing Retarded Actions,” J . Appl. Mech., 9 (1942) 65-71. 7. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. 8. M. Slemrod, ‘The Flip-Flop Circuit as a Neutral Equation,” in Delay and Functional Diflerential Equations and Their Applications, 387-392, Academic Press, New York, 1972.
“Sur la thCorie mathCmatique des ph6noml.nes h&Cditaires,” J . Math. Pures Appl., 7 (1928) 244-298. 10. A. Takayama, Mathematical Economics, 2nd Edition, Cambridge University Press, 9. V. Volterra,
1984. 11. K. J . Arrow, Production and Capital, Collected Papers of Kenneth J. Arrow, The Belknap Press of Harvard University Press, Cambridge, Massachusetts, 1985.
34
Stability and Time-Optimal Control of Hereditary Systems
K. L. Cooke and J. A. Yorke, “Equation Modelling Population Growth, Economic Growth and Gonorrhea Epidemiology,” in Ordinary Differential Equations, edited by L. Weiss, Academic Press, New York, 1972. 13. F. Brauer, “Epidemic Models in Population of Varying Size,” in “Mathematical Approaches to Problems in Resource Management and Epidemiology,” C. CastiUoChavez, S. A. Levin, and C. A. Shoemaker (eds.), Lecture Notes in Biomathematics, 81, Springer-Verlag, 1989. 14. C. CastiUo-Chavez, K.Cooke, W. Huang, and S. A. Levin, “The Role of Long Periods of Infectiousness in the Dynamics of Acquired Immunodeficiency Syndrome (AIDS),” in “Mathematical Approaches to Problems in Resource Management and Epidemiology,” C. Castillo-Chavez, S. A. Levin, and C. A. Shoemaker (eds.), Lecture Notes in Biomathematics, 81, Springer-Verlag, 1989. 15. E. N. Chukwu, “Mathematical Control of AIDS Epidemic,” Preprint. 12.
Chapter 2 General Linear Equations 2.1
The Fundamental Matrix of Retarded Equations
Let E = (-00, co),and E" be a real n-dimensional Euclidean vector space. Let h > 0. Suppose C = C([-h, 01, E"), the space of continuous functions from the interval [-h, 01 into E" with the sup norm 11 defined by 11d11 = sup Iq5(s)I. Consider the linear retarded differential equation 8
E [- h ,O]
(2.1. l )
r>o, where A i , i = 0,. . . , N are constant n x n matrices, f is continuous on E , and 3: is an n vector. If d(0) = 20,and r N = h and E C , then a unique solution x(dIf) of (2.1.1) exists on [-h, 00) and coincides with 6 on [-h, 01. The proof of this is contained in Hale [4, pp. 14,1421. We now represent this solution in terms of the fundamental matrix of the system Z(t) = A o ~ ( t )
c(0) = co
(2.1.2)
obtained from (2.1.1) by setting f = 0 and Ai = 0, i = l , . . ., N . Let X be an n x n matrix function whose column vectors el,... ,en constitute a fundamental set of solutions of (2.1.2) with the property that X ( 0 ) = I , the identity matrix. Then X ( t ) = eAoi and z ( t , z ~= ) eAotco is the unique solution of (2.1.2). Using this we know that the solution of the nonhomogeneous differential equation
(2.1.3) is by the method of variation of parameter given as
(2.1.4)
35
36
Stability and Time-Optimal Control of Hereditary Systems
If x = x(q5,f) is a solution of (2.1.1), which coincides with q5 on [-h,O], then by (2.1.4)
= q5(t), t E [-h,Ol x ( t ) = e Aotq5(0) +
/
t
+
1
A ~ z (s ~ i )f(s) d s .
eAo(t--s)
0
i= 1
(2.1.5a) (2.1.5b)
Clearly (2.1.5) satisfies (2.1.1). That (2.1.5) is unique follows from its explicit calculation by the so-called method of steps: On the interval [0, h] the function x is uniquely represented by
With x defined and continuous on the interval [O,h],we use (2.1.5b) to obtain I on the interval [h,2h]. The process is continued. We now use the method outlined above to calculate the fundamental matrix solution U of the system
.(t) = A o x ( t )
+
c N
A;x(t - i T )
i=l
x ( 0 ) = 10, q5
(2.1.6)
0 on [-h,O)
in terms of i A o t . U is a matrix solution of (2.1.6), which also satisfies the initial condition 0, t < O U(t)= I , t = 0, I identity matrix. Consider a simple version of (2.1.6). The next result is valid:
Consider
Proposition 2.1.1
i ( t )= A o I ( t )
+ Aix(t - h ) ,
x ( 0 ) = Io, x ( s ) = 0, s E [-h,O).
(2.1.7)
Then U ( 0 )= I
u(t)= e A o t , t
t E [o, h]
E (kh,
(k
+ l)h],
(2.1.8)
37
General Linear Equations where we understand that
t +
0
C = 0.
i=l
Thus
.lil
)Aldsk..
l h eAo(t--sk
eAo(aa-al)A1 . e A o ( 8 l - h ) d g l .
We next give an expression of U ( t ) for the general N-delay terms of (2.1.6). Let 20 E E" and U k ( t ) = U ( t define the k-system as follows:
+ (k - l)h),
t E [O, h] k = 1 , 2 , . . . and
where 0 is the zero n x n matrix. Because uk(t)zo is the function translated by (k - 1)h of the solution z ( t ; z o ) of (2.1.6), the kth system is equivalent to step-by-step integration of (2.1.6). By induction,
where Uf(O)zo= U l - l ( h ) z o . From this one obtains the following recursive formula for uk(t) : U , ( t ) = eAot:
U 2 ( t )= eA'J(*+h)+
1
Uk(t) = eAo'Uk-l(h)
t
eAo(t-S)Al&Sds
+
min(k-1,N)
1
E L eAo(t-*)AiUk-lds, k 2 2. i=l
This is a simple induction formula. It is desirable to write uk((t),k 2 1 in terms of eAot and Ai only. For this we let I( = Uc(0)zo l! = 1,... ,k. Then from (2.1.9a),
Stability and Time-Optimal Control of Hereditay Systems
38
The matrices V I ( t ) .. . Vk(t) are defined by induction as
V l ( t )= e A o t , rnin(k-1,N)
C
Vk(t)=
i=l
1 t
eAo('-')AjVk-l(s)ds, k = 2 , 3 , . . . .
These formulas for h ( t ) will enable us to give a simpler expression of Ur;(t) than (2.1.9a), and then for V ( t ) . We now consider (2.1.6). The explicit form of U ( t ) is complicated. We introduce some notation. Define the index A(j, k) for all j = 1 , 2 , . ' and k = 1 , 2 by
-
A ( j , k ) = { ( i l , - - ,. i j ) : 1 5 i l , .. . , i j 5 N and i l + . * . i j= k}. Note that A(j, k) = 4 for j > k. The following finite dimensional analogue of a result of Nakagiri [ll]gives an explicit representation of U ( t )in terms of eAot and Ai i = 1 , . . . , N . Proposition 2.1.2 Define the matrix operator Vk C = 1 , 2 lows: v l ( t ) = eAo*
1 .
as fol-
(2.1.9b)
The fundamental matrix U ( t )t
2 0 of (2.1.6)
is given by
k
U ( t )=
C q ( t - ( i - 1 ) ~ ) t E [(k - l ) ~k,~ ] .
(2.1.1Oa)
i= 1
Writing o u t some expressions of v k , we have
(2.1.10b)
+
1
1
eAo('-'I)A1
1"
eAo('l-s)AleAo('))ds)dsl.
The expression (2.1.10) becomes simpler if Ai i = 1 e A o t . We have:
N commutes with
General Linear Equations
39
Corollary 2.1.1 Assume that for each i and for all t _> 0, Ai commutes with e A o t , i.e., AieAoi = eAotAi i = 1, .. , N. Then
-
For a special case of (2.1.7), namely
i ( t ) = Aoz(t)
+ A l z ( t - l),
(2.1.12)
we give an alternative description [7] of the fundamental matrix U. We note that by definition
a
-U(i at
- S) = AoU(t - S) + A1U(t - s - l),
(2.1.13)
for (t, s) E [slT ] x [0, TI, where
U ( t - s ) = 1 (n x n identity matrix f o r t = s ) = 0, ( t l s ) E [-I,$) x p,q.
(2.1.14)
+
Let Uk(r) = U ( r k) for r E [0,1] k = O , l , . .. , and assume s = 0 in (2.1.13). Then by substituting in (2.1.13) we have
d --Uo(r) = AoUo(7), Uo(0) = 1, dr (2.1.15)
It follows that the solution of (2.1.13) and (2.1.14) over the interval t E [ k , k 13 is given by U ( t ) = Uk(t - k). Set Z k ( r ) = [U,'(T)~... ,U,'(r)IT to convert (2.1.15) into the system
+
d -Zk(r) = BkZk(r) for r E [0,1], dt
Uk(7)= EkZk(T),
(2.1.16) (2.1.17)
Stability and Time-Optimal Control of Hereditary Systems
40
+
where zk(~) is an n(k 1) x n matrix, and Bk and Ek are respectively n(k 1) x n(k 1) and n x n(k 1) matrices given by
+
+
+
Because the unique solution of (2.1.16) is
(2.1.18) (2.1.19)
We note that
Zo(0) = I
Zk(0)=
[
I
.J,
.....
e B r - i Z(O)
k = 1,2,..
(2.1.20)
From the definition U ~ ( Tand ) from (2.1.19), we deduce that
V ( t )= Uk(t - k) = EkeB*('-k)Zk(0)
(2.1.21)
+
for t E [k,k 11. The essential difference between the fundamental matrix solution of the ordinary differential equation (2.1.2) and that of the delay system is that U ( t ) may be singular for some values of t E [0, oo),whereas eAot i s always nonsingular. Indeed, consider the system
k ( t ) = A o z ( t ) - e A o z ( t- 1). We have
It is clear that V(2) = 0, so that the fundamental matrix is singular at t = 2.
Problem 2.1.1: Find fundamental matrixsolution V ( t )using both (2.1.10) and (2.1.8). Check the validity of solutions $ ( t ) = - z ( t ) + z ( t - 1) on (0,3].
41
General Linear Equations
U ( t ) = e-t,
t E [0, 13, U ( t ) = e - ' [ l + e(t - I)],
U ( t ) = e-t[l
+ e(t - 1) + $ e 2 ( t 2 - 4(t + 4)].
42
Stability and Time-Optimal Control of Hemditary Systems
Check validity by substitution. Does V ( t ) satisfy i ( t ) = - z ( t ) + z(t 1); z ( t ) + c ( t ) - z ( t - ~ ) = on [0,1] ~ ( t=) e-', U ( t ) = -e-', ~ ( t - 1 )= 0. Substitute -e-'
+ e-' + 0 = 0 :.
solution valid on [0,1].
Substituting into the equation, we have
The solution is valid on [1,2]. On R31,
Substituting
Therefore the solution is valid on [2,3]. Using (2.1.81 On [ O J I ,
~ ( t=) eAot = e-'.
43
General Linear Equations
We now check the validity of this by substitution. Does U ( t ) satisfy
.(t)
+ z ( t ) - z ( t - 1) = O?
On interval 10, I],
~ ( t=)e-t, ~ ( t=)-e-t, U ( t - 1) = 0. Substituting in the equation, we'have e-t - e-t - 0 = 0. The solution is valid on t E [0, 11. On interval [l,2],
Substituting, e-te - e-'(l+ e(t valid on t E [1,2].
- 1)) + e - * ( l +
e(t - 1)) - e-'e = 0. The solution is
44
Stability and Time-Optimal Control of Hereditary Systems
On interval [2h,3h],
This is the same solution obtained for t E [2,3] using Equations (2.1.10a) and (2.1.10b).
2.2 The Variation of Constant Formula of Retarded Equations Using the fundamental solution U ( t ) of section (2.1), we represent the solution x(+, f ) of the nonhomogeneous equation (2.1.1). We will show that this is given by
The variation of constant formula is presented for a very general linear retarded functional differential equation,
45
General Linear Equations
where f is locally integrable on E. In this case by a solution we mean a function z that satisfies (2.2.3) almost everywhere. We assume f is a function mapping [u,oo)-+ En;L : [O,m) x C + E" is continuous and linear in q ,i.e., for any numbers cr,p and functions 4,$ E C we have the relation L ( t , a$ p4) = a L ( t ,4) PL(t,+). It follows from the Riesz Representation Theorem [6, p. 1431 that
+
+
L ( t , 4) =
J
0
-h
[del7(t,f?)l4(e),
(2.2.4)
where ~ ( 4) t , is an n x n matrix function that is measurable in ( t ,4) E E XE , such that ,
~ ( 0t ),= 0 for e 2 0, V ( t , 0 ) = ~ ( -h) t , for 8 5 -h. Also ~ ( 6t ),is continuous from the left in 8 on ( - h , O), of bounded variation in 8 on [-h,O] for each t. Also there is a locally integrable function m : (-00~00)
-+
E such that
IL(t,4)I 5 m(t)ll4ll,
vt
E (-O0,m), 4 E c.
Included in (2.2.3) is the system N
+J
i ( t ) = [C ~ ~ ( t ) w~k() t k=l
0
-h
~ ( t ,
+qde +f(t),
(2.2.5)
where A k ( t ) are n x n matrices. A(t,O) is an n x n matrix function that is integrable in 0 for each t , and there is a locally integrable function a ( t ) such that
The fundamental matrix of
i ( t ) = L ( t , 2 * ) , 2, = 4 is the n x n matrix U ( t ,s), which is a solution of the equation
-a U ( t ' s ) - L ( t ,U t ( . , s ) ) , t 2 s dt
U(t,S)
(2.2.6)
a.e. in s and t ,
(2.2.7)
=
0 s-h
I
identity map t = s,
For the genera1 system, under the conditions on L , f we have:
Stability and T i m e - O p t i m a l Control of Hewdata y S y s t e m s
46
Proposition 2.2.1 For any u E E, 4 E C there exists a unique solution z(u,6)that is defined and continuous on [u- h, co) and satisfies (2.2.3) a.e. on (6, co). This solution is given as
2 ,
= 6,
where z(u,6,O) is the solution of the homogeneous equation (2.2.6) through d), and V ( t ,s) is the fundamental matrix, which is continuous for t 1 s and measurable in t, s. Equation (2.2.9) gives z(u, 4, f ) ( t )as a point of E". To obtain a solution in C, we write (6,
zt(fl,d,f)= zt(u,d,O)
+J
t
lJt(.,s)f(s)ds, t
3
(2.2.10)
Ql
where i t is understood that the integral equation is an integral in we define zI by zt(u,610)
SfT ( t ,u)4
En.If (2.2.11) (2.2.12)
where
{ I,
-h 5 6 < 0, (2.2.13) 6 = 0, then T ( t , u )is a continuous linear map and Equation (2.2.10) becomes
Xo(6) =
Zt(b,4,f)
2.3
=T(t,fl)4+
O'
t
T(t,s)Xof(s)ds, t 2
6.
(2.2.14)
The Fundamental Solution of Linear Neutral Functional Differential Equations
In this section we consider linear neutral functional differential equations that have the form
for t 2 0,
z ( t ) = 6(t), d E c, t E [-h,O), z(0) = zo.
(2.3.1)
47
General Linear Equations
Here 0 5 hi 5 h2 < _< h~ = h , and A - l j , A o , A l j , j = 1,... , M , are constant matrices. Equation (2.3.1) has a simpler form in
d -[z(t) dt
- A - l z ( t - h)] = Aoz(t) + A l z ( t - h ) + f ( t )
(2.3.2)
with the same initial conditions. A more general form of (2.3.1) is the system
-P+*J = Lzt + f ( t ) , dt d
xu
t 2 0,
= 4 E C, 4(0) = 20 E En,
(2.3.3)
where
(2.3.4) and
D+t = + ( t ) -
Lh 0
dp(8)+(t
+8).
(2.3.5)
In (2.3.4) and (2.3.5) the functions q , p are n x n matrices on [-h,O] of bounded variation that are understood to have been extended on E by requiring, for example, that q(S) = q ( - h ) , for 0 5 -h, and q ( S ) = v(O), for 8 2 0. The integration is given by the Ftiesz &presentation Theorem in the Lebesgue-Stieltjes sense. It is assumed in (2.3.5) that D is atomic at zero in the following sense:
Definition 2.3.1: The map D : E x C variable and which is given by
~
t01 =,
J
+ E", which
is linear in the second
0
-h
deq(tl 4 ~ 6 )
(2.3.6)
is atomic at a on E x C if, whenever
we have det(A(t, a))# 0 V t E E. In (2.3.2), D ( t , 4) = 4(0) - A - l 4 ( - h ) and D is atomic at zero. In (2.3.5) we assume that the variation of p , Var[-,,op 0 as s 4 0.
-
Definition 2.3.2: The fundamental matrix solution of (2.3.2) is the solution V ( t )of V ( t )- A - l V ( t
+
- h ) = AoU(t) AiU(2 - h ) ,
(2.3.7)
Stability and Time-Optimal Control of Hereditary Systems
48
for t # kh h = 0 , 1 , 2 , . . . and U ( t ) = 0, t < 0, U ( 0 ) = I identity. For the system d (2.3.8) -[Dzt] = L q ,
dt
the fundamental matrix is the n x n matrix solution (of bounded variation on compact intervals that is continuous from the right) of the equation
o(Ut) = I
+
t
L(U,)ds, t 2 0, (2.3.9)
0, - h < O < O I, (identity) 8 = 0. It is important to get an expression of the fundamental matrix solution of (2.3.8). Because it is complicated for the general situation, we consider
1
M
dt
j=1
M
+ C AIjz(t - h j ) ,
E(t) - C A - l j z ( t - h j ) = A o ~ ( t )
j=1
(2.3.10)
Our aim is to represent U explicitly in terms of e A o t , A-lj and A1j j = 1,.. . , M . The expression we give is a result of a much wider statement in Banach space setting by Datko [2]. Proposition 2.3.1 Consider (2.3.10). (2.3.10) satisfies the identity
+
M
U ( t ) = eAof x X ( t - h j ) A - l j U ( t
The fundamental matrix U of
- hj)
j=1
where X ( s ) = O if s < 0, and X ( s ) = 1 if s 2 0.
Proof: Let
eAot
be the fundamental solution of
i ( t ) = AOE(t).
49
Genera1 Linear Equations We apply the variation of parameter (2.1.4) to the system
i ( t ) = Aoz(t) +
M
C Aljz(t - hi)
j=1
M j=1
Then the integrated form of (2.3.10) is the equation
if t 2 0 and z ( t , O , + ) = d ( t ) , 4 E C if t E [-h,O]. Let the Laplace transform of this solution z ( t , O , O , 4) be denoted by F(s,4). If we observe that the Laplace transform .C{eAo'}(s)= (sZ - Ao)-', then
We now use the properties of Laplace transform and the finite dimensional version of the reasoning of Datko to deduce from (L) the following expression:
50
Stability and Time-Optimal Control of Hereditary Systems
+
M
[(I - (sI- AO)-'Ao)Alje-"J j=1
Consider now the left-hand side of (L) and the equation in 3 given by
We now invert this expression and transfer all terms except U ( t ) , (where U ( t ) = L - ' ( S ( s ) ) ( t ) ,the fundamental matrix of (2.3.10)), to the right-hand side. This yields (2.3.11). Notice that we have assumed IU(t)l = 0 if t < 0. Assuming eAot is known, Equation (2.3.11) is used to compute U ( t ) by the method of steps over intervals of the form [khl, (k+l)hl]. For example, we consider Equation (2.3.2). Proposition 2.3.2
The fundamental matrix solution of
d - [ ~ ( t )- A - l ~ ( t- h ) ] = A o ~ ( t + ) A l ~ (t h) dt
(2.3.12)
51
General Linear Equations
is given as follows:
~ ( t=)e A o t , t E [ ~ , h ] ,
for t E [2h, 3h]. One can express the solution of the homogeneous system (2.3.12) with initial function 4 E C in terms of U , and furthermore obtain a solution of Equation (2.3.2) in terms of the homogeneous solution and another term. It is contained in the following statement whose proof is given in Hale [4].
Proposition 2.3.3 If U is the fundamental solution of (2.3.12), the solution ~ ( 4 of ) (2.3.12) with zo(4) = 4 is given by z(+,o)(t) = U ( t ) [#to) - A-14(-h)]
+ A1
1
0
-h
V ( t - 8 - h)4(B)de
(2.3.14) The solution z(4,f) of (2.3.2) with zo(4) = 4 is given by the variation of the constant formula rt
Also, if (2.3.8) is considered, and U is the fundamental matrix as defined by (2.3.9) while z(4,O)is the solution of (2.3.8), then the unique solution of (2.3.3) is the function z(4, f) with zg(4, f ) = 4 and
52
2.4
Stability and Time-Optimal Control of Hereditary Systems
Linear Systems Stabiiity 141
In this section we tackle at various levels the problem of global uniform asymptotic stability of linear hereditary systems that was posed in Section 1.9. First we characterize this desirable property by examining the properties of the characteristic equations of the linear systems. Next in Section 3.2 we shall use a Lyapunov-Razumikhin type theorem based on work by Haddock and TerjCki t o derive a computable sufficient condition, in terms of the system’s coefficients, of asymptotic stability of the linear system k ( t ) = qt,Z t ) , (2.4.1) where L is some variant of (2.2.4). We now specify our notion of stability. Note that L(t,O) = 0 V t E E .
Definition 2.4.1: The solution 2 = 0 of (2.4.1) is called uniformly stable if for any u E E c > 0 there is a 6 = 6 ( c ) such that if 11411 5 6, then ((q(c~,4)115 c, V 1 2 c. The solution z = 0 of (2.4.1) is uniformly asymptotically stable if it is uniformly stable and there is a positive constant X > 0, such that for every q > 0 there is a T(q)such that if 11411 < A, then llzt(o,q5)II < q V t 2 B T(q)for each B E E . If y ( t ) is any solution of (2.4.1), then y is said t o be uniformly stable if the solution z = 0 of
+
is uniformly stable. The concept of uniform asymptotic stability is defined similarly. In the definition given above, 6 is independent of c and therefore stability is uniform. If 5 = S(c,o), we will have mere stability. However, the two concepts coincide if L ( t , 4) is periodic in t and L ( t + w , 4) = L ( t , q5), or if L is time invariant. The concept of uniform boundedness is useful. It is defined as follows: The solution z(o,c,b) of Equation (2.4.1) is uniformly bounded if for any a > 0 there is a p(a) > 0 such that for all u E El Q E C and 11+11 5 (Y we have IIz(o, 4)(t)ll 5 for all t >_ u. Recall that L satisfies for some locally integrable A4 : (-m,m) --+ E the inequality
Stated below is a characterization of the two concepts of stability in terms of the fundamental matrix solution U of Equation (2.4.1).
53
General Linear Equations
Theorem 2.4.1 Consider Equation (2.4.1)where M in (2.4.2) satisfies, for some constant M I , the inequality
(2.4.3)
Then the solution x = 0 o f (2.4.1)is uniformly stable i f and only if there is a constant k such that
Also the trivial solution x = 0 of (2.4.1)is uniformly asymptotically stable i f and only if there are constants k > 0 and a > 0 such that for all s E E ,
IV(t,s)l 5 ke-"('-'),
t 2 s.
(2.4.5)
I f the solution x(u,q5) o f (2.4.1)defines the operator T as follows:
then we can replace (2.4.4)and (2.4.5)respectively by
and
IT(t,u)1 5 k e - u ( t - u ) ,
t 2 u.
(2.4.7)
Asymptotic stability can also be deduced from some properties of the characteristic equation of a homogeneous linear differential difference equation with constant coefficients (2.1,7),namely A(X) = I X - A o - A l e - x h = 0,
(2.4.S)
where X is a complex number. Equation (2.4.8)is obtained from (2.1.7)by substituting x = ext< in (2.1.7)where X is a constant and is a nontrivial vector. We have:
<
Proposition 2.4.1 Let V ( t )be the fundamental matrixsolution of(2.1.7). Let (Y = max{ReX : A(X) = 0). I f a < 0, we have
(2.4.9)
54
Stability and Time-Optimal Control of Hereditary Systems
and (2.1.7) is asymptotically stable. In this case if (2.1.7), which coincides with 4 on [-h, 01, then
and all solutions approach zero exponentially as t Consider the more general situation
--t
x(4)
is a solution of
00.
i ( t ) = Jqxt),
(2.4.11)
where L is a continuous linear function mapping C into En and described bY
L ( 4 ) = J0
-h
Pde)l4(e), 0 E c,
where q is an n x n matrix q(O),-h variation. Let
(2.4.12)
5 0 5 0, whose elements are of bounded (2.4.13a)
The characteristic equation for (2.4.11) is
det A(X) = 0.
(2.4.1313)
The following theorem is valid. Theorem 2.4.2 Let all the roots of the characteristic equation (2.4.13) of (2.4.11) have negative real parts. Then there are positive constants Q > 0, k > 0 such that if r(4) is a solution of (2.4.11) with ~ ( 4 =) 4, then (2.4.14) ll.t(d)ll 5 kI1411e-Q', t 2 0, (2.4.15)
The definition of uniform stability and uniform asymptotic stability can be applied to the neutral linear system (2.3.8), i.e., d
-[Ox*] =Lxt, dt
(2.4.16)
where (2.4.17)
55
Geneml Linear Equations and
D t t = t(t) -
JIO,
pp(e)j+
+ el,
and the variation Var[-,,o]p of p satisfies Vaq-,,~] p has no singular part. This means that
where
+0
as s
-
(2.4.18)
0 and p
00
DO4 = 4(0) + x A k # ( - W k ) , k= 1
and
0 < wk 5 h,
00
k=1
lAkl
+/
0
-h
IA(s)fds< 00.
We now give conditions in terms of the characteristic equation det A(X) = 0, A(X) = X D ( e X I )- t ( e X I )
(2.4.19)
for uniform asymptotic stability. Theorem 2.4.3 Suppose in (2.4.16) where L and D are given as in (2.4.17) and (2.4.18), we assume that A(A) in (2.4.19) satisfies
sup{Re X : det A(X) = 0)
< 0.
Then the zero solution of (2.4.16) is uniformly asymptotically stable. Furthermore, if ~ ( 4 )is the solution of (2.4.16) with t o = 4, and if U is the fundamental matrix solution of (2.4.16), then there exist constants k > 0, a > 0 such that
t 2 0. For the autonomous system (2.4.16) stability implies uniform stability, but asymptotic stability does not imply uniform asymptotic stability. It is know in Brumley [l]that we can have the eigenvalues X with Re A < 0 and have some solutions unbounded. To have the same situation as in retarded
56
Stability and Time-Optimal Control of Hereditary Systems
equations where for autonomous systems or periodic systems uniform asymptotic stability is equivalent t o asymptotic stability, we require that D be stable in the following sense:
Definition 2.4.2: Suppose D : C -+ En is linear, continuous, and atomic at 0. Then D is called stable if the zero solution of the homogeneous difference equation, Dyt=O, t > O (2.4.20) Yo=$, D$=O is uniformly asymptotically stable. Let the characteristic equation of D be
Stable operators are characterized by the following theorem: Theorem 2.4.4 The following assertions are equivalent: D is stable. CYD = sup{& A : det AD(A) = 0) < 0. Any solution of the nonhomogeneous equation
D Y ~= W),t 2 0 , where h : [0, co) -+ En is continuous, satisfies
(2.4.21)
-
where a > 0, b > 0 are some constants. Let D+ = d(0) - Jzh[dc((S)]d(S),V a q - s , y --,0 as s 0, and p has no singular p a r t . Then all solutions of the characteristic equation det satisfy Re X
5 -6
[I
-
for some 6
Lh
eA”4(S)]
=0
> 0.
There are simple examples of stable operators: 1. Let D$ = d ( 0 ) . This corresponds t o the retarded functional differential equations. 2. D$ = $(O) -A$(--h), where the eigenvalues of A have moduli < 1. D is stable because the eigenvalues of det[l - Ae-Ah] = 0 have Re A 5 -6 < 0. It is illuminating t o state a corollary of Theorem 2.4.3.
57
General Linear Equations
Consider the system
Corollary 2.4.3
- A - i z ( t - h)] = A o ~ ( l+) A l ~ (-t h).
-d[ z ( t )
dt
(2.4.22)
Suppose oo = sup{Re X : X ( 1 - A - l e - x h ) = Ao
+ A l e - A h } , ao < 0
z(4) is a solution of (2.4.22), which coincides with q5 on [-h, 01. Then Iz(q5)(t>l5 ke-afllq511, t 2 0 for some k 2 0 and o > 0.
and
2.4
Examples
Example 2.4.1 [9]: Consider the delay system
where a , q , k, and h are positive constants. If we assume
b > q,
(2.4.24)
then (2.4.23) is uniformly asymptotically stable. We shall prove this by showing that every root of the characteristic equation
A’
+ b~ + qXe-xh + k = o
(2.4.25)
has negative real part. Indeed, suppose that X = a+ ip is a root of (2.4.25) with a 2 0, and aim a t a contradiction. Suppose j3 = 0. Then the left-hand of (2.4.25) will be positive yielding a contradiction. Hence j3 # 0. Note that
Im[X2 + bA
+ qXe-xh + k)//3] ah sin ph ph ah) > b - q > 0.
2 2 a + b + qeVnh
> b - q e e Q h ( l+
Since the imaginary part of (2.4.25) is zero we have deduced a contradiction. Hence every X has negative real part.
Example 2.4.2: Consider (2.4.23). The following is another sufficient condition that depends on the magnitude of h for the uniform asymptotic stability of (2.4.23): b+q>O,
(b+q+ki)h
(2.4.26)
Stability and Time-Optimal Control of Hereditary Systems
58
To see this, assume t ha t X = a + ifl with 0 = 1X2
(Y
>_ 0. Then
+ bX + qXe-xh + kl >_ IX(’
- (b
+ q ) l A ( - k.
From this we deduce that
Hence
+
As before, X = a ig, lphl < a / 2 . With this,
(Y
lXhl < a / 2 . 2 0, /3 = 0 is impossible for (2.4.23). Hence
+ bX + qXe-xh + k ) / f l = 2 a + b + qe-crh
Im(X2
sin fl h
> 2 a - qe-Uhah= 2 a - qah 2 0.
Ph
This contradiction proves the required result.
Example 2.4.3: The same methods yield the following sufficient conditions for uniform asymptotic stability of
+
i ( t )= a o t ( t ) a l x ( t - h ) : h > 0, + a1 < 0, i ( t ) 1- a l i ( t - h ) : C Z ~> 0 , 0 5 a l h < ~ / 2 . Z(t)
a0
+ b i ( t ) + q i ( t - h i ) + k ~ ( t+) p ( t - h2) = 0 :
+
5 0. (2.4.27) (2.4.28)
b, q , k , ~h l, , h2
> 0;
2.5 Perturbed Linear Systems Stability [lo] In this section we consider conditions that will ensure that the stability properties of the homogeneous linear equation
.(t) = L ( t , q ) will be shared by the system
&(t)= q t ,.t)
+ f(t,.t),
(2.5.1) (2.5.2)
where f ( t ,0) = 0. In (2.5.1) there is a locally integrable function m : (-co,CO) such that
Mi,411 Im(t>ll411, v t E ( - W W ) .
Also the function f : E x C -+ En is continuous. The following lemma is well known:
(2.5.3)
Geneml Linear Equations
59
L e m m a 2.5.1 (Bellman’s Inequality): Let u and a be real valued continuous functions on an interval [u,b],and assume that /3 is an integrable function on [a,b] with P 2 0 and
Then
+
u ( t ) I a(t)
/
a
t
P(s)a(s)[exp / ‘ P ( ~ ) d r ] d t ~ a, S
5 t 5 b.
(2.5.5)
If we assume further that a is nondecreasing, then
Theorem 2.5.1 Suppose the system (2.5.1) is uniformly stable and there is a locally integrable function 7 : [0, 00) + E such that
then (2.5.2) is uniformly stable on [O,oo). Suppose (2.5.1) is (i) uniformly asymptotically stable, so that there exist some constants k 2 1 a > 0 such that every solution of (2.5.1) satisfies
where c
> 0 is the constant c = a / 2 k and x is an integrable function with
n=
JOm
.(t)dt
< 00.
Then every solution z(9, c) of (2.5.2) with z,(c, 4 ) = 4 satisfies
60
Stability and Time-Optimal Control of Hereditary Systems
Remark 2.5.1: Condition (ii) of the second part of Theorem 2.5.1 is not too severe. See [5, p. 861. They are natural generalizations of conditions stated in the famous theorem of Lyapunov on stability with respect to the first approximation. Proof: From Theorem 2.4.1 and condition (2.4.6), IT(t,a)l I k , t 2 u for some k > 0, because (2.5.1) is uniformly stable. Also IT(t,u)XoI I k V t 2 u where X Ois given in (2.2.13). The variation of constant formula (2.2.14) applied to (2.5.2) implies bt(d,,u)ll 5 IIT(t,o)d,)II
+
1 t
IIT(t,s ) x o f ( s ,z,)dsll
t
< ~lldIl+k J 7(S)ll~.dlldS, t 1 0Apply Bellman's Inequality (2.5.6) to this to deduce that
[c lw
5~II~ exp II
r(s)ds]
for all d, E C. This proves that (2.5.2) is uniformly stable on [O,oo). For the second statement, recall that by Theorem 2.4.1 IIT(t,a)ll I /ce-Q(t-u), IIT(t,u)Xoll 5 k e - Q ( t - u ) for t 1u.
It follows from the variation of constant formula applied t o (2.5.2) that
Thus if
tt = e Q t z t ,then
111 t.1
L ~l1411eQU+
J
t [E
+ r(s)lll..dlld..
Using Bellman's Inequality (2.5.6) once again we deduce from this estimate that
61
General Linear Equations
This yields
where c = a/2k, M = kII. The proof is complete. Condition (ii) of Theorem 2.5.1 can be relaxed by using a recent sharper generalization of Bellman's Inequality due to En-Hao-Yang [3]. Lemma 2.5.2 Let C ( J ,E + ) be the Banach space of continuous functions taking J into E+ = [0, co). Let the following conditions be satisfied: (i) ~ ( tE)C ( J ,E + ) is a nondecreasing function with a(t) 2 1, V t E J. (ii) The functions u ( t ) , f i ( t ) E C ( J ,E+). (iii) The inequality
u ( t ) 5 ~ ( t +)
2 J' i=l
f i ( ~ ) ( u ( ~ ) ) r i dt sE, [0, T ] 3 J
0
holds where Ti E (0,1]. Then we also have the inequality u(t) 5 a(t)
fi G ; ( t ) t E J
i=l
where
here in
n Gk(t)= 1 t E J. 0
i=l
Theorem 2.5.2 In (2.5.2) assume that (i) f ( t ,0,O) = 0. (ii) The system (2.5.1) is uniformly asymptotically stable. (iii) For each E C,t 2 u 2 0 we have
+
n
If(tld)I _< Cci(t)lld llr i + C n + l ( t ) , i=l
c ES
Stability and Time-Optimal Control of Hereditary Systems
62
where ri E (O,1] are constants, c j ( t ) E t i o n s , j = 1 , 2 ,... , n + l . (iv) c j ( s ) e a b d s < 00, j = 1 . . . n 1. Then all solutions of (2.5.2) satisfy
Jr
C(E+,E + ) are known func-
+
llCt(u,4)II
for some constant M
I M ( u , 4)e-'(t)
= M ( u , 4) and
Q
> 0.
Proof: Just as before, using uniform asymptotic stability of (2.5.1) and the variation of constant formula, we have
Now apply the lemma to deduce the inequality
here in
n & ( t ) = 1, 0
k=l
t 2 u. Consequently, Ibt(u,4111
where lim D ( t )
t-co
by hypothesis (iv). Hence
I e-atD(u,41,
n:==, R(t)= D ( u , 4 ) <
00,
63
Geneml Linear Equations completing the proof.
Example 2.5.1 [14]: We have seen that under certain conditions on f the stability properties of (2.5.1) are shared by those of (2.5.2), (at least locally). Indeed, if f is smooth enough, a nonlinear system
i ( t ) = f(t,2 1 ) can be written (locally) in the form (2.5.2), where L(t,z t ) = D 2 f ( t ,z,)q and D2f (t, z , ) is the F’rF’chet derivative with respect to zt about the equilibrium I,. This insight enables us to determine stability about equilibrium of nonlinear systems. The next example from Brauer illustrates this. Let + ( t ) be the number of a population susceptible to an infectious disease. We assume that births and deaths other than those caused by the disease are dependent on the number of susceptible members. If g(z) represents excess of births over deaths in a unit time when the susceptible population is z,then ~ ( tsatisfies ) the following:
1
i ( t ) = g ( z ( t ) )- P z ( t ) z(t
- hl
- hz) - z(t - h l )
+
/
1-hi
t-hl-ha
g(z(s)).”]
.
(2.5.7)
Proposition 2.5.1 Suppose in (2.5.7) (i) g(0) = g ( k ) = 0 , g ( z ) > 0 for 0 < z < k where k is some positive constant (the carrying capacity in the absence of disease). (ii) g ( z ) > zg’(z) 0 < z < k . Then (a) The equilibrium z , = 0 is unstable. (b) The equilibrium , 2 = k is stable if (iii) Ph2k < 1, and unstable i f p h z k > 1. 1 (c) The equilibrium x, = -is stable if Phak Ph2
0 and a + ch2
> 0 or Ph2g
Ph2!7(%3)- g‘(z,),
c
(L) -2q’(L) Ph2 Ph2
= -Pzmg
(Pi2)
>
’
0 with a =
‘(Go).
Proof: The equilibria for (2.5.7) are solutions Y(zm)
> 1 or ifg’ -
of
2,
- ~h2Zw9(%0)= 0.
If g ( z , ) = 0 , which implies that
zoo= 0 (and we
1
have extinction), or zoo=
k (disappearance of the disease), or I, = -(endemic equilibrium), then m 2
Stability and Tame-Optimal Control of Hereditary Systems
64
we now linearize about the equilibrium x,:
i ( t ) = [g’(x,) - p h 2 g ( ~ m ) ] t . ( t-) pzcoz(t x
J’+
z(s)ds,
t-hi-h2
and construct the characteristic equation-the z ( t ) = ceXt is a solution of (2.7.8):
“
= (Xb +~)e-’~l At
X,
= 0, the equation (2.7.8) is Z(t)
- hl - h)- Ptmg‘(Xa,) (2.7.8) condition on X such that
J-ha-hi
(2.7.9)
-
= g’(O)z(t).
This is unstable by hypothesis (ii). For the equilibrium 2, = k , we observe that a = -g‘(k) > 0, b = p k , c = -pkg’(k) > 0, since g ( k ) = 0, g’(k) < 0. But the roots of (2.7.9) have negative real part (by [S]) if ph2k < 1 and unstable if Ph2k > 1, since g ( k ) = 0, g‘(k) < 0 and all the roots of (2.7.9) satisfy real A < 0 if hl 2 0, h2 2 0, 0 < Iclh2 < a, lblhz 5 1, and we have
= -g‘(k) > 0, b = p k , c = -pkg’(k) > 0 .
a
At the equilibrium x,
1 1 a = -g Ph2
Ph2
(&)/ h 2 . Thus ($-) > 0. Thus if (&) < 0
b = g
=
1 , c = -g’ h2
0
g‘
(A) > .. ($-) (k)>
0 so that c
stable. If g‘
(&) - cr > so that c
g‘
(L) > 0, by (ii) Ph2
0 if p h z k > 0, so that
> 0,
the equilibrium is
< 0, we have stability
-,
if a
+ c r > 0 or
- 2g’ 0. We sum up: If ph2k < 1, the only stable equilibrium is x, = k , and the disease disappears. If p h z k > 1 , there is 1 an endemic equilibrium x, = I and this is stable if Phzk is sufficiently phzg
Phz close t o 1 and unstable (with oscillation about x,) large.
if ph2k is sufficiently
65
General Linear Equations
REFERENCES 1. W. E. Brumley, “On the Asymptotic Behavior of Solutions of Differential Difference Equations of Neutral Type,” J . Differential Equations 7 (1970) 175-188. 2. R. Datko, “Linear Autonomous Neutral Differential Equations in a Banach Space,” J . Differential Equations 25 (1977)258-274. 3. En-Hao, Yang, “Perturbations of Nonlinear Systems of Ordinary Differential Equtions,” J. Math. Anal. Appl. 103 (1984) 1-15.
4. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 5. J. Hale,
Ordinary Differential Equations, Wiley, New York, 1969.
6. S. Nakagiri, “On the F’undamental Solution of Delay-Differential Equations in Banach
Space,” J. Differential Equations 41 (1981) 349-368.
7. R. B. Zmood and N. H. McClamrock, “On the Point-Wise Completeness of DifferentidDifference Equations,” J . Differential Equations 12 (1972)474-486. 8. F. Brauer, “Epidemic Models in Population of Varying Size,” in “Mathematical A p proaches t o Problems in Resource Management and Epidemiology,” edited by C. Castillo-Chavez, S. A. Levin, C. A. Shoemaker, Lecture Notes in Biomathematics, 81, Springer-Verlag, 1989. 9. R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. 10. E. N. Chukwu, “Null Controllability of Nonlinear Delay Systems with Restrained Controls,” J . Math Analysis and Applications 76 (1980) 283-296.
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Chapter 3 Lyapunov-Razumikhin Methods of Stability in Delay Equations [ 5 ] 3.1
Lyapunov Stability Theory
For linear and quasilinear systems the methods of the previous chapter are adequate. For inherently nonlinear systems the ideas of Lyapunov as extended by LaSalle and Hale are powerful tools. For hereditary systems these methods require the construction of Lyapunov functionals. The main theory is now presented. Consider the nonlinear system
where
f ( t ,0) = 0.
Suppose V : E x C + E is continuous, and We define
(3.1.2) Z ( U , ~ is )
a solution of (3.1.1).
The following result is valid [3, p. 1051.
Theorem 3.1.1 Suppase f : E x C 4 En in (3.1.1) takes E X (bounded sets of C) into bounded sets of E n , and the functions u, v , w : E+ + E+ are continuous, nondecreasing functions with the property that u ( s ) ,v ( s ) are positive for s > 0, u(0) = v ( 0 ) = 0. Suppose there exists a continuous function V : E x C -+ E such that (4 ~(l4(0)1)L V ( t ,4) L 4ll4Il), and V(t7 4) 5 -W(l4(0>)1 then c = 0 is uniformly stable. If u(s) -+ 00 as s 4 00, the equation is uniformly bounded. If w ( s ) > 0 for s > 0, the solution c = 0 of (3.1.1) is uniformly asymptotically stable. Example 3.1.1: Consider the system (3.1.3)
67
68
Stability a n d Time-Optimal Control of Hereditary Systems
We now show that if a > Ibl, then (3.1.3) is uniformly asymptotically stable. We use the Lyapunov functional
where p
> 0.
Note that (3.1.3) is the same as
so that
This is a negative definite quadratic form in d(0) and # ( - h ) if
a
as possible. As a consequence, (61 < a. 2 It is now very easy to see that conditions (i) and (ii) are satisfied with .(l#(O)l) = +#’(O) and .(11411) = V(4), and .i(ld(O)l) 5 for Some 0-> 0. We now apply Theorem 3.1.1 t o an inherently nonlinear system.
Note that if p =
-, (bl is as large
@m)
Example 3.1.2: Consider
i ( t ) = a ( t ) x 3 ( t )+ b ( t ) x 3 ( t - h ) .
(3.1.4)
Assume that there are some constants
6 > 0 and q , such that
a ( t ) 5 -6 < 0, 0 < q < 1,
(3.1.5)
(3.1.6) Then (3.1.4) is uniformly asymptotically stable. Indeed, consider the functional
Lyapunov-Rarumikhin Methods of Stability in Delay Equations
69
which obviously satisfies condition (i) of Theorem 3.1.1. If we differentiate, we obtain
by (3.1.5) and (3.1.6). Thus conditions (ii) of uniform asymptotic stability of Theorem 3.1.1 are verified. For autonomous systems an interesting theory is available for investigating the asymptotic behavior of solutions. Consider
(3.1.7) where f : C
--t
En is continuous on some open set
s = {4 E c : lldll < HI H is constant. We denote the solution of (3.1.7) by z(4) and define a path or motion in C through 4 as the set U z t ( 4 )where the solution z(4) is defined on [0, a).
o
Definition 3.1.1: A point 1c, E C is said to be in I‘+(d), the positive limiting set of 4, if the solution z(4) is defined on [ - h , co) and there exists a sequence of nonnegative real numbers {tk} such that t k + 00 and k -+ 00 and l i l t k (4)- $11 -+ 0 as k -+ 00. A subset M of C is said to be an invariant set if for any 4 E M , the solution z(4) of (3.1.7) is defined on (-00, 00), and xt(+) E M for all t E (-m, co).The next two statements are technical but important. They help validate the so-called invariance principle of LaSalle and Hale for differential equations with delay. Lemma 3.1.1 l f z ( 4 ) is a solution of (3.1.7), which is defined on [-h, co) and satisfies 11zt($)11 5 a1 < a for all t E [0, co), then the set of functions (zt(4) : t 2 0) belongs to a compact subset of C. Proof: Recall that a metric space X is compact if we can select from any sequence {zn}of points on X a subsequence ( i n k } convergent to the space X.In the space of functions, the following Arzela-Ascoli Theorem is used to prove that a set is compa.ct:
70
Stability and Time-Optimal Control of Hereditary Systems
Arzela-Ascoli Theorem: A necessary and sufficient condition for the space S of continuous functions 3: : [a,b] -+ En with the sup norm to be compact is as follows:
(i) There exists a constant Ic such that
(ii) S is equicontinuous, i.e., for every
6
> 0 there is 6 such that
if
It - t‘l < 6, then Iz(t) - z(t’)l < E for all z E S t , t’ E [u, b]. (iii) Condition (ii) is satisfied whenever Ik(t)l
5 M for all t E [ a , b ] , z E S,
where M > 0 is a constant. To conclude the proof of the lemma, we observe that if z(4) is a solution of (3.1.1), which is bounded, then i ( + ) ( t )= f(zt(4)) so that If(zt(4))l5 L for some L since llzt(S)ll 5 a , and f maps bounded sets into bounded sets. Since IIi(4)(t)ll 5 L for all t 2 0, the Arzela-Ascoli Theorem guarantees that { z t ( + ): t >_ 0) belongs to a compact set.
Lemma 3.1.2 Let ~ ( 4 be ) a solution of Equation (3.1.1) which is defined on [-h,00) and 11zt(c$)l15 a1 < a for all t E [O,m). Then the r+(4)the positive limiting set of 4 is nonempty, compact, connected, invariant, and , --+ 0 as t -+ 00. distant ( z t ( 4 )r+(+))
Definition 3.1.2: A map V : C G in C relative to the equation
.--)
E+ is a Lyapunov functional on a set
i ( t ) = f(zt), if V is continuous on
c,the closure of G and,
V ( 4 ) = V ( 4 ) = limsup:[V(zr(4)) r-O+
on G. Let
(3.1.7)
- V ( ~ ) ]0,S
s = {q5 E c : V ( 4 ) = O } ,
and let the set M be the largest set in S invariant with respect to (3.1.7).
Lyapunou-Ratumikhin Methods of Stability in Delay Equations
71
Theorem 3.1.2 [3, p. 1191 Suppose V : C -+ E+ is a Lyapunov functional on G and xt(4) is a bounded solution of (3.1.7)) which remains in G for each 2 . Then xt(4) -+ M as t 00. -+
Theorem 3.1.3 [3, p. 1191. Let Ue = { 4 E C : V ( 4 ) < C}, where V :C E+ is a Lyapunov functional, and C is a constant. Suppose there is a constant k = k(C) such that if 4 E Ue, then I4(O)l < k, for some constant k . Then any solution xt(4) of (3.1.7) with 4 E Ut approaches M ast-+co. -+
Corollary 3.1.3 Let V : C + E+ be continuous. Suppose there are nonnegative functions u(r), v ( r ) such that a(.) + 00 as r 00 and u(lq5(O)l) 5 V ( 4 ) ,V ( 4 ) 5 -v(l4(0)l). Then every solution of (3.1.7) is bounded. If we assume further that v(r) is positive definite, so that v(0) = 0, v(r) > 0, r > 0, then every solution approaches zero as t ---* co. -+
Example 3.1.3: Consider the system of two masses and three linear springs. Let xi denote the displacement of the mass mi from its equilibrium position, with positive displacements measured to the right. The masses are subject to external forces Fj(t). This coupled system is described by the equations
In matrix form this becomes
Mx + Bx = F , where
Obviously M is symmetric and positive definite, B is symmetric and positive semidefinite. The external force F can be taken to be a state feedback that has delay and is given by
H(xt)=
lh
F ( s ) x ( t - s)ds.
72
Stability and Time-Optimal Control of Hereditary Systems
Thus the system considered is
M I ( t ) + B t ( t )=
I”
F ( s ) ~ (-t s)ds;
(3.1.8)
M , B are 2 x 2 symmetric matrices, and F is a 2 x 2 matrix that is continuously differentiable. We now consider a coupled system of n such masses. Then M i , i = 1 , . . . ,n.A, B , F are n x n symmetric matrices. If we set
A =B
-
I
h
F(s)ds,
(3.1.8) can be rewritten as
w = Y(t>,
My(t) = - A z ( t ) -I-
F(S)[Z(t
(3.1.9)
- S) - z ( t ) ] d s .
Theorem 3.1.4 In (3.1.9) assume that A and M are positive definite and F ( s ) is positive semidefinite for s E [0, h], and assume that F has the property that there exists a c E [O,h] such that F ( c ) < 0. Then every solution of (3.1.9) satisfies x ( t ) -+ 0 as t -+ 00. Proof: The proof of this result is due to Hale [3, p. 1241. It uses the following Lyapunov functional:
V(d, $1 = +dT(0)Ad(O) + + V ( O ) M $ ( O )
+
+ g7#(-S)
- d ( o ) l T F ( ~ ) [ d ( - 4- d(O)lds, where T denotes vector transpose and d,$ are the initial values of
3:
and y
of (3.1.9), One verifies easily that
V(d7lCl)= -iEd(-h) - #(O)lTF(h)[#(-h) - d(0)l
+ 4 J;[d(-s) - d(O)lj(s)[d(-s)
- d(O)lds I 0,
(3.1.lo)
by conditions of the theorem. From the hypothesis there exists an interval I, containing c such that k(s)< 0 for all s E I,. It follows from (3,l.lO) that V ( # , $ ) = 0 implies d(--s) = d(0) for s E I,. For a solution (3:,y) to belong to the largest invariant set where V ( ~ , $ J=) 0 we must have z(t - s ) = z ( t ) V t E (-00,m) s E I,. This implies that z ( t ) = b is a constant. From (3.1.9), y = 0 and thus M b = 0. Because M > 0, b = 0. Therefore the largest set where V ( 4 ,$) = 0 is (O,O), the origin. But V satisfies the conditions of Theorem 3.1.3 and its Corollary 3.1.1. The conclusion is valid. Every solution of (3.1.9) approaches (0,O) as t -+ co.
Lyapunov-Rarumikhzn Methods of Stability in Delay Equations
3.2
73
Razumikhin-type Theorems
There is an extensive theory in the application of the Lyapunov stability method for ordinary differential equations. Often the Lyapunov functions constructed for ordinary systems can be fruitfully exploited to investigate the stability of analogous functional differential equations. T h e theorems of Razumikhin are the bridge. These results are based on the properties of a continuous function V : En -+ E whose derivative V ( t , d ( O ) ) along solutions ~ ( 4) t , of the system (3.2.1) is defined to be
The following results are contained in Hale [3,p. 1271. Theorem 3.2.1 Let f : E x C -+ En take E+ (bounded sets of C) into bounded sets of En. Suppose u , v , w : E+ + E+ are continuous, nondecreasing functions u(s),v(s) positive for s > 0 u(0) = v(0) = 0. Let V : E x E n -+ E be a continuous function such that
and
Then the solution 3: = 0 of (3.2.1) is uniformly stable. If w(s) > 0 for > 0 and there is a continuous nondecreasing function p(s) > s for s > 0 such that i . ( t , d ( O ) ) i -W(ld(O)l) if V ( t + e,d(e)) < P(V(t,dJ(O)), 0 E [-h, 01, then thesolution I = 0 of (3.2.1)is uniformly asymptotically stable. If u ( s ) -+ 00 as s + oc), then the solution I = 0 is globally uniformly asymptotically stable. s
Example 3.2.1: Consider the scalar system
.(t) = -az(t) - 6I(t - h).
(3.2.2)
74
Stability and Time-Optimal Control of Heredttary Systems
Proposition 3.2.1 Assume that a , b are constants such that lbl 5 a . Then the solution x = 0 of (3.2.2) is uniformly stable. If in addition there exists some q > 1 such that 1b1q < a, then (3.2.2) is uniformly asymptotically stable. Proof: Consider V = xz/2. V ( z ( t ) )= -az2(t)
- bz(t)z(t- h ) ,
< -[u - lbl]z2(t) if Iz(t - h)l < Iz(t)l. Thus V ( z ( t ) )5 0 if V ( z ( t- h ) ) 5 V ( c ( t ) ) .It follows from Theorem 3.2.1 that x = 0 is uniformly stable. For the second part, for some q > 1, choose p ( s ) = 4’s. If p ( V ( c ( t )> V ( z ( t- h ) ) , i.e., (z(t- h ) ( < qlx(t)l, then V ( t ( t ) )5 -(u - lbIq)z’(t). Since u - lblq > 0, the second part of the theorem guarantees that the solution 3: = 0 of (3.2.2) is uniformly asymptotically stable. Example 3.2.2: Consider
.(t) = ?At), (3.2.3)
Proposition 3.2.2 Assume in Equation (3.2.3) that:
> a > 0, y # 0, for some constant a ; (ii) f(x)sgn z 00 as 1x1 -+ 00, f(x)sgn x > 0; and V x E E , where Lhq < a. Then (3.2.3) is uniformly (iii) Ig(x)l 5 L (i) a ( t , y ( t ) ) Y
4
asymptotically stable.
Proof: Consider
V ( Z ,Y) =
4 t )
+
f ( z : ( s ) ) d s Y2/27
V ( x W , ?/(t))= dt)Y(t)
+ f(+))Y(t)l
Lyapunov-Rarumikhin Methods of Stability in Delay Equations
if a - Lhq
> 0.
75
The result follows.
3.3 Lyapunov-Razumikhin Methods of Stability in Delay Equations [2] The method of Lyapunov and its extensions by LaSalle and Hale require the construction of a Lyapunov functional, which can be very difficult to practice. These functionals are nonincreasing along solutions. Lyapunov functions, which are simpler and which are not nonincreasing along solutions, can be appropriated. This is the fundamental idea of Haddock and TerjCki [2]: the development of an invariance principle using Razumikhin functions. In what follows, the power of the Lyapunov method is married to the simplicity of Razumikhin functions in a theory proposed by Haddock and TerjCki [5]. A quadratic Lyapunov function is shown t o yield good stability criteria. Consider an autonomous retarded functional differential equation of the form
(3.3.1) where f : C -, En is continuous and maps closed and bounded sets into bounded sets. We also assume that solutions depend continuously on initial data.
Definition 3.3.1: Let q5 E C. An element 11, E C belongs to an w-limit set of 4, n(4) if the solution z ( t , O , $ ) of (3.1.1) is defined on [ - h , m ) and there is a sequence { t , } , t , -+ 00 for n -+ oo such that [lzt,,(4)- $11 -+ 0 as n -+ oo and zt,(#)(s) = z(t, s,O,#). A set M E C is called an invariant set (with respect t o Equation (3.3.1)) if for any 4 E M there is a solution z(4) of (3.3.1) that is defined on (--oo,oo) such that zo = 4 and C is positively invariant if for each zt E M V t E (-00,oo). A set M 4 E M, zt(q5) E M V t E [O,w). It is well known [8] that if z(4) is a solution of (3.3.1) that is defined and bounded on [-h, oo), then (i) the set ( ~ ~ ( 4t) 2; 0) is precompact, (ii) Q(4) is nonempty, compact, connected, and invariant, and (iii) q ( 4 ) -+ Q(4)as t -+ 00.
+
76
Stability and Time-Optimal Control of Hereditary Systems
Haddock and TerjCki [2]applied a well known Lyapunov function argument to these stated properties of w-limit sets of solution to obtain an invariance principle that utilizes the easier to construct Lyapunov function instead of the usual Lyapunov functional of HaIe-LaSalle [S].To state the result we need the following definitions:
Definition 3.3.2: A function V : En -+ E that has derivatives is called a Lyapunov function or a Razumikhin function. Let V be a Lyapunov function. The upper right hand derivative of V with respect to (3.1.1) is defined by
gk51= limsuP:(VMO) r-O+
+ .f(4)l-
Vf4(0)1).
(3.3.2)
Because V has continuous first partial derivative, we have
(3.3.3) where fi is the i-component o f f . We note that if x is a solution of (3.3.1),
Definition 3.3.3: Suppose V is a Lyapunov function and G c C a set. Let
4 E G : - hmax V[xt(+)(s)] = - hmax V [ + ( s ) ]V , t20 <s
Theorem 3.3.1 Let V be a Lyapunov function and G a closed set that is positively invariant with respect to Equation (3.3.1). Suppose
V[41 5 0 v 4 such that V[4(0)] = - ~ < o v [ 4 ( s ) l . Then for any 4 E G such that z(o>(.) is defined and bounded on [-h, m), Q(4) C M v( G ) 2 Ev(G) and
Lyapunov-Razumikhin Methods of Stability an Delay Equations
77
Corollary 3.3.1 Consider Equation (3.3.1) and suppose (i) f(0) = 0. Suppose there exists a Lyapunov function V and a constant a that: (ii) V ( 0 )= 0 and V ( x ]> 0 V 2, 0 # 1x1 < a, (iii) V[O]= 0, and (iv) V[4] < 0 for any 0 # 11411 < a such that
> 0 such
Then the solution x = 0 of Equation (3.3.1) is asymptotically stable. We note that if every solution of (3.3.1) is bounded on [ - h , m ) by a, say, then Corollary 3.3.1 gives a global asymptotic stability result. We now state a criterion for boundedness in terms of a Lyapunov function V . Theorem 3.3.2 Suppose u , v , w : E+ E+ are continuous, nondecreasing functions, u ( s ) , v ( s )positive for s > 0, u(0) = v(0) = 0. Let V : En E be a Lyapunov function such that -+
-+
4 4 )I V(Z) I v(I.1) v x E En, and V[43
(3.3.4)
5 -w(14(0)1) for any 4 E C such that (3.3.5)
If u ( s ) + 03 as s
-+
00,
the solutions of (3.3.1) are uniformly bounded.
Proof: Consider the function max V [ x t ( 4 ) ( s ) = ] W[zt]. Just as in Had-h
dock and Terjdki [2], W [ Q ]is a nonincreasing function o f t on [O,m). By (3.3.4) 4l4t)l) I W z t l I v(11.*11) V t E E , 4JE c. Since u ( s ) + 03 as s 00, if a > 0 is any given constant there is a p > 0 such that u(p) = v ( a ) . As a result, if 11q511 I a , the solution x ( 0 , d ) with ~ ( 0 ~ = 4 )4J satisfies Ix(4)(t)l 5 ,B V t 2 0, which implies uniform boundedness. Indeed, -+
4144>(t)I) I V[4t)I = W
Z t )
L W 4 ) L v(11411~5 .(a) =
a.
Therefore Ix(q5)(t)l 5 /3 t 2 0 as asserted. It now follows from Theorem 3.3.2 and Corollary 3.3.1 that the following is true:
78
Stability and Ttme-Optimal Control of Hereditary Systems
Corollary 3.3.2 Consider (3.3.1) and suppose 6) f(0) = 0, (ii) and there are continuous nondecreasing functions u(s), v(s) positive for s > 0, u(0) = v(0) = 0; u(s) -+ 00 as s + 00. Let V be a Lyapunov function such that (iii) V[O]= 0, u(IcI) 5 V ( c )5 v(lzl), V c E En, (iv) V[4] < 0 for any 4 such that max V[~(S)]= V[t$(O)]. -n<s
Then the solution I = 0 of (3.3.1) is globally asymptotically stable and every solution z(6) of (3.3.1) satisfies ~ ( 4 + ) 0 as t -+ 00.
We now use Corollary 3.3.2 to give checkable criterion in terms of the system’s coefficients for global uniform asymptotic stability and the convergence z t ( 4 ) + 0 as t + 00, (3.3.6) for the nonlinear system (3.1 .l) whose linear approximation is
i.(t) = L ( c * ) , (3.3.7) where L(zt) = Dlf(0)ct. Dlf(0) is the FrCchet derivative of f(4) with respect to 6 a t 0, and this is the mapping L(.) : C + En given by
Here 0 5 w1 note that
5 w 2 . .. 5 W N = h , and Ai are n x
n constant matrices. We
f(4) - L(4) - f(0) = d4), where llg(q5)11/11q511 -+ 0 as 11411 -+ 0, so that, given any > 0, there is a a(€) > 0 such that if 11q511 < 6, then Ilg(4)ll < ~11611.Thus any f with F’rkchet derivative L can be written as where for some c > 0
f(4) = L(6) + f(0) + d6),
Is(6)l 5~11611for all 6 E 0,
(3.3.9)
(3.3.10) with 0 = {q5 E C : 114ll < 6). We assume that f given in Equation (3.3.1) is of this form where (3.3.10) holds for all 4 E C, since it is aglobal result that we need. If f(0) = 0 and g(6) 5 0, we deduce conditions on the system’s coefficients that guarantee global uniform stability. If f(0) = 0 but g(6) satisfies (3.3.10) for all 4 E C,we deduce results on stability in the first approximation with acting perturbations. We also obtain the global decay (3.3.6) when f(0) # 0.
Lyapunov-Razumikhin Methods of Stability in Delay Equations
79
Theorem 3.3.3 In Equation (3.3.1) assume that (i) f(0) = 0. (ii) There exists a constant, nonsingular,positive definice symmetric matrix H that satisfies dllz(t)12 5 z T ( t ) H z ( t ) 5 d21z(t)I2, V t E E , di > 0, i = 1 , 2 constants, and the matrix
is negative definite; (I is identity matrix), and q 2 1. (iii) f(4)= L ( 4 ) g(4) where L is given in (3.3.8), and
+
(3.3.12) Q
a constant, and
for some 6 > 0. Then every soh tion z(4) of Equation (3.3.1) with zo = 4 satisfies (3.3.6).
Proof: We take inner products in Equation (3.3.9) and use (3.3.12) to obtain
so that
Since f(0) = 0, we have the estimate
Now consider the function
80
Stability and Time-Optimal Control of Hereditary Systems
But if V [ z ( t ) ]=
max V [ z t ( s ) ]then , by Lemma 3.3.2 below we have
-h<s
11ztll 5 qlz(t)l, where q = value of H . Thus
a2/a1, a 2
i. 5
the greatest and a1 the least eigen4 t ) ) < 0,
where B is the n x n matrix defined by
B = HA0
+ A ; f H + 2&11Hll
N
-
This B is negative definite by (ii). Setting u ( s ) = d l s 2 , ~ ( s )= d2s2 in Corollary 3.3.2, we conclude that ~ ( 4 ) 0 as t -+ cy), and the trivial solution is globally asymptotically stable. Corollary 3.3.3
Consider the linear system
where A ( s ) satisfies (3.3.12). Assume there is a positive definite symmetric matrix H such that the matrix
Lyapunov-Razumikhin Methods of Stability in Delay Equations
81
is negative definite. Then (3.3.17) is globally, uniformly, asymptotically stable and every solution z(4) of (3.3.17) with zO(4) = 4 satisfies
for some k
> 0, a > 0.
Proof: This is an immediate consequence of Theorem 3.3.3, since g(4) implies E can be taken as zero, and Theorem 3.3.2 can be applied.
The estimate B
< 0 may
0
well be the best possible. The case
i ( t )= a z ( t ) + bz(t - h ) , considered by Infante and Walker [ 6 ] ,can be used to test this statement. Indeed. assume H = 1. Then
B = 2a
+ 2(bl < 0
or a
+ Ibl < 0.
Recall that in [6] uniform asymptotic stability is valid for every h E [0, m) if and only if a Ibl 5 0 and a b < 0. If the system
+
+
i ( t )= A ~ z ( t+) A l z ( t - h )
(3.3.18)
is considered, and if H = I is taken where the matrices are n x n , then the condition for global asymptotic stability is B = $(A0 A:) 21(11A111) and is negative definite, which is comparable to a recent result of Mori [7]. The next result can be proved by the methods of this section and the ideas of observability. The proof is contained in Chukwu [l].
+
Proposition 3.3.1
Consider the system
i ( t )= A o z ( t ) + a = h/N
Assume that
(i) B
HA0
+
N
> 0.
N
C Aiz(t - ia), i=l
+ A T H + k = l 2pnMkH
is negative semidefinite, for some
positive definite symmetric matrix H where Mk = m a I(Ak)ijl. Then (S) is asymptotically stable if and only if
82
Stability and Time-Optimal Control of Hewditary Systems
(ii) rank [A(A), B] = n for each complex A, where A(A) = A 1 - Ao N
k=l
A k e e X k a , and
(ii) for some complex number A,
[
Ao-AI
rank
AN
A1
... 0
AN B
0
0
Remark 3.3.1: Condition (i) of Proposition 3.3.1 ensures that (S) has uniformly bounded solutions. Example 3.3.1: The scalar system
has (by Proposition 3.3.1) the following property: If
the scalar system is uniformly asymptotically stable if and only if for some complex number A
This rank condition holds whenever
QN
# 0.
We now state a lemma that was used in the proof of the theorem.
Lemma 3.3.1 Let t E (--oo,oo). llztll
I f V [ z ( t ) ]= max V [ z t ( s ) ] ,then
5 q)z(t)lfor the same t , where q = aZ/a1,
the least eigenvalue of H , and where
-h<s
and a1
83
Lyapunov-Rarumakhin Methods of Stability in Delay Equations
with H a positive definite symmetric matrix, and (., .) denotes the inner product in En. Proof: Because
( H c ( t ) ,z ( t ) ) = - hmax (Hz(t <s
+ s),x ( t + s))
and the maximum is attained at some s* E [-h, 01, we have
+
( H z ( t ) ,z ( t ) ) = ( H z ( t s * ) , x ( t
+ s'))
so that
a1(z(t
+ s*)) 5 ( H z ( t + s * ) , x ( t+ s*) = (H(4th 4 t ) )
where a1 and
a2
L a 2 ( 4 t ) ,4 t ) )
are the least and greatest eigenvalues of H. Thus (x(t
+ s*),z(t + s*)) i Za1( x ( l ) ,z ( t ) )
that is 11x111 5 qlz(t)l. Note that the argument can be reversed. A special case of Equation (3.3.1) is the system
where 0 5 wk < h , k = 1,... , N , h > 0 and where x ( s ) E En, f , g k : En + En are continuous. We require that
(3.3.20) where ,f? is a constant. We assume f , g k take (bounded sets of En) into bounded sets of En and are continuouslv differentiable. Let Dn denote the n x n matrix function a.fi(z(t)), and Dk the n x n matrix function dXi
ag'(x(t - w k ) ) .Let D,, a = 0 , 1 , - . . ,N be bounded. Let H be a constant azj (1 - wk) n x n positive definite symmetric matrix. Define
J,=HD,+DzH
a = O , l , . . . , N.
(3.3.21)
a4
Stability and Time-Optimal Control of Hereditay Systems
Theorem 3.3.4 In (3.1.19), suppose there exists a positive definite constant symmetric matrix H such that the n x n matrix
B
JO
+ (I
N
IlJkll (k=l
+ 211ffllp
)
I
(3.3.22)
is negative definite uniformly on + ( t ) , x(t - W k ) . Assume
f(0) = 0 = g k ( o ) , k = 1 , . .. ,N . Then (3.3.19) is globally uniformly asymptotically stable. Proof: Let H be as stated and consider V = ( H x ( t ) , z ( t ) ) .Then there exist positive finite constants d l , d2 such that
dllX(t)I25
v 2 dzlx(t)I2.
(3.3.23)
Differentiating V along solutions of (3.3.19), and utilizing Lemma 3.1.1 and Lemma 3.1.2 below, we have
By hypothesis,
N
B
3 JO
qI
C
IlJkIl
2PIIHII
(k=l
is negative definite, so that
V 5 ( B d t ) ,~ ( t )<) 0.
(3.3.24)
This inequality (3.3.24) and (3.3.23) suffice for Theorem 3.3.2 to apply. The theorem is proved. To round up we state Lemma 3.3.2.
Lemma 3.3.2 Let t : E x E" + En be a continuously differentiable function, and suppase there is a constant M > --03 such that
( I , the identity matrix) uniformly in x and t > u, u,t E E. Then the scalar products
+
LI = (t(t,x r ) - t(t,x ) , r ) Lz = ( [ ( t ,2 + r ) - [ ( t ,x ) , Y)
Lyapunov-Ratumikhin Methods of Stability i n Delay Equations
85
Proof: The first estimate follows readily from the Mean Value Theorem; and the second follows from the Mean Value Theorem and the Schwartz inequality. Indeed, if H is the n x n matrix hi, where
and Bi = & ( x , t ) satisfies 0 < Bi < 1, the Mean Value Theorem yields the identity L1 = ( H r , r )= ( J Y , ~ ) , and because J is symmetric, ( J r , r ) < M ( r , r ) = Mlrl’. Also
L2 = (Hr,y), and by the Schwartz Identity,
REFERENCES 1. E. N. Chukwu, “Global Behaviour of Linear Retarded Functional Differential Equations,” J . Math. Anal. Appl. 162 (1991)277-293.
2. J. R. Haddock and J. TerjBki, “Lyapunov-Ftazumikhin Functions and a n Invariance Principle for Functional Differential Equations,” J. Differential Equations 48 (1983) 95-122. 3. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 4. J. Hale,
Ordinary Diflerential Equations, Wiley, New York, 1969.
5. J. Hale, “Sufficient Conditions for Stabifty and Instability of Autonomous Functional Differential Equations,” J. Diflerential Equations 1 (1965)452-482. 6. E. F. Infante and J. A. Walker, “A Lyapunov Functional for a Scalar Differential Difference Equation,” Proc. Royal. SOC.,Edinburgh 79A (1977)307-316. 7. T. Mori, “Criteria for Asymptotic Stability of Linear TimeDelay Systems,” IEEE Transactions Aut. Control AC-30 (1985)158-161.
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Chapter 4 4.1
Global Stability of Functional Differential Equations of Neutral Type
Definitions [1,2]
Suppose f,g are continuous functions taking functional difference operator
[T, 00)
x
C -+ En. Define the
D(.,*) : [T,CO) x C -+ En (4.1.1)
where zt E C. We say I is a solution of (4.1.2) with initial value 4 at u if there exists a E [ ~ , o o ) ,a > 0 such that c : [u - h , a a] -+ En is continuous c, = 4, D ( t ) q is continuously differentiable on (6, u a), and (4.1.2) is satisfied on (u,u Q). In this section we shall consider the
+
following systems: (4.1.2);
+
+
d x D ( t ) z t = f(t,
21)
d -[D(t>.* - G(t,.t)I = f ( t , .t) dt
+ q t ,z t ) , + F ( t ,z t ) ,
(4.1.3) (4.1.4)
under the following, basic assumptions: D(t)4 = 4(0) - g ( t , 4 ) is defined in (4.1.1) where g ( t , 4) is linear in is given by g(t,4) =
J
4 and
0
-h
[ d e ~ t , ~ ) ~ ~ ( ~ ) .
(4.1.5a)
Here p ( t , B ) is an n x n matrix function o f t E [T,oo), B E [-h, 01, of bounded variation in B which satisfies the inequality
87
88
Stability and Time-Optimal Control of Hereditary Syslems
Whenever (4.1.5) holds for g , we say g is uniformly nonatomic at zero. We also assume that G in (4.1.4) is uniformly nonatomic at zero. Furthermore, there is a nonnegative constant N such that
(4.1.6) Basic in our discussions below are the assumptions that f ,g, D, G, F are smooth enough to ensure that a solution of (4.1.2) - (4.1.4) exists through each (u,4) E [ T , 00) x C, is unique and depends continuously upon ( c T , ~ ) , and can be continued to the right as long as the trajectory remains in a bounded set in [T, 00) x C. The needed conditions axe given in Section 7.1. Our main interest here is the stability theory of (4.1.2), (4.1.3), and (4.1.4). The basic Lyapunov theory on (4.1.2) is first presented. In addition, if in (4.1.2) f has a uniform Lipschitz constant, a converse theorem t o the basic Lyapunov theory of Cruz and Hale [2] is formulated. All these are applied in subsequent sections. Some definitions are needed.
Definition 4.1.1: The zero solution of (4.1.2) is Uniformly Asymptotically Stable in the Large (UASL) if it is uniformly stable and if the solutions of (4.1.2) are uniformly bounded, and if for any a > 0 and any 6 > 0 and any u E [O,oo) there exists a T(c,a) > 0 such that if 11411 5 a , then 112t(u,4)ll < 6 for all t 2 u "(€,a).The trivial solution (4.1.2) is said to be Weakly Uniformly Asymptotically Stable in the Large (WUASL) if it is uniformly stable and for every u E [ O , o o ) and every 4 E C we have Ilq(u,q5)))4 0 as t 4 00. The zero solution of (4.1.2) is Exponentially Asymptotically Stable in the Large (EXASL) if there exists c > 0 and for any a > 0 there exists a k ( a ) > 0 such that if 11q!~11 5 a ,
+
4)ll I
112t(u,
II4IL v t 2 cT*
The zero solution is Uniformly Exponentially Asymptotically Stable in the Large (UEXASL) if there exist c > 0, k > 0 such that
Il~t(~,4)11 I~e-e(t-")((411, v t 2 6.
(4.1.7)
The trivial solution is Eventually Uniformly Stable (EVUS) if for every 6 > 0 there is a = a ( € )such that (4.1.2) is uniformly stable fort 2 u 2 a(€). Other eventual stability concepts can similarly be formulated by insisting that the initial time u satisfy u 2 a. For example, the trivial solution of (4.1.2) is called Eventually Weakly Uniformly Asymptotically Stable in the Large (EVWUASL) if it is (EVUS) and if there exists a0 > 0 such that for every u 2 a0 and for every 4 E C, we have ~ ~ z t ( u+, 0~as ) t~ -+~ 00.
Global Stability of Functional Diflerential Equations of Neutral Type
4.2
89
Lyapunov Stability Theorem
Consider the homogeneous difference equation D(t)zt
=o
t
u
zu
= 4 D(u)4=0,
(4.2.1)
which was discussed in Section 2.4. There it was stated that D is uniformly stable if there are constants p, cr > 0 such that for every u E [r,co), 4 E C the solution z ( u , 4 ) of (4.2.1) satisfies
I l ~ t ( ~ l 4 5) l pe-a(t-u)l1411, l
t L u.
Cruz and Hale have given the following two lemmas [2]: Lemma 4.2.1 If D is uniformly stable, then there are positive constants a , b , c , d such that for any h E C((r,co), En),the space of continuous functions from [r,co) --t En, and u E [T,co) the solution z(u,4,g) of
D(t)zt = M ( t ) , t 2
6,I,
= 4,
(4.2.2)
satisfies llEt(u,4,
s)ll I e - Q ( t - u ) (bll4ll + c S U P IM(.)I) a
+ d uS
Furthermore, a , b , c, d can be chosen so that for any s E [a, m),
for t 2 s
+ h.
Lemma 4.2.2 Suppose D is uniformly stable, and M : [r,co)+ En is continuous such that M ( t ) --t 0 as t -+ co. Then the solution 1 ( u , 4 , M ) of (4.2.2) approaches zero as t -+ co uniformly with respect to u in [ T , O O ) and 4 in clased bounded sets. Let V : [ ~ , c ox) C + E be a continuous functional and 1(u,4) be a sofution of (4.1.2) through (u,4), i.e., a solution of the integral equation
D(t)zt = D(a)4 Define V ( t ,4) by
+ J,' f ( s ,z 3 ) d s ,
t 2 0,
10
= 4.
(4.2.3)
90
Stability and Time-Optimal Control of Hereditary Systems
Theorem 4.2.1 [2] Let D be uniformly stable, f : [r,00)x C -, En, continuous, and f maps [r,00) x (bounded sets of C)into bounded sets of E" . Suppose there are continuous, nonnegative, and nondecreasing functions u ( s ) , o(s), w(s) with u(s), v ( s ) > 0 for s # 0 and u(0) = v ( 0 ) = 0, and there is a continuous function V : [r,00) x C -+ E such that (4.2.4a) then the solution x = 0 of (4.1.2) is uniformly stable. If u(s) + 00 as s + the solutions of (4.1.2) are uniformly bounded. H w ( s )> 0 for s > 0, then the solution x = 0 of (4.1.2) is uniformly asymptotically stable. The same conclusion holds if the upper bound on V(t, 4) is given by -w( lc$(O)l. 00,
The following converse theorem to Theorem 4.2.1 was stated and proved in Chukwu 111.
Theorem 4.2.2 In (4.1.2)) assume (i) f ( t , O ) = 0 V t, and f has a uniform Lipschitz constant, i.e., there exists a constant L such that for any 4, II, E C and for all t E [r,00)
If(t74)- f(t,+)l I 414 - 1111. (ii) There is a constant N > 0 such that
l W 4 I I "4 v t E [T, a), 4
c.
The zero solution of (4.1.2) is (UEXASL), so that
Il.t(.,4)II
v t L 6,
I ke-""-"'11411
for some k > 0, c > 0. Then there are positive constants M, R and a continuous functional V : [r,00) x C -+ E such that
l W 4 l 5 V(t,4) I Mll4ll,
w,4) 5
IV(4 4) - V(t, $)I L
-c/2V(t9
41,
Rll4 - II,lL
(4.2.4 b)
v t E [7,m), 4,II,E c.
We now use Theorem 4.2.1 to study the system d -[x(t) dt
- A - l z ( t - h)] = f ( t , t ( t ) , z ( t - h)),
(4.2.5)
Global Stability of Functional Differential Equations of Neutral Type
91
where A-1 is an n x n symmetric constant matrix and f : [ r , m ) x En x En -+ En and f : [T,00) x (bounded sets of En)x (bounded sets of En)+ (bounded sets of En).We also assume that f has continuous partial derivatives. If we let D(t)4 = 4(0) - A-1 q5(-h), then (4.2.5) is equivalent to (4.2.6)
In (4.2.6) denote by D, ( u = 1,2) the n x n symmetric matrices +(d,ij where
+
d,ji),
dlij = a f i i t , 4(0), d4-h)) a4j ( - h )
Let
1 J, = Z
7
( ~ +~D ~, A ) , u = 1,2,
(4.2.7)
where A is a positive definite constant symmetric n x n matrix, and Df is the transpose of D,. Our stability study of (4.2.6) will be made under various assumptions on J,. Theorem 4.2.3 In (4.2.5), assume that (i) D4 = d(0) - A - l + ( - h ) is uniformly stable, (ii) f ( t ,0,O) = 0. (iii) There are some positive definite constant symmetric matrices A and N such that the matrix J = ( ;
!),
(4.2.8a)
where
and J , are defined in (4.2.7), is negative definite. (iv) A-1 is symmetric. Then the solution x f 0 of (4.2.5) is uniformly asymptotically stable. Remark 4.2.3: J in (4.2.8) will be negative definite if
92
of Hereditary Systems Systems and Time-Optimal Control Control of Stability and
(i) the eigenvalues of X 2 j of Jz in (4.2.7) satisfy X u 5 - A 2 < 0, (j= 1 , 2 , . . . ,n ) , V q5(0), and t 2 0 where A2 is a constant; (ii) the eigenvalues A3j of the matrix J2+N satisfy A , 5 - A 3 < 0, (j= 1 , 2 , . , n ) , V 4(0), and t 2 0 where A3 is a constant; and (iii) the eigenvalues A y of N-A-1NA-1 satisfy A+ 2 Ad, ( j = 1 , . . . ,n ) where 2A3A4 > IIJ1+J2A-1+NA-111 and 11.11 denote the matrixnorm. +
-
Proof: Let D4 = d(0) - A-14t-h) and define the Lyapunov functional rO
where A and N are positive definite symmetric n x n constant matrices and (-,-) denote inner product in En.Because A , N are positive definite symmetric matrices there are constants a i , n i , i = 1,2, such that
Thus the first inequality of Theorem 4.2.1 is satisfied. That the second inequality of (4.2.4) is also satisfied follows from the following calculations: Because A is symmetric, we have
since f(t,O,O) = 0, and we have used Lemma 3.1.2. Note that 4(0) = Dq5 A-l$(-h), so that
+
Global Stability of Functional Differential Equations of Neutral Q p e
93
since J is negative definite. It now follows that (4.2.4) of Theorem 4.2.1 is completely verified. Because D is assumed to be uniformly stable, all the requirements of Theorem 4.2.1 are met: We conclude that the solution z = 0 of (4.2.5) is uniformly asymptotically stable. We now consider some examples. Example 4.2.1: Consider the scalar equation
where a and c are constants with a
4< > 0 1.1 < 1 and a > c/(l - c’) -
2 Lemma 3.1 of Cruz and Hale [2]implies that the operator Dq5 = #(O)+cq5(-h) is uniformly stable. We now apply Theorem 4.2.3 with A-1 = a = -3a/2 = Q -c, A = 1, J1 = 0 , 5 2 = -a, N = a / 2 , 2 J z + N = -2a+2 c
< 1.
p = J1+ J z A - i + N A - 1 a
ac
= a c - 5 c =3-
is negative definite. We can also use Remark 4.2.3 with u / 2 , A4 = :(I - c2), so that
2 A 3 ~= 4 2 ( u / 2 ) ( a / 2 ) ( 1 - c’)
A2
= a , XB =
> ac - ac/2,
since a2(1- 2 ) / 2 > ac/2 if we assume a ( 1 - c’)
> c.
Example 4.2.2: Consider the equation
d dt
-[z(t)
+ c ( t ) z ( t - h)] + a z ( t ) + b ( t ) z ( t- h ) = 0 ,
where a > 0 is a constant, c ( t ) , b ( t ) are continuous for t 2 0 , and there is a 6 > 0 such that c 2 ( t )5 1 - 6 . Let D ( t ) d = 4(0) + c(t)#(-h), so that by a remark in Cruz and Hale [2,p. 3381, D ( t ) is uniformly stable. Observe that f ( t ,z ( t ) ,z(t - h)) = - a z ( t ) - b(t)z(t- h). On taking A = 1, A-1 = -c(t), N = u / 2 , J1 = - b ( t ) , J2 = -a = -A’, JZ N = -a/2 =
+
94
Stability and Time-Optimal Control of Hereditary Systems
JzAVl = ac, NA-1 = -ac/2, -A3, We observe that
A4
= N - A-1 NA-1 = a / 2 - ac2/2.
ensures uniform asymptotic stability. This inequality will hold if a2/26 > Ib(t) ac(t)/21 for all t 2 0.
+
Example 4.2.3: Consider the n-dimensional autonomous neutral difference equation
where z ( t ) E En.A _ l , A o , A 1 are constant n x n matrices h > 0. Let D1 = i ( A l AT), D2 = 5(Ao 1 AT), Ji = ADi DTA, i = 1,2, A , a
+
+
+
positive definite matrix. Assume 11A-111 < 1 and A-1 to be symmetric. For some constant positive definite symmetric matrix N , Remark 4.2.3 yields the required condition for uniform asymptotic stability.
4.3
Perturbed Systems
The converse Theorem 4.2.2 will now be employed to prove stability results for the system (4.1.3).
Theorem 4.3.1 In System 4.1.3, assume that (i) Ig(tj4)I 5 NII411, If(tp4) - f(t,dJ)I 5 Nll4 V 4,dJ E C,t E [r,00); and D is uniformly stable. (ii) Suppose that the zero solution of (4.1.2) is (UEXASL), so that every solution z ( t , 4) satisfies
for some constants c > 0 k > 0. (ii) The function F in (4.1.3) satisfies F = J'1 FZ with IFl(t,4)( 5 u(t)lD(t)4(V t 1 u, 4 E C,where there is a constant [ > 0 such that for any p > 0
+
t+P
1P 1 and for some 6
>0
we have
v(s)ds
< t , t 2 6,
(4.3.1)
Global Stability of Functional Differential Equations of Neuirul Qpe
95
Then every solution z(u, 4) of (4.1.3) satisfies
uniformly with respect to 5 E [T,oo), and 4 in closed bounded sets. In particular, the null solution of (4.1.3) is WUASL.
Proof: The assumptions we have made in Theorem 4.2.2 guarantee the existence of a Lyapunov function V with the properties of Theorem 4.2.2. Suppose y = y ( u , 4 ) and 3: = z(u,qh) are solutions of (4.1.3) and (4.1.2) respectively with initial values 4 at u. If 64.1.3) and $4,1.2) are the derivatives of V along solutions of the systems (4.1.3) and (4.1.2) respectively, then the relations (4.2.4b) yields the inequality
But then
Because g(t,4) is uniformly nonatomic at zero and (4.1.5) holds, there is an ro > 0 such that llgu+r
for 0 < r
- zu+rll L
-
o+r
1
1 c(r0)
I{f(SJ - f('l Yd)
z8)
+ F(s,
!/d))lds
< rg. With this estimate in (4.3.2) we obtain
V t 2 u, 4 E C where Q =
1 - c( ho) *
It follows from (4.2.4b) that
We now set c = -c/4Q and choose some ( < c/SQ so that from (4.3.1)
Stability and Time-Optimal Control of Hereditary Systems
96 and
and t
(-44
+ Q v ( s ) ) d s I (-f + QE) (t - u) 5 -c/8(t
- 6).
Using all these in (4.3.4), one obtains
and with r ( t ) = V ( t ,zt(u,d)),
The solution of this inequality satisfies
Since (4.2.4b) is valid, this last inequality leads to
Because D is assumed uniformly stable, Lemma 4.2.2 can be invoked to ensure that z(u,$J)-+ 0 as t -+ 00, uniformly with respect to u E [T,co), and 4 in closed bounded sets. To conclude the proof we verify uniform stability as follows: Because D is uniformly stable, there are positive constants u l , u2, u3, a4 such that
For any c > 0 chocse 6 = ( Q ~ + u ~ M + u ~ M and ) -observe ~ ~ , that if 11411 < 6 , then IIZt(u, 4)II < a26 -k Q3M6-k U4M6 < 6 , for t 2 u. This proves that 2 = 0 is uniformly stable, so that the zero solution of (4.1.3) is WUASL.
Global Stability of Functional Differential Equations of Neutral Type
97
Remark 4.3.1: Condition (4.3.1) can be replaced by the assumption M
VE
v(s)ds
< 00
(4.3.5)
to deduce the same results. When f ( t ,4 ) G A ( t ,4 ) is linear in 4 , one can obtain a generalization of the famous theorem of Lyapunov on the stability with respect to the first approximation. It is the global stability result that is most useful when dealing with controllability questions of neutral systems with limited power. Consider the linear system
d
--D(t)zt
dt
(4.3.6)
= A ( t ,2 1 1 ,
and its perturbation
d -dt[ D ( t ) t t - G(t, xt)] = A ( t ,x t ) where A ( t ,4 ) , G(t,4) are linear in
+ f ( t ,z t ) ,
(4.3.7)
4. The following result is valid:
Theorem 4.3.2 In Equations (4.3.6) and (4.3.7) assume that (i) the linear system (4.3.6) is uniformly asymptotically stable, so that for some k 2 1, a > 0 the solution x(u,4 ) of (4.3.6) satisfies
ll+l4)ll
5 ke-a(t-u)l141L
t2
4 E c.
+ F 2 satisfies
(ii) The function F = Fl
where €=
a l-k(X+Mo) -
+ + Ma)
2 L 1 k(X
and where i f ( = 1x14, then for each p
/
1
P t
(iii) The function G = GI
t+P
v(s)ds
+ G2 satisfies
>0
< ( V t 2 U.
(4.3.8)
98
Stability and Time-Optimal Control of Hereditary Systems for t 2 u, q5 E C where n ( t ) is continuous and bounded with a bounded MO such that
n ( t ) 5 Mo, t 2
6,
k ( X + Mo) < 1.
Then every s o h tion z(u,4) of (4.3.7) satisfies
for some N. The proof is contained in [l,p. 8661.
Remark 4.3.2: The requirement of (4.3.8) can be replaced by the hypoth-
Jom
esis
v ( s ) d s < 00.
It is possible to weaken the conditions on F , to deduce global eventual weak stability. Because of its importance it is now stated.
Theorem 4.3.3 For the system (4.3.6) and (4.3.7) assume that (i) Equation (4.3.6) is uniformly asymptotically stable. (ii) The function F in (4.3.7) can be written as
where
for all t
2 u 2 TOfor some sufficiently large To and all q5 E C, where
is a sufficiently small number, and where
1- k ( X + M o ) 41c 1 k ( A Mo)' ff
<=-
+
Also
H(t)=
g(s)ds
+
- 0 as t
00.
Global Stability of Functional Differential Equations of Neutral Type
99
(iii) The function G = G1 + Gz satisfies
fort 2 u, 4 E C , where s ( t ) is continuous and bounded with a bound MO such that T(t)
I Mo, t 2 6,E ( X + Mo) < 1.
Then every solution of (4.3.7) with initial data (a, 4) satisfies
for some TO,u 2 TO.As a consequence the trivial solution of (4.3.7) is (EVUASL).
For the proof consult [l,p. 8691. The above result deals with the linear case f ( t , 4 ) = A ( t , + ) . A similar result holds for general nonlinear f. Indeed, we have the following:
Theorem 4.3.4 Assume that
- $11, v 4,$ E c, t E and D is uniformly stable. (ii) The trivial solution of (4.3.6) is uniformly asymptotically stable. (iii) The function F in (4.3.7) is such that F = F1 F2 + F3 where
(9 Is(tt4)I I Nll4ll, If(t,4) - f ( t , $ ) I 5 "14 [T, 00)
+
for some TO,where
with c , c sufficientlysmall and for u large. Also
M(t)=
J"
m(s)ds-+O as t - 0 0 .
Then there exists some T such that i f u
>T,
Stability and Time-Optimal Control of Hereditary Systems
100
Thus the solution x = 0 of(4.3.7) is (EVWUASL). A proofisindicated in [l,p. 8721. Example 4.3.1: Delay Logistic Equation of Neutral Type [S]. Consider the system
where h l , ha, K , c , r are constants 2 0, with r, h2, K conditions of the type
> 0. We assume initial
If we let N(t)
= 1i-p+ x(t)],
+
A(t) = r[l ~ ( t ) ] , B ( t ) = 2ckA(t)1[1 k2E2(t - h z ) ] ,
+
(4.3.9) is equivalent t o d
A(s
+ hl)r(s)ds] = -A(t + h l ) + B(t)E(t - h 2 ) . (4.3.10)
Set a* = (1
+ c) exp[r(l + c)hl]
b' = (1 - c)exp[hlr(l
pa=--
'
ra*
- c)]
2cka* b* exp(-a') [l+ru*hz+
+ Pckra'hl+
rb* 2(2cka*r)2 exp( -a*)
1
.
b* exp(-a')
Global Stability of Functional Differential Equations of Neutral Type
101
For this we use the Lyapunov functional
V ( t ) = V ( t ,4 t ) ) ,
and calculate the upper right derivative (4.3.10).
D+V of dt
V along solutions of
Proposition 4.3.1 Assume that r, hl, k: E (0,m); hz E [O, m),
> 0, P2 > 0, 2ckra* < 1; 0 < c P1
< 1.
(4.3.11) (4.3.12) (4.3.13)
Then every positive solution of (4.3.9) satisfies
N ( t ) + O as t
+m.
Proof: We need only show that if (4.3.11) - (4.3.12) hold, an arbitrary solution of (4.3.10) satisfies
t ( t )-0
as t +m.
(4.3.14)
The needed details are contained in [5].
4.4 Perturbations of Nonlinear Neutral Systems [S] Consider a nonlinear system that is more general than (4.1.3), namely
d --D(t, dt
xt)
= f ( t ,2 t ) + g ( t , 4.
(4.4.1)
We assume the following as basic: Let h c C be an open set, I1 c E an open interval, P = 11 x A. Denote by En' the set of n x n real matrices. Suppose D, f : r --t En are continuous functions. We assume also that 0 E A. The main result in this section is stated as follows:
Stability and Time-Optimal Control of Hereditary Systems
102
Theorem 4.4.1 Assume (i) D , f in (4.4.1) satisfy the following hypothesis: (a) The Fre'chet derivatives of D , f with respect to 4, denoted by D,, f+ respectively exist and are continuous on r as well as D++,and (b) for each ( t , 4 ) E r, $ E C write ~ + ( 4~ t ,
0
= ~ ( ~t , ( 0 -)
J-h
dep(t,4,
mwi
for some functions
-
with A continuous, p ( t ,4, .) of bounded variation on [-h, 01 and such that the map ( t ,4) s_", dep(t,4, O ) [ $ J ( ~ ) ] is continued for each rl, E
c.
(bf Assume for each ( t , 4 ) E I' D is uniformly atomic at zero compact sets (I C r, i.e., det A ( t ,4)
IL
OR
#0
d e p ( t , 4 , e ) [ $ ( 0 ) 15/ [(S)II$II
s E [O, h] for all ( t , 4 ) E I( for some function l : [0, h] + [O,co) that is
continuous and noijincreasing, [(O) = 0. (ii) g is Lipschitzian in q5 on compact subsets of r. (iii) f ( t ,0,O) = O for all t E I . (iv) For each (a,4)E I', we have
+
IIT(t,a, 4111 I exP(Q(t) b(a)), t
1 0,
where a ( t ) , b(t) are continuous functions from E+ = [0, co) into E+, i.e., are elements of C ( E + ,E+). Here T ( t ,a ;4) is the solution operator associated with the variational equation
d dt
-O+(t,
(v) For each
tda, 4))[&1 = f&, 2*(0,4))[4,
t E [a,u
+ a).
(a,4)E I', N
Ig(t,$)I I Cci(t)II$llri + c N + l ( t ) , t 2 6, i=l
(4.4.2)
Global Stability of Functional Diflerential Equations of Neutral Type
103
+
where r, E (0,1], q ( t ) E C(E+, E + ) , j = 1,2,... ,N 1. (vil cj(s)e‘Ja(a)+b(a), c$ileb(8) E ~ 1 ( r , u )j, = 1,. .. , N . (vii) a ( t ) -00 as t 4 00. Then every solution of (4.1.1) satisfies IICt(fll4)ll as t -+ 00.
-
The proof is contained in Chukwu and Simpson in [6, p. 571. It uses the nonlinear variation of parameter equation
where y t ( a , 4) is the solution of
(4.4.3)
(4.4.4)
This formula (4.4.3) holds when the solution of (4.4.1) is unique.
REFERENCES 1. E. N. Chukwu, “Global Asymptotic Behavior of Functional Differential Equations of the Neutral Type,” Nonlinear Analysis Theory Methods and Applications 6 (1981)853-872. 2. M.A. Cruz and J. K. Hale, “Stability of b c t i o n a l Differential Equations of Neutral Type,” J . Differential Equations 7 (1970)334-355. 3. J. K. Hale, Ordinary Differential Equations, Interscience, New York, 1969.
K. Hale, “Theory of Functional Differential Equations,” Applied Mathematical Sciences 3 , Springer-Verlag,New York, 1977. K. Gopalsamy, “Global Attractivity of Neutral Logistic Equations,” in Differential Equations and Applications, edited by A. R. Aftabizadeh, Ohio University Press,
4. J. 5.
Athens, 1989. 6. E. N. Chukwu and H. C. Simpson, “Perturbations of Nonlinear Systems of Neutral Type,” J. Differential Equations 82 (1989)28-59.
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Chapter 5 S nthesis of Time-Optimal and dnimum-Effort Control of Linear Ordinary Systems 5.0 Control of Ordinary Linear Systems The material in this section is introductory and deals with ordinary differential equations. Our aim is to solve the optimal problem for linear ordinary systems that will then become the basis for generalization to hereditary syst e m . We shall first introduce in this section the theory of controllability of linear ordinary differential equations
where Ao(t), B ( t ) are analytic n x n , n x rn matrices defined on [ O , c o ) . The admissible controls Q* are at first required to be square integrable on finite intervals. We need the following controllability concepts:
Definition 5.0.1: System (5.0.1) is controllable on an interval [ t o , t l ] if, given zo,z1 E En, there is a control u E Q* such that the solution = zosatisfiesz(tl,t~,zo,u)= 11. z ( t , t 0 , z ~ , u ) o f ( 5 . 0 . 1withz(to,to,zo,u) ) It is called controllable at time t o if it is controllable on [to,tl]for some tl > t o . If it is controllable at each t o 2 0, we say it is controllab1e:System (5.0.1) is said to be fully controllable a t t o if it is controllable on [to,tl]€or each tl > t o . It is said to be fully controllable if it is fully controllable at each t o 2 0. If u is a control, the solution of (5.0.1) corresponding to this u is given by the variation of constant formula
= X(t)X-'(to)zo + X ( t )
Z ( t , t 0 1 2 0 ,u)
Lo 2
X-'(s)B(s)u(s)ds,
where X ( t ) is a fundamental matrix solution of
i ( t )= A o ( t ) ~ ( t ) . Define
Y ( s )= X - ' ( s ) B ( s ) ,
105
( 5.0.2) (5.0.3)
106
Stability and Time-Optimal Control of Hereditary Systems
and define the operator l? by
r =where
A ~ + D,
(5.0.4)
d = D. Also define the expression M as follows: dt
M(t0,t)=
la t
Y(s)YT(s)ds=
k
t
x-'(s)B(s)BT(s)X-'*(s)ds,(5.0.5)
where XT is the transpose of X. Then the following theorem is well known: Theorem 5.0.1 The following are equivalent: (i) M ( t 0 , t ) is nonsingular for all t l > t o and all t o 2 0. (ii) System (5.0.1) is fully controllable. Also if A o ( t ) and B ( t ) are analytic on ( t o , t l ) , t l > t o 2 0, then (5.0.1) is fully controllable at t o if and only iffor each tl > t o there exists a t E (to, t l ) such that rank [ ~ ( t )rB(t), , .. . ,r " - l ~ ( t = ) ]n. (5.0.6)
Furthermore, if A0 and B are constant, then (5.0.1)is fully controllable if and only if rank [ B , A o B , .. . ,Az-'B] = n. (5.0.7)
From the definition we note full controllability of t o implies that one can start a t t o , and using an admissible control, reach any point xl in an arbitrarily short time (any t l > to). If the matrices are constant and the system is full controllable, then (5.0.7) holds no matter the time, and this contrasts with the situation in the delay case. (Section 6.1.)
5.1 Synthesis of Time-Optimal Control of Linear Ordinary Systems Minimum-Time Problem Consider the following problem: Find a control u that minimizes
tl
subject to
i ( t ) = A z ( t ) + Bu(t), z(0) = 2 0 , z(t1) = 0, where A is an n x n and B an n x m matrix, and where u EU
(5.1.1)
= { u measurable u ( t ) E Em,Iuj(t)l 5 1, j = 1 , . . . , m}.
Thus we want explicitly to find a function f(.,zo) : [O,tl] -+ U ,f ( t , z o )= u ( t ) that drives System (5.1.1) from xo t o 0 in minimum t l . The following is well known:
Synthesis of Time-Optimal and Minimum-Effort Control
107
Theorem 5.1.1 [l] In (5.1.1) assume that (i) rank [ B , A B , .. . ,A"-'B] = n. (ii) The eigenvalues of A have no positive real part. (iii) For each j = 1,.. . ,rn, the vectors [bj ,Ab j,.. . ,A"-'bj] are linearly independent. Then there exists a unique optimal control u* of the form u*(t)= u ( t,zo)= ~ g n [ c ~ e - ~ ' ~ ] (5.1.2)
almost everywhere, which drives zo to zero in minimum time. Thus if y ( t ) is the Em-vector, cT e-AtB, the so-called index of the control system (5.1.1), then uj(t) = sgn y j j = 1, .. . ,m, where +
sgn yj =
1 ifyj -1 i fyj
>0 < 0.
It is undefined when y j ( t ) = 0. Because (5.1.1) is normal in the sense that (iii) holds, yj(t) has only isolated zeros, where the control switches in the following sense: Definition 5.1.1: If t > 0 and uj'(t - O ; z ~ ) u j . ( t , . , z o< ) 0, then t is a switch time of u* (or u switches at t). Thus each point of discontinuity of u* is called a switch time, and u*(s) = u1 on some interval [ t l , t )and u * ( s ) = u2 on some ( t , t z ) , t l < t < t 2 , where ui # uz in 2.4, then we say that u* switches from u i to uz at time t. It is clear that u* has only a finite number of switch points on [O,tl].If (5.1.2) is valid, then c is said to generate the control u* and -u* (which is the optimal control function for -to). If system (5.1.1) satisfies (iii) of Theorem 5.1.1, then it is said to be normal. Let
R(t) =
{1'
e-A8Bu(s)ds : u ( s ) E 24, u measurable
be the reachable set at t 3 0, and R =
1
u R(t) the reachable set.
Note
t>O
that R(t) is related to C ( t ) , the set of points that can be steered to zero by controls as follows:
~ ( t=) eAtR(t). If
2
is in C =
U C ( t ) and
t2o
u* is its unique time-optimal control function,
then the function
f(.)
= u*(O;.),
2
# 0, f(0) = 0
Stability and Time-Optimal Control of Hereditary Systems
108 defines a map
f
:c+u
that is called the time-optimal feedback control or the synthesis function. Once f is found, the time-optimal problem is completely solved. For example, consider the system i ( t ) = AZ
+ B f (x),
(5.1.3)
z(0) = 20.
If Z(ZO,ZO) is a solution of (5.1.3), then z describes the optimal trajectory from20 t o the origin, and u*(t,za) = f(z(t,zo)) is the time-optimal control function for 20. We shall now describe how Theorem 5.1.1 helps us to construct optimal feedback control for the simple harmonic oscillator. Example 5.1.1: The system (5.1.4a)
is equivalent to where
x = Ax!-+ Bu, 0 A = [-1
1 0]’ B =
(5.1.14b)
[;I, [;I. ic=
It is controllable, normal, and satisfies condition (ii) of Theorem 5.1.1. Because eAt =
cos t -sin2
cTe-At B = -c1 sin t
where c:
]
sin t cost ’
e-AtB
+ c2 cos t ,
=
[- ] ’ sin t cost
c = ( C I ,C Z ) ,
+ c: # 0. Therefore the (unique) optimal control is given by u*(t) = sgn(sin(t
+ 6)),
+
tan6 = -c2
C1
+
for some --a 5 6 5 -a, since -c1 sint c2 cost = asin(t 6), a > 0. We now derive optimal feedback control that drives (20,yo) E E2 to (0,O) in minimum t l . For each initial (20,yo) E E2 there exists an optimal control
Sgn thesis of Time- Optimal and Man imu m-Effo rt Control
109
+
u*(t) uniquely determined by u * ( t ) = sgn[sin(t S)]. When u = f l , the trajectory in E2 is a circle of center (h1,O) with a clockwise direction of motion of increasing t . (In time t the trajectory moves through an arc of the circle of angle t). Indeed, when +1, x 2 = -21 il = x2, dXl 22 d ~ 2 -XI + 1 '
+
+ 1,
Hence (1 - x1)dxl = x2dx2, and xi (1 - ~ 1 =) a2. ~ This is a circle centered at (1,O). Note that 1 - X I = QCOS t, 22 = a s i n t . When u = -1, xi (-1 - ~ 1 =) u2, ~ which is a circle centered at (-l,O), where 21 = acost - 1, x2 = -usint. One way of discovering an optimal control law is to begin at zero and move backward in time until we hit (z0,yo) at time - t l . Since we are moving in circles, the motion is counterclockwise. Begin with u = 1, when 0 < 6 5 x , and move with decreasing t away from the origin in a counterclockwise direction along a semicircle center (1,O). At PIon this semicircle, t = -6, and sin(tt-6) changes sign so that ti switches to -1. With this value the trajectory is a circle with center (-1,O) that passes through PI.For exactly x seconds we move counterclockwise along this circle, and describe a semicircle. (Recall that a full circle takes 2 x seconds.) After ?r seconds, P2 is reached. This point is a reflection of PI onto the circle with radius 1 and center (-1,O). At P2 the control becomes u = +1, and optimal trajectory becomes a circle with center (1,O) that passes P2. After x seconds on this circle we reach P3 and the control switches to u = -1. This is the way that an optimal trajectory that passes through (x0,yo) is generated. Because the system is normal, this trajectory is unique. There is another route. Begin at zero with u = -1, when --?r 5 6 < 0 until at Q t = -6 - x , and sin(t 6) = 0. This is a circle with center ( - l , O ) , and motion is counterclockwise until we switch at Q to u = 1. For x seconds the motion is a semicircle with center (1,O) and at Q2 we switch t o u = -1, etc. Clearly, the switching occurs along two arcs r+ and r-. The plane is divided into two sets M I and M2,and then it40 = (0). Optimal feedback control is described as follows:
+
+
110
Stability and Time-Optimal Control of Hereditary Systems
( -1 if
(z,y) is above the semicircles,
or the arc r- : M Z +1 if (z,y) is below the semicircles oronr+:Ml.
Thus for
+
The optimal trajectories are just solutions of f z = f ( z , i ) with initial point (z0,yo) and final point (0,O)at time t l . See Figure 5.2.1. From this analysis we deduce that there exists a unique function f : E2 -, E such that (5.1.5) i ( t ) = A d t ) Bf(c(t))
+
describes an optimal trajectory of reaching 0 in minimum time. In (5.1.5), A and B are as described following (5.1.4). The semicircles constitute the switching locus: I’+ are arcs of circles of radius 1, centers (l,O), (3,O) .. .; I’- are arcs of circles of radius 1, centers (-1, 0), (-3,O) .. . . They move counterclockwise. These are described as “two one-manifolds.” Above the semicircle is the subset M2 of E2, and below is the subset MI of E 2 . These portions of the plane E2 in some t neighborhood of zero are called “terminal twemanifolds.” The synthesis of optimal feedback control is realized as follows: Given any initial state ( z 0 , Z o ) in some terminal manifold Mk, the optimal feedback function remains constant and optimal solution remains in Mk until the instant the optimal solution enters Mk-1, when a switching occurs. After this it continues t o move within Mk-1 until it reaches the origin, Mo,and the motion terminates. For multidimensional controls ( m > 1) the feedback function can be quite complicated. But Yeung [8] has proved that there are precisely 2rn”-’ “terminal n-manifolds”, and the “switching locus consists of a finite number of (analytic) k-manifolds” 1 5 k 5 n- 1); one for k = 0 and at least two for, k 2 1. Just as before, motion continues within Mk-1 until it reaches M k - 2 and the optimal feedback function switches again. The process terminates when the optimal solution reaches the origin, Mo.This description of the general situation is valid for controllable, strictly normal systems in an 6neighborhood of origin, i.e., in Int R(c), E > 0. In the case of the simple harmonic oscillator, c can be taken to be T , and R(T) contains a circle of
Synthesis of Time-Optimal and Minimum-Effort Conirol
111
radius 2 about the origin. If (z0,li.O)E Mz, one uses the control u = -1 to describe a circle with center (-1,O). This motion hits I?+ at some point. The control switches to u = 1 and motion continues on I"+, which is part of M I , until it terminates at the origin, which is Mo. An analogous situation holds for ( z 0 , l i . O ) E MI. Note carefully that in our circle of radius c = 2 there is at most one switch. In the next section we shall describe Yeung's description of the general situation.
5.2 Construction of Optimal Feedback Controls In this section, in an €-neighborhood of the origin, we construct an optimal feedback control for controllable autonomous strictly normal systems. Using the switching times of the controls, which are among the roots of the index of the control system y ( t ) = cTeA'B in the interval [O,c], we define terminal manifolds. Optimal feedback is constructed by identifying its value as a constant unique vertex of the unit cube in each manifold.
Definition 5.2.1: The system i ( t )=Az(t)
+ Bu(t)
is strictly normal if for any integers rj 2 0 satisfying
(5.2.1) m
C r, = n , the vectors
j=1
A'bj with j = 1,e.a , m , and 0 5 i 5 rj - 1 are linearly independent (whenever rj = 0, there are no terms A'bj).
Remark 5.2.1: If the system (5.2.1) is strictly normal, then rank B = min[m,n]. Also if m = 1 and B is an n x 1 matrix, strict normality is equivalent to normality. The number c > 0 that determines a neighborhood where the optimal feedback is constructed is obtained from the following fundamental lemma of HAjek: L e m m a 5.2.1 System (5.2.1) is strictly normal if and only if there exists an c > 0 with the following property: For every complex nonzero n-vector c # 0 and in any interval of length less than or equal to c, the number of roots, counting multiplicities of the coordinates of the index y ( t ) = cTe-A'B of ( 5 . 2 4 , is less than n . With
E
determined by the above lemma, define the reachable set
112
Stability and Time-Optimal Control of Hereditary Systems
It is in Int R(E),the interior of R(c) relative t o E” that the construction of a feedback will be made. In Int R(E)we shall identify the terminal manifolds. But first recall that optimal controls are bang-bang and are the vertices of the unit cube U.We need some definitions. Definition 5.2.2: Let 0 < B < c, and suppose k is an integer 1 5 k 5 n. For any optimal control u : [O,B] ---t U defined by u(s) = u j for tj-1 5 s < t j with 0 = t o < tl < .. . < t k < E and uj-1 < u j in U ,where uj is a vertex, the finite sequence (u1 -+ . . . -+ U k ) is called an optimal switching sequence. If u is not optimal and u j is not necessarily a vertex, then the finite sequence is called a switching sequence. The numbers t j 1 5 j 5 k are points of
discontinuities of u, and they are called switch times of u. The number of discontinuities of optimal controls is important. The following statement is valid: Corollary 5.2.1 Let E begiven by the Fundamental Lemma, and consider the interval [0, €1. Every optimal control u : [O, €1 + U has at most n - 1 discontinuities. Proof: Let u : [Bl,B2] U be optimal control, where 0 5 81 < 02 and 82 - < E . Then by Theorem 5.1.1, u = sgn cTe-*’B a.e. on [B1,02], for some c # 0 in E”. Since the discontinuities of u are among the roots of the index y(s) = cTe-As B , and this, by HAjek’s Lemma, is less than n, the assertion follows. -+
Definition 5.2.3: For each k = 1,.. .
, R , let Mk denote the points in Int R(c) whose corresponding optimal controls have exactly k - 1 discontinuities. Assume A40 = (0). “terminal manifolds .”
We are now ready to identify and define
Definition 5.2.4: For each k = 1, ... , n , and each k-tuple i = {ul ... -+ uk} of vertices of U with uj-1 # u j , define as terminal manifold the set Mki of points in Int R(E),whose optimal controls have (u1 ---t - .. -+ uk} as the optimal sequence. (Note that i ranges over a finite set I k . ) We will
-+
say that the optimal switching sequence {ul . uk} corresponds to Mki. The terminal manifolds Mki are disjoint since optimal controls are unique.
Next we shall relate the terminal manifolds t o the reachable sets. It is contained in the following statement: Proposition 5.2.1
are valid:
For terminal manifolds, the following disjoint unions
Synthesis of Time-Optimal and Minimum-Eflorl Control Mk =
U
U Mk.
IntIR(€) =
Mkj,
113
k=O
jEIk
For k = n, there are precisely 27nn-l nonvoid terminal manifolds Mki. The set M,i is open and connected (in En). Furthermore, each Mki is the image Of the Set Q k of all 't = ( t i . . . t k ) E Ek with 0 = ti < t i < - . -< t k < 6 of a diffeomorphism r': Mki = F(Qk), where F is defined as the map F : Q k + Int W ( E ) , by k
F(t)=
J'J j=1
e - A s Bujds,
tj-1
a function that is analytic in its variables, and whose Jacobian has rank k a t each point of Q k . F-' is continuous.
The description of Mki contained in Proposition 5.2.1 asserts that each is an analytic k-submanifold of En.The following is a consequence of these results: Mki
Corollary 5.2.2 If ( 2 1 1 + . . . -+ U k } is an optimal switching sequence corresponding toMkj, then { u , + l , . . . , u k } isan optimdswitchingsequence corresponding to Mk-j , i for j = 0 , . . . , k - 1 M k - j , i k nonvoid whenever Mki is nonvoid. Moreover, if we let M o , = ~ Mo, and if an optimal solution in Int EL(€) meets M k , i , then it meets only M k - l , i , . . . ,MO thereafter in this order.
These assertions are used to prove the next theorem. Proposition 5.2.2 The set Pnt R(E) is made u p of a union of 2m"-' disjoint connected nonvoid open sets and a finite number of analytic kmanifolds, 0 5 k 5 n - 1; one for k = 0 and at least two for k 2 1. Proof: From Proposition 5.2.1, Int W ( E ) =
u n
k=O
kfk
= Mn
u
k
Mk,
and
Mn is the union of 27nn-1 disjoint connected nonvoid open sets. We easily k f k validates the last assertion. But Proposition see that the union
u
ksn-1
5.2.1 states that Mni is nonvoid, so that M n - l , . . . ,M1i are also nonvoid, and by symmet,ry - M n - l , . . . ,-M11 are nonvoid as well. Because we have observed before that hfki are distinct analytic manifolds in En by Proposition 5.2.1, the proof is complete. The promised synthesis of optimal feedback control is contained in the proof of the following theorem of Yeung [S].
114
Stability and Time-Optimal Control of Hewditary Systems
Theorem 5.2.1 For the system (5.2.1) where A is an n x n matrix and B is an n x m matrix, assume that (i) the system is strictly normal. (ii) rank[B, ... ,A"-'B] = n . (iii) N o eigenvalues of A have a positive real part. Then there exists c > 0, and a function
f : Int R(€) + Ern, where
{
R(6) = Jd'e-A'Bu(s)ds : u E U
I
,
such that in Int R(c), the set of solutions of (5.2.2)
coincides with theset ofoptimalsolutionsof(5.2.1). Furthermore, f(0) = 0; for 3: # 0, f(z) is among the vertices of the unit cube U and f(3:) = -f(-2). If m 5 n, then f is uniquely determined by the condition that optimal solutions solve (5.2.2). Also the inverse image under f of an open set is an F,,-set, that is: the union of at most countably many closed sets.
Proof of Theorem 5.2.1 An c is obtained from HAjek's Lemma. Let f(0) = 0. If t o # 0, and 20 E Int R(c), find a terminal manifold Mki that contains 10.Suppose {ul + . . --* u k } is its corresponding optimal Because of the control, set f(3:) = u 1 , and note that f(3:o) = -f(-3:0). disjointedness of the union in Proposition 5.2.1, f is well defined. Let 3:o # 0, and 3:o E Int R(c), we can let 3:(-,3:0) be the optimal solution of (5.2.1) in Int R(E) through 20. Suppose now the optimal control u(s) of t o is defined by u(s) = u j for t j - 1 L s < t j , with 0 = to < 1 1 < . . . < t k = T ( i o )< c, where T(+o)is the minimal time function defined by T ( z o )= Inf{t 2 0 : t o E R(t)}. It follows that z0 belongs t o some Mki, so that by Corollary 5.2.2, ~ ( z ( z , c o )= ) u(s). As a consequence, every optimal solution of (5.2.1) in Int R(c) is a solution of (5.2.2). The converse follows from the following assertion: If y : [0,0]+ En is a solution of (5.2.2), and 0 < 0 < c, and if y ( t ) E Mki with 0 5 t 5 0, then also y ( s ) E Mkj for sufficiently small s - t 2 0. Indeed, the assertion implies that f(y(-)) is constant on any interval on which y(.) is differentiable, so that Corollary 5.2.2, y ( . ) coincides with the optimal solution through y ( t ) .
Synthesis of Time- Optimal and Minimum-Effort Control
115
The above assertion is proved by induction on n - k. Continuing the proof, we recall that f : Int R(c) -+ Em, and consider any open set 0 in Em. Then f-’(O) is the union of some {0}, Mki which, because they are manifolds in En,are u-compact. Thus the inverse image of an open set is a a-compact set, the union of at most countably many compact spaces, an F,-set. Dugunji [ll,pp. 74, 2401. To prove that f is unique we let g be another optimal feedback control. Because after reaching zero optimal solutions remain constant, g(0) = 0 = f(0). If zo # 0, and 20 E Int R(c), find Mk(, which contains ZO. Suppose z ( t ) = z(., 20) is an optimal solution through 20, and {ul ..-uk} is an optimal switching sequence corresponding to Mki. Then i ( t ) = A z ( t) - B u l for sufficiently small t 2 0, because of Proposition 5.2.1. Since z(., ZO) also solves zi = A v - W P ) , we have
i ( t ) = Az(t9 - Bul = A z ( t ) - B g ( z ( t ) )
on some (0, a ] , a > 0. Because i(.)is continuous on (0, a],it follows that Bul = B g ( t ) . Since the system is strictly normal and rn 5 n , the columns of B are linearly independent. As a result, g(x) = u1 = f(z). The theorem is proved. In the examples that follow, we shall construct optimal feedback controls for a variety of problems. The switching curves are carefully described, and the terminal manifolds constructed. All this is done in an c-neighborhood of the origin, where the c is given by Hlijek’s Lemma.
Example 5.2.1: We apply Theorem 5.2.1 to our system (5.1.4). From Htijek’s Lemma [3],cTe-A‘b has n - 1 discontinuities on [O,c] where c < and ai iwi are the roots of the characteristic equation IA - AII = 0, w = maxi(w,). In our case, w1 = 1, wi = -1, so that c < T. Since in (5.1.4) n = 2, Int R(c) = M1 U M2 U r+U r-. The switching curves are shown in Figure 5.2.1.
+
3
116
Stability and Time-Optimal Control of Hereditary Systems
U = +i (below Wand on r +)
FIGURE 5.2.1 We may define our terminal manifolds by moving backward in time from the origin (considering all possible control sequences as being moved backwards in time). The terminal manifolds, Mk’s, are the set of points in Int R(E) whose optimal controls have exactly (k - 1) discontinuities. If our final control is u2,the only possible sequence as we approach the origin is (u1-+ u2). If we move backward in time (ccw) in our phase plane, we can describe a terminal manifold M I . Control u1 = +1 corresponds t o a circle centered at ( l , O ) , so if we vary the radius of the circle from 1 t o 3 (and move backward in time until we reach the switching curve again) we may describe that particular terminal manifold. See Figure 5.2.2.
Synthesis of Tame-Optimal and Minimum-Eflort Control
117
u, =+1 u, = -1
FIGURE 5.2.2. Using Definition 5.2.3 we see that MI is defined above. Once the system is within M I , there is exactly one discontinuity before reaching the origin. Terminal manifold M2 is constructed similarly, but with optimal control sequences {u1 -+u g } . See Figure 5.2.3. We will now look at the possible control before M I (the manifolds). Our final control is {u1 + ua}. Moving backward in time, the only possible control is u2. So our optimal control sequence will be { u g + u1 + uz}. Now we will construct the corresponding manifold. Control of u2 = -1 corresponds t o a circle centered at (-1,O). So if we vary the radius of this circle (moving backward in time, counterclockwise) from r = 3 to r = 5 until we reach our switching curve again, the manifold will be defined. See Figure 5.2.4.
118
Stabilily avid Tanre-Optimal Control of Hereditary Systems
UM,
‘Mi
FIGURE 5.2.3.
FIGURE5.2.4.
ur..I urti
Synthesis of Time-Optimal and Minimum-Effort Control
119
So the optimal control sequence of the system, once it reaches the manifold above, is {ua -+ u1 4 212). The rest of the manifolds may be found in a similar manner. The result is shown in Figure 5.2.5.
FIGURE 5.2.5. NOW that our terminal manifolds have been described and drawn and our computed, our time-optimal feedback control is known.
E
Example 5.2.2: Our initial conditions will be represented by the ((t”in our phase plane in Figure 5.2.6. The initial conditions place it in M11,so our initial control will be 211 = $1 until our trajectory takes the states t o r-. Here we enter so the control will be u2 = -1. Our trajectory then reaches r+,at which time the states will be in M11. The control here will be q = +l. The states now enter E so there will be at most n - 1 = 1 more switches. We reach r- where the control switches to 21 = -1. This control takes the states to the origin.
120
Stability and Time-Optimal Control of Hereditary,Systems
FIGURE 5.2.6 We have used Yeung’s Theorem t o construct optimal feedback controls for the simple harmonic oscillator. The diagrams illustrate how we can attain zero from any initial position.
5.3 Time-Optimal Feedback Control of Nonautonomous Systems In this section we extend Yeung’s construction [8] of time-optimal control feedback t o cover the time varying n-dimensional differential equation
+
i ( t )= A ( t ) r ( t ) B(t)u(t),
(5.3.1)
where A is n x n , B n x m matrices, and where t + A ( t ) has n - 1 continuous derivatives and t ---t B ( t ) has n continuous derivatives on the interval [ t o , t l ] . If X ( t , t o ) is the fundamental matrix solution of
i ( t )= A ( t ) r ( t ) ,
(5.3.2)
Synthesis of Time-Optimal and Minimum-Effort Control
12 1
then the function g : [to,tl]-+ Emdefined by g ( t ) = cTX-'(t,to)B(t), with c E En a constant, is the index of the control system. If we assume normality (which is defined for Equation (5.3.1) below), then g is the switching function of the optimal controls in the sense that u ( t ) = sign g ( t ) .
(5.3.3)
By convention, X ( t ,0) = X ( t ) .
Definition 5.3.1: Let the operator I' be defined inductively as follows:
I? = -A(t) + - r k + l = rrk. dt ' The system (5.3.1) is normal in [to,tl]E J (or at all points of J 3 [to,tl]) if for each column b j of B and at each t E J , the n vectors in En,
ro= the identity,
bj,I'bj,. . . ,I'"-'bj,
(5.3.4)
are independent
Remark 5.3.1: If A and B are constant matrices, then normality is equivalent to the linear independence of b j , Abj , . . . , A"-'bj for each j = 1 , . . . , m. The following theorem of Chukwu and HAjek [2] relates the above definition of normality t o those of Hermes and Lasalle [6].
Proposition 5.3.1 System (5.3.1) is normal in [to, t l ] if and only if the following conditions are satisfied: (i) Each index g = cTX-' B of (5.3.1) satisfies the so-called indicia1 equat ion, n- 1
(5.3.5) k=O
with continuous diagonal matrices &(t); and (ii) for each c # 0 and each column bj of B , the set
G~(c) = {t E [to,tl]: CTX-'(t,to)bj(t) = 0) is finite. Proof: Assume system (5.3.1) is normal, and consider any column b j of B , and t E [to,tl].Then at 1 E [to,tl],rnbjmust be a linear combination of vectors from the basis in (5.3.4). Thus n-1 k=O
Stability and Time-Optimal Control of Hereditary Systems
122
with real dk depending continuously on t . If we collect the dk for i = 1,.. . ,m into a diagonal matrix, we find
Because the kth derivative of each index g = c T X - ' B is g k = c T X - ' r k B (5.3.6)leads to (5.3.4),for any c E En, so that (i) holds. Assume that (ii) fails and that so E [ t o , t l ] is an accumulation point of zeros of the component g j = cTX-'b,. Then at s ,
0 = (dk/dtk)cTX-'bj = cTX-'rkbj for k = 0,. .. , n - 1.
(5.3.7)
Because X-' is nonsingular and c # 0, the vector ( c ~ X - ' )#~ 0 and is perpendicular to bj ,... ,I'"-lbj , which contradicts normality. To prove the converse, we assume that normality fails, so that rank [ b j , . . , r n - ' b j ] < n , for some bj at some s E [ t o , t i ] . If we choose a nonzero vector of the form ( c ~ X - ' ( S ) that ) ~ is perpendicular to all the columns above, then (5.3.7)holds at s. With this c form the index
.
g(t)
= cTX-'(t)B(t)
that satisfies Equation (5.3.5). Because Dk is diagonal, p a t multiplication by ej yields that q = cTX-'bj satisfies the equation q(n) =
n-1 k=O
=
$dkj(t).
Clearly this equation has (5.3.7)as initial condition. Hence q 0, so that G j ( c ) = [ t o , t l ] . Gathering results, if normality fails, then (i) implies that (ii) is not valid. This concludes the verification of Proposition 5.3.1. We need the notion of strict normality for nonautonomous systems.
Definition 5.3.2: The system (5.3.1)is strictly normal on J (i.e., on each t E J) if for any integers rj 3 0 with
rn
j =1
r = n , the vectors
are independent for each t E J. Strict normality is equivalent to the "disconjugacy" of the indices of the system at each point. For an elaboration of this concept, see Chukwu and Hajek [l]. With these concepts, one follows the treatment in Section 2 and obtains the following extension of Theorem
5.2.1:
Synthesis of Time-Optimal and Minimum-Eflort Control
123
Theorem 5.3.1 Suppose that (i) A has k - 2 continuous derivatives and B has k - 1 continuous derivatives on the interval [0, t l ] . Assume that (ii) rank [ B ( t ) I'B(t), , -. , B ( t ) ]= n on [0, t l ] . (iii) The system
k(t) = A(t)z
(5.3.8)
is uniformly asymptotically stable. (iv) The system (5.3.1) is strictly normal at t = 0. Then there exists E > 0 and a function
f : Int R ( E )+ Em, such that in Int R(c) the set of solutions of
(5.3.9) 3i(t) = 4 M t ) + B ( t ) f ( Y ( t ) ) coincides with the set ofoptirnal solutions of (53.1). Also f(0) = 0; and for z # 0, f(z) is among the verticesof the unit cube U. Aho, f(z) = -f(-z). If m 5 n, f is uniquely determined by the condition that optimal solution of (5.3.1)solve (5.3.9). Consider the autonomous system
k=Ax+Bu. (S) The construction of time-optimal controls that is implied by Theorem 5.2.1 and Theorem 5.3.1 is dependent on the value of E . On that interval [O,c) every function
= cTeA'bj , j = I,. . . , m, (5.3.10) has a t most n - 1 roots. Here A is an n x n real constant matrix and bj b is an n x 1 matrix. It is important to know what is the largest value of E such that on the interval of length less than E , g(t) has a t most n - 1 g(t)
roots. The solution given by Hiijek [3]will be stated after introducing the following notation. Let r be the number of real eigenvalues c y i of A , and 2k that of the complex ones pj f iw, (counting multiplicities, w, > 0), so that n = r 2k. It is possible to write
+
rn
where pj ,qj , rj are real polynomials of appropriate degrees. Set w = maxwj, with w > 0 chosen arbitrarily if k = 0. The smallest integer M greater than x is denoted by [x]'. Observe that [z]' 2 1 whenever 2 > 0.
124
Stability and Time-Optimal Control of Hereditary Systems
Theorem 5.3.2 Consider g(t) in Equation (5.3.11). Then either g ( t ) E 0 or in any interval [O,T], T > 0, the function g ( t ) has a t mast N distinct zeros, where N 5 ( n - l)[TW/lr]*. Indeed,
N l n - 1 if all the eigenvalues of A are real, and N
5 r + (2k - l)[Tw/lr]'
(5.3.12)
if all the ejgenvaiues are not real, in which case we have the foiiowing special case: Ifg(t) = acos wt+bsin wt (i.e., r = 0, k = 1, n = 2), aciosed interval of length T = Cr/w, in which case the estimate in (5.3.12) (i.e., N 5 1) should be replaced by N 5 ! 1.
+
We have the following: Corollary 5.3.1 Suppose in Equation (5.3.10) g(2) $ 0. Then g ( t ) has at most n - 1 zeros, counting multiplicities in every nonclosed interval of length T/W.
[%I*
Proof: If [ O , T ] , T > 0, and T < T / W , then Note that w = max Iinag (eigenvalues of A).
= 1.
A somewhat better estimate on the number N of the zeros of (5.3.11) is given in the next result that is due t o Voorhoeve. Theorem 5.3.3 Let g(t) be as given in (5.3.11). For T > 0, let N be the number of zeras on [OT]. P u t J = max (Imag wj). Then 1 9 <m
In general, consider
m f(2)
= Cpj(r)exp(wjr),
(5.3.13)
j=1
where pj are complex polynomials, pj # 0, wj are complex numbers such that wi # wj if i # j . If I = min (Imag wj); J = max (Imag wj), then l
N 5
71
l<j
- 1 + T(J - 1 ) / 2 ~ .
Synthesis of Time-Optimal and Minimum-Effort Control
125
Remark: Note that f(z) in (5.3.13) satisfies linear ordinary differential equations. Example 5.3.1: 1: The Simple Harmonic Oscillator Equation of motion: mx + kx = u. For
-,
k
w=
m = l
(5.3.14)
= 1. This is the equation treated in (5.1.4a) and
(5.1.4b). See Figures 5.3.1 and 5.3.2.
...
r-
r-
r-
...
*Xi
r+
r+
U=+l
FIGURE 5.3.1. Note that the switching locus for Example 5.3.1 is I'+ U I?-. On r+,the control is u = 1. On r-, the control is u = -1. The control changes sign on the switching locus.
126
...
Stability and Tame-Optimal Control of Hereditary Systems
r-
r-
r- I
...
>Xi
r+
r+
r+
U=+l below Wand
onr+
FIGURE 5.3.2. It is stable, controllable, and normal, since it is the system
i: = A z + Bu, where A =
[ i] ,
B =
[!].
(5.3.15)
Optimal controls are given by
+
=' sgn[sin(t ~] a)], sgn[g(t,c)] = sgn(cTe-A'B) : u * ( t ) = s g n [ ~ ~ e - ~ where tan 6 = -2, c = (c1, c2). The simple harmonic oscillator is strictly C1
normal, so that there exists E > 0 such that the number of roots (counting multiplicities) of the coordinates of the index of the control system g(t, c ) = cTe-AtB in [0, c) is less than 2 and is at most 1. By Corollary 5.3.1,
Since the eigenvalues of A are fi, w = 1. Hence E < f. Recall that the terminal manifold Mk is the set of points in Int R(E)whose optimal controls have exactly one discontinuity. If we set u1 = 1, and 212 = -1,
127
Synthesis of Time-Optimal and Minimum-Effort Control
and if our final control that steers any initial point to zero is -1, then the only possible sequence is {q 4 .a}. If we move backward in time in a counterclockwise direction in the phase plane, the terminal manifold MI is described. Recall that with 1 ~ 1= +1 we described a circle centered at (1,O). If the radius is increased from 1 to 3 (and we steer backward in time until we hit the switching curve), we obtain all of MI. See Figure 5.3.3.
r-
I U,=tl
u, = -1
FIGURE 5.3.3.
-
In M I , there is exactly one discontinuity for the optimal control that drives any point in M I to the origin. With the optimal sequence ( ~ 2 14)we can analogously describe the terminal manifold in Figure 5.3.4.
128
Stability and Time-Optimal Control of Hereditary Systems
FIGURE5.3.4. TERMINAL MANIFOLDS Ml, M2 Going beyond terminal manifolds to other manifolds before M1 or MZ we can obtain a strategy as follows: Assume our final optimal sequence is (u1-+ ua}. Then if we move backward in time, the only possible sequence is (u2 --+ u1 --+ uz}. To construct the corresponding manifold we note that u2 = -1 corresponds to a circle centered at (-1,O). If we vary the radius of this circle (moving counterclockwise and backward in time) from r = 3 to r = 5 until we hit the switching curve again, the manifold M12 in Figure 5.3.5is determined.
Synthesas of Time-Optinid and Manamurn-Eflort Control
129
FIGURE 5.3.5. Other manifolds may be determined in a similar manner. See Figure 5.3.6.
FIGURE 5.3.6.
130
Stability and Time-Opiimal Control of Hereditary Systems
With terminal manifolds described and c computed, our time-optimal control law is known. For example, if our initial condition is point zo in Figure 5.3.7.
FIGURE5.3.7. zo E A411 so that the initial control is u1 = +1, until we enter Mlz and switch to u2 = -1 at the switching locus I?-. With this control we travel until I'+ is hit and enter MI, where the control is 111 = +l. Since we are within W(c), we will have at most one more switch a t r-, where the control is u = -1. With this control we enter zero optimally.
Example 5.3.2: Rest-to-Rest Rigid Body Maneuver Consider a rigid body such as a crane or a trolley whose mass is m, which moves without friction along a track. If u ( t ) is an external controlling force that is applied t o the rigid body, and if z ( t ) represents the position at time t , we have the following equation of motion: miqt) = u ( t ) , t > 0. (5.3.16) Assuming initial displacement t(0) = 20 and initial velocity i ( 0 ) = 0, we want t o construct a control u that brings the trolley to rest at the origin in
Synthesis of Time-Optimal and Minimum-Effort Control
131
minimum time: x(t*) = 0 i ( t * ) = 0. This is the problem of the rest-torest maneuver. We assume Iu(t)l 5 1, m = 1, and the dynamics become
x = Ax
+ Bu,
(5.3.16a)
where
A=
[
0
1
0]
, B=
[i] ,
z(0)
= (xo,O),
g(t*) = (0,O).
(5.3.16b)
System (5.3.16) is clearly controllable, and strictly normal. The eigenvalues of A are all zero. Therefore there exists a unique optimal control that steers the system from (x0,O) to (0,O) in minimum time t * , and is of the form
u*(t)= ~ g n [ c ~ e - ~ ' B ] . We easily calculate ,-At
= ~ - l {[
I
- A]-'} = L-l
=L-'(-[ 1 2 1] } = L - l { 0 s
s2
Therefore
=
[t s
0
4 1
1
'
S2
[:-El, cTe-A'B = &[-clt
1 +[l
+ c ~ =] g(t,c),
c2 where g(t,c) is a straight line in t and has only one switch at t = -. We examine two possible controls u = +1,
x2
= u ( t ) = 1,
C1
u = -1. If u = +1,
dXl 32 =-
d~2 1
Stability and Time-Optimal Control of Hereditary Systems
132
In the phase plane this is a parabola on its side that opens to the right and is shifted to the right by cl. See Figure 5.3.8. This parabola passes through the origin if c1 = 0.
I
I
*
x1
FIGURE 5.3.8.
FIGURE 5.3.9. 2 For u = -1, z2 = - 2 ( q - c 2 ) . This is a parabola that opens to the left and is shifted to the right by c2. See Figure 5.3.9. To obtain the switching curves, we move backward in time counterclockwise from our final target (0,O). With u = +1, z 2 = &&. With 'u = -1, 2 2 = &-.
Synthesis of Time-Optimal and Minimum-Effort Control
133
Inspection of Figure 5.3.8 with u1 = +1 shows that 2 2 is always increasing and we must use the lower half of the parabola in Figure 5.3.10.
FIGURE 5.3.10. In an analogous situation for u = -1, 2 2 is always decreasing so that we must use the upper half of the parabola in Figure 5.3.11.
FIGURE 5.3.11. The switching locus combines Figure 5.3.10 and Figure 5.3.11. See Figure 5.3.12.
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Stability and Time-Optimal Control of Hereditary Systems
U=-1
- - - - - - - - - - - - - - \ , - - - - - - - - - - - - - - - - - Xl
M1 U=+1
In!IR(e) =M1 UM2 ur- ur+ FIGURE
5.3.12.
To identify the terminal manifolds we use Corollary 5.3.2 to show that for one discontinuity c < 7r/w 00, (w 0). Thus any finite e is possible. Thus Int ~(f) = M 1 U M 2 U I'+ U I' + is the entire plane. Thus optimal feedback is defined by
=
!(Xl,X2)
={
- I
=
if (Xl,X2) lies on M 2 or on
r..
+1 if(Xl,x2)liesonM1oronf+.
Optimal solutions are trajectories of the system Mx(t) = f(Xl,xI). Example 5.3.3: Damped Harmonic Oscillator Many problems in engineering give rise to a damped harmonic oscillator equation. For example, the control surface of a flexible structure that is designed for space exploration is required to rest in a fixed position. If disturbed by an external force, and if nothing is done to restore it, the surface may behave as a damped harmonic oscillator. Let x(t) be a displacement from the desired position. Then the surface satisfies the differential equation mx(t) + 2bx(t) + k 2 x = 0,
Synthesis of Tirne-optamal and Minimum-Efloor2 Control
135
with initial conditions z(0) = 20 and i(0) = ko. The value 20 is an initial displacement, and io is the initial velocity imparted by the external force. The aerospace engineers aim to damp out the oscillatory behaviour of z ( t ) by applying a restoring torque u that should bring the surface to rest (0,O) = (z(i*), k ( t * ) ) in minimum time. Thus the dynamics is given by
m?(t) + 2 b i ( t )
+ k 2 z ( t ) = u(l),
z(0) = 20, i ( 0 ) = ko, z ( t * ) = 0,
i ( t * )= 0,
(5.3.17a)
where t* is minimum. We assume that
The state space representation of (5.3.17a) is &(t) = Ag(t)
+ B t ~ ( t ) ,~ ( 0 =) g o ,
i ( 0 ) = 50,
(5.3.17b)
where
2b k2 where - = 2 a , - = w 2 , We note that m is the mass, 26 is the damping, m m and k 2 is the stiffness. All these constants are positive, so that (5.3.17) is stable. We solve the time-optimal problem using Theorem 5.1.1. We simplify our calculation by setting k = 1, m = 1, so that
*
A = [ -w2
-2a
1, .=[;I.
Then (5.3.17) is controllable: rank [B, AB] = rank
[;
4
=2*
The system is also strictly normal since m = 1. The eigenvalues, A, of A, are solutions of
136
Stability and Time-Optimal Control of Hereditary Systems
Hence no eigenvalues of A have a positive real part. The conditions of Theorem 5.1.1 are satisfied, and therefore there exists a unique time-optimal control that drives our system from zo to 0 in minimum time. This control is given almost everywhere by
There are three cases: The UnderdamDed Case (a2
Let
- w 2 < 0).
p2 = a2- p2 < 0. If we let y ( t ) = cTe-At, then Y ( t ) = -y(t)A, where
Yl = w Y2
which is equivalent to y 2 (set y2 = es*) is s2 - 2as
2
y2,
= -y1
- 2ay2 - w2y2 = 0.
+ w 2 = 0, where
s = cr f & g(t,c)
where D = [c: control is
+ 2ay2,
G r 2
The characteristic equation
= cr f ip,
= y2(t) = ea'[cl co sp t+ c 2 s i n ~ t ] , = Dea' sin(,& + S),
+ ci]a
u * ( t ) = sgn[y&)]
and S = tan-'(-cz/cl).
Thus the time-optimal
= sgn[Dea' sin(@ + S)], Deaf is positive.
Hence u*(t) = sgn[sin(Pt+ 6)J.
The zeros are the t , such that
137
Synthesis of Time-Optimal and Minimum-Eflort Control
i.e., t such that Pt
+ S = Lr,
(k integers),
kn-6 t =P
I
T / P after the first switch. We now determine the switching curves for the underdamped case:
so that controls switch at intervals
= 22,
21
= -w2x1- 2ax2 f 1, 22 dXl- -w2xl 2 ~ ~f x 21 ‘ d ~ z x2
If u = +1, we obtain
- 2 ~ ~ +x 1)dzl 2 = 22d~2, -+w2x; - 2ax1x2 + E l = 32; + c1, 2 2 W 2 1 + 4~221x2- 221 + X f = a:, (-w 2 X I
The curve is described in Figure 5.3.13, and it is a spiral approaching
(-$,O)
as t
+ m.
For u = -1, we deduce the curve
See Figure 5.3.13. If we let 21
=
21
=
22
= a s i n t in the case
u
= +1, then
-ucost-2aesint+l W2
In the case u = -1, -acost -22aasint
-1
W2
In this underdamped case p2 = a’ - w 2 < 0, x1 = 1 2 ,
= - w 2 2 1 - 2ax2 f 1, xl(t) = e-Q*[cl cospt + c2 sin@] + 1/k2, 2 2 ( t ) = e-af[/3c2 cos pt - Pc1 sin Pt + l / k 2 . x2
138
Stability and Time-Optimal Control of Hereditary Systems
All solutions tend to (1/P2,0) as t
-+
00.
I
u=-1 Spiral Approaching$
FIGURE5.3.13. If is increasing, one of these spirals will be traversed c-dckwise (cw). If 1 is decreasing, one of these spirals will be traversed counterclockwise. Let us move backward in time in order to find the switching curves. We know that our optimal control is of the form u*(t) = sgn[sin(Pt - S)]. Control switches at intervals of a/@ after the first switch. Let = 1. Following the same process as for the undamped oscillator, we see that the switching curves correspond with “unfolding” our spirals corresponding to u = -1 and u = +1 from the origin along the x2 axis. The resulting switching curves are seen in Figure 5.3.15. Let w = 1. Let 0 < 6 5 T. Move -6 seconds counterclockwise around the arc (on our spiral) corresponding to u = +l. At 1 = -6, sin(t + 6) changes sign so our control switches to u = -1. With u = -1, the optimum trajectory is centered at (- 1,0) and passes through PI. This arc is traversed for ?r seconds (a seconds between switches) until on a spiral P2 is reached. At P2, t = -6 - ?r, so sin(t + 6) changes sign and the control is once again u = +l. With u = +1, the optimum trajectory is a spiral centered at (1,O).
Synthesis of Time-Optimal and Minimum-Effor2 Control
139
FIGURE 5.3.14. If 6 is varied from 0 to 7~ seconds along the arc shown in Figure 5.3.14, the switching curves begin to take shape. The entire switching curve for our problem can be formed by moving backward in time, beginning with u = +1 as done above, then beginning with u = -1. The switch curves will follow. See Figure 5.3.15.
x2
A
r-
u=-1
(aboveWand on r-)
U=+l
(below W and on r + )
FIGURE 5.3.15. We may now define our terminal manifolds by moving backward in time from the origin (considering all possible control sequences as we move back in time). The terminal manifolds, Mk’s, are the set of points in Int R(E)
140
Stability and Time-Optimal Control of Hereditary Systems
whose optimal controls have exactly ( n - 1) discontinuities. The steps for determining terminal manifolds M I , M2 mirror those for the undamped harmonic oscillator. See Figure 5.3.16 for the Terminal Manifolds, The manifolds are shown in Figure 5.3.17.
t
X2
FIGURE 5.3.16. From Hijek’s Lemma [3], there is for n - 1 = 1, ( n = 2) discontinuity of lr qTe-AtB on [0, c), 6 < =. Indeed, ?F+ zGi roots of IA - Ail, V = mv(Vi) W
1
+ A) + w2 = 0 , X2 + 2 a A + w2 = 0,
X(2a
A=
so i z= d w T , We have See Figure 5.3.17.
w
-2a f-d 2
> a , and c <
J
a W
q
Int Ilk(€) = M1 u M 2u
.
u rl.
Synthesis of Time-Optimal and Minimum-Eflort Control
141
FIGURE5.3.17.
So our optimal feedback control will be defined as
,
f(21 2 2 )
=
-1
if if
$1
on M2 or on r-, lies on M: or on l?.
,
(21 22) lies ( 2 1 , y2)
Our optimal trajectory is simply solutions of the equation
21 or
+ mC + mk --i1
c x + -x m
-21
+ --2mk
= f(Z1,
El),
= f(z, E )
for underdamped system p2 = a2 - w 2 < 0, Critically Damped System (a2- w 2 = 0)
C
27r = -w2
m
k = -.
m
142
Stability and Time-Optimal Control of Hereditary Systems
In this case, s = 0 y,
- 2ay2 - w 2y2 = 0 ,
+
= e='"c1 cztl, k l ( t ) = W2y2(t)= u2eo*[c1 + c z t ] , Y2(t)
i2(t)
= aeat[cl + cat] + c2eat,
+ 2 a y z ( t ) = +(t) + 2aeat[c1 + c z t ] ,
y l ( t ) = -&(t) y l ( t ) = aeat[cl
Because B
=
[;],
+ czt] - c2eof.
we have
g ( t , c ) = c T e - A t B = y ( t ) B = y 2 ( t ) = eclt[cl
+ czt].
Optimal control is given by u * ( t ) = sgn[y2(t)]. There is at most one switch when c 1 + cat = 0 , i.e., t = -5. To obtain the trajectory, c2 2 i 2 = -w 21 - 2ax2 f 1. il = x , ,
This passes through the origin if d x1 dt2
x2
-W2X1
- 2at2 - 1
(u
-
= -l), 22
-a221
The trajectory describes the curve
This corresponds to I'-
- 2ax2 - 1 '
in Figure 5.3.18.
u=-1
U=+l
I \
FIGURE 5.3.18.
(a2 =
2)).
Synthesis of Time-Optimal and Minimum-Effort Control
143
For u = 1.
FIGURE5.3.19. This corresponds to d - w 2 >O)
r+ in
Figure 5.3.18.
For the overdamped case
= -a f p < 0, y2(t) = eQ'[clep' + c2e-o'J. s
1
This is at most one switch when t = --ln(-q/c2). 2P Lemma yields 6 = r/O = m). For u = +l.
This corresponds to I?+ in Figure 5.3.18. For u = -1.
(Note that Htijek's
144
Stability and Time-Optimal Control of Hemdita y Systems
This corresponds to r- in Figure 5.3.18. The terminal manifolds are described in Figure 5.3.19. The curves in both cases are hyperbolas.
Example 5.3.4: Consider the system
which is equivalent to i = Ax
+ Bu, where
The system is normal and controllable since rank [ B ,AB] = rank
[:; -;:]
=2,
and rank It is also strictly normal.
[ b j , Abj] = 2,
For any integers rj 2 0 with 2 =
( n = 2, m = 2), the vectors
are linearly independent. (If following choices:
j = 1,2.
rj
2
C rj, j=1
= 0, there are no terms A'bj.) We have the r1
r2
0 1 2
2 1 0
r1 = 0 implies there are no terms in A'bl, since r1 - 1 = -1 and 0 5 i 5 r j - ? . Also ~1 = 1 implies i = r1 - 1 = 0. r1 = 1 implies r2 = 1, so [A'bl,A'bz] = [Aabl,Aob2]= [ b l , b 2 ] , (Ao is the identity matrix), [bl,bz] =
[ ],
that
and this has rank 2. If r1 = 2, i = 0 , l and rz = 0; and we have
Synthesis of Time-Optimal and Minimum-Effort Control This matrix has rank 2. If vectors are
r2
= 2, then
rl
145
= 0 and the corresponding
with rank 2. We note that r2 = 0 implies no terms in A'bz, since rz-1 = -1 and 0 5 i 5 rj - 1. If 7-2 = 1, i = rz - 1 = 0, and rl = 1. An earlier argument yields linear independence. We have verified the strict normality condition. The equation x = Ax has eigenvalues X = f i , which has zero real parts. All the conditions of the theorem are satisfied: There exists which we shall now determine. The an optimal feedback control f(z1, z~), open loop control is given as u*(t) = ~ g n [ c ~ e - ~ ' o~ 5 ] ,t
5 t*,
where t' is the minimum time. For our system eAt =
= where A =
d-,
cost -sint
[
sin(t CW(t
1
sin t cost '
+ 6) + 6)]
e-At
= cost - sin t sint
1
cost '
I
6 = tu;'(-cz/cl). u*(t) = sgn
[
sin(t cos(t
It follows that
+ 6)
+ 611 .
Because cos u and sin a differ by 7r/2, the sign changes of the optimal control components are 7r/2 apart. Each component switches every 7r seconds. The control U* is bang-bang and are the extreme values
146
Stability and Time-Optimal Control of Hereditary Systems
of the 2-dimensional unit cube. We now construct !(XI, Case 1: u =
[:I,
x1
i 2
= q + 1
$21 *-=-
22).
22+1
+1 + 1' + l)d+l = + l)d22,
= -21
$22
-21
(22
(-21
integrating 1 * -2": +
(21
+ +
-
(22
21
1)2
1 2
= -2;
+
22
+el,
= a2
The equation describes a circle of radius a centered at (1, -1) in the phase pIane. Case 2: u =
[-:I.
so that - p1 1 2
- 21 = 522 1 2
+ + c1, 22
(21
+ + + 1)2 = az1 1)2
which is a circle of radius 0 centered at (- 1, -1). c a s e 3: u =
[-:I.
which is a circle of radius a centered at ( 1 , l ) .
(22
Synthesis of Time-Optimal and Minimum-Effort Control Case 4: u
=
147
[I:]. il = 2 2 - 1 i 2 = -21 - 1
*
(21
- 1)2 + ( 2 2 + 1)2 = a2.
This is a circle of radius a centered at ( - 1 , l ) . Forming the Switching Curves With increasing time t , the circles described on the previous page are traversed clockwise; with t decreasing, the circles are traversed counterclockwise. Let us move backward in time to find the switchin curve.
[ -:]
-62
I-:].
>7r > 62 > 0.
Suppose our final control is uo = Move 2 seconds counterclockwise to PI around the arc corresponding to u1 =
Let 61
(circle centered at (1,l) that passes through the origin). Since we are
moving backward in time, our only possible control is 61
> n/2 > 62 > 0 and
[
sin(t - 61)
u * ( t ) = sgn sin(t
I
Ub
=
[I;],because
- 6,) . The control Ub =
[I:]
corresponds to a circle centered at ( - 1 , l ) and passes through PI in the phase plane. We now traverse this circle counterclockwise for 7r/2 seconds (61 - 52 = 7r/2). Traversing the circle n/2 seconds corresponds to moving a quarter circle counterclockwise from PI to P2. At P z , t = -62-7r/2(= -&), so sin(t-61) changes sign and the control will be u, =
[-:I,
circle centered
at (-1, -1). See Figure 5.3.20. Looking a t Figure 5.3.20, if we vary 62 from t = 0 to t = 7r/2 seconds, we see our switching curves for our problem take shape. The remaining switching curves for our problem can be formed as before by moving backward in time, starting with all possible controls. The switching curves are shown below in Figure 5.3.21. If an optimal trajectory hits the switching locus, it changes by using another value of optimal control and transverses another path of optimal trajectory until it reaches the origin. Detailed descriptions of the terminal manifolds will now be given.
148
Stability and T i m e - O p t i m a l Control of Hereditary Systems
FIGURE 5.3.20. The calculations needed for this diagram and some of the diagrams below were verified by Kenneth Coulter, C. Ukwu, and D. C. Etheridge. The information was made available by a personal communication.
149
Synthesis of Time-Optimal and Minimum-Eflort Control
FIGURE 5.3.21. SWITCHING CURVES Terminal Manifolds and Switching Locus &call the definition. For each k = 1, ... , n , let Mk be the set of points in Int R(6) whose corresponding optimal controls have exactly k - 1 discontinuities. Set Mo = [O]. For each L = 1 , . . . , n and each k-tuple = (211 -+ . . . 4 u k } of vertices of u with uj-1 # u j , let Mki be the Set of points in R ( E )whose optimal controls have (211 -+ . .. -+ I&} as the optimal switching sequence ( M k i may be void; the indices range over a finite set 4 ) .w e may also say that the optimal switching sequence (211 . . . + uk} corresponds to kf&. The Mki are called terminal manifolds. It follows from the uniqueness of optimal controls that the terminal manifolds are disjoint. For k = n there are precisely 2rnfl-' nonvoid sets Mki. The switching curves for this problem is below. -+
150
Stability and Time-Optimal Control of Hereditary Systems
...
...
* x1
FIGURE 5.3.22. We may define our terminal manifolds by moving backward in time from the origin (considering all possible control sequences as we move backward in time). The terminal manifolds, Mk's, are the set of points in Int R(E) whose optimal controls have exactly (n- 1) discontinuities (n- 1 = 2- 1 = 1 for this case). Suppose our final control is uc (1'+2 in Figure 5.3.23). If we move backward in time (ccw) in our phase plane, we can describe the terminal manifold M2. If our final control is u e , the only possible control sequence as we approach the origin is
{ud
-+
tie}.
Control
ud
=
[-;I
corresponds to a circle centered at (-1, -1); so we vary the radius of this curve from 4 to 0(and move backward in time (CCW)until we reach
Synthesis of Time-Optimal and Minimum-Effort Control
151
the next switching curve) to describe that particular manifold. See Figure 5.3.23.
% r=m
(1,-1)
FIGURE 5.3.23. TERMINAL MANIFOLDM2 Using the definition, we see that M 2is defined above. Once the system is within M 2 , there is exactly n - 1 = 1 discontinuity before reaching the origin. Terminal manifolds M3, M4, and M I are constructed in a similar manner but with control sequences {u, -+ Ub}, { U b -+ u,,}, and {ua -, u d } respectively. See Figure 5.3.24 for terminal manifolds.
152
Stability and Time-Optimal Control of Hereditary Systems
FIGURE 5.3.24. Now we may define the other manifolds in our phase plane, by moving backward in time (ccw) from our terminal manifolds Mk's. Suppose our final control sequence is {ud -+ u,} (terminal manifold Mz). If we move backward in time from A42, the only possible control is u,. So our optimal control sequence will be {u, -+ ud + u c ) . Now we will construct the corresponding manifold. The control ua =
[ -:]
corresponds to a circle
centered at (1,l) in the phase plane. So if we vary the radius of this circle (moving backward in time, ccw) from r = 0to r = until we reach our switching curve again, the manifold will be defined. See Figure 5.3.25. in Figure The optimal control sequence of the system once it reaches 5.3.25 is (u1 -+ ud -+ u c } . The rest of the manifolds may be found in a similar manner. The result is shown in Figure 5.3.26.
a,
Synthesis of Time-Optimal and Minimum-Effort Control
153
:I FIGURE 5.3.25. MANIFOLD Mal Some of the calculations needed for diagram 5.3.25 were verified by C. Ukwu, D. L. Etheridge, and Kenneth Coulter (personal communication).
154
Stability and Time-Optimal Control of Hereditary Systems
FIGURE 5.3.26. MANIFOLDS Since the system is strictly normal, there is c > 0 such that in Int It(€), g ( t ) = cTe-AtB has at most one discontinuity. Clearly c < T. Also Int R(c) =
u M~ u r + j . 4
j=1
Synthesis of Time-Optimal and Minimum-Effort Control
4
155
x2
FIGURE 5.3.27. Now, since we have our terminal manifolds described, our time-optimd feedback control is also described. As an example, suppose our initial conditions place the states in M 4 4 as shown by the “t” in Figure 5.3.27. The trajectory of our states and the time-optimal feedback control necessary to reach the origin is known.
As seen from Figure 5.3.27, our control will be 214 ---* ul -+ U Z -+ 213 -, u4 + u1. The manifolds traversed will be M 4 4 , M 4 1 , M 4 Z J M 4 3 , M 4 .Our control switches as we switch manifolds. Our final switch in control, once we are within the €-neighborhood of the origin, will be from u4 to u1.
156
Stability and Time-Optimal Control of Hereditary Systems
Our time-optimal feedback control for this problem will be defined as:
I[-:I
f(XlI22)
r4 and I" or on r',
if
21, t 2
lie between
if
t1,x2
lie between I'1 and
if
t1,22
lie between
r2 and r3 or on r3,
if
21,x 2
lie between
r3 and r4 or on r4.
=
r2or on r2,
Our optimal trajectory is simply solutions of the equation
5.4 Synthesis of Minimum-Effort Feedback Control Systems In this section we present the solution of the minimum-effort/energy control problem. For unrestrained controls, an explicit formula is given for the optimal feedback control. When the controls are restrained, the Neustadt/HAjek [4,7] open-loop solution is presented. With the insight gained from Yeung's optimal feedback solution of the minimum-time problem, we shall attempt to construct optimal feedback control of the system. Define the set U as follows:
U = {u
:
[O,q
---+
E m , u measurable and integrable}.
Suppose J : U -+ E , then J ( u ( T ) )is the effort associated with u(t). The minimum-effort problem is stated as follows: Minimize J ( u ( T ) ) ,
(5.4.0)
u €24
subject to
i ( t )= At(t) + Bu(t), where t(0) = to, z(T,to,u)= z1
(5.4.1)
(5.4.2)
Here t ( . , t o , u ) is the solution of (5.4.1) with t ( O , q , u ) = to. Implicit in this problem is the assumption that with 20,tl E En arbitrarily given,
157
Synthesis of Time-Optiinal and Minimum-Eflort Control
there is an appropriate T such that some control u transfers z from 10 to 2 1 in time T ; that is, the solution satisfies z(0,10, u) = zo and z(T,10, u) = z1. Thus we assume that System (5.4.1) is controllable on the interval [ O , T ] with controls as specified. We now isolate several cost functions J ( u ( T ) )that describe efforts to be minimized, and then state the solutions. A general proof of these theories will be discussed in Section 5.5.
Theorem 5.4.1 where
Consider Problem (5.4.0) subject to (5.4.1) and (5.4.21,
Jo(u(T))=
1' 0
u(t)+R(t)u(t)dt,
and where R(t) is a positive definite n x n matrix. (Here pose.) Suppose rank [ B , A B , .. . , A"-'B] = n.
(5.4.3)
+ denotes trans(5.4.4)
Let
(5.4.5) The control
where is the optimal one.
Remark 5.4.1: Note carefully that there is no constraint on the controls. When R ( t ) I, the identity matrix, then (5.4.8) and the following corollary is valid: CoroIIary 5.4.1 Consider problem (5.4.0) with (5.4.1) and (5.4.2). The optimal control is u*(t)= B+(e-At)+M-'q,
J in (5.4.8) subject
to
(5.4.9)
where
(5.4.10) q
= (e-
AT
z1 - to).
158
Stability and Time-Optimal Control of Hereditary Systems
Proof: It is quite standard that condition (5.4.4) is equivalent to the nonsingularity of W ( T ) ,so that (5.4.5) is well defined. Insert (5.4.5) into the solution of (5.4.1) that is given by the variation of parameter z ( t , z O , u )= eA' [ZO
+
1 t
e-"'Bu(s)ds]
.
(5.4.11)
Thus u' transfers zo to t 1 in time T. Suppose u ( t ) is any other control that transfers 2 0 to 2 1 in time T . We shall prove that
and also that J(U*(T))
= (P, W--lP)I
(5.4.13)
where (., -) denotes inner product. Because
iT
we have
e-A8Bu(s)ds=
1'
e-"'Bu*(s)ds.
(5.4.14)
If we subtract both sides and use inner product, we obtain
(lT
e-A"B[u(s)- u * ( s ) ] d s , W - ' ( T ) [ e - A T z l - 201
We use (5.4.6) and the properties of inner product to obtain
iT(.(.) -
u * ( s ) , u*(s))ds
= 0.
(5.4.16)
Using (5.4.16) and some easy manipulation, one deduces that indeed
159
Synthesis of Time-Optimal and Minimum-Eflort Control
Jr
But J ( u * ( T ) )=
where q = [ e - A T t l Jo(u*(T))=
1
u * + ( t ) R ( t ) u * ( t ) d so t , that
-
T
201.Since
W ( T )is symmetric, (5.4.17) yields
( q , W - ' ( T ) ) . [e-A8B]R-'(~)[e-A"B]+W-'(T)q)ds
The optimal solution given in Theorem 5.4.1 is a feedback one. It is global. But the controls are "big" and unrestrained. When the effort function is defined to correspond to the "maximum thrust" available, there are hard limits that bound the controls. This situation is treated in the next theorem.
Theorem 5.4.2
Consider the problem (5.4.0) with
(5.4.18) where
luj(t)l _< 1, j = l , . . .,m,O5 t 5 T}, (5.4.19) subject to (5.4.1) and (5.4.2), with 21 3 0. Define the function u
E U1 = {u E E m , u measurable,
= e - A t e l ( t ) - 20, g(t, c) = c .e - A t B , y(t)
(5.4.20)
(5.4.21)
where c is an n-dimensional row vector, and g the index of the system (5.4.1). Let
F ( c ,T ) =
cJ m
j=1
T
0
c)ldt.
(5.4.22)
Assume that: (5.4.23) (i) rank [ B ,A B , . . . ,A"-lB] = n. (ii) For each j = 1 , . . . ,m, the vectors { b j , A b j , .. . ,A"-'bj} (5.4.24) are linearly in depen den t . (iii) No eigenvalue of A has a positive real part.
160
Stability and Time-Optimal Control of Hereditary Systems
Then for each 20, and some response time T, there exists a minimum effort control that transfers xo to 0 in time T. If y(T) # 0, i.e., xo # 0, the minimum-effort J1 min = J1 (u*(T)) is given by
1
-
Jlmin
= min F(c, T ) ,
(5.4.25)
~ E P
where P is the plane c y(T) = 1. Furthermore, the optimal control u*(t) is unique almost everywhere and is given by u * ( t ) = Jlminsgn g ( t , c * ) ,
(5.4.26)
where C* is any vector in P for which the minimum in (5.4.25) is attained. If y(T) = 0, i.e., 10 = 0, the control u * ( t ) = 0 is the desired minimum-effort control.
Remark 5.4.2: The statement and proof of Theorem 5.4.2 is given by Neustadt [ 7 ] . He suggests that the method of steepest descent can be used to calculate min F(c, T ) c E P = {c : c y(T) = I}. It is important to observe that for some fixed time T, minimum-effort optimal control u* is given by u*(t) = J 1 rninsgn g(t,c),
2 E [O,
TI,
where J1 = Jlrninis a constant and g is the index of the control SYStem. For the time-optimal problem, the optimal control v * ( t ) is given by v * ( t ) = sgn g(t,c); t E [0, T ] with T the minimum time. That is, the indices are switching functions of optimal controls. Thus, optimal solution of the minimum-effort control of (5.4.1) is a constant multiple of the minimum-time one. Therefore, the treatment of Sections 5.4.2 and 5.4.3 can be appropriated for the construction in a neighborhood of zero, of optimal feedback control of minimum-effort problem. It is stated in the next result . Theorem 5.4.3 Consider the problem (5.4.0) with J l ( u ( T ) )defined in (5.4.19) subject to (5.4.1) and (5.4.2), where x1 = 0. Assume that: (i) rank [ B ,A B , , .. , A"-lB] = n. (ii) No eigenvalue of A has a positive real part. (iii) The system (5.4.1) is strictly normal. Then there exists an X > 0 and a function f : Int R ( E )+ E", where E = min[X,T], and
161
Synthesis of Time-Optimal and Minimum-Effort Control
such that in Int R(c), the set of solutions of
i ( t )= Az(t)
+ Bf(z(2))
(5.4.27)
coincides with the set of optimal solutions of (5.4.1). firthermore, f(0) = 0, and for 2 # 0, f(x) is among the vertices of the cube V
V = { V E Em,Ivj 1 5
Jmin,
j
= 1 , . . . ,m } ,
and f(x) = -f(-2). If rn 5 n, then f is uniquely determined by the condition that optimal solutions solve (5.4.27). Also, the inverse image of an open set in Em is an F,-set.
Remark 5.4.3: If X 2 T , the optimal feedback control constructed from Theorem 5.4.3 is a global one for the given T so that the minimal control strategy f(z) drives any 20 to 0 in time T . If X < T,only a local fueloptimal feedback control is constructed for strictly normal systems. Proof: The only modification needed in the proof of Theorem 5.4.3is to replace the unit cube U by the cube V whose components have dimensions ztJ,i,,. Thus, the optimal switching sequence is the finite sequence (v1 - . . v k } , where v j are among the vertices of V , and optimal control u on [O,B] for B < E is defined by u ( s ) = vj for t j - 1 5 s < tj, and 0 = tl < t l < - - - t k with vj-1 # w j , where k is an integer 1 5 k 5 n. As before, we set Mo = {0}, and for k = 1, .. . , n , we let Mk be the set of points in Int W ( E )whose components have exactly k - 1 discontinuities. For each k = 1,.. . , n , and each k-tuple 1= ( ~ 1 , .. . , V k } of vertices of V with vj-1 # vj, let Mki be the set of points in Int R ( E )whose optimal controls have {vl -+ . . . 4 v k ) as the optimal switching sequence. Because (5.4.1)is strictly normal, there exists E > 0 such that the indices g ( t , c ) of (5.4.1)has at most n - 1 discontinuities on [ O , E ] . With this E ,
define R(E) and note that IntR(E) =
u n
k=O
kfk,
MI, =
u
Mkj
as before.
j € [I,
Set f(0) = 0, and for 0 # 2: E IntR(c), find Mkj containing x and let (v1 4 . . + V k } be the optimal switching sequence corresponding to Mbj. Finally, set f(z) = v1. The rest of the argument follows as in Theorem 2.1.
Theorem 5.4.4 Consider the problem (5.4.0) with the cost function representing a pseudo fuel (energy if p = 2) :
Stability and Time-Optiiital Control of Hereditary Systems
162
where admissible controls are
As usual, the minimization is subject to (5.4.1) and (5.4.2). Assume that (4 rank [B,AB, . .. ,A"-'B] = n. (ii) No eigenvalue of A has a positive real part. (iii) For each J' = 1,... ,rn, the vectors b j , Abj . ,An-lbj are linearly independent. Then for each xo and some T,there exists an optimal control u * ( t ) that transfers 2 from xo to 0 in time T. firthermore, we let g(t, c) = c e - A f B , c E E", a row vector, (5.4.30)
where
-1 + -1 = 1. If 10 # 0, then
-P
given by
9
the minimum-effort Jmjn= J ( u * ( T ) )is
1
1
Mz(T)
Jzmin
- min F ( c , T ) ,
A=---
CEP
(5.4.31)
where
P = { C E En : -CXO = 1).
(5.4.32)
The optimal control is unique almost everywhere and is given by
where p = J2 min[F(c*, T)]-$ and C* E E" is any vector in P where the minimum in (5.4.31) is attained. lfzo = 0, the optimal control is u * ( t ) = 0. The last cost function considered is the measure of absolute fuel defined by
(5.4.34a)
Synthesis of Time-Optimal and Minimum- Effort Control
163
where the set of admissible controls is given by
This is the so-called absolute fuel minimum problem. There is no optimal soIution for this cost function if we assume that u E U3 is integrable. HAjek [4] has proved the following result:
Theorem 5.4.5 In problem (5.4.01,with cost (5.4.34) and constraints (5.4.1)and (5.4.2)with $ 1 = 0, assume: (i) rank [ B ,A B , . . . ,A”-’B] = n. (5.4.35)for each column b, of B (this (ii) rank [Abj, . . . ,Anbj] = n means that (5.4.1) is metanormal). There is no optimal solution with 21 integrable that steers 20 to 0 in some T , while minimizing J3(u(T)). We now admit, in absolute fuel minimization problems, as admissible controls those that are unbounded and “impulsive” in nature, the secalled Dirac delta functions. If such functions are considered optimal minimum, -IIuII1 controls exist.
Theorem 5.4.6 In problem (5.4.0), with the cost function (5.4.34)and constraints (5.4.1)and (5.4.2)with X I = 0, suppose (i) rank [ B , A B ,. .. , A”-’B] = n . (ii) rank [Abj,.. . , A”bj] = n for each j = 1 , . . . , m. Suppose g ( t , c) = c . e-A‘B,
F ( T , c ) = max max Igj(c,t)l.
(5.4.36)
l < j S m OStST
If xo # 0, and if we consider generalized (delta) functions to be admissible, then there exists a minimum-effort control u*(t) that transfers xo to 0 in time T . The minimum fuel J3(u*(T))= J3min is given by 1 = min F ( T ,c ) = F ( T ,c*), Jmin ~ E P
where P = (-czo = 1). The optimal control is given by ? ( t ) = J 3 m i n E ( t , c*), where the boundary control is given by Nj
~jiijt)= C s g n ( g j ( c * , r j i ) ) s (t r i=l
m j i ) / x ~ j , j=1
15 j 5 m.
164
Stability and Time-Optimal Control of Hereditary Systems
Here the maximum in (5.4.36) can occur at multiple j and at multiple instances rji, i = 1,2,. . ,N j , where Nj equals zero if g j does not contain the maxima. I f we replace the control set by
then optimal control is given by
u*(t)= Jminn(t, c*) where the boundary control is given by
where
- 1-
Jmin
and
- min F(T,c ) = F ( c * ) CEP
m
(5.4.37) with rji, i = 1 , 2 , . . . , m j , being where maximal in (5.4.37) may occur at multiple instances rji i = 1,2,.. . , M j , Mj 2 1.
5.5
General Method for the Proof of Existence and Form of Time-Optimal, Minimum-Effort Control of Ordinary Linear Systems
In the strings of results, Theorem 5.4.2- Theorem 5.4.5,on the system
i ( t ) = A ( t ) t ( t )+ B ( t ) u ( t ) , +(O) = 20,
(5.5.1)
one begins with the variation of parameters, +(t,+o,u)
= X ( t ) [ZO
+1 t
X- ' (s )B u (s )d s ] ,
(5.5.2)
where X ( t ) is the fundamental matrix solution of
i ( t )= A(t)z(t),
(5.5.3)
Synthesis of Time-Optimal and Minimum-Effort Control
165
and defines the functions (5.5.4) (5.5.5) Also defined is the map
St(.) =
1'
x-~(s)B(s)u(s)ds,
(5.5.6)
where St : Y -+ En is a continuous linear map with So(Y) = 0. Here Y is the dual space Z* of a separable Banach space 2, or is a reflexive Banach space. It is the space of controls whose elements u influence the state space of (5.5.1) that are elements of E". We assume a uniform bound M in norm of admissible controls:
u = {u E x : llull 5 M } .
(5.5.7)
If zl(t) is a time-varying target that the system will hit, then y(t) describes a reachable set-a desirable state of the system. Thus, at t = 0, y(0) = zl(0) - 20 # 0, and (5.5.1) is not at a desirable state. But if it hits the target at some t , then we have the coincidence y(t)
= x-'(t)21(t) - 2 0 = St(.)
(5.5.8)
for some u E U . As a consequence, y(t) E R(t), the reachable set, defined by
R(t) =
{I'
I
x - l ( s ) B ( s ) u ( s ) d s: u E u .
(5.5.9)
For the time-optimal problem, the minimal time for hitting the target is given by
t* = Inf{t E [O,T]: S t ( u )= y(t) for some u E U } .
(5.5.10)
The admissible control u* E U that has st* (u*) = y(t*) is the time-optimal control. For the minimum fuel problem, the admissible control u* that has St(u*) = y(t) while minimizing J ( u ( T ) )is the minimum fuel optimal control. As suggested by Hrijek and Krabs [ 5 ] , it is useful to study in abstract setting the mapping
Stability and Tame-Optimal Control of Hereditary Systems
166 and its adjoint
Sf : En + Y
with the following insight
If Y = L,([O,T], Em),then Y = 2' with 2 = L1([O,T],Em).St is given in (5.5.6), and its adjoint S: is given by s;(cT) =
0 a.e. (t,T] { (x~-~(T)B(T))~c a.e. E [o, I], T
E
t
(5.5.11)
which obviously maps En into Ll([O,T],Em) Y * so that IlS:(c)ll
=
1'
Il(x-'(s)B(s))Tclllds,
c
E En, a row vector
(5.5.12)
where /1.[/1 denotes the L1 norm in Em. If Y = Lz([O,TJ,Em),then Y = Y' is a Hilbert space. For each t E [ O , T ] , the adjoint operators St : E" + Y of St are given by (5.5.6) so that (lS;(c)II =
(J/ot
I I ( X - ' ( s ) B ( 8 ) ) ' c T ( ( ~ ~ ~,) 'for c E E".
(5.5.13)
We now consider the continuous linear mapping S, : Y -+ En under the following assumptions: (i) = {St(U ) : u E V } is closed. (ii) The mapping t + S, , t E [0, T ] is continuous with respect to operator norm topology of L(Y,En),the space of linear maps from Y into En. (iii) The function y : [0,T ] + En
Wt)
is continuous. (iv) y : [O,T]-+ En is constant and nonzero, i.e., y(t) = yl, V t E [O,T]I Y # 0. (v) For each c E En, c # 0,the function
is strictly increasing in [0, TI.
Synthesis of Tame-Optimal and Minimum-Effort Control
167
Theorem 5.5.1 Assume condition (i), and let t E [ O , q . Then there is an admissible control u E U , such that St(u) = y(t) if and only if cy(t) I M~~s;(c)II, v c E E"*, a row vector
(5.5.14)
where En*is the Euclidean space, cT is a row vector, and S; is the adjoint of St mapping En*to Y * . Proof: From the assumptions there is a u E U with St(u) = y(t), so that
cy(t) = C ( S t ( . u ) ) = s:(C)(.)
I Mlls;(c)ll.
This proves that (5.5.14) is true. Conversely, recall the definition of the reachable set R(t), and note that it is convex since St is linear and U convex. Because it is a closed convex subset of En,if there is no u E U with St(.) = y(t), then y(t) # R(t), and by the separation theorem of closed convex sets in En,[6, p. 331, there is a hyperplane which separates R(t) and y(t). This means there is a c E En*such that cy(t)
> sllp(cT(St(u)) : u E U } = sup{S*(c)(u) : u E U } .
This contradicts (5.5.14).
Theorem 5.5.2 Suppose conditions (i) - (iii) are satisfied. Then there exists a c E En*, a row vector with 1 1 ~ 1 1= 1 such that CY(t*)
= ~llSXC)ll*
(5.5.15)
The proof is contained in Antosiewicz [9].
Theorem 5.5.3 (HAjek-Krabs Duality Theorem) [5] Suppose (i), (ii), (iv), and (v) are valid. Let t* be the optimal time defined in (5.5.10). Then t* = max{t E ( 0 , q such that cy(t) = MllS;(c)Il with llcll = 1).
for some c E En*
Proof: From Theorem 5.5.2 and its corollary, the minimum time is a point in the set from which the maximum is taken. Suppose t > t* where t E (O,"], and where cyl = MllS;(c)ll for some c E En*with llcll = 1. Then by Theorem 5.5.1 and (v),
This is a contradiction.
168
Stability and Time-Optimal Control of Hertditary Systems
Corollary 5.5.1 time. Then
Let
ti*
be a time-optimal control and t* the minimum
= c(St.(u*)) = cy(t*)= Mlls;*(c)ll.
s;.(c)(u*)
Now, consider U c L,([O,t*],Em) as the control space. From the above there exists a* such that cy(t*)
=
1
t*
(cX-1(T)B(T))Tu*(T)dT,
This implies that u* is of the form u;(t,c)
= Msgn(cX-'(t)B(t))j
(5.5.17a)
when (cX-'(t)B(t))j# 0 for each c # 0 and each j = 1,. . . ,m. Thus i f the system is normal, optimal controls are given by u*(t) =
when U C L,([OT],Em). Suppose
M sgn(cX-l(t)B(t)),
u = {. E Lp
: IluIIp 5
(5.5.17b)
M}.
If U C Lp([Olt*], Em),where p > 1, we obtain from (5.5.15)that
(5.5.18)
Synthesis of Tame-Optimal and Minimum-Eflort Control
1 where -
P
169
+ -1 = 1. Recall that Q
u=
{
21
E L, :
(lT
121(s)lpds)
I Ad} .
Let g(s) = cX-l(s)B(s). Then
since by (5.5.18) cy(t*>= M
(lo 5 j=l
1
tgj(s)lq)
we have equality everywhere and the control u* that gives the equality is 4
uJ(s,c) = Mlilgj(s)lPsgn(gj(s)),
j = 1,.. . , m ,
(5.5.19)
where
This is the time-optimal control. It holds when the system is normal. To obtain expressions for the minimum fuel optimal controls, we note carefully that the controls u * ( t , c )in (5.5.17) and (5.5.19) are boundary controls in the sense that if
q,c) =
I'
X-l(s)B(s)u*(s, c)ds,
(5.5.20)
170
Stability and Time-Optimal Contml of Hereditary Systems
then z ( T ,c) is on the boundary of the reachable set R ( t ) , so that cz(T, c) = F ( c , T ) = Mlls;(c)ll, cz(T,c) > C Y , v Y E " T ) , Y # z(T,c).
(5.5.21)
I f y(T) is as defined in (5.5.4) and y(T) E R(T), we can extend y(T) to reach the boundary as follows: Let a = max{P : P y ( T ) E R(T)}.
(5.5.22)
I f y(T) # 0 , a can be taken positive, and clearly ay(T) is a boundary point of R(T), so that ay(T) = 4 T , c ) for some c a row vector, where matters can be so arranged that c * y(T)
= 1.
(5.5.23)
We observe that (5.5.24)
transfers 20 to z l ( T ) in time T while minimizing J ( u ( T ) ) .Also 1 1 - min J(u(T)), a - M(T)
(5.5.25)
where
a=
min
cEP={ cE E":cy(T)=l}
F(c,TI.
(5.5.26)
As a consequence of these observations, minimum energy control is given bY (5.5.27)
where c* is the minimizing vector in (5.5.26). These are the verifications of Theorem 5.4.2 and Theorem 5.4.4. In details for L , control, minimum (5.5.28)
where c' is the minimizing vector. For L, control, minimum pseudefuel controls are
where
Synthesis of Time-Optimal and Minimum-Eflorl Control
171
Thus for minimizing L, controls in U ,
(5.5.30) 1 M ( T )l
a = minF(c,T) = F ( c * , T ) = C€P
(5.5.31)
and optimal control (5.5.31) is given by (5.5.29) where C* is the minimizing vector in (5.5.30) and (5.5.31). I t is illuminating to link up minimum time control with minimum energy problems. In some situations, we show that the optimal controls are the same.
Definition 5.5.1: For each t f (OIZ'l, let
(5.5.32) where
P = { c f En*:cy1 = 1).
Theorem 5.5.4 Suppose the basic assumptions (i), (ii), (iv), and (v) are valid. Then for each t E ( O , T ] ,
M(t)
{ $} A4 t { i) t * ,
(5.5.33)
where M is the bound on the corresponding control set. Proof: From the definition we quickly deduce that CYl
I M(t)llSt(c)ll1
v c f En.
Because E n is finite dimensional, it is routine to verify that for each t E ( O I T ] there exists some c(t) f P such that
(5.5.34) and there is a ut E Y such that St(ut) = y1 with llutll = M ( t ) .
Using this in (5.5.34), there exists c(t) E En*such that
C(t)Yl = M(t)lls;(cT(t)ll = 1. If we invoke Theorem 5.5.1 and Theorem 5.5.2,the result follows at once.
172
Stability and Time-Optimal Control of Hereditary Systems
Theorem 5.5.5 Assume (i), (ii), (iv), and (v). The optimal control u* that transfers xo to 0 in minimum time t* while minimizing the effort J ( u ( t * ) )is given by
u'(t) = sgn[g(t,c*)],
o 5 t 5 t',
(5.5.35)
when U1 C L , with J = J1 and c minimizes (5.5.26) and (5.4.22). Also, isgiven by u p ) = W s j ( t ,c)lfsgn!Jj(t, c), (5.5.36)
U*
where
if U2 c L, with J = J2, where
c
in (5.5.36) minimizes (5.5.31) with F in
(5.4.30).
Proof: If T = t * , the minimum time required to transfer xo to 0, then from Theorem 5.5.4,
-1= - - 1 - inf{ IlSy. (.)I[ M M(t*) c
:c
E P},
where P = {c E En: cTyl = 0}, and where M is the bound on U , i.e.,
Since minimum fuel controls are given by (5.5.27), optimal controls are given by
where cis the minimizing vector of (5.5.26). But if we use ?5*(t)in (5.5.28) or (5.5.29), the results (5.5.35) and (5.5.36) are deduced. Thus, when M = 1, time-optimal control for the system (5.5.1) is also a minimum fuel control. Remarks on the assumptions: The coincidence y ( t ) = S t ( u ) for some u E U is equivalent to controllability with constraints. If y ( t ) = -20, this is null controllability with constraints, which is ensured by the following assumptions: (a) rank [ B ,A B . . . A"-'B] = n. (b) No eigenvalue of A has positive real part.
Synthesis of Time-Optimal and Minimum-Effort Control
173
(c) The condition that for each c # 0, t + IlS;(c)ll is strictly increasing is ensured by the system being proper, i.e., for each t l , t 2 E [0, T ] with 0 5 tl 5 t z 5 T , [ ~ e - ~ ' ( t ) B ( t )=] ~0, V t E [ t l , t z ] if and only if c = 0. This is equivalent to the controllability condition
(4-
(ii) In (5.5.17) the optimal controls are uniquely determined if for each j = 1,.. . , m , and each doE En, [ ~ e - ~ ' ( t ) B ( t ) ]#j 0. We note that theorems on minimum fuel problems, Theorem 5.4.2 Theorem 5.4.4, depend on the &sumption that ensures that the reachable set is closed. Because the reachable sets are also convex, optimal controls are controls that generate points on the boundary of R(t). If, however, the problem is to minimize the L1-cost function ~3(u(~ = )11ulll= )
1c T M
tuj(t)ldt
(5.5.37a)
j=1
subject to z(0) = 20 and z ( T ) = 0 where U = Us is given in (5.5.34), the reachable set (5.5.37b) is open, if we assume that
rank [Abj ,. .. ,A"bj] = n, for each column b j of B . This is the content of the following Lemma by HAjek, [4].
Lemma 5.5.1 Suppose (i) 5 ( t ) = Ao Bu (5.5.1) is metanormaf, i.e., rank [ABj ...A"bj] = n for each j = 1,.. . , m. Then the reachable set R(t) of ( 1 ) is open, bounded, convex, and symmetric.
+
Proof: Since boundedness, convexity, and symmetry are obvious, we give only the proof that R(t) is open. Suppose it is not open, so that rT
is on a boundary point. Let c # 0 be an outernormal to (the closure of) R(t) at E O , and use c to define the index y(t)
= c - e-A'B.
174
Stability and Time-Optimal Control of Hereditary Systems
It is clear that control uo maximizes (5.5.38) But the mapping
rT
is a linear functional on L110, T ] with norm
Since (5.5.38) implies that ilylla, is attained at the element ball, we have
UO(.) of
the unit
(5.5.39) This shows that we have equality throughout, so that
But HAjek has observed [4, p. 4171 that metanormality implies Iyj I is nonconstant a.e. This implies uo = 0 a.e. With this uo = 0, our boundary point is 10 = 0. This contradicts the following containment:
R,(T)
c Int W p ( T ) , Int(l/cr)R,(T)
3
(;)
RP7
whenever 0 < a < _< mT, which is a consequence of metanormality. Thus the set of points -10 which can be steered to 0 a t T by using controls with L1 bound 11~1115 1 is the open set R. If the controls have 11~111 M , the set is MIW for M > 0. From this it follows that minimal M can never be attained unless 10 = 0. This proves Theorem 5.5.5.
<
Because the reachable set is open, its boundary points can only be "reached" by convex combination of delta functions. We can therefore admit delta functions as controls when we consider absolute fuel minimization.
Synthesis of Time-Optima l and Min imu rn-&ffort Control
175
Optimal controls are impulsive in nature. If such functions are considered, optimal minimum -11u111 controls exist. Because Iw is open we can now prove Theorem 5.4.6 as follows:
Proof of Theorem 5.4.6: Let
Then by Htijek [4,p. 4351,
where l / a u = v , and
Iw is open.
Recall that the set of admissible controls is
(5.5.40) and the reachable set
is open. If g(s) = ce-""(s), then cy(T) = g(c,t)u(t)dt. The control that realizes a reachable set on the boundary of IR maximizes equation (5.5.38)over U . We note that
(5.5.41) By inspection, we observe for time-optimal controls
where
176
Stability and Tame-Optimal Control of HeRditary Systems
If equality holds everywhere in (5.5.41), then impulsive functions, applied at the points where gi is largest, maximizes (5.5.38). Note that the maxima in (5.5.43)can occur at multiple j and at multiple instances of rji, i = 1 , 2 , .. . ,Nj, where Nj is taken to be zero if gj does not contain the maximum. As a consequence, impulsive optimal controls are
j=1 i = l
where cij are positive constants. For the minimum absolute fuel problem, we have
F ( c ) = max sup Igj(c,t)l, 1<3<m O(t
Q
= minF(c) = F(c*), CEP
(5.5.45)
u(t c’)
and optimal minimum fuel control is Z’= -, where c’ is the minimizQ ing vector in (5.5.45), and where the boundary controls u ( t , c * ) are given in (5.5.44). If we replace the constraint set U in (5.5.40)by
then from equations (5.5.46)and (5.5.38)we have
The maximum in (5.5.47)occurs as before at multiple instances of time rji, i = 1 , 2 , . . . ,Mj where Mj 2 1. The maximizing impulsive controls that approximate reachable points on the “boundary” of R(t) are
Synthesis of Time-Optimal and Minimum-Effort Control
177
Therefore, if m
F(c*)=
C max
j=1
l
Igj(c,t)l, E = minF(c) = ~ ( c * ) , C€P
(5.5.49)
then optimal control that minimizes the absolute fuel is given by u* = u ( t ' c * ) , where
a
C*
and a are determined as in (5.5.49).
Example 5.5.1: Simple Harmonic Oscillator z(0) = 20, i ( 0 ) = 0 x+2=
U
-x = A z +
z ( T )= 0, k ( T ) = 0 Bu
Appropriate the earlier treatment in Example 5.1.1.
The hyperplane P is defined by P = {c : cy(T) = 1). This implies that c1 = - l / t o where ca is free. The index of the control system is
g ( t , c) = -c1 sin t where 6 = tan-'
(-):
With cost function
J1,
and U = { u : Iu(t)l 5 l ) ,
+ c2 cost = d c : + cE sin(t + 61,
. Therefore
178
Stability and T h e - O p t i m a l Control of Hereditary Systems
Clearly cz = 0 is the minimizer, and this forces 6 = 0. Thus the minimizing vector c* = - 1 , O . It follows from Theorem 5.4.2 that the unique =o optimal open-loop strategy is
(
)
u * ( t ) = J1,insgn[g(t,c*)]
10. I sgn[sin t J . Isin t Idt
=
+
Thus u* = J1 ,,,in or u* = -J1 ,,,in. For u = +J1 min, zg (J1,,,in - z1)2= a'. This is a circle of radius a centered at (Jlmin,O). For u = -Jlmin, (21 Jmin)' = u'. This is a circle of radius a centered at (-Jmin,O). Terminal manifolds are constructed as before. Note that e < 17. Int R(E) = Mi U Afz U +'I U r-. But since u is not f l , but fJ1 min, the circles on the switching loci have centers ( J 1 m i n l O ) and (-Jmin,O). See Figure 5.5.1 and Figure 5.5.2. Optimal feedback control is
+
+
The switching curves may be found as in Figure 5.2.1. See Figure 5.5.1 below.
...
r-
r-
u=-1
...
t
r+
r+
U=+l
FIGURE 5.5.1. SWITCHING CURVES W
X1
Synthesis of Time-Optimal and Minimum-Eflort Control
179
i r-
...
...
+D +
FIGURE 5.5.2. The associate manifolds are constructed as before in Example 5.3.1. See Figure 5.3.5. Thus, the terminal manifolds for the minimum fuel problem have the same form as those of the time-optimal problem. The difference is that the switching curves (in the minimum fuel case) have the center of the circles at J-multiple of the centers of those of the minimum time case.
180
Stability and Time-Optimal Control of Hereditary Systems
FIGURE 5.5.3. Example 5.5.2: Rigid Body Maneuver: mx = u .
We use earlier calculations y(T) = e - A % ( T ) - g(0) = --go.
The hyperplane P is defined to be PI = cy(T) = 1, [cl , c2] c1
[-to]
= -l/;co, where
c2 is free. The index of the control system is defined by
g ( t , ~ ) = ~ T . - A ' B = [ ~ l , ~12 1-t[1] o [l;m]
=--[;+c2]> 1 t
= 1, i.e.,
Synthesis of Time-Optimal and Minimum-Eflori Control
181
on P. Using the notations of Theorem 5.4.2,
--1 - E = min F ( c ,T ) . J Imin
CEP
If we consider the three cases (i) g ( t , c ) > 0, V t E [0,TI. (ii) g ( t , c ) < 0, V t E [O, TI. (iii) g ( t , c ) changes sign, we find that T2 - -. 1 a,=-4mz0 J1min By Theorem 5.5.2, the unique (a.e.) optimal control is u*(t)=
4mZo 7'2 sg*'
[-i &]
for all xo
# 0,
05t
5 T.
Thus Just as in Example 5.3.2 of (5.3.16), we consider the dynamics with these two values of u * . For u = +Jmin,
u
= -Jmin, x; = -2Jmin
(tl
- b2).
Note that if Jmin> 1, our parabola will be spread out more; but if Jmin= 1 (m = 1, to = 1, T = 2), the strategy is the same as the time-optimal control one. But if Jmin< 1, the parabola is more condensed. The switching curve is as described in Figure 5.5.4.
182
Stability and Tame-Optimal Control of Hereditary Systems
u* = -Jdn
U = ,+,J
M1= M21 M2 = M22
r- = M,, r+= M,,
FIGURE 5.5.4. Note that 6 = 00, as we saw before. The terminal manifolds M I and Mz are as shown in Figure 1, M2 is the space above the switching curve, and M I below it. Optimal controls law is f ( z 1 , x ~ ) :
Synthesis of Tame-Optimal and Minimum-Effod Control
183
Switching locus W :
Example 5.5.3: Damped Harmonic Oscillator iqt)
+ 2ai!(t) + W%(t) = U(t).
Recall the following:
P = { c : cTy(tl) = l},
c = (C1,CZ).
1 Hence c1 = --
, c2 is free. The index of the control system is g ( t , c ) = cTe-AtB. If a2 - w 2 < 0 , p2 = w 2 - a 2 , 20
e-AtB = -eQ' 1 m
cos Pt
+
sin Pt
so that
1
1 1 g ( t , c ) = --eat [ p ( - c 1 +crcz)sinpt + c 2 c o s ~ t = Deatsin(Pt +a), m
where
The cost function is
184
Stability and Time-Optimal Control of Hereditary Systems
Our aim is to find optimal feedback control u * ( t ) E U that transfers x(0) to x(t1) in time tl while mininlizing Jl(u(t1)).
This minimum occurs when
= 0, c1 =
c2
[-yxq
-1
Y
1
6 = 0.
By Theorem 5.4.2, the optimal control is u * ( t ) = Jl,i,sgn[sinPt]. u* = J1 ,,,in or -J1
Thus With these controls we have that if u = J1 mint
min.
(wxl
-)
- J 1 min
2
+ x22 =
(Jzin ,0)
02,
W
which is a spiral approaching
as t
+ 00.
The drawing similar
as t
+ 00.
The switching curves
t o Figure 5.3.13.If 'U = - J i m i n ,
which is a spiral approaching
(-,0)
are similar to those given in Figure 5.3.15 and Figure 5.3.17.We now use HGjek 's Corollary to deduce that ?r
€ <
Jn'
In Int R(r) optimal controls have at most one discontinuity, and Illt Our optimal strategy is
= M~ u M~ u r+u r-.
~ ( 6 )
Synthesis of Time- Optimal and Minimum-Effort Control
185
Example 5.5.4:
1 Just as before, c1 = --
20
, c2 is free. cost, -sint sint, cost
g(t,c)
where
[
=
D=
sin(t cos(t
1
+ 6)
+ a,l '
dm,
6 = tan-'--, c2 C1
The minimum is attained when c2 = 0, i.e., 6 = 0. Thus
Just as before, u*( t ) = J1min sgn
[t:]
It follows that optimal controls will be the cube of length JI min ub= [ - J ~ m i n ]
Case 2
, Ua=
[
JI min Jlrnin]
Case 1
Jlmin
[ ] [ -JC
7
uc=
Jm
Case 3
9
W=
-J1 min -Jmin].
Case 4
186
Stability and Time-Optimal Contml of Hereditary Systems
Just as before we have the four cases: Case 1:
which is a circle of radius d centered at (J,-J)
d, in the phase plane.
Case 2:
.=(-;),
k1=22+J,
x2
= -21
- J,
(21
+ J ) 2 + ( 2 2 + J ) 2 = d2.
This is a circle of radius d center at (-J, -J). Case 3:
(21 - J ) 2
+
(22
-
= d2.
This is a circle of radius d center at (J,J). Case 4:
(21
+J)2+
(22
- J ) 2 = d2.
This is a circle of radius d centered at (-J). ,IWe , observe that 6 < A , and with this we define the terminal manifold R(6) = R ( x ) as follows:
where
with
Ik
as the permutation of the controls;
Mo =
w,M1 =
M11
u Ml2 u M13 u M14,
u 4
M2
=
j=1
M2j.
Synthesis of Time-Optimal and Minimum-Effort Control
187
[-;I I";[
Recall that there are 2.22-' nonvoid sets M2i = 4 terminal manifolds.
{ [;]
[I;]} is the optimal sequence. The corre[I;] [;I [-;Isponding centers of the circular arcs are [-;I By definition, MI is the set of points that can be driven t o the origin with Suppose
no switch in control.
M~ = r++u r+-u r-- u r-+ G
4
CivIj,
j=1
[-;I
M2 is the set of points that can be steered t o zero with at most one switch.
[-;I. [;I
M21 corresponds to the control
[I;],and M24 t o
. M22 corresponds to
1
M23 to
Switching occurs at intervals of length */2,
after the first switch, not necessarily including the last switch. See Figure 5.5.5. Optimal feedback control is
With this synthesizer, the optimal responses are just solutions of
where A and B are given in Example 5.5.4. Thus, an observer moving with the system can operate it optimally knowing only 2 1 and 2 2 .
188
Stability and T i m e - O p t i m a l Control of Hereditary Systems
t xz
XI
FIGURE 5.5.5. Note that J depends on xo and on the final time. It has been observed by Ukwu (personal communication) that
J= -
-20
4k
+ 1- cos(T - k n ) + sin(T - k n ) -10
4k
+ 3 - cos(T - k n ) - sin(T - k n )
2k if k n _ < T < -
if E
2
n
+1 2
n9
< T 5 ( k + 1)n.
Synthesis of Tirne-Optimal and Manimum-Effort Control
189
Example 5.5.5: Impulsive Controls We consider the cost function J S ( u ( T ) ) for the damped oscillator with rn = k = 1, 20 = 1, and cr = 0.03. Recall that g ( t , c ) = Deatsin(Pt + S), where
D=
((-;+c2ff,P)2+c;)+,na, c2 -c1
P
+
c2cr
F ( c * , T ) = max g ( c , T ) , O
1 = min F( c, T ) . Jmin
The maximum occurs a t
CEP
T
=
('p - S) / p (approximately) (see Redmond
and Silverberg 1101). F ( c ,T) = Dear I sin(PT
+ 6)l.
F ( c , T ) is minimized over H by letting c2 = 0, so that
1
--
53rnin
-
Pm.0
ear,
ll7r Optimal strategy is u*(t) = Pm~oe-arsgn(sinPr)S(t2P T). As observed by Redmond and Silverberg [lo], Figures 5.5.6A and 5.5.6B show that impulse control allows for a period of free oscillation during which the natural damping present in the system removes energy from the system. Then a t the last instant, when the systems displacement is identically zero, an impulse of magnitude equal t o the system's momentum is applied. This instantaneous change in velocity abruptly transfers the system to the origin at time T. The timing of the impulse is extremely important. In Figure 5.5.6B, the impulse is applied when the potential energy is a minimum and the system's kinetic energy is a local maximum. This observation is of great value in the development of near-optimal control laws of large scale systems. in which
T
= -.
190
Stability and Time-Optimal Control of Hereditary Systems
I
I
I
I
I
I
-1.2 -1.5
-1
-0.5
0
.5
1
x (t)
Time A.
B.
FIGURE 5.5.6A.
IMPULSECONTROL OF
A
FIGURE 5.5.6B. DAMPEDHARMONICOSCILLATOR
We consider another example-the
Rigid Body Maneuver. rn:(t) = u ( t ) .
Considering all three possible ranges of argument (positive, negative, and sign change) the minimum is attained at 1 -=J3min
T
220M'
This minimum occurs both at 71 = 0, and M = 2, the open-loop optimal control law is u*(t) =
~2
= T. From (5.5.37) with
$0 M sgn(T2 - t ) [ S ( t )+ S ( t - T)]. T
This control consists of one initial impulse to impart a velocity to the s y s tern. Then the system drifts until the desired time T, when a final impulse is applied to terminate the motion at rest. See Figure 5.5.7A and Figure 5.5.7B.
1.5
Synthesis of Time-Optimal and Minimum-Effori Control
I
1
u (1)
I
I
I
I
1.2
I
:-4-+--
-1 P
I
I
1
I
-
A:;lt(
I
I
-1
I
-0.2
FIGURE 5.5.7A.
I
I
I
I
I
-
-
.
I
191
I
I
I
I
I
I
FIGURE 5.5.7B.
Redmond and Silverberg have determined that if fuel consumption in propulsive systems is defined by
where u ( t ) represents the thrust and Isp the specific impulsive, which is set to be 1, there are big savings in fuel if impulse controls are used. The result is illustrated in the table below. Continuous controls are those of Theorem 5.4.4;bang-bang controls represent Theorem 5.4.2.
Table 5.5.1 Fuel Consumption Bang-Bang Continuous Impulse Oscillator Maneuver
1.168
4.0
0.921 3.0
0.595 2.0
192
Stability and Tirne-Optimal Control of Hewditary Systems
REFERENCES 1. E. N. Chukwu and 0. Hrijek, A Connection Between Optimal Control and Disconjugacy i n Dynamical Systems, vol. 11, An International Symposium, Academic Press, 1976. 2.
E. N. Chukwu and 0. Htijek, “Disconjugacy and Optimal Control,” J. Optimization Theory and Applications 27(3), 1979 333-356.
3. 0. Hrijek, “On the Number of Roots of ExgTrig Polynomials,” Computing 18 (1977) 177-183. 4. 0. Htijek, “Ll-Optimization in Linear Systems with Bounded Controls,” miration Theory and Applications 29 (1979) 409436.
J.
OptG
5. 0. Hrijek and W. Krabs, On A Geneml Method For Solving Time-Optimal Linear
Control Problems, Preprint No. 579, 1981, Technische Hochschule Darmstadt.
6. H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969. 7. L. W . Neustadt, “Minimum Effort Control Systems,” SIAM Journal on Control 1 (1962) 16-31. 8. D. Yeung, “Synthesis of Time-Optimal Control,’’ Ph.D Thesis, Case Western Reserve University, Cleveland, Ohio, 1974. 9. H. A. Antosiewin, “Linear Control Systems,” Arch. Rat. Mech. Anal. 12 (1963) 313-324.
10. J. Redmond and L. Silverberg, “Fuel Consumption in Optimal Control,” To appear A I A A Journal of Guidance Control and Dynamics.
11. J. Dugunji, Topology, ALlyn and Bacon, Boston, 1966.
Chapter 6 Control of Linear Delay Systems In this section we study some controllability questions of the system
where (6.1.1b) and ~ ( t0), is a matrix-valued function. We assume that r] is measurable in t and of bounded variation in 0 on [-h,0], and that there is an integrable function m : (-00, m) such that
Throughout this section we assume that t
-+
B ( t ) is continuous.
6.1 Euclidean Controllability of Linear Delay Systems In time-optimal control theory, one assumes that beginning from some initial point the target point can be reached in finite time by using some admissible control. The usual initial point is the point 4 in the function space C = C([-h,01, En) of continuous functions from [-h, 01 into E", or the Sobolev space W,(l)([-h,O], En)= W ~ ~ ~ s p swhich m , is the space of absolutely continuous functions from [-h, 01 t o E" with the first derivative square integrable. We first consider the target to be a point in Euclidean space E". It is desirable t o reach this point (from any initial function) in finite time using some admissible controls u that are unlimited in magnitude. We now investigate this Euclidean controllability problem for the system
+
i ( t >= L ( t , st) B ( t ) u ( t ) ,
(6.1.1)
where L and B satisfy all the conditions of Section 2.2.
Definition 6.1.1: The linear control process (6.1.1) is Euclidean controllable on the interval [ g o , 111 if for each 4 € C and 2 1 € E" , there is a square integrable controller u such that to(u4,u)= 4 and s(t1,u,q5,u)= 2 1 . It
193
Stability and Time-Optimal Control of Heredita y Systems
194
is Euclidean null-controllable if in the definition 2 1 = 0. It is Euclidean controllable if it is Euclidean controllable on every interval [ a , t l ] , t l > Q. The conditions for Euclidean controllability have been well studied: See Kirillova and Churakova [ l l ] Gabasov , and Kirillova [ S ] , Weiss [15],and Manitius and Olbrot [13,14]. They are all conditions on matrices representing the system, and are a consequence of the following lemma:
Lemma 6.1.1 In the system (6.1.1), the following are equivalent: (i) The matrix
G ( Q , ~ I=)
l'
U(ti,s)B(s)B"(s)U(t,s)ds
(6.1.2)
has rank n, where U ( t ,s) is the n x n matrix solution of
i ( t ) = q t , q).
(6.1.3)
(ii) The relation C T Y ( s , t l ) B ( s )E 0 , c E E n , s E [ ~ , t l ]
implies c = 0, where Y ( s , t l )= U ( t 1 , s ) a.e. in s is an n x n matrix defined by
Y ( a , t )=
>t, I - J,' Y ( c r , t ) q ( ad, - cr)dcY, Q 5 t . u
0,
(iii) The system (5.1.1) is Euclidean controllable on
[QI,~I],
(6.1.4) 11
>u
As a consequence of this lemma, the next characterization of Euclidean controllability in terms of the system's coefficients is available for the autonomous system
+
- h)+ Bu(~),
Z(t) = A o ~ ( t ) A l ~ ( t
(6.1.5)
where Ao,A1, B are constant matrices. First introduce the determining equations,
+
Q ~ ( s=) A o Q E - I ( s ) AiQE-i(S
Qo(s)=
and define
We have:
= 1 , 2 , 3 , .. . ,
- h),
{
B, s=O, 0,
s
# 0,
s E (--oo,-oo).
(6.1.6)
Control of Linear Delay Systems Theorem 6.1.1 only if
195
System (6.1.5) is Euclidean controllable on [O,tl]if and
rankQn(tl) = n.
Remark 6.1.1: Note that the nonzero elements of Q b ( s ) form the sequence: s=o h 2h Qo(s)= Bo QI (s) = AoB AlBO Qz(s) = A i B (AoAi AiAo)Bo A:Bo Note that if t l _< h, the only elements in the sequence are the terms [B,AaB, . .. ,A:-'B], so that En-controllability on an interval less than h implies the full rank of [B,. . . , A t - ' B ] . If this has less than full rank and t l > h, other terms can be added to &,(tl) and the system may still be controllable on [ O , t l ] . This contrasts with the situation in Section 5.1 for systems without delay, where if the system can be steered to some point of En,it can be steered t o that point in an arbitrarily short time.
+
Example 6.1.1: Consider
+
i ( t ) = A o t ( t ) A l l ( t - 1) + Bu(t), where
Ao=
[0
-1
Q1(s)z 0,
4 0
0 0 A 1 = [1
.=[;I,
s # 1 and s # 0.
Since rank g2(2) = 2, the system is Euclidean controllable on [ 0 , 2 ] .
Stability and Time-Optimal Control of Hereditary Systems
196
Example 6.1.2: Consider
i ( t )= A o ~ ( t+ ) Alz(t
AoB=
- h) + B u ( ~ ) ,
[%I, [i], [HI, AlB=
rankQ(tl),
21
B=
> h is 3.
Hence this system is E3-controllable.
Example 6.1.3: Consider the scalar differential-difference equation
where airbi and c are constants and t(0),
z2 = &)
.. . zn = x(n-1)
n- 1
2,u
are scalar functions. Define z1 =
n-1
Written in matrix form
i ( t )= A o ~ ( t + ) A l t ( t - 1) + B u ( t ) ,
Control of Linear Delay Systems
197
we have
This has full rank. Hence the system is Euclidean space controllable on any [O,tlI, tl > 0.
6.2 Linear Function Space Controllability The true state of solutions of (6.1.1) is an element of some function space. Thus the state a t time t is denoted by 21, and this is a segment of the trajectory s + z(s) t - h 5 s 5 t . Very often C = C([-h,O],E") is used. But W,(')([-h,01, E") = Wi'), the state space of absolutely continuous functions from [-h,O] t o E" with first derivative square integrable and in Lz([-h,O],E") is also natural. Indeed, if 4 E W,(')([-h,O],E") and u E Lz([O,t,],E"), then z(4) = 3: : [O,tl] -, E" is absolutely continuous, and if i ( t ) = L(3:t) u ( t ) , then z e Lz([O,t],E"). Therefore z E W,(')([-h,tl],E"),so that ttE W,(')([-h,O],E")for all t E [O,tl]. In this section we explore controllability questions in W,(').
+
Definition 6.2.1: The system (6.1.1) is controllable on the interval [a,t1] if for each 4,$ E W i l ) there is a controller u E L 2 ( [ a , t l ] , E msuch ) that 0 in the t t 1 ( u , 4 , u )= $, t 0 ( u , 4 , u )= 4. It is null controllable if 4 above definition. It is said to be controllable (null controllable) if it is controllable (null controllable) on every interval [a,t l ] with tl > a+ h. The following fundamental result is given in Banks, Jacobs, and Langenhop [2, p. 6161.
198
Slabilily and Time-Optimal Control of Hereditary Systems
Theorem 6.2.1 In (6.1 . l ) , in addition to the prevailing conditions of this section (for L and B), assume that (i) t -+ B+(t), t E E is essentially bounded on [tl - h, t l ] , where B+(t) denotes the Moore-Penrose generalized inverse of B(t) 1111. Then (6.1.1) is controllable on the interval [a,t1] with tl > a + h i f a n d only if (ii) rank B(t) = n on [tl - h , t l ] . When t -+B(t) is constant, we have the following sharp result: Corollary 6.2.1 Suppose the n x rn matrix B in (6.1.1) is constant. A necessary and sufficient condition that (6.1.1) is controllable on [a,tl] is that rank B = n. The proof of Theorem 6.2.1 is available in [2] where a thorough discussion of the result is made. Manitius also discusses the result [14]. In the investigation of controllability of (6.1.1) in the state space W;'), with L2 controls, it is impossible to handle the situation of controls with pointwise control constraints. In this case controls are taken in the space L,, and the state space is either W 2 )or C. Though several researches are available on the optimal control of variants of (6.1.1) when the state space is C and the controls are L,, (see Banks and Kent [3]and Angel1 [I])there seems to be no general results on the controllability of such systems, a necessary requirement for the existence of an optimal control. There are other problems connected with such treatment; for example, the attainable sets cannot be closed and the multipliers cannot in general be nontrivial. See Manitius [14,pp. 120,1281. We begin to address some of these problems by first formulating function space controllability state conditions of (6.1.1) when the space is C and the controls are L,. The result is also valid for W g )state space of initial functions and C'([-h, 01, En)space of terminal conditions, with L , controls. We assume that the control matrix B(t) is continuous.
Theorem 6.2.2 In (6,f.l),namely,
suppose the state space is C = C([-h,O],E n ) , and B is continuous. Assume: (i) rankB(t) = n o n [ t l - h , t ~ ] . Then(6.1.1)iscontrollableon [ a , t l ] , tl > a t h.
Control of Linear Delay Systems
199
Proof: Assume (i). Then rank B(t1 - h ) = n, and this implies that H(u,t1 - h) =
1
11--h
X(t1 - h , s)B(s)B*(s)X*(11 - h,s)ds
(6.2.1)
+
has rank n. Let t 1 > u h, 4,$ E C([-h,O],E"). It is a consequence of Lemma 2.3 of Manitius [14], (or Lemma 6.1.1) that there is a u E Loo([cr,tl-h],Em) such that z(tl-hlcr,q5,u) = +(-h). Weextend u and t = t(., u,4, u) t o the interval [u, tl] so that $(t --21) = ~ ( t ) ,tl - h 5 t 5 t 1 , and
$(t
+
- 11) = +(-h)
/
t
ti-h
L ( s , +,)ds
+
ll-h t
B(s)u(s)ds,
(6.2.2)
on [tl - h, tl]. Note that the integral form of (6.1.1) is z ( t ) = 40)
+ J0
t
L ( s ,% ) d S
t
+
B(s)u(s)ds,
x ( t ) = #(t), t E [-h, 01, t 2 0,
# E c.
(6.2.3)
Because of (i), rank [B(t)B*(t)]= n,
and
H(t) =
it
B(S)B*(S)dS,
h
- h, tl],
V t E ftl
t > tl - h
has rank n for each t f (tl - h , t1). As a consequence of this rank condition, define for each t E (tl - h , t l ] a control ~ ( s ) ,s 5 t , v(s) = B+(s)H-'(t)[+(t- t l ) - $(A)-
J
t
ti-h
L ( s , z,)ds],
where z(.) is that solution of (6.1.1) or (6.2.3) with
= 4, z(t1 - h, 4, ). = + ( - h ) .
I,(+)
That zi exists follows from the usual argument [9, p. 1411 for the existence of a solution of a linear system (6.1.1). With this zi in the right-hand side of (6.2.2), we have 1-h
= $(I - t l ) ,
t E [tl - k t l l .
200
Stability and Time-Optimal Control of Hereditary Systems
It is interesting t o note the absence of condition t -+ B+(t) being essentially bounded on [tl - h , t l ] , which is required in Banks, Jacobs, and Langenhop [ 2 ] . Indeed, if B ( t ) has rank n on [tl - h , t l ] and is continuous, then B + ( t ) is continuous on [tl - h , t l ] . See Campbell [4, p. 2251. In this case B+(t)is uniformly bounded on [tl - h , t l ] .
Remark 6.2.1: Note that the control that does the transfer from is 1 ' 1 E ~ C o ( [ ~ ,t lhl, Ern), ii= v : [tl - h , t l ] , continuous.
to
t)
{
Thus we can assume that in our situation the subspace of admissible controls is u = {. E L,([O, t1l)'11t1 E C([-h, 01, Ern)).
Remark 6.2.2: Because the proof is carried out in an integrated form, it is easy to consider 111 E C. If, as was done in [ 2 ] , a "differentiated" version of (6.1.1) is used in the definition of control, 111 will be naturally in C 1 ( [ - h , O ] , E " ) . In both cases the full rank of B ( t ) is needed as was formulated by Bank, Jacobs, and Langenhop [ 2 ] . The full rank of B required by condition (ii) of Theorem 6.2.1 is very strong: for W2(')-controllability we must have as many controls as there are state variables. There are very many practical situations when this will fail. For example, the scalar nth order retarded equation with a single control of example 6.1.3 is ruled out. We now study other types of controllability, which we must need to make progress in our study of time-optimal control. Consider the system
.(t) = L ( t ,Z t )
+ B(t).(t),
(6.1.l)
where L is given in (2.2.4) with assumptions on L ( t , 4) stated there. With these assumptions we may decompose L ( t , 4) as follows:
(6.2.4)
Control of Linear Delay Systems
We call the operator t the system
+H(t,
20 1
0) the “strictly retarded” part of L(t,+). In N
i=l
0 < W 1 < W2 < - . -< WN = h
(6.2.5)
N
~ ( 4) t ,= C A i ( t ) 4 ( - w t ) i= 1
We call system (6.1.1) a system with strict retardations if the mapping in (6.2.4) satisfies the condition, there exists a 6 , 0 < 6 < h such that g t , e ) = o for - 6
5 e 5 0,
7
v t E 8.
Note that all systems of the form (6.2.5) are strictly retarded. We have: Proposition 6.2.1 Suppose system (6.1.1) has strict retardation, then (6.1.1) is Euclidean null controllable if and only if G(a,t l - h ) = S,’’-* U(t1 - h , s ) B * ( s ) B * ( s ) U * ( t-l h , s ) . ds has rank n for every choice of u , t l , with tl > u h. With this one proves tlie following [2].
+
Theorem 6.2.3 Suppose: (i) (6.1.1) has strict retardation, and (ii) t + B S ( t ) , t E E is essentially bounded on [tl - h , t l ] for each t i with tl > u h. Then ( 6 1 . 1 ) is null controllable if and only if (iii) Condition (i) of Proposition 6.2.1 is satisfied. (iv) B(t)B+(t)v(t,O) = i j ( t , e ) i f - h 5 0 5 Oforalmosteveryt E [ t l - h , t l ] for every 2 1 > u h . When L , B are independent of time, the following sharp condition is deduced:
+
+
Corollary 6.2.2 Suppose (6.1.1) is strictly retarded. Then (6.1.1) is null controllable if and oiily if (i) Condition (iv) of Tbeorern 6.2.3 is satisfied, and (ii) rank [ B , A o B , .. . ,A:-’B] = n. In particular, if we consider
i ( t ) = A o ~ ( t+)
N
C A i ~ (-t hi) + B u ( t ) ,
(6.2.6)
i=l
where 0 < hl obtains:
< . . . < hN with A i ,
i = 0 , . . . , n n x n matrices, then one
Stability and Time-Optimal Control of Hereditary Systems
202
Corollary 6.2.3
The system (6.2.6) is null controllable if and only if N
and
N
rank[B,AoB,... ,A!-'B] = n.
As a consequence, the nth-order scalar differential difference equation in Example 6.1.3 is null controllable.
6.3 Constrained Controllability of Linear Delay Systems [5] In the last sections the controls are big. In this section we consider controllability of (6.3.1) i ( t ) = L ( t ,.t) B(t)u(t),
+
when the controls are required to lie on a bounded convex set U with a nonempty interior. For ease of treatment, U will be assumed to be the unit cube C"' = { U E E"' : I ~ j 5l 1, j = 1, ... ,m}. (6.3.2) Here u j denotes the j t h component of u E Em. Consistent with our earlier treatment, the class of admissible controls is defined by Uad
= {U E L , ( [ O , t l ] , E m ) : u ( t )E C"' a.e. on
[O,21]}.
It is easy to see that Uad has a nonempty interior relative to L,, and that 0 E Uad. The state space is either W c ) or C. Conditions on L and B of Sections 6.1, 6.2, and 6.3 are assumed to prevail. We need some definitions.
Definition 6.3.1: The system (6.3.1) is null controllable with constraints if for each 4 E C,there is a t l < 00 and a control u E Uad such that the solution z( ) of (6.3.1) satisfies . o ( d J , 4
= 4, %(6,4,U) = 0.
It is locally null controllable with constraints if there exists an open ball 0 of the origin in C with the following property: For each 4 E 0,there exists a t l < 00 and a u E Uad such that the solution x( ) of (6.3.1) satisfies .o(fl,
4,
= 4,
.:I
4,
(b,
= 0.
We now study the null controllability with constraints of (6.3.1). Two preliminary propositions are needed.
Control of Linear Delay Systems
203
Proposition 6.3.1 /5] Suppose (6.3.1) is null controllable on the interval [a,tl]. Then for each 4 E C, there exists a bounded linear operator H : C + Lw([u,t l ] ,Em)such that the control u = Hq5 has the property that the solution z(a,(P, H 4 ) of (6.3.1) satisfies
With this H , one proves the following:
Proposition 6.3.2 [5] Assume that (6.3.1) is null controllable. Then it is locally null controllable with constraints. Proof: Since (6.3.1) is null controllable, we have by Proposition (6.3.1) that there exists a bounded linear operator H : C 4 L o o ( [ u , t l ]Em) , such that for each 4 E C,u = and the solution z ( u , ~Hq5) , of (6.3.1) satisfies
Because H is continuous, it is continuous at zero in C. Here for each neighborhood V of zero in L , ( [ u , t l ] , E m ) there is a neighborhood M of 0 in C such that H ( M ) c V. In particular, choose V to be any open set in L , ( [ a , t l ] , E m ) containing zero that is contained in Uad. This choice is possible since U a d has zero in its interior. For this particular choice, we see that there exists an open set 0 around the origin in C such that H ( U ) c V c &([a, t l ] ,Em). Every 4 E 0 can be steered to zero by the control ti = H 4 . Hence (6.3.1) is locally null controllable with constraints.
Theorem 6.3.1 Assume that: (i) The system (6.3.1) is null controllable. (ii) The system i ( t ) = L ( t , Zt)
(6.3.3)
is uniformly asymptotically stable, so that there are constants k > 0, (Y > 0 such that for each u E E the solution x of (6.3.3) satisfies
Then (6.3.1) is null controllable with constraints.
Proof: Condition (i) and Proposition 6.2.2 guarantee the existence of an open ball 0 c C such that every 4 E 0 can be steered to the zero function
204
Stability and Time-Optimal Control of Hereditary Systems
with controls in U a d in time t l < 00. Condition (ii) assures us that every solution of (6.3.3), i.e., every solution of (6.3.1) with u = 0, satisfies 2 t ( u , + , O )+ 0
as t
+ 00.
Thus, using u = 0 E U a d , the system rolls on as Equation (6.3.3), and there is a finite t o , such that $J = zto(a,4,O) E 0.With this initial data ( t o , $ ) there exists a t l > t o such that for some u E &d, zto(a,$J,u)= $J,q ,( a ,4 , 0) = 0. Thus with the control, v ,
which is contained in &d, 4 is indeed transferred t o 0 in a time tl This concludes the proof.
< 00.
As a consequence we have the following sharp result: Corollary 6.3.1 Consider (6.3.4) where for an n x n matrix ((0) - h
5 0 5 0, L is given by
L(4) = /o [d€(0)14(4) 4 E c. -h
s_Oh
Let A(X) = XI eXBd((0). (i) Let the roots of the characteristic equation det A(X) = 0 have negative real parts. (ii) Suppose .(t) = L ( z t ) B u ( t )
+
(6.3.5)
is null controllable. Then (6.3.5) is null controllable with constraints. Corollary 6.3.2 Consider (6.3.6)
Control of Linear Delay Systems
205
Assume that: (i) There exists a positive definite symmetric matrix H such that the matrix G is negative semidefinite, where
G
HA0
+ A;fH-I-
N
2qnMkH,
k=l
and where p 2 1 is a constant, Mk = max IAkijl. (ii) rank [A(A),G] = n for each complex A where A(A) = A 1 - A0 N
(iii)
k=l
-
Ake-xha.
rank
[
"1
A0 - A I , A, . . - A N , A1 AN 0 AN 0 0
= n+rank
'0
or in place of (ii), (iii) we have that G is negative definite.
(iv) BB'
N
N
i=l
i=l
C Aj = C Ai.
(v) rank [ B , A o B , .. . ,AI;-lB] = n . Then (6.3.6) is null controllable with constraints.
Proof: For the system (6.3.6) conditions (i), (ii), and (hi) are the requirements of uniform asymptotic stability of Corollary 3.2.3 or of Chukwu [6]. Hypotheses (iv) and (v) are needed for null controllability.
Example 6.3.1: Consider n-1
n- 1
j =O
i=O
where ai, bi are constants. If the homogeneous system is uniformly asymptotically stable, then Example 6.3.1 is null controllable with constraints. The following result is implied by the main contribution in Chukwu [5, Corollary 4.11. Theorem 6.3.2
Consider the system
i ( t ) = q t , .t)
+ B(t)u(t),
(6.1.1)
where L is given in (2.2.4) with assumptions on L(t,q5) stated there and with B continuous. Assume that the controls u are L2(L,) functions with
206
Stability and Time-Optimal Control of Hereditary Systems
values in a compact convex subset P of Em and 0 E P . Assume that the trivial solution of (6.3.3) i ( t )= q t ,q ) is uniformly stable, and (6.1 .l)function-space W$’)controllable (Euclidean -Encontrollable) on finite interval [a,t l ] with L2(L,) controls. Then (6.1.l) is function space (Euclidean) controllable with constraints. A s a consequence, uniform stability of (6.3.3) and controllability of (6.1 .l) on some interval [ a , t l ]suffice for global null controllability of (6.1.1), where the controls are measurable with values in P .
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tarded and Neutral Type to Target Sets in Function Space,” SIAM J . Control 10 (1972) 567-594.
4. S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979. 5. E. N. Chukwu, “Controllability of Delay Systems with Restrained Controls,” J . optimization Theory Appl. 29 (1979) 301-320. 6. E. N. Chukwu, “Function Space Null Controllability of Linear Delay Systems with Limited Power,” J. Math. Anal. Appl. 124 (1987) 193-304. 7. E. N. Chukwu, “Global Behavior of Linear Retarded Functional Differential Equcttions,” J. of Math. Anal. and Appl. 162 (1991) 277-298.
R. Gabasov and F. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker, New York, 1976. 9. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 8.
1977. 10. H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969. 11. F. M. Kirillova and S. V. Churakova, “On the Problem of Controllability of Linear Systems with After Effect,” Differential Nye urauneniya 3 (1967) 436-445. 12. D. G. Luenberger, Optimization by Vector Space Methods, John Wiley, New York, 1969. 13. A. Manitius and A. W. Olbrot, “Controllability Conditions for Linear Systems of Linear Hereditary Systems,” SIAM J . Control 13 (1975) 611-649. 14. A. Manitius, “Optimal Control of Hereditary Systems,” in Control Theory and
Topics in Functional Analysis 111, International Atomic Energy Agency, Vienna, 1976. 15. L. Weiss, “An Algebraic Criterion for Controllability of Linear Systems with TimeDelay,” IEEE Trans. on Autom. Control A C - 1 5 (1970) 443.
Chapter 7 Synthesis of Time-Optimal and Minimum-Effort Control of Linear Delay Systems 7.1
Linear Systems in Euclidean Space
Our aim here is to consider the time-optimal control of the linear system N
+ C Ai(t)z(t- hi) + B(t)u(t),
i ( t ) = Ao(t)z(t)
(7.1.1)
i=l
where 0 < hl < hz < - . < h N = h Ai, i = 0 , . . . ,N are n x n analytic matrix functions, and B is an n x m real analytic matrix function. The controls are locally measurable L , functions that are constrained to lie in the unit cube
c" = {u € Ern : lujl 5 1
j = 1, ... , m } .
(7.1.2)
Thus the class of admissible controls is defined by Uad
= {u E L,C[a,t1],
u ( t ) E C" a.e. on [O,t1]}.
In E" we consider the following problem: Minimize
J ( t ,4 t ) ) = t ,
(7.1.3)
where
= 4, ( z ( t ,a , 4 , u ) ,t ) = (0) x [O, I. (7.1.4) Here z ( . , a , + , u ) is a solution of (7.1.1) with zu = 4. This is the timeXu
and continues optimal problem. It is old [17]and very important [13,16,18] to be interesting even for linear ordinary differential systems [19]and [20]. In the economic applications, for example, it is desirable to force the value of capital to reach some target in minimum time while maximizing some welfare function. For example, we may want the economy to grow as fast as possible. In Example 1.4, one would like to control the epidemic of AIDS. It is known [7, p. 1831 that, associated with the delay equations
207
208
Stability and Time-Optimal Control of Hereditary Systems
that model AIDS as a progressive disease is the reproductive number R given by
This system has a unique positive endemic state if and only if R > 1. If R < 1, the infection-free state is globally asymptotically stable, while the endemic state is locally asymptotically stable whenever R > 1. Since R is the number of secondary infections produced by an infectious individual in a purely susceptible population, to control the spread of AIDS as rapidly as possible, R should be reduced t o less than 1 in minimum time. This is a time-optimal problem. Theoretically this can be done by introducing control strategies that will decrease the number AC, the transmission rate per unit time per infected partner, as fast as possible. Conceivably the control ~ ( t= ) b(t)AC(T)S(t)-
T(t)
is introduced, so that the system becomes
The aim is to drive this control system to the infection-free state in minimum time. With this as a motivation, we now study (7.1.1), a linear approximation, by constructing an optimal feedback control in a way similar t o the development in [lo]. Recent advances in the theory of functional differential equations in Section 7.3 [lo], [Ill and [14],and in the constrained controllability questions [9], [15],have made it possible to attempt to solve the time optimal control problem of linear systems (7.1.1) within the classical theory. The development of the theory was long abandoned because of lack of progress in the directions which have now taken place. We shall illustrate our theory with several examples of (7.1.1) from technology, and one from ecology. If z(u,4,O) is a solution of
i ( t ) = A,z(t) + 20
=4 E
c,
N
C A i ( t ) ~ (-t h i ) , i=l
(7.1.5)
209
Synthesis of Time-Optimal and M i n 3 r n ~ ~ - EControl ~o~
in En,the solution of (7.1.1) is given by
z ( t ,u,4,u) = z ( t ,u,4,0) +
J,' U ( t ,s ) B ( s ) u ( s ) d s ,
(7.1.6)
where U ( t , s ) is the fundamental solution of (7.1.5).
Definition 7.1.1: The Euclidean reachable set of (7.1.5) at time t is defined bv
This is the set of all points in E" that can be attained from initial point = 0 using all admissible controls. The following properties of R(t,u> are easily proved:
Proposition 7.1.1 R(t,u) is convex and compact, and 0 E R(t,r), V t 2 the subset of u,,d defined by
6. Consider
U:d
= { u E En' u measurable luj I = 1 j = 1 , 2 , . . . ,m } .
The following bang-bang principle is valid in En:
Theorem 7.1.1 Let
and
Then Ro(t,u)= R ( t , u ) , V t 2 u.
Proof: Because U ( 2 , s )E L , ( [ u , t ] , E f l X " )and , B ( t ) analytic, we have
U ( t , s ) B ( s )E L,([u,t], Enxm).
It now follows from Corollary 8.2 of Hermes and LaSalle [4] that R ( t , o )= RO(t, a),
v t 2 u.
Remark 7.1.1: Theorem 7.1.1 states that the bang-bang principle of LaSalle is valid for (7.1.1); that if, of all bang-bang steering functions, there is an
Stability and Time-Optimal Control of Hereditary Systems
210
optimal one relative to UaOd, then it is optimal relative to Uad. Also if there is an optimal steering function, there is always a bang-bang steering function that is optimal.
Definition 7.1.2: Let F" denote the metric space of all nonempty compact subsets of En with the metric p defined as follows: The distance of a point z from I is d ( z , U ) = inf{lz - a1 : a E U } , N ( O , c ) = {z : E" : d ( z , U ) 5 c},
P( 0 1 , 6 2 ) = inf { c
:0 1
c N ( 0 2 , c) and O2 c N(O1 ,c)}.
Proposition 7.1.2 The set R(.,u) : [u,oo) -+ r" is continuous. Because t 4 U ( t , s ) , t 2 s is continuous the usual ideas such as are contained in Lee and Markus [S, p. 691 can be adapted to prove this using the variation of parameter. Associated with W(t,u) is the Euclidean space attainable set
d ( t , u ) = { x ( t , a , 4 , u ) : u E Uad, z is a s o h tio n of(7.1.1)}. The same type of proof of Proposition 7.1.2 yields that t -+ A(t,a) = z(t,u,0) R ( t , u ) is continuous in the Hausdorff metric, and it is compact and convex. The following properties of R(t,a) as a subset of En are available:
+
Lemma 7.1.1 R ( t ,u) is compact and convex and satisfies the monotonicity relation: (i) 0 E R(t, u) for each t 2 u. (ii) W ( t , s ) R ( s , u c ) R ( t , u ) u 5 s _< t . (iii) The mapping t + R ( t , u) is continuous with respect to the Hausdorff metric .
Remark 7.1.2: Relations (i) and (ii) are proved as in [21]. To motivate the discussion for the form of optimal controls, we note that if an admissible control u drives d, to zero in time t l , then
Synthesis o f Time-Optimal and Minimum-Effort Control
211
Note that U ( t 1 ,u) = T(t1,u)I where I is the identity (see [S,p. 1461). For example, for the system
with solutions
44,0)(l) = U(t)4(O) + A1
JI:
V ( t - s - h)4(s)ds,
we may designate
W ( t ): C --+ E" by
If
Wt)4 = 44,W). 2,
= 4,then W ( t )
W ( t , u ) . Also recall that T ( t , u ) ( b )E z t ( u , 4 , 0 ) .
The coincidence
W(tl)4 E 8 @ 4 t l , U ) , where dR is the boundary of Ilk, is very important in characterizing the form of optimal controls. We need the following definitions. Definition 7.1.3: The control u E U a d is an extremal control if the as sociated solution z ( c , d , u ) corresponding t o (u,4)lies on the boundary of d(t,u),that is,
Definition 7.1.4: System (7.1.5) is pointwise complete at time t 1 , if and only if the solution operator defined by W ( t ,u)$ = z(u,4 , 0 ) ( t )is such that
W ( t l , ~: C ) + En is a surjection: W(t1,u)C = En. Conditions for pointwise completeness are found in [6]. These conditions are stated in Lemma 7.1.2. Lemma 7.1.2
For System (2.1.12), nameZy
a necessary and sufficient condition for the pointwise completeness fort = t l is that, for every nonzero q E En there exists a t 2 t l such that r f U ( t ) #
212
Stability and Time-Optimal Control of Hereditary Systems
0 where U is the fundamental matrix solution of (2.1.12). If we define Fk = Zk(O)Al, where z k is identified in (2.1.16) - (2.1.21), then (2.1.12) is pointwise complete for t l E [k,k l ) , k = 0, 1 , 2 , . . . whenever the matrix
+
M(ti) = [Ek-iF&-i,. .. ,~%-iA;:~pk-i, has rank
la.
Example 7.1.1: A . =
EzZz(0) =
[:: -:I 1
1 -2
M(2)=
&zis(O)I
[;f :]. [; i] 0 0 -1
A1 =
, U(2) =
and this is a singular matrix
[
2 -2 0 2 -4 0 0 -4 0 1 -2 0 0 -2 0 0 0 01. 0 2 0 2 0 0 0 - 4 0
This has rank less than 3 because vT = [l,-2, -11 satisfies v T M ( v ) = 0. The example is not pointwise complete for t l 2 2. The following is also true. Lemma 7.1.3 The system is pointwise complete for all t l E [O,oo)whenever AoAl = A1Ao. The system is pointwise complete for all tl € [0,2). Theorem 7.1.2 The following are equivalent: (i) The system (7.1.1) is Euclidean controllable on [ u , t ] ,1 > u (ii) O E I n t R ( t , a ) , t > u + h . (iii) W(t,s)R(s, u) c Int R ( t , u ) , u < s < t .
+ h;
Proof: Suppose (i) is valid. Then the mapping
H : L,([u, t ] ,Em)
--+
En
given by Hu = x ( t , u ,4, u) is surjective. Clearly H(Uad) = R(t, u). Also
where Iis an open ball containing 0 such that 0 E @ c Uad. Because H is a continuous linear transformation of L , onto En,it is an open map. Hence H ( @ ) is open and contains 0. Thus 0 E Int R(t,u). Conversely, assume that 0 E Int R ( t , u ) . Then 0 E Int R ( t , u ) c H ( L , ( [ u , t ] , E " ) ) . Because
Syn t h es i s of Time-Op 1im a1 and Man irn u m- Ego rt Con2 rol
213
the last set in the inclusion is a subspace, containing a full neighborhood of zero, we have H ( L , ( [ u , t ] , Em))= En.Thus (7.1.1) is controllable. Assume that 0 E Int R(t, u) and W ( t , s ) qE aR(t,u) where q E R(s, u), since by Lemma 7.1.1, W ( t ,s)q E R(t,u), W ( t , s )= T ( t ,s ) I , and aim at a contradiction. Because of the point in the boundary, we have
where (-,.) is the inner product in En. We now observe that
because of the semigroup property of the solution operator. Define U’(7)
Then this admissible
= U*
o
determines a p E R(t, a),and
Thus (c, p ) > (c, W ( t ,s ) q ) , contradicting the assumption that W ( t ,s ) q is the boundary of R(t, u). Hence W(t,s)R(s, u) c Int R(t, u),for u < s < t . This completes the proof of Theorem 7.1.2. Proposition 7.1.3 Suppose (7.1.5) is pointwise complete and w is a t i m e optimal control for (7.1.1). Then w is extremd.
Proof: We first show that if w is a time-optimal control at the first instant
tl of arrival a t 0 E En,then the solution will lie on d ( t 1 ,a),the boundary of the attainable set. We shall then show that if ~ ( u, t ,4 , w ) lies on d(t,u) at any fixed t,, then
Suppose that w is the optimal control that steers 4 to 0 E En in minimum time t l , i.e., z(t1) = z ( t l , u , # , w )= 0, and assume that z(t1) = 0 is not in the boundary, BA(t1,u). Then there is a ball O ( 0 , p ) of radius p about 0 such that O(0,p) E d ( t 1 , u ) . Since d ( t , u ) is continuous in t , we can preserve this inclusion at t near tl by reducing the size of O(O,p),i.e., if
214
Stability and Time-Optimal Control of Hereditary
Systems
there exists a 5 > 0 such that O(O,p/2)C d ( t , o ) , tl - d 5 t _< t l . Thus we can reach 0 at time tl - 6 . This contradicts the optimality of t l . The conclusion is valid that 0 = z ( ~ I , u , ~ ~E, w ad(t,u). ) We now prove the second part by assuming that t* = z(t,, u , 4 , w ) E Int d ( t , cr) for some 0 < t , < t l . We claim that z(t,u,qh,w)E Int
d ( t , a ) for t
> t,.
Since z* E Int d ( t , , a ) , there is a ball U ( z * , S )c d ( t , , a ) such that c E O(z*, 6) can be reached from 4 at time t , using control UO. Now introduce the new system N
+ B(t)w(t)j t 2 t',
~ ( t=) C A i ( t ) z ( t - hi) i=O
(7.1.7)
zt* = 4 0 with z ( t * ) = c = 40(0), t* < t with w fixed. We observe that for the coincidence 4 0 = z*,we have from uniqueness of solutions that z(t,u,q$-,,w) = t ( t , u , d , w ) , t 2 t*. This solution is given in En by U, 4 0 , w )
= W ( t ,0 ) 4 0
that is,
At
Z(t,cr,40,@)
+
b 1
U(t)s)B(s)v(s)&
= W(t,u)do(O)
+ d(t),
where d ( t ) = U ( t , s ) B ( s ) w ( s ) d s . Since (7.1.5) is complete so that W(t,cr)C= E n ,W ( t , a )is an open map. Hence 40 = c -* z(t,cr,do,u) is an open map and takes open sets into open sets. As a consequence the image of the open ball 0 lies in the interior of d(t,a).Thus the image of z*,which is just z ( t ,u,#,u), lies in Int d(t,u).
Theorem 7.1.3 Suppose (7.1.5)is pointwise complete and (7.1.1) is Euclidean controllable. Let u* be an optimal control and t* the minimum time. Then w 2 -T(t*, u)d(O)E bR(t*,cr), the boundary of the reachable set R(t*,u ) OR En if and only if U*(S)
= sgn[y(s)B(s)l,
Y(S)
$ 0 , s E [a,t*I,
where y : [a,t*]is an n-row vector of bounded variation that satisfies the adjoin t equation
(7.1.8)
215
Synthesis of Time-Optimal and Minimum-Effort Control
Proof: For the system N
+ C Ai(t)x(t - h i ) + B(t)u(t),
i ( t )= Ao(t)t(t)
i=l
x(t) = d(t)t
e [-h,O),
t(0)= qq0) = c,
t
1
Q
by definition the En-reachable set is
If (7.1.1) is null controllable with constraints so that optimal control u* exists, then z ( t * , Q, 4, u * ) = 0, and
w z -T(t*, Q)O(O) =
1
t*
U ( t * ,s)B(s)u(s)ds.
U
Consequently w E R(t*,a). Because u* is optimal and (7.1.5) is complete, the solution x(o,4, u*) is extremal. Therefore -T(t*,o)q5(0) = w e aR(t*,m). But by Proposition 7.1.1, R(t*,u) is compact and convex, and because (7.1.1) is controllable, Theorem 7.1.2 yields that 0 E Int R(t*,a). It follows from the separation theorem of closed convex sets [l,p. 4181 that there exists a nontrivial d E En,a row vector, such that
where
(a,
.) defines the scalar product in En.Thus U ( t * , s ) B ( s ) u ( s ) d s )5 ( d , l * U(t*,s)B(s)u*(s)ds
so that [*
dTU(t*,s)B(s)u(s)ds5
l*
drU(t*,s)B(s)u*(s)ds.
(7.1.9)
Recall the definition Y ( s , t ) as the matrix solution of the adjoint equation (7.1.8) as was treated in Hale [8, pp. 147-1531 and the coincidence Y ( s , t )= U ( t ,s) a.e. Then clearly (7.1.9) yields the inequality
l'
dTY (s,t*)B(s)u(s)ds5
I"
dTY (s,t*)B(s)u*.
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Slability and Time-Optimal Control of Hereditary Systems
So that
I J,
t'
t* Y(S,
t*)B(s)u(s)ds
Y(S,
t*)B(s)u*(s)ds,
where y(.,t*) : E En*vanisheson [t*,co),satisfies(7.1.8) on (-co,t*-h] and is such that y(t*,t*) = d # 0. As a consequence, ---+
u * ( s ) = sgn[y(s,t*)B(s)l,
Y(S,t*)
f 0.
The argument can be reversed.
Remark 7.1.3: Note that y(t*,t*) = d
# 0, so that y(s,t*)
f 0.
Remark 7.1.4: The form of optimal control in Theorem 7.1.3 asserts that each component u; of U* is given by u;(s)
= sgn[Y(s, t*,$)bj (a)]
on [ u , t * ] ,j = 1 , . . . ,n,where bj(s) is the j t h component of B ( s ) . Provided we assume completeness, it uniquely defines an optimal control if (7.1.1)is normal in the following sense:
Definition 7.1.5: Define gj(d) = { s : y(s,t*,d)bj(s) = 0, s E [O,t*]}. System (7.1.1)is normal on [a,t*]if gj(d) has measure zero for each j = l , . . ., m , where y(t*,t*,d)= d # 0. If (7.1.1)is normal for each [u,t*],we say (7.1.1)is normal. It follows from the above definition that (7.1.1)is normal on [u,T ] , T > c if for each j = 1 , . .. , m , dTU(T,s)bj = 0 almost everywhere for each j = 1 , . . . , m, implies d = 0. Here b j is the j t h component of B. %calling the definition of determining equations for constant coefficient delay systems,
Qo(s) =
= 0, I (Identity),
I
s
0
S # O ,
and defining
one proves in a way similar to [3,p. 791 that the following proposition is valid.
Synthesis of Tame-Optimal and Manimum-Eflort Control
217
Theorem 7.1.4 A necessary and sufficient condition that a complete system (7.1 .10) is normal on [0, T ] is that for each j = 1 , . ,rn, the matrix
-
Qjn(T)=
{ Q o ( s ) b j , . . ., Q n - l ( s ) b j , s E [ O , T 3 }
has rank n.
7.2
Geometric Theory and Continuity of Minimal Time Function
We note that if (7.1.1) is null controllable with constraints, then z(u,4, u ) ( t ) = 0 for some t 2 u and some u E u a d , that is,
- T ( t , u)4(0) =
w
/
t
U ( t ,s)B(s)u(s)ds.
0
Consequently
if and only if 4 can be steered t o zero at time t by some admissible control. If the system is Euclidean null controllable with constraints, then
d(u) = {4 : ~ ( u4,,u ) ( t )= 0 for some =
26
E Uad and some
t},
c = (4 : T(t,u)+(O)= z ( t , u , 4 , 0 )E R ( t , a ) } ,
where z ( t , n,d,O) solves (7.1.5).
Definition 7.2.1: The minimal time function is the function M : d(u) -+ E defined by
M(4) = Inf(t 2’ u : - T ( t , a)4(O) E R(t, u)}. Thus u 5 M ( 4 ) w t , u).
5
+00
with M ( 4 )
<
00
if and only if --T(t,u)4(0) E
The following result is fundamental.
Theorem 7.2.1 Let (7.1.1) be Euclidean controllable. Then the minimal time function is continuous. Proof: First we show that M is upper semicontinuous. Let (bk +-4 as k -+ 00 so that - T ( t , u ) # ~ k-+ - T ( t , u ) 4 as k -+ 00. Suppose M ( 4 ) < 00, and for some fixed t , t > M ( 4 ) . We know that
218
Stability and Time-Optimal Control of Hereditary Systems
Also, since (7.1.1) is Euclidean controllable and Lemma 7.1.1 (ii) is valid, there is some s, t < s such that
Therefore, for sufficiently large k,
Because T(s,t)[-T(t,6)4e(O)] = -T(s, u)&(O), we have M ( 4 t ) I s. Therefore for all s > M ( 4 ) , limsupM(4k) 5 s. If we let s = M ( 4 ) +,then
+
limsup M(4d
IM(4.
This inequality is still valid if M ( 4 ) = +oo, proving upper semicontinuity. For the proof of lower semicontinuity, suppose
M ( 4 ) > liminf M ( 4 k ) . Since M ( + E )cannot be +oo, there is a subsequence such that M ( 4 k ) converges: M ( 4 k ) s < M(4). Therefore, for k large,
-
M ( 4 k ) < s.
From the definition of M ( $ t ) , M ( + L . ) 5 t whenever -T(t,u)+t(O) E R(t, u). There is some 4 k i such that
Since R(s, a) is closed,
This contradiction proves that M is lower semicontinuous. We conclude that M is continuous. Note that M is always lower semicontinuous.
Synthesis of Time-Optimal and Minimurn-Effort Control
219
Corollary 7.2.1 Let (7.1.1) be controllable and (7.1.5) be pointwise complete. For t 2 u we have the following relations:
Proof: We now observe that
We now show that
03
:]
Let 4(0) E [u,t
+
.
n R
v=l
Then there is an admissible control up :
, such that
+(O) = f + +
u
U
t
=J,
U(t
(L + ;+) B(s)up(s)ds,
+ -P1, s ) B ( s ) u p ( s ) d s +
= + yp.
l++(t + u
,;
.)
2p
We note that 2p
=
J,’ u +
=U Thus zp E U
(t
,:
t)
U ( t ,s)B(s)up(s)ds,
(1+ :,t) J,’ U ( t ,
(1+ t , t ) R ( t , u ) ,or
s)B(s)up(s)ds.
B(s)up(s)ds,
220
Stability and Time-Optimal Control of Hereditary Systems
Hence zp + T(t,t)4(O)= O(0) as p + 00, since yp + 0 as p 4(0) is in the closed set R(t,u).
+ DC),
so that
For the second formula we note that since (7.1.1) is Euclidean controllable and (7.1.5) is pointwise complete, we have that
W ( t , u ): C
+
En, ( W ( t , u ) 4= T(t,u)+(O))
is an open map, and T ,M are continuous. Hence
is open, so that
{-T(t,u)4(O): M ( 4 ) < t } c Int W(t,u). From the proof of Proposition 7.1.1, we know that if
-T(t*,u)4(O)E Int R(t*,u),for u < t* then
< 00,
-T(t, u)4(O)E Int R ( t , u ) ,for t > t*.
From the above we have
M ( 4 ) 5 t* < t .
This completes the proof. As an immediate by-product of the continuity of the minimal time function, we construct an optimal feedback control for the system (7.1.1). In the spirit of HAjek [lo],we present some needed preliminaries. We identify Bo with the conjugate space of C and define a subset of Bo, which we describe as the cone of unit outward normals t o support hyperplanes t o R(t,u) at a point T ( t ,u)4(O)on the boundary of R(t, u).
Definition 7.2.2: For the controllable system (7.1.1), let
d(o)= C = {4 : T ( t ,u)d(O) E R(t, u)}. For each I$ E d(u), let
I((+) = (11 E Bo : 111111 = 1 and such that
v P E WM(4),0)1.
($(()),PI5
($(O),T(M(4),0)4(0:
Note that T ( M ( + ) ,u)4(0)E aR(M(q5),u),the boundary of R(M(I$),u).
Synthesis of Time-Optimal and Minimum-Effort Control
22 1
Lemma 7.2.1 Suppose (7.1.1) is Euclidean controllable. Then K ( 4 ) is nonvoid and I ( ( - + ) = -I<($). Also limsup K ( 4 n ) c
K(4) a~ 4n -+ 4 in d(a).
That is, K is upper semicontinuous a t 4. Proof: From the controllability assumption,
0 E Int R ( t , a ) , t > u
+ h,
by Theorem 7.1.2. Consider K ( 4 ) . Since T ( M ( d ) , u ) 4 ( 0 )E aR(M(4),u ) by the usual separation theorem [l, p. 1481, there exists a 1c, E BO the conjugate space of C,$(O) f 0 such that
(11,(0), P ) 5 (11,(0>, T(M(41,u)4(0)) VP E W W d ) , u). It is clear that $J can be chosen such that ll11,ll = 1. Hence K ( 4 ) is nonvoid. For the second assertion, observe that R(M(#), c) is symmetric about zero. Because t + R(2,a) is continuous [21]and closed, and because 4 -+ M ( 4 ) is continuous, a simple argument proves the last assertion. Because K ( 4 ) is a nonvoid subset of Bo, for each 4 we can choose a y(4) E K(c#J)whenever (7.1.1) is Euclidean controllable. We now prove that this selection can be made in a measurable way. Lemma 7.2.2 Assume that (7.1.1) is Euclidean controllable. There exists a measurable function y : d ( u ) + Bo such that
y(4) E K(4) for all 4 E d(o). Proof: We observe that C is a measurable space. Its conjugate Bo is a Hausdorff space. Also C is separable. Let k : C -+ BO be the mapping given by the collect ion
I((4) = (k(4) = 11, : 11, E Bo, ll$ll = 1 ( $ ( O ) , P ) I (11,(O),T(M(4),u)4(0))
v P E R(M(4),.>I.
The continuity of k is a simple direct consequence of its definition as well as the continuity o f t + R ( t ,u) and of + M ( 4 ) . In McShane and Warfield [23]we identify M = C = d(u), A = Bo, 11, E Bo, (and invoke the continuum hypothesis). Theorem 4 of [23] asserts that there exists a measurable function y : d ( u ) -+ Bo such that y(4) E K ( 4 ) . This proves Lemma 7.2.2. We are now ready for our main result.
222
Stability and Time-Optimal Control of Hereditary Systems
Theorem 7.2.2 In (7.1.1) assume that Ail Bi are real analytic. Assume that (7.1.1) is both Euclidean controllable and normal. Let (7.1.5) bepointwise complete. Then there exists a measurable function f : d ( u ) 3 Em that is optimal feedback control for (7.1.1) in the following sense: Consider the system N
i(t)=
C Ai(t)t.(t- hi) + B ( t ) f ( z t ) ,
zu
= 4.
(7.2.1)
i d
Each optimal solution o f (7.1.1) is a solution of (7.2.1). As a partial converse, each solution of (7.2.1) is a solution (pcssibly not optimal) of (7.1.1). Also the function f is given by
where g ( t , s,.) : Bo
---*
E" is defined by
where y : [a,t]satisfies
-
Proof: Let y : d(u) Bo be the measurable selection described in Lemma 7.2.2. For each 4 E d(a)set y(4) = $ E Bo, and define f(4)as in (7.2.2). We recall that
+
S(t,s,$1 = J P , [ d W l [ ( i ( t 0 , S ) l I where U solves (7.1.5). We recall from Tadmor [25, Theorem 4.11 (see also Corollary 10.2 of [IS])that s -+ U ( t ,s) is piecewise analytic for each s E [c,M ( 4 ) ] ,since A i ( t ) is analytic in t. It follows from the analyticity of B ( s ) that each coordinate of g ( t , s , $ ) B ( s )is analytic on [si-~,si]i = 1 , 2 , . .. , v for each partition c 5 so 5 s1 5 . . . 5 sv = M ( 4 ) . Thus sgn[g(t,s,$ ) B ( s ) ] is piecewise constant on [si-l,si] and therefore on [u,M(4)]. Hence the limit in (7.2.2) exists. Because y(4) is a measurable selection of 4, so is f(4).
Synthesis of Time-Optimal and Minimum-Effort Control
223
Let 3: : [ a , M ( 4 ) ]+ C be an optimal solution of (7.1.1) with c, = 4. We now show that t satisfies (7.2.1). Indeed, take an arbitrary s, 0 5 s < M(+), then t8E d(a).There is an optimal control that transfers 2, to zero. We take it to be
for any choice of y E K ( t d ) ,i.e., -y E K(-t8). This choice follows from Theorem 7.1.3. We choose y = y ( z s ) . Because (7.1.1) is assumed normal, optimal control is unique almost everywhere and it uniquely determines a response on [ a , M ( d ) ] ;that is, the response 2 . Thus for almost all t 2 s and each s we have
Now take the limit as t
-+
s+ (c5 s
5 t ) . We now have
Since u, is piecewise constant, u,(s) = f(ts) for almost all s. Therefore, the response t t o u, satisfies
almost everywhere. Because 2 is absolutely continuous, it is a solution of (7.1.1). Finally, let c be a solution of (7.2.1) that is a response to u ( t ) = f ( z t ) . Clearly u E L,([a,M(+)], E") and v E Uad.
7.3 The Index of a Control System From our analysis it is clear that the properties of the function
with y s ( t , - ) = $, $ E Bo, the Banach space of functions 1c, : [-h,O] ---+ En*of bounded variation on [-h,O], continuous on the left on (-h,O) and vanishing at zero with norm Var $ , are important in determining [--h,Ol
224
Stability and Time-Optimal Control of Hereditary Systems
optimal strategies of delay equations (7.1.1). We call t the index of the control system. To determine it explicitly, one first finds the fundamental matrix U ( t ,s) of (7.1.1). We consider its autonomous version, namely N
i ( t ) = A,z(t)
+ C A;z(t - ih), i=l
(7.3.1)
~ ( 0= ) 2 , E En;X ( S ) = 4 ( ~ ) ,s E [-hN,O), h > 0. The determination of the fundamental matrix U of (7.3.1) was treated in Section 7.2.1. With the explicit formula for U , one determines the index of the controi system using g , given as
(7.3.2a)
Thus,
(7.3.2b)
Consider N = 1, that is, the system
i ( t )= A o ~ ( t+) A l t ( t - h ) + B ( t ) u ( t ) .
(7.3.3)
The index of this control system is given as follows: On [0,2h]
(7.3.4a)
Synthesis of Time-Optimal and Minimum-E8or-i Control g ( t , 6, $)
=
1
0
225
[dG(6)][eAo(t+e-S)
-h
t - s E [h,2h]. We now observe that k ( s ) is an rn-vector function, the sign of whose compe nents can be determined and thus an open loop control constructed. Thus, once the disconjugacy properties of k are analyzed, an optimal feedback control of (7.3.1) can be constructed, as was done in [20]. Applications
For linear autonomous systems, we have developed some theory about optimal control that, when applied t o simple examples, will give us insight to the construction of optimal control laws. Since optimal control exists when the complete system is Euclidean controllable with constraints, and is of the form sgn(drU(t - s ) ) = u ( s ) , the usual methods of ordinary differential equations can be applied. Using the values of U on [kh,(k+)h] beginning at the origin, integrate backward with controls u ( s ) , and find all optimal trajectories to the Euclidean origin. Since in the first interval [0, h ] , U ( t ) = eAot, the analysis is relatively routine, though complicated. The application of this method, though feasible, is definitely limited, because of the complicated nature of U. It does, however, provide insight into possible general methods of construction of optimal controls of systems with limited controls. Very many numerical methods which have so far been developed [28,29] are for La controls, and can reasonably be modified to help solve the synthesis problem of time-optimal control systems with L, controls whose components are bounded, a problem long abandoned by researchers more brilliant than I. This book is an affirmation of hope that they will return! Example 7.3.1: We want to find a time-optimal control u * ( t )such that z ( t ) = $ J ( t ) , t E [-l,OI, and z ( t * ) = 0, with t* minimum when
.(t) = - z ( t )
+ z(t - 1 ) + u ( t ) ,
)u(t)l5 1.
(7.3.5)
It is easily proved that optimal control exists. We use the fundamental matrix solution .(t) = - z ( t ) z(t - l ) ,
+
Stability and Time-Optimal Control of Hereditary Systems
226
of Problem 2.1.1
U ( t )= e-*,
= e-'[l+ e(t - I)], 1
1
+ e ( t - 1) + -21e 2 ( t 2 - 4t + 4)
to deduce the index of the control system, g(4
t E [O, 11, t E [1,21, ,t E [2,3]
= g ( t , s , + ) , ($(O) = d # 01, = d U ( t - s) = d e a - t * , s E [t* - l,t*], - deS-t' [I + e(t* - s - I)], s E it* - 2,t* - I], 1 2
- ded-f' [I + e(t* - s - 1) + -e2(t* s E [t*- 3,t' - 21.
-s ) ~ 4(t'
+
- s ) 41,
Optimal control is given by sgn d , since all factors of d are positive. The time-optimal control will be either 1 or -1 on the entire interval [O,t*].
Example 7.3.2: Antirolling Stabilization of a Ship A ship is rolling in the waves. The linear dynamics of the angle of tilt z from the normal upright position is given by
i ( t )= Y ( t ) , Y(t) = - b y ( t ) - qy(t - h ) - k ( t )
+
i ( t ) = - b y ( t ) - qy(t - h ) - k . Z ( t )
+ uz(t),
w = Y(t>,
UZ(t),
(7.3.6) (7.3.7)
System 7.3.7 is obtained when a servomechanism is introduced and designed to reduce ( 2 ,y) to (0,O) as fast as possible. What the contrivance does is to introduce an input to the natural damping of the rolling ship, a term proportional to the velocity at an earlier instant t - h : qy(t - h). Also introduced is a control with components (0,212) yielding the equation above. Thus
i ( t ) = A2g(t) + A l ~ ( t h ) + Bu(t),
(7.3.S)
227
Synthesis of Time-Optimal and Minimum-Effort Control
x =Angle of tilt
FIGURE 7.3.1. with k = 2, b = 3, q = 1. The parameter values correspond to one particular operating point. In this example,
e-'(e(Zt-6))+e-"(2+ea(Zt)),
e-'(2e(Z-t))+e-'*(-4+ez(-6-4t)),
e-'(e(t-4))+e-2'(l+ea(Zt)) e-'(e(5-t))+e-2*(-2+ea(2-4t))
1
, (7.3.10)
228
Stability and Time-Optimal Control of Hereditary Systems
and if h is arbitrary,
'I
-1
(7.3.11)
on [h,2h], and so on. The index of the control system is
g(s) = g ( t * , s,4 ) = #U(t* d
- s),
= $(O) = d = ( d 1 , d z ) .
Our problem is to find a control u* that drives the initial position 3 = (z0,yo) = q5(s), s E [-h,O] t o the Euclidean terminal point g(t*) = ( z ( t * ) ,y(t*)) = 0 in minimum time t * . Because
we invoke Theorem 7.1.4 to show that the system is normal and Euclidean controllable. Uniform asymptotic stability is assured by the analysis of Example 2.4.1 since 3 = b > q = 1. The system is null controllable with constraints, and optimal controls exist and are uniquely determined (almost everywhere) by u*(s> = sgn d s ) , where the index of the control system is g ( s ) = $rUl(t* g(s)
= #U2(t*
g ( s ) = #U3(t*
- s), - s),
- s),
E [O,h], 0 5 s 5 t* 5 h , s E [O,h], s h 5 t* _< 2h, s
+
s E [O,h], s
+ 2h 5 t* 5 3h,
(7.3.12)
and so on. In general, g ( s ) = drUk(t* - s),
s
E [O, h], s
+ (k - l ) h 5 t* 5 kh,
where Uk = U is as defined in Equation (2.1.8) or Equation (2.1.10). We now use U , which has been determined. In [0, h], g ( s ) = g l ( s ) = ( 2 ( d l - d 2 ) e - s + ( 2 d 2 - d l ) e - 2 5 , (dl - ~ f 2 ) e - ' + ( 2 d z - d l ) e - ~ ' ) .
(7.3.13)
Synthesis of Tirne-Optimal and Minirnurn-Effort Control
g ( s ) = g2(s) = (2(dl - d z ) , ( d l - d2))(e-(t'--d)+ (t - e -2(t'--~-h) @d2, - M e - ( t ' - h - S ) 1
229
- s - h)e-(t-8-h) 1
+ + ( 2 4 - d l ) ( l , l)(e-2(t*-B)- 2(t* - s - h)e-2(t-"h) s E [O, h], + h 5 t* 5 2h.
)I
5
(7.3.14) To obtain optimal trajectories in E 2 , we start at the origin and integrate backwards in time. To do this we replace 2 by -7 in our system (7.3.7) and obtain the dynamics (7.3.15) where 4 7 )
= sgn[(dl
- d 2 ) + e'(2d2 - d ~ ) ] .
(7.3.16)
Taking dl < 0, d2 < 0 , we begin a t the origin (z(0) = y(0) = 0) along the trajectory S ( r ) with u z = -1. This control may change sign at some 7 1 E ( 0 , h] where (dl - d z ) 71 = In 2dz - dl
(-
)
Therefore, starting at 7 = 0 a t the point ~ ( 7 1 )of S(7) = LY with u2 = 1, we leave LY along a new curve p(7) that begins at ~ ( 7 1 ) .There is no further change of sign if 7 E [0, h] until 72 = 7 E [h,2h], and then the control is u ( s ) = sgn g2(0). Also, if 71 E [h,2h],the control u(s) = sgn gz(s) is used. In both situations, and under the assumption dt < 0, dz < 0,
= Sgn[(dl - dz)(e-('*-S) + (t - S - h)e(t-a--h) - dl(e-,(t'-~-h) - e - Z ( i ' - ~ - h ) )
~ 2 ( 5 )
+ (2d2 - d1)(e
-(t'--s)
The new control then is
-qt*
- - h)e-2(-h)
(7.3.17)
11.
Stability and Time-Optimal Control of Hereditary Systems
230
Thus we begin at the point ,L?(T~)with u ~ ( s= ) the second component of g2(0), where dl < 0 d2 < 0, and move along the trajectory S(-T), which , this control until possible switches at the zero of the second is C ( T ) with component of g2(s) on [h,2h]. The earlier process is repeated. Next we start at the origin. With dl < 0, d2 > 0, we have u2 = 1. The process is repeated, and the control law is deduced, by considering the situation dl > 0, d2 > 0 and d2 > 0, and d2 < 0 as well.
Example 7.3.3: We consider a two-dimensional retarded system
+ +
+ +
k I ( t ) = -221(t) - 22(t - h ) .l(t) .2(t), .2(t) = -22(t) 2 l ( t - h ) U l ( t ) 2u2(t),
+
1.11
L 1,
1.21
5 1,
(7.3.19) where x l , x 2 stand for population densities of two species (and so they are nonnegative). This system may describe the linear dynamics of a predatory). If z and jj are the population prey system around the equilibrium (X, densities of the prey and predator respectively, then 21
=r-X,
22
=V-Y.
The controls 21 = (u1,u 2 ) are harvesting/seeding strategies. We want an optimal uC that will drive the populations to the equilibrium (0,O) E E2 in minimum time starting from any initial population density function (10, yo) 5 q5 E C ( [ - h ,01, E 2 ) . The system can be written in matrix form as k ( t ) = A o ~ ( t ) A l ~ (-t h ) B u ( t ) ,
+
where
+
0 -2
We observe that i(2)
= Aox(t) + A l ~ (-t h)
is Exponentially Asymptotically Stable (EAS) for all h 2 0. Indeed, we can use a result in a very recent book [27,pp. 98-99] that the system
kl(t) = -allzl(t) & ( t ) = -a22zz(t) is EAS for all h 2 0 if
- bl222(t - h ) , + b21z(t - h )
(7.3.20)
Synthesis of Tirne-Optimal and Minirnurn-Effort Control
23 1
or if
(7.3.22) We can also use Proposition 3.3.3. The first condition is 3 > 1, while the 1 3 second is 3 > - or 3 > -1 and 2h < -. The system is also Euclidean 4 2 controllable, since by Theorem 6.1.1,
and this has rank 2 for any t > 0. Theorem 6.3.1 assures us that (7.3.19) is null controllable with constraints. Optimal controls exist, and since it is normal for each t > 0, the optimal control is uniquely determined by
s g n ( g U k ( t - s ) B ) , s E [O, h ] , k: 5 t 5 (k
+ l)h,
where u k is the fundamental matrix which we now calculate the methods of Section 2.1:
(7.3.23)
232
Stability and Time-Optimal T i m e - O p t i m a l Control of Hereditary Systems
Hence
(7.3.25) .CL".,"
'.,.
For each point of the Euclidean state space (the plane), there is an optimal control. We first assume that the delay h is sufficiently large. Thus
drUb(t - s)B, s E [0,h], L 5 t _< (k+ 1)h
is given as follows:
d
\-----.I
To obtain the optimal trajectories, we start at the origin and as usual integrate backwards. This is done by replacing t with -T and considering the dynamics (7 -3.28) where
(7.3.29)
Synthesis of Tarne-Optimal and&finirnurn-Effort Control
233
+
We now assume that the optimal time t is known and that dle' d2 < 0, diet 2d2 < 0, and d2 > 0 and begin a t (0,O) with controls ul = -1, 212 = -1 and move along the parabola defined by Equation (7.3.28), which is a : q ( 7 ) = e2' - 1 on [ ~ , h ] , (7.3.30) 2 2 ( 7 ) = 3(e' - 1).
+
Clearly u2(7) will switch t o a new value at
and u l ( 7 ) will change sign at
72
= t + en
(i:). - Hence
72
- 7 1 = tn2. It
follows then that if we begin at T = 0 at the point ( z 1 ( ~ 1 ) , 2 2 ( 7 1 ) of ) the curve a with 211 = -1 and u2 = 1, we depart from a along the parabola
Clearly u1 changes signs at
These are the calculations of Hermes and LaSalle [4, p. 831 for ordinary differential systems. They are valid here because we have assumed that h is so large that on [0, h] all the indicated switches occur, and the dynamics is that of e A o + .Thus the last equations above tell us how t o transform a to obtain the curve p where u1 will change sign. For each component there is only one sign change. We complete the analysis by assuming now that
Now begin at (O,O), with control u1 = -1, u2 = 1, with a sign change of u1 not later than T = l n 2 apart. The trajectory is along the line (0,O) to ( - l , O ) , which is the switching curve y. The others are obtained by symmetry. Thus within [0, h], h large, the complete optimal control law is deduced. This is given in the diagram.
234
Stability and Time-Optimal Control of Hereditay Systems
FIGURE 7.3.2. 14, P . 831. If the switch time T > h , we have to go to the next interval [h, 2h] before switching. Thus we consider u(t), t E [h,2h]in [h,2h]; and then U(T)
= sgn[dTUl(s - t ) B ] , t E [h,2h] with s = 0, i.e.,
- 4(e+(t-h)- e+2(t-h)11 + dz{e+(lSh) - e+2(t-h)+ 2e+'} u ~ ( T= ) sgn[dl{e2' - 2(e+t-h - e+2(t-h)
u ~ ( T= ) sgn[dl{e+"
+ dz{e+' + el-'
11
- 2e2('-h)}],
where dle'+& < 0, d1et+2dz < 0, and dz > 0, t E [h,2h].Thus we begin with this control at the point a ( q ) and move along this trajectory until switches occur at the zero of 91 and the zero of 9 2 . The earlier process is
Synthesis of Time-Optimal and Minimum-E$od Control
235
repeated] after going to the next interval, [2h, 3h];the analysis is completed. In this way the optimal control law is deduced.
Example 7.3.4: Wind Tunnel Model The problem concerns a time-optimal control of a high-speed closedcircuit wind tunnel. We are concerned with the so-called Mach number control. A linearized model of the Mach number dynamics is a system of three state equations with a delay. The state variables z1,zz,z3 represent deviations from a chosen operating point (equilibrium point) for the following quantities: z1 = Mach number, 2 2 = actuator position (guide vane angle in a driving fan)] and 2 3 = actuator rate. The delay represents the time of the transport between the fan and the test section. We assume that the control variable has three components that constitute an input to the rate of the Mach number. The model has an equation of the form
1 with a = - k = 0.117, w = 6, E = 1.6, and h = 0.33s. 1.964 ' The parameter values correspond to one particular operating point. We write the equation as
k ( t ) = A , z ( t ) + A l ~ (-t h ) + B U ( t ) , where
-a
0
0
-w2
-2Ew
] [!5 !]
B = [k\],
]A1=
The eigenvalues of A , are A 1 = -0.509165, -2.106024. The fundamental matrix is
A2
k3=36.
= -17.0939, and
A3
=
236
Stability and Time-Optimal Control of Hereditary Systems
The system is Euclidean controllable on [ O , t ] , V t > h , since O,O, 0, 0,2.144603
36, -6912,11975.05,0,0 and rank Q3(h) = 3. The system
i ( t ) = Aoa(t) + A l z ( t - h )
is uniformly asymptotically stable since the eigenvalue A of the characteristic equation A+&
A(A) =
= (A
0
A
A0
36
+
0
0.117e-Xh
-1.964
A) + (A2
A 19.22A
+ 19.2
+ 36) = 0,
and A1
= -0.50916,
A3
= -2.106.
A2
= -17.094,
It follows that the system is controllable with constraints. The fundamental matrix is
U(t)=
+
u*( s ) = sgn[dl( .00129 - .003603e-2~6152('-a) .07527e-17.6032(t-a) )
+ d2(.06672(e-z.lo6(t-3) - e-17.044('-8) 1)
+ d3(-0.1405e-2.'06f + 1.1405e-'7.044t)]. 7.4 Time-Optimal Feedback Control of Autonomous Delay
Systems
In this section we complete the investigation that is reported in Theorem 7.2.2 of the problem of the construction of an optimal feedback control needed to reach the Euclidean space origin in minimum time for the linear system N
i ( t ) = A o ~ ( t+) C A j z ( t - ~ j+)B u ( ~ ) .
(7.4.1)
j=1
Here 0 < r < 27 < . . . < r N = h; Ai are n x n constant matrices, and B is an n x m constant matrix. The controls are L , functions whose values on any compact interval lie in the rn-dimensional unit cube
C m = { u E E m : J u j l _ < lj, = l l . . . l m } .
We shall show that the time-optimal feedback system N
k(t)=
C A j ~ (-t rj) + B f ( ~ ( t ) )
(7.4.2)
j=O
executes the time-optimal regime for (7.4.1) in the spirit of HAjek [30] and Yeung [32,33]. The construction o f f provides a basis for direct design, and it is done for strictly normal systems, which we now define.
238
Stability and Time-Optimal Control of Hereditary Systems
Definition 7.4.1: Let Jo={t=j7,j=0,1,2
,... },
and assume Jo is finite. Suppose U ( Et, ) is the fundamental matrix solution of
i ( t )= Aot(t)
N
+
Ajz(t
-rj)
(7.4.3)
j=l
on some interval [0,€1, E > 0. Note that U ( 6 ,t ) is piecewise analytic, and its analyticity may break down only at points of JO (see Tadmor [25]). System (7.4.1) is strictly normal on some interval [0, c) if for any integers rj
2 o satisfying
M
Crj = n,
j=1
the vectors Qij(6)
j = l , ... ,m, s E [ O , C ] - J O , O s i < r j - 1
are linearly independent. (If rj = 0, there are no terms Here, as before in (7.1.10),
Q,j
(7.4.4) in (7.4.4).)
N Qltj(S)=CA;Qb-lj(S-~j), j=O
k=1,2,
SE(-~,W),
j = 1,... , m ,
and B = (bl . . . bm).
It follows from Theorem 7.1.4 that a complete, strictly normal system is normal, and has rank B = min[m,n]. Indeed, choose any column bj of B and set r, = n , r, = 0 for i # j in the definition. Clearly Qoj(s),
Qn-l.i(s)I
s
E [O,c] - JO
are linearly independent. Because of Theorem 7.1.4 we have normality since bj is arbitrary. The second assertion is obvious since bj is linearly independent and we can take rj = 1 or rj = 0. If the system is an ordinary one, (7.4.5) i ( t ) = A o t ( t ) Bu(t),
+
our definition reduces to that of Htijek [31].
Synthesis of Tame-Optimal and Minimum-Effori Control
239
Lemma 7.4.1 (Fundamental Lemma). System (7.4.1) is strictly normal if there exists E > 0 with the following property: for every n-vector c # 0 and in any interval of length 5 6, the sum of the number of roots counting multiplicities of the coordinates of the index g of the control system (7.4.1) g ( t , c ) = CTU(E,t)B
is less than n . Proof: Suppose that such an E > 0 exists and (7.4.1) is not strictly normal on [Ole)
m
- Jo. Then there exist integers P j 3 0 such that C rj j=1
Qkj(iT),
(1 I j
= n, and
> 0, i = 1 , 2 , . - . Hence there exists c # 0, c E Ensuch that
5 m,
are linearly dependent.
CTQkj(iT)
0 _< k
= 0,
(1
5 rj-i),
E -
it
5 j 5 m,o 5 k. 5 rd-1).
(7.4.6)
Supposeaisazeroofgj(t,c) = ~ ~ ~ ( ~ , t () j b= j I ,, . . . , m ) o n [ o , E ) - J ~ . Then analyticity yields
where g j is expressed as the Taylor series at a particular time ( t = a+, or t = a - ) , and g ~ k ) ( Q +C)l
k = 0 , 1 , 2 , ... ,
-(u+,c), dkg
dtk
etc. It now follows that 00
0=
(1 A g j k ) ( u ,c)-
k=O
- Q)k k!
'
where Agjk) z g j k ) ( u - , )- g ! ( u + , ) . If u = 6 - ih > 0, W
( t - (€ - i - T))k
Agj(')(~- i ~ )
0= k=O
E( W
0=
k=O
i = 1,2, ... , then
k!
- (€ - ih)) - l)kcTQkj ( i ~ ) k!
1
(7.4.7)
(t
9
240
Stability and Time-Optimal Control of Hereditary Systems
where we have used (7.4.7). It follows from (7.4.6) that (7.4.7) becomes
( t - (€ - T i ) ) k 0 = [t - ( E - i ~ ) ] "C ~ ~ & k + ~ ~ , j ( i ~ ) k=O ( k + rj)! ' 00
Hence for each j = 1,. .. , rn, t = E - iT is a zero of g j ( t ) of multiplicity rj , and consequently the sum of the number of roots in [0,4 - JO counting multiplicities of the coordinates of g ( t ) = c*U(€,t)B is at least n. This completes the proof.
>
It is very desirable to have information on the largest that has the property of the Fundamental Lemma. Results on this do not seem to be currently available. It is conjectured that the converse of the Fundamental Lemma is valid. In all that follows, we assume as basic that (7.4.1) is strictly normal and (7.4.3) complete in the sense that the operator W ( t ,u) : C 4 En,defined by W ( t , c ) 4= z(u,4,O)(t), where z ( u , ~O)(t) , is a solution of (7.4.3), is a surjection: W ( t , a ) C= En. We shall also retain
E
as that described by the Fundamental Lemma.
Let k 5 n. Suppose there are distinct times t l < * . < t k with tk - t l 5 E and integers t i among 1 , . . . , m. If (7.4.1)is strictly normal, the vectors U(f,tl)bll, * . . I U ( f , t k ) b l , Corollary 7.4.1
are linearly independent.
Proof: Since a subcollection of linearly independent vectors are also linearly independent, we shall prove the statement for k = n. Suppose the
assertion is false, there exists c
# 0, c E En such that
Thus the sum of the number of roots, counting multiplicities of the coordinates of g i , is at least n in [ t l , t l €1; and this is a contradiction.
+
Corollary 7.4.2 Suppose (7.4.1) is strictly normal, and 0 t , < E , any n terms among rti
< t l < ... <
24 1
Synthesis of Time-Optimal and Minimum-Egod Control
are linearly independent. Proof: Suppose that statement is invalid. Then there exists a nonzero vector c E E n that is perpendicular to n terms among
Observe that t = 0 is a root of M ( t ) , so that the sum of the number of roots in
[O, €1 of
Jd'
a ( € ,t ) ,Bdt
+
is at least n rn. The first derivative of this integral is ~ ? U ( Ct,) B , so that its coordinates have at least n zeros in [0,€1. Corollary 7.4.3 Assume that (7.4.1)is pointwise complete and strictly normal. Then every optimal control on an interval of length less than c has at most n - 1 discontinuities almost everywhere. Proof: Consider the interval [&,&I, with 0 5 81 < 82 and 82 - 81 < c. If u is an optimal control on [01,82], then by Theorem 7.1.3, for some c#O, CEE", ~ ( t=)sgn c T u ( c , t ) ~
almost everywhere on [61,f?2]. It follows that the points of discontinuity of u are among the roots of the coordinate of cTU(c,2)B. This is at most n - 1 because of the Fundamental Lemma.
Definition 7.4.2: Let k be an integer with 1 5 k
5 n, and let
and for the k-tuple i = (u1 . . . uk) of vertices of C"' with uj-1 the mapping Fki : -+ Int %(€)
# u j , define
by
(7.4.8b)
Definition 7.4.3: The minimum time function M : 1+ E' is defined by
242
Stability and Time-Optimal Control of Hereditary Systems
where x ( t , 6,4, 0) is the solution of N
i(t) = c A j z ( t -
Tj),
20
= 4,
(7.4.9)
j=O
and R =
U R(t).
Suppose
t>o
0 =to
< i!l < . . . < t k < 6 . Then M ( z ) = t k .
Definition 7.4.4: Let k be any integer 1 5 k 5 n . For any (optimal) control u defined by U ( S ) = U j on [ t j - I , t j ) for ( t i , .. . , t k ) E Q k and U j - 1 # u, in C" , we define the sequence (211 --+ . . . + U k } its (optimal) switching sequence. Corollary 7.4.4 Suppose 1 5 k 5 n . Suppose that 4 E C, 4 # 0 is steered to the Euclidean origin in time t k by an optimal control u whose switching sequence is given by (u1 + . U k } . Let --+
Vi
= B ( U i - U i + l ) for i = 1,.. . , k - 1,
vk
=BUk.
Then the vectors U(tk,ti)vi>
(1 _< i
5 k)
are linearly independent.
Proof: T h e statement is trivially valid for k = 1. We assume k
2 2 and
use induction on k and Corollaries 7.4.1 and 7.4.2 to prove the corollary using the arguments corresponding to [32,pp. 12-13]. Now recall that in Section 7.2 we proved the following. Theorem 7.4.la
The minimal time function
M:E"--+E is continuous if (7.4.1)is Euclidean controllable. Consequently,
is continuous.
S y n t h e s i s of Time-Optimal and Minimum-E8or-t Control
243
Lemma 7.4.2 Let Q k be as defined in (7.4.8a), and F k i as defined in (7.4.8b). If i = (u1 -+ + . u k } is an optimal switching sequence, then the Jacobian matrix of F k i ( t ) = F k i ( t o . . . t k ) has rank k at each point of Q k . Fki is an analytic function of its variables. -+
Proof: We observe that
tl
1”
U(tk,t)BuldS+
-k * . , $
lk
U(tk,i)B(212
- 213)dS
U(tk,t)BUkdS,
which is analytic. It is clear that the column vectors of its Jacobian matrix are constant multiples of U ( t k , t ) B u k , or
These vectors are linearly independent since Corollary 7.4.4 is true. Therefore the rank is k. If u is an optimal control on an interval [ O , t l ] , t 1 E 6 , then its values are extreme points of C”’, the m-dimensional unit cube. It is therefore bang-bang and piecewise constant. Since Corollary 7.4.3 asserts that 21 has at most n - 1 discontinuities, if there are exactly k discontinuities, k 5 n- 1 at t i . . . t k With t o = 0 < t i < . . . t k < 0, U ( l ) = u k for t E [ t k - l , t k ) , and if these switching times are changed, but the switching sequence (u1 --* - - . u k } is retained, then the resulting control is still optimal. This is the content of the next theorem.
-
Theorem 7.4.lb Let i = (u1 -+ . . + U k } be a switching sequence, and such that for some t E Q n , x 6 En is given by
x =Fni(t) =
n
j=l
with t o = 0
1
tj
U(tn, s ) B u j d s
tj-1
< t i < .. . < t , < c; M ( x ) = t,.
Then for any
Stability and Time-Optimal Control of Hereditary Systems
244
has T ( y ) = s,. Proof: Let
F = { r e Q n: z =
klJ:,
U ( r n ,s)Bu(s)ds has M ( r ) = r n } .
j=1
By assumption, t = ( t l , . . . , t n ) E F , so that F is nonempty. If we prove that F is both open and closed in &nl then F = Qn.Let a sequence T~ in E converge to r E Q,. Suppose
Since rp + r , we have xp --P z . Now rp E F , so that M ( x p ) = Tp,n and from continuity of M , M ( z ) = r,. Hence r E F . We have proved that F is closed in Q,. We now assume that F is not open in Qn and deduce a contradiction. Because of our assumption, there is a sequence T~ E Q, - F and a T E F such that rp ---* T . Let
From rp + r we have x p + z in Int R ( 0 . As a consequence of this, xp can be driven to the Euclidean origin by an appropriate optimal control (7.4.10) kp
5 n,
t p , = ~ 0
< t p l , . .. , t p , k , <
€7
M(Zp)
= Tp,kp.
Clearly vpj are among the vertices of C". Because C" has finitely many vertices and 1 5 kp 5 n , we can take kp = k and up, = v ,, independent of p . Also since 0 < t,,, < E , by taking subsequences we can assume t P j --+ U j . Because x p + z , we conclude that
Synthesis of Tame-Optanzd and Minimum-Effort Control
245
Because optimal controls are unique, we have k = n, v, = uj, and u, = rj. Therefore Fni(rp)
= z p = Fni(tp),
Tp,j
+
rj
+
tp,j*
Invoke (7.4.10) to obtain
M(xp) = tp,n
(7.4.11)
rp,n-
But if Fni corresponds to the optimal switching sequence {UI + ... + un}, then Fni is locally one-to-one by Lemma 7.4.2. This is true in a neighborhood of r. But then rp,j + rj E t p , j . Therefore for sufficiently large p , rP = t p , so that in particular rp,n= tP,+.This equality contradicts (7.4.11). The proof is complete. We shall now follow the development of the report in Section 5.2 and generalize Yeung’s thesis in [32] and [33]as follows: Theorem 7.4.2
Consider the system N
i ( t ) = A,z(t) + C A j z ( t - ~
j
+) Bu(t),
(7.4.1)
j=1
and assume: (i) System (7.4.1) is Euclidean controllable. (ii) System
(7.4.2) is uniformly asymptotically stable. (iii) System (7.4.1) is strictly normal. Then there exists an E > 0 and a function f : Int R(E) + Em that is an optimal feedback control of (7.4.1) in the following sense: If N
i ( t )=
C A j ~ (-t ~ j+ )B f ( Z ( t ) ) ,
z
E Int R ( E ) ,
(7.4.12)
j=O
then the set of solutions of (7.4.12) coincides with the set of optimal solutions of (7.4.1) in Int JR(c). Also f(0) = 0 for z # 0, f ( z )is among the vertices of the unit cube U . Furthermore,’ f ( z )= - f ( - 2 ) . If m 5 n, f
246
Stability and Time-Optimal Control of Hereditary Systems
is uniquely determined by the condition that optimal solutions of (7.4.1) solve (7.4.12). Proof: We follow closely the treatment of Section 5.2, and define terminal manifolds Mki as follows. Definition 7.4.5: For each k = 1 , . . . , n and each switching sequence i = {ul + ... ---* u k } of vertices of C M with uj-1 # uj, the terminal Mkj is defined to be the set of points in Int R ( E )whose optimal controls have i as optimal switching sequence. We set MO = (0). The set of points in Int R ( E )whose optimal controls have exactly k - 1 discontinuities is designated as Mk, and the switching locus is
u Mk.
n-1
k=O
The following proposition is easily proved as is done for ordinary systems in Proposition 5.2.1.
Proposition 7.4.1 (i) Mk =
u
jEIk
Mkjl
Int R ( E )=
u Mk, and these are disjoint unions. n
k=O
(ii) There are exactly 2M"-' nonvoid sets Mni. (iii) Each nonvoid Mki is an analytic k-manifold of En, and M,, is open and connected. (iv) Int R(c) is an open and dense set that is the union of 2rn"l-I disjoint connected nonempty open sets, and a finite number of analytic kmanifolds, 0 5 k 5 n - 1, one for k = 0, and at least two for k 2 1. With this proposition, one proves the following: Proposition 7.4.2 Suppose on an interval [O, 01 - JO with 0 < 0 < E , i = ( ~ 1 ... u k } is an optimal switching sequence. If t~ has the same switching sequence i, but not necessarily the same switching times, then v is also optimal. Also if i = (u1 -+ . . . -+ U k } is an optimal switching sequence corresponding to Mk,i and ( u j + l -+ ... + u k } corresponds to M k - j , i J' = 0,. . . , k - 1, with Moj = M o , then if an optimal solution in Int R ( E )meets Mk,j then thereafter it meets only M k - l , i , .. . ,Mo in this order. We now use these preliminary results to prove Theorem 7.4.2. Set f (0) = 0. Let 4 # 0 be an initial point that is driven to 0 E En in time t 5 E , so that t = t ( t , 4, 0) E Int R ( E ) , -+
-+
where x ( t , 4,o) is a solution of (7.4.3). Now find Mki containing x. Suppose the corresponding optimal switching sequence is {u1 + . . . + U k } , set
Synthesis of Tirne-Optimal and Minirnum-Egod Control f ( x ) = 21. Because
fbfk
=
u
Mk,,
j Elk
Id R(r) =
u n
Mk,
k=O
247
are disjoint unions,
f is well defined. Let x # 0 be in Int R(c), and let z(+) be the optimal solution of (7.4.1)in Int Ips(€) through x . Suppose the optimal control U ( S ) of x is given by U(S) = uj on [ t j - l , t j ) with
0= Then
2
< ti <
< t k = M ( 2 ) < c.
belongs tosome Mki. It follows from Proposition 7.4.2that f(z(8)) =
u(s), proving that any optimal solution of (7.4.1)in Int R(c) is also a solu-
tion of (7.4.2)or (7.4.12). We indicate a proof of the converse. Just as in [32],if y : [0,8]-* En is a solution of (7.4.12)with 0 < 6 < c, and if y(t) E k f k j and 0 _< t < 8, then for s - t 2 0 sufficiently small y(s) E Mkj. The proof of this assertion is given by induction on n - E . For E = n , the assertion is true because y(-) is continuous and Mni is open in En.Details of the consequence of the inductive assumption is as outlined in p, p. 791 for ordinary systems. With this proved, it will follows that f ( y ( . ) ) is constant on any interval on which y(.) is differentiable. Because of Proposition 7.4.2, y(-) coincides with the optimal solution through y ( t ) . The rest of the proofis as in Theorem 5.2.1. Remark: In [32,Theorem 4.11 HAjek’s Proposition 12 [lo] was used. An analogous result is valid in our situation when (7.4.1) is normal: Proposition 7.4.3 Every optimal solution of a normal system (7.4.1),on [0, +m) can be extended to an optimal solution on (-m,oo). The proof is analogous to H&ek [lo, p. 3461. See Hale [8, p. 681.
7.5
Minimum-Effort Control of Delay Systems
In Sections 7.2 - 7.4 we studied the time-optimal control of linear delay systems. In this section we study the minimum-effort control of the system N
+ C Ai(t)x(t- hi) + B(t)u(t),
i ( t )= A o ( t ) ~ ( t )
(7.5.1)
i=l
where 0 < hl 5 h2 5 . . . h ~= h , Aj i = 0, ... , N are n x n analytic functions, and B is an n x m real analytic matrix function. The controls u are bounded measurable functions. We consider the following problem:
Stability and Time-Optimal Control of Hereditary Systems
248
Find a u (subject t o some constraint, i.e., u E V ) that minimizes an “effort” function E(u(t1))subject to (7.5.1), where
Here I = I(.,u,$,u) is the solution of (7.5.1) with We identify the following effort functions:
I~
= 4.
(7.5.3a) where R ( t ) is a continuous positive definite m x m matrix. When R(t)E I, the identity matrix, then
(7.5.3b) (7.5.4) where
5
1, j = 1 , . . . , m ,
u 5 t 5 tl}. (7.5.5) In (7.5.4) E l ( u ( t 1 ) )describes the maximum thrust available to the system u E U = {u E Em, u measurable Iuj(t)l
where p
> 1, and
u E U = {u E Em, u measurable Ilu)Ip= Ez(u(t1))5 1).
(7.5.7)
If p = 2 in (7.5.6), then Ez(u(t1)) = IIu112 represents the energy or power of the system, and this is to be minimized. (7.5.8) In (7.5.8) we may sometimes assume a constraint u E U = (u measurable, u E Em, luj(t)l
5 1).
(7.5.9)
Synthesis of Time-Optimal and Minimum-Eflort Control
249
Without constraints (7.5.9) in E3, the optimal controls are impulsive in nature. With constraints (7.5.9), the optimal controls are “bang-of-bang”, as we shall see. T h e solution of (7.5.1) in the state space Enis given by the variation of parameter
(7.5.10) where U ( t ,s) is the fundamental matrix solution of N
i ( t )= A o X ( t ) 2,
+ C A i ( t ) ~ ( t- hi), j=1
= 4 E c,
with
V ( t ,s) = I , t = s,
(7.5.11)
U ( t ,s) = 0 s > t .
In (7.5.10), set Y ( t ,s) = V ( t ,s ) B ( s ) .
(7.5.12)
This is an n x rn matrix function which, because B is analytic, is piecewise analytic in 2 , s [ll]and at least measurable in t , s [8, p. 1451, and continuous in t for t 2 s for each fixed s. We now state the solutions of the minimum-effort problems for the various efforts.
Theorem 7.5.1 Assume that (i) (7.5.1) is Euclidean controllable on [a,tl],and this holds if rankQn(tl) = n,
(7.5.13)
where
Qn(tl)
with
Qi
= {Qo(s,tl),.*.
,Qn-l(S,tl),
s E (o,tl]}
defined by tlie determining equations
Qo(s,t) =
{ B(t), t
E [n,t1] f o r s = hi, other wise.
(7.5.14)
Stability and Time-Optimal Control of Hereditary Systems
250
(ii) Let
tl
Y ( t 1 ,S)R--'(S)[Y(tl,S)ITds.
(7.5.16)
The control u* is defined by
u*(t)= R - ' ( t ) ( Y ( t l , t ) T ) W - l q , where q
E [o,t1],
= [ t l - z(t1, o,4,0)1
(7.5.17) (7.5.18)
is the optimal control that minimizes Eo(u(tl)),i.e.,
Proof: Because (7.5.13) holds, W-' exists and u * ( t ) is well defined. If we use u * ( t ) in (7.5.10), we obtain
+J
X ( t l , ~ , d , , U= * )~(t1,(7,d,,O)
tl
Y(tl,s)R-'(s)(Y(t,,s))TW-lqdt,
0
= 2 ( t l , B , d , O )+ q = 2 1 . Thus, indeed Y * transfers to 11 in time t l . That u* minimizes Eo follows standard arguments. Indeed, let 7i be any other control that transfers d, to 11 at time t l . Since u* in (7.5.17) also steers the system from d, to 2 1 , we have the equality
l1
U ( t 1 ,s)B(s)u'(s)ds =
l1
U(t1,s)B(s)E(s)ds.
Using the inner product on both sides of this equality, we obtain
(J,
tl
Y(tl,S)(E(S)- u * ( s ) ) d s , w - l q ) = 0 .
Using (7.5.17) and the properties of the inner product, we obtain J,"(E(.)
- u*(s)),
U*(S))dS
= 0.
We now use this equality to derive
Eo(u*(tl))=
1''
u*(s)R(s)u'(s)ds
n
5
?i(s)R(s)E(s)ds= E(E(t1)).
Synthesis of Time-Optimal and Minimum-Effort Control
25 1
This completes the proof. Remark 7.5.1: We can easily show that
Observe that there are no constraints on u except that it is measurable and integrable. The controllability assumption enables one t o infer that I$ is steered to 2 1 in time t l . The rank condition is available in Manitius [6]. If the controls are constrained to lie on a bounded set U ,then some stability condition on (7.5.11) is required. For the solution of the minimum-effort problem with effort defined by E l ( t ) in (7.5.4), set (7.5.19) Y ( t )= 2 1 - z ( t ,g , I$,O), and call it the reachable state. Let
g(t1,c) = C T Y ( t l , t )= CTU(tl,t)B(t),
(7.5.20)
where c is an n-dimensional vector. Note that g is an rn-vector. Suppose
(7.5.21)
Theorem 7.5.2 In (7.5.1)assume: (i) that (7.5.13)holds. (ii) The system (7.5.11)is uniformly asymptotically stable. En defined by (iii) Also (7.5.11)is complete, i.e., the map T ( t ,u ) : C
where x(t,u,q5,0) is a solution of (7.5.11),satisfies
T ( t , a ) C= E".
(7.5.22)
(iv) The system (7.5.1)is normal and this holds if for each j = 1,... ,m, the matrix
252
Stability and Time-Optimal Control of Hereditary Systems
has rank n where Q,j is defined by
k = 1 , 2 ,..., n - 1 ,
and B = ( b l .. . b j
tE[a,tl],
. . .bm).
(7.5.23b)
Then there exists a minimum-effort control u * ( t ) such that E l ( U * ( t l ) )I E l ( 4 t l ) ) ,
v 21 f u,
with U defined in (7.5.5)subject to (7.5.1) and (7.5.2),with 11 E 0. Furthermore, if y(t1) # 0, the minimum-effort Elmin= Ei(u'(t1)) is given by 1 -- - rninFl(c,tl), (7.5.24) El min ~ E P where P is the plane c T y ( t l ) = 1. The optimal control u * ( t )is unique almost everywhere and is given by
u*(t)= El minsgn g ( t , c * ) ,
(7.5.25)
where c* is any vector in P for which the minimum in (7.5.24) is attained. If y(t1) = 0 , the control u * ( t ) z 0 is the m'nimumeffort control. Theorem 7.5.3 define
If the effort function Ez(u(t1)) is as defined in (7.5.6), (7.5.26)
1 -+
1
- = 1. For (7.5.1)assume conditions (i) - (iv) of Theorem 7.5.2. p q Then for each 4 E C, 0 = x1 E E n , and some t l , there exists an optimal control u*(t) that minimizes E z ( u ( t l ) ) ,ie., E z ( u * ( t l ) )5 Ez(u(t1))for all u E U defined in (7.5.7),subject to (7.5.1) and (7.5.2). Furthermore, if y ( t 1 ) # 0, the minimum-effort E,min= E 2 ( u * ( t l ) )is given by where
(7.5.27)
Synthesis o f Time-Optimal and Minimum-Effort Control
253
where P = {c E En : cTy(tl) = 1). The optimal control u* is unique almost everywhere and is given by uj*(t) = pIgj(t,c*)I1'Psgn gj(t,c*),
(7.5.28)
where P
= ~ z m i n [ ~ z (c*)]-~'P, tl,
and C* 6 En is any vector in P where the minimum is attained. Ify(t1) = 0, the minimum-effort control is u * ( t ) E 0. The solution of the minimum-effort problem when E 3 ( ~ ( t l ) is ) as defined in (7.5.8) (and it is not constrained) does not exist among integrable functions. If impulsive functions, the so-called Dirac functions, are admissible, then an optimal solution exists. These observations are contained in the next theorem. Theorem 7.5.4 Consider the minimum-effort problem with effort function &(U(tl)) defhed in (7.5.8), where u E U = { u : 11~1115 1) is defined in (7.5.9).Assume that for (7.5.1),the following conditions exist: (i) (i) - (iii) of Theorem 7.5.2. (ii) Assume metanormality for (7.5.1), i.e., rank Gn+lj = n, for each .i= I , . . . , m , where Q n + l j = {Qoj(s,tl)...Qnj(s,tl), s E [a,tl]) with Qij defined in (75.236). Then there is no optimal solution with u an integrable function that steers 4 (unless z(t1, a,q5,0) = 0) to 0 in some 2 1 , while minimizing E 3 ( ~ ( t l ) ) .But if impulsive controls are admissible and
(7.5.29) then there exists a minimum-effort control u * ( t ) if y(t1) minimum-effort is given by E 3 ( ~ * ( t l )= ) E 3 m i n and
# 0. The (7.5.30)
where P = { c E E" : cTy(tl) = 1). The optimal control is given by u*, where
(7.5.31)
254
Stability and Time-Optimal Control of Herzditary Systems Here 6 ( t ,-rji) is the so-called Dirac delta function. The suprema in (7.5.29) may occur at multiple j and a t multiple instances of times r-i, i = 1 , 2 , . . . , N , , whereNj equalszeroifg, doesnotcontain the suprema. Thus, rji E [a,t1] are the finite number of times a t which lgj(t1,c*)l= F3(ti c*). 9
In Theorem 7.5.4, the components of the controls are not bounded. If
U = {U measurable ~ ( tE)Em (uj(t)l 5 1,
j
= 1,. .. ,m,
(lull1
5 a},
then the optimal controls are “bang-of-bang.”
Theorem 7.5.5 Assume in (7.5.1) the following: (i) Conditions (i) and (ii) of Theorem 7.5.4. Suppase the problem is to minimize 11u111 subject to u E U and (7.5.1) and (7.5.2). Then there exists a unique minimum fuel control u * ( t ) . This control is “bangof-bang” in the sense of having only the values fl,and 0, with no switches +1 to -1 or back (however, 1, 0 , l is possible) unless a = rntl and y(t1) t a R ( t , ) where
R(tl) =
{l1U ( t ,
s)B(s)u(s)ds: llU1lm
}
51 .
(7.5.32)
Optimal control is given by u* where
(7.5.33) where
rj
2 0 is some constant.
To prove the above theorems on minimization of effort, we recall the variation of parameter in (7.5.10) and Y in (7.5.2), and define the function
which maps the control space L into the state space En : St : L -+ En. This map is continuous and linear with & ( L ) = 0. If U c L is the control constraint set, then the reachable set is the set
R ( t ) = {S,(u) : 21 E U } =
{l
Y ( t , s ) u ( s ) d s: u E
1
u .
(7.5.35)
Synthesis of Time-Optimal and Minirnum-Eflori Control
255
Thus if y ( t ) is a reachable state defined in (7.5.19), i.e., y ( t ) = t ( t , B, 4,0), the coincidence
21
-
for some u E U ,implies that y ( t ) E R(t), and therefore 4 can be driven to t l by the control u E U . We hit 2 1 in minimum time t* if
t* = Inf{t E [O,tl]: S,(u) = y ( t ) for some
z1
EV}.
(7.5.37)
The time-optimal control u* E U is such that y ( t * ) = S p ( u * ) , The minimum-effort control is the admissible control u* E U such that Stl(u*)= d t l ) , with E ( u * ( t i ) )5 E ( u ( t i ) ) , V 11 E U. In what follows we assume L is either L , or L p , 1 5 p < 00. In this case, the map St : L 4 En has defined in (7.5.34) its adjoint as S; : E" -+ L , represented by
If L = L ,
I
S,* maps En into L1([0,t l ] ,Em)c L , so that
(7.5.39)
E", where ] I . by (7.5.38) with
cE
111 is
the L1 norm in Em. If L = L p , then S: is still given
(7.5.40) for c E En. We now impose some conditions on St : L -+ En and then show what conditions on the system's coefficients ensure that the assumptions hold.
Prevailing Assumptions: I. The reachable set is closed.
R(2) = {St(u) : u E U }
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Stability and Time-Optimal Control of Hereditary Systems
With U in (7.5.5) or (7.5.7), the closure of R(t) is automatic and is proved using weak compactness argument. See also Proposition 7.1.1. 11. The map t + St, t E [0, T ] is continuous with respect to the operator norm topology of 1(L,E n ) ,the space of bounded linear transformations from L into En.See Proposition 7.5.2. 111. The function y :[ ~ , t + l ] En defined by Y(t)
= Z l ( 4 - z ( t ,r,d,O)
in (7.5.19) is continuous. Here z l ( t ) is a continuous point target. In the next assumption we maintain it is constant. IV. y : [ a , t l ]-+ En is constant and not zero: y ( t ) = y1 for all t E [ u , t l ] . V. For each c E En,c # 0 the function t -+ I]S;(cT)]] is strictly increasing. This condition is guaranteed by the following condition: For each
r 1 , E~ [ u , t l ] , c T U ( t l , t ) B ( t )= 0,
VtE
if and only if c = 0.
[~1,72]
(7.5.41)
This condition (7.5.41) is equivalent to Euclidean controllability. See Manitius [37,pp. 77-86]. In terms of the system’s coefficients, Euclidean controllability is assured by the rank condition (7.5.13). VI. The system (7.5.1) is normal in the following sense: S;(cT)is not , c # 0. identically zero for t E [ u , t l ] and In view of (7.5.38), we have that System (7.5.1) is normal on [ a , t l ]if for each c E E n , c # 0, and each j = 1 , . . . , m , the set
{t > u : C T U ( t l , t ) b j ( t )E 0, t E [ U , t l ] ) has measurable zero. In this case, llSr(cT)II> 0 in (7.5.39) or in (7.5.40), for each c # 0, t > u. Conditions on the systems’ coefficients for normality are given in (7.5.23). We state and prove this for the autonomous simple system i ( t ) = A o ~ ( t ) A l ~ (t h) B u ( ~ ) , (7.5.42)
+
+
where the “determining equations” are given by Qkj(S)=AOQL-l(S)+A1Qk-1(S-h),
k:= 1 , 2 , 3 , . - -, s E [-m,m), (7.5.43)
Syn t h es is of Time- Opti i n a 1 a n d Min im IJ m- Effo rt
Con1ro 1
257
Set
-
Qnj(t1) = {Qoj(s),Qlj(s),... ,Qn-lj(s),
6
E [o,tlJ}.
(7.5.44)
Theorem 7.5.6 The system (7.5.42) is normal on [O, t l ] if and only if for each j = 1 , . . . , m , rank Q v j ( t l )= n.
Proof: This is an easy adaptation of Manitius [37,p. 791. Because of its utility in subsequent discussions we shall outline it in some detail. We note that s + U(t1,s) is the fundamental matrix solution to the adjoint equation:
The function gj (s,c>= cT ~
(1 7 s)bj t t
the j t h component of the index of the control system, has the following properties: s + g,(s,c) is defined on [0, w); it vanishes on (t1,oo) and is piecewise analytic in (0, m). The isolated exceptional points are at points s = t1,tl h , t l - 2h,. . . ,tl - hi, where i is such that t1 - ih > 0 (see Tadmor[ll]). Define
-
A g f ( t , C ) = gf(t - 0 , C ) - $ ( t
+ 0 ,c ) ,
t E ( 0 ,OO),
(7.5.46)
+
and we have designated g f ( t - O,c),g;(t 0,c) as the left- (respectively right-) hand side limit of the kth derivative of g j at s = t , k = 0,1,2,. . . . Clearly.
(7.5.47) and the jump at t l occurs because of the jump discontinuity of the fundamental matrix U(t1,s) a t s = 11. Also
(7.5.48) otherwise. We can prove by induction that, in general,
Agjk)(t1- ih) = (-l)kcTQkj(ih) :
i = t1 - ih > 0.
(7.5.49)
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Sfability and Time-Optimal Control of Hereditary Systems
Assume that rank G n j ( t r )= n , but (7.5.42) is not normal on [0 t l ] . Then there is a c # 0 c E En such that for some j = 1,... ,rn, g j ( s , c) = 0 on [0, t l ] . Because of this g j ( s ,c) E 0 on [O,m],so that on differentiating,
- ih > 0 = (-l)kcT&k(ih), for k = 0, 1,2, . . . , i = 0 , 1 , 2 , . .. ,tl - ih > 0.
0 = cTAgk(tl - i h ) , t l
We deduce that c E E" is orthogonal t o all the vectors of g n j ( t 1 ) ,which is a contradiction. We have proved normality. We now prove the necessity for this. The constant coefficient case is proved exactly the same way as was done in [37, p. 801. For the general situation of (7.5.1), it is far from clear whether the rank condition is necessary. The next prevailing assumption on (7.5.1) is that of metanormality. The notion was introduced by Hfijek [35, p. 4161. VI. The system (7.5.1) is metanormal on [ u , t l ]if and only if every index g(t, c)
= CTV(tl,V ( t )
with c # 0 has each component g j ( t ,c ) constant only on sets of measure zero: The set
{ t > o : C*V(tl,t)bj(t)= 0) has measure zero for each column b j of B , constant cr E E , c # 0 in E". Though metanormality is a condition on S;, we characterize it as the full rank of some of the systems parameters. We restrict our discussion to (7.5.42). Lemma 7.5.1 The system (7.5.42)is metanormal if and only if each j = 1 , . . . , m ,r a n k { Q l j ( s ) ,. . . ,Qnj(s), s E [O,tl)} = n where Q k j is as defined in (7.5.43).
Proof: Suppose a j t h coordinate $ j ( t , c ) = c*V(tl,t)bj
Q,
cr a constant on a set of positive measure. Then t -+ g,(t,c)-cr 5 M ( t ) = 0 on a set of positive measure. Since it is piecewise analytic except at the points s = t 1 , t l - h , t l - 2 h , . . . , t l - ih where i is such that tl - ih > 0, the function m ( t ) E 0 on [0,m). Hence it follows that
0 = AM'(t1 - ih), t i - ih
= (-l)'~'Q'j((ih)
> 0,
(7.5.50)
Synthesis of Tame-Optimal and Minimum-Effort Control
259
for k = 1 , 2 , . . . ,n, a' = 0 , 1 , ... , t l - ih > 0. Hence c is orthogonal to all vectors & l j ( s ) ,... , Q , l j ( s ) s E [O,tl]. Therefore (7.5.42) is not metanormal. We now prove necessity. We show that metanormality implies that for each j = 1 , . . . , rn rank Qoo, = n where Qoojdenotes a matrix with infinite number of columns given by Q l j ( s ) ,Q 2 j ( s ) ,. . . , S E [0, t l ] ;and the assertion rank Q-j = n means that there are n linearly independent columns in the sequence {Qk.j(s), s E [O,ti), k = 1 , 2 , * * . ) . Suppose this is false. Then there is a c
# 0, c E E"
such that
(7.5.51 ) Since Ao, A l , B are constant,the function
is piecewise analytic except at isolated points of [O,t1],which are seen to be t 1 , t l - h , t l - 2h, etc. But g j ( t , c ) vanishes for t > t l , hence M ( t ) E g j ( t ,c ) - a -a for t > t1. It follows that M ( k ) ( t l + O )= 0 for k = 1 , 2 , . . . . Since (7.5.47 - 7.5.50) holds and AAdk)(t)= M(k)(t-O)-M(k)(t+O), 2 E (O,oo), we deduce that M ( k ) ( t l - O ) = g jk( t , - O , c ) = 0, k = 1 , 2 . Because g j ( t , c ) is piecewise analytic on [O,tl], M(1) f 0 on [tl - h , t l ] . In this same way we obtain M ( t ) 0 on [ t l - 2 h , tl - h ] , and by induction M ( t ) f 0 on [ O , t , ] , i.e., gj(1,c) = a on [O,tl]. This contradicts metanormality. To complete the proof it can be shown that rank Q - j = n implies rank Q n = n , where Q n j = [ Q l j ( s ) , . *,.Q n j ( s ) E [o,t1]1-
=
7.6 Proof of Minimum-Effort Theorems With the six general assumptions I - VI stated, and conditions for their validity in terms of the system's coefficients deduced, we are now prepared to state three preliminary results on which the solution of the problem of the minimum-effort control strategies are based. Theorem 7.6.1 In (7.6.1), let t E [ u , t l ] , and consider U in (7.6.5) or (7.6.7). Then there exists an admissible control u E U such that
Stability and Time-Optimal Control of Hereditary Systems
260
if and only if
CTY(t) I Ils;(cT)II,
v c E E",
(7.6.1)
T
where S; is the adjoint of St, a map of En to L. Proof: It is assumed that there is a u E U such that St(u) = y ( t ) . It follows that
Therefore (7.6.1) is valid. To prove the converse, we recall that the Euclidean reachable set R ( t ) is closed. It is also convex, being a linear image of the convex set U . If there is no u E U such that S*(u) = y ( t ) , then y ( t ) 4 R ( t ) . Because R(t) is a closed and convex subset of En,the Separation Theorem 14, p. 331 asserts that there is a hyperplane that separates R ( t ) and y ( t ) : This means there exists a c E E", such that
c T y ( t ) 2 sup{cT(S,(u)) : u E U } = sUp{sf(CT)(u) : 21 E U } . This invalidates (7.6.1). The assumption that there exists some u E U such that St(u)= y ( t ) is a statement on the constrained controllability of (7.6.1) at some t . The optimal (minimum) time t* for hitting the target is defined as t* = Inf{t E [ g , t l ] : S t ( u )= y ( t ) for some u E U } .
(7.6.2)
The admissible control u* E U that ensures the coincidence St*(u*) = y ( t * ) is the time-optimal control. For the minimum-effort problem, the optimal strategy u* that ensures that Stl(u*) = y ( t 1 ) while minimizing E ( u ( t 1 ) ) is the (minimum) optimal control. Inspired by the ideas of H&jekand Krabs [36],we propose the following:
Theorein 7.6.2 In (7.6.1) assume that the point target q ( t ) E En is continuous. Then there exists a c E E" with llcll = 1 such that
C T d t * ) = llSXcT)Il,
(7.6.3)
so that St.(CT)(U*)
= cT(S*. (u*))= c T y ( t * ) ,
(7.6.4)
where u* is the time-optimal control, and
Ilu*ll= 1,
(7.6.5)
Synthesis of Time-Optimal and Minimum-Effort Control
26 1
if the system is normal. Remark 7.6.1: Because z l ( t ) is continuous,
is continuous. The proof as pointed out in [36,p. 51 is a little adaptation of the argument in the proof of Theorem 6 in [34]. Because the next result links the time-optimal control and the minimum fuel strategy, it is described by HAjek and Krabs in another setting [36]as the Duality Theorem. It is simple but fundamental.
Theorem 7.6.3 In (7.6.1) assume null controllability with constraints. This is satisfied if (7.6.11) is uniformly asymptotically stable and (7.6.1) Euclidean controllable. If t* is the minimum time, then
t* = max{t E
( O , t , ] such that cTy(t) = IIS;(cT)((
for some c E En : llcll
= 1).
(7.6.6)
Proof: Because (7.6.1) is null controllable with constraints, we set q ( t ) 0; for each nontrivial 4 E C, y(t) = z l ( t ) - z ( t , u , 4 , 0 )becomes y(t) = - z ( t , u,q5,O); and if -z (t l, (T,4,O) # 0 and if we set y(t) z - z ( t l , 6,4, 0) = y1, V t 2 (T,then (IV) is satisfied. Because of Euclidean controllability, assumption (V) is satisfied. Since Theorem 7.6.2 is valid, the minimum time t* is a point over which the maximum in (7.6.6) is assumed. Suppose there is a t > t* such that
for some c E En with llcll = 1. Then
Th e first inequality follows from Theorem 7.6.1, the second from assumption V. The obvious contradiction proves our assertion. We now designate U to be as in (7.6.5) or (7.6.7), and derive from Theorems 7.6.1 - 7.6.3 optimal controls for minimizing effort and time.
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Stability and Time-Optimal Control of Hereditary Systems
Theorem 7.6.4 In (7.5.1), assume: (i) (7.5.1) is Euclidean controllable. (ii) (7.5.11) is uniformly asymptotically stable. (iii) (7.5.1) is normal. (iv) System (7.5.11) is complete. UU, = L,([O,tl],Cm), t * is the minimum time, and u* is the time-optimal control, then u* is uniquely determined by
for some e # 0, and each j = 1,. . . , m . If Up C
1,
Lp([~,t1], Em), p >
1 1 -+= 1 such that u E U implies [lullp 5 1, then the time-optimal
P
Q
controls are uniquely determined by
where g(t, c) = CTU(t*,t)B(t),
(7.6.9)
Proof: From Theorem 7.6.3, if u* is a time-optimal control and t* the minimum time, then s;.(cT)(u*) = cT(St*(u*)) = cTy(t*) = Ils;.(cT)II. But then the u* E U , is such that cTy(t*) =
l*
C T U ( t * , t ) B ( t ) U * ( t ) d t= cT
I"
U ( t * ,t)B(t)u*(t)dt
(7.6.1 1a) This implies that u* is of the form a.e. u j ' ( t ) = sgn[cTU(t*,t)bj(t)],
u 5t
5 t',
Synthesis of Time-Optimal and Minimum-Eflort Control
263
when c T U ( t * , t ) b j ( t )# 0 for c # 0, and each j = 1,e.O , m , which is true since the system is normal and (7.6.11) complete. For U p , we obtain from (7.6.3) in Theorem 7.6.2 that cTy(t*)
1 where P
t*
= [IS,'.
(?)I1
=
IlcTU(t*,t ) B ( t ) p d t ) t ,
I"
=
J,
CTU(t*,t)B(t)'U*(t)dt, (7.6.11)
Jo
+ -1 = 1. Since g ( t , c ) = cTV(t*,t)B(t), P
But then (7.6.11) is valid, i.e.,
and so we have equality everywhere in the above estimates. The control u* that gives the equality is (by inspection)
. This is the time-optimal control, and as observed'before, it is uniqueiy determined when the system is normal. The proof is complete.
Stability and Time-Optimal Control of Hereditary Systems
264
Proof of Theorem 7.6.2: Because of the Euclidean controllability and the stability assumptions, there is indeed a tl such that the solution 2 of (7.5.1) satisfies
We observe that in this case y(t) defined in (7.5.19), y(t) = zl(t)-z(t, u,4,O) = -z(tl,u,c$,O) y1 # 0 and y(t1) E R(t1). Observe that u* in (7.6.7) is a boundary control in the sense that if
Z(t1,C)=
tl
V(tl,t)B(t).*(t,C)dt,
then r(t1,c ) is on the boundary of the reachable set R(t),so that
and
cTz(t1,c ) > C T Y With y(t1) 3 follows: Let
y1
v Y E W(tl),
Y
# Z(tl,C)
E R ( t l ) , we can extend this to reach the boundary as a = m={P : Py(t1) E R(t1)).
(7.6.13)
Since y(t1) # 0, a can be assumed to be positive, and obviously ay(t1) is a boundary point of R(t1). This means that ay(t1) = z(t1,c)for some c, where (7.6.14) cTy(t1) = 1 = -cTz(t1, u,4,O). It is easy t o verify that the control (7.6.15) steers 4 t o 0 in time tl while minimizing E1(u(tl)).Also (7.6.16) where
a = min{I;;(t,c):
c E P = {c E En : cTyl = 1)).
(7.6.17)
Synthesis of Time-Optimal and Minimum-Effort Control
265
Details of the verification are the same as in the case of ordinary linear differential equations (see Neustadt [38]).We conclude that (7.6.18) is the minimum energy control, where c* E E" is vector in P on which the minimum in (7.6.17) is attained. Because u*(t) = sgn(g(t,c)), we deduce
that (7.6.19) where c* is the minimizing vector in (7.6.17). Normality ensures that '?i is uniquely determined. If y(t1) = y1 = 0, then the choice ii 0 is appropriate. This completes the proof of Theorem 7.6.2. Proof of Theorem 7.6.3: In this case, Up in (7.5.7) is used to define the reachable set
which, as we have noted earlier, is closed. Just as in the proof of Theorem 7.5.2, the time-optimal control u * ( t , c) in (7.6.8) is a boundary control that is uniquely defined. Thus if z(t1,c)
=
1
tl
U(t,,t)B(t)u*(t,c)dt,
z(t1,c) E dR(t1).
With a as defined in (7.6.13), it is easy to verify that the minimum-effort (7.6.20) where (7.6.21) and (7.6.22) and Fz(t1c) is as given in (7.5.26):
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Stability and Time-Optimal Control of Heredita y Systems
The vector c* is that which minimizes the function in (7.6.22). Because u*(t,c*) in (7.6.20) is given by (7.6.8), the minimum-effort strategy is given by Uj(t,C*) = Igj ( t ,c*) :Sgn(gj ( t , c*) 1 (7.6.23) a where
Through Theorem 7.6.2 and the general ideas of HAjek and Krabs [36], we can link up the time-optimal controls and the minimum-effort controls. In some situations we see that they are the same. We observe that IIS;(cT)II is given by (7.6.11) and it is to be minimized in (7.6.22) and (7.6.17) over the hyperplane P. We are led to the following definition: For each t E [0,t l ] let
a(t)= inf{llS,'(cT)II : c E P } where P = {c E En : cTyl = 1). (7.6.24) Theorem 7.6.5 Let the prevailing assumptions I - V hold for (7.5,l). Then for each t E ( O , t , ] , we have that ift* is the minimum time, then (7.6.25) Remark 7.6.2: Note that 1 is the norm bound of the control set U , and is defined in the minimum-effort problem in (7.6.2).
tl
Proof: Quite rapidly from (7.6.24), one has that 1 (7.6.26) cTy1 5 - ~ ~ S ~ ( C ~ V) c~E~En. , 4t) From an elementary approximation theory argument we can verify that for each t E (0,111 there exists some c(t) E P such that llsr(cT(~))ll= 4 t ) > 0.
Also there exists a
ut
(7.6.27) 1 a ( t ) . From
E L such that St(.,) = y1 and 11utll = -
(7.6.27) we argue that there exists a c(t) E Ensuch that
Invoke Theorem 7.6.1 and Theorem 7.6.2 to validate (7.6.25).
Synthesis of Time-Optimal and Minimum-Eflort Control
267
Theorem 7.6.6 Let the prevailing assumptions I - VI hold for System 7.5.1,in which (7.5.11)is complete. The optimal control u* that steers 4 E C to 0 in minimum time t* while minimizing El(u(t*)) is given by u * ( t ) = sgn[g(t,c*)],
d
5 t 5 t*,
(7.6.28)
where c* is the vector in P that minimizes (7.6.17),i.e., a = min F(t*,c ) CE P
(7.6.29)
and
(7.6.30) g ( t , c ) = C T V ( t * ,t ) B (t).
(7.6.31)
The optimal strategy that is time-optimal and minimizes Ez(u(t*)) is u* with componen t ti;(t>= klsj(t,c*)l:sgn gj(t,c*), (7.6.32) where
(7.6.33) with c* the minimizing vector in (Y
= min F1(t*,c), cEP
(7.6.34) (7.6.35)
Proof: By assumption, the feasible t l of (7.5.2) in the minimum-effort problem is the minimum time t*. But then
a(t*)= inf{llSy.(cT)II} = 1 CEp
by (7.6.25) in Theorem 7.6.5. Because the minimum fuel controls are the functions
-
u * ( t ,C * ) u ( t ) = -= U * ( t , C * )
ff(t*> by (7.6.18) or (7.6.20), the corresponding expressions in (7.6.7) and (7.5.8) for time-optimal controls of Theorem (7.5.4) prove that (7.5.8) and (7.5.32) are correct. The following fundamental principle is now obvious: In our dynamics (7.5.i), the time-optimal control i s also the control that minimizes the effort function.
268
7.7
Stability and Time-Optimal Control of Hereditary Systems
Optimal Absolute Fuel Function
The solutions of the minimum time/minimum fuel problems contained in Theorem 7.5.1, Theorem 7.5.2, Theorem 7.5.3, and Theorem 7.6.4 depend on the closure of the reachable sets. Since the reachable sets are also convex, optimal controls are boundary controls in the sense that they generate points on the boundary of R(t1). But if the effort function is defined by ES(u(t1)) in (7.5.8), as the absolute fuel function
and if U is defined by
U = {u measurable llulll 5 11,
(7.7.2)
then the reachable set R(t1) = I[{
U(t,,t)B(t)u(t)dt : u E
u
I
(7.7.3)
is not closed, but open. Because the reachable set is open, its boundary points can only be "reached" by convex combination of delta functions. Optimal controls that yield boundary points must therefore be delta or dirac functions, which are impulsive in nature. If we admit such functions as controls in absolute fuel minimization problems, optima1 controls exist. If we rule them out and R(t) is open, there is no integrable optimal control. We consider i ( t ) = Aoz(t) A l ~ (-t h ) Bu(t). (7.7.4)
+
Lemma 7.7.1
+
Suppose (7.7.4) is rnetanorrnal, i.e., for each j = 1 , . . . , rn,
where Qkj is as defined in (7.5.43). Then the reachable set R(t) of (7.7.4) is open, bounded, convex, and symmetric.
Proof: The symmetry, boundedness, and convexity of R(t) are easy to verify. We prove that R(t) is open. Assume on the contrary that IW(t1) is not open and t h at
Synthesis of Time-Optimal and Minimum-EBort Control
269
is a boundary point. Let c # 0 be an outer normal to (the closure of) R(t) a t y1. With this c, define the index of (7.7.4) g(t,c) = c T U ( t l , t ) B . Obviously, uo is a control that maximizes
l1
g(t, c)u(t)dt
But the mapping u( ) with norm
I
l"
--+
g(t,c)uo(s)ds whenever [lull1 5 1.
(7.7.6)
J"'g(t, c)u(t)dt is a linear functional on Ll([u,tl])
The inequality on (7.7.6) implies that the value llgllm is attained at the element uo of the unit ball in L1. Therefore
This shows that we have equality throughout, so that
It follows that (Ilgllm - lgj(t,c)l)- luoj(l)l = 0 a.e. for each j = l , . . . , m , t E [u,tl].
Because the system is metanormal, lgjl is constant (= Ilgllm) only on a set of measure zero. Hence, uo = 0 a.e. on [c,tl]. But with uo = 0, our boundary point is y1 = 0. This contradicts the following containment:
(7.7.7) whenever 0 < a < /3 5 m t l , which is an easy consequence of metanormality and completeness as can be proved by the methods of Hijek [35, p. 4321. Recall that (in Hijek's notation)
270
Stability and Time-Optimal Control of Hereditary Systems
and
Thus the reachable set R(t1) is open. Proof of Theorem 7.5.4: Because IW(tl), the reachable set using L1 controls with bound 1, is open, then
<
where k u = v is an open set. We conclude that if the controls u are measurable with llull 5 k, then the reachable set is k R ( t 1 ) for le > 0. Since this is open, the minimal k can never be attained unless y1 = - z ( t l , c , q 5 , 0 ) = 0. Continuing, if g is the index, i.e., g ( t ) = g ( t , c ) = c T U ( t l , t ) B ,then cTy1 = cT y ( t 1 ) =
l1
s(t,c)u(t)dt.
(7.7.10)
The control that realizes a boundary point of R(t1) definitely maximizes
l1
g(s)u(s)
whenever llulll
I/"
(7.7.11)
g(s)uo(s)
I 1, i.e., over u E U . We have that
If t* is a minimum time for the time-optimal control problem with u E U , then C T Y ( t * ) = 11s .; (CT)llm (7.7.13)
where
IlSY. (CT)ll@J = ly<m ,?$.
ISi(Ct
t>l.
(7.7.14)
If equality holds in (7.7.12), then impulse functions applied at the points where gj is largest maximizes (7.7.11). The maximum values in (7.7.14)
Synthesis of Time-Optimal and Minimum-Effori Control
27 1
occur at multiple j and at multiple instances s j i , i = 1 , 2 , . . .N j , where Nj is taken to be zero if g , does not contain the maximum. Because of these, the time-optimal controls (which are “boundary” controls) are u* ,
For the minimum-fuel problem we deduce as before that the optimal control is u*(t,c’) (7.7.16) u(t) = -, Q where
1 ---- 1 Q
M(t1)
- min ES(u(t1))
(7.7.17)
and
a ( t ~=) min F(t1,c ) , P = { c E En : c T y ( t l )= l}, CEP
(7.7.18) (7.7.19)
In (7.7.16) c* is the minimizing vector in (7.7.17). The proof is complete. Remark 7.7.1: If U in (7.7.2) is replaced by u measurable u ( t ) E E” :
l1
Iuj(t)ldt 5 1,
j = 1 , . .. , m
1
,
(7.7.20)
then
(7.7.21) The maximum in (7.7.21) occurs as before at multiple instances of time rj;, i = 1 , 2 , . . . , M j , where Mj 2 1. The maximizing impulsive controls that approximate reachable points on the boundary of R(t) are
Stability and Time-Optimal Control of Hereditary Systems
272 Therefore, with
M
(7.7.23a)
a ( t l )= min F(t1, c) = F(c*), CEP
(7.7.23b)
the optimal control that minimizes the absolute fuel is
U*(t,C*) 'u= -,
(7.7.24)
Q(t1)
where
C*
and a are determined by (7.7.23b).
Proof of Theorem 7.5.5: Recall the following definitions:
(7.7.25)
where a 2 0. These sets are nonvoid, convex, and symmetric about 0. Clearly R, c R(t1) and equality holds if a 2 rntl. Indeed, if a 2 rntl and y E R ( t l ) , then y E R(t1) corresponds to some admissible control u such that y=
l1
U ( t 1 ,t)Bu(t)dt,
IIulla, L 1-
This shows that y E Rmtl c R,. We also observe that for 0 < ct 5 p,
R,
c Rp,
3 p-Rp.
(7.7.27)
The first is obvious from the definition. The second follows from the following containment:
XIW,
+ (1 -
c h,+(l-A)a for all y , 6 2 0, 0 5 X 5 1. If 6 2 0, y = p, and X = alp, then the second containment follows. From (7.7.27) we deduce that
R, 3 (cl./rntl)rW(tl), 0 5 ff 5 rntl. If (7.5.1) is controllable, then R(t1) has a nonvoid interior, and this forces R, t o have nonvoid interior for each a > 0. From the usual weak-star compactness argument we can prove that R(t1) is compact. Other properties of R(t1) are contained in the next lemma.
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273
Lemma 7.7.2 For each q5 E C such that r(t1,u,4, 0) E R(t1) (and for no other points) there is an admissible control u that steers q5 to 0 a t time tl while minimizing E 3 ( ~ ( t l = ) ) llulll.Also t ( t l , u , q 5 , 0 )E 3lW~ for 8 = 11~11. Proof: By definition, q5 can be steered to zero at time tl by a control if and only if 2(tI,fl,4,0) E Wh).
Let
8 = inf{a 3 0 : z(tl,u,q5,0)E R}, where W l )
=
u Rr.
(7.7.28) (7.7.29)
a>O
It is clear that there is a sequence of admissible controls u( ), each steering q5 to 0 at time t l with 11.111 4 8+. The usual weak compactness argument yields an optimal control E with 8 = llVll1 = &(E(t1)) and
E 3 ( z ( t l ) )5 E3(.(tl)),
v 21.
To see this, interpret admissible controls as points in Lz-space, i.e., L2([u,tl], E m ) .Thus the conditions
l.lm 5 1,
l11 .11
5 8+
€7
determine a convex set that is bounded in L2 norm. It is weakly closed since if a sequence converges weakly, then a sequence of convex combination converges strongly in L z , and as a result a subsequence converges pointwise a.e. Thus r(tl,u,q5,0)E Re. We now verify that t(tl,u,q5,0)E dRe. We assume this is false and aim a t a contradiction: z ( t 1 ,u,q5,O) E Int Re. This implies that for sufficiently small A - 1 > 0, Xz(tl,u,q5,0)E Re. But if 0 < a 5 0, then by what was proved before
c (e/a)R(tl). Let a = 0/A, then AZ(tl,fl,d,O)
even though a complete.
<0
E A&,
Z ( t l , U , 4 , 0 ) E Re,
and 0 is minimal. If 0 = 0,
RO = 0. The proof is
The next lemma is the Maximum Protoprinciple of Hiijek, which corresponds to our System (7.7.4).
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Stability and Time-Optimal Control of Hereditary Systems
Lemma 7.7.3 If 4 E C is such that t ( t l , u , d , O ) E aR,, and c is an exterior normal to R, a t z ( t l , u,4, 0), and if ?LO(.) is any admissible control steering 4 to 0 a t t l , then
I"
g(t)u(W
5
Jo"
g(t)uo(W,
(7.7.30)
V u(.) with llulloo 5 1, llulll 5 a, where g is the index of the control system: g ( t ) = c T u ( t 1 , t ) ~ . It is not necessary that Iluolll 5 a. Conversely, if c # 0, and lluollo0 5 1, lluolll 5 a , and if (7.7.30) holds for all controls described there, then uo(.) reaches the point z ( t l , u,4, 0) E dR, for some qi E C, a t which c is an external normal. Proof: Let 4 E C be a point for which z(t~,u,qi,O) is on the boundary of R, with c an external normal. If c # 0, then this is equivalent to cTy(tl) 5 cTz(tl,~,Ql,O),Vy(t1) E R,. But then
and
The proof is complete. Corollary 7.7.1
Assume that the boundary control uo satisfies
Then each coordinate uoj of uo satisfies
(7.7.31) Proof: Define a control u with coordinates uj
= (sign g j ) . Iuoj I.
This control satisfies the assumptions. Since (7.7.30) is valid,
275
Synthesis of Time-Optimal and Minimum-Effort Control Because of this,
so that gj uoj
= lgj uoj I 2 0
a.e., since the summands are nonpositive. We know from the lemma that optimal controls exist as boundary controls of R,. We obtain more information by applying the extension of Pontryagin's maximum principle reported by Manitius [37,p. 981. Define the Hamiltonian
for some constant q 2 0, where the adjoint equation may be written as N
G(t) = - y ( t ) A o ( t ) - C y ( t + h j ) A j ( t + h j ) a.e. in [u,t l ] ,
y ( t ) = 0,
t
> tl.
j=1
(7.7.33) The optimal control that maximizes H is the u ( t ) , which maximizes the expression
(7.7.34) But y ( t ) = c T U ( t l , t ) is the solution of the adjoint equation [S, pp. 147-1491 with y ( t l ) = c T . Thus with the index of the control system
g(t,c)
= y(t)B = CTU(tl,t)B,
we desire that u that maximizes
over all rn-vectors u whose coordinates satisfy 1ujl 5 1. If q = 0, then the optimal control would coincide with that for the time-optimal problem, and
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Stability and Time-Optimal Control of Hereditary Systems
we should therefore have to assume summand in (7.7.35) separately:
r]
> 0. In this case we maximize each (7.7.36)
provided that g j f r ] . The coordinates of u have values f l and 0 only. As t varies, there is no switch from 1 t o -1, nor vice versa. Just as in ordinary differential systems studied by Hfijek, optimal controls either always use all the available capacity or are completely dormant. The expression (7.7.36) uniquely determines optimal controls provided g, $ r] on set of positive measure, i.e., provided (7.5.1) is metanormal, which is assumed in the theorem. The expression for optimal control depends on a careful choice of 1. The following construction follows very closely those of HAjek for ordinary linear systems. First, choose any r] 2 0, such that (7.7.36) holds and hunt for some ii: llTilloo 5 1. But
We select some suitable r] for which /lull1 = a. Indeed, since (7.5.1) is metanormal, gj(t, c) is nonconstant a.e., and the mapping rn(r]):
is continuous, strictly decreasing, 0 5 rn(r]) 5 t l . Because 0 there exists a unique value of 7 2 0 such that
With this
so that
r]
set
5
a
5
rntl,
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277
With this, define the control E( ) as follows
This a has the following properties:
It is easy t o see that
for all controls u(.) with
Because (7.7.4) is metanormal, ? isiunique a.e. The theorem is completely proved.
7.8
Control of Nonlinear Functional Differential Systems with Finite Delay [9,40]. Existence, Uniqueness, and Continuity of Solutions
In this section we give some known result on the existence, uniqueness, and continuous dependence on initial data and parameter of the system
.(t) = f ( t , z t ) , t 2 2,
= 4,
Q(S)
= z(t
6,
+s),
-h
I s 5 0.
(7.8.1)
Also considered is the control system monitored by
As usual, C is space of continuous functions from [-h,O] + En with the sup norm. If 0 is a subset of En, and f : E x C([-h, 01, 0 )-.+ E n , we say z is a solution of (7.8.1) on [u - h , u u ) if there is an a > 0 such that x : [u- h , u + u ) + En is continuous and z ( t ) satisfies (7.8.1) for all t E [u,u u ) . We assume i(a)= f ( t l 4). For a subset D of E x C x Em given by D = E x C([-h, 01 0 )x Em and F : D + En, a solution z of (7.8.2) is defined similarly. For the system (7.8.1) we have the following:
+
+
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Stability and Time-Optimal Control of Hereditary Systems
Theorem 7.8.1 Suppose 0 c En is open, 0 = E x C([-h,O],O),and f :0 En is continuous and satisfies a local Lipschitz condition in the second variable. If (a,4) E 0, then there exists a unique solution for (7.8.1) on the interval [a, u u ) for some Q > 0. This solution varies continuously with initial data, and satisfies the integral equation -+
+
The solution I given by Theorem 7.8.1 is such that xt E C. There are various function spaces where the solution of (7.8.2) can be considered to lie. It depends on the choice of the control space from which u is selected. For the existence problems, we select the control space to be the Banach space C([a,71, Em)of continuous functions mapping [a, 71, (a 5 7 < 00) into Em with the sup norm. The results we shall state are valid for controls in L, spaces (2 5 p 5 co) treated in Rudin [39]. Thus we can replace C([~,T Em) ] , by L m ( [ a , r ]Em), , whose members are essentially bounded measurable functions defined on [ a , ~with ] values in Em. We adopt the following notation in the theorem: t h e Fre'chet derivative ofg a t 4 is denoted by Dg(r#J)where 4 E C, a Banach space. IfC is the product ofseveral Banach spaces, then the partial derivative with respect to the ith variable will be denoted by D;g(q5).
Theorem 7.8.2 [40] Consider System (7.8.2), where F : [ ( T , T ]x C([-h,O],O)x Em -+En satisfies the following conditions: (i) F ( t , ., .) is continuously differentiable for each t . (ii) F(.,d,w) is measurable for all 4 and w. (iii) For each compact k C 0 there exists an integrable function M I : [a, T] [0, cm)and square integrable functions Mi : [a, T ] -+ [0, m) i = 2,3, such that ---+
for all t and w and all 4 E C ( [ - h , 0 ] , 0 ) . (iv) F ( t , 0,O) = 0 for all t . Let the function c(u,4, u ) E C([a- h , 71, En) be the solution of (7.8.2) for any pair of (4, u ) such that the solution exists.
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279
Then there exist open neighborhoods N1 and N2 of the origin in C([-u,01, E") and C([u,71, Em)such that, for each (5 E N1, u E N2 (7.8.2) has a unique solution z = z(u,(5, u ) and such that z is continuously differentiable on N 1 x N 2 . Remark 7.8.1: If a measure theoretic analysis is undesirable, we can formulate the following version of Theorem 7.8.2: Theorem 7.8.3 In (7.8.2), assume that: (i) F ( t , ., is continuously differentiable for each 1. (ii) F(.,(5,w) is continuous for all (5 and w . (iii) For each compact k c 0 there exists continuous functions Mi : [u,r] [0, cm) i = 1 ,2,3, such that a)
IP2F(t,(5,w)lI I Ml(t) IID3F(t, 4,u)ll
-+
+ M2(t)lUI,
I M3(1)' v t , w and (5 E C([-h,01'0).
(iv) F(t,O, 0) = 0 for all t . Let the function z ( u , d , u ) be the solution of (7.8.2) for any pair of ((5, u ) such that the solution exists. Then there exist open neighborhoods N1 and N2 of the origin in C([-h,01, E" and C ( [ a- h,7],Em)such that for each 4 E N1, u E N2, (7.8.2) has a uniquesolution z = z(u,(5, u), and such that z is continuously differentiable on N1 x N2. Remark 7.8.2: Theorem 7.8.2 is valid with u E Lp([u,73, Em),though this seems to be more useful when p = 00, since difficulties are encountered when differentiating on L, 2 _< p E 00.
Proof of Theorem 7.8.2: We first observe that if z is any continuous map from [a- h , r ] to 0 and u E Lm([u,.r],Em),there exists a compact convex set k c 0 containing the origin such that z ( t ) E k , u - h 5 t 5 r. For this k let A l l and M3 be as in hypothesis (iii) of Theorem 7.8.2. It follows that for all t E [n,71, (p E C([-h, 01, k), w E Em we have
If'(t,4,w)l
5 Mi(t)II(5II + Wdt)bl
so that F ( t , z t , u ( t ) ) is an integrable function of
t.
Next we note that there is at most one solution z of (7.8.2) that satisfies = 4. The Gronwall's type argument of Hale [8, p. 421 is applicable for its proof. We express (7.8.2) in integrated form as follows: Define the operator T : C C ( [ u- h, E") by to
-+
TI),
(7.8.4)
Stability and Time-Optimal Control of Hereditary Systems
280 Then
1: is
a solution of (7.8.2) on [u- h, T ] if and only if
+
z ( t ) = (Tf$)(t) /""IF ( s , zs, 21(6))dS,
u-h
5 t 5 7,
0
wherecr(t) = max(u,t). Ifwe define H : C ( [ ~ - ~ , T ] , N ) ~ L ~ ---* ([U,~], C([U - h, 4 E n ) by
then H is continuously differentiable, and
where t E [u- h , T ] ,
so that
for all 1: E C ( [ u- h , 71, E n ) , u E L,([u, 71,Em),and t E [u- h , 71. Detailed proof of these formulae require the use of the Dominated Convergence Theorem, which is permissible because of condition (iii) of T h e e rem 7.8.2. To prove continuity of DiH i = 1,2, we show that
by proving for example that for i = 1
Synthesis of Time-Optimal and Minimum-Egod Control
28 1
The argument involves extraction of subsequences, and the use of dominated convergence theorem. With H defined in (7.8.5), the solution 2 of (7.8.2) can be rewritten as z = H ( t , u ) 7'4, where H(0,O) = 0, since condition (iv) is valid. Now define G ( t , U , 6 ) = x - H ( x , U ) - T6. (7.8.6)
+
Clearly G(O,O,0) = 0, and G is continuously differentiable on
and DlG(O,O,O) = I - DlH(0,O). Carefully note that if DzF(t,O,O)I# = A ( t , 6 ) and conditions (i) and (iii) hold, then the operator I(, defined by
( I ( z ) ( t )= ~ a ( * ) A ( s , t s ) dd~-, h 5 t 5 T , is such that ( I exists. This is true since {Kt : t E C([uh , r ] , E " ) lltll 5 1) is equicontinuous, and therefore Ii' is compact. Furthermore, since
has a unique solution on [a- h , TI, the unique solution of x = Kx is t = 0. The assertion that ( I - K ) - l exists follows readily from F'redholm's alternative [40]. As a consequence (I-D1H(O, O))-' = (I-G(O,O, O))-' exists. We are now assured by the Implicit Function Theorem that there exists a continuously differentiable function defined on an open neighborhood of the origin i n C ( [ - h , 01, En)x &,([a,4, Em),such that
G ( q 6 ,u),U , 6 ) = 0,
v (6,).
E
x?.
By choosing x? appropriately small, the proof is complete.
7.9
Sufficient Conditions for the Existence of a Time-Optimal Control [9]
Consider the general delay differential control system (7.9.1)
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Stability and Time-Optimal Control of Hereditary Systems
whose time-optimal control problem is formulated in Section 1.9. The initial state 4 is given in the space C of continuous functions from [-h, 01 into En with the sup norm. The state space is either the n-dimensional Euclidean space En,the space C , or the Sobolev space W;'), 1 5 p 5 00, of absolutely continuous functions t : [-h,O] -+ En whose derivatives are &-integrable. If En, C, or W g ) is the state space, admissible controls are measurable functions u : [0, m) -+ Em whose values u ( t ) E U ( t ,z t ) , where U ( t ,q )are compact, convex, set-valued function with V ( t , z t ) c Em. Thus if 2E"I is the collection of nonempty, compact, convex subsets of Em,the set map U : E x C -+ 2Em is the control set. In particular, we shall consider the fixed control set U that is the m-dimensional unit cube, C", of Em,that is
C" = {U : u E Em, l
5 1, j =
~ j l
1 , e . e
7
m).
When controls are taken as measurable functions with values in U , we say the controls are constrained. Thus we shall study constrained controls for En,C, and W g ) state spaces. If we restrict our attention to the state space W;'), we use the control space L z ( [ a , r ] , E m of ) square integrable functions u : [a,T ] -+ Em. The problem resolved in this section is to show the existence of a constrained control u* such that the solution z ( a , 4 , u * ) of (7.9.1) reaches a continuously moving target zt E C([-h,O],En) = C in minimum time t* 2 a. Thus we shall investigate the existence of optimal control in C . We assume that f is smooth enough t o guarantee global existence and uniqueness of a solution passing through each (a,4) E [0, m) x C for each u E Lm([a,m),Em).We also assume that f satisfies the inequality
(7.9.2) for some Ii' > 0, all t E E , zt E C,and u ( t ) E U ( t , z t ) . This second assumption on f is giveii to avoid finite escape times. We also assume for any bounded set N c C that there exists an almost-everywhere bounded measurable function 6~ : E -+ En such that
for all zt E N , u(t) E V ( t , z g ) . Associated with (7.9.1) is the contingent equation, (7.9.4) i ( t ) E F ( t , z t ) , 20 = 4,
Synthesis of Tirne- Opt am a1 and Minima m-Eflort Control
283
where
F ( t ,4) = {f ( k 4 ,u ) : u E U ( t ,4)).
(7.9.5)
Since f is continuous and U ( t ,4) nonempty and compact, F ( t , 4) is nonempty and compact. When U is not fixed, we assume it is upper semicontinuous with respect to inclusion in t , 4 in the following sense:
Definition 7.9.1: The set map, F(., : E x C -+ 2En is said to be upper semicontinuous with respect to inclusion in t , t t ,if for any 6 > 0 there is a S > 0 such that F ( t l , x t l ) c F,(t,zt)whenever It1 - t l < 6 llztl - ~ t l 5l 6. Here F, represents the +neighborhood of F in En.A solution t of (7.9.4) is an element x E C ( [ a- h , 7.1, En)such that (i) t is absolutely continuous on [a,T ] , (ii) to = 4, and (iii) i ( t ) E F ( t , xt). We deduce from McShane and Warfield's generalization [42]of an important result of Filippov [41]that every solution of (7.9.4) can be viewed as a solution of (7.9.1). Indeed, we have: a)
Lemma 7.9.1 Let F ( t , d ) be convex for each ( t , 4 ) E E x C. A function is a solution of (7.9.4) if and only if x is a solution of (7.9.1) for some admissible control u. Definition 7.9.2: A subset A of C is called the attainable set if it is given by A ( t , u ) = { x t : x is a solution of (7.9.1) for some u ( t ) E V ( t , t t ) } . This is the same as A ( t , u ) = { z t : i ( t ) E F ( t , x t ) : t 2 u, 2, = 4}. The closure of A ( t , a) in C is needed to prove the existence of an optimal control. Indeed the following is true: Theorem 7.9.1 Suppose (i) f : E x C x Em --t En is continuous and is continuously differentiable in the last arguments. (ii) f verifies the inequalities (7.9.2) and (7.9.3). (iii) Let F ( t , 4), as defined in (7.9.5), be convex for each t , 4. (iv) Suppose U : E x C --+ 2Em is upper semicontinuous with respect to inclusion in the two arguments, and is compact valued. Then d ( t , a ) ,t 2 u, is compact. Proof: Because f : E x C x Em --t En is continuous and U upper semicontinuous with respect to inclusion, F ( . , .) is upper semicontinuous with respect t o inclusion. Because U is compact, F is closed valued. Let {t$}
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Stability and Time-Optimal Control of Hereditary Systems
be a sequence of points in d(T,a) with x: = 4. Then x k is a solution of (7.9.4) : i k ( ( T ) E F ( T ,XT), T 2 U , 2: = 4. We now prove that { z k }is equicontinuous and equibounded, so that {z$} is equicontinuous and equibounded. Since (7.9.2) and (7.9.3) are valid,
We have proved that the function { z k }is equibounded. Because {zk} is equibounded, (7.9.3) guarantees that there exists an almost-everywhere bounded function t$ such that Iik(t)l 5 I<(t)l 5 M a.e. in t . It follows is equicontinuous and that {Iik(t)l} is equibounded a.e. As a result, { t k } therefore compact. By the Ascoli Theorem there is a subsequence (and we can take it to be the original sequence) zk converging uniformly as k + 00 on [.,TI to some 2. Note that E, = 4 and Iz(t)l 5 M , and t is absolutely continuous. We now prove that x is a solution of (7.9.4). Suppose a t the point to E [c,T],i ( t 0 ) exists. Then
= lim k-oo
For a given c
1
i k ( ( t- t0)s)ds.
> 0 we can choose 6 > 0 such that (7.9.6)
whenever It - to1 < 6, lltt - ttoll < 6 . The inclusion in (7.9.6) follows from the upper semicontinuity of F ( t , q ) . Thus for almost all T E [ a , t ] , it(.) E
Synthesis of Time-Optimal and Minimum-Effort Control
285
F ( T , x ~ ) .Since x k converges uniformly t o z so that for N _> No (say), llzt - zkll 5 6/2. Therefore ik(r) E FE(to,zto) for almost all T E [to,t]. Thus i k ( t o (t - t o ) s ) E F,(to,zt,) a.e. s E (0,l). Because Fc(to,zto) is convex, the Mean Value Theorem yields that
+
J,' z k ( t o + (t -
to)s)ds E
F,(to,zr,)
for N _> N O ,say. Thus
Therefore,
i(t0)
E FZE(tO,z t o ) ,for arbitrary c
i ( t o ) F ( t o ,z t o ) =
n
~
> 0. Since F ( t ,q )is closed, ~
~
(
t
~
E>O
Since this holds a.e. on [a,Tl, we have indeed verified that z satisfies (7.9.4). Thus ZT E d(T,a), and d(T,a) is closed. Summing, we have shown that x$ is an equibounded sequence which converges to a point of d(T,a). Hence d(T,a) is compact.
Definition 7.9.3: Let zt E C be a target function that is time varying and for each t is an element of the function space C. Suppose ZT E d(T,a),T 2 B , then the system (7.9.1) is controllable to the target. In other words, for each 6, E G there exists a T 2 a and an admissible control u , u ( t ) E U ( t , ~ t )t, E [a,T]such that the solution of (7.9.1) satisfies zo(U, 6,,u)=
41 zT(a,6,)u)= Z T .
In this case we also say that (7.9.1) is function-space z-controllable on [a, T] with constraints. System (7.9.1) is null controllable with constraints if for each 6, E C , there exists a tl < 00 and an admissible control u E U such that the solution z( ) of (7.9.1) satisfies zq(a,4, u) = 4, zt,(a,4, u) = 0. In what follows it will be very convenient t o work in the space En, C , or W:) as the state space with controls L , that are constrained to lie in the unit cube U in Em. Controllability concepts can also be defined without assuming constraints.
+
Definition 7.9.4: System (7.9.1) is controllable on [a,t~], tl > a h, if for each 6,,$ E C there is a control u E L,([a,tl],E") such that the
~
Stability and Tarne-Optamal Control of Hereditary Systems
286
solution z ( u , + , u ) of (7.9.1) satisfies z u ( o , 4 , u ) = 4, zt1(u,4,u) = $. It is null controllable on [ a , t l ] , t l > a h , if for each q5 E C there is a control u E L , ( [ a , t l ] , E m ) such that the solution of (7.9.1) satisfies z , ( o , ~ ,U) = 4, Z t l ( u , 4, u) = 0. The qualifying phrase “on the interval [a,t l ] ” is dropped if they hold on every interval with t l > h. In the above definition, we consider Euclidean controllability by replacing @ by 21 € En. We now investigate the existence of optimal control in the state space
+
+
C. Theorem 7.9.2 (Existence of a Time-Optimal Control) (i) Assume that in (7.9.1), conditions (i) - (iv) of Theorem 7.9.2 hold. (ii) For some T 2 u, System (7.9.1) is function-space z-controllable on [a, T ] with constraints. Then there exists an optimal control. Proof: Let
t* = inf{t : T 2 t 2 u, zt E d(T,u)}. Because of (ii) there is at least one T such that ZT E d(T,a);as a consequence t* is well defined as a finite number. We now prove that zt= E d ( t * , a ) . Let { t , } be a sequence of times that converge to t’ such that zt, E d ( t n , u ) . For each n , let Z” be a solution of the contingent equation (7.9.4) with zt, = z:~. Then
where, by condition (7.9.3), ( is bounded almost everywhere. Since the target zt is continuous in t , zt + zt* as t , + t * . Also f.”<(r)dr+ 0 as t , -+ t * , since is bounded a.e. and is measurable. I t follows that .:x + zt* as n -+ 00. Because z.: E d ( t * , a ) ,and this is a closed set, zt E d ( t * , u ) . This completes the proof.
<
Corollary 7.9.1 Consider the linear system
( 7.9.7)
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287
where
with 0 = wo 5 w1 5 ... < W N 5 h, A k is a continuous n x n matrix, B ( t ) is a continuous n x m matrix function, A ( t , s ) is an n x n matrix function continuous in t and integrable in s for each t with the following property: There is a locally essentially bounded measurable function a(t) E (-m,m) 4 E such that
I
A ( t , s)+(t
+ s)dsl F a(t)ll.tIl
(7.9.9)
for t E E, xt E C. The admissible controls u E U are measurable L , functions with values in the unit cube
c" = (uE Ern, 1.j I 5 1,
j = 1, . . . , m } .
(7.9.10)
Suppose that in (7.9.7) where (7.9.8), (7.9.9), and (7.9.10) hold, there is some tl 2 u such that (7.9.7) is function-space r-controllable with constraints on [ u , t l ] ,tl > u h, where zt is continuous in t and is in C. Then there exists time-optimal control.
+
Proof:
f(t, .t, .(t)) = L(t,.t)
+ B(t)u(t)
is a map f : E x C x Em -+ En is continuous and is continuously differentiable in the last two arguments. Now, F ( t , z t ) = L ( t ,t t ) + B ( t ) U is convex for each t , z t , because it is linear in u . Since U is compact, and constant, it is upper semicontinuous, and conditions (i), (iii), and (iv) of Theorem 7.9.1 are satisfied. That (ii) holds as well follows from the following argument:
for some continuous functions c ( t ) , P ( t ) , and a locally integrable ~ ( 2 ) . If 11xtll is bounded,
( 4 t )+ 4t))llQll+ P ( t ) I r ( t ) ,
where [ ( t ) is essentially bounded measurable function on (-m,oo), and therefore satisfies (7.9.3). Condition (7.9.2) is needed to prevent finite escape times for solutions, i.e., to ensure that the solution of (7.9.7) is bounded
288
Stability and Tame-Optimal Control of Hereditary Systems
on every finite time T . From Proposition 2.2.1 any solution z(n, 4) of (7.9.7) is defined and continuous on [u- h , a). It is therefore uniformly bounded on any finite interval [u,T] T 2 u. It follows now that Theorem 7.9.3 can be invoked for the system (7.9.7), and the existence of a time-optimal control can be inferred. Theorem 7.9.2 is of fundamental importance in time-optimal control theory. It asserts that one must make the controllability assumption and have conditions that guarantee constrained controllability before the timeoptimality questions can be resolved. Controllability results with controls constrained t o lie in C"' are therefore very useful in the resolution of the time-optimal control problems of delay equations. This is tackled in Chapters 6 and 8. Questions of controllability with unlimited power are next taken up.
7.10 Optimal Control of Nonlinear Delay Systems with Target in En Consider the following problem: Minimize J(u) =
subject t o
1'
f O(z(t),.c(t - h ) ,4 t ) )
(7.10.1)
where f ( 2 ,y, u ) and f o ( z ,y , u ) are continuously differentiable on En x En x E m . f o : E n x E n x E " ' + E , w h i l e f : E n x E n x E m - + E n . Theclassof admissible controls are measurable functions u with values u ( t ) E U , where U is the unit m-dimensional cube. For each ( z , y , u , p ) E En x En x Em x En, and p o E E, define the Hamiltonian
where p ( t ) is a row-vector function. Theorem 7.10.1 [46] (Maximum Principle) Let u * ( t ) be an optimal control and z * ( t , u*)the corresponding optimal trajectory. For this optimal
Synthesis of Time-Optimal and Minimum-Eflort Control
289
pair (u*, z*) there necessarily exists a nonzero function p ( t ) , and a p ; such that
0)
H ( z * ( t ) , z * ( t - h ) , u * ( t ) , p * ( t= ) ) sup H(z*,z*(t-hh),u,p*)(7.10.6) UEU
holds for all t E [0, TI. (ii) At T,the relation
H(z*(T),z'(T - h),u*(T),p*(T)) =0
(7.10.7)
is satisfied. Also the vector p*(T)is orthogonal to the tangent plane to the manifold (which defines the target) a t the point z * ( t l ) . (iii) p: 5 0, wherepo constant. Furthermore,
8H i ( t ) = - = f ( z ( t ) , z ( t- h ) , u ( t ) ) , dP i ( t ) = --p(t)Ao(t) - p ( t
p(t)
0,
Vt
+ h)Al(t + h ) - po(t)f;
(7.10.8) a.e. on [O, T I , (7.10.9)
> T,
p ( T ) satisfies (ii). Here Ai(t), i = 0, 1, are given by
Remark 7.10.1: Because p ( t ) 0, V t > T ,the advanced term in the adjoint equation vanishes on [T- h, T ] . The time-optimal control problem is a special case of the situation in Theorem 7.10.1. Indeed, let f o ( z ( t ) , z(t - h ) ,u ( t ) ) E 1. In Theorem 7.10.1 one assumes the existence of optimal control. This existence was proved for the time-optimal situation in Theorem 5.2.3. If the state space is C, the resulting general cost functional situation was treated by Angell in [44] and Angell and Kirsch [45]. It is treated in Section 7.3. An important problem to be resolved is to find when the maximum principle is a necessary and sufficient condition for optimality. The following general result for linear systems is available.
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Stability and Tame-Optimal Control of Hereditary Systems
Consider
i ( t ) = A o ( t ) z ( t )+ A l z ( t - h ) + B ( t ) u ( t ) , t 2 0, e(t>= 4(t), where h u(t)
>0
constant
t E [-h, 01,
(7.10.11) (7.10.12)
4 is a continuous function, u is measurable with
E C" = {u E Em : Iujl 5 1, j = 1 , . . . ,m},and Ai := 0 , l B(.) are
continuous matrices. Let G c En be a closed target set. The problem is t o minimize
subject to z(.) satisfies (7.10.11), (7.10.12), and the terminal condition z(Ti,(6, u)E G.
(7.10.14)
In (7.10.13), assume that 2: -+go(z) is continuously differentiable and convex, and fo(., .) and ho(.,.) are nonnegative continuous functions that are convex with respect to z for all t E [O,T]. We assume 2 + f o ( z , t ) is continuously differentiable for all t E [ O , T ] . The following theorems are valid.
Theorem 7.10.2 (Existence) If there exists an admissible control u E U that transfers the response z ( t ) in (7.10.11) from the initial function 4 to the target G a t time T , then there exists an optimal controller that steers the response to G a t time T and minimizes J ( u ) . The optimal controller is given as follows: There exists a nontrivial function t -+ ( p o ( t ) , p ( t ) ) satisfying the equation (i) po(t) f constant < 0, (ii) ;(t) = -p(t)Ao(t)-p(t+h)Al(t)-po(t)f:(z(t),t) a.e. on [0, T ]such that
Conversely, if u*,an admissible controller, is given as above then it is an optimal controller.
Remark 7.10.2: The proof is similar to the reachable set approach of Theorem 7.1.1, and is contained in Manitius [37,pp. 104-1081.
Synthesis of Time-Optimal and Minimum-Effort Control
29 1
7.11 Optimal Control of Delay Systems in Function Space For h > 0 let C = C([-h, 0), En) be the Banach space of continuous functions defined on [-h,O] with values in En equipped with the sup norm. If c is a function on [-h,T], define a Banach space C ( [ - h , q ,En), analogously, where T > h, and for each i E [O,T] let ct E C be defined by e*(s) = c ( t s), - h 5 s 5 0. Consider the following problem: Minimize
+
rT
(7.11.1)
subject t o the constraints i ( t ) = f ( t ,ct,u ( t ) ) a.e. on [O, TI, 20
(7.1 1.2)
= 4 E W$([-h,O], E n ) ,
= 1c1 E CQ)([-h,01, E n ) , u ( t ) E C" a.e. on [O,T], 2,
(7.1 1.3)
(7.11.4)
where
and 3:
E WL([-h,T],E"),
E J5m([O,TI,Ern).
(7.11.6)
For this problem we assume that W g ) ( [ - h , T ] , E " )is the state space of absolutely continuous n-vector valued functions on [-h, T ] with essentially bounded derivatives, and L,([O,T], Em)is the set of functions that are essentially bounded rn-vector valued functions. Also, 4 E C'([-h,O],En) if 4 is continuously differentiable. The functions f O , f ,f' are mappings f o : [ O , T ] x C x Em + E ,
f : [ O , T ] x C x Em + En, f1 : [O,T]x
C + EP,
which we assume t o satisfy the following basic conditions: f,f o are continuous, FrCchet-differentiable with respect to their second argument, and
292
Stability and Time-Optimal Control of Hereditary Systems
continuously differentiable with respect t o their third argument, the derivative being assumed continuous with respect t o all arguments. Also f' : [0, T ] x C -+ EP is continuous and continuously F'rdchet-differentiable with respect t o its second argument. Denote by NBvk(to,bl)and NBV#xp(([a,b]) respectively the k-vector and the k x pma t ri x valued functions whose components are of bounded variation left continuous at each point of [a, b] and normalized t o zero at t = b. Associated with (7.11.2) is the linear equation i ( t ) = L ( t ,z t ) 20
= 0,
+ B(t)U(t), 0 5 t 5 T,
(7.1 1.7) (7.1 1.8)
where u E U , x E X , with these subspaces U , X defined as follows:
x = {z E W g ) ( [ - h , T ] , E " ): z T E C'([-h,O],E")}, u = { u E Lm([O,T],Em) : U T E C([-h,O],E")}. In (7.11.7),
L ( t , z t ) = [f&,zT,U*(t))lzt,
B(t)= fu(t,zf,u"(t)) are the Frdchet derivatives with respect to 4 and u respectively. Thus there t ,E NBV,,,([-h, 01) such that L ( t ,xt) = s_", deq(t,@)z(t 6). exists ~ ( .) Angel1 and Kirsch [45] have proved the following fundamental result:
+
Theorem 7.11.1 (i) Let ( x *, u*) be the optimal solution pair of the problem (7.11.1) (7.11.6), (ii) Assume that all the smoothness conditions are satisfied. (iii) Suppose the map t -+ B+(t)is continuous on [T - h , T ] , where B + ( t ) is the generalized inverse o f B ( t ) ,and rank B ( t ) = n , V t E [T- h, TI. Then there exists ( X , a , p , v , q )E E X Lm([O,T],E")x NBVp[O,T]x NBVn[O,T]x En,
(A a, P,21, d f (O,O, O,O,O), such that X 2 0 , p = ( p l , .. . , p ) , p , is nondecreasing on [O,T],pj is constant on every interval where fjl)(t,x;) < 0, a(t) =
A1
T VO(S,t
- s)ds
+
6'
Vl(S,t
- s)Tdp(s) (7.11.9)
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293
and T
5X
1'
BF(t)u*(t)dt+
L-h T
-
1'
uT(t)B(t)u*(t)dt
dVT(t - h)B(t)u*(t)
(7.1 1.lo) for all u E U a d . The multiplier can be taken to be 1. We now state a Kuhn-Tucker type necessary optimality theorem on which the maximum principle is based. The notations are first stated.
Definitions: Let X be a real Banach space. (i) Y C X is a linear subspace if z, y E Y,Alp E E implies Az py E Y . (ii) 2 C X is an affine manifold if c , y E 2, A E E implies (1- X)z Ay E
+
+
2. (iii) H c X is convex if x,y E H , X E [0, 11 implies (1 - X ) z + Xy 6 H . (iv) M c X is a cone (with vertex at 0) if z E MI A 2 0 implies Az E M. Definition 7.11.1: Let X be a real Banach space and S c X a subset. The (i) linear span of S , denoted by span ( S ) ,is the smallest linear subspace of X that contains S . (ii) the affine hull (or affine span) of S is the smallest affine manifold containing S. Definition 7.11.2: Let X be a real Banach space. A transformation f : X --+ E is a functional. The space of all bounded linear functionals f : X E is called the dual of X and is denoted by X * . We recall that X * is a Banach space with norm --f
Let X be a real Banach space with X * as dual, and let Y c X . The interior of Y relative to X will be denoted as Y o ,while its interior relative to its closed affine hull will be denoted by YO0. Let Z c X be a convex subset of X . Define = {e E : t(z) 2 0, vz: E X}.
z+
x*
Let f : X -+ Y be a map of the Banach space X into the Banach space Y . The F'rCchet derivative of f at c* E X is denoted by D f ( z * ) . We use
294
Stability and T i m e - O p t i m a l Control of Hereditary Systems
Di f(z) to denote the partial F'r6chet derivative o f f with respect to the ith variable. Theorem 7.11.2 Let Iw, 21, and 22 be Banach spaces, I/ c R a closed convex set with nonempty interior relative to its closed affine hull W , and let Y c 21 be a closed convex cone with vertex at the origin and having a nonempty interior in 21. Let go : R
-+
E , g1 : R
and let
M = (2 E
-+
21,
g2 : R
+22,
v : g l ( x ) E -Y,g2(z) = 0).
Suppose that go takes on a local minimum at x* E M , and that g l , g l , and g2 are continuously Frkchet differentiable a t x*. Suppose that Dg2(x*)W is closed in 2 2 , but either of the following two conditions fails: ( j ) There exists an ' 2 E V o owith D g 2 ( z * ) z o= 0 and (ii) gl(z*) Dgl(z*)zo E -Yo. Also (iii) Dg2(+*)(W) = Zz. Then there exists a nontrivial triple (A,!l,!,) E E x 2; x 2; such that (i) A 2 0, I! E Y + , (ii) t o o g'(z*) = 0, (iii) XDgo(z*)z !,Dg'(z*)z t2Dg2(z*)x 2 0, V 2 E V - 2*.
+
+
+
To apply the multiplier rule to the optimal problem, we define follows:
21,22,
as
Also
R =x x
u, W = {(z, u ) E R I to = $}, v = ((2,u ) E W ( uE V } .
Consider the three mappings go : X x U U 2 2 defined respectively by
g2 : X x
-+
E , g1 : X x U
-+
21, and
-+
(7.11.11)
Synthesis of Tame-Optimal and Minimum-Eflort Control
295
Under the basic prevailing assumptions on fo, f', f,the mappings go, g1,g2 are continuously F'rCchet differentiable at each point (z*, u * ) E X x V , and the derivatives are of the forms:
Dzgi(z*,u')z: = where Li(t)zt
I'
&(t)ztdt,
I
E x , i = 0,1,
= [fi(t,zt,u*(t))]~t, I t E C,
D"gi(z*,u*)u =
1'
B;(t)u(t)dt, u E u,
where & ( t ) = [fi(t,zt,u*(t)). Here Also where
where Now set
We note that gi, i = 0 , 1 , 2 , satisfy the conditions of the multiplier rule of Theorem 7.11.2 relative to the spaces X, U , 21, and 2 2 . Denote by L& = (2 E L,([O,T],E") : IT E C } and observe that by Theorem 7.11.2 there exist t! E [L&]*, the dual of L&, v E NBVn([-h,O]), q E En, p E NBV,([O,T],and A 2 0, ( A , l , v , q , p )# (O,O,O,O,O)such that p is nondecreasing and
(7.11.14)
(7.11.15) T +
L - h
dv(t - q T 2 ( t )
+ qT+)
= 0,
296
Stability and Tame-Optimal Control of Hereditary Systems
V x E X with x o = 0, and
That the map D l g 2 ( z * ,u*), (2,u) -+ (i(.)- L; .z(.) - B ~ ( . ) u ( . )z~( TT),) is a surjection is a consequence of condition (iii), which is the criteria for the controllability of (7.11.7) in the space C ' ( [ - h , O ] , E " ) . The proof is essentially contained in Theorem 6.2.2. We now affirm that (7.11.14) (7.11.16) hold. The function v is defined on [-h, 01. Outside this interval, we let v(0) = form
{ i(-h), o < - h t(2)
. We claim that the functional l is of the
= 1'aT(t)z(t)dt -
k-h T
dv(t - T ) T Z ( t ) ,
for z E L k , a E Lm([O,q). Indeed, consider (7.11.15). Let z E L&. The solution of
i ( t ) - L 2 ( t ) z t = r ( t ) in [ O , T j , is given by c ( t ,2 ) =
J/d'
10
=0
(7.1 1.17)
X ( t , s ) z ( s ) d s , t E [O, TI,
where X is the fundamental matrix solution of
.(t) = L z ( t ) z t . Thus the system (7.11.15) becomes
T +
L - h
dv(t - T ) T L 2 ( t ) x t ( . 2, )
+ qTz(T,
%)
= 0.
If we use the representation of Li and of z(.,z) and change the order of integration, we deduce that
Synthesis of Tame-Optimal and Minimum-Effort Control
297
for all t E L&,, where a ( t ) depends on A , q , X , p , v , q o , q 1 , q 2 .With this we rewrite (7.11.15) as follows:
dT
L o ( t ) z t ( . ,z)dt T - 1 - h
+
1
T
dp(qTL1(t)zt
+
Jd
d v ( t - T)[Z(t)- L 2 ( t ) z t ]
T
a T ( t ) [ W- J52(t)zt]dt T
+ 1 - h dv(t - T)"[i(t)]
+qTz(t) = 0,
for all z E X with
+
1'
= 0. This is the same as
+
+
a(t)T[Z(t) q 2 ( t , s - t)i(s)ds]dt
T +
10
L - h
-
6'
a ( t ) T q 2 ( t ,-h)z(t - h)dt
dv(t - T)"q2(t1-h)z(t - h )
J" J T-h
t
t-h
d v ( t - T ) T q 2 ( t 1s - t)i(s)ds.
Now define a function ,u as follows:
and for any y E L& extend y by zero and then define
z ( t )=
J,'
y(s)ds for
t E E.
298
Stability and Time-Optimal Control of Hereditary Systems
for all y E L&([O,TJ).Because the equality in (7.11.18) holds for all y, the integrand has to vanish pointwise. As a consequence we have
-A
lT-t qo(e +
h,-h)Tds - A
1
s+h
qo(e,
- e)Tde
T- h
l
T-h
+
+(e
+ h)Tq2(e+ h , - h ) +
which yields
lT + LT -A
qo(e,s - eldo -
I'
qT
= 0,
dp(e)TV1(e, - 0)
+
4 0 ) ~ 7 7 ~ (s0, ope
-
d v ( e ) ~ q 2 ( se ,- e)
+ q~ = 0.
This proves the equation (7.11.9). To show that (7.11.10) is also valid, we substitute the form of t into (7.11.16). We observe that p = ( P I , . . . ,p l ) pj is nondecreasing on [0, TI. Because (A, t, v,q , p ) = (O,O,O,O, 0) is impossible, (A, a,p, v , q ) cannot vanish simultaneously. The proof is complete. As an immediate consequence of this, we state in the next section the solution of the time-optimal control problem.
Synthesis of Time-Optimal and Minimum-Eflort Control
299
7.12 The Time-Optimal Problem in Function Space Theorem 7.12.1 (The Time-Optimal Problem) Consider the following problem: Minimize T subject to the constraints
i ( t )= f(t,xt,u(t)) a.e. on [ O , q , xo = 4 E W&([-h,01, E"),
(7.12.1)
= $ E c'([-h,O],En), u ( t ) E C" a.e. on [ O , T ] , ZT
where
C" = {u € Ern: I U j l _< 1, j = 1,... ,m}.
(i) We assume that f is continuous, and is Fr6chet differentiable with respect to its second argument and continuously differentiable with respect to its third argument, the derivative being assumed continuous with respect to all arguments. Associated with (7.12.1) is the linear equation
i ( t ) = L(t,xt)
+ B(t)U(t),
0 5 t 5 T,
20
= 0,
(7.12.2)
where u E U , x E X where these subspaces are defined as follows:
x = {x E W i ) ( [ - h , T ] , E " ) ,ZT E C'([-h,O],E")}, = {u E Lm([O,T],E"), U T E C([-h,O],E")}. In (7.12.2),
are the Fre'chet derivatives with respect to
usual, q t , xt) =
Ih
4 and u respectively. As
deq(t,q+(t
+ 0).
(ii) Suppose the map t +. B+(t)is continuous on [T- h , TI, where B f ( t ) is the generalized inverse of B ( t ) , and rank B ( t ) = R , V t E [T - h,T]. Then there exists
Stability and Time-Optimal Control of Hereditary Systems
300 such that
a(t) = -qand
1’
lT
q ( s , t-s)Ta(s)ds+
aT(t)B(t)u(t)dt-
I’
I’
T
v t E [O, TI,
dv(t - T ) T B ( t ) u ( t )
L - h
aT(t)B(t)u*(t)-
~(S,t-s)Tdv(s-T),
T
L - h
dv(t - T)TB(t)U*(t),
for all u E Uad.
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18. H. T. Banks and M. Q. Jacobs, “The Optimization of Trajectories of Linear Functional Differential Equations,” SIAM J . Control 8 (1970) 461-488. 19. H. J. Sussman, ”Small-Time Local Controllability and Continuity of the Optimal Time Function of Linear Systems,” J. Optimization Theory and Applications 53 (1987) 281-296. 20. E. N. Chukwu and 0. Hfijek, “Disconjugacy and Optimal Control,” Theory and Applications 27 (1979) 333-356.
J . Optimization
21. H. T. Banks and G. A. Kent, “Control of Functional Differential Equations of Retarded and Neutral Type to Target Sets in Function Space,” SIAM J. Control 10 (1972) 567-594. 22. H. T. Banks, M. Q. Jacobs, and C. E. Langenhop, “Characterization of the Controlled States in W$’) of Linear Heredity Systems,” SIAM J. Control 13 (1975) 611-649. 23. E. J. McShane and R. B. Warfield, “On Filippov’s Implicit Function Lemma,” Proc. Amer. Math. SOC.18 (1967) 4 1 4 7 . 24. 0. Hhjek, “On Differentiability of the Minimal Time Functions,” Funkcialaj ,?hacioj 2 0 (1976) 97-114. 25. G. Tadmor, “Functional Differential Equations of Retarded and Neutral Type: Analytic Solutions and Piecewise Continuous Controls,” J. Digerential Equations 51 (1984) 151-181. 26. L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Diflerential Equations with Deviating Arguments, Academic Press, New York, 1973. 27. G. Sthpan, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific and Technical, New York, 1991. 28. H. T. Banks and J. A. Burns, “Hereditary Control Problems: Numerical Methods on Averaging Approximations,” SIAM J. Control 16 (1978) 169-208. 29. H. T. Banks and K. Ito, “A Numerical Algorithm for Optimal Feedback Gains in High Dimensional Linear Quadratic Regulator Problems,” SIAM 3. Control and Optimization, 29 (1991) 499-511. 30. E. N. Chukwu, “Time Optimal Control of Delay Differential System in Euclidean Space,” Preprint .
302
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31. 0.Hhjek, “Terminal Manifold and Switching Locus,” Mathematical Systems Theory 6 (1973)289-301. 32. D. S. Yeung, “Synthesis of Time-Optimal Controls,” Case Western Reserve University, Ph.D Thesis, 1974. 33. D. S. Yeung, “Time-Optimal Feedback Control,” plications 21 (1977)71-82.
J . Optimization Theory and Ap-
34. H. A. Antosiewicz, “Linear Control Systems,” Arch. Rat. 313-324.
Me& Anal. 12 (1963)
35. 0.HAjek, “L1-Optimization in Linear Systems with Bounded Controls,” J . @timizadion Theory and Applications 29(3) (1979)409-432. 36. 0.HAjek and W. Krabs, “On a General Method for Solving Time-Optimal Linear Control Problems,” Preprint No. 579,Technische Hochschule Darmstadt, Fachbereich Mathematik, Jan. 1981. 37. A. Manitius, “Optimal Control of Hereditary Systems,” in Control Theory and Top-
ics in Functional Analysis, Vol. III, International Center for Theoretical Physics, Trieste International Atomic Energy Agency, Vienna, 1976.
38. L. W. Neustadt, “Minimum Effort Control Systems,” 16-31.
SIAhf J . Control 1
(1962)
39. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974. 40. R. G. Underwood and D. F. Young, “Null Controllability of Nonlinear Functional Differential Equations,” SIAM Journal of Control and Optimization 17 (1979) 753-768. 41. A. F. Filippov, “On a Certain Question in the Theory of Optimal Control,” SIAM J. Control 1 (1962) 76-84. 42. 0.HAjek, “Control Theory in the Plane,” Springer-Verlag Lecture Notes in Control and Information Sciences, New York, 1991. 43. T. S. Angell, “Existence Theorems for a Class of Optimal Problems with Delay,” University of Michigan, A n n Arbor, Michigan, Doctoral Dissertation, 1969. 44. T. S. Angell, “Existence Theorems for Optimal Control Problems Involving Functional Differential Equations,” J. Optimization Theory and Applications 7 (1971) 149-169. 45. T. S. Angell and A. Kirsch, “On the Necessary Conditions for Optimal Control of Retarded Systems,” Appl. Math. Optim. 22 (1990)117-145. 46. H. T. Banks, “Optimal Control Problems with Delay,” Purdue, Doctoral Dissertation, 1967.
Chapter 8 Controllable Nonlinear Delay Systems 8.1 Controllability of Ordinary Nonlinear Systems Consider the autonomous linear control system
k(t)= Az(t)
+ Bu(t),
z(0) = 20,
(8.1.l)
with controls-measurable functions whose values u(t)lie on the m-dimensional cube C" = { U E Em : Iujl 5 1 j = 1, ... , m } . (8.1.2) The solution of (8.1.1) is given by
(8.1.3) Definition 8.1.1: The attainable set of (8.1.1) is the subset d(t,zo) = { z ( t , z o , t i ) : u measurable u ( t ) E C", z is a solution of (8.1.1)} of En. If 20 = 0, d(t,zo) d(t).
Definition 8.1.2: The domain C of (Euclidean) null controllability of (8.1.1.) is the set of all initial points 20 E En,each of which can be steered to 0 in the same finite t l with u measurable u(t) E C", t E [O,tl]:
C = (20 E E"
: z(t1, z o , u ) = 0 for some t l some u
E C"}.
Definition 8.1.3: If 0 E Int C,
(8.1.4)
then (8.1.1) is locally (Euclidean) null controllable with constraints. Lemma 8.1.1 fjr
Assume (8.1.1) is controllable, and this holds if and only rank [B,A B , . . . ,An-'B] = n.
Then 0 E Int C. Proof: Consider the mapping 21
+
z ( t ,0, u) = eAt
it
e-AbBti(s)ds,
T : L,([O,t], Em)-+ En, T u = z(t,O,ti).
303
(8.1.5)
Stability and Time-Optimal Control of Hereditary Systems
304
Equation (8.1.1) is controllable if and only if
T(L,([O,t],E")) = E n . But T is a bounded linear map since u -+ T u is continuous. By the open mapping [5,p. 991 T is an open map. Therefore if U is an open ball such that U C L,([O,t],C"), then T ( U ) is open and T ( U ) c T ( L , ( [ O , t ] , C m ) = d ( t ) . But 0 E T ( U ) C d ( t ) . As a consequence,
0 E Int d ( t ) .
(8.1.6)
It readily follows from (8.1.6) that (8.1.4) is valid. Indeed, assume that 0 is not contained in Int C. Then there is a sequence {zon}?, 20n E 0 as n 00, and no xOn is in C. The trivial solution is a E", ZOn solution of (8.1.1), so that 0 C. Hence xOn # 0 for any n. We also have that 0 # z(tlzon,u ) for any t > 0 and any u E L,([O,t],Cm). Thus -+
-+
y,,
-eAtxOn
# eAt
lo rt
e-A5Bu(s)ds
for any n, t > 0 and any u E L,([O,t],C"). It follows that y,, $2 in d ( t ) for any n. But yn -+ 0 as n -+ 00. Thus the sequence {y,,}? has the property: Yn 0 as n -+ 00, y, $! d ( t ) for any t > 0, yn # 0 for any n. This means 0 $2 Int d ( t ) , a contradiction. --+
Remark 8.1.1: The necessity and sufficiency of the rank condition (8.1.5) for controllability is old and due to Kalman, (see [2, p. 741). The statement in Lemma 8.1.1 can be generalized to nonlinear systems
where f : E x En x Em -+ En is continuous and in the second and third argument is continuously differentiable. We assume all solutions x ( t , z, u ) of (8.1.7) exist and ( t ,$0, u ) -+ z ( t , 2 0 , u) are continuous. The assumptions on f ensure this. In addition t o (8.1.7), consider the linearized system
+
i ( t ) = A(t)z(t) B(t)u(t),
(8.1.8) (8.1.9)
Controllable Nonlinear Delay Systems
305
Theorem 8.1.1 Consider (8.1.7) in which f : E x En x Em + En (i) is continuous, and continuously differentiable in the second and third arguments. (ii) f ( t , 0,O) = 0, V t >_ 0. (iii) System (8.1.8) with A ( t ) and B ( t ) given by (8.1.9) is Euclidean controllable on [0, t l ] . Then the domain C ofnull controllability of (8.1.7) has 0 E Int C.
Proof: The solution of
is given by the integral equation
Consider the mapping
defined by
T u = z ( t ,u). Because of the smoothness assumptions on f , the partial derivative of z ( t , u) with respect t o u is given by
t 2 0. On differentiating this with respect to t , we have
d dt
--Duz(t, u ) v = W ( t ,Z ( t , u),u(t))-Duz(t,u)v
+ D 3 f ( t , z ( t ,.I,
.(t))v.
(8.1.11) Because (ii) holds, the function z ( t , O , 0) = 0 is a solution of (8.1.10). We deduce from these calculations that T'(0)v = Duz(t,O)v = t ( t , v ) is a solution of (8.1.8). Therefore T'(0) : L,([O,tl],E") -+ En is a surjection if and only if the system (8.1.8) with z(0,v) = 0 is controllabie on [ O , t l ] . It follows from Graves' Theorem [3,p. 1931 that T is locally open: There
Stability and Time-Optimal Control of Hereditary Systems
306
is an open ball radius p, Bp c L,([O,tl],C") center 0 , and an open ball radius T center 0, Br c En, such that
where
d(t)= { z ( t , u ) : u E L , ( [ O , t l ] , C m ) z is a solution of (8.1.10) with z(0, u) = 0). We have proved that 0 E Int d(t). With this result one proves, as in the linear case, that 0 E Int C. The proof is complete. Theorem 8.1.1 is a statement on the constrained local controllability of (8.1.7). It is interesting to explore conditions that will yield the global result of C = En.Such a statement is contained in the next theorem.
Theorem 8.1.2 For (8.1.7), (i) Assume that hypotheses (i) - (iii) of Theorem 8.1.1 are valid. (ii) The solution z ( t ) of
.(t) = f ( t , .(t), 01,
4 0 ) = zo,
(8.1.12)
tends to X I = 0, as t -+ m. Then C = En,that is, (8.1.7) is globally (Euclidean) controllable with constraints.
Proof: From Theorem (8.1.1) there is a neighborhood 0 of zero in Enthat is contained in C. Because of (ii), each solution z(t,zo) for any zo E En glides into 0 in time to: z(to,xo) E 0 C C. This point z ( t o , z o ) can be brought t o the precise zero in time t l . Thus the control u E L,([O, t l ] ,C"),
drives
10
into 0 in time t l . This completes the proof.
Remark 8.1.2: Conditions are available for the behavior z(t,zo) + 0 as
needed for (ii).
t
-+ 00
(8.1.13)
307
Controllable Nonlinear Delay Systems
Proposition 8.1.1 In (8.1.12), assume: (i) There exists a symmetric positive definite n x n constant matrix A such that the eigenvalues X k ( x , t ) , k = 1 , 2 , . . . , n of the matrix 1 -(AJ JTA) 2 satisfy & 5 -6 < 0 k = 1 , 2 , . . . , n ,
+
V ( x , t ) E Entl, where 6 is a constant and
J=
af ( 4 X , O ) ax
.
(ii) There are constants T > 0, p, 1 5 p 5 2, such that
4
ttr
lf(T,O,o)yd7+0
as t + m .
Then every solution of (8.1.12) satisfies (8.1.13). The proof is contained in [l]. Example 8.1.1: Consider the model of a mass spring system
(8.1.14)
Let g = ( x l ,x 2 ) where becomes
x=
-
[; I
x1
= 2, x 2 = i,so that the mass spring equation
=
[
22
-bX1
- a22
Here
+ g(u)
1
= f (c, u).
A(t)= A = We may write (8.1.8) as x = Ax
+ B u where B
I:[
=
,
v is in the un-
bounded closed convex cone of g([-l,l]). The rank of [ B ,AB] =
[;
-3
is 2. In (i), A has its characteristic values negative. All the hypotheses of Theorem 8.1.2 are seen to be satisfied. The system (8,1.14) is (globally) Euclidean null controllable with constraints.
308
Stability and Time-Optimal Control of Hereditary Systems
8.2 Controllability of Nonlinear Delay Systems In this section we resolve the problem of controllability of general nonlinear delay systems in function space on which the solution of the optimal problem had rested. In [ll],for example, the problem of minimizing a cost functional in nonlinear delay systems with Wil) boundary conditions was explored under the condition of controllability. The timeoptimal control theory of nonlinear systems requires controllability (see Theorem 7.9.3). As has been argued, if the controls are L, functions, the natural state space is W;'). For general nonlinear systems, a natural way of investigating this problem requires the Frgchet differentiability of an operator F : WJ1) x L, + W;'), which is very difficult to realize for nonlinear systems. To overcome this difficulty we use weaker concepts of differentiability in W;') and more powerful open-mapping theorems. The results obtained are analogous to those of ordinary differential systems. We shall first treat the problem in C, the space of continuous functions. Consider the general nonlinear system (8.2.1)
where f : E x C x Em --+ En is continuous and continuously FrCchet differentiable in the second and third argument. We assume also that there exist integrable functions Ni : E ---t [0,m), i = I, 2 , 3 , such that the partial derivatives D i f ( t ,4, w) satisfy
for all t E E , w E E m , 4 E C . Under these assumptions we are guaranteed existence and uniqueness of a solution through each (a,4) E E x C for each u E Loo([a,m),Em), (see Underwood and Young [15],Chukwu [6], and Section 7.8). Thus for each (r,4) E E x C and for each u E &,([a,oo),Em) there exists a unique response x(a,q5,u) : Et = [a,m) + C with initial data x u ( r l 4, u ) = 4 corresponding to u . The mapping x(r,4, u ) : E+ ---t C defined by t -+ xt(u,4,u) represents a point of C. We now study the mapping u -+ xt(u, 4, u ) : xt : &([a, m),Em)-, C. Lemma 8.2.1 For each v E L,([a, T ] , Em),T > a,we have that the partialFre'chet derivativeoftt(ao,4O,u0) with respect t o u , Dutt(uo140,uo)(v)=
Controllable Nonlinear Delay Systems
309
zt(uo,&,,u0,v) where the mapping t -+ z ( t ,O O , & , U O ,u ) is the unique absolutely continuous solution of the linear differential equation
i ( t ) = DZf(t, Zt(O0,40,uo), uo(t))rf+ DBf(t, Z t ( f l 0 , 4 0 , U o ) , u o ( t ) ) W , ~oo(~o,40,~o =,0.~ )
Proof: The solution I,
I
(8.2.2)
of (8.2.1) is a solution of the integral equation
= 4 in [-h,O],
z(t,u,4, u ) = 4(0)
+ J,'
f(S1 I s ( U , 4 ,
u ) ,u ( s ) ) d s , 1
2 6.
Let D, denote the partial derivative of q ( t ,u,4, u ) with respect to u. Then D,c(t, u,4, u ) = 0 in [ - h , 01, and
Duz(t,694, ). =
J,'
DZf(S, % ( U , 4, u ) , u(S))DuZs(u, 4, u)ds t
+J, D3f(S,
Zs(6,
4, u ) , U ( S ) ) d S , t L
6.
(Difdenotes the FrCchet derivative with respect to the ith argument.) These differentiation formulas follow from results in Dieudonne [9, pp. 107, 1631. On differentiating with respect to t we have d dt
).0,(I).
-[Dd(t,U,
= D2f(t,Zf(b14, u ) , u ( t > ). Du2t(b14, u)v
+ DBf(t,
Q(U,
4, u ) , u ( t ) ) v .
This proves the lemma. Lemma 8.2.2 For 4 E C, u E L , ( [ u , t l ] , E m ) , tl > u + h , let z ( u , ~ , u ) be a solution of (8.2.1) with z,(u, 4, u ) = 4. Consider the mapping u + Z t l ( U , 4 , U ) , F : -L([a,t1I,Ern) C([-h,OI,E") = F(u)= ~ t l ( W w . Then the Fre'chet derivative +
d D F ( u ) = -F(u) du
c,
: L c a ( [ u , t l ] , E m+ )
c
-
has a continuous local right inverse (is a surjective linear mapping if, and onlyif, the variational controlsystem (8.2.2) along the responset zt(c,4, u ) , namely
+
i ( t >= D z f ( t ,Z t , u(t))zt DBf(t, Z t , u ( t ) > v ( t ) ,&((T 4, u ,
= 0,
310
Stability and Time-Optimal Control of Hereditary Systems
is controllable on [ u , t l ] t l
> u + h.
Proof: System (8.2.2) is controllable on [ u , t l ] if and only if the mapping u -+ ~ ( u0,4, , u , v ) t E [a, t l ] is surjective. The lemma follows the observation in Lemma 8.2.1 that
In what follows, we assume in (8.2.1) that
f(t, 0,O) = 0. As a consequence, if u = 0.
(8.2.3)
4 3 0 in (8.2.1), we have a unique trivial solution when
Definition 8.2.1: The C-attainable set of (8.2.1) is asubset of C ( [ - h , 01,En) given by
d ( t , 4 , u ) = { Z t ( U , 4 , U ) : '11 E L,([u,t],Ern), is a solution of (8.2.1)}. If
2,
= 4,
2
4 5 0, we write d ( t , 0, u) E d ( t , u). Let C" denote the unit cube
The constrained C-attainable set is the subset of C defined by a ( t , 4, u) = { z ~ ( u 4,, u): u E L m ( [ u ,t ] , C"),
1:
is a solution of (8.2.1)
with z, = d}. Whenever q5 E 0, we simply write a ( t , 0, u) = a ( t , u).
Definition 8.2.2: System (8.2.1) is proper on [ u , t ] ,t > u + h , if 0 E Int a ( t , 6).
(8.2.5)
Controllable Nonlinear Delay Systems
31 1
Proposition 8.2.1 System (8.2.1) is proper on [ a , t l ] ,t l > u ever
i ( t ) = D z f ( t ,0 , O ) Z t is controllable on [ a , t ~ ]tl,
+ D 3 f ( t ,O , O ) V ( t )
+ h, when(8.2.6)
> u E h.
Proof: Consider the response z(u,4, u)to u E L,([u, t l ] ,Em) for Equation (8.2.1), and the associated map u + ztl(u,O,u) given by Fu = q l ( u , O , u ) , where F : L , ( [ u , t l ] , E m ) + C. Evidently F(L,([u,tl],Cm) = ~ ( u). t , It follows from Lemma 8.2.1 and Lemma 8.2.2 that
is a surjective mapping of L , ( [ u , t l ] , E " ) + C. Therefore, (Lang [ 3 , p. 193]), F is locally open: There is an open ball 1 , c L,([u,tl],Em) containing zero, radius p, and an open ball Br c C containing zero, radius r such that Br c F ( B p ) . Because L , ( [ u , t l ] , C m ) contains an open ball containing zero, r > 0, p > 0, can easily be chosen such that
Thus
Er c F(Lm([u,tl],C"')) = a ( t t u ) , so that 0 E Int ~ ( t , u )This . completes the proof.
Denote by U the set of admissible controls
where C" is as defined in Equation (8.2.4).
Definition 8.2.3: Consider Equation (8.2.1). The domain I) of null controllability of (8.2.1) is the set of initial functions € C such that the solution z(u,#, u ) of (8.2.1) for some tl < 00, and some u E L,([u, t ~ Cm), ], satisfies z,(u, 4, u)= 4, z t l (u,4, u ) = 0. If 2) contains an open neighborhood of z t l = 0, then (8.2.1) is said t o be locally null controllable with constraints when the initial data is restricted to a certain small neighborhood of zero in W;').
+
312
Stability and Time-Optimal Control of Hereditary Systems
Proposition 8.2.2 In Equation (8.2.1), assume that
(4
f : E x C x Em + E" is continuous and continuously differentiable in the second and third argument; and there exists integrable functions N i : E --+ [0, oo), i = 1 , 2 , 3 such that
I P z f 4, w)ll 5 N l ( t ) + W
t ) IwI, f o r al l t E E , w E E m , 4 E C .
I I W ( t , 4, w)11 5 N3(t),
(ii) f(t,O,O)= 0, V t 2 a. (iii) System (8.2.6) is controllable in [a, t l ] , tl > u + h. Then the domain of null controllability D of (8.2.1) contains zero in its interior, that is, 0 E Int D ,so that (8.2.1) is locally null controllable with constraints. Proof: Assume that 0 is not contained in the interior of D ,and aim at a contradiction. By this assumption there is a sequence { & } O O , 4" E C, q5n --+ 0 as n --+ 00, and so 4" is in D. Because the trivial solution is a solution of (8.2.1) an account of (ii), 0 E D ,therefore 4,, # 0 for any n . Also z t ( a , 4 , , , u )# 0 for any t > a h , and any u E LM([a,t],Cm)= U . Thus 03, a h. This contradicts the implication of Proposition (8.2.1) that (8.2.1) is proper, since (8.2.6), by (iii), is controllable. The proof is complete: 0 E Int D. RRcall that tests for the controllability of the linear system
+
--+
+
cn
--+
+
.(t) = L ( t ,z t )
+ B(t)u(t)
( 8.2.8)
in the space C are available in Theorem 6.2.2. Thus, for (8.2.8) the domain of null controllability is open if B is continuous and rank B ( t ) = n on [tl - h , t l ] ,tl > a h.
+
Proposition 8.2.2 is a local result. To obtain a global result in which
D = C , we need a global stability result for the system .(t) = f ( t , z t , 0). It is contained in the next theorem.
(8.2.9)
Controllable Nonlinear D e l a y Systems
313
Theorem 8.2.1 In (8.2.1), assume that: (i) Conditions (i) - (iii) of Proposition 8.2.2 are valid. (ii) System (8.2.9) is uniformly exponentially stable, that is, each solution of (8.2.9) satisfies
Il4Ql4)Il I Wll exp[--cr(t - a)], L b, for some k > 1, a bility V = C.
(8.2.10)
> 0. Then (8.2.1) has its domain of null controlla-
Proof: By Proposition 8.2.2, 0 f Int V ,so that whenever (8.2.10) is valid, every solution of (8.2.1) with 0 = 21 E U ,that is every solution of (8.2.9), satisfies xt(ul4, 0) -+ 0 as t at. Thus there exists a finite t o < at,such that x t o ( u ,4,O) 11, E 0 c V ,where 0 is an open ball contained in V with zero as center. With this t o as initial time now and $ as initial function in 0,there exists some control u E U such that the solution z(t0,11,, w ) of (8.2.1) satisfies z t o ( t O , l l , , u )= $ , x t l ( t O , $ , v ) = 0. Using -+
w(s) =
{
0,
[a,toI1 u(s>, s E [tOItll, sE
we have w E L,([u,tl], Cm),and the solution x = z(u,4,w) satisfies 4, xtl = 0. The proof is complete.
I,
=
We now restrict our attention to the state space W;') and use the control space L2([u,tl],E m )for some tl > u + h. We have: Theorem 8.2.2 In (8.2.1), assume that: (i) f : E x C x Em -+ En is continuous and continuously differentiable in the second and third arguments; and there exist integrable Ni : E [O,oo), i = 1,2,3, such that
-+
IIDzf(t, 4, w)ll L Nl(t) + Nz(t)lwl, ( I D 3 f ( t 1dl w)ll 5 N3(t), V t E El w E Em, 4 E C. (ii) f(t, 0,O) = 0, V t 2 u. (iii) The system
+
i ( t ) = Dzf(l,O,O)~t D3f(tr0,0)v(t), with u E L z ( [ u , t l ] Em), , is controllable in the state space w ~ ( ~ ) ( [ - ~ , o I , E ~ ) . Then the system .(t) = f ( t ,x t 7 4 t ) )
(8.2.6)
(8.2.1)
314
Stability and Time-Optimal Control of Hereditary Systems is locally null controllable with constraints; that is, there exists an open ball 0 center zero in Wil' such that all initial functions in 0 can be driven to zero with controls in
v = {.
E L2([U,tlI,Em): 11.112
5 1).
Proof: The proof parallels that of the state space C except that here the map F may not be FrCchet differentiable. More precisely, if u E L 2 ( [ 0 , t l ]En), , then the corresponding solution of (8.2.1) z ( u ) is an absolutely continuous function with derivative in L2([O,tl],En),so that %(ti) E W,(')([O,tl],En). Consider the mapping
F : L Z ( [ u , t l ] , E " -+ ) W;')([-h,O],E"), defined by Fu = z t ( u ) . To proceed as in the previous case, we need Fu to be Fr6chet differentiable. Because of the norms of L2 and W;", this requirement will place very stringent conditions on f . Unless u appears in an affine linear fashion in f , FrCchet differentiability is impossible because of Vainberg (191.We use Gateaux derivative, which does not require such stringent conditions, and we shall see that this will suffice. For this, let F'(u) denote (formally) the Gateaux derivative of q(u) E W;'), with respect to u. Then we have F'(u) : L2 --t W;') exists and is given by
F'(u)v = D,z(t, u ) ( v ) = Z(t,U, ?I), where the mapping t ~ ( t , u , vof ) E into En is the unique solution of (8.2.2). This assertion follows from the following argument: The solution z ( u ) = z(u, 4, u)of (8.2.1) is given as the integral --f
.(t) = d ( t > , t E [ - h , 01, (8.2.11)
315
Controllable Nonlinear Delay Systems We note that
where 6f denotes the Gateaux differential o f f . Now
where 6f denotes the Gateaux differential off. Since f : E x C x Em is continuously FrCchet differentiable
6f ( . l l o ( t ) , h ( t ) ) =
Z t ( U o ) , uo(t))h
where Duf denotes the F'rCchet derivative with respect t o u ( t ) ([14, Lemma 6.31). Thus we obtain
6T(uo,h)= lim(T(u0
1' T'O
=
+ 7-h)- T(uO)>/7-,
Duf(t,21(uo)r u o ( t ) ) h ( t ) d t ,
since condition (i) of Theorem 8.2.2 holds and the Lebesgue Dominated Convergence Theorem is available. We now prove that 6T(.,-) is continuous at (uo, 0) E L2([0,TI) x L 2 ( [ 0 ,TI), so that by Problem 6.61 of [14], 6T(uo,h ) = T'(uo)h, T'(u0) E B ( L 2 , Will) : T'(u0) : L2 + Wi') is a bounded linear map, from L2 into Wi'). Suppose (uk,hk) (u0,O). Then +
since Duf(S,z,(uk),'Ilk(S))hk(S) + 0 as k + 00 and assumption (i) of Theorem 8.2.2 and the Lebesgue Dominated Theorem is valid.
Stability and Tame-Optimal Control of Hereditary Systems
316 We note that
so that from the continuity of D , f in u we have: for every r > 0 and u E B,(uo) = {u : IIu - uollp 5 T } , there exists some finite constant L < 00 such that
- T'(u0)ll I L.
[IT'(.)
(8.2.12)
We now observe that the solution z ( u ) of (8.2.11) is Frdchet differentiable with respect to u ( t ) E Ern. Indeed, if D, denotes this partial derivative, then by Dieudonne [9]
Using the assertions previously proved,
6T(uo,v) = T'(u0)v = D,z(u)w, and
d dt
-[quo)v]
= D 2 f ( t ,X i ( U ) , u(t))Duzt(u)v+ D 3 f ( t ,z t ( u ) ,u ( t ) ) v .
> h and u -+ xi(.) are the mapping F : L2([0,T],Em) --+ W,(')([-h,O]E")given by F u = zi(u) where z ( u ) is
If u E L2([0,TI,E m ) , T
the solution of (8.2.1), then by what we have proved
F'(u)v = Duzt(u)v = yt(u, v), where y is a solution of the variational equation (8.2.2) and
q,Ern)
F y u ) : L2([0,
+
W,(')[-h,01.
Evidently F'(0) is a bounded linear surjection if and only if the control system (8.2.2) is controllable on [ a , t l ] .To sum up, consider the mapping
F : L 2 ( [ u , t l ] , E m+ ) W,('). F'(0) : L z ( [ u , t l ]Em) , + W2 (1) ,
Controllable Nonlinear Delay Systems
317
is a surjection by condition (iii), so that F satisfies all the requirements of Corollary 15.2 of [lo, p. 1551, an open-mapping theorem. Thus for uo 0 E L z ( [ a , t l ] , E m ) ,F(uo) = F ( 0 ) = 0 E Wj'), to every r > 0 and ) 0 E L2 and radius r , there is an open ball B(0,r) c L Z ( [ a , t l ] , E mcenter open ball B(0,p) c Will center 0 of radius p such that
=
where , X ( t 1 ,a ) = { Z t l ( U , O , U ) : 21 E L 2 ( [ ~ , t l 1 , E r n ) llullz
5 T,
~ ( ua)solution of (8.2.1)}.
(8.2.13)
We have proved that for any r > 0, 0 E I n t x ( t l , a ) where the attainable set is as defined in (8.2.13) above with controls in
Since T is arbitrary, we can select it t o be 1. What we have proved implies that 0 E Int V ,where Z, is the domain of null controllability, i.e., the set of all initial functions 4 such that the solution z(a,q+,u) of (8.2.1) with u E u a d satisfies
Indeed, suppose not. Then there is a sequence
as n + co and no 4,, is in V . Since c(a,O,O) = 0 is a solution of (8.2.1), 0 E V . We can therefore assume that 4n # 0 Vn. Thus, et,( a ,$ n , u)# 0, for any u E U a d l and any t i > h. I f t n Zt1(6,4n,u),tn = ~ t l ( ~ , 4 n , -+o ) e t , ( a ,0,O)
=
(from continuity and uniqueness of solution).
Because
318
Stability and Time-Optimal Control of Hereditary Systems
E d ( t l , ~ ) u. We now have a sequence {&} E Wi') that has the following property: ~ t ~ ( c , O ,= o )0
&,
-+
0 as n
+ 00,
&, # 0 for any n , <, $ d(t1,a) for any tl > h.
We conclude that 0 @ Int d(t1,e)for any t l > h, a contradiction. This concludes the proof that (8.2.1) is locally null controllable with constraints.
Remark 8.2.1: If D 3 f ( t , 0,O) 3 B ( t ) is continuous, a necessary and sufficient condition for Wil) controllability on [ u , t l ] , t l > u + h is rank B ( t )= n on [ t l - h , t l ] . See Theorem 6.2.1. This condition is strong. It is interesting to know whether the controllability condition (iii) of Theorem 8.2.2 can be weakened. Indeed, it can be replaced by the closure of the attainable set of (8.2.6). One such sharp result is obtained for the simple variant of (8.2.6), namely
where
Theorem 8.2.3 In (8.2.1), assume conditions (i) and (ii) of Theorem 8.2.2. Also assume that its linearized system is (8.2.14) where t + Ao(t), A l ( t ) , B ( t ) are analytic on [O,tl]. The rank of B ( t ) is constant on [tl - h , t l ] . ImAl(t)ri(t)B(t) c ImB(t), i = 0 , . . . ,n- 1, for all but isolated points in [e,tl],where the operator r is defined by
+d
r(t)= - ~ ~ ( t )-, dt and Im H denotes the image of H . Then Equation (8.2.1) is locally null controllable with constraints when the initial data is restricted to a certain subspace Y of W,'). Proof: The proof is as before, but because (iii) - (iv) replaces the controllability condition, the mapping F : L ~ ( [ u , t lE] m , ) W;') does not have its Gateaux derivative F' a surjection. Instead, F'(L2([a,t l ] , E m ) Y -+
319
Controllable Nonlinear Delay Systems
is closed as guaranteed by (iii) and (iv) and Theorem 2 of [13]. As a consequence, Y is a subspace of W,(')([-h,01, En).Consider the mapping F : L2([0,tl],Em) 4 Y of one Banach space into another. We have that F' : L2([u,tl],Em) --i Y is a surjection and satisfies all the requirements of Corollary 15.2 of [lo, p. 1551. The proof is concluded a8 in the previous proof of Theorem 8.2.2. The controllability assumption of Theorem 8.2.2 was relaxed in Theorem 8.2.3 by requiring the closure of the attainable set of the linearized equation. The theory of ordinary differential control systems suggests a further relaxation. In this theory, the linear system
i ( t ) = Az(2) + Bu(2) is locally null controllable with constraints, and the domain of null controllability is open if and only if rank [ B ,A B , . .. ,A"-'B] = n. Here the controls are taken t o be in the unit cube C"'. This rank condition, which is equivalent to controllability with Lz unrestrained controls, is also equivalent to null controllability. But in functional differential systems, controllability is not equivalent t o null controllability. It is interesting to assume the weaker condition of null controllability. Thus the problem posed for Equation (8.2.1) can be stated as follows: Suppose (8.2.6) is null controllable. Does the same version of Theorem 8.2.2 hold? The next result in the space C states further conditions that will guarantee this. In (8.2.6) we let
D2f(t, 0,O)Zr =
lh
cs,rl(t,S ) Z ( t
+ s),
(8.2.15)
where ~ ( -) t ,is of bounded variation on [-h, 01, left continuous on ( - h , 0), and ~ ( 2 0) , = 0. We need the following notation: Let the integer j E [l,n] be fixed. Let z E En. Let xlz be the projection of z onto its first j components, and let x2z denote the projection of z onto its last n - j components. We write z = (z1,z2) where z1 = r l z , x 2 = x22. Define f l
: 2 + C([-h,O],E'),
by
(%i+)(s)
T2 : C([-h,O],
E") + C([-h,O],E"-')
= T ; + ( s ) , i = 1,2.
The following result is extracted from the statement and proof of Theorem 2.1 of Underwood and Young [15]. It is the main contribution of Chukwu [7].
Stability and Tame-Optimal Control of Hereditary Systems
320
Theorem 8.2.4
Consider the system
and its linearization (8.2.6):
.(t) = Dz f ( t ,0,O)z
+ B(t)u(t).
(8.2.6)
Assume: (i) that conditions (i) - (ii) of Proposition 8.2.2 hold. (ii) For each t and s , the range o f f is contained in the null space of ~ ( 81, t, the function defined in (8.2.5). (iii) F o r a n y u E L z ( [ u , t l ] , E m satisfyingu(t) ) = O f o r f s t 5 t l , andany y E C ( [ u- h , t l ] ,En) satisfying y ( t ) = 0 for T- h 5 t 5 tl there exists nosolution z o f i ( t )= f ( t , y t + . z t , ~ ( t ) ) - - D a f ( t , O , O ) y t - B ( t ) . l L ( t )on [u- h,tl] that satisfies both z ( t 1 ) = 0 and z t , # 0. (iv) The only solution z E C ( [ a- h , t l ] ,E n ) of 2 ( t ) = D 2 f ( t ,O,O)zt, u 5 t 5 tl are z ( t 1 ) = 0 that is constant on [u- h, a] is z = 0. (v) Suppose that tl - t o > 3h, and suppose there exist functions f i :E
x EJ x C([-h,O],E"-j) x Em 4 E J ,
f2 : E
x En-j x Em-+ E n - j ,
such that
(vi) Let (8.2.6) be null controllable on [ t o , tl - 2h]. Then (8.2.1) is locally null controllable with restraints on [ t o , t l ] ,i.e., with controls in a unit closed sphere o f LZ([to,tl],E m ) with center the origin. Furthermore,
if (vii) the trivial solution of
.
= f(t,zt,O)
(8.2.16)
is globally, uniformly, exponentially stable so that for some M 2 1, cr > 0 , the solution of (8.2.15) satisfies
32 1
Controllable Nonlinear Delay Systems (viii) In (i) - (iii) above, t o is sufficiently large so that 11 - t o (8.2.1) is globally null controllable with constraints.
> 3h.
Then
Remark 8.2.2: If (8.2.6) is null controllable on every interval [to,t l ] , tl > t o + h , then condition (iv) of Theorem 8.2.4 prevails if t l - t o > 3h. Condi-
tions are available in Theorem 6.2.3 and its corollaries. Thus if we assume that (8.2.6) is null controllable, we need not assume t o large as in (viii). There are various criteria for the stability requirement of condition (viii). They are contained in [S]. They are needed for the global result. In Theorem 8.2.2, a global constrained null controllability theorem can be deduced by imposing the required global stability hypothesis.
8.3
Controllability of Nonlinear Systems with Controls Appearing Linearly
We now consider the special case of (8.2.1) of the form
.(t) = g ( t , z t , .Il(t))+ B(t,zt).(t).
(8.3.1)
Here g : E x C x C" -+ En is a nonlinear function. B(.,.) : E x C is an n x m matrix function. We assume that f ( t , Z t , .Il(t))
-+
EnXm
= d t , Q , 4 t ) ) + B ( t ,zt).Il(t)
satisfies all the hypotheses of Section 8.2 for the existence of a unique solution. It is summed up in the following statement: Proposition 8.3.1 In (8.3,1), assume that (i) B : E x C -+ E n X mis continuous; B ( t , .) : C ---t E n x m is continuously differen tiable. (ii) There exist integrable functions Nj : E [0,m) i = 1 , 2 such that -+
IIDZB(t14)II 5 N l ( t ) , IIB(tl4)ll 5 W t ) , V t E E , and 4 E C ( [ - h , 01, En). (iii) g ( t , ., .) is continuously differentiable for each t . (iv) g ( . , 4, w ) is measurable for each 4 and w . (v) For each compact set I< c E n , there exists an integrable function Mi : E -+ E+ and square integrable functions Mi : E [O,m), i = 2 , 3 such that -+
322
Stability and Time-Optimal Control of Hereditary Systems
Under assumptions (i) - (v), for each u E L2, I$ E C, there exists a unique solution x = x(u,4, u) of (8.3.1); that is, an absolutely continuous function 1: : .[ - h , w ) + En such that (8.3.1) holds almost everywhere and xu = 4. The proof is given in Underwood and Young ~51. ) xt(u,I$,u)E C is continuously They also show [15,p. 7611 that ( 4 , ~ + differentiable. Let the matrix H be defined as follows:
H =
l1
B ( s ,4)B*(s,4 ) d s ,
tl
> 6,
(8.3.2)
where B* is the transpose of B , and (b E C([-h,O],E"). The full rank of H is needed to prove that (8.3.1) is Euclidean controllable.
Theorem 8.3.1 In (8.3.1), assume the following: (i) Conditions (i) - (v) of Proposition 8.3.1 on f and B are valid, and there is a continuous function N,'(t) such that
IIB*(t,9)ll 5 N f(t), vt E El 4 E c. (ii) The matrix H in (8.3.2) has a bounded inverse. (iii) There exist continuous functions Gj : C x Em E+ and integrable functions a, : E + E+, j = 1 , . ' . , q, such that I g O , 4, u(t))I i
P
C aj(tIGj(4, ~ ( t ) ) ,
j=1
for all ( t , 4, u ( t ) ) E E x C x Em,where the following growth condition is satisfied: 4
Then (8.3.1) is Euclidean controllable on [a, t l ] , tl
> u.
Remark 8.3.1: If g is uniformly bounded, then condition (iii) is met. Proof: Let E Will, xl E En.Then the solution of (8.3.1) is the solution of the integral equation 4t
+
= 4(t),
t E [-h,OI,
6)
z ( t ) = 4(0)
+J
0
t
g ( s , 2 5 , u(s))ds
+J
U
t
B(s, xs)u(s)ds,
t2
6.
(8.3.3)
Controllable Nonlinear Delay Systems
323
A control that does the transfer is defined as follows: [XI - 4(0)
u(t) = B * ( t , z t ) H - '
-1 tl
g(s,z#,u(s))ds]
,
(8.3.4)
U
where z(.) is a solution of (8.3.1) corresponding t o u and 4. We now prove that such a u exists. It is obviously an L-2 function, since t B ( t , 4 ) is continuous. We need t o prove that such a u exists as a solution of the integral equation (8.3.4). --$
Proof: Introduce the following spaces:
x = C([-h,t11, with norm
Il(4, .)I1
En)x LZ([U,tll, Ern),
+ I I ~ I zwhere ,
= 11411
We show the existence of a positive constant, ro, and a subset A(r0) of X such that A(ro) = Al(t1, r o ) x A2(tl, ro), where
Al(t1,ro) = {€ : [-kt11
--+
En continuous
tq = 4, IICII 5 T O ,
Az(t1,ro) = {u E Lz((0,t1l,Em),(i), lu(t)l 5
To
t E [a,tl]) 8.e. in t E [u,tl],
J'' Iu(t + s) - u(t)lZdt + 0
as s + 0 uniformly with respect to u E Al(tl,ro)}. It is obvious that the two conditions for A2 ensure that A2 is a compact convex subset of the Banach space L2 ([12,p. 2971). Define the operator T on X as follows: and
where
v ( t ) = B * ( t , z t ) H - l [ z l - 4(0) -
J U
11
g(s,z,,u(s))ds .
(8.3.6)
324
Stability and Time-Optimal Control of Hereditary Systems
Obviously the solutions z(-) and u(.)of (8.3.3) and (8.3.4) are fixed points of T , i.e., T ( z ,u)= (2, u). Using Schauder’s fixed-point theorem we shall prove the existence of such fixed points in A . Let
Because the growth condition of (iii) is valid there exists a constant ro such that ro-
4 i= 1
ciFi(r0) 2 dor
4
>0
C ciF,(vo)+d 5 ro. See a recent paper by
i=l
Do [18,p. 441. With this rg, define A(r0) as described above. To simplify our argument we introduce the following notation:
If (z,u ) E A ( r o ) ,from (8.3.5) and (8.3.6)’we have
Controllable Nonlinear Delay Systems
325
Also
TO 2r0 <+ - = -.
- 3
3
3
We now verify that
uniformly with respect to v E Az(t1,ro). Indeed,
where
€ = [Zl- d(0) Because t
-+
B*(t,xt) and t
+ ~t
l1
d s , Is,u(s))ds]
are continuous, we assert that indeed
This proves that v E A2, and we have completed the proof that T maps A(r0) into itself. We next prove that T is a continuous operator. This is obvious if
B(tld), ( t , A u ) g ( t , 4, u ) are continuous since u z(., u) is continuous. To prove continuity in the general situation, we argue as follows: Let (2,u ) , (d, u’) E A(ro), and
( t ,4)
+
+
-+
T ( z , = ( Z ( t ) , u ( t ) ) , T(z’,.’> = ( z ’ ( t ) v’(t)), , ( z ( t ) ,v(t)) = T ( z ,u), ( ~ ’ ( t~) ,’ ( t = ) )T ( d ,ti’).
326
Stability and Tame-Optimal Contml of Hereditary Systems
Then
Because u + B * ( t , t y )and u ( t ) -+ g ( t , t t , u ( t ) )are continuous, given any c > 0 there exists an g > 0 such that if lu(t) - u ( t ) [ ,then l B * ( t , t y )- B*(t,tY‘)I < 6 , Ig (t , t y, u )-g (t ,t y ’ , u ) l
< 6 , V t E [r,ti].
Divide [ D , t l ]into two sets el and e2; put the points at which )zt(t)-u’(t)l g to be e l and the remainder e2. If we write 1121 - 21’112 = y. then
so that mes
e2
<
5 y2/g2. Consider the integral I =
l1
lg(s,t$,u’(s))- g(s,t;,u(s))l2ds.
Then
, (t’, u’)E A(r0). If we insert this last estimate in for some R since ( t u), (8.3.7), we deduce that if lu(t) - u’(t)( < g, then
Iv(t)- v’( t )l2
5 c2 1x1 - $(0 )l2
+c2(t - r)R2+ N,’ (t)[c2(t- +4y2R2/g2]. 1
1
0)
Controllable Nonlinear Delay Systems
327
Thus
+ 2 ( t l - a)R2 +
- u)
l1
+4yR2 v2
N;(s)ds.
Since y2 = 11u - ~ ’ 1 1and ~ N2+ is integrable, v , v ’ can be made as close as possible if u, u’ are sufficiently close. We next consider the term J z ( t ) - z ’ ( t ) l .
Because of this inequality and an argument similar to the above, z , z ’ can be made as close as possible in A1 if u,u’ are sufficiently close. We have proved that T is continuous in u. It is easy t o see that T is continuous in x , the first argument, and thus, by a little reasoning based on the continuity hypothesis on g and B , that T ( x ,u ) is continuous on both arguments. To be able to use Schauder’s fixed-point theorem we need to verify that T(A(r0))is compact. Since Az(t1, ro) is compact, we need only verify that if (2,u) E A(r0) and ( z ,v ) = T ( z ,u), then z as defined in (8.3.5) is equicontinuous for each ro. To see this we observe that for each ( x , u ) E A(r0) and s1 < s2, we have ~ 1 , Es [a,ti], ~
(8.3.8)
328
Stability and Time-Optimal Control of Hereditary Systems
In the above estimate we have used the fact that
and
It now follows that the right hand of this last inequality does not depend on particular choices of ( c , u ) . Hence, the set of the first components of T ( A ( r 0 ) ) is relatively compact. Thus T(A(r0))is compact, which by an earlier remark proves that T is a compact operator. Gathering results, we have proved that T : A(r0) 4 A(r0) is a continuous compact operator from a closed convex subset into itself. By Schauder's ) T(z,u), given by fixed point theorem, there exists a fixed-point ( t , ~= (8.3.3) and (8.3.4),
Euclidean controllability is proved. Our next effort is t o obtain criteria for controllability in Wil) for System
(8.3.1). Theorem 8.3.2 In (8.3.1), assume that (i) Conditions (i) and (iii) of Theorem 8.3.1 are valid. (ii) rank [ B ( t , ( )= ] n OR [tl - h , t l ] for each 6 E C, t E [tl - h , t ~ ] . (iii) ( + B + ( t , ( ) is continuous for each t where B+ is the generalized in verse. Then (8.3.1) is controllable on [ u , t l ]with tl > u h.
+
Proof: The first step is to show that (8.3.1) is Euclidean controllable on [ a , t l - h]. Indeed, let 4 E Wi'), 2 1 E En.Then the solution 2 of (8.3.1) is given by (8.3.3). Because of the rank condition (ii), B ( t , ( ) B * ( t , 0 such that
Controllable Nonlinear Delay Systems
329
for 0 5 s < 6 , B ( t l - h - s,xtl-h-,)B*(tl Because of this,
- h - s,t t l - h - * )
has rank n.
H(t1 - h ) = B ( s ,z,)B*(s,z,)ds
+
/tl-h ti-h-c
B ( s ,z b ) B * ( szd)ds ,
has rank n , since the last integral is positive definite and H(t1- h ) is positive semidefinite. By Theorem 8.3.1, System 8.3.1 is Euclidean controllable on there exists [a,tl - h ] , tl > a h , so that for given any 4,$ E a u E Lz([a,tl - h ] , E m )such that the solution of (8.3.1) satisfies z, = 4, z(t1 - h,a,q5,u) = $(-h). To conclude the proof we extend u and z(.,a,4,u ) = z(.)to the interval [ u , t l ] ,tl > u h, so that
+
+
$(t--1) = g ( t , t t , u ( t ) ) + B ( t , z t ) u ( t ) a.e. on [tl - h , t l ] where z(~) = + ( r - t l ) , tl - h a control u can be defined as follows:
(8.3.9)
5 T 5 t l . Since (ii) holds, (8.3.10)
for tl - h 5 t 5 t l . That such a u exists can be proved as follows: We define the following set:
Al(r-0)= { u E L2([tl - h , t l ] ,Em): I l u ( t ) l l ~_<~ro and with 11
lu(t + s) - u ( t ) J d t
-+
k - h
0 as s -+ 0 uniformly with u E Al(r0)).
It follows from [12,p. 2971 that A = Al(r0) is compact. Let T be a map on A defined as follows:
where
v ( t ) = B+(t>z;)[ll(t - t l ) - g ( t , z;, u(t))l. We shall prove that there is a constant ro such that with A = Al(ro), T : A A , where T is continuous. Because of [12,p. 2971 and [12,p. 6451, T is guaranteed a fixed point, that is -+
330
Stability and Tame-Optimal Control of Hereditary Systems T ( u ) = (u)E A ,
which implies that (8.3.9) and (8.3.10) hold. Observe that A is a compact and convex subset of the Banach space L2. Because of a result of Campbell and Meyer [17,p. 2251 and hypotheses (i) and (ii) of the theorem, the generalized inverse t + B+(t,<)is continuous and therefore uniformly bounded on [tl- h,tl]. Since the growth condition (iii) is valid, there exists an ro > 0 such that 9
CcjFj(r0) + d <
PO
j=1
for some d. With this rg, define A = A l ( r 0 ) . Now introduce the following not at ions :
Controllable Nonlinear Delay Systems where
Therefore
Therefore
llT(u)Il L To. Hence T : A
+ A,
if we can verify the second condition. Now
33 1
332
Stability and Time-Optimal Control of Hereditary Systems
The function u ( t ) E B + ( t , t t ) ( ( t )is measurable in t and is L 2 . We can therefore choose a sequence { k , ( t ) } of continuous functions such that
as n + 00. Therefore
We choose n large so that the last and first integral on the right-hand side of this inequality are less than an arbitrary E > 0. Also s can be made small enough for the second integral t o be less than E > 0. This completes the verification of the first part that T : U -+ U . We now turn t o the problem of continuity. Let (u), (u’)E A(ro), T ( u )= (v), T(u’) = (v’). Then
Since u -+ Bt(t,zy) and u ( t ) + g ( t , z t , u ( t ) are continuous, given there exists an 7 > 0 such that if lu(t)- u’(t)l < v, then
6
>0
333
Controllable Nonlinear Delay Systems
Divide I into two sets el and e2, and put the points at which lu(t)-u’(t)l q to be el and the other t o be e 2 . If we set Ilu - ~ ’ 1 1 2= y, then
so that mes e2
<
5 y2/v2. A simple analysis shows that
n
It follows from these estimates that llw - ~ ‘ 1 1can ~ be made arbitrarily small if IIu - ~ ’ 1 )is~ small. This proves that T : A --t A is a continuous mapping of a compact convex subset of L2 into itself. By Schauder’s fixed-point theorem [12, p. 6451, T has a fixed point. We invoke Schauder’s fixed-point theorem to conclude that
proving that u is well defined. With this u (8.3.6) is satisfied. As a result of Theorem 8.3.2 and Theorem 8.3.1, the following conclusion is valid:
Theorem 8.3.3 In (8.3.1), assume that (i) g(t,O,0) = 0, V t 2 0. (ii) Conditions (i) - (iii) of Theorem 8.3.1 hold. (iii) The equation i ( t )=g(t,zt10)
(8.3.11)
is such that each solution satisfies
xt-0
as t--+cQ.
(8.3.12)
Stability and Time-Optimal Control of Hereditary Systems
334
Then (8.3.1) is locally null Euclidean controllable with constraints, i e . , with controls in
Proof: From the proof of Theorem 8.3.1, for each d, E W;'), z1 E E", there is a control that transfers d, t o 2 1 in time t l . The control that does the transfer is given in (8.3.4):
Since q5
+
xt(q5, u) is continuous, u is a continuous function of d, : H :
is continuous. It is continuous at zero. For each neighborhood V of zero in L z ( [ a , t l ] , E m )there is a neighborhood 0 of zero in such that , : IIu112 5 1). Choose an H ( 0 ) C V . Let Uad = {u E L % ( [ a , t l ]Em) open set V , a neighborhood of zero in Lz that is a subset of Uad such that H ( 0 ) c V . Every q5 E 0 can be steered to zero by the control
u = H ( 4 ) = B * ( t ,2t)H-'F, where
Hence there is a neighborhood 0 of zero in Will such that a control u E Uad drives every point of 4 E 0 to zero in some time t l . Local null controllability with constraints is established. Now use the control u = 0 E U a d in (8.3.1). Then every solution of (8.3.11) is driven into 0 in some finite time to. With this point xto E 0 as an initial point of W;'), one reaches zero in E" in time t l .
Remark 8.3.2: Since Theorem 8.3.2 is also valid, a theorem analogue to Theorem 8.3.3 can be formulated and proved as above. We note that
is implicitly a function of before.
4; 4 --+ Hq5 = u is continuous and one argues as
Controllable Nonlinear D e l a y S y s t e m s
335
REFERENCES 1. E. N. Chukwu, “Finite Time Controllability of Nonlinear Control Process,” SIAM J. Control 13 (1975)807-816. 2. H. Hermes and J. P. LaSalle, Functional Analysis a n d Time Optimal Control, Academic Press, New York, 1969. 3. S. Lang, Analysis 11,Addison-Wesley, Reading, MA, 1969. 4. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, New York, 1967. 5. W. Rudin, Real a n d Complex Analysis, McGraw-Hill, New York, 1974.
6. E. N. Chukwu, The Time Optimal Control of Nonlinear Delay Equations an Operator Methods for Optimal Control Problems, edited by Sung J. Lee, Marcel Dekker, 1988. 7. E. N. Chukwu, “Global Null Controllability of Nonlinear Delay Equations with Controls in a Compact Set,” J. Optimization Theory and Applications 53 (1987) 43-57. 8. E. N. Chukwu, “Global Behavior of Retarded Functional Differential Equations,” in Differential Equations a n d Applications, Vol. I,edited by A. R.Affabizadeh, Ohio University Press, Athens, 1989. 9. J. Dieudonne, Foundations of Modern Analysis, Vol. I, Academic Press, New York, 1969. 10. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. 11. M. Q. Jacobs and T.-J. Kao, “An Optimum Settling Problem for Time Lag Systems,” J . Math. Anal. Appl. 40 (1972)687-707.
12. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press Book, Macmillan, New York, 1964. 13. S. Kurcyusz and A. W. Olbrot, “On the Closure in Wp of Attainable Subspaces of Linear Time Lag Systems,“ J . Differential Equations 24 (1977)29-50. 14. G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York, 1972. 15. R. Underwood and D. Young, “Null controllability of nonlinear functional differential
equations,” SIAM J . Control Optim. 17 (1979)753-772.
16. J. P. Dauer, “Nonlinear Perturbations of Quasilinear Control Systems,” Anal. Appl. 54 (1976)717-725.
J. Math.
17. S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations, Pitman Publishing, London, 1979. 18. V. N. Do, “Controllability of semilinear systems,” J . Optimization Theory and Applications 65 (1990)41-52. 19. M. M. Vainberg, “Some problems in differential calculus in linear spaces,” Uspehi Mat. Nauk 7(4), 55-102 (in Russian).
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Chapter 9 Control of Interconnected Nonlinear Delay Differential Equations in Wf) 9.1 Introduction Our main interest in this section is the resolution of the problem of controllability of interconnected nonlinear delay systems in function space, from which the existence of an optimal control law can be later deduced. We insist that each subsystem is controlled by its own variables while taking into account the interacting effects. This is the recent basic insight of Lu, Lu, Gao, and Lee [5] for ordinary differential systems. Controllability is deduced for the overall system from the assumption of controllability of each free subsystem and a growth condition of the interconnecting structure. Two applications are presented. In one, the insight it provides for the growth of global economy has important policy implications. For linear-free systems, criteria for Wil) controllability have been provided and is reported in Section 6.2. For nonlinear cases, a similar investigation was recently carried out in [2], and presented in Section 8.3. Recently Sinha [7]treated controllability in Euclidean space of composite systems in which the base is linear. We extend the scope of the treatment in [7]by treating interconnected systems with delay when the state space is W j l ) and the base system is not necessarily linear. We state criteria for controllability of the free subsystem by defining an L2 control that does the steering both in the linear and nonlinear case. We prove that such a control exists as a solution of an integral equation in a Banach space. For this we use Schauder's fixed-point theorem. Assuming the free subsystem is controllable and the interaction function has a certain growth condition, we prove the controllability of the overall interconnected system. We begin with a simple system. A linear state equation of the ith subsystem of a large-scale control system can be described by
i i ( t ) = A l i z ' ( t ) + A2izi(t - h ) + A 3 i y i ( l ) + A 4 i y i ( t - h ) + Biu'(t), (9.1.1) where ~ ' ( tE) En' is the ni-dimensional Euclidean state vector of the ith subsystem, ui E Em*is the control vector, and Ali A2i Bi A B A4i ~ are timeinvariant matrices of appropriate dimensions. Also, y'(t) is the supplementary variable of the ith subsystem and is a function of its own Euclidean
337
Stability and Time-Optimal Control of Hereditary Systems
338
state vector z i ( t ) and other subsystem state vector z J ( t ) j = 1,.. . , L We express this as follows:
where Mii, Mij ( j = 1 , 2 , . . . ,el j # i ) are constant matrices. By substituting (9.1.2) into (9.1.1) we obtain the state equation of the overall interconnected system
+
i i ( t )= H ; z i ( t ) Gizi(t - h ) where
Hi = Ali rni(t>=
+ Biu'(t) + m;(t)+ e;(t - h ) ,
(9.1.3)
+ A3iMii, Gi = Azi + A4iMji,
e
e
j=1
j=1
C ~ 3 i ~ i j z j ( ei(t t ) , - h) = C Mijd(t - h) j#i
j#i
In (9.1.3), rni(t) and ei(t - h ) describe the interaction, the effects of other subsystems on the ith subsystem. This can be measured locally. The decomposed system (9.1.3) can be viewed as an interconnection of t-isolated subsystems
i i ( t )= Hiz'(t) + Gixi(t - h )
+ Biui(t),
with interconnection structure characterized by
which does not depend on the state variables z i ( t ) . We now consider the more general free linear subsystem
i i ( t ) = L i p , xf)+ BiU'(t), = ($, and the interconnected system
(Si)
Control of Interconnected Nonlinear Delay Differential Equations
339
where
We assume & ( t ) is an ni x mi continuous matrix. The linear operator 4 + L,(t,4) is described by the integral in the Lebesgue-Stieltjes sense, where ( t , 8 ) +-. q , ( t , 8 ) is an ni x ni matrix function. It is assumed that t + v i ( t ,6), t E E , is continuous for each fixed 8 E [-h,O], and 8 --+ v i ( t , # ) is of bounded variation on [-h,O] for each fixed t E E. Also, vi(t,O) = 0, 8 2 0, v i ( t , 8 ) = qi(t, - h ) , 0 2 -h, and 8 -, qi(t,O) is left continuous on (-h,O). It is assumed that var vi(tY4) I pi(t), t E E ,
BEE
where p i ( t ) is locally integrable. These conditions also hold for v i j . An easy adaptation of the argument in [ l ]yields the following result on the system ( I j ) :
Theorem 9.1.1 Consider the interconnected decomposed system ( I i ) in which B:(t) is essentially bounded. Suppose r ankB i( t) = nj on [tl - h , t l ] , so that B '
is continuous. Then ( I i ) is controllable on
[ c r , t l ] , 11
>n +h
Proof: Let X i ( t , s) be the fundamental matrix solution of
2 ( t ) = Li(t,Z f ) . Then
J,
tl-h
G i ( a , t l )=
Xi(t1 - h , s ) B i ( s ) B i ( s ) * X ; ( t l- h , s)ds
has rank ni, so that ( I i ) is Euclidean controllable. Here B* is the algebraic adjoint of B. This is proved by letting 8 E W i l ) E W$')([-h,01, Pi), zf E E n i , and by defining a control
d ( t ) = [Bf(s)Xf(tl- h , t ) ] G - ' ( ~ , t-l u ) [ x ~ ~'(t1,~,4,0)
Stability and Time-Optimal Control of Hereditary Systems
340
where x i ( t ,o,d,0) is the solution of ( L i ) with u' G 0. Using the variation of parameter, one verifies that ui indeed transfers 4' to zi in time tl -h. Thus there is a control ui E L2([a,tl - h],E".) such that z'(t1 - h , a , 4 , u i )= qbi(-h). We extend ui and xi to the interval [a,tl],so that
+'(t - t l ) = z ' ( t ) , tl - h
and
G'(t - t l ) = L&f)
5 t 5 tl,
+ Bj(t)U'(t)+ g ' ( t )
a.e. on [tl - h , t l ] . To do this note that
L'(t, g+) = -qi(t, -h)&h)
-
J
ti-h
1-h
r)i(tl,CY
..
t
-L
and
- h
- t)i'(a)da
q i ( i , a -r)+'(a)da, tl-h
5 t 5 tl,
L
s i ( t ) = jC= 1J _ : d ~ q ~ j ( ~ , ~ ) x j ( t + ~ ) , j#i
=-
{
T i j ( t, -4)s(t - h )
j=1 j #i
t
+
qij(t,CY - t ) i J ( a ) d a
V j j ( t , CY - t ) i j ( a ) d e
'11-h
Now define
for t 1- h 5 t 5 t l . Because of the smoothness properties of x i , xJ , and @, ui is indeed appropriate. Thus the controllability of the large-scale system can be deduced from that of the subsystems so long as the interconnection is as proposed. We now turn our attention to the nonlinear situation.
Control of Interconnected Nonlinear Delay Differential Equations
9.2
341
Nonlinear Systems
Consider the general nonlinear large-scale system,
where f : E x C x Em En is a nonlinear function, g : E x C x Em +. En is a nonlinear interconnection, and the n x m matrix function B : E x C -+ Enxmis possibly nonlinear. Conditions for the existence of a unique solution z ( . , u , d , u ) , when u E L z , E~ C([-h,O],E"),are given in Proposition 8.3.1. It is shown there that ( 9 , ~ )+ z t ( - , u , u ) E C is continuously differentiable. These conditions are assumed t o prevail here. -+
Remark 9.2.1: Note that conditions of Proposition 8.3.1 imply that for all
t E [a, t l ] ,d E C([-h, 01, I<), w E Ern,
we have
If(t7 d l Q ) l 5 Ml(t)lldll + Mz(t)lwl.
System (9.2.1) may be decomposed as
where fi
:E x C([-h,O],Eni)x Em' -+ Eni,
gij
:E x C([-h,O],Eni) x Em' +. Eni,
Bi :E x C([-h,O],Eni)
+ .
EniXmi.
342
Stability and Time-Optimal Control of Hereditary Systems
Let
j=1
j#i
g(t,Z t , +))
= [g1(t,Z t , Ui(t))T,.. . , ge(t,zt,u'(t))T],
B ( t , c t ) = diag[Bl(t,z'), . . . ,Be(t,ze)J. Then we can view (9.2.1) with decomposition 9.2.2 as an interconnection of! isolated subsystems ( S i ) described by the equation
i'(t) = fa@,&
u'(t))
+ B,(t,Zcf)U'(t),
1
(si
with interconnecting structure characterized by
Conditions for the existence and uniqueness of solutions are assumed. In particular, t + gj(t,xt,u i ( t ) )is assumed integrable. Conditions for controllability of each isolated subsystem (S,) are stated in Theorem 8.3.1 and Theorem 8.3.2. We defined a matrix H i :
(9.2.3) for each
4,
=
E C([-h, 01, Pi) C"'.
Theorem 9.2.1 In (S,), assume that: (i) There is a continuous function N;,(t) such that IIBT(t,@)I15 N;{(t),
v& E en'. (ii) Hi in (9.2.3) h a s a bounded inverse.
Control of Interconnected Nonlinear Delay Differential Equations
343
(iii) There exist continuous functions Fij : C x Em;+ E+ and integrable functions a, : E + E+, j = 1 , . . . , q, such that
for all ( t , 8 ,u i ( t ) )E E x C x Em' where the following growth conditions are satisfied: : ~ ~ ( q 5 ,5 u )r} ~~
r-cu
Then (Si) is Euclidean controllable on [ a , t l ] . We now consider criteria for controllability in Wil) for System ( S i ) .
Theorem 9.2.2 In (Si), assume: (i) Conditions (i) and (iii) of Theorem 9.2.1. (ii) rank [ B i ( t , ( ) ] = R; on [tl - h , t l ] for each ( E C ( [ - h , O ] , E n S ) ,t E [tl - h , t l ] . (jii) The Moore Penrose generalized inverse of Bi , B: ( t ,() is essen tially uniformly bounded on [tl - h , t l ] , for each ( E C, t E [tl - h , t l ] . (iv) E + B + ( t , ( ) is continuous. Then (Si) is controllable on [ a , t l J with , t l > CT + h . In Theorem 9.2.2 we have stated conditions that guarantee the controllability of each isolated free subsystem ( S i ) . Next we assume these conditions and give additional conditions to the interconnection g i that will ensure that the composite system (9.2.2) is controllable. It should be carefully noted that L
g i ( t , x t , ~ ' ( t )=) C g i j ( t ,4 : v i ( t > ) j=1 j#i
is independent of xi the state of the i t h subsystem, though it is measured locally in the ( S i ) system.
Theorem 9.2.3 Consider (9.2.1) and its decomposition (9.2.2). Assume that: (i) Conditions (i) - (iii) of Theorem 9.2.1 are valid. Thus each isolated subsystem is Euclidean controllable on [ a , t l ] .
Stability and Time-Optimal Control of Hereditary Systems
344
(ii) For each i, j = 1 , . . . , l ,
i
#j, L
gi(t,X:f,v')
= Cgij(t,diu'(t)) j=1 j#i
satisfies the following conditions: There are continuous functions
C'j : En'x Em'-+E+ and L' functions Pj : E
-+
E+ j = 1 , . . . ,q, such that
Igij ( t ,4'9 v')I
P
5 C Pj Cij($', u') j=1
e,u i ) ,
for a11 ( t ,
< mi, where
Then 9.2.1 is Euclidean controllable on [u, tl]. The proof is contained in [14]and is similar t o the proof of Theorem 9.2.1. Theorem 9.2.4 Consider (9.2.1) and its decomposition (9.2.2). Assume that: (i) Conditions (i) - (iv) of Theorem 9.2.2 bold. (ii) Condition (ii) of Theorem 9.2.3 holds. Then (9.2.1) is controllable on [U,tl],
tl >
+ h.
Proof: Just as in the proof of Theorem 9.2.2, define H(t1- h) =
J,"-" B ( s ,Xt)B*(S,
Xt)dS
in place of Hi(t1 - h ) , and conclude Euclidean controllability on [r,tl - h]. Define u as
for tl - h 5 t <_ tl in place of u i ( t ) , and deduce controllability. The growth condition (ii) of g comes in handy in the argument t o conclude the proof as before.
Control of Interconnected Nonlinear Delay Differential Equations
345
General Nonlinear Systems
9.3
In (9.2.1) it is very important that the system is of the form in which some term is linear in u. Here we consider the more general situation (9.3.1) .(t) = f(CZt u ( t ) ) , En is continuously differentiable in the second 9
where f : E x C x Em 4 and third arguments, and is continuous, and also satisfies all the conditions of Proposition 8.3.1. Details of the proof of the following are contained in Proposition 8.2.1.
Theorem 9.3.1 In (9.3.1), msume that: (i) f ( t , O , O ) = 0, V t 2 cr. (ii) The system Z ( t ) = L ( t ,% t ) B ( t ) v ( t ) is controllable on [cr, t l ] ,where tl 2 cr h, and where
+
(9.3.2)
+
D2f(4 0 , O ) Z t = q t , D S f ( t ,0,O)v = B ( t ) v . Then
(9.3.3)
0 E Int d(t,G), where
A(t, u) = { x t ( g , 0, u ) : u E L 2 ( [ 6 , t ] Em) , I I u l l ~_<~ 1 is a solution of (9.3.1) with I , = 0)
I ( . )
(9.3.4)
is the attainable set associated with (9.3.1).
We shall now investigate the large-scale system
+
L
.'(t) = f i ( t y z : , u i ( t ) ) E g i j ( t , g { , u J ( t ) ) , where
fi
(9.3.5)
j=1 i #j
and gi, are as defined following (9.2.2). Thus if L
g i ( t , zt
v ( t > )=
C gij ( t ,4
,
uj (t)>
j=1 j f i
and g ( t , zt,v ( t ) ) = bl(t,z t ,~ ( t ) ). .~. , g ~ ( tz, t , ~ ( t ) ) ~ by] considering , (9.3.5) we are investigating (9.3.6) .(t> = f(t,I t , .(t)) + g ( t , I t , v ( t ) ) , where f is identified following (9.2.2). We state the following result:
346
Stability and Time-Optimal Contml of Hereditary Systems
Theorem 9.3.2 Consider the large-scale system (9.3.6) with its decomposition (9.3.5), where (i) fi(t, 010) = 0, gij(t,470) = 0. (ii) f i , gi satisfy all the requirements of Proposition 8.3.1. (iii) Assume that the linear variational system i ' ( 2 ) = L i ( t , zf)
+ B'(t)u(t),
(9.3.7)
of
i i ( t ) = f&, z:,u'(t)),
(9,3.8)
where
+
is controllable on [u, t l ] , t l > u h. Then the interconnected large-scale system (9.3.5) is locally null controllable with constraints. Corollary 9.3.1 Assume: (i) Conditions (i) - (iii) of Theorem 9.3.2. (ii) The system
i y t ) = fi(t, zf, 0)
(9.3.9)
is globally exponentially stable. Then the interconnected system is (globally) null controllable, with controls in ui = {u' E ~ 2 ( [ u , t lE"'), I, 1 I u i I I ~ a 5 1). Proof of Theorem 9.3.2: By Theorem 9.3.1,
0 E Int d i ( t ,u) for t > u
+ h,
(9.3.10)
where di is the attainable set associated with (9.3.8). Let x i be the solution of (9.3.5), with Z$ = 0. Then
Control of Interconnected Nonlinear Delay Differential Equations
347
Thus, if we define the set
we deduce that di(2,a) C
fJi(t1,
a).
Because fi(t,0,0) = g , j ( t , 4 , 0 ) = 0, and because zi(t,O,O) = 0 is asolution of (9.3.5), 0 E Hi(t1,a).As a result of this and (9.3.10), we deduce
0 E Int d i ( t 1 , u )C
Hi(t1,~).
(9.3.11;
There is an open ball B(O,r), center zero, radius r , such that
The conclusion
0 E Int Hi(t1,u)
(9.3.12)
follows at once. Using this, one deduces readily that 0 E Int V ,the interior of the domain of null controllability of (9.3.5), proving local null controllability with constraints. Proof of Corollary 9.3.1: One uses the control u* = 0 E Ui, i = 1 , . . . , C to glide along System (9.3.5) and approach an arbitrary neighborhood 0 of the origin in Wit'([-h, 01, Pi). Note that
Because of stability in hypothesis (ii) of (9.3.9), every solution with ui= 0 is entrapped in 0 in time a 2 0. Since (i) guarantees that all initial states in this neighborhood 0 can be driven t o zero in finite time, the proof is complete.
Remark: Conditions for global stability of hypothesis (ii) are available in [lo, Theorem 4.21.
Chukwu
Remark 9.3.1: The condition (9.3.11) from which we deduced that
0 E Int Hi(t, a),
(9.3.12)
348
Stability and Time-Optimal Control of Hereditary Systems
is of fundamental importance. If the condition
(9.3.13)
O E Int d i ( t , a)
fails, the isolated system is "not well behaved" and cannot be controlled. Condition (9.3.12) may still prevail and the composite system will be locally controllable. To have this situation, we require
0 E Int G i ( t ,a),
(9.3.14)
where : uj
E U,
In words, we require a sufficient amount of control impact (i.e., (9.3.14)) t o be brought to bear on ( S i )that is not an integral part of S;. Thus knowing the limitations of the control ui E U i , a sufficient signal
L
C g i j ( t ,z i , u j ) =
j=1 j+i
g ; ( t , z t , u )is dispatched t o make (9.3.14) hold. And (9.3.12) will follow. Remark 9.3.2: The same type of reasoning yields a result similar to Theorem 9.3.2 if we consider the system
where (i) f ( t , O , O )= 0. (ii) g , ( t , z i )... ,z:-1,o,z:+1,...
,":,Ul(t)
,... , u ' - ' ( t ) , O , u ' + ' ( t ) ,... , J ( t ) ) =o, g . ( t , z t ,... ,z:,o,o (...,0 )
Also conditions (ii) and (iii) of Theorem 9.3.2 are satisfied. If we consider
instead of (9.3.51, we can obtain the following result:
=o.
349
Control of Interconnected Nonlinear Delay Daflerentaal Equations
Theorem 9.3.3 In (9.3.16), assume (i) (9.3.7), L;(t,2 : ) and B’(t) are defined as
(iii) of Theorem 9.3.2. But in
-
D2fi(t,0, o>.f = L;(t,Z f ) ,
D3(fi(t,0,O)) + S i ( t , x t , 0)) =
W).
Then (9.3.16) is locally null controllable with constraints. The proof is essentially the same as that of Theorem 9.3.1. We note that the essential requirement for (9.3.16) to be locally null controllable is the controllability of
+
.(t) = J q t , r t ) (Bl(t)
+ B2(t))U(t),
(9.3.17)
where
& ( t ) = D3f(t10,0), B2(t) = D39(t,r t , 0). If the isolated system (9.3.8) is not “proper” (and this may happen when B l ( t ) does not have full rank on [ a , t l ] ,t > u + h ) , the solidarity function g; can be brought to bear to force the full rank of B = B1 B2, from which (9.3.16) will be proper because (9.3.17) is controllable. Even if B1 has full rank and (9.3.8) is proper, the interconnected system need not be locally null controllable. The function has to be so nice that B2 B1 has full rank. An adequate proper amount of regulation is needed in the form of a solidarity function g;. In applications it is important to know something about g; and to decide its adequacy. It is possible to consider gi as a control, and view
+
+
+
.i(t) = f i ( t t x f , u i ( t ) ) gi(t) as a differential game. Considered in this way, we must describe how big the control set for g; must be. In the linear case see Chukwu [13]. The
linear case is treated in Section 9.5.
9.4
Examples
Example 9.4.1: Fluctuations of Current The flow of current in a simple circuit in Figure 9.4.1 is described by the equation d2x(t) dx(t) 1 ( R R1)- x ( t ) = o. L- dt2 dt c
+ +
+
350
Stability and Time-Optimal Control of Hereditary Systems
FIGURE 9.4.1. The voltage across R1 is applied to a nonlinear amplifier A . The output is provided a special phase shifting network P . This introduces a constant time lag between the input and output of P . The voltage across R in series with the output of P is
e( t) = q g ( i ( t
- h)),
q is the gain of the amplifier to R measured through the network. The equation becomes
L-d2x(t) dt2
+ R i ( t ) + q g ( i ( t - h ) ) + -1x ( t ) = 0. C
Finally, a control device is introduced to help stabilize the fluctuations of the current. If i ( t ) = ~ ( tthe ) ~ "controlled" system is now given by
i ( t ) = Y(t> + % ( t ) , R q Y(t)= --y(t) - -g(y(t L L
- h))
1 - -CL x(t)
+
(9.4.1) U2(t)
Control of Intercon.nec2ed Nonlinear Delay Differential Equations
351
L
C
t-
FIGURE 9.4.2. [CHAPTER 1 , REF. 61. The question of interest is the following: With the control u = ( u l , ~ ) , which is “created” by the stabilizer, is it possible to bring any “wild fluctuations’’ of the current (any initial position) to a normal equilibrium position in finite time? Is (9.4.1) controllable?
352
Stability and Time-Optimal Control of Hereditary Systems
Example 9.4.2: Fluctuations of Current in a Network
A six-loop electric network is connected up as in Figure 9.4.1 and Figure 9.4.2. Each loop is a simple circuit as in Figure 9.4.1, and is described by the equation
The voltage across Rl; is applied t o a nonlinear amplifier Ai. The output is provided t o a special shifting network Pi. This introduces a constant time lag between the input and the output of Pi. The voltage across Ri in series with the output of Pi is ei(t) = qig(ii(t- h ) ) ,q; is the gain of the amplifier t o R, measured through the network. The equation becomes
+ R1i”t) + qigi(i”t Li 7 d2xi(t)
- h))
+ -12 ’ (‘t ) Ci
= 0.
For this ith loop, a control device is introduced to help stabilize the fluctuations of current. If i ’ ( t ) = y i ( t ) ,the “controlled” i-subsystem is
i q t ) = y”t)
8 y”t) = --y“t) Li
+ u!(t), 1 . . - -4ig i ( y y t - h ) ) - ---z“t) ’
Li
ci Li
+ .‘zi)(t).
The question of interest is the following: With a control ui = ( u i , u i ) , which is “created” by the stabilizer, is it possible to bring any “wild fluctuations” of current (any initial posit,ion) t o a normal equilibrium position of the ith subsystem in finite time? Is the ith-subsystem controllable? Furthermore, we study the overall large-scale system in Figure 9.4.3, which we consider as a decentralized control problem, where each subsystem is influenced by interaction terms gi(t,x ) = C g;j(t,x j , u i ) that are monitored (measured) j=1
j#i locally. Is the large-scale system controllable? Proposition 9.4.1 Assume (i) R, L i , q ; , and C; are positive and q i ( 4 ) is continuous; (ii) gij(t,@ , ui) satisfies the growth condition of Theorem 9.2.4. Then the interconnected system is controllable, and each subsystem is controllable. Proof: Note that the matrix
Control of Interconnected Nonlinear Delay Differential Equations
gii*o
FIGURE 9.4.3.
353
354
Stability and Time-Optimal Control of Hereditary Systems
has full rank.
Example 9.4.3: Let z ( t ) denote the value of capital stock at time t . Suppose 2 can be used in one of two ways: (i) investment; (ii) consumption. We assume the value allocated t o investment is used t o increase capital stock. Let u(t) 0 5 u ( t ) 5 1 denote the fraction of x(t) allocated to investment. Capital accumulation can be said t o satisfy the equation
For example, g ( z ( t ) , u ( t ) ) h ( t ) z ( t ) ( E constant) shows that the rate of increase of capital is proportional t o investment. Now if we take depreciation into account, and if at time a after accumulation the value of a unit of capital has decreased by a factor p ( a ) ,
the equation of capital accumulation can be represented by
so L
where f ( t - s , z ( t - s))p‘(s)dsis the value of capital that has “evaporated” per unit time at time t . Implicit is the assumption that equipment depreciates over a time L (the lifetime of the equipment) to a value 0. More generally, we have
(9.42) where we use a Riemann-Stieltjes integral. Equation (9.4.2) is a special case of ( 9 . 3 4 . Along with (9.4.2) are an initial capital endowment function q5 E Wil) and $ E W;’), its long run “desired stock of capital function”. In time-optimal control theory, one minimizes T (time) subject t o (9.4.2) and 20
= 4,
ZT
= $.
(9.4.3)
One can also minimize consumption rT
(9.4.4)
Control of Interconnected Nonlinear Delay Differential Equations
355
subject t o (9.4.2) and (9.4.3). System (9.4.2) is an isolated one. It may form part of a large-scale system
dxi -(t) dt
= gi(X’(t),U i ( t ) )-
J
0
L
f i ( t - S , ~ ‘ (-t s))dP,(s) (9.4.5)
and the problem of controllability can be investigated. From our analysis we have the following insight: If the isolated system is controllable and the interconnection “nice” then the large-scale system is also controllable. Detailed investigations of (9.4.5) will be pursued in Section 9.5, where comments on global economy will be made. See [ll] and [12,p. 6881. It is seen from Remark 9.3.1 that a sufficient amount of control regulation is needed from outside the subsystem t o make the large-scale system controllable. When this is lacking, and the subsystem is not well behaved, the controllability of the large-scale system is not assured. This has grave policy implications for global economy. Deregulation may not be a cure. It may lead t o global depression, or lack of growth of the economy.
9.5 Control of Global Economic Growth Introduction
In Section 1.8 we formulated a nonlinear general model of the dynamics of capital accumulation of a group of n firms whose destiny are linked with l other systems t o form a large-scale organization. As argued by Takayama [12],time lags are incorporated into the control procedure in a distributed fashion: The state and the control history are taken into account. Our aim is t o control the growth of capital stock from an initial endowment t o some target point, which can be taken t o be the origin. From the theory, some qualitative laws are formulated for the control of the growth of capital stock. They help validate the correctness of the new economic policies of movement away from centralization of firms of the East; they raise questions on the wisdom of deregulations in the West. The principles cast doubt on the popular myth that to grow, the firms of the Third World should dismantle the “solidarity function”.
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Stability and Time-Optimal Control of Hereditary Systems
In Section 1.8 we designated z ( t ) t o be the value of capital stock at time t , and then postulated that the net capital formation i ( t ) is given by
i ( t ) = k u ( t ) z ( t )- 6z(t),
(9.5.1)
where t and 6 are positive constants and -1 5 u(t) 5 1, u being the fraction of stock that is used for payment of taxes or for investment. A more general version of (9.5.1) is the system
i ( t ) = L(t,zt,'lLt)et+B(t,ct,u*)ut.
(9.5.2)
Later we denote z ( t ) = (zl(t), . . . , cn(t))to be the value of n capital stocks at time t , with investment and tax strategy u = (211,". ,un), where -1 5 u j ( t ) 5 1. We therefore consider (9.5.2) as the equation of the net capital function for n stocks in a region that is isolated. They are linked to t other such systems in the globe, and the interconnection or solidarity function is given by Si(Zlt,." , z i t , Ul t,... ,w). This function describes the effects of other subsystems on the ith subsystem as measured locally at the ith location. Thus, i i ( t ) = Li(t,lit,uit)zit
a = l
,... ,e
+ B i ( f , z i t , u i t ) ~ i+t g i ( t , z l t , . . . , ~ e t , ~ l t , " ,uet), ' (9.5.3)
is the decomposed interconnected large-scale system whose free subsystem is ii(t)= L i ( t , x i t , u i t ) ~ i t B i ( t, zit, uit) uit. (9.5 Si)
+
We now introduce the following notation: e e Let C ni = n , C mi = m. i=l
i=l
95 = [h,. . . ,954 E E",
Then (9.5.3) is given as
2
= [Xl,.. . ,ze] E E " ,
u = [Ul,.. . ,ue] E E",
Control of Interconnected Nonlinear Delay Diflerential Equations
357
Preliminaries The problem of controllability of (9.5 S) will be explored in this section. Conditions are stated for the controllability of the isolated system (9.5.2). Assuming that the subsystem (9.5 Si) is controllable, we shall deduce conditions for (9.5 S) to be controllable when the solidarity function is “nice.” The following basic assumptions will be maintained: In (9.5.2),
L ( t , 4, $ ) Z t =
1, 0
M t , s>4, $ ) 4 t
+ s),
where q(t, s, 4,$) is measurable in ( t , s ) E E x E normalized so that
q ( t , s, +,$) is continuous from the left in s on (-h, 0), and V ( t ,s, 4, $) has bounded variation in s on [-h, 01 for each t , 4, $, and there is an integrable rn such that M t , s,4, $ ) Z t l 5 m(t>llxtll for all t E (-m, m),4, 4,zt E C. We assume L ( t , ., .) is continuous. The assumption on B ( t ,4, $) in (9.5.2) is continuity on all the variables. Also continuous is the function g : E x C x Em -+ En. Enough smoothness conditions on L and g are imposed t o ensure the existence of solutions of (9.5 S) and the continuous dependence of solutions on initial data. See [19] and [S], and Section 8.3.
Main Results To solve the Euclidean controllability problem for the system (9.5.2), we consider the simpler system
.(t) = q t , 2 , v)zt
+ B(t,
2,
v)v,
(9.5.4)
where the arguments x t and ut of L and B have been replaced by specified functions z E C([-h,O],E”) C , v E L,([-h,O], Em). Here C is the space of continuous En-valued functions on [ - h , 01, and L , is the space of essentially bounded measurable Em-valued functions on [-h, 01. For each ( z , v) E C x L,, the system (9.5.4) is linear and one can deduce its variation of parameter by the ideas of Klamka [23]. Let X ( t , s ) = X ( t , s , z , v) be the transition matrix for the system
i ( t ) = L ( t , z , V)Zt,
(9.5.5)
Stability and Time-Optimal Control of Hereditary Systems
358 so that
ax(t, at
= q t , ., z , .)X,(.,
where
0,
s-h
X ( t , s )= and where X,(.,s)(B) = X ( t (9.5.4) is given by
.(t> = .(t, c,9,O)+
(9.5.6)
s),
+ e,s),
J, X ( t , t
-h
< 0 5 0.
Then the solution of
[/:
+
1
d e H ( s , Z ( e ) , v(e)>u(s 0) ds,
i t i tl, ~ ( t=)4(t), t E [c- h , ~ ] ,~ ( t=)~ ( t ) ,t E [C - h , ~ ] . c
(9.5.7) Here ~ ( Ct T, , ~0), is the solution of (9.5.5). The last term in (9.5.7) contains values of ti on [c- h , 01. As usual in such matters 171,[16],[20],and [23], we use Fubini’s Theorem to write (9.5.7) as
.(t) = .(t, c,4,O) +
{ J_, ~ ( t , e)de[H(s 0
/ O
O+S
- e , z ( e > ,V ( ~ ) I V ( ~ ) ~ S }
s-
Thus (9.5.8) can be written as .(t)
= z ( t , c , $ , O )+ q +
rt
Ju S ( t ,
s, z,v)u(s)ds.
(9.5.11)
The controllability map of (9.5.4) at time t is
W ( t )= W ( e , t ,z , #) =
J,’ S ( t ,
s , z , V>S*(t, s,z, u p s ,
(9.5.12)
Control of Interconnected Nonlinear Delay Differential Equations
359
with W(t1)= W , where the star denotes the matrix transpose. Define the reachable set R(t) of (9.5.4) as follows:
where
= {u measurable
u
E Em Iuj(t)l
5 1,
j = 1 , . . . ,m}.
Thus the reachable set is all solution X(U) of (9.5.4) with q5 = 0, r] = 0 as initial data and with controls u E L,([a,tl], Cm),where Cm is the unit rn-dimensional cube. The Euclidean attainable set of (9.5 S) is the subset of En defined as follows:
d ( t )= { x ( u ) ( t ): u E
Uad : 3:
is a solution of (s) with zu = 0, r] = 0).
The domain D of null controllability is the set of initial 4, such that there exists a t l and and a u E L([a,t13, Cm)such that the solution c of (9.5 S) satisfies c,(u) = 4, z ( t 1 , u ) = 0. The following lemma is fundamental in the sequel: Lemma 9.5.1 In (9.5.4) (i) Maintain the basic assumptions of L and B. (ii) There exists a constant c > 0 such that det W ( a , t l ,v) 2 c ( z , v) E C x L,. Then 0 E Intd(t).
> 0 for all (9.5.13)
Also for (9.5.2), 0 E Int D.
(9.5.14)
Proof: From standard arguments, the nonsingularity of W = W ( t 1 ) f W ( a , t l ,z , v) is equivalent to the Euclidean controllability of (9.5.4) [22, p. 921 and of (9.5.4) being “proper” [22, p. 771. As usual, this is equivalent to
0 c Int R(t,z , v).
(9.5.15)
) C x L,, 0 E Int IR(t,z,v) for all Because (ii) is uniform for all ( 4 , ~ c That (9.5.15) holds follows from the fact that the map
( 4 , ~ E) C x L,.
T : L , +En,
( T u ) ( t )=
J,’ X ( t ,
7, z ,
v ) B ( r ,z , v)u(.r)d.r
(9.5.16)
Stability and Time-Optimal Control of Hereditary Systems
360
is a continuous linear surjection for each ( 2 ,w) E C x L,, which is therefore an open map for each ( z , w). The transfer of any arbitrary I$ E C to any z1 E En is accomplished by the control u ( t ) = S * ( t l , t , z , V ) W - l [ Z 1-.(t,4,0) u ( t ) = rl(t),
t
- Q(t,rl,Z,V)],
t E [u,t11,
E [c- h t l l ,
(9.5.17) which when inserted into (9.5.11) gives an expression of the solution (9.5.4):
+ d t , rl,
4t) = 4 t , 6,4,0)
-1,
).
+ W ( u , t ,~ , V ) W - l ( u , t lz ,, u ) [ 2 1 - Z(tl,tO,4,O) - Q(t1,rl, z , .)I.
(9.5.18) With initial d a ta w(u) = (z(u),q+,q)and z(t1) = 21, controllability of (9.5.4) is established. We now show controllability for (9.5.2). Suppose ( 2 ,u)solves the set of integral equations u ( t ) = S * ( t l , t , 2 , U ) W - l [ ~-1Z ( t l , f l , 4 , 0 ) - d h , r l , z , u ) ] for
t E [U,tll,
u ( t ) = v ( t ) , t E .[ - h a ] ,
(9.5.19)
and
+ q ( t , u , z ,u)+
z ( t ) = z(t,u,4,0)
z ( t ) = 4(t),
t S(t,S,Z,U)U(S)dS, t
2
6,
(9.5.20)
t E [u- h , u ] .
Then u is L,([u - h , t l ] )and 2 solves (9.5.2) with this u and with initial data ~ ( u=) (z(c),4, v), where u, = 7). Also z(t1) = 2 1 . Next we prove that a solution pair indeed exists for (9.5.19) and (9.5.20). Let I be the Banach space of all functions ( z , u ) : [u - h , t l ] x [u- h , t l ] -+ En x Em where z E C ( [ u- h , t l ] ,E n ) u E &([a - h , t l ] , E " ) , with norms defined as IKz, .>I1 = 1141 II~lloo,where
+
Define the operator P : IB -+ IB by P ( z ,u)= (y,w), where
Control of Interconnected Nonlinear Delay Diflerential Equations y(t) = ~ ( u, t ,4,O)
+ q ( t , 71,x,v) + /
361
t
S ( t ,s,z, u)v(s)ds and for E [u,111,
0
and y(t) E 4(t) for t E [u- h, u]. (9.5.22) By a tedious but standard analysis (see Sinha [7], and Balachandran and Dauer [15,16],we can identify a closed bounded and convex subset Q(X) of I5 where Q(X) = {(x,u)E IfB : 1141I A, l141caL X I 1 and P : Q(X) + Q(X), and P is a relatively compact mapping. We are then assured by Schauder's fixed-point theorem that P has a fixed-point: P(xl u)= (c,u).Indeed, let a1 = SUP{llS*(tl,t,2tl,~tl)ll, t E [u,t11}, a2
= IIW-1(u,t1,211,~tl)ll,
a3 =SUP{14t,u,4,0,O)I a4
= s.P{lls(t,
+ Idt,7I,~t,ut)l+1211,
s, Z t , W)1lr
t E [~,tlIl, ( t ,s) E [(.,tlI x [a,till,
b = max{(tl - c ~ ) u 4 1). , Then
5 alaZa3, t E [(7tl], Mt)l = 17I(t)l I S U P Il?(t>l= a5. Iv(t)l
t €lo- h ,ul
If ro = m a x [ a l a z a 3 ~ a ~ ] , then Iv(t)l
Also
L To, t E .[
- ktll,
II~IICUI To
Stability and Time-Optimal Control of Hereditary Systems
362
we have proved that T ( Q ( A ) ) c Q(A). It is clear that Q(A) is closed, bounded, and convex. Also P is continuous since (t,q5,v) + B(t,q5,v) is continuous. To prove that the set of y ( t ) defined in (9.5.22) is equicontinuous is routine. Observe that u defined in (9.5.21) is relatively compact 0 as since all such v are uniformly bounded in L , norm, and 11v, - v11, s -+ 0 uniformly, where --+
[4, pp. 296, 6451. Indeed, since v ( t ) is measurable in t and in L,, there exists a sequence {vn(t)}of continuous functions such that IIv - v,,ll, --+ 0 as n 00. Hence -+
The last and first terms can be made arbitrarily less than c by selecting n very large. The second term can be made less than E if s is selected small enough. This proves that llw, - wll, + 0 as s + 0. All this analysis verifies that P(&(A))is relatively compact. Invoke Schauder's fixed-point theorem t o conclude that there is a fixed point (2,u ) E P(&(A))such that P(z,u ) = ( 2 ,u ) . With this fixed point established, we verify (9.5.13) by arguing as follows: Since T is an open map for each (r,v) E C x L,, and in particular for (2, u ) = (z, u ) , we have that T defined by
Tu =
1
$1
S ( t , s , z , u ) u ( s ) d s , T : L,([t,,tl],Em)
+
E"
is open. Because of this, 0 E Int T(L,(a,tl),Cm) = d ( t l ) , where d ( t l ) is the reachable set of (9.5.2), so that
0 E Int d ( t 1 )
(9.5.23)
0 E Int V .
(9.5.24)
and Details of the needed arguments are contained in [19, p. 1111. Theorem 9.5.1 Consider the large-scale system (9.5.5S ) with its decomposition (9.5.3), and assume that: (i)
.
= 0, gs(t,I l t , . . . , z t t , 0 , . . . , O ) = 0.
g i ( t , z l t , Z i - l t , O , z i + l t , . . . , t t ( t ) , u l t , . . r u i - ~ t , O , ~ i + l t , " .rutt)
363
Control of Interconnected Nonlinear Delay Differential Equations (ii) The system
i i ( t ) = Li(t,
$7
v)xt
+ Bi(t,$ , v ) u ( t )
(9.5 Si)
satisfies all the conditions of Lemma 9.5.1 for each +,v E C x L,([a,tl],Em). (iii) gi : E x En x Em -+ En is continuous. Then (9.5 S ) is locally null controllable on [a,tl] with constraints. I f , in addition, the system ii(t)= L;(t,Z1t)Zit (9.5.25) is globally exponentially stable, then (9.5.5 S ) is (globally) null controllable , for some t l , llzlillo0 5 1, j = with controls in Ui = { u i E L M ( [ a , t l ]Em') 1,... , e } . Proof: Because of Lemma 9.5.1,
0 E Int d;(tl) for tl > u
+h,
(9.5.26)
where Ai is the attainable set associated with (9.5 Si). Let xi be the solution of (9.5.3) with xb = 0, and U: = q = 0. Then
Jo
Jo
(9.5.27)
If we define the set .
.
H,;(tl)= { ~ i ( u ' , ~ ) ( Et iEn* ) : U' E U i ,
U'
E U;, j
# i,
j = 1,.. .
,t},
we deduce that
di(ti)C Hi(ti). Because g i ( t , x l t , . . . , 0 , . . . ,zit, 211,. . . , 0 , . . . , u;e) = 0, and because zi(a,O,O)(t) = 0 is a solution of (9.5.3), 0 E Hi(t1). Because (9.5.26) holds, 0 E Int di(t1) c Hi(t1) so that
0 E Int H i ( t l ) .
(9.5.28)
Using this, one deduces readily that
0 E Int D.
(9.5.29)
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Stability and Time-Optimal Control of Hereditary Systems
The rest of the proof is routine. Appropriate the control ui E U;, ui z 0, i = 1 , . . . , l , and use this in (9.5.3); then the solution of (9.5.3) is that of (9.5.25). There is a time u < 00 such that qb x(u,4,0) E 0 where 0 c V ,and 0 is an open ball containing zero. With qb as initial state, there is a time tl > u and an n-tuple u = (211,. .. ,ue)control ui E Lm([u,t1],Cmi)= Ui such that the solution of (9.5.3) satisfies
4, ui) = 4, ~ ( t u, l , 4, ui) = 0.
lo(.,
The proof is complete. Note that Theorem 9.5.1 generalizes the main results in [7,15,18]. It removes the growth conditions of the nonlinear functions present in the equations. All that is needed here are the usual smoothness conditions required for the existence of solutions, and bigger “independent” control sets.
Remark 9.5.1: Our analysis describes Euclidean null controllability of large-scale systems. The results hold also for arbitrary point targets in En. Indeed, if 2 1 is a nontrivial arbitrary target and y ( t , 4 , u o )= y ( t ) is any solution of (9.5.30) .(t> = f(t,21 , 4 t ) ), with y , = 4 and with uo admissible, and y(t1) = 2 1 , one can equivalently study the null controllability of the system
44 = f ( t , z t + Y t , % )
- f(t,Yt,u(t)),
where z , = 0 so that 2, = yo = q5 and ~ ( t=) z ( t ) integrate (9.5.31) we obtain
Note that
z ( t , u,f$,uo,2 )= 0, tr t
(9.5.31)
+ y(t). Indeed. if we
2 0.
If we can show that there is a neighborhood 0 of the origin in z space such that all initial 4 E 13 can be brought to z ( t 1 , u , 4 , u 0 )= 0 by some admissible control u at time t l , then % ( i lu l , 4 , u O )=
that is, c ( t l )= y(t1) = 2 1 .
“ ( t l ,4,
- Y(t1) = 0,
365
Control of Interconnected Nonlinear D e l a y Diflerential Equations
Universal Laws for the Control of Global Economic Growth The main theorem and some assertions in its proof provide very useful broad policy prescription for the control of a large-scale economic system’s growth. It describes qualitatively when it is possible to control the growth of capital stock from an initial endowment to the long-run desired stock of capital, and when this control is impossible. Indeed, from the proof we deduce that if the isolated system (9.5 Si) is “proper”, and this is valid if (9.5.26) holds, then the large-scale system (9.5.3) will also be proper, i.e., (9.5.28) will hold provided the “solidarity function” gi is nice in the sense of being integrable, etc. From (9.5.28) one proves that the domain ID of null controllability of the large-scale system is such that 0 E Int D.
(9.5.32)
0 E Int Ai(t),
(9.5.33)
O E Int H i ( t )
(9.5.34)
The condition which implies when the interconnection is nice, yields the following obvious but fundamental principles for the control of large-scale organizations. Fundamental Principle 1 If the isolated system is “proper,” which means that it is locally controllable with constraints, and if the interconnection that we will call a “solidarity function” is nice and measured locally at the i t h subsystem and is a function of the state of all subsystems and their controls, then the composite system is also proper and therefore locally null controllable with constraints.
Some isolated system may fail to have property (9.5.33), and is not well behaved. Condition (9.5.34) on which (9.5.32) is deduced can still be salvaged. Obviously what is needed is a condition that ensures that
0 E Int Gj(t1)
(9.5.35)
where
With (9.5.35), one easily deduces (9.5.34) since 0 E di(t).A criteria such as in (9.5.35) can be deduced from a controllability of a linearization of the nonlinear control system ii(t)
= gi(t, z l t r . . . , Z i t , . . . , Z t t ( t ) , u l ( t ) ,. . . , .ili(t),. . . , u t ( t ) ) .
366
Stability and Time-Optimal Control of Hereditary Systems
In other words, we require a sufficient amount of control impact from external sources gi that is monitored locally a t the i-subsystem t o be “forced” on (9.5 Si). It works as follows: The external source of power senses the control ui available t o (9.5 Si) and constructs a sufficient controlling signal gi that is a function of all the states of the subsystems and their controls including u i . With this the right behavior is enforced. It is formalized in the following principle: Fundamental Principle 2 If some subsystem is not proper and is not locally controllable with constraints, the large-scale system can be proper a n d be made locally controllable provided there is an external source of suficient power available to the subsystem to enforce controllability and proper behavior. There is no theorem a t the present time that ensures that the large-scale system i s proper when some system is not proper, and there is no compensating proper behavior from an external source of power that i s monitored locally.
We observe that the solidarity function does not ignore the subsystems controls ui or initiative. Were it t o be ignored, for example, and
to be still operative and nontrivial, the condition 0 E H i ( t ) and subsequently (9.5.34) will fail. The large-scale system will not be proper and locally null controllable. Fundamental Principle 3 If subsystems’ control initiatives are ignored in the construction of a nontrivial solidarity function that is monitored and applied locally, the interconnected system will fail to be proper and locally null controllable.
Failure t o acknowledge individual subsystem’s control initiative in highly centralized economic systems that are described by our dynamics may lead to lack of growth from initial endowment 4 to the desired economic target (which is an equilibrium in this case) in centralized systems. On the other hand, failure t o enforce regulation bringing to bear the right force gi when some subsystems misbehave and are locally uncontrollable in individualistic systems may make the large-scale system locally uncontrollable and may trigger an economic depression. These observations seem t o have a bearing as an explanation of what is happening in the East and what may happen in the West as a consequence of the 1980-89 U S . dismantling of regulations (see [17,p. 2641 and [26]) on the economy. In the East, economic targets cannot be reached because of too much centralization: inflexible g i : t o make firms controllable, market forces are being advocated. In the
Control of Interconnected Nonlinear Delay Differential Equations
367
Third World, the popular policy prescription of dismantling the “solidarity function”, which rides the wave of deregulations of the 1980s in the West, is at work with fury. Aside from its hardships, it does not seem t o have a solid theoretical foundation. A certain amount of solidarity function is effective for economic growth. In other words, carefully timed government interventions (or solidarity functions) are crucial for economic growth. This can be provided by central governments that cherish individual initiatives. A qualitative description of the solidarity function will be explored in Section
9.6.
9.6 Effective Solidarity Functions In Equation (9.5.3) the dynamics of the ith subsystem are described by an equation of the form
where g ( t , z t , v t ) is the interconnection or the solidarity function. We now study simpler versions of (9.6.1) and in the process isolate the basic properties of effective solidarity functions. We consider them “controls,” “disturbances,” “quarry controls,” and appropriate an earlier formulation from ~31. The system we study is the linear one,
4 4 = q t ,2 1 ) + B(t)u(t)+ g ( t , v ( t ) ) ,
(9.6.2)
which can be written as
where
B ( t ) u ( t )= - P W , q(t>= g ( t , 44).
(9.6.4)
We assume P E L,([cr,tl], P ) , P C En,q E L,([cr,t~],Q), Q c En,and where for each t E E , the linear operator C$ + L ( t , C$), C$ E C, has the form
Here the integral is in the Lebesgue-Stieltjes sense, and q ( t , s ) is an n x n matrix. Basic assumptions on q are contained in Chukwu [13, p. 3271.
368
Stability and Time-Optimal Control of Hereditary Systems
Viewed in our setting, Q in [13]is now called a solidarity set, and P is a control set. Define the set
U : U = P.Q=
(. :
u
+
p ~)
~
(9.6.4)
(the Pontryagin difference of P and Q). Associate with (9.6.3) the control system .(t) = L ( t , .t) - U ( t ) , u ( t ) E U ( t ) . (9.6.5) Let X ( t , s) be the fundamental matrix of
i.e., the solution of
d - X ( t , s ) = L ( t , X t ( . , s ) ) , t 2 s a.e. in s , t , at 0 s-h
+
where X,( , s)(8) = X ( t 8 , s) - h 5 0 5 0. The statement and proof of the following theorem is available in [13].
Theorem 9.6.1 Assume 0 E Q and P compact. System (9.6.3) is Euclidean controllable a t time t l (on the interval [ o , t l ] )if and only if the associated linear control system (9.6.5) with u ( t ) E U ( t ) ,
U ( t )= ( P is Euclidean controllable
OR
p ( t ) E g ( q ,t )
+ ker X ( t 1 ,t)).Q,
(9.6.7)
[o, t l ] . Furthermore,
= u ( t )+ q
modulo ker X ( t 1 ,t ) .
(9.6.8)
For all q E Q, t E [ a , t l ]can be used to determine a suitable strategy from u E L,([o,tl], Cr) in (9.6.5) and vice versa. It is important to describe how the control p ( t ) that drives into a target x1 E En is achieved. From the knowledge of each initial endowment 4, the subsystem constructs a vector-valued function t -+ u ( t ) . Next, an unpredictable but observable regulation or solidarity function t -+ q ( t ) is measured locally by the subsystem. The controller of the subsystem responds simultaneously via a stroboscopic strategy using the solidarity function by taking p ( t ) = u ( t ) q(t) modulokerX(t1,t) 5 T ( q ( t ) , t ) .
+
Control of Interconnected Nonlinear Delay Diflerential Equations
369
One might object to this instantaneous observation and response. But if the subsystem is allowed to observe and retain the past history of its control p ( t ) and of the state variable 3: in (9.6.3), then as was argued by Hhjek [25, p. 561, it already has available to it almost everywhere the current value q ( t ) of the “regulation”. The problem then is that of the controllability of (9.6.5). This problem is solved in [20,21]. The applied problem of economic growth of the large-scale economy is reduced to the solution of the controllability questions of (9.6.5). The desirable solidarity function q ( t ) is constrained to lie in Q. Further properties of Q can be deduced from HAjek’s studies [25, p. 61, pp. 110-1481. I t is possible to take into account delays in the transmission of information [25, p. 951. Note also that the minimum-time problems for (9.6.3) and (9.6.5) coincide if P and Q are “nice”.
The Limitations of Q It is very important to determine how “big” the solidarity set Q should be. To determine it, we need the following definition and corollary to Theorem 9.6.1: Definition 9.6.1: Consider a mapping 7 : Q x E -+ P, ( ~ ( q , tis) a point in P whenever q is a point of Q and t E E ) . Let u be an induced mapping on the collection of Q of solidarity functions q ( . ) , which is defined by a[q](t)= ~ ( q ( t 1) ),. The function u is called stroboscopic if it preserves measurability. Suppose t
.+ u (t ) is
given in advance. Then U ( Q ) ( t )=
+44
is a stroboscopic strategy, if u is measurable and Q have:
+ u (t) C P for all t . We
Corollary 9.6.1 In (9.6.3), assume that 0 E Q, P is compact, and P,Q are nonvoid, convex, and symmetric. Then Q C P+kerX(tl,t), u _ < t L t ~
is necessary and sufficient for the presence ofinitial q5 that can be transferred to zero in time tl when the subsystem uses stroboscopic strategy.
Proof: Theorem 9.6.1 asserts that the set of functions q5 that can be forced to zero at time t l via (9.6.3) and the set of initial functions that can be steered t o zero within the control system 9.6.5 are the same. But
U ( t ) = ( P + kerX(t1,t))EQ
370
Stability and Time-Optimal Control of Hereditary Systems
is convex and symmetric. It is nonvoid if and only if 0 E V(t), i.e.,
Q C P+kerX(tl,t). Conversely, if Q c P + k e r X ( t l , t ) , then 0 E ( P admissible control steering 0 to 0. Corollary 9.6.2
+ kerX(t1,t))EQ is an
The condition
Q C Int(P + kerX(tl,t)), u 5 t _< tl is necessary and sufficient for the domain of null controllability of (9.6.3) to have zero in its interior. The given condition is valid if and only if
0 E Int U = Int(P+ kerX(t1,t)ZQ) But 0 E Int U is needed for (9.6.5) to have zero in the interior of the Euclidean reachable set, and therefore in the interior of the domain of null controllability of (9.6.5) that coincides with that of (9.6.3). Note that the Euclidean reachable set is
R(tl) =
{ l1
1
X(t1,s)u(s)ds : u ( s ) E U ( s )
These two corollaries give the last rule of thumb and fourth universal principle for the control of economic growth of interconnected systems. Fundamental Principle 4
No initial endowment can be controlled to the
origin in our dynamics unless
To guarantee control to the equilibrium of each initial endowment, one must require that
Q c I n t ( P + kerX(t1,t)). I n words, to guarantee economic growth from a n y initial endowment to a target it is necessary that subsystem’s control set, together with the system’s internal power for waste, dominate the solidarity set or external power. Loosely translated, the subsystem’s power should dominate the effects of government ‘‘power’’. Power may be understood to be investment and or consumption/taxation.
371
Control of Interconnected Nonlinear Delay Differential Equations
REFERENCES 1. H. T. Banks, M. C. Jacobs, and C. E. Langenhop, “Characterization of the Controlled Stat,es in Wi’) of Linear Hereditary Systems,” SIAM 2. Control 13(3) (1975) 611649. 2. E. N. Chukwu, “Control of Nonlinear Delay Differential Equations in W;’),’’ Preprint. 3. J. P. Dauer, “Nonlinear Perturbations of Quasilinear Control Systems,” Anal. Appl. 54(3) (1976).
2. Math.
4. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normal Spaces, Macmillan, New York, 1964. 5. Qiung Lu, J . N. Lu, Jixans Gao, and Gordon K.F. Lee “Decentralized Control for Multimachine Power Systems,” Optimal Control Application and Methods 10 (1989) 53-64. 6. N. B. Mirza and B. F. Womack, “On the Controllability of Nonlinear Time Delay Systems,” ZEEE Trans. Automatic Control AC-17(16) (1972) 812414. 7. A.S.C. Sinha, “Controllability of Large-Scale Nonlinear Systems with Distributed Delays,” in “Control and States” Proceedings of the 1988 International Conference on Advances in Communications and Control Systems, 11, October 19-21,1988, Baton Rouge, LA, edited by W. A. Porter and S. C. Kak. 8. R. G. Underwood and D. F. Young, “Null Controllability of Nonlinear Functional Differential Equations,” SIAM J. Control and Optimization 1 7 (1979) 753-772. 9. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. 10. E. N. Chukwu, “Global Behaviour of Retarded Functional Differential Equations,”
in Diferential Equations and Applications, edited by A. R. Affabizadeh, I, Ohio University Press, Athens, 1989.
11. K . L. Cooke and J. A. Yorke, “Equations Modelling Population Growth, Economic Growth and Gonorrhea Epidemiology,” in Ordinary Dzflerential Equations, edited by L. Weiss, Academic Press, New York, 1972. 12. A. Takayama, Mathematical Economics, Cambridge University Press, 1974. 13. E. N. Chukwu, “Capture in Linear Functional Games of Pursuit,,” .f. Math. Anal. Appl. 7 0 (1979) 326-336. 14. E. N. Chukwu, “Control of Interconnected Nonlinear Delay Differential Equations in Preprint.
~i’),”
15. K. Balachandran and J . P. Dauer, “Relative Perturbations of Nonlinear Systems,” Optimization Theory and Applications 63(1) (1989) 51-56.
J.
16. I<. Balachandran and J. P. Dauer, “Null Controllability of Nonlinear Infinite Delay Systems with Distributed Delays in Control,” J . Math. Anal. Appl. 145 (1990) 274-281.
17. R. Batra, Surviving the Great Depression of 1990, Dell Publishing, New York, 1988. 18. E. N. Chukwu, “Global Null Controllability of Nonlinear Delay Equations with Controls in a Compact Set,” J. Optimization Theory and Applications 5 3 (1987). 19. E. N. Chukwu, “The Time Optimal Control of Nonlinear Delay Equations,” in o p erator Methods for Optimal Control Problems, edited by Sung J . Lee, Marcel Dekker, New York, 1988.
372
Stability and Time-Optimal Control of Hereditary Systems
20. E. N. Chukwu, “Euclidean Controllability of Linear Delay Systems with Limited Controls,” IEEE Trans. Automatic Control AC-24 (1979). 21. E. N. Chukwu, “Function Space Null Controllability of Linear Delay Systems with Limited Power,” J. Math. Anal. Appl. 124 (1987) 293-304. 22. H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969. 23. J. Klamka, “Controllability of Nonlinear Systems with Distributed Delays in Control,” International J . Control 31 (1980) 811-819. 24. A. Manitius, “Optimal Control of Hereditary Systems,” Control Theory and Topics in Functional Analysis, Vol. I, International Atomic Energy Agency, Vienna, 1976. 25. 0. HAjek, Pursuit Games, Academic Press, New York, 1975. 26. K. Phillips, The Politics of Rich and Poor, Random House, New York, 1990. 27. E. N. Chukwu, “Control of Global Economic Growth: Will the Center Hold?,” Preprint .
Chapter 10 The Time-Optimal Control of Linear Differential Equations of Neutral Q p e In this chapter we study the time-optimal control problem of various forms of 1.9.1. Controllability questions are answered. For linear systems, the existence and forms of an optimal control are established in both Euclidean and in function space. The theory developed here was reported in [13]and
~41.
10.1
The Time-Optimal Control Problem of Linear Neutral Functional Systems in En: Introduction
In this section, the time-optimal problem is explored for the linear neutral system of the form
+
d (10.1.1) 21) = L ( z ,Z t ) B ( t ) u ( t ) , t 2 u, 2, = 4, dt where the control set is a unit rn-dimensional cube and the target is a continuous set function in an n-dimensional Euclidean space. Necessary and sufficient conditions for the existence and uniqueness of optimal controls are stated and proved. The bang-bang form of optimal control is given. For zero target, easily computable conditions are stated for the system t o be proper or normal. Also proved is a sufficient condition for (10.1.1) t o be Euclidean null controllable. As an application we consider a differential pursuit game described by the equation
-D(t,
d
-dtq t ,
= L ( t ,.t)
21)
- P ( t ) + q ( t ) , t 2 u,
q E L ! X u ,m),Q ) , P E L & W , a), PI, P, Q
2,
= d,
C En.
(10.1 .lb)
We show that the differential game is equivalent to a control system of the form (10.1.1) where the control set U is a Pontryagin difference of sets. In ( l O . l . l ) , B E Ll([u,v],EnX"), u 5 v , and D ( t ,4) = d(0) - g ( t , 4 ) ,g ( t , d), L ( t , .) are bounded linear operators from C into En for each fixed t in [0, m), g ( t , 4) is continuous for ( t ,4) E [0, m) x C . g ( i , 4) = 19(t!4)I
J
0
-h
dbCL(t, s)d(s), L ( t , 4) =
J
0
-la
d S V ( t , s)4(s),
5 q1411, lqt,4)I 5 L(t)ll4ll, (t,4) E [O,m) x
373
c,
374
Stability and Time-Optimal Control of Hereditary Systems
for some nonnegative constant k, continuous nonnegative £ and J.l(t, .), 1](t,') are n x n matrix functions of bounded variation on [-h, 0]. We also assume that 9 is uniformly nonatomic at zero, that is, there exists a continuous, nonnegative, nondecreasing function £(s) for s E [0, h] such that £(0) = 0,
11~
d,J.l(t, T)¢(T)!
~
£(s)II¢II·
We shall consider the set of admissible controls that are measurable functions u : [T, t] _ C'", t ~ T, where C'" is the unit cube in Em given by c m = {u E Em : IUil ~ I}. Such controls will simply be denoted by u E C'", Finally we assume that g(t)
== B(t)u(t)
is locally integrable from [0",00) into En, where u E Lp([O", 00), En). Under the above hypotheses, there exists a continuous function x : [0" h,oo) - En such that Xu =' ¢, which is a solution of system (10.1.1) for all t ~ 0". If we denote the solution x = x(O", ¢, u), then for each t ~ 0", ¢ E C, u E Lp([O",t]'E n) Xt(O",¢,u) = Xt(O",¢,O) + Xt(O",O,u). Define the operators
T(t,O") : C _ C, K(t,O"): Lp([O", t], Em) - C by
Xt(O", ¢, u) = T(t, O")¢ + K(t, O")u.
Then T, K are bounded linear operators. The solution of System (10.1.1) is also given by
x(t, 0", ¢, u)
= Y(O", t)D(O", ¢) + 1~~
d.//(t, s)¢(s) +
(10.1.2)
1 t
Y(s, t)B(s)u(s)ds, (10.1.3a)
where
//(t, s) == -
ft+
J
u
da(O', t)J.l(O', s)
ft
+J
u
Y(O', s)1](O', s)dO'
and Y is an n x n matrix defined by:
Y(O", t) Y(t, t)
= I + Ju~t+
= I,
daY(O', t)J.l(O', 0") _
Y(O", t)
=0
for 0" > t.
t.~t Y(O',t)1](O', O")dO',
(10.1.3b)
The Time-Optimal Control of Linear Differential Equations
375
All integrals are understood to be Lebesgue-Stieltjes integrals. The function Y(O", t) is Borel measurable subject to the conditions given in System (10.1.1). It is left continuous on its first argument and
IIY(O",t)lI:::; {3, Var[O",t](Y(-,t)):::; {3 for (O",t) E [O",td x [O",td, where {3 < 00 and is independent of (0", t). (See Henry [16] or Banks and Kent [3].) Because of these properties of Y, we can easily study the adjoint of the operator T(t, 0") in equation (10.1.2). We need some properties of the adjoint of the operator T(t, 0"), which is defined as follows: Let B o denote the Banach space of functions of bounded variation tf; : [- h, 0] -+ Eno< (the row vectors) that are continuous from the left on (-h, 0) and vanish at 0; we may use the norm 1Itf;1I = Var tf;. We identify B o with the conjugate space C with the pairing
[-h,O]
(10.1.4)
The adjoint of T(t, 0"), T*(O",t) : B o -+ B o with domain in Bo is defined by
(T*(O",t)tf;,¢) = (O",T(t,O")¢), whenever 0" :::; t, tf; E Bo, ¢ E C. We now state the following lemma: Lemma 10.1.1
Let tf; E B o , tf;
-Yes, t) Then s
-+
=f. O.
Define
= [T*(s, t)tf;](O-), 0:::; s :::; t.
(10.1.5)
yes) satisfies the adjoint equation
(10.1.6)
This follows from Henry [16]. Remark: T*(s, t) is characterized in [16, Theorem 3] in a way similar to [9, Theorem 4.1, p. 152]. We now designate the strongly continuous sernigroup of linear transformation defined by (10.1.7)
Stability and Time-Optimal Control of Hereditary Systems
376 by T(t, er), t
a , so that
~
T(t, er)¢ = Xt(er, ¢, 0). n 2
Let the function U(t,s) E L oo([er,t],E
)
(10.1.8)
for each t be defined by
_ aW(t,s) . a.e. III s, U( t,s ) as where Wet, s) is the unique solution of (a) W.(·, s) = 0, t (b) Wet,s) = get, Wt(-,s)) + J. L(A,HS,(-,s))dA - (t - s)I, 0 ~ s ~ t. Then the variation of constant formula for a more general version of System (10.1.1), namely, d
dt [D(t)Xt - G(t)Xt]
= L(t, Xt) + B(t)u(t),
(10.1.9)
is given by
xt(er, ¢, u) = T(t,er)¢ + it d.U t(·, s){G(er)(¢) - G(s)(X.,)} (10.1.lOa)
+ it Ut(-, s)B(s)u(s)ds. Here G : E x
e~
En is a linear operator. If we set
0, X o«() = { t, we are justified in writing
-h
< () < 0,
-.. () = 0, I identity,
T(t, er)X o = Ut(-, s).
(10.1.10b)
Here U(t, t) = I, the identity matrix. If we consider the system in En, we have
x(er, ¢, u)(t)
= T(t, er)¢(O) + it s.u«, s)[G(er)(¢) -
G(s)(t» (1O.1.10c)
x it U(t, s)B(s)u(s)ds.
The Euclidean constrained reachable set is given by
= {it U(t, s)B(s)u(s)ds : u E Loo([er, t], em)} , and the constrained reachable set in e is given by aCt,er) = {it Ut(·, s)B(s)u(s)ds : u E Loo([er, t], em)}, R(t, er)
(10.1.11)
(10.1.12)
We summarize the basic properties of these sets in the following proposition.
The Time-Optimal Control of Linear Differential Equations
377
Proposition (lO.l.la) Assume that D and L satisfy the basic assumptions in the introduction, and furthermore there exist t > 0, L > 0 such that (10.1.13)
Then
(i) 0 E R(t, IT) for each t
IT. (ii) W(t, s)R(s, IT) ~ R(t,lT), IT ~ S ~ t, where W(t,8) : e -+ En is the operator, W(t, 8)1/> = x(t, IT, 1/>, 0) is a solution of (10.1. 7). (iii) R(t,lT) is compact in En, and convex. ~
Also, (iv) 0 E o:(t,lT) for each t
(v) T(t, S)0:(8, IT)
~
IT. o:(t, IT), IT ~
~ S ~
t.
(vi) o:(t, IT) is compact in C, and convex. (vii) Also A(t) is compact an,d convex where A(t) is defined as follows: Let 5 be a compact and convex subset of C. The attainable set A(t) of (10.1.1) at time t is defined by
A(t)
= {x(t) : I/> E 5,
u E cm}
C En.
Proof: Convexity of A(t) follows trivially from that of 5 and em. That of :w.(t) follows from the convexity of em. Because 5 is compact and x(t, IT, ·,0) continuous, x(t,lT,5,0) is bounded. Since U(t,s)B(s) is integrable and u(s) E cm, A(t) is bounded in En. Also :w.(t) is bounded. We use a weak compactness argument and the compactness of 5 to deduce that A(t) is closed in En. The same weak compactness argument proves the closedness of :w.(t). It now remains to prove (10.1.7). Let r E :w.(s); then for some E r= U(s, r)B(r)u(r)dr. Define
u em,
J:
u*(r) Then u*(r) E C m .
= { u(r), 0,
IT
~
r
~
8
~
s,
t.
Stability and Time-Optimal Control of Hereditary Systems
378
Consider the point
q = Wet, s)r
1" = 1"
=
=
1"
Wet, s)U(s, r)B(r)u(r)dr,
W(t,s)U(s,r)B(r)u(r)dr+O,
1
U(t, r)B(r)u(r)dr +
It
U(t, r)B(r) 0 dr,
t
=
U(t,r)B(r)u*(r)dr E lR(t).
Hence
W(t,s)lR(s) Theorem 10.1.1
Let CO'
~
lR(t),
= {u E c
m
:
a
s s s t.
lUi I
= I}.
If
AO(t) = {x(t) : ¢ E S, uO E CO}, lRO(t) then lRo
=
{1 U(t, s)B(s)uO(s)ds : UO CO} , t
E
= lR(t) and AO(t) = A(t) for each t ~
Proof: Because U(t,') E Loo([u, t], E have that
n2
and
)
a.
BO
E Lt([u, t], E n x m
),
we
U(t, s)B(s) E Lt([u, t], E n x m ) . It follows from Lemma 2 of LaSalle [2] that
lRO(t) = lR(t) for each t. Hence A(t)
= AO(t).
Remark 10.1.2: The bang-bang principle of Theorem 10.1.2 is not valid for aCt, u) in function space as shown by the following example of Banks and Kent [3]. Consider the neutral system
= x(t - 1) + u(t), t E [0,2], U = {v E E: Ivl ~ I}, ~(t) = 2 -t, t E [1,2]. x(t)
The assumption that u is bang-bang and SIOn
1¢(t)1 = 0
~
or ¢(t)
is attained leads to the conclu-
= -3,
The Time-Optimal Control of Linear Differential Equations
379
where
e
a proper subset of the set attained using all admissible controls. Definition 10.1.1: Let r" denote the metric space of all nonempty compact subsets of En with the metric p defined as follows: The distance of a point x from A(td = d(x,A(tt}) = inf[]» - al : a E A(t)},
N(A(tt},f)
= {x E En: d(x,A(td) ~
e},
p(A(h),A(t2)) = inf'[e : A(td ~ N(A(t2),f) and A(t2) ~ N(A(td, f)}. The target in our system (10.1.1) is the continuous set function: G : [T, (0) ---+ r". The problem of reaching G in minimum-time will be called the general problem. For this we want at some t ~ a to have x(t, a,
it
U(t, s)B(s)u(s)ds,
(10.1.14)
where wet) = z(t) - x(t, a,
> a.
Remark 10.1.3: Theorem 10.1.2 states the bang-bang principle of LaSalle is valid for (10.1.1): that if of all bang-bang steering functions there is an optimal one relative to CD, then it is optimal (relative to cm). Also, if there is an optimal steering function, there is always a bang-bang steering function that is optimal. Proposition 10.1.1b The attainable set AU : [u,oo) ous. Also t ---+ ~(t, o ); t ---+ aCt, u) are continuous.
---+
r" is continu-
Remark 10.1.4: The proof of this proposition is essentially the same as the assertion which is contained in Theorem 4.1 of Banks and Kent [3]. One proves that there exists an M > 0 such that IIx,(u,
A
= {x(u,
is an equicontinuous subset of C([u ~ h, ttl, En). As a consequence, t ---+ x(t, a,
380
Stability and Time-Optimal Control of Hereditary Systems
follows readily from this. As pointed out in [3], because the fundamental matrix t ----> U (t, s) is not continuous one cannot prove A continuous directly using the variation of parameter formula (10.1.3) as is done for the example in [4]. Theorem 10.1.2 Iffor the general problem there exists a pair ¢J E S, u E em such that x(t, ¢J, u) E G(t) for some t, then there is an optimal pair ¢J" E S, u" E em. Proof: By assumption there is some t
~
A(t) n G(t)
a, ¢J E S such that
i= 0.
If we now define the minimal-time function
tOeS) = inf{t
~
a : A(t) n G(t)
i= 0},
where A(t) = A(t, S), we can prove first as is done by Strauss [5, p. 63] that A(t*) n G(t") i= 0. We use the compactness of G(t), A(t), and their continuity. Definition 10.1.2: Let
D(t,¢J) =
l°h
d.TJ(t,s)¢J(s),
and suppose
= 7J(t,,8+) - TJ(t,,8-). Then D(t,¢J) is atomic at ,8 on Ex e if detA(t,,8) =f 0 for all tEE. In particular, if ,8 =f 0, ,8 E [-h,O], and D(t, ¢J) = ¢J(O) + M(t)¢J(,8), then M(t) = A(t,,8) and D(t,¢J) is atomic at,8 on Ex e if det M (t ) i= 0 for all A(t,,8)
tEE. See Hale [9, p. 50]. For the linear system (10.1.1), a fundamental theorem of Hale shows that the solution operator
T(t,a):
e
---->
C,
defined by T(t, a)¢J = xt(a, ¢J), where x is a solution of system (10.1.1), is a homeomorphism, provided D is atomic at 0 and at -h [9, p. 279]. As asserted by Hale, since L(t, xt} = f(t, Xt)
The Time-Optimal Control of Linear Differential Equations
381
is linear and therefore continuously differentiable in the second argument, the solution operator is a diffeomorphism. It follows from an argument similar to Hale [9] that the solution operator
is also a homeomorphism. Definition 10.1.3: The control u E em is an extremal control on [0", tll if for some ¢ E and each t E [0", ttl the solution x(O", ¢, u) of (10.1.1) through 0", ¢ belongs to the boundary BA(t) of A(t).
e
Assume that conditions of Proposition (1O.1.1a) hold. Assume the solution of 10.1.7 is pointwise complete. Let u* be optimal on [0", t*]. Then there is a nonzero n-dimensional row vector c* depending on ¢* and t* such that
Theorem 10.1.3
{u*(t)} for each 1 ~ j
~
= sgn{ c* tur, t)B(t)}j,
0"
~
t
~
r,
(10.1.15)
m for which
{c*U(t*, t)B(t)}j
=f. O.
Proof: To prove that any optimal control is extremal, one uses the method of Strauss and the following proposition which is proved exactly as in [5). Proposition 10.1.2 If 0' is an interior point of A(t), then there is some f> 0 such that the e-neigbborbood N(a, f) of 0' satisfies
N(a, f)
~
A(s)
for all s E [t - e, t]. To prove the last assertion, we note that x(t*, 0", ¢*, u*) E BA(t*). Because A(t") is convex and closed, there exists a support plane 71"" through x(t", 0", ¢* ,u*) = z" such that A(t) lies on one side of 71"". Let v be a unit normal to 71"" directed away from A(t"). Clearly, for each
uE Hence (v, x" - x) equivalent to
~
em,
x = x(t",¢*,u) E A(t").
O. From the variation of parameter (10.1.3a), this is
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Stability and Time-Optimal Control of Hereditary Systems
for all u E em. Let A(S) = vTU(t*,s)B(s). Then
Hence we see that, on any interval of positive length where A(s) must be that uj(s) = sgn Aj(S). We have proved the theorem.
#
0, it
Definition 10.1.4: System (10.1.1) is said to be normal on [0', t] if no component of cTY(t,s), c # 0, vanishes on a subinterval of [O',t] of positive length. This definition is equivalent to the following: Let Yj(t, s) be the jth column vector of Y(t, s). For each j = 1, ... ,m the functions Yjl(t,S), ... ,Yj,,(t,s) are linearly independent on each subinterval of [O',t] of positive length. Corollary 10.1.1 If System (10.1.1) is normal on [0', t], then u*(t), the optimal control, is uniquely determined by (10.1.9) and is bang-bang. Definition 10.1.5: Let S ~ En be a set. S is strictly convex if for every Xl and X2 in S, Xl # X2, the open line segment {Xl
+ (1 -
A)X2 :
0<
A
< 1}
is in the interior of S. Following the methods of Strauss [5, p. 70] we can prove the following: Theorem 10.1.4 Assume conditions of Theorem 10.1.3. Suppose (10.1.1) is normal and G : [T,OO) ----> r n continuous and strictly convex valued. Then if there exists a pair
10.2 Forcing to Zero In this section, we consider System (10.1.1) with a fixed target G(t) == {O}, where the aim is to start at any initial state
e
Definition 10.2.1: System (10.1.1) is proper on an interval [0', t*], if cTU(t*, s)B(s) == 0 almost everywhere on [0', t*] implies c = O. If (10.1.1) is proper on [0',0'+ 6] for each 6 > 0, we say that (10.1.1) is proper at time
The Time-Optimal Control of Linear Differential Equations
a. If (10.1.1) is proper on each interval [u, t), t proper.
> a,
383
we say the system is
Definition 10.2.2: System (10.1.1) is Euclidean null controllable if given any initial state if> E C = C([-h, 0], En) there exists a t1 and an admissible control u E L oo([u,t1],Em) such that the solution x(u,if>,u) of (10.1.1) satisfies xq(u,if>,u) if>,X(t1,u,if>,U) O.
=
=
Definition 10.2.3: The domain N of null controllability of(1O.1.1) is the set of all initial functions if> E C for which the solution x = x(u,if>,u) of (10.1.1) with Xq if> satisfies x(t 1 ) 0 at some t 1 using an admissible control u E Loo([u, t 1 ], cm).
=
=
Theorem 10.2.1 The domain N of null controllability contains zero in its interior whenever (10.1.1) is proper. To prove Theorem 10.2.1 we need the following proposition: Proposition 10.2.1 System (10.1.1) is proper on [u, t) if and only if the origin is an interior point ofR(t). Proof of Proposition 10.2.1: Observe that 0 is always in R(t) for each t ~ a . Suppose (10.1.1) is proper but zero is not in the interior of R(t) for some t ~ a, Because R(t) is a closed and convex set and because the constraint set C'" is the unit cube, 0 being on the boundary of ~(t) is equivalent to cT
it
U(t, s)B(s)u(s)ds = 0
=
for some nonzero c E En and for u E L1([U,t],cm) given by u(s) sgn(cTU(t,s)B(s)) where c is the outward normal to the support plane to ~(t) at O. Thus
cTU(t, s)B(s) == 0, almost everywhere s E [u, t), where c # 0, which contradicts the fact that (10.1.1) is proper on [u, t). We now reverse the argument to complete the proof. Proof of Theorem 10.2.1: With 0 = u E C'": x(t) = 0 is a solution of (10.1.1) so that 0 E N. Suppose 0 is in the interior ofR(t) for each t, but 0 is not in the interior of N. Then there is a sequence {if>m)1" ~ C such that if>m --;. 0 as m --;. 00 (the convergence is in the sup norm of C([-h, 0], En))
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Stability and Time-Optimal Control of Hereditary Systems
and no ¢m is in N (so ¢m -:j:. 0). From the variation of constant formula we deduce that
for any i l > U and any u E C'", Hence Ym = T(il, u)¢m(O) is not in R(il) for any il' We therefore obtain a sequence {Ym)r' ~ En, Ym (j. R(ir) for any iI, Ym -:j:. 0 such that Ym ---+ 0, as m ---+ 00. We conclude that 0 is not in the interior of lW.(tr) for any iI, a contradiction. Hence 0 E Int N. Theorem 10.2.2 If (10.1.1) is proper and (10.1.2) is uniformly asymptotically stable, then (10.1.1) is Euclidean null controllable. Proof: Let N be the domain of null controllability of (10.1.1). Since (10.1.1) is proper we have from Theorem 10.2.1 that 0 E Int N. In (10.1.1) consider the trivial admissible control u = 0 on [u,oo). Then the solution x(u, ¢, 0) of (10.1.1) with u = 0 (that is the solution of (10.1.2)) satisfies
lIXt(u,¢,O)II ~ ke-c(t-u)II¢II,
t 2: a, ¢ E C,
for some k > 1 and c > O. This behavior of (10.1.2) follows from Cruz and Hale [1]. Hence Xt(u,¢,O) ---+ ¢o = 0 E Int N, as t ---+ 00. Therefore there exists a tr > U .such that Xt, (u, ¢, 0) E B where B is a ball in N with the origin as center and radius sufficiently small. With Xt, (u, ¢, 0) E N as an initial point, there exists some t2 and some u E Ll([tl, t2], cm) such that the solution x(il,xt"u) of (10.1.2) satisfies x(il,xt"u) = O. This proves the theorem. We conclude this section by considering a more general class of controls" in which we only require them to be integrable on finite intervals. If a control u satisfies this condition, we simply write u E C*. Definition 10.2.4: System (10.1.1) is Euclidean controllable on [0", i l ] if for each ¢ E C and each Xl E En there exists a control u E C* such that the solution x(u, ¢, u) of (10.1.1) satisfies xu(O", ¢, u) ¢ and X(il, U, ¢, u) Xl.
=
=
Theorem 10.2.3 System (10.1.1) is proper on [0", td ifand only if(10.1.1) is Euclidean controllable on [0", td. Remark 10.2.1: The ideas behind the proof are standard in the literature. See, for example, Zmood [7, Theorem 2], and Lemma 6.1.1 here.
The Time-Optimal Control of Linear Differential Equations
385
10.3 Normal and Proper Autonomous Systems In this section an autonomous special case of (10.1.1) will be considered. Sharp necessary and sufficient conditions for normality and for the system to be proper are deduced. The method of Gabasov and Kirillova [12] is used to obtain results that generalize the recent normality conditions of Kloch [8], and that specialize to well-known conditions for autonomous linear ordinary differential equations. Consider the system
d dt (x(t) - A_Ix(t - h))
= Aox(t) + AIx(t -
h) + Bu(t),
t ~ 0, (10.3.1)
x(t) = ¢(t) for t E [-h,O], where A j are n x n constant matrices and B is an n x m constant matrix. Here u E c m and ¢ E C([-h, 0], En) == C. It is well known that for the above assumptions, for each admissible control u E L10C([0, 00), cm), there exists a unique solution to (10.3.1) on [-h,oo) through ¢. Furthermore, if A-I # 0, this solution exists on (-00,00) and is unique. (See-Hale [9, p. 26].) The fundamental matrix of
d
dt (x(t) - A_Ix(t - h»
= Aox(t) + AIx(t -
h)
(10.3.2)
is a solution of (10.3.2) with initial data X o() t
={
0, t
I, t
< 0,
= 0 (Identity),
(10.3.3)
°
for which U(t) - A_IU(t - h) is continuous and satisfies (10.3.2) for t ~ except at kh, k = 0,1,2, .... Indeed, U(t) has a continuous first derivative on each interval (kh, (k + l)h), k = 0,1,2, , the right- and left-hand limits of U(t) exist at each kh, k = 0,1,2, , so that U(t) is of bounded variation on each compact interval and satisfies
U(t) - A_IU(t - h) = AoU(t) + AIU(t - h), t # kh, k = 0, 1,2, ....
(10.3.4)
Also if U(t) is the fundamental matrix solution of (10.3.2) described above, then the solution x(¢, u) of (10.3.1) is given by
x(t, ¢, u)
= x(t, ¢, 0) +
it
U(t - s)Bu(s)ds,
(10.3.5)
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Stability and Time-Optimal Control of Hereditary Systems
where x(t, 4>, 0)
= U(t)(4>(O) - A_ I4>( -h)) + Al lOh U(t - s - h)4>(s )ds - A_IjO dU(t - s - h)4>(s). -h
In order to introduce computational criteria to check when (10.3.1) is proper, or normal, we introduce the following notation by defining Qk(S) k
= AOQk-I(S) + AIQk-I(S - h) + A-IQk(S - h),
= 0, 1,2, ... , s = 0, h, 2h, ... ,
Qo(O) = I identity matrix, Qo(s)
== 0 if s < O.
Theorem 10.3.1 A necessary and sufficient condition for System (10.3.1) to be proper on the interval [0,T) is that the matrix II(T) = {Qk(s)B, k = 0,1, ... ,n -1, s E [O,T)} has rank n.
Proof: The proof is exactly as in [12, pp. 51-60]. Corollary 10.3.1 In (10.3.1), assume that A-I = O. Then a necessary and sufficient condition that (10.3.1) is proper on [0, T] is that II(T)
= {Qk(S), k = 0,1, ... ,n -
1, s E [O,T]}
has rank n where
= AoQk-I(S) + A1Qk-I(S - h), Qo(O) = B, Qo(s) == S '# O.
Qk(S)
This is Theorem 6.1.1, located in this book.
Corollary 10.3.1 is the algebraic criterion for complete controllability given by Gabasov and Kirillova for the delay system x(t) = Aox(t) + A1x(t - h)
+ Bu(t),
(10.3.6)
when the controls are not restrained to lie on a compact set but are only required to be integrable on compact intervals. We note that an algebraic
The Time-Optimal Control of Linear Differential Equations
387
criterion for the delay equation (10.3.6) to be proper is given by Corollary 10.3.1. This is a generalization of the fundamental result of LaSalle in Hermes and LaSalle [6, p. 74] on the autonomous system
x=
Ax + Bu(t).
(10.3.7)
Recall that (10.3.1) is normal on [0, T], T> 0, iffor any r = 1, ... ,m,
r{U(T - s)B,. = 0 almost everywhere s E (0, T], implies lJ = O. If we follow the idea of Theorem 10.3.1, we deduce the following theorem:
Theorem 10.3.2 A necessary and sufficient condition for (10.3.1) to be normal on the interval (0, T] is that for each r = 1,2, ... , m, the matrix
= {Qk(s)B,.,
II(T)
,.
k=O,l, ... ,n-l, SE[O,T]}
has rank n. We now apply our result to the general nth order scalar autonomous neutral equations of the form n
xn(t) =
I: bixCi)(t i=O
where
lui :S
1) +
n-l
I: a;xC;)(t) + u(t),
(10.3.8)
i=O
1. Define
B=
1
o
o 1
o
o o o o
o o o bn -
1
(10.3.9) Then System (10.3.8) is equivalent to (10.3.1) with A j , B so defined.
Stability and Time-Optimal Control of Hereditary Systems
388
Corollary 10.3.2 T>O.
The scalar control system (10.3.8) is proper for every
Proof: The result follows immediately from (10.3.9) and Theorem 10.3.1.
It is fairly obvious from Theorem 10.3.1 that if (10.3.1) is proper on [0, Ttl,
then it is proper on [0, T] for T 2: T 1 •
Consider the pointwise complete system (10.3.1). Suppose (i) for each T 2: 0 and for each r = 1,2, ... ,m, the matrix
Theorem 10.3.3
I1(T)
= {Qk(s)B r , k = 0,1, ...
,n - 1,
5
E [0, T]}
(10.3.10)
r
has rank n, and (ii) suppose a = sup{R E A :det
= A(I -
~(A)
~(A)
= O} < 0 where
A_1e- Ah )
-
A o - A1e- Ah .
Then there is precisely one time-optimal control that drives any E C([-I, 0), En) to the origin in minimum-time t/". It is given by
uj(t)
= sgn(c T X(t·
- t)B)j, j
= 1, ... ,m,
0 ~ t ~ t",
(10.3.11)
Proof: Because of (i), (10.3.1) is normal and a fortiori proper. Because of (ii), (10.3.2) is uniformly asymptotically stable. Hence (10.3.1) is Euclidean null controllable; see Theorem 10.2.2. Because (10.3.1) is null controllable, Theorem 10.1.3 guarantees there is an optimal control that is extremal by Theorem 10.1.4 and uniquely determined by (10.1.9) because of Corollary 10.1.1.
10.4
Pursuit Games and Time-Optimal Control Theory
In this section we show that a class of differential pursuit games is equivalent to some time-optimal control problems. We can therefore appropriate the earlier theory that we have developed for System (10.1.1) to solve this later problem. The emphasis in this section is exactly as in Hajek [10]: the emphasis is on the reduction rather than on the consequences. We consider the linear neutral differential game described by d
dt D(t, Xt) = L(t, Xt) - p(t) Xq
=
+ q(t),
t 2:
5,
(10.4.1)
The Time-Optimal Control of Linear Differential Equations
389
where pet) E P, q(t) E Q with P ~ En, Q ~ En is the pursuer control and quarry control constraints. The functions p E Loa ([u, t], P), q E Loa ([u, t], Q) are called pursuer and quarry controls. They are said to steer the initial function ¢ E C([-h, 0], En) to the origin in En in time tl if the solution x(u,¢,p,q) of (10.4.1) with xq(u,¢,p,q) = ¢ satisfies X(tl,U,¢,P,q) = O. The information pattern of our game can be described as follows: For any quarry control q, (i) there exists (that is, "the pursuer can choose") a pursue control p such that for each s E [u, t] the value of p(s) depends only on q(s) (and of course on ¢, D, L). (ii) The pair of controls p, q steer ¢ to 0 E En. (iii) This is done in minimum-time. Associated with the game (10.4.1) is a linear control system
d dt D(t, Xt) Xq
= L(t, xd ~
t ~ a,
vet),
(10.4.2)
¢,
where vet) E vet). Here D and L are as given in (10.4.2), but the control constraint set is defined by
v = P:!.Q = {x: x + Q ~
P}
(10.4.3)
where j, denotes the Pontryagin difference of P and Q. It is important to observe that we can define
pet) = B(t)u(t),
q(t) = C(t)v(t),
where B is an n x m matrix function and C is an n x r matrix function, provided we assume u(t) E Em, vet) E E" with the constraint sets PI ~ Em and Ql ~ E", Viewed in this way there is nothing in (10.4.1) that suggests that the control functions have the same dimensions as the state space. These same comments are valid for (10.4.2). Definition 10.4.1: A point ¢ E C is said to be in position to win in time t 1 > o if, for any quarry control q, there is a pursuer control subject to the
information pattern (i) - (iii) in the introduction.
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Stability and Time-Optimal Control of Hereditary Systems
°
Theorem 10.4.1 Assume E Q and P is compact. The game (10.4.1) is equivalent to the associated time-optimal control problem (10.4.2), where V is given by
= (P + ker U(tl, t))±Q,
Vet) Furthermore,
f(q,t)
= u(t) + q
for some tj.
modulo kerU(tl,t)
(10.4.4)
(for all q E Q, t E [u, tl]) can be used to obtain a suitable strategy from u E Loo([u, til, V). In detail, ¢ is in position to win in time tl if and only if ¢ can be steered to 0 in time tl within the control system (10.4.2), and the corresponding minimum-times coincide. Proof: Assume ¢ E C is in a position to win at time tl. Then from the definition, given any quarry control q : [u, til -+ Q, the mapping f : En X [u, til -+ En exists with values p(t) = f(q(t), t), t -+ pet) is integrable, pet) E P, and
(,
0= x(t l, ¢, 0, 0) -}" U(t l, s)(f(q(s), s) - q(s))ds or
(,
x(tl,¢,O,O) = }"U(tI,S)(f(q(s),s) - q(s))ds,
(10.4.5)
°
where X(tl,¢,O,O) is the solution of the homogeneous equation (10.1.7). Because E Q, we can consider the quarry control to be 0. Then
(,
x(tl,¢,O,O) =}" U(tl,s)v(s)ds,
(10.4.6)
where v(s) = f(O, s). Now take any point q E Q and a time t E [u, t l ], and consider the piecewise constant quarry control qo, qo
=
{° q
Apply (10.4.5) to obtain
(,
x(t1,¢,0,0)=)" U(tl,s)u(s)+
in [u,t], in [t, til.
it' t
U(t1,s)(f(q,s)-q)ds.
(10.4.7)
The Time-Optimal Control of Linear Differential Equations
39 1
On subtracting (10.4.7) from (10.4.6),
for all t E [ u , t l ] . Observe that the integrand is independent o f t , so that by differentiation
+
U ( t l , t ) ( v ( t ) q - f ( q , t ) )= 0 almost everywhere. Use the kernel t o interpret this as
v ( t )+ q E f ( q ,t )
+ ker U ( t 1 , t ) almost everywhere t E [ u , t l ] .
(10.4.8)
Since f has values in P ,
v(t)
+ q E P + ker U ( t 1 , t ) .
From the continuity of solutions of (10.1.1) with respect to initial conditions, ker U ( t 1 , t ) is closed. Hence ( P lter U ( t 1 , t ) ) is closed. Because of this, Hzijek's Lemma [ll,p. 591 yields
+
v(t)
+Q
P
+ ker U(t1, t ) almost everywhere,
or
~ ( tE)( P + ker U ( t 1 , t ) ) z Q V ( t ) almost everywhere. Hence 'u E L , ( [ u , t 1 ] , V ) where V is as defined in (10.4.4). It now follows that v is an admissible control for (10.4.2), and hence from (10.4.6)
or
Observe that z,(u,4,0,0) = 4 , so that z(o,4 , 0 , 0 ) is a solution of (10.1.2). Note that (10.4.8) proves the last assertion of the theorem. Hence 'u steers 4 to 0 in time t l . Next 'u and 0 are pursuer and quarry controls (0 E Q , PzQ c P ) ; for this choice the dynamical equations of the game and the control systems coincide; so do their solutions. Hence the optimality problems are the same.
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Stability and Time-Optimal Control of Hereditary Systems
Conversely, let an admissible control v steer ¢J to 0 at t1 within the control system (10.4.2). To show that ¢J is in position to win at t 1 , take quarry control q. Then t1
0= X(t1,¢J,0,0) _ l V(t1,s)v(s)ds
(10.4.9)
where v(s) E V(s) is such that v(s)+q E P+ker V(t1, s). We now construct a pursuer-control strategy as follows: by Filippov's Lemma (see the form in [11, p. 119]) there exist measurability preserving maps
f : Q x [0", ttl A: Q x [O",td for each 0"
~ S ~
-+
P,
-+
kerU(t 1,s)
t1 with A(q(S), s) E ker V(t1, s) such that v(s)
+q=
f(q, s).
Since f(q, s) E P is a compact set, we have f E Loo([O", ttl, P). We now verify that to any quarry control q, f thus constructed, will force ¢J to 0 E En. Indeed, if q E Loo([O", ttl, Q), f -q = V-A so that the solution of (10.4.1) at time t 1 with the pair of f, q and initial function ¢J is
x(t 1,¢J,0",q) = X(t1,¢J,0,0) + = X(t1, ¢J, 0, 0)
1
_1
=X(tl,¢J,0)-l
1
t1
t1
V(t1,S)(f(q(s),s) - q(s))ds, V(t 1, s)(u(s) - A(q(S),s))ds,
t 1 U(t1,s)v(s)ds
t1
+
V(t1, s)A(q(s), s)ds
= 0,
from (10.4.9) and the definition of A. Hence on taking f and q as controls we find again that the dynamical equations coincide, so that steering to 0 in t 1 and the minimal times are preserved. This completes the proof. For more general targets
that are continuous we do not have the duality of Theorem 10.4.1. Instead we have the following:
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The Time-Optimal Control of Linear Differential Equations
Theorem 10.4.2 Let the pursuer-constraint set be compact and let G : [0-,00) ---. r n be continuous. An initial position tP E C can be forced to a target G (using the usual information pattern) at time t 1 within the game (10.4.1) whenever in the associated control system (10.4.2) tP can be steered to the target in time t 1. Furthermore,
f(q,t)
= v(t)+q
modulo kerU(t1,t)
determines a control strategy for (10.4.1) that counters any quarry action q E Loo([o-, ttl, Q) where v is an admissible control for (10.4.2).
Proof: Assume that in (10.4.2) tP can be steered to G in time t1. Then there exists v E £1([0-, ttl, V) such that the solution x = x(o-, tP, v) of (10.4.2) satisfies (10.4.10) Because v : [0-, t1] ---. V,
v(s)
+ q E P + ker U(t1, s)
for s E [0-, t1] and every q E Q.
With v fixed, we now construct a pursuer-control strategy Filippov's Lemma. We obtain measurable functions
f by applying
f: Q x [0-,00) ---. P, .\: Q x [0-,00) ---. kerU(t 1,s),
.\(q(s), s) E ker U(it, s) such that
v(s) + q(s)
= f(q(s), s) + .\(q(s)s).
Just as in Theorem 10.4.1, f E £00([0-, it], P). For any quarry action q,jq = v -.\ so that the solution of (10.4.1), with this f and q and initial data tP, satisfies
X(t1,tP,j,q) = X(t1,tP,0,0) -
J(7t'
U(t1,S)(J(q(s),s) - q(s))ds,
= x(t 1,tP, 0, 0) - J(7t' U(t1, s)v(s) + J(7t' U(t1, s).\(q(s), s)ds, = X(tl,tP,O,O) - J(7t' U(t1,s)v(s)ds E G(tt}
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Stability and Time-Optimal Control of Hereditary Systems
by (10.4.10). This proves the theorem. We now examine the quantitative properties of P and Q from a close examination of Theorem 10.4.1. Definition 10.4.2: Consider a mapping r : Q x E -+ P, where r(q, t) is a point in P whenever q is a point of Q and tEE. Suppose is an induced mapping on the collection of Q of functions q('), which are defined by e[qJ(t) = r(q(t), t). Then e is called a stroboscopic function if it preserves measurability. We describe the following example: Suppose t -+ u(t) is given in advance. Then
e
e(q)(t) = q(t) + u(t) is a stroboscopic strategy if u is measurable and P :::> Q + u(t) for all t. Theorem 10.4.3 In (1004.1), assume that 0 E Q, P is compact, P, Q are nonvoid, convex, and symmetric. Then the set inclusion (10.4.11) is necessary and sufficient for the presence of initial ¢J, which can be controlled to zero in some time t 1 when the composite system (10.4.1) uses stroboscopic strategy. The inclusion (10.4.12) is necessary and sufficient for the domain of null controllability (10.4.1) to have zero in its interior. Proof: Theorem 10.4.2 states that the set offunctions ¢J that can be driven to zero at some time h using the dynamics (10.4.1) is the same as the set of initial functions that can be controlled to zero when System (10.4.2) is used. Because Vet) = (P + ker V(t1, t)) -* Q is the control set for (10.4.2), a set which is convex and symmetric, we deduce that Vet) is nonvoid if and only if 0 E Vet), i.e., P
+ ker V (t 1, t)
But then with
1R(td
=
:::> Q,
r o
a 5: t 5: t 1 .
V(t1' s)V(s),
The Time-Optimal Control of Linear Differential Equations
395
there are initial positions that can be forced to zero stroboscopically at time t l if and only if 1R(td is nonvoid. By Filippov's Lemma (see also Chukwu [17, pp. 437-439]), 1R(tl) is nonvoid if V(t) is nonvoid, for a ~ t ~ t l, so
that
P
+ ker U(tl, t) => Q.
For the converse, if P + kerU(tl,t) => Q, then 0 E (P + kerU(tl,t),O is an admissible control steering zero to zero. The second assertion is valid if and only if
o E Int V(t) = Int«P + ker U(tl, t)) -• Q).
But 0 E Int V(t) is a requirement for (10.4.2) to have zero in the interior of the Euclidean reachable set, and therefore in the domain of null controllability of (10.4.2), which by the duality Theorem 10.4.2 coincides with that of (10.4.1). For ordinary differential linear systems, Hajek has obtained the following results [11, pp. 60-87]: Consider the system x(t)
= Ax -
p +~;
(p(t) E P, q(t) E Q)
(10.4.13)
in En, where A is an n x n matrix. Proposition 1004.1 In (10.4.13) assume that P, Q are non void, convex, and symmetric. Then (10.4.14) P=>Q is a necessary and sufficient condition for presence of initial positions that can be forced to zero stroboscopically in strictly positive time. If Int P => Q,
(10.4.15)
then for every t > 0 the set of positions that can be forced to 0 stroboscopically at time t is a neighborhood of O. If G
= {x : x E En;
Mx
= Mb}
(10.4.16)
where M is an m x n matrix and bEEn, then
M(P - P) => M(Q - Q)
(10.4.17)
is necessary, and if (Q is compact),
IntMP=>MQ
(10.4.18)
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Stability and Time-Optimal Control of Hereditary Systems
is sufficient for the presence of positions that can be forced to G at strictly positive times. Suppose the pursuer's (firm's) control order k is defined as that k such that MAi(p - P) = 0, (10.4.19) (the set consisting of the zero vector, not the empty set) holds for j = 0, ... , k - 2 but not for j = k - 1. Assume P is compact. A necessary condition for the presence of initial points that can be forced to G at t > 0 is that (1) firm's control order (pursuer's) ~ solidarity control order; and if k is the solidarity (quarry) control order then (2) (10.4.20) (i) Suppose P, Q are compact, convex, and symmetric, and (ii) firm's control order ~ solidarity control order; (iii) (10.4.21)
Then for sufficiently small it' > 0, the set of initial endowments that can be forced to G at time t i has a nonempty interior. Let (10.4.13) describe the growth of capital stock with any initial value x(O) = Xo and target 0 or target G in (10.4.16). If the target is 0, we can define the firm's initiative (investment, consumption) as the control set P, and the set Q as solidarity (e.g., government taxation or subsidy). If G is the target, M A k - 1 P is defined to be the firms control initiative and M A k - 1 Q is solidarity. With this terminology we deduce the following universal principle: Principle 10.4.1 No initial value of capital stock can be controlled to the target unless the firm's initiative contains solidarity as a subset. To ensure growth of any initial value of capital stock to the target that is sufficiently close, the firm's initiative must dominate solidarity. These principles provide a broad policy prescription for national economies, which will be explored in a subsequent communication.
Fundamental principle 10.4.1 emerges from studies of linear systems. Consideration of general nonlinear dynamics in Theorem 3.2 of [15] show that the solidarity assumption is the generalization offundamental principle 10.4.1.
The Time-Optimal Control of Linear Differential Equations
397
10.5 Applications and Economic Growth From the discussions of Section 1.8 that yielded (1.8.5) and the insight of (1.8.4), it is not unreasonable to postulate that the dynamics of capital function of n stocks in a region may be described by the equation
3:;(t) - A;x;(t - h) = L;(t, xt} - p;(t) + q;(t)
(10.5.1)
where p;(t) == B;(t)u;(t), q;(t) = C;(t)v;(t) where q;(t) is the solidarity function that is the effect of government intervention (taxation, subsidy) in economic growth. Equation (10.5.1) describes the net capital formulation of the ith subsystem. The effects of government acts are measured locally. We now interpret Theorem 10.4.2. Let
3:;(t) - A i3;;(t - h)
= u«, xt} -
u(t)
(10.5.2)
can grow from
Q c Int(P + ker U(tl, t)). Principle 10.4.2 To guarantee economic growth from any initial endowment to a target, it is necessary that the firm's capacity for investment and its internal power for waste dominate whatever government can do. Government intervention is needed, but it should not be too big. We now consider System (10.4.1) very carefully. First, consider the system d
dt D(t, Xt)
= L(t, xt} -
pet),
pet) E P.
(10.5.3)
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Stability and Time-Optimal Control of Hereditary Systems
If (10.5.3) is controllable on [0-,t 1 ] with p(t) E P, then
o E IntA(tI) where
A(tI)
(1O.5Aa)
= {x(p)(tI) : p(t) E Pl.
The attainable set of (10.4.1) is then
A(t)
= {x(p, q)(t) : p(t)
E P, q(t) E Q}.
If
Qc P
or
Q C Int P,
System (1004.1) is such that
o E IntA(tI) C A(t),
(10.5Ab)
provided (10.5.3) is controllable. Thus with
-p(t)
= B 1(t)u(t)
and
q(t)
= B 2u(t),
for example, the system (10.5.5) is made controllable. It is possible that the intervention of q may make matters worse. Indeed, even if (10.5.6) is controllable, the composite system (504.1) may not be controllable. A "proper amount" of q is needed. We can formulate an economic interpretation with the following remarks: To ensure growth from an initial endowment to the target we may assume (10.5.5) is Euclidean controllable. Hence matters should be so arranged that the solidarity function q brought to bear on the isolated system ensures controllability. These observations are formalized in the following principle:
The Time-Optimal Control of Linear Differential Equations
399
Principle 10.5.3 If an isolated system is not proper and is not locally controllable with constraints, the composite system may be made ''proper'' and locally controllable provided there is an external source of power or initiative q available to enforce controllability and proper behavior. It is possible that the intervention of solidarity can make matters worse. Only a "proper" amount is needed. There is no theorem at the present time that states that the interconnected system is well behaved when it is misbehaving in isolation, and there is no compensating external solidarity q, We observe that the intervention of q can make matters worse. If (10.5.4a) holds and we consider (10.5.1) with q(t) == qo i- 0, then 0 need not be in A(t), the attainable set of (10.4.1) and controllability may fail. In this case, the solidarity function is not flexible. This seems to be the case in centralized economies. Principle 10.5.4 If the solidarity function is inflexible and rigid, or if the isolated systems controlling initiatives are ignored in the construction of a nontrivial solidarity function, the interconnected system will fail to be proper and locally null controllable. We have isolated null controllability in En as our objective in the analysis of the growth of capital stock. Though zero target at the final time seems artificial, the theory incorporates nontrivial targets. Indeed, if Xl is a nontrivial arbitrary target and y(t) = y(t, ¢J, uo) is any solution of
x(t) - A_lx(t - h) = f(t, Xt, ut} with Yu = ¢J and uo admissible such that y(tl) study the null controllability of the system
= Xl, one
can equivalently
where Zu = 0, so that Xu = Yu = ¢J and x(t) = z(t) + y(t). If we can show that there is a neighborhood 0 of the origin in z space such that all initial ¢J E 0 can be brought to Z(tl' u, ¢J, uo) = 0 by some admissible control at time tl, then so that
X(tl,¢J,U) = y(td = Xl·
Stability and Time-Optimal Control of Hereditary Systems
400
The remarks we have made on the consequences and insights of Theorem 10.4.1 - Theorem 10.4.2 justify a qualitative description of the solidarity function q and its constraint Q. How big should it be? This q can be viewed as a control disturbance, and we can then study System (1004.1).
10.6 Optimal Control Theory of Linear Neutral Systems In this section we study the optimal problem of the system
t,
:, [X(I) -
A_,;x(1 - h;)] (10.6.1)
m
= Aox(t) + L
Ajx(t - hj)
j=1
x(o)
= gO E En,
x(t)
+ B(t)u(t),
= gl(t),
t > 0,
t E (-h,O],
gl E C.
Define the fundamental matrix W of 10.6.1 by
°_
W t {x(t,O,gO,O), t 2:: 0, ()g 0, t < 0, where x(t,u,gO,y!) is the solution of (10.6.1). Hence Wet) is the unique solution of Wet) = { e A ot + i t e A oCt -
°
m
+ Lx(t where Xes)
°
j=1
=
.) .
xes - h j )(A j
+ A oA_ 1j)
. W(s - hj )]ds
j=1
hj)A_ 1jW(t - hj)
if s < 0, xes)
Proposition 10.6.1
+ Lm
j=1
it
= I if s 2:: 0, I identity.
(10.6.2)
The variation of constant formula for (10.6.1) is
x(l, u, g' ,g') = [W(t) -
+
f
t,
Wet - h; )A-d] 9'
fO . Wet - s -
hj)[A jg 1(s) + A_ 1jg 1(S)]
-h J
Wet - s)B(s)u(s)ds,
== x(t,O,gO,gl) +
it
Wet - s)B(s)u(s)ds.
(10.6.3)
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401
We study the optimal control and the adjoint system. Consider (10.6.1), where
u E Uad C Lp([O, Tj, En), p E [1,00),
BE Loo([O,T],En xm)
Assume Uad closed and convex set in L p • We denote the solution of (10.6.1) by x(t, u). Let the integral cost function be given by J=J(u,x), J=o(x(T))+ 0 :
En
ko : L p
-+ X
R,
[O,T]
iT
fo(x(t),t)
fo: En
X
+ ko(u(t),t),
(10.6.1b)
1-+ E,
E.
-+
We study the following problems: P1 Find a control u E Uad that minimizes the cost J subject to (10.6.1). P2 Find optimality conditi(:ms for the optimal pair (u*,x*(u*)) E Uad X
C([O,T], En) such that' InfJ(u,x) = J(u*,x*(u*)), u E
Us«.
We shall prove the existence of optimal controls for P 1 , and solve P2 by deriving necessary optimality conditions. Let H be a compact target set in En. Suppose
Uo = {u E Uad: x(u,t) E H for some t E [O,Tj}, and suppose Uo # 0. This is the constrained-controllability assumption. We now formulate the time-optimal problem. P3 Find a control u* E U« such that t*(u*) :S t(u) for all u E U» subject to (10.6.1). The number t*( u*) is the first time x(t*, u*) E H, i.e.. t* is the optimal time. We now consider the adjoint system. Let qo E (E n)* be a row vector,
402
Stability and Time-Optimal Control of Hereditary Systems
The adjoint system for (10.6.1) is
.r;
[m
d y(t) dt
y(t
+ hi )A- li ] + y(t)Ao(t) +
.r;m
y(t
+ hi )Ai
- ql(t)
= 0,
a.e. tEl,
y(T)
= -qo,
y(s) = 0, a.e. s E [T,T + h]. (10.6.4a)
The solution of the adjoint equation is given by the adjoint state
y(t) = W*(T - t)( -q~)
+
iT
W*(s - t)( -qr(s))ds,
where W*(t) is the adjoint of W(t), t E [0, T], and is the fundamental matrix solution of (10.6.4a), which is unique. We note that
Y(s, t) == W*(t - s) = W(t - s) a.e. in s, where W(t) is the fundamental solution of (10.6.4a). Thus
Y(s,t)
=
1+1
s E
Y (t, t)
t+
daY(a,t)p(a,s)
[IT, t],
= I,
Y ( s,t)
=
°
for s
-1
t
Y(a,t)1](a,s)da,
> t,
where
1 Ih
M
t
dQP(t,s)x(s) == LA_liX(tl - hi)' i=l
-h t
M
d1](s)x(s) == t;Aix(t - hi)'
We next state conditions for the existence of optimal controls for problem Pl. Theorem 10.6.1 Assume that: (i) 0 : En -+ E is continuous and convex. (ii) fo : En X [0, T] -+ E is measurable in t for each x E En, and continuous and convex in x E En for a.e. tEl, and assume further
403
The Time-Optimal Control of Linear Differential Equations
for each bounded set K C En there exists a measurable function E L 1 ([0, T], E) such that
u,
suplfo(x,t)l::; Mk(t) a.e, t
xEK
e t.
(iii) k o : Lp([O, T], Em) X 1---> E is such that for any tJ E Uad, ko(u(t), t) is integrable on I and the functional I' 0 : Uad --+ E defined by
fo(u)
=
iT
ko(u(t),t)dt
is continuous and convex. (iv) Uad is bounded. Then there exists a control Uo E Uad that minimizes the cost J J =
1>0 (x(T)) +
iT
fo(x(t), t)
+ ko(u(t), t)dt.
(1O.6.4b)
Proof: The proof is an easy adaptation of its infinite dimensional analogue developed recently by Nakagiri [27, Theorem 4.1]. Proof: Let {un} be a minimizing sequence of controls for J and Xn the corresponding trajectory:
Since we assumed that Uad is bounded and since it is weakly closed, there is a subsequence (again denoted by {Un}) of {un} and a Uo E Uad such that
(10.6.5) Suppose Xo is the trajectory corresponding to Uo. Let c be a row vector in En", and tEl = [0, T] be fixed. Because the fundamental matrix W is such that W(t) = 0 if t < 0, we have that (xn(t), c)
= (x(t, U, gO,gl), c) +
Recall that BE Loo(I, E n x m
)
it
(un(s), B*(s)W*(t - s)c)ds.
(10.6.6)
and W is by Tadmor piecewise analytic
404
Stability and Time-Optimal Control of Hereditary Systems
It now follows from (10.6.5) and (10.6.6) that
(xn(t), c)
-+
(x(t, u, s", g1), c) +
= (x(t,u,gO,g1),c)+ = (xo(t),c) as n
i
-+ 00
it
(UO(s), B*(s)W*(t - s)c)ds (10.6.7)
t
(W (t - S)B (S)UO(S),C)dS, (weakly) in En.
(10.6.8)
But if a function is continuous and convex, it is weak lower semicontinuity. Therefore Assumption (i) and (10.6.8) imply that (10.6.9)
lim cPo(xn(T)) ~ cPo(xo(T)).
n-+oo
In the same way,
lim fo(xn(t), t)
n-+oo
~
fo(xo(t), t), a.e. i
e i.
(10.6.10)
By standard arguments that use Holder inequality, the set J(
= U{xn(t): tEl, n = 1,2, ... }
is bounded in En. Since there exists an mk E £1 (I; R) such that
Ifo(xn(t), t)1 :S mk(t),
a.e. tEl,
(10.6.11)
Lebesgue-Fatou Lemma yields the following assertion, which follows from (10.6.10) and (10.6.11): lim n~oo
lim fo(xn(t), t))dt, io[T fo(xn(t), t)dt ~ 10[T(n-+oo ~ faT fo(xo(t), t)dt.
The following estimate is available for the terms lim rO(Un) 2: ro(uo)
n-+c:::o
= ir
T
o
f:
(10.6.12)
ko(un(t),t)dt:
ko(uo(t),t)dt,
(10.6.13)
The Time-Optimal Control of Linear Differential Equations
405
since hypothesis (iii) is valid. On gathering the above results in (10.6.9), (10.6.12), and (10.6.13), we have
M o = InfJ 2: lim
+ n--+oo lim fo(u n),
2:
= J(uo, xo) > -00. We have proved that M o = J(uo,xo), i.e., the pair (uo,xo) is the optimal solution for J. Remark: Note carefully that the set of admissible controls is bounded. Theorem 10.6.2 For problem P2, assume that: (i) 0, such that (a) a1kO is measurable in t for each u E L 2 and continuous in u E L 2 for a.e. t E 1, and the value a1kO( u, t) is the Gateaux derivative of k o(u, t) in the first argument for (u, t) E L 2 X 1, and (b) 101 k o(U, t)IEn* ::; Oa( t) + M411ull2 for (u, t) E L 2 X 1; (iv) Us« = {u E L 2 (1, E m ) : Ilulb::; a}. Let (u, x) E Uad X C([1, En]) be an optimal control solution for J In (10.6.15). Then the optimal control u is characterized by
where
1\
is the canonical isomorphism of L 2 (1, Em) into £2(1, E m* ), and
K(u)(t)
= ihko(u(t),t) -
BT(t)y(t)
a.e,
t E 1,
Stability and Time-Optimal Control of Hereditary Systems
406
and yet) satisfies the equations d dt
[M Myet + h )A yet) - f; yet + h )A_ ] + y(t)A o + f; j
j
1j
j -
odo(x(t)t)
= 0,
a.e. tEl,
y(T)
= -d<po(x(T)),
yes)
= 0,
s E (T, T
+ h). (10.6.14)
Proof: Since the cost function is Gateaux differentiable it follows from [17, p. 10] that the necessary optimality condition is given by the variational inequality (10.6.15a) J'(u)(v - u) ~ 0, V v E Uad
when J is differentiable. Because of the hypothesis, and since Lebesgue's Dominated Convergence Theorem is valid, we have
J'(u)(v - u)
=
(iT + iT (is + iT
W(T - s)B(s)(v(s) - u(s))ds, d
(v(s) - u(s), olko(u(s), s))ds.
(1O.6.15b) All the integrands are well defined because of the hypothesis. The first term in (10.6.15) can be transformed by using Fubini's Theorem:
iTis =iT
(W(s - T)B(T)(V(T) - U(T)), odo(x(s), S)dT)ds (v(s) - u(s), B*(s)
iT
(10.6.16)
W*(T - S)Od(X(T), T)dT)ds.
If we let
y(t)
= -W*(T -
t)d<po(x(T))
-iT
W*(s - t)odo(x(s), s)ds,
then from (10.6.15a) - (10.6.16) the following inequality follows:
I
T
(V(t ) - u(t), olko(u(t),t) - B*(y(t))dt
~
0,
for all v E
u.;
The Time-Optimal Control of Linear Differential Equations
407
and this is reduced to
= 0,
<'ltko(u(t),t) - B*(t)y(t) Since Ua/J. = {u E L 2(1, Em) : easily deduce that
lIuli ::; _0'
e t.
(10.6.17)
O'} and our hypothesis is valid, we I«u) I«u)1I2
1\-1
111\-1
u=
a.e. t
where 1\ is the canonical isomorphism of L 2(1, Em) into L2(1, E m )* and
I«u)(t) = 0IkO(U(t), t) - B*(t)y(t), Example 10.6.1: Let
Uad
= L 2([0, T], Em). +
Ji = (x(T),Nx(r))
I
a.e. tEl.
The cost
T
(X(t ), M (t )X(t ))
+ rQu
where
r Q(u) =.!. 2
fT (u(t), Q(t)U(t))Emdt.
Jo
We assume N is n x n matrix MC) E Loo([o,T],Enxn), Q E Loo([O,T]Emxm); and N,M,Q, are positive and symmetric for each t E [0, T]. There is a constant c> such that
°
(u, Q(t)u) 2::
cllul1 2 ,
a.e. t E [0, T].
Thus r Q is strongly continuous and strictly convex. Then there exists a unique optimal control for J 1 , and the following is a consequence of Theorem 10.6.2: Proposition 10.6.2 Consider the cost function in Example 10.6.1. There exists a unique optimal solution
for h. The optimal control is given by
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Stability and Time-Optimal Control of Hereditary Systems
where m
m
x(t) - LA_ 1jx(t - hj) j=l
= Aox(t) + LAjx(t -
hj) j=l + B(t)Q-1(t)BT(t)y(t) + J(t),
x(O)
= l,
xes)
= gl(s),
a.e. s E [-h, 0),
~
[y(,) -
t,
y(t
+ h; )A_.;] + y(')Ao
m
+ Ly(t + hj )Aj
j=l yeT) = -Nx(T), a.e. s E (T,T E h).
- M(t)x(t)
yes)
= 0,
= 0,
The proof follows from Theorem 10.6.2 and Condition 10.6.9.
10.7 The Theory of Time-Optimal Control of Linear Neutral Systems In this section we study the time-optimal problem: Minimize
J(t, Xt) = t
(10.7.0)
subject to: N
x(t) - A_ 1(t)x(t - h) Xu
= Ao(t)x(t) + LAj(t)x(t -
hj) + B(t)u(t), j=l (10.7.1) =
(10.7.2)
where A j are analytic n x n matrix functions and B is an n x m analytic matrix function. The controls are constrained to lie in U
= {u
measurable, u(t) E Em, IUj(t)l:::; 1, a· e j
= 1, ...
,m}. (10.7.3)
For conditions for the existence of analytic solutions of
x(t) - A_ 1x(t - h) = Aox(t) +
N
L
j=l
Ajx(t - hj)
(10.7.4)
The Time-Optimal Control of Linear Differential Equations
409
see Tadmor [29]. If we designate the strongly continuous semigroup of linear transformation defined by solutions of (10.7.4) by T(t,u), t 2:: a so that
T(t, u)
= T(t, u)
Xt(u,
where X is defined as follows: Let
Xo(O) =
0, (
-h
I,
~
0
<
(10.7.5)
0) .
0=0
We are justified in writing
T(t,u)X O= Xt(-,s) where X t(-, s )(0) = X(t + 0, s) 0 E [-h, 0], and where X is the fundamental matrix solution of (10.7.4) or
x(t)
= W(t)l +
where
Ih
0'
Ut (s)g1(s) +
( ) °= {x(t;U,gO,O)
W t 9
and x(t, u,
=
°
1 t
W(t - s)B(s)u(s)ds
°
ift 2:: gO E En, if t < 0,
(10.7.6)
(10.7.7)
s". l) is a solution of (10.6.1) with
x(O) = gO(s)
= x(t) = g1(s),
a.e. s E [-h,O],
gO E En, g1 E W~1).
The solutions of the time-optimal problem. We now solve the timeoptimal problem as formulated by reinterpreting N akagiri in Euclidean ndimensional space. Thus in our case the state space is En, and the target is a fixed convex compact subset H of En with nonempty interior. Define Uoo = {u E L 2 (1, En) : u(t) E em, a.e. t E I}, where
em
is the unit m-dimensional cube, i.e.,
em c Define Uo
= {u E Uad
Em,
IUj I ~ 1 j
= 1, ...
,m.
(10.7.9)
: x(t, u) is a solution of (10.7.1) and x(t, u) E T}
for some t E I}. (10.7.10)
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Stability and Time-Optimal Control of Hereditary Systems
Theorem 10.7.1 Suppose the system is controllable, i.e., Uo "#
(10.7.0)
where
Jor (v(s),B*(s)W*(t* - s)q*)ds t"
max
IiEU. d
1 t·
=
(10.7.11)
(u(s), B*(s)W*(t* - s)q*)ds,
where (".) denotes inner product in En. If Uad is as defined in (10.7.10) and U is the unit m-dimensional cube,
max(v, B*(t)W*(t* - t)q*) = (u(t), B*(t)W*(t* - t)q*), a.e. t E [0, t*]. «eo
(10.7.12)
Consider the adjoint system N
y(t)+ Ly(s+hj)Aj(s+h j) j=l
=-
N
Ly(s+hj)Aj(s+hj)-q;(t), a.e. t E I, j=l
(10.7.13) yeT) = -qo, yes) = 0, a.e. s E (T, T + h). It is said to be regular (or proper) if whenever there is a set of positive measure h C I such that
The Time-Optimal Control of Linear Differential Equations
=
411
=
mUd> 0 and y(t; q~, 0) 0 for all t E I, then q~ 0 E s«. Systems (10.7.4) that are pointwise complete for all t > 0 are regular. Recall that (10.7.4) is pointwise complete if
is a surjection, a.e. Theorem 10.7.3
Consider the optimal problem with cost J
= ¢o(x(T)).
(10.7.0)
Assume that the adjoint system is regular and the matrix B T (t) is one to one, a.e. t E I. Suppose the Gateaux derivative d¢o(Y) # 0 in E n * for all y in the reachable set lW.(t) = {y E En : y = x(t,u) : u E Uad}. Then the optimal control u(t) is bang-bang: that means that u(t) satisfies
u(t) E 8U(t), a.e. t E I,
(10.7.14)
where 8U(t) denotes the boundary of U(t), the m-dimensional unit cube. Proof: Because of our definition of J, the maximum principle is stated as max (u, B*(t)y(t))
vEU(t)
= (u(t), B*(t)y(t)),
a.e. t E I,
(10.7.15)
where y(t) = y(t; d¢o(x(T)), 0) and x(t) is the trajectory corresponding to the optimal control u(t). It suffices to show that
B*(t)y(t)
#0
in
e«, a.e. t E I.
(10.7.5)
Suppose to the contrary that there is a set of positive measure finite I, mUd> 0 with B*(t)y(t) = 0, V t E 11 , Since B*(t) is one-to-one and the system is regular, we have that d¢o(x(T)) = 0. But then x(T) is in the attainable set, this condition d¢o(x(T)) = 0 is impossible. As a consequence of these results, we deduce the following solution of the timeoptimal problem: Theorem 10.7.4 Consider the optimal problem Pa with t" the optimal time. Let the target H be a closed convex subset of En with nonempty interior. Suppose the adjoint system is regular and B*(t) one-to-one. Then the time-optimal control u(t) is bang-bang on 1* = [0, to], i.e., u(t) is one of the vertices of the unit m-dimensional cube. In addition, suppose U(t) =
412
Stability and Time-Optimal Control of Hereditary Systems
{u E Em : lu - y(t)IE m
~ vet)} tEl, and this replaces the m-dimensional unit cube. Also the time T = t* is the optimal time. Then the optimal control is given by
u(t)
I\E~B*(t)z(t)
= yet) + vet) 11\-1 EmB*(t)Z(t)IEm'
a.e, i
e t«,
where z(t) = W*(t* -t)q* is the solution of the adjoint equation t E 1*, and q* is as given in Theorem 10.7.3. Here I\Em is the canonical isomorphism of En onto En'.
10.8 Existence Results Definition 10.8.1: Let Zt E G([-h, 0], En) be a target point function that is time varying. System (10.1.9) is controllable to the target if for each ifJ E G there exists a t1 ~ (J' and an admissible control u E L oo([(J',t1],Gm) such that the solution of Equation (10.1.9) satisfies
Xq((J',ifJ,u) == ifJ,
Xt,((J',ifJ,u)
= Xt,·
Theorem 10.8.1 Assume that System (10.1.9) is controllable to the target. Then there exists an optimal control. Proof: The variation of constant formula for system (10.1.10a) is
Xt((J', ifJ, u) = T(t, (J')ifJ + it d.X t(-, s)[G((J')(ifJ) - G(s)(x.)]
+ it X t(-, s)B(s)u(s)ds. Controllability to the target is equivalent to
Xt,((J',ifJ,u)
= Zt"
for some t 1 ,
that is,
t' s.x.,(., s)[G((J')( ifJ) -
Wt, ~ Zt, - T(t, (J')ifJ -}q
= l;t' Xt,(-,s)B(s)u(s)ds.
G(s)(x.)]
The Time-Optimal Control of Linear Differential Equations
413
This is equivalent to Let
t* = inf{t : Wt E a(t,O")}. Now 0" ~ t" ~ it. There is a nonincreasing sequence of times t n converging to r , and a sequence of controls un E Loa ([0", tl], cm) with
Also
where I
~
Ill
tn XtJ,s)B(s)unds
I IIXtn (., s)B(~)un(s)ds ~ l: t
+
-r
XtJ,S)B(S)Un(S)dSII
-I
t X t.(-, s)B(S)Un(s)dsll
n
IIXtJ, s)B(s)un(s)dsll
+
1 t'
Il(XtJ, s) - X t·(-, s)JB(s)u n(s)lIds.
Because X tn(-, s)B(s)un (s) is integrable and [tn,t*] < the r .h.s. of the inequality tends to zero as t n ---> i", We know from Henry [25] that
IIXtJ,s)11 ~
{3
< 00,
00,
the first term on
for all tn,s, for some {3;
also X tn(-, s) ---> Xt. (-, s) in the uniform topology of C. Hence by the bounded convergence theorem, the second summand on the l.h.s. tends to zero as n ---> 00. From the continuity of solution in time and the continuity of the target, Ilwt. - Wtnll---> 0 as t« ---> t", Hence Wt' = lim n -+ oo y(t*, un). Because a(t*, 0") is closed and y(t*, un) E a(t*,O"), w(t*) = y(t*, u*), for some u* E Loa ([0", tl], Cm), and by definition t/", u* is optimal.
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Stability and Time-Optimal Control of Hereditary Systems
A controllability assumption was made in Theorem 3.1 for System (10.1.9). When the target is the zero function, what is required is the assumption of null controllability only. To get conditions for this, we need two preliminary results and precise definitions. We work in the space W~). The argument is also valid in C. Definition 10.8.2: System (10.1.9) is controllable on [0', ttl, t1 > 0' + h if for each t/J, ¢ E Woo there exists a control u E Loa ([0', ttl, Em) such that the solution of (10.1.9) satisfies Xu (0', ¢, u) ¢ and Xt 1 (0', ¢, u) t/J. If System (10.1.9) is controllable on each interval [0',ttl,t 1 > 0' + h, we simply say that it is controllable. It is null controllable on [0', ttl if t/J == 0 in the above definition.
=
=
Definition 10.8.3: System (10.1.9) is null controllable with constraints if for each ¢ E W~) there exist a t1 ;:::: 0' and au E Loo([O',td,C m) such that the solution x(O', ¢, u) of system (10.1.9) satisfies xu(O', ¢, u) = ¢ and Xtl (0', ¢, u) = O. Proposition 10.8.1 Suppose System (10.1.9) is null controllable on [0', ttl· Then for each ¢ E Woo, there exists a bounded linear operator H : W~) ~ L oo([0',t1],Em) such that
u=H¢
has the property that the solution x( 0', ¢, H ¢) of System (10.1.9) satisfies
xu(O',¢,H¢) = ¢,
xt,{O',¢,H¢) = O.
Proof: From the variation of constant formula (10.1.10),
Xt(O', ¢, u)
= T(t, O')¢ + C(t, O')¢ + 8(t, O')u
where
T(t, O')¢
= Xt(O', 0', 0),
C(t, O')¢
= it ds{X.(-, s)}[G(O')¢ -
8(t,0')u = it X.(-,s)B(s)u(s)ds,
G(s)x.(O', ¢)], t E [O',td.
The null controllability of System (10.1.9) is equivalent to the following statement: For every ¢ E Woo there exists a t1 and there exists a u E Loo([O',td,En ) such that
T(t1,0')¢
+ S(t 1,0')u + C( t1,0')¢ = 0,
t1 >O'+h.
The Time-Optimal Control of Linear Differential Equations
415
This is in turn equivalent to (10.8.1) The condition (10.8.1) is now valid by hypothesis. Denote by N the null space of S and by Nl. the orthogonal complement of N in Loo([O',td,E n). Let
So: Nl.
-+
S(tl, O')(Loo([O', til, En)
be the restriction of S(t 1,0') to u», Then Sol exists and is linear though not necessarily bounded, since S(td(Loo([O',td, En) is not necessarily closed. Define a mapping H : W~) -+ Loo([O', tIl, En) by H ¢
= -SOl [T(t l , O')¢ + C(t l, O')¢].
Xt(O', ¢, H ¢)«()) = x(O', ¢, H ¢)(h
Then
+ ()),
=T(t10')¢«()) + C(tl, O')¢«())
+ S(h, ())[-Sol(T(t1' O')¢( ()) + C(t 1, 0' )¢(()))] = 0,
-h
s () ::; O.
=
=
Since u H¢ E Loo([O',tl],En), we deduce that Xt,(O',¢,u) O. We now prove the boundedness of H as follows: Let {¢n} be a convergent sequence in W~) such that {H ¢>n} converges in Loo([O', til, En), and let
¢
=
lim ¢>n,
n~oo
u
= n-+oo lim H ¢>n,
Since Nl. is closed in Loo([O',td, En),
U
E
Un
»» and
T(t 1, O')¢> + C(h, O')¢> + S(t1, O')u = lim (T(t1, O')¢>n n--+oo
= O. Thus, u
= H ¢>n. + C(tl, O')¢>n + S(tl, O')u n
= -Sol[T(tl'O')¢> + C(t1,0')¢>]'
By the Closed Graph Theorem, H is bounded. The proposition is proved. Definition 10.8.4: System (10.1.9) is locally null controllable with constraints if for each ¢ E 0,0 an open neighborhood of zero in W~), there exists a finite t 1 and a control u E Loo([O',td,cm) == U such that the solution x(O', ¢>, u) of Equation (10.8.1) satisfies
xq(O', ¢, u)
= ¢,
Xt,(0', ¢>, u)
= o.
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Stability and Time-Optimal Control of Hereditary Systems
Proposition 10.8.2 Suppose that System (10.1.9) is null controllable. Then System (10.1.9) is locally null controllable with constraints. Proof: Because System (10.1.9) is null controllable, from Proposition (10.8.1) there exists a bounded linear operator H : W~)
such that for each
-+
Loo([O", til, En)
and control u = H
Since H is a bounded linear map, it is continuous at a E W~). Therefore, for each open set containing zero in Loo([O",td, En), there is an open neighborhood U of a E w~) such that
H(U) C V. Clearly Loo([O",td,em) has zero in its interior. We can choose V open and contained in L oo ([0", t 1 ] , em). For this particular choice there exists an open set 0 around zero in W~) such that
Every
e,
Suppose in System (10.1.9), with state space W~)
or
(i) The system (10.1.9) is null controllable. (ii) The solution x = 0 of (10.8.2)
is exponentially stable. Then System (10.1.9) is null controllable with constraints. Proof: The first assumption yields an open ball 0 C W~) such that every initial
The Time-Optimal Control of Linear Differential Equations
417
= 0, i.e.,
every solution
By condition (ii), every solution (10.1.9) with u of Equation (10.8.2) satisfies
so that
Xt(O",.p,O)--O as t
s-s
oo.
Therefore, there is a finite to < 00 such that tP = Xto(u,.p,O) E O. With (to, tP) as initial data, there exists tl > to such that some control
gives a solution x(to,tP,u) such that Xto(to,tP,u) = tP, Xtl(O"'tP,U) Thus, system (10.1.9) is null controllable. The control w
= 0, = U,
0.
1D [0", to], 1D [to, tl]
is contained in U and does the transfer of.p to
°in time
<
tl
00.
Definition 10.8.5: System (10.1.1) is said to be proper on [0", t] if and only if E Int net, 0"), t ~ 0" + h.
°
It is proper if it is proper on every [0", t], t > 0" + h. In the above definition, net, 0") is assumed to be a subset of C or of W?) if the controls are L p • More precisely, if we work on C or W~), we use
controls in Loa ([0", t], cm) == U. However, if we are in WP), then Uad is a closed and bounded convex subset of Lp([O", ttl, En) with zero in its interior. The next result is stated for the space WJ1}. It is equally valid for W~) or C. Theorem 10.8.3 System (10.1.1) is proper if and only if it is functionspace controllable. Proof: Suppose system (10.1.1) is controllable on [0", t], t
~
0" + h. Then
is onto
H(U) H(B) C H(U)
= net, 0"), = net, 0") = {Xt(O", 0, u) : u E Uad } ,
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Stability and Time-Optimal Control of Hereditary Systems
°
where B is an open ball containing with Be U. Because H is a continuous linear transformation of Lp onto WP), H is an open map. H(U) is open and contains zero. E Int a(t, u). Hence System (10.1.1) is proper. Assume System (10.1.1) is proper, i.e.,
°
°
E Int a(t, u).
°
Because E Int a(t,u) C Int A(t,u), where A(t,u) = {Xt(u,O,u): u E L p } is a subspace this implies A(t,u) = WP). For the linear system (10.1.9) we have proved that uniform asymptotic stability of System (10.8.2) and null controllability of System (10.1.9) suffice for constrained null controllability of System (10.1.9). It is clear that any test for controllability, though perhaps too strong, will guarantee null controllability. Indeed, we will now show that when D(t)Xt in System (10.1.1) is atomic at both and -h, then null controllability and controllability are equivalent.
°
Proposition 10.8.3 System (10.1.1) is null controllable if and only if it is controllable, provided D is atomic at a and at -h.
°
Proof: It is given that D(t, ¢J)is atomic at and at -h, for (t, ¢J) E Ex C. By a theorem of Hale [9, p. 279] the operator T(t, u) given by the solution Xt(u, ¢J) = T(t, u)¢J of Equation (10.1.9) is a homeomorphism for t 2: a . As a consequence of similar arguments,
= W~),
T(t,u)W~)
t
2: a,
If system (10.1.1) is null controllable, then for each ¢J E W~), T(t 1, u)¢J + it!
With 5(t 1 , u) : L oo
W~)
-+
x t !(-, s)B(s)u(s)ds = 0.
defined by
5(t 1,u)u
= it! Xd.,s)B(s)u(s)ds,
null controllability is equivalent to
T(t1'U)W~)
m
C 5(t 1,u)(L oo[u,td,E ) .
Therefore, this implies W~)
C 5(t1,U)(L oo[u,t1],E
m
)
so that 5(t1, u) : L oo -+ W~) is a surjection. However, 5(t1, u) is surjective if and only if System (10.1.1) is controllable. We have proved that null controllability implies controllability. The converse is trivially true: controllability always implies null controllability.
The Time-Optimal Control of Linear Differential Equations Corollary 10.8.1
419
The system
d
dt [x(t) - A_Ix(t - h)] = Aox(t) + Alx(t - h) + Bu(t),
(10.8.3)
with det(A_d :I 0, is null controllable if and only if rank [Ll(A), B) = n, 't/ A E C, rank [U - A_bB] where
= n,
't/ A E C,
Ll(A) = fA - AA_Ie- A - A o - Ale-A.
Proof: D(t)xt - A_Ix(t - h) is atomic at 0 and at -h if det(A_d :I O. The rank conditions suffice for controllability by a result of Salamon [18, p. 157], which we now state. 'I'heorern 10.8.4 System (10.8.3) is controllable in the state space W~1), 1 ~ p ~ 00, if and only if rank [Ll(A), B) = n 't/ A E C and rank [U A_I,B] = n, 't/ A E C where Ll(A) = D; - AA_Ie->' - A o - Ale->'.
10.9 Necessary Conditions for Optimal Control We now return to our original problem of hitting a continuously moving point target Wt E C in minimum time. At the time of hitting, in System (10.1.9)
Xt(O', ¢, u)
= T(t, O')¢ + it [d.XtL s)][G(O')(¢) -
G(S)(Xt)]
+ i t XtLs)B(s)u(s)ds = Wt, or equivalently,
Wt - T(t,O')¢ - C(t,O')¢
=z(t) = it XtLs)B(s)u(s)ds,
where
C(t, O')¢ = it [d.X t(-, s)][G(O')(¢) - G(s)(x.)). Thus, reaching Wt in time t corresponds to
Wt - T(t, O')¢ - C(t, O')¢
=z(t)
E a(t, 0').
We shall prove shortly that if u* is the optimal control with optimal time t*, i.e. if u* is the control that is used to hit Wt in minimum-time t*, then
Wt* - T(t*, O')¢ - C(t*, O')¢
= z(t*) E 8a(t*, 0'),
that is, z(t*) is on the boundary of the constrained reachable set.
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Stability and Time-Optimal Control of Hereditary Systems
Theorem 10.9.1 Let u* be the optimal control with t* minimum-time. Then z(t*) E oa(t*,u). Proof: Suppose u* is used to hit
z(t*) =
Wt
in time i":
T(t*, u)4> - C(t*, u)4> E a(t*, u).
Wt- -
Suppose z(t*) is not in the boundary of a(t*, e ):
z(t*) E Int a(t*, o ),
t" > a.
Therefore, there is a ball B(z(t*),p) of radius p about z(t*) such that
B(z(t*),p) C a(t*,u). Because a( t, u) is a continuous function of t, we can preserve the above inclusion for t near t* if we reduce the size of B(z(t*),p), i.e., if there is a 8 > 0 such that
B(z(t*),p/2) C a(t,u), t* - 8 Thus, z(t*) E a(t, u), t" - 8 are led to conclude that
~t.
~
t
~
r.
This contradicts the optimality of t", We
z(t*) E oa(t*, u). Theorem 10.9.2 Let D and L satisfy the basic assumptions and inequality (10.1.13). Suppose System (10.1.9) is controllable and the solution operator is a homeomorphism, and
z(t)
= Wt -
T(t, u)4> - C(t, u)4>.
(10.9.1)
If u* is an optimal control and t* the minimum time, then z"
= z(t*) E oa(t*,u) C C
if and only if u* is of the form u*(t)
= sgn[-y(t, t*)B(t)],
a ~ t ~
r,
(10.9.2)
where y(t) is a nontrivial solution of the adjoint equation (10.1.6). Proof: Because of Proposition 10.1.1, a(t, u) is closed and convex. If u* is an optimal control and t* the minimum time, then by Theorem 10.9.1, z" E oa(t*, u). Also from controllability, and Theorem 10.8.3,
o E Int a(t*,u).
The Time-Optimal Control of Linear Differential Equations
421
By the Separation Theorem of Dunford and Schwartz [26, p. 148], there exists a 'l/; E B o such that 'l/; #- 0, and
('l/;, p)
~
('l/;, z") for every p E a(t", 0').
It follows from this that
('l/;,
itO T(t", S)XoB(S)U(S)dS) ~
for every
U
('l/;,
itO T(t" ,S)XOB(S)U"(S)dS) ,
E Loo([O', t"], C m). On using Equation (10.1.4), we deduce that
('l/;,
itO T(t"S)XoB(S)U(S)dS) a
= lh [d'l/;(O)] = =
1 to
T(t", s)Xo(O)B(s)u(s)ds,
itO {IOh [d'l/;(O)]T[(t", s)Xo](O)} B(s)u(s)ds,
1 to
-y(s, t")B(s)u(s)ds,
where s ---+ y(s, t") is the solution of the adjoint equation (10.1.6). (See Henry [25].) Thus
1 t·
[-y(s, t")B(s)u(s)]ds
~
1 t·
[-y(s, t")B(s)u"(s)]ds
(10.9.3)
for every admissible u. We see from inequality (10.9.3) that u.. is of the form u"(s) = sgn[-y(s, t")B(s)], (10.9.4) where y( s, t") is the solution of the adjoint equation. We note that u.. maximizes the l.h.s. of inequality (10.9.3). Thus, with t" fixed and u .. of the form (10.9.4), the point z" is on the boundary of a(t", 0') and ('l/;, p) ~ ('l/;,z") for every p E a(t" ,0'), 'l/; E B o. In Theorem 10.9.2, we have the form of optimal control for the timeoptimal problem in the space of continuous functions C. We now show that this form is also valid in En, without the controllability assumption. This is true since the Separation Theorem is true in En for any closed and convex set, that is, through each boundary of a closed convex set in En there is a support plane. We now state the following theorem:
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Stability and Time-Optimal Control of Hereditary Systems
Theorem 10.9.3 Let D and L satisfy the basic assumptions and Inequality (10.1.13). Let w(t) be a point function that is continuous in En. Let
z(t)
= w(t)
- T(t, o-)¢(o) - C(t, o-)¢(O).
(10.9.5)
If u" is an optimal control and t* the minimum time, then z(t*) E 8R(t*, 0-) C En if and only if
u*(s) where 11'(0)
= sgn[-B"(s)y(s)] = sgn[B"(s)[T*(t,s),p](O)]jUj(s),
:f. 0
and y is a nontrivial solution of the adjoint equation.
10.10 Normal Systems It is important to know when the necessary condition for optimal control in Theorems 10.9.2 or 10.9.3 uniquely determines an optimal control. This occurs when System (10.1.1) is normal in the sense to be made precise below. First observe that the necessary condition for optimal control in C states that
u*(t) = sgn[-B"(s)y(s)l, 0-::; s::; to,
(10.10.1 )
where y(s) is a nontrivial solution of the adjoint equation. This condition states that
uj(s)
= sgn[-b*j (s)y(s)]
on [0, tjl,
sgn[ -b*j [T* (s, t*),p](0)], for j
= 1, ...
,m, where b"j(t) is the jth row vector of B"(t). Define
gj(,p(O)) = {s: b*j(s)y(s) = 0, s E [O,t*]}
= {s: b"j(s)[T*(s,t*),p](O) = 0,
s E [O,t"]}.
Define
Yi(,p) = {s: y(s,t*,,p)bi(s)
= 0,
s E [o-,t*]},
gij = {s: -y(s,t",,p)bi(s) = 0, s E [o-,t"]}. Definition 10.10.1: We say that System (10.1.1) is En normal on [0-, to] if Yj(,p) has measure zero for each j = 1, ... ,m, and for each nontrivial 11' with 11'(0) :f. 0. The system is En normal if it is normal on every interval
[0-, to]. Similar definitions can be made for C -normality if we replace Yj( 11') with gij (11') in the above definition. Utilizing the ideas of Gabasov and Kirillova [12] we are led to a necessary and sufficient condition for normality that is easily computable. It is Theorem 10.3.1.
423
The Time-Optimal Control of Linear Differential Equations
10.11 The Geometric Theory of Time-Optimal Control of Linear Neutral Systems In this section we study the time-optimal problem: Minimize
J(t, Xt) = t
(10.7.0)
subject to: N
X(t) - A_1(t)x(t - h)
= Ao(t)x(t) + L
Aj(t)x(t - hj)
+ B(t)u(t),
(10.11.1)
j=l
Xt7 = ¢, (t,Xt(u,¢,u) E [0,00) x H,
He C, (10.11.2)
where Aj are analytic n x n matrix functions, and B is an n x m analytic matrix function. The controls are constrained to lie in
Uad = {u E L~C([u,
00), Em) : u(t) E U a.e.
t E [u,oo); U C E'!', compact and convex;
°
E Int U.
Here L~C([u, 00), Em) represents the space of measurable functions defined on each interval [u, t] having values in Euclidean n-space En and having essentially bounded values. We work in the space C = C([-h, O], En) of continuous functions from [-h,O] ---> En with the sup norm. For these two spaces, the appropriate control set is U
= {u
measurable, u(t) E Em, IUj(t)1 ~ 1, a.e. j
= 1, ... ,m}.
(10.11.3) In what follows, Xt E C is defined by Xt(s) = x(t + s), s E [-h,O]. For the existence, uniqueness, and continuous dependence on initial data of the solution xC a, ¢, u) of (10.11.1), see Hale [9, pp. 25, 301]. See also Henry [16]. For conditions for the existence of analytic solutions of N
x(t) - A_1x(t - h)
= Aox(t) + L
Ajx(t - h j),
(10.11.4)
j=l
see Tadmor [29]. If we designate the strongly continuous semigroup of linear transformations defined by solutions of (10.11.4) by T(t, o ), t ~ a so that T(t, u)¢ = Xt(u, ¢, 0),
424
Stability and Time-Optimal Control of Hereditary Systems
then the solution x(u,,u) of (10.1.1) with x,,(u,,u) = relation
satisfies the
(10.11.5) where X is defined as follows: Let
X o(8) =
(
0,
-h~8<0).
I,
8
=
°
We are justified in writing
where X t (-, s)( 8) = X (t + 8, s) 8 E [-h,O] and where X is the fundamental matrix solution of (10.11.4).
Definition 10.11.1: The reachable set of time t is defined by
a(t, u)
=
{it T(t, s)XoB(s)u(s)ds : u
E
Uad} ,
and it is a subset of C. The following results are easily proved:
Theorem 10.11.1 (i) System (10.1.1) (ii) T(t,s)a(s,u) C (iii) E lnt a(t,u),
°
The following are equivalent: is controllable on [u, t], t 2: a lnt a(t,u), a ~ s < t. t 2: a + h.
+ h.
Proof: The equivalence of (i) and (iii) is established in Theorem 3.3 of [4]. To see that (ii) and (iii) are equivalent, we first note that E a(t, 0-) for each t 2:: a , so that
°
° T(t, s)a(s, u) E
C Int
a(t, o ),
°
~ s
if we assume (ii). Conversely, assume (iii). Suppose T(t, s)q E oa(t, e ), the boundary of a(t, e ) where q E a(s, u), and aim at a contradiction, since indeed
The Time-Optimal Control of Linear Differential Equations
425
By a separation theorem of Dunford and Schwartz [26, p. 418], there exists
1/; t 0, 1/; E B o the conjugate space of C, such that (1/;,p)::; (1/;,T(t,s)q)
(10.11.6)
for every p E a(t, IT), where (.,.) is the outer product in C. From (10.11.6),
(1/;,p- T(t,s)q)::; 0, V P E A(t, IT). But for some u E U,
(1/;, T(t, ~)q)
= (1/;, = (1/;,
it it
T(t, s)T(s, r)XoB(r)u(r)dr) , T(t, r)XoB(r)u(r)dr) .
If we define
u*(r)
=
{
u(r), '
o~
sgn[g(t, r, 1/; )B( r)],
s
r
~
s,
< r::; t,
where
g(t, r, 1/;)
= (1/J, T(t, r)Xo) = lOh [d e1/;(8)][U(t + 8, r)],
(10.11.7)
then u* E U determines apE a(t, IT), so that
it = (it {lOh it
(1/;,P - T(t, s)q) = (1/;,
=
T(t, r)XoB(r)U(r)dr) , de[1/;(8)][T(t, r)Xo(8)]} , D(r)u*(r)dr) ,
Ig(t, r ; 1/;)ldr > O.
This contradicts our assumption that T(t, s)q is in the boundary of a(t, IT) and (-/!, p-T(t, s)q) ::; O. Hence T(t, s)q E Int a(t, IT), so that T(t, s)a(s, IT) C Int a(t, IT).
426
Stability and Time-Optimal Control of Hereditary Systems
Theorem 10.11.2 [14, Theorem 4.2] Suppose System (10.1.1) is controllable. If u* is an optimal control and t" the minimum time, then u* is of the form u*(s) = sgn[g(t* ,s, tI')B(s)] (10.11.8) where g(t*, s,·) : B o -. E'"' is the row-vector function defined in (10.11.7). It is easy to verify that s -. g(t*, s, tI') satisfies a linear differential equation [16]. It follows from Tadmor [29] that s -. g(t*, s, tI') is piecewise
analytic if the initial conditions are analytic. Remark 10.11.1: In En the optimal control is given by u*(s)
= sgn[-y(t)B(t)],
a ~ t ~
r ,
where y : [u, t*] is an n-row vector function of bounded variation, which is a nontrivial solution of the adjoint equation N d ds[y(s)-y(s+h)C(s+h)] = y(s)A(s) + .?=y(s+hj)Bj(s+hj). (10.11.9) J=1
The form of optimal control in (10.11.8) states that each component uj of u* is given by uj(s) = sgn[g(t*,s,tI')ll(s)] on [O,tj] j
= 1, ...
,m, where bi(s) is the j-th column of B(s).
Remark 10.11.2: It is easy to see from (10.11.8) that if (10.11.1) is normal, then optimal control is uniquely determined by (10.11.8) and it is piecewise constant and bang-bang. This implies that optimal trajectories are umque.
10.12 Continuity of the Minimal-Time Functions From the definitions of Section 10.1.1, we deduce that if (10.11.1) is controllable with constraints, then:
A(u)
= UU) : Xt(u, ¢J) = 0
for some t, some u}
If (10.11.1) is null controllable with constraints, then:
-T(t, u)¢J E a(t, u)
= C.
The Time-Optimal Control of Linear Differential Equations
427
for some t. In this case, can be steered to zero at time t by some admissible control. Definition 10.12.1: The minimal-time function M is defined by
M(
M(
tr ~
~
~ a :
-T(t, rT) E a(t,rT)}.
+00, with M( E a(t,rT)
for some t
~
a, We observe that
When (10.11.1) is controllable so that A(rT) = C, then M is continuous. We have: Theorem 10.12.1 function
Let (10.11.1) be controllable. Then the minimal-time
is continuous. The proof is similar to that of Theorem 7.2.1. Corollary 10.12.1
For t
~ tr,
we have
{-T(t,rT): M(
~
t} = a(t,rT).
Also
{-T(t, rT) : M(
aCt, rT) C {-T(t, rT) : M(
~
t} C
n
a (t
p=l
We now show that
a(t,rT):J
n
a (t
p=l
+ ~,rT). P
+ ~,
P
rT) .
428
Stability and Time-Optimal Control of Hereditary Systems
n (t + ~,U').
Indeed, let ¢ E
Then there is an admissible control up ;
a
p=l
[U', t + ~]
such that
t+l. T (t + p'1) s XoB(s)u(s)ds,
¢ = I~
P
r ("»: 1)
= J<1 T
t+l. T (t+p's 1) XoB(s)u(s)ds,
XoB(s)u(s)ds+ J<1
P
== x p + Yp' But then
xp =
it
T (t
+ ~,
t) T(t, s)XoD(s)u(s)ds,
= T (t
+ ~,t)
it
E T (t
+ ~,t)
a(t,U').
T(t,s)XoD(s)u(s)ds,
Therefore,
Xp
-+
T(t,t)¢ = ¢ as p -+
00.
Because Yp -+ 0 as p -+ 00, the elements ¢ form the closed set a(t, U'). For proof of the second formula, we observe that
T(t,
U') :
C
-+
C
is an open map, since in (10.11.1) detA_ 1 (t ) =f. 0 for each t, [9, p. 279]. Indeed, T(t,U')¢ = Xt(U',¢,O), t 2: U' where x(U',¢,O) is a solution of (10.1.7). Theorem 2.6 of [9, p. 279] insures that
T(t,U')C = C, t 2:
U'+
h,
so that T(t, U') is an open map. Since T(t, U') is also continuous, we have {-T(t,U')¢: M(¢) < t} is open, so that
{-T(t, U')¢ : M(¢) < t} C Int a(t, U'). We know that if
-T(t*,U')¢ E Int a(t*,U'),
U'
< t* <
00,
The Time-Optimal Control of Linear Differential Equations
429
then -T(t,u)¢Elnta(t,u), for t
o t":
Therefore M (¢) ~ t" < t. This completes the proof. Indeed, suppose z" = -T(t*, u)¢ E Int A(t*, u), a < t* < 00. Then there is a ball B = B(x*, 8) of radius 8 about x* such that Be aU*, o ). Thus T(t, t*)x*
= T(t, rvri«, u)¢,
= -T(t, u)¢ E T(t, t*)(B) C T(t, t*)a(t*, u)
C a(t, u).
Since T(t, t*) is open, -T(t, u)¢ E Int a(t, u) for t > t", Just as in Hajek [30] and Chukwu [31], we use the continuity of the minimal-time function M to construct an optimal feedback control for (10.11.1). We need a special subset of C* = B o, the conjugate space of C, which may be described as the cone of unit outward normals to support hyperplanes to a(t, u) at a point ¢ on the boundary of a(t, u).
=
=
Definition 10.12.2: For each ¢ E A(u), let I«¢) 'l/J E B o : 11'l/J11 1 such that ('l/J,p) ~ ('l/J,¢), V p E a(M, (¢),u) where C·) is the outer product in C. Remark 10.12.1: Note that ¢ E aa(M(¢),u). We now outline key properties of the set I«¢). Let 5(B o) denote the collection of subsets of Be: Then I< : C --+ 5(B o)
is the mapping defined above. Definition 10.12.3: We say that I< is upper semicontinuous at ¢ if and only if limsupJ{(¢n) C I«¢) as ¢n --+ ¢. We have: Lemma 10.12.1 If (10.11.1) is controllable, then I«¢) is nonvoid and J{(-¢) = -I«¢). Also, I«¢) is upper semicontinuous at ¢.
Proof: Let (10.11.1) be controllable. Then 0 E Int a(t,u) by Theorem 10.11.1. Because ¢ E aa(M(¢), u), there exists a 'l/J E B o, 'l/J =I=- 0 [26, p. 418] such that ('l/J,p) :::; ('ljJ,¢) for all p E a(M(¢),u).
430
Stability and Time-Optimal Control of Hereditary Systems
The choice of t/J can be made such that 1It/J1l = 1. Hence K(
Let (10.11.1) be controllable. There exists a measurable y :
a(u)
-+
Bo
such that
Y(
K(
= {k(
From the continuity of t -+ k(
M
(t/J,p):S; (t/J,
aCt, u) and of
-+
= C = A(u),
A
= B o,
t/J E B o,
theorem 4 of [32] asserts that there exists a measurable function y : A( u) -+ B o such that y(
g(t,s,t/J) =
lOh de[t/J(B)]X(t + B,s)
(10.12.0)
as a function of s is important. This is determined by the same property of s -+ X (t, s). For this we now recall that the fundamental matrix X (t, s) of System (10.11.4) satisfies as a function t of the equation n
X(t, s) - A_ 1(t)X(t - h, s)
= Ao(t)X(t, s) + L j=l
Aj(t)X(t - hj , s)
for t # kh, k = 0,1,2, ... ,t ~ a . We now assume A j are real analytic functions. It is a consequence of Theorem 4.1 of Tadmor [29] that, for each t,s -+ X(t,s) that is piecewise analytic in s on [O,t], it follows that s -+ get, s, t/J) is also piecewise analytic on [0, t] and more generally on any interval [u, a + T], for T > o.
The Time-Optimal Control of Linear Differential Equations
431
Theorem 10.12.2 In (10.11.1), assume that A j are each analytic functions, det A_ 1 (t ) f: 0 for each t. Let (10.11.1) be controllable and normal. Then there exists a measurable function f : A( e ) - Em that is an optimal feedback control for (10.11.4) in the following sense: Consider the system n
i(t)-A_ 1(t)i(t-h)
= Ao(t)z(t)+ EAj(t)z(t-hj)+B(t)f(zt).
(10.12.1)
j=l
Then each optimal solution of (10.11.1), is a solution of (10.12.1) and each solution of (10.12.1) is a solution (possibly not optimal) of (10.11.1). Proof: By Lemma 10.12.2, select a measurable function y : A(u) B o , y(¢) = t/J E B o , and define
= t lim -sgn[g(t,s,y(¢))B(s)], ....... +
f(¢)
(10.12.2)
where g is defined in (10.12.0). Because B is analytic, each coordinate of
h(sJ == g(t, s, y(¢))B(s) is piecewise analytic on [u, M(¢)]. That is, h(s) is analytic on i = 1,2, . " ,v for a partition a
=:
[Si-1, Sil,
So < s, < ... Sv = M(¢).
Therefore
sgn[g(t, s, y(¢))B(s)] is piecewise constant on each (Si-1,Si], i = 1,2, ... ,v, and therefore on [u, M(¢)]. Because of this, and because the right- and left-hand limits of X(t,s) exist at each kh, k = 0,1,2, ... , the limit (10.12.2) exists. Since y(¢) is a measurable selection, so is f(¢). Let x : [u, M(¢)] - C be an optimal solution of (10.1.1) with Xu =: ¢. It can be verified that x satisfies (10.12.1). Indeed, take an arbitrary s, a :s: s < M(¢); then x. E A(u), from controllability of (10.11.1). There exists an optimal control through x., a control that can be taken to be
u.(t)
=:
-sgn[g(t, s , y)B(s)]
for any choice of y E K(x.), i.e., -y E K(-x.). Our choice is y = y(x.). This choice is possible by Theorem 10.11.2. But then, because (10.11.1) is
432
Stability and Time-Optimal Control of Hereditary Systems
normal, optimal control is unique and determines uniquely a response x. Therefore for almost all t ~ S we have
Uq(S)
= u.(t) = -sgn[g(t, s, y(x.))B(s)].
On taking the limit as t
s+(O":::; s :::; t),
-+
Uq(s) = u.(t)
= lim-sgn[g(t, s, y(x.))B(s)],
we obtain uq(s+) = f(x.), s E [0", M(¢)], since Uq is piecewise constant U q (s) = f (x.) for almost all s. The response x to U q satisfies n
x(s) - A_1(s)x(s - h)
= Ao(s)x(s) + L
Aj(s)x(s - h) + B(s)f(x.)
j=l
a.e.,
Xq
= ¢.
Clearly x is a solution of (10.11.1). Now let x be a solution of (10.12.1), which is a response to v(t) = f(xt}. Then v(t) E U. The proof is complete. Remark 10.12.2: If in Theorem 10.12.2 we work in En, we replace g in (10.12.2) by y where y solves the adjoint equation (10.11.9). For En we use the usual inner product (-, .).
10.13 The Index of the Control System In our main Theorem 10.12.2 for C, the function
k(s)
= g(t,s,1/;)B(s)
(10.13.1 )
is fundamental in determining optimal control strategy for our time-optimal problem. It is designated as the index of the control system. To study the index, a thorough study of the fundamental matrix X(t, s) of (10.11.4) is required. We initiated such a study in Proposition 2.3.1 by considering the autonomous version of (10.11.4), namely m
x(t) + LA_1jx(t - hj) j=l
if t
~
m
= Ax(t) + LAjx(t -
hj),
j=l
0 and
x(t) =
C, t E [-h,
0)
x(o) =
XQ.
(10.13.2)
The Time-Optimal Control of Linear Differential Equations
433
Here 0 ~ h 1 ~ h 2 ~ ... ~ h m = h. With X determined, the index of the control system is calculated using g given as
The index of the control system for (10.13.3) is given by
k(s) =
lOh d
g[1P(O)][X(t
+0-
s)]B,
0
s s s t.
Remark 10.13.1: In the space En, g is given by
y(t, s, xo)
= xl;X(t -
s),
which is the solution of the adjoint equation. The index becomes
k(s)
= xl;X(t -
s)B,
0 ~ s ~ t.
10.14 Examples Example 10.14.1: We consider linear models of the equation obtained from lossless transmission line problems, namely
d
dt [x(t) + cx(t - h)] = -ax(t) where a > 0,
c 2 ~ 1 - 8, 8 >
+ bx(t -
h) + u(t),
(10.14.1)
o.
Variants of (10.14.1) were obtained by Slemrod [37] and Lopes [19, p. 200]. Hale [38, p. 349] derived conditions for (global) uniform asymptotic stability of the system
~
[x(t) + cx)x(t - h) = -ax(t)
Hale showed that if (i) a> 0, c2 ~ 1 - 8, for some fJ > 0,
+ bx(t -
h).
(10.14.2)
434
Stability and Time-Optimal Control of Hereditary Systems
(ii) and ifthere exists a j3 E [0, 1] such that
then (10.14.2) is uniformly asymptotically stable. In (10.14.1) we assume lui ~ 1.
(10.14.3)
Physically the function x is the voltage at the end of the line, and u(t) is related either to the current or to the voltage at the source. We use u to control fluctuation of the current at the end of the line to its equilibrium position (0) as fast as possible. That (10.14.1) is controllable follows from Theorem 10.8.4. When conditions (i) and (ii) of Hale hold, (10.14.2) is uniformly asymptotically stable. Therefore (10.14.1) is null controllable with constraint. It follows that a time-optimal control exists (Theorem 10.8.1). It is given by
u*(s) = sgn[g(t, s,
~)B]
(open loop form), or by
f(¢) = lim -sgn[g(t,s,y(¢))B(s)] t-+.+
in (10.12.2), where g is given by (10.11.7). With an initial choice of y(¢) = ~, II~II = 1, let us calculate f(¢) for t E [0,2h]. We select the (measurable) function ~
= y(¢) on [-h, 0].
Then for System (10.14.1) where a
= 2, C = -~,
= e- 2t , x(t) = e- 2t + ~e-2(t-h)[1
b = ~,
t E [O,h],
x(t)
so that k(s)
k(s)
1° dll~(B)[e-2(t-I -.)], = 1° dll~(B)[e-2(t-I -')
=
-
(t - h)],
t E [h,2h],
t - s E [0, -h],
-h
-h
t - s E [h,2h].
+ .!e- 2(t-II-.-h)j[1 2
(t - B - s - h)],
The Time-Optimal Control of Linear Differential Equations
435
Example 10.14.2: A two-dimensional neutral system was derived by Lopes in [40J, in a transmission-line problem. Its linear version is given by
x(t) - A_lx(t - h)
= Aox(t) + Alx(t -
h),
(10.14.4)
where
and u is related to the voltage vex, t) as follows: vel, t) = u(t) - gu(t - h), and i(t) is the current. In (10.14.4),
We consider the time-optimal control of the system
x(t) - A_lx(t - h) = Aox(t) where
B = -(
~
+ Alx(t -
J 4
)
,
h)
+ Bu(t)
(10.14.5)
# O.
di
We recall from Chukwu [39, Example 5] that the conditions for uniform asymptotic stability are: (i) lei < 1. (ii) Let
1
T
o, = "2(Ao + A o ), 1 T = "2(Al + Al ), J, = HDj + DTH,
D2
where H is a positive definite symmetric matrix. Suppose the eigenvalues Ai of J, satisfy
Ali < -8 1 < 0,
i
= 1,2,
where For the control system (10.14.5), we assume (iii) IUjl~l, j=I,2.
436
Stability and Time-Optimal Control of Hereditary Systems
We examine 10.14.4 when
For stability we find that if H = I, the identity matrix J1
= A o + ATo =
Hence the eigenvalues are All be 2.
= -6,
(
- ~
(-6 0) 0
A12
-2
= -2,
.
and we can choose 81 to
1)
10'
Using the matrix norm
lIall =
'~P
{t.laiil}
we deduce that
Hence conditions (i) and (ii) of uniform asymptotic stability are satisfied. We know from Salamon [6, p. 151] that (10.14.5) is exactly controllable on wJ1) if and only if 2 V A where ~(..\) A(I -A_1e-rh)-Ao-A1e-r, (iv) rank [~(A),B] (v) rank [..\I - A_ 1 , B] = 2 V A. Since B has full rank, conditions (iv) and (v) are satisfied; and therefore the system is exactly controllable. It follows from arguments contained in Chukwu [14, Theorem 3.2) that (10.14.5) is null controllable with measurable controls that satisfy (iii). We now deduce optimal control law
=
=
The Time-Optimal Control of Linear Differential Equations
437
in the space En. To do this we need to determine the fundamental matrix X:
X(t)
t -2t = eAot = ( e -2tte:?' - e
We select
(t - s)e- 2(t-.)) + C2(t - s)e - 2(t - S), - Cl(t - s )e- 2(t-.) + C2( e -2(t-.) + (t - s)e- 2(t-. ))),
(Cl' C2)X(t - s) = (Cl(e- 2(t -
y(s)B
= y(s) [d al
.) -
0]2 '
d
(t - s)e- 2(t-.)) + C2(t - s)e- 2(t-.)), d 2( -Cl(t - s)e- 2(t-.) + C2(e- 2(t-.) + (t - s)e- 2(t-.))),
= (d l(Cl(e- 2(t -
.) -
t - s E [0, h], (Ul,U2)
= sgn(y(s)B),
+ C2(t - s)), s)) + c2(1 + (t - s)).
Ul = sgn(dl(Cl(l - (t - s)))
U2 = sgn(d 2( -Cl(t One can obtain values of plicated.
U
on [h,2h]. The calculations become more com-
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Stability and Time-Optimal Control of Hereditary Systems
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&
Banach Space,"
37. M. Slemrod, "Nonexistence of Oscillations in a Nonlinear Distributed Network," J. Math. Anal. Appl. 36 (1971) 22-40. 38. J. K. Hale, "Stability of Functional Differential Equations of Neutral Type," J. Diff. , Equat. (1970) 334-355. 39. E. N. Chukwu, "An Estimate for the Solution of a Certain Functional Differential
Equation of Neutral Type," in Nonlinear Phenomena in Mathematical Science, edited by V. Lakshmikanthan, Academic Press, New York, 1982. 40. O. Lopes, "Stability and Forced Oscillations," J. Math. Anal. Appl. 55 (1976) 686698. 41. E. N. Chukwu, R. G. Underwood, and L. D. Kovari, Global Constrained Null Con-
trollability of Nonlinear Neutral Systems, Preprint.
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Chapter 11 The Time-Optimal Control Theory of Nonlinear Systems of Neutral Type 11.1
Introduction
This chapter introduces the problem of reaching a continuously moving target z by a trajectory of the control system described by the nonlinear functional differential equations of neutral type,
d
-[D(t)Xt] = f(t,Xt,u(t)), dt Xq
=
in minimum time. The state space is either En, the Euclidean n-dimensional vector space; the space C = C([-h, 0], En) of continuous functions from [-h,O] into En with the sup norm, or the Sobolev space WJ!) = WJ!)([-h, 0], En) of functions [-h,O] -+ En whose derivatives are L p integrable. The admissible controls are measurable functions u : [t,oo] -+ Em whose values u(t) E U, a compact convex set. The existence theory in C is established in 11.3; the corresponding constrained controllability questions are answered in Chapter 12. The necessary condition for time-optimal control, the maximum principle, is stated in 11.4. It is important to refer to Section 1.6 for the motivation from electric current. Before the sixties, the stability of electric power systems was seldom threatened. This was because of very conservative system's design based on fairly constant and predictable load growth. The situation has now changed. Environmental concerns, economic realities and other factors have led to the development of a new generation of equipment that is prone to cause network voltage collapse and acute instability. Power systems planners and operators have become increasingly concerned with stability problems. As the region of stability of the equations of motion of such systems becomes better understood, it is predicted that the time-optimal control of voltage and current fluctuations will assume greater importance. The natural models of fluctuations of voltage and current in problems arising in transmission lines are functional differential equations of neutral type. See Cruz and Hale [10, p. 5], Salamon [5, p. 151], and Lopes [21] for derivation and other references. One is interested in driving fluctuations of voltage to
441
442
Stability and Time-Optimal Control of Hereditary Systems
its stable equilibrium state as rapidly as possible. This is the time-optimal problem. It is old, very important, and continues to be interesting even for linear ordinary differential systems (see [22]). Our aim here is to present a comprehensive theory that will be needed for the construction of an optimal feedback control similar to the development in [23]. We consider the problem of reaching, in minimum time, a continuously moving target Zt, in function space, by a trajectory of the control system described by the nonlinear functional differential equation of neutral type, d dt [D(t)xt] = f(t, Xt, u(t)), t ~ 0, Xq
(11.1.1)
= cp.
In (11.1.1), D is a continuous function
DO : [T,OO) x
C
-+
D(t)Xt = x(t) - g(t, Xt),
En given as (11.1.2)
and f : [T,OO) X C x Em -+ En is a continuous function. Here T is a real number. The target Z may be taken as a point of En, or of C, or of wJ1), 1 ~ p ~ 00, where wJ1) is the Sobolev space of all absolutely continuous functions x : [-h, 0] -+ En whose derivative x(t)
= ~;
belongs to the
p-integrable space L p ([ -h, 0], En). The admissible controls are measurable functions u : [T,OO] -+ En with u(t) E U(t, Xt), where U : [T, 00) XC -+ 2E" is a set-valued map that is nonempty and compact. We can also consider the set of admissible controls that is a closed and bounded subset of Lp([O',oo), En) with zero in its interior, when the state space is WP). The time-optimal problem is to determine an admissible control u* such that the solution x( 0', cp, u*) of (11.1.1) hits a continuously moving target Z (in the appropriate space) in minimum time t* ~ 0'. Such a control is called time-optimal, and t* is the optimal time. This problem has three parts: (i) Does there exist an admissible control u such that X,.
(0', cp, u) = z,. for some
T.
(11.1.3)
This is the problem of controllability. With the optimal time fined by TO
= inf'{r :
T
such that (11.1.3) holds},
TO
de-
(11.1.4)
(ii) Does there exist an optimal control u such that (11.1.5)
The Time-Optimal Control Theory of Nonlinear Systems
443
Finally: (iii) What are the form, uniqueness, and general properties of the optimal control if it exists. Part three of the problem is inferred from the necessary condition for an optimal control, the so-called maximum principle. Though a very large body of research is available on the control of neutral systems, most are concerned with optimal control of systems with quadratic cost functions, [1-2]. The time-optimal problem was implicitly touched upon in [1], as a consequence of a result on minimizing a general cost function. Here a necessary condition was formulated. For linear systems with unrestrained controls, controllability questions were investigated by Gabasov and Kirillova [3], Rodas and Langenhop [4], and Salamon [5] and more recently in [7] for linear systems. The present treatment gives a comprehensive theory for the nonlinear system (11.1.1). In Section (11.2), we establish the existence and uniqueness and some regularity properties of solutions of the differential equations (11.1.1). We introduce some conditions that will prevent finite escape times of trajectories. This will enable us to restrict our attention to a closed and bounded set of the (t, xt)-space. If x(t, (J', u) is a solution of (11.1.1) and Dd(',',') is the Frechet derivative of f on E x C x Em with respect to ith variable, then the linear approximation of (11.1.1) about x(t, (J', u) is d
dt[D(t)Yt] = L(t,Yt) + B(t)v(t)
where
= Dd(t, Xt, u)Yt,
L(t, Yt)
B(t)
(11.1.6)
= D3f(t, Xt. u),
Some of the properties of the solution map of 11.1.16 is outlined in 11.2. In Section 11.3, we prove the closure of the attainable sets of (11.1.1). Under the constrained controllability assumption, we prove the existence of a time-optimal control. Sufficient condition for controllability for (11.1.1) is taken up in Chapter 12. For completeness we include the necessary condition for an optimal control, a maximum principle in Section 11.4. There the theory is applied to linear systems.
11.2 Existence, Uniqueness, and Continuity of Solutions of Neutral Systems In (11.1.2), assume that g(t,·) is a bounded linear operator from C into En for each t E (r,oo), g(t,
get,
=
1 0
k
[dojj(t, B)]
(11.2.1)
444
Stability and Time-Optimal Control of Hereditary Systems
for some nonnegative function k E G((T,oo),E), and /J(t,') is an n x n matrix function of bounded variation on [-h, 0]. We assume 9 is uniformly nonatomic at zero; that is, there exists a continuous, nonnegative, nondecreasing function L( s) for s E [0, h] such that
1°.
I
L(O) = 0,
[dlJ/J(t, ()]iP(()
I:: ; L(s)lliPll
(11.2.2)
for all t E (T,oo), iP E C. We shall have cause to require that D in (11.1.2) is uniformly stable in the following sense of Definition 11.2.1: Definition 11.2.1: The operator D is uniformly stable if there are constants, Q, f3 > 0 such that for every (1' E [(1',00), iP E G the solution x((1', iP) of the homogeneous difference equation
= 0 t ~ (1', Xf7 = ip, D((1') = 0
D(t)Xt
(11.2.3)
satisfies
IIXt((1', iP)1I :::; f3lliPIIC- a ( t -
f7 \
t > O.
(11.2.4)
In (11.1.1) we assume that
f : [T, 00) X G x Em
---+
En
is continuously differentiable and is G 1 . We now establish the existence of a solution of (11.1.1) for each initial data ((1', iP) E Ex G([-h, 0], 0), where o E En is an open convex set containing the origin, and where
f : E x G([-h, 0], 0) x Em
---+
En,
g: E x G([-h, 0], 0)
---+
En
are continuous and 9 satisfies (11.2.1) and (11.2.2). The existence result of Cruz and Hale [8] does not quite cover the situation here. One can use the ideas of Melvin [9] to prove existence and uniqueness with initial data ((1', iP) E E x WP), and this is done by Chukwu and Simpson [24]. Theorem 11.2.1 Suppose in (11.1.1), (i) fe-,·,') : Ex G([-h, 0], 0) x Em ---+ En is continuously differentiable. (ii) For each compact convex set I< ~ 0 there exists an integreble function M 1 : E ---+ [0,00) and integrable functions M; : E ---+ [0,00), i = 2,3, such that
IIDd(t, 'P, w)1I :::; M 1 (t ) + M 2 (t )llw ll , IID 3f(t, 'P, w)1I < M 3 (t ) (11.2.5)
445
The Time-Optimal Control Theory of Nonlinear Systems
for all t, w, and
jL(t,
(11.2.6)
(t,
= Cr([O"-h, T], En), and define the function G : Ex Cr --+
Cr by G(x)(t)
={
0,
if
0" -
g(t, xd - g(O",
if
0"
Define H : Cr x Loo([O", r]' Em)
H(x, u)(t) = { Finally, define I : CT
--+
h :S t <
0",
(11.2.7)
:S t :S T.
Cr by
0,
f" f(s, x s , u(s»ds, t
if
0" -
if
0"
h :S t <
0",
:S t :S T.
(11.2.8)
--+ CT
I
< t < 0", if 0" :S t :S T. if 0"
0"),
-
h
(11.2.9)
Written in integral form, Equation (11.1.1) is equivalent to
x"
= ip E [-h, 0],
x(t) =g(t,Xt)
= g(O",
I
t
! (s , X., u(s))ds , t>O".
(11.2.10)
Using this fact and the operators G, H, and I, we see that x satisfies (11.1.1) if and only if x = Gx + H(x, u) + Lip (11.2.11)
446
Stability and Time-Optimal Control of Hereditary Systems
for x E CT. It is easy to see that (11.2.10) is well defined, since Ie.,·,') is assumed continuous and since by the mean value theorem (we use the convexity of
K),
II/(t,
+ Ml(t)II
(11.2.12)
for all t E [00, tIl,
Lemma 11.2.1 Let H : C(ru - h, T], 0) X L oo -. CT be as defined in (11.2.8), G be as defined in (11.2.7). Then H is continuously differentiable and
(D1H(0, O)x)(t) = {
(DaH(O,O)u)(t) = {
0,
J; D2f(s, 0, O)x.ds, 0,
J; D2f(s, 0, O)x.ds,
00- h:::; t < 00,
u:::;t:::; T,
(11.2.13)
00 - h:::; t < 00, 00
(11.2.14)
for all x E C T , u E Loo([u, T], Em), t E [00 - h, T). Also G is continuously differential and for any x E CT([U - h, tl], En),
(DGX)x = G».
(11.2.15)
Proof: By [12, p. 177],
it = it
(DlH(x,u)x)(t) = (D 2H(x,u)u)(t)
D2f(s,x.,U(s»x.ds, (11.2.16)
Da/(s,x.,U(s))u(s)ds (11.2.17)
where t E [00 - h, T), x E C([u - h, TJ, 0) x E CT = C([u - h, T], En), u,"IT E L oo ([00, T), Em). The results follow quickly from these. Also, because of the linearity of g(t,
447
The Time-Optimal Control Theory of Nonlinear Systems Define
Lemma 11.2.2
by
K x = DG(O)x + D1H(0, O)X.
(11.2.18)
Then (1 - K)-l exists as a bounded linear map. Let q be any function in C([r, 00), En). Consider
where xa
= It' = qa·
(11.2.19)
This is the same as d dt [D(t)xt - q(t)J = L(t, Xt).
(11.2.20)
By assumptions (iv) and (v) and Theorem 2.1 of Hale and Cruz [10, p. 333], there exists a function x(O", rp, q)(t) with initial value rp at 0" uniquely defined and continuous on [0"- h, 00) and satisfying (11.2.20). Furthermore, for any fixed 0" and t 2: 0", x(O",·,O)(t) is a continuous linear operator from C into En. The integrated form of (11.2.20) is
x(t) = get, Xt) - g(O", rp)
+ q(t) + rp(O) +
it u»,
x.)ds.
But this is x(t) = DG(O)x(t) Since x"
= q" = rp,
+ rp(O) + q(t) + (D1H(0, O)x)(t) q(O")
= rp(O),
x = DG(O)x
q(O"), t E [0", TJ.
we have that
+ D1H(0,O)x + q(t)
has a unique solution for any q E CT. Therefore x Kx DG(O)x + D1H(O, O)x has the unique solution x == O. Hence 1- K is one-to-one mapping of CT onto CT. Also 1- K is a bounded linear operator. As a consequence of the open mapping theorem ([11, p. 99]) (I - K)-l is a bounded linear mapping OfCT onto CT.
=
=
448
Stability and Time-Optimal Control of Hereditary Systems
We now complete the proof of Theorem 11.2.1. Consider (11.1.1) in its integrated form (11.2.11). By Gronwall-type argument we can easily show that its solution ~(ip, u) E CT is necessarily unique for any pair of ip, u such that the solution exists. Define M(x, u, ip) = x - Gx - H(x, u) - lip. By hypothesis (iii), M(O, 0, 0) differentiable on
= 0.
By Lemma 11.2.1, M is continuously
C([O" - h, TJ, 0) x Loo([O", TJ, Em) with D 1M(0, 0, 0)
X
C([-h, 0], En), (T
= 1- DG(O) -
~
(0)
D 2H(0, 0).
By Lemma 11.2.2, D 1M(0, 0,0)-1 exists. Therefore by the Implicit Function Theorem [12, p. 270] there exists a continuously differential mapping ~(ip, u) defined in an open neighborhood N of the origin in
=
°
=
u), u, ip) for all (ip, u) EN. Furthermore, x ';(ip, u) such that M(~(ip, satisfies (11.2.11) or (11.1.1). We now choose OcT,Ou so that Oc x Ou C N to complete the proof. It is clear from Theorem 11.2.1 that the solution x(O", ip, u) of (11.1.1) is continuously differentiable for each (0", ip) E E x C and each u E Loo([O", (0), Em). Corresponding to the point (t, 0", ip, u) E Ex E x C x L oo, the mapping x : E x E x C x L oo --> C defined by x(O", ip, u)(t) = Xt(O", ip) represents a point in C. We now give some very useful properties of this mapping. Lemma 11.2.3 Let (t, 0", ip ; u) E E x E x C x L oo. Assume all the conditions of Theorem 11.2.1. Let Dx(t,O",ip,u) denote the partial derivative of x(t, 0", ip, u) with respect to its last two variables at the point (t,O",ip,u). Then for every ('ljJ,v) E C x L oo we have Dx(t,O",ip,u)('ljJ,v) = )..(t, 0", ip, U, 'ljJ, v), where the mapping t --> )..(t, 0", ip ; U, 'ljJ, v) of E into En is the unique solution of the linear differential equation of neutral type d dt [D(t)Yt]
= D2f(t, Xt(O", ip, u), u(t»Yt + D3f(t, Xt(O", ip, u), u)v(t),
Y" ='ljJ.
(11.2.21)
The Time-Optimal Control Theory of Nonlinear Systems
449
Also for v E L oo ([17, OO ), E ffi ) we have D 4x(t,17,If',u)(v) = y(t,17,If"U,v) where the mapping t --+ y(t, 17, If', u, v) of E into En is the unique solution of (11.2.21) satisfying y,,(17, If', U, v) = O. Proof: For each (17, If') E Ex C, the unique solution of (11.1.1) with initial data (17, If') is given in (11.2.10) by
x"
x( 17, If', u )(t)
= If'
in [-h,O],
= x(t, 17, If', u) = g(t, Xt(17, If', u)) +
1 t
g(17, If') + 1f'(0)
f(s, X.(17, If', u), u(s))ds, t 2: 17.
Let D 4x(t, 17, If', u) = D u be the partial derivative of x(t, 17, If', u) with respect to the fourth argument. Then by [12, pp. 107, 173) D 4 x" = 0 in [-h,O),
D 4x(t, 17, If', u)
+
1
=
1 t
D2f(s, X.(17, If', u)), u(s)D uX.(17, If', u)ds
t
Daf(s, X.(17, If', u), u(s))ds ,
dXt
+ D 2g(t, Xt(17, If', u)) d; (17, If', u). Thus, by the same reasoning in (11.2.15), we have
Thus,
D 4x(t,17,If',u) - D 2g (t,Xt(17'If',U)~:t(17,If"U))
1 t
D 2f(s, X.(17, If', u), u(s))D uX.(17, If', u)ds +
=
1 t
Daf(s, X.(17, If', u)), u(s)ds.
On differentiating with respect to t, d
dt [Dux(t, 17, If', u)v - D 2g(t, Xt(17, If', u))Du Xt(17, If', u)v]
= D2f(t, Xt(17, If', u), u(t) )Duxt (17, If', u)v + Daf(t, Xt(17, If', u), u(t) )v, or equivalently, d
dt [Dux(t, 17, If', u)v - g(t, D u Xt(17, If', u)v)]
= D2f(t, Xt(17, If', u), u(t))DuXt(17, If', u)v + D4f(t, Xt(17, If', u), u(t))v.
450
Stability and Time-Optimal Control of Hereditary Systems
This proves the second assertion. For the first assertion, note that
Dx(t, a,
Dax(t,rr,
+1+ D4X(t, a,
ip,
u)
=
- D 2g(u,
1D2f(s,x.(r ,
1 1 t
xt(rr,
D 2f(s, x.(rr,
+
Daf(s, x.(rr,
Thus
dXt ) -g(rr"p) Dx(t ... ) Dx(t,rr,
+ "p + + +
1 1 t t
1 t
u
dx D2/(s, x.( a,
D2f(s, x.(rr,,,p, u), u(s))Dux.(rr,
D2/(s, x.(rr,
Now take the t derivative to obtain
~
[Dx(t,rr,
= D2/(t, xt(rr,
~~"p+DuXtV)]
[~:"p
+ Dux.(rr,
+ Daf(t, Xt (rr,
Dxu(rr,
is said to be upper semicontinuous with respect to inclusion in t, 0 there is a {j > 0 such that
U(tl,
~
U,(t,
whenever It 1 - tl ::; {j, lI
~
(Dxt) satisfies (11.3.2),
(iii) Xu =
2:
0"
is a
A(t) = {Xt E C, x is a solution of (11.3.1) for some u(t) E U(t, Xt), (t, Xt) EM}. By Lemma 11.3.1, this is the same as:
A(t) = {Xt E C, x is a solution of (11.3.2), i.e.,
~[D(t)xt]
E F(t,xt),
Xu=
454
Stability and Time-Optimal Control of Hereditary Systems
In (11.3.1) assume that: (i) f : E x C([-h, 0], En) X Em -+ En is continuously differentiable. (ii) D is uniformly stable and satisfies (11.2.1) and (11.2.2). (iii) For any measurable y(t) satisfying y(t) E F(t, Xt), we have /y(t)l :::; m(t), a.e., mE L 1 ([lT,T ],E ). (iv) F(t,cp) is compact and convex for each t,cp. (v) U(t,cp) is uppersemicontinuous with respect to inclusion. Then the attainable set A(t) of (11.3.1) is closed.
T'heorern 11.3.1
Proof: To prove the closure of the attainable set, let xf E A(t) n = 1,2, .... Let xf -+ Xt as n -+ 00. We prove Xt E A(t). Because xf A(t), for each n,
d [D(t)x~] dt By condition (iii),
E F(t,
.
x~),
i
s t.
I~ D(t)x~ I:: ; m(t)
lrt
where m E L 1(I,E). From the above inequality, we deduce that
ID(t, x~)
it Li
- D(lT, cp)1 :::;
so that
m(s)ds,
D(s,
x~)ds
ID(t, x~)
-+
0,
- D(t)xrl :::;
m(s)ds
t -+ 00
uniformly with respect to n for each decreasing sequence {E;}, E; ~ I with void intersection. Therefore, by [15, p. 292] there is a sequence (we retain the same notation weakly convergent in L 1 (I, En) to a function ~ E L 1(I,E n ) ) . Then for each tEl,
D(t)xt
= n-oo lim [D(t)x~]
t
= n-+oo lim D(lT)cp+l D(s)(x~)ds (J
t
= D(lT)cp+l ~(s)ds q
It now follows from page 422 of [15] that there is a sequence {~d of convex combinations of the functions {D(t, xnD(t, X;+l) ... } converging in L 1 (I , En) norm to~. From this sequence (6) select a subsequence that converges to ~ a.e .. Thus, almost everywhere on I
The Time-Optimal Control Theory of Nonlinear Systems
455
where co(M) is the closed convex hull of M E En. Hence
Hence A(t) is closed, since also (t,Xt) E M from the closure of M. This concludes the proof. Note that we have used the upper semicontinuity of F in (11.3.3). The so-called property Q of Cesari, a much milder condition, could have replaced upper sernicontinuity [see 16, p. 7]. Condition (iv) could be replaced by the closure and convexity of F(t, rp). The existence of an optimal control for (11.3.1) requires a controllability assumption. This concept is defined as follows:
=
Definition 11.3.4: Let Zt E C([-h,O], En) C be a target point function that is time varying. Suppose ZT E A(T) for some T> a, then the system is controllable to the target. Thus for each rp E C there exists a T 2:: a and an admissible control u(t) E U(t, Xt), t E [u, T], such that the solution of (11.3.1) satisfies xO'(u, rp,u)
=
=
Definition 11.3.5: System (11.3.1) is null controllable with constraints if for each rp E C, there exists a tl < 00 an admissible control u(t) E U(t, Xt), t E [u, tl] such that the solution of (11.3.1) satisfies
The main result of this section is now stated. Theorem 11.3.2 In (11.3.1), assume that: (i) f: E x C([-h, 0], En) X Em -+ En is continuously differentiable. (ii) D is uniformly stable and satisfies (11.2.1) and (11.2.2). (iii) For each measurable y(t) satisfying y(t) E F(t, x,), we have ly(t)1 ::; m(t) a.e, mE L 1([u,T],E]. (iv) F(t, rp) is closed and convex for each t, rp. (v) U(t, rp) is upper semicontinuous with respect to inclusion. (vi) System (11.3.1) is function space z-controllable with constraints. Then there exists time-optimal control for (11.3.1). Proof: By assumption, there exists some t 1
::;
T such that
456
Stability and Time-Optimal Control of Hereditary Systems
where A(td is the attainable set of (11.3.1) at time t1. Let
t" = inf{t : Zt E A(t)}. It follows that a ~ t* ~ t 1 ~ T. There is a sequence of nonincreasing times t n E [u, ttl converging to t* and a sequence u" of admissible controls, un(t) E U(t, x~), such that
Since
Zt
is continuous and t n
-+
t; as n
-+ 00,
we have
By condition (iii),
Let YCt)
=
== xn(t) - get, x~).
=
The set H {yn : yn(t) xn(t) - get, x~)} is an equicontinuous family. Indeed, for every measurable subset E of [u, T] we have
Let c > O. By integrability of m we conclude that there is a 8 > 0 such that if meas(E) < 8, then m(t)dt < e. For this E we have
IE
so that the functions
~yn(t), dt
yn E Hare equi-absolutely integrable. It
now follows that H is an equi-absolutely continuous set of functions. Since M is bounded and g(M) bounded, we readily deduce that H is bounded. By Ascoli's theorem there is a subsequence, which we again call
The Time-Optimal Control Theory of Nonlinear Systems
457
{Yn} and which converges in the metric form of C([O", 1'], En) to a function C([O", 1'], En), that is absolutely continuous on [0",1']. By condition (iii), we deduce Iyn(t) - yn(I1)1 :::; m(s)ds, Iyn(t) - y(t)l :::; J{ m(s)ds, so that y E
f;
f
lEi
dd[yn(s)]ds---,O as t
n---.oo
uniformly with respect to each decreasing sequence {Ed, E; ~ [11,1'] == I with void intersection. Therefore [see 15, p. 292] there is a sequence {yn} (we retain the same notation) weakly convergent in L 1 (I, En) to a function ~ E L 1 (1, En). Since the functions u" converge uniformly on [O",T] to u, certainly they converge pointwise almost everywhere to y and hence by absolute continuity of these functions we have y(t) = ~(t) a.e. Thus, for t« E I, t.; ---. t*,
We note
x~
= <.p for each n, so that lim n _ oo x~ = <.p.
(tn,x?J ---. (t*,Xt.) E Mas n ---.
Because M is closed,
00.
It now follows from [15, p. 422] that there is a sequence {~n} of convex combinations of {yn, yn+!, ... } that converges in L 1 (1, En) norm to ~. From this sequence {~n}, select a subsequence which converges a.e. to ~. Thus, almost everywhere on I,
~(t*)
E
~
151 co CQ yk(t*)) , nQ1
co C9n r«; xU) ~ r«; Xt.)
where co(N) is the closed convex hull of N in En. Hence
We have already seen that Xu
= sp ;
(t*, Xt.) EM, so that
(11.3.4)
458
Stability and Time-Optimal Control of Hereditary Systems
This completes the proof. We observe that we used the upper semicontinuity of F in (11.3.4). It could be replaced by using the weakened version of Cesari's property Q. See Angell [16, p. 7].
11.4
Optimal Control of Neutral Systems in Function Space
In this section we present a maximum principle for optimal control of nonlinear systems of neutral type with function space initial and terminal conditions. In our setting we guarantee the existence of regular nontrivial multipliers when the controls are constrained to lie on a unit m-dimensional cube. This problem was broached in [2] and [26], but not completely resolved. In [26] the nontriviality of the multipliers could not be guaranteed and pointwise control constraints were ruled out. This section continues the exciting recent work of Angell and Kirsch [25] on delay equations, which was reported in Section 7.3. We maintain the basic definitions and notation of the corresponding treatment in 7.11.2, and consider the following problem: Find a control u that minimizes (11.4.1) subject to the constraints N
x(t) -
L
A_li(t)X(t - hi) = f(t,
Xt,
u(t)),
(11.4.2)
i=l XQ
= ¢>,
XT
= t/J,
(11.4.3)
u(t) E O" a.e. on [0, T],
(11.4.4)
C'" = {u E Em: IUil:S 1, j = 1, ... ,m},
(11.4.5)
where
fl(t,xt):SO,
i=1, ... ,p,
tE[O,T]
(11.4.6)
and N
x E C([-h, T], En),
x(t) -
L: A_li(t)X(t -
hi) E £00([0,T], En),
i=l
(11.4.7)
459
The Time-Optimal Control Theory of Nonlinear Systems
We assume further that the initial data ¢ and the terminal function t/J satisfy with (11.4.8) We observe that if the admissible control u is such that then by equation (11.4.2) x(t)-
N
L
j=1
UT
is continuous,
Aj(t)x(t-h j) is continuous on (T-h, T),
since we have assumed the same properties of Thus the mappings
f°,f, P,
as are in 11.7.3.
t" :[0, T) x C
x Em -. E, f :[0, T) x C X Em -. En,
fl :[O,T)
X
C -. EP
are continuous; t. fO are Frechet differentiable with respect to the second argument and continuously differentiable with respect to its third argument, the derivatives being assumed continuous with respect to all arguments. Also, fl : [0, T) X C -. EP is continuously Frechet differentiable with respect to its second argument. We associate with (11.4.2) the linear equation N
x(t) -
L
Ajx(t - hj)
= L(t, Xt) + B(t)u(t),
Xo
= ¢,
(11.4.9)
j=1
where u E U, x E XI; and these subspaces are defined as follows:
(11.4.10)
U = {u E Loo([O, T], Em) : UT E C([-h, 0], Em)},
(11.4.11)
Yl
(11.4.12)
= C([O, T), EP),
Y2 = ((z,u,,B) E Loo([O,T],E n)
X
C([-h,O],E n) X
En:
ZT
E C([-h,O),E n)}.
(11.4.13) In (11.4.9), L(t, Xt) B(t)v
= D2f(t, x;, u*(t))Xt, = D 3f(t, x;, u*(t))v,
(11.4.14)
460
Stability and Time-Optimal Control of Hereditary Systems
where D;f(t, x;, u*) is the Frechet derivative with respect to the ith argument at (t, x;, u*(t)) and x;, u*(t) are the optimal trajectory and optimal control respectively. We introduce the following maps: gO
:X 1 x U
--+
E,
gl :X1 xU --+ Y1,
g2
:X 1 x U --+ Y2,
which are defined as follows: gO(x,u)
=f o
T
(11.4.15)
JO(t,x"u(t»dt,
(11.4.16)
g'(x,u) = J(-,x(-), g2(X ,u) = (x O- j t AjOx(-- hj)- J(-,x(-) ,uO)- C,),(0)+i:(T)-
j~l
Aj(T),),C - h j»
j~l Aj(T)X(~-hj)'X(T)-.p(T»)
Observe that if g2(x, u)
(11.4.17)
= 0, then
N
xC) - LAjC)x(. - hj) j=l N
x(T) - L Aj (T)x(T - hj) j=l
= f(-'xo,u(.)), = ~(O)
N
- L Aj (T)t/J( -hj), j=l
x(T)
= t/J(T).
(11.4.18) Because of our previous assumptions on t, fO, P, the functions s", l, g2 are continuously differentiable at each (z", u*) E Xl X U. Indeed,
1l(x*,u*) = iT D2fO(t,x;,u*(t))x x E Xl, D 2l(x*,u*)u = iT D3fO(t,x;,u*(t))u(t)dt, u E U, D
tdt,
D 1 g 1(X*,u*)y = D2f'(t,xnY, D 2 g 1(X*, u*)
= o.
y E C,
(11.4.19) (11.4.20) (11.4.21) (11.4.22)
The Time-Optimal Control Theory of Nonlinear Systems
461
Also
D ly 2(X*, U*)X
= xO -
N
L
AjOx(- - hj ) - D2f(-, x(.), u*(.»xC),
j
N
x(T) - LAj(T)x(T - hj),
x(T»,
where x E Xl, (11.4.23)
j=l
D2y(x*,u*)v(t)
= D3f(t,xt,u(t»v(t),
v E U.
(11.4.24)
Because of the validity of the Riesz Representation Theorem, there exist
"11(t,') EN BVpxn([-h, 0]), '1]o(t,') E N BVnxn([-h, 0]), '1](t,') E N BVnxn([-h, 0]) such that
Lo(t)¢ LI(t)¢ L(t)¢
l°h = l°h
= D2fo(t, x;, u*(t» = = D2f l(t,'x;,u*(t»¢
= D2f(t,x;, u*(tȢ =
In all these representations, these functions by defining
'1], tt«,
'1](t,8)
=0
dO'1]o(t, 8)¢(8),
i
= 0,1
i
dO'1]I(t,8)¢(8),
(11.4.25)
(11.4.26)
dO'1](t,8)¢(8). (11.4.27) are measurable. We extend
for t > T.
To match up our notation in the statement of the multiplier rule in Theorem 7.11.2, we set X V
= Xl xU, W = {(x,u) EX: Xo = ¢}, = {(x, u) E W : u E Uad},
Uad = {u E U : u(t) E
where
em
= {u E
Em;
em
a.e. on [0, T]},
IUjl::;
1,
j = 1, ... ,m}.
Note that Uad has a non empty interior relative to U. We now state a local maximum principle.
(11.4.28)
Stability and Time-Optimal Control of Hereditary Systems
462
°
Theorem 11.4.1 Let u· be the optimal solution of (11.4.1) - (11.4.8) and x; ~ t ~ T the corresponding optimal solution. Assume that Dg 2 ( x· , u·) = D 1g 2 ( X · , u") + D 2g 2 ( X · , u·) has a closed range in Y2' Then there exists
p.,a,p,v,q) E E x Loo([O,T]) x NBVp([O,T)) x NBVn([O,T]) x En, ('\,a,p,v,q) '1= (0,0,0,0,0) such that P = (pl' ... ,pp), Pi is non decreasing in [0, T]
,\ ~ 0,
(11.4.29)
and Pi is constant on every interval where f;(t, x;) T
a(t)=,\l TJo((},t-(})d(}+
iT
< 0;
TJt((},t-(})dp((})-q
N
- L a (t + h )A _1k (t + h k
+
iT
1
T+
k
)+
-iT
k=l
(11.4.30)
TJ_l((},t-(})dv((}-T)
t
TJ((},t - (})a((})d(}
TJ((}, t - (})dv((} - T)
for all t E [0, T]. (11.4.31)
Also
- ,\
i
T
o
Bo(t)u(t)dt +
i
T
a(t)B(t)u(t)dt -
t
iTT-h
dv(t - h)B(t)u(t)
~
_,\ Jor Bo(t)u·(t)dt + Jor a(t)B(t)u·(t)dt _ JT-h ( dv(t - h)B(t)u·(t), T
T
(11.4.32)
for all u E Uad .
Corollary 11.4.1 In addition to all the conditions of Theorem 11.4.1, assume that the mapping Dg(x·, u·) : (W - [z"; u·))
---+
Y2
The Time-Optimal Control Theory of Nonlinear Systems
463
is a surjection. Then we can take the multiplier A = 1. Proof: Carefully note that the functions gO, s', g2 of the Multiplier Rule of Theorem 7.11.2 satisfy the same assumptions ofthe mappings of(I1.4.15)(11.4.17) and the spaces (11.4.10) - (11.4.13). We are guaranteed the existence of the multipliers of Theorem (11.7.32). Thus if we denote the space E by
and its dual by E"; we have the existence of f E E", v E N BVn([-h, 0]), q E En, p E NBVp([O,T)), and A ~ 0, (A,f,v,q,p)::j:. (0,0,0,0,0) such that p is nondeereasing and
loT dp(t)fl(t, x*(t)) = 0,
>.I
T
+
Lo(t).,dt +
£-h
I
T
dp(I)L, (t).,
dv(t - T)[x(t) -
+e
(XO - ~
A; (.)x(·) - L(·
&
T N .
for all x E Xl,
(11.4.33)
Aj(t)x*(t)] + qx(T) = 0,
Xo = 0; and
loT Bo(t)u(t)dt + f[-BC)(uC) -
A
u*(-))]
~
)'0)
(11.4.34)
°
(11.4.35)
for all u E Uad. Recall the definition of 11, 11i and their extension outside their intervals of definition, and extend the definition of v by constancy and continuity outside its original interval of definition. We need an expression for the linear functional f. It is contained in the next lemma.
Lemma 11.4.1
fez)
The linear functionall has the form
=
fT a(t)z(t)dt _ fT dv(t - T)z(t),
Jo
(11.4.36)
JT-h
for all z E £. and a E Loo([O,T],En). Proof: Consider (11.4.34). Let z E E and consider the equation N
x(t) - EA_lj(t)x(t - hj) - L(t)xt = z(t) in [O,T], j=l
Xo = 0. (11.4.37)
464
Stability and Time-Optimal Control of Hereditary Systems
The variation of the constant formula of Hale and Meyer (27] gives the solution as
X(t, Z) =
1 xu, T
s)z(s)ds,
t E [0,71,
(11.4.38)
where X(t,s) E Loo([O, 71, E n 2 ) is the fundamental matrix defined by
X( t, s )
= O. N W(t, s) = L
= oW(t, os s)
a.e. in s.
(a) W'(', s) (b)
0< 8 < t.
A(t)W(t - s - hj) + I. L(>., W>.(·, s))d>. - (t - 8)1, t
j=l
With this expression, (11.4.34) takes the form
>.
i
T
a
Lo(t)Xt(-,z)dt+
f
Jo
T
dp(t)L 1(t)Xt(-,Z)+£(Z)+
+ fT dv(t - T)L(t)Xt(-, z) JT-h
f
T
JT-h
dv(t-T)z(t)
+ qx(T; z) = 0
(11.4.39) for all z E£.. But then La, L 1, L, are as given in (11.4.25) - (11.4.27), so that (11.4.34) yields
£(z)
= ->.I
T
(lOh d"'7o(t, s)x(t +
1 0
-
( dp(t)
T i
I,
-h
d.1]l(t, s)x(t + s, z)
dv(t - T)z(t) -
T-h
s, z)) dt
iT
dv(t - T)
1 a
T-h-h
d.1](t, s)x(t + s, z)
- qx(T).
(11.4.40) We now substitute the expression x(t; z) in (11.4.38) into (11.4.40), and change the order of integration to deduce (11.4.36), where a E E depends on >.,q,X,V,1]0,1]1,1]. With (11.4.36) established, we rewrite (11.4.39) as follows:
The Time-Optimal Control Theory of Nonlinear Systems
iT
A
i
Lo(t)Xtdt
T
+
iT
465
dp(t)Ll(t)Xt+
N
a(t)[x(t) - L A_lj(t)x(t - hj) - L(t)Xt]dt
o
j=l
T
N
- [
dr(t - T)[x(t) - L A_lj(t)x(t - hj) - L(t)Xt]
JT-h
+ [
T
j=l N
dr(t - T)[x(t) - LA_lj(t)x(t - hj)] + qx(T) = 0,
JT-h
j=l
for all Xl E Xl. Once more we use the expressions for TJO,TJ1,TJ to deduce that
A
[T it deTJo(t, B_ t)x(O)dt + [T it dp(t)deTJ1(t,O - t)x(B)
Jo
+
t-h
i
o
+ (
Jo
T
t-h
t
N
a(t)[x(t)- LA_lj(t)x(t-hj)-i
JT-h
dBTJ(t,B-t)x(B)]dt
t-h
j=l
it dv(t - T)deTJ(t, B- t)x(B) t-h
+ qx(T) = 0,
for all x E Xl with Xo = o. We consider each of the first four terms above, beginning with
A [T it de TJo(t , B- t)x(B)dt,
l«
t-h
and ending with
[T it dv(t _ T)deTJo(t, 0 - t)dt, JT-h
and integrate by parts:
t-h
466
Stability and Time-Optimal Control of Hereditary Systems
71o(t, -h)x(t _ h)dt _
_ oX {T
Jo
x (T Jo
it
71o(t, s - t)x(s)dsdt
t-h
T -iT dP(t)71I(t,-h)x(t-h)-l dp(t)71I(t,s-t)x(s)ds
1 -I
T N T
+
L A_1j(t)x(t -
a(t)[x(t) -
o
hj)]dt +
j=l
t, A_'i
T
a(t) [;,(t) -
r
(t);'(' -
( a(t)71(t, -h)x(t - h)dt
l«
hi)] dt
+ (T dv(t _ T)71(t, -h)x(t - h)
JT - h
+ qx(T) - {T JT-h
dv(t _ T)71(t, s - t)x(s)ds
t-h
= 0,
for all x E Xl with Xo = O. As in Angell and Kirsch [25], use the following abbreviation:
J.l(t)
- .f a(s)ds,
={
0 ~ t ~ T - h, T
-v(t-T)-ft a(s)ds,
T-h~t~T.
For any y E £. that is extended by zero, set
x(t) =
it
y(s)ds for tEE.
Use this above and change the order of integration:
-,\ Jo
I
(
{T-,
{T
s
T- h
{T l'+h
- Jo J,
(I s
T
-
h
(j'+h
dp( ())711 (() + h, -h )y(s)ds - Jo
t r'
{T a(t)y(t)dt + J + Jo + Jo
,
71o(()+h,-h)d()y(s)ds-oX Jo
o
t:
s
t
71o((),s-())d()y(s)ds dp(())711 ((), s - ())d()y( s)ds
dJ.l(())71((),S - ())y(s)ds
dJ.l(() + h)71(() + h, -h)y(s)ds
(T y(s)ds + + qJo
-h
k
N 1k (t ).:i:(t aCt) {;A_
+ hk)dt = a
The Time-Optimal Control Theory of Nonlinear Systems
467
for all y E E, Note that
Because the integrand above must vanish pointwise, we have for s E [0, Tj,
r: TJo(B, s - B)dB o r: -j . dp(B)TJl(8+h,-h)- Jo dp(B)TJl(O,s-B) +a(s)+. j .+h dp(O)11(O,S-O)+.t dp(O+h)TJ(B+h,-h)+q j
- A.
T- .
TJo(B + h, -h)dB - A J
T- h
N
+ La(s + hk)Ak(s + h k) k=l which yields
a(s)
iT -iT
=A
= 0,
TJo(O,s - O)dO a(O)11(B, s -
+
~)dB
iT -
dp(O)TJl(O,S - 0)
iT
dv(B)TJ(B, s - 0)
N
-q- L ak(S+h k)A_ 1k(s+h k). k=l We have proved that (11.4.31) holds. To prove (11.4.32), we insert fI(z) in (11.4.36) into (11.4.35):
iT -iT
A
Bo(t)[u(t) - u·(t)]dt +
it
a(t)[-B(t)(u(t) - u·(t)]dt]
dv(t - T)[-B(t)(u(t) - u·(t))] 2
°
for all u E Ua d . The result follows at once, since p is monotonic. Note that (A, a, p, v, q) do not vanish simultaneously, since otherwise (A,E, v, q, p) == (0,0,0,0,0). The proof of Corollary 11.4.1 and Theorem 11.4.2 follows as in [25]. Remark: The properties of the range of Dg 2 are conjectured as follows:
468
Stability and Time-Optimal Control of Hereditary Systems
Closure: Proposition 1.1 of Section 12.1. Surjectivity: Controllability of the linear system N
x(t) - L Ai(t)x(t - hi) i=I
= L(t, Xt) + B(t)u(t)
in W£.;). See for example Theorem 10.8.4. We now formulate the necessary conditions of the time-optimal control problem. Theorem 11.4.2 to the constraints
Consider the following problem: Minimize T subject
N
x(t) - LA_li(t)X(t - hi) ;== 1 XQ
= f(t,Xt,u(t)),
(11.4.2)
= eP E W£.;)([-h,O],E n ) ,
(11.4.3)
XT=.,p,
D(T).,p E CI([-h, 0], En),
u(t) E C" a.e, on [0, Tj,
where
C'" = {u E Em: lUi I ::; 1,
(11.4.4)
j = 1, ... ,m}.
(i) We assume that f is continuous, and is Frecbet. differentiable with respect to its second argument and continuously differentiable with respect to its third argument, the derivative being continuous with respect to all arguments. Consider the linear approximation (11.4.9) of (11.4.2), and assume that: (ii) This system N
x(t) - LAix(t - hi) = L(t,Xt) i=I
+ B(t)u(t),
XQ
= 0,
(11.4.9)
with attainable set A defined by
A = {XT(U) E C([-h, 0], En) DXT E C'([-h, 0], En) : U E U} where
The Time-Optimal Control Theory of Nonlinear Systems
469
is such that
= {tP E C([-h, 0]) : D'lj; E C'([-h, OJ, En)}.
A
Then there exists
(a, v, q) E Loo([O, T], En)
X
N BVn([O, Tj)
X
En
(a, v, q) "# (0, 0, 0) such that N
L
a(t) = -q -
a(t
+ hk)A_ 1k(t + hk)
k=l
-1
T
TJ(S, t - s)a(s)ds+
t
iT TJ(S, t - s)dv(s - T) for all
t E [0, T], and
a(t)B(t)~(t)dt
fT
Jo
s for all u E
_ fT dv(t _ T)B(t)u(t)
JT - h
fT a(t)B(t)u*(t) _ fT dv(t _ T)B(t)u*(t)
Jo
JT - h
u,«.
Proof: We need to prove that the attainable set A is all of V = {'lj; E C([-h,O]): D'lj; E C'([-h, 0], En)} if and only if the required map
Dg(x*, u*) : (W - [z ", u*» = Y2. Suppose that A = {tP E C([-h, 0]) : D'lj; E C'([-h, aD}, and fix a point (z, v, p) E Y2 . We shall prove that there exists a pair (x, u) E Xl X U such that if i = x - z" and u = u - u", then N
i(t) -
L
A_Ij
(t)i(t - h j
) -
j=1
DiT = v
and
i(T)
= p.
L(t, it) - Bu(t) = z(t),
470
Stability and Time-Optimal Control of Hereditary Systems
Now define tf; E G([-h, 0], En) with Dtf; E G'([T - h,T],En) by tf;(t) = Pv(s)ds T - h ~ t ~ T. By assumption, there exists a control u such that
ItT
T
X(t;O,U)=tf;(t)-l U(t,s)z(s)ds,
T-h~t~T.
It follows from the variation of constant formula in Proposition 2.3.3 that
x(t; z, u)
= x(t, 0, u) +
it
U(t, s)z(s)dt.
The required choice is the triplet x = (x; z, u), u(t) = u(t). To see that the converse is also valid, we note that if tf; E G([-h, 0], En), with Dtf; E G'([T - h, T], En), then the fact that Dg is surjective implies there exists a pair (x,u) that is the preimage of the triple (O,tf;,tf;(T». We conclude that (ii) holds if and only if the required map D 2 g is surjective. But this is equivalent to the controllability of the linear system (11.4.9) with controls in U. Theorem 11.4.1 and its corollary can therefore be invoked to conclude the proof. Remark 11.4.1: For autonomous systems, necessary and sufficient conditions for exact controllability on the interval [0, T) are given by Salamon [5, p. 155]. There we can identify our interval to be [0, T - h] with controls u E £00([0, T - h], Em) and state space W~([-h, 0], En). On the interval [T - h,T] one can use a control that is continuous. Thus it can easily be proved that in the autonomous case Salamon's conditions suffice for the surjectivity conditions. REFERENCES 1. G. A. Kent, Optimal Control of Functional Differential Equations of Neutral Type, Ph.D. Thesis, Brown University, 1971. 2. H. T. Banks and G. A. Kent, "Control of Functional Differential Equations to Target Sets in Function Space," SIAM J. Contral10 (1972) 567-593.
3. R. Gabasov and F. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker, New York, 1976. 4. H. R. Rodas and C. E. Langenhop, "A Sufficient Condition for Function Space Controllability of a Linear Neutral System," SIAM J. Control and Optimization 16 (1978) 429-435. 5. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. 6. E. N. Chukwu, "The Time Optimal Control Problem of Linear Neutral Functional Systems," J. of Nigerian Mathematics Society 1 (1982) 39-55.
The Time-Optimal Control Theory of Nonlinear Systems
471
7. E. N. Chukwu, The Time Optimal Control Theory of Linear Differential Equations of Neutral Type, Proceedings, Second Bellman Continuum, Georgia Institute of Technology, Atlanta, Georgia, June 24,1986. 8. M. Cruz and J. K. Hale, "Existence, Uniqueness and Continuous Dependence for Hereditary Systems," Annali Mathematica Pura ed Applicata 85 (1970) 63-82. 9. W. R. Melvin, "A Class of Neutral Functional Differential Equations," J. Differential
Equations 12 (1972) 524-534.
10. M. A. Cruz and J. K. Hale, "Asymptotic Behavior of Neutral Functional Differential Equations," Archives of Rational Mechanics and Analysis 34 (1969) 331-353. 11. W. Rudin,
Real and Complex Analysis, McGraw-Hill, New York, 1974.
12. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1969. 13. M. A. Cruz and J. H. Hale, "Stability of Functional Differential Equations of Neutral Type," J. Differential Equations 7 (1970) 334-355. 14. K. Kuratowski and C. Ryll-Nardzewski, "A General Theorem on Selectors," Bull. Acad. Polon. Sci. 12 (1965) 397-403. 15. N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience, New York, 1958. 16. T. S. Angell, "Existence Theorem for Hereditary Lagrange and Mayer Problems of Optimal Control," SIAM J. Control and Optimization 14 (1976) 1-18. 17. S. Lang,
Analysis II, Addison-Wesley, Reading, MA, 1969.
18. E. N. Chukwu, "An Estimate for the Solutions of a Certain Functional Differen-
tial Equation of Neutral Type," Proceedinqs of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Academic Press, 1982, edited by V. Lakshmikantham. 19. E. N. Chukwu, "Global Asymptotic Behavior of Functional Differential Equations of the Neutral Type," J. Nonlinear Analysis Theory, Method and Applications 5 (1981) 853-872. 20. J. Hale, Theory 1977.
of Functional Differential Equations, Springer-Verlag, New York,
21. O. Lopes, "Forced Oscillation in Nonlinear Neutral Differential Equations," J. of Applied Mathematics 29 (1975) 196-207.
SIAM
22. H. J. Sussmann, "Small-Time Local Controllability and Continuity of the Optimal Time Function for Linear Systems," J. Optimization Theory and Applications 53 (1987) 281-296. 23. E. N. Chukwu and O. Hajek, "Disconjugacy and Optimal Control," Theory and Applications 27 (1979).
J. Optimization
24. E. N. Chukwu and H. C. Simpson, "Perturbations of Nonlinear Systems of Neutral Type;' J. Differential Equations 82 (1989) 28--59. 25. T. S. Angell and A. Kirsch, "On the Necessary Conditions for Optimal Control of Retarded Systems," Appl. Math. Optimization 22 (1990) 117-145. 26. H. T. Banks and M. Q. Jacobs, "An Attainable Sets Approach to Optimal Control
of Functional Differential Equations with Function Space Terminal Conditions," J.
Differential Equations 13 (1973) 129-149.
27. J. K. Hale and R. R. Meyer, "A Class of Functional Equations of Neutral Type," Mem. Amer. Math. Soc., No. 76, 1967.
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Chapter 12 Controllable Nonlinear Neutral Systems Introduction Criteria for linear systems controllability and constrained controllability are contained in 10.1 and 10.2. There Euclidean and function-space targets are considered. In this chapter we treat the general nonlinear situation in WJ!).
12.1 General Nonlinear Systems We now consider the general system (12.1.1a) where f : E x C x Em -> En is continuously differentiable in the second and third arguments, and is continuous, and where (12.1.1a) satisfies the basic assumptions on D and f in Section 11.2. In particular, we assume that there exists an m E L 2 ( [0;, 00, E) such that
IID.p/(t, ¢, u)11 and
+ IID,J(t, ¢, u)11 :::; m(t),
(12.1.1b)
D(t, Xt) = x(t) - g(t, xd
where
Ig(t,¢)I:::; k(t)II¢11. V ¢ E C, t 2:
a
(12.1.1c)
for some k continuous, g(t,¢) is linear in ¢. We need some preliminary definitions from analysis. We work in the space WJ!). Definition 12.1.1: Let X and Y be real Banach spaces and F a mapping from an open set S of X into Y. If for each fixed point Xo E S and every hEX, the limit lim[F(xo
t-O
+ th) - F(xo)]jt = 8F(x oh)
exists in the topology of Y, then the operator 8F( Xo, h) is called the Gateaux differential of F at Xo in the direction of h. If for each fixed
473
474
Stability and Time-Optimal Control of Hereditary Systems
Xo E X the Gateaux differential 8F(xo,·) is a bounded linear operator mapping X into Y, we write 8F(xo, h) = F'(xo)h, and F'(xo) is called the Gateaux derivative of F at Xo. If F has a Gateaux derivative at Xo, we say F is G-differentiable at xo, and F'(xo) E L(X, Y), the space of bounded linear operators from X into Y. Definition 12.1.2: F : X ---+ Y is weakly G-differentiable at Xo if there exists a bounded linear map F'(xo) E L(X, Y) such that
([F(xo
+ th)
- F(xo)]jt - F'(xo)h, y*) ---+ 0
as t ---+ 0 for all hEX, y* E Y*. As a consequence, if F is G-differentiable, it is weakly G-differentiable. Consider
d dt [D(t, Xt)]
= f(t,xt,u(t)),
Xq
=
(12.1.1a)
Corresponding to the point (t, a,
is defined by
x(rJ',
Let (t,rJ',
Lemma 12.1.1
F'(u)(v) where the mapping t of
= Dux(t,rJ',
---+
y(t, rJ',
yet, a,
yq =
(12.1.2)
475
Controllable Nonlinear Neutral Systems
with ¢J == O. Also, for every (t/;, v) E WP) x L p we have that if F'(u, v) denotes the G-derivative of x(t, a, ¢J, u) with respect to its last two variables at the point (t, a, ¢J, u), then
Dx(t,rJ,¢J, u)(t/;,v)
= A(t,rJ,¢J,u,v),
where t -+ )..(t, a, ¢J, U, v) is a mapping of E into En, which is the unique solution of (12.1.2) with Yq = t/;. Proof of Lemma: For each (rJ, ¢J) E E x WP), the unique solution of (12.1.1a) with initial data (rJ, ¢J) is given by Xq = ¢J in [-h,O],
x(rJ, ¢J,u)(t)
= x(t, a, ¢J, u) = g(t, Xt(rJ, ¢J, u» -
1
g(rJ, ¢J) + ¢J(O)
t
+
(12.1.3)
f(s,x.(rJ,¢J,u),u(s»ds.
We claim that x( a, ¢J, u) is Frechet differentiable with respect to u(t) E Em. Indeed, if DfJ denotes this partial derivative, then by Dieudonne [10],
DfJX q
=0
DfJx(t, a, ¢J, u) =
in [-,h,O],
1 1
+
t
D2f(s, x.(rJ,¢J, u), u(s»DfJx.(rJ, ¢J, (u»ds t
D 3f(s, x.(rJ, ¢J, u), u(s)]ds
dXt +D 2g(t, Xt(rJ, ¢J, u»-d (rJ, ¢J, u). u
Because of the linearity of g,
Thus
dx DfJx(t, a, ¢J, u) - D 2g(t, Xt(rJ, ¢J, u» du (rJ, ¢J, u) t = D2f(s, x.(¢J,u), u(s»DfJx.(rJ, ¢J, u)ds +
1
1 t
D3f(s, x.(rJ, ¢J, u), u(s»ds.
Furthermore, on differentiating with respect to t we obtain
d
dt [DfJx(t, a, ¢J, u)v - D 2g(t, Xt(rJ, ¢J, u»DfJxt(rJ, ¢J, u)v]
= D2f(t, Xt(rJ, ¢J, u), u(t»DfJxt(rJ, ¢J, u)v + D3f(t, Xt(rJ, ¢J, u), u(t»v,
476
Stability and Time-Optimal Control of Hereditary Systems
or equivalently d
dt [Dux(t, a,
= D2f(t, Xt(u,
F'(u)v
= Dux(t, a,
to complete the first assertion. Consider
u
-+
F(u) = x(t,u,
which is a mapping F '. L p
F(uo
+ rh) -
-+ W(l) p'
F(uo) = g(t,Xt(u,
+ it f(s, x.(u,
f(s, x.(u,
6f(uo(s), h(s»)
= Duf(s, x.(u,
Also
g(t,Xt(u,
-+
6g(uo, h) as r
-+
0,
and from linearity
get, (Xt(u,
-+
get, 6xt(u, Uo, h»
as r
-+
0,
Controllable Nonlinear Neutral Systems
477
so that We now have
c5F(uo, h)
= r--O lim(F(uo + rh) -
F(uo))jr
= g(t, Duxt(u, 4>, uo)h) +
it
Du!(s, x.(u, 4>, uo), uo(s))h(s)ds.
We now prove that c5F is continuous at (uo,O) E Theorem 2 [9, p. 336].
~
X
L p , so that by
c5F(uo,h) = F'(uo)h, F'(uo E L(L p , WJ1»): F'(uo) : Lp is a bounded linear map. Suppose (uk,h k) 1imk_oo
+
--+
--+
WJ1)
(uo,O). Then
II g(t,D.x,( <1,q"uk»h k - g(t,D"x,( <1,q"uo)O)1I
1imk_oo
J: II D . ! ( . ,x . ( <1 ,q" u k ),u k( . » h k( . ) -
D.!(.,x.(<1,q"uo),uo(.»Olld.
=0
since Du!(s, x.(u, 4>, Uk), uk(s))hu(s) --+ 0 as k --+ 00 and the L p bound assumption coupled with the use of the Dominated Convergence Theorem. Since
DuXt(u, 4>, u), Du!(s, x.(u, 4>, u), u(s)) are continuous in u, and k(t) (in 12.1.1c) is continuous on [u,oo), for every finite r > and u E Br(uo) = {u : lIu - uoll p ~ r} there is some L < 00 such that if t E [u, til, t 1 < 0, then
°
11F'(u) - F'(uo)1I ~ L <
00.
(12.1.5)
We note that
IF'(uo) - F'(uo)1 ~ Ig(t, D"xt(u, 4>, u)) - g(t, D u(xt(u,4>, uo)))1
it + it + it +
~
~
IDu!(s, x.(u, 4>, u), uo) - Du!(s, x.(u, 4>, uo), uo)ds),
Ig(t, DuXt(u, 4>, u)) - Du(Xt(u, 4>, uo))! ID,,!(s,x.(u,4>, u), u(s)) - D,,!(s,x.(u,4>,uo),uo(s))lds,
k(t)IID u Xt(U, 4>, u) - DuXt(u, 4>, Uo)1I ID,,!(s, x,(u, ¢, u), u(s)) - Du!(s, x.(u, ¢, uo), uo(s)) Ids.
Stability and Time-Optimal Control of Hereditary Systems
478
If X(U, ¢, u) is a solution of (12.1.1a), the associated variational control system along the response t -> Xt(u, ¢, u) is the system (12.1.2). In particular, if (12.1.6) f(t,O,O)=o
°
so that the trivial solution x(u, 0, 0) = is a solution of (12.1.1a) the variational control system along the trivial solution is (12.1.7a) The following theorem is reproduced from [9, p. 155]. Theorem (Open Mapping Theorem) Let X, Y be Banach spaces and Br(xo) an open ball of radius r about Xo EX. Suppose F : Br(xo) C X -> Y is weaklyG differentiable such that R(F'(xo)) = Y and IF'(x)-F'(xo)1 ~ k in Br(xo) for some k > 0. Then Bp(Fxo) C F(Br(xo)) for some p > 0, provided k is sufficiently small. Theorem 12.1.1 tem along x == 0,
Consider System (12.1.1a) whose linear variational sysd
dt D(t)z(t)
= L(t, Zt) + B(t)v(t),
(12.1.7b)
is controllable on [17, tl] where tl > 17 + h and where
D2f(t, 0, O)Zt
= L(t, Zt),
D3f(t, 0, O)v = B(t)v.
Assume that
f(t,O,O) =0, vt Then
°
E IntA(tl,
e:«. (12.1.8a)
17)
where
A(tl,U)
= {Xt,(u,O,u): u E Lp([u,td,Em), is a solution of (12.1.1a) with
X/7
lIuliLp ~ 1, x(u,O,u)
= O}
is the attainable set associated with (12.1.1a). Proof of Theorem: Let ¢ E W~l), U E Lp([u, td, Em), tl > 17 + h. Let u -> Xt,(u,¢,u) be the mapping F : Lp([u,td, Em) -> WP)([-h, OJ, En)
Controllable Nonlinear Neutral Systems
479
given by Fu = Xtl(U,¢, u), where x(u,¢,u) is a solution of (12.1.1a). Then by Lemma 12.1.1
F'(u)v
= DuXtl(U,¢,U)(v) = Yt(u,¢,u,v),
where Y is the solution of the variational equation (12.1.2b) and F'(u) : Lp([u,t1], Em) -+ WP). Evidently F' is a bounded linear surjection if and only if the control system (12.1.2b) is controllable on [u,td. As a consequence, F'(Lp[u, t1],Em) = WP); and Lemma 12.1.1 guarantees that all the requirements of the open mapping theorem are met for the map
°
°°
°
Thus for Uo == E Lp([u,td), F(uo) = F(O) = E WP), for every r > and open ballll£(O, r) C Lp([u,td, Em) of center E Lp and radius r there is an open ball ll£(0, p) C WP) center of radius p such that
°
ll£(O,p) C ll£(F(uo),p) C F(ll£(uo,r)) = F(ll£(O,r)). Thus
It follows from this that
° IntA(t1,u) E
C
wJ1)
where
A(h,u) = {Xtl(U,O,U): u E Lp([U,t1],Em), lIuIlL.:S r, x(u,O,u) is a solution of (12.1.1a)}. We have proved that for any r
> 0,
°
E
IntA(t1,U)
(12.1.8b)
(12.1.9)
where A(t1,U) is defined in (12.1.8). Since r is arbitrary and the set of admissible controls Uad with constraints is (12.1.10) (12.1.3) is valid.
480
Stability and Time-Optimal Control of Hereditary Systems
Corollary 12.1.1 Assume all the conditions of Theorem 12.1.1. Then System (12.1.1) is locally null controllable with constraints. That is, there is a neighborhood 0 of zero in WP) such that every initial point of 0 can be driven to zero in some finite time tl using some u E Uad .
Condition 12.1.3 will prove not only local null controllability but constrained null controllability as well, because (12.1.9) implies that
o E IntD
(12.1.11a)
where D is the domain of null controllability, i.e., the set of initial functions
=
=
Indeed, suppose (12.1.11) is false. Then there is a sequence { 0' + h. Thus, on setting ~n == xt,(O',
=
=
for any t l ~ 0' + h, a contradiction. Thus (12.1.11) is valid. Therefore the local null controllability of (12.1.1a) is proved, and, as remarked, so is the local null controllability with constraints. Corollary 12.1.2
Consider the autonomous system
d
dt[x(t) - A_lx(t - h)] = f(xt,u(t)). Let
DII(O,O)Yt and assume that: 0) rank [A('\), B] A_l'\e-'u.
= Aoy(t) + Aly(t = n,
h),
(12.1.l1b)
D2f(0, O)u
V'\ E C where A(,\)
= l,\ -
= Bu, A o - Ale- h
-
481
Controllable Nonlinear Neutral Systems
(ii) rank [U - A-I, B] = n, V A E C. Then (12.1.12) is locally null controllable and locally null controllable with constraints. Proof: Conditions (i), (ii) are the necessary and sufficient conditions for the controllability of the linear variational equation
d dt [x(t) - A_Ix(t - h)] = Aox(t) + AIx(t - h) + Bu(t).
(12.1.11c)
Theorem 12.1.1 yields the required assertion. In Theorem 12.1.1, the controllability of the linear system (12.1.11c) is of fundamental importance. Can this be weakened? The next result asserts that the closure of the attainable set of the linear system (12.1.11c) can replace the controllability requirement. The result is stated for linear autonomous system (12.1.12) and the associated linear system
x(t) - A_Ix(t - h)
= Aox(t) + AIx(t -
h) + Bu(t),
(12.1.13)
where
Dtf(O,O)Xt
= Aox(t) + AIx(t -
h),
D2f(O,O)v
= Bu.
Following [1] we introduce the following notation: Let s be fixed. Define the n x n-matrices
o {I, 6..= J 0,
6.J =
j = 0, j = 1,00' j = 0,
A o,
+ Ad, j = 1, ...
A~-;.I(A_IAo j
6.~
(12.1.14)
,s,
= L6.~-I6.J_k'
i
= 2'00'
,s,
,n(s+ 1) -1, j
= 0,
00'
,s,
k=O j
zj = L6.~A~lkB, k=O
r:j--Zi A Zi-I j-Oj' i r = [r~, ,r~]. 00
•
i=O,oo. ,n(s+l)-l, j=O,oo. ,s, (12.1.15) i
= 1, 00. ,n(s + 1) -
1, j
= 0, ...
,s (12.1.16)
482
Stability and Time-Optimal Control of Hereditary Systems
The matrices K, are defined as follows:
K."
= ri
i-1 -
"r;-i(ZO)+ ~
+ K, '" , t. = 1, ...
,n (s
+ 1) - 1,
(12.1.17)
j=l
where (ZO)+ is a fixed right inverse of ZO, i.e., ZO(ZO)+ tosiewicz [1] has proved the following:
Proposition 12.1.1 A(t) =
{Xt
I.
Bar-
Let
E wP)([-h, 0], En) : X is a solution of (12.1.13)
for some u E Lp([O, til, Em)}. The set A(t 1), t 1 = (s
+ l)h,
is closed in WP) if and only if
K;(kerZO) C ImZo, i= 1, ... ,n(s+ 1)-1. H t ; > nh, then put s
= n in
(12.1.18)
(12.1.18).
As an immediate consequence of Proposition 12.1.1, the following is valid:
Corollary 12.1.3
A(t 1 ) is closed in WP) whenever
imr; C Im ZO, i = 1, ... ,n( s or
+ 1) -
1,
(12.1.19)
rank ZO
= n,
(12.1.20)
rank B
= n.
(12.1.21)
or
Note that rank ZO = rank [A: 1B, ... ,B].
(12.1.22)
Thus (12.1.22) is equivalent to closedness and finite codimensionality of A(t1) in WJ1) [1, p. 181]. Theorem 12.1.1 can now be weakened.
483
Controllable Nonlinear Neutral Systems
Theorem 12.1.2 In (12.1.12), assume that all the conditions of Proposition 12.1.1 are fulfilled. Then there is a closed subspace Y of WJl) and a neighborhood 0 of zero in Y such that every initial point of 0 can be driven to zero in some time t l using some u E Ua d . Proof: The prooffollows the same pattern as that of Theorem 12.1.1, with the new hypothesis replacing the controllability condition. The mapping
F: Lp([u,td, Em) Fu
->
WP),
= Xt,(u,O,u), = g(Xtl (u, 0, u)) -
g(u, 0)
+ Jut ' f(s, x.(u, u), u(s))ds,
t
>a
°
where x(u, 0, u) is a solution of (12.1.12) does not have a surjective Gateaux derivative F' at E Lp([u, tl], En). Instead its range is closed: F'(O)(Lp([u, tl], Em)) == Y is closed. Therefore it is a closed subspace of WJ!)([-h, 0], En). We can therefore study the mapping F : L p ----> Y of the Banach space L p into the, Banach space Y, whose Gateaux derivative F'(O) : Lp([u, td, Em) ----> Y is a surjection. It satisfies the conditions of Corollary 15.2 of [7, p. 155]. The proof ends in the same way as before. Global controllability can be proved by imposing a stability condition on
Theorem 12.1.1 and Theorem 12.1.2 are local controllability results. We are now ready for the global constrained null controllability result.
Theorem 12.1.3 In (12.1.1a) assume that (i) f is smooth enough for uniqueness, existence, and continuous dependence on initial data of solutions of (12.1.1). (ii) f(t, 0, 0) = for t 2: a . (iii) System (12.1.2b) is controllable. (iv) The system
°
(12.1.23)
is uniformly exponentially stable, that is, every solution of (12.1.10) satisfies (12.1.24)
484
Stability and Time-Optimal Control of Hereditary Systems for some k > 0, a > O. Then (12.1.1) is null controllable with constraints, i.e., with controls in Uad of 12.1.10 for some t ~ a + h.
Remark 12.1.1: Conditions that guarantee the behavior (12.1.24) for System (12.1.23) are contained in Section (4.2). We now consider a special case of System (12.1.1a), namely (12.1.25). The results obtained are sharp. We now study the nonlinear in the state difference-differential control system d dt [x(t) - A_1x(t - h)]
= f(x(t), x(t -
h» + Bu(t),
(12.1.25)
where A_ 1 is an n x n constant matrix, f is continuously differentiable, and B is an n x m constant matrix. Let H 1 ~ D1f(x(t), 0), H 2 ~ D2f(x(t), x(t - h» where D;/ is the ith partial derivative of f. Let
A o = D1f(O, 0), A 1
= D 2 f (0,0),
and denote by D a the symmetric n x n matrices
Dr
A, where A is a positive definite Define J« as follows: J; = ADa + symmetric n x n matrix. We have the following special result for (12.1.25): Theorem 12.1.4 In (12.1.25) assume: (i) f(O, 0) = 0. (ii) rank (A(~), B] n, for all ~ E C where A(~) A1e->'h, rank [AI - A_1B] = n for all ~ E C. (iii) All the roots of the equation
=
=I -
~A_le->'h
- A o-
have moduli less than 1; and for some positive definite n x n matrix A we have where, for some q > 1,
485
Controllable Nonlinear Neutral Systems Then (12.1.25) is null controllable with constraints.
Proof: Conditions (ii) and (iii) are the necessary and sufficient conditions for the linear approximation of (12.1.25):
d dt [z(t) - A_1z(t - h)J = Aoz(t) + A1z(t - h) + Bu(t) to be controllable. Condition (iii) is the required condition for the system d
dt [x(t) - A_1x(t - h)J = I(x(t), x(t - h)) to have all its solution x( 0", If') satisfy
Ilxt( 0", 1f')11 - t 0 12.2
as t
-t
00.
Nonlinear Interconnected Systems
We now investigate the ith interconnected subsystem described by d . . . . dt [Di(t)X~] li(t, xL u'(t)) + ki(t, Xt, v'), i 1, ... ,f, (12.2.1)
=
=
where k, describes the action of the whole system d
dt D(t, xd = I(t, Xt, u(t))
+ k(t, Xt, v(t))
(12.2 S)
on its interconnected system (12.2.1), which when "free" and "disconnected" is given by d . . .
dt [Di(t)X~]
= !i(t, x~,
u'(t)).
(12.2.2)
In (12.2.1), u i E L 2([0", tIl, Em.), and vi E L2([0", tIl, Em). We shall sometimes restrain the control u i to lie in the set
and vi E Qi Let
= {fi(t, ¢J, ui) : ui E Wi}, Ki(t,¢J) = {ki(t,¢J,V i): vi E Q;}.
Fi(t,
Ki(t, xt} C Int Fi(t,X~). The following result is valid.
486
Stability and Time-Optimal Control 01 Hereditary Systems
T'heorern 12.2.1 In (12.2.1) assume that: (i) /;(t, 0, 0) = 0, ki(t, ¢, 0) = O. (ii) Ii, k i , and D; satisfy all the smoothness conditions of existence, uniqueness, and continuous dependence on initial data. (iii) Assume that the linear variational system
d
..
..
dt [D'(t)z'(t)] = Li(t, xD + B'(t)w(t)
(12.2.3)
of (12.2.2), where
is controllable on [0", td, h > 0" + h. (iv) Ki(t, ¢) C Int Fi(t, ¢i). Then the interconnected subsystem (12.2.1) is locally null controllable on [0", td with constraints. Corollary 12.2.1 Assume (i) Condition (i) - (iii) of Theorem 12.2.1. (ii) The system
d
dt [Di(t)X~]
.
. = li(t, x~, 0),
(12.2.4)
i = 1, ... ,R,
is globally exponentially stable. Then the interconnected subsystem is (globally) null controllable with constraints, that is, with controls
= {u i E L 2 ([0", h], Em,) : Iluill2 ::; I}, E a. = {vi Vi E L 2 ([0",td , E m ) : IIv i ll2::; I}.
ui E IP' vi
Proof: Since the hypothesis of Theorem 12.2.1 holds
o E IntAi(t, 0")
for t
> 0" + h,
(12.2.5)
where Ai(t, 0") is the attainable set associated with (12.2.2) where
Ai(t,O") = {xi(O", 0, ui) : x solves (12.2.2) with Xo
X
o
= 0:
= O}.
This is also the solution of (12.2.1) with u i as the control of the interconnected subsystem and with v = 0 as the control from the large-scale system.
Controllable Nonlinear Neutral Systems Thus xi(u,O,ui,O) x~ = 0, then
==
487
xi(u,O,u i). Suppose xi is a solution of (12.2.1) with
for t 2:: a . On defining the set
we deduce (on account of the hypothesis (i) and the fact that xi(t, 0,0,0) = Since (iv) is valid and (12.2.5) holds,
o is a solution of (12.2.1), that 0 E A;(tI,U) c Hi(tl, u).
(12.2.6) and
o E Int Hi(t l, u)
(12.2.7)
as well. If D denotes the domain of null controllability of (12.2.1), one proves readily that o E Int D, proving local null controllability with constraints. Proof of Corollary 12.2.1: One uses the control u i = 0 E Ui, 1, ... ,£ to glide along System (12.2.4) and approach an arbitrary neighborhood of the origin. Since (i) guarantees that all initial states in this neighborhood can be driven to zero in finite time t l, we are done. Remark: Conditions for uniform asymptotic stability are available in [3]. Remark 12.2.1: It is very important to study the condition (12.2.6) from which (12.2.7) was deduced. Suppose
o E IntAi(tl, u)
(12.2.8)
fails, and the isolated system is not even controllable locally and is "not well behaved". Condition (12.2.7) can still be salvaged and the interconnected system made controllable. What may be needed is a criteria that ensures that (12.2.9)
Stability and Time-Optimal Control of Hereditary Systems
488
where
G;(t 1,0")
= {it k;(s,x.,vj(s))ds: v j E Qj}'
In other words we require a sufficient amount of control impact from external sources «k;), which is monitored locally at the i-subsystem) to be "forced" on (12.2.2). The source of power senses the control u; available to (12.2.2) and sends a sufficient controlling signal k; to the large-scale system, which is a function of the power's state variable x and the control v j • With this the right behavior is enforced. Policy implications of this insight will be pursued elsewhere in [7]. There we observe that the interaction does not ignore the subsystems' (initiatives) v j E Em; it utilizes them. For example, were they to be ignored, and
k; = k;(t, xt) still be operative and nontrivial, the condition 0 E H;(t,O") and subsequently (12.2.7) will fail. The interconnected system will not be null controllable. Failure to acknowledge individual subsystems' initiatives (vj ) in highly centralized interconnected organizations monitored by our dynamics leads to lack of growth from initial endowment ¢ to desired economic target 'l/;. On the other hand, failure to enforce regulations or use subsidy, bringing k; to bear when same subsystem misbehaves or is depressed, and is uncontrollable, in "individualistic" systems may make the interconnected system uncontrollable and trigger an economic depression. In place of (12.2.1), consider the system
~[Di(t)X~]
=
l
f;(t,x~,u;(t»
+ ?=k;j(t,~,u;(t)).
(12.2.10)
3=1 ;~j
The following can be proved: Theorem 12.2.2 For (12.2.10) assume (i) - (iii) of Theorem 12.2.1. But in (12.2.2), L;(t, z;) and Bi(t) are defined as
= u«, z;), D3(Ji(t, 0, 0) + k;(t, Xt, 0» = B;(t) D2f;(t, 0, O)z;
.
with ki(t,xt,u') =
l
L j=l ;~j
'.
kij(t,xLu'(t». Then (12.2.10) is locally null con-
trollable with constraints.
Controllable Nonlinear Neutral Systems
489
We note that condition (iii) of Theorem 12.2.2 requires the controllability of
. . .
d
.
= Li(t, z;) + (Bli(t) + B 2i(t))U'(t),
dt [D'(t)z~]
(12.2.11)
where
Bli(t) B 2i(t)
= D3/i(t, 0, 0), = D3ki(t, Xt, 0).
Suppose (12.2.5) is invalid for the isolated system (12.2.3), and this can happen when (12.2.3) is not controllable with the control matrix Bli(t), the "solidarity function" ki can be brought to bear to force the controllability of (12.2.11) with B i = Bli + B 2i to ensure (12.2.5). If the system is autonomous, the requirement would be (i) and (ii) of Corollary 12.1.2. Even if (i) and (ii) hold with Bli and (12.2.5) is assured, the composite system need not be locally null controllable unless (12.2.11) is controllable with B i = Bli + B 2i. An adequate proper amount of intervention in the form of a solidarity function ki is needed. The linear case was explored in Chapter 11. It is quite possible to extend this treatment to the situation d
..
dt [Di(t)x~]
.
= li(t, x~) - pi(t)
+ ki(t),
(12.2.12)
xD
where li(t, is independent of control Pi, which enters the dynamics in an additive fashion. One can then deduce the four universal principles for the control of an interconnected organization as are contained in Section lOA.
12.3
An Example: A Network of Flip-Flop Circuits
The basic element in a digital computer is the flip-flop circuit, a model of which is given in Figure 12.3.1, [17]. The section between x = 0 and x = 1 is a lossless transmission line with specific inductance L. and specific capacitance C.. The voltage v across the line and the current flowing through it are both functions of x and of time t. They are described by the following partial differential equations:
av __ ai •at - ax'
C
0< x < 1, t > O.
(12.3.1 )
490
Stability and Time-Optimal Control of Hereditary Systems
R
E-
.. x
o FIGURE
12.3.1
The boundary conditions at the ends of the line are 0= E - v(O, t) - Ri(O, t), -Cl
dv~~,t)
(12.3.2)
= i(l,t)+f(v(l,t),
where f( v(l, t)) is a nonlinear function of v and gives the current in the displayed box in the direction shown. If we set the left-hand side of (12.3.1) and (12.3.2) to zero and solve, we obtain a single equilibrium (v*, i*). If it is assumed that (v", i*) is globally asymptotically stable, then the flip-flop circuit can be shown to operate as a memory device. To investigate conditions on the system's parameter to guarantee this stability assumption (12.3.1), (12.3.2) is converted into a functional differential equation of neutral type:
where
and also vo
= v(O,t),
V1
= v(l,t),
ia
= i(O,t),
il
= i(l,t).
491
Controllable Nonlinear Neutral Systems
One can pose the following question in place of the stability one: Is it possible to drive all fluctuations of (v(x,t), i(x, t» to the equilibrium (v*, i*) in finite time by introducing a control device (e.g. a stabilizer)? The same question can be posed for the circuit in Figure 12.3.2, whose control equation is
CI
- qu(t - h)] = -~u(t)
~[u(t ) dt
- !u(t - h) - g(u(t» - qu(t - h) - i(t)
2
2
+el(t),
~
[Li(t)] = -RI i(t)
+ u(t) -
qu(t - h) + e2(t), (12.3.4)
lossless transmission line
i =g(v)
I(t)
x=o
x= I FIGURE
12.3.2. [17]
=
=
where u is related to the voltage at x L by veL, t) u(t) - qu(t - h), and L 1 , R, and CI are the parameters of the line. The function e = (el, e2) is a control function generated by the control device at x = 0, and it is related to E(t). Equation (12.3.4) can be put in vector form:
d
dt [x(t)-A_Ix(t-h)] where x(t)
A-I Al
= Aox(t)+ AIx(t-h)+CI(x(t-h»+ Be(t),
(12.3.5)
= (u(t),i(t»)T, e = (el,e2)T,
= [ -0q 00] ' =
[-!t
~],
A 0--
[-2b, I
t:
G(x(t~h»=
-c,I
]
-4; , [g(u(t)-qO(U(t-h»].
Suppose each such circuit in Figure 12.3.1 or Figure 12.3.2 is connected up in a complicated fashion with other such circuits to form an interconnected system. The problem of controllability can also be fruitfully posed.
492
Stability and Time-Optimal Control of Hereditary Systems
lossless transmission line +
I(t)
x=o
x=1
..
+
I I gil
..- ••
+ Yj
• • • •
-8 FIGURE
12.3.3
For example, suppose there are £ = 1, ... ,10 such circuits of Figure 12.3.3. We can consider the overall system as a decentralized one in which each circuit is influenced by the interaction terms
Yi(t,x,e i)
10
= Lgij(t,:d,e i ) j=l
itcj
as follows: d . . . dt (x'(t) - A_lix'(t - h)] = AOix'(t)
.
+ Gi(X'(t -
h))
.
.
+ Bie'(t) + Yi(t, x,e') (12.3.6)
is the ith-control system. In (11.4.6), set Ali = DGi(O),
BZi = D3Yi(t, 0, 0),
493
Controllable Nonlinear Neutral Systems and consider the companion system d·.
dt [x' - A_lix' (t - h)]
. .
= AOix'(t) + A 1iX'(t -
a) + (Bli
.
+ B 2i)e'.
(12.3.7)
We have the following result:
Proposition 12.3.1 System 12.3.6 is locally null controlJable on [0, td with constraints ify,(t,x,e') is such that (12.3.7) is controllable on [O,td. REFERENCES 1. Z. Bartosiewicz, "Closedness of the Attainable Det of the Linear Neutral Control System," Control and Cybernetics 8 (1979) 179-189.
2. E. N. Chukwu, "On the Euclidean Controllability of a Neutral System with Nonlinear Base," Nonlinear Analysis, Theory, Methods and Applications 11 (1) (1987) 115-123. 3. E. N. Chukwu, "Uniform Asymptotic Stability of Large Scale Systems of Neutral Type," IEEE Proceedings of the Eighteenth Southeastern Symposium on Systems Theory, April 7-8, Knoxville, TN, Computer Society Press, 1986. 4. E. N. Chukwu, "The Time Optimal Control Theory of Linear Differential Equations of Neutral Type," Comput. Math. Applic. 16(10/11) (1988) 851-866. 5. E. N. Chukwu and H. C. Simpson, "Perturbations of Nonlinear Systems of Neutral Type," J. Differential Equations 82 (1989) 28-59. 6. E. N. Chukwu, "Some Controllability Questions of Linear and Nonlinear Neutral Functional Differential Equations in WJl) ," Preprint , 7. E. N. Chukwu, "Global Economic Growth: Will the Center Hold?" Preprint. 8. J. P. Dauer, "Nonlinear Perturbations of Quasilinear Control Systems," J. Math. Anal. Appl. 54(3) (1976). 9. K. Deimling, Nonlinear Functional Analysis, Springer- Verlag, New York, 1985.
10. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960. 11. J. Hale and M. A. Cruz, "Existence, Uniqueness and Continuous Dependence for Hereditary Systems," Ann. Math. Pura. Appl, 85 (1970) 63-82. 12. J. Hale, "Forward and Backward Continuation of Neutral Functional Differential Equations," J. Differential Equations 9 (1971) 168-181. 13. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 14. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan Company, New York, 1964. 15. G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York, 1972. 16. A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, New York, 1977.
494
Stability and Time-Optimal Control of Hereditary Systems
17. M. Slemrod, "The Flip-Flop Circuit as a Neutral Equation," in Delay and Func-
tional Differential Equations and Their Applications edited by K. Schmitt, Academic Press, 1972. 18. V. N. Do, "Controllability of Semilinear Systems," J. Optimization Theory and Applications 65 (1990) 41-52. 19. D. Salamon, Control and Observations of Neutral Sylltems, Pitman Advanced Publishing Program, Boston, 1984. 20. H. T. Banks, M. Q. Jacobs, and C. E. Langenhop, "Characterization of the Controlled States in WJ1) of Linear Hereditary Systems," SIAM J. Control 13 (1975) 611--{;49. 21. M. Kalecki, "A Macrodynamic Theory of Business Cycles," Econometrica 3 (1935) 327-344. 22. R.O.D. Allen, Mathematical Economics, Macmillan, London, 1960. 23. K. J. Arrow, Production and Capital, Collected Papers of Kenneth J. Arrow, Belknap Press of Harvard University Press, Cambridge, Massachusetts, 1985. 24. E. N. Chukwu, "Stability and Time-Optimal Control of Hereditary Systems,"
N. C.S. U. Lecture Notes, 199<'>
Chapter 13
Stability Theory of Large-Scale Hereditary Systems
Introduction In Sections 2.4 to 4.4 we studied the stability properties of isolated systems and their perturbations, which are described by
x(t) = f(t, Xt) + g(t, Xt), or
d
dt [x(t) - A_1x(t)]
(13.1.1)
= f(t, Xt) + g(t, Xt).
(13.1.2)
But several systems of practical interest, such as the economic models of Sections 1.8 and 9.5 or nonlinear electric circuits described in 1.6 and 12.3, may often be viewed as interconnected or composite systems. We now investigate the stability properties of such hereditary systems in terms of the same properties of the subsystems and the growth conditions of the interconnecting structure. The analysis will then be used to provide an interesting policy prescription for the growth of large-scale systems. The definitions ofthe various stability concepts of Section 2.4 will be maintained.
13.1
Delay Systems
To deal with the problem, we let en; denote the set of all continuous functions 'l/Ji : [-h,O] -+ En, with norm lI'l/J i ll = max{I'l/Ji(t)1 : -h ~ t ~ O}, where I'l/J i (t) I is the Euclidean norm of system
'l/Ji. Consider the interconnected
l
L
ii(t)=fi(t,Z~)+
gij(t,z1),
i=1, ...
.e,
j=li~j
where
h .E
x en,
-+
En"
gij: E
495
X
C"!
-+
En"
(13.1.3)
496
Stability and Time-Optimal Control of Hereditary Systems
are nonlinear functions. If we let
xT
l
I: ni = n,
i==1 = [(z1)T,
f(t,xf = [h(Zlf,
, (zl)T) E En, ,ft(Zl)T),
l
gi(t,X)
= Li;ejgij(t,zj), j==l
and
[g(t,X)]T = [(gl(x)f ... [gl(x)f), (13.1.3) can be written as
x(t)
= f(t, Xt) + g(t, xt}.
(13.1.4)
System (1.4) can be viewed as an interconnection of i-isolated subsystems (13.1.5) with interconnection described by l
gi(X) = Li;ejgij(zj). j==l
We call (13.1.4) a composite system and (13.1.3) its decomposition. The basic assumptions for the existence of unique solutions are valid for (13.1.1), (13.1.3), (13.1.4), and (13.1.5). We recall Theorem 3.1.1 for uniform asymptotic stability of (13.1.4) in en, which is rephrased for (13.1.5) in the space en. as a definition of Property Fl for (13.1.5). Definition 13.1.1: In (13.1.5), suppose Ii : E x en; -. En. takes Ex (bounded sets of en.) into bounded sets of En,. System (13.1.5) is said to have Property Fl if there exist continuous functions Ui,Vi,Wi : E+ -. E+ that are non decreasing with the property that Wi(S),Ui(S), Vi(S) are positive 0, Ui(S) -. 00 as S -. 00; and there exists a for S > 0, Ui(O) Vi(O), Wi(O) continuous functional Vi : E x en. -. E such that
=
=
(i) ui(l¢i(O)I):5 Vi(t, ¢i) :5 Vi(II¢ill), (ii) V;(t,¢i):5 -CiWi(l¢i(O)l), and (iii) Vi(t, ¢i)- V(t, ¢~)I :5 Ldl¢i -¢~I\' and all these hold for all ¢i, ¢i, ¢~ E en" where L, is a constant. Note that Property Fl is the required criteria for uniform asymptotic stability of (13.1.5).
Stability Theory of Large-Scale Hereditary Systems
497
Theorem 13.1.1 For the composite system (13.1.4) with decomposition (13.1.3), assume that the following conditions hold: (i) Each isolated subsystem (13.1.5) possesses Property Fl. (ii) For each i,j = 1, ... ,.e, if. j there are constants k;j ~ 0 such that
(iii) All successive principle minors of the test matrix S = [s;j] (n arrow) defined by C;, if i = j, s·· - { I) -L.k.. ifi.,J. J' , ')' rare positive. Then the trivial solution of the composite system (13.1.4) is uniformly asymptotically stable. Proof: Let V; be the functionals of hypothesis (i), and let a; > 0, 1, ... .E, be arbitrary constants. Choose V as follows:
f
=
£
V(t,
= LaiV;(t,
for 0, s > 0, Ui(S) > 0 for S> 0, Ui(S) --+ 00, u(s) --+ 00, and Ui(O) Vi(O) 0, and such that
=
=
£
£
;=1
i=l
u(I
V(t,
= La;V;(t,
S; L
;=1
£
a;{ -c;wi(l=1
£
:S La;{-C;Wi(I
;=1 = aT Sw,
;=1
where aT = (a1,'" ,at), and w6 = (W1(W(O)l), ... ,Wl(W(O)I)), and S is the test matrix contained in hypothesis (iii).
498
Stability and Time-Optimal Control of Hereditary Systems
From the treatment of M -matrices in Michel and Miller [2], the constant can be chosen such that aT S > O. Hence we have proved that there is a function w : E+ -+ E+, w(s) > 0, s » 0 such that a
V(t, ¢) :::;
_aT Su
:::; -w(I¢(O)l).
Therefore the trivial solution of (13.1.4) is uniformly asymptotically stable.
Remark 13.1.1: The interconnection may be regarded as delay-free in the sense that there are functions G i j such that
where We consider systems 13.1.3 - 13.1.5 that are time invariant, i.e., fi : en. E?», gij: en. -+ En -.
-+
Definition 13.1.2: The isolated autonomous system (13.1.5) is said to possess Property F2 if fi : en. -+ En. takes (bounded sets of en;) into bounded sets of En., and furthermore: (i) fi(O) = O. (ii) There are continuous nondecreasing functions Ui(S), Vi(S), Wi(S), which are positive for S > 0, Ui(O) = Vi(O) = 0 = Wi(O), Ui(S) -+ 00 as S -+ 0, and constants Ci, Li; and there is a Lyapunov function V; : En, -+ E such that (iii) V;(O) 0, ui(lzil):::; V;(zi) :::; vi(lzil), V zi E En,. (iv) ~[
=
-h~.~O
V; [¢i(O)]. (v) 1V;(zD - V;(z;)I:::; Lilzf - z;1,
V zj E E?', If the isolated system
(13.1.6) possesses property F2, then the trivial solution zi = 0 of (13.1.6) is globally asymptotically stable and every solution zi( ¢i) of (13.1.6) satisfies z;(¢i) -+ 0 as t -+ 00. See Theorem 3.3.2 and Corollary 3.3.2.
Theorem 13.2.2 tem
The trivial solution of the composite autonomous sys-
(13.1.7)
Stability Theory of Large-Scale Hereditary Systems
499
with its decomposition l
ii(t)
= f;(z;) + Li;tj9ij(Z!),
(13.1.8)
j=:l
°
is globally uniformly asymptotically stable and every solution x( ¢) of (13.1.7) satisfies Xt(¢) --+ as t --+ 00 if: (i) Each system (13.1.6) possesses Property F2. (ii) All interconnected gij satisfy the inequalities Igij(¢i)1 ~ kij wj (I¢i (O) 1) for ¢i E (iii) A11 the successive principle minors of the test matrix S = [Sij] defined by for i = i. Ci, s" - { 'J -L.k .. if i =I j I J}'
en-.
are positive. Proof: Just as before, define l
V.{x)
= LaiV;(zi) i=: 1
for x = [(z'f, ... ,[zljT], and observe that there are functions u, v such that
u(s) > 0,
u(lxl) ~ V(x) ~ v(lxl), and u(s)
--+ 00
as s
--+ 00,
s> 0,
u(s) >
°as s > 0,
V(O) = 0. Differentiating V we deduce that
l
V = LaiV;(i), i=:l 1
1
~ Lai[-Cil¢i(O)1 i=:l 1
+ L#i9ij(¢j)], j=l 1
~ La;{-c;j¢i(O)1 i=:l
+ Lj;tiLikijwj(I¢i(O)ln j=:l
= _aT Su == -w(lx/) < 0,
=
=
where aT (a1,'" ,al) uT (w1(1¢(0)1),·" ,wl(l¢l(O)I», and S is the test matrix given in the hypothesis. The above estimates are valid for max V[¢(s)]
-h':;'':;O
= V[¢(O»).
500
Stability and Time-Optimal Control of Hereditary Systems
It follows from Corollary 3.3.2 that the assertions of Theorem 13.1.2 are
valid. Remark 13.1.2: If the interconnecting structure in g;(t, z) is replaced by l
g;(t, z ) = L: g;j(t, zj), j=l
and the estimate in condition (ii) of Theorem 13.1.1 is replaced by
e Ig;(t,¢)/ ~ L:k;jwj(I¢i(O)I), j=l
then the test matrix S in (iii) defined by Sij
={
fri(-Ci+Lik;i, 1
"2(friL;aij
i=j,
+ frjmjaji),
"
.
If z =f J
may be assumed to be negative definite. If c, > 0, the isolated system is stable. If Ci < 0, it is unstable. In this case, to ensure stability of the composite system we must have L;k i i < 0 and IL;kid > Ci. This has important policy implications.
13.2
Uniform Asymptotic Stability of Large-Scale Systems of Neutral Type
In Section 4.2 we reported the Lyapunov theory of stability of the isolated functional differential equation of neutral type d
dt [D(t)xtJ = f(t, Xt) + get, xd,
=
(13.2.1)
=
where Xt E C C([-h, 0], En) is defined by Xt(s) x(t + s), s E [-h, OJ. Here En is the Euclidean n-space with norm I. I, and C is the space of all continuous functions mapping [-h, OJ into En with the sup norm /1./1. The operator D( ): [0',00) X C ~ En is given by
D(t)
=
get,
tEE,
(13.2.2)
The function g in (13.2.2) is given by
get,
=
l°h
[dell(t, B)J
(13.2.3a)
Stability Theory of Large-Scale Hereditary Systems where J.l(t,8) is an n x n matrix function of t E [r.oo), bounded variation in 8 that satisfies the inequality
501 0 E [-h,O] of
for all t E [r, 00), c/J E C, where i(s) is a scalar continuous nondecreasing function of s E [0, h] with i(O) = 0. We now consider equations of the form
~
l
[Vi(t)z;l = fi(t, z;) + L
(13.2.4)
kij(t, z!),
j=li-lj
where
E
X
z;
en;
E en'([-h,O],En,) -+
E?', Letting
== en.,
l
L: ni =
and fi
E x en,
-+
En"
gij
n,
i=l
xT
= [(z'l, ...
,(zl)T] E En,
[f(t,x)f ='[(f1(t,z')T, ... ,ft(t,zl)]T] l
ki(t,x) = Li-ljkij(t,zj), j=l
and we may rewrite (13.2.4) as (13.2.5) where We assume that
f(t,O)
= k(t,O) = 0,
Vt E E.
(13.2.6)
The composite system (13.2.5) with decomposition (13.2.4) may be regarded as an interconnection of i-isolated subsystems (13.2.7)
502
Stability and Time-Optimal Control of Hereditary Systems
with interconnecting structure described by t
ki(t,x)
= L:i~j9ij(t,zj).
(13.2.8)
j=l
We assume that conditions on D, f, and k are sufficient for the initial-value problem to have a unique solution. We assume also that D is a uniformly stable operator in the sense of Section 4.2. The conditions of Theorem 4.2.1 that ensure uniform asymptotic stability for (13.2.7) are now isolated and called Property A.
Definition 13.2.1: The isolated subsystem (13.2.7) is said to possess Property A if there exist a continuous functional V;(t,4i) on E x C'"; and three continuous functions Ui, Vi, Wi mapping [0,00) -+ [0, 00), which are strictly increasing with Ui(S) -+ 00 as S -+ 00, Ui(S), Vi(S), Wi(S) > 0 for S > 0; and there exist constants c, > 0, M, > 0 such that (i) Ui(ID i (t )4Ji I) ::; V;(t, 4Ji) ::; Vi(lI4Ji 11), (ii) V;(t, 4Ji) ::; CiWi(JDi(t)4Jil) or V;(t, 4Ji) ::; CiWi(l4Ji(O)I), (iii) 1V;(t,4Ji) - V;(t,4J;)I::; Mill
en..
Theorem 13.2.1 The trivial equilibrium of the composite system (13.2.5) is globally uniformly asymptotically stable if the following conditions hold: (1) Di(t) is a uniformly stable operator. (ii) Each isolated subsystem (13.2.7) possesses Property A. (iii) For each i,j = 1, ... ,£(i f:. j) there are constants aij 2: 0 such that Ikij (t, 4Jj)I ::; aij Wj (IDj (t)4Ji I)
en
for all 4Ji E j , j = 1, . .. ,£, tEE. (iv) The constants Ci, M, in (ii) and aij in (iii) satisfy the following conditions: There exists a vector aT = (a1,'" ,at) with aj > 0 such that S = [Sij] specified by ai(ci Sij = {
+ Miaij/(1-
~(aiMiaijl(1-
Li(h o)) ,
i = j,
Li(h o)) + ajMjaj;j(l- Lj(h o)) ,
if:. j
is a negative definite matrix. The constant Li(h o) is associated with gj as in (13.2.3b).
Proof: Let V; be given by hypothesis (ii), and let ai, arbitrary constants. Define t
V(t,Xt)
= LaiV;(t,Z;). i=1
i
= 1, ... ,£
be
503
Stability Theory of Large-Scale Hereditary Systems
Because of Property A, there exist u, v, W with the same properties as
Ui, Vi, Wi l
s La,u,(ID'(t)z;D,
u(ID(t)XtD
i=l
l
s Vet, Xt) ~ L ai v,(lIz;ID, i=l
~ v(IIX t ID·
For the second inequality in the derivative of V, we use Cruz and Hale Equation 7.8 of [3, p. 354] as was done by Chukwu (4, p. 354] to evaluate the derivative of V along solutions of (13.2.5). Indeed, l
V(t,xt)
= LaiV(t,Z;), i=l
~
l
l
L aic,wi(IDi(t)z;D + ~aLi. i=l 1 l
L i;t!j Ikij(t, z!)1 'j=l
,
~ L ai[c,w,(IDi(t)z;D i=l
.
M,ai
.
l ] + 1- L. Li;t!ja,jwj(I1Y(t)ztD .
,
[
j-1
Now let R = [rij] be the £ x £ matrix specified by
r, _ { a,[ci + M,ai;/(l - Li)] i
') Then
aiM,a,j/(l- Li),
Vet, Xt) = wT Rw
= i.
i::f j.
= wT (R + R T)/2,
W
= wTSw,
where S = [Sij] is the test matrix in (iv), and where
wT = [w1(ID'z;D,··· ,wl(IDlz:l)]. Because S is symmetric and negative definite, all its eigenvalues are negative, so that for some ,X > 0 l
Vet, xd ~ -'xwT w = -,X L w,(ID i z; D.
,=1
Hence Vet, Xt) is negative definite. Global uniform asymptotic stability follows at once.
504
Stability and Time-Optimal Control of Hereditary Systems
13.3 General Comments The parameter Ci in Definition 13.2.1 is the so-called degree or margin of stability. As observed by Michel and Miller [2] in ordinary differential equations, for the system to satisfy (iv) of Theorem 13.2.1, it is necessary that Ci+Miaii 0 so that (13.2.7) is unstable, we must insist that Miaii < 0 and IMiaii I > Ci. Thus, to ensure stability of the large-scale system when a subsystem is unstable, we must provide a sufficient amount of stabilizing feedback from outside the subsystem. This has very great policy implications, and is now restated as a universal principle. Principle 13.2.1
If a subsystem is misbehaving and unstable, the large-scale system can be made stable provided there is sufficient external feedback (external force) brought to bear on the subsystem to ensure stability. The dismantling of external feedback on a subsystem can create instability. This principle is analo-
gous to those of controllability of large economic systems studied in Section 9.5. REFERENCES 1. E. N. Chukwu, "Uniform asymptotic stability of large scale systems of neutral type,"
IEEE Proceedings of the 18th Southeastern Symposium on System Theory, 1986. 2. A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, New York, 1977. 3. M. A. Cruz and J. K. Hale, "Stability of Functional Differential Equations of Neutral Type," J. Differential Equations 7 (1970) 334-355. 4. E. N. Chukwu, "Global Asymptotic Behaviour of Functional Differential Equations of the Neutral Type," Nonlinear Analysis, Theory, Method and Application I> (1981) 853-872.
Index Controllable with constraints, 33,
A
202-204, 455, 484
Absolute fuel minimum problem, 163 Acquired immunodeficiency syndrome, 12-17
Controller, 368 Control switches, 107, 112 Converge uniformly, 457
(AIDS), 208
Convex, 174, 181, 369
Active immunization, 10, 17
Cost, 27, 32, 156--157, 401, 408
Actuator position, 18, 235
Coulter, K., 153
Adjoint equation, 214-215, 289
D
Adjoint system, 214-215, 402, 406-408, 420,425,
D'Alamberto solution, 23
Affine hull, 293
Damped harmonic oscillator, 1-2, 7
Affine span, 293
Depression, 488
Analytic, 105, 243, 430
Deregulations, 366-367
Analytic matrix function, 207
Determining equations, 194, 216, 249
Arzela-Ascoli Theorem, 69-70
Digital computer, 18-19, 505
Atomic at zero, 47, 56, 102, 380, 418
Dirac functions, 254
Attainable set, 359, 363, 378, 453
Discontinuities, 241 Disturbances, 367
B
Domain of null controllability, 370, 480
Banach space, 360
Dominated Convergence Theorem, 280,
Bang-bang principle, 209,378-379
315, 477
Bang-of-bang, 249, 254
E
Bellman's Inequality, 59 Bounded, 59, 95, 173
Economic interpretation, 495, 508
Bounded linear operator, 414, 429
Economic target, 365-367
Bounded variation, 45, 87
Effort, 156 Eigenvalues, 53-58
c
En-Hao-Yang, 61
Canonical isomorphism, 405, 412
Epidemic, 9-17,207
Capital accumulation, 25-30, 354, 368
Epidemics, 9-17
Cesari's property, 458
Equi-absolutely, 456
Characteristic equation, 53, 54, 55, 56,
Equibounded, 284
57
Equicontinuous, 70, 284, 327
Closed affine hull, 293-294
Essentially bounded, 198
Codimensionality, 482
Etheridge, D., 153
Compact, 69, 361, 377
Euclidean controllability, 193
Compact convex subset, 330, 377
Euclidean controllable, 193-194, 214,
Computational criterion, 382
262
Control, 1-3, 5-6,7-9,17-18,27-30,32
Euclidean null controllable, 203
Controllable, 30, 105, 285, 303
Euclidean reachable, 359, 370
505
506
Index H
Extremal, 211, 213, 381
F Filippov's Lemma, 392 Finite escape times, 450 Flip-Bop, 18 Flip-Bop circuit, 18 Flip-Bop circuits, 489-493 Fluctuation of the current, 6-9, 434 Frechet derivative, 102, 278, 299, 309, 445
Frechet differentiable, 292, 316, 475 Fredholm's alternative, 281
Haddock and Terjeki, 75 Hajek and Krabs, 165 Hajek-Krabs Duality Theorem, 167 Hajek's Lemma, 114 Hamiltonian, 275, 288 Harvesting/seeding strategies, 5, 230 Hausdorff metric, 210 Health policy, 11 Holder inequality, 404 Homeomorphism, 380 Homogeneous system, 35-36
Fubini's Theorem, 358 Fully controllable, 105
Implicit Function Theorem, 281, 448
Functional, 67-72, 89-91, 308
Impulse control, 176, 189, 191
Functlon-space controllable, 207
Impulsive controls, 176, 253
Functional, 67-72, 89-91, 293
Index of the control system, 111, 223,
Function space controllability, 197-200 Fundarnent al lemma of Hajek, 111
432
Individual initiatives, 367
Fundamental matrix, 35-40
Infectious disease, 9--17
Fundamental matrix solution, 36-44,
Information pattern, 389
48-51
Initial endowment, 27-29,355, 366
Fundamental principles, 365 G
Gateaux derivative, 314,318, 405, 474 Gateaux differential, 315, 474, 476 Generalization of Bellman's Inequality, 61
Generalized (delta) functions, 163 Generalized inverse, 292, 299, 328 Global controllability, 483 Global economy, 354 Globally asymptotically stable, 52-u5, 208
Initiatives, 367, 488 Inner product, 79, 83 Integral equations, 16 Interconnected, 339, 349, 486 Interconnected system, 337 Interconnection, 337-338, 340
K Kuhn-Tucker type, 293
L Laplace transform, 49 Large scale systems, 341, 364-365
(globally) null controllable, 363
Lebesgue-Stieltjes sense, 339
Globally null controllable with
Linear differential equation of
constraints, 203, 320
neutral type, 448
Gronwall, 59-ul, 279,448
Linear functional, 463
Growth condition, 322, 337, 343
Linearly independent, 111
Index
507
Local maximum principle, 461 Logistic equation, 100
Lyapunov-Razumikhin type theorem, 76-85
M Maximum principle, 275, 288, 443 Maximum Protoprinciple, 273 Maximum thrust, 159 Measurable function, 107, 222 Metanormal, 173, 258, 276 Metanormality, 174, 258 Michaelis-Menten, 13 Minimal control strategy, 161 Minimal time function, 217, 426--429
o Open loop control, 32 Open map, 304, 360 Open Mapping Theorem, 478 Operator, 121,323 Optimal control, 3, 31-32 Optimal control function, 32, 107 Optimal control strategy, 432 Optimal feedback control, 33, 106-110, 123-126,149,222,430-431 Optimal problem, 32, 134 Optimal switching sequence, 242 Oscillations, 2 Outer product, 425, 429
p
Minimal time functions, 217 Minimum effort, 252
Piecewise analytic, 403, 430
Minimum effort control, 163, 252-254
Point of discontinuity, 107
Minimum effort problem, 156, 160
Pointwise complete, 212-214
Minimum energy control, 265
Pointwise completeness, 388
Minimum fuel controls, 165
Policy, 488
Minimum fuel optimal controls, 169
Positive definite symmetric, 79, 91, 205,
Minimum pseudo-fuel controls, 170
329
Minimum time, 266
Positive semidefinite, 72, 329
Moore Penrose generalized inverse, 343
Predator-prey, 3--5
Mortality, 15
Proper, 365, 373, 386, 417
Multiplier Rule, 295, 463
Pursuit game, 373, 388
Multipliers, 461
N
Q Quadratic cost functions, 443
Negative semidefinite, 81, 205
Quarry, 367
Neutral system, 46-51
Quarry control, 390
Nonlinear, 473
R
Nonsingular, 106 Nonsingularity, 359
Razumikhin function, 75
Nonvoid, 369
Reachable set, 107, 173, 359
Normal, 107,251,262,373,382,387
Redmond and Silverberg, 189
Normality, 385
Regulations, 488
Null controllability, 359
Rest-to-Rest Maneuver, 130
Null controllable, 285-286
Retarded Equations, 35--44
null controllable with constraints, 416'
Riesz Representation Theorem, 47
Indez Rigid body maneuver, 130,180
S Schauder's fixed-point theorem, 324-333,361-363 Secondary infections, 208 Servomechzhsm, 1-3 Simple harmonic oscillator, 108, 177 Singular part, 55 Sobolev space, 31, 193 Solidarity function, 349,366-370,489 Solution operator, 102,221 Square integrable, 105,193 Stability, 33, 52-64 Stable, 67,88-89 uniformly asymptotically stable, 53-57,88,90-94,123,261,434 uniformly, exponentially stable, 54-55, 59, 62,320,483 uniformly stable, 444 Strictly convex, 407 Strictly normal, 111, 238, 240 Strictly normal system, 111, 238 Strict retardations, 201 Stroboscopic strategy, 369,394 Strongly continuous semigroup, 409. 423 Suitable strategy, 368 Surjection, 296,317,319,479 Susceptible population, 208 Switching curve, 181 Switching curves, 138-142 Switching locus, 133, 148-151 Switch time, 107 Symmetric, 173,369 Synthesis of Optimal Control, 106
T Targets, 473 Terminal manifolds, 110, 112,128-134, 246-247
T i m e - o p t i d control, 169,455 Transition matrix, 357 Transmission-line problem, 19,435 Transmission rate, 208
U Ukwu, 188 Universal prinaple, 370 Upper semicontinuity, 217,283, 454 Upper semkontinuous, 217, 453
V Variational equation, 102 Variation of constant formula, 105, 400
W Weakly convergent, 457 Weakly G-differentiable, 478 Wind Tunnel Model, 17-18, 235
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