Differential Equations: Classical to Controlled (Mathematics in Science and Engineering, 162)

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Author: Dahlard L. Lukes

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Differential Equations Classical to Controlled

This is Volume 162 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Differential Equations Classical to Controlled Dahlard L. Lukes Department of Applied Mathematics and Computer Science University of Virginia Charlottesville, Virginia

1982

ACADEMIC PRESS A subsidiary of Harcourt Brace Jovanovich, Publishers

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©

COPYRIGHT 1982, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

•

ACADEMIC PRESS, INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. {LONDON) LTD. 24/28 Oval Road, London NWI

7DX

Library of Congress Cataloging in Publication Data lukes, Dahlard l. Differential eq~ations. (Mathematics in science and engineering) Bibl iography: p. includes index. I. Differential equations, Linear. 2. Differential equations, Nonlinear. 3. Control theory. I. Title. I i. Series.

QA372.L84 515.3'5 ISBN O-12-459980-X

82-6797 AACR2

PRINTED IN THE UNITED STATES OF AMERICA

82 83 84 85

987654321

To my parents, Lawrence and Josephine

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Contents

xi

Preface

Chapter 1 Introduction 1.1. 1.2. 1.3. 1.4.

Chapter 2

Origin and Evolution Sources of First-Order Equations Classical Questions Control Questions

1

4

12 15

Matrix Algebra-The Natural Language of Linear Systems 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

Terminology Addition, Multiplication, and Scalar Multiplication The n-Vector Spaces ?}in Systems of Linear Algebraic Equations Matrix Inversion Large-Scale Matrix Computations Using Computers Determinant Functions Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials 2.9. The Jordan Form of a Matrix

19

20 23

25 29

31 35 39

43

Chapter 3 Linear Constant-Coeffident Equations: Computation 3.1. The Initial-Value Problem 3.2. The ABC Algorithm 3.3. A Vector Method for Computing the Matrix C

47 48

54

viii

Contents 3.4. Explicit Formulas for C(c) and eIl(t) 3.5. Additional Comments about Computing and Changing Variables

Chapter 4

59 60

Linear Constant-Coefficient Equations: Theory 4.1. Preliminary Remarks 4.2. Some Elementary Matrix Analysis 4.3. Existence, Uniqueness, and Solution Representations Using etA. 4.4. The Structure of etA 4.5. The Theoretical Foundation of the ABC Algorithm for Computing etA. 4.6. Explicit Formulas for C(c) and etA.

64 65 72 77 78

83

Chapter 5 Linear Constant-Coefficient Equations: Qualitative Behavior 5.1. Stability and Periodic Oscillation 5.2. The Lyapunov Approach to Asymptotic Stability

92 95

Chapter 6 General Linear Equations with Time-Varying Coefficients 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8.

Widening the Scope Preliminaries The Fundamental Matrix Series Defining ell Existence and Uniqueness of Solutions to Vector Differential Equations and Their Duals Variation-of-Parameters Formulas for Vector and Matrix Differential Equations Estimates of Solution Norms Fundamental Matrices of Equations with Time-Varying Coefficients Continuous Time, Discrete State Markov Processes

106

107 108 112 114 115 128 132

Chapter 7 Commutative Linear Differential Equations 7.1. Introductory Comments 7.2. The Definition and an Example of Commutative Equations

138 139

ix

Contents 7.3. Computation of Closed-Form Solutions to Linear Commutative Equations 7.4. Some Sufficient Conditions for Commutivity 7.5. Matrix Functions and Operations Preserving Commutivity 7.6. The Special Case n = 2 7.7. A Criterion for Exponential Decay of Solutions

140 145 148 157 160

Chapter 8 Periodic Linear Equations 8.1. Periodic Homogeneous Equations 8.2. Qualitative Behavior 8.3. Nonhomogeneous Equations 8.4. Periodic Commutative Equations

Chapter 9

162 167

172

175

Local Existence and Uniqueness Theory of Nonlinear Equations 9.1. Complications That Can Occur 9.2. Local Existence and Uniqueness

Chapter 10 Global Solutions

10.1. Introduction 10.2. Maximal Solutions and Extensions of Local Solutions 10.3. The Behavior of Solutions on Maximal Intervals

181 182

190 190

192

Chapter11 The General Solutio~Dependence of Solutions on Parameters 11.1. Mathematical Preliminaries 11.2. A Space of Curves 11.3. The General Solution (D, 'f) to the Cauchy Problem (D, f) and Continuous Dependence on Parameters 11.4. Differential Dependence of the General Solution (D, 'f) on Parameters and Related Variational Equations

200

202

205

212

Chapter12 Limit Properties of Solutions 12.1. 12.2. 12.3. 12.4. 12.5.

Limit Sets and Invariance Stability of Nonlinear Equations Partial Stability Local Behavior and Linearization about Rest Points Planar Systems

223

231

236

243

250

Contents

x

Chapter 13 Applications in Control Theory 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13.9.

The Controlled and Uncontrolled Equations Transient and Steady-State Parts of the Solution The Frequency Response Function The ABC Approach to Computing Frequency Response Matrices Controllability Altering System Dynamics by Linear Feedback Decoupling and Recoupling-Noninteractive Design Controllability of Nonlinear Equations The Domain C of Null Controllability

261 262 263 266 269 275 284 307 313

References

317

Index

319

Preface

This book deals with the classical theory of differential equations, showing some of the impact that computers and control theory are having on the subject. It is based on notes that I developed in recent years while teaching an advanced-undergraduate beginning-graduate course aimed at applied mathematics and engineering students. These notes succeeded better than the numerous textbooks that I tried in dealing with the somewhat heterogeneous background of my students. All had studied calculus and beginning differential equations and had some familiarity with matrix manipulations. Additional courses in advanced calculus or analysis proved helpful but were not crucial. The book provides a basis for a onesemester course for such students. To complete the text, say in a secondsemester course, the students should first have acquired some knowledge of analysis, linear algebra, and topology. The historical development of differential equations and control theory is sketched in Chapter 1, in which motivation and an overview of the subjects are also provided. Chapter 2 contains a brief review of matrix theory and notions of linear algebra that are drawn on throughout much of the book. It can be skipped and used only as a reference if the instructor wishes to require a course in matrix theory as a prerequisite. On the basis of a heuristic matrix approach, which I have dubbed the ABC method, I introduce in Chapter 3 the computational aspects of linear constant-coefficient systems. The theoretical underpinning and the move to progressively more general equations are found in succeeding chapters, in contrast to the usual approach of treating nonlinear equations first. The loss in efficiency seemed to be outweighed by the pedagogical advantage gained, wherein the applied science students first see the theory building upon real analysis at a less abstract level and only after they are well into xi

xii

Preface

the course. At the same time the more mathematically inclined students get a better picture of where the loss of theoretical detail occurs during the transition to more general equations and more abstract concepts. In recent years, great progress has been made in "the use of computers to solve large systems of linear algebraic equations and to compute eigenvalues. The ABC approach developed in this book reflects my efforts to reduce the problem of generating the analytical (closed-form) solutions to large systems of linear .differential equations to the first problem and hence to exploit this progress . The applications of the ABC formula made here demonstrate its theoretical as well as computational value. The study of the variable coefficient systems that I call commutative equations, which appears in Chapters 7 and 8, grew out of this renewed interest in computing. The theorem concerning linearization of nonlinear equations about a hyperbolic rest point, first studied by Poincare in the context of analytic systems and later settled by Hartman, is given an elementary proof in Chapter 12in order to add further significance to linear equations. In Chapters 9-12, I cover many of the standard topics dealing with nonlinear equations. Dependence of solutions on parameters and associated variational equations are given particular attention. Although Chapter 13 is intended to provide only an introduction to control theory, the material covered is of fundamental importance to anyone with limited opportunities for studying modern control theory. In many respects the feedback theory presented in this chapter goes far beyond the old-fashioned root locus and transfer function methods still being taught in many courses: I believe it is necessary to devote special attention to the computational aspect of the decoupling problem since the abstract theory of the past ten years does not seem to have percolated down into the engineering texts, in spite of the recognized importance of the problem in applications. It should be noted that the sections of Chapter 13 on control theory (1-7) could be covered after finishing Chapters 1-5, and in general the book can be used in several ways. For example, a one-semester introductory level systems-oriented course might cover Chapters 1-6 and 8, and Sections 1-7 of Chapter 13. If students have had matrix theory and some complex function theory, then Chapter 2 could be skimmed over and Chapter 7 inserted into the course coverage. If more emphasis were to be placed on nonlinear equations, using Chapters 1-6,9, and 10 would leave some time to pick and choose from the topics in Chapters 11 and 12. If the text were to be used for a two-semester sequence, the chapters would be covered in the normal order, with the option of skipping Chapter 7. Numerous examples and exercise sections have been inserted at appropriate points throughout the book.

Preface

xiii

I wish both to thank Ruth Nissley for so graciously contributing her energy and professional skill in typing the final manuscript and to express my gratitude to Carolyn Duprey and Faye O'Neil for their help in preparing the earlier drafts.

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Chapter

1

Introduction

This chapter begins with a historical sketch of events that shaped the development of the subject. Its simple examples illustrate the well-established applicability of differential equations. They also provide motivation for the choices of the model equations studied in later chapters and indicate some of the new directions that control ideas have given to the classical theory. 1.1

Origin and Evolution

Since its invention by Isaac Newton circa 1666, the theory of ordinary differential equations has occupied a central position in the development of mathematics. One reason for this is its widespread applicability in the sciences. Another is its natural connectivity with other areas of mathematics. Newton conceived the idea of a gravitational field and concluded that the rate of change of the momentum of a free particle of mass in such a field must equal the gravitational force exerted upon it by that field. His equality is an example of what is now called a differential equation. The necessity for defining the meaning of the rate of change of a non uniformly changing variable prompted his discovery of differential and integral calculus. With these new concepts he was able to explain the tides and place the motion of the moon and planets upon a firm mathematical foundation. The phenomenal success of extensions of his ideas into engineering, physics, mathematics, and other branches of science now has a rich history. The profundity of Newton's original discovery is reflected in the fact that the continually expanding notions of derivative, integral and differential equation still underlie much of modern mathematical research. Following Newton, a number of great mathematicians, including Cauchy, Peano, and Riemann, contributed to what is now referred to as the classical 1

2

1.

Introduction

foundations of differential equations-a major topic of this text. The theory provides an elegant and powerful treatment of existence and uniqueness problems as well as other questions. In effect, it established the logical legitimacy of differential equations as models for scien.tifi.c phenomena.The scientist's concern is thus reduced to selecting the appropriate equations. Control. theory, a relatively modern development, attempts to go a step further. Control phenomena, in contradistinction to control theory, have their primordial origins in nature. The biologist Richard Dawkins regards the gene as a fundamental control element in his view of Darwin's theory of natural selection [6]. He envisions a gene as arising from a special configuration of organic molecules that somehow appeared in the primordial soup with the stability property of attracting sequences of elementary molecules from the environment and then splitting into two copies of the original after a threshold of complexity had been reached-a kind of primitive cell division. Another biologist, Lyle Watson, entertains the idea that these seed molecules might have originated from beyond the earth, possibly from the tails of comets through whose debris the earth passed [26]. At any rate, this replicator soon populated and in the process altered its environment by depleting the supply of elementary molecules. The variability in the replication produced a family of competing replica tor types. Some complexes of replicator molecules stumbled upon adaptive control processes that allowed them to survive the competition and the changing environment. Those discovering reproductive behavior and possessing other strong stability characteristics persisted through" successive generations. Thus, nature advanced toward creating a biological invariant called life. After some 4 billion years, this scenario culminates in the fundamental problem of biology; to determine how the information stored in the DNA of the cell's genes is used in controlling the cellular development of the surviving organisms. Dawkins regards living organisms as elaborate, highly successful survival structures built about their genes. Proceeding from this view, he goes on to offer new explanations for altruism and other unusual forms of behavior exhibited by some species. Watson suggests that man may have reached such a complex state of development that, in some profound manner, the genes' control is linked with and perhaps is superseded by the subconscious mind. Some scientists speculate that man's behavior, institutions, perception of himself, and possibly physical reality itself evolve according to some higher levels of Darwinian natural selection. Whatever the origin and evolution of living things, the present world abounds with organisms, each operating sophisticated control mechanisms. The human body's system for maintaining its temperature within the narrow

1.1 Origin and Evolution

3

range necessary for survival is a good example. Another is its immunological system for identifying and destroying invading bacteria and other foreign substances. Watson discusses the possibility that unstable cell growth is a common occurrence in physiology and that cancer develops when the identification process fails or when for some other reason the stabilizing immunological feedback system malfunctions. Examples of such automatic control systems can easily be multiplied. Civilization has long been concerned with control at various levels of consciousness. Childrearing, education, religion, and politics all involve a high degree of psychological control. The manifold institutions of government-including the maverick education-are highly evolved control devices intended to help manage (but unfortunately often dictating) man's affairs. Thus it appears that control phenomena have pervaded all aspects of life since its beginning and undoubtedly will continue to do so. Only in relatively recent times have engineers finally injected the element of control into their equations. The manner in which this came about is rather interesting. In 1769 James Watt, a Scottish engineer, invented the governor for steam engines. This early example of a feedback control system was a device for maintaining a specific speed of the flywheel of an engine operating under variable load. Before that- time, the applications of the steam engine were quite limited. The Watt governor dramatically transformed the steam engine into a feasible mechanism for delivering steady power and consequently was an important factor in the occurrence of the industrial revolution. Control phenomena attracted very little mathematical interest before 1900, with the exception of J. C. Maxwell's 1868 study of the steady-state error which occurred in applications of the Watt governor [17]. A great period of-industrialization took place from about 1900 up to World War II. It witnessed the development of large-scale power generation and transmission, aeronautics, the chemical industry, communications, and electronic engineering. This activity gave rise to the design of many pneumatic, hydraulic, electromechanical, and electrical regulators. The period, sometimes referred, to as the era of handbook engineering, was narrowly directed toward a few specific applications, relied heavily upon low-order linear models in Laplace transform form, and generally employed rather limited areas of mathematics. This status began to change rapidly during World War II, stimulated by such problems as fire control (which involves the rapid and accurate positioning of guns). The great advances in electronics, followed by the emergence of analog and digital computers, greatly expanded the horizons of control. The upgrading of the mathematical training of engineers was another

4

1.

Introduction

important factor involved in the transformation of control engineering into a science. These engineers were better prepared to define problems mathematically and to develop new concepts of control. This is tum allowed mathematicians to bring to bear upon the problems the power of abstract mathematics. Control theory is now regarded as one of the major areas of mathematics. Increasing numbers of scientists and laymen are recognizing the controltheoretic aspect of many of the serious problems of mankind that arise from the expansion and growing complexity of the world's evolving systems with their seemingly inextricable coupling. A major difficulty is that the actual dynamics and coupling of these systems are often poorly understood and sometimes ignored for economic, political, or other reasons. One important example of this is the problem of controlling insect damage to crops [1 J. It is estimated that in 1977 at least 250 million pounds of pesticides were dumped on California alone in an effort to combat insect damage to farm crops. (The complexity of the problem is indicated by the fact that in spite of these chemicals there remain 70 major insect pests that plague corn.) With Darwin's natural selection operating as predicted, the intended victims often develop resistant varieties, and these survivors are even more difficult to control. Moreover, the chemicals often create harmful imbalances in the ecosystem that trigger other infestations, contaminate drinking water, and generate many other problems. Again, the examples can be multiplied. The critical nature of these problems sorely awaits applications of control science outside classical areas of engineering. During the past 15 years, a great deal has been learned about the mathematical control of finite-dimensional systems, although the theory is far from complete. Currently there is considerable interest in the control theory of systems with time-delay and in systems modeled by partial differential equations (see [20J). Many of these problems can be treated by the theory of functional analysis as ordinary differential equations in infinite-dimensional (function) spaces. In this book, we treat primarily the classical theory of differential equations at the more modest finite-dimensional level where, ordinarily, more detailed results are available. This provides a firm foundation for more advanced study of the control aspects of such equations for those wishing to pursue the subject. 1.2 Sources of First-Order Equations

Although only limited space can be devoted to discussing practical applications in this book, some feeling for simple kinds of problem situations

5

1.2 Sources of First-Order Equations

-

rl rl

Tank

u(t)1 - -

I

rz

Tank 3

Tank 2

f---

I--

rl

Fig. 1.2.1 A heat-flow system with input.

which can be modeled by differential equations is desirable. The following examples serve that purpose.

Example 1.2.1 Flow and Mixing. Suppose that water is pumped between three thermally insulated tanks through connecting pipes at rates '1 = 1, '2 = 2 gallons per minute, configured as in Fig. 1.2.1. Initially each tank contains 100 gallons of water at specified temperatures. The temperatures of the water in the tanks change with time due to the pumping between tanks. Tank 1 has a heat exchanger which allows heat to be injected or extracted directly at a rate of u(t) British thermal units (Btu) per minute. Thus u(t) is regarded as an input to the system and the output of interest is the temperatures of the water in the tanks. Since the temperature of a gallon of water is proportional to the heat that it contains, it is sufficient to work with xlt), the amount of heat in the ith tank at time t (i = 1, 2, 3). By taking into account the heat flow both into and out of each tank and using the fact that the heat flow rate through a pipe equals the water flow rate times the heat per unit volume of water, it is easy to see that the appropriate system of differential equations satisfied by the x;(t) (assuming the heat loss and the contents of the pipes to be negligible) is

2[x100 + u(t), 2(t)] dX2(t) = l[x (t) ]_l[x100 100 ' dt dX3(t) = l[x (t)] 1[x2(t)] _ 2[x 3(t)] 100 + 100 100 . dt

dx 1(t) . dt

= 2[x 3(t)] _ 100 1

1(t)]

(1.2.1)

1

Letting x(t) denote a column-vector variable with scalar coordinate variables X2(t), X3(t), system (1.2.1) can be rewritten as a single vector-matrix

Xl (t),

1. Introduction

6

differential equation

x=

Ax

+

f(t), where

- lo A=

f(t) =

[

1

0

50

1

TOO

-TOO

1 100

1

TOO

o1 ] , --10

U(t)] [~ .

(1.2.2)

(1.2.3)

The same model would apply to other mixing problems such as the flow of pollutants in a water supply system, with the tanks being lakes or reservoirs. Example 1.2.2 Mechanics. Consider the forced spring-mass system with friction indicated in Fig. 1.2.2. The mass m moves in response to the spring force - k 2 x generated by displacement of the mass x units from equilibrium, friction force - k 1 x, and external driving force u(t). The k 1 and k 2 are nonnegative constants of proportionality associated with the assumed standard models of springs and friction. Application of Newton's law of motion leads directly to the second-order scalar equation

(1.2.4) By introducing variables Xl = x and X 2 = dx.ldt, Eq. (1.2.4) can be transformed into the equivalent first-order system (1.2.5) This system can likewise be written in the vector-matrix form x = Ax

u(t)

Fig. 1.2.2

A forced spring-mass system with friction.

+ f(t),

1.2

7

Sources of First-Order Equations

where (1.2.6) (1.2.7) Example 1.2.3 Electrical Circuits. Differential equations are often used for modeling electrical circuits. Recall that the voltage drop across an idealized resistor, inductor, or capacitor is vR = Ri, vL = L dildt, or uc = (1/C) i dt, respectively, where R, L, and C are physical constants and i(t)is the current passing through the circuit, Thus application of Kirchhoff s law, which says that the total voltage drop around a simple closed circuit must be zero, shows that the current i ( t ) in the circuit of Fig. 1.2.3, containing a timevarying voltage source e(t), must satisfy the-equation di Ldt

+ Ri +

AJ:

i(z)d z = e(t)

(1.2.8)

(assuming the charge on the capacitor to be zero at t = 0). Introducing the variables x 1 = i, x 2 = di/dt, and u = (1/L) deldt, the differentiated form of (1.2.8) transforms into the first-order system (1.2.9)

which once more has the vector-matrix form f

=

0

Ax

+ f (t),this time with (1.2.10) (1.2.11)

More complicated forms of Examples 1.2.1-1.2.3 can easily be obtained by R

C

I

Fig. 1.2.3 A-imple circuit.

8

1. Introduction

considering more tanks, springs and masses, or circuit elements connected in countless complex patterns. All the resultant systems of differential equations as well as the equations arising from many other areas of application are subsumed by the general equation of the form

x=

Ax

+ f(t),

(1.2.12)

in which x is a column-vector of n real variables, A is an n x n real matrix, and f(t) is a prescribed column-vector of n real-valued functions of t. In Chapters 3- 5, we develop the classical theory of linear differential equations with constant-coefficient matrix as given by (1.2.12). Obviously a substantial amount of matrix algebra is involved in such a treatment and this is anticipated by the review of matrix algebra that is presented in Chapter 2. The classical theory of (1.2.12) presented will be found to be in a rather satisfactory state of completion. It provides, amongst other results, an effective method for computing all solutions to (1.2.12) in a closed form. A moment of reflection makes it clear that only slight changes in the physical problems underlying Examples 1.2.1-1.2.3 produce systems of equations requiring a model more general than (1.2.12). For example, suppose that the value of the flow rate rl of the system presented in Example 1.2.1 were changed from r 1 = 1 to r 1 = 2. It is easy to check that this would require replacement of the real matrix (1.2.2) in the model equation by the matrix function of t, . [- 2/(50 - t) 1/(50 - r) 1/(50 - t)

0

-lo

1/(50 + t) ] 0 . 1/(50 + t)

to -

(1.2.13)

Clearly, something as simple as an error in a pumping rate or valve setting could turn the original system into one with time-dependent coefficients. One can as well envision the situation for which in Example 1.2.2 heat generated from the mass sliding along the surface or from other sources causes the' coefficient of friction k 1 to change with time. Likewise the resistor in Example 1.2.3 might heat up with the operation of the circuit and thereby result in a time-dependent value of resistance R. Again, these time-dependent effects would appear in the matrix coefficient of (1.2.12) and thus it would seem appropriate to expand the theory to cover the more general equation

x=

A(t)x

+ f(t),

(1.2.14)

in which the coefficient A(t) is an n x n real matrix-valued function of t. This indeed is on the program. It turns out that important questions concerning the behavior of many nonlinear differential equations as well can be studied

9

1.2 Sources of First-Order Equations

through associated (variational) linear time-dependent equations. This provides additional cause for interest in equations of type (1.2.14). Differential equations of the form (1.2.12) and, more generally, (1.2.14) are called linear first-order equations. These equations are sufficiently general to cover many important applications. However there are problems where they are not appropriate. For example, in the study of large-amplitude oscillations of the system in Example 1.2.2, some springs might be better modeled with the spring force - k2 x replaced by - k2 x 3 • The resultant differential equations replacing (1.2.5) would. then be (1.2.15) which is no longer of the linear type (1.2.14). Similarly, if the smoothness of the surface upon which the mass slides were not uniform, then the model of the friction force might require alteration; e.g., - k 1 Xmight be replaced by -k 1x2 X. This too would destroy the linearity. The same kinds of remarks apply to Example 1.2.3. In fact the famous Van der Pol equation .2 d2i dt 2 + (I

-

di.

1) dt

+1=

u(t)

(1.2.16)

arose from attempts to account for self-excited electrical oscillations in circuits by assuming the circuit resistance dependent upon the current. The first-order form of this equation, of course, is again not linear. Examples of such nonlinearities led researchers to the study of equations of the type

d2x dt2

dx

+ f(x) dt + g(x) = u(t)

(1.2.17)

called Lienard equations and, more generally, 2x

d dt2

+f

(dX) dx x, dt dt + g(x) = u(t),

(1.2.18)

the Levinson-Smith equation, in which the coefficient function f is allowed to depend on the derivative dxldt as well as on x. Discussion of these and other nonlinear equations will reappear in later chapters. For some applied problems a linear or slightly altered (perturbed) linear model is altogether inappropriate; e.g., long-term motion of bodies in gravitational fields (Newton's equations were nonlinear) or population models. In the final design stage of engineering it is frequently necessary to include numerous nonlinear terms in the simulations in order to fix the hardware specifications accurately.

10

1.

Introduction

Fortunately a substantial part of the classical theory applies to equations of the very general form dx (1.2.19) dt = f(t, x), with only rather mild assumptions imposed on f; for example, that f be a real n vector-valued function satisfying some continuity or differentiability condition. The precise details and consequences of the nonlinearities will emerge in later chapters. EXERCISES

1. Rewrite the following equations in first-order form, identifying the matrix coefficient and forcing term of the form (1.2.14) for those cases in which the system is linear. (a) x + f(x) X + g(x) = u(t), (b) x + f(x, x) x + g(x) = u(t), (c) x + (sin t) x + 2x = u(t), (d) {m 1 ~1 + k 3 ~1 + (k 1 + k 2 ) X l - k 2 X2 = ul(t),

m2 X 2 + k 3 X 2 - k 2X l + k2X2 = U2(t). 2. Find a scalar differential equation whose first-order form is x = Ax f(t) in which

+

y const.

Verify that the matrix function A(t) given by (1.2.13) is correct. Show that if x(t) is a solution to the homogeneous vector-matrix equation x = Ax, where A is an n x n constant matrix, then J& x(t) dt = 0 implies that x(T) = x(O). (Although its proof requires a result not yet derived, the stronger conclusion that x(t + T) = x(t) for all t is valid, i.e., x(t) is periodic . of period T.) 5. A small drop of oil is placed at the rest point of the mass in Fig. 1.2.2. Which of the following alterations of (1.2.4) best reflects that phenomenon? (a) m x + 2k l X + k2 x = u(t), X2 (b) m x + k l (1 - e- ) x + k 2 x = u(t), (c) m x - k l X + k 2 x = u(t), (d) m x + k 1 X + k 2 x 3 = u(t). 6. Derive the differential equations model for the forced spring-mass system indicated in Fig. 1.2.4, letting Xl and X2 denote the displacement of 3. 4.

11

1.2 Sources of First-Order Equations

k3

Fig. 1.2.4 A double forced spring-mass system with friction.

the respective masses from equilibrium and using linear modeling as in Example 1.2.2. 7. Three tanks are connected by pipes through which salt brines flow with xi(t)denoting the number of pounds of salt in tank i ( i = 1,2,3) at time t . The variables u1 and u2 are pure salt flow rates. Determine the configuration of pipes, flow rates, and volume of water initially in each tank that would be modeled by the equations

What is the significance of the negative coefficient of u2 in the equation for i2? 8. Derive the differential equation model for the currents i l ( t ) and i z ( t ) passing through the respective resistors in the electrical circuit indicated in Fig. 1.2.5. 9. The populations x1and x2 of two predator-prey species of fish occupying a lake are modeled by the nonlinear differential equations i 1

=

~ 1 x 1

klXlxz,

i 2

=~

2 x 2 k2XIX2,

in which p l , p z , k l , k2 are positive constants. Discuss the possible interpretation of the terms in the equations in context of reproduction and losses due to predation. Think of situations in which alternative models of birth and predation might be appropriate.

Fig. 1.2.5 A two-stage electrical network.

12

1. Introduction

1.3 Classical Questions

A major part of the classical theory is concerned with the pair of equations

dx

dt = f(t, x),

X(t) = ~,

(1.3.1)

called the Cauchy or initial-value problem. Roughly speaking, the function f and parameters r, f are presumed given, with a solution x( .) being any continuous function defined on an open interval that contains t and satisfying (1.3.1) on that interval. In the text, we deal with the existence and uniqueness of solutions to (1.3.1) and study their dependence on the parameters r and ~. The general results obtained are particularly significant since, with the exception of the special equations of type (1.2.12), it is usually impossible to represent solutions to such problems in terms of elementary functions. In Chapter 3 we present a method (the ABC method) for constructing an explicit representation of the solution to the initial-value problem associated with (1.2.12) using only the operations of finding roots of a polynomial (i.e., the eigenvalues of A), solving systems of linear algebraic equations, and performing elementary integration. Even for this simplest class of equations, an independent existence proof is needed to substantiate critical steps in the algebraic construction. In applications of differential equations, any information about the qualitative behavior of the solutions is usually extremely valuable. For equations of type (1.2.12), a rather complete picture is presented in Chapter 3. The solution is shown to be the sum of a forced solution and the unforced solution. In particular, it turns out that for a fixed matrix A, a coordinate function Xi(t) of the unforced solution x(t) to (1.2.12) can exhibit one of the forms indicated in Fig. 1.3.1 (neglecting transients), and computations on A will predict which can be expected. In applications in which xi(t) denotes the difference (error) between the actual value and the ideal value of some variable in an operating system, forms (a) or (c), having decreasing amplitude, are usually the most desirable, with the periodic form (e)sometimes tolerable but unstable forms (b) and (d) assuredly disastrous. The theory also provides estimates of the rate of attenuation of (a) and (c) as well as information concerning the frequencies of oscillation that might occur. Another property of the solutions that is of importance in applications is the extent of the coupling between the coordinate variables of the solution; i.e., the extent to which a change in initial values of some of the xi(t) willaffect the others. (For example, a well-designed automatic control system for an airplane would not allow one

13

1.3 Classical Questions

I

(el

Fig. 1.3.1 Qualitative response forms of unforced solutions.

orienting variable to respond to a disturbance in another.) Here again the theory is helpful. The computational and' qualitative theory of the time-varying equation (1.2.14) is less complete, as might be expected due to the increased generality of the equation. Although it turns out that a certain finite dimensionality of the solutions persists, the variety of possible qualitative behavior is so rich that it is difficult to select any one set of properties upon which to develop the theory and it appears impossible to derive a comprehensive notion of closedform solution that approaches the relative simplicity of that of constantcoefficient equations. Specifications arising from applications provide some insight into which properties might be most important, but presently that source likewise appears overwhelmed by the possibilities. It is clear that any results which will predict periodicity in the solutions or detect the stability property that x ( t ) 0 as t 00 are of utmost interest. Some rather detailed results concerning the structure and qualitative behavior of the solutions are provided in this book for the special cases in which A@)is periodic or satisfies a certain commutivity property. In applications it is often necessary to resort --+

--f

14

1. Introduction

to numerical integration or computer simulation of the equations to obtain an adequate understanding of the behavior of the solutions. The broader generalization from linear time-varying equations to nonlinear equations (1.3.1) creates much greater difficulties. Even for autonomous equations for which f in (1.3.1) is a function of the variable x alone, the finite dimensionality disappears and new complications arise. One of these is what we call the immortality problem; that is, the phenomenon in which even though f in (1.3.1) is defined for all values of its argument, the solutions to the initial-value problem can fail to exist for t outside a finite interval containing r. An example would be the nonlinear equation (1.3.1) associated with the second-order nonlinear equation ii

+ uu

- u 3 = 0,

(1.3.2)

which has a solution u(t) = 1/(t - 1) that cannot be extended beyond the interval - 00 < t < 1. In spite of the incomplete state of the theory, later chapters of this book do contain some often useful results dealing with the immortality problem and other aspects of qualitative behavior. These results are largely based upon perturbation techniques, differential inequalitites, and linear variational equations. In passing, it should be mentioned that modern mathematicians have made serious attempts to deal with the question of what form the qualitative theory should take when the equations are generalized beyond the linear constantcoefficient type modeled by (1.2.12). One idea considered by Smale [24] and others is to define some choice of equivalence relation on, roughly speaking, all differential equations on some region of (t, x)-space (more generally, on a manifold) which in some sense preserves the qualitative behavior of the solutions. Next some choice of a subset of differential equations is made, satisfying two conditions: The subset must be sufficiently rich to provide approximations to all differential equations, on one hand, and, on the other hand, if a differential equation is approximated sufficiently accurately by another from the prescribed set, then both will be in the same equivalence class. Ideally, all the choices would be made so that the equivalence classes could be distinguished by numerical and algebraic invariants. This would be a reasonable compromise with the unrealistic goal of trying to in some sense solve and completely understand the behavior of the solutions to all differential equations. Thus far, such a program has not proved to be of much practical value in studying specific types of nonlinear equations. To a considerable degree the technical difficulties encountered in the study of nonlinear equations have turned the attention of many engineers and mathematicians toward new questions about linear equations, namely, toward the control theory of linear systems. Needless to say, nonlinear equations reappear as an obstacle in control theory.

15

1.4 Control Questions

I

--J----..

s

u - -........

... x

Fig. 1.4.1 An input-output system.

1.4 Control Questions

A natural setting for the application of control theory is any situation involving one or more input-output systems, as depicted in Fig. 1.4.1, involving input or control variable u and output or state variable x. The intuitive notion of an input-output system is quite general and ordinarily applies to any phenomenon, real or imagined, which conforms to a causeeffect relationship. Examples would include problems modeled by difference, algebraic, ordinary differential, partial differential, or functional-differential equations. There is no need for a formal definition here, since the example to be discussed shortly will serve the purpose adequately. Control theory deals with the problem of connecting compatible inputoutput systems to create new input-output systems satisfying some criterion of performance or otherwise directed toward some goal. As discussed in earlier sections, nature seems to be highly successful in this process. To illustrate the idea, consider two input-output systems Sl and S2 of compatible types, connected as indicated in Fig. 1.4.2. The resultant system S3 with input variable U3 and output X 3 is described as the consequence of wrapping a feedback system S2 about Sl' An example of why and how this might happen will now be discussed. The control-theoretic nature of Examples 1.2.1-1.2.3 becomes apparent under closer examination of the term f(t) in their differential equation models. In Example 1.2.1,

J

U(t)l

[ ~

f(t) =

= u(t)

[1]

(1.4.1)

~ .

The differential equations (1.2.1) can be written as

d [Xl]

dt

X2 X3

=

[-lo °:~o lo0-L][Xl] + 1~0 -

100100

X2

-50

X3

[1]

u(t) 0 . 0

(1.4.2)

Here the scalar control or input function u(t) provides an opportunity to influence the vector output or state vector variable X as it evolves with time. Observe that the value of the control function u(t) at time t directly affects the

16

1-------,

"'1 C

1"'·1

5,

5,

L

1. Introduction

hr" J

Fig. 1.4.2 Connecting input-output systems using feedback.

Xl

rate of change of (t), whereas the numbers in the coefficient matrix A direct the manner in which this control influence on Xl (t) indirectly feeds into the other state variables. Control theory attempts to illuminate the consequences of the manner in which the control variables enter the equations and their capabilities for altering the uncontrolled dynamics of the system. It tends to focus upon a limited number of mathematical problems abstracted from the study of applied problems. Undoubtedly other important concepts and fundamental problems will emerge as the field of applications expands. It will be sufficient, in this introductory chapter, to mention a few which occur in the continued discussion of Example 1.2.1. One fruitful concept is that of system controllability. Roughly speaking, the idea is to assess those output states which can be achieved by applying all allowed inputs. For Example 1.2.1, a typical controllability question would ask whether, over a prescribed finite interval of operation to ::s;; t ::s;; t l and for arbitrarily prescribed initial and final temperature values of water in each tank, there exists some appropriate control input function u(t) which will drive the initial temperatures at time to to their final prescribed values at time t i - It turns out that if the only constraint upon u(t) is that it be continuous, then it is in fact always possible, whatever the values of to < t 1 and prescribed temperatures, although this is not at all obvious. Similar questions could be asked concerning Examples 1.2.2 and 1.2.3, whose respective control differential equations are

[Xl] [0 1 ][Xl] + u()t[OJi ' -k /m -kdm d [Xl] [0 1 J[XlJ +ut()[OJLl' dt = -1/LC -R/L

d dt

X2

X2

=

2

X2

X2

(1.4.3) (1.4.4)

Again the answer is in the affirmative, and thus the mass in Fig. 1.2.2 can be driven from any state of position and velocity to any other such state over any

17

1.4 Control Questions

finite time interval by appropriate choice of input function u(t). From the similarity between (1.4.3) and (1.4.4) it is clear that the result translates into a corresponding valid statement about the driven circuit of Fig. 1.2.3. Other examples complicated by more state variables and multiple control inputs are easily imagined. Obviously the controllability can fail for those systems involving subsystems that are totally disconnected from inputs and controllable state variables, and in more subtle ways whose discussion is deferred to later chapters. The concept of controllability has strong mathematical appeal and obvious utility in mapping out an upper limit of the extent to which a system might be controlled. Interest in the phenomenon of an even more compelling sort has been stimulated through discovery of its remarkable implications concerning other important aspects of control. Stabilizability is one such notion, and automatic regulation is another. The general meaning of these terms is most easily conveyed by further discussion of Example 1.2.1. Intuition suggests that if the input u(t) into the system of Fig. 1.2.1 eventually became and remained zero, then the heat in the system would tend to a uniform distribution due to the circulating flow. Hence the water in each tank would approach the same limiting temperature

Too = _1_ [x? 300e

+ x~ + x~ + roo u(a) daJ Jo

(1.4.5)

as t - 00, in which e is the specific heat of water in the compatible units of calories per gallon per degree. Suppose that the intended purpose of the system were to drive the temperature of the water in the tanks to some prescribed value Too' If the initial temperatures were measured, the values of x? (i = 1, 2, 3) would be known, and then upon assuming some particular form of u(t) (such as constant for an appropriate initial interval and then forever zero), Eq. (1.4.5) could be employed to solve for the appropriate u(t). Each choice of Too and each "run" of the system would require recalculation of u(t) and an operator to turn the heat-exchanger on and then off. This raises the question of whether it might be possible to design a (regulator) device that would sense the tank temperatures and operate upon these along with a prescribed temperature command to drive the temperatures toward the prescribed value automatically. A simple form of regulator that might be considered would be of the linear type (1.4.6) in which the k, (i

= 1,2,3) are parameters to be determined and

Uc

is a

18

1. Introduction

constant input command. Thus the control problem is transformed into the study of the system with parameters kl> k 2 , k 3 , ue

~ [Xl] _ [- (k~ dt

X2 X3

-

100 1

TOO

lo)

3] [Xl] ~k12 fa -k 0 100 1

TOO

X2

-

1 50

X3

+

Ue

[1]0

(1.4.7)

0

resulting from substitution of (1.4.6) into (1.4.2). This equation is a concrete example of an input-output system with input Ue and output X = (Xl' X2' X3) obtained by wrapping a feedback system around the system of Example 1.2.1 in order to solve a control problem. It fits into the abstract diagram of Fig. 1.4.2 with U3 = Ue , 8 1 the input-output system of Fig. 1.2.1,and 8 2 the affine operator defined by Eq. (1.4.6). The stabilization problem is that of determining which values of the parameters k, (i = 1,2,3) in (l.4.7) will ensure that each x;(t) approaches a limit as t --+ 00 for every initial value X O of x. One elegant result of control theory states that the possibility for stabilization is ensured by the system's controllability. Hence the search for the appropriate values of the k;s is not futile. This is but one of many interesting relationships between various concepts of control theory. The question of how the limiting values ofthe X; depend on the kis and Ue is also of vital interest. For example, if the regulator were intended to drive all tank temperatures to the same prescribed value automatically, then it would be desirable to arrange for the limiting temperature to be some scalar multiple of U e if possible. The qualitative behavior of the responses of the feedback system to the initial X O and Ue , as well as certain quantitative properties such as the rate of convergence, also would likely bear upon the final choice of the values of the k;s in completion of the design. It is hoped that this discussion of the examples has given the reader at least a glimpse of the manner in which control problems can lead to interesting questions about differential equations. It should also be evident that a firm foundation in the classical theory of differential equations is a logical prerequisite for the pursuit of such questions. Hence, in Chapter 2 we shall begin the study of the classical theory by reviewing some basic matrix algebra.

Chapter

2

Matrix Algebra-The Natural Language of Linear Systems

The invention of vectors and matrices arose from the need to manage large numbers of variables and equations efficiently. Although eventually they must be pursued to a somewhat higher level of abstraction, the basic ideas are quite visible in the problem of solving systems of linear algebraic equations. That problem is taken as the starting point for a rapid review of the standard theory discussed in the context of finite-dimensional real and complex number spaces. This chapter paves the way for the elementary treatment oflinear systems of differential equations with constant coefficients that appears in Chapters 3 and 4. It also provides a convenient intermediate point from which to complete the transition to the abstract axiomatic view of vector spaces as its need begins to materialize at the end of Chapter 4. 2.1 Terminology

An m x n matrix A over afield!F is defined as any function with domain {(i,j)ll ~ i ~ m, 1 ~j ~ n} and range in a field fli. In this book, the fields

employed in the discussions of differential equations are all subfields of the complex number field re, i.e., subsets of re closed under the operations a - b and ab - 1, b :1= 0 on its elements a, b. The most frequently occurring examples are fli = re or the field of real numbers !F = fJl. Elements of fli are called scalars.

The standard notational convention denotes the value of A at (i,j) by A;j' called the (i,j)th element of A. When the elements of A are arranged in the 19

20

2. Matrix Algebra-The Natural Language of Linear Systems

rectangular array

(2.1.1) A m2· · ·

the index (variable) i is interpreted as a row position and j as the column position. Conversely, every such array offield elements determines a matrix A over IF. If m = n, the A is called square. The positions in A occupied by Au (i = 1, 2, ... , n) are referred to as the main diagonal. A square matrix is called a diagonal matrix if all its elements off the main diagonal are zero, and a diagonal matrix A with all A ii = c is called a scalar matrix, which is identified with the scalar c in IF. An upper-triangular matrix is one whose elements Aij satisfy Aij = 0 for i > j, and a lower-triangular matrix is one satisfying that equality for i < j. 2.2

Addition, Multiplication, and Scalar Multiplication

Problems of working with large numbers of scalar variables and numerical data motivated development of matrix algebra as an extension of the one-dimensional scalar case. The following are the standard operations which arose:

Definition Let A and B be m x n matrices over a field IF. The matrix sum + B is the m x n matrix defined by

A

(A

+

B)ij = Aij

+

Bij'

(2.2.1)

It is a simple matter to verify the following consequences of definition (2.2.1): A A

+ (B

+B = + C) =

+ A; (A + B) + C B

(2.2.2) (2.2.3)

for all m x n matrices A, B, C over IF; A+Z=A

(2.2.4)

has a unique solution for each m x n A, namely, Zij = 0 for all i.j; Z is called the m x n zero matrix and is denoted by 0; and A+N=O

(2.2.5)

has a unique solution N for each m x n matrix A, namely, N i j = - A ij for all i,j; N is called the additive inverse of A. The standard notation for N is - A.

21

2.1 Addition, Multiplication, and Scalar Multiplication

Definition A second operation on an m x n matrix A over f/i called scalar multiplication by c in f/i is defined by

for all

(CA)ij = cA ij

(2.2.6)

i,j.

As a simple consequence of the definitions it follows that (cIC2)A

=

(2.2.7)

CI(c2A),

+ c2)A = cl(A + B) =

cIA

(CI

cIA

+ C2 A, + c .B,

(2.2.8) (2.2.9) (2.2.10)

lA = A

for all m x n matrices A, B over f/i and scalar

Cl' C2

in f/i.

Definition Let A and B be m x rand r x n matrices, respectively, over f/i. The matrix product AB is the m x n matrix defined by (AB)ij

=

r

I

k=1

(2.2.11)

AikB kj

= 1,2, .. 0' m;j = 1,2,. 0" n). Note the compatibility requirement, namely, that A must have the same number of columns as B has rows.

(i

The appropriateness of the definitions of matrix addition and multiplication for producing extensions of familiar algebraic operations on scalar variables to matrices is supported by the following consequent properties: (2.2.12)

(AB)C = A(BC),

+ B)C = C(A + B) =

(A

AC CA

+ BC, + CB

(2.2.13) (2.2.14)

for all matrices over f/i of appropriately compatible sizes. The manner in which algebraic properties of f/i extend to matrices via definitions (2.2.1) and (2.2.11) is illuminated by the following calculation: [(A

+ B)C] ij =

I

(A

k=l

+ B)ikCkj

r

=

I

k=l

(A ik

+ Bik)Ckj

r

=

I

k=l

= (AC

AikCkj

+

+ BC)ij'

I

k=1

=

I

k=1

BikC kj

(AikCkj

=

(AC)ij

+ BikCkj) + (BC)ij (2.2.15)

22

2. Matrix Algebra- The Natural language of linear Systems

Since (2.2.15) holds for all i, j, obviously the corresponding elements of (A + B)C and AC + BC are the same elements of:F and (2.2.13) is thereby proved. There is a matrix generalization of the multiplicative identity 1 in :F. That is, (2.2.16)

IA = A = AI

has a unique n x n matrix solution I over:F holding for all n x n A over :F; namely, (I)ij =. bijwhere bij takes on the value 1 in :F when i = j and the value 0 in :F when i '" j. I is called the n x n (multiplicative) identity matrix over:F and is denoted by In when its size needs to be emphasized. For n x n matrices A over:F and c in:F, cl; is a scalar matrix and (2.2.17) Thus it is natural to identify each c in:F with the matrix cl.; Scalar matrices commute, under multiplication, with all square matrices of corresponding size and, more generally, for all scalars c in :F, (2.2.18)

(cA)B = A(cB)

for all matrices A, B over :F of compatible size. Thus the definitions of matrix addition, multiplication, and scalar multiplication provide extensions of many of the formal rules of algebra in :F to matrices over :F. A word of warning: Even with square matrices, not all the formal manipulations and inferences of elementary algebra carry over to matrices. For example,

(2.2.19)

AB", BA

for some square matrices A and B; and AB = 0

(2.2.20)

does not imply that at least one of A, B is the zero matrix. Hence while a factorization such as (2.2.21) is valid for all n x n matrices A over gj, the factorization A2

-

B 2 = (A - B)(A

+ B)

(2.2.22)

fails for matrices satisfying (2.2.19), and it would be erroneous to conclude

23

2.3 The n-Vector Spaces 9'"

from the equation (2.2.23) that A = ±2In • EXERCISES

1.. Prove the following matrix identities directly from the definitions of matrix sum, product, and scalar product: (a) A + B == B + A, (c) A(BC) = (AB)C, (e) (- c)A = - (cA),

(g) cO

(b) (d) (f)

= O.

A + (B + C) = (A + B) C(A + B) = CA + CB,

+ C,

OA = 0,

for matrices A, B, C and scalars c in fF. 2.. A nonstandard product A * B of an r x n matrix A with an m x r matrix B, both over a field fF, is defined by (A

* B)ij =

r

L AkjB ik· k=l

(a) Do the identities corresponding to (c) and (d) of Exercise 1 hold for the nonstandard product? (b) Solve the equations X * A = A and A * X = A for the unknown n x n matrix X with A likewise n x n. (c) Find matrices A, B for which A * B i= AB. (d) Find an example 'for which A * B i= B * A. 3. Find an example of a square matrix A for which A 4 i= 0 but AS = O. 4. Evaluate (A - 2B)2 - A 2 + 4AB - 4B 2 in which A =

101]

[o -1

1 2, 1 3

B=

[~

-1

~ ~].

1 0

2.3 The n-Vector Spaces 17-

The n x 1 matrices over a field fF are called n-uectors over fF, with the collection of all such n-vectors being denoted by fFn. As matrices, the elements of fFn can be added and multiplied by elements of fF to again produce n-vectors in fFn. These two operations thus conform to Eqs. (2.2.1)(2.2.10). fFn together with the two operations is called the n-oector space

24

2. Matrix Algebra-The Natural Language of linear Systems

over :IF. Often n-vectors are called column-vectors or simply vectors. The superfluous column index 1 is dropped, and thus the equations defining addition and scalar multiplication of n-vectors are written as

Xl] [Yl] [Xl +-Yl] X2 Y2 X + Y2 ·· + .. = 2 .. , . . [· X" Yn X" + Y" C

Xl] X2

.. •

[

Xn

=

[CXl] CX2 .

..

,

(2.3.1)

(2.3.2)

CX"

respectively, in which Xi' Yi' and c are all elements of:IF (i = 1,2, ... , n). If v1 , V 2' ... , Vm are n-vectors over :IF and c l' C2, ... , Cm are scalars in :IF, then the vector ClV l + C2V2 + .. , + CmV mis unambiguously defined in:IF" (since matrix addition was found to be associative) and is called a (finite) linear combination ofvl' V 2, ••• , V m with respective coefficients Cl' c 2 , ••• , Cm • A subspace of :IF" is any nonempty subset of :IF" containing all linear combinations of vectors taken from the subset. (The subset is said to be closed under linear combinations.) A subset!/' of distinct vectors in .'F" is called linearly independent if the only linear combinations of distinct elements taken from!/' that yield the zero vector are those with all coefficientszero. If!/' is not linearly independent, then it is called linearly. dependent. If !/' is a subset of :IF", then the collection of all linear combinations of vectors taken from!/' is called the span of !/' and is denoted by span(!/'). Note that if !/' is not empty, then span(!/') is a subspace of :IF". Let!/' be a subspace of :lFn • A subset of!/' is called a basis for!/' if it is linearly independent and spans !/'. A basis for!/' is called an ordered basis if its elements have been placed in a one-to-one correspondence with a collection of consecutive positive integers, including the integer 1. Important facts concerning the above concepts in :IF" are summarized by the following exercises. EXERCISES

Prove the following statements: 1. A collection {Vl' V2" .., vm} of distinct vectors in:IF" is linearly dependent ifand only ifat least one of the vectors in the collection is a linear combination of the others.

25

2.4 Systems of linear Algebraic Equations

2. If g is a subset of distinct vectors in a subspace of IF n , then it is a basis for the subspace if and only if each vector in the subspace is a linear combination of distinct vectors in g with the choice of nonzero coefficients being unique. 3. The single zero n-vector over IF constitutes a subspace of IF n called the trivial subspace of IF n. It has no basis and is said to have dimension zero. Each nontrivial subspace of IF n has a basis; the basis can be ordered; and any two ordered bases for the subspace have the same (finite) number of elements, called the dimension of the subspace. 4. IFn is a subspace of itself having the consecutive columns of the identity matrix In = (en e2' ... , en) as an ordered basis called the standard ordered basis for IF n.

2.4

Systems of Linear AlgebraicEquations

A system of m linear algebraic equations all xl + al2x2 + a21 xl + a22x2 +

+ alnXn = bl , + a2nXn = b2,

(2.4.1)

with the aij, b, (i = 1,2, ... , m; j = 1,2, ... , n) prescribed elements of a field IF, can be written as the single matrix equation Ax

= b.

(2.4.2)

The m x n matrix A with elements Au = aij is called the coefficient matrix;

(2.4.3)

is regarded as an n-vector variable in IFn; and the m-vector b is defined as

(2.4.4)

Equations (2.4.1) and (2.4.2) are called homogeneous if b = O. It is possible to write down examples of (2.4.2) that have no solution, precisely one solution, or an infinite number of solutions. Generally, counting

26

2.

Matrix Algebra-The

Natural Language of Linear Systems

equations and unknowns provides no information about which is the case. Fortunately, there exists a systematic method of computing that generates any and all solutions and detects the case for which there is no solution. The technique involves performance of elementary row operations on the augmented matrix (Alb),obtained by adjoining b as an (n -t 1)st column to A . The three allowed operations are (I) addition of any scalar multiple of one row to any other row, (11) multiplication of any row by any nonzero scalar, and (111) interchanging any two rows. (It is an easy exercise to show that a type I11 operation can be accomplished by an appropriate sequence of elementary row operations of types I and 11.) The method is based on the fact that the solutions to the system (2.4.2) are preserved (invariant) under the three elementary row operations. Hence an appropriate finite sequence of elementary row operations is applied to (Alb), transforming it into its so-called row-reduced echelon form.A matrix is in its unique row-reduced echelon form if the following conditions are met: 1. All the zero rows (if there are any) occupy the bottom rows. 2. The leftmost nonzero term in each nonzero row is a one, called a leading one. 3. Each column containing a leading one has all its other elements zero. 4. The column occupied by a leading one is to the right of the column occupied by the leading one in the row (if there is one) directly above it.

Example 2.4.1 The solution to the system with row-reduced echelon form of its augmented matrix 0

1 0 - 1 (2.4.5)

0 0 0 is x2 = x4

+ 3,

0

x3 = -2x4 - 1,

x1 and x4 arbitrary.

(2.4.6)

That is, the solutions are all vectors of the form

[[

:[2:

ll

a,b arbitrary].

1]

(2.4.7)

2.4

27

Systems of Linear Algebraic Equations

Example 2.4.2 The unique solution to the system having row-reduced echelon form of its augmented matrix (2.4.8)

[:x ;],

(2.4.9)

0 0 0 0

It is easy to show that the row space of A is invariant under elementary row operations on A . Hence the nonzero rows of the row-reduced echelon form of A are a basis for the row space of A and the number of such rows is the rank of A . In particular, row reduction provides a method for testing for the It . is also true linear independence of a finite collection of m-vectors in 9'" but not obvious that rank ( A T )= rank ( A ) .

CRITERIA FOR THE SOLVABILITY O F SYSTEMS

The following results are quite transparent when (2.4.2) is looked at in the form for which its augmented matrix is in row-reduced echelon form. With A an m x n matrix, the homogeneous equation AX = 0

(2.4.11)

has a nontrivial (i.e., nonzero) solution if and only if rank(A) < n. A sufficient

28

2. Matrix Algebra-The Natural Language of Linear Systems

condition for (2.4.1 I) to have a nontrivial solution is that m < n, i.e.,that there be fewer equations than unknowns. In general, (2.4.2) will have at least one solution if and only if rank(Alb) = rank(A).

EXERCISES

1. Find all solutions to the following systems by row reducing the associated augmented matrices: (a)

(b)

(c)

3Xl + 2X2 + 3X3 + 12x4 = 9, -Xl + X2 - 3X3 - 3X4 = -5, Xl + X2 + 3X4 = 2, Xl + X2 + X3 + 5X4 = 3. Xl + X2 + 4X3 = 3, 3Xl + 2x 2 + IOx3 = 4, 2Xl + X2 + 6X3 = 2. Xl - X2 - X3 = - 2, Xl + 2x 2 + X3 = -3, Xl + 2x 2 + 2X3 = 1, Xl + 4X2 + 2x 3 = -5, Xl + 3x 2 + X3 = -6.

2. Compute the ranks of the following matrices. (a)

(c)

[1 2-1]

1 1 2, 231

[ 12]u: -1 0 1 1 1

(b)

[l

2 0 1 2 1 3

(d)

[~

3 2 1 2

~l -~l

234 3. Determine all the subsets of rows which are bases for the row space for the matrix (c) of Exercise 2. 4. Prove that the transpose operation has the following properties: (a) (b) (c)

(AT)T = A, (AB)T = B TAT, (A + B)T = AT

+ BT.

29

2.5 Matrix Inversion

2.5 Matrix Inversion

In the scalar case for which in (2.4.2) m = n = 1, an elementary method of solving (2.4.2) is to multiply that equation by the solution L to LA

= In

(2.5.1)

(at least in the case for which the coefficient of x in (2.4.2) is not zero.) Observe that if (2.5.1) has a solution L, the same technique solves (2.4.2) for general n. By writing out (2.5.1) as n column equations, it is apparent that (2.5.1) is equivalent to a system of type (2.4.2) but of a special form and larger size. Equation (2.5.1) plays such a prominent role in matrix theory that its solution is given the status of a definition.

Definition If A is an n x n matrix over a field:#' for which (2.5.1) has a matrix solution Lover :#', then A is called invertible or nonsingular and L, ordinarily denoted by A-I, is called the (multiplicative) inverse of A. Remark 2.5.1 It is possible to prove that if L is a solution to (2.5.1), then it is also a solution to AL

= In

(2.5.2)

and conversely, if a matrix L satifies (2.5.2), then it satisfies (2.5.1) as well. Remark 2.5.2 As a corollary of Remark 2.5.1, it is an easy exercise to show that if A has an inverse, then the inverse is unique and is itself invertible with (A- 1 ) - 1 = A. COMPUTING A-'

The problem of inverting a matrix is one of solving a system of linear algebraic equations. To see that this is indeed true, let A be an n x n invertible matrix and denote the columns of A-I by Cl' C 2, ... , Cn and those of In by e 1 , e2' ... , en' Equation (2.5.2) can then be written as (2.5.3) which is the same as (2.5.4) Thus the problem of computing A-I = (c., C2, problem of solving the system of vector equations Ac; = e;

(i

... ,

= 1,2, ... , n).

cn) is precisely the

(2.5.5)

30

Matrix Algebra-The

2.

Natural Language of Linear Systems

But all these equations have the same coefficient matrix A. Hence by row reducing (Ale,, e2, . . ., en),all equations (2.5.5) are solved simultaneously, and clearly the row-reduced form of the matrix (Ale,, e2, . . ., en)will be U,IA

-

(2.5.6)

).

Hence for any n x n matrix A, either row reduction of (AII,) will produce a matrix with the first n columns constituting I,, in which case A is invertible and A-' can be read off the reduced form (2.5.6), or else the row-reduced form will have a string of n zeros at the left end of the bottom row, indicating that A is not invertible. This method for computing A-' is called the Gauss-Jordan method.

Example 2.5.1 Consider the matrix A=

[ ' :]. -1 -1 -1 -2

(2.5.7)

-2

To study its invertibility, row reduce the associated augmented matrix

[ [

1 2 3 -1-1 2 -1 -2 -2

1 2 0 0 1 0 0 0 1

1 0 0 0 1 0 0 0 1-

I

(2.5.8)

- 2 0 - 3 - 4 1 - 5 1 0 1

The conclusion drawn is that A is invertible and the inverse is read off to be

(2.5.9)

Example 2.5.2 Consider the matrix A=

['

-1 -1 2 5

:],

11

(2.5.10)

31

2.6 Large-Scale Matrix Computations Using Computers

whose augmented matrix row reduces as follows:

H

2 3 -1 2 5 11

1 0 0 1 0 0

~]-[~ -[~

2 3 1 5 1 5 2 3 1 5 0 0

1 0 1 1 -2 0 1 1

0 1 -3 -1

~]

(2.5.11 )

~]

and the calculation can be terminated, since the three zeros in the bottom row indicate that A is not invertible.

2.6 Large-Scale MatrixComputations Using Computers

In recent years the capability for doing numerical matrix computations on digital computers has progressed to the point where large-scale problems can now be solved quickly and accurately. This success is attributable to the numerical analyst's careful attention to the accuracy and efficiency of the algorithms developed for doing the computing and the programming effort expended in implementing those algorithms. Any scientist contemplating extensive numerical work is advised to consult the references to learn about the capabilities and limitations of available packaged routines such as ElSPACK and UNPACK (see [5, 18]). Much of Chapter 3 concerns derivation of appropriate forms of the equations for treating large-scale systems of differential equations. For a discussion of the large-scale systems of linear algebraic equations to which they are reduced, the reader should consult the cited references. Only one aspect of the latter problem is mentioned here. Programs for solving large systems of linear algebraic equations ordinarily do not reduce the augmented matrix to the echelon form described earlier in this chapter. The more widely used method of Gauss elimination row reduces the matrix to an upper-triangular form and then completes the computation by back substitution, starting with the bottom nonzero row, rather than proceeding with back elimination to reach the echelon form. To compare the efficiency of algorithms, suppose that each division and each multiplication-addition is counted as one operation. For an n x n nonsingular system, it is easy to show that Gauss elimination takes the order

32

2,

Matrix Algebra-The Natural Language of Linear Systems

of !-n3 operations to reach the upper-triangular form and the order of another in2 operations to complete the back substitutions-a total still of the order tn3 • (The back substitution takes fewer operations, since it is done to a triangular matrix.) Clearly the multiplication of two n x n matrices done in the standard way takes n 3 multiplications-the same order of operations required to invert an n x n matrix using the Gauss-Jordan method described in Section 2.5. Thus Gauss elimination compares favorably to the method of solution based on inverting the coefficient matrix. Suppose that the task were to solve the system of equations Ax;

= b,

(2.6.1)

= 1,2, ... , m) in which A is an n x n invertible matrix. A novice might make the mistake of applying Gauss elimination to reduce each matrix (Alb;), with a total operation count of order !-mn 3 • If all the data b, were available at the start, a more efficient strategy would be to apply the Gauss elimination to the augmented matrix (Alb 1 , b2 , ••• , bm ) , which could be done with the order of j-n3 + mn 2 operations-a worthwhile improvement if n were large and m large but relatively small compared with n. If m were much larger than n, then the latter method again would be more efficient. Of course, in an application in which some of the initial XiS had to be computed before the later bjs arrived, the reduction of (Alb 1 , b2 , ••• , bm ) just one time would not be possible. However, there is another efficient solution to the problem that the uninitiated most likely would overlook! Suppose that (2.6.1) were to be solved for m relatively large compared with n. Obviously the only reason for row reducing A in (2.6.1) is to generate the row operations that need to be applied to each b.. This information is available after solving (2.6.1) for i = 1. The problem reduces to the question of how best to record those operations so that they can be repeated on the other biS in whatever order and whenever they arise. A solution to this problem is well-known in numerical analysis. It is called triangular (i

factorization.

Triangular factorization is particularly well adapted to problems of type (2.6.1) in which the coefficient matrix A is common to all the equations. The idea is to compute a lower-triangular matrix L and an upper-triangular matrix U such that A = LU. Then each equation (2.6.1) splits into the pair LYi = b.,

(2.6.2)

Ux, =Yi

(2.6.3)

(i = 1,2, ..., m). The solution Xi to (2.6.3)-and hence to (2.6.I)-is achieved by first solving (2.6.2) to get Yi and then substituting Yi into (2.6.3) to get Xi'

33

2.6 Large-Scale Matrix Computations Using Computers

The advantage gained by the pair is that each takes but i n 2 operations, since L and U are triangular. The computation of L and U is accomplished once and for all in i n 3 operations. Thus the total count for LU factorization is about i n 3 + mn'. Notice the improvement in efficiency over the first approach, which took the order of 3mn3 operations. An advantage of the LU approach over the second method is that one equation in (2.6.1) can be solved at a time with no penalty in computation. The matrix U is the one obtained by doing Gauss elimination in reducing A to upper-triangular form. L is simply a bookkeeping matrix that records the multipliers involved in the triangularization of A and requires no operations. For a more detailed discussion of LU factorization and related topics see [ 5 ] .

Example 2.6.1 This example carries out the solution of (2.6.1)by the method utilizing L U factorization for A=

[ J 2 1 3 - 2 - 10 1

bl =

[-;I9 [;I. b,

=

" ' 3 " ["3

(2.6.4)

By Gauss elimination A is reduced to upper-triangular form to get 1 3 -2 0 1 - 0 1 4 + 0 1 4 = u , 4 -1 2 0 - 3 -4 0 0 8

(2.6.5)

and the multipliers used are recorded in the corresponding positions below the diagonal of the identity matrix to get 1

L=

.]

0 0 1 2 -3 1

[-'

(2.6.6)

Note that LU = A as intended. The solution to (2.6.1) for i = 1 is accomplished by doing the forward and backward substitutions that solve (2.6.2) and (2.6.3):

[-:

0 0 1 0 2 -3 1

[

2 1 3 0 1 4 0 0 8

(2.6.7)

(2.6.8)

34

2. Matrix Algebra-The Natural Language of LinearSystems

The reader is invited to repeat the substitutions following replacement of the augmented column in (2.6.7) by b 2 to get X2 = (-i, -1, !)T. The problem of efficiency is just one of those that must be dealt with in computerized numerical linear algebra. A. more extensive discussion of it and other problems such as roundoff errors can be found in the cited references. EXERCISES

1. Determine which of the following matrices are invertible and compute the inverse in those cases where it exists.

(a)

H

(b)

~],

-1

(c)

2.

[~

(d)

[~

1

3

5

o 1 o -1 o 0

fl

Prove the following statements: (a) If A and B are invertible n x n matrices, then AB is invertible and

(AB)-l = B-IA- I. (b) If A is invertible, then so is A-I and (A-l)-l = A. (c) If A is invertible, then so is AT and (AT)-l = (A-I)T. (d) rank(AA T ) = rank(A) for every m x n matrix A. (e) An n x n matrix A is invertible if and only if rank(A) = n. 3. Compute all the matrix solutions X to the matrix equation AX = B,

where

1 - 1 1

2]

A= 3 02 -1, [2 1 1 3 (Hint:

4 0 4]

B= 2 6 7 .

[ 409

minimize the calculations by row reducing the appropriate matrix.)

35

2.7 Determinant Functions

2.7

Determinant Functions

In the following discussion A is an n x n matrix variable over a field !F written out in terms of its columns A = (al> a2'" ., an)'

Definition A determinant function det(A) is defined to be an fF-valued function (of the columns) of A, det(A) = dia«, az, ... , an), with the properties (1) d(el' ez,· .. , en) = 1, where In = (el' ez,"" en);

d(a 1, az,"" an) = -d(al' az,.··, an), where the matrix (ai, a z, , an) arises from the interchange of any two adjacent columns of (ai' az, , an); (3) dia«, az,···, an)la,=ktbt +k2 b2 = k 1d(al> az,···, an)!a,=bt + k zd(al' az,···, an)la,=b2 for every kj in IF, bj in IFn (j = 1,2; i = 1,2, ... , n). (2)

A function satisfying (2) is called alternating and one satisfying (3) is said to be n-linear. Hence a determinant function is an n-linear, alternating IFvalued function of an n x n matrix variable A over fF with value 1 at the identity matrix. Case n = 1. For n = 1 the function det(a) = a is clearly a determinant function and there is no other, since if d(a) is a determinant function, by (1)-(3), d(a) = d(a'l Case n

+ 0,1) = ad(l) + Od(l) =

a.

= 2. For n = 2 and A

11 12] det[a a a21 a22

==

[aau11 a22 a12 ] ,

the standard function,

= a11 aZ2 - a12a21

can easily be checked to be a determinant function, and there is but one, since if dia-, a2) is a determinant function, properties (1)-(3) can be applied to compute d[:::

:::]

= d[ a11 =

a11d[~

[~]

+

: ~]

a2{~J[: :] +

a21d[~

: ~] (continues)

36

2. Matrix Algebra-The Natural Language of Linear Systems

=

+

a2Id[ ~1

a12

[~J

all

{a12d[~

~J

+ a22d[~

+ a21

{a12d[~

~J

= alla22d[~

~J

= alla22-

+ a22

[~J] .~J}

+ a22d[~

~J}

+ a2IaI2d[~

a2Ia12d[~

~J

~J =

alla22 - a21 a12'

(2.7.1)

Case n General. Expansion by Minors Formulas. Let A be an n x n matrix and let AOJj) denote the(n - 1) x (n -l)matrixobtainedbystriking out the ith row and jth column of A. Let D be a determinant function on (n - 1) x (n - 1) matrices. Then the formulas n

det(A) =

I

( - l r j AijDA(iU)

(2.7.2)

i= I

can easily be shown to define a determinant function on n x n matrices for each fixedj (j = 1,2, ... , n). In this way, by induction, a determinant function can be shown to exist for each n = 1,2, .... With some work it can be shown that for each n there is but one determinant function, which is then simply called the determinant of order n. Hence in (2.7.2) D can be replaced by det and the resultant formula is one of the expansion by minors formulas. Thus the right-hand side of (2.7.2) is actually independent of j. This and other consequent results concerning determinants can now be summarized. DERIVED PROPERTIES OF DETERMINANTS

det(a l, a2""'~)

=

0 if a, = aj

for some

det(al' a2' ... , an)la, =kb = k det(al' a2' ... , an)1 a, =b for each k in IF, b in IFn (i

i "# j.

(2.7.3) (2.7.4)

= 1,2, ... , n).

det(AR) = det(A) deteR). det(A- I } = [det(A)]-1

(2.7.5) (2.7.6)

37

2.7 Determinant Functions

for nonsingular A. det(AT ) = det(A).

(2.7.7)

n

L (-l)i+iAijdetAUIj)

det(A) =

(j = 1,2, .. . ,n).

(2.7.8)

(i = 1,2, ... , n).

(2.7.9)

i= 1 n

det(A) =

L (-l)i+iAijdetA(ilj)

i> 1

(2.7.10) in which the sum is over all permutations a of {I, 2, ... , n} and sgn(a) takes on the value 1 or -1 as a is even or odd, respectively. Formula (2.7.10) is sometimes taken as the definition of the determinant function and is logically equivalent to that employed in Section 2.7. Frequent use will be made of the following standard result concerning determinants. Theorem 2.7.1 For A an n x n matrix over afield iF, the followinq statements are equivalent:

det(A) = O. Ax

=0

has a nontrivial solution A

is not invertible.

(2.7.11)

x in

(2.7.12) (2.7.13)

COMPUTING DETERMINANTS

Formula (2.7.10) shows that the determinant of a matrix variable is a very special polynomial in the elements of the matrix. Note that if the matrix is triangular, then the value of its determinant is just the product of the diagonal elements. Determinant functions are used primarily in theoretical work. In numerical work their use should be avoided when possible, since the number of arithmetical operations involved in their evaluation tends to grow rapidly with the size of the matrix. When the determinant of a sizable matrix must be computed, it is generally advisable to base the calculation on triangularization of the matrix using elementary operations on rows and columns. It is apparent that every square matrix can be triangularized by row operations. The calculations often utilize (2.7.4) and (2.7.7) as well as the invariance of the value of a determinant under type I row operations (which follows readily from the defining properties of the determinant function). Another useful operation is the interchange of two rows or columns, which simply alters the

38

2. Matrix Algebra-The

Natural Language of Linear Systems

sign of the value of the determinant. The latter is a consequence of the alternating property. For notational convenience det(A) is occasionally denoted by [ A [ . -

Example 2.7.1 0 1 3-1 2 1 3-1

1-2 2 2 3 1 2 - - _1 0 1 2 1 3 3 2 2 2 1 4 3 - 1 1

4 3 2 4

1-2 2 4 - - _1 0 1 2 3 2 0 5 -1 - 6 O 5 - 5 -8

1 0

0 0 -15

1 0 0 0 1 1 0 1 0 1 0 0 = - 2 0 2 0 0-11 1 0 0 -15 7 0 1 - _1 0 2 0 0

0 0 1 0 0 6 2 0 0

0

0 1 0 0 6

0 0 0 2

0

-23 0 0 1 7

0 0 = 31. 7 1

(2.7.14)

EXERCISES

1. Derive the formula det(A) = a l l % ~ a 3 3+ a13a2lU32+ a12a23a31 - a11a23a32

- a12a21a33 - 413u22u31

for a 3 x 3 matrix: (a) directly from the defining properties (1)-(3), (b) directly from formula (2.7.10), (c) directly from (2.7.9) and the standard formula for a 2 x 2 matrix. 2. Prove that an interchange of any two rows of a matrix results in the change in sign of the value of the determinant. 3. Evaluate the determinant of each of the square matrices appearing in Exercise 1, Section 2.6.

39

2.8 Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

4. Show that (2.7.5) implies (2.7.6). 5. Apply the defining properties (1)-(3) of a determinant to prove (2.7.3) and (2.7.4). 6. Deduce (2.7.7) from (2.7.10). 7. Prove Theorem 2.7.1. 8. Verify that (2.7.2) defines a determinant function of an n x n matrix.

2.8

Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

Definition Let A be any n x n matrix over a field iF and v be an n-vector in iF n • The scalar A in iF is called an eigenvalue (or characteristic value) of A if (2.8.1)

Av = AV n

for some v :I: O. If Ais an eigenvalue of A, then any v in iF satisfying (2.8.1) is called an eigenvector (or characteristic vector) associated with A. The collection "11/;. of eigenvectors associated with an eigenvalue Ais clearly a subspace of iF n• Note that if 1 is an eigenvalue of A, then the problem of computing "11/;. is the problem of finding all solutions v in iF n to the system of linear algebraic equations

(2.8.2) (Un - A)v = O. A polynomial p(x) = Co + CtX + ... + cnxn of degree n is said to be over the field iF if its coefficients are elements of iF. Such a polynomial is called monic if c; = 1.

Definition The characteristic polynomial p..i x) of an n x n matrix A over a field iF is the monic polynomial defined as PA(X) = det(xl n

-

A).

(2.8.3)

In view of (2.8.2) and Theorem 2.7.1, it is apparent that the eigenvalues of A are precisely the-roots of PA in iF. By the fundamental theorem of algebra, for iF the complex field, A will have n eigenvalues, possibly with repetitions. A real matrix may be regarded as over the real field or over the complex field. If A is an n x n matrix with real elements, it might not have any eigenvalues when regarded as a matrix over the real field, e.g., A = [_? aJ. The main point to be made is that the problem of computing the eigenvalues and eigenvectors of a matrix over a field is a problem of computing the roots of a polynomial and solving the associated systems of linear algebraic equations.

40

2. Matrix Algebra-The Natural Language of Linear Systems

This is not to say that eigenvalues should be computed by numerically computing with PA(X). Numerical analysts have a variety of algorithms which work more directly with A, each with its virtues and disadvantages. For f(x) = Co + c ix + .. , + cmxm a polynomial over a field ff define f(A) = col + ciA + '" + cmAm called a matrix polynomial. If f(A) = 0 then f is said to annihilate A. These notions appear in the following theorem. Theorem 2.8.1 Cayley-Hamilton Let A be any n x n matrix over afield ff and PA be its characteristic polynomial. Then PA(A) = 0, i.e.; PA annihilates A.

Definition The minimal polynomial qA of an n x n matrix A over a field ff is defined to be the monic polynomial over ff annihilating A and ofleast degree. It can be shown that qA divides PA and that both polynomials have the same distinct roots in ff. Thus qA can be computed by finding PA which is then factored over ff into the product of powers of distinct prime polynomials. The minimal polynomial is then obtained by the finite process of deleting one prime factor at a time and testing to see whether the resultant polynomial annihilates A. The proof of these facts concerning PA, qA, and Theorem 2.8.1 can be found in [11].

A demonstration of the calculations involved in obtaining the characteristic polynomial, eigenvalues, and associated eigenvectors is given for Example 2.8.1

A =

2 2 -2] -4 3 1, [ 413

(2.8.4)

where A is regarded as a matrix over the real field ~. First it is necessary to compute the characteristic polynomial PA(X) = Ix13

=

-

AI =

(x - 4)

x-2 2 -2 -1 4 x-3 -1 x-3 -4

x-2

-2

4

x-3

o

1

= (x - 4)[(x - 2)2 + 16].

=

2 x-2 -2 -1 4 x-3 0 x-4 x -4

2 -1 = (x - 4)

1

x - 2

-4

4

x-2

0

o

2 -1 1

(2.8.5)

From PA it is apparent that A has but the single eigenvalue A = 4 in ~. The associated eigenvectors are obtained by row reducing the matrix of (2.8.2)

41

2.8 Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

[

2 -2

-44 - 11

3-t

2

1 - 1

-;

0

1

5 - 50 0.

3 (2.8.7)

which is noted to be one-dimensional.

Example 2.8.2 Consider the same problem as in Example 2.8.1 but with A now regarded as a matrix over the complex field V. The characteristic polynomial computes exactly as in (2.8.5) but now factors into a product of linear polynomials over V, PA(x)

= (X -

4 ) [ ( ~- 2)’

= (x - 4)[x - (2

+ 161

+ 4i)] [x - (2 - 4i)l.

(2.8.8)

Hence the eigenvalues of A are I = 4, 2 f 4i. From the calculations of Example 2.8.1 it is possible to immediately write (2.8.9)

4i 4 [-4

-2 -1+4i -1

’

2 -1 -1+4i

0

1

0

(2.8.10)

42

2. Matrix Algebra-The Natural Language of Linear Systems

From (2.8.10) it follows directly that (2.8.11) which also has dimension 1. Similar calculations show (2.8.12) Remark 2.8.1 If A is a real n x n matrix with a complex eigenvalue A. = a + ico, the associated eigenvectors can be computed by real arithmetic. Let v = x + iy. Equation (2.8.1) splits into the pair of equations over fJl, (rxI - A)x

=

(2.8.13)

wy,

(2.8.14)

(rxI - A)y = - wx,

which can be written as [(IXI - A)Z

+ wZI]x = 0,

(2.8.15) (2.8.16)

Y = (Ilw)(rxI - A)x.

The former is a homogeneous linear system over fJl that can be solved for x, and then y can be computed by substitution into (2.8.16), doing only real arithmetic.

EXERCISES

1. Find the characteristic and minimal polynomials of the following matrices:

. (a)

(c)

[~ [~

21]

-1

o

1 2 0

1, 3

(b)

[~,

1 0 kz k3

~l

n H ~J (d)

2 1 0 0

0 0 2 1 . 0 2

43

2.9 The Jordan Form of a Matrix

2. Compute the eigenvalues and associated eigenspaces of the matrices in Exercise 1, first regarding the matrices as over 9 and then over W. 3. Let A be a real n x n matrix having eigenvalue u + iw with associated eigenvector x + iy where x and y are in P.Show that u - i o is likewise an eigenvalue of A with associated eigenvector x - iy.

2.9

The Jordan Form of a Matrix

Let A be an n x n matrix over a field 9 in which the characteristic polynomial pA “splits.” That is, assume that PA(X)

= (x - n,)ml(x- J-2)m2..

‘

(x - n

~)~‘

(2.9.1)

with the Ai in 9distinct. The positive integer mi is called the multiplicity of Li. As remarked earlier, the minimal polynomial qA will have the same linear factors but with possibly lower multiplicities qA(X) = (x - n’)l’(x - n2)12 ’. ( x - &)r., 1 5 ri 5 mi

(i = 1 , 2 , .. .,s).

(2.9.2)

Theorem 2.9.1 Jordan Form. There exists a nonsingular n x n matrix P over 9 such that PAP-’ has the block diagonal form J

=

diag[J,, J z , . . ., Js]

(2.9.3)

in which block Jiassociated with ,Ii is mi x mi. Each Ji is itselfa block diagonal matrix whose subblocks are of the form 1 J-i

0

1

. .. .

(2.9.4)

with all diagonal positions occupied by li,ones along the superdiagonal, and all other elements zero. ‘The uppermost subblock of Ji of type (2.9.4) has size ri x ri where ri is the multiplicity of Ai in the minimal polynomial qA of A. IThe sizes o f t h e subblocks (2.9.4) of Ji form a nonincreasing sequence down the diagonal of Ji. IThe number of subblocks of Ji equals the dimension of the eigenspace W n ,associated with ,Ii.

44

2. Matrix Algebra-The Natural language of Linear Systems

Remark 2.9.1 If the multiplicity of A.i in qA is one, then dim "I(/' Ai = m., and if this is true for each i = I, 2, ... , S, then PAP - 1 is diagonal. This then gives a necessary and sufficient condition for the diagonalizability of A by a similarity transformation over :F. A sufficient condition for diagonalizability is that PA have no repeated roots. . Remark 2.9.2 The characteristic polynomial PA always splits over :F = 'IJ. (This is a consequence of the fundamental theorem of algebra.) Sometimes P can be taken over fJl or some other subfield of ce. For example, if the elements of A are all real and PA has only real roots, then P can be taken to have real elements. Other examples of subfields of'IJ are [J,ta, the rational numbers, the subfield obtained by adjoining to [J,ta, etc.

±.J2

Remark 2.9.3 The Jordan form of a matrix A as defined by Theorem 2.9.1 is unique up to the order of the numbering of the distinct eigenvalues AI' A2' ... , As of A. Any such ordering can be selected. Generally, the characteristic and minimal polynomials of a matrix do not carry enough information about the matrix to determine its Jordan form. In those cases where they do or where J is somehow known, P can be determined by solving the system of linear algebraic equations JP - PA = 0

(2.9.5)

for a nonsingular P. Generally, such a P is not unique. Example 2.9.1 Suppose that A is a real matrix with pix) = (x - 2)2(X - 1) and qA(X) = (x - 2)2(X - 1). In this case the Jordan form of A can be written as

J =

Moreover, dim

"1(/'2 =

dim

"1(/'1 =

2 1I 0] . [~--=-l~ o 0 I 1

(2.9.6)

1.

Example 2.9.2 Consider a matrix A whose characteristic and minimal polynomials are PA(X) = (x - i)3(X + i?(x + 2)2 and qA(X) = (x - i)2(X + i)2(X + 2), respectively. Here again, Theorem 2.9.1 determines

4.5

2.9 The lordan Form of a Matrix

J , which would be

(2.9.7)

EXERCISES

1. Consider the matrix

A=

I

1 -2 0 0 -1

0 0 3 -1 0 2 0 1 1-1

1 1 0 2 0

0 1 1 0 0

over the rational field. Compute (a) the characteristic polynomial pa, (b) the minimal polynomial qa , (c) the Jordan form J , (d) dim WA,. 2. Suppose that all that is known of a matrix A over the field W is its characteristic polynomial and minimal polynomial

respectively. What can be said concerning the Jordan form of A? What can be said about the dimensions of the eigenspaces? If A itself were given, how could it be used in solving the problem? 3. Consider the matrix

'1

A = [ ' k2 k l

46

2. Matrix Algebra-The Natural language of linear Systems

in which k 2 = - (a 2 + ( 2 ), k1 = -20(, with a and ca real and t» #- O. (a) Determine J, the Jordan form of A. (b) Determine all nonsingular 2 x 2 matrices P for which PAP- 1 = J. 4. Suppose that A is a matrix with pix) = (x - Ar, qA(X) = (x - A)3, and dim 11/;. = 3. Is this enough information to determine the Jordan form of A ? 5. If A and Bare n x n matrices over a field /#', then they are called similar if A = PBP- 1 for some n x n nonsingular matrix P over /#'. Prove that if A and B are similar, then (a) det(A) = det(B), (b)

PA = PB'

(c) tr(A) = tr(B), where the trace of a matrix A is defined by tr(A) = I:?= 1 Au. 6. Prove that if A is an n x n matrix over ~, then (a) tr(A) = I:i = 1 m;Ai' (b) det(A) = ni= 1 ).i', where Al' ... , As are the distinct eigenvalues of A and m, is the multiplicity of Ai in the characteristic polynomial PA • 7. Prove that each n x n matrix A over ~ is similar to its transpose AT.

Chapter

3

Linear Constant-Coefficient Equations: Computation

In this chapter, we present several versions of a step-by-step procedure (the ABC algorithm) for computing the closed-form solution to the general,

linear, first-order differential equation with constant-coefficient matrix. In solving the homogeneous equation, all forms of the computation require determination of the eigenvalues of the coefficient matrix (more precisely, the roots of the characteristic polynomial with their multiplicities). One version involves the solution of a finite system of recurrent linear algebraic vector equations. Another is based on solution of a single matrix equation with nonsingular coefficient. Both require a certain amount of matrix-vector multiplication and addition. All operations are adaptable to standard computer routines using either real or complex arithmetic. The solution to the nonhomogeneous problem is accomplished by an additional integration. Although heuristic derivations of the methods are given, their rigorous justification is deferred to Chapter 4.

3.1

The Initial-Value Problem

The procedure to be discussed deals with the initial-value problem in $'",

x = Ax + f(t), x(O) = x",

(3.1.1) (3.1.2)

in which A is an n x n matrix over $' and f: fYt -- $'" is assumed continuous with Xo in $'" prescribed. In a step-by-step manner it constructs a closed-form 47

48

3.

Linear Constant-Coefficient Equations: Computation

solution to (3.1.1) and (3.1.2) in terms of elementary functions and an integral involving f. The procedure includes the option of avoiding complex arithmetic when F = fJl. The following discussion treats the real case first and requires only real arithmetic. The problem in which A is complex or A is real but the facility for doing complex arithmetic is available is covered by Remark 3.2.2.

3.2

The

ABC Algorithm

Step 1. Compute the eigenvalues {/l'l' A2' ... , An} of A (some possibly repeated and/or complex) regarded as a matrix over re. This is equivalent to finding all the real and complex solutions to the equation (3.2.1)

in which PA is the characteristic polynomial of A. This is done analytically or numerically. More often than not, the eigenvalues would be found by using a computer. A number of programmed algorithms are available, several of which work with A directly rather than with (3.2.1). A numerical analysis or text should be consulted about which is most appropriate and about difficulties that can occur (see [5]). When A is real, complex eigenvalues will occur in conjugate pairs. Step 2.

Apply the real correspondence rule

(3.2.2)

which assigns to the collection of eigenvalues a collection of n distinct realvalued functions arranged as an n-vector function according to the scheme 2

m-l

' ).t t Al t Al t Al rea 1I\. - - e , 1! e '2! e , ... , (m _ I)! e , complex A =

Ct.

± uo -- e" cos tot, e" sin cot; I t e~ Ism ' cot . -t e~ cos cot I! ' 1! '

tm - l tm - l ~I t ~/' t (m _ I)! e cos w '(m _ I)! e sm w .

(3.2.3)

3.2

49

The ABC Algorithm

In (3.2.3) m is the multiplicity of the eigenvalue A. The algorithm adopts the convention that in writing down y(t), (3.2.3) is applied to each distinct real eigenvalue and then to each distinct complex conjugate pair. Otherwise the order is immaterial.

Step 3.

Determine the real n x n matrix B such that

By(t)

= y(t).

(3.2.4)

This can be done by inspection, due to the fact that the derivative of each Yi(t) assigned by the correspondence rule is a linear combination of the coordinate functions composing y(t).

Step 4. With A given and B computed in step 3, solve the linear homogeneous matrix equation (3.2.5) AC - CB = 0 to obtain its general n x n matrix solution C. Equation (3.2.5) can be regarded as a system of n2linear algebraic equations in n2 unknowns, and the standard technique of solution discussed in Chapter 2 applies. However, B is a sparse matrix (i.e., it has many zero elements) with special structure. More practical techniques for solving (3.2.5) which utilize the special form of B are presented later in this chapter. Whatever approach is followed, the general solution C to (3.2.5) will have elements that are linear combinations of n arbitrary constants C 1, C2' ••• , Crt which can be arranged as the n-vector (3.2.6)

Hence, by inspection it is possible to read off a unique n x n matrix function Y(t) such that Y(t)C = Cy(t). (3.2.7) (The matrix function Y(t) is an example of what will later be called a fundamental matrix function for (3.1.1).)

Step 5. Using the matrix function Y(t) obtained in step 4, evaluate Y(O) and invert to obtain the matrix function (t) = Y (t)Y (0)- 1. The solution to the homogeneous form of (3.1.1), (3.1.2), oX = Ax,

(3.2.8) (3.2.9)

is now x(t) = (t)xo.

50

3.

Linear Constant-Coefficient Equations: Computation

Step 6. The matrix function (t) likewise provides a solution (later shown to be unique) to the nonhomogeneous problem (3.1.1), (3.1.2),

x(t) = (t)XO

+ f~

(t -:- a)f(a) da,

(3.2.10)

called the oariation-of-parameters formula for the solution. Example 3.2.1 The six steps of the ABC algorithm are demonstrated by applying the procedure to solve the simple example problem

:t[;:] = [-~ [;:](0) [;~J

~J[;J

+

[e~J

(3.2.11) (3.2.12)

=

In this example, (3.2.13) Step 1.

The characteristic polynomial is computed as

PA(X)=IXI2-AI=I~

-~I=X2+1,

whose roots are the eigenvalues {A'l' ),2} Step 2.

(3.2.14)

= {± i}.

Application of the real correspondence rule provides

yet) = [c~ssm rtJ.

(3.2.15)

Step 3. The equation for B is

B[cosm. stJt = [-sintJ, cos t

(3.2.16)

whose solution is read off to be B

=

[°1 -olJ.

(3.2.17)

Step 4. With A given in (3.2.13) and B as found in (3.2.17), Eq. (3.2.5), written out in terms of the elements of C, appears in the form

51

3.2 The ABC Algorithm

The resultant scalar equations reduce to the pair (3.2.19) It follows directly that the general solution is

(3.2.20) involving two arbitrary constants now computed, Eq. (3.2.7) reads Y(t)[cIJ Cl

=

Cl

and

[C I Cl

Cl

as anticipated. With y(t) and C

ClJ[C~S

-CI

(3.2.21)

tJ'

sm t

from which is read off Y(t) = [ Step 5. Setting t

c~s t sin tJ. -sm t cos t

(3.2.22)

= 0 and inverting give Y(O)-l =

[~ ~rl

=

[~ ~J

(3.2.23)

The matrices of (3.2.22) and (3.2.23) produce
sintJ. cos t

(3.2.24)

Step 6. Substitution of the computed
[=~~:G

=

[-:~: +

:~: J[=n

ft [ cos(t Jo -sin(t -

0) u)

U)J[e:"0 Jda,

sin(t cos(t - u)

(3.2.25)

which integrates to give the sought closed-form solution XI(t) = (x~ - i) cos t + (x~ + o I 0 Xl(t) = (Xl + ~) cos t - (Xl -

t) sin t + ie- t, 1

~)

1

sin t - "Ie-t.

(3.2.26)

Remark 3.2.1 The rather simple idea leading to the equations of the ABC algorithm is easily explained. Consider first the homogeneous problem (3.2.8), (3.2.9). The basic strategy is to attempt a solution of the form x(t) = Cy(t) with the matrix C to be determined. Substitution of this form into (3.2.8) imposes the condition that Cy(t) = ACy(t), but because of (3.2.4), the equation reduces to (CB - AC)y(t) = O. The linear independence of the

52

3.

Linear Constant-Coefficient Equations: Computation

functions composing y(t) allows y to be canceled, and this accounts for the appearance of the resultant matrix equation (3.2.5) for the unknown C. With the general solution to (3.2.8) now forced into the form x(t) = Cy(t), Eq. (3.2.7) (which in the present heuristic derivation is taken on faith) further transforms the representation into the form x(i) = Y(t)c. Since c is arbitrary, it can be selected so as to realize the initial condition x(O) = x", and the result is the equation x(t) = cI>(t)XO, in which cI>(t) = Y(t) y- 1 (0). It turns out that the existence of the inverse that appears follows from the linear independence of.the elements of y(t). Once the solution x(t) = cI>(t)XO to (3.2.8), (3.2.9) has been obtained, the solution (3.2.10) to the nonhomogeneous problem (3.1.1), (3.1.2) can be discovered using the standard uariation-of-parameters technique; i.e., attempt a solution of the form x(t) = cI>(t)u(t) with the vector function u(t) to be determined. Since (t) = AcI>(t), substitution of the proposed form into (3.1.1) leads to the requirement on u(t), cI>(t)u(t) = f(t), which can be written as u(t) = cI>-l(t)f(t) and then integrated, providing the candidate

u(t)

= XO +

L

cI>-l(u)f(u) da.

(3.2.27)

But it turns out that cI>(t)cI>-l(U) = cI>(t - e), and (3.2.27) then leads to (3.2.10), which can be verified to be the sought solution.

Remark 3.2.2 If A is a real n x n matrix, then the procedure described in steps 1-6 involves doing only real arithmetic on the real matrices A, B, and C. Some computers can do complex arithmetic, although generally it is rather awkward. An alternative procedure is available to those willing to do complex arithmetic. It replaces (3.2.3) by the complex correspondence rule, 1

2

AttAttAt

('1-1

At

JI.-e 'TIe '2!e ""'(m_l)!e ,

(3.2.28)

applied to each distinct eigenvalue whether it be real or complex. The remaining steps are the same, although they require working with complex numbers, since the resulting complex exponential functions will produce B and C with complex elements. If A is real, then of course the matrix cI>(t) will be real. If A itself is complex, then (3.2.28) can likewise be applied and the ensuing complex arithmetic performed.

Remark 3.2.3 Undetermined Coefficients. It has been stated that once the homogeneous equation has been solved, the solution to the nonhomogeneous equation is attainable by the additional integration appearing in the variation-of-parameters formula (3.2.10). In the special case for which f(t)

S3

3.2 The ABC Algorithm

is itself a solution to a constant-coefficient homogeneous linear system, the integration can be avoided by embedding the problem in a higher-dimensional homogeneous problem; i.e., suppose that in the equation

x=

Ax

+

(3.2.29)

f(t)

the coordinate functions /;(t) are linear combinations of functions of the type arising from one of the correspondence rules. Then f(t) = Fy f(t),

(3.2.30)

= Bfy/t)

(3.2.31)

YJ(t)

for appropriate matrices F and Br- Hence the problem of solving the initialvalue problem (3.1.1), (3.1.2) for f(t) of the special type under discussion is reduced to solving the homogeneous problem

z=Az, z(O)

(3.2.32)

=z",

(3.2.33)

where

z=[~J A=[~

(3.2.34)

~J

(3.2.35)

EXERCISES

1.

Compute the matrix function w(t) for the matrix A =

[_~ -~J

by applying the complex correspondence rule (3.2.28) to get y(t) and doing the complex arithmetic required to solve (3.2.5). 2. Solve the initial-value problem

x = 2x + 3t,

X(O)

=4

without integrating by embedding it as a homogeneous problem in the method of Remark 3.2.3. 3. What is the dimension n of fJl" in which the system Xl = -Xl + 2X2 + et sin 2t + t",

x2 = Xl + X2 + te 2tcos2t could be embedded as a homogeneous system?

~3

by

54

3.

Linear Constant-Coefficient Equations: Computation

4. If variation of parameters were selected as the technique for finding the closed-form solution of the problem in the previous exercise, approximately how maJ?y integrations by parts would be involved? 3.3 A VectorMethodfor Computing the Matrix C

In the ABC algorithm, A is given and B is just an intermediate matrix used to obtain the matrix C as the general solution to the equation AC - CB

= O.

(3.3.1)

For even moderately large n, it is not realistic to solve (3.3.1) by regarding it as a system of n2 scalar equations in n2 scalar unknowns. By exploiting the special block diagonal structure of B, this section derives a vector method for computing C directly from A. The resultant equations produce a vast decrease in the computational complexity and allow passage directly from step 1 to Eq. (3.2.7) of step 4. The discussion again first assumes A to be real and the calculations are set up for doing real arithmetic. Treatment of the problem with A complex or by employment of complex arithmetic is covered by Remark 3.3.2. By writing out the matrix variable C in (3.3.1) in terms of column variables and computing the form of B, it is easily shown that each distinct real eigenvalue A. of multiplicity m contributes m successive columns Um , Um-I"'" UI of C obtained by successively solving (k

= 1,2,

00

.,

m)

(3.3.2)

with Uo = O. Similarly, each distinct complex conjugate pair of eigenvalues A. = oc ± iw of multiplicity m contributes m successive ordered pairs of columns (vm , um ), (Vm-l, um-d,· .. , (VI' ud of C obtained by successively solving [(ocI - A)2 + w 2I ]uk = -2(ocI - A)Uk-I - Uk- 2 (k = 1,2,oo.,m) (3.3.3) with

Uo

= U-l :;:: 0 and then computing

Vk

= w-1[(ocI - A)uk + Uk-l]

(k = 1,2,oo.,m).

(3.3.4)

Remark 3.3.1 Note that the first equation (k = 1) in (3.3.2) is that for the real eigenvectors U 1 associated with the real eigenvalue A. of A, and the first equations in (3.3.3) and (3.3.4) are those for the complex eigenvectors U 1 + iVl associated with the complex eigenvalue oc + ico. In solving (3.3.2) and (3.3.3), the equations at each stage k must be solved to obtain the general solution Uk' It would be erroneous to conclude that for A. a real eigenvalue with associated eigenspace 11';. of dimension d there will

3.3 A Vector Method for Computing the Matrix C

55

be d arbitrary constants in the part u1 of the solution to (3.3.2). Certainly when (3.3.2) is solved with k = 1, the resultant solution will contain d tentative arbitrary constants, but as the computation proceeds, some of the constants will be forced to be zero to maintain the consistency of the equations at successive stages. In other words, the solution u1 to (3.3.2) is a subspace of the eigenspace WL,and u1 results from successive extraction of subspaces, starting with W A as , the calculation proceeds to stage m. This phenomenon is illustrated by Example 3.3.1.

Remark 3.3.2 If complex arithmetic is to be employed by choice, as discussed in Remark 3.2.2, or because A is complex, then the computation of C associated with (3.3.1) can be done by using (3.3.2) alone. The literature sometimes calls the vectors that are generated generalized eigenuectors.

[i 3.

Example 3.3.1 Let A be the matrix A=

(3.3.5)

The eigenvalues obviously are 1, 1, 1. The general solution (eigenspace Wl) of (3.3.2) with k = 1 obtained by the row reduction (3.3.6)

is read off as (3.3.7)

with pl, Bz arbitrary real numbers. The solution of (3.3.2) with k = 2 is accomplished by the reduction (3.3.8)

with Bz forced to be zero to maintain consistency. Thus u1 is cut down to the one-dimensional subspace of Wl , (3.3.9)

56

3.

Linear Constant-Coefficient Equations: Computation

and the resultant solution, read off (3.3.Q is u2

(3.3.10)

=

/I1, /I3, p4 arbitrary. The equation (3.3.2) with k

=3

is solved as (3.3.11)

with consistency forcing /Il = /I4 = 0 and the resultant solution being (3.3.12)

/I3, &, /I6 arbitrary. After the /Is have been renamed as cs, the general solution to (3.3.1), for A given by (3.3.5), is thereby computed to be

c=

[:6' 3 c2

0

0 .

(3.3.13)

The remaining steps in the ABC algorithm can now be computed:

=

[

d td 0 0 d 0 0 e'

c1

= Y(t)c,

@(t)= y(t)Y-'(O) =

Example 3.3.2

(3.3.14)

c3

(3.3.15)

Consider 2 = Ax in which (3.3.16)

3.3

57

A Vector Method for Computing the Matrix C

The characteristic polynomial computes as pA(x) = (x - 1)’(x2 + 2x + 2), from which it is apparent that the eigenvalues are {Al,,l2,A3,,l4} = {1,1, -1 lL- i}. The row reduction (11- All4) + (El M ) with 1= 1 produces the matrices

.]

1 0 0-2

0 0 1 - 1 ’ 0 0 0 0

2-i M = [ : -1

-1

:].

(3.3.17)

2 0

(3.3.18)

1 1

The general solution u1 to (3.3.2) for k = 1 is read off E as (3.3.19)

and substitution of (3.3.19) into the row-reduced form of (3.3.2) with k = 2, (3.3.20)

provides (3.3.21)

Since the pair of eigenvalues -1 & i has multiplicity one, (3.3.3) is solved directly by the row reduction

58

3.

linear Constant-Coefficient Equations: Computation

with the solution being (3.3.23)

Application of (3.3.4) results in

V1

=

(J)

-1

+

tC4 C3J 1 --Z C4 + C 3

«(XI - A)u 1 = [

2

-

C3 -

4C3

C4

(3.3.24)

.

+ C4

Since the cs are arbitrary constants, C4 can be replaced by 2c4 throughout for the convenience of eliminating fractions. With all columns computed, the solution C is 2C

2 -

5C1

-C 1

C= [

C2 -

C1

C2

1

2

C3

+

0C

C3 -

C1

-2C3 -

C1

4C3

C4 C4

-C3

J

+ 2C4

3C3

2C4

2C3

+ 2C4

2C4

.

(3.3.25)

The ABC algorithm can now be continued by jumping to Eq. (3.2.7) of step 4 with C as computed in (3.3.25) and

y(t) =

By inspection,

Y(t)

=

~:)e'

[

(2t (t _ 1)e' tet

2;' e'

e'

e' ] te' [ e cos t «: sin t -r

t

(3.3.26)

•

.c:«: e-'(2:~~_~o~tcost)]. 2e-'(sin t - cos t) 4e-' cos t

(3.3.27)

cos t 2e-t(cos t + sin t) - 2e

!

The precise manner in which the general solution x(t) = Y(t)c depends on t is vividly displayed by (3.3.27). A routine calculation yields

Y-

1(O)

=

[=; =~ ; 50 7

"5O-so 23

12

30 -"50 -"50

50 22

so 1.QJ

8 50 2 -50

.

(3.3.28)

59

3.4 Explicit Formulas for C(c) and «II(t)

The matrix «1>(t) = Y(t) y- 1(O), needed for the solution of the initial-value problem and for the nonhomogeneous form of the differential equation, can be obtained from (3.3.27) and (3.3.28) by multiplication.

3.4 Explicit Formulas for C(c) and Cl»W)

Let y(t) and B be defined as in Section 3.2. Let x = Ax, x(O) = c, and x(t) = C(c)y(t). Denote y(O) by yO. Since y = By, it follows formally from these equations that

.

(3.4.1) for all n-vectors c. The claim that the inverse in (3.4.1) exists and the presumption that (3.4.1) provides the general solution to (3.2.5) as well as the consequent formula (3.4.2) in which (3.4.3) (i = 1,2, ... , n), In = (el' e2"'" en), will be rigorously established in Chapter

4. In particular, it follows from (3.4.1) that the solution x(t) to (3.2.8) and (3.2.9) could be obtained as

(3.4.4)

x(t) = C(XO)y(t),

where the matrix

C(XO)

is computed by solving the system of equations

(yO, By", . . ., Bn-lyO)TCT(xO)

=

(XO, Ax o, ... , A n-

1xO)T.

(3.4.5)

Since the coefficient matrix in (3.4.5) is nonsingular, that equation could be solved most efficiently by doing a triangular factorization of the coefficient matrix and applying it to solve (3.4.5) as a system of vector equations. It should be noted that on the order of 2n 3 multiplications would be required to compute the two 'data matrices in (3.4.5), about ~n3 would be required to do the factorization, and about n 3 more would be required to do all the back substitutions. Thus this version of the ABC algorithm has an operation count of the order of n3 , which is better by at least one order of magnitude than are most other methods! It is important to note that if all that were required were the calculation of x(t) for a few values of XO with n large, then the calculation (based on (3.4.5)) just described would be much more efficient than computing «1>(t). This is apparent from (3.4.3), which would require on the order of n4 operations.

60

3.

Linear Constant-Coefficient Equations: Computation

EXERCISES

1.

Apply (3.4.4) and (3.4.5) to solve the initial-value problem Xl = X2 + X3' X2 X3

= Xl + X3' = Xl + X2

with Xl(O) = 1, X2(0) = -1, xiO) = 2. 2. Apply (3.4.4) and (3.4.5) to solve the initial-value problem

3.5 Additional Comments about Computing and Changing Variables

Earlier in this chapter it was shown that once the homogeneous problem (f = 0) has been solved, that is, the fundamental matrix (t) is computed, then the solution to the nonhomogeneous problem is obtained by an integration. Thus attention is returned to the homogeneous problem X

=

Ax,

(3.5.1) (3.5.2)

The change of variable z = r :». involving any nonsingular matrix P, transforms the equations into i =

Az,

z(O) = z",

(3.5.3) (3.5.4)

in which A = r:' AP and Zo = Px". Thus the change of variables induces a similarity transformation on the coefficient matrix. The rigorous analysis of (3.5.1) and (3.5.2) presented in the next chapter will utilize such a transformation. The approach will be to select P in a manner rendering the behavior of (3.5.3) transparent. This will be adequate since most dynamical behavior of interest is invariant under such a trans-

3.5 Additional Comments about Computing and Changing Variables

61

formation. Frequently, there is considerable freedom anyway in the choice of variables selected to model a physical system. One instance of this occurs when an engineer or scientist decides which system of physical units to use. To give some indication of the role of coordinate transformations in the subject, consider the special class of matrices A in (3.5.1) that have a full linearly independent system of eigenvectors WI' W2,"" Wn, corresponding to respective eigenvalues AI, A2, ... , An (not necessarily distinct or real). Recall that these are precisely those matrices for which each distinct eigenvalue has multiplicity one in the minimal polynomial qA' By the independence, it follows that the matrix (WI, W2, ..., Wn) is nonsingular. Thus suppose that the change of variable made in (3.5.1) and (3.5.2) uses P = (WI' W 2, .•• , wn ) :

A = P-1AP = P-I(AwI,Aw2,oo.,Awn) = P-1(AIWI,A2W2, ... ,AnWn)

= P- 1pD(AI,A2,oo.,An) (3.5.5)

in which D(AI' A2,"" An) is the diagonal matrix with AI' A2"'" An along the main diagonal. From the calculation (3.5.5), it is now clear that a solution to (3.5.3) and (3.5.4) is (3.5.6) and hence a solution to (3.5.1) and (3.5.2) is computed to be x(t) = (WI> W2,"" wn)D(e"'I, e"2 1, ... , e"nl)zO -- (e"t1w 1e"2l ' . w2'···, e"n1wn )ZO

(3.5.7) Moreover, it is apparent that the fundamental matrix (t) for (3.5.1), as discussed in Section 3.2, could be taken to be (3.5.8) Observe that if A is real, then although some of the AS and ws may be complex, (t) will remain real. Computation of (t), using (3.5.8) for those matrices to which it applies, is sometimes called the method of eigenvectors.

There is a notable case in which (t) can be computed with minimal effort. Consider the nth-order scalar constant coefficient equation

Remark 3.5.1

(3.5.9)

62

Linear Constant-Coefficient Equations: Computation

3.

The coefficient matrix of the first-order form of (3.5.9) is . 0

0

0 . ... 1 0 ...

1 0

0

0 0

(3.5.10)

A=

0 -a,

. ... -anFl . . . . . 0

1

0 -a2

0 1 -a,

For such a matrix, it will be shown in Chapter 4 that @ ( t ) = Y(t)Y-'(O),where Y ( t )is the matrix function whose first-row elements are those assigned to A by the appropriate correspondence rule (written in any order) and whose successive rows are generated by differentiation of the row immediately above it.

Remark 3.5.2 The solution to the initial-value problem in which the value xo of x is prescribed at t o , i= A x X(t0) =

+ f(t),

(3.5.11) (3.5.12)

xo,

is given by the modification of (3.2.10)

+

x ( t ) = @(t- t O ) x o

@(t - o)f(o)do.

(3.5.13)

EXERCISES

1. Verify that (3.5.13) satisfies (3.5.11) and (3.5.12). 2. Show that Eqs. (3.5.11) and (3.5.12) in complex n-space V" with A a matrix over 3 ' can be converted into a problem in g2"with a real 2n x 2n coefficient matrix. This points out that equations in complex n-space could be treated by the ABC approach, using real arithmetic. 3. Solve the initial-value problem (3.1.1), (3.1.2) in which

using the method of Section 3.3. 4. Solve the problem of Exercise 3, using the method of Section 3.4.

3.5 Additional Comments about Computing and Changing Variables

5.

63

Solve the initial-value problem (3.5.1), (3.5.2) in which 1

~J,

o

-8 -5

(a) by the ABC algorithm, employing Eq. (3.3.2), (b) by the method employing Eqs. (3.4.4) and (3.4.5), (c) by the method of Remark 3.5.1. Does the method of eigenvectors, employing (3.5.7), also provide the solution? 6. Compute the closed-form solution to (3.5.1), (3.5.2) in which

A

~

U_~ !]

(a) by the ABC algorithm employing Eqs. (3.3.2), (3.3.3), and (3.3.4), (b) by the method of Section 3.4, (c) by the method of eigenvectors (3.5.7). 7. Find the solution to x = Ax, x(1) = XO with A as in Exercise 6. 8. Solve the initial-value problem eb = AlP, lP(O) = 14 in which

A

=

0 1 1-1] [-~ -~ -~ ~. -4 -4 -8

9.

Solve the matrix differential equation problem eb = AlP + lPA with

A=[_~ and

0

~J

Chapter

4

Linear Constant-Coefficient Equations: Theory

The heuristic reasoning which led to the equations purportedly computing the closed-form solution as discussed in Chapter 3 will now be given a rigorous foundation. As is the case with many existence theorems, a solution to the initial-value problem will be shown to exist by means of a limit process, namely, through the exponential function of a matrix. This solution is proved to be unique. The requisite matrix solution C to the equation AC - CB = 0 is constructed from the solution to the initial-value problem, and a representation of the solution of the type computed by the ABC algorithm is thereby shown to exist. . The Jordan form of a matrix applied to the exponential representation of the solution illuminates the choice of elementary functions assigned by the correspondence rules and at the same time reveals precisely those characteristics of the coefficient matrix which govern the qualitative aspects of solution behavior of primary interest in applications. 4.1 Preliminary Remarks

A rigorous and thorough treatment of Eqs. (3.1.1) and (3.1.2) of Chapter 3 will now be developed. It is the homogeneous equation that requires the most attention since the nonhomogeneous problem involves only an additional integration. 64

65

4.2 Some Elementary Matrix Analysis

Assuming that the construction of the closed-form solution presented there is correct, one is struck by the fact that such a limited class of elementary functions is involved. A clue as to why this is the case is already present in the one-dimensional problem in which in beginning calculus the solution to the scalar equation x = ax is found to be the exponential function ce", usually defined as a power series. Hence, the notions of convergence of series, their differentiability, etc., arise. It will soon become apparent that one ofthe main reasons for working with differential equations in first-order form (at least for linear equations) is that the higher-dimensional problem can be treated along somewhat similar lines. (The analogy actually extends well beyond finite-dimensional problems.) The first step will be to extend some familiar notions of real analysis to matrices.

4.2 Some Elementary Matrix Analysis In this chapter, :F denotes any subfield of the complex numbers and, as before, :F" is the collection of n-vectors over :F with the standard algebraic operations. When there is a need to be more specific, :F is replaced by ~ when considering the complex numbers and by 9t when dealing with real numbers. Recall the basic properties of the absolute-value function on :F:

o s Ix I

with equality if and only if

lexl = lei I~I, [x - yl ~ Ixl + Iyl

x = 0,

(4.2.1) (4.2.2) (4.2.3)

for all e, x, and y in :F. A real-valued function with these properties can be extended to higher dimensions. Definition

For x in :F" with coordinates

Ixl =

max 1 ~i~n

Xi'

lXii,

define (4.2.4)

called the norm of x. Remark 4.2.1 Note that when n = 1, the norm of x and its absolute value agree. Moreover, the norm function defined by (4.2.4) continues to satisfy (4.2.1)--(4.2.3), where in (4.2.2) e denotes any element in :F while x is in :F". (The student should verify these claims.) To a considerable extent, the notion can be extended one step further to m x n matrices.

66

4. linear Constant-Coefficient Equations: Theory

Definition

For A an m x n matrix over

IAI

= max

1"1,, 1

§',

define

IAxl,

(4.2.5)

in which x denotes a variable in §'n. IAI as defined by (4.2.5) is called the operator norm of A relative to the norm defined on §'n.

Proposition 4.1

The operator norm of an m x n matrix A over §' is a realvalued function and satisfies the conditions

o ~ IAI if and

with equality holding

(4.2.6)

if A = O.

only

leAl

=

IcllAI

(4.2.7)

for every c in §'.

IA - BI s IAI + IBI

(4.2.8)

for all m x n matrices A and B over §'.

IAxl

~

IA/lxl

(4.2.9)

for all x in §'n.

IABI ~ IAIIBI

(4.2.10)

for all matrices A, B over §' of compatible size.

JAI = IIn l = Proof.

For all x in

§'n

n

L IAul·

max l~i~mj=l

l

(4.2.11)

(n=1,2, ...).

satisfying

Ixl

~ 1,

n

n

n

j;l

j;l

j;l

I L Auxjl ~ L IAul ~ L IAi.jl (i

=

(4.2.12)

(4.2.13)

1,2, ... ,m), where i* is chosen so that

max l~i~m

Defining y* in

§'n

n

n

j=l

j=l

L IAul = L IAiJ·

(4.2.14)

by

I

yj =

sgn(A i• j

(4.2.15)

)

(j = 1,2, ... , n), note that y*/ ~ 1, and furthermore n

n

n

L1 IA jI = L1 A,.jyj = Ii>L1 Ai.jYjl·

j=

i•

j;

(4.2.16)

67

4.2 Some Elementary Matrix Analysis

Thus it follows from (4.2.13) and (4.2.16) that n

max l~i~m

IL

n

j=l

Ai';xjl::; max l~i~m

IL

(4.2.17)

Aijyjl

j=l

for all xin fFn with Ixl ::; 1. From (4.2.17) it is apparent that the left-hand side achieves a maximum over x and moreover from (4.2.13) on one hand and (4.2.16) on the other it follows from (4.2.17) that ,

n

n

L

max max I Aijxj! Ixl,;;l l,;;i,;;m j=l

=

L IAi.jl, j=l

(4.2.18)

which, according to (4.2.2), (4.2.5), and (4.2.14), is the required inequality n

L IAijl·

IAI = max

(4.2.19)

j=l

l~i~m

Properties (4.2.6) and (4.2.7) are obvious consequences of (4.2.19). Since (4.2.20) for numbers n

L IAij j=l

n

Bijl ::;

n

L IAijl + j=l L IBijl, j=l

(4.2.21)

and thus by formula (4.2.19), etc., IA -

BI =

n

max l~i~m

s

L

j=l

IAij - Bij!::; max l~i~m

n

max l~i~mj=l

L IAijl +

n

n

j=l

j=l

[L IAijl + L IBijl]

n

max

L

IBijl = !AI

l~i~mj=l

+ IB!,

(4.2.22)

proving (4.2.8). To prove (4.2.9), apply (4.2.2) and (4.2.5) to get

IAx'l =

IxllA

-,xII::; Ixl lyl,;;l max IAyl x

=

IxllAj,

(4.2.23)

which is valid for x#-O in fFn. (Note that the maximum in (4.2.23) was shown to exist in the previous part of the proof.) Thus (4.2.9) follows from (4.2.23) for x#-O and also holds (with equality) for x = O. This finishes the proof of (4.2.9). For

Ixl s

1, by applying (4.2.9) twice,

I(AB)x/

= /A(Bx)

I::; IAllBxl s IAllBllxl ::; IAIIBI,

(4.2.24)

68

4.

which maximized over

Linear Constant-Coefficient Equations: Theory

Ixl :5: 1 yields IABI :5: IAIIBI,

(4.2.25)

concluding the proof of (4.2.10). Equality (4.2.11) was already proved and (4.2.12) is a direct consequence of (4.2.11). This' concludes the proof of Proposition 4.1. The main reason for introducing the absolute-value function on /F is for convenience in talking about convergence and continuity. It is assumed that the reader is familiar with these ideas in the case of real or complex numbers; i.e., in dimension one. Norms are introduced in order to expand these notions to higher dimensions, where it soon becomes clear that formally there is little difference from the one-dimensional case.

Definition For A k (k = 1; 2, ...), a sequence of m x n matrices over /F, the sequence is said to converge to an m x n matrix Aoo as k - 00 if (4.2.26) as k - 00. The matrix Aoo is denoted by lim k _ oo Ak> and one writes A k as k- 00.

Aoo

Remark 4.2.2 Recall that a sequence of numbers ak is called Cauchy if for each real e > 0 there exists a real K. such that k ~ K. and p ~ K. implies that lak - api < s, The Cauchy criterion (theorem) says that a sequence of numbers is convergent if and only if it is Cauchy. The result is important because it can be used to prove a sequence convergent without the difficult task of finding the value of the limit. It is easily seen from formula (4.2.11) that a matrix sequence A k - Aoo as k - 00 if and only if (Ak)ij - (A oo )ij as k - 00 for each i, j. That is, convergence of a matrix sequence is equivalent to elementwise convergence. Thus it is clear that the Cauchy criterion is also valid for matrix sequences.

Definition the type

If A k is a sequence of m x n matrices arising as a matrix sum of

Ak =

k

L e,

(4.2.27)

i=O

and is convergent, then its limit is denoted by L~

0

B;.

The text follows the standard abuse of notation wherein the same symbol is used to denote the series, whether or not it converges. Series can be used to define matrix functions in much the same way as they are used to define scalar functions. The exponential function plays a particularly dominant role in this chapter.

4.2

69

Some Elementary Matrix Analysis

Definition For A any n x n matrix over of A is defined by the series

§,

the exponential function e A

(4.2.28)

The series is convergent for all A. This can be seen by computing the difference between two terms in the sequence of sums to get

Ai PAil IAli IAli "--"-<" L...., L...·f L... -<" L... - - 0 I k

k

i;O l.

i;O l.

00

i;p+l

'f

-

l.

i;p+l

as

'f

l.

P':"

00,

(4.2.29)

since the series for eX converges for all x. Thus convergence of the series (4.2.28) follows by the Cauchy criterion. The next lemma proves a more general result. Definition

An m x n matrix series over

§,

(4.2.30)

is called absolutely convergent if 00

L I(Bk)ijl k;O

(4.2.31)

is convergent for each i = 1, 2, ... , m; j = 1, 2, ... , n. Lemma 4.2.1 Let P and Q be matrices over § of size p x m and n x q, respectively. If the series (4.2.30) is absolutely convergent, then PSQ is likewise absolutely convergent. Proof. i andj.

Let

IX = max!Pij!, fJ =

maxlQijl with the maxima taken over

Since the series (4:2.30) is assumed to be absolutely convergent, for each real e > 0 there exists an integer K. such that 00

L I(Bk)ijl k;.

< e

(4.2.33)

for K. ::0:; s (i = 1,2, ... , m; j = 1,2, ... , n). From (4.2.32) and (4.2.33) it follows that 00

00

L I(PBkQ)ij I ::0:; IXfJ L L L J(Bk)'!'21

k=s

S2

SI

k==s

::0:; IXfJ

LL(. = 52

SI

IXfJmn(..

(4.2.34)

70

4.

Linear Constant-Coefficient Equations: Theory

Since e > 0 is arbitrary, the required absolute convergence follows directly from (4.2.34).

Theorem 4.2.2 Suppose that a power series <Xl

fez)

=

L

akzk

(4.2.35)

k=O

over ((j has a radius of convergence p. Let A be an n x n matrix over ((j. A sufficient condition for the absolute convergence of the matrix series f(A) is that either AP = 0 for some positive integer p (in which case A is called nilpotent) or IAI O. Let A = PJP- 1 where J is the Jordan form of A. If f(J) is absolutely convergent, then by Lemma 4.2.1 it follows that f(A) will be absolutely convergent, since Pf(J)P- 1 = f(PJP- 1 ) = f(A). Thus without loss of generality, assume that A is in Jordan form. Furthermore, since power series preserve the block diagonal structure, it is sufficient to assume that A is an elementary Jordan matrix

(4.2.36) in which Z is the n x n nilpotent matrix with all zero elements but for a superdiagonal of ones as in (2.9.4). By the binomial theorem, which applies since AIn and Z commute, (4.2.37)

(4.2.38) Since a scalar power series can be differentiated termwise within its circle of convergence, f(S)(z)

= s!

f ak (k)s ~-s

(4.2.39)

k=s

for Izi < p. Consequently the last term and hence the first term in Eqs. (4.2.38) have a limit as K -+ 00 if IAl < P and, moreover, the limiting equation

71

4.2 Some Elementary Matrix Analysis

shows that (4.2.40) The absolute nature of the convergence is a consequence of that of power series and of the position of the ones in ZS. This concludes the proof of Theorem 4.2.2. Since the radius of convergence for the series defining eZ is p = 00, the convergence of the series for eA for every n x n A also followsas a direct corollary to Theorem 4.2.2. Remark 4.2.3

EXERCISES

1. Compute

(a)

[~-~

3 -2

(c)

1

AI, the norm of A, for the following matrices:

-~],

(b)

2

[_~ ~ _~ ~l, 0 0 -2

[1 2 -2 3J,

(d)

nl

dJ

2. Applying the Cauchy criterion, show that for any n x n matrix over with IAI < 1, . (a) Lo A k is convergent, (b) In - (In - A)L~ A k - 0 as K - 00, (c) LO A k = (In - A)-1. Determine whether or not (c) is valid for

A =

[~ ~J

and compute IA I. ·Is your conclusion consistent with Theorem 4.2.2? 3. Consider the power series fez)

=

1

+ 2z + 24 z2 + ... + 2n2zn + ....

(a) Determine the radius of convergence of the power series. (b) Compute f(A) for this power series where the matrix is

A=[~ ~ ~J. 000

C(j

72

4.

linear Constant-Coefficient Equations: Theory

(c) Are your results for (a) and (b) consistent with Theorem 4.2.2? 4. In the following statements, A denotes any n x n matrix over the complex field except where otherwise indicated. Determine which are true. Justify your conclusions. (Hint: Apply Theorem 4.2.2 where appropriate.) (a) leAl::::;; e lAI. (b) (eAr = e'A (r = 0, ± 1, ±2, ...). (c) (d)

(eA)-1 = e- A. A) det(e ¥erlAer2A = e(r1+r2)A, all real r l and r2' eAeB = ~+B where A = ~], B = [~

°

(e) (f) [g gl 5. The Euclidean norm on fJi" is defined by the formula )xl e = ~ all x in fJi". Find the smallest positive numbers PI and P2 such that

for

for all x in fJi". Note that a sequence in fJi" is convergent relative to one norm if and only if it is convergent relative to the other. The operator norm of a real n x n matrix A induced by the Euclidean norm is defined by the formula IAle = maxlAxl e, where the max is over x in fJi" with [xIe ::::;; 1. Can you find a formula explicitly computing IA/ e from the elements of A? Show that there exist positive numbers rl and r2 such that for all real n x n matrices A. Can you compute values of r l and r2?

4.3 Existence, Uniqueness, and Solution Representations Using etA.

If M(t) is anm x n matrix-valued function defined on an interval (ex, /3), then the derivative dM(t)jdt at a point t in (ex, /3) is defined to be the limit dM(t) = lim M(t dt h-O

+ h) h

- M(t)

(4.3.1)

when it exists. Note that differentiation can be performed elementwise and that this is consistent with the elementwise differentiation of n-vector functions appearing in earlier chapters. An integral of a matrix-valued function can likewise be defined as a matrix limit and integration can be done elementwise. Many of the standard formulas of elementary calculus continue to hold for matrix-valued functions such as

4.3 Existence, Uniqueness, and Solution Representations Using

73

etA

They will be applied without further mention since they are all easy exercises for the reader. By replacing A in (4.2.28) by tA, the matrix-valued function of t, (4.3.2) is defined for all t in ( - 00, (0). The next theorem shows that this matrix function can be differentiated by using the same formula that applies in the scalar case. For any n x n matrix A over C(j, the function etA has a derivative satisfying the equation

Theorem 4.3.1

(4.3.3) for all t in (- 00, (0). Proof.

~[e(t+h)A h

For all real t and h =I 0, _ etA] _ Ae tA = !etA[e hA _ I _ hA] = ~etA h h =

~ et~hA)2

~

I (hA)k k! 2

(hA1: - 2

(4.3.4)

Restricting Ihl < 1, it follows that 1 Ih[e
~

IhlletAleIA'_O

(4.3.5)

as h - 0, and the proof is finished. Remark 4.3.1 It follows as a corollary to Theorem 4.3.1 that differentiated any number of times with k

d _etA = AketA

r

drk

e'" can be (4.3.6)

'

and if A is invertible, then

etA dt = A- 1[ebA

-

eaAJ.

(4.3.7)

Note that Theorem 4.2.2 can be applied to obtain an alternative proof of Theorem 4.3.1 as well as (4.3.6) and (4.3.7).

74

4.

Linear Constant-Coefficient Equations: Theory

Corollary 4.3.2 Let A be an n x n matrix over .fF and f: (ct, P) _.fF n be

continuous. For each to in (e, P) and initial-value problem

x=

X

O

Ax

in .fFn there exists a solution to the

+

(4.3.8)

f(t),

x(to) = xo,

(4.3.9)

given by the formula x(t)

=

e(t-,to)AxO

+

rt e(/-lT)A f(a) da,

ct < t <

Jlo

p.

(4.3.10)

Proof. Certainly (4.3.9) is satisfied, and (4.3.8) follows directly from differentiation and application of Theorem 4.3.1. The continuity ensures the existence and differentiability of the integral in (4.3.10).

Corollary 4.3.3

There exists a solution to the initial-value problem

x=

(4.3.11)

Ax,

(4.3.12) given by - 00

< t<

(4.3.13)

00.

Remark 4.3.2 The next task will be to prove that the solutions to the problems in Corollaries 4.3.2 and 4.3.3 are unique. Although it has not yet been proved, upon comparing (4.3.10) with (3.2.10) one might suspect that the
Lemma 4.3.4 Gronwall's Inequality. every continuous solution to x(t) ,;$; ct +

p

f:

x(a) da,

o ;$; t <

T

p~

0,

(4.3.14)

;$; 00,

satisfies the inequality

o ;$; t <

(4.3.15)

T.

Proof. There is nothing to prove if p = 0, so assume that plication of (4.3.14) by e-(Jt gives

~[e-(Jt

I

x(a) daJ

;$;

cte-(Jt,

0

;$;

t < T.

p>

O. Multi-

(4.3.16)

75

4.3 Existence, Uniqueness, and Solution Representations Using e'A

Integration of (4.3.16) shows that

I

e- Pt

~

x(oJ da

~(e-Pt

-

-

(4.3.17)

1),

and applying (4.3.14) to eliminate the integral in (4.3.17) gives x(t) -

p

Q(

< _ ~(1

-

_ elit)

p

,

os t <

T,

(4.3.18)

from which (4.3.15) follows directly to conclude the proof. Theorem 4.3.5

The solutions given in Corollaries 4.3.2 and 4.3.3 are unique.

Proof. It is sufficient to consider Corollary 4.3.2. Let z(t) = x(t) - y(t) where x(t) and y(t) are both solutions to (4.3.8)and (4.3.9). It is easy to check that z(t) satisfies :i

=

(4.3.19)

Az,

z(to) = 0,

Q(

< t < p.

(4.3.20)

By integration of (4.3.19) and (4.3.20) it follows that z(t + to) =

for 0 ~ t < T

= P-

I

Az(u + to) da

(4.3.21)

to, and hence that

/z(t +. to)1

~ IAI S: jz(u

+ to)1 do,

(4.3.22)

Application of Lemma 4.3.4 to (4.3.22) shows that z(t + to) = 0 for o ~ t 0 any n x n matrices over~, to the initial-value problem

d> =

A,

(t o) = <1>0

Corollary 4.3.7 i.e., for which AB

the unique solution

(4.3.23) (4.3.24)

For A and B any two n x n matrices over <{j which commute; = BA,

(4.3.25)

76

4.

Linear Constant-Coefficient Equations: Theory

EXERCISES

1. Find three proofs of the uniqueness concluded in Corollary 4.3.6. 2. Prove Corollary 4.3.7 by considering the initial-value problem = A, <1>(0) = eB • 3. Prove that the unique solution to the matrix equations = A

+ B,

(4.3.26)

<1>(0) = <1>0 (4.3.27) is (t) = etA°etB. 4. Compute etA for the following matrices, using the series defining the exponential of a matrix: (a)

(c)

A =

A =

[_~

~J

(b) A =

[

00 0I 0]I ,

000

[-~ -~ ~]. o

0-2

5. Compute etA for the following matrices by solving the appropriate initial value problems.

6.

(a)

A =

[_~

~J

(c)

A =

r~ ~ ~l,

L1

0

oJ

(b)

A =

(d)

A =

[-1 1J

-1 -1 '

[~o -1~ 0~J

.

Consider the second-order, initial-value problem in X

{)In,

+ A x = 0, 2

x(O)

=

x(O)

= VO

(4.3.28)

0,

(4.3.29)

where

VO = Ac,

cE

{)In.

(4.3.30)

(a) Define sin A as a power series. (b) For which n x n matrices A does the series converge? (c) Prove that x(t) = [sin(tA)]c is a solution to (4.3.28)-(4.3.30). (d) Prove that the solution is unique (two ways). 7. Compute the elements in the matrix function !(A) = cos(itA), where A =

[=~ -~ ~]

002

(Suggestion:

cos cot = -!<e iwt

+ e- iwt ). )

and

4.4

4.4

77

The Structure of elA

The Structure of e'"

This section takes a deeper look into the problem of determining, to the extent possible, the manner in which the elements in the matrix function etA depend on the matrix A. Consider an n x n matrix A over a subfield 9of W in which the characteristic polynomial pa splits. By Theorem 2.9.1 A

=

(4.4.1)

PJP-'

in which J is the Jordan form of A and P is a nonsingular matrix over 9, From the series definition of the exponential of a matrix it is easy to see that

(4.4.2)

etA = Pe'Jp-'.

In view of (4.4.2) and the fact that J has a block diagonal form J = diag[J,, J 2 , . .., J , ] ,

(4.4.3)

it follows that erJ = diag[eZJ', e l J 2 , .. ., e l J s ] .

(4.4.4)

Each Jk is itself block diagonal with blocks of the type LZ + Z as indicated in (2.9.4). Hence computation of elJk reduces to computing the elements of the matrix

(4.4.5)

e'(kkk'+Z)= etkkelz

Since Z is a nilpotent matrix with all zero elements but with the superdiagonal terms ones, it folIows readily that

I - t - -t2 t3 l ! 2 ! 3! t t2 1 I ! 2!

0 r-1

...

+k

... (4.4.6)

0 0

1

... 0

0

in which r is determined by the size of Z which is r x r. It follows from (4.4.5) and (4.4.6) that written out in terms of its columns, er(kkr+z)= et l Y e l ,t e ,

f - 1

+ e 2 , .. ., (r - l)!el + (r - 2) ! e2 + ... + en), (4.4.7) ~

tr-2

~

78

4.

linear Constant-Coefficient Equations: Theory

in which the e, are the columns of the identity matrix I., Some important observations are summarized in the following remarks. Remark 4.4.1 The elements of etA consist of linear combinations, with coefficients in ff, of functions of the type

tk _etA)

k!

(k=O,l, ... ,rj-l;j= 1,2, ... ,s),

(4.4.8)

in which the Aj are the distinct eigenvalues of A and r j is the multiplicity of Aj in the minimal polynomial of A. Remark 4.4.2 For those eigenvalues Aj = IY.j + ico, which are complex, the functions in (4.4.8) can be split into real and imaginary parts. Hence if A is real, then although Pin (4.4.1) might contain complex numbers, the elements of etA must be real and consequently will consist of linear combinations, with real coefficients, of the functions

(4.4.9)

= 0, 1, ... , rj -

(k

4.5

1; j

=

1, 2, ... , s).

The Theoretical Foundation of the ABC Algorithm for Computing etA

In Section 4.3, we showed that the problem of computing the unique closed-form solution to the problem in PAn,

x=

Ax,

(4.5.1) (4.5.2)

reduced to that of computing etA, and the solution of the nonhomogeneous form of(4.5.1)could then be gotten by an additional integration. The solution of (4.5.1) and (4.5.2) was shown to be x(t) = etAxO.

(4.5.3)

The rigorous details needed to substantiate the algebraic procedure presented in Chapter 3 for computing cI>(t) = etA can now be given. Recall the procedure: Step 1. Compute the eigenvalues of A (possibly repeated and complex). Step 2. Determine the real n-vector function y(t) by applying the corre-

79

4.5 Theoretical Foundation of the ABC Algorithm for Computing erA

spondence rule to the eigenvalues according to the scheme

I ' ...1 t At t At "'t. rear x -e e 'De '2!e '''''(m_1),e, 2

complex A = oc

t'"-1

± ico --+- e" cos rot, e" sin rot; It -t e~ cos rot I! ' I!

tm -

e~

Ism ' rot .

'

tm -

1

1

... ; (m _ 1), e" cos rot,(m _ 1), e" sin rot ; in which m is the multiplicity of A in the characteristic polynomial. Step 3. Determine the real n x n matrix B such that By(t) = y(t). Step 4. Obtain the general n x n matrix solution C to the equation AC - CB = 0 in terms of an n-vector of arbitrary constants c. Step 5. By inspection, determine the n x n matrix function Y(t) such that Y(t)c = Cy(t). Step 6. Invert Y(O) to compute (t) = Y(t)y- 1(0) = e": The appropriate lemmas to be used in the argument will now be stated and proved. Definition A finite collection of $'-valued functions cl>1(t), cl>it), ... , cl>m(t), defined on ( - ex), oo), is called linearly independent over $I' if (4.5.4) for all t and fixed c, in $I' implies Cl

Lemma 4.5.1

=

C2

= ... =

Cm

= O.

(4.5.5)

The collection offunctions

f'e"' j l

(4.5.6)

(k = 0, 1, ... , rj :- 1; j = 1, 2, ... , s), with the Aj distinct elements of re, is linearly independent over re. Proof. Note that the equations resulting from setting linear combinations of the functions (4.5.6) equal to zero are precisely the equations of the type pl(t)e... ,1

+

pit)e... ,1

+ '" + p.(t)e...•1 = 0, t over re, and this class

(4.5.7)

in which the Pi are polynomials in of equations is invariant under differentiation. In fact, since exponential functions never take on the value zero, and since the Ai are distinct, by multiplying by e" . . ,/,

80

4.

it can be assumed that A1 any equation of the type

=

Linear Constant-Coefficient Equations: Theory

O. Thus the problem reduces to showing that

Po(t) + P1(t)eA1t +

0

••

°

+ pit)eAnt =

(4.5.8)

for all t, with Pj polynomials in t over C(j and the Aj # 0 and distinct, implies that all Pj = O. The proof is carried out by induction on n. Clearly the result is true for n = O. Thus assume that (4.5.8) implies that all Pi = 0 whatever the Aj as long as they are nonzero and distinct. The objective is to prove the corresponding statement for poet) + P1(t)eA1t + ... + pit)e Ant + Pn+1(t)e An+ 1' = O. (4.5.9) Let no be the degree of Po in (4.5.9) and then differentiate (4.5.9) no + 1 times to get the equations p~l(t)

(k

+

p~(t)eAlt

+ ... +

p~+1eAn+lt

=

°

(4.5.10)

= 0, 1, ... , no + 1). Note that (4.5.11)

where o

Pj (j = 1,2, ... , On

=

(4.5.12)

Pj

+ 1). But in particular, since pg'0+ 1)

P'i°+ 1(t) + Pzo+ 1(t)el'2t + ... + p~'?t.+/(t)elln+l1

=

0, = 0,

(4.5.13)

in which the J1.j = Aj - A1 are nonzero and distinct. Hence by the induction hypothesis (4.5.8), (4.5.14) (j = 1,2, ... , n

+ 1), and thus from (4.5.11), (pjoPl + AjpjD = 0

(4.5.15)

°

= 1,2, ... , n 0+ 1). But the unique solutions to the linear first-order differential equations (4.5.15) are exponential functions, and since Aj # and P~;o is a polynomial, it follows that p'r = O. By repeating the same argument, it follows by induction on k that

(j

P' =

0

(4.5.16)

°

= 1,2, ... ,n + 1;k = no, no - 1, ... ,0),andhenceby(4.5.12),allpj = in (4.5.10). This completes the induction, and the proof of Lemma 4.5.1 is complete.

(j

4.5 Theoretical Foundation of the ABC Algorithm for Computing etA

Corollary 4.5.2

81

The collection offunctions

(4.5.17) (k = 0, 1, ... , rj - 1; j = 1, 2, ... , s) with !Xj + ico, distinct is linearly independent relative to the real coefficient field 91..

Proof:

The result follows from Lemma 4.5.1 and the fact that 1. 2

t

cos Wot = -(e""j! + e-,roj ) J

0

'

. wot = -1 (e'"! t-O t e-,roj ) J 2i .

SIn

0

(4.5.18) (4.5.19)

The details are left to the reader. Remark 4.5.1

pendent over

Obviously, the collection (4.5.17) remains linearly indeC(J.

Remark 4.5.2 (Those readers who have studied matrix theory in the context of linear algebra can skip this remark.) The basic idea of a linear combination of objects has appeared in no less than three contexts up to this point; namely, linear combinations of n-vectors over $', m x n matrices over $', and $'-valued functions have been defined. Mathematicians have abstracted this common structure into an axiomatically defined algebraic system called a vector space over a field. The three mentioned examples and many others fall into this general framework. Important maps between the prototype vector spaces such as multiplication of n-vectors by a fixed m x n matrix or time differentiation of functions are abstracted by the notion of linear transformation of one vector space into another. Linear algebra develops the important concepts and results pertaining to vector spaces and linear transformations between vector spaces. This has the advantage of efficiency in that definitions of bases, linear independence, etc., are made once and for all, and when a scientist or mathematician encounters a system bearing the structure, he can apply the abstract theory and concepts to it without doing much additional work. Essentially, all matrix theory fits neatly within the framework, and the part of the standard theory that is nothing but convention becomes more apparent. Thus the uninitiated should now be well-motivated to take a few hours to work through an elementary text such as [12] and, it is hoped, include in his program of studies a course based at the level of, say, [11J. The material is important for understanding some of the details presented in the remainder of this section, and it has an even more important bearing on certain parts of the remaining chapters.

82

4. linear Constant-Coefficient Equations: Theory

Lemma 4.5.2

Consider the equation Cy(t) = etAv

(4.5.20)

for all t in f:Jl, in which y( t) is the n-oector function determined from the eigenvalues of A by the correspondence rule (3.2.2). For each v in f:Jln (4.5.20) has a unique n x n matrix solution C and the equation thereby determines a nonsingular linear transformation of f:Jln onto an n-dimensional linear subspace of the real n x n matrices. Proof. The existence of at least one solution C for each v in f:Jln follows directly from Remark 4.4.2. 1f(4.5.20)had two solutions C 1 and C2 for some fixed v in f:Jln, then it would follow that

(4.5.21) for all t, contradicting the linear independence proved in Corollary 4.5.2. Thus (4.5.20) defines a mapping, and it is linear; for if both C1y(t) = etAvb

(4.5.22)

C 2y(t) = etAv2

(4.5.23) (4.5.24)

for all t in f:Jl. The map is nonsingular, for clearly in (4.5.20) C = 0 implies that v = O. Since f:Jln has dimension n, the nonsingularity implies that the map has rank n, and this completes the proof. Proof of the ABC Algorithm.

A real n x n matrix C satisfies (4.5.20) for

some v in f:Jln if and only if AC - CB

= O.

(4.5.25)

Equation (4.5.25) is a consequence of the fact that y = By, which is seen by differentiating (4.5.20) and appealing to the linear independence of the coordinate functions of y(t). To show the reverse implication, multiply (4.5.25) by y(t) to get d

d/Cy) - A(Cy) = O.

(4.5.26)

But this equation has the unique solution Cy(t) = etAv

(4.5.27)

where v = Cy(O). This proves that (4.5.25) has the same n-dimensional linear space of solutions as does (4.5.20). Thus if (bb b 2, ... , bn ) is any real non-

83

4.6 Explicit Formulas for C(c) and e'A

singular n x n matrix (i.e., the b, constitute a basis for 9l n ), then the general solution C to (4.5.25) can be written as a linear combination C = C1Ci

for arbitrary real scalar Eq. (4.5.20)

Ci

+ C2C2 + ... + c.C;

(4.5.28)

in which the Ci are a basis of matrices defined by (4.5.29)

(i = 1,2, . 00' n). That is,

Cy(t) = CietAbl + C2etAb2 + ... + cnetAbn = etA(b 1 , b 2, ..• , bn)c = Y(t)c,

(4.5.30)

in which (4.5.31)

and c is the n-vector over 9l with cT = (c., C2"'" cn)' Hence all of the steps in the ABC algorithm are substantiated, showing that it computes (t) = Y(t)y- i(O) = e tA(bi, b2, . 00' bn)(b1 , b 2, ... , bn)-i = etA (4.5.32) as intended. EXERCISES

1. Show that for any n x n matrix A over

rc,

dette") = etr(A).

(4.5.33)

2. Let Ai' A 2, and <1>0 be any n x n matrices over ~. Consider (t), the unique matrix solution to the initial-value problem

=

A 1 <1>

<1>(0) = <1>0.

+

A 2 ,

(4.5.34) (4.5.35)

(a) Compute det (t). (b) Show that det (t) = 0 if and only if det <1>0 = O. (c) Show that if det <1>0 :F 0, then det (t) - 0 as t - 00 if and only if tr(A 1

4.6

+

A2 ) <

o.

Explicit Formulas for C(c) and etA

For a given matrix A, once the decision has been made as to whether to use correspondence rule (3.2.2) or (3.2.28), the associated yet) and Bare automatically determined.

84

4.

Linear Constant-Coefficient Equations: Theory

Assuming that the choice has been made, the dependence of the general solution C(c) of

AC - CB

=

0,

(4.6.1)

defined by the equation (4.6.2) on the n-vector c, and of etA on y(t), is rather explicitly shown in the following theorem. Theorem 4.6.1 C(c) = (c, Ac, ... , An-1c)(yO, Byo, ... , B n- 1yO)- \

in which yO

(4.6.3)

= y(O), and = (A1y(t), A 2y(t), ... , Any(t»,

(4.6.4)

= (ei, Aei"'" An-1ei)(Yo, Byo, ... , Bn-1yO)-1,

(4.6.5)

etA in which Ai

(el,e2, ... ,en) = In(i = 1,2, ... ,n). Proof. First the matrix inverse appearing in (4.6.3) and (4.6.5) is shown to exist. Select any n-vector ~ and consider (4.6.6) z(t) = (y(t), By(t), ... , Bn- 1y(t»~. It is apparent that z(t) is the unique solution to the initial-value problem z';" Bz,

(4.6.7) (4.6.8)

Thus if z(O) = 0, then z(t) = 0 for all t in f7t, and moreover, by computing successive dot products of(4.6.6) with the columns of In' it would follow that ~ = O. This proves that the matrix appearing in (4.6.8) is nonsingular. From successive differentiation of (4.6.2) at t = 0, the matrix equation

CCVo, By", ... , Bn-1yO) = (c, Ac, ... , An-1c)

(4.6.9)

is obtained, and thus (4.6.3) follows from (4.6.9). Substitution of (4.6.3) into (4.6.2) and the definition (4.6.5) show that

etAc = (c, Ac, ... , An-1c)(yO, Byo, ... , Bn-1yO)-ly(t) n

=

L i= 1

c;(ei,Aei, ... , An-1e;)(yO, Byo, ... , Bn-1yO)-ly(t)

n

=

L

i= 1

ciAiy(t) = (A1y(t), A 2y(t),.··, Any(t»c

(4.6.10)

4.6

85

Explicit Formulas for Ck) and erA

for all n-vectors c, implying (4.6.4), and Theorem 4.6.1 is established.

Remark 4.6.1 An elementary calculation shows that formula (4.6.4) is invariant under the transformation y(t) -+ My(t), B --t MBM-' for any nonsingular n x n matrix M over the complex field. It follows directly from this observation that the correspondence rule (3.2.2) applying to real matrices A could be replaced by {Al, A 2 , . . ., A,} --+ My(t) for any real n x n nonsingular matrix M and that the correspondence rule (3.2.28) applying to complex matrices A could be replaced by {Al, A 2 , . . ., A,} + My(t) for M any n x n nonsingular complex matrix, with the consequence that all the derived results would remain valid. The two original correspondence rules have the virtue that their associated B matrices are particularly simple. Recall that for any monic polynomial p ( x ) = x"

+ clx"-I + c2x,-2 + ... + c,

(4.6.11)

over a field F, the associated matrix

(4.6.12)

is called the companion matrix of p .

Remark 4.6.2 By computing the matrix B arising from each of the two correspondence rules (3.2.2) and (3.2.28),it is easy to check that the matrices A and B have the same characteristic polynomials. By Remark 4.6.1 it follows that this relation would be preserved as well for all other correspondence rules of the type discussed. Corollary 4.6.2

For any real or complex n x n matrix A etA = (z,<(t), z 2 t ( t ) ,* *

*,

zne
(4.6.13)

in which

zi = (ei,A e i , .. ., A"-'ei)

(4.6.14)

( i = 1, 2,. . ., n), I, = ( e l , e 2 , . . ., e,), and

<(t) = efAcel,

(4.6.15)

where A, denotes the companion matrix of the characteristic polynomial p A

86

4.

linear Constant-Coefficient Equations: Theory

of A. Moreover, e(t) can be computed as the unique solution to (yO, Byo,. 00' Bn-lyO)e(t) = y(t). Proof.

(4.6.16)

In view of Theorem 4.6.1, the problem reduces to showing that (yO, Byo, ... , Bn-ly0)etAcel = y(t)

(4.6.17)

for all t in rJf. Note that the proposed equality (4.6.17) does hold at t = O. Since y = By, it would be sufficient to show that the left-hand side of (4.6.17) satisfies the same differential equation. Thus the problem further reduces to showing that (yO, Byo, ... , Bn-ly0)Ac

= B(yO, Byo, ... , Bn-lyO).

(4.6.18)

But this is a valid equality since (4.6.19) and - cn! -

Cn _ I B

-

00' -

CI B

n-

I

= Bn

since in Remark 4.6.2 the matrices A and B were noted to have the same characteristic polynomial (4.6.20) which by the Cayley-Hamilton theorem annihilates B. This concludes the proof of Corollary 4.6.2. Remark 4.6.3 The representation of etA given by (4.6.13)-(4.6.15) is interesting because of its invariance relative to correspondence rules. Moreover, if A were perturbed so as to maintain the characteristic polynomial PA invariant, then the perturbation of etA would be fully reflected in the Z, matrices. The significance of(4.6.16)is that it provides a means for computing and hence etA. Corollary 4.6.2 establishes the rather remarkable fact that the general solution to x = Ax, which is an n-dimensional vector space, can be constructed out of the solution to the single initial-value problem in n-space: = Ace, e(O) = e l·

e

e

Theorem 4.6.3

= (pl(A)c, pz(A)c, etA = (A1y(t), A 2y(t),

C(c)

, piA)c),

(4.6.21)

, Any(t))

(4.6.22)

for all n-vectors c and all t in rJf in which A j = (pl(A)ej, p2(A)ej"'" piA)e),

(4.6.23)

4.6 Explicit Formulas for GC) and

87

etA

where the coefficients of the polynomials Pj(x) =

PI j + P2r + ... + Pnr n- I

(4.6.24)

are the elements of(yO, Byo, ... , Bn-IyO)-1 (j = 1,2, ... , n). In the situation in which y(t), yO, and B result from the complex correspondence rule (3.2.28), alternative formulas for the polynomials are

n (x -

m,-t

pix) =

L

s

rJ.jk(X - Ad

i=2

k~j-1

rJ.jk

=

1 (j _ 1)!(k _j

ak - j + 1

+ 1)' aA~-j+1

(4.6.25)

A;)m.,

I\ (AI s

A;)-ml

(4.6.26)

(k =j - l,j, .. . ,ml - 1; j = 1,2, ... ,m1)' The succeeding m, polynomials are given by (4.6.25) and (4.6.26) by the interchange mi +-+ m" Al +-+ A" etc. (r = 2, 3, ... , s). Proof. Equations (4.6.21) and (4.6.22) are just restatements of (4.6.3) and (4.6.4). Since B = diag(Bt> B 2, ... , B s ), and accordingly

y

° --

[:~J

.

(4.6.27)

,

es

where in (4.6.27) e, denotes the first column of the identity matrix 1m " el

s,«,

B2e2 (y,° B y,0 ... , tr:' y 0) = e2 .. . . . . [

e,

s»,

B~-1ell B~-le2 ..

.

B:-

1

...

.

(4.6.28)

e.

The matrix in (4.6.28) was already proved invertible, and the inverse has unique polynomials associated with its columns via (4.6.24). Thus the polynomials associated with the first m1 columns of the inverse are precisely those satisfying [piBdet]k = bi pj(B;)e i = 0

(k = 1,2, (i = 2,3,

, md, , s; j = 1,2, ... , md.

(4.6.29) (4.6.30)

The proof reduces to showing that these polynomials are the ones given by (4.6.25) and (4.6.26). (The remaining columns of the inverse matrix can be dealt with by repetition of the same argument upon performance of the interchanges mentioned in the statement of Theorem 4.6.3.)

88

4.

Linear Constant-Coefficient Equations: Theory

Since each Bi is an mi x mimatrix of the elementary Jordan type

Bi =

0

*..

1 Ri 0 1

-..

0

-

9

-0

1 li 0 . . . 0 1 Ai

0

A

it follows (see Theorem 4.2.2) that (4.6.29) and (4.6.30) are equivalent to the equations (4.6.32) (4.6.33) i = 2, 3 , . . ., s; j = 1 , 2 , . . .)ml).Note that the poly(k = 1, 2 , . . .,mi; nomials of (4.6.25) and (4.6.26)are of degree n - 1 or less since (ml- 1) + m 2 + m 3 + . . . + m, = n - 1. Since the polynomials p j of degree n - 1 or less satisfying (4.6.29) and (4.6.30),and hence (4.6.32) and (4.6.33))exist and are unique (since the matrix in (4.6.28)has a unique inverse), it is sufficient to show that the polynomials of (4.6.25)and (4.6.26)satisfy (4.6.32)and (4.6.33). From formula (4.6.25) it is clear that those polynomials satisfy

(4.6.34)

(4.6.35) .-a

Thus (4.6.32)holds for k = 1. To perform a proof by induction, assume that (4.6.32) is valid for 1, 2, . . .) k c m l . The task is to show that it is likewise valid for k + 1. Rewriting (4.6.25) as mi-

1

1

ajXx -

r=j-l

n,)r

n (x S

=

pj(x)

(4.6.36)

-

i=2

and differentiating that equation k times relative to x produce the equation ml - 1

r=j-1

r(r - 1).. . (r - k k

=

1 C,kpjyx)

r=O

+ l)aj,(x - Aly-k

"n (x i=2

I' ( k -0

-

li)-m'

(4.6.37)

4.6

Explicit Formulas for GC) and

89

e'A

in which C~ = k !/r !(k - r)! is the binomial coefficient. The resultant equation, obtained by setting x = Al in (4.6.37) and invoking the induction hypothesis, is s

bl+lk!(ljk

=

pJkl(Al)

IT (AI

- ~i)-m,.

(4.6.38)

i=2

Hj t= k + 1, then certainly (4.6.38) implies that pjkl(Ad = O. For j = k formula (4.6.26) reduces to

+ 1,

(4.6.39) which when substituted into (4.6.38) gives p)kl(Ad = 1 in this case. Thus the inductive proof of (4.6.32) for polynomials (4.6.25) is achieved. The fact that those polynomials satisfy (4.6.33) is obvious because of the product term in (4.6.25). The proof of Theorem 4.6.3 is now complete. Since the minimal polynomial of a matrix A divides the characteristic polynomial, it is clear upon distribution of the multiplication in (4.6.25) that piA) = 0 for j ~ rl + 1, where rl denotes the multiplicity of the eigenvalue Al in the minimal polynomial of A. A similar remark applies to the polynomials associated with the remaining distinct eigenvalues. Each matrix A j in (4.6.22) is simply the matrix whose columns are thejth columns of p;(A) (i = 1, 2, ... , n). The net effect is that the active terms of yet) in (4.6.22) associated with a multiple eigenvalue Ak = (lk + iWk are limited to those containing maximal factors t", P ::;; r k - 1, where rk is the multiplicity of Ak (k = 1, 2, ... , s) in the minimal polynomial. Remark 4.6.4

Remark 4.6.5 The polynomials (4.6.25) and (4.6.26) provide generating formulas for the elements of the inverse matrix (4.6.28); i.e.,

pJil(O)

Po = -i!-

(4.6.40)

t. = 1,2, ... , md. The remaining elements of the inverse can be computed in similar fashion from the polynomials obtained via the specified interchanges.

(i = 1,2, ... , ml;

Example 4.6.1

The matrix A =

2 2 -2] 0 3 1 [ o 1 3

(4.6.41)

has characteristic polynomial PA(X) = (x - 2)2(X - 4), and thus the

90

4.

Linear Constant-Coefficient Equations: Theory

eigenvalues are Al = 2, A2 = 4, with multiplicities m, = 2, m2 respectively. Direct substitution into (4.6.25) and (4.6.26) provides

(4.6.42)

Pl(X) = - lx(x - 4),

= - t(x - 2)(x - 4):

P2(X)

The interchange ml remaining

+-+ m2,

Al

+-+

= 1,

(4.6.43)

A2 in (4.6.25) and (4.6.26) produces the

pix) = l(x - 2)2.

(4.6.44)

Substitution of A as given by (4.6.41) into (4.6.42)-(4.6.44) leads to p,(A)

~ [~

p,(A)

~

p,(A)

~ [~

j -tJ'

[H -~l

(46.45)

(4.6.46)

i ~,

(4.6.47)

from which can he read off Al

=

[~ ~ ~l, o

A2 = [

~J

0

t ~ j],

-2

A3 =

(4.6.48)

0

? -20

[!

-2

0

(4.6.49)

2

~]2

!

.

(4.6.50)

Substitution of (4.6.48)-(4.6.50) into (4.6.22) finishes the computation: 2t 2t e 2te2t _ 2te ~ 4t) 2t 4t etA = 0 t(e + e t(e _ e 2t) . (4.6.51) [ 4t) 2t 4t 2t) o !(e _ e !(e + e

4.6 Explicit Formulas for C(e) and

91

erA

EXERCISES

1.

In the representation elA

=

Z2~(t),

... , Zn~(t))

(Zn(t), Z!~(t),

... , Z:~(t))

(Z1~(t),

of Corollary 4.6.2, show that elA T

=

in which (i = 1, 2, ... , n).

2. As in Exercise 1, discuss the action induced upon the remaining representation formulas appearing in Corollary 4.6.2 and Theorems 4.6.1 and 4.6.3 under the transformation A __ AT. 3. Study the question dealt with in Exercises 1 and 2, but for the transformation A -- M AM -1 for M fixed. Can you say anything about A -- M A with M fixed?

Chapter

5

Linear Constant-Coefficient Equations:

Qualitative Behavior

In many applications of differential equations the dependent variables denote errors or displacement of the state of a system relative to some reference or equilibrium state. It is important in such situations that the errors converge to zero appropriately fast with increasing time. Sometimes it is extremely useful to know whether a given differential equation has one or more periodic solutions. The results in this chapter provide answers to such questions based upon somewhat less computational effort than is involved in finding closed-form solutions. 5.1 Stability and Periodic Oscillation

This section concentrates on the qualitative behavior of the solution to the Cauchy problem in £Jt",

x=

Ax,

(5.1.1) (5.1.2)

in which A is an n x n matrix over £Jt. All results can be extended to complex space in a rather obvious fashion. Definition Equation (5.1.1) is called stable if for each XO in £Jt" the solution x(t) to (5.1.1) and (5.1.2) has !x(t)! bounded on 0 :::; t < 00. If for some solution Ix(tk)l-+ 00 as tk -+ 00 for some sequence tk> then (5.1.1) is said to be unstable. 92

93

5.1 Stability and Periodic Oscillation

Definition Equation (5.1.1) is called asymptotically stable if each solution x(t) to (5.1.1) has x(t) - 0 as t - 00. The characteristic polynomial PA and minimal polynomial qA of A contain a great deal of information about the qualitative properties of the solutions to (5.1.1). Such results are summarized in the following theorem. Theorem 5.1.1

Equation (5.1.1) is asymptotically stable (Xj

< 0

(j = 1,2, ... , s) for all distinct eigenvalues satisfying (5.1.3) is called a stability matrix). Instability occurs if

if and

only

if (5.1.3)

Aj

=

(Xj

+

uo, of A. (A matrix

(5.1.4) for at least one j in {1, 2, ... , s}, No conclusion can be drawn if (Xj :-::;; 0 (j = 1,2, ... , s) without further information: If (Xj :-::;; 0 (j = 1,2, ... , s), then stability occurs if and only !f those Aj = (Xj + ico, with (Xj = 0 have multiplicity one in the minimal polynomial qA; i.e., dim "If/;.} equals the multiplicity of Aj in PA' A sufficient condition for stability is that (Xj :-::;; 0 (j = 1,2, ... , s) and that each Aj = (Xj + ico, with (Xj = 0 have multiplicity one in the characteristic polynomial pA of A. Theorem 5.1.2 Equation (5.1.1) will have a nonconstant (oscillating) periodic solution offrequency W if and only if at least one of the distinct eigenvalues A j = (Xj + ico, (j = 1,2, ... , s) of A has (Xj = 0 and Wj = w.

Both Theorems 5.1.1 and 5.1.2 are easily verified consequences of the analysis of the structure of etA that was made in Section 4.4, using the Jordan form of a matrix. The next theorem is useful in estimating the rate of decay of the solutions to an asymptotically stable linear system and provides an upper bound on the rate of growth of an unstable system. Theorem 5.1.3

For A an n x n matrix over

~

n

letAI:-::;;

L IpiA) IIYk(t) I

(5.1.5)

k= 1

for 0 :-::;; t < 00 in which the polynomials Pk(X) are those defined in Theorem 4.6.3 and the Yit) are the coordinate functions of yet).

94

5.

Proof.

linear Constant-Coefficient Equations: Qualitative Behavior

Definition (4.6.23) and elementary estimation give n

I t>

n

1

n

I I kI

I(AjY)il =

j= 1

= 1

n

I

n

s I I

j= 1 k = 1

n

=

(Pk(A»ijyit)

I(Pk(A»ijIIYit)1

n

I I I(piA»ij IIYit) I

k = 1 j= 1

n

I

=

k= 1

O::;t
IpiA)IIYk(t)l,

(5.1.6)

Maximizing (5.1.6) over i produces the inequality n

I(A1y(t), A 2y(t),· .. , Any(t»I::;

I

k=l

Ipk(A)IIYit)l,

(5.1.7)

which, according to (4.6.22), concludes the proof.

Example 5.1.1

Let A be the matrix

- 1 A =

1

o -1 0

[

0

0 0-1

000

which clearly has eigenvalues A.l

~l, -2J

(5.1.8)

= -1, A.2 = - 2 of respective multiplicities

ml = 3, m2 = 1. Application of the formulas of Theorem 4.6.3 provides

the polynomials

= (x 2 + X + l)(x + 2), P2(X) = -x(x + l)(x + 2),

Pl(X)

P3(X) = (x pix)

=

+ 1)2(x + 2),

-(x

+

1)3.

(5.1.9) (5.1.10) (5.1.11) (5.1.12)

Computation of the corresponding matrix polynomials of A can be checked to produce matrices with Ipl(A)1 = IpiA)1 = IpiA) I = 1 and P3(A) = O. The estimate (5.1.5) is then (5.1.13)

95

5.2 The Lyapunov Approach to Asymptotic Stability

for all t 2 0, whereas computation of eta gives e-'

le'*l

=

0 0 0

te-' e-' 0 0

0 0 0 e-"

0 0 e-' O

= .e-[

+ te-'.

(5.1.14)

Remark 5.1.1 At the cost of accuracy in the estimate, evaluation of the matrix polynomials occurring in (5.1.5) can be avoided by dominating their . in Example 5.1.1, IAl = 2, norms by the estimated polynomials in ] A [ Thus and from (5.1.9)-(51.12) it follows immediately that Ip,(A)I I28, Ip2(A)I I 24, Ip,(A)I I18, Ip,(A)/ I 27, producing the less accurate estimate

lefAl I28e-'

+ 24te-' + 9t2e-' + 27e-".

(5.1.15)

Remark 5.1.2 It follows from the analysis of era made in Section 4.4 that if A is a stability matrix, then

o It

[eta/Ice-",

< co,

(5.1.16)

for a any positive number satisfying a

<

-ak

(5.1.17)

for each eigenvalue3.k = t l k + iokof A. The c is a positive number depending on the choice of M and, of course, A. The approach taken there falls short of providing a numerical value of c since the Jordan theorem is an existence theorem alone. Theorem 5.1.3 likewise provides the estimate (5.1.16), but it has the advantage of providing easily computable values for c and is not restricted to stability matrices. Since the information about the possible presence of asymptotic stability is present in the characteristic polynomial pa ,the problem of merely detecting asymptotic stability is a question about the coefficients of pa. The RouthHurwicz criterion effectively deals with the question (see Exercises, pp. 103105), as do other methods. In the next section we discuss an alternative method originated by Lyapunov. In conjunction with the Cetaev estimates, it leads to alternative estimates of the rate of convergence of asymptotically stable linear equations and has the virtue of being extendable to certain nonlinear equations. 5.2

The Lyapunov Approach to Asymptotic Stability

The Lyapunov approach is best introduced by means of a few simple examples in 9'. The first is XI =

x2,

i2 = -XI.

(5.2.1)

96

5.

Linear Constant-Coefficient Equations: Qualitative Behavior

Note that the eigenvalues of the associated matrix A are A. = ± i. By dividing one equation in (5.2.1) by the other and separating variables, it follows that along the solution curves, (5.2.2) where the constant r is determined by the initial values of Xt and X2' Thus the solution curves of (5.2.1) determine a clockwise motion about the origin in ~2 along circles of radius determined by the initial values of Xt and X 2, as indicated in Fig. 5.2.1. The time derivative of the real-valued function on [J/2, V = xi + x~, composed with the solution (Xt(t), xzCt)) of (5.2.1), is d dXl(t) dt v(x t (t), x 2(t)) = 2x t(t) ---;{t

dxzCt)

+ 2x zCt)---;{t for all

t. (5.2.3)

Equation (5.2.3) reflects the fact that the motion generated by (5.2.1) follows the level curves of v. The function vex1, X2) provides a measure of the distance of (x t, X2) from the origin. Suppose that (5.2.1) is altered by addition of a term to give (5.2.4) Although the separation-of-variables approach no longer works, insight into the nature of the motion generated in ~2 by (5.2.4) is attained by again differentiating the composition v(Xt(t), xzCt)) to get :t V(Xt(t), X2(t)) ,,; 2Xt(t)xzCt) + 2X2(t) [ -Xt(t) - x 2(t)] = - 2x~(t)

::::;; 0,

for all

t.

(5.2.5)

Inequality (5.2.5) indicates that the solution curves associated with (5.2.4) are transverse to the level curves of v = xi + x~ at all points where X2 #- O. Thus the motion conforms to that indicated in Fig. 5.2.2. Notice that the

--+--+-+-__1--+--- XI

Fig. 5.2.1 Periodic motion of(5.2.1).

97

5.2 The Lyapunov Approach to Asymptotic Stability

--+-+---1--+--1-- XI

Fig.5.2.2 The asymptotically stable motion of (5.2.4).

conclusion drawn from (5.2.5) was attained without solving the differential equations (5.2.4). The basic Lyapunov approach avoids solving the differential equations but instead selects an appropriate function v which, when differentiated along the solutions of (5.1.1), produces an inequality illuminating the stability-theoretic aspect of the motion. Before abstracting the approach for treating the general equation in (jpn, consider another system in (jpz, (5.2.6) Differentiating v = xi the inequality

+

x~

along the solution curves of (5.2.6) produces

d dt V(Xl(t), xz(t)) = 2x lxz

+ 2xz( -Xl + Xz) =

z 2xz ~ 0,

(5.2.7)

which suggests that the solution curves are of an unstable character, as indicated in Fig. 5.2.3. The final example, (5.2.8)

--+---1r--+--+--+-- XI

Fig.5.2.3 The unstable motion of(5.2.6).

98

5.

Linear Constant-Coefficient Equations: Qualitative Behavior

illustrates an important point leading to the appropriate generalization. Differentiation of v = xi + x~ along the solutions of (5.2.8) results in d dt V(XI(t), xit))

= 2XIX2 + 2xi -2x I - X2) (5.2.9)

= -2XIX2 - 2xt

which provides little information. The problem lies with the choice of v, for an alternative choice v = xi + ix~ gives :t v(xl(t), ~2(t)

= 2XIX2

+ X2( - 2XI - X2)

=

-x~ ~

O.

(5.2.10)

The level curves of the latter, v = xi + ixt are ellipses centered at the origin. Inequality (5.2.10) indicates that the motion is transverse to the level curves, inward, and again indicates the likely possibility of asymptotic stability. The approach and basic phenomena appearing in the above twodimensional examples are abstracted in the Lyapunov treatment of (5.1.1) that follows. Definition A real-valued function f with domain &in of the type n

f(x) =

n

L L sijxiXj,

(5.2.11)

j= I i= I

in which the Xi are the coordinate variables of x in &in and the Sij are fixed in &i, is called a real quadraticform. The Sijcan be regarded as elements of an n x n symmetric matrix S, and then in terms of the dot product on &in (5.2.11) can be written as . f(x)

= x·Sx.

(5.2.12)

Example 5.2.1

f(x)

= Xl2

- 2XIX2 + 3X22

= [Xl] X2 . [ -11

-31][XX21].

(5.2.13)

Observe what happens to S under a linear change of variables x = My, with M any real n x n matrix and y in &in: g(y)

=

f(My)

=

My' SMy

=

y' (MTSM)y.

(5.2.14)

Note that MTSM is symmetric and g is a quadratic form in y. The induced transformation on symmetric matrices S - MTSM is called a congruence transformation. (Recall that S - M - I SM is called a similarity transformation.) Definition A real nonsingular matrix N is called orthogonal if NT = N- I .

99

5.2 The Lyapunov Approach to Asymptotic Stability

Theorem 5.2.1 Let S be any real, symmetric n x n matrix. There exists a real orthogonal matrix N such that (5.2.15)

The A.i are real and are the eigenvalues of S. N can be taken to be of the form (5.2.16)

in which the columns c 1 , C2,"" c" constitute an orthonormal and hence linearly independent collection of eigenvectors of S. Remark 5.2.1 Since Theorem 5.2.1 is a standard result of linear algebra, the proof is omitted. It is the basis for diagonalization of quadratic forms; i.e., if N is applied to make the change of variables x = Ny in (5.2.12), the resulting quadratic form (5.2.14) in y becomes

g(y)

" = L A.iyf.

(5.2.17)

i= 1

Corollary 5.2.2

If S is a real symmetric n x n matrix, then

alx/ 2

~

x· Sx

.Blxl 2

~

Ixl

in which the norm is the Euclidean norm = ~ the smallest and the largest eigenvalues of S.

(5.2.18) and a,

.B are, respectively,

Definition If S is a real symmetric n x n matrix such that x . Sx > 0 for all nonzero x in rJl", then S is called positive-definite. If x . Sx ~ 0 for all nonzero x in rJl", then S is called positive-semidefinite. The notation S > 0 or S ~ 0 is adopted to signify that S is positive-definite or positive-semidefinite, respectively. Definition

Let A be a real n x n matrix. The linear matrix equation

ATp

+ PA =

-In,

(5.2.19)

in which P is regarded as a variable matrix, is called Lyapunov's linear equation. Theorem 5.2.3 If A is a stability matrix, then Lyapunov's equation (5.2.19) has a unique matrix solution P with P > O. Conversely, if(5.2.19) has a solution P with P > 0, then it has no other such matrix solution and A is a stability matrix.

Proof.

Suppose that A is a stability matrix. According to Remark 5.1.2,

o~

t <

00,

(5.2.20)

for some positive numbers c and a. Since A and AT have the same eigenvalues,

100

5.

Linear Constant-Coefficient Equations: Qualitative Behavior

an inequality (5.2.20) with A replaced by AT (with possibly a different value of c) also holds. It follows that the integral

:1

P =

(5.2.21)

etATetAdt

is convergent. This matrix is clearly symmetric. By integration by parts and (5.2.20), elementary computation gives

=

-I,

-

ATP,

(5.2.22)

which shows that (5.2.21) defines a symmetric solution to (5.2.19). To show that it is positive-definite, select arbitrary y in W nand employ definition (5.2.21) to compute

(5.2.23) Since letayl2 is a continuous function of t, equality y * Py = 0 in (5.2.23) can hold if and only if

(5.2.24)

etAy = 0,

for all 0 It < co,and hence precisely when y = 0. Thus P > 0. Suppose Pl were another solution to (5.2.19) with Pl > 0. Consider the real-valued functions on W", u(x) = X - P Xand ul(x) = x * Plx. Along every solution x(t) of (5.1.1) and (5.1.2); d dt

- u(x(t)) = i ( t ) . Px(t)

+ X(t). P q t )

= A x ( t ) *P x ( t ) = X(t).(ATP

+ x(t)-PAx(t)

+ PA)x(t) = - IX(t)12,

(5.2.25)

where the last equality is due to (5.2.19). A similar calculation yields d dt

- u&(t))

=

- lx(t)12,

05t<

(5.2.26)

0O

Subtraction of (5.2.26) from (5.2.25) and integration lead to u(x(t)) - u,(x(t)) = u ( x 0 ) - u1(x0),

0 It <

00.

(5.2.27)

But since x(t) + 0 as t -+ co,it follows from the last equation that u(xo) = ul(xo) for all xo in 9". This can happen only if P = PI. The proof of the

converse part of Theorem 5.2.3 will follow as a corollary to the next theorem.

101

5.2 The lyapunov Approach to Asymptotic Stability

Theorem 5.2.4 Cetaev Estimates. Let A be a real n x n matrix and P be a solution to Lyapunov's equation (5.2.19)for which P > O. Denote the smallest and the largest eigenvalues of P by ex and p, respectively. Then F/iie- t/2a ::;; letAI ::;; ~e-t/2(J for 0::;; t < 00. (5.2.28)

(The norm is the operator norm induced by the Euclidean norm on

~n.)

Proof. In terms of P, an assumed positive-definite solution of (5.2.19), define vex) = x· Px. According to Corollary 5.2.2, alx(tW ::;; v(x(t)) ::;; Plx(tW

(5.2.29)

along each solution x(t) to (5.1.1) and (5.1.2). As in (5.2.25),

dv~(t)

o ::;;

= -lx(tW,

t <

00.

(5.2.30)

From (5.2.29) and (5.2.30) it follows that dv(x(t)) dt

v(x(t))

s --p-,

(5.2.31)

which can be multiplied by et/(J and integrated to get 0 ::;; t <

v(x(t)) ::;; e-t/(Jv(XO),

00.

(5.2.32)

Application of both inequalities of (5.2.29) to (5.2.32) shows that exlx(tW ::;; e-t/(JPlxoI2,

(5.2.33)

which is divided by a and the square root taken, with the result that Ix(t)1 ::;; ~ e- t/2(J IxoI, 0 ::;; t < 00. (5.2.34)

By taking the sup of (5.2.34) over Ixol = 1 and recalling that x(t) = etAxO, the second inequality of (5.2.28) is established. The remaining inequality can be derived in a similar fashion, and Theorem 5.2.4is considered proved. Conclusion of the Proof of Theorem 5.2.3. Let A be a real n x n matrix for which (5.2.19) has a solution P > O. Thus inequality (5.2.34) holds for each XO in ~n. If A were not a stability matrix, then according to Theorem 5.1.1 the condition x(t) - 0 as t - 00 would fail for some solution to (5.1.1). Thus a contradiction is reached and the proof of Theorem 5.2.3 is complete. Remark 5.2.2 In the literature, Lyapunov's linear equation (5.2.19) sometimes appears in the form ATp

+ PA

=

-Q,

(5.2.35)

where Q > 0 is given. However, upon multiplication of (5.2.35) on the left

102

5.

Linear Constant-Coefficient Equations: Qualitative Behavior

and right by Q-1 /2, where QI /2 is a positive-definite square root of Q, and replacement of the variable P by QI /2PQI /2 the equation becomes ATp

+

PA = -In'

(5.2.36)

in which A = Q1/2 AQ -1/ 2. That is, under a change of variable y = Q1/2 X, Eq. (5.2.35) is transformed into the form (5.2.19), and hence there is no loss in taking Q = In. Remark 5.2.3 Since both Theorem 5.1.3 and Theorem 5.2.4 provide estimates of IetA I, their relative merits in applications deserve some discussion. Suppose that the matrix A in (5.1.1) is prescribed (e.g., arises from some engineering design process). If all that is required is verification that A is a stability matrix and that Ix(t) within some rate specification as t - 00 for each solution x(t) of (5.1.1) and (5.1.2), then the most appropriate procedure would be to solve (5.2.19) as a system of linear algebraic equations to obtain P and then compute its eigenvalues. (Actually only the largest and smallest eigenvalues are needed.) If A is indeed a stability matrix, then according to Theorem 5.2.3 the solution will automatically be symmetric and unique. Thus to cut down the dimension of the system of algebraic equations to be solved, the variable P in (5.2.19) should be written in symmetric form. If it turns out that the smallest eigenvalue of P is positive, the asymptotic stability of (5.1.1) is ensured. The inequalities in (5.2.28) provide estimates of the rates of convergence in terms of the smallest and largest eigenvalues of P. The advantage of working with P is that its symmetry implies that its eigenvalues are all real, meaning that P is generally more easily treated using numerical methods. If all that is required is confirmation that A is a stability matrix, then it is enough to check that P is positive-definite. The theory of matrices provides a standard determinant test of a symmetric real n x n matrix P for positivedefiniteness:

1- °

(5.2.37)

(k = 0,1, ... , n - 1), where Pk is the symmetric matrix obtained from P by striking off the 'last k rows and k columns of P. The appeal of this test diminishes for very large n, since even the most efficient computation of determinants involves considerable arithmetic. Of course, (5.1.5) and (5.2.28) are only estimates, which can be quite conservative. The estimate given by (5.1.5) would generally be expected to be the less conservative of the two, and it makes the dependence of the estimate upon the eigenvalues and elements of A more explicit. Moreover (5.1.5) is more comprehensive in that it holds whether or not A is a stability matrix. In short, there is a trade-off between the estimates, involving the amount

5.2

103

The Lyapunov Approach to Asymptotic Stability

of information given on the one hand, and the computing required on the other. Conceivably the situation might arise in which failure of the system specification to meet the estimates would force closed-form computation of etA and explicit calculation of 1 etA1.

Example 5.2.2 Consider the matrix A of Example 5.1.1. From its block diagonal form it is apparent that P can be taken to be of the form P = diag[P,, P2, P 3 ] in which

Substitution of this form of P into Lyapunov's equation (5.2.19) leads to the 1 1 solution x1 = x4 = z, x3 = x 5 = a, x2 = 1. The eigenvalues of the resultant matrix P are $ - a f i , $, +, $ + S f i . For the matrix under consideration, the second' inequality of (5.2.28) is thereby

letAl I 2.62e-0.38',

0I

t

<

(5.2.38)

00.

Although the values of the estimates (5.1.13) and (5.2.38) cannot be compared directly at t = 0 since the norms involved are not the same (only equivalent), the more conservative nature of the decay rate of (5.2.38) is obvious.

a,

1

D, = a,,

Dz'= 11'

1

a2

.. .,

Dk

=

0 0

... 0

a3 a5 ... a 2 a4 a1 a3 1 a2 0 a1

...

...

0

0

a2k- 1

a2k-2

9

ak

104

5.

Linear Constant-Coefficient Equations: Qualitative Behavior

(a) find the symmetric matrix S such that j(x) = x - Sx, (b) find an orthogonal matrix N such that NTSN is diagonal, (c) let x = Ny and compute the quadratic form g(y) = j(Ny). 3. Prove Corollary 5.2.2. 4. Compute letAI as well as the estimates' of the norm appearing in Theorems 5.1.3 and 5.2.4 for the matrix

A 5.

[_~ -~J

=

Compute letAI and its estimate, given in Theorem 5.1.3, for the matrices

(b)

A

=

[

~ ~ ~].

-1 6.

0 0

Prove that each matrix of the type A = P- 1(K - tIn),

(5.2.39)

where P is a real n x n positive-definite matrix and K is an n x n real skew-symmetric matrix (i.e., K T = - K), is a real stability matrix. Show that each real n x n stability matrix A can be factored into the form indicated for unique choice of P > 0 and skew-symmetric K; namely, P is the solution to Lyapunov's equation

ATp

+ PA =

-In'

P = fo'" etATetA dt

(5.2.40) (5.2.41)

and K is the solution to

ATK

+ KA

= _(A -; AT).

K = fo'" e

tAT

(A -; AT)e tA dt.

(5.2.42) (5.2.43)

Note that P and K can be computed by solving a system of linear algebraic equations, i.e., Lyapunov's equation, since K = PA + tIn. 7. Evaluate the integral

105

S.2 The Lyapunov Approach to Asymptotic Stability

in which

- 1 A = 8 [ -1

-~ ~],

M

o -3

=

[

~ -~ ~].

-1 -3

0

8. In the unique factorization of stability matrices (5.2.39), show that the transformation A - A -1 on stability matrices induces the transformation P-(K - tIn)Tp- 1(K - tIn), K - -K, on the symmetric and skew-symmetric parts, respectively. 9. Gershgorin's Theorem. By working with the defining equation of an eigenvector, prove that all eigenvalues of an n x n matrix A over the complex field lie in the union of disks in the complex plane n

U{z:lz -

i= 1

aiil

:s;;

rd,

(5.2.44)

in which the radii are computed as

r,

n

=

L laijl

(i

=

1,2, ... ,n).

(5.2.45)

j= 1.Ni

10. Gershgorin's theorem provides an easily computable localization of the eigenvalues of a matrix which sometimes provides useful information. (a) Which of the following are stability matrices?

1 73] .[-3 1-1] -3 2 1] [1-1 2 1 0-2 [ 3 1-1] r~ ~ _~], 2 1 OJ [ [o l~ 5 -1

0,

2 -4

1,

1 -4 O. -1 1-2

(b) Which of the following are positive-definite matrices?

1 -2 -1 , -1 -1 2

1

-2

4

3 -1 . -1 2

11. Three tanks, each containing v gallons of water, are connected in a loop by pipes through which the contents of each tank is pumped to the next, with all pumping rates equal. Initially arbitrary amounts of salt are instantaneously dissolved in each tank. Derive an appropriate differential equations model of the system and apply it to prove mathematically that the total amount of salt in the system will tend to a uniform distribution amongst the tanks as t - 00.

Chapter

6

General Linear Equations with Time-Varying Coefficients

The class of linear equations treated in previous chapters is expanded to include those with time-varying coefficients. A unique solution to the initialvalue problem is constructed by means of a matrix series (which is no longer a matrix power series) along lines somewhat similar to those taken in the constant-coefficient case. The notion of a fundamental matrix persists, reflecting the fact that the solutions to the homogeneous equation continue to possess a finite-dimensional vector space structure. The major properties of fundamental matrices are developed and applied to obtain representations of the solutions to vector and matrix differential equations and their duals. Estimates of the growth (or decay) rates of the solutions are derived. 6.1 Widening the Scope

A great deal more attention could be spent in pursuing detailed analyses of specific equations of the constant-coefficient type studied in Chapters 3-5. Those equations playa limited but important role in applications, and the results presented are quite useful in such investigations. However, there are compelling reasons for taking an opposite path leading to a more comprehensive theory. Examples in Chapter 1 indicated that only slight changes in problems modeled by linear constant-coefficient equations can produce equations with time-varying coefficients. An even more important source of such equations arises in the theory of nonlinear systems. In Chapter 11, we show that the 106

6.2

107

Preliminaries

derivatives of solutions to nonlinear equations relative to initial and other parameters satisfy linear equations with time-varying coefficients. These linear variational equations are often useful in studying the nonlinear equations from which they originate. Thus there is good reason to study a more general linear model. The question concerning the appropriate form of the theory for such an extension deserves some thought. One approach would be to attempt to follow the program that was completed for the constant-coefficient equation, namely, to develop (1) an existence and uniqueness theory, (2) a computational method for representing the solutions in some closed form that illuminates the underlying structure of the solutions, (3) an efficient method for assessing important aspects of the qualitative behavior of the solutions. We shall find that this admittedly optimistic approach does achieve some level of success. 6.2 Preliminaries

Attention is focused on the equations in J x :Fn,

x= x(to)

=

A(t)x

+ f(t),

(6.2.1) (6.2.2)

x",

where J is a nonempty open subinterval of ~, to E J, :F is either the real or the complex field, and A(' ) and f( .) are matrix-valued functions defined on J of respective sizes n x nand n x 1, generally over :F. Recall that a matrix-valued function, say A('), is locally integrable on J if each of its elements is measurable (relative to the Borel subsets of J) and the norm IAOI is (Lebesgue) integrable on each compact subinterval of J; i.e., lIA(t)1 dt <

00

(6.2.3)

for each closed and bounded subinterval I c J. The text always assumes that both A( .) and f( .) are locally integrable on J. This condition is automatically fulfilled if A(' ) and f(·) are continuous or only piecewise continuous. In this more restrictive setting, the integral is then just the Riemann integral of elementary calculus. The added generality allows modeling systems in which extensive switching can occur without requiring more than the standard theorems of real analysis in effecting the proofs. A solution x(-) to (6.2.1) is defined to be any absolutely continuous map from J into :F n satisfying (6.2.1) on all of J but for possibly a subset of mea-

108

6,

General linear Equations with Time-Varying Coefficients

sure zero; i.e., almost everywhere (a.e.). Of course if A(') and f(·) are continuous on J, then solutions will actually satisfy (6.2.1) at all t E J. Just as for equations having constant coefficients, the variation-of-parameters approach makes it possible to pass from a solution of the homogeneous equation to one for the nonhomogeneous equation by a single integration, and hence it is sufficient to begin by concentrating on the homogeneous equation. 6.3

The Fundamental Matrix Series Defining.

Some experimental integration by parts, etc., leads to the appropriate generalization of the exponential matrix series lI>(t, ..)

=

L [L~(In)](t, CXJ

(6.3.1)

r),

k=O

in which the matrix function A( .) is defined on an open interval J c fA, t and .. are in J, and L~ is the k-fold composition of the linear integral operator defined by [LA(M)](t,

t)

=

f

M(t, a)A(a) da,

(6.3.2)

In (6.3.1) L~ denotes the identity operator, and thus L~(In) = In. It is observed that for A(') locally integrable the terms of the series (6.3.1) are continuous n x n matrix-valued functions defined on J x J. To keep the notation simple, L~(In) will be denoted simply by which, remember, is an operator product, not a matrix product. Thus (6.3.1) is written as

u,

ex>

lI>(t, ..) =

L

L~t,

r ).

(6.3.3)

k=O

Lemma 6.3.1 If A(') is locally integrable on J, then

IL~t, ")1 s

:! If

IA(a)1 dar.

IlI>(t, ")1 ~ k~O IL~t, ")1 s exp (IS)A(a) 1do I) all (t, ..) E J x J (k = 1, 2, ...). lI> is continuous on J x J

(6.3.4) (6.3.5)

for and its series (6.3.3) is absolutely and uniformly convergent on compact subsets of J x J. Proof.

To minimize the number of absolute value signs, the proof is

109

6.3 The Fundamental Matrix Series Defining
written out for r < t. The first step is to prove (6.3.4), ILk(t, t)1

s

:! [f

jA(u)! do

l

k E {1, 2, ...},

(6.3.6)

by induction. Certainly (6.3.6) holds for k = 1 by the standard inequality, taking the norm under the integral. Now assume that (6.3.6) holds for fixed k E {1, 2, ...}. By the induction hypothesis (6.3.6) and the definition of L, etc., IV+ l(t, t)1

~ s

If f [f I

L't, u)A(u) dul :::;;

k\

f

IL't, u)IIA(u)1 do

I JIA(U)I do

A(ro) dco

1

II [II [f J+ 0

= - (k

+ 1)!

t

= (k :

1)!

!A(u)j do

ou

Jk+l do

"IA(ro)1 do:

(6.3.7)

1,

which completes the proof of (6.3.6) by induction. Inequalities (6.3.5) follow directly from (6.3.4) as do the absolute convergence and uniform convergence on compact subsets. The continuity of is a consequence of the uniform convergence and continuity of the L'·,·). This concludes the proof of Lemma 6.3.1.

Theorem 6.3.2 Let A( .) be a locally integrable n x n matrix-valued function defined on an open interval J c f!It. Then the matrix function defined by (6.3.1) is the unique matrix solution to the initial-value problem on J x J,

o

ot (t, r) = A(t)(t, r),

a.e.

t E J,

(6.3.9)

r E J.

for each

(6.3.8)

Moreover is the unique matrix solution to the initial-value problem

0<1>

a:r(t, r) $(t, t)

= -(t, t)A(t),

= In

for each

a.e. t E J.

r

E

J,

(6.3.10) (6.3.11)

As well as being absolutely continuous in each variable, is continuous on J x J. $(t, r) is invertible for each (r, r) E J x J and $-l(t, r) = (t, t).

(6.3.12)

110

6.

General Linear Equations with Time-Varying Coefficients

If A(') is continuous on J, then
k~O

f u«,

(6.3.13)

a)A(o) da.

From (6.3.4) of Lemma 6.3.1 it follows readily that (6.3.14) for , ~ t, and consequently Fubini's theorem applies, allowing the order of summation and integration in (6.3.13) to be reversed. The latter equation can then be written as

J,t k~O e«. a)A(a) do, 00

(6.3.15)

and similar reasoning shows that (6.3.15) is valid for all (t, r) E J x J. By (6.3.4) the series in (6.3.15) is locally integrable for each fixed t E J, and hence (6.3.15) can be differentiated relative to " giving

a -a
L 00

= -

k=O

Lk(t, ,)A(,) = -
a.e.

rEJ

(6.3.16)

for each t E J. This proves (6.3.10). It is asserted that each Lk(t, r) is absolutely continuous in t and that

a I:k+l (r, r)

at

k{

= A(t)r, \t, r),

a.e.

tEJ

(6.3.17)

for each r E J(k = 0, 1, ...). The proof of this assertion is left as an exercise for the reader. To complete theproof of (6.3.8), apply (6.3.17) to rewrite the sum

(6.3.18)

111

6.3 The Fundamental Matrix Series Defining «I>

From (6.3.4) it is clear that !A(W)

:t~

Lk(w,

r)1 ~

W

for r

IA(w)1 exp(i IA(a)1 da)

~

co. (6.3.19)

The series defining l1> was proved convergent. Moreover, the inequality (6.3.19) allows the Lebesgue dominated convergence theorem to be invoked to take the limit as N -+ (f) under the integral in (6.3.18), with the consequent equality l1>(t, r)

= In +

f

A(w)l1>(w, r) dco,

(6.3.20)

and moreover, because of (6.3.19), Eq. (6.3.20) can be differentiated to conclude that

o

ot l1>(t, r)

= A(t)l1>(t, r),

a.e. t e J

(6.3.21)

for each r ~ t in J. The case in which r > t is treated with the obvious minor change. The uniqueness claims are easy consequences of Gronwall's inequality (see Exercise 1 at the end of this section). Equation (6.3.12) can be proved as follows: Consider '¥(t) defined as '¥(t) = l1>(t, r)l1>(r, r),

t E J,

(6.3.22)

with arbitrary r E J fixed. Applying (6.3.8) and (6.3.10), Eq. (6.3.22) can be differentiated to get d

dt'¥(t) = A(t)'¥(t) - '¥(t)A(t),

a.e. t E J.

(6.3.23)

Thus '11(.) is a solution to (6.3.23) satisfying the initial condition '¥( r) = In as is clear from (6.3.9) and (6.3.22). But the matrix function '¥(t) = In, t E J, is another solution to the same initial-value problem. Application of Gronwall's inequality again shows that the solution must be unique, and the conclusion that 'P(t) = In, t E J, is inevitable. This finishes the proof of (6.3.12). If A( .) happens to be continuous, then by inspection of the proofjust given it is evident that (6.3.8) and (6.3.10) hold everywhere in J x J and moreover that l1>(t, r] and its first-order partial derivatives are continuous in J x J. It follows that l1> is thus once continuously differentiable on J x J. This concludes the proof of Theorem 6.3.2. Notation. Occasionally the matrix function l1> will be denoted by l1>A when its dependence upon A(' ) needs to be emphasized.

112

6.

General Linear Equations with Time-Varying Coefficients

Remark 6.3.1 From Theorem 6.3.2 and results derived in Chapter 4 it is apparent that A is just e(l- t)A in the case in which A( .) = A is independent of tEJ. EXERCISES

1. Prove the following form of Gronwall's theorem: Each continuous solution x( .) to the inequality x(t)

s

lX(t)

+

f

also satisfies the inequality x(t)

s

lX(t)

+

I

p(U)IX(U)

p(u)x(u) da,

exp(f

pew)

tE

[a, b),

dW) do,

t E [a, b),

where IX(') is continuous and 13(') is nonnegative and locally integrable on

[a, b).

2. Apply Gronwall's theorem to prove the uniqueness claims concerning Eqs. (6.3.8) and (6.3.9), (6.3.10) and (6.3.l.1), and (6.3.23) that were made in the proof of Theorem 6.3.2. 3. Prove the assertion concerning (6.3.17) that was made in the proof of Theorem 6.3.2.

6.4 Existence and Uniqueness of Solutions to Vector Differential Equations 'and Their Duals The results of Theorem 6.3.2 can be applied to vector differential equations. As often occurs in mathematics, it is natural to discuss some of the results in terms of the dual space of ~n rather than in terms of ~n itself. The appropriate definitions are recalled. Definition An tional on ~n if

~-valued

function I, defined on .

~n

is called a linear func(6.4.1)

for all Cl' Cz in ~ and all Vl, Vz in ~n. The collection of all linear functionals on ~n is denoted by ~". and can be turned into a vector space relative to the addition and scalar multiplication defined by

(11

+ Iz)(v) =

11(v)

+ Iz(v),

(c/ 1)(v) = c/1(v)

(6.4.2) (6.4.3)

113

6.4 Existence and Uniqueness of Solutions and Dual Systems

for all 11 ,12 e $'"*, v e $'", and c e $'. The resultant vector space, still denoted by $'"*, is called the dual space of $'".

Remark 6.4.1 $'"* has dimension n and its elements x* can be identified with I x n (row) vectors x over $' if x*(v) is defined to be the matrix product xv, for all v e $'". Remark 6.4.2 Any n x n matrix function A( .) defined on an interval J can have associated with it the differential equation in J x $'", x

= A(t)x,

(6.4.4)

as well as a differential equation in J x $'"*, (6.4.5)

x* = x*A(t).

Some interesting relationships between the solutions of the two equations appear in Theorem 6.4.1.

Theorem 6.4.1 Let A(·) be a locally integrable n x n matrix-valuedfunction (over $') defined on an open interval J c fit. Then there exists a unique solution x A to the initial-value problem on J x J x $'", x

= A(t)x,

x(t)/t=t =

(6.4.6)

a.e. te J,

e

(6.4.7)

namely,

(6.4.8) There also exists a unique solution x~ JxJx$'"* x* x*(t)lt=t

to the initial-value problem on

= -x*A(t), a.e. t e J, = e* for each (r, e*)eJ x

(6.4.9) $'"*,

(6.4.10)

namely,

(6.4.11) Moreover, the solutions to the two equations satisfy the invariance relation

e

x~(t,

r, e*)xA(t, r, e) = e*e

(6.4.12)

for all (t, r) e J x J, e $'", e* e $'"*. If A(·) is continuous on J, then x A and x~ are once continuously differentiable relative to their argument (t, r) e J x J, and Eqs. (6.4.6) and (6.4.9) hold for all (t, r) e J x J. Proof.

All statements are rather obvious consequences of Theorem 6.3.2.

114

6.

General linear Equations with Time-Varying Coefficients

Remark 6.4.3 Equation (6.4.9), transposed so as to appear as a differential equation in :Fn , is called the adjoint of (6.4.6) and appears frequently in control theory and other areas of mathematics involving differential equations or optimization. Remark 6.4.4 If A(') is assumed to be k-times continuously differentiable on J, then since A satisfies (6.3.8) and (6.3.10), it is clear that A will be (k + Il-times continuously differentiable relative to (t, T) E J x J (k = 0, 1, 2, ...). Thus the solutions (6.4.8) and (6.4.11) will likewise be (k + Il-times continuously differentiable on J x J.

6.5 Variation-of-Parameters Formulas for Vector and Matrix Differential Equations

Theorem 6.5.1 Let A(') and f(·) be locally integrable matrix-valued functions (over :F) of sizes n x nand n x 1, respectively, both defined on an open interval J c flt. Then there exists a unique solution x A to the initial-value problem on J x J x :Fn

x=

A(t)x

+

f(t),

a.e.

for each

x(t)lt=< = ~

(6.5.1)

tEJ, (T,

x J

~)EJ

(6.5.2)

given by the variation-of-parameters formula

xA(t, T,

~)

=

A(~' r)~ +

f

4> A(t, (1)f(a) da,

(6.5.3)

Proof. A proof can be based upon application of Theorem 6.4.1. The details as well as the proof of the following theorem are left to the reader.

Theorem 6.5.2 Let Al ('), A 2 ( ' ) , and F(') be locally integrable n x n matrixvalued functions on an open interval J with arbitrary r E J and n x n matrix Z fixed. Then there exists a unique solution to the initial-value problem

'P =

A1(t)'P - 'P A 2(t)

+ F(t),

a.e.

t EJ,

(6.5.5)

'Plt=< = Z, namely,

'P(t, r, Z)

Example 6.5.1

= 4>Al(t, r)Z4>A2(T, t) +

(6.5.4)

f

4>A/t, a)F(a)A,(a, t) da,

(6.5.6)

An important example of (6.5.4) is the equation a.e. t

E

J,

(6.5.7)

6.6

115

Estimates of Solution Norms

arising from A 1(t) = -AT(t), A 2(t) = A(t), and F(t) = In. Here A(') is assumed to be locally integrable on an interval J. In the special case in which A(') is independent of t, the (steady-state) equation, gotten by replacing P in (6.5.7) by zero, is Lyapunov's algebraic equation (5.2.19). Thus it should be not altogether surprising to find that in the next section (6.5.7) plays an important role in the problem of estimating rates of growth or decay of norms of solutions to differential equations associated with A(·). The solution to (6.5.7), as represented by the variation-of-parameters formula (6.5.6), is pet, r) =

t)Po
f
t)
(6.5.8)

in which Po = plt=<.

6.6

Estimates of Solution Norms

The problem of estimating the rate of growth or decay of solutions, studied earlier in the context of constant-coefficient equations, continues to be of interest, but understandably it is more complicated for equations with timevarying coefficients. Again the (real) n x n matrix-valued function A(') is given on an open interval J for which it is assumed to be locally integrable. The principal equations are

x=

A(t)x,

a.e. t E J,

xlt=t = ~,

(6.6.1)

(6.6.2)

(r, ~) E J x !!An. The primary goal is to obtain estimates of the solution Ix(tW and its integral. Throughout this section the norm employed on !!An is the Euclidean norm and the induced operator norm IMj = maxlMxl, max over Ixl = [x] = 1, is applied to a matrix M. The (r, ~) appearing in (6.6.2) should be regarded as fixed although arbitrary. As already announced, the problem will be dealt with by the device of studying (6.5.7). In explaining the connection, first note that in (6.5.8), if Po is symmetric, then so is pet, r). Moreover if Po > 0 (positive-definite), then likewise pet, r) > 0 for r ~ t and t appropriately near r. Consequently there is no difficulty in fulfilling the assertion that

.vx:x

o< for all nonzero y E !!An, r

a(t)IYI2 ~ y' P(t)y ~ {3(t)lyI2 ~

(6.6.3)

t and t near r for at least some of the solution

116

6.

General linear Equations with Time-Varying Coefficients

curves pet) to (6.5.7). The scalar functions 0(') and {3(-) are assumed to be continuous and could be taken to be, respectively, the smallest and largest eigenvalues of pet), but they are not necessarily so chosen. The problem of selecting the appropriate solution curve P(·) and computing 0(') and {3(') from A(' ) will be discussed later. The next theorem provides the first justification for interest in inequality (6.6.3).

Theorem 6.6.1 Suppose that P(·) is a symmetric solution to (6.5.7) satisfying (6.6.3) for all t in a subinterval ofJ with t :S t. On that subinterval O(t)

[1 -

exp( -

f ~:»)

] \e1

2

:S

s for all

eE

~n.

f

IX(CTW dCT

{3(t) [1 - exp( -

f ct~:»)

]

Jel

2

(6.6.4)

e

Proof. Fix arbitrary E ~n. Since the solution to (6.6.1), (6.6.2) has the representation x(t) = <1JA(t, t), and pet) has a representation given by (6.5.8), by substitution of the latter into (6.6.3) along with y = x(t), it follows that O(t)lx(tW :S

f

e· p(t)e - e· <1J~(CT,

which is equivalent to O(t)lx(tW

+-

f

IX(CTW dCT:S

t)<1JA(U, 7:) do

,·P(t)'.

e,

(6.6.5)

(6.6.6)

Upon division of (6.6.6) by x(t), application of the rightmost inequality of (6.6.3), and multiplication by expm dCT/ct(CT)], it is easy to see that

for t :S t. Integration 'of (6.6.7) from r to t and division by the exponential term lead directly to the second inequality of (6.6.4). Similar operations, starting with the second inequality of(6.6.3),provide the remaining inequality of (6.6.4) to complete the proof of Theorem 6.6.1.

Remark 6.6.1 One immediate consequence of Theorem 6.6.1 is that the second inequality in (6.6.8)

117

6.6 Estimates of Solution Norms

will hold if iX(t) > 0 on

['t,

00),

and if moreover

roo

J.

du f3(u)

=

00,

(6.6.9)

then the first inequality of (6.6.8) is likewise valid.

Remark 6.6.2 The proof of Theorem 6.6.1 relied on the assumption that iX(t) remains positive over the range of t being considered. (In applications, iX(t) is usually obtained, as a lower approximation to the smallest eigenvalue of P(t).) Examples demonstrate that the smallest eigenvalue of P(t) can become zero as t is increased from r to some T < 00. Theorem 6.6.1, for the given P('), will then not apply outside the interval ['t, T] since (6.6.3) must necessarily fail. One remedy for extending the interval determined by some approximating iX(·) would be to find a better solution P(·) to (6.5.7). Another would be to find a better lower approximation than iX(·). An effective procedure for accomplishing these objectives is provided in the next section.

DETERMINATION OF 01(') AND P(·)

To convert Theorem 6.6.1 into a result having practical as well as theoretical interest, two obstacles must be surmounted. The first is to single out solutions P(·) to (6.5.7) which are positive-definite over the interval of concern. The other is to find an effective procedure for approximating the extreme eigenvalues of such' P(·) to within any prescribed accuracy. The former is handled by restricting attention to any finite interval of the type r ~ t T(U, t)cI>(u, t) do

(6.6.10)

on the interval. The latter can then be approached through the series expansion appearing in the next theorem.

Theorem 6.6.2 Let A(') be locally integrable on J and consider any compact interval ['t, b] c: J. The integral (6.6.10) provides a symmetric positive-definite matrix solution to (6.5.7) on the interval ['t, b) and P(t)

=

r

Ao(t)

+ At(t) + ... + Ak(t)

+

cI>T(U, t)[AT(u)Ak(u)

+ Ak(u)A(u)]cI>(u, t) da,

(6.6.11)

118

6.

in which

General Linear Equations with Time-Varying Coefficients

r

(6.6.12)

Ao(t) = (b - t)I n , A k+ 1 i

::;;

=

+ Ak(a)A(a)] do,

(6.6.13)

[1'J~(~)]k

(6.6.14)

Ak(t),

(6.6.15)

[AT(a)Ak(o)

t ::;; b (k = 0, 1,2, ...). Moreover,

IAit) I ::;; (b -

t)

in which

In particular, co

pet) =

L

k=O

with the series uniformly convergent on

[i, b]

and

pet) ::;; (b - t)eqA(t)I n •

(6.6.16)

Proof. Routine calculations show that (6.6.10) is a matrix solution to (6.5.7) that is symmetric and positive-definite at each t E [i, b). Formula (6.6.11) with k finite follows by repeated integration by parts applied to (6.6.10). The estimate (6.6.14) is easily established using induction on (6.6.13). Equation (6.6.15), the uniformity of the convergence, and (6.6.16) are elementary consequences of(6.6.14) applied to (6.6.11). The reader is invited to check the details, and Theorem 6.6.2 is thereby proved.

Lemma 6.6.3 Let S(· ) be a real n x n symmetric matrix-valued function defined on an open interval J. The minimum and maximum eigenvalues of Sea) at each point a E J define real-valued functions on J which are continuous at each point in J at whicn S( .) is continuous. Proof. Since Sea) is symmetric at each a E J, its eigenvalues are real, and thus the smallest eigenvalue ac(a) of Sea) defines a function ac(') on J. Let t E J be a point at which S(·) is continuous, and in order to reach a contradiction suppose that ac(') were not continuous there. Then there would exist a sequence t k -- t as k -- 00 with t k E J such that ac(tk) -f+ ac(t) as k -- 00. Thus by extracting an appropriate subsequence, it can be assumed that either

(k = 1,2,

)

(6.6.17)

= 1,2,

)

(6.6.18)

or (k

6.6

119

Estimates of Solution Norms

for some number p. From the symmetry of S(·) it follows that there exist unit vectors ~k' ~* in [JItn such that Ct(tk) = ~k' S(tkRk and Ct(t) = ~*. S(t)~*. By extracting a further subsequence, it can be assumed that ~k ~ ~oo as k-r CXJ for some ~ 00 E [JItn that necessarily will have I~ 00 I = 1.Thus inequalities (6.6.17) and (6.6.18) could be written respectively as (k

= 1,2, ...)

(6.6.19)

(k

= 1,2, ...).

(6.6.20)

and By the continuity of S(·) at t, it follows upon taking the limit as k ~ (6.6.19) that

00

in

(6.6.21) over the collection of which is a contradiction since ~ * minimizes ~. S(t)~ unit vectors in [JItn. In the complementary case, (6.6.20) obtains and can be rewritten as - ~*'S(tk)~*

~k'S(tk)~k

+ ~*'[S(tk)

- S(t)]~*:2: (k

P> 0

= 1,2, ... ). (6.6.22)

By the continuity of S( .) at t, etc., I~*

.[S(t k) -

s(t)]~*1

~ IS(tk) - S(t)II~*12

= IS(tk) - S(t)1 ~O

(6.6.23)

as k ~ 00. However, for all sufficiently large k, inequalities (6.6.22) and (6.6.23) imply (6.6.24) over the collection which is again a contradiction since ~k minimizes ~. S(tk)~ of ~ E [JItn of unit length. This concludes the proof that the continuity of S(.) at a point t E J implies the continuity of Ct( •) at t. The proof of the remaining conclusion of Lemma 6.6.3 regarding the largest eigenvalue of S(· ) uses the same ideas and similar arguments and is therefore omitted. Lemma 6.6.4 Let Sk(-) and Soo(') be symmetric real n x n matrix-valued functions, continuous on a compact interval [a, b] (k = 1, 2, ...). Denote the minimum and maximum eigenvalues of these matrices by Cti'), CtooC) and Pi'), Poo('), respectively. If Sk(-) ~ Soo(-) on [a, b] as k ~ 00, then Ctk(') ~ Ct oo ( ' ) and Pk(') ~ Poo(') uniformly on [a, b] as k ~ 00.

Proof. To prove that Ctk(') ~ Ct oo ( ' ) uniformly, it is sufficient to prove only pointwise convergence since by Lemma 6.6.3 the Ctk(') and Ct oo ( ' ) are continuous real-valued functions on a compact interval. To effect a proof by contradiction, assume that, at some point t E [a, b], Ctit) fr Ctoo(t) as k ~ 00.

120

6.

General Linear Equations with Time-Varying Coefficients

By extracting an appropriate subsequence, there is no loss in assuming that either

= 1,2,

)

(6.6.25)

(k = 1,2,

)

(6.6.26)

(k

or for some number p > O. As in the proof of Lemma 6.6.3, inequality (6.6.25) can be rewritten as (k = 1,2, ...)

(6.6.27)

for appropriate unit vectors ~k' ~oo in fJtn. By assuming a further extraction of an appropriate subsequence, it can be assumed that ek -+ e* as k -+ 00 for some ~* E rJin of unit length. The same elementary manipulations as those that appeared in the proof of Lemma 6.6.3, along with the hypothesis that Sk( .) -+ S 00 ( • ) on [a, b] as k -+ 00, show that (6.6.27) leads to the contradiction eoo .Soo(t)e 00 >

e*· Soo(t)~*

(6.6.28)

and (6.6.26) leads to the contradiction of the fact that ~k minimizes ~. Sk(t)~ over the collection of unit vectors in fJtn (k = 1,2, ...). In any case, the appropriate contradiction is achieved, thus completing the proof of the statement that if Sk(') -+ Soo(') on [a, b] as k -+ 00, then OCk(') -+ oc oo(') uniformly on [a, b] as k -+ 00. The conclusion concerning Pk) and Poo(') can be argued in a similar manner and is thus left to the reader. Lemma 6.6.4is now established. The next theorem is a continuation of the results obtained in Theorem 6.6.2 and uses the same notation. It provides a means for approximating the extreme eigenvalues of P(·) without requiring the computation of cI>A' and in a form (6.6.3) needed for application of Theorem 6.6.1. The following string of definitions is involved:

I -t; a.e. a E ['t, b]}, = inf{s I AT(a)Ak(a) + Ak(a)A(a) ~ si; a.e. a E ['t, b]},

"Ik = sup{y AT(a)Ak(o) + Ak(a)A(a) ~ ~k

k

Ak(t)

= max{A I L

i=O

A;(t) ~ AIn},

k

f.Lk(t)

= min{f.L I L

i=O

(6.6.29)

A;(t) ~ f.LI n },

OCk(t) = Ak(t)/(l - Yk),

Pk(t) = f.Lk(t)/(l - ~k)'

ocoo(t) = max]« I P(t) ~ ocln }, Poo(t)

= min{PI P(t)

~ PIn},

r ~ t ~ b.

6.6

121

Estimates of Solution Norms

Remark 6.6.3

If

lim inf Yk

~

0,

k ....oo

then Yk -- 0 as k -- 00. To see this, for arbitrary e > 0 consider Yk = 1, 2, ...). For this choice of Y in the definition of Yb

= Yk - e

(k

AT(IT)Ak(IT)

+ Ak(IT)A(IT)

~ (Yk - £)In ,

(6.6.30)

a.e. IT E ["t, b], which implies that lim sup k-C(l

~. (b cIlT(IT, t)[ AT(IT) Ak(IT) + Ak(IT)A(IT)]cIl(lT, t) dIT ~ Jt

~ [lim sup Yk - £]~'P(t)~ k-+ co

(6.6.31)

.

for each nonzero ~ E f1tn, t E ["t, b). From (6.6.14) of Theorem 6.6.2 and the fact that pet) > 0, inequality (6.6.31) reduces to lim sup Yk ::s; e.

(6.6.32)

k ....oo

But

£

> 0 is arbitrary, and this is consistent with (6.6.32) only if lim sup Yk k ....oo

= O.

(6.6.33)

°

This concludes the demonstration that Yk -- as k -- 00. A similar argument shows that bk - - 0 as k -- 00 if lim sup, .... 00 bk ::s; O. The functions Ak(-), Jlk('), IXk('), Pk('), 1X 00 ( - ) ' and Poo(-) are continuous on ["t, b]-a consequence of Lemma 6.6.3. Both IX 00 ( . ) and Poo(') are strictly positive on ["t, b) since pet) > 0 on that intervaL They are the extreme eigenvalues of P(·).

Remark 6.6.4

Theorem 6.6.5 Let A(') be locally integrable on J and consider any compact interval ["t, b] c J. Assume that

lim infYk ~ 0,

(6.6.34)

k ....oo

lim sup k ....co

s, s

O.

(6.6.35)

Then for each e > 0 there exists a K. such that

o < IXk(t)In s 1X00(t)In s pet) s Poo(t)In ::s; Pit)In for

"t ::s; t ::s;

(6.6.36)

b - e, all k ~ K •. Moreover, assumptions (6.6.34) and (6.6.35)

122

6.

General Linear Equations with Time-Varying Coefficients

imply that

both uniformly on [""t", bJ as k -

(6.6.37)

and

IXk(-) - 1X 00(-) 00.

Proof. Lemmas 6.6.3 and 6.6.4and Theorem 6.6.2imply that Ak(' )-1X 00(') and Pi") - 1300(-) uniformly on [""t", bJ as k - 00. As discussed in Remark 6.6.3, assumptions (6.6.34) and (6.6.35) ensure that Yk - 0 and £>k - 0 as k - 00. The uniform convergences (6.6.37) thus follow from the equations defining lXi') and Pk(')' On applying the inequality defining Yk' it follows from (6.6.11) of Theorem 6.6.2 that for every f. > 0,

k

=

L

Ai(t)

i=O

+ (Yk

- f.)P(t),

(6.6.38)

r ~ t ~ b,

for all k sufficiently large that 1 - Yk > O. Since gives

f.

> 0 was arbitrary, (6.6.38)

""t" ~

t

s

b,

(6.6.39)

for all k large enough that 1 - Yk > O. In similar fashion it is easy to see that ""t" ~

t

s

b,

(6.6.40)

for all k so large that 1 - £>k > O. In summary, IXk(t)I" ~ P(t) ~ f3k(t) I" ,

r ~ t ~ b,

(6.6.41)

for all appropriately large k such that 1 - Yk > 0 and 1 - £>k > O. In view of Remark 6.6.4 and the uniform convergence (6.6.37), it follows immediately that for each f. > 0 there exists a K. such that (6.6.42) on

""t" ~ t ~

b -

f.

for all k

~

K •. This completes the proof of Theorem 6.6.5.

6.6

123

Estimates of Solution Norms

Remark 6.6.5 Theorem 6.6.5 and the definitions leading up to it were developed to provide computationally feasible approximations Q(i.), Pk(') to the extreme eigenvalues 01 00 ( -}, Poo(-) of P(·) needed in (6.6.3) for the application of Theorem 6.6.1 in computing upper and lower estimates of the integral norm of the solutions to the initial-value problem (6.6.1) and (6.6.2). The precise manner in which the theory accomplishes this goal deserves some reiteration and further comment. The extreme eigenvalues 01 00 ( ' ) , Poo(-) would provide the best possible choice of 01(') and P(·) in (6.6.3) were it not for the fact that their calculation would require the computation of A or some equivalent to get P(· ). Theorem 6.6.5 advocates the less accurate Q(k(' ), Pk(') on a subinterval T :::; t :::; b - e as appropriate choices of Q( .) and P(·). The computations of the Q(k(') and Pk( .) work more directly with A(') itself, and this of course vastly enhances the applicability of Theorem 6.6.1. The uniform convergence concluded by Theorem 6.6.5 ensures that any desired level of approximation can be achieved (at least in principle) by increasing k at the cost of the additional computation. Remark 6.6.6 In some applications the computational effort can be reduced by sacrificing accuracy. That is, in some cases, rather than compute 'Yk> Ok' Ak(-), and J.lk(· ) for some fixed (perhaps small) value of k, it might be easier to find numbers Yk' bk, E 0, 1 -

I

i=O

(6.6.43)

EJ,

(6.6.44)

lJJ,

(6.6.45)

a.e. a E [T,

k

k(t)I n :::;

a E [T, EJ,

all

Ai(t):::; f1.k(t)I n,

t E [T,

bk > 0. Then application of(6.6.11) leads to for

t E [ r, lJ),

(6.6.46)

where iik(t) =

l k(t}/(l

- Yk),

(6.6.47)

Pit)

f1.it)/(1 - J'k)'

(6.6.48)

=

Thus (6.6.46) provides an estimate of the extreme values of P(·) of the type needed for application of Theorem 6.6.1. The acceptability of the estimates must be judged in the context of the application. If they are adequate, the question of arbitrary high degree of approximation is irrelevant.

124

6.

General Linear Equations with Time-Varying Coefficients

Remark 6.6.7 Hypotheses (6.6.34) and (6.6.35) are rather mild assumptions. For example, they are met if I AT(t) 1 + IA(t)1 is bound on [T, b] since, according to Theorem 6.6.2, A k ( • ) as k - 00 uniformly on [T, b].

1 1- °

The following corollary follows readily from Theorem 6.6.1 and Remark 6.6.6. Corollary 6.6.6 Let A(') be an n x n integrable real matrix-valued function on a compact interval [a, bJ. Assume that for some numbers y and b < 1, a.e.

t E [a, b

J.

(6.6.49)

Then x(t, ~), the unique absolutely continuous solution to the initial-value problem in [a, b] x PAn:

x = A(t)x, x(a, ~)

=

a.e.

t E [ a, b],

(6.6.50) (6.6.51)

~,

satisfies the inequalities

(6.6.53) Proof.

T

The conclusion follows directly from computing, using k =

= a in Theorem 6.6.1 and Remark 6.6.6.

Example 6.6.1

°

and

In the scalar equation

v + a(t)v =

a.e.

0,

t e [0,1],

(6.6.54)

[0, 1],

(6.6.55)

assume that a(' ) is measurable and satisfies

°s

aCt)

s

1,

tE

with strict inequality holding on a subset of [0, 1] of positive measure. The claim is that for each solution v(' ) to (6.6.54) on [0, 1], the norm (6.6.56)

125

6.6 Estimates of Solution Norms

satisfies the inequality

~(O)

[1 ~ f~

~

exp ( -

]

f~ ~:»)

Ix(O)J2

Ix(a)J2 da

13(0)

[1 -

f~ ~~:»)

exp ( -

]I

tE

x(O)J2,

[0, 1],

(6.6.57)

in which ~

1 - t - f/ (1 - a)[1 - a(a)] da 1 + 2 fA (1 - a)[1 - a(a)] do '

(t) -

(6.6.58)

~-::--;;-;----,------:-;>,=--------:--:-=i~

pet) = 1 - t + f: (1 - a)[1 - a(a)] da 1 - 2 fA (1 - a)[1 - a(a)] da

(6.6.59)

To verify this claim, write out (6.6.54) in first-order form to obtain the matrix coefficient A(t)

=

[-~(t)

~J.

(6.6.60)

With formulas (6.6.12) and (6.6.13), compute (6.6.61)

Ao(t) = (1 - t)12'

J1 (1 -

A 1(t) =

a)[1 - a(a)]

da[~

~J,

(6.6.62)

and hence A T(t)A 1(t)

+ A 1(t)A(t)

=

2

5,1 (1 -

a)[1 - a(a)] do [ -

~(t)

~J.

(6.6.63)

From (6.6.63) it is apparent that ")1112

s

A T(t)A 1(t)

")11

=

-2

+ A 1(t)A(t) ~

(6.6.64)

(;112'

where

(;1 =

fa1 (1 -

a)[1-a(a)] da,

(6.6.65)

2 fa1 (1 - a)[1 - a(a)] da

because of (6.6.55). That restriction on a( .) also ensures that 1 -

(6.6.66) ")I 1

> 0 and

126

6.

General Linear Equations with Time-Varying Coefficients

1 - <>1 > O. The eigenvalues of 1

L

Ai(t) = Ao(t)

i=O

=

+ A 1(t)

[1

Jtr (l -

11

1- t

(l - 0')[1 - a(O')] dO'l

0')[1 - a(O')] do

~~

1- t

compute to be Al(t)

=

1- t

fll(t) = 1 - t

-11 + 11

(l - 0')[1 - a(O')] da,

(6.6.68)

(1 - 0')[1 - a(O')] da.

(6.6.69)

Now it is clear from (6.6.65), (6.6.66) and (6.6.68), (6.6.69) that (6.6.58), (6.6.59) arise from the choices and and the claim (6.6.57) is a direct consequence of Theorem 6.6.1. The preceding theory and technique for approximating the extreme eigenvalues of the solution PO to (6.5.7) can be applied to obtain estimates of Ix(tW as well as of its integral.

Theorem 6.6.7 Let PO be a symmetric solution to (6.5.7) satisfying (6.6.3) for all t in a subinterval of J with r ~ t. On that subinterval the solution x(' ) to (6.6.1), (6.6.2) satisfies

~~:~

exp( -

f~~:»)

1~12 ~

Ix(tW

~ ~~;;

fP~:»)

exp( -

1~12

(6.6.70)

for all t in the subinterval. Proof. Let vet) = x(t)· P(t)x(t). Since pet) satisfies (6.5.7), it is easy to check that

d~;t)

= -lx(tW.

(6.6.71)

Since

o < ~(t)In

~

pet)

~

P(t)I n

for

r

~

t,

(6.6.72)

6.6

127

Estimates of Solution Norms

successive applications of (6.6.72) to (6.6.71) imply

dv(t) vet) -

dt pet),

--<-which integrates to give

vet)

~

vCr) exp( -

f P~:)

Then another application of (6.6.72) yields

a(t)lx(tW

s

p(r)I~12

(6.6.73)

exp( -

(6.6.74)

.

f p~:) ,

(6.6.75)

and the second inequality of (6.6.70) is clear. A similar approach produces the remaining inequality in (6.6.70) and can easily be checked by the reader. Thus the proof of Theorem 6.6.7 is secured.

EXERCISES

1. In Chapter 5 we estimated the norm of the solution to equations with a constant-coefficient matrix that is at the same time a stability matrix. The techniques and theory given in this chapter provide alternative methods and also cover problems in which the constant matrix is not necessarily a stability matrix. (a) Show that if A(') is a constant stability matrix, then the integral (6.6.10) has a limit as b --+ 00, and that the matrix limit is the one used in deriving the Cetaev estimates of Theorem 5.2.4. (b) Compare the computations involved in the methods of this chapter with those required for application of Theorem 5.2.4 to equations with constant stability matrix coefficients. (c) Compute the estimates in Theorem 6.6.7 on an interval ~ t ~ 2 based on values of k = 0, 2, 4 in Theorem 6.6.5 for the constant-coefficient matrix of Example 5.1.1 and compare the results with that appearing in (5.2.38). 2. In Eq.(6.6.54), choose aCt) = t to get the Airy equation. Compute the terms in the estimates (6.6.57). Compare the results with those obtained by computing the terms in the estimates (6.6.70) and integrating the resulting inequalities. 3. Derive estimates of Ix(tW and its integral for the equation

°

v + t- 1 v = on J

=

(0, 2).

°

128

6. General Linear Equations with Time-Varying Coefficients

4. Derive estimates of Ix(t)l' and its integral for the Bessel equation of zero order

ii

6.7

+ t-'d + u = 0.

Fundamental Matrices of Equations with Timevarying Coefficients

Theorem 6.7.1 The determinant of any n x n matrix sofution 'I' to the diferential equation

Y = A(t)$,

a.e. t E J ,

(6.7.1)

satisfies the equation

det Y ( t ) = e x p ( l tr A(u) d o ) det Y(z),

(6.7.2)

t EJ, z E J.

Proof. Using subscripts to denote rows, in terms of its rows write

1-

(6.7.3)

Then by the n-linearity of the determinant function relative to rows, it follows that

d dt

- det

Y = det

+

... + det

,

(6.7.4)

a.e. t E J . Since Y satisfies the equation -d' I ' = A Y = A dt

a.e. t E J ,

(6.7.5)

129

6.7 Fundamental Matrices for Time-Varying Coefficients

in which Ii is a linear combination of rows r1' r2' ... , rn , excluding rio Hence, on substitution of the derivatives (6.7.6) into the determinants of (6.7.4), by the n-linearity along with the fact that the determinant of a matrix having a repeated row is zero, the resultant equation

d

dt det 'P(t)

= [tt A(t)] det 'P(t),

a.e.

(6.7.7)

tEJ

obtains. This is a linear differential equation whose unique solution is clearly the function given by formula (6.7.2), and the proof is complete.

Remark 6.7.1 It is evident from (6.7.2) that det 'P(t) = 0 at some point t E J if and only if det 'P(') is identically zero on J. Theorem 6.7.2 If 'P and 'i' are n x n matrix solutions to (6.7.1) and 'P is nonsingular, then 'i'(t) = 'P(t)C,

(6.7.8)

tEJ,

for some appropriate time-independent n x n matrix Cover fF. Moreover, each nonsingular matrix solution qi to (6.7.1) is given by (6.7.8) for appropriate nonsingular C. Proof. Let 'P and 'i' be matrix solutions of (6.7.1) with the former nonsingular. Differentiation of the matrix function (t) = 'P- 1(t)'i'(t), (6.7.9)

using the fact that 'P and qi are solutions of(6.7.1), leads to

~ = (_'P- 1d'P '1'-1) qi + 'P- 1~ 'i' dt

dt

=

-'¥-1A'P'¥-1'i'

dt

+ '¥-1A'i' = 0,

a.e.

tEJ.

(6.7.10)

But this implies that is a (constant) matrix over fF; call it C. That is, (6.7.11)

130

6.

General Linear Equations with Time-Varying Coefficients

which proves (6.7.8). The proof of Theorem 6.7.2 is concluded by noting in (6.7.8) that det 'f1 = 0 if and only if det C = O. Remark 6.7.2 Any nonsingular solution '¥(t) to (6.7.1) is called a fundamental matrix of A(·). It follows from Theorem 6.7.2 that the fundamental matrix cD(t, r) of Theorem 6.3.2 can be computed from any fundamental matrix 'P of A( .) according to the formula (6.7.12)

EXERCISES

1. Verify that the solutions to (6.4.6) constitute a finite-dimensional vector space. Determine an ordered basis for the vector space and determine the dimension. 2. Show that if Yl> Y2, ... , Yn is a linearly independent collection ofabsolutely continuous n-vector functions defined on an open interval J, then its elements constitute an ordered basis of the vector space of solutions to the equation x = A(t)x for some n x n locally integrable matrix function A(') oti J,

3. Verify that $'n., the dual of $'n, is a vector space and show that its dimension is n by exhibiting an ordered basis. 4. Prove that cDA(t, 0) = cDA(t, 't)<1lA('t, u) for all t, r, U in J. 5. Verify Remark 6.7.2. 6. For the matrix function A(t) =

[~ t~

J

tEJ =

s,

show that cD A(t, r) ¥= exp(J~ A(u) du) by computing both sides of the claimed inequality. 7. Show that cD~ = <1l AT if A(t) is independent of t and give an example demonstrating thatthis equality can fail for time-varying A(·). 8. Consider the nth-order scalar differential equation x(n)

+ a1(t)x(n-l) + ait)x(n-2) + ... + an(t)x = 0,

in which the coefficient functions a;(t) are locally integrable $'-valued functions defined on an open interval J c ~ (i = 1,2, ... , n). (a) Prove that the scalar equation has a linearly independent collection of solutions
6.7

131

Fundamental Matrices for Time-Varying Coefficients

scalar equation as a first-order system in J x

Show that

ff'n.

cPn(t) ]

cP~I)(t)

cP~n ~ l)(t) is a fundamental matrix for A(·). 9. Let cPI' cP2"'" cPnbe real- or complex-valued functions, (n - Ij-times continuously differentiable on an open interval J. Assume that the matrix function <11('), computed from the cPi as in Exercise 8, is nonsingular on J. (a) Show that <11 is a fundamental matrix for the continuous matrix function A(') associated with the first-order form of a scalar equation

having continuous coefficients a;(') on J. (b) Show that the cPi constitute a basis for the vector space of solutions to the scalar equation. 10. Apply Theorem 6.7.1 to compute det <1I(t) in Exercise 8. 11. Apply Exercise 8 to compute a fundamental matrix and <11A(t, r) for the differential equation tX(2)

+ 5X(I) + (13/t)x =

0

(a) for t E J = ( - 00, 0), (b) for tEJ = (0,1). (Hint: Look for solutions of the type cP(t) = jtj). and split them into real and imaginary parts.) Generalize the problem to the Euler equation tX(2)

+ {3X(I) + (y/t)x

= 0

in which {3, yare real numbers. 12. Dual equations are occasionally used in control theory to simplify models of input-output systems. Consider the equation (6.2.1), x = A(t)x + f(t), for some family of inputs f E f. Let <11* be the n x n matrix solution to the dual equation
1

11

132

6.

General Linear Equations with Time-Varying Coefficients

6.8 Continuous Time, Discrete State Markov Processes

Problem situations involving some element of uncertainty frequently arise in applied work. A significant subset of these can ultimately be modeled, using techniques of applied probability, as special linear differential equations of the type discussed in this book. The brief introduction given here is intended to make the reader aware of this important field of application and to show by example that it is possible to treat some of these problems with little special knowledge. The exposition is kept intuitive and informal. Probability theory systematically attaches a number p[E], between zero and one, called the probability of E, to each event E of concern. This number in interpreted as a degree of likelihood that the event would happen in a trial. It is done in such a manner that (6.8.1) if the occurrence of E 1 has no bearing on the occurrence of E 2 (i.e., if E 1 is independent of E 2 ) , and vice versa. Furthermore, (6.8.2) and, in particular, if it is not possible for both E 1 and E 2 to happen together, then it follows from (6.8.2) that P[E1 or E 2 ]

= P[E 1 ] + P[E2 ]'

(6.8.3)

E 1 IE 2 denotes the event E 1 under the assumption that E 2 has happened, and the probability of that event p[Ell E 2] is called the conditional prob-

ability of E 1 , given that E 2 has happened. Now imagine a situation in which a real scalar variable Set), called the state of the system at time t, evolves with time t E £Ft. It is assumed that Set) takes on only integer values. The Markov property assumed is the following: For each flnite set of times r, < t 2 < ... < t n < tandintegersi 1,i2,···,in,i, p[S(t) = =;:

il S(td = p[S(t)

t., S(t 2) = i2, ... , S(tn) = in]

= i I S(tn) = in]'

(6.8.4)

This property might be described roughly as short memory. The conditional probabilities in (6.8.4) must be considered in modeling, but in applications it is usually the functions Xi(t)

= p[S(t) = i]

(6.8.5)

that are of interest. Differential equations for these functions can be derived for problems in which it can be assumed that (a) p[S(t + L\) = i ISet) = j] = aiit)L\ + O(L\2), (i =1= j) and (b) p[Set + L\) = i ISet) = i] = 1 - aii(t)L\ + O(L\ 2)for all i,j and small L\ > O.

6.8

133

Continuous Time, Discrete State Markov Processes

The coefficient functions a;/t) must be selected to model the specifics of individual problems. The only restraints upon them a;;(t) =

L ait),

j>#i

(6.8.6)

.

provide the probabilistic interpretation that aij(t).::\ is the probability that starting at time t the state makes a transition from state j to state i during an interval of length .::\. The probability that more than one such transition occurs during that interval is assumed to be 0(.::\2), which is to say, (1/.::\)0(.::\2)--+-0 as .::\--+-0. In numerous problems the aij(t) are independent oft.

DERIVATION OF THE DIFFERENTIAL EQUATIONS FOR THE x,(tl

By breaking up the eventS(t + .::\) = i into the conditional events S(t + .::\) = i IS(t) = j (j = 0, ± 1, ... ) and applying (6.8.1), (6.8.3), and (6.8.4), it is

easy to see that p[S(t

+ .::\) = i] =

p[S(t

+L

j>#i

+ .::\) = i I S(t) = i]P[S(t) = i] p[S(t + .::\) = i IS(t) = j]p[S(t) = j].

(6.8.7)

The terms involving P in (6.8.7) can be eliminated by substitution, using (6.8.5) and assumptions (a) and (b), with the result being xi(t

+ .::\)

=

[1 - a;;(t).::\]X;(t) +

L a;/t)x/t).::\ + 0(.::\2).

(6.8.8)

j'i'i

By subtracting x;(t) from both sides of (6.8.8), dividing by .::\, and letting .::\--+- 0, we obtain the differential equations x:;(t) = -a;;(t)x;(t)

+

L a;/t)x/t).

(6.8.9)

j>#;

In summary, we conclude that in applications for which the Markov property (6.8.4) and assumptions (a) and (b) are appropriate, the probabilities (6.8.5) satisfy linear homogeneous equations of the special type (6.8.9). Conversely, a linear homogeneous system x = A(t)x in which A;;(t) = -aii(t) and A;/t) = aij(t) for i 1= j with the aij(t) satisfying (6.8.6) can be given a probabilistic interpretation. The matrix A(t) associated with (6.8.9) need not be offinite size. In problems in which S(t) can take on only a finite number of values with nonzero probability, A(t) can be taken to be finite. Thus the modeling problem is reduced to determining the aij(t). They are usually quite readily obtained from the specific problem considered. Example 6.8.1 Simple Epidemic Model. In a population of size N, assume that there is a probability of ex.::\ that one contact will occur between two in-

134

6.

General Linear Equations with Time-Varying Coefficients

dividuals in the population during time A and that the probability that more than one contact will occur during that time is relatively negligible. Which two individuals meet is determined by drawing them at random from the entire population. An epidemic starts with one infected. person in the population. Whenever an infected person and a noninfected person have a contact, there is a probability p that the noninfected person will become infected. Once a person becomes infected, he stays infected. The problem is to determine the aij(t)swhere S(t) is the number of infected individuals at time t. The solutions to the resultant equations (6.8.9) are then to be computed and examined. In general, the approach to finding the aij(t)sis through determination of what is called the infinitesimal transition scheme; namely, select an arbitrary state j , time t, and interval length A , then consider the states to which a transition is possible, and compute the probabilities of each. As mentioned above, this will determine the aij(t)s. For the problem under consideration, the possible state values (with positive probability) are restricted to j = 1, 2, . . ., N , since initially one individual is infected and no more than the entire population N can become infected. The parameters of the problem p, a, and N are time independent. The transitions that can occur from an initial state j and their respective probabilities can easily be computed with the resultant infinitesimal transition scheme: Transit ions

j

=

Probabilities

1

N-tj

In view of assumptions (a) and (b), which define the aij(t),it is a simple matter to read off from the infinitesimal transition scheme

aij(t)=

jfN-j)

2pa N(N - 1)

lo

for

1 < j < N - 1, i = j ,

(6.8.10)

otherwise

( j = 1, 2, . . ., N ; i = 1, 2, . . ., N ) . The last part of the problem being considered will be solved only for a

6.8

Continuous Time, Discrete State Markov Processes

135

population with N = 3. In this case (6.8.6) and (6.8.10) produce the coefficient matrix (6.8.1 1)

of the system 1 = A x having x e B 3 where it is kept in mind that xi(t) = P [ S ( t ) = i] = P[number infected = i] (i = 1, 2, 3). Since initially only one individual is infected, the initial condition is x,(O) = 1, x2(0) = 0, x3(0) = 0. The ABC formula can be employed to obtain the solution ~ ( t=) [x', AX', A 2 x 0 ] [ y o ,By', B 2 y 0 ] - l y ( t ) .

(6.8.12)

Clearly the eigenvalues of A are Iz = 0, - $pa,- $pa to which is applied the correspondence rule, yielding

This in turn determines yo

=

[;I,

(6.8.14)

and (6.8.15)

With

this provides all the data needed for (6.8.12), which can be computed with

136

6.

General Linear Equations with Time-Varying Coefficients

the result that Xl(t) = exp( -

2~rx

t),

(6.8.16)

(23Prx t) ,

2 X2(t) = 3Prxt exp -

X3(t)

=

1 - exp( -

2~rxt)

-

(6.8.17)

2~rxtexp(

_

2~rxt).

(6.8.18)

Notice that Xj(t) ~ 0 (i = 1,2,3) and xl(t) + X2(t) + X3(t) = 1, as is required of probabilities. Initially Xl(t) is one and X2(t), X3(t) are zero; as t -- 00, Xl(t) monotonically decreases to zero and x 2 (t) builds up for a short time and then converges to zero, as does x 3 (t). This behavior is compatible with the epidemiological interpretation of the problem. The manner in which the parameters rx and P enter the solution also agrees with one's intuition. Remark 6.8.1 The special stochastic processes touched upon in this section of the text (Markov processes) play an important role in diverse fields of application, including birth-death processes and queueing theory. A number of elementary texts such as [2J are available for those interested in learning more about the subject.

EXERCISES

1. Verify that if the conditions' xj(t) ~ 0, L Xj(t) = 1 are satisfied by a solution of the system (6.8.9) at t = to, then they continue to hold for to < t. 2. Compute the explicit solutions to the differential equations (6.8.9) for Example 6.8.1 when N = 2 and N = 4. Correlate the behavior of the solutions with the epidemiological interpretation of the equations. 3. A head of household repairs the home's single dishwasher whenever it breaks down. Assume that the time required before the machine fails again after being repaired each.time is exponentially distributed with rate parameter A. That is, the probability that the working machine will continue to work for at least t more units of time is e-.l.t. Further assume that when the machine fails, the time required to repair it also is exponentially distributed but with rate parameter JI.. Let the state S(t) of the system be the number of machines broken down at time t (i.e.,zero or one). Determine the infinitesimal transition scheme for the problem. Then compute the probability that the machine is operating at time t if it is known to be operating at time zero. 4. In a chemical reaction a molecule of a substance A can react with a

6.8 Continuous Time, Discrete State Markov Processes

137

molecule of a substance B to form a molecule of a compound C. Assume that during a time interval of length L\ the probability that a molecule of A and one of B are in close enough proximity for the reaction to take place is proportional to the product of the numbers of molecules of type A and type B present. Assume that initially there are N molecules of each type, A and B, present. Let the state be the number of molecules of type A left at time t. Write out the infinitesimal transition scheme. Compute the probabilities (6.8.5)for the problem if N = 4. What procedure could be taken if N = loo? N = 108? 5. In a store with one cashier, let j be the number of customers waiting in line (this includes the customer currently being served). Assume that in time L\ the probability of the arrival of a new customer is 2L\. The arrival of more than one customer during that time is assumed to have relatively negligible probability. Similarly, assume that the customer currently being served has probability PL\ of finishing during time interval L\. Assume that whether or not a customer arrives is independent of whether or not service is completed for the customer presently being served. Determine the infinitesimal transition scheme if the state S(t) is taken to be the number of customers waiting in line at time t.

Chapter

7

Commutative Linear Differential Equations

In this chapter, we define and study a class of linear time-varying differential equations (including the constant-coefficient ones) that we call commutative equations. (Another such intermediate class, periodic equations, is dealt with in Chapter 8.) Commutative equations are shown to be solvable in closed form, using the same operations as those required for the constant-coefficient case. Thus the constant-coefficient theory of Chapters 3-5 is found to have an impact well beyond the original motivating applications. The judicious restriction of the class of equations studied permits refinement ofthe general results obtained in Chapter 6 and thereby leads to a more detailed picture of solution behavior.

7.1 Introductory Comments

Linear constant-coefficient equations constitute an important but relatively meager subset of the full class of differential equations with variable coefficients discussed in Chapter 6. It would prove fruitless to seek a method for generating the solutions to the extended class of equations in terms of elementary functions using the finite operations employed in the constant-coefficient situation. More often than not, differential equations themselves, in conjunction with the existence and uniqueness theory, are used to define the functions recurring in applied mathematics. A few special equations, such as Euler's equation, have been solved in closed form. However, a number of those equations are simply constant-coefficient equations disguised by transformations of variables. 138

7.2

139

The Definition and an Example of Commutative Equations

When the elements of the coefficient matrix are analytic in the region of interest, power series methods are sometimes useful. Of course a series is a limit, as opposed to a finite process such as solving a system of linear algebraic equations. If limit processes were to be admitted in computation, then it should be recalled that the existence proof of the previous chapter has already provided a series uniformly approximating the solutions over compact intervals, and, moreover, it requires only local integrability of the coefficient matrix. It becomes apparent, upon leaving the realm of constant-coefficient equations, that the concept of closed-form solution requires further development. One step in that direction is taken in the next section, in which commutative time-varying systems are defined and analyzed. This class of equations significantly extends the constant-coefficient class while continuing to allow construction of the solutions using, basically, only the operations of solving systems of linear algebraic equations and finding roots of polynomials-operations already required for constant-coefficient equations.

7.2

The Definition and an Example of Commutative Equations

Definition Let A(') be a locally integrable n x n matrix-valued function over fF, defined on an open interval J c r!lt. Call A(') commutative on J if there exists an absolutely continuous matrix function B(') on J such that

dB(~) and moreover

dt

= A(t),

A(t)B(t) - B(t)A(t)

a.e. t

=

E

J,

a.e. t E J.

0,

(7.2.1)

(7.2.2)

Such a B( .) is called a commuting antiderivative ofA( .). A differential equation dx at = A(t)x,

(t, x)

E

J x fFn,

(7.2.3)

is called commutative if its coefficient matrix A(') is commutative on J.

Remark 7.2.1 If A(') is commutative on J, then it has more than one commuting antiderivative. This is apparent on noting that the difference between any two such antiderivatives is a constant matrix C for which CA(t) - A(t)C = 0, a.e. t E J. If B(') is a commuting antiderivative for A('), then so is B(') + C where C is any constant matrix (such as In) that satisfies the latter equation.

140

7.

Commutative Linear Differential Equations

Remark 7.2.2 All constant-coefficient equations are commutative. This is demonstrated by the choice B(t) = tA, t E J, for those A(' ) independent of t. Example 7.2.1

The matrix function A(t)

= [ c~s t sin tJ -sm t

cos t

(7.2.4)

is commutative on J = f71. The reader should verify this by selecting an appropriate definite integral of A(') as a commuting antiderivative. The commutative differential equation (7.2.3) associated with (7.2.4) is explicitly solved in the next section to demonstrate the applicability of the theory. The reader might wish to attempt solving the problem by his own devices before reading further. EXERCISES

1. Show that if A(') is commutative on J, then so are the following:

A T( . ), PA(' )p-l, P a nonsingular matrix. (c) CA('), C a constant matrix commuting with A(t), a.e. t E J. 2. Show that if A(') is commutative on J with commuting antiderivative B( '), then A(t) = A(t)B(t) is likewise commutative on J. (Hint: Consider (a) (b)

B(t)

= tB 2(t).)

7.3

Computation of Closed-Form Solutions to Linear Commutative Equations

Theorem 7.3.1

If A(') is commutative on J, then (t, T) E J x J,

(7.3.1)

in which B(') is any commuting antiderivative of A(·). Proof.

The proof commences with an inductive proof of the equations d dt B"(t)

=

kA(t)B"-l(t),

(7.3.2)

a.e. t E J (k = 1,2, ...). By assumption (7.2.1), Eq. (7.3.2) is valid for k = 1. Now assume that (7.3.2) holds for fixed k E {I, 2, ...}. Since B(' ) is assumed to be absolutely continuous, application of (7.2.1), (7.2.2), (7.3.2), and the

141

7.3 Computation of Solutions to Linear Commutative Equations

product rule of differentiation yields

d

d

+ A(t)Bk(t)

dt B"+ let) = B(t) dt Bk(t)

= B(t)kA(t)Bk-l(t)

= (k +

+ A(t)B~t) a.e.

1)A(t)B~t),

t E J.

(7.3.3)

This concludes the proof of (7.3.2) by induction. Since the exponential function is an entire function and (7.2.1) holds at a.e. t E J, it follows that it is possible to do term-by-term differentiation of the series eB(t) =

f

B"(t),

(7.3.4)

k=O k!

using (7.3.2) to get

~

dt

eB(t)

=

f.! ~

k= 1 k! dt 00

= k~l (k -

Bk(t)

1

00

1)! A(t)Bk-l(t)

= A(t)eB(t),

a.e.

1

= A(t) k~O k! B~t)

t E J.

(7.3.5)

From (7.3.5)it readily follows that the matrix function defined by the product in (7.3.1) satisfies (6.3.8) of Theorem 6.3.2 and clearly (6.3.9) as well. Thus the proof of Theorem 7.3.1is completed by appealing to the uniqueness statement in Theorem 6.3.2 regarding Eqs. (6.3.8), (6.3.9).

Theorem 7.3.2 If A(') is locally integrable on J, then it is commutative there

if and only if for

each fixed

BoA(t) - A(t)B o

't" E J

= A(t)

f

there exists a matrix B o over ff such that A(O") do -

f

A(O") do A(t), a.e. t E J. (7.3.6)

(Generally the choice of B o depends on r.) Proof.

Assume that (7.3.6) holds for some B(t) =

s,

+

f

A(O") da,

't"E

J and B o over ff. Define

t E J.

(7.3.7)

From (7.3.6), (7.3.7), and the assumed local integrability of A(') it follows that (7.2.1) and (7.2.2) hold for the B(') defined in (7.3.7). Hence A(') is commutative on J. Conversely, if A(') is commutative on J, then there exists an

142

7.

Commutative Linear Differential Equations

absolutely continuous matrix function B(') satisfying (7.2.1) and (7.2.2). Moreover, B(') must satisfy (7.3.7) for each r E 1 when one takes B o = B(r). Substitution of (7.3.7) into (7.2.2) leads directly to (7.3.6), and the proof is complete. Remark 7.3.1 The characterization of commutative matrix functions given by Theorem 7.3.2 provides a test for commutivity that is sometimes useful in applications. (Other criteria will appear later in this chapter.) Note that computing B o is a .problem of solving a system of linear algebraic equations. Such a rand B o could be used in the formula B(t)

= Bo +

f

(7.3.8)

A(s) ds

to produce an antiderivative B(') as required for computation of A via formula (7.3.1) if A(') turns out to be commutative. Remark 7.3.2 If A(') is commutative on 1 and a commuting antiderivative B(') is given or else computed according to Remark 7.3.1, then the matrix function A providing the general solution to (7.2.3) can be computed by solving a linear differential equation with constant-coefficient matrix containing a parameter, namely, (7.3.9)

A(t, r) = 'P(1, t)'P( -1, r)

in which 'P(a, t) is the unique solution to the initial-value problem

aaa 'P

=

B(t)'P,

(7.3.10)

a E [ -1,1],

for each fixed value of the parameter t e J, (The continuous function 'P that deforms the initial matrix In into the desired function 'P(l, r) = eB(t) as a is varied from to 1 is an example of what in topology is called a homotopy.)

°

The applicability of Theorems 7.3.1 and 7.3.2 is demonstrated by the following example. Example 7.3.1 The problem considered is that of computing the closedform solution to the equation

:t[:J

=

[-:~~:

;~: J[: J

teJ

=~,

(7.3.11)

associated with the time-varying matrix (7.2.4). For this problem, Eq. (7.3.6) reduces to BoA(t) - A(t)B o = 0,

(7.3.12)

7.3

143

Computation of Solutions to Linear Commutative Equations

a.e. t E &t, for each, E &t. Hence Theorem 7.3.2 substantiates the commutivity of(7.3.11) since (7.3.12) is satisfied by B o = 0 or In, for example. Thus choose , = 0, B o = O. Equation (7.3.8) is then applied to compute the appropriate commuting antiderivative B(t) =

rJot A(u) do = [ cost-1 sin t

1 -. cos tJ

(7.3.13)

smt

to be used in (7.3.10). Pursuing the solution of (7.3.10) by the procedure of Chapters 3 and 4 requires computing the eigenvalues of B(t), which are

A.

sin t

=

± i(l

- cos t).

(7.3.14)

As long as 1 - cos t i= 0, the correspondence rule gives y(u)

e lf sin t =;

[

. t elf Sin

cos[u(1 - cos t)]J • [ sin u(1 - cos t)] ,

(7.3.15)

which is differentiated relative to a to produce sin t [ 1 - cos t

cos .t - 1J y (a) = By (o) sin t

(7.3.16)

(where B is the matrix of the ABC method and is not to be confused with B(t)). According to Theorem 4.6.1, elfB(t) = (A l y(o'), A 2y(u)),

(7.3.17)

in which Ai = (e, B(t)e;)(y(O), By(O))-l

(i = 1,2).

(7.3.18)

With B(') computed in (7.3.13), B in (7.3.16), and y(.) by (7.3.15), the Ai in (7.3.18) are computed by elementary matrix calculations and are substituted into (7.3.17) with the result that lfB(t) = e

elfsint [

cos[er(I - cos t)] - sin u(1 - cos t)

[

]

tr

sinEu(1 - cos t)]J cos u(1 - cos t)

[

(7.3.19)

Note that (7.3.19)remains valid for 1 - cos t = 0 by continuity. Substitution of (7.3.19) into (7.3.1) finally provides the fundamental matrix cI> A(t, r) =

e sint - sinr

[c~S( cos

t - COS,) sm(cos t - cos r)

- sinecos t - cos r) J . (7.3.20) cos(cos t - cos r)

In particular, cI>A furnishes the solution to (7.3.11), satisfying Xl(') = x 2(r) = ~2: X1J = [ X2

esint-sint

= [~l ~l

~l'

c~s(cos t - cos r) - ~2 sin(cos t - cos r)J. (7.3.21) sm(cos t - cos r) + ~2 cos(cos t - cos r) (7.3.22)

144

7.

Commutative linear Differential Equations

EXERCISES

1. Verify that the following system is commutative on qj by application of Theorem 7.3.2, and determine a commuting antiderivative B(') of its coefficient matrix A(·). X2

=

+ (1 + 2t)X2'

-tx 1

2. Check the commutivity of the following system on qj and then compute its general solution by the, method discussed in Section 7.3.

3.

Consider the matrix function of qj,

A(t)

O1J {[1 0'

t

~ [~ ~J

irrational, rational.

Find an appropriate commuting antiderivative to demonstrate that A(') is commutative on qj. 4. The matrix function B(t) = sin 2t [

(0. ) sgn sin t

olJ

is absolutely continuous on qj. Why? Compute its derivative A( .) and confirm the commutivity of A(') on qj. Find those (t, r) E qj2 for which A(t)A(-r) - A(-r)A(t) "# 0.

5. Show that if A(') is commutative on J with commuting antiderivative B( '), then for each -r E J, B(-r)A(t) - A(t)B(-r) = A(t)

f

A(u) do -

f

A(u) da A(t),

a.e. t E J. Apply this result to conclude that the equality A(t)

t

A(lT) do

=

fo~(U)

do A(t),

a.e. t E qj, fails for the commutative system of Exercise 4.

145

7.4 Some Sufficient Conditions for Commutivity

7.4

Some Sufficient Conditions for Commutivity

The results of this section provide easy verification of commutivity where they apply.

Theorem 7.4.1 A locally integrable n x n matrix-valued function A(') on J is

if either of the following equivalent conditions are A(t)A(r) = A(r)A(t), a.e. t E J for a.e. r E J, A(t) J: A(O") da = J: A(O") da A(t) for a.e. t E J, all r E J.

commutative

(a) (b)

Proof.

met:

Clearly, (a) implies (b). Now assume (b), which is to say that A(t)

f

A(O") da -

f

A(O") do A(t)

=0

(7.4.1)

for a.e. t E J, all r E J. At each point t E J at which (7.4.1)holds, the left-hand side is an absolutely continuous function of r: therefore the equation can be differentiated relative to r and (a) results. Thus (a) and (b) are equivalent. If (b) is assumed, then, upon setting Bo = 0 in (7.3.6), Theorem 7.3.2 applies and A(') must be commutative. This concludes the proof.

Corollary 7.4.2 Let A o be any n x n matrix over CC and f(z) be a power series centered at the origin, as in Theorem 4.2.2, with radius of convergence p, If Ao is nilpotent, then the formula (7.4.2)

A(t) = f(tA o)

defines a matrix function A(') that is commutative on 9f. If P > 0, then (7.4.2) determines a commutative matrix function on the interval - plr < t < otr in which r is the maximum absolute value of the eigenvalues of A o.

Example 7.4.1 According to Corollary 7.4.2, A(t) =

ao

f'

L: I" A~ k.

= e tAo,

t

E

9f,

(7.4.3)

k=O

is commutative fo.r any n x n matrix Ao over CC. The commutivity of Example 7.2.1 is now transparent when it is recognized to be of type (7.4.3) with Ao the matrix [; ~ A].

Example 7.4.2 series is

Another commutative matrix function defined by a power ao

A(t) =

L:

k=O

t

E (

f' A~

= (In -

tA o)- t,

(7.4.4)

-llr, l/r), where Ao is any n x n matrix over CC whose eigenvalues

2 each satisfy 121

< r.

146

7.

Commutative Linear Differential Equations

If, in lieu of the series in (7.4.4), the inverse matrix in that equation is taken as the defining formula, then the interval can sometimes be extended. In fact, for A o having no real eigenvalues, commutivity occurs on (- 00, (0), and if OCl < OC2 < ... < oc, are the reciprocals of the real nonzero eigenvalues of a matrix A o over re, then intervals over which commutivity occurs can obviously be taken to be anyone of (- 00, ocd, (OCl' O(2), ••. , (OC'-l' oc,), (OC., (0). Although the inverse is defined for a.e. t E ~, the OCi must be excluded from the domain of A(') to maintain the local integrability. A more complete description of the manner in which analytic functions can be used in constructing commutative A(') is given in Section 7.5. Theorem 7.4.3 Let {Ak}O' be a commuting family ofn x n matrices over re and {OCk(')} 0' be a sequence of complex-valued measurable functions defined on an open interval J. If the series CXJ

L

(7.4.5)

locit)IIA k !

k~O

is convergent for a.e. t

E

J and is locally integrable, then the series

A(t)

converges for a.e. t

E

=

CXJ

L

k=O

(7.4.6)

ocit)A k

J and defines a commutative matrix function A(') on J.

Proof. The assumed convergence and local integrability of (7.4.5) imply the same for (7.4.6). Equation (a) of Theorem 7.4.1 follows from writing A(t) and A('l:) as series and applying the commutivity of the Aks. Thus the conclusion of Theorem 7.4.3 follows as a corollary to Theorem 7.4.1.

Series that are not necessarily power series serve as useful devices for producing commutative matrix functions. One example that illustrates the point is the series

Example 7.4.3

•

A(t) =

L CXJ

k~

(1 +-

e- kt [sin ktl I

1

k

)k

Ao ,

t E (0, (0), which clearly satisfies the hypothesis of Theorem 7.4.3. EXERCISES

1.

Consider any matrix function A(') of the type 00

A(t) =

L k~O

ocit)A k,

t

E

[0,

(0),

(7.4.7)

7.4

147

Some Sufficient Conditions for Commutivity

in which the n x n matrices A o' At> A 2 , .•. satisfy AiA j = AjA i for all i,j and the scalar-valued ak(') are locally integrable. (a) Show that if

k~O Loo lak(a) I do IAk ! <00 for some k o E {O, 1,2, ... }, then A(') is commutative on (0, 00). (b) Show that if A o is a stability matrix and as

tao(a)da-oo

t-oo

with

k~lLoolak(a)ldaIAkl
then all solutions to x(t) = A(t)x(t) satisfy lim, ... co x(t) = O. 2. Apply Exercise 1 to prove that all solutions to the system

approach zero as t written as

1: t[ -

3.

Note that the coefficient matrix can be

Hint:

00.

~ - ~] + 1 ~

t2 [

-

~ ~ -

J

Apply Theorem 7.4.1 to prove that A(t) = (tin - A o)-l(tIn + A o)

is commutative on an interval J if the n x n matrix A o has no eigenvalues in J. 4. Investigate the behavior of the solutions x(t) as t - 00 for the system x = A(t)x in which (a)

A(t)

= Lk=o e- kr cos kt At+ 1 with

~ ~],

Ao = (b)

A(t)

-2 -2

= Lk=O cos kt At+ 1 in which Ao

= [

~

0.1

J

0 0.1. -0.1 -0.2 -0.2

o

148

5.

7. Commutative linear Differential Equations

t

Find the closed-form solution to the equation A(t) = A o

when (a)

7.5

Ao = [ _

~ ~

J

(b)

Ao =

x=

A(t)x in which

eO"Ao do

G~l

Matrix Functions and Operations Preserving Commutivity

It is generally not a trivial matter to determine whether or not a given matrix function is commutative. Hence the extent of the class of commutative systems is of interest. One approach is to study the class of all commuting antiderivatives, i.e., all absolutely continuous n x n matrix solutions B(') to the equation B(t)B(t) - B(t)B(t)

=

0,

(7.5.1)

a.e. t E J. Their first derivatives produce all commutative A(' )s. The extent of the solutions to (7.5.1) is illuminated by finding operations that preserve its solutions. For example, all functions of the type B(t)

=

PA(t)P- 1

(7.5.2)

in which P is a constant n x n nonsingular matrix and in which the diagonal matrix A(t) = diag[ A,l(t),A,2(t), ... , A,n(t)] has the A,i(') absolutely continuous on J are solutions since A(·) is obviously a solution and the operation B(') - PB(,)p-l preserves solutions to (7.5.1). In Section 7.4, we have already indicated that functions of matrices are useful in constructing examples of commutative systems. Proper treatment of the subject draws on the elementary theory of functions of a complex variable. That theory provides an elegant solution to a problem that began to emerge in Example 7.4.2, namely, that of defining a function f(A) of a matrix A for which some eigenvalues of A fall outside the open disk of convergence for the power series defining fez). The matrix form of Cauchy's integral formula is now stated along with one of its corollaries. (For further results and proofs see [22]')

rc

Theorem 7.5.1 Let A be an n x n matrix over and consider any complexvalued function fez) analytic in an open set Q containing the collection a(A) of all distinct eigenvalues of A. There exists an open set w whose closure iiJ

149

7.5 Matrix Functions and Operations Preserving Commutivity

satifies wen and whose boundary aw consists of a finite number of simple closed curves that do not intersect and such that u(A) c co. A matrix f(A) is defined by the line integral f(A) =

2~ i. tu

j~w

f(z)(zI - A)-l dz,

(7.5.3)

in which the path of integration is oriented in the standard way. The integral (7.5.3) is otherwise independent of the choice of wand agrees with the power series definition off(A). Corollary 7.5.2 If fez) and g(z) are analytic in an open set containing u(A) and h(z) = f(z)g(z), then h(A) = f(A)g(A).

The Cauchy integral formula is a useful device in the study of commuting matrix functions. Theorem 7.5.3 Let fez) be analytic on an open set nand B(') be an absolutely continuous solution to (7.5.1) on the open interval J. If

u(B(t)) c

n

(7.5.4)

for all t E J, then B( .) = f(B(')) is also a solution that is moreover a commuting antiderivative of A defined as A(t) = !,(B(t))B(t),

t E

J,

(7.5.5)

where the prime denotes the complex derivative. Proof. Fix any t E J. The Cauchy integral formula as described in Theorem 7.5.1 provides the equation

1

f(B(t)) = -2. 1tl

i.j~Wt

f(z) [ zI - B(t)] -1 dz.

(7.5.6)

Generally the open set co, depends on t, but since B( .) is continuous and W t is open, co, can be held fixed for small changes in t. The analyticity of fez) and absolute continuity of B(') then provide sufficient conditions for application of a standard theorem for taking the derivative of (7.5.6) past the integration. The differentiation, aided by (7.5.1), proceeds as 1

i.

d f(B(t)) = -2. dt 1tl j~Wt

= -2 i. 1. 1tl

j~Wt

f(z)[ zI - B(t)] -1 B(t)[zI - B(t)] -1 dz

f(z)[zI - B(t)]-2 dz B(t),

(7.5.7)

a.e. t E J. As with Cauchy's formula for a scalar variable, the integral in the

150

7.

Commutative Linear Differential Equations

last term of (7.5.7) can be written as f'(B(t». The equation

d -

dt B(t)

.

= f'(B(t»B(t),

(7.5.8)

a.e. t E J, is thereby derived. It can be noted in (7.5.7) that f'(B(t» commutes with B(t) since B(t) commutes with B(t), a.e. t E J, and consequently

BB - BB =

f(B)f'(B)B - f'(B)Bf(B)

= 0,

(7.5.9)

a.e. t E J. The absolute continuity of B(') follows from (7.5.8) and the proof is complete. Corollary 7.5.4 Assume that B(') is a commuting antiderivative on J. Then the classical adjoint B*(') inherits the same property. IfB(t) is moreover nonsingular for each t E J, then B- 1( . ) is likewise a commuting antiderivative on J. Proof. By choosing fez) = liz, it follows directly from Theorem 7.5.3 that if B( .) is a commuting antiderivative on J with B(t) nonsingular for each t E J, then B- 1( . ) is a commuting antiderivative on J. To prove the remaining conclusion, recall that the classical adjoint of a matrix B satisfies

(7.5.10) where jBj is the determinant of B. Now assume that B(-) is a commuting antiderivative on J and thus B(t)B'(t)· - B'(t)B(t)

= 0,

(7.5.11)

a.e. t E J. First consider the case in which B(t) is nonsingular at each t E J. Multiplication of (7.5.11) on the left and right by B*(t) and application of (7.5.10) lead to (7.5.12) a.e. t

E

J, and hence to

B'(t)B*(t) - B*(t)B'(t)

a.e. t

E

E

+ B~(t)B(t)

E

- B(t)B~(t)

- B'(t)B.(t)

= 0,

(7.5.14)

J, and in view of (7.5.13) this is the same as B~(t)B(t)

a.e. t

(7.5.13)

J. Differentiation of (7.5.10) gives

B*(t)B'(t)

a.e. t

= 0,

- B(t)B~(t)

= 0,

(7.5.15)

J. Multiplication of (7.5.15) on the left and right by B. and a final

7.5

151

Matrix Functions and Operations Preserving Commutivity

application of (7.5.10) show that

= 0,

- B~(t)B*(t)]

IB(t)I[B*(t)B~(t)

(7.5.16)

and thus B*(t)B~(t)

-

B~(t)B*(t)

=

0,

(7.5.17)

a.e. t E J. All that remains to be done is to show that (7.5.17) would likewise hold at points where IB(t)1 = o. Suppose that IB(r)1 =:= 0 at a point r E J at which the equation in (7.5.11) holds. Then by the continuity of B(' ), for each f. > 0 appropriately small the matrix function Bi«, t) = B(t) + el; will satisfy IB(f., t)1 #- 0 on some subinterval J, of J containing r. But a trivial calculation shows that B(e, t )

dB(f., t) _ dB(f., t) ( )-0 dt dt B f., t - ,

(7.5.18)

a.e. t E J" including the point t = r. It follows from what has already been proved that (7.5.19) a.e. t E J" including the point t = r, Now in view of the fact that the elements of B*(f., t) are polynomials in both f. and the elements of B(t) it is possible to let f. -+ 0 in (7.5.19), with the result that at least at t = r, (7.5.20) Since the argument applies to a.e. r

E

J, as restricted, the proof is complete.

Remark 7.5.1 A matrix function B(') on J is called analytic if it has a convergent power series

B(t) =

Lco k=O

(t

- rl B(k)(r) k!

(7.5.21)

about each point r ~ J. If B(') in Theorem 7.5.3 is analytic, then f(B(')) will likewise be analytic on J. Example 7.5.1 If fez) is any entire function such as eZ , sin z, any polynomial p(z), p(z)eZ , etc., then e", sin B, pCB), p(B)eB, etc., will be commuting antiderivatives of the consequent commutative matrix functions eBB, B cos B, p'(B)B, [pCB) + p'(B)]eBB, etc., for BO any commuting antiderivative. One allowable choice of B( .) would be B(t) = tAo, where A o is any n x n matrix. Theorem 7.5.5

Let B o be any n x n matrix over CC and f(t, z) be defined and continuous on J x E, where J is an open interval and E => a(B o) is an open

152

7.

Commutative linear Differential Equations

subset of~. Further assume that f(t, z) is absolutely continuous relative to t for each fixed z E E and that bothf(t, z) andfl'/, z) are analytic in zfor each fixed t E J. Then the matrix function B( .) defined as B(t) = f(t, B o) is a commuting antiderivative on J and moreover, BoB(t) - B(t)B o = 0, Proof.

t

E

(7.5.22)

J.

The assumptions allow application of the Cauchy integral formula 1 B(t) = -2. tu

1.

jll"',

f(t, z)(zI - BO)-l dz,

t E J.

(7.5.23)

The details are left as an exercise. Corollary 7.5.6 Iff(z) is analytic on an open set nand A o is an n x n matrix over ~ for which a(tA o) c: for all t

E

n

(7.5.24)

J, J an open interval, then the matrix function

A(t) = f(tA o)

(7.5.25)

as well as its derivatives A(k)(t) =

Pk)(tAo)A~

(7.5.26)

(k = 1, 2, ...) are analytic commutative matrix functions on J. Proof.

Apply Theorem 7.5.3 to verify condition (a) of Theorem 7.4.1.

Theorem 7.5.7 Let A o be any n -x n matrix over ~ and f(t, z) be defined on J x E where J is an open interval and E::J a(A o) is an open subset of~. Assume thatf(t, z) is locally integrable relative to t for fixed z and analytic in z for a.e. fixed t E J. Then the matrix function A('), defined as A(t) = f(t, A o), is commutative on J. Proof.

The assumptions allow application of the Cauchy integral formula A(t)

= -2. 1 1

1tl

jll"',

f(t, z)(zI - AO)-l dz,

(7.5.27)

The details are omitted. The next theorem is another example of the many interrelations between linear and nonlinear differential equations and linear algebra.

Theorem 7.5.8 Let B( .) be an n x n matrix-valued commuting antiderivative over

~

on an open interval J. If the characteristic polynomial of B(t),

(7.5.28)

153

7.5 Matrix Functions and Operations Preserving Commutivity

has distinct roots at a.e. t

E

J, then B(') satisfies the differential equation

a.e.

t

E

J.

(7.5.29)

(Subscripts denote partial derivatives.) Conversely, for every monic polynomial pet, 2) - over ((5 with absolutely continuous coefficients, each n x n matrix solution B(') to (7.5.29) will be a commuting antiderivative. If pet, 2) annihilates B(t) and has distinct roots at a.e. t E J, then the roots of the minimal polynomial of B(t) are also distinct at a.e. t E J and the eigenvalues of B(t) are roots of pet, 2) for a.e. t E J. Proof. If B(') is an n x n absolutely continuous matrix function over ((5, then its characteristic polynomial (7.5.28) will obviously have absolutely continuous coefficients. By the Cayley-Hamilton theorem, pet, B(t))

=

t

0,

E

(7.5.30)

J,

which can be differentiated, and when B(') is a commuting antiderivative, that differentiated equation will be of the form pit, B)B

+ pit, B)

=

0,

a.e.

t E J.

(7.5.31)

By a standard theorem of algebra the assumption that p(t,2) has distinct roots implies that pit, 2) #- 0 at those 2 in ((5 where pet, 2) = O. Hence since a(p(t, B)) = p(a(t, B)), the matrix pit, B(t)) is nonsingular at a.e. t E J. This allows (7.5.31) to be rewritten as .

B

=

-pit, B)

-1

a.e.

pit, B),

t

E

J.

(7.5.32)

To prove the last half of the theorem, let pet, 2) be a monic polynomial over absolutely continuous coefficients. Then the right-hand side of Eq. (7.5.29) is defined. Now let B(') be an absolutely continuous solution to that equation. (The existence and uniqueness problems for such nonlinear equations are treated later in the text and are of no direct concern here.) Since B(t) commutes with the right-hand side of(7.5.29), it satisfies (7.5.1) at a.e. t E J and hence is a commuting antiderivative. Suppose that ((5 with

per, B(r)) = 0

at some r

E

(7.5.33)

J. Since B(') is assumed to satisfy (7.5.29),differentiation leads to

d dt pet, B(t))

=

pit, B)

.

+ pit, B)B

= p,(t, B) - pit, B)p;: let, B)p,(t, B) a.e. t E J. = 0,

(7.5.34)

154

7.

Commutative Linear Differential Equations

The only continuous function satisfying (7.5.33), (7.5.34) is pet, B(t)) = 0,

all

t

E

(7.5.35)

J.

If (7.5.35) holds and pet, A) has distinct roots at a.e. t E J, then the minimal polynomial of B(t) divides pet, A) at each t E J, and hence that minimal polynomial must have distinct roots at a.e. t E J. Obviously the eigenvalues of B(t) are roots of pet, A) (with not necessarily the same multiplicity) a.e. t E J. This completes the proof. Corollary 7.5.9 Let B(') be an n x n matrix-valued commuting antiderivative over on an open interval J. If the characteristic polynomial pet, A) of B(t) has n distinct roots Ai(t) (i = 1,2, ... , n), a.e. t E J, they satisfy the scalar differential equation

rc

Z = - Pr(t, z)j pz(t, z), The eigenvalues of A(t) = B(t) are ~.lt)

a.e.

t

E

J.

(i = 1,2, ... , n), a.e. t

(7.5.36) E

J.

Proof. The conclusion that the A;(t) satisfy (7.5.36) on the intervals where they are distinct follows from differentiation of the equations pet, A;{t))

= 0.

(7.5.37)

Since B(') will satisfy (7.5.29), A(t) = -p;,(t,B)-lPrCt,B),

(7.5.38)

which in tum implies that the eigenvalues of A(t) are - p;,(t, ..1;)-1 PrCt, A;) (i

=

i;

(7.5.39)

= 1,2, ... , n), a.e. t E J. This proves Corollary 7.5.9.

Remark 7.5.2 Commuting antiderivatives exhibit a great deal of geometric regularity; i.e., they show a proclivity to be solutions to differential equations. Theorem 7.5.8 concludes this to be true even for those whose eigenvalues collide (but do not stick together). The differential equations are quite nonlinear.

The next theorem provides a clear picture of what is happening during the intervals between collisions. Its proof must borrow some knowledge about nonlinear differential equations from subsequent chapters. In this instance the maintenance of subject continuity seems to warrant the break in the traditional order of writing. Theorem 7.5.10 Let A(') be a continuous commutative n x n matrix-valued function over C(j on an open interval J and suppose that A( .) has a commuting

155

7.5 Matrix Functions and Operations Preserving Commutivity

antiderivative B(' )for which B(t) has n distinct eigenvalues at each t E J. Then B(t)B(I:) - B(I:)B(t) = 0, Proof.

all (r, 1:) E J x J.

(7.5.40)

Assume the hypothesis and consider the characteristic polynomial pet, z) = det[ zI - B(t)J,

t

E

(7.5.41)

J.

The proof revolves around the nonlinear differential equation in J x i

= - plt, z)/pit, z).

~,

(7.5.42)

As stated by Corollary 7.5.9 the eigenvalues A.;{t) of B(t) are each solutions to = 1,2, ... , n). The argument advanced in the proofof Theorem 7.5.8 explained that the denominator pit, z) in (7.5.42) is not zero along the paths {(t, Ai(t)) It E J} (i = 1, 2, ... , n). From the once continuous differentiability of B(') it is readily shown that for each compact subinterval [a, bJ c J the corresponding segments of those paths {(t, Ai(t)) It E [a, bJ} (i = 1,2, ... , n) are contained in open tubular neighborhoods on which the right-hand side of (7.5.42) is continuous in (t, z) and analytic in z for fixed t. The existence of the solutions Ai(') to (7.5.42) and this analytic behavior ofthe differential equation in their neighborhoods is enough to allow the classical theory of analytic nonlinear differential equations to apply (see Theorem 11.4.6). This theory states that (7.5.42) has a unique general solution ntt, 1:, z), regarded as a complex-valued scalar function of the initial parameters (I:, z) as well as t satisfying 11(1:,1:, z) = z. In the situation being considered I: is regarded as fixed in J. The classical theory concludes that for each fixed t E J the function l1(t, 1:, z) is analytic. in z on some open neighborhood of the eigenvalues Ai(I:)(i = 1, 2, ... , n) of B(')' in the complex plane. The uniqueness aspect of the theory ensures that l1(t, 1:, Ai(I:)) = A;(t), t E J (i = 1,2, ... , n). More could be said, but it is primarily the existence of the general solution l1(t, 1:, z) and its analyticity relative to z that are germane in the present proof of Theorem 7.5.10. They allow application of the Cauchy integral formula. The claim is that (7.5.42)onJ(i

B(t) =

~J:

2mjow,

l1(t,1:, z)[zI - B(I:)]-l dz,

t

E

J.

(7.5.43)

In (7.5.43) the open set co, is centered about the eigenvalues Ai(1:) of B(1:) = 1,2, ... , n). Note that the integral in (7.5.43) does define a matrix function of B(I:) for t E J fixed and that (7.5.43) is valid at t = I: since 11(r, r, z) = z. As a consequence of the fact that (as a function of t) l1(t, r, z) satisfies (7.5.42), it is easily verified that the integral (7.5.43) does indeed provide a matrix solution to the differential equation (7.5.29). But the anti(i

156

7.

Commutative linear Differential Equations

derivative of A(') hypothesized does satisfy (7.5.29)-one of the conclusions of Theorem 7.5.8-and the classical theory moreover ensures the uniqueness of the solution to (7.5.29), taking on the value B(7:) at t = 7:. This proves the equality (7.5.43). With (7.5.43) established, the commutivity claimed in (7.5.40) is now transparent. Since initially 7: E J was arbitrary, the proof is complete.

Remark 7.5.3 For those A(') satisfying the hypothesis of Theorem 7.5.10, Eq. (7.5.40) can be differentiated relative to t and 7: to get (t,7:) E J x J.

A(t)A(7:) - A(7:)A(t) = 0,

(7.5.44)

Thus at least for continuous A(') with commuting antiderivatives B(') satisfying the conditions imposed in Theorem 7.5.10, it is now clear what is happening on the open intervals between collisions of the eigenvalues of B(.).

EXERCISES

1. Show that the matrix function A(') on

- 2t A(t) =

[

~

defined by

2t]

1

o1

2t

1 -2t

1

is analytic and commutative. 2. Verify that B(') defined on ~ by

m,) ~ [,;

'~1

:;]

is a commuting antiderivative. Let A(') = (B*)'(') where * denotes the classical adjoint. Why is A(') commutative on ~? Compute A(·). 3. Show by direct computation that if B(') is a commuting antiderivative on J with B(t) nonsingular for each t E J, then B- 1( .) is also a commuting antiderivative. 4. Explain what is wrong with the following reasoning: For a scalar z, j(z) = det(z) is analytic in the entire plane. Hence for any n x n matrix A, 1

.i.

det(A) = -2

mj"w

det(z)[zl - AJ-l dz.

7.6 The Special Case n

5.

=

157

2

Without doing any extensive computing, show that the matrix function A(t) =

[2~2

-4t

3

~

t2

2t

~

t ] 3 _t 2

is commutative on &l and that on those intervals in which it is invertible, its inverse A - 1(.) is likewise commutative.

7.6 The Special Case n - 2

Clearly, all absolutely continuous scalar functions are commutative. This section analyzes 2 x 2 systems in detail.

Theorem 7.6.1 If a commutative 2 x 2 matrix function A(') on an open interval J is analytic, then A(t)A(-r) - A(T)A(t) = 0

for all

(r, T) E J x J.

(7.6.1)

On the other hand, if A(') is but Coo, then (7.6.1) can fail on a subset of J x J of positive measure. Proof. Let A(') be analytic on J. If A(') is assumed to be commutative, then there exists a commuting antiderivative B(') of A(') and B(t) = B(T)

+

f

A(s) ds

(7.6.2)

for each T E J and all t E J. Since A(') is analytic on J, it follows from (7.6.2) that B(') is likewise analytic there and that the equation (7.6.3)

B(t)B(t) - B(t)B(t) = 0

must hold for all t

E

J. Write out B(t) in terms of its elements as

B(t) - [bit) - bz(t)

b 1(t) ] bit) - bit) .

(7.6.4)

It follows upon substitution of(7.6.4) into (7.6.3) that the latter is equivalent to the system b 1(t)bz(t) - b 1(t)bz(t) = 0,

(7.6.5)

bz(t)b 3(t) - b 2(t)b 3(t) = 0,

(7.6.6)

b 3(t)b 1(t) - bit)b 1(t) = 0

for all

t

E

J.

(7.6.1)

158

7.

Commutative Linear Differential Equations

If btU, bi'), and bk) are all identically zero on J, then

= bit)Iz,

B(t)

t

E

(7.6.8)

J,

and hence

= bit)Iz.

A(t)

(7.6.9)

Certainly in this case A(t) satisfies (7.6.1). Now consider the remaining case in which at least one of btC-), bi'), bi') is nonzero on some open subinterval (ex, f3) c J. Suppose that it is b l (- ) that is nonzero on (ex, f3). Multiplying (7.6.5) and (7.6.7) by b I Z(t) and integrating produce the equations bz(t) = c Zb l(t),

(7.6.10)

bit) = C3bt(t)

(7.6.11)

for some constants Cz, C3 for all t E (ex, f3). But since b t('), bi'), and bi') are analytic on (ex, f3), it follows that (7.6.10) and (7.6.11) hold for all t E J. Substituting (7.6.10) and (7.6.11) into (7.6.4) and differentiating show that A(t)

. [1° OJ + . [0 1J

= bit)

1

bt(t) Cz -C3 '

(7.6.12)

and again it is a trivial matter to check that (7.6.1) holds. A similar argument applies in the two remaining possible cases for which it is b z or b 3 that is nonzero on a subinterval of J. This concludes the proof of the first part of Theorem 7.6.1. To establish the last part, consider the matrix function A(t)

=

a(t{~

~J

+ a( -t)[~

~J

(7.6.13)

on J = (- 1, 1), in which a( .) is the classical example of aero but nonanalytic function .

{

aCt) =

°,

2

e -1/t ,

0< t < 1, -1 < t ::::;; O.

(7.6.14)

Note that

_ { A(t) -

aCt)

[~ ~l

[0 IJ

a( - t) 1 0 '

0::::;; t < 1,

(7.6.15) -1 < t ::::;; O.

159

7.6 The Special Case n = 2

The next task is to verify that A(') is commutative. Consider B(t) =

J:

t

A(s) ds,

E

(7.6.16)

(-1,1).

Certainly B(t) = A(t)

(7.6.17)

for all t E( -1, 1), and B(') is not only absolutely continuous but Coo. By looking at the two cases, first that in which 0 ~ t < 1 and second that in which -1 < t ~ 0, it follows readily from (7.6.15) and (7.6.16) that A(t)B(t) - B(t)A(t) = A(t)

J:

A(s) ds -

J:

A(s) ds A(t) = 0 (7.6.18)

for all t E ( -1, 1). The commutivity of A(') is thus established. The last item of concern is calculation of the term involving A(') in (7.6.1). Restrict attention to -1 < , < 0 < t < 1. Applying (7.6.15), compute A(t)A(,) - A(,)A(t) =

a(t)[~

~Ja( -,{~

- a(

-,{~

~Ja(t{~

= a(t)a( -,{~ = exp( -

~J

t~

-

-

,~)[~

~J

~J -

~J

# O.

(7.6.19)

The proof is complete. Theorem 7.6.2 Every 2 x 2 commutative matrix function A(' ) on an interval J has a commuting antiderivative of the form B(t) , =

et(t{~

~J

+

ezet)[C2~t)

~:~:~J

(7.6.20)

in which etC'), ek) are absolutely continuous on J and CtC'), ck), ck) are constant over the open intervals at which ezet) # 0 (i.e., the c i ( · ) are step functions). Every such matrix function is a commuting antiderivative. Hence the 2 x 2 commutative matrix functions are those of the form a.e.

t

E

J.

(7.6.21)

160

7.

Commutative Linear Differential Equations

Remark 7.6.1 Not every absolutely continuous cl(.), c2(.) and step functions cl(-), cz(-), c3(*)in (7.6.21) will produce a commutative A(.). The switching must be synchronous with the zeros of tz(t) to maintain the continuity of B ( - ) .The proof of Theorem 7.6.2 is left as an exercise for the reader.

7.7

A Criterion for Exponential Decay of Solutions

For applications it is useful to know conditions on the coefficient matrix A( of (7.2.3) which will ensure that all solutions to the differential equation approach zero as t + 03. One such criterion for the variable-coefficient case is provided by the following theorem. a )

Theorem 7.7.1 Let A( *)be a locally integrable, commutative, complex n x n matrix-valued function on an interval (a, co) with commuting antiderivative B( .). If for some positive z E (a, co) (7.7.1) and the set (7.7.2) is contained in the half-plane Re z c v, then p t ) I 5 ce"t-" z

(7.7.3)

< t < co,for some c E a. Moreover, c can be chosen so that Jx(t)lI cIxoJeu(t-.F),

(7.7.4)

< co,for all solutions of (7.2.3) with x(z) = xo. In particular, ifu < 0, then each solution x ( - )of (7.2.3) will approach zero as t + co.

z 5 t

Proof. Assume the hypothesis. It is sufficient to prove (7.7.3) since each solution to (7.2.3) can be represented as x(t) =

eB(t)e-B(r)XO.

(7.7.5)

Since B ( * )is assumed to satisfy (7.7.1), it follows that the set K defined by (7.7.2) will be compact. Thus K lies in the interior of the closed half-disk with center at z = u + Oi, side on the line Re z = u and radius p for some p > 0 appropriately large. With the boundary ao of this half-disk oriented in the

161

7.7 A Criterion for Exponential Decay of Solutions

counterclockwise direction, the Cauchy formula provides the representation

eB(tl = -1. 2m

t. [ tJ(fJ

e'" zI - -B(t}]-l dz, t

(7.7.6)

t E [T, co). For fixed t, estimation of (7.7.6) gives

B(t)]-l/ ds

< -etut. /[ zI - - 2n tJ(fJ t

'

(7.7.7)

in which ds is the differential element of arc length. From the fact that zl - B(t}jt is invertible for each z E OW, t E [T, (0), the compactness of OW, and the boundedness (7.7.1), it follows that

/[zI for some b e flt, all z E OW, t is the inequality

B(t)]-l/ 5, b <

-t-

E [T,

, B(tl' e

00

(7.7.8)

(0). A direct consequence of (7.7.7), (7.7.8)

5,

bp(n

+ 2}

2n

ut

e,

(7.7.9)

t E [T, oo}. Inequality (7.7.3) now follows from (7.7.9) by an obvious choice of C, and Theorem 7.7.1 is proved. Remark 7.7.1 Note that the special case in which A(-) is a constantcoefficientmatrix is covered by Theorem 7.7.1 if B(') is taken to be B(t) = tAo Remark 7.7.2 The hypothesis of Theorem 7.7.1 does not rule out the possibility that 'A(t)1 is unbounded on [T, (0).

Chapter

8

Periodic Linear Equations

8.1

Periodic Homogeneous Equations

The initial analysis is restricted to the homogeneous equation

x = A(t)x,

(8.1.1)

(t, x) E fJ.t x S;". As in previous chapters, the n x n matrix function A(') over

is assumed to be locally integrable. In many applications the coefficient matrix A(') is in fact continuous, but such a restriction need not be imposed. This chapter explores the consequences of the assumption that A(') is periodic; i.e.,

S;

A(t

+ p) =

a.e. t E fJ.t

A(t),

(8.1.2)

for some fixed number p > 0 called the period. To save words, a periodic function with period p is called simply p-periodic. The period is not unique; if p is a period, then so is kp (k = 1,2, ...), and if A(') is constant, then it is p-periodic with p > 0, P otherwise arbitrary. As intuition might suggest, a certain aspect of constant coefficient behavior is prominent in tile solutions to all linear periodic equations. A precise account of this phenomenon is described by the Floquet theorem and is amplified in the applications to commutative systems. Lemma 8.1.1 For each nonsingular n x n complex matrix M there exist complex n x n matrix solutions C to the equation

M

=

eC •

(8.1.3)

Proof. Since M is nonsingular, there exists a simple closed curve ow bounding an open set to that contains the eigenvalues of M with the origin in the complement of its closure w. It follows that each branch In(z) of the 162

163

8.1 Periodic Homogeneous Equations

logarithm function is analytic on an open set containing 6.Consequently Theorem 7.5.1 can be applied to determine a matrix (8.1.4)

In(z)[zZ - MI-' d z .

The matrix obtained in (8.1.4) depends on the branch of the logarithm chosen. However, irrespective of the choice, eln(') = z and a second application of Theorem 7.5.1 then gives

This proves Lemma 8.1.1.

Lemma 8.1.2 For each real nonsingular n x n matrix M there exists a real n x n matrix solution R to the equation (8.1.6)

M Z = eR.

Proof. According to Lemma 8.1.1 there exists a complex matrix C = h(M) that satifies (8.1.3). The claim is that CC = CC. To show that this is true, compute r r (8.1.7) Thus it is sufficient to show that C M a real matrix it is clear that

=

MC. From the assumption that M is

(8.1.8)

ln(z)[zZ - MI-' dz M = M c and the equation

cc = Cc

(8.1.9)

is established. The commutivity (8.1.9) and reality of the matrix M facilitate the final calculations MZ

where R = C

= M R = [eC][Tq =

[ e c ] [ g ] = ec+T

=

,

(8.1.10)

+ C is a real matrix. This concludes the proof of Lemma 8.1.2.

Theorem 8.1.3 Floquet. Zf A ( * )is a locally integrable n x n matrix function over F on Se and moreover is p-periodic, then each fundamental matrix @( -) of A( .) can be factored as @(t) = P(t)efc

(8.1.1 1)

164

8.

Periodic Linear Equations

in which the n x n matrix function P(·) is absolutely continuous and p-periodic and C is an appropriate complex n x n matrix. Proof.

Let cD( .) be a fundamental matrix of A(·). Hence
(8.1.12)

a.e. t E rJt,

and det cD(t) '# 0,

all

(8.1.13)

t E rJt.

+ p). The calculation = A(t + p)cD(i + p) = A(t)'P(t),

Consider 'P(t) = cD(t 'P(t)

a.e. t E rJt,

(8.1.14)

using (8.1.2), (8.1.12), and the fact that det 'P(t) '# 0, which follows from (8.1.13), shows that 'P(' ) is also a fundamental matrix of A( .). Consequently, cD(t

+ p) =

cD(t)M,

all

t E rJt,

(8.1.15)

for some nonsingular matrix Mover !F (see Theorem 6.7.2). According to Lemma 8.1.1 there exists a complex matrix C such that (8.1.16) Taking (8.1.15) and (8.1.16) together, cD(t

+ p) =

cD(t)ePC,

all

(8.1.17)

t E rJt.

Now define P( .) by pet)

= cD(t)e- tc.

(8.1.18)

The required periodicity of P(·). follows from (8.1.17), (8.1.18) in the calculation pet

+ p) =

cD(t

+ p)e-(t+p)C

= cD(t)ePCe-(t+ p)C = pet),

all

t E rJt.

(8.1.19)

Thus (8.1.11) is a consequence of (8.1.18). The absolute continuity is clear from (8.1.18) and the available absolute continuity of cD('). The proof is complete. Remark 8.1.1 From (8.1.11) and the study of the structure of matrix functions tA. made in Chapter 4, it' is apparent that each of the coordinate functions of the solutions x(') to (8.1.1) will be a linear combination of scalar functions of the type

(8.1.20) in which a(') is absolutely continuous and p-periodic, A. is an eigenvalue (possibly complex) of C, and 0 :$ k :$ m - 1, where m is the multiplicity of A. in the minimal polynomial of C.

165

8.1 Periodic Homogeneous Equations

Remark 8.1.2

Neither C nor its eigenvalues are uniquely determined by

A('); e.g.,

cI>(t) = P(t)e tC = [el-21ti/P)tp(t)]et[C+I2lti/p)I] = P(t)e tc.

(8.1.21)

However, it will shortly be shown that the eigenvalues of ePc are uniquely determined by A(·). Remark 8.1.3 The eigenvalues of ePc, denoted by tTl' tT2"'" tTn' are called characteristic roots of (8.1.1) or of A(·). If AI> A2"'" An are the eigenvalues of C, then the eigenvalues of eC are e)'l, e),2, , e),· and thus, when properly enumerated, give tTl = eP),I, tT2 = eP),2, , an = eP), •. The numbers AI, A2' ... , An (determined only mod(2nijp)) are called characteristic exponents of (8. 1.1). Note that (j = 1, 2, ... , n)

Re Aj = p -1 Re In(aj)

(8.1.22)

are uniquely determined by A(')' These numbers will be shown to have a strong influence on the qualitative behavior of the solutions to (8.1.1). Remark 8.1.4 The matrix M that appears in (8.1.15) should be recognized as the matrix of the linear operator x(') - x(' + p) on the vector space of all solutions x(·) to (8.1.1), relative to the ordered basis provided by the columns of the cI> appearing in (8.1.15). Generally, if A(') is p-periodic, then

M = cI>-l(t)cI>(t + p),

(8.1.23)

although dependent on the' fundamental matrix cI>(.) of A(') selected, is independent of t E fll and, in particular, (8.1.24)

.«

It is easy to show that M is similar to cI> + p, t), which is a p-periodic function of t E fll (see Exercise 8). The next section shows that M carries a great deal of useful information about the solutions to (8.1.1). Corollary 8.1.4

Ij'A(') is a locally integrable real p-periodic n x n matrix function, then each fundamental matrix cI>( •) of A(') can be factored as cI>(t) = Q(t)etR (8.1.25)

in which the n x n matrix function Q(') is real, absolutely continuous, and periodic with period 2p, and R is an appropriate real matrix. Proof. Let cI> be any fundamental matrix of A(·). Then, as argued in the proof of Theorem 8.1.3, cI>(t

+ p) = cI>(t)M,

all

t E fll,

(8.1.26)

166

8.

Periodic Linear Equations

for some (now real) nonsingular matrix M. From (8.1.26) and application of Lemma 8.1.2, for some real matrix R, ~(t

+ 2p) =

~(t

+ p)M

= ~(t)M2

all

= ~(t)e2pR,

t E gpo

(8.1.27)

By defining Q(' ) as Q(t) =

(8.1.28)

~(t)e-IR,

(8.1.27) can be applied to check that Q(t

+ 2p) = =

~(t

+ 2p)e-(1+2p)R

~(t)e2pRe-(1+2p)R

= Q(t),

all

t E gpo

(8.1.29)

The conclusion of Corollary 8.1.4 now follows directly from (8.1.28) and (8.1.29).

Remark 8.1.5 Theorem 8.1.3 and Corollary 8.1.4 are existence theorems and as such are useful in theoretical work. However, the computations of P( '), C, Q('), and R that occur in the factorizations are usually difficult at best. A notable exception is presented in Section 8.4. EXERCISES

1. Let P(·) be any absolutely continuous p-periodic nonsingular n x n matrix-valued function on gp and let C be any complex n x n matrix. Show that ~(t) = P(t)e'c is a fundamental matrix of some locally integrable p-periodic n x n matrix function A(') on gpo Is such an A(') unique? Show that if C and P( .) are real, then ~( .) is the fundamental matrix of some real matrix function of period 2p. 2. Let A(·) be a locally integrable real p-periodic n x n matrix-valued function on gpo Consider the associated matrix differential equation problem 'i'(t) = A(t)'P(t) - 'P(t)K,

'P(O) = 'Po.

(8.1.30)

Show that for each nonsingular real n x n matrix 'Po there exists an appropriate real n x. n matrix K such that the given initial-value problem has a solution 'P(') that is periodic of period 2p and has det 'P(t) '# 0 for all t E fYI. 3. Show that if an n x n matrix function A(') is not periodic, then it has no periodic fundamental matrix. 4. Suppose A( .) is an n x n matrix function that factors as A(t) = etOAoe- rn (8.1.31) (in which A o = A(O» for some matrix n whose eigenvalues all have zero real parts and are of multiplicity one in the minimal polynomial of n. Show that

8.2

167

Qualitative Behavior

such an A( .) has a fundamental matrix <1>(') that has the Floquet factorization (t)

= P(t)e tC

(8.1.32)

in which pet) = eta and C = Ao - Q. Compute the Floquet factorization of all fundamental matrices of such an A(·). 5. Prove the following generalization of the result of Exercise 4: If A(') is a locally integrable n x n matrix function that has a factorization of type A(t)

=

(8.1.33)

P(t)AO(t)P-1(t)

T

T

in which p ( . ) is a periodic fundamental matrix of AI;(t) - C for some matrix C, then A(') has a fundamental matrix <1>(') which has the Floquet factorization (t) = P(t)e tc . (8.1.34) No explicit mention of the periodicity of Ao( ' ) or of A(') was made. Why will Ao( ' ) and A(') automatically be periodic with the same periods as P(· )? 6. Compute the 2 x 2 periodic matrrx functions A(' ) that have the Floquet factorization (t) = P(t)e tC of their fundamental matrices when (a)

pet)

= [

(b) P(t)=[ (c)

pet) = [

c~s

- sin

t

t

co~t

-I sm t 1

~t e~itJ.

tJ'

sin cos t

ISintlJ, cos t

C

=

[-1°-OJ

2 '

° - OJ2 ' C [1 ~ 1~ iJ C

[-1

=

i

=

7. Is the factorization given by Theorem 8.1.3 unique? 8. Verify the claims made in Remark 8.1.4. Show that A moreover satisfies the relation t E (lJi.

(8.1.35)

8.2 Qualitative Behavior The characteristic roots of a periodic matrix function A(') are well-defined; i.e., they are independent of the choice of the fundamental matrix appearing in their definition. (Recall Remark 8.1.3.) To prove this claim let ('} and '1'( .) be two fundamental matrices for A( '). Then all

t E (lJi,

(8.2.1)

168

8"

Periodic Linear Equations

where p is a period of A( .) and the matrix C is appropriately selected according to Theorem 6.7.2 and Lemma 8.1.1. Likewise, (t) = 'P(t)T,

all

(8.2.2)

t E !!It,

for some (constant) nonsingular matrix T. Combining (8.2.1) and (8.2.2) shows that (8.2.3) 'P(t + p)T = (t)e PC, and it is then clear from (8.2.2) and (8.2.3) that 'P(t +p) = 'P(t)[Te PCT- 1] ,

(8.2.4) all t E !!It. 1 P cTBut the matrices ePc and Te are similar and thus have the same eigenvalues. This demonstrates that calculation of the characteristic roots of A(' ), using either the factorization (8.2.1) based on <1>( .) or the factorization (8.2.4) based on '1'('), leads to the same eigenvalues. Recall that the characteristic roots a, and characteristic exponents Ai arose in the factorization (t

+ p) = (t)M,

(8.2.5)

with the a, defined to be the eigenvalues of M and (8.2.6)

(i=1,2, ... ,n).

The next corollary summarizes some important relationships between the a, and the qualitative behavior of the solutions to (8.1.1).

Corollary 8.2.1 If the locally "integrable n x n matrix function A(') is p-periodic, then (a) all solutions x(·) of (8.1.1) have x(t) --+ 0 as t --+

00

(i = 1,2, ... , n),

if

and only

if (8.2.7)

(b) some solution x(·) of (8.1.1) is a nontrivial (i.e., nonzero) p-periodic solution if and only if

(jj=l,

forsome

iE{1,2, ... ,n}.

(8.2.8)

Proof. The proof is based on Theorem 8.1.3. In view of the relationship (8.2.6), condition (8.2.7) is equivalent to the matrix C in (8.1.11) being a stability matrix. Since P(· ) is periodic and continuous, it has Ip(')1 bounded on !!It. Hence if etC --+ 0 as t --+ 00, it follows that likewise P(t)e tC--+ 0 as t --+ 00. Since P(·) is continuous, periodic, and nonsingular on !!It, Ip- 1(·)1 is bounded. This fact along with the inequality tE!!It,

(8.2.9)

8.2

169

Qualitative Behavior

shows that, conversely, e'c -+ 0 as t -+ 00 if P(t)e 'C-+ 0 as t -+ solution x(·) of (8.1.1) can be represented as x(t)

= (t)~,

00.

Since every (8.2.10)

tEfJI,

n

for some ~ E !F , the conclusion of part (a) is now apparent. Condition (8.2.8) translates, via (8.2.6), into the equivalent statement that at least one of the eigenvalues Aj of C in (8.1.11) be an integral multiple of 2ni/p. By the theory of constant-coefficient equations, the latter condition is noted to be equivalent to the statement that e'c~ be periodic with period p for some nonzero E !F n • But in conjunction with (8.1.11) this implies that

e

(8.2.11)

is a nonzero periodic solution of (8.1.1) of period p. Conversely, if (8.1.1) has a nonzero periodic solution x(·) of period p, then by (8.1.11) x(t

+ p) = x(t)

P(t

=

+ p)e(I+P)C~

= P(t)e(I+P)C~,

(8.2.12)

P(t)e'Ce,

(8.2.13)

and as a consequence P(t)e'Ce = P(t)e(l+p)Ce

(8.2.14)

for some nonzero ~ E !Fn , all t E fJI. Since P(·) is nonsingular on fJI, (8.2.14) can hold if and only if

[In -

ePcJ~

= O.

(8.2.15)

Equation (8.2.15) obtains precisely when one of the eigenvalues of ePc is equal to 1, which is just condition (b) of Corollary 8.2.1. This proves the corollary. Example 8.2.1 Consider the problem of studying the qualitative behavior of the solutions to the periodic system

d .dt

[Xl] [0 X2

1 ][xl]

-sin t cos t - 1

=

X2'

(8.2.16)

which is the first-order form of U + [1 - cos tJ U + [sin tJu = O.

(8.2.17)

Note that one solution of (8.2.17) is u(t)

=

e-I+sint.

(8.2.18)

Using the "reduction of order method," i.e., letting u(t) = e-t+sintv(t),

(8.2.19)

170

8.

Periodic Linear Equations

one finds on substitution of (8.2.19) into (8.2.17) that vet) =

f~

et-Sint dt;

and consequently a second solution is u(t) = e-t+sint

J:

(8.2.20)

et-sint dt .

(8.2.21)

Thus the general solution of (8.2.16) is (8.2.22) xz{t) = Xl(z).

(8.2.23)

Reading off a fundamental matrix from (8.2.22) and (8.2.23) by writing those equations in the form (8.2.24) we get e-t+sint

eD(t) = [ [cos t

-

e-t+sint

f~

]

et-sint dt

1]e-t+Sint 1 + [cos t - 1]e-t+Sint

r

J et-sint dt t

•

(8.2.25)

0

The theory says that eD(t + p) = eD(t)M, where p = 2n, and it is clear from (8.2.25) that eD(O) = 12 ; consequently, 2 2" 2 t- sint _ [ee- 1< 1< e d-C] (8.2.26) M0 . o 1

1

Therefore the characteristic roots of (8.2.16) are /11 = e- 2 " , /12 = 1. Thus, according to Corollary 8.2.1,that differential equation has a periodic solution of period p = 2n. To compute one, set u(O) = u(2n) to get Cl =

C2e-2" J~" et-sint d-c (1 _ e 27<)

(8.2.27)

The resultant periodic solutions obtained are then u(t) =

C

2

e-t+sint

e- 2"J21< et-sint dt 0 1 - e -2" [

+ It et-sint dt] 0

'

(8.2.28)

8.2

171

Qualitative Behavior

which can easily be checked to satisfy u(O) = u(21t) as well as (8.2.17). This is enough to imply that (8.2.28) are periodic solutions for arbitrary C2 (see Theorem 8.3.1). Can you see directly that (8.2.28) is periodic? EXERCISES

1. On the basis of the detailed theory of constant-coefficient equations developed in the text, other conclusions could be added to Corollary 8.2.1. For example, if at least one of the characteristic roots Uj satisfies IUjl > 1, then at least one solution x(·) of (8.1.1) would have Ix(tk ) / - 00 as k - 00 for some sequence tk - 00 as k - 00. Justify this statement. Explain why the condition Iud::; 1 (i = 1, 2, ... , n) is not enough to guarantee IX(')I bounded on [0, 00) for each solution x(·). 2. Compute the characteristic roots Ul' U2 of the periodic matrix function A( '), defined by . A(t) = [

c~s

t

-sm t

sin tJ . cos t

(8.2.29)

(Recall Remark 8.1.4 and the calculation of ellA in Chapter 7.) 3. Investigate the qualitative behavior of the solutions to the differential equation d

dt

[XIJ = [ X2

0 cos t

-(1

+1 cos t)

J[XIJ

X2'

(8.2.30)

Follow the procedure of Example 8.2.1.) 4. Compute characteristic' exponents and the characteristic roots for the matrix function A(t) = Al(t)AoA l( -t) and describe in detail the behavior of the solutions to the associated vector differential equation x = A(t)x, where (Suggestion:

Al(t)

= [ c~s

t sin tJ -sm t cos t

(8.2.31)

if A o is the matrix: (a) [

- 1 -lJ . 1 -1'

(b)

[

1 1J -1 0 '

(c)

[1 3J 2 2 .

5. Consider Eq. (8.1.1) in which A(') is locally integrable and periodic. Assume that on proper enumeration the characteristic roots of A(') satisfy Iud < 1 (i = 1,2, ... , s) and Iuil > 1 (i = s + 1, s + 2, ... , n). Let

y = A(t)y

(8.2.32)

be the equation resulting from the change of coordinates y = Nx, N a con-

172

8.

Periodic Linear Equations

stant nonsingular matrix. Split Y into the two coordinate vectors Zt = (Yl' Y2,"·' Ys), Z2 = (Ys+t, Ys+2"'" Yn)' Show that for some choice of N each nontrivial solution y( .) of (8.2.32) will have IZ i (t)l- 0 as t - 00 if the initial z2(-r) = 0 and, on the other hand, will have IZ2(tk)l- 00 for some sequence tk - 00 as k - 00 if initially zt(r) = O. Further show that if A(-) is real, then N can be taken to be real. What can you say about the qualitative behavior of every solution to (8.2.32)? 6. Let A(') be a real or complex p-periodic and locally integrable n x n matrix function. Consider the boundary-value problem x(t)

=

0:::;; i

A(t)x,

s:

Mx(O) - Nx(p)

p,

= 0,

in which the n x n matrices M and N are prescribed. Let ~(.) be any fundamental matrix of A(·). Show that the collection of solutions to this boundary-value problem constitutes a vector space of dimension equal to n - rank[M~(O) - N~(p)]. .

8.3

Nonhomogeneous Equations

We return our attention to the nonhomogeneous equation in x

=

A(t)x

+ f(t),

a.e..

~

x /Fn,

(8.3.1)

tE~.

Recall that (8.3.1) has a unique (absolutely continuous) solution through each initial point over any interval in which the matrix function A(') and vector function f( .) are locally integrable. Theorem 8.3.1 If in (8.3.1) both A(') and f(·) are p-periodic as well as locally integrable, then a solution x(·) to that equation will be p-periodic if and only if

(8.3.2)

x(p) = x(O).

Proof. If a solution x(·) of (8.3.1) is p-periodic, then (8.3.2) obviously must hold. Now assume that (8.3.2) is satisfied by a solution x(·) of (8.3.1) in which A( .) and f( .) are as described in the hypothesis. Consider the difference ~(t)

=

x(t

+ p) -

x(t).

(8.3.3)

Equation (8.3.1) and the assumed periodicity underly the calculations &(t)

= =

[A(t

+

p)x(t

[A(t)x(t

= A(t) ~(t),

+

p)

+

f(t

+ p) + f(t)] a.e. t E~.

+

p)] - [A(t)x(t)

- [A(t)x(t)

+

f(t)]

+ f(t)] (8.3.4)

8.3

173

Nonhomogeneous Equations

Moreover, (8.3.2) and (8.3.3) imply that l1(0) = O. By uniqueness it follows that the solution to the homogeneous initial-value problem must be M.) = O. The periodicity of x(') then follows directly from (8.3.3), and Theorem 8.3.1 is proved.

Consider only locally integrable A(') and f(·) in (8.3.1). Assume that A(') is p-periodic. Then (8.3.1) has a p-periodic solution for each p-periodic f(·) if and only if the homogeneous equation (8.1.1) has no p-periodic solution other than the zero solution.

Theorem 8.3.2

Proof.

According to Theorem 8.3.1, Eq. (8.3.1) has a p-periodic solution

x(·) if and only if (8.3.2) is valid. By the variation-of-parameters formula, the

solutions can be represented as

x(t) = <1>(t)XO

+ <1>(t) f~

<1>-1(0")f(0") da,

(8.3.5)

where <1>(-) is the fundamental matrix with <1>(0) = In. Substitution of (8.3.5) into (8.3.2) shows that the latter is equivalent to

[In - <1>(p)JXO

=

<1>(p)

f:

<1>-1(0")f(0") da,

(8.3.6)

Thus for each given f( .) the solution (8.3.5) is p-periodic if and only if (8.3.6) has a solution x'', Therefore, if both A(') and f(·) are p-periodic, then it can be concluded that (8.3.1) has a p-periodic solution if det[In - <1>(p)J "# O.

(8.3.7)

<1>(t) = P(t)e'C

(8.3.8)

By Theorem 8.1.3 in which P( .) is p-periodic. Hence

<1>(p) = P(p)e PC = P(O)e Pc = ePc.

(8.3.9)

Thereby (8.3.7) translates into the inequality det[In - ePcJ "# O.

(8.3.10)

But (8.3.10) says simply that 1 is not a characteristic root of A(·). In short, according to (b) of Corollary 8.2.1, it can be concluded that (8.3.6) has a solution XO and hence that (8.3.1) has a p-periodic solution if (8.1.1) has no nontrivial p-periodic solution. To complete the proof of the converse, it is sufficient to show that if (8.3.6) has a solution XO for each p-periodic f('), then (8.3.7) must hold. Consider

174

8.

Periodic Linear Equations

the special nonhomogeneous functions of the type

°of<

f(t)

= ~-T(t)~,

(8.3.11)

t < p, ~ E :F , extended to be p-periodic on ~. (c:I> - Tdenotes the transpose For such functions n

~-1.)

~(p)

f: ~-l(O")f(O")

da = c:I>(p)

f: ~-l(O")~-T(O")

do

e.

(8.3.12)

The last integral in (8.3.12) is noted to be a symmetric matrix and is moreover, nonsingular. To demonstrate the nonsingularity, compute the quadratic form

e fP c:I>-l(O")c:I>-T(O") do ~ = fP 1~-T(0")eI2 °

.

°

°

da 2: 0,

(8.3.13)

which by continuity is zero if and only if ~-T(O")e = 0, 0" E [0, p]. This of course is possible precisely when ~ = since fundamental matrices are nonsingular. This shows that the integral of (8.3.12) under consideration is in (8.3.12) is also positive-definite and therefore nonsingular. Since ~(p) nonsingular and is free to vary in :Fn , it is now obvious that the right-hand side of (8.3.6) takes on all values of :Fn under all choices of p-periodic and locally integrable f(·). Consequently for each such f(·) (8.3.6) will have a solution x? only if(8.3.?) obtains. This completes the proof of Theorem 8.3.2.

e

Remark 8.3.1 With A(') p-periodic it is possible for (8.1.1) to have a nontrivial p-periodic solution and yet for (8.3.1) to have nontrivial p-periodic solutions for some p-periodic nontrivial f(·) (see Exercise 1). EXERCISES

1. Why does the equation ii

+ u = sin 2t

(8.3.14)

fail to produce a counterexample to Theorem 8.3.2? 2. Consider the differential equation

x=

A(t)x

+ B(t)u(t)

(8.3.15)

in which A(') and B(') are continuous matrix-valued functions of size n x n and n x m, respectively. If A(') and B(') are defined on an interval [to, t 1 ] and for each choice of'(x", x') E:Fn x :Fn there exists a continuous m vectorvalued function u(') (called a control function) on [to, t 1 ] such that the resultant nonhomogeneous equation (8.3.15) has a solution x(·) on [to, t 1 ]

175

8.4 Periodic Commutative Equations

that satisfies the boundary conditions x(t o) = x",

(8.3.16)

x(td = x ',

(8.3.17)

then (8.3.15) is called controllable on [to, t 1l In particular, suppose that A(') and B(') are p-periodic on f7t. Call (8.3.15) p-controllable if for each to E f7t and x? E /Fn there exists a p-periodic control function u(·) on f7t such that the resultant solution x(·) of (8.3.15) satisfying (8.3.16) is likewise p-periodic. Prove that (8.3.15) is then p-controllable if and only if it is controllable on [0, pJ.

8.4

Periodic Commutative Equations

The classical Floquet theory of Sections 8.1 and 8.2 establishes a close tie between the characteristic roots of a p-periodic matrix function and the qualitative behavior of the solutions to the associated linear differential equation. This elegant theory fails to dispense with one important obstacle to its practical applicability, namely, the problem of computing (or at least estimating the absolute values of) the roots. The hitch is that their definition involves knowledge of the generally unknown fundamental matrix at two points. (Recall Remark 8.1.4.) In this section we present one significant class of time-varying periodic systems for which this obstacle can be overcome-commutative periodic equations.

Theorem 8.4.1 Let A(') be a locally integrable, commutative, p-periodic n x n matrix function on f7t and let B(') be one of its commuting antiderivatives. Then the characteristic roots of A(') are the eigenvalues of the matrix M

=

(8.4.1)

e-B(OleB(pl.

If B(') satisfies the additional condition B(p)B(O) - B(O)B(p)

= 0,

(8.4.2)

then

(8.4.3) in which C is the average matrix

C = -1

p

fP A(u) da. °

(8.4.4)

176

8.

Proof.

Periodic Linear Equations

Theorem 7.3.1 implies that

=

(8.4.5)

eB(t)

is a fundamental matrix of A(·). The conclusion concerning (8.4.1) is a direct consequence of (8.4.5) and Remark 8.1.4. When (8.4.2) holds, the first equality in (8.4.3) follows from (8.4.1). Since B = A, a.e. t E~, B(p) - B(O) =

f:

A(a) da,

(8.4.6)

and thus the second equality in (8.4.3) is valid since the exponent matrices are equal. This concludes the proof of Theorem 8.4.1.

Remark 8.4.1 In Section 8.2, we discussed the impact that the eigenvalues of the matrix (8.4.1) have on the stability properties of the differential equation with coefficient A(')' When (8.4.3) applies, even more information is available, e.g., in cases for which one or more of the characteristic roots a of A(·) have lal = 1. The matrix C contains, amongst other things, the information about whether solutions of the differential equation can be unbounded on s; t < 00. This is evident from the following corollary.

°

Corollary 8.4.2 Let A(') be a locally integrable, commutative, p-periodic n x n matrix function on ~ with a commutative antiderivative B(') that satisfies Eq. (8.4.2). Then the fundamental matrix e B ( . ) of A(') has the Floquet factorization (8.4.7)

in which C is the average matrix (8.4.4) and B p(') is the p-periodic matrix Bp(t)

=

B(t) - tC.

(8.4.8)

Proof. Since B( .) is a commuting antiderivative of A('), the p-periodicity of A(' ) can be applied to compute

d

dt [B(t

+ p) -

B(t)] = A(t

+ p) -

A(t) = 0,

a.e.

tE

~,

(8.4.9)

which implies that C

=

p-l[B(p) - B(O)]

=

p-l[B(t

+ p) -

B(t)],

all

tE

~.

(8.4.10)

Furthermore, because of (8.4.10), A(t)C - CA(t)

=

p-l[A(t)B(t - p-l[B(t

= p-l[A(t

+ p) - A(t)B(t)]

+ p)A(t) - B(t)A(t)]

+ p)B(t + p) -

B(t

- p-l[A(t)B(t) - B(t)A(t)]

+ p)A(t + p)] = 0,

a.e. tE~.

(8.4.11)

8.4

177

Periodic Commutative Equations

Application of (8.4.11) gives

d

dt [B(t)C - CB(t)] = A(t)C - CA(t) = 0,

a.e.

t E~.

(8.4.12)

This shows that B(t)C - CB(t) is independent of t, and because of assumption (8.4.2), it is apparent that B(t)C - CB(t) = B(p)C - CB(p) = B(p){p-l[B(p) - B(O)]} - {p-l[B(p) - B(O)]}B(p) = -p-l[B(p)B(O) - B(O)B(p)] = 0,

all

tE~.

(8.4.13)

This commutivity is sufficient to support the calculation all

tE~.

(8.4.14)

One consequence of(8.4.IO) is the equation B(t

+ p) =

B(t)

+ pC,

(8.4.15)

which leads to the required p-periodicity Bit

+ p) = B(t + p) - (t + p)C = B(t) + pC - (t + p)C = B(t)

- tC = Bp(t) ,

all

t E~.

(8.4.16)

This proves Corollary 8.4.:2. Remark 8.4.2 An important consequence of Corollary 8.4.2 is that it reduces the detailed qualitative analysis of a class of time-varying equations to that of the constant-coefficient type to which the extensive theory of earlier chapters applies. Generally a fundamental matrix for A(·) need not be computed. Remark 8.4.3 That A(') is p-periodic and commutative does not by itself imply that its commuting antiderivatives are p-periodic (see Example 8.4.1). Those A(·) with p-periodic commuting antiderivatives are distinguished by having averaged values C = O. Theorem 8.4.3 If A(·) is a p-periodic commutative n x n matrix function on having an analytic commuting antiderivative B(·) for which B(r:) has n distinct eigenvalues for at least one 'r E ~, then B( .) satisfies (8.4.2), and both Theorem 8.4.1 and Corollary 8.4.2 apply.

~

Proof.

Theorem 8.4.3 follows as a consequence of Theorem 7.5.10.

178

8.

Periodic Linear Equations

Example 8.4.1 The matrix function sin t 1 sin t

A(')~[+

Si~

,]

°-

(8.4.17)

is commutative and 2n-periodic. Although its commuting antiderivative B(t)

[

°

-cos t

t

-cos t

= -cos t

t

t ] -cos t

°

(8.4.18)

is not periodic, it does satisfy (8.4.2). The matrix

c

~ (1/2.)[8(2.) -

R(O)]

~ [~ ° °1] 1

°°

(8.4.19)

has eigenvalues 1, 1, -1. From Corollary 8.4.2 it is now clear that the homogeneous differential equation associated with (8.4.17) will have some solutions x(t) with Ix(t)1 unbounded on ~ t < 00.

°

Example 8.4.2 The matrix function A(t) = 2(cos 2t) [

(0, ) 01J sgn sm t

(8.4.20)

is locally integrable, 2n-periodic, and commutative. It has the commuting antiderivative B(t) = (sin 2t) [

°

(0. ) 1J. sgn sin t

°

(8.4.21)

°

Although both A(t)A(r) - A(-r)A(t) = and B(t)B(-r) - B(-r)B(t) = fail at some (t, r) E [Jf2, it is of no concern since B(2n)B(0) - B(0)B(2n) = 0, which is enough for Corollary 8.4.2 to apply. Since B(') is obviously 2n-periodic, C = and all solutions to the homogeneous differential equation having (8.4.20) as coefficierit will be 2n-periodic.

°

EXERCISES

1. Find an example of a 4 x 4 nonconstant p-periodic commutative matrix function A(') having a nonperiodic commuting antiderivative B(') on Bt. 2. Let A(') be a p-periodic commutative matrix function on Bt with commuting antiderivative B( .).

8.4

Periodic Commutative Equations

179

(a)

Show that B(t) = B(t

+ p) defines another commuting antiderivative

(b)

For C defined by (8.4.4) show that

B(.) of A(' ).

eapCB(t)e-apC (c)

= B(t) + up[ CB(t) - B(t)C]

t E iJi, a E iJi.

for all

Verify that

C[CB(t) - B(t)C] - [CB(t) - B(t)C]C = 0

for all

t E iJi.

3. Let B(') be a commuting antiderivative of a matrix function A( .) on an open interval J. Show that if T is a nonsingular matrix such that T A(t) - A(t)T = 0,

a.e.

tEJ,

a.e.

t E J,

then there exists a matrix K such that

KA(t) - A(t)K = 0, with

for all

t E

J.

4. Let A(') be a locally integrable commutative p-periodic n x n matrix function on iJi with commuting antiderivative B( -), Show that the following are equivalent conditions; (a) B(p)B(O) - B(O)B(p) = 0, (b) Jg {A(t) So A(u) do - So A(u) duA(t)} dt = 0, (c) CB(t) - B(t)C = 0, all t E iJi. (C is defined by (8.4.4).) 5. Investigate the qualitative behavior of the solutions to the vector differential equation x = A(t)x where (a)

A(t) = (sin 2t) [

(b)

A(t)

=

O. - sgn(sm 2t)

sgn(sin t) + 2 cos 2t [ sin t 1

sgn(sin t)] , 1

sin t sgn(sin t) 1 + 2 cos 2t

1

+ 2 cos 2t] sin t sgn(sin t)

.

Chapter

9

Local Existence and Uniqueness Theory of Nonlinear Equations

Many natural phenomena of the physical world, including gravity, friction, and electromagnetic forces, give rise to nonlinear differential equations. These equations cannot be dealt with directly by the theory of the preceding chapters. (Nevertheless the linear theory will bear upon them indirectly through their linear variational equations, discussed later in this book.) In spite of the availability of a substantial body of results concerned with nonlinear equations, the variety and complexity of the models encountered by engineers and scientists working on applied problems often force heavy reliance upon computer simulations and numerical experiments. This is to be expected. Relative to the general class of linear time-varying equations, those with constant coefficients constitute a limited subclass; transition to the larger class is understandably accompanied by a loss in theoretical detail. The transition from general linear to nonlinear equations is even more radical. Hence, far richer variety in possible qualitative behavior with a proportionate reduction in complete detail should be expected. This indeed turns out to be the case, but fortunately the topic of this chapter-existence and uniqueness-survives the expansion as a relatively complete branch of the theory. A preview of new phenomena that any extended theory must necessarily assimilate is given by examination of a few simple examples.

180

181

9.1 Complications That Can Occur

9.1 Complications That Can Occur

To illustrate one sense in which a nonlinear equation can fail to have a solution, consider the initial-value problem in PA, x(O) =

~,

(9.1.1)

O~t~1.

Separation of variables and integration lead to the formula x(t, ~)

=

~/(1

-

(9.1.2)

t~)

which can be verified to satisfy the equations in (9.1.1) at all points, with a possible exception at t = 1/~, ~ =I: O. Thus for ~ < 1 the problem (9.1.1) has a solution but the formula fails for ~ ~ 1. It turns out that in the latter case the difficulty is not due to failure to find the correct formula, but to the fact that none exists. The singularity showing up in (9.1.2) cannot be attributed to a singularity in the differential equation since x 2 is in fact an analytic function of x. Note however that the initial-value problem would always have a solution for t restricted to an appropriately short interval about t = O. The phenomenon that shows up in this example is quite typical of many nonlinear equations. Observe that the singular behavior is not present in linear equations, e.g., in the equation in which x 2 is replaced by x. Also note the more complicated way in which the initial parameter ~ enters the solution in comparison to that in the linear equation. The uniqueness story is also more complex for nonlinear equations. This statement can be demonstrated by the problem

x=

os

x(O) = 0,

Ixll/~,

t

< co,

(9.1.3)

It is easy to check that the formula x(t) =

{1(t -0, c)2, 4

o~

t ~ c, c < t < co,

(9.1.4)

defines a solution to (9.1.3) for each choice of 0 ~ c < co. This example shows that continuity of the function defining the differential equation (here, Ixj1 /2) is not enough to guarantee the uniqueness of solutions to the initial-value problem. Although the above examples show that an existence and uniqueness theory for nonlinear differential equations is necessarily more complicated than that for linear equations, there is no need for undue pessimism since the problem can be dealt with quite successfully. The more challenging

182

9.

Local Existence and Uniqueness Theory of Nonlinear Equations

problem turns out to be that of assessing the qualitative behavior of any given nonlinear differential equation or system of equations. The first topics to be discussed are existence and uniqueness. 9.2

Local Existence and Uniqueness

The equations (9.2.1)

x(t) = f(t, x(t)),

xl/=t

=

(9.2.2)

~,

where ('I:,

~) E D, with D a nonempty open subset of f7t x f7tn (or f7t x C1J") and f: D - ~ (or f: D - C1Jn), are called a Cauchy problem or initial-value problem. A solution to the Cauchy problem, or solution curve of (9.2.1) through ('I:, e), is defined to be any pair (1, ifJ) in which 1 is an open subinterval of f7t containing '1:, ifJ: 1- f7tn (or ifJ: 1- C1Jn) is absolutely continuous, (t, ifJ(t)) E D for all t e I, and ifJ satisfies (9.2.2) and also satisfies (9.2.1) at a.e. t e I,

The norm on f7tn (or C1Jn) in the next theorem is the one defined by (4.2.4). As in previous chapters the integral employed is the Lebesgue integral. The hypothesis is stated in terms of the rectangular subset of f7t x f7tn (or f7t x C1Jn) centered about ('I:, ~), R a•b = {(t,x):lt - '1:1 ~ a, [x - ~I ~ b},

a> 0,

b > O.

(9.2.3)

Theorem 9.2.1 Caratheodory.. The Cauchy problem (9.2.1), (9.2.2) has a solution if for some R a ,beD centered about ('I:,~) the restriction of f to R a b is continuous in x for fixed t, measurable in t for fixed x, and satisfies

.

If(t, x)1 ~ met), for some m integrable over the interval Proof.

(9.2.4)

(t, x) E Ra,b' ['I: -

a, 'I:

+

a].

Assume the hypothesis and define M(t)' =

f

m(s) ds,

'I:

~ ts

'I:

+

a.

(9.2.5)

It is clear that M(') is continuous and nondecreasing and that M('I:) = O. Therefore (t, ~ ± M(t)) E R a •b for some interval 'I: ~ t ~ 'I: + (J( ~ 'I: + a where (J( is some positive number. Choose any such 0( and define the approximations ifJ i (j = 1, 2, ...) by the formula ifJP) =

{

~ ~

t~t~'I:+(J(h

+J

(I-ali

f(s, ifJi s)) ds,

'I:

+ 0(/ j <

t ~ ..

+ (J(.

(9.2.6)

9.2

183

Local Existence and Uniqueness

e.

Note that cP1 is defined on T :::;; t :::;; T + rx, for it is the constant vector For any fixed integer j ~ 1, formula (9.2.6) defines cPj on T :::;; t :::;; T + rx/j and since (t, e) E Ra,b for T :::;; t :::;; T + rx/j the second part of that formula defines cPj as a continuous function on the interval T + a] j < t :::;; T + 2rx/ j. Further, on this interval . IcP/t) -

el : :; M(t -

rx/ j)

(9.2.7)

by virtue of(9.2.4) and (9.2.5). Assume that cPj is defined on T :::;; t :::;; T + ka]j for 1 < k < j. Then the second part of (9.2.6) defines cPj for T + ka]j < t :::;; T + (k + l)rx/j, since knowledge of the measurable integrand is required on only T:::;; t:::;; T + kaf], Also, on T + krx/j < t:::;; T + (k + l)rx/j, the function cPj satisfies (9.2.7) because of (9.2.4) and (9.2.5). Therefore, by induction (9.2.6) defines all cPj as continuous functions on T :::;; t :::;; T + rx that satisfy

e,

+ «[], IcP/t) - el : :; M(t - rx/j), T + «[] < t 2 are any two points in [T,T + rx], then cP/t) =

T :::;;

t :::;;

(9.2.8)

T

If t 1 and (9.2.5), and (9.2.6),

t :::;;

T

+

rx.

(9.2.9)

on account of (9.2.4),

/cPj(t 1) - cP/t2)/ :::;; /M(t1 - «li) - M(t 2 - rx/j)/.

(9.2.10)

Since M(' ) is continuous on [T, T + rx], it is uniformly continuous there. This implies by (9.2.10) that the set {cPj} is an equicontinuous sequence of continuous functions on [T, T + rx]. Also by (9.2.8) and (9.2.9), the sequence is uniformly bounded on [T, X + rx]. Consequently it follows from the ArzelaAscoli theorem (see [19]) that there exists a subsequence {cPjJ which converges uniformly on [T,T + rx] to a continuous function cP as k -- 00. From (9.2.4) (9.2.11) and since

f

is continuous in x for fixed t, (9.2.12)

f(t, cPik(t» -- f(t, cP(t»

as k -- 00 for every fixed t E [T, T + rx]. Therefore by the Lebesgue dominated convergence theorem (see [19]), lim k-«

for each t

E [T, T

IX)

+ rx].

cPik(t) =

It t

f(s, cPjk(S» ds = t f(s, cP(s» ds J'I:

(9.2.13)

Note from (9.2.6) that

e+

tf(s, cPjk(S» ds -

J.

it . f(s, cPik(s»

Jt-a/Jk

ds

(9.2.14)

184

9.

local Existence and Uniqueness Theory of Nonlinear Equations

in which it is clear that the latter integral tends to zero as k letting k - 00 and using (9.2.13), it follows that 4J(t) =

~

+

f

00.

Therefore,

(9.2.15)

f(s, 4J(s)) ds

for each t E [.,. + a]. A similar existence argument applies to the situation in which t ::;; r, and the conclusion of Theorem 9.2.1 is thereby established. Corollary 9.2.2 Cauchy-Peano, Iff is continuous on D, then there exists a solution (1, 4J) to the Cauchy problem (9.2.1), (9.2.2) with I = (r - a,. + a) where a, b are chosen appropriately small so that the rectangle R a•b centered at (r, ~) satisfies Ra,b c

(9.2.16)

D,

a = min{a,

blM},

M = max If(t, where the maximum is over (t, x) points t E I.

E

(9.2.17)

x)l,

R a •b • Moreover,

(9.2.18)

4J

satisfies (9.2.1) at all

Remark 9.2.1 There is a variety of methods for producing sequences of functions directed toward proving the existence of a solution to the Cauchy problem or approximating its solutions. The traditional proof of the CauchyPeano existence theorem (Corollary 9.2.2) uses Euler's method to produce a sequence of piecewise linear functions {4J j} and then applies the ArzelaAscoli theorem to conclude the existence ofa subsequence that converges to a solution. In problems in which the solution to the Cauchy problem on an interval is unique, it can be shown that no subsequence need be extracted for convergence. Hence for such problems, letting the step size h in Euler's method approach zero provides one of the simplest numerical methods for approximating the solution to any prescribed accuracy. There exist examples in which the solution is not unique yet the sequence of Euler approximations converges to a solution without extraction of subsequences. For an alternative proof of the Cauchy-Peano theorem, using piecewise linear approximations, as well as related remarks and examples, see [4]. Remark 9.2.2 In comparing the proof of Theorem 9.2.1 with that of Theorem 6.3.2, we noted that the latter, which established the existence of solutions to linear equations, has the advantage of avoiding the subsequence problem, thus providing a sequence (series) of computable approximations. Moreover, the approximations converge to the solution on the entire interval for which the differential equation is defined.

9.2

185

Local Existence and Uniqueness

The following definition is often useful for discussing the existence and uniqueness properties of differential equations. Definition The function f in the Cauchy problem (9.2.1), (9.2.2) is said to satisfy a Lipschitz condition on D relative to x if .

clx -

If(t,x) - f(t,Y)1 ~

r].

(9.2.19)

some c, all (t, x) and (t, y) in D. The constant c is called a Lipschitz constant. Suppose that f is continuous on D and has a continuous partial derivative f2 relative to x on D. If the sets Dr = {xl(t, x) E D} are all convex and Remark 9.2.3

(9.2.20)

If2(t, x)1 ~ c

on D for some constant c, then f satisfies the Lipschitz condition (9.2.19) on D. This becomes obvious on writing

Jr

1

f(t, x) - f(t, y) =

0

f:

=

0

os f(t, sx f2(t, sx

+ (1 -

+ (1

s)y) ds

- s)y) ds (x - y).

Theorem 9.2.3 Suppose that f satisfies a Lipschitz condition relative to x on a rectangle about each point in D. lithe Cauchy problem (9.2.1), (9.2.2) has a solution (1,
Proof.

Assume the hypothesis and suppose that there are two solutions,

(1,
loss of generality that
If(s, x)

- f(s, y)1 ~

CR

[x - yl

(9.2.21)

for all (s, x) and (s, y) in R. Since
=

f

[f(s,
(9.2.22)

(9.2.21) can be applied to (9.2.22) to conclude that i\(t)

f

~ I CR

i\(s) ds

I

(9.2.23)

186

9.

Local Existence and Uniqueness Theory of Nonlinear Equations

on an interval about r, in which (9.2.24)

Li(t) = IcP(t) - 1/1(t)I·

Applying Gronwall's theorem (Section 6.3, Exercise 1) to (9.2.23), we find that Li(t) = 0 on a neighborhood of r. This is a contradiction, which concludes the proof.

Remark 9.2.4 If (1, cD) and (T, cP) are two solutions to a Cauchy problem satisfying the hypothesis of Theorem 9.2.3, then cP(t) = cP(t) for all t e I n T. Theorem 9.2.4 Picard-Lindelof, satisfies the Lipschitz condition If(t, x) -

Assume that

t« Y)I

f is continuous on D and

:s; cRlx - yl

(9.2.25)

on each rectangle ReD. Then there exists a unique solution to the Cauchy problem (9.2.1),(9.2.2) on I = (t - ex, t + ex), where ab are chosen appropriately small so that for the rectangle centered at (r, ~), Ra,b c D,

(9.2.26)

ex = min{a,b/M},

(9.2.27)

and where (9.2.28)

M = maxlf(t,x)l,

the maximum over (t, x) E Ra,b' Proof.

The approach is to show that it is possible to define

cPo(t) = cPk+ l(t) (k = 0, 1,2, ...) for t e l =

=

[t -

~,

~

+

ex, r

f

(9.2.29)

f(s, cPk(S)) ds

+ ex]

(9.2.30)

and derive the estimates

. IcPk+it) - cPit)1 :s; Mex (k

(:):)!

e"

(9.2.31)

for some c > O. This would prove that {cPk} is a Cauchy sequence in the normed linear space of continuous afn-valued (or ~n-valued) functions normed with the uniform metric norm (see [19]). But this space is known to be complete (i.e., it is a Banach space) and hence

at;

for some

cP E C(l),

(9.2.32)

9.2

187

local Existence and Uniqueness

i.e., uniformly, as k --

00.

Thus taking limits in (9.2.30), it would follow that

'+ f

(9.2.33)

/(s,
It is a simple matter to see that

It is now necessary to attend to the rigorous details. Defining
'I

l
(9.2.34)

~ Mit - 'OJ,

all t e l . Hence, because of (9.2.27), the curve (t,
.«:

(9.2.35)

Now
I
f

~ I

I /(s,

~

(9.2.36)

Mit - '0/.

The next goal is to establish the estimate

c!'-1 M

IcPk(t) --.:
rl

k

(9.2.37)

(k = 1, 2, ...), t E 1, where c is the Lipschitz constant associated with R = Ra,b' Note that (9.2.37) is valid for k = 1 because of (9.2.35). Now

assume (9.2.37). From (9.2.30) and the Lipschitz inequality it follows that

I

s

If elf

s

~~lf(S

~

[/(s,
(k

!
-

r)kdSI

ckM

+

-

I

1)! t -

'0

Ik + 1

(9.2.38)

188

9.

Local Existence and Uniqueness Theory of Nonlinear Equations

and the proof of (9.2.37) by induction is complete. Now reapplication of (9.2.37) gives

!
- (k

+ p)!

+ ... + < Mrx(crx)k - (k + 1)!

Mit - rl + P ck

(k

+

[1 +

.

1)!

Mit -

(crx) 2!

+

+

(k

+ P-

1)!

Mit - rlk+p-l

-rl k + l

...

+

(crx)P-IJ

p!

Mrx(crx)k Cat ~ (k + 1)! e .

(9.2.39)

This supplies the details that complete the proof of existence. The uniqueness follows directly from Theorem 9.2.3.

Remark 9.2.5 The technique used to generate the approximating sequence {cPk} in the proof of Theorem 9.2.4 is often called the method of successive approximations. EXERCISES

1. Compute an interval of. existence I of Corollary 9.2.2 for the Cauchy problem i 4 - 3 - (t 2 + x 2 )' 2.

x(O)

= O.

Show that Theorem 9.2.4 applies to the Cauchy problem of Exercise 1

by showing that IfzCt,x)1 s 8 on Ra •b = {(t,x)/Ixl 3. Analyze existen.ce and uniqueness for (a) i = It - x211/2, x(O) = 0, (b) i = It - x211/2, x(l) = 2.

s

1, It I ~ 1}.

4. Derive the estimate for the error in the successive approximations of Theorem 9.2.4:

189

9.2

Local Existence and Uniqueness

5.

Compute the successive approximations for

x = x,

x(o) = 1.

Apply the inequality in Exercise 4 to estimate the error IcP(t) - cP3(t)1

on

[-I, 1].

6. Determine a continuous function f(x) on (0, 1) that is not Lipschitz continuous (i.e., does not satisfy a Lipschitz condition) on any subinterval of(O, 1).

Chapter

10

Global Solutions

In this chapter, we show that, for continuous differential equations, there passes through each initial point a solution curve defined on a maximal interval. Such maximal solution curves are shown to connect points on the boundary of the domain of the differential equation. Criteria are developed for ensuring that all solutions to general classes of equations will have solutions extending to infinite or semi-infinite intervals-the immortality property. 10.1

Introduction

In this chapter, we study the Cauchy problem x(t) = f(t, x(t)),

xl

t = !

= ~,

(10.1.1) (10.1.2)

where (1:, e) ED, D is a nonempty open subset of fJI x fJln, and f: D - fJln is continuous. In this setting a solution curve of (10.1.1) through (1:, ~), as defined in Chapter 9, is a pair (I,
The Cauchy-Peano theorem (Corollary 9.2.2) ensures the existence of at least one solution curve (I,
191

10.2 Maximal Solutions and Extensions of Local Solutions

that f is continuous. However, as examples demonstrate, the interval 1 is usually conservatively short. This gives rise to the topic of extending solutions. Recall from elementary set theory that a partially ordered system consists of a nonempty set on which is defined a reflexive, anti symmetric, and transitive binary relation. A chain is any subset of the partially ordered set having all of its elements pairwise comparable. Zorn's lemma concludes that if each chain in a partially ordered system has an upper bound, then the system has at least one maximal element. Discussion of the extensions of solutions to differential equations falls neatly within this framework, as follows. Let (10, ¢o) be a solution to (10.1.1), (10.1.2). A solution (I, ¢) is called an extension of (10, ¢o) if 1° eland ¢IIO = ¢o. Suppose that (Ii, ¢i) (i = 1, 2) are any extensions of (10, ¢o). Write (II, (1) < (12, (2) to signify that (12, (2) is an extension of (It, (1). It is an easy exercise to check that the latter relation partially orders the collection of all extensions of (10, ¢o). Moreover, each chain of extensions {(I", ¢")}, IX E A of (10, ¢O), has an upper bound, namely, (I, ¢), where 1 is the union of the I" over IX E A and ¢ is well-defined by ¢II'" = ¢". Thus by Zorn's lemma it follows that each solution (10, ¢O) has a maximal extension. Of course the maximal extension need not be unique. A solution (I, ¢) is called maximal if it is its own maximal extension. For such an extension, 1 will be denoted by «(,(1-, w+), and occasionally the notation /('r, ~) or w±(r, ~) will be employed when it is desirable to emphasize the dependence on (7:, ~) E D. Moreover, (w_, w+) will be referred to as a maximal interval ofexistence. The simplest situation is the one in which for each (7:, ~) E D each solution (I, ¢) through (7:, ~) is the only one with interval I. This will occur, for example, if, in addition to being continuous, f is locally Lipschitz relative to x for (t, x) E D. Then all maximal extensions of all solution curves through (7:, ~) are one and the same.

EXERCISE

Assume that f is continuous on D. Show as a consequence of Corollary 9.2.2 that for each compact KeD the Cauchy problem (10.1.1), (10.1.2) has a solution (I, ¢)(7:, ~) for each (7:, ~) E K for which 1 = (7: - IX, 7: + IX), some IX independent of'(r, ~). In particular, show that if the maximal solution through each (7:, ~) E K is unique, then

for some positive number IX independent of (7:,

~)

E K.

192

10. Global Solutions

10.3 The Behavior of Solutions on Maximal Intervals

The following terminology is convenient for describing the behavior of solutions on maximal intervals: For g: 1- N where I is an open subinterval in f!Il and N is an open subset of f!Ilm, the notation g(t) - aN as t - aI signifies that for each compact KeN there exists a compact subinterval IKe I such that g(I - I K ) c N - K. (The minus denotes complement.) Theorem 10.3.1

If f is continuous on D, then (t, ¢(t») - aD

t - aI

as

(10.3.1)

for each maximal solution (1, ¢) of (10.1.1), (10.1.2). Proof. Let (1, ¢) be a maximal solution through (r, ~). It will be sufficient to examine the behavior of ¢(t) as t - co , and as t - w_. Since the arguments are essentially the same, consider only the former. If co , = 00, then for each compact KeD, it, ¢(t» E D - K for all sufficiently large t since K is a bounded subset of f!Il x f!Il". Now suppose that w+ < 00. To obtain a contradiction, assume that for some compact KeD there exists a sequence tk [co, as k - 00 with all (t k , ¢(tk») E K. In view of the compactness of K, by first extracting an appropriate subsequence, it can be assumed that ¢(tk ) is convergent and hence (tk> ¢(tk» -(w+, ~(x,) E K as k - 00 for some ~<X) E f!Il". Consider the open ballBrofradiusrabout(w ,~<X»inf!ll x f!Il".SinceDisopenand(w+,~<X»)EK (hence ED), +

Br c

Er

(The bar denotes closure.) But since

If(t, x)1

some

c D,

~ b,

f

r> O.

(10.3.2)

is continuous on D and Er is compact, all

(r, x) E En

(10.3.3)

some bE f!Il. Select arbitrary (. subject only to 0 < e < r. Since ¢(tk ) - ~<X) as k - 00, there exists a k* such that (t k , ¢(tk») E B
~ I¢(t) - ¢(T1)1

I¢(t) - ~<X)[

~

r JT t

If(a, ¢(a) I do

+

+ t(.

[¢(Td - ~<X)I (10.3.5)

1

by the triangle inequality and the fact that (T 1 , ¢(Td) E B
193

10.3 The Behavior of Solutions on Maximal Intervals

The next step is to show that

f t

Tl

If(lT, 4>(lT»/ da

S

c

"2

(10.3.6)

Certainly (10.3.6) will follow from (10.3.2)-(10.3.4) if (t, 4>(t» E B. for T, S t < co , . Thus suppose that (Tz , 4>(Tz » E qj X qjn - B. for some T z satisfying T, < T z < «i , . By continuity of 4> there exists a largest T < «i., such that (t, 4>(t» E Bn all t for which T, S t S T; call it T 3 . Thus

If(t, 4>(t» I S b

(10.3.7)

for T, S t S T 3 < T z < w+. Since (T l , 4>(Td) E B./ z by the choice of T l , (T3 , 4>(T3» E B. by the choice of T 3 , and by (10.3.4)and (10.3.7) it follows that

~

15: f'ia, 3

-

4>(Tl )! =

3 -

Tli S

< !4>(T3 ) S

bjT

4>(lT»

dlTl

b(;b) =~,

(I0.3.8)

which is a contradiction. This proves (10.3.6). Combining (10.3.5)and (10.3.6) shows that

14>(t) - ~<Xli

(10.3.9)

Sf,

and since e was arbitrary, we conclude that (10.3.10) Since (w+, ~<Xl) E D, by the continuity of f on D, the Cauchy-Peano theorem ensures the existence of a solution (1 co s 4><Xl) of(1O.1.1)through (w+, ~<Xl). The claim now made is that the curve

co : < t < w+, «i.,

S tE1<Xl'

(10.3.11)

extends (1, 4». Certainly $ is continuous because of (10.3.10) and satisfies (10.1.1) for co : < t, < w+ and w+ < t e I <Xl' Since 4><Xl satisfies (10.1.1) at w+' all that remains to be shown is that (10.3.12) as fLO. Because of(10.3.1O) and the facts that 4> satisfies (10.1.1) on (aL, w+) and that $(o» +) = ~ co s it follows that $(w+ - f) - $(w+) = 4>(w+ - f) - ¢<Xl = -

f:++-. felT, 4>(lT» da;

(10.3.13)

194

10.

Global Solutions

and hence (10.3.12) reduces to showing that 1

£"

t+"co+

-r

c

feu,
~<Xl)

as

f~O.

(10.3.14)

By the mean-value theorem,

f:++-< fi(u,
ffi(ui>
(10.3.15)

for some a, satisfying 00+ - f::O:;; a, ::0:;; «i., (where by a), then
Proof.

[c, d]

Suppose that co , < b. Fix any cECa, w+) and dE(w+, b). Then

x C is a compact subset of D for each compact subset C c E. By

Theorem 10.3.1, (t, a can be handled similarly. Corollary 10.3.2 is established. R.emark 10.3.1 In the special case in which E = rJtn in Corollary 10.3.2, the conclusion translates into the statement that if ce., a), then 1
Let f be continuous and satisfy an inequality /f(t, x)1

::0:;;

«(r)

+ P(t)lxl

(10.3.16)

on the product D = (a, b) x ~, where (X(') and P(·) are nonnegative and locally integrable on (a, b). If

195

10.3 The Behavior of Solutions on Maximal Intervals

in which case there is nothing to prove, or w + . < b. In the latter situation Corollary 10.3.2 applies and hence 1¢(t)l- 00 as tjw+, as discussed in Remark 10.3.1. On the other hand, integration of (10.1.1) gives c/>(t)

=

~

+

f f

(lO.3.1?)

f'ia, c/>(a)) da, .

co: < t < w+, from which application of (10.3.16) yields

Ic/>(t) I : :; r :::;; t <

W+,

(;(1

+

where (;(1

=

I~I +

p(a) Ic/>(a) I da,

f'+

(10.3.18)

(l0.3.19)

(;«(a) da,

As a consequence of Gronwall's inequality and (10.3.18),

Ic/>(t) I:: ; c, r :::;; t < w+, where c=

(;(1

+ (;(1

f'+

(10.3.20)

pes) exp(f'+ pea) da) ds <

00.

(10.3.21)

Thus (10.3.20) contradicts the conclusion that Ic/>(t)!- 00 as tjw+, and therefore it is impossible for co , < b when co., < 00. A similar argument can be repeated to treat the case in which - 00 < co : , and Theorem 10.3.3 is declared proved. Remark 10.3.2 Note that Theorem 10.3.3 applies to the special case in which f is continuous with If(t, x)1 bounded on D = (a, b) x ~n. Remark 10.3.3 Corollary 10.3.2 and Theorem 10.3.3 are often quite useful in proving that the local solutions to specific classes of nonlinear differential equations have extensions to semi-infinite or infinite intervals. Such an application is made in proving the next theorem. Equations with co , = 00 are called immortal. Theorem 10.3.4 the type

Forced Lienard equations are scalar differential equations of it

+ feu) u + g(u) =

e(t, U,

u)

(10.3.22)

in which f, g, and e are assumed to be continuous on a region (a, (0) x ~2. Suppose, in addition, that f, g, and e satisfy

(a) lim sUPx-+oo

g g(a) da =

lim sUPx-+oo - J~x

g(a) do =

00.

196

10. Global Solutions

(b) For each! E (a, (0) and some corresponding p, q E rJe the union over t ;;::: ! of the sets of points (x, y) that satisfy

~

y2 - ye (t, x, y -

< -g(x)

f:

f:

f(u) dU)

f:

f(u) do - p

g(u) do - q

(10.3.23)

is a bounded subset of rJe2. Then each maximal solution u(·) on(w_, w+) to (10.3.22) will have co , = Proof.

x

=

y-

00.

The argument is based on the first-order system

f:

f(u) da,

y=

-g(x)

+e

(t,

x, y -

f:

f(u) dU),

(10.3.24)

arising from (10.3.22) written in first-order form and from the transformation

y = u+

x=u

f:

(10.3.25)

f(u) da.

It is easy for the reader to check that each solution u( .) to (10.3.22) on a subinterval of (a, (0), when substituted into (10.3.25), produces a solution (xf-), y('» to (10.3.24) on that subinterval. Thus it is sufficient to prove that all solutions to (10.3.24) on maximal intervals (co , , w+) c (a, (0) will have w+ = 00. To obtain a proof by contradiction, suppose there were a solution (x('), y(.» to (10.3.24) with «i., < 00. Then according to Corollary 10.3.2, as

(10.3.26)

tjw+.

Consider the "energy" function v(t)

y2(t)

= -2- +

IXII) 0

(10.3.27)

g(u) da,

t E (co..; w+). By the intermediate-value theorem it follows from assumption (a), (10.3.26), and (10.3.27) that lim sup v(t) =

(10.3.28)

00.

If
On the other hand, computing v(t), using (10.3.24) and (10.3.27), leads to v(t) = y(t)e ( t, x(t), y(t) -

XII»

I

0

f(u) do

)

- g(x(t»

IXlt) 0

f(u) do,

(10.3.29)

10.3

197

The Behavior of Solutions on Maximal Intervals

As a direct consequence of assumption (bj.for some p, q E

~

yZ - ye

(t,

x, y

> -g(x)

f:

f:

f((J) do)

f((J) do - p

f:

gJ,

g((J) da - q

(10.3.30)

is satisfied by the tail of (xC), y(.» as tj w + . Combined with (10.3.29) this yields

+ q,

vet) :5: pv(t)

(10.3.31)

all t < «i., near w+. Solution of inequality (10.3.31) shows that vet) remains bounded as tjw+, which is inconsistent with (10.3.28). Hence Theorem 10.3.4 has been proved by contradiction. Remark 10.3.4

Condition (a), imposed on g, is fulfilled if

LX> g((J) da

= -

f~

00

g(a) do =

(10.3.32)

00

but is not that stringent. An example of a function that satisfies (a) but not (10.3.32) is g(u) = dldu[ U Z sin? u]. Note that the standard linear spring g(u) = ku, k a positive constant, satisfies the more stringent condition as well as (a). Recall that the standard linear model of friction uses feu) a positive constant. Readers familiar with elementary classical dynamics will recognize the integral term in v = yZ + 2

IX g((J) do

(10.3.33)

0

as a generalization of what is referred to as a potential energy function for equations modeling spring-mass systems. The function v is the total energy (kinetic plus potential). It is interesting to reflect about the extent to which conditions (a) and (b) allow for generalizations of the linear friction and spring models. The nature of the forcing terms e allowed by (10.3.23) is also of interest. These points are explored further in the exercises. EXERCISES

1. Compute all the maximal solutions to (10.1.1), (10.1.2) for the problem in which f(t, x) = tx Z on D = gJ x gJ. Compare their qualitative behavior with that predicted by Theorem 10.3.1. Determine W±(T, 2. Consider the Cauchy problem

n

Xl = -xt/t - x-ft", on D = {(t, x) I t < 0, x =F O} with

Xz = Xl

(tZXI

+ tXz)/(t4x i + xD

= 0, Xz =

lin att = -lin.

198

10. Global Solutions

(a) Verify that Xl = sin(1/t)/t, X2 = t cos(1/t) is the unique solution. (b) Compare the qualitative behavior with that predicted by Theorem 10.3.1 and by Corollary 10.3.2. Does Theorem 10.3.3 apply? 3. Two species of fish that occupy a lake reproduce and prey upon one another in a fashion such that their respective populations Xl and X 2 can be modeled by the differential equations (a) An appropriate choice of D would be D = ( - 00,00) x (0,00) x (0,00). Explain why this choice is appropriate. (b) Discuss the physical interpretations of the constants rJ.i > 0, Pi > 0 (i = 1, 2), and the terms in the differential equations. (c) Analyze the existence and uniqueness questions for the problem. (d) Investigate the behavior of the solutions to the differential equations, applying the theory of Section 10.3, to conclude that w+(r, e) = 00 for each (r, e) ED. (Hint: Analyze the differential equations about the boundary of the square {(X1X2) I0 < Xl < s,O < X2 < s} with s appropriately large.) (e) Determine a Ir, e)ED for which w_(r, e) = -00. 4. Suppose that the real-valued functions Adefined on [r, 00) x ~2 and p defined on ~ satisfy: (a) p(x) < inf, SUPt A(t, X, y), all Ixi large, (b) sup, p(x) < inf, SUPt A(t, X, y), alllyllarge. Show that {(x, y)lsUPt A(t, X, y) < p(x)} is then a bounded subset of ~2. 5. Assume that the forcing term in the Lienard equation (10.3.22) also satisfies le(t, x, Y)I ~ b < 00 for some b E~, all (t, x, Y) E (a, 00) x ~2. Apply Exercise 4 to show that condition (b) is then fulfilled if I and g satisfy -g(x)

r

I(a) do - p

S:

g(a) do

~

q

for all

X

E~,

some nonnegative p E ~. Show that if I and g satisfy the latter inequality with p > 0 and g satisfies (a) of Theorem 10.3.4, then

-c~'I:g(a)da some c > O. 6. Assume that

forall

XE~,

I and g are continuous on ~ and that all

XE~,

in which the integers r, s as well as the real coefficients /1' gl are positive with

10.3

199

The Behavior of Solutions on Maximal Intervals

for some f3 E fjl and all x E fjl. Show that for each bounded and continuous forcing function e on [0, 00) X fjl2, Eq. (10.3.22) will then have all of its solutions extendable to co , = 00. 7. A classical example of a nonlinear equation satisfying the conditions imposed in the result of Exercise 6 is the Van der Pol equation it + (u 2 - 1) u + u = e(t, u, u). Exercise 6 expands this example to include, in particular, all polynomial equations with polynomial coefficients f(u)

=

+ fm um,

fo + flU + f2U2 +

+ gnun,

g(u) = go + glU + g2u2 +

with m even, n odd, and fm' gn positive coefficients. Verify these claims. To see that the conditions imposed upon the polynomials f(u), g(u) are not superfluous, verify that the equation it - u

u-

u

3

=

°

has a solution u(t) = 1/(1 - t) for which w+ < 8. Consider the equation it

+ [(hg)'(u)] u + g(u) =

00.

e(t, u, u)

(in which the prime denotes the derivative relative to u). Assume that e is continuous and bounded on (a, 00) x fjl2, both hand 9 are once continuously differentiable, h is nonnegative, and (hg)(O) = 0. Show that the conclusion of Theorem 10.3.4 applies when 9 moreover satisfies condition (a) of that theorem. 9. Show that all equations of the type it

+ f(u)u + h(u)

f:

f«(J) da = e(t, u, u)

have the immortality property if f and h are continuous on fjl, h is nonnegative, g(u).= h(u) So f«(J) do satisfies (a) of Theorem 10.3.4, and e is continuous and bounded on (a, 00) x fjl2. 10. Verify that the equation it

+ g'(u) u + g(u) = 0,

in which the prime denotes derivative and g(u) = u cos u + sin u, is an example of an immortal equation that is capable of taking on negative potential energy of arbitrarily large magnitude. Compute the potential energy function. Will the immortality persist if the equation is forced with any continuous e bounded on [0, 00) x fjl2? Is w_ = - 00 for each maximal solution to the unforced equation?

Chapter

11

The General Solution-Dependence of Solutions on Parameters

In previous chapters we established the existence of a maximal solution curve through each initial point (r, ~) in an open subset D of fJIl x fJIl" on which is defined the differential equation x = f(t, x), with f: D -- fJIl" continuous. In this chapter we topologize the collection of maximal solution curves eJ>/D) in such a manner that for those equations with unique solutions, the map, called the general solution, that assigns to an initial point (r, ~) E D the maximal solution curve through it, turns out to be continuous relative to the topology imposed on eJ>f(D). From this geometric approach such classical results as the semicontinuous dependence of the maximal interval of existence on the initial point, openness of domain, and the uniformly continuous dependence of solutions over compact time sets relative to parameters are derived as corollaries. The differential dependence of solutions on parameters is developed in the same framework and the results are used to derive related variational equations. As an application of the main theorems it is shown that eJ>/D) is an n-dimensional (topological) manifold; moreover, it has the richer structure of orientable C'-manifold when f is r times continuously differentiable relative to x for (t, x) in D (r = 1, 2, ...). 11.1

Mathematical Preliminaries

The notions of vector space and linear transformation provide a natural context within which the theory of linear differential equations can be 200

11.1

Mathematical Preliminaries

201

developed. In deference to the student, in previous chapters we held the level of abstraction to a minimum by the extensive use of matrices. One would expect that the transition to nonlinear differential equations would require a more general mathematical framework. The generalization adopted here is topological space and continuous map. A topological space is a nonempty set 8 (called the underlying set) together with a distinguished family !F of subsets of 8, satisfying the following conditions: !F contains the empty set, (2) every collection from !F has its union in !F, (3) the intersection of any two elements in !F is again in !F, (4) the union of all elements of!F is 8. (1)

The elements of !F are called open sets. Thus a topological space is the pair 8, !F. (This text frequently follows the standard abuse of notation in which 8 is used to denote the topological space as well as the underlying set.) A map f : 8 1 - - 8 z from one topological space into another is called continuous at p E 8 1 if for each open set Oz containing f(p), f(Ol) c Oz for some open set 0 1 containing p. If f is continuous at each point in 8 1 , then it is simply called continuous on 8 1 , A collection fJ6 of subsets of a nonempty set 8 is called a basis if the union of its members is 8; and whenever p is a point in the intersection B 1 (\ B z of two elements B 1 , B z offJ6, then p E B 3 C B 1 (\ B z for some B 3 E/!J. Note that all possible unions of elements drawn from fJ6 produce a family of open sets for 8. fJ6 is called a basis for the topological space defined by that family. Familiar examples of topological spaces are 8 = PAn (n = 1, 2 ... fixed), in which the open sets are generated by the basis fJ6 = {B~ I x E PAn, r > O} where B~ = {y IY EPAn, Ix - yl < r}, called the open ball of radius r centered at x. Similarly, cc n is a topological space. A standard way of producing a topological space 8 1 , !F1 from a given topological space 8, !F is to let 8 1 be any non empty subset of 8 and choose !F1 to be the family of subsets of 8 1 obtained by all intersections of elements of !F with 8 i- The topological space 8 1 , !F1 is called a subspace of 8, !F and 8 1 is said to be given the induced or subspace topology. A subset K c 8 of a topological space 8 is called closed if the complement of K in 8 is open. Note that this generalization agrees with the use of the term in PAn. The axioms of a topological space strip away the irrelevancies that occur in the first appearance of the notions of continuity, connectedness, compactness, etc., in PAn. Many important topological spaces are function spaces. For example, the topological space 8 = {c/J I c/J: [a, b] -- PAn continuous} with basis fJ6 = {B~ I c/J E 8, r > O}, where B~ = {t/J I t/J E 8, max1a,bllc/J(t) - t/J(t) I < r},

202

11. The General Solution-Dependence of Solutions on Parameters

already provides the setting for the proof of the Picard-Lindeloff theorem (Theorem 9.2.4 in Chapter 9). It is hoped that the treatment of differential equations appearing in the remainder of this chapter will encourage the student who is not yet acquainted with the basic notions and theorems of topology to engage in some outside reading.

11.2 A Space of Curves

For the moment, we suspend direct concern with differential equations in order to develop an ambient space within which to view the collection of solution curves of differential equations. Let D continue to denote a nonempty open subset of PA x PAn and define (D) to be the collection of ordered pairs (1, ¢) with I an open subinterval of PA, ¢: I - PAn continuous, and (r, ¢(t» E D for all t e I, The space of curves (D) will be topologized, using as a base for the topology the family of subsets of (D),

N'f:!) = {(J,t/!)/(J,l/J)e(D),J:::l K,!¢(t) -

t/!(t)! < e for all r s K}, (11.2.1)

where (1, ¢) E (D), K c I is compact, and e > O. Although mathematicians frequently topologize collections of maps from one space into another, the topic under consideration involves maps that have individual domains.

Lemma 11.2.1 The family of sets (11.2.1) is the base for a topology of (D) relative to which a sequence Uk' ¢k) converges to a point (1, ¢) in (D) if and only if both I k :::l K

for all large

00,

(11.2.2) (11.2.3)

¢klK -¢IK'

uniformly as k -

k,

all compact K c I.

Proof. It is clear from the definition (11.2.1) that the subsets cover (D) since (1, ¢) e N'f:t J• Now consider any

N'f:t J

(11.2.4) It is necessary to show that

(11.2.5) for some (13' ¢3) E (D), compact K 3

c

13 , and £3 > O. Consider the

11.2

203

A Space of Cu rves

tentative choices

= (1, cP), (11.2.6) K3 = K 1 U K2, (11.2.7) £3 = min ls, - maxlcP(t) - cP1(t)l, £2 - maxJcP(t) - cPit)I}. (11.2.8)

(13' cP3)

K,

K2

Observe that (13' cP3) E cJ)(D) since (1, cP) E cJ)(D), and K 3 c 13 since 13 = 1, and due to (11.2.4)both 1 =:> K 1 and 1 =:> K 2 • Moreover, £3 > 0 since t

E

K; (i = 1, 2),

(11.2.9)

which by the continuity of the functions involved and the compactness of the K; implies via the extreme-value theorem that maxlcP(t) - cP,{t) I < £; K,

(i = 1,2).

(11.2.10)

This shows that at least N~;:t.3) is a subset of type (11.2.1). The first part of (11.2.5) follows from the choice (11.2.6). Now the last part will be verified. Thus select arbitrary (14' cP4) E N~;:t.3). Consequently, (14' cP4) E cJ)(D), 14 =:> K 3 =:> K 1 because of (11.2.7), and obviously

cPit) - cP1(t)

= [cPit) - cPi t)] + [cPit) - cP1(t)] = [cPit) - cP3(t)] + [cP(t) - cP1(t)],

t E K 1, (11.2.11)

in view of (11.2.6). From (11.2.11) and the definition (11.2.8) it follows that

IcPit) - cP1(t)1 ~ IcPit) - cPit)1

< £3

+ IcP(t) -

+ maxlcP(t) K,

cP1(t)!

cP1(t)1

= min{£l - maxlcP(t) - cP1(t)l, £2 - maxlcP(t) - cPit)j} K,

+ maxlcP(t) .K,

K2

cP1(t)1 ~ £1'

(11.2.12)

all t E K 1 • Note that (11.2.12) uses the fact that K 3 =:> K 1 • Thus it has been verified that (h, cP4) E N~::~tl and since the choice of (14' cP4) in N~;:t3) was arbitrary, the conclusion (11.2.13) follows. By symmetry it is clear that by repeating essentially the same argument one gets (11.2.14)

204

11. The General Solution-Dependence of Solutions on Parameters

and thus verification of (11.2.5) is complete. This proves that the subsets (11.2.1) constitute a base for a topology on (D). Now consider any convergent sequence (Ik' c/Jk) - (I, c/J) as k - 00 in (D) relative to the topology established. Select arbitrary compact K c I. Then for each e > 0, (Ik> c/Jd E N~:t) for all large k. But then (11.2.2) and (11.2.3) follow directly from definition (11.2.1). Conversely, if(11.2.2)and(11.2.3) hold for a sequence {(Ik' c/Jd (k = 1,2, ...) in (D) and (I, c/J) E (D), then (Ik' c/Jk) eventually enters and remains in each basic open set centered on (I, c/J) for large k and thus does likewise for all open sets containing (1, c/J). This completes the proof of Lemma 11.2.1. With the topology on (D) vividly in mind, we shall close this section with a lemma that will be needed later in this chapter. Lemma 11.2.2

If the map D - 9l"I(t, x) -

(11.2.15)

f(t, x)

is continuous, then the induced map

(11.2.16) is likewise continuous. Proof. The proposed map (11.2.16) is defined and maps into (91 x 9l") since f is assumed continuous on D, (t, c/J(t» E D for all t e I, and c/J: I _ 9l" is continuous for each (1, c/J) E (D). Select arbitrary (Io, c/Jo) E (D) and denote fe-, c/JoC-» by fo. Consider arbitrary e > 0 and compact K c ID" It is sufficient to show that the proposed map carries the basic open set

into

(11.2.17)

N(Io.!o) K,f

for some compact eel0 and 1J > O. Consider the candidate C = K. The problem then reduces to showing that there exists a <5(£) > 0 such that for each (I, c/J) E (D) with I ::::J K and Ic/J(t), - c/Jo(t) I < 1J(£),

t E K,

(11.2.18)

it necessarily follows that I!(t, c/J(t» - f(t, c/Jo(t»

I<

e,

tEK.

(11.2.19)

To argue existence of the required <5(£), consider the family of tubular sets T p = {(t,x)l(t,X)E91 x 9l",lx - c/Jo(t) I ~ p, tEK},

(11.2.20)

p > O. Since c/Jo is continuous and K compact, it follows that each T p is compact and P1 > P2 > 0 implies TP 1 ::::J T p 2 ' The claim made is that

r;

c D

(11.2.21)

205

11.3 The General Solution and Continuous Dependence

for some p. > O. If this were not the case, then there would exist a sequence {91 x 9l n - D} ( l T 1/ k (k = 1,2, ...) that, by extracting a subsequence, can be assumed to converge to a point, say (too' X(.,). But too E K since K is compact and Xoo - 4>o(t oo ) = 0 by continuity of 4>0' This places (too'x oo) E D. Thus the sequence (t k , xd E 9l x 9l n _ D converges to a point (too,x oo) E D, which contradicts the assumption that D is open. This proves (11.2.21). Since! is given to be continuous on D, its restriction to Tp * is continuous and hence is uniformly continuous on this compact set. This uniform continuity provides the required <5(£) for (11.2.18). The proof of Lemma 11.2.2 is now complete.

«; Xk) E

Definition The collection
Remark 11.2.1

11.3

The General Solution W, 11) to the Cauchy Problem (D, f) and Continuous Dependence on Parameters

The Cauchy problem (10.1.1), (10.1.2), with parameters identified with the map D

--+

9l n j(t, x)

--+

!(t, x)

(r,~)

E

D, can be

(11.3.1)

presumed given, in which recall that D is a nonempty open subset of 9l x 9l n• For brevity this map (often called a vector field) will be denoted by (D, f). For (D, f) continuous, it turns out that through each point (r,~) E D there passes a maximal solution curve (1,4» which is now assumed to be always unique. Of course both I and 4> depend on the choice of (r, ~) E D. Thereby, for each (T,~) E D there is associated an element (1(r, ~), 1]( " r, ~)) E
--+

--+

(1('!,

0.

1](', r, ~))

(11.3.2)

will be called the general solution to the Cauchy problem (D, f) and for brevity will be denoted by (D, 1]).

206

11.

The General Solution-Dependence of Solutions on Parameters

Remark 11.3.1 The literature contains an assortment of views and terminology regarding (10.1.1), (10.1.2). One might alternatively define the general solution of (D, to be the map

n

9) -

9l" I(t, r, ~) - /1(t,

~),

(11.3.3)

e D, t e 1(.,~)}

(11.3.4)

'r!

where !J) =

{(t,., ~)I(.,~)

denotes the maximal interva1.) See, for example, Hale [8], who (Again 1(.,~) defines the general solution as a map into 9l". However, the author feels that the definition given by (11.3.2) is more basic and leads to greater unity. The following theorem presents a concise description of the dependence of the solutions of (10.1.1), (10.1.2) on the initial parameters (r, ~) e D.

Theorem 11.3.1 The general solution (D, /1) to a continuous Cauchy problem (D,

n,

(11.3.5) as well as the map

(11.3.6) is continuous. Proof. Since f(D) has the subspace topology, (11.3.5) is proved continuous first by regarding it as a map into (D). To reach a contradiction ultimately, assume that the map under consideration fails to be continuous at some point (r, ~) e D. Then there is a convergent sequence (.k' ~k) -(., ~) as k - 00 in D for which(1ko ¢k) -1+ (1, ¢). (Why?) Here the notation denotes that /1(', .k' ~k) = ¢k' /1(-, .,~) = ¢, and 1ko 1 are the respective maximal open intervals. By extracting an appropriate subsequence it can be assumed that each (1ko ¢k) is outside some open subset of (D) containing (1, ¢) which can be taken to be a basic open set. Thus assume that

(k

= 1,2, ...)

(11.3.7)

for some compact K c I and £ > O. By expanding K and shrinking e, it can be assumed that K = [a, b] is a closed interval containing r in its interior and that the tubular set T= {(t,x)l(t,x)e9l 1 x 9l",

I(t,x) - (t,¢(t))1 < £,teK}

(11.3.8)

satisfies TeD. (Here the overbar denotes closure.) For all sufficiently large k, (11.3.9)

11.3

207

The General Solution and Continuous Dependence

since rP in continuous at r, ~ - rP(r) = 0, and (rk' ~k) - (r, 0 as k - 00. By discarding a finite number of elements it can be assumed that (11.3.9) is valid for k = 1, 2, .... For an element of cI>(D) to fail to be in a given basic open set of the type defined in (11.2.1), one of the two remaining conditions in that definition must be violated. In particular, (11.3.7) can occur if and only if I k ::J K fails or IrP(t) - rPk(t) I < e, t E K, fails. But from Theorem 10.3.1 it follows that IrP(t) - rPk(t) I < £, t e K n I k, must fail for each k E {I, 2, ... }. By the continuity of each rPka!1d the compactness of the boundary aT of T, it follows that for each k E {I, 2, ...} there exist unique points (IXk' xd, (13k> Yk) in aT such that (IXk' 13k) Elk, (11.3.10)

rPiIXk) = xi, rPk(f3k) = Yk' and (r, rPk(t» E T for IXk < t < 13k' Once again, by extracting a subsequence, it can be assumed that (IXk' Xk) - (IX, x) and (13k> Yk) - (13, Y) as k - 00. Note that IX < r < 13 (see Exercise 10.2.1). Since IX < 13, certainly IX + lin < 13 - lin for all large n. On the other hand, since IXk - IX and 13k - 13 as k - 00, IXk < IX + lin and 13 - lin < 13k for each n E {I, 2, ... } for all large k. Thus it is apparent that IXk < IX

+ lin <

13 - lin < 13k

(11.3.11)

for all (independently) large k and n. In particular, for each fixed integer n > 2(13 - IX),

{rPk I[a + I/n.p -lin]}

(11.3.12)

for all large k, is an equiuniformly continuous sequence and is uniformly bounded. Consequently the Arzela-Ascoli theorem [19] applies, thus assuring that each subsequence of {rPd, k E {I, 2, ... }, has in turn a subsequence that is uniformly convergent on [IX + lin, 13 - lin]' In particular, starting with the smallest integer value n > 2(13 - IX) and working inductively, one can extract for each n > 2(13 - IX) a subsequence {rPn.k}, k E {I, 2, ... }, of {rPk} that is uniformly convergent on [IX + lin, 13 - lin] such that {rPn.d, kE{1,2, ... }, is a subsequence of {rPn-l.d, kE{1,2, ... }. Now consider the diagonal sequence t/fk = rPk.k' It follows that {t/fd is a sequence of continuous functions, uniformly convergent on each compact subset of (IX, 13), and its pointwise limit, denoted by t/f, is defined and continuous on (IX, 13). From the definition of t/fk observe that E

(IX

Since IX < r < 13, as already shown, r

E

t

+

11k, 13 - 11k).

(IX

+ 11k, 13 - 11k) for all large k.

(11.3.13)

208

11.

The General Solution-Dependence of Solutions on Parameters

Integration of (11.3.13) shows that

!/tk(t) - !/tk(T:)

f

=

Ito, !/tlu») de,

(11.3.14)

r

t E (a + 11k, fJ - 11k), for all large k. By the uniform continuity of Jon and the uniform convergence of!/tk on compact subsets of («, /3), it follows on taking limits as k --* 00 in (11.3.14) that

f

!/t(t) ., !/t(T:) =

f'to, !/t(u» do,

t

E

(a, fJ).

(11.3.15)

It was proved that !/t(t) = lim i --+ oo 4Jk.(t), t E (a, fJ), for some subsequence k, --* 00 as i --* 00. Since the convergence is uniform on compact subsets of (iX, fJ), 4Jk,(T:kJ = ~k" and T:kl --* T: as i --* 00, it therefore follows that

!/t(T:) = lim 4Jk,(T:k.) = lim ~k, i-r o:

i-v co

=

~'

(11.3.16)

Thus by differentiation of(11.3.15), along with (11.3.16), it has been shown that

~ = J(t, !/t), !/t(r)

=

t E

(ex, /3),

e,

(11.3.17) (11.3.18)

and, in particular, by the (assumed) uniqueness of the solutions to the Cauchy problem (D, f),

!/t(t) = 4J(t),

t

E

(ex, fJ).

(11.3.19)

For each i E {I, 2, ... }, by definition of the aks and fJkS, at least one of l4Jk,(akJ - 4J(iXk,) I, l4Jk,(fJk.) - 4J(/3kJ Imust equal £. Hence it can be assumed,

by extraction of a subsequence, that one of the following holds for all large i:

l4Jk'(ak.) - 4J(iXk.) I = e,

(11.3.20)

l4Jki(/3k,) - 4J(/3kJI =

(11.3.21)

£.

Suppose that (11.3.20) were the case. For each tJ satisfying a < ex

+ tJ < /3,

+ tJ) - 4J(i/.k, + tJ») = !4Jk,(ak, + tJ) - !/t(exk, + tJ)!--* 0

(11.3.22)

!4Jk,(exk,

as i --* 00 because of the convergence of 4Jk, to !/t on (a, /3). Since 4Jki and 4> satisfy the same differential equation

¢ki

=

J(t,4JkJ,

(11.3.23)

¢

=

J(t,4J),

(11.3.24)

209

11.3 The General Solution and Continuous Dependence

subtraction and integration give

cPki(ak,) - cP(ak,)

=

cPk,(aki + £5) - cP(rxk,

-f:~'

+0

+ b)

[J(O", cPk,(O"» - f(O", cP(O"»] da. (11.3.25)

Application of (11.3.19) and (11.3.20) to (11.3.25) shows that

e = IcPk;(rxk,) - cP(ak,) 1 ::;; IcPk;(ak; + b) - cP(ak, + 15)/ +

r.

IbJc

for all large i, where IJ(t, x)1 ::;; c for (r,x) E Thus letting i -+ and taking into account (11.3.22), it can be concluded that

(11.3.26) 00

in (11.3.26) (11.3.27)

which provides a contradiction since 1151 could be chosen arbitrarily small independently of e and c. The case in which it is (11.3.21) that obtains can be treated similarly, and thus the long-sought contradiction establishing the claimed continuity of the map (11.3.5) is reached. Since 1'/lt, T,~) = f(t,1'/(t, T, ~», t E l( T, ~), by definition of 1'/ the map defined in (11.3.6) is the same as the map of D into (~ x ,!~") given by (T,

~) -+

(I( T,

~),

f(· , 1'/(., T, ~»).

(11.3.28)

But the latter map is the composition of the map D -+ (D) that was just shown to be continuous, followed by the continuous map of (D) into (~ x ~") given by (11.2.16). Thus (11.3.6) is continuous and Theorem 11.3.1 is proved. . Theorem 11.3.1 encompasses a number of results of the classical theory of differential equations. Recall some of that terminology: An extended-real-valued function b(x) is called lower semicontinuous at X o if for each e > 0, b(xo) - e < b(x) for all x in a neighborhood of Xo' Similarly, a(x) is called upper semicontinuous at Xo if for each e > 0, a(x) < a(x o) + e for all x in a neighborhood of Xo' For such functions, if a(x) < b(x), then the interval-valued function lex) = (a(x), b(x» is said to be semicontinuous at Xo' Amongst other consequences, Theorem 11.3.1 implies that for each compact K c I(TO' ~o), I(T,~) ::;) K for all (T,~) in a neighborhood of (To, ~o) E D. This translates into the statement that 1(', .) is semicontinuous at each (T,~) E D, i.e., on D. In particular CO+(T,~) is lower semicontinuous and CO_(T,~) is upper semicontinuous on D. As a final note about classical terminology, it should be pointed out that a lower semicontinuous function is simply a continuous function if its range is retopologized by the topology generated by the base {(a, 00] 1 a E ~}. A similar statement applies to an upper semicontinuous function. Remark 11.3.2

210

11.

The General Solution-Dependence of Solutions on Parameters

Coronary 11.3.2 Let (D,17) denote the general solution to a continuous Cauchy problem (D, f). Then the subset P)

=

{(t, r, ~)I('t,~)

ED,

t E I('t,~)}

c 9t x D

(11.3.29)

is open. Furthermore the maps £1) -

9t n l(t, 't,~)

-17(t, r, ~),

(11.3.30)

p) -

9t n l(t, 't,~)

-17,(t, 't,~)

(11.3.31)

are continuous and Ie· ,..) is semicontinuous on D.

To prove £1) open, choose arbitrary (to, 'to, ~o) E £1). Then which is open, and hence toE(a,b)c[a,b] cI('to,~o) for some a, bE 9t. By the continuity of (D,17) proved in Theorem 11.3.1, I('t,~) =' [a, b], all ('t,~) in a neighborhood of ('to, ~o)E D. Since (to, 'to, ~o) E P) was arbitrary, it follows that p) is open. To prove the continuity of (11.3.30) - ('to, ~o), so that by the continuity let (t, 't,~) - (to, 'to, ~o) in p). Then ('t,~) of (D, 17) provided by Theorem 11.3.1, 11(', r, ~) - 11(', 'to, ~o) uniformly on compact subsets of I('t o, ~o). Hence 17(t, r, ~) - 11(to, 'to, ~o) as (t, 't,~) (to, 'to, ~o). This proves the continuity of (11.3.30). In tum it follows that £1) - DI(t, 't,~) - (t, 11(t, r, ~» is continuous. But since f is continuous on D and 11lt, 't,~) = f(t, 11(t, r, ~», it is clear that (11.3.31) is continuous because it is the composition of continuous maps. The semicontinuity of 1(', .) was already pointed out in Remark 11.3.2 to be a corollary of Theorem 11.3.1. This completes the proof of Corollary 11.3.2. Proof,

toEI('to,~o),

Coronary 11.3.3

Assume the same hypothesis and notation as in Corollary 11.3.2. Then for each (t, r, ~o) E p), (t, 11(t, r, ~o»

r

E

E

~o»

11('t, t, 11(t, r, ~o»

(11.3.32) (11.3.33)

I(t, 11(t, r, ~o»,

(r, t, 11(t, r, In particular, h(~) with inverse h-l(~)

D,

E £1),

= ~o.

(11.3.34) (11.3.35)

= 11(t, 't,~) defines a local homeomorphism h about ~o = 11('t, t, ~).

E

9t n

Proof. Statements (11.3.32)-(11.3.35) follow successively from the definitions of £1), 11, and I and from the assumed uniqueness of maximal solutions to the initial-value problem. For fixed t and r the semicontinuity of I('t, .) implies that (t, r, ~) E p) for all ~ in a neighborhood of ~o. Hence (11.3.32) remains valid for ~ in a neighborhood of ~o and (11.3.33)-(11.3.35)

11.3

211

The General Solution and Continuous Dependence

remain valid as well. The continuity of h( .) and h- 1( . ) is provided by Corollary 11.3.2 and the proof is complete.

Remark 11.3.3 Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3 illuminate the nature of the dependence of the solution curves of (10.1.1), (10.1.2) on the initial parameters Cr,~) E D. In both the theory and the applications of differential equations one frequently encounters differential equations with additional parameters, say Z E Bim, entering the function! directly, and the nature of the dependence of the solutions on z is likewise of interest. In most instances the following device reduces z to the status of initial coordinates of ~ and therebyplaces the questions within the purview of Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3: Regarding f as a function of (t, z, x) E 15 c Bi x Bim x Bin, pass to the Cauchy problem (15, f) in which f(t, z, x) = [!(t,Oz, X)]

(0

E

(11.3.36)

Bim).

If 15 is open and! is continuous, thereby making f continuous on 15, then Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3 apply to (15, f). The conclusions concerning ft, i, and 15 then translate directly into statements about the original problem since 1](t, t, z,~) consists of the last n coordinate functions of ft(t, t, Z, ~); e.g., 15 -<11(15)I('t', Z, ~)-(I('t', Z, ~), '1(', r, z,~)) is a continuous map, 1(·,',·) is semicontinuous on 15, ~ - Bin let, r, z,~) 1](t, r, Z, ~) is a continuous map, etc.

EXERCISES

1. A topological space S, lJ' is called Hausdorff if for each pair of distinct points s; E S (i = 1,2) there exists a pair of open sets O, E lJ' such that Si E 0; (i = 1,2) and 0 1 n O2 is empty. Show that the space of curves
=

{slsES, d(s,so) < r},

r> 0,

So

E S,

constitutes a base for a topology on S. Show that the resultant topology on S is Hausdorff.

212

11. The General Solution-Dependence of Solutions on Parameters

3. Show that there is no distance function generating the topology that is defined on
11.4 Differential Dependence of the General Solution (D, 11) on Parameters and Related Variational Equations

In this section we discuss the extent to which differentiability ofthe Cauchy problem (D, f) induces differentiability of the general solution (D, '1) and derive the variational equations involved in computation of those derivatives. Notation for Derivatives. The derivatives of maps between subsets of Euclidean spaces will be identified with their Jacobian matrices. Variables in subscript position will denote differentiation relative to those variables; e.g., [Jx]ij = afdaxj and [ft]i = afdat. Square brackets will be used to help identify matrices; e.g., the full n x (n + 1) Jacobian matrix f(t,x) is on occasion written as [ft, fx]' A Cauchy problem (D, f) with [f, fx] continuous on D has a general solution (D, '1) for which the map

Theorem 11.4.1

D--
__ (I(r,~),

['1(', r,

0,

'1(t,~)(-,r,~)])

(11.4.1)

is continuous. Moreover, the Jacobian matrix function '1(t,~)(" r, ~) is the unique x 9l n (n + 1), solution to the linear Cauchy problem in I(r,~)

~) = fx(t, '1(t, r, ~)) and In denotes the n x n identity matrix. is the maximal interval over which '1(-, r, ~) is defined.)

where A(t, r, (I(r,~)

Proof. The existence of a general solution (D, '1) to (D, f) follows from the theory of Chapters 9 and 10 since [f, fxJ is continuous. Using Corollary 11.3.2 and the assumed continuity, the reader can easily check that for each

213

11.4 Differential Dependence and Related Variational Equations

fixed (r,~) E D the matrix function A(-, T,~) is continuous on leT, ~). The continuity of the matrix coefficient ensures that the linear Cauchy problem (11.4.2) has a unique solution. Completion of the proof is now just a matter of showing that this solution satisfies the conclusion of Theorem 11.4.1. To show that (11.4.1) defines a map into <1>(Bl X Bln(n + 2», all that remains to be verified is that J is the derivative that is claimed to exist. For notational convenience denote (T, ~) by y and define L\( h) t,

I'/(t, Y + h) - I'/(t, y) - J(t, y)h Ihl '

=

(11.4.3)

for all h =1= 0 in some appropriately small ball centered about the origin in Bl x Bln. The problem is to show that 1L\(t, h)l- 0 as Ihl- 0 for each fixed t E ley). Applying the equation I'/t = f(t, ",) and (11.4.2), differentiate (11.4.3) to get ~ A( h)

ot

IJI

t,

= f(t, I'/(t, Y + h)) - f(t, I'/(t, y)) - A(t, y)J(t, y)h (11 44) ..

Ihl

and then by the fundamental theorem of calculus, the chain rule, etc., rewrite this equation as :t L\(t, h)

=

L l

fit, O"I'/(t, y

+ h) + (1 -

I'/ (t, Y + h) - I'/(t, Y)J

.[ .

Ihl

-

O")I'/(t, y)) do

h A(t,y)J(t,y)lhr"

(11.4.5)

Applying (11.4.3) to substitute out the difference quotient appearing in (11.4.5) and recalling the definition of A(t, y) reduce Eq. (11.4.5) to the form j)

ot L\(t, h) = B(t, h)L\ + aCt, h),

(11.4.6)

where

fo

l

B(t, h) =

fx(t, O"I'/(t, y

+

h)

+ (1

- O")I'/(t, y») da, h

aCt, h) = [B(t, h) - fx(t, I'/(t, y))]J(t, y) pt1.

(11.4.7) (11.4.8)

Integration and estimation of (11.4.6) give

IMt, h)1 s (1.(t, h) +

If

P(O", h) IL\(O", h) I dO"l ,

(11.4.9)

214

11.

The General Solution-Dependence of Solutions on Parameters

where a(t, h)

=

!A(-r, h)! + If! a(O", h)! dO"/'

(11.4.10) (11.4.11)

(3(0", h) = !B(O", h)[.

By the continuity of j~ it follows from (1 zA.7) and (11.4.8) that la(O", hl0, as Ihl- 0, uniformly in 0", and hence the integral in (11.4.10) approaches zero as Ihl-O. It will now be argued that the other term IMr, h)!- 0 likewise as Ihl- 0: . Denoting h as (Jr, JO E 9l x 9l", from (11.4.2) and (11.4.3) compute A( h) _ l1(r, y Ll't",

+ h) -

-

'1(r, y) - J(r, y)h

Ihl

- ['1(r + Jr,y + h) - '1(r,y + h)]

f(r, ~)Jr

Ihl

f(r, ~) Jr - J~Ht

=

f(O", '1(0", Y

Ihl

+ h)) dO"

(11.4.12)

By applying the mean-value theorem to each of the coordinate functions in the integral occurring in (11.4.12), it follows readily that IA(r, h)!- 0 as Ihl- O. This shows that a(t,h)-O

as

Ihl-O.

(11.4.13)

By Gronwall's theorem it follows from (11.4.10) and (11.4.13) and continuity that IA(t, h)l- 0 as Ihl- O. This proves the existence of the derivative of '1(t, r, ~) relative to (r, By the continuity of [f, fx] it follows that A(t, r, ~) is continuous relative to t, and since J(t, r,~) = '1(r.~)(t, r, ~) satisfies (11.4.2) it is apparent that '1(t.~)(t, r, ~) is continuous in t. Hence (11.4.1) defines a map into «I>(9l x 9l"(" + 2». The final task is to argue that this map is continuous. Remark 11.3.3 points out circumstances under which the general solution to a Cauchy problem with parameters entering the differential equation is in turn continuously dependent on those parameters (relative to the topology on «1». This argument applies to the Cauchy problem (fj, ]) defined by

n

](t, r, ~, x, y) =

f(t, x) A(t, r, ~)Yo A(t, r, ~)Yl

(11.4.14)

,

A(t, r, ~)y"

fj = {(t,r,~,x,y)l(t,r,~)E.@,

(t,x)ED,

YiE9l"+l}.

(11.4.15)

215

11.4 Differential Dependence and Related Variational Equations

Certainly B is open since it is the intersection of a finite number of open sets, and J' is continuous since both f and A are continuous. Thus according to Remark 11.3.3, the general solution fj(., z, 5, ?, E ) to (b,f)has parameters (z, <) entering it continuously (relative to the topology on @(a x Wn("+l)). Then letting ? = z and

where ei is the ith column of Z,(i = 1,2, . . ., n), the map (1 1.4.1) is obtained

as the composition of continuous maps, and the proof is complete.

Corollary 11.4.2 A Cauchy problem (0, f ) with [ f , fx] continuous has a general solution (D, q ) for which the map 9

+

gn(n+z)I(t, z, 5 )

+

[ ~ ( 7, t , O,q(r,d)(t,z,

t)]

(11.4.17)

is continuous. Moreover, the matrices of partial derivatives qr and qr:are unique solutions to the respective linear Cauchy problems with parameters (t, t) E D :

a

-q r =

at

qrlt=r

=

f.x(t7~ ( t7,, t ) ) q z ,

-f(z, t),

t E I(z3 t),

(1 1.4.18)

( I 1.4.19) qclt=r = In.

Proof. Corollary 11.4.2 is just a summary of results already proved.

Remark 11.4.1 Suppose ( Z a , x(., a)) is a one-parameter family of solution curves to a differential equation 2 = f(t, x).

(11.4.20)

By formally differentiating (11.4.20)relative to the parameter u and reversing the order of differentiation, one sees that the derivative x,(t, a ) satisfies the linear differential equation

3 = A(t, a)y,

(11.4.21)

where A(t, a) = f,(t, x(t, a)). A linear equation obtained from a nonlinear one in this manner is called a variational equation of (11.4.20). (Examples

216

11. The General Solution-Dependence of Solutions on Parameters

f directly, then

appeared in Theorem 11.4.1.) If the parameters enter differentiation of the differential equation

x=

(11.4.22)

f(t, a, x)

produces the nonhomogeneous linear variational equation

y=

A(t, a)y

+

get, a)

(11.4.23)

in which A(t, a) = fit, a, x(t, a) and get, a) = f2(t, a, x(t, a)). As might be expected, variational equations are involved in important aspects of the theory of nonlinear equations, and results of the type stated in Theorem 11.4.1 and Corollary 11.4.2 are fundamental in such studies. Remark 11.4.2 Special (linear) differential equations (D, f) of the type in which D = (a, /3) x f!An and f(t, x) = A(t)x + get) with A(') continuous

(or at least locally integrable) were discussed in earlier chapters. For such equations '7it, T, e) = (t, T), the fundamental matrix for which <1>( T, T) = In. The homogeneous equation is distinguished by the fact that it is its own variational equation. Corollary 11.4.3 The general solution (D, '7) to a Cauchy problem (D, f) satisfies the equations I(t, '7(t, T,

en =

'7(t, a, '7(a, T, ~n (t, a)

E

I( T, ~) x I( T, ~),

(t,

'7(t, r, c;) '7it, T,~)

T, ~) E

=

e~

= - '7~(t,

I( T, ~),

(t,

T, ~) E

E0,

(11.4.25)

= '7(t, T, e),

E0. Moreover,

f '7~(t, T,

(11.4.24)

if [f,

a, e)f(a,

fxJ is continuous, then

e) da,

e)f( T, ¢),

(t,

T,

(11.4.26)

0

E

E0.

(11.4.27)

Proof. Equation (11.4.24) follows directly from the uniqueness of maximal solution curves through each point (r, e) E D. Equation (11.4.25) also follows from uniqueness since both sides satisfy the same differential equation and are equal at a = t; hence the result is ultimately based on Corollary 11.4.2. Equation (11.4.27) results from applying (11.4.18) and (11.4.19) to show that the right-hand side of (11.4.27) is the solution to (11.4.18). The relation (11.4.26) follows directly from integrating (11.4.27) relative to T and from the equation '7 (T, T, ¢) = This concludes the proof.

e.

Higher-Order Differentiability. Theorems 11.3.1 and 11.4.1 and Corollaries 11.3.2, 11.3.3, 11.4.2, and 11.4.3 give a precise account of the sense in which and extent to which continuity and differentiability properties of (D, f) are reflected in those of (D, '7). We shall now show Remark 11.4.3

217

11.4 Differential Dependence and Related Variational Equations

that the results are sufficiently comprehensive to cover higher-order differentiability with no real additional work Observe what happens when, using the embedding technique introduced in Remark 11.3.3, one passes from (D, f ) to (8,f),using the notation and definitions

(11.4.28)

(11.4.30)

in which A(t, z, C;) = ,f,(t, q(t, z, 5)). Observe that application of Theorem 11.4.1 to (B, $) requires continuity of [f,f?] and hence of [f,f,,f,,] and nothing more. Thus, on recovering [?,(., z, <), q&-, z, <)] from (6,f))as the last n(n + 1) coordinate functions of f ) ( * , 5, [,z, 5 ) evaluated at 5 = z and

(11.4.31)

it is apparent from Theorem 11.4.1, the chain rule, and Lemma 11.2.2 that

D +@(a x ~ n 3 ) 1 ( ~5 ,)

+

5)

(11.4.32)

qrr(t,z, 5 )

(11.4.33)

qcc(.,z,

and

9

+

~3

~ ( tz,, 5)

+

are continuous. Note that because of the occurrence of the term f(z, C;) in (11.4.31), the same argument fails to produce continuity of the maps asso-

218

11.

The General Solution-Dependence of Solutions on Parameters

ciated with '1~T and '1« (at the point where the chain rule is to be applied) unless one makes the additional assumption that Ilt, x) exists and is continuous. Thus observe that to ensure that the degree of differentiability of '1 relative to t or l' is raised by one unit, the degrees of differentiability of I relative to t and x must each be raised by one level. On the other hand, to raise the degree of differentiability of '1 relative to ~ by one unit, it is sufficient to increase the differentiability of I relative to x by one level. Induction thereby leads to the following results. Corollary 11.4.4 Let iX, /3, and y be any fixed integers in {a, 1, 2, ... }. A general solution (D, '1) to a Cauchy problem (D, f) has continuous maps

°SiS

iX,

D - <J)(~

x

D-

2l

g.ln(n+

Os j S

/3,

Oi+ j+k

g.ln(n +

2»j(1', ~) - at; 01'j O~k '1(., 1', ~),

(11.4.34)

ai+j+k

l(t, 1', ~) - oti 01'j a~k '1(t, 1', e), o s k S y,

(11.4.35)

if the derivatives

Oi+j

°

in which S j S

°+ /3 + SiS

iX

at; iX

+ /3 - 1 when

ox j l ,

iX

+ /3

~ 1, and i =

y, exist and are continuous.

°

when

(11.4.36) iX

+ /3

= 0,

AUTONOMOUS EQUATIONS

The differential equation x = I(x) in which I is independent of t on g.l x E, E an open subset of g.ln, is called autonomous, as is the associated Cauchy problem (D, f).

D =

Corollary 11.4.5 II the Cauchy problem (D, f) is autonomous and has a general solution (D, 1'/), then

= =

w±(1',~)

I'/(t, 1',~)

+ 1',

(11.4.37)

I'/(t - 1',0,

(11.4.38)

w±(~)

in which are defined w±(e) = w±(O,~) w+(~), ~ E E. Furthermore, if [I, Ix] is continuous, then

(t, 1',~)

lor

E~,

w_(~)

and I'/(t,0

< t <

I'/(t,~)

=

~

'1t(t,~)

= I'/~(t,

+

f~ I'/~(U, ~)/(e),

e) do

I(~),

= I'/(t, 0, e)

(11.4.39) (11.4.40)

219

11.4 Differential Dependence and Related Variational Equations

f: '1~(u, for some ~ '1(t + p,~)

E

=

E and 0 < P E (w_(~), '1(t, 0 for all t E f1l.

0 do w+(~))

f(~)

=

(11.4.41)

0

if and only if w±(~)

=

± 00

and

Proof. The proofs are easy applications of the previous results and are left for the reader.

ANALYTIC PROPERTIES

Most of the results of Chapters 10 and 11carryover with minor changes to differential equations on open subsets D c fYt x ~n. New questions arise when the differential equations are specialized to those having real or complex convergent power series. Here we shall only touch on the topic. A reference such as [14J should be consulted by those readers interested in extensive use of power series techniques. Call a scalar-valued function g, defined on an open subset Q c ~n, analytic if it is representable by an absolutely convergent power series about each point in Q. (Note that if a function is representable about each point in an open subset of fYtn by such a power series, then that power series extends it to an open subset of ~n.) An m vector-valued function is said to be analytic on Q if each of its coordinate functions is analytic on Q. An analytic differential equation will always have a general solution since the vector field will be locally Lipschitz. Theorem 11.4.6 Consider the Cauchy problem (D, f) in which D is a nonempty open subset of fYt x ~n. If f(t, z) is analytic in z as well as continuous on D, then there exists a general solution (D, '1) to the Cauchy problem with '1(t, L, ~) analytic relative to ~ as well as continuous on its domain f0. If f(t, z) is analytic on D, then '1 is analytic on f0. Remark 11.4.4 The embedding technique discussed earlier in this chapter can be applied to show that '1 will be analytic in any parameter on which f is analytically dependent. A proof of Theorem 11.4.6 can be based on the uniform convergence of the approximating sequence that appears in the proof of Theorem 9.2.4 and the analyticity of those approximations (see [14J for details).

220

11.

The General Solution-Dependence of Solutions on Parameters

AN APPLICATION

The topological structure imposed on tD/D) was found useful in providing a unified geometric view of the theory of dependence of solutions on parameters. This chapter concludes by showing that tDfeD) admits a richer geometric structure that is consistent with that topology-namely, the structure of differential manifold. Theorem 11.4.7 For D c f!ll x f!ll" open and nonempty with (D, f) continuous, tDf(D) is an n-dimensional topological manifold. If f is, moreover, r-times continuously differentiable relative to xfor(t, x) E D, then tDfeD) is an orientable C -manifold (r = 1, 2, ...).

Proof. Assume that (D, f) is continuous and select arbitrary (l0, t/J0) tD/D) and T E 1°. As a chart consider (1jJ, V) where V

= tDiD) n U

N~~'q,°l,

E

(11.4.42)

<>0

1jJ(l,t/J) = t/J(T)

for

(l,t/J)E V.

(11.4.43)

Since (I", t/J0) E tDf(D), the collection of all (1jJ, V) defined by (11.4.42) and (11.4.43) constitutes an open cover of tDfeD). The task now is to show that IjJ is a homeomorphism of V onto an open subset of f!ll". By the definition (11.2.1) of basic open sets, (11.4.42), and (11.4.43), the range of IjJ is g I(T, ~) E D}, which is noted to be open in f!ll" since D is open in f!ll x f!ll". IjJ is one-to-one by the definition of solution curve and by the assumed uniqueness of solutions to (10.1.1) and (10.1.2). Since ljJ-l(~) = (l(T, ~), 11(', T, ~)), its continuity follows from Theorem 11.3.1. Finally, IjJ is continuous since from (11.4.42) and (11.4.43) we observe that ljJ-l carries open sets into open sets. This proves that the collection of charts of the type (11.4.42), (11.4.43) determines a topological manifold structure on tDfeD) that is compatible with the original topology imposed on tDf(D). To prove the last part of Theorem 11.4.7, assume that f is r-times continuously differentiable relative to x on D, r E {I, 2, ... } fixed. As the candidate C" n-subatlas on tD /D), consider the collection of charts of the type defined in (11.4.42) and (11.4.43).To show that the collection is C-related, select any pair (ljJi' VJ (i = 1,2) with Vi n V 2 nonempty, (11.4.44) all

(l, t/J) E Vi'

(11.4.45)

where Ti E I, (i = 1,2). Since the choice of charts in the proposed collection was arbitrary, it is sufficient to prove that IjJ 20 IjJ "1 1(-) is C where defined,

221

11.4 Differential Dependence and Related Variational Equations

namely, at each

~ E

1/11(U 1 n U 2). But 1/11 1 = (1(T 1, ~), '1(',

m

T1, ~)),

(11.4.46)

0

(11.4.47)

and consequently by (11.4.45),

1/12

01/11

1

(0 = '1(T2' T1 ,

n U 2). Since (T2,Tl,~)E~, the required continuous differfor ~EI/11(Ul entiability follows directly from (11.4.47) and from (11.4.35) of Corollary 11.4.4. Moreover, since '1~ satisfies (11.4.19), which is linear, it follows that

det'1~(T2,T1>~)

= exp

(f2 fX(U''1(U,T1>~))dU)

>

tr

° (11.4.48)

and the orientability now is an immediate consequence of(I1.4.47), (11.4.48). This concludes the proof of Theorem 11.4.7.

EXERCISES

In the following exercises, unless otherwise specified, let (D, '1) denote the general solution to the Cauchy problem (D, f) having [f, fx] continuous at all (t, x) E D.

1.

Derive the formulas tr fx(u, ni«, T, ~») dos, (a) det '1~(t, T,~) = exp(J~ (b) '1~(t, T, ~)-1 = '13(T, t, '1(t, T, ~)), all (t, T,~) E ~. 2. (a) Show that '1(t, T, ~) is a solution to the problem '1ft

+ '1f~'1Z

l'1t = 0,

'1(T, T,

0 =

~,

(t, T,

0

E

~.

(b) In the case in which (D, f) is autonomous, let '1(t,~) = '1(t, 0, ~) and w±(~) = w±(O, ~), (T,~) E D = [)l x E. Show that the equations of part (a) then reduce to tl« -

'1f~'1

~-1 '1f

=

°,

3. Fix any (T, ~)'E (a)

D. Derive the equation

z(t) - '1(t,

T,~)

=

f

'1z(t, a, z(u»

[d~~)

- feu, Z(U»] da,

t E I( T, ~), for any C 1 curve z: I( T, ~) -+ [)In, satisfying both (1) Z(T) = ~ and (2) (t, a, z(u») E~, all a in the interval with ends T, t. (Hint: (11.4.27), replacing (T,~) by (e, z(u»).)

Apply

222

11. The General Solution-Dependence of Solutions on Parameters

(b) Verify that the derived equation of (a) reduces to (11.4.26) with the special choice of z being z(t) = ~, all t E I(r, ~). (c) Let (D, z) be the general solution to the "perturbed" Cauchy problem (D, f + h). Apply the derived formula of(a) to obtain the Alekseev"variationof-parameters formula" z(t,

r,~)

= rt(t, r,~)

+

f

rtzCt, (J, z((J, r,

~))h( J,

z((J, r,

~))

da.

For which hand (t, r, 0 is it valid? 4. Show that if D = (to, tr) x PJtn and f satisfies

fit, x) + f':(t,

x) :s:; yIn

for some y E PJt, all (r, x) E D, then w+(r,~) = t 1 for each (r,~) ED. Further such that prove that there exists abE PJt (generally (r, ~)-dependent) Irt~(t, r, ~)I :s:; b for r :s:; t < t 1 • (Hint: Derive the inequality

Irt~(t,

r,

~W

s

1

+y

f

Irtl(J, r,

~W

da,

and apply (11.4.26) and Corollary 10.3.2.) 5. As a consequence of Exercise 4, show that if II, I is bounded on PJt x then w±(r,~) = ± 00 for all (r, ~). 6. Prove that if(D, f) satisfies the hypothesis of Exercise 4 and if [h, hxJ is continuous with Ihxl bounded on D, then (D, f + h) will likewise have w+(r,~) = t 1 for all (r,~) in D. 7. An autonomous Cauchy problem (D, f) has solution curves given by the formula

ee:

rt(t,~)

= (1-

t~d

1

+ t2~~

[~1

- t(~i~2

+ ~DJ

n

(~1'~2)EPJt.

2

Determine f and the general solution rt(t, r, Compute the 2 x 2 matrix function A(t, r, ~) that appears as the coefficient in the variational equations (11.4.18) and (11.4.19).

Chapter

12

Limit Properties of Solutions

Although the results of Chapters 9-11 have the virtue of applying to a broad class of equations, by themselves they generally do not provide the detailed description of solution behavior needed in applications. By building upon those foundations the material covered in this chapter leads to a more complete structure.

12.1

Limit Sets and Invariance

We shall restrict attention to those autonomous Cauchy problems (D, f) that have a general solution (D, ,,). That is, D = f1l x E, E an open subset of f1l etc. (See Corollary 11.4.5 for a reminder about the notation.) Hence ,,(t, ~) is uniquely determined by the equations ft

,

~ ,,(O,~)

= 1(,,),

(12.1.1)

= ~

(12.1.2)

for ~ E E, t E «(JL(~), w+(~». The definition of a general solution given in Chapter 11 assumes local existence and uniqueness of solutions to the initial-value problem (12.1.1), (12.1.2). As was shown in Chapter 9, a sufficient condition for ensuring the existence of a general solution in the autonomous case is that I be locally Lipschitz on E. Throughout this chapter I is understood to be continuous onE. For (E E define (12.1.3) 223

224

the union over 0

12. Limit Properties of Solutions

~

t < w+W. Similarly,

(12.1.4)

the union over w-W < t ~ O. The sets y+W, negative semiorbits of ~, respectively, and

y-:-(~)

are called positive and (12.1.5)

is called simply the orbit of ~. The notation is shortened to y when there is no need to distinguish a particular point on the orbit and to y+(y-) when any respective semiorbit of y will do. A subset SeE is called positively (negatively) invariant if it contains y+(~)(y-W) for each ~ E S. S is called invariant ifit contains y(~) for each ~ E S. The general solution Yf to an autonomous Cauchy problem satisfies the equation

Remark 12.1.1

Yf(t, Yf((J,

~))

= Yf(t + (J,

~)

(12.1.6)

when both (J and t + (J are in (w_(~), w+(~)). This relation, sometimes called the semigroup property, is a direct consequence of Corollary 11.4.3. Note that the semigroup property substantiates Y+W, y-W, and y(~) as examples of positively invariant, negatively invariant, and invariant sets, respectively. Further observe that a point ~ E E at which fW = 0, called a rest point, equilibrium point, or critical point, is its own orbit. More generally, a point ~ E E is called periodic if Yf(', ~) is periodic on (- 00, (0). Then y(~) is called a periodic orbit and is invariant. The positive limit set n+(y) of an orbit y is the set of points in E of the type lim, .... co Yf(t k , ~), some ~ E y, some t k - 00 as k - 00. The negative limit set n-(y) is similarly defined as those points in E of the type lim, .... co Yf(t k , ~), some ~ E y, some tk - - 00 as k - 00. The limit set n(y) consists ofthe points in E that are limits of Yf(tb ~) for some ~ E y, where either tk - 00 as k - 00 or tk - - 00 as k:+ 00. Remark 12.1.2 Some authors, at the outset, restrict the notions of orbit, limit set, and invariance to those differential equations having Iw±WI = 00 for all ~ E E. This condition can always be achieved by a reparametrization of the solution curves; for example, by replacing f(x) with [1 + If(xWJ- 1f(x). The approach taken in this book avoids that restriction and seems more natural. Thus the differential equations considered can have an orbit y(~) with w-W > - 00 or w+W < 00, but of course it then follows from Corollary 10.3.2 that n-(y) = 0 or n+(y) = 0, respectively.

12.1

225

Limit Sets and Invariance

Proposition 12.1.1 n+(y) =

ny+W,

(12.1.7)

~EY

n-(y)

=

n

(12.1.8)

y-W,

~EY

(12.1.9) (The overbars in (12.1.7) and (12.1.8) denote closure relative to E.) Proof. To prove (12.1.7), consider arbitrary x E n+(y). Thus x = limk--+oo ,,(tb '0) for some '0 E Y and t k - - 00 as k -- 00. Select arbitrary, E y. Then '0 = "(0', e) for some 0' E (w-(e), w+(e)), and according to (12.1.6),

(12.1.10) with t k + 0' E [0, w+W) for all large k since 00 = w+('o) = w+W - 0' and tk - - 00 as k -- 00. This shows that x E Y+W, and since E y was arbitrary, the conclusion reached is that

e

n+(y) c

n y+W.

(12.1.11)

~EY

To establish the reverse inclusion, suppose that some x E E satisfies x E y + (e) for all eE y. Thus for some fixed eE y, ,,(tk , e) -- x as k -- 00 for tk E [0, w+W)· Ify is periodic, say with period p > 0, then ,,(Tk, e) = ,,(tb ,), where T k = t k + kp -- 00 as k -- 00; this says that x E n+(y). The same conclusion follows trivially if y is a rest point. Suppose, on the other hand, that 't is not periodic. If tk can be selected so that tk - - w+(e) as k -- 00, then clearly w+W = 00 and x E n+(y), as required. Now consider the complementary case which would imply that for some e > 0 and {} > 0, I,,(t, ,) - xl > e for all t E [{}, w+W). In this situation, by first extracting a convergent subsequence, it could be assumed that tk - - too as k -- 00 for some too E [0, (}). Then x = limk--o oo ,,(tk , , ) = ,,(too, ,) E y. But f(x) i= 0 since y is not periodic, and consequently x ¢ y+(,,({}, ')). The preceding argument shows that x ¢ y+("({}, ')), the desired contradiction of the assumption that x belongs to the closure of every positive semiorbit of y. This completes the proof of (12.1.7). The remainder of the proof of Proposition 12.1.1 proceeds similarly and therefore is omitted. Amongst other uses, the next lemma provides a precise statement of the sense in which the general solution will be shown to carry semiorbits to limit sets.

226

12. Limit Properties of Solutions

Lemma 12.1.2 Let (M, d) be a metric space. (Recall Exercise 2 following Section 11.3.) g is taken to be the collection of nonempty compact subsets 5 of M. Denote the open ball in M of radius e and center m e M by B'tm). Then g together with the formula c5(5 1 , 52)

= inf{£.IB£(sd n

52 #

0, B£(S2) n

51 #

0, s, E 5 i (i = 1,2)} (12.1.12)

for (5 1,52) E g x g defines a metric space (g, c5). (The distance function c5 is called the Hausdorff metric.) Proof. The function c5 is real-valued since compact subsets of a metric space are bounded. Suppose that c5(5 1 , 52) = O. Fix SI E 51 and let 0 < £.k - 0 as k - 00. There exists a sequence ~ E B£k(SI) n 52 and ~ - SI as k - 00. But 52 is compact and hence closed, which puts lim, .... co s~ E 52' Therefore SI E 52' and since SI E 51 was arbitrary, 51 c 52' By symmetry 52 c 51 and thus 51 = 52' Conversely, if 51 = 52 = 5 E g, then B£(s) n 5 # 0, all s E 5, all £. > 0, which says that c5(5 1 , 52) = o. To continue the proof, define d(m, 5)

= inf{d(m, s) Is E 5}

(12.1.13)

for mE M, 5 c M. As a first step in showing that c5 satisfies the triangle inequality, it will be shown that c5(5 1, 52) = J(5 1 , 52) where 8(5 1,52)

= max{max des!> 52), max d(s2' 51)} 81

$2

(12.1.14)

in which s, E S, E g (i = 1, 2). By compactness there is an s! E 52 with d(SI' s!) = d(SI' 52) ~ max" d(SI' 52) ~ $(5 1, 52), which implies that BP(sdn5 2 # 0,all SIE51' where p =3(5 1,52), By symmetry BP(s2)n5 1 # 0, all S2E 52' which proves that c5(5 1 , 52) ~ J(5 1 , 52)' Clearly the reverse inequality likewise holds and the proof reduces to showing that 3 satisfies the triangle inequality. The inequalities (12.1.15) imply that

d(SI' 52) ~ d(SI' S3)

+ d(s3' 52)

~

d(sl' S3)

+ max '3

d(s3' 52),

(12.1.16)

and consequently max d(SI' 52) ~ max d(SI' 53) $1

for

Si E S, E

81

+ max d(s3' 52)

(12.1.17)

+ max

(12.1.18)

83

g (i = 1, 2, 3). By symmetry,

max d(s2' 5d ~ max d(s2' 53) b

82

83

d(S3' 5d·

12.1

227

Limit Sets and Invariance

As a direct consequence of (12.1.15) and (12.1.17), (12.1.18) it follows that S(SI, Sz) ~ max{max d(sl' S3), max d(s3' SI)} 81

83

+ max{max d(S3' Sz), max..d(sz, S3)} 83

=

82

J(Sl' S3) + J(S3, Sz)

(12.1.19)

and Lemma 12.1.2is proved.

Remark 12.1.3 It can be shown that if (M, d) is a complete metric space, then the metric space (i/, b) of compact subsets is complete. If (M, d) is compact, then (i/, b) is likewise compact. Theorem 12.1.3 The sets n-(y), n+(y), n(y) are invariant and closed(relative to E). If y-, y+, or y has compact closure in E, then, respectively, n-(y), n+(y), or n(y) is nonempty, compact, and connected. Compactness of y+(e) implies that 17(t, y+W) - n+(y) in the sense of the Hausdorff metric as t - 00. Likewise 17(t, y-W) - n-(y) as t - - 00 if y-W is compact in E. Proof. The sets n-(y), n+(y), and n(y) are closed since, according to Proposition 12.1.1, they are the intersections of closed sets. In proving the invariance ofn+(y) there is nothing to do ifn+(y) = 0. Now suppose there is an element x E n+(y). Then by the definition of positive limit set,

x = lim 17(tk> e) k .... co

e

(12.1.20)

for some E y, tk E [0, w+W) with t k - W+W as k - 00. By openness of domain, continuity, and the semigroup property it follows from (12.1.20) that for each t E (w_(x), w+(x)), 17(t, x) = lim 17(t, 17(tk> e)) = lim 17(t + tk , e)· k-oo

k-oo

(12.1.21)

Since x E E, (12.1.20) along with Corollary 10.3.2 ensure that W+W = 00 and consequently t + tk - W+W as k - 00 in (12.1.21). This proves that 17(t, x) E n+(y), and the invariance of n+(y) is established. The similar proof of the invariance of n-(y), and hence of n(y), is omitted. Now assume that y+(e) is a compact subset of E. Obviously n+(y) is then not empty, and n+(y) is compact since it is a closed subset of y+W. Since n+(y) is compact, it follows readily that for any open set S (of E) containing n+(y), 17(t, e)E S for all large t e [0, (0). The conclusion that 17(t, y+(e)) c S for all large t now follows from (12.1.6). In particular, since n+(y) is compact, for arbitrary £ > 0 the open set S can be taken to be the union of a finite number of open balls of radius £ centered on points in n+(y). But 17(t, y+(~))

228

12. Limit Properties of Solutions

is compact since it is the continuous image of a compact set, and consequently it too will be contained in the union of the finite collection of e-balls for all large t. Then the relation B£(s) n g+(y) '1= 0, SE 11(t, y+(~» obviously holds for all large t, and moreover, B£(s) n 11(t, y+(~» '1= 0, s E g+(y), for otherwise the defining property of g+(y) would be violated. Since £ was arbitrary, we con-- g+(y) relative to the Hausdorff metric as t -- 00. clude that 11(t, y+(~» The proof that 11(t, y-W) -- g-(y) as t -- - 00 when Y-W is compact is similar and therefore is omitted. Recall that a set is connected if it is not the union of two nonempty disjoint open sets. Again consider the situation in which y+(~) is a compact subset of E. To reach a contradiction, suppose that g+(y) = Sl U S2 with S 1, S2 nonempty open subsets of E with S 1 n S2 = 0. From the first part of the proof, 11(t, y+(~» c Sl U S2 for all large i. But 11(t, y+(~)) is a connected set and hence must lie wholly within one of Sl or S2' This is impossible since both Sl' S2 are nonempty and consist of points of the type lim k -+ oo 11(tk , ~), tk E [0, w+W), t k - - W+W as k -- 00. Thus the proof that g+(y) is connected is complete. The proofs of the corresponding statements concerning g-(y) and g(y) proceed in a similar fashion and thus are omitted. The next theorem takes a Lyapunov approach to deriving an upper estimate of the sets g+(y). The examples following its proof demonstrate how the theorem can be applied to obtain detailed information about the limiting behavior of solutions to some nonlinear differential equations.

Theorem 12.1.4 Let f: E--~" and v: E-fYi be once continuously differentiable on a nonempty open subset E of fYi". For fixed c E fYi define V = {XIXEE,v(x) < c},

(12.1.22)

= {x IXE V, f(x)'vx(x) = O}, Z* = {the largest invariant subset of Z for (D Z

Assume that V is bounded and f(x)'vix) positive-invariantfor (D; f), W+W = 00, c

g+(y(~»

and 11(t, y+W) - g+(yW) as t -

00

~

°

= E x fYi,

fn.

(12.1.24)

throughout that set. Then V is

Z1 c Z,

for each ~

(12.1.23)

E

(12.1.25)

V.

Proof. Assume the hypothesis. To prove the positive invariance, consider arbitrary ~ E V. Then the unique solution (D, 11) to (D, f) has 11(t, ~) E V on some open interval containing t = since V is open. On that interval,

°

(12.1.26)

12.1

229

Limit Sets and Invariance

and hence v(I'/(t, ~» is monotonically nonincreasing relative to increasing t and v(I'/(t, ~» v(~) < c. Suppose that v(I'/(t, ~» < c were violated at some t 1 E [0, w+(~». Let t* be the infimum of those t E (0, w+(~» at which v(I'/(t, W ~ c. By continuity it would follow that v(I'/(t*,~)) = c. Since v(I'/(t, ~» E V for 0 ::;; t < t*, v(I'/(t, ~» is nonincreasing on that interval; hence v(I'/(t, ~» v(~) < c on 0::;; t < t*, which contradicts the continuity of v(I'/(t, ~». This establishes the positive invariance of V. Moreover, it shows that y+<e) c V, and by Corollary 10.3.2 we conclude that w+<e) = 00. Since V is bounded, y+<e) must be compact in E and Theorem 12.1.3 applies. The next task is to show that the now compact and nonempty n+(y(~» c z. By the boundedness of V, its positive invariance, and the continuity of v(·), the monotone nonincreasing v(I'/(t, ~» is bounded below and hence has a lim, 00 v(I'/(t, = Co < c. By continuity of v(·) it follows that vex) = Co for all x E n+(y(~». But according to Theorem 12.1.3, n+(y(~» is invariant, and thus

:;

:;

m

V(I'/(t, x)

= f(I'/(t, x»· v~(I'/(t,

x)

=0

(12.1.27)

which says that n+(y(e» c Z. However, since n+(y<e) is for all x E n+(y(~)), invariant, n+(y(~» c Z* as well.The remaining conclusion that I'/(t, y+(~» n+(y(~» is a direct consequence of Theorem 12.1.3, and the proof of Theorem 12.1.4 is now complete. The function f(x)'vix) that appears in (12.1.23) is sometimes called a Lyapunov derivative or directional derivative ofv in the direction f. Its sign at a point x E E gives some indication of the direction in which the solution curve is moving across the level set of v through x. Theorem 12.1.4 clarifies the solution behavior in certain regions that contain points at which the Lyapunov derivative is zero. The significance of Theorem 12.1.4 is that it provides a useful device for in applications. (This will be brought out in the helping to locate n+(y(~» example following this remark.) It should be noted that the proof of the as being inside Z* but also actually shows theorem not only locates n+(y(~» that n+(y<e) lies inside some level set of v. The theorem imposes no requirement on the sign of u; however, any v satisfying the hypothesis could be converted into one that is positive by adding an appropriately large constant to v. The literature sometimes imposes the condition that vex) - 00 as x - 8E. That condition would ensure that V is bounded, but it is not necessary. Remark 12.1.4

Example 12.1.1

The pair of coupled nonlinear second-order scalar

equations ii

+ (1 - v2 ) U + U =

0,

jj

+ (1 - u2 ) v +

V =

0

(12.1.28)

230

12. limit Properties of Solutions

has the first-order form in [)f4 (12.1.29)

X4 = -X 3 + (xi - 1)x 4.

X3 = X4'

The function v(x) = t[xi + x~ + x~ the direction of f determined by

+ xi]

has the Lyapunov derivative in

+ x 2[ -Xl + (x~ - 1)x 2] + X3X4 + X4[ -X3 + (xi - 1)x4] - [(1 - x~)x~ + (1 - xi )xi]

f(x)'vx(x) = XlX2

::;;0

(12.1.30)

on the set

v

I

= {x v(x)

I + x~ + x~ + xi < I}.

= {x xi

(12.1.31)

By noting that 1 - x~ > 0 and 1 - xi > 0 for x E V, it is apparent from

(12.1.30) that

Z = {x Ixi

+ x~ <

1, X2 = X4 = O}.

(12.1.32)

Further examination of the f(x) determined by (12.1.29), at the points in Z, reveals that a single point.

Z* = {OE[)f4},

(12.1.33)

Application of Theorem 12.1.4 provides the conclusions that n+(y(~)) = of each solution x(-) = 11(', ~) of (12.1.29) initiating at ~ E V and that Ix(t)!- 0 as t - 00 for each such solution.

oE [)f4 for the orbit y(~)

Example 12.1.2

The scalar equation it

+ u + 4(u 2

-

1)u = 0

(12.1.34)

has the first-order form (12.1.35)

The energy function v(x)

= ~x~

+

f:'

4(0'2 - 1)0' do

=

!x~ + (xi -

1)2

(12.1.36)

has Lyapunov derivative relative to (12.1.35) f(x)'vx(x) = X2[ -4(xi -1)x l

-

X2]

+ 4x l(xi -

1)X2 = -x~::;;

0 (12.1.37)

at all points x E [)f2. Clearly, the set V = {x Ix E [)f2, v(x) < c} is bounded and can be made to include any bounded subset of [)f2 by selecting c appro-

231

12.2 Stability of Nonlinear Equations

priately large. For small c the set V consists of two disjoint topological open disks about ( -1,0) and (1, 0), respectively, which expand and merge into one such disk as c is increased. Thus Theorem 12.1.4can be applied with Z that part of the Xl axis lying in V. Inspection of (12.1.35) on that set reveals Z* to be the three rest points ( -1, 0), (0, 0), and (1,'0). The conclusion is that each initial point in fJl2 is carried toward one of the points ( - 1, 0), (0, 0), (1, 0). The points in the disjoint level disks centered about ( -1, 0) and (1, 0) are carried to the respective centers. Although the appropriate theorem has not yet been proved, a careful analysis of the system about (0,0) reveals that all initial points are attracted to either ( - 1, 0) or (1, 0) except for a pair of orbits each with heads and tails approaching (0, 0), one circling ( - 1, 0) once and the other circling (1, 0) once. The points on these orbits along with (0, 0) itself are the only ones attracted to (0, 0). The reader should sketch the family of level curves and the orbits crossing them. EXERCISES

1. Analyze the limiting behavior of the solutions to the following equations by applying Theorem 12.1.4. (Take v to be the energy function appearing in the proof of Theorem 10.3.4.) U2)u (a) u + Ii + (1 + e= 0, U2 (b) u + (1 - e- ) Ii + u = O. 2. Examine the qualitative behavior of the system in fJl2 using vex) = !(xi + xn. 3. Investigate the limiting behavior of the solutions to the equation

u + Ii + (u 2

-

3u

+

l)u

=0

by applying Theorem 12.1.4.

12.2

Stability of Nonlinear Equations

The terminology used in the discussion of the stability of linear systems must be expanded to allow for the more complex behavior inherent in the solutions to nonlinear equations. Let j be defined and continuous on (a, (0) x E, where E is an open set containing the origin in fJl". Assume that jet, 0) = 0 for all t E (a, (0). The zero solution to

x=

jet, x)

(12.2.1)

232

12. Limit Properties of Solutions

is called Lyapunov stable if for each or E(a, (0) and f. > 0 there exists a b = b(or, e) such that each right-maximal solution ¢(t) to (12.2.1) with I¢(or) I < b has co , = 00 and satisfies 1¢(t)1 < f. for all t E [or, (0). If the zero solution is Lyapunov stable and, moreover, the b can be chosen independently of or, then the solution is said to be uniformly stable on (a, (0). If the zero solution is Lyapunov stable and, moreover, the b(or, s) appearing in that definition can be taken so that 1¢(t)l- 0 as t - 00, then the zero solution is called Lyapunov asymptotically stable. The above stability-theoretic notions extend to any solution defined on an interval of type (a, 00): For x(t) any such solution apply the definitions to the zero solution of the equation

y = ](t, y),

(12.2.2)

where ](t, y) = f(t, y + x(t» - f(t, x(t», that arises from the change of variable y = x - x(t) in (12.2.1). In passing, it is worth noting the similarities and differences between the definition of Lyapunov stability and the continuity of the map ~ -Y/(', or, ~) (with or fixed) as discussed in Chapter 11. The former involves the uniform metric topology of bounded continuous functions on [or, (0), whereas the latter works with the topology of uniform convergence on compact sets. Both are concerned with the same problem, i.e., continuity. Although the terminology had not yet been introduced at that point, the results of Chapter 5 contain the fact that the zero solution to

x=

Ax,

(12.2.3)

in which A is an n x n stability matrix, is uniformly as well as Lyapunov asymptotically stable on (- 00, (0). The next theorem proves that those properties are preserved for a general class of perturbed nonlinear equations

x = Ax + h(t, x).

(12.2.4)

Theorem 12.2.1 Let A be an n x n stability matrix and h in (12.2.4) be defined and continuous on a set D = (a, (0) x E, where E is an open subset of ~n containing the origin. Further assume that for each f. > 0 there exists a b(f.) > 0 such that Ih(t, x)1 ::;;;

f.lxl

(12.2.5)

Ixl

for < b(f.), t eta, (0). Then the zero function is a solution to (12.2.4) and is uniformly and (Lyapunov) asymptotically stable. Proof. Since A is a stability matrix, Theorem 5.2.3 ensures that Lyapunov's linear equation (5.2.19) has a symmetric, positive-definite matrix

233

12.2 Stability of Nonlinear Equations

solution P whose associated quadratic form vex) = x· Px then satisfies (12.2.6) for all x E fYt", where ex, Pare the (positive) extreme eigenvalues of P. For arbitrary f. > 0, it follows from (12.2.6) that p = p(f.) can be chosen appropriately small so that vex) < p(f.) implies that Ixl < c5(I/4I p j), Ixl < ~, and x E E. For arbitrary r E (a, 00) consider any ~ E fYt" that satisfies v(~) < P and let c/J(.) be any solution curve of (12.2.4) on a right-maximal interval [r, co ,) with c/J(r) = ~. Then by continuity v(f/J(t» < P on some open interval containing r, and hence by (5.2.19), (12.2.4), and the choice of p it follows that d

dt v(f/J(t»

= 2f/J'P[Ac/J +

h(t, c/J)J

= f/J'(ATp +PA)f/J + 2f/J'Ph(t, f/J) = -1f/J1 2 + 2f/J' Ph(t, f/J)

s

-1f/J1

2

+ 21pllc/J1 (41~1)

:$;

-1f/J12

+ 2Ipl/f/Jllh(t, f/J)I

If/JI = -

~ 1f/J1 2 s O.

(12.2.7)

By continuity it can be concluded from (12.2.7) that v(f/J(t» :$; v(~) < p for all t E [r, w+), and this inequality along with the first in (12.2.6) allows application of Corollary 10.3.2, which shows that ce, = 00. Now it can be concluded that (12.2.7) is valid for all t E [r, 00). By the same argument found in the proof of Theorem 5.2.4, it follows from (12.2.6) and (12.2.7) that

If/J(t) I s

~

(12.2.8)

e-(t-t)/2PI~I,

for t E [r, 00), vW < p. Since ex, P depend only on A and since p = p(f.) is independent of r, inequality (12.2.8) provides the subsequent inequalities

Ic/J(t) I :$; e-(t-t)/2Pf. :$; f. (12.2.9) I s c5 o(f.) where c5 o(f.) = min{c5(I/4I p l),J p(f.)/P}. This establishes

for If/J(r) I= I~ the claimed uniform and asymptotic stability since r E (a,

00)

was arbitrary.

Remark 12.2.1 The definition of Lyapunov asymptotic stability requires more than just the solutions f/J(-) with f/J(r) = ~ having 1f/J(t)1-0 as t - 00 when I~I is small. There are examples of nonlinear equations with this property that are nevertheless not Lyapunov asymptotically stable since their maximum value If/J(t) I over r :$; t < 00 does not approach zero as the initial ~ - 0 (cf. [7J). Remark 12.2.2 Note, from the estimate (12.2.9) establishing the uniform and asymptotic stability of the zero solution of (12.2.4), that for solutions

234

12. limit Properties of Solutions

with initial ~ satisfying I~I ~ c5 o(£) there is a uniform length of time during which I
Remark 12.2.3

Ih(t, x)1 ~ (1/4f3)lxl

(12.2.10)

for t E (a, (0) and all x in a neighborhood of the origin. An example of an equation to which this remark would apply is oX

where B(') has limH

oo

B(t)

= B(t)x

(12.2.11)

= A, a stability matrix. By rewriting (12.2.11) as oX

= Ax + h(t, x)

(12.2.12)

in which h(t, x) = [B(t) - A]x, it is clear that (12.2.10) is satisfied on an interval (a, (0) for a appropriately large, and consequently on that interval the zero solution to (12.2.11) is uniformly and asymptotically stable. A similar analysis would apply if a term of type (12.2.5) were added to (12.2.11). Remark 12.2.4 The constraint (12,2.5) on h can be satisfied in a number of ways, e.g., if Ih(t, x)1 ~ ylxl~ for numbers IX, y with oc > 1. In particular, this

occurs in the following corollary. Corollary 12.2.2 If the autonomous system(D, f) on D = 9l x E, E an open subset of 9l n , is twice continuously differentiable and f(p) = 0 at some point pEE, then the rest solution x = p will be uniformly asymptotically stable if the Jacobian matrix fx(p) is a stability matrix. Proof. As remarked earlier, there is no loss in assuming that p is the origin in 9l n • Rewriting the equation as oX = f(x) = Ax

+ hex),

(12.2.13)

where A = fx(O) and hex) = f(x) - Ax, by Theorem 12.2.1 and Remark 12.2.4 the proof is reduced to showing that (12.2.14)

12.2

235

Stability of Nonlinear Equations

on a neighborhood of the origin for some number y. To this end, rewrite h as l

hex) = f(x) - f'(O)x =

fo

Io f'(ax) do x -

:a f(ax) do - f'(O)x

I

=

(12.2.15)

f'(O)x

in which f'(x) denotes the Jacobian matrix. Similarly, rewrite f'(ax) =

Io 8~ I

f'(wax) dw

+ f'(0) =

L I

a f"(wax)x dco

in which f" involves second-order partial derivatives of (12.2.16) into (12.2.15) leads to

=Jo [L af"(wax)x dwJ xdo l

hex)

+ f'(0) f.

(12.2.16)

Substitution of

I

(12.2.17)

and standard estimation of(12.2.17) using the continuity of the second partial derivatives of f produces (12.2.14) to complete the proof.

EXERCISES

1. Discuss the geometric interpretation of the definition of a Lyapunov asymptotically stable rest point for a two-dimensional autonomous system by drawing an appropriate sketch in the plane. 2. What can be said about the geometric behavior of the solution curves of Eq. (12.2.13) if all the eigenvalues of A = fx(O) have positive real parts? 3. Apply Corollary 12.2.2to determine the asymptotically stable rest points of the system (12.1.36) occurring in Example 12.1.2 and correlate the results with the analysis presented there. 4. Find all the rest points of the system

and check them for asymptotic stability. 5. Let E be an open subset of fYt" that contains the origin, and denote (oc, (0) x E by D<>. for IXE fYt. Assume that for some a E fYt the continuous Cauchy problem (Da, f) has a general solution and that f(t,O) = 0 for all t E (a, (0). Prove that if, for some b > a, (Db' f) has the zero solution Lyapunov stable (asymptotically stable), then (D a , f) has likewise. Note that the conclusions remain valid for a = - 00. 6. Justify the claim that the zero solution is asymptotically stable on

236 ( - 00,

12. limit Properties of Solutions

(0) for the system

+ x 2 + (sin t)xt x2 = te- Ix 2 + X3 + 2X 1X3 , Xl

=

e-Ix l

x. 3

=

-Xl -

2t

2

- - 2 X2 -

1+ t

2x 3

2 + e - 21(cost) Xl'

7. Prove that if either (12.2.1) is autonomous or f(t, x) is periodic in t, then each stable (asymptotically stable) solution to (12.2.1) is uniformly stable (uniformly asymptotically stable).

12.3 Partial Stability

Although the literature contains numerous stability-theoretic definitions, those of the Lyapunov variety discussed in Section 12.2 are the most prominent. The concepts are firmly rooted in applied problems of interest, and as the next theorem indicates, they serve as useful hypotheses in the description of more general aspects of stability.

Theorem 12.3.1 Suppose that the continuous system

x = f(x, y),

y = g(x, y)

(12.3.1)

on an open subset E c f7l" x f?im has a general solution, and let (x(· ), y(. » be one ofits solution curves whose orbit is denoted by y. Assume that y+ is compact in E, y(t)- Yoo as t - 00, and g(x, Yoo) = 0 for all (x, Yoo)EE. Then

n+(y) c where

r

r

x Yoo'

(12.3.2)

consists of those points of the limit system

I .»

x=

f(x, Yoo) = f 00 (x)

(12.3.3)

on E oo = {x (x, y.oo) E E} whose orbits have compact closure in E oo . If (x(· ), y( is, moreover, Lyapunov stable, then

n+(y) = lim n+(Yoo(x(t») x Yoo' I .... 00

(12.3.4)

(Note: In (12.3.4) yoo(x(t» denotes the orbit of (12.3.3) passing through x(t) which is the projection of (x(t), yet»~ on f?i", not a solution of (12.3.3).) Proof.

co +

The assumption that y + is compact assures that (x('), y(-» has

= 00. The invariance of E oo follows from the condition that g(x, y 00) = 0 for (x, Yoo) E E. The fact that yet) - Yoo as t - 00 implies that n+(y) = S x Yoo

237

12.3 Partial Stability

for some subset S c E oo. Application of Theorem 12.1.3 shows that S is a compact invariant set for (12.3.3), and hence S c T. Now further assume that (x(' ), y(' )) is Lyapunov stable. The proof reduces to showing that S = limt-+oo n+(Yoo(x(t))). Let 11(t, x, y) denote the general solution of (12.3.1) with 11(0, x, y) = (x, y) for (x, y) E E. Note that the general solution to (12.3.3) is then just the projection 111 (t, x) of 11(t, x, Yoo) onto ffr. From the assumptions and Theorem 12.1.3 it follows immediately that for arbitrary e > 0 there exists a T = T(£) such that t5(S, x+(t)) <

!£

(12.3.5)

for all t ~ T, where in (12.3.5) t5 is the Hausdorff metric and x+(t) denotes the projection of the semiorbit Ua~O 11((1, x(t),y(t)) onto {jfR. On the other hand, since (x(· ), y(.)) is Lyapunov stable and yet)-- y 00 as t -- 00, it follows that there exists a T = 'r(£) (which is taken to be ~ T(£)) such that Ix(t

+ T) - 111(t, x('r)) I < !£

(12.3.6)

for all t ~ O. By the definition of the Hausdorff metric t5 given by (12.1.13), it is easy to see that (12.3.6) implies that (12.3.7) for t ~ T where yoo(x(t)) is defined in the note following (12.3.4). From (12.3.5) and (12.3.7) a standard application of the triangle inequality shows that t5(S, y~(x(t)))

s

t5(S, x+(t))

+ t5(x+(t), y~(x(t))) (12.3.8)

which concludes the proof of Theorem 12.3.1 since e > 0 was arbitrary.

Remark 12.3.1 Simple examples show that n+(y(x)) generally is not continuous at all points x E E. Hence the proof of (12.3.4) does not reduce to a triviality. Remark 12.3.2 For equations to which it applies, Theorem 12.3.1 reduces the determination of the limit set of certain semiorbits to the analysis of a lower-dimensional equation. Examples are forthcoming. Remark 12.3.3 It should be apparent that Theorem 12.3.1 can be extended to equations in which the invariant manifold approached is more complicated than the linear "slice" Eooof E.

238

12.

Limit Properties of Solutions

Example 12.3.1 The problem is to analyze the limiting behavior of each solution x(· ) to the system in 81 4

(12.3.9) X4 = -X 3 - x 4[(xi

+ xD + (x~ + xD]

for increasing t. Let x(·) be any solution to (12.3.9) defined on a right-maximal interval [0, w+) and denote the orbit of x(·) by y. From (12.3.9) it is easy to check that the functions vl(x) = xi + x~ and V2(X) = x~ + x~ defined on 814 have derivatives along the solution xC) of (12.3.9), Vl(X) = 2x~[1 v2(x) = -2x~[(xi

+ (x~ + x~) - (xi + xD], + xD + (x~ + x~)] :::;; O.

(12.3.10)

(12.3.11)

From (12.3.11) it is evident that V2 is bounded along y +. This fact along with (12.3.10) makes it clear that Vl is likewise bounded along y+. Thus with y+ bounded, w+ = 00 and n+(y) =F 0. The boundedness ofv 2(x(·» on [0, w+) and the monotonicity provided by (12.3.11) ensure that lim.; 00 V2(X(t» exists. Therefore V2 is constant on n+(y), and the invariance of n+(y) implies that at each point (E n+(y) the derivative in (12.3.11) must be zero. This can only happen if (4 = O. But from (12.3.9), note that at a point at which (4 = 0, (3 must likewise be zero to maintain the invariance of n+(y). The preceding argument shows that V2(X(t» - 0 as t - 00, which is to say that X3(t)- 0 and X4(t) - 0 as t - 00. Observe that Theorem 12.3.1 applies to (12.3.9), with the limiting form of the system being

+ x 2[1 - (xi + xDJ. (12.3.12) Relative to (12.3.12), the function v(x) = xi + x~ has the Lyapunov derivative v(x) = 2x~[1 - (xi + x~)], (12.3.13) Xl =

X2'

X2 =

-Xl

from which it follows that each point in 81 2 , other than the origin, which is a rest point, has the periodic orbit Ixl = 1 as the positive limit set of its orbit. Moreover, that periodic orbit is Lyapunov asymptotically stable, as indicated in Fig. 12.3.1. The orbit y of (12.3.9) in 814 has n +(y) = S x Y00' where Yoo is the origin in 81 2 and S is a connected invariant subset of r. But r is just the union of the periodic orbit Ixj = 1 with the origin in 81 2 • Thus we conclude from Theorem 12.3.1 that n+(y) = S x (0,0),

(12.3.14)

239

12.3 Partial Stability

---t-t----i----i+----Xl

Fig. 12.3.1 The limit system.

in which S is either the origin in f7t2 or the circle Ixl = 1 in f7t2, respectively, depending on whether or not y is the rest point x = 0 in f7t4. The computation of .o+(y) completes the description of the limiting behavior of each solution x(t) of (12.3.9) as t - 00. Remark 12.3.4 A differential equation (D, f) with D = (a, (0) x E is called asymptotically autonomous if Iim; 00 f(t, x) = f oo(x) exists for each x E E. Many such equations can be analyzed by application of Theorem 12.3.1. The requirement is that its positive semiorbit to be analyzed have compact closure. The next example demonstrates the point. Example 12.3.2

Although the system

. X2

= .-

(Xl

+ x 2 )[ 1 - t 2 (x I 1 + t2

- 1)]

(12.3.15)

on ( - 1, (0) X f7t2 is not autonomous, it is clearly asymptotically autonomous. By addition of another coordinate, say y = (2 + t) - 1, it can be embedded into the autonomous system

(12.3.16)

y=

_y2

in f7t3 for which Theorem 12.3.1 applies. (Certainly other embeddings are possible since in (12.3.16) only the region 0 < y < 1 reflects the dynamics of the original problem.) For (12.3.16), Yoo = 0 and the limiting system is Xl = x 2,

X2 = -(Xl

+ x2)(1

- xi).

(12.3.17)

Thus the behavior of bounded solutions to (12.3.15) on [0, (0) can be studied

240

12.

Limit Properties of Solutions

by means of the positive limits sets of (12.3.17). To see that (12.3.15) does indeed have such solutions, rewrite the equations as

. X2

=

t 2xi(Xi + x 2) X2 1 + t2

-Xl -

(12.3.18)

and note that the coefficient matrix of the linear part of(12.3.1S) is a stability matrix. Application of Theorem 12.2.1 now shows that the zero solution of (12.3.18) is Lyapunov asymptotically stable. The detailed question concerning which solutions to (12.3.18) are bounded on [0, (0) and the determination of the positive limit sets of (12.3.17)are left to the reader. The point is that the limit system (12.3.17) is useful in the analysis of (12.3.15).

Example 12.3.3 Diffusion-Reaction Equations. Various problems in the theory and application of partial differential equations provide a rich source of ordinary differential equations and related questions. For example, equations of the type

AU at

=

D,1u

+ VF(u)

(12.3.19)

often occur in the study of heat conduction, chemical reactions, population dynamics, and other diffusion processes. The dependent variable u is an n vector-valued function of(t, x) E PIt x Plt m, D is a positive-definite real n x n matrix, and ,1 is the Laplacian operator in Pltm acting componentwise on u. F is a twice-continuously differentiable real-valued function on PIt", and V is the gradient operator. In applications in which x is a scalar variable, special solutions of the form u = u(x - ct), called traveling waves, are of interest. They are called forward or backward traveling waves, depending on whether the wave speed c is positive or negative. Most applications are concerned with solutions u( .) bounded on PIt or on a semi-infinite subinterval. Substitution of the special form u = u(x - 'ct) into (12.3.19) yields a system of second-order ordinary differential equations that under the renaming of variables u - Xl' X - ct- t, and Xl - X2' can be checked to produce the first-order system Xl

=

X2'

X2

=

-CD-lX2 - D-lVF(xd

(12.3.20)

°

with (Xl' X2) E PIt" x PIt". The equation obtained by setting au/at = in (12.3.19) generally is called the steady-state form of the equation and, in the case under discussion in which the space variable has dimension one, the solutions are called steady-state or standing waves. They are governed by the

241

12.3 Partial Stability

equation that results from setting c = 0 in (12.3.20). A Lyapunov function of considerable value in the study of (12.3.20) can be taken to be vex) = !x z, Dx z

+ F(x 1 ) .

(12.3.21)

Its derivative along solutions of (12.3.20) computes to be vex) = -clxzI

Z

(12.3.22)

•

It seems worthwhile to develop some intuition concerning the terms in (12.3.19). Suppose the coordinates of u denote population sizes of n species. Dropping the term containing L\u leaves an ordinary differential equation that would describe the responses of each individual population to the others. Clearly, the response would be to locally maximize F(u). Reinstatement of the diffusion term allows the populations to diffuse through space. If the space is one-dimensional, the question of the existence of a traveling wave of some speed c would be of interest. Such a question translates into a question about the system of ordinary differential equations (12.3.20). Note that the rest points of the latter system are independent of c and are in fact the critical points of F. A solution of (12.3.20) that connects two rest points corresponds to a particular type of traveling wave solution of (12.3.19), namely, a shift in population levels in each region of space. The rest point that is being approached having a zero coordinate would be interpreted as local extinction of a species. Periodic and other bounded solutions likewise have population dynamical interpretations. Thus the question ofthe existence of traveling waves reduces to the study of not one but an entire one-parameter family of systems of ordinary differential equations. More concrete examples can be found in the exercises at the end of this section. Although the situation can be more complicated, the notion of wave solution can be considered for systems in which the space variable x of (12.3.19) has dim(x) = m > 1. For example, a spherically symmetric wave is any solution of (12.3.19) of the form u = u(r - ct) in which r = Ixl is the Euclidean norm of x. After a bit of elementary differentiation, substitution of such a function into (12.3.19) results in the second-order vector ordinary differential equation

dZu [m - 1 JdU dr + - r - 1n + cD- dr + D1

2

1

VF(u) = O.

(12.3.23)

There is no problem with backward traveling waves if in (12.3.23) r is restricted to positive values. Special devices are needed to account for the singular coefficient in the equation if c > 0 since the argument of u(r - ct) can then become zero as t increases. Aside from this complication, by again renaming variables u - Xl' r - t, and Xl = Xz, the resultant first-order form

242

12. limit Properties of Solutions

of (12.3.23) is the system

(12.3.24) which we note agrees with (12.3.20) for m = 1. In view of the interpretation of the original variable r, Eq. (12.3.24) is of interest for t > O. In particular, the steady-state analysis of(12.3.19) translates into the study of(12.3.24) with c set to zero. Thus (12.3.24) can be taken as a starting point and looked at afresh from the point of view of the notation and theory developed in preceding sections of this chapter. Since for m > 1 the generalization (12.3.24) of (12.3.20) to a higherdimensional space variable is no longer autonomous but clearly is asymptotically autonomous, the embedding device introduced in Example 12.3.2 is in order. Thus, introduction of an additional coordinate variable, say y = r", embeds the equations into the autonomous system

X2

= - [(m - 1)y1n + cD-1JX2 - D-hvF(xd,

y=

(12.3.25)

_y2

with (t, Xl' X2, y) E 9f X 9f2n+ 1. The original problem is concerned with the solutions in the subregion in which t > 0 and y > 0 with the initial values of those variables related by the equation y = t- 1 • The study of the limit sets n+(y) of the orbits of those solutions provides a geometric approach to investigation of the limit behavior of solutions to (12.3.24). In the context of Theorem 12.3.1 it is interesting to note that the limit system of (12.3.25) is just (12.3.20)! In other words the theorem provides a way to use information about nonlinear diffusion along one space dimension to study diffusion through higher-dimensional regions of space. For detection of bounded positive sem'iorbits of(12.3.25) that are needed for application of Theorem 12.3.1, it is useful to work with the Lyapunov function (12.3.21), regarded as a function defined on the domain of (12.3.25). Its derivative relative to the latter system can be checked to be (12.3.26) Detailed analysis of examples is left to the reader. Similar topics can be pursued for other classes of partial differential equations.

243

12.4 local Behavior and linearization about Rest Points EXERCISES

1. Find a simple example of an autonomous differential equation in 9f2 that substantiates Remark 12.3.1. 2. Analyze the limiting behavior of the solutions to the following system for increasing t: Xl =

X2 = X3

=

+ X 2X3, - X l - X2 + X3 + x2x I + xt

X2

[sin

Xl -

2]x 3 .

Investigate the limiting behavior of the system

3.

• Xl

=

3 X 2,

4. Consider the diffusion-reaction equation in one space variable

au a2u at

=

ax2

+ 4(u

2

- l)u.

Derive the first-order system of ordinary differential equations governing the traveling wave solutions u = u(x - ct) and verify that the results agree with (12.3.20) if D = 1 and F(u) = (u 2 - 1)2. Note that the system obtained with c = 1 is (12.1.35) of Example 12.1.2. Describe the behavior of the bounded forward traveling wave solutions u(x - t) of speed c = 1 as Itl- 00.

12.4

Local Behavior and Linearization about RestPoints

In this section we shall concentrate on autonomous equations of the form X = Ax

+ h(x)

(12.4.1)

in which A is an n x n matrix and the n-vector function h is once continuously differentiable on a neighborhood of the origin and of higher order in a sense to be made precise. Such an equation arises from an autonomous equation with a rest point when the rest point is translated to the origin under a suitable change of coordinates. The rest point-which is to say, the origin for (12.4.1)-is called hyperbolic if all the eigenvalues of A (regarded as a matrix over the complex field) have nonzero real parts. Such a matrix will likewise be called hyperbolic. For example, a stability matrix is an extreme example of a hyperbolic matrix. Throughout the literature an extensive and successfuleffort has been made to relate the solution behavior about the origin of(12.4.1) to that of the linear equation resulting from deletion of the perturbation term h. An example of

244

12. Limit Properties of Solutions

one such result is Corollary 12.2.2. The rather remarkable Theorem 12.4.1 accounts for much of this success. (See [9] for a historical account of its development and for related results.)

Theorem 12.4.1 Suppose that in (12.4.1) the n x n matrix A is hyperbolic and that h is of class C 1 on a neighborhood of the origin in :)In. If as

Ih(xWlxl-O

Ixl-O,

(12.4.2)

or, equivalently, h(O) = 0

(12.4.3)

and

then there exist open sets U, V in :)In that contain the origin and a homeomorphism g: U - V such that the general solution '7 of (12.4.1) factors as

(12.4.4) for each ~ E U and corresponding t in the open interval about the origin in over which etAg(~) E V.

:)l

Proof. By a preliminary nonsingular linear change of coordinates, it can be assumed that A is of a block diagonal form A = diag[ A 1 , A 2 ] in which the nl x nl matrix A 1 has eigenvalues all with negative real parts and the n2 x n 2 matrix A 2 has eigenvalues all with positive real parts. (One of A 1 , A 2 might not occur.) Then there exist positive numbers c, 0(1' and 0(2 such that

(12.4.5)

and - 00

< t s; O.

(12.4.6)

Moreover there exist matrix solutions Pi > 0 (i = 1, 2) to the Lyapunov equations (12.4.7) (12.4.8) (Recall Theorem 5.2.3.) The associated quadratic forms are Vi(X i ) = Xi' Pix, = 1,2) and vex) = v 1(x d - V2(X2) for x = (Xl' X2) E :)In! X :)ln2 • Let = diag[ P 1> P 2] and with the aid of (12.4.2) choose f. > 0 sufficiently small so that

(i P

2I p llh(x)//lxl < i for all x

E

BE' the open ball of radius

f.

(12.4.9)

centered about the origin. Note that (12.4.10)

245

12.4 Local Behavior and Linearization about Rest Points

for all xEB•. Define H = {xllv(x)1 < c5} n B. and 8H = {x I xEH, Iv(x)1 = c5}. It is left as an exercise to show that c5 > 0 and r > 0 can be chosen appropriately small so that B zr c H c B. and, moreover, so that for each ~E U = B" y+W n 8H:F 0 ify+W n(,gfR - B.):F 0, and y-Wn 8H:F 0 ify-W n (,gfR - B.) :F 0. (In other words the seniiorbits must exit B. across 8H only.) The above choices of £,15, and r remain valid if hex) is replaced by A.(x)h(x), where A.(x) is any function on ,gfR with 0 ~ A.(x) ~ 1. By a classical construction such a A.(x) can be selected to satisfy the further requirements that A.(x) = Ion V and A.(x) = 0 on,gfR - B zr . Hence there is no loss in assuming that the domain of hex) is ,gfR and that hex) = 0 on 8H. The general solution l1(t, ~) of (12.4.1) with 11(0, ~) = ~ is then defined for all t E,gf and ~ E ,gfR. The required map g(~) = (glW, gzW) on the set U defined above is to be determined by the formulas

Coo

(12.4.11)

(12.4.12)

in which h(~) = (h1W, hzW)E,gfRl X ,gfR2 and ~ = (~l' ~Z)E,gfRl X ,gfR2. Inequalities (12.4.5) and (12.4.6) and the boundedness of Ih(x)1 ensure the uniform convergence of the integrals on ,gfR. It follows that g is continuous on ,gfR. Note that g(O) = O.: The next step is to establish the validity of the equation (12.4.13)

By referring to (12.4.11) and (12.4.12), applying the semigroup property of 11, and then making a simple change in the variable of integration, it is easy to check that g(l1(t,~))

=

- (f~oo

(111(t, ~), e(t-a)

m h (I1(u, en da,

I1z(t, A1

1

f:

e(t-a)

A2

hz(I1(U,

e» dU).

(12.4.14)

Since (12.4.15)

246

12.

Limit Properties of Solutions

Eq. (12.4.9) can be differentiated, with the result that d dt g(y/(t, ~»

=

(Algl(y/(t,

m, Azgz(Y/(t, ~»),

(12.4.16)

which immediately implies (12.4.13). Completion ofthe proof of Theorem 12.4.1 requires only the demonstration that the restriction of g to U is one-to-one. Because of(12.4.10), each nonzero element of U lies in the level set A(t) =:= {x Ix = y/(t, ~), ~

(12.4.17) E oH} n U Z for some t E &t. Select any nonzero x! and X in U. Then Xl = 1](t l, ~l) and XZ = Y/(tz, ~Z) for some t i E &t, ~i E oH (i = 1,2). Observe that Iv(~l)1 = Iv(e)1 = b. Since h = 0 on 8H, it is clear from (12.4.11) and (12.4.12) that the restriction of g to oH is the identity. From this and (12.4.13) we have ~l = gee) = e-t1Ag(y/(tl, ~1» = e-ttAg(x l), (12.4.18) ~z = g(~Z) = e- t2Ag(1](tz, = e- 12Ag(xZ). (12.4.19)

e»

If g(x l)

= g(XZ), then certainly (12.4.20)

But v is strictly monotone along the solutions of x = Ax. Equation (12.4.20) then violates the earlier observation that Iv(e)1 = Iv(e)1 unless t l = Vz» which is to say that ~l = ~z and hence that Xl = xZ. Our calculations further show that x#-O implies g(x) #- O. This shows that g is one-to-one on U, and Theorem 12.4.1 is established. Remark 12.4.1 Examples show that it is not always possible to choose the function g of Theorem 12.4.1 to be C l .

The conclusion of Theorem 12.4.1 says that the solution curves of (12.4.1) near the origin are mapped onto the solution curves of the linear system y = Ay by the coordinate transformation y = g(x) with the parametrization preserved. The conclusion of Theorem 12.4.1 can fail if the matrix A has an eigenvalue with zero real part. (See the exercises following this section.) Although an extensive treatment of the marginal case wherein A is not hyperbolic will not be presented, the following result gives some indication of its delicate nature. Remark 12.4.2

Theorem 12.4.2

In the scalar differential equation

i = iwz + h(z),

(12.4.21)

12.4

247

Local Behavior and Linearization about Rest Points

assume that the real co #- 0 and that h is a complex analytic function on a neighborhood of the origin in the complex plane with h(O) = h'(O) = O. Then there exists an open set U containing the origin and a bijective, analytic map g: U - V = {z Ilzl < p}, some p > 0, for which the general solution 1/ to (12.4.21) can be written as 1/(t,z)=g-l(e i wtg(z))

forall

ZEU

and tE81.

(12.4.22)

Proof. Let Uo be an open disk about the origin in which h(z) is analytic. Since h(O) = h'(O) = 0, then h(z) = Z2 fez) for some fez) analytic in a neighborhood of the origin, and thus by making the radius of U 0 small it can be assumed that 1

1

icoz

1

h(z)

fez)

+ h(z) - -itoz - iwz[iwz + h(z)] = iwz - iw[iw + zf(z)] (12.4.23)

is analytic on the complement of the origin in U 0 • Fix any Zo #- 0 in Uo . For z #- 0 in Uo let C be a path from that does not meet the origin. Now define g(z)

=

(i

Jiz exp(iW)j~iww

Zo to

dW) dW) exp(F(z)) + hew) = exp Jzo -;-

z in Uo

Z

(12.4.24)

where

_i few) dw . Jzo uo + wf(w) z

F(z) =

But independent of C,

(f

exp

Z

-dW) = -z ,

Zo, W

(12.4.25)

Zo

and it is further noted that the integral defining F(z) is independent of C. Thus g(z)\ =

-z

(12.4.26)

eF(z)

Zo

is an analytic function on U 0 witn g(O) = 0 and g'(O) #- O. By the inverse function theorem, g is therefore bijective on a neighborhood of the origin with g-l likewise analytic. Thus it is possible to choose a disk V c g(Uo ) centered about z = 0 and an open set U c U o containing the origin such that the restriction g: U - V is a bijection with analytic inverse. For any z E U, g(z) E V. Now restrict t to the open interval about t = 0 where 1/(t, z) E U. Then

(f

g(1/(t, z)) = exp u»

q (t .Z)

zo

.

dw

lWW +

) h( ) w

(12.4.27)

248

12.

Limit Properties of Solutions

in which C is a path connecting Zo to I'/(t, z) and can be taken to be C = C I + C2 where C I is a path from Zo to z and C 2 is a path from z to I'/(t, z) that will be taken to be 1'/(-, z). Therefore (12.4.27) can be rewritten as g(I'/(t, z)) = eXP(iW

f~

iww

() (. rt

a;

= g z exp IW Jo iwI'/(u,~) =

heW)) eXP(iW

f;
I'/,,(u, ~) du ) + h(I'/(t, ~))

g(z) exp(iwt) Eo V

(12.4.28)

since g(z) E V and the disk V is invariant under rotation; that is, (12.4.29) But from this it follows that I'/(t, z) remains in U for all t, and the proof is complete. Corollary 12.4.3

Consider the system in ~2

x = Ax + hex)

(12.4.30)

with A = [~ - ~], real w =1= 0, and h of class CIon a neighborhood of the origin with h(O) = 0, hx(O) = O. If the coordinate functions hI, h2 of h satisfy the Cauchy-Riemann equations h~, = h;2' h~2 = - h;" then there exists an analytic diffeomorphism (bijective map with it and its inverse representable in convergent power series) g: U --+ V mapping a neighborhood U of the origin onto an open disk V centered at the origin such that the generalsolution 1'/ to (12.4.30) is factorable as for all

~ E

U, t E ~.

(12.4.31)

Proof. Introduce a complex variable z = Xl + iX2 in which Xl' X2 are the real coordinate variables of X E ~2. By a standard theorem of complex variables (see [16]), the assumptions ensure that the coordinate functions hI, h2 of h satisfy hl(x) + ih2(x) = F(z) for some complex analytic function F about the origin of the complex plane. Note that F(O) = F'(O) = O. It follows that the real system (12.4.30) can be written as a single complex scalar equation i = ioiz

+ F(z),

(12.4.32)

and thus Theorem 12.4.2applies to conclude the proof. Example 12.4.1

The real scalar system

Xl = - WX2 + xi X2 = WXI + 2XIX2'

xL W =1= 0,

(12.4.33)

249

12.4 local Behavior and linearization about Rest Points

+ Z2. An appropriate choice of g(z) is (12.4.34) cz/(z + iw)

is the real form of the equation i = icoz g(z) =

for any C =f O. (See the exercisesat the end ofthis section.) It is easy to compute (12.4.35)

g-1(Z) = iwz/(c - z),

and then the general solution can be computed, according to Theorem 12.4.2, as (12.4.36) Note that for Izj appropriately small the solutions are periodic as predicted but that not all orbits are periodic. The solution to (12.4.33) can be computed either by separating (12.4.36) into real and imaginary parts or, alternatively, by computing the real functions g1, gl for which 9 = g1 + ig l and then applying (12.4.31). The computations are left as exercises.

EXERCISES

1. Consider the equation x = g(x) in an open subset of qJ" or ~" where 9 is of class C 1 and is invertible. Show that its general solution 1'/ can be written e)gm). Interpret the latter equation in the factored form 1'/(t, = g-1(1'/~(t, in terms of linearization. Note that as a corollary it follows that if for some the matrix function 1'/~(t, of such a system is p-periodic, then 1'/(t, will likewise be p-periodic. 2. (a) For the equation in qJ1, X = ax + h(x) (a > 0), make the change of variable of integration s = e"Og(e) in (12.4.2) and then differentiate the resultant equation to show that gm satisfies the differential equation

e)

e

e)

(b) For the particular equation with h(x) = in (a) to get

e)

Xl,

solve the equation derived

c =f 0 a constant. Using the latter equation, compute

and then in accordance with Theorem 12.4.1, compute the general solution

250

to X = ax

12.

Limit Properties of Solutions

+ x 2 as Yf(t, ~) = g-l(etag(~»

= a

a~eta

+ ~(1

_ eta)'

Note that the solution is independent of c, as it should be. Verify that the formula obtained is indeed the solution. Substitute the expression obtained for Yf(t, ~) into the integral in (12.4.12) to get g2(~) = O. How do you explain that? Further note that the above formulas also apply to the case in which a < O. 3. In Example 12.4.1, why is the choice of the function made in (12.4.34) appropriate as claimed? Compute the general solution to (12.4.33) by the two methods proposed in the discussion of that system. 4. Show that the conclusion of Corollary 12.4.3 is not valid for the system Consider the function v = xi + x~.) In what way does this system fail to satisfy the hypothesis of Corollary 12.4.3? 5. Let f be a real-valued function that is locally Lipschitz continuous and nonzero on an open interval I. Fix any X o E I and define (Hint:

g(~)

=

exp(L:

f~:)

for ~ E 1. Show that the general solution Yf to the equation X = f(x) on I can be represented as Yf(t,

0=

g-l(etg(~).

6. Return to Example 12.1.2and, with the knowledge of Theorem 12.4.1 in mind, justify the description given there of the local behavior of the solution about the origin.

12.5 Planar Systems

Nature abounds in periodic solutions and motion about such solutions. Although linear equations can have periodic solutions, with the exception of isolated rest points that periodicity can be destroyed by the slightest perturbation in the equation or initial state. This indicates that the subject is primarily a concern of nonlinear equations. A rest point is an especially simple type of periodic solution. In this section, we shall concentrate on the complementary type that has positive minimal period. The global nature of the latter makes its study comparatively difficult.

251

12.5 Planar Systems

This difficulty is less severe for a planar system

x =f(x),

(12.5.1)

where x is in [H2 and here f: [H2 - [H2 is taken to be once continuously differentiable. The theorems that follow demonstrate one of several approaches that sometimes apply to establishing the existence of a periodic solution. Fundamental in such studies is the notion of a simple closed curve and the Jordan curve theorem. A map y: [a, b] - 81t 2 is called a curve. It is called closed if y(a) = y(b) and simple if the restriction of y to (a, b) is one-to-one. Theorem 12.5.1 Jordan Curve Theorem. Let y: [a, b] - [H2 be a simple, closed, continuous curve. Then the complement of the image set y([ a, b]) in [H2 can be written uniquely as the disjoint union of two open, connected sets (regions) I; and O; such that L, is bounded. The region L, is called the inside of y. Region I; is simply connected and y([ a, b]) u L, is contractible to each of its points. The boundary of each of the two regions is y([a, b]). Proof.

The proof of Theorem 12.5.1 can be found in Whybum [27]'

A curve satisfying the hypothesis of Theorem 12.5.1 is called a Jordan curve, and it is called a C 1 Jordan curve if y is, moreover, once continuously differentiable. A closed, bounded line segment T is called a transversal to (12.5.1) if f(x) # for x E T and the direction of f(x) at points x E T is not parallel to T. Note that all crossings of T by a solution x(t) of (12.5.1) are in the same direction with increasing t.

°

°

Lemma 12.5.2 Let q E T, f(q) # 0, T a transversal through q. Then there exists a small neighborhood N of q and an f. > such that any solution x(t) of (12.5.1) with x(o) E N exists for I tl :$; f. and crosses T exactly once for 1 tl :$; e. Proof.

Let

1] be

the general solution to (12.5.1), and consider the equation 1](t, e) - (q

+ O"V)

=

°

(12.5.2)

in which v is a unit vector in [H2 parallel to T. The scalar t and 0" are regarded as unknowns with a vector parameter in [H2. Certainly the equation has the solution t = 0, 0" = for = q. Computation of the Jacobian of the term composing the left-hand side of (12.5.2) relative to t and 0" at t = 0, 0" = 0, = q gives

e

°

e

e

de{;:~:~

=~:J

= _[Jl(q)V

2

-

j2(q)v

1

]

#

°

(12.5.3)

252

12. Limit Properties of Solutions

since T is a transversal to f at q. By the implicit function theorem there exists a unique solution t = t(~), (J = (J(~) to (12.5.2) for ~ in a neighborhood of q. This completes the proof.

Lemma 12.5.3 If T q is a transversal of(12.5.1) containing a point q E n+(y), then either the orbit y is periodic or else y + crosses T q at a sequence of times t 1 < t 2 < .. " and the corresponding crossing points ~k converge monotonically to q along T q as k -

00.

Moreover, q is the only point of n+(y) on T q •

Proof. If y is not periodic, then certainly Lemma 12.5.2 applies. Hence, since q E n+(y), the semiorbit y+ intersects Tq at times forming a sequence t 1 < t 2 < " " with corresponding crossing points ~ 1, ~ 2, .• '. Certainly ~ 1 , ~2' ... and q are distinct, for otherwise y would be periodic. Consider the Jordan curve determined by the arc of y" connecting ~1 to ~2' together with the segment of Tq connecting ~2 to ~1' There are three cases to consider: The point q lies on, outside, or inside the Jordan curve. If q lies inside, then since all crossings occur in the same direction, ~2 must be between ~1 and q, for otherwise the tail of y+ would be outside and would not be able to reenter the inside, which, however, it must do since q E n+(y) (see Fig. 12.5.1). There is also the possibility that q lies outside the Jordan curve, in which case ~2 would again have to be between ~1 and q, for otherwise the tail of y+ would enter the inside and would remain trapped away from q. Finally, q cannot lie on y+; consequently, the only place it might occur on the Jordan curve would be on the segment of Tq between ~1 and ~2' But that too is impossible since all crossings of Tq are in the same direction, and once the tail of y+ enters the inside or outside it cannot get back to the outside or inside, respectively, to reapproaeh q. In conclusion, ~2 always lies between q and ~1' The argument can be repeated to show that ~3lies between ~2 and q, etc. Since Tq can be identified as a compact subinterval of Yl, ~k converges as k - 00 because it is a monotone bounded sequence. Moreover, the limit of ~k is q. This follows directly from the fact that q E n+(y) and from Lemma Tq

Fig. 12.5.1 Two cases in the argument leading to monotonicity of crossing points.

12.5 Planar Systems

253

12.5.2. The only point of g+(y) on Tq is q, for any other would produce crossing points other than ~ 1 , ~ 2' •... Theorem 12.5.4 Poincare-Bendixson Theorem. If y + is a bounded positive semiorbit of(12.5.1) and g+(y) does not contain a rest point, then either

(a) g+(y) = g(y) = v or (b) g+(y) = y+ - y+. In either case, g+(y) is a periodic orbit. Proof. Let g+ satisfy the hypothesis of Theorem 12.5.4. Since g+(y) is not empty, it is possible to select one of its points q 1, which necessarily is not a rest point. Denote the orbit of q1 by Y1' Then g+(Y1) c g+(y) since g+(y) is closed. Select any q E g+(Y1) and let T q be a transversal of(12.5.1) containing q. As a consequence of Lemma 12.5.3, Y1 must be periodic. Otherwise Y1 would intersect T q at a sequence of distinct points which necessarily are in g+(y); but that is ruled out by the uniqueness statement in Lemma 12.5.3. Thus g+(y) contains at least one periodic orbit Yl' The remaining task is to show that g+(y) = Y1' Suppose that g+(y) - Y1 were not empty. Since g+(y) is connected, it would be possible to choose a point q2 E Yl which is a limit point of g+(y) - Y1' Let Tq be a transversal through q2' Any small disk about q2 contains a point q3 ~ g+(y) - Y1' For such a q3' (12.5.1) has a solution x(t) such that x(O) = q3 and the orbit Y3 of q3 is contained in g+(y). If q3 is sufficiently close to q2' then Y3 crosses Tq • By the uniqueness statement of Lemma 12.5.3 the crossing point is ne~essarily q2' However, since q3 ¢ Y1' a contradiction of the uniqueness of solutions to (12.5.1) is achieved. Thus g+(y) - Y1 is empty and the proof is complete.

The conclusion of Theorem 12.5.4 remains valid under slightly weaker assumptions. The differential equation need only have f defined on an open subset E of 911 2 and then y+ is required to have compact closure in E. The once continuous differentiability of f can be relaxed to f continuous as long as the initial-value problem has. unique solutions. For example, it is sufficient that f be locally Lipschitz on E. The more general result, using essentially the same proof, along with an extensive treatment of many related subjects, can be found in Hartman [9]. Those interested in a comprehensive treatment and references to the extensive literature should consult Hale [8] as well as Hartman [9] and Lefschetz [14]' Remark 12.5.1

The following corollary gives a detailed description of the manner in which the tail of y+ approaches the periodic orbit shown to exist in Theorem 12.5.4 (see Fig. 12.5.2).

254

12. Limit Properties of Solutions

Fig. 12.5.2 The approach of the tail of y'

to R+(y)

Corollary 12.5.5 Let y satisfy the assumptions of Theorem 12.5.4 for the case in which y is not periodic. Fix any q on the periodic orbit R+(y) and let T, be a transversal at q. 7hen.T, n R+(y) = q. y + crosses T, at a sequence of points ( k ( k = 1, 2, . . .), converging strictly monotonically to q as k ---t co. The respective crossing times t , < t , < * * * o fthe solution x(t) to (12.5.1) for which x ( t l ) = have (12.5.4) the smallest period of the solution x,(t) of (12.5.1) satisfying x,(O) = q. The sequence of functionsgk(t)= x(tk + t )converges uniformly to x,(t) on 0 I tI p as k + c o . Proof. All but (12.5.4) and the uniform convergence have already been proved. The g k s are all solutions of (12.5.1) with the same orbit y, and since y + is bounded they are all defined on 0 I t < co.Because of the convergence of the initial values gk(0) = tk+ q as k + co, the uniform convergence of &(t)to x,(t) on 0 I tI p as k--+ 03 is ensured by Theorem 11.3.1. In particular, g k ( p ) x,(p) = q as k + co. The proof is completed on noting that with Lemma 12.5.2 the negation of (12.5.4) would violate the uniform convergence of gk@ to x,(t) for 0 It Ip. --+

THE INDEX OF A CLOSED PATH

There is a very useful formula that computes how many times a curve y winds around a given point zo. This number is called the index of y with respect to zo (see Fig. 12.5.3). The definition is given in terms of complex variables, from which it naturally arises.

Dehition Let y be a closed curve in the complex plane V and zo be a point not on y. Then the index of y with respect to z,, (also called the winding

255

12.5 PlanarSystems

I

(Y

,zo) " 0

Fig. 12.5.3 The index of a curve about a point.

number) is defined by I(y,zo) =

~

2m

r ~.

Jy Z

-

Zo

It is relatively easy to show that I(y, zo) is an integer.

Rather than work with y directly, a complex number (vector) f = fl + if2 might be attached to each point along y to get a new curve yf = f 0 y and select Zo = O. The latter is the situation of interest in the study of (12.5.1). Since solutions to differential equations ordinarily do not cross, y will be a Jordan curve J and thus the index of J f will be denoted as IiJ). Thus I f(J) counts the net rotation of f as a point moves around J in counterclockwise fashion. The following basic properties of I f(J) are stated without proof. For more details and derivations see [16].

Theorem 12.5.6 If J is a Jordan curve surrounding a finite number of rest points PI' P2"'" Pn of f, then If(J) =

n

I

If(p;}·

;=1

If f has no rest points inside or on J, then I f(J) = O.

Theorem 12.5.7 If J is a C 1 Jordan curve and .f is a continuous unit tangent vector field along J, then I f(J) = 1. Corollary 12.5.8 . If Y is the orbit of a periodic nonconstant solution to (12.5.1), then Iiy)

=

1.

Corollary 12.5.9 A periodic orbit contains at least one rest point in its interior.

APPLICATION TO L1ENARD EQUATIONS

The Poincare-Bendixson theorem finds many uses in the study of periodic solutions in the plane. A demonstration of this is provided by the proof of

256

12. Limit Properties of Solutions

the next theorem, taken from Hale [8]. It considers the Lienard equation

u + f(u) u + u =

0,

(12.5.5)

written in the equivalent first-order-system form

x= yin which F(x) =

Y = -x

F(x),

(12.5.6)

H f(u) da. (See the proof of Theorem 10.3.4.)

Theorem 12.5.10 The planar system (12.5.6) has a periodic solution circling the origin if the continuous function f has F satisfying the conditions (a) F(x) is an odd function of the variable x. (b) F(x) - 00 as x - 00, and for some p > 0 it is positive, monotone increasing for x > p. (c) F(x) < 0 on 0 < x < ex, and F(ex) = 0 for some ex > O. Proof.

It will be useful to consider the scalar equation dy dx

-x

y - F(x)

(12.5.7)

for the orbits of (12.5.6) (refer to Fig. 12.5.4). The approach taken will be to show that for A = (0, a) on the y axis, with a sufficiently large, the positive semiorbit initiating at A enters the right half-plane and returns to the y axis, as indicated, to a point D = (0, d) with 0 < -d < a. Since F(x) is an odd function, any solution (x(t), y(t» to (12.5.6) will also have ( - x(t), - y(t» as a solution. Consequently, any positive semiorbit initiating at a point on the y axis strictly between the origin and D will be bounded. Examination of (12.5.6) and (12.5.7) shows that in the right half-plane solution orbits have negative slope above the curve y = F(x) and positive slope below it, and they cross the curve at vertical slope. Moreover, they cross the y axis horizontally from left to right above the x axis and from A

o Fig. 12.5.4 Starting a bounded positive semiorbit.

257

12.5 Planar Systems

right to left below the axis for t increasing. Hence, with assumption (b) there is no problem in concluding that for a sufficiently large an orbit initiating at A = (0, a) will return to the y axis at some point D = (0, d) on the negative y axis. The problem now is to show that 0 < - d < a for a appropriately large. . Consider the function vex, y) = t(x 2 + l). Along solution curves of (12.5.6) and (12.5.7), dv dt = -xF(x),

(12.5.8)

dv dx

(12.5.9)

-xF(x) - F(x)'

=y

dv dy = F(x).

(12.5.10)

From these three equations it follows that v(D) - v(A)

=

f

=

(f

dv

ABECD

+

AB

r )[y--

J +r

xF(x) dx F(x)

JCD

F(x) dy

(12.5.11)

JBEC

along orbits of (12.5.6). Clearly, the integrals in (12.5.11) over AB and CD remain bounded as a - 00.. By fixing any G on the x axis between (P, 0) and E, assumption (b) can be invoked to estimate the integral over BEC as

-r

JBEC

F(x) dy =

r

JCEB

F(x) dy >

r

JEK

F(x) dy >

IGHI x IGKI,

(12.5.12)

where IGHI and IGKI are the lengths of the line segments indicated in Fig. 12.5.4. As a- 00, IGHI remains fixed, whereas it is apparent from (12.5.7) that IGKI- 00. 'rhus for a sufficiently large, the inequality v(D) - veAl < 0 is implied by(12.5.11), which establishes the required inequality 0 < -d < a. Thus there exists a bounded positive semiorbit y+ of (12.5.6) that crosses the negative y axis. The system (12.5.6) has the origin as its sole rest point. However, y remains bounded away from the origin since (c) and (12.5.8) imply that v > 0 on a deleted neighborhood of the origin in al 2 • Thus the Poincare-Bendixson theorem ensures the existence of a periodic solution. Its periodic orbit must circle the origin-a consequence of Corollary 12.5.9 and of the fact that the origin is the only rest point. This concludes the proof of Theorem 12.5.10.

258

12. limit Properties of Solutions

Remark 12.5.2 Further results concerning the uniqueness and location aspects of the periodic solutions to the Lienard equation (12.5.5) exist (see [8]). Example 12.5.1 The Van der Pol equation U + k(u 2

1) Ii

-

+ u = 0,

(12.5.13)

k a positive constant, has a periodic solution circling the origin. (See Exercise 1 at the end of this section.) Remark 12.5.3 At this point the global and topological nature of the Poincare-Bendixson theory should be rather apparent. The subject of differential equations naturally extends to equations on two-dimensional surfaces and, more generally, on differential manifolds. There, under the title of differential equations on manifolds, topological dynamics, and differential topology, the constraints imposed on the solutions by the underlying manifold are an active element of the theory. We end this discussion with the remark that there exist periodic-free differential equations on the two-dimensional torus for which each orbit has the entire torus as its positive limit set. Thus there is no simple generalization of the Poincare-Bendixson theorem to manifolds in general. A natural generalization of a periodic orbit is a minimal set (a closed invariant set containing no proper such subset). A famous result of Schwartz [23] proves that minimal sets are either rest points, periodic orbits, or tori. Thus the search resumes.

EXERCISES

1. Verify that the Van der Pol equation (12.5.13) satisfies the hypothesis of Theorem 12.5.10. 2. Let y be an orbit of (12.5.1) in fJt2. Suppose that there is a continuous curve g: [a, b] - n+(y) which is onto and that Q+(y) contains no rest points. Show that n+(y) is a periodic orbit. 3. Let y: [a, b] - f!l.2 be a Jordan curve. Suppose that (12.5.1) has a solution which enters and remains inside y for all large t. Show that (12.5.1) has at least one rest point inside or on y. 4. Show that it is impossible for the positive limit set of each orbit initiating inside a given Jordan curve y to be outside y. 5. Starting with the definition of index of a closed curve, derive the formula I (J) = f

~

211:

f J

11 dI2 - 12 dI1

Ii + I~

259

12.5 Planar Systems

for the index of a real C 1 vector field f = (fl' f2) in ~2. (J is a C 1 Jordan curve oriented in the counterclockwise direction.) 6. Prove Corollaries 12.5.8 and 12.5.9 by applying Theorem 12.5.7. 7. Let J: [a, b] -- ~2 be a Jordan curve and let f(t), g(t) be two continuous vector fields on J that can be continuously deformed into one another without vanishing; that is, there exists a continuous function H: [a, b] x [0, 1] __ ~2 such that H(t, 0) = f(t), Hit, 1) = g(t), H(a, s) = H(b, s), and H(t, s) #- 0 for all t and s. Show that IiJ) = IiJ). 8. Show that IiJo) = If(J 1 ) if J o and J 1 are C 1 Jordan curves that can be deformed into one another without crossing rest points of f. 9. Prove that if f has div(f) of one sign not zero inside a Jordan curve J, then (12.5.1) has no periodic solution other than possible rest points. (Hint: Use Green's theorem.) 10. The problem of determining rest points is the problem of computing the zeros of f(x). A periodic solution is just a rest point in another dynamics. That is, a Jordan curve y in ~2 is carried along by the general solution to get the trajectory of Jordan curves ~(t, y(. )): [a, b] -- ~2 definded by ~(t,

y(s))

= l1(t, y(s)), S E [a, b]'

(a) Show that the rest point set of (12.5.1) inside y is invariant along the trajectory ij. (b) Denoting the inside of a Jordan curve J as I J for y a Jordan curve, derive the area formula

and apply it to obtain the result that ify is a periodic solution of(12.5.1),then

which says that

r div(f)(O d~l

Jry

u, =

O.

(Note the agreement with Exercise 9.) (c) Let y: [0, A] -- ~2 be a C 1 Jordan curve parametrized by arc length.

260

12. Limit Properties of Solutions

Derive the curve length formula

to, y) =

fij

ds

=

f"l7uCt , y(O'»/ da

and apply it to show that if y is a periodic solution, 'then

fy T'[Jh) + f~r(y)]T

ds

= 0,

where T is the unit tangent vector field to y and ds is the element of arc length.

Chapter

13

Appl ications in Control Theory

This chapter is a brief introduction to a few of the important mathematical notions that appear in the control of differential equations. It demonstrates how some of the results of previous chapters bear on the subject. 13.1 The Controlled and Uncontrolled Equations

The system in !J.ln

x = dx,

(13.1.1)

in which d is an n x n matrix, hereafter taken to be real, has been treated extensively. The results revolved about the solution formula x(t) = [XO, dxo, ... , d

n

-

1xO]

[yO, Byo, ... , B n - 1 yO] - l y(t), (13.1.2)

in which the n x n matrix Band n-vector function yet) are derived entirely from the eigenvalues of d, and yO = yeO). This formula proves useful in the theoretical and numerical calculations that follow. Control problem situations in which the appropriate model is the control differential equation

x=

dx

+ (J4u,

(13.1.3)

were described in Chapter 1. The term with the n x p matrix coefficient (J4 accounts for control variable u E!J.lP entering the system. (Note that in conformity with contemporary notational usage the letters d and (J4 are used for the coefficients in (13.1.3). Script letters are employed to avoid confusing the coefficient (J4 of u in (13.1.3) with the matrix B in (13.1.2), which also occurs in a number of previous chapters.) 261

262

13. Applications in Control Theory

When the control variable u in (13.1.3) is replaced by a control function u = u(t), the resulting nonhomogeneous equation can be written in the solved forms provided by the variation-of-parameters formula x(t) = etdxO

+ f~

e(t-a)dBlu(a)da = etdxO

+ {e"'dBlU(t

- a)da. (13.1.4)

13.2 Transient and Steady-State Partsof the Solution

Suppose that d is a stability matrix and that the control input function u(t) is bounded; that is, it satisfies /u(t)1 s b for some b e 9f for - 00 < t S O. (For example, it might be zero there.) Then the integral terms in (13.1.4) can be rewritten to obtain the decomposition (13.2.1 ) where x'(r) = etdxO,

1 00

x 2(t)

=

-

XS(t) =

f~

(13.2.2) ead Blu(t - a) do,

and 00

e(t-a:dBlu(a) do =

r

(13.2.3)

ead Blu(t - a) do.

(13.2.4)

In (13.2.1) the first part, x 1(t), is the solution to the uncontrolled equation and is a transient term associated with the initial value of the output x". Methods for computing x 1(t) in closed form, e.g., formula (13.1.2), methods for estimating the decay rate of Ix 1(t)I, and the possible qualitative behavior of that part of the output were covered earlier in the text. The second part, x 2 (t ), is likewise a transient term but it arises from the input u(t). The techniques developed also provide estimates of its decay rate, Ix 2(t)1

s

1 00

blBlI

leadl da,

(13.2.5)

for example, with leadl estimatable by methods applicable to x 1(t). The last term, XS(t), in (13.2.1) is called the steady-state solution. (Note that it is a solution to the differential control equation, although XS(O) i= xo.) It is the part of the solution that persists after the transients have died out. Equation (13.1.4) defines a linear integral transformation !I', the steady-state

263

13.3 The Frequency Response Function

~

u (t)

0--------

xs(t)

Fig. 13.2.1 The steady-state operator.

operator, which maps whatever linear space of control inputs it might be

allowed to act on onto the resultant linear space of steady-state responses (see Fig. 13.2.1). The relative importance of the three parts of the outputs x(t) to (13.1.3) depends on the application at hand. In some applications all three need to be considered to some extent. Most of the important information about the steady-state response of interest in the control of(13.1.3) is readily extractable from the frequency response function-the next item up for discussion.

13.3

The Frequency Response Function

In the typical situation (13.1.3) is composed of connected subsystems, some to be controlled and others built in by the engineer to do the controlling. A major question of interest concerns the modification of the amplitude of a control input as it passes through the system. In some situations it is desirable for a small-amplitude control input u i( . ) to produce a large value xit) of one of the output variables. An example would be a thrust control on an airplane in which the pilot inputs a low-energy command to increase the thrust of the engines as needed in takeoff. On the other hand, some of the inputs might carry high frequency noise from the equipment or other sources which must be attenuated (filtered out) as it passes through the system. Intuition would suggest that delays exhibited by the system in responding to the inputs would also influence important matters such as stability. These concerns lead to the notion of a frequency response function In (13.2.4) let j E {I, 2, ... , p} be fixed and consider the special control input u(t) = eirotej, with co real and ej the jth column of the p x p identity matrix. Since d is' assumed to be a stability matrix, the response is the n-vector function xj(t)

=

foro e
= eirot(iwln

-

d)-l:14 j

(13.3.1)

in which :14j is the jth column of :14. The vector function Fiiw)

=

(iwl n

-

d)-l:14j

(13.3.2)

264

13.

Applications in Control Theory

is called the frequency response function between thejth input variable the output x", Thus the response can be rewritten as xj(t)

= eiwtFiiw).

Uj

and

(13.3.3)

If the complex coordinate functions of Filw) are written in polar form,

i91J a 1 w )e (W ] a j~w)ei92j(W) 2 : '

i

Fiiw) = [

(13.3.4)

aniw)ei9nj(W)

which can be split into real and imaginary parts with the conclusion being that the equation xz/t) = akiw) sine cot + Okiw)]

(13.3.5)

describes the steady-state response to an input uj{t) = sin cot with the other input variables of u held at zero (k = 1,2, ... , n). The nonnegative function akiw) is called the gain, or amplitude response, and Okiw) is called the phase shift. Note that the column-vectors F/iw) could be arranged into a matrix F(iw) = [F1(iw), F 2(iw), ... , Fiiw)]

(13.3.6)

called the frequency response matrix. The frequency response matrix F(iw) provides a useful representation of the linear steady-state integral operator (13.2.4), which is dealt with at great length in a design course. The. classical approach to computing F(iw) is through the Laplace transform operator defined as the integral operator (.!l'f)(z)

=

t.() e-ztf(t)dt

(13.3.7)

which produces from the n-vector function f(t) the associated n-vector function .!l' f of the complex variable z. Using the easily derived operational property describing the action of .!l' on a derivative, (.!l' j')(z)

= z(.!l'f)(z) - f(O),

(13.3.8)

the Laplace transform of (13.1.3) produces the equation X(z) = (zI n

-

d)-lXO

+ (zI n

-

d)-l~U(Z)

(13.3.9)

that exhibits the dependence of X(z) (the Laplace transform of the response x(t) with initial value XO to the control input u(t» on XO and the Laplace transform U(z) of the input u(t). The matrix of rational functions of z G(z)

= (zI -

d)-l~

(13.3.10)

265

13.3 The Frequency Response Function

that appears in (13.3.9) is called the transfer function (matrix) of(13.1.3). The name derives from the relation X(z)

=

(13.3.11)

G(z)U(z)

that holds when XO = 0 in (13.3.9). In view of definition (13.3.2) the frequency response matrix is then classically computed as (13.3.12)

F(iw) = G(Z)lz=iro'

Example 13.3.1 The scalar equation

x + ax+

(13.3.13)

bx = u(t)

can be written in the first-order form (13.1.3) in which

.91=[0-b

1J

14 =

-a '

[~J

and u is one dimensional. For these matrices (13.3.10) computes to be G(z)

=

Z2

1 [lJ

(4)

+ az + b z '

13.3.1

and then application of(13.3.12) to (13.3.14) produces the frequency response matrix

in which and Equation (13.3.5) can be applied to read off the steady-state solution to the first-order representation of (13.3.13), S( ) _ t -

Xt

sin[wt

J (b -

W

+ 2)2

et(w)J

+ a 2 w2 '

S( ) w sin]'cot + et(w) X2 t = -r~===='~=~~ 2)2

J(b -

W

+

+ inJ a2 w 2

(13.3.16)

'

The amplitude dependence on w, the frequency of the input u(t) = sin cot, and the phase shift that occurs are apparent in (13.3.16).

266 13.4

13. Applications in Control Theory

The ABC Approach to Computing Frequency ResponseMatrices

For a number of reasons, formulas (13.3.10) and (13.3.2) or (13.3.12) are not very effective computationally for anything but low-dimensional systems. The main difficulty is that they involve inversion of a matrix containing a variable-a task that generally entails a considerable amount of algebra. Moreover, the fullest reduction in that algebra is not realized in problems requiring the computation of only a few elements of G(z) or F(iw). An alternative method for computing the elements of G(z) (and hence of F(iw» is derived in Theorem 13.4.1 by application of the ABC formula (13.1.2). To a considerable degree it overcomes the mentioned difficulties and enhances the possibilities for exploiting the use of computers in the treatment of large-scale systems. Theorem 13.4.1

Separation Theorem. G(z)

=

... , Pp~(z)],

(13.4.1)

,s;1n-l,qjj][YO, Byo, ... , Bn-lyO]-I,

(13.4.2)

[Pl~(Z),

P2~(Z),

in which

r, = [,qjj' ,s;1,qjj"'"

where ,qjjis the jth column of ,qj and (13.4.3)

is the Laplace transform ofy(t) (j = 1, 2, ... , p). Proof.

From (13.1.2) and (13.4.2) it is clear that (13.4.4)

In view of the definition of Band y(t), we further have (13.4.5) The Laplace transformed forms of equations (13.4.4) and (13.4.5) imply that (13.4.6) in which ~(z) is defined by Eq. (13.4.3) (j = 1, 2, ... , p). Equations (13.4.6) are the column-wise statement of (13.4.1), and the proof is complete.

13.4 The ABC Approach to Computating Frequency Response Matrices

267

Remark 13.4.1 Theorem 13.4.1 separates the arithmetic from the algebra required in computing the transfer function G(z). The ABC formula implies that x(t) = Cy(t) is the solution of (13.1.1) where the matrix C is produced by the row reduction

In a similar fashion the matrices Pj required in the representation (13.4.1) of G(z) would be computed by the row reduction

Columns of the augmented matrix associated with those elements of G(z) of no interest would be deleted from the matrix before doing the row reduction to avoid unnecessary arithmetic. No numerical computation is required to determine ( ( z ) in (13.4.1). Its coordinates are the Laplace transforms of the coordinate functions of y(t), which are available in standard tables of Laplace transforms and which could in fact be written down from the ordered eigenvalues of d. Example 13.4.1 To demonstrate the computational procedure, based on Theorem 13.4.1, for determining G(z) and F ( i o ) , consider the system (13.1.3) with coefficient matrices

The eigenvalues of d are A = -2, -2, -2 ? 2i. They determine y ( t ) = [e-", te-", e-" cos 2t, e-" sin 2~1'. Differentiation of this vector function

268

13. Applications in Control Theory

determines B, and its Laplace transform provides

~(z)

1

z+2

1

JT

2

+ 2)2 + 4' (z+ 2)2 + 4 . (13.4.10)

= [ Z _ 2 ' (z - 2)2' (z

Suppose that the problem is to compute the frequency response of X 4 to input Ul' Thus only G41(Z) is to be found. For this only the first two blocks of the data matrix (13.4.8) have to be computed, providing the matrix

[

-1]

o o o

1 o4 4 -4 4 -4 0-8 12 -16 -8 . 40 0 -8 12 16 16 8 -32

- ~ ~ -~ ~

(13.4.11)

Since U2 is not involved, only the last column of the second block of (13.4.11) needs to be retained for doing the row reduction

[1 0 I

0 -2 1 -2 2 4 -4 0-8 -8 12 16 16

0 1 0 0 0 0

-~]-[~

-8

o

0 0 1 0

0 0 0 1

-fl

(13.4.12)

giving G41(Z) = [Pl~(Z)J4

2 (z - 2)2

=

[0

2 -1

0J~(z)

z+2 (z + 2f + 4'

(13.4.13)

which in turn produces the frequency response function F 41(iW) = G4 1(iw). Remark 13.4.2 The frequency response matrix can be split into real and imaginary parts according to the equation F(iw)

=

-d(w 2In

+d

2 ) - l fJ1 -

iw(w 2In

+d

2 ) - l fJB .

(13.4.14)

Remark 13.4.3 Although the definition of a frequency response matrix was motivated by the notion of a steady-state solution with the assumption that d is a stability matrix, the definition (13.3.2) makes sense for any matrix d at all real «o for which ico is not an eigenvalue of d. Formula (13.4.1) is, of course, valid for any d.

269

13.5 Controllability

EXERCISES

1. Derive Eq. (13.4.14). 2. Compute the amplitude responses, phase shifts, and frequency response matrix by the formulas appearing in Theorem 13.4.1 for the first-ordersystem form of the equation x + a x + bx = u that uses each of the sets of parameter values (a) (b)

(c)

a = 3, b = 2; a = 2, b = 1; a = 4, b = 13.

Check your results against the conclusions of Example 13.3.1. 3. Compute the amplitude response and phase shift between the input and output x 1 in the system

4.

U2

Using the notation of Section 13.4, derive the equation _ B)(yO,B yO, ... ,Bn- 1yO)]-lyO =

(iw - d)(gjj,dgjj, ... ,dn-1~j)[(iw

~j

(j = 1,2, ... ,p).

13.5 Controllability

The differential control equation in (13.1.3) is called controllable if for each x'', Xl in tYr, and each compact interval [to, t 1] there exists a continuous control function u(t) on [to, t 1 ] for which the response x(t) satisfies the boundary conditions x(to) = XO and x(t 1) = x'. Since the coefficients d and gj are constant, to can be taken to be zero and t 1 will be denoted simply by T. It turns out that the controllability condition is a property of the pair d, ~ and is independent of T. Theorem 13.5.1 . The differential control equation in f!lin with control variable u in f!liP

x= is controllable

equals n.

dx

+ ~u

(13.5.1)

if and only if the rank of the n

x np matrix 1gj] n[gj, dgj, ... , d

(13.5.2)

270

13.

Applications in Control Theory

Proof. Assume that the matrix in (13.5.2) has rank n. It will first be shown that, consequently, the symmetric matrix tld• ST = f: etld f!4f!4*e da

(13.5.3)

is nonsingular. (In (13.5.3), * denotes the transpose of a matrix.) It is sufficient to show that ST is positive-definite. Suppose that it were not so that for some ~ #- in f7ln

°

~'ST~

=

f:If!4*etld·~12dq

= 0.

(13.5.4)

By continuity this would be possible only if f!4*e tld• ~ =

°

(13.5.5)

for all a E [0, T]. Differentiation of (13.5.5) n - 1 times and evaluation at = produce the equations

°

o

(13.5.6) (k

= 0, 1,2, ... , n - 1). But equations (13.5.6) say that the coordinates

of

~

~

provide a linear combination of the rows of (13.5.2) that is zero. Since

#- 0, this contradicts the assumption that (13.5.2) has full rank, and thus

ST is nonsingular. Select any Xo and Xl in 9(n. By application of the variationof-parameters formula it can easily be checked that the control function u(t) = f!4*e(T-t)d·Si l[x l - eTdxO] produces a response x(t) to (13.5.1) that satisfies the boundary conditions x(o) = x", x(T) = x'. To prove the converse, assume that the matrix in (13.5.2) has rank r less than n. Therefore equations (13.5.6) have a solution ~ #- in f7ln. Choose XO = and consider the response at t = T to any continuous control function u(t) on [0, T],

°

°

x(T) = f: etldf!4u(T - rr] do,

Computation of the inner product of this vector with x(T)'

~

(13.5.7) ~

= f: u(T - rr}: f!4*e tld• ~ de.

gives (13.5.8)

By writing out the exponential series and applying the Cayley-Hamilton theorem and then (13.5.6), Eq. (13.5.5) is seen to hold on [0, T]. This applied to (13.5.8) implies that x(T) . ~ = 0, independently of the choice of control function u(t). This proves that (13.5.1) then fails to be controllable since

271

13.5 Controllability

x(T) is contained in the orthogonal complement of~, which necessarily has dimension greater than zero since ~ #- O. This concludes the proof.

The matrix (13.5.2) is called the controllability matrix associated with Eq. (13.5.1). If fJl = b is n x 1, then the rank condition for controllability given in Theorem 13.5.1 is equivalent to the condition Remark 13.5.1

(13.5.9) This situation arises if all control variable coordinates but one are set equal to zero-the scalar control situation. Note that if (13.5.1) is controllable relative to the jth control variable alone, then the corresponding matrix P, in Theorem 13.4.1 will be nonsingular. The next theorem describes the extent to which controllability can fail by examining the set K T of end points of trajectories initiating from XO = O. Theorem 13.5.2 Let C; be the linear space ofcontinuous real p vector-valued control functions on £1l and for fixed T > 0 consider the set KT

= {x =

f:

eadfJlu(T - a) dalu(')

E

Cp }

.

(13.5.10)

Then K T , called the reachable set from the origin, is a linear, invariant subspace of £1l n, independent of T. Furthermore, K T = the column space of[fJl, dfl4, ... , d n - 1f14 ] , (13.5.11) dim K = rank[fJl, dfJl, ... , d n - 1f14 ] , (13.5.12) T

and the restriction of(13.5.1) to K T is controllable. Proof. To show that K T is linear, select any two points from that set. They are then x 1(T ), x 2( T ), where x'(r) and x 2(t) are solutions to (13.5.1), with initial value XO = 0, that correspond to respective control functions u1 , u2 in Cpo For arbitrary real coefficients c1 , C2 the computation CIX1(T)

+ c 2x Z(T) =

f:

e adfJl[cI U1(T - a)

+ C2 U2(T

- a)] da

(13.5.13)

shows that [C 1x 1(T) + c zx 2(T)] E K T since [CIU1 + czUZ ] E Cpo Thus K T is a linear subspace of £1ln. One consequence of this is that £1ln = K T EB K}, where K} is the orthogonal complement of K T in £1ln. Therefore for some nonsingular matrix PT each x E £1ln can be written uniquely as (13.5.14)

272

13. Applications in Control Theory

In terms of these coordinates the differential control equation (13.5.1) is

d

dt

=

X2

= PildPT

r

in which d T accordingly, [=:Jet)

[Xl]

=

and fJlT

eUNTfJlTu(t - e) da

But

[Xl] + fJlTu,

d T

GJ

= PilfJI.

=

(T)

r[::i:G

E

KT

By

u(t - o) da.

(13.5.16)

;

hence x2(T) = 0, which says that

SOT
(13.5.15)

X2

- e) da =

°

°

(13.5.17)

for each u E Cpo This implies that 0. Furthermore, we conclude that K T is invariant (that is, if a solution curve to (13.5.1) passes through a point of K T , then it remains in K T for all values of t E 9f). We have already shown that eUNTfJlT =

for all a

E

[
°

(13.5.18)

9f. Partitioning

and setting a = partitioning

°

in (13.5.18) imply that fJl2 = 0. Finally, by accordingly d

T

=

Id

u .91 1, .91 2 1 .9122 12

Eq. (13.5.15) can be written as the pair of vector equations

Xl dUXl + d12X2 + fJllu, d2lXl + d x + Ou. =

X2

=

22

2

(13.5.19) (13.5.20)

273

13.5 Controllability

If a solution has xiO) = 0, then X2(t) = 0 for all t, independent of the choice ofxl(O). In view of (13.5.20) this can happen only if .91 2 1 = O. Thus, in terms of the Xl' X2 coordinate system, (13.5.1) reduces to

+ PA 1u + d. 1 2 X 2 ,

Xl = d u x 1

x2 =

(13.5.21) (13.5.22)

d 22x 2'

We shall next show that (13.5.23) = r, rank [PAl , d uPA 1, ... , d~11PA1] where r = dim K T . Since it is easily shown that the rank of the controllability matrix is invariant relative to similarity transformations on f}fn, this will imply that rank[PA, dPA, ... , d r - 1PA] = r. (13.5.24)

Consider (13.5.21) with X2(0) = 0 and hence xit) = 0 for all t. The controllability of the resultant equation (13.5.21) already established by construction will be used to prove (13.5.23). To prove the contrapositive proposition, suppose that there exists a "# 0 in f}fr such that

e

k = 0,1,2, ... , r - 1.

PAt(dtl)ke = 0,

(13.5.25)

But again, by the Cayley-Hamilton theorem, it follows that etld:.

=

p-l

L

(13.5.26)

fi(J)(dtd

k=O

for some continuous scalar functions fk' Therefore, PAtetld:·e = 0

(13.5.27)

for all (J. Taking the inner product of u(T - (J) with (13.5.27) and integrating lead to the equation

e· faT etld llPA 1u(T -. (J) da = 0

(13.5.28)

for all U E Cpo This implies that dim K T < r, a contradiction. The last detail is to show that the column space[PA, dPA, ... , d n - 1PA] = K • T

(13.5.29)

By the Cayley-Hamilton theorem, etld

=

n-l

L

k=O

IXk«(J)d k

(13.5.30)

274

13. Applications in Control Theory

for continuous, scalar ociu). Now for arbitrary

re Jo T

a.9lI!4u(T

_ e) do =

U E

C p compute

r ociu)u(T Jo T

ni,l

d

kI!4

k=O

u)du.

(13.5.31)

Equation (13.5.31) shows that K T c column space[I!4, dI!4, ... , d

n

-

lI!4].

(13.5.32)

But it was already proved that both linear subspaces in (13.5.32) have the same dimension. Hence the subspaces are the same, and Theorem 13.5.2 is established. . Remark 13.5.2 The subsystems (13.5.21) and (13.5.22) are sometimes referred to as the controllable and uncontrollable parts, respectively, of (13.5.1). No control variables whatsoever enter into the uncontrollable part. If its coefficient matrix d 22 is a stability matrix, then in applications it ordinarily creates no serious problem-other than being uncontrollable. A few of the many far-reaching implications of system controllability will be discussed in the next section of this chapter.

EXERCISES

1. Find an example of a linear differential control equation in fJ1l3 having two control variables which is not controllable but has a control variable entering each scalar equation explicitly. 2. Test the following to see which are controllable:

Xl = {~2=Xl-U, =

{Xl

X 2,

(a)

X3

Xl

(b)

+

U.

=

X2 -

X3'

~2=X3+Ul' X3

=

-X2 -

u2 ·

3. Show that system controllability is invariant relative to a nonsingular linear change of state coordinates. 4. Verify that the first-order-system form of an nth-order scalar equation x(n) + alX(n-l) + .. , anx = u is controllable. 5. Find a basis for the reachable set from the origin and thus find the dimension of the controllable part of the system

+

+ Uh

X3 =

X2

X4 =

U2 -

ul

·

275

13.6 Altering System Dynamics by Linear Feedback

6. By choosing a suitable coordinate system, compute the differential equations for the controllable and uncontrollable parts of the system Xl

=

X2 = X3

7.

=

+ U, -Xl + X 2 - X 3 + U, Xl - X2 + X 3 - U. X2

Consider the scalar control system X = dx + bu, X E Derive the formula for the frequency response function

[}In,

b E [}In,

U E [}l.

F(iw)

=

[b,db, ... ,dn-lb][yO,ByO, ... ,Bn-lyO]-I(iwln - B)-lyO,

where yO and B are defined in (13.1.2). 13.6 Altering System Dynamics by Linear Feedback

The matrix d in (13.1.1) determines the nature of the solutions to that equation. For systems (13.1.3) with control variables entering, there is the possibility of altering various aspects of the system behavior by introducing an additional equation u = Kx

+

v

(13.6.1)

in which K is a real p x n matrix. Substitution of(13.6.1)into (13.1.3)produces another differential control equation X= d

KX

+

PAv

(13.6.2)

in which d K = d + fJlK and v is the new control (input) variable. Equation (13.6.1) is called a linear feedback controller. (See Fig. 1.4.2, in which Sl is identified with (13.1.3) and S2 with (13.6.1).) For the initial condition fixed, each choice of control function v(t) in (13.6.2) produces a response x(t) that, when substituted into (13.6.1) along with v(t), generates a control input u(t) to the original system (13.1.3) having the same response x(t). Thus the input u(t) at time t is partly due to v(t), which enters directly into (13.6.1) and is also dependent on the current or "fed back" value x(t) of the output state. In particular, if the input v(t) to (13.6.2) is taken to be identically zero, then (13.6.2) is of the homogeneous type (13.1.1) with new coefficient d K • Now it is possible to influence the eigenvalues of «« and hence the dynamic behavior of (13.6.2)-called a closed-loop form of (13.1.3)-by the choice of the feedback matrix K. A substantial segment of the extensive theory of feedback controller design centers about the extent to which the feedback matrix K influences the eigenvalues (also called closed-loop poles) of d k and the frequency

276

13. Applications in Control Theory

response matrix Fk(iw) = [iwl, - d K ] - l & . The following results show that the extent is considerable for systems that are controllable. It is convenient to begin with the special case in which the matrix 549 in (13.1.3) is a real n-vector b and u is a scalar control variable.

Theorem 13.6.1

Assume that the scalar control system in A?" i= d x

+ bu

(13.6.3)

is controllable (e.g., (13.5.9) holds). Then for each real monic polynomial q of degree n there exists a unique k E ansuch that the characteristic polynomial of the closed-loop coefficient matrix d k = d + bk* resulting from the feedback controller u = k * x + v has q as its characteristic polynomial. (Here, k* denotes the transpose of k.) Proof. In light of the controllability assumption, the n-vectors b, d b , . . .,

d"'b constitute a basis for 9". Hence it is possible to make the nonsingular change of variables

x

= y,d"-'b

+ y2M"'2b + ... + y,b.

(13.6.4)

By substituting (13.6.4) into (13.6.3) and applying the Cayley-Hamilton theorem, it is a simple matter to check that in terms of the'n-vector y, with the yi as coordinates, system (13.6.3) transforms into

j=J y

+ e,u,

(13.6.5)

where

(13.6.6)

in which the aiarise from the coefficients in the characteristic equation for d , An - u n - l A n - l - a,-2 1"-2 - ... -ao = 0, and en is the last column of the n x n identity matrix, Keep in mind the fact that a nonsingular linear change of state coordinates preserves both system controllability and the characteristic polynomial of the state coefficient matrix. Consider the first-order form of the scalar equation w(n)

-

a n - 1W f n - l )

- *.

i =doz

. - MOW

+ enu

= u,

(1 3.6.7) (13.6.8)

277

13.6 Altering System Dynamics by Linear Feedback

which has

r

0 0

1 0

0 1

-.. 0 1 (13.6.9)

In view of the coefficients in (13.6.5) that resulted from the transformation (13.6.4) applied to f13.6.3), the (nonsingular) transformation z = y1dc-'en

+ y 2 d ; f - ' e n + ... + y,en

(13.6.10)

must produce the same equation (13.6.5). (The controllability of (13.6.7) appeared as Exercise 4, Section 13.5.) Thus it is sufficient to prove Theorem 13.6.1 for system (13.6.8). But from (13.6.7) it is clear that if q = 1" - pn -

in-1 1

-

... - Po

is the characteristic polynomial to be realized, then the feedback control function u = koz, + k , ~ + , ... + kn-lzn in which ki = /?, - a, ( i = 0, 1,2, . . ., n - 1) does produce a matrix d o enk* with that q as its characteristic polynomial. This completes the proof.

+

Remark 13.6.1

The Pole Relocation Formula.

1" - an - 1 .p

- 1

Let

- a n - 2 2 - 2 - ... - a. = 0

be the characteristic polynomial of d, dobe the matrix (13.6.9) written fromitscoefficients,andq = 1" - pn-lAn-l - pn - 2 An-2 - Po. Denote as fl - CL the column n-vector with elements

( B - a)i = pi - a,

( i = 0, 1,.

,

., n

-

1).

Examination of its proof shows that the formula for the k that was concluded to exist in Theorem 13.6.1 is k = [ d " - ' b , ~ d ~,..., - ~ bb ] * - 1 [ d : - 1 e n , d " , 2 e n,...,e,,]*(P - a). (13.6.11)

An obvious corollary of Theorem 13.6.1 is that a controllable scalar control system (13.6.3) is stabilizable by linear feedback; that is, there exist feedback control functions u = k - x + v for which d,= d + bk* is a stability matrix. It is worth noting that the proof of Theorem 13.6.1 showed that a controllable scalar control system (13.6.3), relative to a nonsingular linear change of coordinates, is the first-order form of an nth-order scalar control

278

13. Applications in Control Theory

equation of type (13.6.7). Finally, observe that the choice of the feedback controller u = k· x + v referred to in Theorem 13.6.1 is unique. The potential that exists for altering the dynamical behavior of a controllable system by using feedback is explored next for the case in which the system has multiple control variables. Remark 13.6.2 Remodeling the Control Term. The control variables in (13.5.1) are just the coefficients for the column space of 81. If the columns of 81 = [bt, b2 , ••• , bp ] are linearly independent, then 81*81 is ap x pinvertible matrix. If the columns are orthogonal, then 81*81 = A, a diagonal matrix, and, in particular, if the columns of 81 are orthonormal, then 81*81 = I p • By renumbering the control variables, the columns of 81 can be permuted into any order. In particular, any maximal linearly independent subset of q columns can be made to occupy the first positions 81 = [bt, b2 , ••. ,b q , ••• ,b p ] . Then the special change of control variables u = (Ut,U2,""Uq,O,oo.,O)* remodels the control term as 81u = [ju in which [j = [bl> b2 , ••• , bq ] and 17 = (Ul> 17 2 " " , Uq )* with [j*[j invertible and no loss in controllability. Thus there is little lost in assuming that 81 has full rank. If 81 is of full rank, then by the Gram-Schmidt orthogonalization process a nonsingular p x p matrix Q can be computed for which the change of variable u = Qu gives 81u = [ju where [j = 81Q satisfies [j*[j = I p , and again there is no loss in controllability although the Ui are now "mixed together" to produce u-a point that is sometimes of physical significance. Example 13.6.1 Suppose that the term 81u bearing the control variables in (13.5.1) were the one following. The remodeling could proceed as

The relationship between the two sets of control variables is

Note that

is invertible as anticipated. Any requirement that the columns of the coefficient matrix be orthonormal

279

13.6 Altering System Dynamics by Linear Feedback

could be met by the second round of remodeling

s«

=

[~o ~1] [UIJ [~0 ~1] [10 -II1/ fiJ [~IJ fi =

Uz

Uz

0

=

[

llfi] U_I

o 1/.Ji [uJ 1

0

Note that now u = Qii, ~ = fJIQ, ~*~

-

= fJIii

=

.

/z for nonsingular

l fi J _[1 -lllfi

Q- 0 which mixes iiI' ii z.

The following theorem is from Heymann [10]' Theorem 13.6.2 Reduction to Scalar Control Using Feedback. that (13.5.1) is controllable. Choose any nonzero column b.from

Suppose

81 = [bl,bz, ... ,bp ]. Then there exists a feedback controller of the type

(13.6.12) for which the resultant scalar control system

x=

(d

+ fJIKdx +

biDi

(13.6.13)

is controllable. Proof. Renumber the control variables so that i becomes 1. Rearrange the columns of the controllability matrix (13.5.2) to get the matrix [bl' .9/.b l, ... , d n- 1b l l " ·Ib p , db p , ••• , dn-Ib p ] (13.6.14)

that likewise has rank n because of the controllability assumption. Passing through the columns of (13.6.14) from left to right, select certain columns according to the following procedure: Select b i but throw out any column that is a linear combination of the columns previously selected. After an appropriate renumbering of the control variables U z, u 3 ' ••• , up, the resulting matrix is of the type P = [br.dbl, ... ,dr.-Ibll"·lbq,dbq, ... ,drq-lbq]

(13.6.15)

280

13. Applications in Control Theory

in which b l, b2, ... , bq , but not necessarily all of b., b2, ... , bp , appear and dr;b i is a linear combination of those columns appearing to the left of and including dr;-Ib i (i = 1,2, ... , q) in (13.6.15). By the procedure used in making the selection, the matrix P has columns that are linearly independent. Moreover, P has rank n since the only columns of (13.6.14) thrown out were those linearly dependent on those retained. In particular, r l + r 2 + ,.'. + r q = nand P is nonsingular. Consider as the candidate matrix K in the controller u = Kx + b, v that is being sought the matrix K = EP- I

(13.6.16)

in which E is the p x n matrix whose columns E k (k = 1,2, ... , n) are defined to be E kJ = ej+1 for j = 1,2, ... , q - 1 in which j

kj =

L r,

(13.6.17)

i= 1

and ej + 1 is the (j + 1)st column of the identity matrix Ip and Ek otherwise. Since KP = E, it is easy to see that (j

= 1,2, ... , q

- 1)

=0

(13.6.18)

and Kdkb j = 0,

(13.6.19)

where 0 S k S rj - 2 for j = 1,2, ... , q - 1 and 0 s k s "« - 1 for j = q. Let d * = d + f!JK and let P* be the controllability matrix associated with the scalar control system x = d *x + bID, P* = [b l , d *b l

, ... ,

d:-Ib i

J.

(13.6.20)

To see that P* has rank n, apply (13.6.18) and (13.6.19) along with the definition of the riS to compute bi

=

b.;

d *~I

= (d + f!JK)b l = dbl.

d;b l

=

(d

= (d

d~I-lbl

d~lbl

= (d

+ f!JK)db l

= d

2b l,

+ f!JK)d rl- 2bl = d r, -lb l, + f!JK)d r, -ib i = drlb i + f!Je2

(13.6.21)

= b 2 + "', d~'

+ Ib i = (d

+ f!JK)b 2 + ...

= db 2

+ .. "

(Eq. (13.6.21) cont.)

13.6 Altering System Dynamics by Linear Feedback

281

= (d + PlK)(d'P- 2b p) + ...

d~-lbl

= drp-1b p + ... , in which the dots (... ) denote linear combinations of preceding vectors. It is apparent that the vectors generated in (13.6.21) are linearly independent since (13.6.15) is of full rank. This concludes the proof. Remark 13.6.3 The conclusion of Theorem 13.6.2is quite remarkable. The closed-loop system (13.6.13) is the one that results from installing the feedback controller u = Kix + v and then setting all inputs but Vi equal to zero. In other words, the steering that needs to be done for controllability can be accomplished by introducing direct control through only one and anyone control variable so long as the responses to that input are properly fed back into the original control slots. A familiar application of the above idea appears in automobiles that have automatic transmissions. Roughly speaking, the mechanism combines the two input variables-acceleration position and gear shift of a conventional car-into a single accelerator variable. In principle, through feedback all the control variables, including steering wheel position, brakes, etc., could be reduced to only one control input. Of course, human ingenuity had long ago already created one such device-the chauffeur. It should be noted that the proof of Theorem 13.6.2 contains an algorithm for computing K i • This is significant, for it provides a basis for doing the necessary computations in pole relocation. (See the proof of the following corollary.) Corollary 13.6.3 Pole Relocation. Assume that (13.5.1) is controllable. Then corresponding to each real monic polynomial q of degree n there exists a linear feedback controller u = Kx + v such that the characteristic polynomial ofthe closed-loop coefficient matrix d K = d + fJlK has q as its characteristic polynomial. Proof. Choose any i E {I, 2, ... , p} for which b, #- O. Application of Theorem 13.6.1 to the scalar control system (13.6.13) shows that the controller u = (K i + eik*)x + v satisfies the requirements.

The controller performing the pole relocation in Corollary 13.6.3 is obviously not unique.

Remark 13.6.4

Example 13.6.2

Consider the system

x=

y,

y = -x + u2 ,

(13.6.22)

282

13.

Applications in Control Theory

which is controllable. Note that the system is not controllable relative to u1 alone. Thus, consider the problem of computing a feedback controller that transfers control to ul. Application of the formulas appearing in the proof of Theorem 13.6.2 gives

P rl

=

1, r2

=

=

1I ;;]

= P-1,

2, q = 2, kl = 1, k , = 3,

The controller is u = K

3 Z

+u=[z:.j

and the resulting closed-loop system

1

= y,

i=u,

j = --x+z+u,,

(1 3.6.23)

is now controllable relative to u l . The eigenvalues of the state coefficient matrix d of (13.6.23) can be checked to be 1 = 0, k i, which is to say that the characteristic polynomial is 1(12 + 1) = A3 + 1.Let the second problem be to relocate the poles to 1 = 1, 1,2. Thus the desired characteristic polynomial is q

=

(A - 1)2(A - 2)

In addition to the matrices

[.

= A3 -

41,

+ 5A - 2.

1. b=[l c: 3

0 1 0 d = -1 0 1 0

that are read off (13.6.23), the data needed for computing k from formula (13.6.11) are

d,= 0

0 1 ,

283

13.6 Altering System Dynamics by Linear Feedback

read off from the characteristic polynomial of .91, and

p-.~

HJ HJ Hl

computed from q and the characteristic polynomial of d. Application of formula (13.6.11) gives k* = ( - 2, - 4,4) and the required controller is then VI = - 2x - 4y + 4z + WI' Substitution of this controller into (13.6.23) can be checked to yield a coefficient matrix with eigenvalues A. = 1, 1,2, as required. Note that in Example 13.6.2 the original system (13.6.22) naturally splits into a pair of uncoupled subsystems-one of dim 2 and the other of dim 1. The feedback controller computed for transferring control to Ul tied these subsystems together. In many applications dealing with multi-input systems it is necessary to take an opposite point of view; that is, it is important that certain input variables interact only with specific output variables. For example, it would be extremely dangerous to drive an automobile in which flexing of the body due to braking or turning interacted with the accelerator mechanism. The problems of decoupling and, more generally, of noninteractive control will be discussed in the next section. Transfer of control and pole relocation, using feedback, will be important parts of that process.

EXERCISES

1. Compute a linear feedback controller that relocates the poles of the system

x = y,

y=z+U,

z=x

to A. = - 1, -1 ± i. 2. Find a feedback controller for the system (13.6.22) of Example 13.6.2 that transfers control to the second control variable U 2• Then compute further feedback that restores the resulting poles to those of the original system. . 3. Consider the span of the vectors b., db;, ... , d r i - 1b; in which r, is the largest integer for which that collection is linearly independent. This subspace of f7tn is often called a controllability subspace. What is the control-theoretic significance of that subspace and its dimension insofar as the feedback capabilities of the corresponding control variable U i are concerned? 4. In the matrix P of (13.6.15) not all of the columns of PA might appear. Of what significance is that? 5. If(13.5.1) is controllable and consists oftwo decoupled subsystems, then,

284

13.

Applications in Control Theory

as pointed out in the discussion of Example 13.6.2, a feedback controller u = Kx + v can be found such that the resulting system x = (d + flIK)x + flIv is controllable relative to Vt alone. Verifythat, by repetition ofthis process it is possible to find a K such that the mentioned system is controllable relative to each of the ViS alone. Thus the system is tightly tied together. If all but one control variable Vi were "lost," controllability would still be maintained. Note that any feedback inserted can be subtracted back out by additional feedback. Thus while addition of feedback will always preserve system controllability, it can destroy controllability relative to individual input variables.

13.7 Decoupling and Recoupling-Noninteractive Design

There are intrinsic limits to the extent to which one subset of inputs and outputs of a system can be decoupled from another such subset. On the basis of intuition one might concede that the position and velocity coordinates of a mass in a forced spring-mass system could not be decoupled. However, in a more complex system, as depicted in Fig. 1.2.4, involving two such systems that are connected, it seems conceivable that a feedback system could be devised to measure the force exerted on the first mass by the second. This quantity could be fed into U t with a reversal of direction to counteract the influence of the second mass on the first, and then the closed-loop response of mt should resemble that of a single mass system. In this section it will be shown that such possibilities exist in much more subtle situations through indirect feedback. Once a system has been split properly into noninteracting subsystems, these subsystems can be tied back together in new ways with their dynamics readjusted to satisfy additional requirements of the overall system design. This is the subject matter of the present section. Before assessing the possibilities for decoupling by using feedback, it is useful to consider the question of reducing the number of control variables without using feedback. .The proof of the following theorem can be found in Lee and Markus [13]' Theorem 13.7.1 Let the system (13.5.1) be controllable. Then there exists a real constant vector c E f7lP such that the scalar control system

x = dx + (flIc)il(t)

(13.7.1)

that results from the (open-loop) controller u = cp. is controllable if and only if each pair ofdistinct elementary Jordan canonical blocks ofd contains unequal / eigenvalues of d. Remark 13.7.1

There is an important application of Theorem 13.7.1.

13.7

285

Decoupling and Recoupling-Noninteractive Design

Suppose that one has two controllable systems of type (13.5.1): (1 3.7.2)

(13.7.3) with x i E Pi,uiE g P(i i = 1,2). The question is whether or not these systems can be tied together into a single controllable scalar control system without using feedback. Consider using the open-loop controller

I:[ [:I. =

in which ciE g P i( i = 1,2) is a constant vector and jx is a scalar control variable. It is easy to see that Theorem 13.7.1 implies that the answer is in the affirmative if and only if d l and d,have no common eigenvalue and if each of d , ,d , satisfies the condition stated in the hypothesis of the mentioned theorem. A sufficient condition would be that each of d l ,d 2is similar to an elementary Jordan block with the two blocks containing unequal eigenvalues. The necessary and sufficient conditions for tying together a larger number of systems are now obvious. If a controllable system were to be split into a collection of controllable subsystems, decoupled at both the input and output ends, then it is clear that at least one input variable would be needed for each subsystem in order to maintain control. The next theorem states that the number of decoupled subsystems can always be taken to be as great as the number of (independent) input variables.

Remark 13.7.2 To a considerable extent the Jordan form of the closed-loop matrix can be set by the insertion of feedback in a controllable system. According to Theorem 13.7.1 the coefficient d W K i of Theorem 13.6.2 has one elementary Jordan block associated with each distinct eigenvaiue. Moreover, that property persists for the coefficient d + @Ki bik* that results from the insertion of further feedback vi = k * x w. But Corollary 13.6.3 says that the eigenvalues along with their multiplicitiescan be regulated through k. €n particular, the sizes of the elementary Jordan blocks can be set by the choice of the multiplicities.

+

+

+

Theorem 13.7.2 Decoupling. Assume that (13.5.1) is controllable and that the p columns of @ are linearly independent. Then there exist positive integers r l , r,, . . ., rp satisfying rl + r, + . . . + rp = n, a linear feedback controller u = K x + w,and real nonsingular linear changes of state coordinates y = P x and control’coordinates w = Qv relative to which (13.5.1) splits into a collection of decoupled scalar control subsystems in W r i ,

3’

=

S,yi

+ eiui

(i = I , & . . ., p).

(13.7.4)

286

13. Applications in Control Theory

In (13.7.4) biis the ri x ri matrix 0 1 0 0 0 1

'.. 0

o...;]

/:[;

-

(13.7.5)

0 0 0 ... and e' is the last column of the identity matrix I r i . The ri are uniquely determined upon imposition of the order condition rI I rz I . . . I r p . (by d and 9)

The canonical form (13.7.4), (13.7.5), generally attributed to Brunovsky [3], has a complicated pre- and posthistory. Proofs of Theorem 13.7.2 can be found in the works of Wonham and Morse [28], Warren and Eckberg [25], and others in the fashionable setting of abstract algebra and vector spaces. Rather than settle for the addition of another existence proof, this book goes beyond to what seems to be missing in the literature, namely, a relatively efficient matrix algorithm for computing the canonical form and the associated feedback controller that does the decoupling. The proof of Theorem 13.7.2, as well as other useful information concerning the decoupling, follows automatically from the derivation of the algorithm.

I,

33

d9

...

(13.7.6)

As the computation proceeds, this basic block structure persists although the contents of the subblocks change. The first step in the computation will be to perform on that part of (13.7.6), excluding the right-end block, elementary row operations that transform the controllability matrix into row-reduced echelon form. This process, to be called row reduction, will be repeated a number of times throughout the calculation.

287

13.7 Decoupling and Recoupling-Noninteractive Design

As a consequence of the assumptions that the columns of 9 are linearly independent and that d,9 is controllable, it follows that the (row-) reduced form of (13.7.6) can be written as I,

...

d9

&?

I

Pl

I

I,

I

I I

I

0

d"l 9 I I I I

IP

I

Fl

I

(13.7.7)

I

10000 00100 01000

100 010 001

000 100 000

000 000 021

000 000 000

311 020 000

00001 00010

000 000

010 001

001 001

001 001

001 001

100 010 001

(13.7.8)

which has three cycles, including a 1-cycle and a 2-cycle with the third cycle not being closed. The Diagonal Case. It is convenient to begin with the special case in which the matrix Fl in (13.7.7)has all its leading ones falling along a diagonal. The general problem will later be reduced to working with the diagonal case in a slightly different context. In the special case under consideration, (13.7.7)can be written in the refined

288

13. Applications in Control Theory

block form

P1

t,

0

H1

T1

0

I n- p

G1

L1

Q1

I

(13.7.9)

in which G1 is (n - p) x p and H 1 is p x p. The matrices T 1 and L 1 will be of little interest in the diagonal case. At this point the matrix Q1 is just I p • Note that in the case of scalar control p = 1, the controllability matrix in (13.7.7) must be computed one more term to dnfJI in order that there be "room" for the matrix G1 • The operations that need to be performed on (13.7.9) are listed in the following steps: Step 1. Column reduce. Step 2. Insert the first stage of feedback, recompute the controllability matrix, and then row reduce. Step 3. Repeat the process of step 2 for each of the successive stages of feedback. Step 4. Read off the ordered cycle basis and cycle lengths. Step 5. Read off P and Q and then compute K

=

(fJI*fJI)-1fJ1*[p- 1,2"P -

.91].

(13.7.10)

The main objective of steps 1-3 is to force the matrix in the block of (13.7.9) initially occupied by G1 to be zero. Clearly, that would close all

the cycles. Step 1 is bypassed if p divides n. Hence assume that n = (q + l)p + P with q a nonnegative integer and 0 :s;; p < p. The immediate goal of step 1 is to get the bottom left p x (p - p) corner subblock of G1 in (13.7.9) to be zero. This is accomplished by making the last p columns of the matrix I n - p in (13.7.9) operate upon the first p - p columns ofG 1 . This is done as follows: +---;-

) p

1

P

,--------------- ---------, 11 0 ... 0 *1 * I

:0

I. I· I' I I I I I

10

G1

I

1 I ·1

1

:I I

1

1 0

0 1

--------------P

...

I I I

*1 - -*- ______1

(13.7.11)

13.7 Decoupling and Recoupling-Noninteractive Design

289

In the diagram the asterisks denote elements of G, to be made zero. The desired block of zeros in GI is attained by restricted column operations that add appropriate scalar multiples of the columns of the matrix (13.7.9) with feet in the p x p block of(13.7.11) to columns of G I . For an explanation of a restricted column operation, refer to (13.7.7). A res-trictedcolumn operation is any standard column operation that takes place within any of the blocks labeled by d k g(k = 0, 1, . . ., n - 1) along with the right end p x p block, but when a column operation is performed within one of these blocks the same operation must be performed on each of the other blocks. After the desired corner block of zeros has been generated, the resulting matrix is again row reduced, and thus (13.7.9) has been transformed into

in which

K, G, =

(13.7.13)

consists of a vertical stack of q blocks K i ,each p x p, sitting atop the block of zeros just generated alongside K O ,with K O then p x p. This concludes step 1. In the case in which p divides n, only the matrices K , , K,, . . ., K , occur in G 2 . The immediate objective of step 2 is to drive the subblock of G , in (13.7.12), presently occupied by K O ,to zero. To understand the procedure, consider the matrix (13.7.14)

290

13.

Applications in Control Theory

extracted from (13.7.12). The (feedback) blocks of (13.7.13) are to be loaded into the zero block of this matrix according to the scheme -K o

...

-K 1

0

-K q

Hz

(13.7.15) In-

Gz

p

However, this will be done in q + 1 stages. That is, in the first stage only the first subblock is inserted into (13.7.15), with the other feedback subblocks left at zero to obtain the matrix -K o

0

...

0 I n-

0

Hz

(13.7.16)

Gz

p

and the block containing Hz is set to zero as well. Let fAo denote the matrix of zeros and ones occupying the block of (13.7.9) and (13.7.12) that initially contained fA. Generate the controllability matrix of do, fAo, attach the matrices Pz and Qz of (13.7.12) to it on the left and right, respectively, and then row reduce the resultant matrix to get

P3

t,

0

H3

T3

0

In-

G3

L3

p

Q3

I

(13.7.17)

This completes step 2. Note that it is not necessary to compute the entire controllability matrix for do, fAo since theblocks T and L (and H) are not used. Of course, the first p columns of the controllability matrix of do, fAo are just fAo, and it turns out that the next p columns are the first p columns of do, so that those columns can be written down without computing. Step 3 begins by breaking G3 up into the same block structure as that of Gz in (13.7.13). Although the contents of the blocks will have changed, to keep the notation simple the same symbols that were used for Gz will be used to denote the blocks ofG 3 . If no errors were made, the block K o should now be zero. No column operations are needed. The contents of K 1 (of G3 ) are

291

13.7 Decoupling and Recoupling-Noninteractive Design

inserted according to the scheme (13.7.15) to define

(13.7.18)

d,= In--p

G3

in which the block containing H3 is set to zero. (When p divides n, the zero blocks to the left of - K1 in (13.7.18) are absent.) Then the controllability matrix of d l ,gois computed and P3 and Q3of (13.7.17) are attached with the resultant matrix row reduced to give

which completes the repetition of the process of step 2 for the second stage of feedback. The procedure is repeated, each time using the next higher K i in the stack(13.7.13)of the most recently computed G. The process is terminated after G becomes zero, which occurs no later than after insertion of the last feedback K,. Denote the resulting matrix by (13.7.20) All the information needed for steps 4 and 5 is available in (13.7.20). To begin step 4, read off the ordered cycle basis, which is defined as the nonzero elements of the closed cycles of (13.7.20) in the order in which they appear there, {el, e 2 , . . ., ep12e,, . . ., 2 e p l . - .12qe,,

. . ., 2 g e p 1 2 q + ' e l , .. ., i @ + l e,L (13.7.21)

where, as in the discussion of step 1, the numbers q and p are defined by writing n = ( q 1)p p. (If p divides n, take p = 0.) Clearly, the respective cyclelengthsareri = q + 2fori = 1,2, . . . , p andri = q + l f o r i = p + 1, p 2,. . ., p. Note that rl r2 + rp = n, as anticipated. This ends step 4. For step 5, we continue reading off information from (13.7.20). The sought matrices P, Q, and K of Theorem 13.7.2 are obtained as follows: The matrix

+

+

+

+ +

292

13. Applications in Control Theory

Q isjust Qof(13.7.20). P is obtained from Pby applying the same permutation to the rows of P as need be applied to the ordered cycle basis (13.7.21) to rearrange it in the order jq+ 1 jq { .>4t et>.l4< e1,

I

jq+ 1 jq ... ,el.>4t e2• .>4< ez, ... ,ez ,..

I j q ep'.l4<j q - l ep, ... ,ep'}

'.>4<.

(13.7.22) Finally, the feedback matrix that accomplishes the decoupling can be computed by formula (13.7.10) in which the asterisk denotes matrix transposition and fL = diag[fL 1 , fLz, ...• fLpJ is block diagonal with fL i an r i x r, matrix of type. (13.7.5). (Note the role of step 4.) This completes the discussion of the operation of the decoupling algorithm for the diagonal case. Before presenting the further details involved in the general case, an example of the diagonal type is worked out. Example 13.7.1

Consider the system with coefficient matrices

.91=

0 1 0 -1 0 1 0 1 0 1 o -1 0 0 1

0 0 1 1 1

0 0 0 0 1

1 0 0 0 0

0 1 f!A= 0 0 0

(13.7.23)

The matrix (13.7.6) computes to be 1 0

0 0 0

0

1

1

o

-1

0

0

0 0

0

1

0

1

0

1 0 0 0

1 0

o -1

0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

0 0 0 0 0 0

1

0

0 0

0

0 0 0 1

0 0

0 1 0 0

0

1

1

o

-1

1

1 2

1

3

1

1 0

1

0

1

(13.7.24)

The row-reduced form (13.7.7) in this example can be checked out to be 0 1

1

0

0 0 0 0 0 0

1 0

0

0

0

0 0

1 0

0

0 0 0 0 0 0

0 0

o -1

0 0

1 0

0

1

.1

0

0 0

1 0

0

1

o -1

0 1 0 -1

0 0

0 0

0 0

0

1

1

0 0

0

1

0

o -1

0

1

1

1

2

1

3

1

1 0

0

1

(13.7.25)

A glance at (13.7.25) shows the example to be of diagonal type. With + l)p + p gives q = 1, p = 1, and step 1 requires a corner block of zeros of size p x (p - p) = 1 x 1 at the

n = 5 and p = 2 the equation n = (q

293

13.7 Decoupling and Recoupling-Noninteractive Design

position occupied by the middle one in the bottom row of (13.7.25). The appropriate column operation is thus the subtraction of the first column from the second in each of the appropriate blocks in (13.7.25). By following this with a row reduction, the matrix derived as the completion of step 1 can be checked to be 1 1 -I 1 o -I 0 0 0 0 0 0

1 0 0 0

1 0 0 1

0 0 0 0

o

0 -I

o

0 -I

1 1 0 0 1 0 0 0 1

0 0 0 0 0 0

1 0 0 1 0 0

0 0 1

1 1 0

0 0 1

0

0

0 0

0 0

1

0

0 1

o

0 -I

1

2

1 -I 0 1

(13.7.26)

The block (13.7.14) of interest in this example is the one in (13.7.26) containing the numbers 0 0 0 0 0 0

0 -1

0 0

1 0 0 0 1 0

1 1

0 0

0 0

0

1

I

(13.7.27)

1

which contains the matrix

G2 =

1 1

0 0

(13.7.28)

Thus K o = 1 in the notation of (13.7.13), and according to (13.7.16) -1 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 1 1 0

0 0 0 0 1

and

:!lo =

1 0 0 1 0 0 0 0 0 0

, (13.7.29)

294

13.

Applications in Control Theory

the latter obtained from (13.7.26). Computation of the controllability matrix of do, PAD' attachment of the ends of(13.7.26), and a row reduction complete step 2 with the resulting matrix being 1

1

2

0

1 0

0

0

0

1

0

1 -1

0

0

0

1

0

0

0

0

0

0

1

1

1

0

0

1 0

0

1

1

0

1 0

0

0

0

1

0

1

0

0 0

0

0

1 0

0

0

1

o -1

0

0

0

0

0

0

0

0

1

1

(13.7.30)

(The blanks in (13.7.30) emphasize the fact that the matrices T and L in (13.7.17) need not be computed in the diagonal case.) The second-stage feedback matrix K1 =

G~J

(13.7.31)

is read off(13.7.30) and substituted into the appropriate subblock of(13.7.30) in the prescribed fashion, and H is set to zero to get

o -1 o -1

-1 0 0 0 1 0 0 1

.91 1 = 1 0 0

0 0 1 1 0

0 0 1 0 0

(13.7.32)

Computation of the controllability matrix of .91 1, PAD, attachment of the ends of(13.7.30),and row reduction can be checked to give the result from insertion of the second and final stage of feedback 1

1

1

o

0

0

1

1

1

0

0

0

0

1 0

0

0

0

0

0

1

0

0

0 -1

3

1

1 0

1 0

0

0

0

1

0

1 -1

1

0

0

0

1

0

0

0

1 0

0

0

0

0

0

1

0

0

0

0

0

1 0

0

0

1

(13.7.33)

Note that the G matrix in (13.7.33) is zero as anticipated and hence both cycles of (13.7.33) are now closed. Proceeding with step 4, the ordered cycle basis of (13.7.33) is read off to be e 1 , e2, del, de2' d 2e1 , and the lengths of the two cycles in (13.7.33) are noted to be r 1 = 3, r 2 = 2, respectively.

295

13.7 Decoupling and Recoupling-Noninteractive Design

The instructions for step 5 say that the right end matrix of (13.7.33) gives

[1-lJr r

(13.7.34)

Q= 0

By noting the permutation required to reorder the ordered cycle basis as dZel> del' el' dez, ez, the rows of the left end matrix P of (13.7.33) are rearranged accordingly to get 0 0 P= 1 0 1

0 0 0 1 1 0 0 0 0 -1

0 1 3 1 1

1 1 1 0 0

(13.7.35)

The cycle lengths r l = 3, rz = 2, obtained in step 4, dictate thatthe canonical matrix fZ in formula (13.7.10) is

fZ=

0 1 0:0 0 0 0 1 : 0 0 0 0 oio 0 --------4---0 0 o :0 1 0 0 o:0 0

(13.7.36)

which, along with the original .91, Bl, and P found in (13.7.35), is substituted into (13.7.10) to finally compute

K= [-1 -1 0-4 -lJ. -1

0

0

1

0

(13.7.37)

It can be verified that the computed P, Q, and K satisfy the equations imposed by the conclusion of Theorem 13.7.2, P(d

+ BlK)P- l

o o PBlQ

=

(13.7.38)

0 0

= 1 0 o 0

o

fZ,

(13.7.39)

1

Remark 13.7.3 The decoupling algorithm makes all the control variables work in breaking up the system into the maximal number of decoupled (independent) controllable subsystems by inserting appropriate feedback.

296

13. Applications in Control Theory

In the system U 1 and U2 control the same two-dimensional subspace of fJl3. Theorem 13.7.2 says that with proper feedback the system can be- decoupled into three independent subsystems with each input variable controlling one of the subsystems. In some problem situations an intermediate mode is appropriate whereby distinct control variables control different subspaces but those subspaces are allowed to intersect In a nontrivial fashion. In other words the subsystems need not be totally decoupled. In Exercise 4 at the end of this section, a simple formula is derived for computing a universal control function that achieves this mode. It has the advantage of involving no change of state variables. Conceivably, it might be taken as the first step in numerous design procedures since it provides another way of strongly "tying in" all the control variables into the system.

The General Case. The first and last segments of the algorithm for solving the general problem are the same as those in the diagonal case. Thus, every problem is started by writing out the augmented controllability matrix (13.7.6) and then row reducing it to get (13.7.7). As in the diagonal case, a finite sequence of row, column, and feedback steps is then performed. For the nondiagonal case it is the choice of this sequence that remains to be described. At any rate the sequence leaves the augmented matrix in a form in which the computations can be completed by following the same procedure as that in the diagonal case, beginning with step 4. The ultimate goal of the sequence of matrix operations is to get the cycles to produce an ordered cycle basis. This is the requirement that all cycles be closed and that the vectors of each cycle preceding the first vector in the cycle with zero foot, taken together, constitute a permutation of the columns of the identity matrix In. A vector in a cycle can be pictured as moving through the row in which it has a 1 as the cycle is traversed. Those elements, with the order that they are given by the choice of basis in the controllability matrix (in the diagonal case, the natural order from left to right) are defined as the ordered cycle basis. In the diagonal case the ordered cycle basis arose from the columns containing the diagonal of leading ones in the row-reduced form, and all cycles were closed by a single sequence of feedback insertions that forced the feet of the p columns at the end of the diagonal (the G matrix) to be zero. The matrix (13.7.14) was just that of the state coefficient matrix (viewed as an operator on fJln) relative to the standard unit basis sitting in the first n columns of the controllability matrix. It was the diagonal of leading ones in that matrix that carried the feedback matrices inserted above the foot down

297

13.7 Decoupling and Recoupling-Noninteractive Design

along the diagonals to close the cycles in G under the repeated multiplication by the closed-loop matrix that occurs in regeneration of the controllability matrix. In the non diagonal case the (ordered) basis cannot be made to consist entirely of columns containing leading ones because there are not enough of them. Fortunately, other columns of the controllability matrix are always available to complete the basis due to the assumed controllability. This alternative choice of basis implies that the procedure for reading off the open-loop matrix into which the feedbacks are to be inserted must be altered accordingly. The riondiagonal case allows additional coupling of the cycles that does not occur in the simpler case. This is no insurmountable problem. It means that only one cycle at a time can be cut loose, as opposed to the diagonal case in which all are closed simultaneously by a single sequence of feedback insertions and the disconnecting accompanies that closing of the cycles. The complications that must be dealt with in the non diagonal case will be discussed in terms of an example rather than by working abstractly. The system is one with p = 2 control variables and n = 6 state variables. The augmented controllability matrix is

Example 13.7.2

1 0 0

I

0 0 0 0 0 0 0

1 0

0

0

I

0 0 0 0

0 0 0 0

0 I 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I

0 0 0 0

0 0 0 0

I 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 1

0 I 0 1 0 I I I

0 I 0 I 0 I I 2

I I

0 0

I 1 I 0

I 0 0 I

(13.7. 41)

For a good reason soon to be discussed, the state coefficient matrix ,r;I is not given explicitly. (Neither is !!4, but of course that is just the second block.) The system is not diagonal because of the bottom zero element of the sixth column of the controllability matrix (not to be confused with the augmented controllability matrix). The diagonal of ones that does not reach the bottom of the matrix will be called a short diagonal (of ones). The short diagonal in (13.7.41) prevents the annihilation of the foot of the fifth block by insertion of feedback using the technique of the diagonal case. Notice that a short diagonal allows nonzero terms to remain, after row reduction, in the foot of the matrix above the diagonal line. (In (13.7.41) they are the two Is in the sixth column ofthe controllability matrix.) These nonzero terms "tie" or "connect" the cycles together and must be eliminated by row and column operations and feedback if the cycles are to satisfy the definition of an ordered cycle basis.

298

13. Applications in Control Theory

The strategy of the algorithm is to start untying the cycles at the first point at which they come together, using feedback. To do this the open-loop matrix must be made to appear so that the feedback can be inserted. As already explained, in the diagonal case this matrix is already found sitting, after row reduction, in the controllability matrix by shifting over p columns from the left end. The reason why this matrix (13.7.14) is the open-loop matrix should be clear from the facts that the standard unit basis for f!Jln occupies the first n columns of the controllability matrix in the diagonal case and that the latter is generated by multiplication by .91. However, if a short diagonal occurs, say oflength s, then only the first s columns of the open-loop matrix will be found by shifting p units over. The remaining columns must be recovered from other columns of the controllability matrix. Thus in (13.7.41) the first five columns of the open-loop matrix can be read off by shifting two columns to the right, and it is the seventh column that can easily be made to produce the remaining column, namely, the ninth column, since the last base vector appears in the seventh. The matrix just read off is then

0 0

o o

0 0

1 0 o 1

o o

1 1

o

o o o

0 0

1 0

o

0

o

o o o o

0 0 0 0 0

(13.7.42)

1 1

To help locate the feedback that needs to be inserted to eliminate the Is in the sixth column of the controllability matrix that tie the cycles together, drop the row beneath the short diagonal of Is. Since there are two control variables, G consists of the foot of the two columns that follow the end of the short diagonal. Thus the problem looks like a diagonal one with p = 2, n = 5. Proceeding with the method for treating diagonal problems, we find that the two zeros allow insertion of the second stage of feedback into (13.7.42) with no column reduction required, and hence the closed-loop form of (13.7.42), following from insertion of feedback, is

o -1 o -1 1

o o o

o

0 0

o o

0 0

o

1 1

o o

0 0 0

o

0 1

o

0 0

o

1 0

0

o

1 1

(13.7.43)

299

13.7 Decoupling and Recoupling-Noninteractive Design

Regeneration of the controllability matrix, by using this matrix, and row reduction produce the matrix I 0 0 I 0 I 0 I

0 0 0 0

0 0 0 0

0 I 0 0 0 0 I 0 0 0 0 I 0 0 0 0

0 0 0 I

0

0 I

0 0 0 0

0 0 0 0

0 0 0 ·0

0 0 0 0

0 0 0 0

I 0 0 I

0 0 0 0

0 0 0 0

I 0 0 0

0 0 I 0

0 I 0 I 0 0 I 0

0 0 0 I

0 0 I 0

0 I 0 I 0 0 I I

(13.7. 44)

I

0 I

0 0

0 0 0 0

(After the row reduction that recorded the insertion offeedback, the contents of the second through the sixth blocks ofthe controllability matrix above the foot in (13.7.44) were set to zero. This is permissible since the direct feedbacks are taken into account in the formula that computes K.) Notice that the insertion untied the cycles in the third block and closed the second cycle. This disconnected closed cycle must be "gotten out of the way" so that the remaining cycle can be worked on. The process, to be called casting out closed (and disconnected) cycles, simply relocates the rows through which the cycle passes to the bottom positions of the matrix as I

0 0

I

0 0

0

0 0

0 0

0 0

0 0

0 0

I

0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 I

0 0 0 0 0 0

I 0 0 0 0 0

0 0 I 0 0 0

0 I 0 0 I 0

0 0 0 I I 0

0 I 0 0 I I

0 I

0 0 0 0

0 I 0 0

0 0 0 I

0 0 0 0

0 0 0 I

0 0 0 0

0 0 0 I

0 I 0 I 0 0 0 I

I

0

(13.7.45)

The remaining (first) variable is in control in the problem with p = 1. n = 4 above the bold-face line in (13.7.45). Note that this problem is of diagonal type and that the cycle is not yet closed. The basis to use to make the insertion of feedback work in the top problem is el' del' dle l, d3el> el' del' Relative to this choice the open-loop matrix is 0 1 0 0 0 0

0 0 1 0 0

0 0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.46)

300

13. Applications in Control Theory

To close the first cycle of(13.7.45) (see the ninth column), the insertion of the first stage of feedback in (13.7.46) gives the closed-loop matrix -1 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0

(13.7.47)

The resulting row-reduced augmented controllability matrix is then

o

0

1 0

o

0

0 0

0 0

o

0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1

0 0 0 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 1 0 1 0 0 0 0 0 1 0 0

0 1 0 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 1 1

o

0

1 0

0 0 1 0 0 0

0 0 0 0 1 0

0 1

0 0 0 0

0 0 0 0

0

(13.7. 48)

Relative to the same basis el> del> d2el> d3el> ez, dez, the open-loop matrix read off (13.7.48) is 0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.49)

To close the first cycle, inspection of (13.7.48) shows that feedback can be inserted at the second level with the resulting closed-loop matrix being

o -1 1 0 0 0 0

0 1 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.50)

301

13.7 Decoupling and Recoupling-Noninteractive Design

Its row-reduced controllability matrix computes to be 1 0 1 1 1 1

1 0

o

0

0 0

o

0

1 0

0 0

1 0

0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1

0 0 0 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1

0 1 0 1 0 0 0 0 0 1 0 0

0 1 0 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(13.7.5 1)

Inspection of the ninth column of the controllability matrix in (13.7.51) shows that the first cycle is now closed. (The 1 in that column is in a line of direct control; hence it can be set to zero but it need not be.) With the cycles closed and disconnected, the algorithm now finishes up with steps 4 and 5. Thus an ordered cycle basis is to be read off. In the diagonal case the basis elements occurred in the natural order. In (13.7.51) they were selected in the order et> del' dZel, d3el' ez, dez, and this then is the ordered cycle basis. The cycle lengths are r l = 4, rz = 2. Continuation with step 5 permutes the rows of Pin (13.7.51) in accordance with the ordered cycle basis to give 0 0 0 p= 1 0

a

0 0 0 0 0

0 0 1 1 0 1 0

0 a 1 0 1 1 0 1 1 1 1 1 1 0 0

(13.7.52)

a a

1

Q is read off(13.7.51)to be 1z . The cycle lengths determine the canonical form 0 1 0 0 0 0 1 0 0 0 0 1 !!Z= 0 0 a 0 -----------~-----

0 0 0 0 0 0 0 0

I I I I I I I I I I

I I I I I

0 0 0 0

0 0 0 0

(13.7.53)

0 1 0 0

Formula (13.7.10) is applied to compute the decoupling feedback matrix

302

13. Applications in Control Theory

coefficient K

=

[-1o -1 -1-2-1-IJ -1

0 -1

0

(13.7.54)

0'

and the problem is solved. Remark 13.7.4 In a high-dimensional problem that is not diagonal, a large number of cycles can occur. Each time one is closed and disconnected it is cast to the bottom of the. controllability matrix by row operations and the remaining cycles are worked on until the ordered cycle basis is achieved. This puts the coefficient matrix in block diagonal form. The Mathematical Basis of the Algorithm. The reasons for a number of steps in the algorithm become quite transparent when nonsingular changes of state and control coordinates y = Px, v = Q- 1U are made and the induced transformation of the controllability matrix is examined. The coefficients of the differential equation undergo the transformation .91 ---. PdP- 1, fJI---. PfJlQ and it follows that [fJI, .91fJI, ... , d

n

-

1

fJI] ---. p[fJlQ, .91fJlQ, ... , d

n

-

1

fJlQ]

(13.7.55)

is the induced transformation on the controllability matrix. A standard elementary theorem of matrix theory says that elementary row operations on a matrix can be realized by performing those operations on an identity matrix and then doing left multiplication by the resulting matrix. Hence the n x n matrix attached to the left end of the controllability matrix in (13.7.6) records the operations that are later (in step 5) converted back into a change of state coordinates. Similarly, elementary column operations on a matrix can be realized by performing those operations on an identity matrix and then doing right multiplication by the resulting matrix. Thus the right end matrix of (13.7.6) records column operations on fJI. In short, the augmented controllability matrix, along with row and restricted column operations, provides a systematic and convenient device for changing state and control coordinates (which, alternatively, could be viewed as changing bases in the column spaces of the controllability matrix and of fJI, respectively). The full rank assumed of f!J and induced in the controllability matrix by the assumed controllability is of fundamental importance. One significant consequence is that the basis used to read off the open-loop matrix needed for insertion of feedback can always be taken to be the columns of fJI together with other columns of the controllability matrix. This ensures that the numbers in the associated p rows of the row-reduced controllability matrix can be chosen arbitrarily. In the diagonal case the remaining leading Is in the diagonal are picked up by the open-loop matrix, and under regeneration

303

13.7 Decoupling and Recoupling-Noninteractive Design

of the controllability matrix they translate the inserted feedbacks downward along the diagonals to close the cycles in the G matrix. Under repeated matrix multiplication, mixing can occur, and this is prevented by the column reduction and by the insertion of feedback in stages. The verification that this happens is left to the reader as an easy exercise. In the nondiagonal case, by the full rank available, a short diagonal of Is is always present, which allows one cycle at a time to be closed and disconnected. The row reductions that follow the insertion of feedback are not similarity transformations since they are guided by new numbers that have been inserted in the matrix. The decoupling matrix does not have to be explicitly computed and updated during the process because of the uniqueness of the solution to the equation that it must satisfy (see Exercise 9). By selecting a basis from the controllability matrix and writing out any solution to (13.5.1) in terms of coordinates relative to that basis, it is easy to see that the closed property of all the cycles prevents the coordinates of the solution associated with one cycle from interacting with those of another. Finally, the permutation of the matrix P in (13.7.20) to get P amounts to reordering the ordered cycle basis in conformity with the order needed for the matrices of the decoupled subsystems to come out in the prescribed canonical form.

EXERCISES

1. Find a feedback controller u = Kx coefficients

+

v for the system (13.5.1) with

0 1 OJ [

.5#=000, 000 such that the Jordan form of the closed-loop coefficient matrix .5# as indicated. (a)

2.

[-~ _.~

~l,

° o-d

(b)

[-

+ fJlK

is

~ _ ~ ~l. ° ° -dJ

Consider the pair of controllable decoupled systems

X2 =

AtX2

+

u,

Y2 = A2Y2

Under what conditions on the parameter values

At

+

v.

and A2 does there exist a

304

13. Applications in Control Theory

control function of the type u = clp, u = czp for some scalar cl, c2 such that the joint system is controllable relative to the control variable p? 3. Apply the decoupling algorithm to compute the matrices P, Q, and K of Theorem 13.7.2 for each of the diagonal systems (13.5.1) determined by the following coefficients.

1, j, 1, [;d9

0 1 0 0 (c) .=[l 0 0 10 0

0 0

a=k

0 0 1 0

1-1 2 1 (d) d = [ 2 0 1 1 2 1-1

0 -1 a=[!-; Jl

1 1

[.

0 0

0 1 0 0

&=

0 0 1 0 0 0 1]’ .

a=

0 0 0 0

0 0

(f) d

=

1’ j-1. 1

0 0 (g) d = 0 0 0

’

1 0 0 0 0

0 1 0 0 0

0 0 1 1 0 0 0

a=b

0

0

1-1

Assume that in (13.5.1) the columns of B are linearly independent. This exercise does not require d,B to be controllable. (a) Show that the equation d29 = 0, in which d , = d BK, has a unique solution. Find an explicit formula for the solution K in terms of

4.

+

13.7 Decoupling and Recoupling-Noninteractive Design

305

.91 and fJI. For the solution show that .91K =

[I -

fJI(fJI*fJI)-lfJ1*Jd.

(b) Show that the process in (a) that takes .91, fJI - .91K takes .91K' fJI- .91K' What control-theoretic interpretation can you give this property? (c) What simplification occurs in the above formulas for K and .91K if the columns of fJI are orthonormal? (d) Show that bi'(dK)kb j = 0 (k = 1, 2, ... , n - 1; i = 1, 2, ... , p; j = 1,2, ... , p) if K is the solution of (a). Note that the relationship also holds for k = 0, i '" j if the columns of fJI are orthogonal. (e) Assume that the columns of fJI are not only linearly independent but also orthogonal. Let no be the dimension of the controllable subspace of .91, fJI. Show that there exist positive integers SI' S2' ..• , sp with SI + S2 + ... + sp = no such that the column-vectors of the n x no matrix [b 1, dKb b ... , dr1bl/" '!bp, dKb p, ... , d~p-lbpJ constitute a basis for being a linear combination of the controllable subspace of .91, fJI with d~bi the column-vectors d~bj' 0 :s; k :s; Sj - 1, 1 :s; j :s; i for i = 1,2, ... , p. (The point here is that all of the columns of fJI appear in the basis.) 5. Application of Exercise 4. Compute the controllable subspaces of the individual control variables in the system

to show that Ul and U2 control the same subspace. Then compute the matrix K of Exercise 4, substitute in the feed-controller u = Kx + v, and examine the subspaces controlled by the individual variables VI' V 2, V3' Investigate the extent to which this controller decouples the system by finding the basis discussed in part (e) of Exercise 4 and rewriting the state vector in terms of that basis. 6. A simple geometric interpretation underlies the controller discussed in Exercise 4. Let .A.P be the vector space of real n x p matrices endowed with the inner product (M1IM2) = tr(Mt M 2 ) for M 1 , M 2 in .A.p • In terms of the columns of fJI :;= [bb b2 , ••• , bpJ, define fJI(i,j) to be the matrix in .A.P with jth column b, and the other columns zero. Then fJI = L fJI(i, i), and if the b, are orthonormal (or orthogonal), which is to say that fJI *fJI = I p(or diagonal), then the fJI(i,j) constitute an orthonormal (orthogonal) system in .A.p' That system is of interest since in terms of it one can write fJlK = Li,j KijP(i,j). It is natural to ask whether or not some K in .91K = .91 + fJlK makes (.91 KIfJI(i, j)) = 0 for all i, j. The unique K that solves these equations is the one found in Exercise 4, and the equation drfJI = 0 is simply the statement that .91K .1 fJI(i, j) for all i, j. Verify the statements made. 7. Consider the situation described in Exercise 6 under the assumption that the columns of fJI are linearly independent and orthogonal. From the

306

13. Applications in Control Theory

facts that the g ( i , j ) constitute an orthogonal set and that B K lies in the spanned by that subset for each K , it follows from the subspace of A,," general properties of inner product spaces (see, for example, [111) that the solution K of Exercise 4 is the unique solution to the minimization problem minld, 1, where the minimization is over K E Mp,,and the norm is the one induced by the inner product, i.e., IdK = tr(d;dK). Using the formula for K found in Exercise 4,show that

l2

rninId,l

=

Jtrd*[z

-B(~*B)-'B*]~

and note the simplification that occurs if g has orthonormal columns. 8. Apply the decoupling algorithm to the following systems:

j7

0 1 0 0 0 0 1 0 (a) & = [o 0 0 0 0 0 0

(b) d =

[%j

1 '

:

1,

0 0 & = [l 0

1 0 0 1

0 0 1 0 0 0 0

0 0 (d) d = 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 , 1 0

B=

0 0 (e) d = 0 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 1 1

0 0 0 , 0 0

g=

1 "1 1 0

0

9. Assume that d,B, P,and the canonical matrix 2 in Theorem 13.7.2 are known. Derive formula (13.7.10) for the decoupling feedback matrix K.

307

13.8 Controllability of Nonlinear Equations

13.8 Controllability of Nonlinear Equations

The significance of the controllability hypothesis is evident in the study of linear control systems that appeared in the preceding sections of this chapter. Generally, controllability theory attempts to define and isolate the theoretical limits to which various types of dynamical systems can be controlled. It turns out to have a fundamental bearing on many aspects of control (see, for example, [13,21]). The discussion here focuses on nonlinear equations of the type

x=

f(t, x, u) = .9Ix

+

fJlu

+ h(t, x, u)

(13.8.1)

with the standing assumption that the term h(t, x, u)is defined and continuous at all (t, x, u) E PAl X PAn X PAP. The real matrices .91, fJI are of respective sizes n x n, n x p. The following theorem is a generalization of Theorem 13.5.1, which was first proved by the author in [15]. Theorem 13.8.1 In Eq. (13.8.1) let h(t, x, u) be continuous and Ih(t, x, u)1 be bounded on PAl X PAn X PAP. Further assume that h(t, x, u) is periodic in t with period T > 0 for each fixed x, u. Finally assume that

Then for each ()(, control function u(·)

p E PAP . croCCO,

E

X o , Xl E PAn . and 0 < t I < T there exists a T], PAP) such that

(a) u(O) = u(T) = ()(, u(t I) = p, and (b) a corresponding solution xC) of(13.8.1) for which x(O) = both X(tl) = x., x(T) = ·Xo.

Xo

satisfies

T his control function extends to a continuous periodic function on ( - 00, 00) with period T and with a response agreeing with the periodic extension of x(t). Remark 13.8.1 In Theorem 13.8.1, T need not be a minimal period. Hence that theorem applies to the special case in which h = hex, u) is independent of t (autonomous systems). In particular, it applies to the case in which h is the zero function (linear autonomous systems). It says not only that the response can be steered through any two states but also that the values of the control function can be prescribed as well. In particular, the response can be steered through the prescribed states with any attainable velocities. Remark 13.8.2 If the periodicity requirement on h(t, x, u) is dropped altogether, then T > 0 can be taken to be arbitrary and all conclusions of Theorem 13.8.1 remain valid, with the exception of the last sentence concerning the periodicity, which is lost.

The proof of Theorem 13.8.1 will appeal to the following lemmas.

308

13. Applications in Control Theory

Lemma 13.8.2

Define
=

So ead P4 do and let T

> O. If

then the matrix ST = f:
~ (faT
(13.8.2)

is symmetric and positive-definite. Proof.

For nonzero x

E ~"

y

~ JoT 1
(f: I
~

f: 1 doi f: /
2

f: 1
~

2

by Schwarz's inequality, in which equality holds only if I 0 for all x '# O. It is a triviality to check that S~ = ST, and Lemma 13.8.2 is thereby established The following notations will prove to be convenient:

. fT t CT(t) = T~I
J: ell-a)d

JoiT
flJCrCO') da.

(13.8.4)

(13.8.5)

(It will be shown that SrCT) = ST') Lemma 13.8.2 states a condition under

which ST is nonsingular, and hence it is possible to write down the following system of equations whose importance will become apparent in the next

309

13.8 Controllability of Nonlinear Equations

lemma:

x(t) = etdx o + ljJ(t)u o + -1 It ljJ(w)dw(uT - Uo)

T

u(t) = y

+ ST(t)y + f~

e(t-W)h(w, X(W), u(w)) dco,

(1 - ~)uo

(~)

+

= ST-1 [XTT- a e l Xo

- s; 1 L

T

(13.8.6)

0

UT

-

+

(13.8.7)

CrCt)y,

ljJ(T)uo - T1

Jr ljJ(w) dro(UT 0

e(T-w)alh(ro, x(ro), u(ro») dos.

]

uo)

(13.8.8)

Lemma 13.8.3 Any solution x(t), u(t) to the nonlinear functional equations (13.8.6)-(13.8.8) provides a solution to the boundary-value problem

x(t) = .9Ix(t)

+

PJu(t)

+

h(t, x(t), u(t)),

x(O) =

X o,

x(T)

= XT,

u(O) =

Uo,

u(T)

=

UT'

(13.8.9) (13.8.10) (13.8.11)

Proof. Let x(t), u(t) be a solution to (13.8.6)-(13.8.8). Since ljJ(O) = 0 and ST(O) = 0, it is clear that x(O) = Xo' By noting that CT(O) = CT(T) = 0, we conclude from (13.8.7) that u(O) = U o and u(T) = UT' as required. The following computation shows that ST(T) = ST: SrCT) = f: e(T-a)dPJCT(O') da

=

fT e(T-a)aIPJ [IT ljJ*(ro) dro - !!- fT ljJ*(ro) dro] do Jo T-a T Jo

= LT e(T-a)dPJ f:-a ljJ*(ro)drodO' -

~ (f: ljJ(ro)dro)(f: ljJ*(ro)dro). (13.8.12)

310

13. Applications in Control Theory

Interchanging the order of integration gives T e(T-a)d f!l IT ¢*(w) dta do = IT IT e(T-a)d f!l da ¢*(w) dco T-a 0 T-w Io

= foT fow ead f!l do ¢*(w) do» =

foT ¢(w)¢*(w) dco.

(13.8.13)

Substitution of (13.8.13) into (13.8.12) and application of definition (13.8.2) show that ST(T) = ST' With this fact it is now possible to compute, from (13.8.6) and (13.8.8), x(T) = eTdx o

+I

[

+ ¢(T)uo +

T1 IT 0 ¢(w)dw(u T - Uo)

~ XT - e t:« X o - ¢(T)uo - T1 IT 0 ¢(w) dW(UT - uo~

~

f:

+

foT e(T-w)dh(w, x(w), u(w)) doi

e(T-w)dh(w, x(w), u(w)) doi

(13.8.14) Therefore all boundary conditions are satisfied. All that remains is to show that x(t), u(t) satisfies (13.8.9). Using (13.8.7), compute

f~

ell-aIdf!lu(a) do

= ¢(t)u o + T1

II 0

¢(w) dW(UT - uo)

+

ST(t)y.

(13.8.15)

13.8

311

Controllability of Nonlinear Equations

Combining this equation with (13.8.6) shows that x(t) - [e td X o

+ f~

e(t-u\d£!Bu(rr) de

+

f~

e(t-U)dh(rr, x(rr), u(rr» drrJ = 0. (13.8.16)

But this equation can be differentiated, with the result being x(t) - [dx(t)

+ £!Bu(t) + h(t, x(t), u(t»] =

0,

(13.8.17)

and Lemma 13.8.3 is proved. Remark 13.8.3 Any solution x(t), u(t) to (13.8.6)-(13.8.8) will have u(t) analytic in t and x(t) continuous along with its first derivative. In general, the differentiability will be limited only by the differentiability of h.

Proof of Theorem 13.8.1. Substitute (13.8.8) into (13.8.6) and (13.8.7) so that Eqs. (13.8.6) and (13.8.7) can be dealt with in the form (x, u)(t) = N(x, u)(t)

(13.8.18)

in which N is a nonlinear operator on the Banach space E of continuous maps t ~ (x, u)(t) from [0, T] into fJln x r7t P with the sup-norm. A preliminary translation of the origin in E allows N to be taken to be of the form N(x, u)(t) = [N(x, u)(t)

+ D(t)N(x, u)(T), C(t)N(x, u)(T)]

t

(13.8.19)

in which the matrices C(t), D(t) are continuously differentiable in t, N(x, u)(t) =

(13.8.20)

e(t-OJ)dh(w, x(w), u(w»dw,

and h is the induced translation of h (inheriting the continuity and boundedness properties of h). The following facts can now be verified: (a) IIN(x, u)11 is bounded on E. (b) N: E ~ E is continuous. (c) For all sufficiently large c, N maps E into the subset of E, E; = {(x, u) E E :

I(x, u)(t + ~) -

(x, u)(t) I :::; cl~l;

all t, M.

(d) E, is a compact and convex subset of the Banach space E. The proof of (a) follows from the continuity of C(t), D(t) on [0, TJ and the boundedness of Ihl on fJI x r7t n x r7t p • Statement (b) is based on the continuity and hence the uniform continuity of h on compact subsets of fJI x fJln X r7t p • The invariance of E, under N for sufficiently large c can be shown by writing e(t) and D(t) as integrals of their derivatives in the formula defining N(x, u) and then once again applying the boundedness of Ihl. Showing that E; is closed and convex is a simple exercise. Since E, is a bounded subset of E,

312

13. Applications in Control Theory

once its closed character has been established, then by noting that it is an equiuniformly continuous family, the Arzela-Ascoli theorem can be invoked to conclude its compactness. A direct consequence of (a)-(d) is that for all sufficiently large c the restriction of N to E, provides a continuous map of a compact and convex subset of a Banach space into itself. Hence the Schauder fixed point theorem applies, establishing the existence of the required fixed point. This in turn determines a solution to (13.8.6)-(13.8.8) and, in light of Lemma 13.8.3, concludes the proof of Theorem 13.8.1. Theorem 13.8.4 in [}In,

In the one-parameter family of differential control systems

x=

f.(t, x, u)

= dx + !JIu +

ehit, x, u),

(13.8.21)

yl + lu - vlJ

(13.8.22)

let h be continuous and satisfy Ih(t, x, u) - h(t, y, v)1 ~ L[lx -

for some L E [}l and all (r, x, u) and (t, y, v) in [}l x f!Jln X f!JlP. Further assume that h(t, x, u) is periodic in t with period T > Ofor each fixed x, u and,finally, that rank [ !JI, d!JI, ... , dn-1!JIJ = n. Then the conclusion of Theorem 13.8.1 and the remarks following it are valid for all e in a neighborhood of £ = O. Proof. As in the proof of Theorem 13.8.1, the problem reduces to one of showing the existence of a solution to Eq. (13.8.18) in the Banach space E, with the difference that (13.8.2,0) is now replaced by N(x, u)(t)

=

e

r

e(t-W),,",'ii(w, x(w), u(w)) do:

(13.8.23)

since the previous role of h is now played by (h. It is easy to see that N does define a map of E into itself. From application of (13.8.22) to (13.8.19) and (13.8.23) and from routine estimation, it follows that there is aCE [}l such that

IIN(x,~)

- N(y, v)11 ~ !£lcLII(x, u) - (y,

v)11

(13.8.24)

for all (x, u), (y, v) in E and all e in some neighborhood of e = O. Thus for 1£/ < (CL)-l the map N: E- E is a contractive operator on a complete metric space E. Thus by the standard theorem which says that such an operator has a unique fixed point (see, for example, [19J), it follows that (13.8.18) has a solution. Application of Lemma 13.8.3 concludes the proof of Theorem 13.8.4. Remark 13.8.4

The controllability of a linear system can be destroyed by

313

13.9 The Domain C of Null Controllability

addition of a term h satisfying (13.8.22). An example is

x=

f.(x, u) = u

+ c J u2 +

x2.

(13.8.25)

J

For it h(t, x, u) = x 2 + u 2 does satisfy (13.822) for some L and is continuous. However, if 1£1 ~ 1, then £ x(t) ~ 0 for all t. This shows that the requirement in Theorem 13.8.4 that 1£1 be sufficiently small is not superfluous. It is worth noting that if the square root in (13.8.25)is dropped, the resultant equation (13.8.26) fails to satisfy the conclusion of Theorem 13.8.4 for all e but e =

o.

13.9 The Domain C of Null Controllability

In Section 13.8 we were concerned with finding conditions on .91, PA, and h in (13.8.1) which assure that it is possible to steer the system from any state to any other in finite time. However, in applications in which x denotes the error in some controlled physical system, the point of interest is the capability for steering all initial states to one fixed state, the origin. This leads to the definition of the domain of null controllability that follows. The discussion is restricted to autonomous systems in !?An of the type

x=

f(x, u) = dx

+ PAu +

h(x, u),

(13.9.1)

with the standing assumption that h and its first-order partial derivatives are continuous in !?An X ~P and are zero at the origin in !?An X !?Ap.

Definition The domain C of null controllability for (13.9.1) consists of all initial states in !?An that can be steered to the origin in finite time using continuous controls. Theorem 13.9.1 1f(13.9.1) has . rank [ PA, dPA, ... , dn-1PA] = d ~ n,

(13.9.2)

then C contains ad-dimensional submanifold of !?An that contains the origin. Proof.

The approach taken is to consider the equation

x=

- f(x, u)

(13.9.3)

and to study the collection of states that can be attained by steering from the origin as the initial state. This set is contained in C. By an appropriate nonsingular linear change of state coordinates, (13.9.3)

314

13.

Applications in Control Theory

is transformed into the system

+ !?IIU + d 12 x 2 + X2 = d 22X2 + h2(Xl , X2, u) in which dimtx.) = d, dim(x2) = n - d, and = d

Xl

11 X l

rank [!?II' d

11 fJI1> ••• ,

h l ( X l , X 2 , U),

(13.9.4) (13.9.5)

dft 1!?II J = d.

(13.9.6)

Now introduce the d-parameter family of control functions u(~,

=

t)

~

1

UI(t)

+

+ '" -:t-

~2U2(t)

~duit).

(13.9.7)

For I~I restricted to small values there exists the corresponding unique solution of (13.9.4), (13.9.5) defined on 0 :s; t :s; 1:

for which '1t(u(~, ·),0) (13.9.5) with respect to

°

ot (Ortl) ~ =

(13.9.8)

'12(U( ~, .), t)

(13.9.9)

= rtl(U(~,

X2

=

= 0, 112(U(~, ~

'),t),

Xl

·),0)

it follows that

(art ~ 1) + d (art2) ~

d

11

+

!?II [Ul (r),

e

+ o~.

= O. By differentiation of (13.9.4), 12

hl(rtl,

U2(t), ... , uaCt)J

rt2, u(~,

t))

(13.9.10)

(13.9.11)

(These are the variational equations discussed in Chapter 11.) In particular, at ~ = 0, (13.9.12)

(13.9.13)

In view of (13.9.6) the

U I (r), U2(t),

... , ud(t) can be chosen so that (13.9.14)

315

13.9 The Domain C of Null Controllability

By the implicit function theorem, it follows that (13.9.8) at

t

= 1, (13.9.15)

can be solved for ~ = ~(xd for all Xl near the origin with tion of ~(Xl) at t = 1 into (13.9.9) shows that

~(O)

=

O. Substitu(13.9.16)

which defines the required d-dimensional submanifold of attainable states. Generally, the manifold discussed in the proof of Theorem 13.9.1 is a proper subset of C. Furthermore, the manifold is not always unique since the control function selected to satisfy (13.9.14) is not unique. This remark is further illuminated by Exercise 2 at the end of this section. Remark 13.9.1

Remark 13.9.2 Any equation satisfying the hypothesis of Theorem 13.8.1 or Theorem 13.8.4 clearly has domain of null controllability C = ~n.

Controllability criteria for the nonautonomous equation that arises when d, Pi in (13.5.1) are allowed to be time-varying can be derived in terms of the fundamental matrix of the uncontrolled system (see, for example, [13]). Likewise, the coefficients d, Pi in (13.8.1) can be taken to be functions of t satisfying such criteria, and then analogous results to those of Theorems 13.8.1 and 13.8.4can be derived by using similar methods. Remark 13.9.3

Chapter 13 was written with the intention of giving some indication of the directions that control theory has imparted to the classical theory of differential equations. The material selected lies very much at the foundation of the subject and the treatment avoids elaboration. For example, it is often assumed that not all the state information is available for generating the feedback control functions. This gives rise to the notion of observability, which was not covered. To keep this introductory glimpse of the subject simple, less emphasis was placed on nonautonomous equations than is found in texts devoted solely to control theory. In situations in which the mathematical model describes the dynamics of the system to be controlled to a high degree of accuracy, various notions of optimality can be defined to determine values of free parameters in the controllers. Sometimes an optimization problem is formulated and solved only to see what level of improvement is available in the design of an existing control system. The topic of optimization has received a great deal of attention in the literature, particularly by the mathematically inclined, to whom the notion of optimization is an old and familiar one in mathematics. For coverage of this topic the reader is referred to [13] and [14] as a starter.

316

13. Applications in Control Theory

EXERCISES

1. Verify the claims made in Remark 13.8.4. 2. For the system compute explicitly the solutions (13.9.8), (13.9.9) and apply them to examine the manifold (13.9.16). By varying the choice of the control function u('), show that this family sweeps out C = f7t2. 3. For the system

show that the manifold determined by (13.9.16) is X2 = txi, which is precisely C. 4. An initial state Xo of(13.5.1) is called returnable if there exists a continuous control function steering x(O) = Xo to x(T) = Xo for some T > O. Find necessary and sufficient conditions on .91, f!4 for the collection of returnable points to be a linear subspace of f7tn. What is its dimension? 5. Show that there exists a periodic and continuous forcing function J(t) for which the nonlinear scalar differential equation

x + [sin(x)] x + e-

x2

=

J(t)

has a periodic solution x(t) that satisfies the boundary conditions x(O) = 0,

x(O) = 1,

x(l) = 1,

x(l) = O.

References

I. 2.

3. 4. 5. 6. 7. 8. 9. 10. I I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Boraiko, A. A. (1980). The pesticide dilemma. Natl. Geogr. 157, 145-183. Breiman, L. (1969). "Probability and Stochastic Processes." Houghton, Boston, Massachusetts. Brunovsky, P. (1970). A classification of linear controllable systems. Kybernetika (Prague) 173-188. Coddington, E. A., and Levinson, N. (1955). "Theory ofOrdinary Differential Equations." McGraw-Hill, New York. Conte, S. D., and de Boor, C. (1980). "Elementary Numerical Analysis." McGraw-Hill, New York. Dawkins, R. (1976). "The Selfish Gene." Oxford Univ. Press, London and New York. Hahn, W. (1967). "Stability of Motion." Springer-Verlag, Berlin and New York. Hale, J. K. (1980). "Ordinary Differential Equations." Krieger, Huntington, New York. Hartman, P. (1964). "Ordinary Differential Equations." Wiley, New York. Heymann, M. (1968). On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control, AC-l3, 748-749. Hoffmann, K., and Kunze, R. (1971). "Linear Algebra." Prentice-Hall, Englewood Cliffs, New Jersey. Kolman, B. (1976). "Introduction to Linear Algebra." Macmillan, New York. Lee, E. B., and Markus, L. (1967). "Foundations of Optimal Control Theory." Wiley, New York. Lefschetz, S. (1977). "Differential Equations: Geometric Theory." Dover, New York. Lukes, D. L. (1972). Global controllability of nonlinear systems. SIAM J. Control, 10, 112-125; errata 186. 11, (1973). Marsden, J. E. (1973). "Basic Complex Analysis." Freeman, San Francisco, California. Maxwell, J. C. (1868). On governors. Proc. Roy. Soc. 16,270-283. Ortega, J., and Poole, W. (1981). "An Introduction to Numerical Methods for Differential Equations." Pitman, Marshfield, Massachusetts. Phillips, E. R. (1971). "An Introduction to Analysis and Integration Theory." Intext Educational, Scranton, Pennsylvania. Russell, D. L. (1978). Controllability and stability theory for linear partial differential equations: recent progress and open questions. SIAM ReI'. 20,639-739. Russell, D. L. (1979). "Mathematics of Finite-Dimensional Control Systems." Dekker, New York. 317

318

References

22. Schechter, M. (1971). "Principles of Functional Analysis." Academic Press, New York. 23. Schwartz, A. J. (1963). A generalization of a Poincare-Bendixson theorem to closed two dimensional manifolds. Amer. J. Math. 85, 453-458; errata (1963), 85, 753. 24. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73,747-817. 25. Warren, M., and Eckberg, A. (1975). On the dimensions of controllability subspaces. SIAM J. Controll3, 434-445. 26. Watson, L. (1979). "Lifetide: The Biology of the Unconscious." Simon & Schuster, New York. 27. Whyburn, G. T. (1964). "Topological Analysis." Princeton Univ. Press, Princeton, New Jersey. 28. Wonham, W. M., and Morse, A. S. (1972). Feedback invariants of linear multivariable Systems. Automatica-J. lFAC 8, 93-100.

Index

A

function, 174 history, 3 variable, 15 Controllability, 16,269,271,279 nonlinear systems, 307. 312 null,313 Controllability matrix, See Matrix subspace, 283 Controllable part, 274 Controller, decoupling, 283, 285 feedback, 275, 279 recoupling, 285 regulator, 17 universal, 296, 305 Cycle, 287, 299

ABC algorithm, 47, 48, 78 in computation of matrices, 266 and correspondence rule, 48, 52, 85 formula, 59, 84, 135, 261 matrix equation, 54 Absolute convergence, 69 Adjoint equation, 114 Airy equation, 127 Alekseev formula, 222 Amplitude response, 264 Analytic matrix function, 151

B Boundary, 192

c

Caratheodory, 182 Cauchy criterion, 68, 69 Cauchy integral formula, 148 Cauchy-Peano existence, 184 Cauchy problem, 12,47, 190,205 Cauchy sequence, 68 Cayley-Hamilton, 40 Characteristic roots, exponents, 165 Closed cycle, 287, 29&, 300 Closed loop poles, 275 Closed loop system, 275 Closed set, 201 Column space, 27 Commutative equation, see Differential equation Commuting antiderivative, 139 Continuous map, 201 Control differential equation, 261 feedback, 15

D Decoupling algorithm, 286, 288 Dependence, on parameters, 200 analytic, 219 continuous, 205 differential, 212 Differential equation analytic, 219, 246 asymptotically autonomous, 239 autonomous, 14, 218, 223 commutative linear, 139 constant-coefficient linear, 9, 47,64,92 control, 261 diffusion-reaction, 240 dual, 112, 131 general solution, 200, 205 history, 1 linearization, 244 matrix, 76, 109, 114, 153 nth order, 130

319

320

Index

periodic, 93, 162, 172, 307 periodic commutative, 175 sources, 4 time- varying linear, 8, 11, 106 variational, 212, 215, 216

E EISPACK,31 Electrical circuits, 7, 11 Embedding, 53 Epidemic model, 133 Estimates, solution norms Cetaev, 95, 101, 115 commutative equations, 160 constant-coefficient equations, 12 exponential function, 93, 95 variable coefficient equations, 115 Euler equation, 131, 138 Existence, local, 182, 184, 186 Exponential decay, 93, 160, 168 Extension of solution, 191

F Feedback, 15 Field,I9 Floquet, 163 Flow and mixing, 5 Fundamental matrix, 49, 59, 75, 77, 108, 128, 129, 130, 131, 140, 163, 165, 166, 167, 176

G Gain,264 Gauss-Jordan elimination, 30, 31, 33 Generalized eigenvectors, 54, 55 Gershgorin's theorem, 105 Gronwall's inequality, 74, 111, 112

H Hausdorff metric, 226, 227 Hausdorff space, 211, 212 Homeomorphism, 210, 244 Homogeneous system, 25, 27 Hyperbolic rest point, 243

I Index (winding number), 254 Infmitesimal transition scheme, 134 Initial-value problem, 47 Insertion of feedback, 290, 291, 298

J Jacobian matrices, 212 Jordan curve, 251

L Laplace transform operator, 264 Laplacian, 240 Levinson-Smith equation, 9 Lienard equations, 9, 195 Limit set, 224, 236 Linear algebraic systems, 25 Linear functional, 112 Linear independence, 51, 79, 81 Linear transformation, 81 Linearization, 244 UNPACK,31 Lipschitz continuity, 185 Locally integrable, see Matrix Lyapunov approach, 95, 228 asymptotic stability, 168, 232, 234 derivative, 229 energy function, 196 equation, 99 function, 241 partial stability, 236 stability, 92, 232 uniform stability, 232

M Manifold, 200, 220 Markov process, 132 Matrix 19 algebra, 20 characteristic polynomial, 39,45, 89, 93, 276 classical adjoint, 150 computing, 31 congruence, 98 controllability, 269, 271 convergence, 68 determinant, 35, 36, 37, 38, 46 diagonalization, 44, 99 differential equations, 114, 153 eigenspace, 39, 43, 45 eigenvalue, 39 eigenvector, 39 equation, 49,51,54,59, 82, 84,99, 101, 104, 105 exponential, 69, 75, 77, 78, 84, 85, 86, 91 feedback, 275

Index

321

frequency response, 264 function, 70, 148 hyperbolic, 243 inverse, 29 Jordan form, 43, 44, 45, 64, 77 locally integrable, 107 logarithm, 162, 163 minimal polynomial, 40, 44, 45, 89, 93 norm, 65, 66 positive definite, 92, 102 power series, 70 row-reduced echelon form, 26 similarity, 98 stability, 93, 99, 104, 232, 234 symmetric, 98 trace, 46 transfer function, 265, 266 transpose, 27, 30, 37 triangular factorization, 32 Maximal interval, solution, 191 Maxwell, J. C., 3 Mechanics, 6, 10, 11 Metric space, 211 Method of eigenvectors, 61

N Newton, I., 1,9 Norm equivalent, 72 Euclidean, 72, 99 matrix, see Matrix vector, 65 n-vectors, 23

o Open set, 201 Optimization, 315 Ordered cycle basis, 291, 292, 296

p. Periodic equation, See Differential equation Periodic orbit, 253, 255 Phase shift, 264 Picard-Lindelof theorem, 186 Pole relocation, 277, 281 Poincare-Bendixson theorem, 253, 255 Probability, 132

Q Quadratic form, 98, 99

R Reachable set, 271 Reduction, scalar control, 279, 281, 284 Regulator, 3, 17 Rest point, 224 Restricted column operation, 289 Routh-Hurwicz criterion, 95, 103 Row reduction, 286 Row space, 27

s Semicontinuity, 209 Semigroup property, 224 Semiorbit, 224 Short diagonal, 297 Solution, 47, 107, 182 general, 200, 205 global, 190 local, 182 maximal extension, 190, 191 periodic, 93, 168, 172, 173, 175, 176, 177, 253, 254, 255, 256, 258, 259 steady-state, 240, 262 transient, 262 Stability, 92, see also Lyapunov Stabilizability, 277 Stabilization, 18 Steady-state solution, 240, 262 Submanifold, 315 Subspace, 24 Successive approximations, 188 Systems linear algebraic equations, 25

T Time-varying coefficients, 106, 128 Topological space, 201 basis, 201 subspace, 201 Transfer function, see Matrix Transient solution, 262 Transversal, 251 Traveling wave, 240

u Undetermined coefficients, 52 Uniqueness, 185

v Van der Pol equation, 9, 199, 258

322

Index

Variation of parameters, 50, 52, 74, 114, 173, 222,262 Vector field, 205 Vector space, 81

w Watt, J., 3

z Zorn's lemma, 191

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