GEOMETRIC CONTROL AND NONSMOOTH ANALYSIS
This page intentionally left blank
Series on Advances in Mathematics for Applied Sciences - Vol. 76
GEOMETRIC CONTROL AND NONSMOOTH In Honor of the 732 Birth2uy of H. Hermed an2 of the 7 h t BirthJay of R. Rockufellar Edited by
Fabio Ancona Universita di Bologna, Italy
Alberto Bressan Penn State University, USA
Piermarco Cannarsa Universita di Roma “Tor Vergata”, Italy
Francis Clarke Universite Claude Bernard Lyon I, France
Peter R Wolenski Louisiana State University, USA
r pWorld Scientific N E W JERSEY
*
LONDON * SINGAPORE
BElJlNG
*
SHANGHAI
*
HONG KONG
*
TAIPEI
*
CHENNAI
Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link,Singapore 596224
USA ojice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ojice: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-PublicationData Geometric control and nonsmooth analysis / edited by Fabio Ancona ... [et al.]. p. cm. -- (Series on advances in mathematics for applied sciences ; v. 76) Includes bibliographical references and index. ISBN-13: 978-981-277-606-8 (hardcover : alk. paper) ISBN-10: 981-277-606-0(hardcover : alk. paper) 1. Control theory--Research. 2. Nonsmooth optimization--Research. 3. Systems engineering-Research. I. Ancona, Fabio, 1964QA402.3.G4362008 5 15'.642--dc22 2008017673
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
Copyright Q 2008 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoJ; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher
Printed in Singapore by B & JO Enterprise
V
PREFACE Applied mathematics and engineering abound with situations in which one wishes to influence the behavior of systems that are modeled as trajectories of differential equations. This has given rise to the large and multivalent subject known as control, the mathematical theory of which is thought of as having been born in the 1950’s. The goal of controlling a system can be (roughly speaking) of two sorts: for optimality (as in the minimization of a cost), or for positional purposes (for example, driving the state to an equilibrium). The first of these is directly in the line of thought of the classical calculus of variations, while the second has its origins primarily in engineering systems design. An important feature of control theory in recent decades has been the convergence of these two types of control at the level of the underlying theory. Thus the distinction between them has blurred in recent decades. A principal reason for this has been the development of tools which successfully address issues of both types, especially those stemming from geometry and from nonsmooth analysis. The interaction between geometry and control theory rests upon the use of classical differential-geometric tools, in particular Lie algebras of vector fields, free and nilpotent Lie algebras, filtrations, symplectic geometry, exterior differential systems, infinite dimensional manifolds. Also involved are such concepts of analytic geometry and dynamical systems as subanalytic sets, stratifications, center and stable manifolds, among others. Somewhat more recently, the use of nonsmooth analysis has revealed itself to be essential in both optimal and positional control. Nonsmooth calculus and its related circle of ideas (variational principles, penalization, differential inclusions, generalized tangency and normality. ..) can now be viewed as a relatively mature subject. The theory has been applied to control issues with great effect, in combination with such tools as viscosity solutions, dynamic programming, Hamilton- Jacobi inequalities, duality, discontinuous feedbacks, convexity and semiconcavity, and nonsmooth Lyapunov functions.
vi
Geometric methods and nonsmooth analysis have together given rise to many recent developments in control theory. Among these are: new and unified formulations of necessary conditions in optimal control, the synthesis and classification of optimal or stabilizing controls, and new techniques for optimal and stabilizing feedback design, in particular sliding modes. The conference Geometric Control and Nonsmooth Analysis, which took place in Rome in June 2006 at the Istituto Nazionale di Alta Matematica (INdAM), celebrated the leading contributions to control theory of two eminent researchers, Henry Hermes and R. T. Rockafellar (often known as Hank and Terry to their many friends). The topics covered a wide range of the issues mentioned above, from the calculus of variations to optimal control, from controllability to stability, from invariance to impulse systems, from approximation to feedback design. This volume, which assembles the contributions of the speakers of the conference, provides an overview which illustrates well the vitality, the range, and the importance of the subject that is control theory. Soon after the conference, we learnt that Wijesuriya Dayawansa, whom we had just seen in Rome, had abruptly passed away. This volume is dedicated to his memory.
Fabio Ancona Albert0 Bressan Piermarco Cannarsa Francis H. Clarke Peter W. Wolenski April 2008
Bologna, Italy Penn State, USA Roma, Italy Lyon, France Baton Rouge, USA
PROFESSOR HENRYHERMES BOULDER
UNIVERSITY O F COLORADO AT
PROFESSOR R. TYRRELL ROCKAFELLAR UNIVERSITY O F WASHINGTON
viii
SCIENTIFIC COMMITTEE A. Bressan P. Cannarsa F . H. Clarke G. Stefani
Penn State University, USA Universith di Roma “Tor Vergata”, Italy -Universit6 Claude Bernard Lyon I, France - Universith di Firenze, Italy -
ORGANIZING COMMITTEE F. Ancona M. Kawski B. Piccoli P. R. Wolenski
Universitb di Bologna, Italy Arizona State University, USA - IAC - CNR, Roma, Italy - Louisiana State University, USA -
SUPPORTED BY
0 0
0
0
0
INdAM - Istituto Nazionale di Alta Matematica “F. Severi”. MIUR PRIN 2004 “Controllo, Ottimizzazione e Stabilith di Sistemi non Lineari: Metodi Geometrici ed Analitici” MIUR PRIN 2005 “Metodi di viscosith, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari” National Science Foundation grant ‘‘Conference on Optimal Control and Nonsmooth Analysis”, DMS 0612807. Dipartimento di Matematica, Universith di Roma “Tor Vergata” .
ix
CONTENTS Preface Conference Committees
V
viii
Multiscale Singular Perturbations and Homogenization of Optimal Control Problems 0. Alvarez, M. Bardi and C. Marchi Patchy Feedbacks for Stabilization and Optimal Control: General Theory and Robustness Properties F. Ancona and A. Bressan
28
Sensitivity of Control Systems with Respect to MeasureValued Coefficients Z. Artstein
65
Systems with Continuous Time and Discrete Time Components A. Bacciotti
82
A Review on Stability of Switched Systems for Arbitrary Switchings U. Boscain
100
Regularity Properties of Attainable Sets under State Constraints P. Cannarsa, M. Castelpietra and P. Cardaliaguet
120
A Generalized Hopf-Lax Formula: Analytical and Approximations Aspects I. Capuzzo Dolcetta
136
Regularity of Solutions to One-Dimensional and MultiDimensional Problems in the Calculus of Variations F.H. Clarke
151
X
Stability Analysis of Sliding Mode Controllers F.H. Clarke and R.B. Vinter
164
Generalized Differentiation of Parameterized Families of Trajectories 177 M. Garavello, E. Girejko and B. Piccoli Sampled-Data Redesign for Nonlinear Multi-Input Systems L. Griine and K. Worthmann On the Definition of Trajectories Corresponding to Generalized Controls on the Heisenberg Group P. Mason Characterization of the State Constrained Bilateral Minimal Time Function
206
228
243
C. Nour Existence and a Decoupling Technique for the Neutral Problem of Bolza with Multiple Varying Delays
248
N.L.Ortiz Stabilization Problem for Nonholonomic Control Systems L. Rifford Proximal Characterization of the Reachable Set for a Discontinuous Differential Inclusion V.R. Rios and P.R. Wolenski
260
270
Linear-Convex Control and Duality R.T. Rockafellar and G. Goebel
280
Strong Optimality of Singular Trajectories G. Stefani
300
High-Order Point Variations and Generalized Differentials H. Sussmann
327
List of Partcipants
359
Author Index
361
1
MULTISCALE SINGULAR PERTURBATIONS AND HOMOGENIZATION OF OPTIMAL CONTROL PROBLEMS 0. ALVAREZ UMR 60-85, Universitd de Rouen, 76821 Mont-Saint Aignan cedex, France E-mail: olivier. alvarezQuniv-rouen.fr M. BARD1 Dipartimento di Matematica Pura ed Applicata, Universitb di Padova, via lPneste 63, 35121 Padova, Italy E-mail: bardiQmath. unipd. it C. MARCH1 Dipartimento di Matematica, Universita della Calabria, Ponte P. Bucci 30B, 87036 Rende (CS), Italy E-mail: marchiQmat.unica1. it The paper is devoted to singular perturbation problems with a finite number of scales where both the dynamics and the costs may oscillate. Under some coercivity assumptions on the Hamiltonian, we prove that the value functions converge locally uniformly to the solution of an effective Cauchy problem for a limit Hamilton-Jacobi equation and that the effective operators preserve several properties of the starting ones; under some additional hypotheses, their explicit formulas are exhibited. In some special cases we also describe the effective dynamics and costs of the limiting control problem. An important application is the homogenization of Hamilton-Jacobi equations with a finite number of scales and a coercive Hamiltonian. Keywords: Singular perturbations, viscosity solutions, Hamilton-Jacobi equations, dimension reduction, iterated homogenization, control systems in oscillating media, multiscale problems, oscillating costs.
2
1. Introduction
The controlled system z o = 2,
ks = f ( ~ s , Y s , Z s , Q s ) , EYS
2
E i s
=g(~s,Ys,zs,as),
Yo
= r(zs,ys,zs,as),
zo = z ,
= y,
where Q is the control and E > 0 a small parameter, is a model of systems whose state variables evolve on three different time scales. Consider the cost functional
PE(t72,Y,Z,QI) := and the value function
I"
+
~(~s,Ys,zs,as)ds h(zt,yt,zt)
uE(t,z, y , z ) := inf P"(t,z, y, z , a). a
The analysis of the convergence of uE as E + 0 gives informations on the optimization problem after a sufficiently large time, namely, when the fast variables y and z have reached their regime behaviour. For two time scales, i.e. r = 0, the problem has a large mathematical and engineering literature, see the books by Kokotovid et al.24and Bensoussan,14 the references therein, and the more recent contributions by Artstein and GaitsgorylO and the a u t h 0 1 - s .For ~ ~ more than two time scales Gaitsgory and Nguyen21 extended the method of limit occupational measures. In this paper we follow a method based on the Hamilton-Jacobi-Bellman equation satisfied by the value function, that is - f a
a
DyuE DZuE D,u" - g . -- r . -1 &
E2
in
}=o
(0,T)x R" x Rm x RP.
It is based on the theory of viscosity solutions (for an overview, see the book Ref. 12) and was used for two-scale problems by the first and the second named author^.^?^ They also developed it further to stochastic systems and differential games.5 In a companion paper6 the authors extended the method to stochastic problems with three and n scales; in those cases the value functions solve some 2nd order degenerate parabolic equation. In the present paper, we show how to apply our method6 to deterministic control problems and 1st order H-J-B equations with a finite number of scales.
3
An important advantage of our PDE approach is that it applies naturally in the generality of differential g a m e s , namely, problems governed by two conflicting players, a and ,L?, acting on the system x s = f(.s,Ys,Zs,Qs,Ps), EYS
E2iS
= g(.s,Ys,zs,as,Ps), = r ( z s ,Ys, z s , as, P S I .
If the game is 0-sum, i.e., the second player's goal is the maximization of the cost P E ,the (lower) value function of the game satisfies the Cauchy problem for the Isaacs PDE
{
&ue
+ minmax { -f P a
+
DxuE- g .
-r
.
in U E ( O ,z,
9- l } = 0 (0,T)x R" x R" x RP,
Y,z ) = h ( z ,Y,z ) .
Throughout this paper, in fact, we will never assume the Hamiltonian be of the Bellman form, that is, convex in the gradient variables. Therefore, our results apply, for instance, to the robust optimal control of systems with bounded unknown disturbances. Our goal is proving that the value functions uE converge locally uniformly to the viscosity solution of a new Cauchy problem (called the eflective problem)
{
i3,u+ E(z,D,u) u(0,). = h(z)
=0
in (0, T ) x R" on B".
(HJ)
For the two-scale case, the effective Hamiltonian and the effective initial condition h are obtained, respectively, as the ergodic constant of a stationary problem and by the time-asymptotic limit of the solution to a related Cauchy problem. For the multiscale case, this construction must be done iteratively. Moreover, owing to this procedure and to the coercivity assumption, the effective (and the i n t e r m e d i a t e ) operators inherit several properties of the starting ones, in particular those ensuring the Comparison Principle (we refer the reader to Ref. 5 for the case of intermediate or effective operators lacking these properties). An interesting issue is to represent the effective solution u as the value function of some new control problem. In some special cases, we shall prove that the effective PDE is associated to a limit control problem whose dynamics and effective costs can be described explicitely. An example of Hamilton-Jacobi equation that fits within our theory is
4
in (0, T ) x R" x Rm x RP, where F is a standard Bellman-Isaacs operator, and, for some u > 0, (PI 2 u and (p2 2 u. In this special case we can compute the effective Cauchy problem, which is
dtu
+
{ 4 0 ,x )
F ( x ,y, z , Dzu) = 0 = miny,, h(x,Y,2)
in ( 0 , T )x R" on R".
,
An important byproduct of the previous theory is the homogenization of systems in highly heterogeneous media with more than two space scales. Now the system is
and the cost
In this case, the value function Cauchy problem
{
V"
satisfies (in the viscosity sense) the
d t w E + m i n m a x { - f ( x , f , ~ , a , P ).D,w" - L ( x , f , $ , a , P ) } = O P
a
in ( 0 , T )x Rn,
~ ' ( 0 X, ) = h ( x ,f , $) .
Here the oscillations are in space. By setting y = X / E , z = X/E' this problem can be written as a singular perturbation one. Motivated by this, we will call throughout the paper x the macroscopic variables, y the mesoscopic ones, and z the microscopic variables. Under an assumption of coercivity of the Hamiltonian in the gradient variables we prove that the solution V" converge locally uniformly to the solution of an effective problem for a suitable construction of the effective operator and initial data. Our result applies to the homogenization of the eikonal equation
(m),
with 'p 2 u > 0. In the framework of viscosity solutions, the study of the two-scale homogenization, initiated by Lions, Papanicolaou, Varadhan26 and improved by Evans,l8>l9has been extended to related questions: see, e.g., Capuzzo-Dolcetta and Ishii17 for the rate of convergence, Horie and Ishii22 and the first author2i7 for periodic homogenization in perforated domains, I ~ h i iArisawa,' ,~~ and Birindelli, Wigniolle15 for non-periodic homogenization, Rezakhanlou and T a r ~ e r , ~S' o ~ g a n i d i s Lions, ,~~ Souganidi~~~ for t~' stochastic homogenization, the book Ref. 12, Artstein, Gaitsgory,l' the
5
first two author^,^ and the references therein for singular perturbations in optimal control. Let us stress that all the aforementioned papers consider only two scales and that, as far as we know, fully nonlinear problems with multiple scales have been attacked for the first time in our paper Ref. 6; in fact, iterated homogenization was addressed only in the variational setting, starting with the pioneering work of Bensoussan, J.L. Lions and Papanicolaou13 for linear equations and, afterwards, for semilinear equations, using the r-convergence a p p r o a ~ h l (see ~ > ~also ~ and references therein) or Gconvergence t e ~ h n i q u e s . l > l l > ~ ~ The plan of the paper is as follows. The standing assumptions are listed in Section 2. Section 3 recalls the notions of ergodicity and stabilization for a Hamiltonian. Section 4 is devoted to the regular perturbations of two-scale problems because they are of independent interest and for later use. We address the multiscale singular perturbations and the multiscale homogenization respectively in Section 5 and in Section 6. Some examples arising from deterministic optimal control theory and differential games are collected in Section 7. One of them replaces the coercivity of the Hamiltonian with a non-resonance condition and allows to show that exchanging the roles of E and E~ may produce a different effective PDE. 2. Standing assumptions
We consider Bellman-Isaacs Hamiltonians
for the family of linear operators L " ' P ( ~ , Y , P , , P , ) := -Pz . f ( z , y , a , P )- P , * g ( z , y , a , P ) - l ( z , y , a , P ) . The following assumptions will hold in Sections 3 and 4: 0 0
0
0
0
The control sets A and B are compact metric spaces. The functions f , g and 1 are bounded continuous functions in Rn x Rm x A x B with values, respectively, in Rn, Rm and R. The drift vectors f and g are Lipschitz continuous in (z,y), uniformly in (a, p). The running cost 1 is uniformly continuous in (z, y), uniformly in (Q,P). The initial condition h is bounded and uniformly continuous.
6
0
The functions f , g, h and 1 are Zm-periodic in the fast variables y. H is coercive in p , : there exist v,C E R+ such that:
H(z,Y,Pz,P,) L vlpyl
-
C ( 1 + lPzI>
for every z,Y,PZ,Py
Let us observe that the last assumption holds provided that:
Brn(0,v) c m M z ,Y, a ,P )
I QI E A }
'dz, Y, P
where B,(O, v) is the ball centered in 0 with radius v in the space R". In the deterministic control theory, this relation entails a strong form of small-time controllability of the deterministic fast subsystem, i.e. any two points can be reached from one another by the player acting on a , whatever the second player does, and within a time proportional to the distance between the points. We introduce the recession function (or homogeneous part) of H in p , by
We note that H' is positively 1-homogeneous in p , , namely H'(z, y, Xp,) = XH'(z, y , p y ) for X 2 0 and that, for every ZIPZ E Rn, there is a constant C so that
IH(z,Y,Pz,P,) -H'(z,y,p,)I
Ic
'd(Y,P,) E Rrn x Rrnl
(3)
for every (z,pZ)in a neighborhood of ( T , P Z ) . In the case of three scales, treated in Section 5.1, the Hamiltonian depends also on z and p,. Then the linear operators La)P have the additional term -p, . r(x,y , z , a ,P), whereas f , g, and 1 may depend on z as well. We make the same assumptions on the dependence of the data from t as from y, namely, periodicity, Lipschitz continuity of f , g, T , and uniform continuity of 1. The obvious analogous assumptions are made in the general case of j 1 scales studied in Section 5.2.
+
3. Ergodicity, stabilization and the effective problem
The aim of this Section is to recall from Ref. 5,6 the notions of ergodicity and of stabilization that are crucial in the definition of the effective problem We establish some properties of the effective operators and in some cases we provide their explicit formulas.
(m).
7
3.1. Ergodicity and the effective Hamiltonian This Subsection is devoted to recall the definition of ergodicity introduced in Ref. 4. For @,p,) fixed, by the standard viscosity solution theory, the cell 6-pro blem
6wg
+ H(Z, y , p X ,Dywg) = 0
in Rm,
wg periodic,
(CPS 1
has a unique solution. We denote the solution by w g ( y ;Z, p,) so as to display its dependence on the frozen slow variables. We say that the Hamiltonian is ergodic in the fast variable at (Z,p,) if
6wg(y;Z, p,)
4
as 6 4 0, uniformly in y .
const
In this case, we define
-
H(Z,p,) := - lim Swg(y;Z,p,); 6-0
the function is called effective Hamiltonian. We say that H is ergodic if it is ergodic at every ( Z , p , ) . In the next Proposition we collect some properties of and, in some special cases, also its explicit formula. Proposition 3.1. Under the standing assumptions there holds
(a) the Hamiltonian H is ergodic. is regular: there are C E I% and a modulus of continuity w such that: (b) I q W P ) -mZZ,P)I
5 ClZl -221(1+
IZ(ZlP1) - m Z 1 P Z ) l I C(lp1 - P 2 l )
lpl) + W ( l Z l VZlPi
-mI) V'ZilP E Rn;
E R";
in particular, the Comparison Principle holds for the efective problem
(m). (c) If
H@,YlF,,PY) 2 H ( ~ ~ Y l F , , O ) VYlP, E R"1 then
(4)
has the explicit formula: -
H(Z,p,) = maxH(Z, yljjx1O). Y
Proof. The proofs of (a ) and ( b ) are slight adaptations of the arguments used in [4, Proposition 91, [3, Proposition 121 and [19, Lemma 2.21 so we omit them. (c) The Comparison Principle for the cell &problem (CPg) entails: 6wg 2 -supy H ( Z ,y , p x , 0); as 6 --+ 0, we infer: -
H ( Z , F Z )5 supH(~,y,p,,O). Y
8
In order to prove the reverse inequality, we shall argue by contradiction, assuming: p ( T , p , ) < H ( T , y,p,, 0 ) in a open set U . Therefore, the cell &problem reads
=o
~ ~ b + ~ o ( Y , ~ y W b ) + ~ ( : , Y , P , , O )
where Ho(y,q) := H(T,y,Tj,,q) - H(T,y,p,,O). The ergodicity of H and the relation (4) entail
As 6 -+ 0, we obtain the desired contradiction.
0
Remark 3.1. Let us observe that condition (4) is satisfied if the control a splits into (alla2),where ai belongs to the compact Ai (i = 1 , 2 ) , the drift f and the running cost 1 do not depend on a2, while the drift g = g ( x ,y , a2,P ) fulfills: &Z(O,v) c = = { g ( z , y 1 a 2 , P )
I a 2 E A21
k Y l P .
3.2. Stabilization and the eflective initial data
The stabilization to a constant for degenerate eqs. was introduced by the first two author^.^ For T fixed, the cell Cauchy problem for the homogeneous Hamiltonian H'
+ H'(T, y , D,w)
w(0,y ) = h(T,y ) on R" (CP') has a unique bounded viscosity solution w(t,y;:). Observe that by the positive homogeneity of H ' , the constants llh(T,.)/Im and -Ilh(Z, .)llm are respectively a super- and a subsolution. Furthermore, the Comparison Principle yields the uniform bound: Ilw(tl .)llm 5 llh(zl.)llm for all t 2 0. We say that the pair ( H ,h ) is stabilizing (to a constant) at zif &w
=0
in (0, +oo) x EXm,
w ( t , y ; T )-+ const
as t -+ +oo, uniformly in y.
(5)
In this case, we define -
h ( z ) := lim w ( t ,y ; T ) . t-++cc
We say that the pair (HIh) is stabilizing if it is stabilizing a t every ? Ei? R". The function 5 is called the effective initial data.
Proposition 3.2. Under the standing assumptions, the pair ( H ,h ) is stabilizing. Moreover, the effective initial datum is continuous and has the
9
form: -
h ( T ) = min h(T, y ) . Y
The proof is a slight adaptation of the arguments used in [4, Proposition 101 and in 13, Theorem 81 and we shall omit it. 4. Regular perturbation of singular perturbation problems
This Section is devoted to a convergence result for the regular perturbation of a singular perturbation problem
i
atu"
+ H E (x,y , D
y) o
~ ~ E ,
=
in (0,T ) x
~n
on R" x Rm.
u"(0,x,Y ) = h E ( ZY, )
x ~m
(HJ")
Regular perturbation means that H" -+ H and h" -+ h as c 0 uniformly on all compact sets and that H , h and every H E , h" satisfy the standard assumptions of Sec. 2. For example, we have a regular perturbation when the control sets A and B are independent of E and the functions f E , g" and 1" converge locally uniformly to f ,g and 1. We suppose also that
c
IH"(Z, Y , 0,O)l I
q x ,Y ) ,
(7)
for some constant C independent of c small (this assumption is satisfied for instance if the running costs 1" are equibounded). Let us note that the problem (HJ") has a unique bounded solution (that is also periodic in y ) and fulfills the Comparison P r i n ~ i p l e . ~ > ~ The next result and the arguments of its proof will be used extensively in the next Sections.
Theorem 4.1. Assume that H E and h" converge respectively to H and to h uniformly o n the compact sets and that the equiboundedness condition (7) holds. Then, U" converges uniformly on the compact subsets of ( 0 , T )x Rn to the unique viscosity solution of where the eflective Hamiltonian and the effective initial datum are defined respectively in Subsec. 3.1 and 3.2.
(m)
Proof. The proof of this Theorem relies on [6, Corollary 11 (see also [4, Theorem 11)and on the ergodicity and stabilization results stated in Proposition 3.1 and in Proposition 3.2. For the sake of completeness, let us sketch the main features of the argument. The family {u"} is equibounded; indeed,
10
the Comparison Principle gives: IIu'(t, .)lloo 5 supEllh"llw therefore define the upper semilimit ii of U' as follows -
u(t,x) :=
-
u ( 0 , x ) :=
supu"(t', x', y)
limsup E'O,
(t',x')+(t,z)
> 0,
if t
2/
ii(t',x') if t = 0.
limsup (t',X')'(O,X),
+ Ct. We can
t'>O
We define analogously the lower semilimit g by replacing limsup with liminf and sup with inf. The two-steps definition of the semilimit for t = 0 is needed to avoid a possible initial layer. Let us notice that, under our hypotheses, the effective problem satisfies the Comparison Principle and admits exactly one bounded solution u.If ii and are respectively a super- and a subsolution to then the proof is accomplished. Actually, the Comparison Principle ensures: ti 5 u 5 u.On the other hand the reverse inequality g 5 ii is always true. Whence, we have ii = g = u and, by standard arguments, we deduce: uE-+u locally uniformly as E + 0. Let us now ascertain that ii is a subsolution to being similar, the other proof is omitted. We proceed by contradiction assuming that there are a point (3, ?iE ?()0 , T )x Rn and a smooth test function cp such that: u(3,z)= cp(3, z),(3, ?f)is a strict maximum point of ii - cp and there holds
(m)
(m)
(m);
dtcp(3,z) for some v > 0. For every r
+ q z ,D,cp(t, 3 ) )2 37
> 0, we define
H,E(y,py):= min(H'(x,y,D,cp(t,x),p,)
1 It - 31 5 r,
Ix - zI
5 r}.
px = D,cp(I,Z) and we fix ro > 0 so that as It - 31 < T O , Ix - ZI 5 ro. l&cp(t,x) - dtcp(3,~)I5 7 Now we want to prove that, for every r > 0 small enough, there is a parameter E' > 0 and an equibounded family of functions {x' I 0 < E < E ' } We put
:= g ( T , p x )with
(called approximated correctors) so that
2 77 - 2~7 in R ~ . (8) To this aim, taking into account the ergodicity of H , we fix a parameter 6 > 0 so that the solution wg to the cell &problem (CPg) fulfills: ~ , " ( yD,x') ,
I I h + f7lloo 5 7. Since H,E(ylpy)-+ H(z,y,p,,p,) as ( E , T ) -+ (0,O) uniformly on the compact sets, the stability property entails that the solution w : , ~of
Sw:,,
+ H,E(y, Dywi,,) = 0
in
Rm,
w;,,periodic,
11
converges uniformly to wg as 0 < r' < min{ro,Z}, we get
5 2q
IIdw;,,
(E,T) +
(0,O). In particular, for e'
> 0 and
< E' and 0 < r < r'.
when 0 < E
The function X" = w;,~ is a supersolution of (8). Moreover, by the Comparison Principle, the family {x"} is equibounded: llxdlloo 1. 6-1sup{lH,E(y,O,O)l I y E R", 0 < e < e'}. Hence, our claim is proved. We consider the perturbed test function
$"(t,2 , Y) := d t ,).
+EX"(9).
In the cylinder QT =]t-r,Z+r[xB,(Z) xR", $" is a supersolution of (HJE) (see Ref. 4 for the rigourous proof). Since {+"} converges uniformly to cp on we obtain
a,
sup(u" - $")(t',x',y)
limsup "-0,
t'-t, 5 ' - x
= ii(t, x) - 'p(t,x).
y
But (?,Z) is a strict maximum point of ii - 'p, so the above relaxed upper limit is negative on 8QT.By compactness, one can find rl' > 0 so that uE5 -q' on 8QT for E small. Since $" is a supersolution in Q T , we deduce from the Comparison Principle that $' 2 uE r]' in QT for E small. Taking the upper semi-limit, we get 'p 2 ii q' in (t - r, t r ) x B ( Z ,r ) . This is impossible, for 'p(3,:) = ii(Z,:). Thus, we have reached the desired contradiction.
+"
+
+
+
We now check that ii satisfies the initial condition, that is ii 5
h. Let
wF be the unique solution of the following Cauchy problem
{
8twF
+ H,E>'(y,DywF) = 0 in (0, +co) x R", on R",
WF(0, Y) = h 3 y )
wF periodic in y
where the Hamiltonian H,Ei' and the initial datum hF are given by H;)'(y,py) := min{HE7'(x,y,p,)
I 1x - ZI 5 r } ,
hF(y) := max{h"(x, y) I Ix - 51 5 r } . Let us claim that T-0,
limsup E+O,
t+CC
sup IwF(t,y) - h ( ~ )=l 0. y
(9)
Fix Q > 0. The stabilization ensures that the solution w to the cell Cauchy problem (CP') fulfills l l w ( T .) - m)IIoo
5 rl
12
for some T > 0. Letting ( E , T ) 4 ( O , O ) , we have HF?' H'(z,.) and h: 4 h(z,.)uniformly on the compact sets; therefore, by the stability properties of viscosity solutions, we know that w: + w' locally uniformly. Whence, there are E' and r' so that -+
llwF(T, .) - z(?E)llm5 277
for all 0
< E < E', 0 < r < r'.
Since HFJ(.,0) = 0, the Comparison Principle entails that for all t 2 T , 0
IlwF(t, .) - 8(:)Ilw 5 277
< E < E', 0 < r < r'.
This gives (9). Let r > 0, E' > 0 and T > 0 be such that the last inequality is satified. For Q,+(:) := ( 0 , r ) x BT(Z)x R", we fix A4 so that M 2 ~ ~ u ' ~ ~ L ~ for all E < E' and we construct a bump function +O that is nonnegative, smooth, with $o(E) = 0 and +O 2 2M on Finally, we choose the constant C > 0 given by (3) so that
a&.(:).
IH'(x,Y,Dz+o(~),Py) - HC1'(x,y,Py)l5 for every (y,py), z E B T ( T )0, < E
$"(t,Ic, y )
c
< E'. We introduce the function
:= w;(E-lt,
y)
+ $o(z) + Ct
and we observe that it is a supersolution of
at$" + H " ( z , y, D2$E,~-1Dy$E)= 0 $' = h"
on (0) x B T ( z )x R",
in Q:(E) $" = M on [O,r)x aB,(z) x R".
By the Comparison Principle, we deduce that u E ( tz, , y)
5 $'(t, x,y)
= wF(c-'t,y)
+ $o(z) + C t
Taking the supremum over y and sending E
E ( t , x) I @)
+ 277 + +o(x) + Ct
4
in
Q:(Z).
0, we obtain the inequality
for all t
> 0,
z E B~(:).
z(:) +
Sending t -+ O+, z -+,: we get E(0,T) _< 7. Taking into account the arbitrariness of v, one can easily accomplish the proof.
Remark 4.1. Let us stress that the coercivity assumption has been used only for establishing the following properties: i) the starting Hamiltonian H is ergodic, ii) the pair ( H ,h) is stabilizing, iii) the effective Hamiltonian H is sufficiently regular to fulfill the Comparison Principle. It is worth to recall that there exist non-coercive Hamiltonians that enjoy properties (i)-(iii)(e.g., see Sec. 7 below). It is obvious that Theorem 4.1 applies also to these Hamiltonians.
13
5. Singular perturbations with multiple scales
This Section is devoted to the study of singular perturbation problems having a finite number of scales. For the sake of simplicity, in the first Subsection we shall focus our attention on the three scales case, which is the simplest one, providing a detailed proof of our result. After, in the second Subsection we shall briefly give the result for a wider class of problems.
5.1. The three scale case We consider the problems:
{
%,9 )= o
+
in (0, T ) x I W x~
atu& H E (2, y,z , D ~ ~ E , uE(O,x,Y,2) = h"(2,Y,2)
on
Rn x IW"
IW"
x RP
x RP
(10) where H E and h" are 1-periodic in y and z . Each variable corresponds to a certain scale of the problem: z is the macroscopic (or the slow) variable, y is the mesoscopic (or the not so fast) variable and z is the microscopic (or the fast) variable. Roughly speaking, we shall attack this problem iteratively: by virtue of the different powers of E , one first considers both x and y as slow variables, freezing them and homogenizing with respect to z and after, still with z frozen, one shall homogenizes with respect to y. In other words, in a first approximation, problem (10) is a singular perturbation problem only in the variable z ; under adequate assumptions of ergodicity and stabilization with respect to z , we shall achieve a mesoscopic effective Hamiltonian H I and u"(t,z, y, z ) should converge to the solution wE(t,2,y) of the mesoscopic problem
i
atve
+HI
(z, y, D,V",
y)o
v"(O,z, Y) = hl(? Y)
=
in (0, T ) x
IW"
x
~m
on R" x R".
This problem falls within the theory of Sec. 4 (see also RefU4); V" will converge to the solution u of the limit problem provided that H1 is ergodic and ( H I ,h l ) is stabilizing. In conclusion, we expect that u E ( tz, , y, z ) will converge to u(t,z) where the effective quantities are defined inductively. For the sake of simplicity, we shall assume as before that the operator H is given by
(m)
H ( 2 , Y,z , Pz 7 P y ,Pz ) :=
g;z2z Layo
(2, ! I , 2, Pz
,P, ,P Z )
7
14
for the family of linear operators
La%, Y,21 Px,P,, Pz) := -Pz . f(z1Y
7
-Pz
P ) - P, g(x,Y 2, P ) . T ( Z , Y , z , a ,P ) - q x , Y,z , Q!,P).
2 , Q,
*
I
Q,
We shall require the following assumptions:
0
0
H E -i H and h“ -+ h as E + 0 uniformly on the compact sets. We also suppose that HI h and every H E , hE satisfy the standard assumptions of Sec. 2, i.e. they are 1-periodic in (y, z ) and H E ,H are HJI operators with the regularity in the coefficients suitably extended to the additional variable z. The Hamiltonians are equibounded: IHE(x,y, z , O , O , 0)l I C , for every x,y, z and E . Microscopic coercivity The Hamiltonian H is coercive in p , : there are v,C E R+, such that H ( ~ , Y , ~ , P z , P , , P z2)
Mesoscopic coercivity
~~,Y,z,PmP,lPz. -C(1 + lPxl + IPyl) For some v,C E R+, there holds
VIP21
H ( ~ , Y , ~ , P Z , P ,L, VlPYl O)
- C(1
+ IPZI)
~~,Y,zlPxlP,.
It is worth to observe that the first two assumptions ensure that problem (10) admits exactly one continuous bounded solution uEthat is periodic in (y, z ) . For instance, the second assumption is guaranteed by the equiboundedness of the running costs while the last two assumptions are satisfied if there is v > 0 such that, for every ( 2 ,y, z , P), there holds
I Q! E A } , B,(O, v) c COI~V{T(Z,y, z , Q, P ) I a E A}.
Bm(0,v) c = i 3 g ( z 1 Y,z , Q, P )
(11) (12)
We denote by HI the recession function of H with respect to the variables (y,z):
H ’ ( z , y , z , ~ y , ~:= z )minmax{-~y * g(x7 Y, z1 Q ~ P- )P Z D E B aEA
* T ( Z , Y ,~
1
PI). ~
The function H’ is positively 1-homogeneousin (py,p z ) and for every (Z, pZ) there is a constant C such that
IH(~,v,z,PZ,P,,Pz) - H1(~,Y,~,Py1Pz)l Ic
V Y , P , E Rrn,Z , P Z E EXp (13) for every (x,pz)in a neighborhood of (z,px). We introduce also the recession function with respect to z for x and y frozen:
H ’ ’ ( z , Y , ~ , P:= ~ )H ’ ( z , y , z , O , ~ z =minmax{-~z ) P E B (YEA .r(x,~,z,~!,P)).
1
15
We observe that H" is positively 1-homogeneous in p, and for every (F,g,px,py) there is a constant C such that
IH(z,Y,z , Px, P,, Pz) - HYz, Y,z , Pz)l 5
c
vz, Pz E Itp
(14)
for every ( ~ c , y , p ~ , pin , ) a neighborhood of (z,g,pz,py). Let us now state some results on the construction of the effective Hamiltonian B and the effective initial datum E. For some special cases, explicit formulas are also provided.
Proposition 5.1. Under the above assumptions, we have: (a) The Hamiltonian H is ergodic in the microscopic variable z . The effective Hamitonian H1 = Hl(z,y,px,p,) is regular: there are C E R and a modulus of continuity w such that:
I H l ( ~ , Y , P l , q l ) - H l ( ~ , y , P 2 , 4 2 )I I C(lP1 -P2l+ 141 -421)
f o r eve? z i , p i E Rn, y i , q i E R". (b) HI i s coercive in p,; moreover, it is ergodic in y and its effective Hamiltonian fuljills the Comparison Principle. (c) Assume that for each ( j j , q ) there holds H(Z,V,z7Fx,Fy,~z) -H(Z,g,z,Pz,FyrO) 20
Vz,pz E R p , (15)
and H(:,y,z,p,,py,O)-H(Z,y,z,iT,,O,O)
20
VY,P,
E
R",
v z E Rp*
(16)
Then, p(z,px) can be written as -
H(Z,px) = max H(T, y, z,px, 0,O). Y,Z
( d ) (6,Lemma 21 The recession function H' is ergodic in the microscopic variable z ; its effective Hamiltonian is the recession function Hi of HI.
Proof. a ) By virtue of the microscopic coercivity and Proposition 3.1-(a), H is ergodic in z with an effective Hamiltonian HI. The regularity of H1 is an immediate consequence of Proposition 3.1-(b) (with z and px replaced respectively by (5, y) and (pz,py)).
16
b) In order to prove that H1 is coercive in p,, we first observe that the solution wg to the microscopic &cell problem satisfies: Swg 5 - inf, H ( z ,y, z , p , , p , , 0). As S -+ 0, we get: i;f
H ( z , y7 z>pZ,p,,O) 5
(z7
y, p Z 7 p y )
and, by the mesoscopic coercivity, we deduce that H1 is coercive in p,. Applying Proposition 3.1, one obtains the second part of the statement. c) Proposition 3.1-(c) yields: H 1 ( z , y , p z , p y )= max, H(z,y, z,p,,p,,O). For each (Z, p,), relation (16) ensures:
Applying again Proposition 3.1-(c), we obtain the statement. d ) For the sake of completeness, let us recall the arguments of [6, Lemma 21. Arguing as in ( a ) , one can prove that H ~ ( z , y , z , p ~ , := p~) X - l H ( z , y, z , 0 , Xp,, Xp,) is ergodic in z with the effective Hamiltonian H x ( z , y , p Y ):= X-lHl(z,y,O,Xpy). Since H A + H' as X -+ +oo uniformly (for z bounded), by the Comparison Principle on the cell &problem, one obtains that H' is ergodic in z with H i := limx++co X - l H l ( z , y, 0, Xp,) as effective Hamiltonian. Let us now check that H i is the recession function of H I . We first note that the positive 1-homogeneity of H' entails the one of H i . By estimate (13), the Comparison Principle yields: IHl(z,y,p,,Py) - Hi(z,y,Py)l 5
c
VY,P, E R"
for every ( z , p , ) in a neighborhood of (Z7pZ),namely H: is the recession function of H I . 0
Proposition 5.2. Under our assumptions, the pair ( H " , h) is stabilizing at ( T , g ) to rnin, h(T,g, z ) =: hl(T,g). Moreover, the pair ( H I ,h l ) is stabilizing at T to min,,, h(T,y, z ) =: E(T). The proof of this Proposition relies on the iterative application of Proposition 3.2 and we shall omit it.
Theorem 5.1. Under the above assumptions, the solution u" to problem (10) converges uniformly on the compact subsets of (0, T )x Rn to the unique viscosity solution of where the effective Hamiltonian and the effective initial datum are defined respectively in Propositions 5.1 and 5.2.
x
(m)
x
Proof. The proof of this Theorem is based on [6, Theorem 21 and on the properties of ergodicity and stabilization established in Propositions 5.1
17
and 5.2. For the sake of completeness, let us just stress the crucial parts. We shall argue as in the proof of Theorem 4.1: we set -
u(t,x) :=
-
u(0,z) :=
sup uE(t’,z’, y, z ) if t
lim sup E’O,
> 0,
(t’,z‘)+(t,z) Y J
~ ( t z’) ‘ , if t = 0,
limsup (t’,z‘)-(O,z), t’>O
(m).
and we want t o show that E is a subsolution t o As before, it suffices t o prove that, for every r > 0 small enough, there is a parameter E’ > 0 and an equibounded family of continuous correctors {x‘ I 0 < E < E ’ } so that
F) L
H E ( z , y ,z , D,cp(t, z), DyxE,
- 277
in QT(?,Z):= (?-r,?+r) x BT(z)x R” x RP, for every E we consider the mesoscopic &cell problem
< E’. To this aim,
=0
6W6 +Hl(:,y,F,,DyWb)
(17)
with pz := D,cp(?,z). For 6 > 0 sufficiently small, the ergodicity of (established in Proposition 5.1-(b)) ensures
I I h+
mx 5
where E is defined as before. For every problem
E
77
H1
(18)
> 0 and r > 0, we consider the
withH,E(y,z,py,pz):= minzEB,(Z) HE(x,y,z,Dzcp(xC),py,pZ). Wenotethat H,E(y,z,pll,pZ)--t H ( ~ , y , z , ~ ~ , p , , plocally ,) uniformly as ( E , T ) (0,O) and that the limit Hamiltonian H is ergodic in z with effective Hamiltonian HI. Hence, applying Theorem 4.1 for the stationary eq., we obtain that { w ; , ~uniformly } converge t o w6 as ( E , T ) + (0,O). By (18), we deduce that there are small E’ and r’ so that -+
for all o < E < E’, o < r < r’. IISwg,, +Ell, 5 2r] Finally, as in Theorem 4.1, it suffices to define x E ( y , z ) := w ~ , ~ ( Y ,(for z) 0 < r < r’ fixed). Now, let us check that E ( 0 , T ) 5 E(Z).We introduce the following notations: fy(Y,z,Py,Pz) := I X -m& XI
HE%,Y,Wy,Pz),
hF(y, z ) := max h E ( zy, , z). IX-TI
18
It can be easily checked that, as ( E , T ) -+ (0, 0), H,"?' and h: converge locally uniformly respectively to H ' ( 5 , .) and to h(z,.). Let w: be the unique solution of the Cauchy problem
atwF
+ H,Ei'(y,z , Dyw:,~-lD,w:)
wE(O,y, z ) = hF(y, z ) ,
=0
in (0, +oo) x R" x Rp
on R" x R p , w: periodic in y and in z.
By Proposition 3.1-(d) and Proposition 5.2, the limit Hamiltonian H ' ( T , .) is ergodic with effective Hamiltonian H i and the pair (H", h ) stabilizes with respect to the microscopic variable z. Thus, Theorem 4.1 ensures that, as ( E , T ) -+ ( O , O ) , the solution w: converges locally uniformly to the one of the mesoscopic cell Cauchy problem
&w
+ H i @ , y, D,w)= 0
4 0 , Y) = hl@, Y)
in (0, +GO) x Rm, on Rm, w periodic in y.
Since ( H i ,h l ) is stabilizing (still by Proposition 5.2), for every 17 > 0, there exists T > 0 such that Ilw(T, .) - x(T)Iloo5 7. Hence, for every 17 > 0 and for T sufficiently large, there exist E' and r' so small that
llwF(T,+,.) - x(z)II,
5 217 for every E 5 E',
T
5 r'.
Therefore, by the Comparison Principle, we obtain:
IlwF(t, ., .) - ~ ( ? E ) / I m5 17 for every E 5 E',
T
5 T', t 2 T
and we conclude as before.
0
Remark 5.1. This Theorem also applies to non coercive Hamiltonian that are microscopically stabilizing and ergodic with a mesoscopic Hamiltonian that fulfills the Comparison Principle, stabilizes, and is ergodic with an effective Hamiltonian that also fulfills the Comparison Principle (e.g., see Sec. 7 below). We refer the reader to the paper Ref. 6 for the case when the Comparison Principle fails either for the mesoscopic Hamiltonian or for the effective one. 5 . 2 . The general case
We consider the problem having j &uE
+ 1 scales:
+ H E(Z,YI,.. . ,y j , D , U ~ , E - ~ D ~.~. .U, E' , - ~ D ~ ~=U0" )
(0,T)x Rn x Rm' x . . . x Rmj Rn x Rm' x . * ' x Rmj. U E ( O , X , y1,. . . , y j ) = h"(z,y1,. . . , y j ) We assume the following hypotheses:
(19)
19
H E -+ H and h' -+ h locally uniformly as E -+ 0; these functions are periodic in ( y l ,. . . , y j ) ; the initial data h" and h are BUC; the H E are equibounded: IH'(z, y1,. . . , y j , 0 , 0 , . . . , 0)l I C ; iterated coercivity: there are'v, C E It+,such that, for every k = j , . . . , 1, there holds H(zl
911 ' * * 7yjIpZ>pyl? ' * '
> P Y k ,O,
'
2
7
"
'IPYk
I
for every z 7 Y17 ' ' * > Y j > p Z , p y l >' ' * I P Y k ; there exists a recession function HE?'= H E > ' ( yz 1, , . . . , y j , p y l , . . . ,pyj), positively 1-homogeneousin ( p y l ,. . , ,pY3),which satisfies, for some constant C > 0
IH%,
Y1,
. . . ,Y j , P z , P y 1 ,
'
*.
I
pyj
-
HE%,
Y1, *
. .,Y j , P Y l
I * * ' I
Py3)
I Ic
for every y i l p y , E IW". (i = 1 , . . . , j ) , for every ( z , p z ) in a neighborhood of ( T , p Z )and for every E . Let us observe that the equiboundedness of the running costs ensures the third assumption; furthermore, the Hamiltonian
is iteratively coercive whenever there holds %,(O,
v) c m { g i ( z , y l i . . .
1
~
j a1 i P)
I
E
A}
for every z, y 1 , . . . , y j , P (i = 1 , . . . , j ) . We write Hj = H and hj = h. For each i = j , . . . , 1, Hi fulfills the Comparison Principle and is ergodic with respect to yi. We denote by Hi-1 := Ho. its effective Hamiltonian. We set As before, one can prove that the Hamiltonian H has a recession function H' which is the uniform limit on the compact sets of HE>'as E + 0. Moreover, as in Proposition 5.1-(d), we obtain that Hi has a recession function H,!, that every H,! is ergodic and that its effective Hamiltonian is Hi.-l (for every i = j , . . . , 1). For H[ := H , ! ( z , y l , .. . ,yi,O,. . . ,O,pi), the pair ( H [ , hi) is stabilizing with respect to yi at each point (z, y 1 , . . . , yi-1) (for i = j , , . . ,1).We denote by hi-1 its effective initial data and we put = ho.
20
Theorem 5.2. Under the above assumptions, U" converges uniformly on the compact subsets of (0, T ) x R" to the unique viscosity solution of
(m).
We shall omit the proof of this Theorem: actually, it can be easily obtained by using iteratively the arguments followed in Theorem 5.1.
Remark 5.2. This result can be immediately extended to pde with non power-like scales: dtu"
with ~1 (19) for
-+
+ H"(Z,y 1 , . . . ,y j , D , U " , E ~ l D y l u " I . .,&jlDYjU")= 0 , 0 and
~i =
E ~ / E ~ - I-+0
(i
=
2 , . . . , j ) . The above eq. encompasses
2.
6. Iterated homogenization for coercive equations In this Section we address the study of the iterated homogenization of first order equations with multiple scales. For the sake of simplicity, we shall focus our attention on the three-scale case:
d t z f + F E ( x , ~ , f , D z v "= )O X) = h " ( z ,,: f )
{~"(0,
in ( O , T ) x R n on Rn.
(20)
The operator F" = F " ( ~ , y , z , p and ) the function h" = h " ( x , y , z ) are periodic in y and in z ; moreover, they are respectively a regular perturbation of F and h , namely, F" -+ F and h" -+ h locally uniformly as E -+ 0. We assume that F is a HJI operator
F(X,Y,Z,P,) := minmax{-pz PEB aEA
'
f ( G Y , z , a , P )- l ( G Y , Z , ~ I P ) )
where the drift f and the costs 1 and h fulfill the requirements stated in Sec. 2. We assume also that F is coercive with respect to p : for u, C E EXf there holds
F(X,Y,Z,P) L VIP1 - c
vJz,Y,zlP;
(21)
(for instance, this condition holds if B,(O, v) c E%iV{f(z,y , z , a , P) I a E A ) for every (z, Y l Z l P)). Let us emphasize that problem (20) encompasses the problem studied by Lions, Papanicolaou and Varadhan.26 Actually, we extend the previous literature in two directions: we consider regular perturbations F" and h" of F and of h and, mainly, we address the three-scale problem both for the HJI eq. and for the initial condition. Our purpose is to apply Theorem 5.1 by proving that (20) is a particular case of (10). To this aim, we introduce the shadow variables y = X / E and z =
21
z / c 2 and consider the solution u'(t, z, y, z ) of (10) with the Hamiltonian
H given by H ( z ,Y,z , P z , P,,
P z ) = F(z1 Y,z , P I
+ P, + P z ) .
The Hamiltonian H clearly satisfies the assumptions of Sec. 5. By uniqueness, one sees immediately that
v"(t,z)
= U"(t, X , Z / E , 2/E2).
By the periodicity in y, Theorem 5.1 ensures that u" converges uniformly on compact subsets to the unique solution of Therefore, the following result holds:
(m).
Corollary 6.1. Under the above assumptions, v" converges uniformly on the compact subsets of (0, T ) x R" to the unique viscosity solution of
(m).
Remark 6.1. Arguing as in the Subsec. 5.2, one can easily extend this result to the homogenization with an arbitrary number of scales. 7. Examples
In this Section we discuss some examples arising in the optimal control theory and in deterministic games. For simplicity, we shall only address t hree-scale problems. The first Subsection is devoted to a singular perturbation of a deterministic game; in some special cases, the effective problem is still a deterministic game and we shall provide the explicit formulas for the effective quantities (dynamics, pay-off, etc.). The second Subsection concerns the homogenization of a deterministic optimal control problem. In the third Subsection the coercivity of the Hamiltonian is replaced by a non-resonance condition introduced by Arisawa and Lions.g Here we show that the effective problem may change if the roles of E and E' are exchanged. 7.1. Singular perturbation of a differential game
Fix T > 0 and, for each
E
> 0, consider the dynamics
22
for 0 5 s 5 T . The admissible controls a, and Ps are measurable function with value respectively in the compact sets A and B . They are governed by two different players. We consider the cost functional
I"
P E ( t , ~ l Y 1 ~ l ~ := 1 P ) ~(~s,Ys,z,,a,,Ps)ds+h(zt,y,,zt). The goal of the first player controlling a is to minimize P", wheras the second player wishes t o maximize P"by controlling P. Consider the upper value function u E ( tz, , y, z ) := sup
inf P'(t, z, y, z , a, P [ a ] ) ,
PEB(t)a E 4 t )
where d(t) denotes the set of admissible controls of the first player in the interval [ O , t ] and B ( t ) denotes the set of admissible strategies of the second player in the same interval (i.e., nonanticipating maps from d(t) into the admissible controls of the second player; see Ref. 20 for the precise definition). Under the assumptions of Sec. 2 the upper value function is the unique viscosity solution of the HJI eq. (10) with2'
:>:
H E= H ( z ,y Iz I pz py ,p z ) =
{-Pz
*
f -P y
*
SJ - P z r -
I
Let us assume (11)-(12), so the Hamitonian H is microscopically and mesoscopically coercive. Suppose in addition that H has the properties (15)-( 16). By Theorem 5.1, the upper value u" converge locally uniformly to the solution u to the effective problem
{
&u
+ max min { -D,u y,z,a P
*
f(z, y, z , a , P ) - l ( z ,y, z , a, P ) } = 0,
u(0,x) = min{h(z, y,z ) } . YJ
Using again the theory of Evans and Souganidis,20one can see that u is the upper value of the following effective deterministic differential game. The effective dynamics are
5 s = f(zs,Ys,zs,~s,Ps),
z o =z,
where y and z are new controls, while the effective cost is
P(t, z, Yl z ,
a 1
P ) :=
I'
Z(zsl
Ys, zs, a s , P s ) ds
+ min q z t , Y,2). Y,Z
In the effective game the first player wants to minimize P by choosing the controls y, z , a, and the second player wants to maximize P by choosing P.
23
We end this Subsection by giving a simple condition on the control system that implies (15)-(16). Suppose the controls of the first player are in the separated form a = ( a s ,a F ,av)E AS x AF x AV, where as is the control of the slow variables x, aF of the fast variables y, and av is the control of the very fast variables z . More precisely
f = f (X,Y,Z,(*IS,P)
1 = 1 (x,Y,z,as,P),
7
9 = 9 (x,Y,z , Q F , P)
1
?-
= ?-
(x,Y , z , av,P) .
In this case the conditions (11)-(12) become
Bm(O,v) c m { g ( x , Y,z , a F ,P ) I Q F E A F ) , B,(O, v ) c ~ { r ( x , yz ,,a',@) I av E A'} for every x,y,z,P. Then it is easy to check that (15)-(16) hold, and the effective Hamiltonian is
Note that in the further special case
AF = &(O, I), cpi(x,Y,z ) 2 v > 0, r = PZ(Z,Y, z ) a V , AV = Bp(0, I ) , cpz(x,y, 2) 2 v > 0, = cpi(x, Y,z ) a F ,
the PDE in (10) becomes the model problem (1) presented in the Introduction. 7 . 2 . Homogenization of a deterministic optimal control problem For each
E
> 0, we consider dynamics having the form
for 0 5 s 5 T . The admissible controls a. are measurable function with value in the compact A. We denote by d(t)the set of admissible controls on the interval (0, t ) . Our goal is to choose the control in order to minimize the payoff functional:
By standard theory12 the value function w B ( t , x) := inf orEd(t)
P'(t, x,y, a),
24
is the unique viscosity solution of the HJ eq. (20), provided that f, 1 and h fulfill the assumptions of Sec. 2. Moreover, let us recall that the Hamitonian F is coercive provided that
B,(O, v) c m { f ( z ,y, z , a ) I a E A }
for all (z, y, z ) .
Under these assumptions, Corollary 6.1 ensures that w" converges locally uniformly to the solution of problem Here h(z) = miny,zh ( z ,y, z ) , but B does not have an explicit representation. Note that the eikonal equation (2) is a special case of this example. It is enough to take
(m).
A = &(O,
f(z,Y,z , a ) = cp(z, Y,Z ) Q I ,
I),
cp(z, Y,z ) 2 v
> 0.
7.3. Multiscale singular perturbation under a nonresonance
condition This Subsection is devoted to the case of a nonresonance condition, introduced by Arisawa and Lionsg (see also the first two authors5),that ensures the ergodicity for a class of non-coercive Hamiltonian with an effective operator that fulfills the Comparison Principle and can be written explicitly. As a byproduct, we show that the roles of E and c2 can not be exchanged in general. To this aim, we first consider a three scales perturbation problem that is nonresonant in the microscopic variable z and coercive in the mesoscopic variable y, and then a problem that is coercive in z and nonresonant in y. The two effective Hamiltonians are different. lStcase: For each
E
> 0, consider the dynamics
for 0 5 s _< T with initial conditions zo = 2 ,
yo = y,
20
= z.
The admissible control a = ( a s ,a F )splits in two control: one for the slow and one for the fast variable (the superscript recall this fact). Furthermore, a: and af are measurable functions with value respectively in the compacts AS and A F .By the choice of a , one wants to minimize the payoff functional:
p E ( t , z , y , Z , a ):=
I"
~ ( % , Y s , Z s , ~ : ) d s +h(zt,yt).
25
Then the value function u" solves the Cauchy problem:
[
&ue
+
f(z, y, z , a') - l(z,y, z , as)}+ max { - ~ D ~ U . g(z, & y,a F ) }- +(z, y) . D,U€
max {--o,u' d € A S
*
=o
CXFEAF
u'(o,z,Y,z)
= h(z,y)
We require that the microscopic dynamic is nonresonant and that the mesoscopic one is coercive, namely: T ( Z , y)
*
k # 0 Vk E Zp\ {0},
B,(O,
V) C
conV{g(z, y,a F )I aF E A F }
for every (z,y). The Hamiltonian H is ergodic in the fast variable5i9 and the mesoscopic Hamiltonian H1 has the form:
One can easily check that H1 fulfills the Comparison Principle, is ergodic with respect to y and satisfies (4). Therefore Theorem 5.1 ensures that U & converge locally uniformly to the solution of problem where the effective quantities are given by:
(m)
-
H(z,p,) = max Y
/
max
{-Pz
&€AS
*
f(z,Y,z , a')
- l ( z ,Y,z , as>}dz
(071P
-
h ( z ) = min h ( z ,y). Y
2nd case: For each
E
> 0, consider the dynamics
for 0 5 s 5 T with initial conditions zo = z ,
yo = y ,
20
= 2.
The control a is a measurable function with value in the compact A. As before, by the choice of a, one wants to minimize the payoff
26
The value function uE is the viscosity solution of the following Cauchy problem:
{
+ max{-Dzue. aEA
&ue
f(z,y, z , a ) -
S D , ~~ (~z ,.yz,, a ) - l(z,y, z , a ) } -;DYuE. g(z) = 0
U E ( O , z,
Y,2 ) = h ( z ,2 )
Assume that the dynamics are microscopically coercive and mesoscopically nonresonant: g(z)
9
Ic # 0 VIc E Zm \ { 0 } ,
Bp(0, V ) C COIIV{T(Z, y, Z ,a ) 1
Q!
E A]
for every (z,y,z). We require also that condition (15) is satisfied. Hence, H is microscopically ergodic and stabilizing and the mesoscopic quantities are given by:
Hl(z,Y,Pz,P,)= ??X{-Pz.
f ( z , y , z , a ) - l ( z , y , z , a ) }- P ,
.dz)
hl (z, y) = min h(z,2 ) . 2
Arguing as before, one can prove that the H I is ergodic, stabilizes, and fulfills the Comparison Principle. Whence, by Theorem 5.1, the value function uE converge locally uniformly to the solution of with:
(m)
-
h ( z )= min h(z,2 ) . z
References 1. G.Allaire and M. Briane, Proc. Roy. SOC.Edinburgh sect. A 126,297 (1996). 2. 0. Alvarez, J. Differential Equations 159,543 (1999). 3. 0. Alvarez and M. Bardi, SIAM J. Control Optim. 40,1159 (2001). 4. 0. Alvarez and M. Bardi, Arch. Ration. Mech. Anal. 170,17 (2003). 5. 0. Alvarez and M. Bardi, Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations, tech. rep., Dipartimento di Matematica P.A., Universith di Padova (Padova, Italy, 2007), available at http://cpde.iac.rm.cnr.it/ricerca.php.(January 2007). 6. 0. Alvarez, M. Bardi and C. Marchi, J . Differential Equations 243, 349 (2007). 7. 0. Alvarez and I. Ishii, Comm. Partial Differential Equations 26,983 (2001). 8. M. Arisawa, Adw. Math. Sci. Appl. 11,465 (2001). 9. M. Arisawa and P.-L. Lions, Comm. Partial Differential Equations 23,2187 (1998). 10. Z. Artstein and V. Gaitsgory, Appl. Math. Optim. 41,425 (2000). 11. M. Avellaneda, Comm. Pure Appl. Math. 40,527 (1987).
27
12. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhauser, Boston, 1997). 13. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for periodic Structures (North-Holland, Amsterdam, 1978). 14. A. Bensoussan, Perturbation methods in optimal control (Wiley/GauthiersVillars, Chichester, 1988). 15. I. Birindelli and J. Wigniolle, Comm. Pure Appl. Anal. 2, 461 (2003). 16. A. Braides and A. Defranceschi, Homogenizations of multiple integrals (Clarendon Press, Oxford, 1998). 17. I. Capuzzo-Dolcetta and H. Ishii, Indiana Univ. Math. J. 50, 1113 (2001). 18. L. Evans, Proc. Roy. SOC.Edinburgh Sect. A 111,359 (1989). 19. L. Evans, Proc. Roy. SOC.Edinburgh Sect. A 120,245 (1992). 20. L. Evans and P. E. Souganidis, Indiana Univ. Math. J. 33,773 (1984). 21. V. Gaitsgory and M.-T. Nguyen, SIAM J. Control. Optim. 41,954 (2002). 22. K. Horie and H. Ishii, Indiana Univ. Math. J. 47,1011 (1998). 23. H. Ishii, Almost periodic homogenization of Hamilton- Jacobi equations, in International conference on differential equations. Proceedings of the conference, Equadiff '99 (B. Fiedler, ed.), vol. 1 , (World Scientific, Singapore, 2000). 24. P. V. KokotoviC, H. K. Khalil and J. O'Reilly, Singular perturbation methods in control: analysis and design (Academic Press, London, 1986). 25. J.-L. Lions, D. Lukkassen, L. E. Persson and P. Wall, Chinese Ann. Math. Ser. B 22, 1 (2001). 26. P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton Jacobi equations, Unpublished, (1986). 27. P.-L. Lions and P. E. Souganidis, Ann. Inst. H. Poincare' Anal. Non Line'aire 22, 667 (2005). 28. P.-L. Lions and P. E. Souganidis, Comm. Partial Differential Equations 30, 335 (2005). 29. D. Lukkassen, C. R . Acad. Sci. Paris Sdr. I Math. 332,999 (2001). 30. F. Rezakhanlou and J. E. Tarver, Arch. Rational Mech. Anal. 151, 277 (2000). 31. P. E. Souganidis, Asymptot. Anal. 20, 1 (1999).
28
PATCHY FEEDBACKS FOR STABILIZATION AND OPTIMAL CONTROL: GENERAL THEORY AND ROBUSTNESS PROPERTIES F. ANCONA Dipartimento di Matematica and C. I.R.A.M., Universita di Bologna, Piazza di Porta S. Donato 5, Bologna 40126, Italy E-mail:
[email protected] A. BRESSAN Department of Mathematics, Penn State University, University Park, PA 16802, U.S.A. E-mail:
[email protected] This paper provides a survey of the theory of patchy feedbacks, and its applications to asymptotic stabilization and optimal control. It also contains two new results, showing the robustness of suboptimal patchy feedbacks both in the case of (internal and external) deterministic disturbances, and of random perturbations modelled by stochastic Brownian motion. Keywords: Optimal Feedback Control; Discontinuous Feedback; Robustness; Stochastic Disturbances.
1. Introduction
Consider a nonlinear control system on
R",of the form
j: = f(x,u).
(1) Here the upper dot denotes a derivative w.r.t. time. We assume that the control u takes values in Rm and that that the map f : Rn x Rm H R" is smooth. In connection with (l),two classical problems have been extensively studied in the control literature.
(I) Asymptotic Feedback Stabilization: Construct a feedback control u = V ( x )such that all trajectories of the resulting O.D.E. j: =
approach the origin as t
g(x)
--+ 00.
f(Z, V ( x ) )
(2)
29
(11) Optimal Feedback Control: Given a compact set K c R" of admissible control values, construct a feedback control u = U ( z ) E K such that all solutions of (2) reach a given target set S c Rn in minimum time. In an ideal situation, these goals would be achieved by a C1 feedback control z H U ( z ) .In this case, for every initial state
x(0) = z ,
(3)
the solution of the Cauchy problem (2)-(3) has a unique classical solution, continuously depending on the initial data. However, smooth feedback controls exist only in a limited number of cases.
(I) For the problem of asymptotic stabilization, several examples of control systems are known, where every initial state can be steered toward the origin as t -+ 0;) by an open-loop control t H u(t).However, due to the presence of certain topological obstructions, no smooth (or even continuous) feedback control function z H U ( x )can accomplish the same t a ~ k not even locally in a small neighborhood of the origin. In all these examples asymptotic stabilization can be achieved only by a discontinuous feedback. (11) For nonlinear optimization problems, even in few space dimensions, it is known the optimal feedback can be discontinuous, with very complicated structure, while optimal controls can have infinitely many switchings. Moreover, a discontinuous optimal feedback may not be robust w.r.t. perturbations. In other words, one may find arbitrarily small functions el, e2 such that the solution to the perturbed equation j. = f(z,U ( z
+ el(t))) + e 2 ( t )
(4)
achieves a much worse performance than the optimal one. The use of discontinuous feedback controls z H U ( z ) leads to the theoretical problem of how to interpret solutions for the O.D.E. (2), whose right hand side is discontinuous w.r.t. the state variable z. Different approaches can be followed: 0
Consider only solutions in a strong sense. We recall that a Curathe'odory H x ( t ) which
solution of (2)-(3) is an absolutely continuous map t satisfies (2) at a.e. time t. Equivalently,
,
~
30
In general, we expect that these strong solutions will have the required asymptotic convergence properties. However, if one does not impose any regularity assumption of the feedback U ( z ) ,there is no guarantee that any Carathkodory solution will exist. 0
Define weaker concepts of solutions. For example, one can consider trajectories of the differential inclusion (see Refs. 6,18,19)
It is not difficult to show that solutions in this relaxed sense always exist. However, too many solutions are now allowed. Not all of them may have the desired properties. Of course, there is a wide variety of technique^^^^^^^^^ for constructing “generalized solutions” to the Cauchy problem (2)-(3). One of the paths followed in recent literature is t o consider approximate solutions (or limits of approximate solutions) obtained by robust approximation methods, such as the “sample and hold” technique introduced in Ref. 13, originated from the theory of positional differential games.22 In the present paper we discuss an alternative approach, introduced by the authors in Ref. 1. It involves a particular class of feedback controls U (.), called patchy feedbacks. These are piecewise constant controls, whose discontinuities are sufficiently tame in order to guarantee the existence of Carathkodory solutions forward in time. At the same time, this class is sufficiently broad to solve a wide class of stabilization and optimization problems. Section 2 contains the basic definitions of patchy vector fields and patchy feedbacks, together with a brief discussion of their basic properties. In Section 3 we show how to solve the asymptotic stabilization problem by means of a patchy feedback. Section 4 describes the construction of a patchy feedback which provides a close-to-optimal solution to a minimum time problem. In Section 5 we discuss the robustness of patchy feedbacks, showing that they still perform well also in the presence of small external perturbations, or small measurement errors in the state of the system. Finally, in Section 6 we analyze the performance of a patchy feedback in the presence of random perturbations, modelled by stochastic white noise. For the problem of reaching the origin in minimum time, we provide an estimate of the probability of entering the &-ballBE around the origin within a given time T .
31
2. Patchy vector fields and patchy feedbacks
For a general discontinuous vector field g, the Cauchy problem for the
O.D.E.
x
(7)
= g(x)
may not have any Carathkodory solution. Or, on the contrary, it may have infinitely many solutions, with wild behavior. We shall now describe a particular class of discontinuous vector fields g, whose corresponding trajectories are well defined and have nice properties. This is of particular interest, because a wide variety of stabilization and optimal control problems can be solved by discontinuous feedback controls (2) , within this class. Throughout the paper, the closure and the boundary of a set R are denoted by and aR, respectively.
Definition 2.1. By a patch we mean a pair (R, 9 ) where R c R" is an open domain with smooth boundary and g is a smooth vector field defined on a neighborhood of which points strictly inward at each boundary point x E dR. Calling n(x) the outer normal at the boundary point x,we thus require
n,
( g ( x ) , n(x)) < 0 Clearly, in the case where R vector field g.
= W",no
Vx E 8 0 .
(8)
condition is required on the smooth
Definition 2.2. We say that g : R H Rn is a patchy vector field on the open domain R if there exists a family of patches { (a,, ga); a E A } such that (see Fig. 1) - A is a totally ordered set of indices, - the open sets R, form a locally finite covering of R, - the vector field g can be written in the form g ( x ) = &(.)
if
z E Ra
\ (JRP. P>a
By defining
a*(x)A max { a E A ; x E a,}, the identity (9) can be written in the equivalent form
(9)
32
We shall occasionally adopt the longer notation (R, 9, (a,, ga),GA) to indicate a patchy vector field, specifying both the domain and the single patches.
Remark 2.1. In general, the patches (R,, g,) are not uniquely determined by the patchy vector field g. Indeed, whenever a < p, by (9) the values of g, on the set R, n Rp are irrelevant. In the construction of patchy vector fields, the following observation is often useful. Assume that the open sets R, form a locally finite covering of R and that, for each a E A, the vector field g, satisfies (8) at every point x E dR, \ Up,, 02p. Then g is still a patchy vector field. To see this, it suffices to construct vector fields & which satisfy the inward pointing property (8) at every point x E dR, and such that 3, = g, on the closed set \ Rp .
n, up,,
a
Fig. 1. A patchy vector field.
Remark 2.2. In some situations it is useful to adopt a more general definition of patchy vector field than the one formulated above. Indeed, one can take in consideration patches (a,, g,) where the domain R, has a piecewise smooth boundary (cf. Ref. 3). In this case, the inward-pointing condition (8) can be expressed requiring that
33 0
where T n ( x ) denotes the interior of the (Bouligand) tangent cone to R at the point x , defined by (cfr. Ref. 14)
VEEP: liminf
d ( x + t v , 0)
t
tl0
=O}.
Clearly, at any regular point x E dR, the interior of the tangent cone T a ( x )is precisely the set of all vectors v E Iw" that satisfy (v, n ( x ) ) < 0 and hence (12) coincides with the inward-pointing condition (8). One can easily see that all the results concerning patchy vector fields established in Refs. 1,5 remain true within this more general formulation. The main properties of solutions to the O.D.E. (7) when g is a patchy vector field are collected below.
Theorem 2.1 (Trajectories of a patchy vector field). I n connection with a patchy vector field (0, g , (a,, the following holds.
(i) If t
x ( t ) is a Carathe'odory solution of ( 7 ) on an open interval J , then is piecewise smooth and has a finite set of jumps o n any compact subinterval J' c J . The function t H a * ( x ( t ) )defined at (10) is piecewise constant, left continuous and non-decreasing. Moreover there holds
t
-+
+ x(t)
4t-> = g(x(t))
Q t E J.
(14)
(ii) For each Z E R, the Cauchy problem for (7) with initial condition x ( 0 ) = 3 has at least one local forward Carathe'odory solution and at most one backward Carathe'odory solution. (iii) The set of Carathe'odory solutions of (7) is closed. More precisely, assume that x , : [a,, b,] +-+ R is a sequence of solutions and, as v + 00, there holds
a,
+ a,
b,
--f
b,
x,(t)
-+
2(t) V t € ] a ,b [ .
(15)
Then it(.)is itself a Carathe'odory solution of (2.1). Sketch of the proof. We outline here the main arguments in the proof. For details see Ref. 1. 1. To prove (i), observe that on any compact interval [a,b] a solution x ( . ) can intersect only finitely many domains R,, say those with indices a1 < a2 < ... < a,. It is now convenient to argue by backward induction. Since R,, is positively invariant for the flow of g,, , the set of times { t E
34
[a,b] ; z ( t )E Ram} must be a (possibly empty) interval of the form It,, b]. is an interval of the form Similarly, the set { t E [ a , b ] ; z ( t ) E ]tm-l,t,]. After m inductive steps we conclude that
t Eltj, tj+l[ for some times t j with a = tl 5 t 2 5 . . . 5 tm+l = b. All statements in (i) now follow from this fact. In particular, (14) holds because each set 0, k ( t ) = gaj (.(t)>
is open and positively invariant for the flow of the corresponding vector field ga. 2. To prove the local existence of a forward Carathkodory solution, consider the index 6 = max{a E
A; 3E
na}.
Because of the transversality condition (8), the solution of the Cauchy problem z(0) = 3 ,
j: = g d x ) ,
(16)
remains inside R B for all t 2 0. Hence it provides also a solution of (7) on some positive interval [O,6].
3. To show the backward uniqueness property, let XI(.), z2(-) be any two Carathkodory solutions to (2.1) with q ( 0 ) = ~ ( 0 =) 3. For i = 1,2, call
a:(t) = max{a
EA ;
zi(t) E R,}.
By (i), the maps t H a f ( t ) are piecewise constant and left continuous. Hence there exists 6 > 0 such that
a;(t)= aa(t)=
= max
{ a E A ; 3 E R,}
W E ] - 6, 01.
The uniqueness of backward solutions is now clear, because on ] - 6, 01 both z1 and x2 are solutions of the same Cauchy problem with smooth coefficients
x = g&), 4. Finally, let z, : [a,, b,]
H
z(0) = 3 .
R" be a sequence of solutions satisfying (2.6).
To prove that the limit Z(.) is also a Carathkodory solution, we observe that on any compact subinterval J c ] a , b [the functions u, are uniformly continuous and intersect a finite number of domains a,, say with indices a1 < a2 < . . . < a,. For each v , the function
a:(t) A max{a E A ; z,(t) E 52,)
35
is non-decreasing and left continuous, hence it can be written in the form aY:(t) = aj
if t ~ ] t ; , t;+,].
By taking a subsequence we can assume that, as u -+ co,Y t -+ & for all j. By a standard convergence result for smooth O.D.E's, the function i provides a solution to li: = g,, (z) on each open subinterval Ij A ] i j , ij+l[. Since the domains S l p are open, there holds ?(t) $ 4 3
'dD > a j 7 t E I j .
On the other hand, since gaj is inward pointing on the boundary a i l a j ,a limit of trajectories kV = g,,(zv) taking values within R,, must remain in the interior of R,, . Hence a*( 2 ( t ) )= aj for all t E I j , completing the proof of (iii).
Remark 2.3. For a patchy vector field g , in general the set of Carathkodory solutions to a given Cauchy problem is not connected, not acyclic. In spite of appearances, this fact is a distinct advantage. Indeed, it avoids the topological obstructions toward the existence of stabilizing feedbacks which are otherwise encountered working with continuous O.D.E's, or differential inclusions such as ( 6 ) , with upper semi-continuous, convex valued right hand side.
Example 2.1. Consider the covering of R = R2 consisting of three patches:
Moreover, consider the family of inward-pointing vector fields
Then the vector field g on R2 defined as
36
t t
l " 1
:Y/.-- *I.---!-------
~
'.
Fig. 2.
Trajectories of a patchy vector field.
is the patchy vector field associated with (0, , ga),=1,2,3 (see Fig. 2). Notice that in this case the O.D.E. (7) has exactly three forward Carathkodory solutions starting from the origin a ttime t = 0, namely
& ) ( t ) = (t,O),
X q t ) = (-t,O),
d 3 ) ( t )= (0,t)
t 2 0.
The only backward Carathkodory solution is
d l )( t )= (0, t )
t 5 0.
Moreover, for every initial point of the form jj = (5, A) with 5 > 0, there exist two forward Carathkodory solutions, but no solution backward in time. On the other hand, starting from the origin, the relaxed differential inclusion (6) has infinitely many forward and backward solutions. Indeed, since
W , O ) = m{ (0,1), ( L O ) , (-1, O ) } for every r
7
< 0 < r', the function 4 t )=
i
(0, t - ).
if
t
(0,O)
if
t
if
t > r' ,
(t - r', 0)
E
[r,r'],
provides a solution to (6).
Because of the nice properties described in Theorem 2.1, in connection with several control problems one may seek a feedback u = U ( x ) such that
37
the resulting map g ( z ) = f (z, V(z)) is a patchy vector field. This leads to the following:
Definition 2.3. Let (0, g , (R,, ga),,A) be a patchy vector field. Assume that there exists control values k , E K such that, for each (Y E A
gcr(z)
f(z, k a )
\
vx E 52,
u
Rp.
(17)
B>a
Then the piecewise constant map
V(z)=k,
if
u
ZER,\
Rp
(18)
D o :
is called a patchy feedback control on R. Defining a*(z)as in (lo), the feedback control can be written in the equivalent form
wc)ka*(z) =
XEUR,.
(19)
a
3. Stabilizing feedback controls
In this section we describe the use of patchy feedbacks to solve a problem of asymptotic stabilization. Throughout the following we assume that the control set K c Rm is compact. We first recall a basic d e f i n i t i ~ n . l ? l ~ > ~ ~
Definition 3.1. The control system
i = f(z,u)
u(t)E K
(20)
is said to be globally asymptotically controllable to the origin if the following holds. 1 - Attractivity. For each y E R" there exists an open-loop control t uY(t)such that the corresponding solution of
j.(t) = 2
-
f(W, uY(t)),
40) =Y
H
(21)
either tends to the origin as t + 03 or reaches the origin in finite time. Lyapunov stability. For each E > 0 there exists 6 > 0 such that the following holds. For every y E Rn with IyI < S there is an admissible control UY as in 1, steering the system from y to the origin, such that the corresponding trajectory of (21) satisfies Iz(t)l < E for all t 2 0.
38
Fig. 3.
Lyapunov stability.
A major result established in Ref. 13 is that if a control system is globally asymptotically controllable then there exists a feedback u = U ( z ) for which all trajectories converge asymptotically to the origin. Since this stabilizing feedback may well be discontinuous, rather than strong solutions in Ref. 13 the authors consider approximate solutions obtained by a “sample and hold” technique. As proved in Ref. 1, the asymptotic stabilization can also be achieved by a patchy feedback. In this case, one can work with solutions defined in the classical Carathkodory sense. Theorem 3.1 (Stabilization by a patchy feedback). Assume that the control system (20) is globally asymptotically controllable t o the origin. Then there exists a patchy feedback control U : Rn \ ( 0 ) H K such that every Carathe‘odory solution of ( 2 ) either tends asymptotically to the origin, or reaches the origin in finite time. Sketch of the proof. The main part of the proof of Theorem 3.1 consists in establishing the semi-global practical stabilization of system (20), i.e. in showing that, given two closed balls B‘ c B centered at the origin, there exists a patchy feedback that steers every point 3 E B inside B‘ within finite time. The basic idea of the construction is rather simple (see Fig. 4-5). Consider an open-loop control t H uY(t)steering a point y E B to the interior of B’. By continuity, we can construct a “flow tube” ?I around the trajectory z(., u y ) , steering all points of a neighborhood of y into B’. Finally, we can patch together a finite number of these tubes ?I covering the entire ball B . The basic steps of this construction are sketched below. Further details can be found in Ref. 1.
39
.-........
: -
Fig. 4. Constructing a Row tube around a given trajectory.
1. By assumption, for each point y E B , there exists an open-loop control t H uY(t) that steers the system from y to a point y' in the interior of B' in finite time r Y (Fig. 4-a). By a density and continuity argument, we can replace uy with a piecewise constant open-loop control i 2 (Fig. 4-b), say
P ( t )= k, E K
if t ~ j t , - l , t,],
(22)
for some finite partition 0 = t o < tl < . . . < t , = r Y . Notice that it is not restrictive to assume that the corresponding trajectory t I--+ 3 ( t ) A x ( t ,3 ) has no self-intersections.
.
We can now define a piecewise constant feedback control u = U ( x ) , taking the constant values k,, , . . . ,k,, on a narrow tube I' around the trajectory (23)
40
so that all trajectories starting inside I' eventually reach the interior of B' (Fig. 4-c). 3. By slightly bending the outer surface of each section of the tube I', we ka) point strictly inward can arrange so that the vector fields gcy(z)2 f(z, along the portion of boundary aI', \ ra+l.Recalling Remark 2.1, we thus obtain a patchy vector field (Fig. 4-d) defined on a small neighborhood of the trajectory ( 2 3 ) , which steers all points of a neighborhood of y into the interior of B'.
Fig. 5 . Patching together several flow tubes.
4. The above construction can be repeated for every initial point y in the compact set B. We now select finitely many points y l , . . . , YN E B and gZ,cy),EA,)defined on the corresponding patchy vector fields, (Oz, gz, tube-like sets F Y S , with the following properties. The union of all domains eventually I'Yr covers B , and every trajectory of every vector field gz on reaches the interior of B'. 5 . Finally, we define the patchy feedback obtained by the superposition of the gz,a, arranged in lexicographic order:
41
This achieves a patchy feedback control (Fig. 5), defined on a neighborhood of B \ B‘, which steers each point of B into the interior of B’ in finite time. 6 . For every integer v , call B” the closed ball centered at the origin with radius 2-”. By the previous steps, for every v there exists a patchy feedback control U, steering each point in B, inside B,+l, say UV(.)
= k/,a
if
2 E %,a
u
\
Q,p.
(25)
The property of Lyapunov stability guarantees that the family of all open sets {OV+; v E Z, (Y = I , . . . ,N , } forms a locally finite covering of Rn \ (0). We now define the patchy feedback control
uv(z)= kv,a
if
z E %,a
\
u
flp,p,
(26)
(P,P)*(W)
where the set of indices (v,a)is again ordered lexicographically. By construction, the patchy feedback (26) steers each point z € B” into the interior of the smaller ball BV+l within finite time. Hence, every trajectory either tends to the origin as t + 00 or reaches the origin in finite time. 0
4. Nearly optimal patchy feedbacks
Consider again the nonlinear control system
k = f(z, u) where
4 t )E K ,
(27)
f is a smooth map with sublinear growth If(z,u)II c ( 1 + 1x1)
for all u E K ,
(28)
and the set of admissible control values K c Rm is bounded. For sake of definiteness, in this section we discuss a specific optimization problem, namely, how to steer the system to the origin in minimum time, where it is not restrictive to assume that K is compact and that the sets of velocities {f(z, u); u E K } is convex. Indeed, allowing the set of controls to range in the closure of K does not affect the minimum time function. Moreover, if the sets of velocites are not convex, we can replace the original system (27) by a chattering one (see Ref. 7), so that the resulting time optimization problem yields exactly the same value function. In general, a feedback control u = U ( z ) that accomplishes this task can have a very complicated structure. Already for systems in two space
42
dimensions, an accurate description of all generic singularities of a time optimal synthesis involves the classification of eighteen topological equivalence classes of singular point^.^^^^ In higher dimensions, an even larger number of different singularities arises, and the optimal synthesis can exhibit pathological behavior such as the the famous "Fuller phenomenon", where every optimal control has an infinite number of switchings. In alternative, one may construct a more regular feedback, which still achieves a close-to-optimal performance. As shown in Refs. 5,10, this task can also be accomplished by patchy feedbacks. For each initial state y E Rn, call T ( y ) the minimum time needed to steer the system from y to the origin. As usual, BE denotes the closed ball 0
centered at the origin with radius E , while B E denotes its interior. The following result was established in Ref. 5:
Theorem 4.1 (Nearly time optimal patchy feedback). Under the above assumptions, let E > 0 , r > 0 be given. T h e n there exists a suboptimal patchy feedback U ( x ) = / c , * ( ~ ) , as in (19), defined on the sub-level set
such that the following holds. For every y E R,, IyI > E , a n y solution of 4 0 )=Y 1 reaches a point in the ball BE within t i m e T ( y ) + E . Remark 4.1. For each y E R,, one can consider a time-optimal trajectory steering y to the origin. As in the proof of Theorem 3.1, we could then cover the compact set R, \ BEwith finitely many tubes rl, . . . ,r N and, for each a = 1 , .. . N , construct a patchy feedback U, : r, H K steering every point y E r, into the ball BE within time T ( y ) E . Unfortunately, when we patch together all these feedbacks U, (say, using a lexicographic order), there is no guarantee that the resulting feedback
+
4.)
= Ua*(z)
Q * ( x )A max{a; x E r,}
is nearly-optimal (Fig. 6). Indeed, call T,(y) the time taken by the control U, to steer the point y E 0, inside BE.Let t H x ( t ) be a trajectory of the composite feedback
43
that reaches the ball BEwithin time T . Assume a*(t)= a for t E ]t,-1 , t,] . The near-optimality of each feedback U, implies T ( z )5 T,(z) 5 T ( z ) E for every z E Fa. Moreover
+
T,(z(ta-1)) - T,(z(tw))= ( t a - t,-1)
'
Unfortunately, from the above inequalities one can only deduce
T(z(t,-1)) - T(z(t,)) 2 (t, - ta-1) - E * and hence T 5 T(y) + N E. This is a useless information, because we have no control on the number N of tubes. Indeed, we expect N -+ 00 as E 0. -+
Fig. 6 . Patching together optimal trajectories does not yield a near-optimal feedback.
Sketch of the proof. As a starting point, instead of open-loop optimal controls, it is convenient to use here the value function z H T(y). We describe here the main steps in the proof of Theorem 4.1. For all details we refer to Ref. 5. 1. We begin by relaxing the optimization problem. For E > 0, define the function
44
and consider the following &-approximate minimization problem
subject to 40) =Y1
li: = f ( z , u )
u ( t )E K .
(33)
For every initial point y E R", we denote this infimum by V(y),and refer to y H V(y) E [0,m] as the value function for (32). Observe that
v y E R".
V(Y) I T(Y)
(34)
Thus, for a fixed time r > 0, the set of points that can be steered to the origin within time T is contained in the sub-level set
A,
{y E R" ; V(y) 5 r } .
Hence, to establish Theorem 4.1 it will be sufficient to construct a suboptimal patchy feedback on the set
AT,&2 {YE R";
V(Y) 5 7 ; IYI 2 E } .
(35)
The assumptions on the dynamics (27) imply that the sub-level set A, (and hence is compact. Moreover, by the regularity of the functions f and pE,it follows that the restriction of the value function V t o A, is Lipschitz continuous and locally semi-concave (see Ref. 5, Lemma 1). 2. It is convenient to further approximate the value function V ( . ) with a semi-concave, piecewise quadratic function of the form
(36) with the following properties: (i) For every z E A,,,, there holds
T ( z )I ?(z)
I V(z) + EO ,
for some positive constant EO < E . (ii) For every x E AT,&where ? is differentiable, one has
By the special form of the function
?, it follows that
(37)
45
The set of points where ? is not differentiable is contained in a finite union of hyperplanes, namely
Hij-{ai+bi.z=aj+bj.z 0
1.
Each level set Ch A {z ; ?(z) = h } is contained in the union of finitely many spherical surfaces, say Si, ,. . . Si, .
Moreover, observe that by (37) one has
{
AT0+ C z ; ?(z)
I TO; 1x1 2 c},
TO
A
T
+
€0
.
(39)
-
The construction of the patchy feedback is now performed by induction, progressively working on larger sub-level sets of T(.).Namely, assume to have already constructed a patchy feedback U = U ( x ) on the sub-level set
- { z;
Ah-
?(z)Ih},
h
(40)
so that
VF(2). f(z,U ( z ) ) I -1
+
Eo
VX
E xh
\ BE.
(41)
We wish to define a patchy feedback with the same property around the level set c h . 3. If the surface {z ; ?(z) = h } is smooth, i.e. if ?(z) = aj
for some j , the level set
Fig. 7.
+ bj 2 + M .
Ch
*
V’zECh,
is exactly a sphere Sj. Then, because of (38),
Construction of the suboptimal patchy feedback around a sphere.
for each point y E Sj we can find a lens-shaped patch
RY
containing y in
46
its interior, and a constant control wY E K such that
V?(?/) f ( y , w”) a
+
I -1
Eo
.
By continuity, possibly choosing a smaller domain RY, we Ljduce that (see Fig. 7, left)
V?(Z) . f ( 2 , W Y )
5 -1
+3 2EO
VZ E RY.
(43)
Moreover, we construct the outer surface of the patch RY so that the vector field 3: H f(x,wy) points strictly inward on the upper boundary dRy \ Sj. By compactness of the sphere S j , we can cover it with finitely many patches fly1 ,. . . ,RYN constructed as above. On the union of these patches we define the feedback if z E Rya \
V(Z) = wya
u
RY@, ?(z) > h ,
while we assign an higher index order to the patchy already constructed on the set &. Recalling Remark 2.1, this provides a patchy feedback defined on a larger sub-level set &+6, 6 > 0, enjoying the property (41). 4. If the boundary
ch is not smooth, the above construction requires more
care. Indeed, consider a point y where ? is not differentiable. For sake of illustration, assume that y lies on the intersection of exactly two spheres, say I
T(y) = ai
+ bi . y + M . IyI2
= a2
+ b2 y + M
*
IyI2,
while aj
+ bj - y + M . IyI2 > ?(y)
j 2 3 .
Consider the half-spaces
ri = {z; r2
+bi
A {z; a2 +b2
.Z< a2 +b2 * Z< a1
.z},
+ b i .z},
and the dividing hyperplane
H
A
{xi
(a1
-Q)
+ (bl - b2) . z = 0 } .
For i = 1 , 2 we can again choose a lens-shaped patch Ri and a constant control wi E K such that, as in (43), there holds
47
Y Fig. 8. Construction of the feedback near a corner point. Two different cases.
Notice that by construction one has (bi + 2M Z) = V?(Z) in Ri n ri . In order to combine these two patches and construct a feedback in a whole neighborhood of the point y, we need to consider two different cases.
CASE1: The vector f ( y , w1) points toward r2.
rl, while f ( y , w2) points toward
More precisely,
In this case, we can choose the hyperplane H as boundary between the two patches. Setting
U ( x )=
~a~nr,,
w1
if
Z
w2
if
z ER
~ ~ T , ,
because of (44), and by Remark 2.1, we thus obtain a patchy feedback, defined on a small neighborhood of the point y (see Fig. 8, left), that satisfies the estimate (41).
CASE2: One of the two vectors points toward the opposite half-space. To fix the ideas, assume
By possibly choosing a smaller lens-shaped neighborhood R1 so that,
48
and relying on (44), one achieves
VF(2) f(z,~ 1
1=) (bi
+ 2M z) . f(z,
~
<-1+-
VT(z) . f(z,W I )
=
(bl
2EO
3
i
vz E fil
n rl , (45)
+ 2M z) . f(z,W ) +
I -1+eo
)
( b 2 - bi)
. f(z,~
1 )
vz~52~nT~.
In other words, the same control w1 used on 521 nl?l now works well also in the other region r2. In this case, we can cover a neighborhood of the point y with the single patch 521 (see Fig. 8, right) in which the estimate (41) is verified.
Fig. 9.
Construction of the suboptimal patchy feedback around a surface with corners.
The remainder of the construction is performed as before. We can cover a neighborhhood of each point y E by one or more patches. By compactness, the entire level set s h can be covered with finitely many patches (Fig. 9). This yields a patchy feedback defined on a larger sub-level set &+6 satisfying the decrease property (41). 5. A key step in the analysis is to show that, at each inductive step, we can construct a patchy feedback, enjoying the property (41), on a sub-level
set x h + & with some 6 > 0 uniformly positive. This ensures that the inductive procedure terminates after a finite number of steps. One can accomplish this task by providing an accurate estimate of the size of lens-shaped domains constructed around a finite collection of spheres (associated to a family of inward-pointing vector fields) and cutted along the hyperplanes passing through their intersections (cf. Ref. 5, Lemmas 3-4).
49
If now t H x ( t ) is any trajectory of the closed-loop system determined by our feedback control passing through points of thanks to (41) we deduce that
xTo,
as long as Ix(t)l > E . Therefore, if x(.) starts at a point y and reaches the ball B E ,say at a point z E ,in a time t,, one has
F(Y)- q x E ) L (1 - E o ) t,.
(47)
Hence, (47) together with (37), (39), implies that every initial point y E AT0+ is steered into the ball BEwithin time
Since one can choose EO < E so that the right hand-side of (48) is less than V(y) E for all y E and because of (34), this proves the theorem.
+
xTo,
5. Robustness
For practical applications, one has to take into account the presence of several perturbations, which may degrade the performance of the feedback control. For example: (i) The model equation, described by the function f in (l),may not be precisely known. (ii) The evolution of the system may be affected by (possibly random) external perturbations. (iii) While implementing the feedback, the state of the system may not be accurately measured.
As a result, instead of the planned dynamics j. = 9 ( z ) A
f(.,
W ) ,)
(49)
the system will actually evolve according to j: = f ( x , U ( x
+El@))) +
EZ(t),
(50)
for some small perturbations E ~ , E Z . Here the “inner perturbation” ~1 accounts for measurement errors, while the “outer perturbation” E Z models external disturbances.
50
In the above framework, it is important to design a feedback control which still accomplishes the desired task in the presence of (sufficiently small) perturbations. Since we are dealing with a discontinuous O.D.E., one cannot expect the full robustness of the feedback U ( x ) with respect to errors in the measurement of the state vectors because of possible chattering behaviour that may arise at discontinuity point^.'^^^'^^^ For trajectories constructed by the “sample and hold” technique in Ref. 13, a “relative robustness” with respect to measurement errors was shown in Refs. 12,27, meaning that the magnitude of the error must not exceed the maximum step size of the underlying discretization of the sampling solution taken in consideration (in this case, one is not concerned with the robustness property of the limit solution as the step size vanishes, but only of the approximate sampling solution with a fixed step size). Instead, for patchy feedbacks, we have established in Ref. 2 a robustness property of the Carathkodory trajectories with respect to measurement errors which have sufficiently small total variation (to avoid possible chattering phenomena around discontinuities) , while the external disturbances were assumed to be small in L1norm. We describe here the main results in this d i r e ~ t i o n . ~Together -~ with the O.D.E. determined by a patchy feedback f =
s(z) =
f(z,kY*(z)),
(51)
we thus consider the perturbed equation
We recall that the total variation norm of a BV function defined as
4 : [0,T ]H Rn is
The Cauchy problem for (52), with initial point ?j in the interior of the domain of g, admits a forward Caratheodory solution y ( t ) = g+ [g(y(s) e , ( s ) ) e z ( s ) ] d s ,t 2 0 , whenever Ilelllsv, llezll~lare sufficiently small.2 Notice that, since the Cauchy problem for (51) does not have forward uniqueness and continuous dependence, one clearly cannot expect that a single solution of (51) be stable under small perturbations. What we extablished in Ref. 2 is a different stability property, involving not a single trajectory but the whole solution set: if the perturbations e l , e2 are small in the BV and L’ norm, respectively, then every solution of (52) is close to some solution of (51). This is essentially an upper semicontinuity property
+
s,”
+
51
of the solution set. Namely, the main robustness result for the flow of a patchy vector field can be stated as follow^.^-^
Theorem 5.1 (Flow stability of a patchy vector field). I n connection with a patchy vector field g : R H Rn,given any compact subset CO c R , and any T , E > 0 , there exists 6 > 0 such that the following holds. If y : (O,T]H R is a solution of the perturbed system (52)) with Y ( 0 ) E CO, and
Ile1 IIBV 5 6 lle21IL1 5 6 , (53) then there exists a solution x : [O,T]++ R of the unperturbed equation (51) with 7
Ix(~)
- y ( t ) ]< E
V t E [O,T].
(54)
Proof. An outline of the main arguments of the proof can be found in Ref. 4.For details we refer to Ref. 2. The meaning of the theorem is illustrated in Fig. 10. Remark 5.1. As shown in Fig. 10, the solution z ( . ) to the initial value problem
z(0)= z = y ( 0 ), (55) may be unique, but very different from a solution y ( . ) of the perturbed system (52) with the same initial data. In order to find a trajectory x(.) of the original system (51) which remains always close to y(.), one may need to start from a different initial point. = 9 ( z >>
Remark 5.2. If we consider a solution y ( . ) of the perturbed system (52) on a time interval [O,Ty[,with Tg 5 T , y ( 0 ) E CO,and e1,e2 satisfying (53), by the same arguments of Theorem 5.1 one can find a solution x(.) of the unperturbed equation (51) on a interval [O,T,[ so that (54) holds on [O,min{Tz,Ty}[. If we assume that the boundary of the domain R is (piecewise) smooth, we can choose x(.) so that either T, = Ty, or IT, - Tyl< E and lim z ( t ) E a R . t-+Tz-
Relying on Theorem 5.1, one can show that a stabilizing patchy feedback as the one provided by Theorem 3.1 still performs well in the presence of small (internal and esternal) disturbances. Namely, we obtain the following robustness property formulated within the context of a semi-global practical stabilization p r ~ b l e m . ~ ~ ~
52
Fig. 10. The flow stability property stated in Theorem 5.1.
Theorem 5.2 (Robustness of a stabilizing patchy feedback). Let u = U ( x ) be a patchy feedback for the control system (l), defined on an open set R C R" containing the anular region {x E R"; r 5 1x1 5 r'}, and assume that every solution of (49) starting inside a compact set CO c R reaches the closed ball B, within time T. Then, given E > 0, there exists 6 > 0 such that, if the perturbations € 1 , ~2 : [O,T]H Rn satisfy
5 6,
56
(56) then any solution x"(.) of (50) with initial data x E ( 0 )E CO enters the ball BT+€within time T . l l 4 3 "
IIE2IIL'
Proof. To analyze the performance of a patchy feedback U ( x ) in the presence of (internal and external) disturbances, for every given solution t H x E ( t ) ,t E [O,T],of the perturbed system (50), writing
+
f(x, U ( x ~ l ( t ) + ) ) d t ) = g(x
+ e l ( t ) ) + e2(t) ,
(57)
where
e1(t) A E l ( t ) 7 e2(t) = 4 t )
+ [ f ( x " ( t )u(xE(t) , +El(t))) - f (z"(t)+ E l ( %
U(x'(t) + E d t ) ) ) ]
(58) 7
we find that x"(.) results to be a solution of the perturbed patchy vector field (52), with e1,e2 as in (58). Observe that, by the regularity of f
53
and since all trajectories of (50) starting from Co and defined on [0,TI are uniformly bounded, one can choose 6 sufficiently small so that, if E I , E ~ satisfy (56) then e1,e2 satisfy the bounds (53) stated in Theorem 5.1. Hence, the conclusion of Theorem 5.1 guarantees the existence of a solution t H x ( t ) ,t E [O,T],of the unperturbed system (49) such that
Iz"(t)- 4 t ) l I: E Since, by assumption, x ( ~ E) B, for some claimed.
vtE T
[O,T].
(59)
E [O,T],we conclude that
~ " ( 7E)B,+&,as
0
With the same arguments used in the proof of Theorem 5.2, one can apply Theorem 5.1 to derive the robustness of a sub-optimal patchy feedback. In fact, consider the patchy feedback U ( x ) constructed in Ref. 10 for the control system (27), whose trajectories are all nearly optimal solutions for the general optimization problem
The following result shows that all the trajectories of the perturbed system (50) remain nearly optimal with the same order of accuracy as the ones of the unperturbed system (49), provided that the disturbances are sufficiently small. As in Section 4 we assume here that f is smooth and satisfy the sublinear growth condition (28), and that the set of admissible control values K is compact, while the cost functional $J,L are smooth, and the running cost is strictly positive L ( x , u ) 2 2 > 0. For every initial point y E Rn, we let V(y) denote the value function for the optimization problem (60), where the minimization is taken over all T 2 0 and over all solutions x(.) of (27) starting from y and corresponding to a measurable control u taking values in K . A solution t H x ( t ) defined on a maximal lim x ( t ) (E 8 0 ) . interval [0,T [ , is extended by continuity setting X ( T ) t+r-
Theorem 5.3 (Robustness of a sub-optimal patchy feedback). Under the above assumptions, let E > 0 and a compact set CO be given. Then, there exist a patchy feedback U ( x ) defined on the domain
R
A
{x E Rn;V ( x ) < $J(z)- E } ,
(61)
and a constant 6 > 0 such that, if the perturbations €1, c2 : [0,a[ H Rn satisfy (56), then any solution x"(.) of (50), with initial data ~ " ( 0 E) Co,
54
enters the terminal set S = Rn \ R within a finite time T & ,for which there holds $(z'(~&))+ / T c
t
L ( z " ( s )U , (z'(s))) ds I v ( z E ( t )+)E
V t E [O,T&]. (62)
Proof. We start by observing as in Ref. 10 that we can replace f (2, u)with h ( z ,u) L - l ( z , u )f (z, u),so that the optimization problem (60) takes the equivalent form
with dynamics 40)
j: = h ( z ,u)
=Y
(64)
1
where h satisfies a sublinear growth condition as f because of the uniform lower bound on L. Hence, without loss of generality we shall assume throughout the following that the running cost is L ( z ,u)= 1. 1. Let still V denote the value function for the minimization problem (63), with dynamics (27). Observe that V is locally Lipschitz continuous (cfr. Ref. lo), and by possibly enlarging the set COassume that
CO= {Z E JR"; V ( Z )5 for some
K,
C
K}
BT ,
(65)
\ BT
(66)
r > 0. Moreover, we can assume that V ( z )L
K
+
2E0
V x E R"
for some positive constant EO < E . By the same arguments in the proof of Theorem 4.1 (cfr. Ref. 10) one then constructs a Lipschitz continuous, piecewise smooth approximation ? of V , and a patchy feedback U ( z ) on the domain
-
R
{X E
BT; ?(x) < $(z)} ,
(67)
with the following properties: (i) For every z E
6 there holds
+
V ( z )5 ?(z) 5 V ( z ) (ii) For every z E
€0
(68)
*
6 one has
c .f
( 2 ,U ( z ) )
I -1
+ €0
V
c E a?'(zC),
where a?(z) denotes the generalized gradient of
(69)
? at z (see Ref. 14).
55
v along any solution t z ( t ) , t 2 0, of (49) as = <(t) . f ( W ,ww) C(t) E mm) (70)
Notice that, differentiating
% dt ) )
H
7
because of (ii) we find
v(z(t")) - v(z(t')) =
1"''
I (t"
C(s)
. f ( z ( s ) ,U ( z ( s ) ) )ds
- t')(-1
+
Q 0 5 t' < t"
€0)
.
Hence, relying on (65), (68), (71), we deduce that every trajectory z(.) of the unperturbed system (49) starting from a point z(0) E C0 n 6 reaches in finite time the set
-
s A {z E IWn;
V ( z ) 2 $(z)},
(72)
since otherwise it would cross the boundary of B, a t some time r > 0 and we would have F(X(T))
< P(z(0))I V(z(0)) + E o 5 fc + co
which is in contrast with (66). From (71) and (65), (68) it follows that the minimum time needed by any trajectory of (49), starting from C0, to reach the set g is certainly less than T A
e.
2. Consider now a solution t H z'(t), t 2 0, of the perturbed system (50), with z E ( 0 )E CO,which by (57) can also be seen as a solution of the perturbed patchy vector field (52), with e1,e2 defined in (58). Choose the bound 6 in (56) on the perturbations ~ 1 , c 2sufficiently small so that the quantities e l , e2 are, in turn, small enough to guarantee, by Theorem 5.1, that zc"(.)lies in an co-neighborhood of a trajectory of the unperturbed patchy vector field g in (51). Let [0,rE[ be the maximal interval of definition of zE,so that one has z E ( r EE) B(a2, co), which implies V(z"(7,))2
$(XE(T&))
- LiP(V)
.€0,
(73)
letting Lip(v) denote a Lipschitz constant for over the ball B,. Then, recaling Remark 5.2, we deduce that there is a solution z : [0,T [ H fi of the unperturbed system (49), such that
In particular, (74) yields
56
with M A 11g11L"(B7) . To fix the ideas, assume that given time t E [O,T,]
+
+(x€(T,))
(T€
, relying on
-t)i
T€
5 T . Then, for any
(68), (71), (73)-(75), we find
V(X€(T,)) +Lip#) . EO + (7,
-
t)
i V ( X ( T )+) (1 + Lip(V)(M + 1)) . EO
+ -t) 5 V ( x ( t ) )+ t ) + 1 + Lip(V)(M + 1)) I v ( x E ( t )+) 2( T + 1+ Lip(V)(M + 1)). , (T
((T -
EO
EO
(76) which establishes (62) (when L = l),provided that EO be sufficiently small so that the second term on the right-hand side of the last inequality in (76) is < E . This completes the proof of the theorem observing that, by (65), (68),_comparing the definitions (61), (67), (72), we derive CO\ 6 C C S,
s
R c R.
0
6. Stochastic p e r t u r b a t i o n s
Aim of this section is to analyze the performance of a patchy feedback in the presence of random perturbations. The basic setting is as follows. We consider the problem discussed in Section 4 of reaching the origin in minimum time, for the control system (27). We assume that the map f : R" x K H Rn is smooth and satisfies the growth conditions (28), while K c R" is compact. We denote by T(y) the minimum time needed to steer the system from y to the origin, and consider the sub-level set
RT' A
{y E R";
T(y) 5 T * },
(77)
which can be assumed to be compact as observed in Section 4. We know by Theorem 4.1 that we can construct a near time-optimal feedback u = V ( X ) on the set R p ,so that every initial point y E Rp,IyI > E , is steered by the resulting O.D.E. (30) into the closed ball BEwithin time T ( y ) E . We wish to investigate now the effect of a stochastic disturbance on the trajectories of the closed-loop system (30). Namely, we shall consider a stochastic differential equation, obtained by adding a random perturbation to the equation (27), say
+
dX = f ( X ,U ) d t
+ A ( X )dB ,
(78)
57
where B = (B1,.. . ,B n ) is an n-dimensional Brownian motion, while A = A ( x ) is an n x n matrix valued function, locally Lipschitz continuous, and satisfying the sublinear growth restriction IA(x)l 5 c(l Here and in the following ( A ( (TY.{AA'})'/2denotes the Euclidean norm of a matrix A , where A' is the transpose of A and Tr.{A} is its trace. Under the above assumptions, for a given initial condition
+ 1x1).
X ( 0 )= Y, (79) and for any measurable control input u ( t ) with values in K , the Cauchy problem (78)-(79)admits a unique (stochastic process) solution t H X ( t ) defined for all t 2 0 on a certain complete probability space ( O , F ,P ) , with the filtration {Ft}generated by the Brownian motion B (see Ref. 23). In connection with the patchy feedback u = U ( x ) ,for every given time T > T(y) E , we wish to estimate the probability that a solution t H X ( t ) of the corresponding closed-loop system
+
dX = f ( X ,U ( X ) )dt
+ A ( X )d B ,
(80) starting from y, reaches the ball BEwithin time T . The next result shows, in the same spirit of Ref. 17, that we can provide an estimate of the distance from 1 of such a probability in terms of the supremum sup IA(z)I2of the X
infinitesimal covariance function of the driving noise AdB. For any fixed
Theorem 6.1. (Stochastic stability of a nearly time optimal patchy feedback). Under the above assumptions, let T* > 0 be given, and fix T' > T * . Then, there exist a patchy feedback U ( x ) defined on the sublevelset R p , and constants c1,c2,77 > 0, so that, if IIAIIT, < 77 the following holds.
(i) For every y E R p , IyI > satisfies
for some time
E
> 0 , any solution X,(.)of (79)-(80)
ryIT ( y ) + c211AIIT, . 1/3
(ii) There exists a continuous, strictly increasing function [O, co[vanishing at zero, such that
{' I
I } 2 1 - ("
lim Prob. inf t X,(t) 5
IYI--'E+
E
E
x
+x(E))
: [O,CO[H
.
(82)
58
Proof. 1. Fix some constant T' > T*.By the proof of Theorem 4.1 we can construct a semiconcave, piecewise quadratic Lyapunov function T as in (36), and a patchy feedback U ( z ) on the compact sub-level set
-
I
AT! A
F(z) 5 T'} , For every point II: E XT,,
{II: E R"
;
with the following property. where ? is differentiable, there holds
(83) with 1x1
+
T ( z )I F(II:)I T(II:) EO ,
V F ( 2 ). f(z,U ( z ) ) I -1
+
Eo
> E , and (84)
,
(85)
for some positive constant EO << min{c, T' - T*}.Observe that by (84) one has R p c AT^, for TO= T* EO, and X T ~c R p . Hence, to establish the theorem it will be sufficient to prove that statetments (i)-(ii) hold when y varies in X T ~ and , llAllTl is redefined as IIAIIT, sup IA(z)I2.
-
+
x E A ~
2. Next, for a given initial point y E
t
H
XT,,,
IyI
>
E,
consider a solution
X ( t ) of the stochastic differential equation (80), with initial condi-
tion (79). Observe that, since the feedback is not defined on the whole space R", while we have to take into account also the possibility that the trajectoty X ( t ) touches the boundary d x p , it will be appropriate to introduce the stopping time min { t 2 0;
lx(t>l= c or
F ( X ( t ) )= T I } .
(86)
Our main goal is to provide an estimate of the probability that X ( . ) reaches the ball B, within any given time t > 0, i.e. of Prob.{T 5 t , IX(T)I = & } .
(87)
To this end, consider the scalar random variable Y ( t )= ? ( X ( t ) ) , t 5 T . According to (36), at every point where F is smooth, the n x n Hessian matrix of second order partial derivatives of ? is 2 M I , where I denotes the n x n identity matrix. By ito's formula (see Ref. 23, p. 48), as long as t < T and X ( t ) remains on an open region where ? is smooth, we have
dY(t)= V F ( X ( t ) )* f ( X ( t ) ,U ( X ( t ) ) d) t
+ V F ( X ( t ) ). A ( X ( t ) )dB(t)
+ M . T r . ( A ' ( X ( t ) )A ( X ( t ) ) )dt . (88)
59
Indeed, since the set of points where ? is not differentiable is contained in a finite union of n - 2 dimensional manifolds of R”, the probability that F be differentiable at X ( t ) ,t < T , is one, and hence we may assume that Y ( t ) is a solution of (88) for all t < T . We can extend the random process Y ( t ) for all t 2 T by letting Y ( t ) ,t 2 0, be the solution of
dY(t)= V F ( X ( tA T ) ) . f ( X ( t A T ) , V ( X ( tA T ) ) ) dt
+V?(X(t A T ) )
A ( X ( t A T ) ) dB(t)
(89)
+ M . Tr.( A ’ ( X ( t A T ) ) A ( X ( t A 7))) d t , where t A T 4 min{t,T}. Notice that this extension is well defined, again because the probability that ? is not differentiable at X ( T ) is zero. 3. Observe now that, from the definition (86) it follows
Prob.{r 5 t , ( X ( T ) = ( E}
2 Prob. { Y ( t )5 0 -Prob.
T
5 t , Y ( T )= TI).
(90) Towards an estimate of the first term on the right-hand side of (go), to simplify the computations it is convenient to introduce a further random variable 2, such that
Z ( t ) = Y ( t )- (1 - eo - M . IIAllT,)t -
I” {
V ? ( x ( s A 7 ) ). f ( x ( sA T ) , ~ ( x (A s7)))
(91)
+ M . Tr.(A’(X(s A T ) ) A ( X ( s A T))) } d s , where Ic
{
1SUP Iv?(Z)I ;
2 E AT)}.
(92)
By the above definitions, thanks to (85) we deduce Z ( t ) 2 Y ( t ) ,so that
{
Prob. Y ( t )5 0
I2
p ( t ) 4Prob.{Z(t)
5 0}
V t 2 0.
(93)
Moreover, from (89) it follows that Z provides a solution to the linear stochastic differential equation
dZ(t ) = ( - 1
+ + M . IIAIIT~)dt + P ( t ) d B ( t ) , EO
(94)
with P ( t ) 1V ? ( X ( t A T ) ) . A ( X ( t A T ) ) .
(95)
60
Therefore, the expectation and the variance of Z ( t ) y(t) A Var.{ Z ( t ) },
a(t)= E [ Z ( t ) ] 7
(96)
satisfy the differential equations b(t) =
( - 1 + EO + M . ~ ~ A ~ ~ T ~?(t) ) ‘= , ( p ( t ) ) 2I n2 . llAll~/, (97)
with initial conditions
4 0 ) = F(Y)
$0)
7
= 0.
(98)
~ ( t5)(K’ . llAll~i)t
(99)
Relying on (97)-(98), we derive
a ( t )= F ( y ) - (1 - EO -
llAll~/)t ,
Hence, observing that when t > T(y)/(l - EO - A4 . IlAll~l)one has Prob. Z ( t ) > 0, IZ(t) - a(t)I 2 la(t)l}= 1 ,
{
from (99) we deduce
. (1 - p ( t ) ) 5
[ q y ) - (1 - E o - M . llAllT+]2
(K2
*
IlAllT,) t ,
(100)
which, in turn, yields
for all t
> ?(y)/(l
- EO
-
M * llAll~c).
Concerning the probability that the trajectory hits the outer boundary
{
1
p AProb. Y ( T )= T’ ,
(102)
observe that, by (99) one has
(T’ - a ( T ) ) 2 pI (n2 . llAllT+. Thus, assuming
since y E
x~~from (99), (103) we derive
(103)
61
Hence, ( g o ) , (93), (101), (105) together yield
n
(106) for all t > ~ ( ~ ) / ( ~ - E ~ - M . I I A I I TSince I ) . we may repeat the same reasoning for an arbitrary small EO < E , we thus obtain from (106) the estimate
(107) Then, taking
and recalling (104), we deduce from (107) that
which proves the statement (i) of the theorem, since
TI. One can easily see that the approximate minimum time function ? defined as in (36), (with a possible slight modificaton) beside (84)-(85) enjoys the further property: 4. Concerning (ii), let BR be a ball containing
-c3(XI'1 for some constants
5 F(x) 5 ~ c3
+
3 1 ~ 1EO
V 1x1 5 R ,
(110)
> 0, 1 < q < 2. Next, observe that setting pE = max {?(XI ; 1x1 5 2 ~ ,)
(111)
62
by (110) we have pi = pa
B2&C
+
Xpc
E
< 2(C3 -k l ) E ,
-
C
C
(112)
,
BCqEl/q
with c4 A C3(2(C3+l))1/q.Hence, letting Lip(A) denote a Lipschitz constant for A over the ball B R , we find IA(z)I 5 I A ( O ) I + L ~ ~ ( A ) . C ~ Eb’z€&;, ~/~
(113)
which yields
where 2/3
XI(&)
A (lA(O)/+Lip(A).c4~’/~)
-
IA(0)12/3,
E>O,
defines a continuous, strictly increasing function vanishing a t zero. Finally, applying the estimates (107)-(108), with p c ,p: in place of T * ,T’, respectively, and relying on (112), (114) (and on IIAllp; < l), we derive
for suitable constants ci , cy, c y , c p
> 0, and with
This establishes (82) thus completing the proof of the theorem.
0
63
References 1. F. Ancona and A. Bressan, ESAIM - Control, Optimiz. Calc. Var. 4, 445 (1999). 2. F. Ancona and A. Bressan, SIAM J . Control Optim. 41,1455 (2002). 3. F. Ancona and A. Bressan, ESAIM - Control, Optimiz. Calc. Var. 10,168 (2004). 4. F. Ancona and A. Bressan, Stabilization by patchy feedbacks and robustness properties, in Optimal control, stabilization and nonsmooth analysis, eds. M.S. de Queiroz, M. Malisoff and P. Wolenski, Lecture Notes in Control and Inform. Sci., Vol. 301 (Springer, Berlin, 2004), pp. 185-199. 5. F. Ancona and A. Bressan, Ann. Inst. H. Poincare' Anal. Non Line'aire 24, 279 (2007). 6. J.-P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory. Grundl. der Mathemat. Wissensch., Vol. 264 (Springer-Verlag, Berlin, 1984). 7. L. D. Berkovitz, SIAM J . Control Optim. 27,991 (1989). 8. A. Bressan and B. Piccoli, SIAM J. Control Optim. 36,12 (1998). 9. A. Bressan and B. Piccoli, Introduction to the mathematical theory of control, AIMS Series on Applied Mathematics, Vol. 2, first edn. (AIMS, Springfield MO, 2007). 10. A. Bressan and F. Priuli, Nearly optimal patchy feedbacks for minimization problems with free terminal time, Discrete Contin. Dyn. Syst. (to appear), available at http://cpde.iac.rm. cnr.it/ricerca.php (July 2007). 11. R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential geometric control theory (Houghton, Mich., 1982), eds. R. W. Brockett, R. S. Millman and H. J. Sussmann, Progr. Math., Vol. 27 (Birkhauser Boston, Boston, MA, 1983), pp. 181-191. 12. F. H. Clarke, Y. S. Ledyaev, L. Rifford and R. J. Stern, SIAM J . Control Optim. 39,25 (2000). 13. F. H. Clarke, Y . S. Ledyaev, E. D. Sontag and A. I. Subbotin, IEEE Trans. Automat. Control 42,1394 (1997). 14. F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, Vol. 178 (SpringerVerlag, New York, 1998). 15. J.-M. Coron, Systems Control Lett. 14,227 (1990). 16. J.-M. Coron and L. Rosier, J . Math. Systems Estim. Control 4,67 (1994). 17. H. Deng, M. KrstiC and R. J. Williams, IEEE Trans. Automat. Control 46, 1237 (2001). 18. A. Filippov, Transl., Ser. 2, A m . Math. SOC.42,199 (1964). 19. A. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), Vol. 18 (Kluwer Academic Publishers Group, Dordrecht, 1988). Translated from the Russian. 20. 0. HAjek, J . Dzfferential Equations 32,149 (1979).
64
21. H. Hermes, Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P. R., 1965), eds. J. K. Hale and J. P. LaSalle (Academic Press, New York, 1967), pp. 155-165. 22. N. N. Krasovskii and A . I. Subbotin, Positional Differential Games [in Russian] (Izdat. Nauka, Moscow, 1974). Revised English translation: GameTheoretical Control Problems (1988). Springer-Verlag, New York. 23. B. Bksendal, Stochastic differential equations. An introduction with applications. Universitext, sixth edn. (Springer-Verlag, Berlin, 2003). 24. B. Piccoli, SIAM J. Control Optim. 34, 1914 (1996). 25. E. P. Ryan, SIAM J . Control Optim. 32,1597 (1994). 26. E. D. Sontag, Mathematical control theory. Deterministic finite-dimensional systems. Texts in Applied Mathematics, Vol. 6, second edition (SpringerVerlag, New York, 1998). 27. E. D. Sontag, Stability and stabilization: discontinuities and the effect of disturbances, in Nonlinear analysis, differential equations and control (Montreal, QC, 1998), eds. F. Clarke and R. Stern (Kluwer Acad. Publ., Dordrecht, 1999), pp. 551-598. 28. E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, in Proc. IEEE Conf. Decision and Control, (Albuquerque, NM, 1980): 916-921. 29. H. J. Sussmann, J . Differential Equations 31,31 (1979).
65
SENSITIVITY OF CONTROL SYSTEMS WITH RESPECT TO MEASURE-VALUED COEFFICIENTS Z. ARTSTEIN * Department of Mathematics The Weizmann Institute of Science Rehovot 76100, Israel E-mail: zvi. artsteinOweizmann. ac.il Measurevalued coefficients arise when limits of rapidly varying parameters of control systems are examined. The rapidly varying parameter is considered then a perturbation of a measure-valued parameter. We provide quantitative estimates for the sensitivity of the optimal value and near optimal solutions, with respect to such perturbations. As a particular case, which is of interest for its own sake, we establish quantitative estimates for the sensitivity of ordinary control systems with respect to relaxed controls.
1. Introduction This paper examines control systems of the form minimize
dx 4 . 1 = xo. dt The state variable is x E Rn, the n-dimensional Euclidean space. The control variable u belongs to a set U in R"; the admissible controls are measurable functions u = u(.) from [a,b] into U . The function p(.) is a prescribed measurable map from [a, b] into a complete separable metric space r; the interpretation is that p ( t ) : [a,b] + r is an external parameter of the system. subject to
- = f(x,t ,u , P ( t ) ) ,
We are interested in the dependence of the value and the optimal solutions of the system on this external parameter. Of particular interest to us in this paper is to get, under appropriate conditions, quantitative estimates in the *Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics. Research supported by the Israel Science Foundation.
66
form of Lipschitz estimates, on this dependence. Furthermore, the case we concentrate on is where the external parameter p ( . ) may change abruptly and frequently, for instance, when it oscillates very rapidly. A natural approach to address sensitivity issues is to identify a nominal system, for instance, to identify a parameter PO(.) and examine the influence on the value and on the optimal solutions when a perturbation p(.) of PO(.) occurs. This approach does not fit the case of rapidly oscillating parameters since a sequence of rapidly oscillating functions cannot be represented as a perturbation which tends to zero from a fixed function. Indeed, a sequence of more and more rapidly oscillating functions does not converge strongly, while a weak limit does not capture the oscillatory nature which may, in turn, play a role in the optimization. An appropriate nominal system, however, is one with the parameters being measure-valued, namely, Young measures. A theory of such nominal systems was offered in Artstein,2 and was applied in various frameworks, including the case where the fast parameter is ignited by a singularly perturbed control system. See also Artstein3v4 and Artstein and Gaitsgory5i6 and references therein. We briefly recall the main ingredients of the theory in Section 3. Young measures play a similar role in other theories; a non exhaustive sample of references is Balder,7 Pedrega1lo-l2 Tartar,14 Valadier,15 Warga16 and Young;17 the latter two references concentrate on relaxed controls and generalized curves. The available theory in the applications of Young measures is primarily qualitative, due to the need to work in the highly nonsmooth space of probability measures (see though Pedregal" for a variational theory). The present paper is devoted to the derivation of the quantitative estimates of the sensitivity problem. The paper is organized as follows. In the next section we display some technical assumptions on the data; these are used throughout the paper. In Section 3 the chattering parameters model is recalled. The sensitivity estimates are established with respect to the Prohorov metric which is displayed in Section 4. In Section 5 we establish the sensitivity in the particular case of relaxed controls without external parameters. A matching lemma which is of interest for its own sake, is displayed in Section 6; it serves as an important tool in verifying the general sensitivity result in Section 7. The closing section offers few comments and concrete examples illustrating the main results.
67
2. Standing hypotheses We display here the technical assumptions employed throughout the paper. In order not to blur the main contribution of the paper, no attempt is done to display the most general assumptions possible; see, though, the closing section for some remarks in this respect.
Assumption 2.1.
(i) The set of controls U in Rm and the space J? in which the parameter function p ( . ) takes values are compact; the space r is endowed with a prescribed metric d(., .). (ii) The functions Q ( x , t , u , y ) and f ( x , t , u , y ) are Lipschitz in the four variables x , t ,u and y,with Lipschitz constant, say, 6. Under the Standing Hypotheses, standard considerations guarantee that for any fixed control function u(.)and any parameter function p ( . ) , the differential equation in (1) has a unique solution defined on the entire interval [a,b]. Furthermore, there is a common bound on all these trajectories. Consequently, the functions f ( x ,t , u,y) and Q ( x ,t , u,y) are uniformly bounded on all these trajectories. In particular, the cost is well defined.
Notation 2.2.
(iii) W e denote by p the common bound, guaranteed by Assumption 2.1, on the values ( x ,t , u,y)l when evaluated o n the trajectories generated by feasible parameter and control functions.
If
3. The chattering parameters model
We briefly review in this section the relevant ingredients of the chattering parameters model. We start with the definition of Young measures (newcomers to the Young measures paradigm may wish to look at Balder,7 Valadier;15 the examples in the closing section may also help in clarifying the structure). Let M be a complete metric space, with a metric which we again denote by d(., .). A probability measure on M is a a-additive set function, say p ( . ) , assigning to each Bore1 subset B of M a value p ( B ) E [0,1] with p ( M ) = 1. The family of all probability measures on M is denoted P ( M ) . We later consider P ( M ) as a metric space. We consider mappings p ( t ) : [a,b] 4 P ( M ) . We write p ( t , B ) for the value which the measure p ( t ) assigns to the set B . A Young measure into M (pertinent to the framework in (1)) is a mapping p ( t ) : [a,b] -+ P ( M )
68
which is measurable in the sense that for each fixed Bore1 set B in M the real-valued map p(.,B ) : [a,b] ---$ [0, 11 is measurable. The space of such Young measures is denoted by y = y ([a,b ] ,M ) . We consider a convergence structure on y given by: The sequence p j ( . ) converges as j 4 00, in the sense of Young measures, to PO(.)if (here m is the independent variable in
M)
whenever h(t,m ) is a bounded and continuous real-valued function defined on [a,b] x M . (Notice that in accordance with the previous notation we write p ( t , dm) for the integration with respect to the variable m with respect to the measure p ( t ) for a fixed t.) The Young measure p ( t ) : [a,b] ---t P ( M ) is be denoted either by p(.) or in bold face, namely, p. We consider in this paper three spaces of Young measures: into the parameter space r, into the control set U , and into I? x U . The space of Young measures, namely, the space of probability measurevalued maps with the aforementioned convergence, arises in a variety of applications. A sample of applications can be found in the following papers and monographs, and references therein: BalderI7 Ball18Pedrega110-12 Tartar,14 Valadier,15 Warga,“ Young.17 In particular, it was established that when M is compact, the space y ( [ a , b ] , M with ) the convergence in the sense of Young measures is induced by a metric which makes it a compact space (in the sequel we concentrate on a specific metric). The space of point-valued functions m(t) : [a,b]-+ MI is a subspace of Y ( [ ab, ] ,M ) , where m(.)is interpreted as a Young measure whose values are Dirac measures, namely m(t) is interpreted as a probability measure supported on the singleton { m ( t ) } . The subspace of Dirac-valued measures is dense in y ( [ a , b ] , M )(this property relies on the Lebesgue measure, which is the base measure on [a,b ] , not having atoms). The following definition introduces Young measures parameters into the systems (1);it was introduced in Artstein.’
Definition 3.1. The parameter in (1) may be a chattering parameter, namely a Young measure p(.) into r; the optimization problem takes then
69
where the optimization is done over all admissible controls (namely, measurable function into U ) of the form u(t,y) : [a,b] x r + U . It is clear that (1) is a particular case of (3), namely, when the pointvalued parameter p ( . ) is interpreted as a Young measure. In fact, the motivation for introducing the more complicated model (3) is that the elements with a chattering parameter form appropriate limits of sequences of problems with ordinary parameters, thus providing an appropriate closure of the space. As in the standard control problem, the optimal control problem (3) may not have an optimal solution even if the control set U is compact. Relaxed controls, as introduced by J. Warga (see Ref. 16) may yield a solution. The induced structure is as follows.
Definition 3.2. A relaxed control for the system (3) is a mapping, say v = v(t,y), which assigns to each pair (t,y) a probability measure on U , and as such it is measurable. The resulting cost function and differential equation are then
We refer to the cost given via the integration of Q ( x ,t , u,y) in (4)and the solution of the associated differential equation as the cost and, respectively, the trajectory generated by the parameter p ( . ) and the control v(.,.) Relaxed controls were developed by Warga, see Ref. 16, and were used in the chattering model in Artsteix2 Notice that a relaxed control is a Young measure into U . A key observation is as follows.
Definition 3.3 (and observation). Let p ( . ) be a chattering parameter of (3) and let v(.,.) be a relaxed control. The effective input induced by p ( . ) and v(.,.) on (3) is the Young measure which assigns to each t E [a, b]
70
the probability measure, say p ( . ) , on
P ( t ) ( C )=
r x U defined by
Jr v ( t ,7 ,C,)P(t, d y )
(5)
(where C, is the y-section of C in U , recall that w ( t , r, C,) is the value the measure w ( t , y) assigns to (7,). What counts in regard t o solving the control equations and computing the cost when a relaxed control is applied to a chattering parameter is the effective input. Indeed, the system (4) takes then the form
Jo” l,,
Q ( x ,t ,u,y ) p ( t ,d u x d r ) dt
dx =
f lXU
(6)
( x ,t , ‘1L, T ) P ( t ,d u x d y ) , .(a) = 50,
where p ( . ) is the effective input.
Proposition 3.4. Let p ( . ) be a chattering parameter in (3) and let w(.,.) be a relaxed control. There exists a sequence of ordinary controls u i ( t , y ) : [a,b] x r -+ U such that the effective inputs induced by ui(s,.) converge, in the sense of Young measures, t o the effective input induced by w(.,.). Consequently, the trajectories xi(.) and the costs ci, generated by applying ui(.,.) t o (4), converge unzformly o n [a,b] and in, respectively, R, t o the trajectory and cost generated by applying v(.,.) t o (4). Proof. We consider the Young measure p = p(.) as a base measure on the space [a,b] x I?. It is an atom-less measure. It follows from the basic properties of Young measures that the Young measure v(.,.) : [a,b] x r P ( U ) from the base space [ a ,b] x r into U can be approximated in the sense of Young measures by a sequence of ordinary functions ui(., .) : [a,b] x r -+ U . It follows directly from the definition (2) of convergence according to Young measures and the definition ( 5 ) of the effective input, that the induced effective inputs also converge in the sense of Young measures. The effective inputs can now be viewed as relaxed controls of the control system (without parameters); thus the convergence of the trajectories and the costs follows from the classical work of J. Warga, see Ref. 16. 0 -+
4. The Prohorov metric
We introduce here the metric on the space of Young measures, with respect to which the quantitative estimates are established. It is based on the Prohorov distance between two probability measures, see e.g., Billing~ley,~
71
hence we use the same term. We first recall the Prohorov metric in general spaces (although the definition in Ref. 9 refers to probability measures, it is clear that it is applicable to all finite measures). In what follows M is a metric space with a metric d(., .); the q-neighborhood of the set B in M is denoted by B(q).
Definition 4.1. Let M be a complete metric space. The Prohorov distance between two finite measures p and u on M is the infimum of all q > 0 satisfying u ( B ) 5 p ( B ( q ) ) q and p ( B ) 5 u ( B ( q ) ) q for every Borel set B of M . We denote the Prohorov distance by Proh(p, v).
+
+
The Prohorov distance reflects the weak convergence of measures. It is known that with the Prohorov distance the space of all finite measures on M is a complete metric space. These properties are established in Ref. 9 for the space P ( M ) of probability measures, but the arguments go through for all finite measures. We now identify the space of Young measures y ([a,b ] ,M ) as a space of measures to which the Prohorov distance applies. As a preliminary step, with any p ( . ) E y ( [ a ,b],M ) an equivalent measure p on [a,b] x M is associated, as follows. The measure of a Borel subset C of [a,b]x M is given by
where Ct is the t-section of C. Notice that then p ( T x M ) = X(T) for every subset T of the interval [a,b] (and where X is the Lebesgue measure). It is easy to see that this property characterizes the space y ([a,b],M ) . In particular, when M is compact, the space y ( [ a ,b ] , M )is a compact subset of the space of measures on [a,b] x M .
Definition 4.2. The Prohorov distance on y ( [ a , b ] , M )is defined to be the Prohorov distance on the associated measures on [a,b] x M , where the metric on [a,b] x M is the L1-metric, namely, the distance between ( t l ,m l ) and (tz,mz)is It1 -tzI+d(ml,m2). We use Proh(p,v), or Proh(p(.),u(.)), to denote the Prohorov distance between p and v. In the sequel the Prohorov distance on the three spaces of Young measures, namely, into r, into U and into r x U , will be invoked.
72
5 . Sensitivity for relaxed controls
This section establishes the promised sensitivity estimate in a particular case, namely, the classical situation of control systems with relaxed controls as follows. minimize
JIRb
&(x, t ,u ) d t
(8) dz = zo . dt Such estimates seem not to be available in the literature. The general case of systems with chattering parameters, treated in the next section, will in fact be reduced t o the setting in (8). Recall that a relaxed control for (8) is a Young measure w ( t ) : [a,b] -+ P(U),denoted either by v or by w(.). We write then f ( z , t , w ( t ) )for J ,f (z, t , u)v(t,d u ) . subject to
- = f (z, t , u), .(a)
Theorem 5.1. Under Assumption 2.1 there exists a constant r ] , such that given any two relaxed controls v1 and v2, the resulting trajectories and costs in (8), say X I ( . ) and z2(.), and c1 and c2, satisfy Izl(t) - z2(t)l 5 r] Proh(v1, va), and Ic1- c2 I 5 r] Proh(v1, v2). The constant r] depends only
o n the length b - a of the interval, the Lipschitz constant ,8 guaranteed by the assumption.
K
and the bound
Proof. On the space C( [a,b ] ,R") of continuous functions from [a,b] to R" consider the distance induced by the norm
where K. is the Lipschitz constant given in Assumption 2.1. The trajectories X I ( . ) and Q(.) are fixed points of, respectively, the operators from C( [a,b],R") to itself given by
T z ( z ( * ) ) (=t )2 0
+
/f t
(z(s), 3, uz(s))ds
(10)
a
for i = 1 , 2 . Each of these two operators is a contraction on C ( [ a ,b ] , R n ) with respect to the norm (9); this is a classical result, see Reid [13, p. 561. Let the contraction factor be denoted by a , namely, a < 1 and llTi(.(.)) - Ti(Y(.))llKIaII4.) - Y ( * ) l l K .
It follows from Reid [13, p. 561 that a depends only on b - a and
(11) K.
73
First we give an estimate for the distance 11z1(.)- z2(.)lln.Since xi(.) is a fixed point of the corresponding operator it follows that 11x1(.) - z2 (.) Iln I llTl(21(*)I- TZ(Zl(.))IIK.
+ llT2(z1(.)) - T2(z2(.))
Iln.
(12)
The contraction property (11) implies that last term in (12) is less than or equal to a11z1(.)- z2(.)lln.Hence, by shifting this estimate to the left of the equation, we get that (1 - a)llz1(.) - z2(.)1In is bounded by the first term in the right hand side of (12). Since Izl(t) - zz(t)l is bounded by en(b-a)((zl(.)- z2(.)lln,and since the norm in (9) is less than or equal to the sup norm, it follows that Izl(t) - zz(t)l 5 enE(b-a)(l - a)-'IlTl(zl(.)) - T2(51(.))11
(13)
where the norm in (13) is the sup norm on continuous functions. The claimed estimate for the trajectories would follow from (13) once we verify that there exists a constant 770 (depending only on b - a, K and p) such that ITl(Zl(.))(T)- Tz(z1(*))(7)1 I110Proh(v1,v2)
(14)
for all T E [a, b]. Verifying (14) is what we do next. Since P , Jis a bound on If(., t ,u)I it follows that any trajectory generated by applying a relaxed control to (8) is Lipschitz in the time variable with Lipschitz constant p. We show that (14) holds when X I ( . ) can in fact be replaced by any P-Lipschitz map, hence in the sequel we drop the subscript from XI(-). Recall that, in general, when h(E) : M -+ [0, GO) is a bounded nonnegative measurable function on a metric space M , say bounded by 2p, and when p is a probability measure on M then
IM
28
P({E : h ( l ) 2
h(t)P(dE) = 0
XI dX,
(15)
where dX indicates integration with respect to the Lebesgue measure; see, e.g., Billing~ley.~ We apply this representation when M = [ a , ~x ]U and when h(t,u) = f j ( z ( t ) , t , u ) p where the subscript denotes the j - t h coordinate (the addition of p makes it nonnegative; it is clearly bounded by 2p). We deduce therefore that the difference between the j - t h coordinates of T~(z(.))(T) and Tz(z(.))(~) (appearing in (14)) is
+
74
+
where, as mentioned, h(t,u) = fj(z(t),t ,u) P. The Lipschitz constant guaranteed in Assumption 2.1 and the P-Lipschitz property of z(.) imply that h(t,u)is Lipschitz, with Lipschitz constant KP. The inequalities in Definition 4.1 which define the Prohorov distance imply that vi({(t,u) : h ( t , u )2 A}) 5 vz({(t,u) : h ( t , u )2 A}(@) + 6
(17)
with S = Proh(vl,vz), and the similar inequality where v1 and v2 are swapped. The KP-Lipschitz property of h(t,u)together with (17) imply the inequality
+ KPS L X} + 6
vi({(t,u) : h(t,u)2 A}) I ~ 2 ( { ( t , u :) h(t,u)
(18)
and the similar inequality with v1 and v2 swapped. Replacing the first term of the integrand in (16) by the right hand side of (18), changing then the variable of integration (from X to X - @S), also taking care of the extra interval of integration (of length KPS) and the error term 6 from (17) integrated over the 2 p interval, implies that (16) is less than or equal to (2 K ) ~ S Applying . the same argument with v1 and v2 swapped implies that the absolute value of (16) is bounded by (2 K ) ~ S . The argument can now be repeated for the n-coordinates j = 1,. . . , n. Since 6 = Proh(v1, v2) the existence of 70 in the estimate (14) and its dependence on b - a, K and /3 only is established; hence the claimed estimate on the distance between the trajectories is verified. We turn now to the claimed estimate on the distance between the costs c1 and c2. We need to show that
+
+
for some constant q which depends only on b - a, K and p. When x2(.) in (19) is replaced by XI(.),the same estimate follows from the argument we used t o verify (14); namely, we get I
for an appropriate ql. When v1(+) in (19) is replaced by u2(.) the same estimate follows from the estimate on the distance between the trajectories and since Q ( z , t , u)is 6-Lipschitz; namely, we have
75
for an appropriate 7 2 . The latter two estimates combined with the triangle inequality verify that (19) holds with 7 = ql + 7 2 . This completes the proof of the theorem. 0
6. A matching result We display here a matching result, related to the Monge-Kantorowitz transport problem; it assures, roughly, the existence of a relaxed transport which is of the order of the Prohorov distance. The result will serve in the derivation of the estimate established in the next section. Recall the variational distance, which we denote by llpl - pzll, between two probability measures 1-11 and p~ on a metric space M ; it is the maximum of I p l ( B )-p2(B)I Ipl(M\B) -p2(M\B)I over the Borel subsets B of M . Recall that P ( M ) denotes the space of probability measures on the space M . If p is a measure on M and p ( . ) : M -+ P ( M ) is a Young measure then (in full analogy with the point-valued case) p ( p ) is the induced measure,
+
namely, P(PL)(B)= JM p ( s ) ( B ) p ( d s ) .
Definition 6.1. Let M be a complete separable metric space with metric d(., .). Let p and v be two probability measures on M . A Young measure p(s) : M P ( M ) is a relaxed 7-matching from p to v if for every s the measurep(s) is supported on the 7-neighborhood of s and if Ilv-p(p)II 5 27. Remark 6.2. When p ( s ) : M -+ P ( M ) is a relaxed 7-matching from p to v , then the equivalent measure (see Section 4) p on M x M is supported on {(sl,sz) : d ( s 1 , s z ) 6 7 ) . Furthermore, the disintegration of p with respect to the second coordinate is a relaxed 7-matching from v to p. We may say then that p is a relaxed 7-matching between v and p. Proposition 6.3. Let M be a compact metric space with metric d(., .). Let p and v be two probability measures o n M such that Proh(p, v) 5 7 . T h e n there exists a relaxed 7-matching between p and v. Proof. We rely on a result established in Artstein.' We add to M an isolated point, say SO, and let both p and v assign to it the weight 7. (Now both p and v are not probability measures, but this will not change the argument). With each s E S we associate the set F ( s ) = { s } ( q ) U {SO}, namely, F ( s ) is the union of the closed 7-neighborhood of s and the singleton SO. Then F ( s ) is compact. Denote by F ( B ) the union of { F ( s ) : s E B } . It is easy to see that for every Borel set B in M U {SO} the
76
inequality P(B) I v(F(B))
(22)
+
holds. Indeed, since Proh(p, v) I 77 it follows that p ( B ) I v ( B ( 7 ) ) 77; then (22) follows since SO}) = 7. By [I,Theorem 3.11, a mapping p(.) : M U {SO} 4 P ( M U {SO}) exists such that the equality p = p ( v ) holds on M U {SO} We modify now p ( . ) as follows. Whenever p ( s ) assigns a positive weight to {SO} we shift this weight to {s}. Now p ( . ) is from M to P ( M ) . Since both measures assign to {so} a weight 77, it is clear that the variational distance between p ( p ) and v is less than or equal to 277; this completes the proof. 0 Observation 6.4. It is clear that the previous result holds also when the two measures, rather than being probability measures, satisfy p ( M ) =
4M).
7. Sensitivity for chattering parameters Here is the quantitative sensitivity estimate for the system (1) with measure-valued coefficients, namely, the system (3).
Theorem 7.1. Under Assumption 2.1 there exists a constant 0 such that: Let p1 be a chattering parameter for (4) and let v1 be a relaxed control that when applied with p1 it results in the trajectory X I ( * ) and the cost c1. Let p2 be another chattering parameter. Then there exists a relaxed control v2, that when applied with pz it induces a trajectory x2(.) and a cost c2, such OProh(p,,p,), and Ic1 - c21 I 0Proh(pl,p2). The that Izl(t) - xZ(t)l I constant 0 depends only o n the length b - a of the interval, the Lipschitz constant K and the bound p guaranteed by the assumption. Proof. Consider the space M = [a,b] x r with the L1-metric on the product, namely, the distance between ( t l , y 1 ) and ( t 2 , y z ) is It1 4 2 1 + d ( y l , y z ) . Consider the Young measures p1 and pz as measures on M . Denote their Prohorov distance by 77. In view of Proposition 6.3 (see Observation 6.4) there is a relaxed 77-matching pz(.) : M + P ( M ) from p2 to pl. (We put the subscript in pz(.) in order to emphasize that it is a matching from p2 to pl). We use this matching in the definition of v2. For the simplicity of the argument we first work out the case where the control v1 is an ordinary control, i.e., the values wl (s) are points in U (recall that s E M is a pair (t,y) E [a,b] x I?). We define ~ ( s to) be the probability
77
measure on U which is the image by the function w 1 ( . ) of the probability measure p 2 ( s ) on M ; namely, 212 ( s )(C) = p2 ( s )({ c : w1 (c)E C}). We claim that then the effective inputs to (4) (see Definition 3.3) induced by ( p l ,v1) and ( p z , v 2 )are at most 271-apart in the Prohorov distance. To this end we first compare the effective inputs of ( p Z , v z ) and ( p 2 ( p 2 ) , v l )namely, , when v1 is applied to the chattering parameter which is the image of p 2 by p 2 ( . ) . Denote the two effective inputs by e2 and ei (each of these is a measure on [a,b]x I’ x U ) . Let D be a Borel subset of [a,b] x x U . The q neighborhood of D includes the set, say D‘, given by {(t’,r’,u) : d((t’,r’),(t,r))5 71 for some ( t , r , u ) E D } . Since p z ( s ) is supported on an 71-neighborhood of s it is clear that e 2 ( 0 ) I e i ( D ’ ) . In particular, e z ( 0 )I ei(D(q)).
(23)
The estimate can be reversed by applying the disintegration of p2 with respect to the second coordinate (see Remark 6.2); since the latter is also supported on the same set, it follows also that for each Borel subset D the inequality
e i ( D ) Ie
2 ( ~ ( ~ ) )
(24)
holds as well. The two displayed inequalities form a stronger version of the Prohorov distance, between, however, p 2 and p2(p2)as chattering parameters. Now we take advantage of the fact that the variational distance between p2(p2)and p1 is less than or equal to 271. Hence on each set D the values of the effective input, say e l , of ( p l ,v l ) ,and the effective input ei may differ by a t most 271. This results in the inequalities
el(^(^)) + 271
e 2 ( ~I )
(25)
and
+
el(D)I e z ( D ( q ) ) 2r]
(26)
(compare with (23) and (24)); these establish the claimed Prohorov distance. Once the Prohorov distance between the effective inputs is less than or equal t o 271, Theorem 5.1 together with Definition 3.3 imply the existence of the claimed estimate 8. This completes the proof in the special case of a point-valued control. To establish the existence of 8 in the general case where q ( s ) may be a relaxed control we may proceed in several ways. One possibility is t o approximate the relaxed control by an ordinary one (see Proposition 3.4)
and apply the previous considerations to the approximation. (The resulting effective inputs may then be of distance slightly greater than 277; a limit argument would produce a relaxed control with 277 being the distance among the effective inputs.) Another possibility is to extend the previous construction to relaxed controls. For instance, use the same construction when q ( s ) is interpreted as a function to the metric space P ( V ) . Then vz(s) is a Young measure into P ( V ) ,namely, a probability distribution on measures. The desired relaxed control should then assign to each set C the expectation of p ( C ) according to this probability distribution. One way or another, the proof is complete. 0
Corollary 7.2. The optimal value of problem (3) is continuous with respect to the chattering parameter p when the latter is endowed with the Prohorov metric; in fact, the optimal value is Lipschitz with respect to the Prohorov metric.
8. Remarks and examples
In this section we provide some comments on the conditions in Assumption 2.1, and examples demonstrating the main results.
Remark 8.1. The Lipschitz dependence on the time variable demanded in Assumption 2.1 cannot be replaced by, say, continuity (this in contrast to many developments in ordinary differential equations). The reason is that the Prohorov metric on Young measures is linear with respect to shifts in time. As an example consider the scalar control system = t i u , z(0) = zo. Consider the control u,(.) defined by u,(t) = 0 if t 5 T and u,(t) = 1 if t > T. Then the Prohorov distance between u,(.)and u,(.)(say T > (T) is T - (T,while the distance between the respected solutions is t f dt. Clearly, the Lipschitz dependence on the Prohorov metric near t = 0 is violated.
9
s,'
Remark 8.2. The requirement that Q ( z , t , u , y ) be Lipschitz can be relaxed to being locally Lipschitz. Indeed, as noted in Section 2 boundedness of the relevant trajectories is guaranteed by f(z,t , u,y) being Lipschitz. The Lipschitz condition on f (z,t , u,y) cannot be replaced by its local analog since then the solutions may not stay bounded. We illustrate now the main result concerning relaxed controls.
79
Example 8.3. Consider the scalar system minimize
ll(x(t)'
+ Ju(t)- 11')dt
(27) dx subject to - = f(x,t)u, x(0) = 0, dt with f(x,t) satisfying Assumption 2.1 (we employ here the observation in Remark 8.2). A direct inspection reveals that in general an optimal control of (27) must be relaxed; indeed, the Young measure whose value at each t is the probability measure which assigns equal probabilities to (1) and {-1}, is optimal. A classical approximation to this relaxed control is the ordinary control, say uh(.), which alternates between the values 1 and -1 on intervals of length h. The Prohorov distance between the optimal relaxed control and uh(.) is h. Now, Theorem 5.1 assures that the error in the cost and in the trajectories, resulting when the approximation uh(.) is used, is of order h. (This can be inspected directly in the classical version of the example, namely when f(x,t) = 1.) Next we illustrate the developments concerning the chattering parameters. In particular, the discussion serves as a motivation to the structure presented in Section 3.
Example 8.4. Let [a,b] = [0,1]and let the parameter function depend on a positive parameter E and given by P&(t) =g(W&-lt),
(28)
where g(t) is a prescribed continuous positive function, and p ( t ) : [0, 00) + lR is periodic. In particular, the parameter function oscillates, having amplitude varying with time. We are interested in the case where c is small (and its exact quantity may not even be known). Then, a nominal parameter function, say po(t), of which p d ( t ) is a relevant perturbation, does not exist. The Young measure limit captures the limit distribution of the values of the parameter function. Denote by p the probability measure which is the distribution of the values of p ( .) over one period. The Young measure limit, as E 4 0, of the functions in (28) is the map v(t) with values being probability measures on the real line EX, given by v ( t ) ( B )= p(g(t)-lB) (here r B = {rx : z E B } ) .As a particular example consider p E ( t ) = sin(&-lt)
Then the limit distribution is
(29)
80
Consider now an optimization problem based on the example, say, the scalar system minimize
ll(xz
+d ) d t
subject to
-=x
dx dt
+ sin(&-lt)u,
(31) X(O)
= 1,
with u in a bounded interval. For a fixed E the optimization problem is timevarying, yet can be solved using standard techniques. However, as E --t 0 the computations become difficult to execute due to the oscillations of the parameter function. The chattering parameters limit is given by
(we examine a simple version of (28), namely, with
where v(dr) = 7r(l-r2)
g ( t ) = 1). Notice that the chattering limit in this example is time-invariant; (the solution u = u(t,r ) may, however, depend on time).
The theory presented in Ref. 2 guarantees that the value of (32) is the limit, as E -+ 0, of the values of (31). It is easy to see that the Prohorov distance between the limit Young measure v and the rapidly oscillating parameter is ~27r.Hence, Theorem 7.1 implies that the error caused by first solving (32) and then obtaining an appropriate approximation, rather than solving (31), is of order E .
References 1. Z. Artstein, Israel J. Math. 46, 313 (1983). 2. Z. Artstein, Forum Math. 5, 369 (1993). 3. Z. Artstein, The chattering limit of singularly perturbed optimal control problems, in Proc. 39 IEEE Conf. Decision and Control (Sydney, Australia, 2000): 564-569. 4. Z. Artstein, On the value function of singularly perturbed optimal control systems, in Proc. 43 IEEE Conf. Decision and Control (Paradise Island, Bahamas, 2004): 432-437. 5. Z. Artstein and V. Gaitsgory, SIAM J. Control Optim. 35, 1487 (1997). 6. Z. Artstein and V. Gaitsgory, Appl. Math. Optim. 41, 425 (2000). 7. E.J. Balder, Rend. Istit. Mat. Univ. R e s t e 31, suppl. 1, 1 (2000).
81
8. J.M. Ball, A version of the fundamental theorem for Young measures, in Partial Differential Equations Models of Phase D-ansitions, (Nice, France, 1988), eds. M. Rascle, D. Serre and M. Slemrod, Lecture Notes in Physics, Vol. 344 (Springer, Berlin, 1989): 207-215. 9. P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, second edition (Wiley, New York, 1999). A Wiley-Interscience Publication. 10. P. Pedregal, Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and their Applications, 30 (Birkhauser Verlag, Basel, 1997). 11. P. Pedregal, SIAM J . Control Optim. 36,797 (1998). 12. P. Pedregal, Bull. Amer. Math. SOC.36,27 (1999). 13. W.T. Reid, Ordinary Differential Equations (Wiley, New York, 1971). 14. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot- Watt Symposium, Vol. I V , eds. R.J. Knops, Research Notes in Mathematics, Vol. 39 (Pitman, Boston, 1979): 136-212. 15. M. Valadier, Rend. Istit. Mat. Univ. W e s t e 26,suppl., 349 (1994). 16. J. Warga, Optimal Control of Differential and Functional Equations (Academic Press, New York, 1972). 17. L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Foreword by Wendell H. Fleming (Saunders Co., Philadelphia, 1969).
82
SYSTEMS WITH CONTINUOUS TIME AND DISCRETE TIME COMPONENTS A. BACCIOTTI
Dipartimento d i Matematica, Politecnico di Torino Corso Duca degli Abrzlzzi 24, 10129 Torino, Italy E-mail: andrea.
[email protected] We introduce a class of differenceldifferential equations which is sufficiently large to include systems of interest for applications, but at the same time sufficiently easy to handle. In this framework, we give in particular a detailed and rather complete study of the asymptotic behavior of pairs of oscillators. We finally introduce appropriate notions of stability and extend Liapunov first and second theorem.
Keywords: Hybrid systems; Stability; Liapunov functions.
1. Introduction
Motivated by a wide range of industrial and technological applications, there has been a rapid growth of interest in the recent engineering literature about hybrid systems. The term "hybrid" often is applied informally t o denote systems which combine subsystems of different nature, and whose time evolution is characterized by discontinuities in the state (impulses) and/or in the velocity (switches). A large variety of systems with very complex behavior fall in this class. For this reason, it is hard to figure an axiomatic definition of hybrid system. Recently, there have been some interesting attempts, see Refs. 2,8,10,13,15,17.However, we remark that all these definitions turn out to be extremely formal and abstract (this is the obvious price to be payed if we want to include more and more general classes of systems), and hence difficult to handle. In this paper we consider a class of systems, large enough to include, as particular cases, finite dimensional continuous time systems, discrete time systems, open loop switched systems, feedback systems with quantized control, sample data systems, certain types of delayed differential equations and certain types of hybrid systems with timed automata. They will be
83
called systems with continuous time and discrete time components. Systems with continuous time and discrete time components admit a simple mathematical representation and hence, although their generality is limited (for instance, they cannot account for impulse effects), they have the advantage of being rather concrete. On the other hand, we notice that certain aspects of systems with continuous time and discrete time components are not covered by the definitions given in Refs. 10 (where switches can be interpreted as state discontinuities but not as changes of the dynamical rules), and in Refs. 2,8,13,15 (where the state space of the discrete time component is finite). As in Refs. 10,13,17, we address the stability problem. Far from being surprising, our results are natural generalizations of Liapunov first and second theorems. Nevertheless, their proofs are not completely obvious and cannot be deduced from the existing literature. In particular, we note that our results, compared with Ref. 17, are more precise and require less conservative assumptions, in spite of a less general setting. We now shortly explain the organization of the paper. Section 2 contains the definition of system with continuous time and discrete time components, comments on its generality and further comparisons with analogous definitions available in the literature. In Section 3 we discuss in detail an example: we will see that in spite of a very simple structure, the asymptotic behavior of a system with continuous time and discrete time components may be very complex. The notions of stability and asymptotic stability are stated in Section 4. Section 5 is devoted to the extension of Liapunov first theorem: we give a proof of it and some other remarks. Two slight different extensions of Liapunov second theorem are finally presented in Section 6.
2. Description of the model We are interested in objects defined by the following set of data: an integer n 2 1; a locally compact metric space Q a continuous map f ( z , q ) : R" x Q + R"; a continuous map g(z, q ) : R" x Q 4 Q; a sequence {dk}, such that dk > 0 for each k = 0 , 1 , 2 , . . . and k limk++oo di = +m. Throughout this paper, such objects are called systems with continuous time and discrete time components (in short, CTDTC-systems), and conventionally represented by writing
84
For a n y g i v e n f e W,we set 70= f,7 1 = 70+do, 7 2 = ~ 1 + d l , .. . , ~ k + l = a solution of (1) corresponding to the initial condition (f,1,Q) E W x W" x Q, we mean any pair (cp(t),{uk}) such that:
7k
0
0 0
+ d k , . . .. By
cp(t): [f,+oo) + Rn is a curve with p(f) = 3,which is assumed t o be continuous at every t 2 f; { U k } is a sequence in Q, with uo = Q; for each k = 0 , 1 , 2 , . . . and each t E ( ~ k7 k, + 1 ) , p ( t ) is differentiable and
Uk+l
= g(Y'(%+l), u k )
.
The idea underlying this notion of solution can be intuitively described in this way. Starting from the point 3, the continuous time component evolves according to the differential equation
on the interval [ T O , T ~ while ], the discrete time component remains unchanged. At the instant 7 1 , the discrete time component is updated, according to ui = g ( ( p ( T i ) , u o ) .
Then, on the subsequent interval [ T I ,721 the continuous time component evolves according to the new equation
li: = f (z, u1) and so on. To clear up the notation, it is convenient to introduce the map
85
so that a solution can be written as a curve
t
H
(dt), ~ ( t
) [f, ) +m)
+
Rn x
Q.
Note that such a curve is not continuous, in general. Note also that in order to guarantee existence of solutions which are actually defined for each t 2 t, the continuity of f is not sufficient in general, due to the possible finite escape time phenomenon. To prevent it, we assume that the vector field f(.,q ) is complete for each q E Q (sufficient conditions for completeness are well known and can be found on the more popular handbooks about ordinary differential equations). In what follows, uniqueness of solutions plays no role at all.
Remark 2.1. The following remarks illustrate the generality of CTDTCsystems. (a) If Q reduces to a singleton, then (1) reduces to a finite dimensional, time-invariant system of ordinary differential equations. (b) If n = 0 and Q = Em,then (1)reduces to a finite dimensional, discrete time dynamical system. (c) If d k = 1 for each k = O , l , 2 , . . ., Q = a", g(x,q)= x and f = 0 , then (1) reduces to
where [t]denotes the greatest integer less than or equal to t. Note that (2) is a retarded differential equation of the type considered in Ref. 7. (d) If g ( z , q ) = g(z) : En --f Q, then (1) describes a system with a feedback connection, where the actuator is a digital device which is able to change its value only at the prescribed instants ~ 0 ~ . .~.. This 1 , situation is similar to what happens in the so-called quantized control problems, where one initially starts with a feedback g(x) which can vary continuously and free of constraints, but then the levels of quantization must be found in such a way to preserve the achievement of the control goal. (e) If d > 0 is a fixed time size and d k = d for each k = 0 , 1 , 2 , . . ., a CTDTC-system can be thought of as sample-data system^.^ (f) If Q is a finite set (endowed with the discrete metric) and g(x,q) = g ( q ) : Q --+ Q, then the sequence {uk} is independent of the evolution of x and can be computed in advance. Then (1) reduces to a switched system of the type considered in Refs. 6,16.
86
Remark 2.2. One feature of our definition of CTDTC-system is that changes in the continuous time dynamics can occur only at the prescribed instants TO,71,. . .. This feature is shared by similar notions available in the literature (see for instance Refs. 2,8,10,13).We point out that if Q is finite, our definition of CTDTC-system can be viewed as a special case of the definition of hybrid system studied in Refs. 8,13, the differences being that here the discrete state transitions are uniquely determined by a function, rather than by a relation (or by a set valued map as in Ref. l o ) , and the reset map is the identity (which implies in particular that systems with impulsive effects are not comprised in (1)).On the other hand, the generalization to sets Q which are not necessarily finite sets, enables us to include a wider range of applications, as indicated by the examples above. Remark 2.3. Functions f and g do not depend explicitly on time. Nevertheless, because of the constraint on the updating times, the translation of a solution (p(t T ) ,u ~ ( ~ + in T )general ) is no more a solution. In other words, the semigroup property does not hold. Accordingly, we should not expect that a CTDTC-system behaves as a time-invariant one.
+
3. Oscillatory systems: an example In spite of its simplicity, a CTDTC-system may exhibit very complex and unexpected behaviors. In this section we discuss with some details the system
where n = 2, Q = { w , $}, and w > 1 is a given real number. Equivalently, we can look at (3) as a switched system formed by the pair of harmonic oscillators
{
i = -wy y=;
(4)
87
3-3
-2
-1
0
I
2
Fig. 1. Some trajectories of system on the right (w = 2 )
(4) on the left and some trajectories of system
(5)
and
Some trajectories of these systems are plotted in Figure 1. Note that both (4) and ( 5 ) are stable a t the origin. For simplicity, we limit ourselves to the case where ?i = 0, d k = T (i.e., Tk = kT),T being a fixed real number, T E (0,2./r]. Moreover, without loss of generality we agree that 4 = w . It is clear and well known6 that if we choose Tk = k./r/2, then the behavior of the solutions depends on the initial condition (3,j j , Q). For instance, if (3,j j , Q) = (1,0,w ) then it is natural to guess that ( z ( t )y(t)) , converges to the origin for t -+ t o o , while if (3,ji,Q) = (0,1, w ) then ( z ( t )y, ( t ) ) becomes larger and larger as t +a. In fact, examples similar to the present one are often invoked in order to show that a switched system may exhibit features which are not recognizable in the singular subsystems. Here, our purpose is to analyze how the behavior of the system actually depends on the choices of w and T. We show in particular that for “many” values of w the system is actually stable’, and that for the remaining values of w the behavior of the solution corresponding to a fixed initial condition is extremely sensitive to the choice of T : in particular, we will see that the .--)
*Stability of systems with continuous time and discrete time components will be formally defined later; for the moment, the term “stable” is used in the obvious heuristic meaning.
occurrence of trajectories convergent to the origin is extremely rare, and practically impossible to simulate in machine experiments. We start by computing, for t = T, the fundamental matrix of system (4)
)
(
cos T -w sin T
@(4)(T)
zz
sinwT
COST
-
and the fundamental matrix of system (5)
The idea is to look at (3) as a discrete time system of R2, whose state is updated at the instants 0,2T, 4T, 6T,. . .. More precisely, we study the system
where sin2T 1 + w2 COS~ -T - -sin T cos T W2
W
@(TI= @(5)(T)@(4)(T) = +
w2
W
sinT cosT cos2T - w2 sin2 T
It is clear, and not difficult to prove, that the stability properties of (3) can be deduced from those of (6) (see [4,Ch. 81). To compute the eigenvalues of @ ( T )we , must solve the equation
pa(,(X)
= ,A2
sin2 T] X
-
whose discriminant is
W2
+ (l + w2)2 sin2 TCos2T w2
89
Note that A ( T ) = 0 only in the following two cases:
(Cl) sin2 T = 0, that is T = n or T = 2n. (C2)
0
< sin2T =
4w2
+
(1 w2)2 solutions in (O,2n).
Fig. 2.
< 1, which gives rise to exactly
4 distinct
Graph of A ( T )
The graph of A ( T ) is plotted in Figure 2 for w = 1.5. Let us examine first the case A < 0. It is not difficult t o check that in this case @ ( T has ) a pair of conjugate (distinct) eigenvalues, lying exactly on the boundary of the unit circle of the complex plane. Hence, system (3) is stable. Now, we pass to consider the case A ( T ) > 0. Here, we have real eigenvalues. It is not difficult to see that one of them is always less than -1, while the other is inside the interval (-1, 1).Hence in this case the system is unstable. More precisely, system (6) has a saddle point at the origin: the stable manifold coincides with the x-axis and the unstable manifold with the y-axis. If we assign an initial condition on the x-axis, we therefore expect that the trajectory converge toward the origin. Surprisingly, this prediction seems to be contradicted by numerical experiments: see Figure 3, where T = n/2. What actually happens is that, n being an irrational number, round off errors are inevitable in machine computations; as a consequence, it is impossible to keep a simulated trajectory inside the stable manifold
90
when k becomes larger and larger. Note that A(7r/2)> 0 for every w > 1, and that the measure of {T E ( 0 , 2 7 ~:)A ( T )> 0 } goes to zero as w -+ 1+.
Trajectory with w = 1.5,T = ~ / starting 2 from (1,O)
Fig. 3.
Finally, when A ( T ) = 0 the eigenvalues of @ ( T )coincide: they are both equal to 1 in case (CI),and equal to -1 in case (Cz).Moreover, the eigenvalue is simple in case (CI), so that the system is stable, but not in case (Cz),so that the system is not stable. l
o
.
.
.
.
.
.
.
.
.
,
I
10
8-
.e-
2-
024-
44-
0
1
-
-10
4
4
-4
-2
0
2
.
8
Fig. 4. Trajectory with w = 2.4, T = ~ / starting 4 from (1,O)
Of course, we can look at the problem from an other point of view; for instance we can fix T and take w as a parameter. Let us considered for instance the choice T = 7r/4. Our investigation reveals that with this
91
choice, the system is stable only if w and 5).
m = 2.4142. . . (see Figures 4
Remark 3.1. We know that for certain values of T there are trajectories of system (4) which diverge. Because of periodicity, the same happens of course if we replace T by T 2k7r (for each k). This shows that for simple stability we do not have an analogue of Lemma 2 in Ref. 14.
+
0-051-
-1.5
-
2-2.5
-
4. -35
Fig. 5.
"
"
'
"
Trajectory with w = 2.5, T = s/4 starting from (1,O)
4. Stability notions
Motivated by the example of the previous section, we give some definitions of stability which seem to be appropriate for CTDTC-systems. Let M be a compact subset of Q. Let us denote dM(q) = minpEMd(p, q ) , where d is the distance function of &. The euclidian norm of z E R" is denoted by 1 ~ 1 Moreover, ~ ~ . we set E = R"x Q. For (2, q ) E El let us denote N ( z ,q ) = max(lz1p , d M ( q ) } . We also denote B R( T~) = {z : I z ( ~ n < T}, B Q ( T= ) { q : djw(q) < T } and B E ( T = ) { ( z , q ) : N ( z , q ) < T } , where T > 0. The subscripts R", Q and E will be dropped out, when there is no risk of ambiguity. From now on, we assume (A1)f ( 0 , p ) = 0
(A2) g ( 0 , M )
for each p E M
cM
that is, the origin of R" is an equilibrium point for the continuous time component of the system for each p E MI and M is a positively invariant set for the discrete time component, when z = 0.
92
Definition 4.1. A CTDTC-system is uniformly stable with respect to (0) x M if for each E > 0 there exists 77 > 0 such that
a)
N ( 3 , < 77 ==+ N(cp(t)lU h ( t ) ) < E for each ?i E R, each t 2 f and each solution (cp(t),uh(t)) corresponding to the initial condition (i?, 3,Q). A CTDTC-system is uniformly-uniformly stable with respect to (0) x M if it is uniformly stable for each choice of the sequence { d k } . Definition 4.2. A CTDTC-system is (locally) uniformly-uniformly asymptotically stable with respect to (0) x M if it is uniformly-uniformly stable and, in addition, there exists SO > 0 such that for each 5 E R N ( z , Q )< 60
* t++m lim N(cp(t),uh(t)) =0
for each solution (cp(t),uh(t)) corresponding to the initial condition
(f,3,Q).
We emphasize that the definitions above depend on the choice of the origin, as a special steady state of R", and of the set M C Q: in what follows, we omit t o mention them explicitly for the sake of simplicity, since no ambiguity is possible. We remark also that Definitions 4.1 and 4.2 could be referred t o a more general set Mo x M , where MO is a compact subsets of R" not reduced to the origin. Again, the origin has been chosen to simplify the exposition. Instead, as far as the discrete dynamics are concerned, an analogous simplification is not convenient. The reason is that in some applications Q might be a finite set with no distinguished elements. If in addition Q is endowed with the discrete topology, then by ( A 2 ) , M plays no role at all in checking stability: in particular, when M is a singleton, the problem becomes trivial. Note that if we are interested in stability of the continuous time component alone, we can take M = Q and look at the discrete time component as a stabilizing device.
5. A sufficient condition for stability The following result is the natural extension of Liapunov First Theorem to nonlinear CTDTC-systems. With respect to the well known classical case, the interplay between the continuous time dynamics and the discrete time one requires some more care in the proof.
Theorem 5.1. Let the CTDTC-system (1) be given. Assume that there exists r > 0 and a continuous map V : BE(r) -+ R such that:
93
(i) V(z, q ) is positive definite a t ( 0 ) x M , that is V ( x ,q ) 2 0 and V(x, q ) = 0 for each ( z , q ) E B E ( T )implies ( z , q ) E (0) x M ; (ii) for each q E B Q ( T )the , m a p z ++ V(z, q ) i s of class C1 o n B p ( r ) ; (iii) V,V(z, q ) f (z, 4 ) I 0, for each (2, q ) E B E ( T ) ; (2.) V(z, g(z, 4 ) ) I V ( z ,q ) , for each (z, 4 ) E B E ( T ) . T h e n , the s y s t e m i s uniformly-uniformly stable. Proof. Let 0 < R < r so that V is defined and continuous on B E ( R ) .Let r n= ~ i n f N ( t , y ) = ~ V ( z , q It ) . is clear that the set ( ( z , q ) : N ( z , q ) = R} is compact. Hence, r n is~ actually a minimum and, by (i), mR > 0. The set
a = { ( z , q ) : V(z,q) < mR) is open, and (0) x M c 0. Let Ro be the connected component (i.e., the largest connected subset) of R which contains (0) x M . Of course, Ro c B E ( R ) .Let us consider the continuous map j ( x , q ) = (2, g(z, q ) ) : E E. We claim that g(C20) c 00.Indeed, from ( z , q ) E 520 and (iv) it follows --f
V ( ~ g(zcl4)) C, 5
v(z,q ) < mR
which in turn implies g(z, q ) E R. Moreover, if p E M , then by virtue of (A2) we have g ( 0 , p ) = (O,g(O,p)) E (0) x M ; on the other hand, O ( 0 , p ) E O(Ro), since (0) x M c 00.This means that
The claim is proven, since the continuous image ~ ( R o of ) the connected set Ro is connected. Pick now E > 0 such that B E ( &c) Ro c B E ( R ) .Let
Again, we have that m, is a minimum and m, > 0. Let 6 > 0 such that
N h q ) < (5 ===+ V(2,q) < m, . Of course, 6 < E . Let t E IR, ( 2 , ~E) BE(^), and let (cp(t),uh(t)) be any solution of (1) such that ((P(To),uo) = ( 2 , Q ) . We want to prove that
94
(cp(t),uh(t)) 6 B E ( &for ) each t 2 T O . To this purpose, using the mathematical induction principle, we show that the statement ( c p ( t ) , U h ( t ) )E
B E ( & ) for each t E
[7k,Tk+l)
is true for each k = 0 , 1 , 2 , . . .. We proceed according to the following pattern. First step (it = 0). Using the fact that V ( 3 ,Q) < m,, we prove that
((P(TI),UI)E
> 0). Assuming that < m,, we prove that
Inductive step (k V ( c p ( ~ k )u,k )
B E ( & ) and V ( ( P ( T I ) ,
(8)
B E ( & )and
Proof of the first step. Let T E (TO, 711 be such that (cp(t),U O ) E B E ( &for ) each t E [TO,TI. Because of (iii), we obviously have
V(cp(t),uo)5 V ( Z , C ) < m,
(11)
for each t E [TO,T]. Since cp(t) is continuous, using (11) and arguing by contradiction, it is immediate to check the validity of (7) (note in particular that since d ~ ( u 0<) E , Icp(T0)l= E for some TOimplies N(cp(To),uo)= E ) . In fact, we have V ( ( P ( T ~ ) < , Um,. O) From ( ( P ( T ~ ) , u oE) B E ( & c ) 0 0 it follows ( ( P ( T ~ ) , uE~ )00.Next, by (iv)
95
This in turn implies that validity of (8) is so proven.
( ( P ( T ~ ) , ucannot ~)
belong to Ro \ B E ( & )The .
Proof of the inductive step. Taking into account the inductive assumption, the proof that (cp(t),u k ) remains in B E ( &for ) t E [ ~ kT,~ + I can ] be carried out as in the case k = 0. In particular, we can conclude that
and v(cp(Tk+l),uk) < m&.
(13)
From (12) it follows that ( ( P ( T ~ + I ) , U ~ +E~Ro, ) and from (13) and (iv) it follows V ( ( P ( T ~uk+l) + ~ )<, me. Hence, ( ( P ( T ~ + I )u,k + l ) cannot belong to a 0 \ B E (&)* The statement is proven, taking into account that no special role is played in the proof by the sequence { d k } . 0
Remark 5.1.
Conditions (iii), (iv) imply that for each solution
(cp(t),u h ( t ) )the , map y ( t ) = V(cp(t), ~ h ( ~is)nonincreasing, ) at least as far as (cp(t),u h ( t ) )remains in B E ( T ) .This fact will be used later, in the proofs of Theorems 6.1 and 6.2. However, it does not allow us to obtain a simpler proof of Theorem 5.1 (on the contrary of what happens for the classical Liapunov first theorem) since in general (cp(t), u h ( t ) )is not continuous.
Remark 5.2. Basically, stability is a local notion. It is transformed in a global one when combined with Lagrange stability.' A corresponding global version of Theorem 5.1 can be stated under the additional assumption
(v) for each q E Q , the function x and radially unbounded.
H
V(x,q )
is defined for each x E Rn
It is worthwhile to mention that assumption (v) is actually made in Ref. 17 to obtain a merely local result: it also enables the authors to bypass the more involved aspects of the proof (due to the co-existence of discrete and continuous dynamics) and all become simpler. As in the classical Liapunov theory, the assumption about the differentiability of x H V(x, u)can be weakened; in fact, it is sufficient to ask that it is lower semicontinuous; accordingly, an appropriate notion of generalized derivative must be used in (ii) (see Ref. 3).
96
In a topological context, even the monotonicity condition (ii) can be relaxed. l7 We finally point out that, by allowing time-varying Liapunov functions, a converse of Theorem 5 . 1 has been obtained in Ref. 17.
Remark 5.3. We can interpret V(z, q) as a family of Liapunov functions indexed by q E Q, and condition (iv) as a kind of compatibility condition for multiple Liapunov f u n c t i o n ~ . ~ To~ this ~ * respect, ~~>~~ we point out that the condition imposed in Refs. 5,6,11,12 ore formally more general but they require an explicit knowledge of the solutions. Our condition is more conservative, but easier to apply in practice. Remark 5.4. For the case where Q is finite, conditions (iii) and (iv) of Theorem 5.1 are the same as the conditions 1) and 2) of Theorem IV.1 of Ref. 13. To this respect, we remark that in Ref. 13 the authors are mainly interested in attractivity of certain compact sets (generalization of LaSalle invariance principle); in a sense, our Theorem 5 . 1 complements Theorem IV.l of Ref. 13, showing that the selected compact set is not only attractive, but also stable. Example 5.1. We can apply Theorem 5.1 in order to prove stability for the following system:
p
=
- ( 1 + q2)y
where n = 2, Q = JR and M = ( 0 ) . The continuous time component of this system can be viewed as a family of harmonic oscillators (however, we notice that the eigenvalues here are different from those of example treated in Section 3). Define V(z, y , q ) = z2 (1 q2)y2 q2. It is not difficult to check directly then the assumptions of Theorem 5 . 1 are satisfied (a trajectory is shown in Figure 6).
+ +
+
6. Sufficient conditions for asymptotic stability
In order to obtain sufficient conditions for asymptotic stability, we need strengthened forms of conditions (ii), (iii) and (iv). Recall that a function a : [O,ro) -+ [0, +m) (where TO is some positive real number possibly dependent on a ) is of class Ic if it is continuous, strictly increasing and
97
1-
05-
0-
4 5 .
I -
-1 51
-I
-1.5
Fig. 6.
-0.5
a5
0
Trajectory of system ( 1 4 ) with
I
1
dk E 1r/2
1.5
starting from (0,1,1)
such that a(0) = 0. We are able to prove two theorems under alternative assumptions.
Theorem 6.1. Let a CTDTC-system (1) be given, and assume that (a), (ii), (iv) hold. Assume further that:
(ii’) for each q E B Q ( ~ the ) , map x H V ( x , q )is of class C1 on B ~ n ( r ) moreover, the function W(xlq ) = V,V(x, q ) f ( x ,q ) = is continuous on BE(r); (iii’) W ( x ,q ) is negative definite i.e., W ( x ,q ) 5 0 for each (xlq ) E B E ( r ) and W(xlq) = 0 if and only if ( x , q ) E (0) x M . Then, the CTDTC-system is uniformly-uniformly asymptotically stable. Proof. According to Theorem 5.1, the system is uniformly-uniformly stable; thus, for any positive fixed number ro < r we can find 60 > 0 such that for each triple (f,1,Q) and each solution (cp(t),qt)) such that cp(0 = 3, uo = one has
al
a) <
N(3,
60
===+N(cp(t),% ( t ) )
< TO
for each t L f. Let let y ( t ) = V ( c p ( t ) , ~ h (Because ~ ) ) . of (iii), (iv), y ( t ) is nonincreasing on [fl+a), so that lim y ( t ) = L
t++m
2O
98
and y ( t ) 2 L for each t 2 f. Assume by contradiction that L > 0. Using (i), we can find p E (0,So) such that V(z,q) < L for N ( x , q ) < p. Of course, we must have CL I N(cp(t),U h ( t ) ) I ro
for each t 2 .?i Let now c = sup,IN(z,q)lro W ( z ,q). By (v), c is actually a maximum and c < 0. Now,
+
/
t
W(cp(s), U h ( t ) )ds < Y(Q
+ ct
.
Th(t)
This would imply limt--r+ooy(t) = -00, which is impossible since y ( t ) 2 L > 0. 0 Theorem 6.2. Let a CTDTC-system (1) be given, and assume that (i), (ii), (iv) hold. Assume further that:
(ivy there exists a map p E K: such that for each ( x , q ) E BE(r) V(z, dx,4 ) )
i V(x, 4 ) - P(N(z,4 )
*
Then, the CTDTC-system is uniformly-uniformly asymptotically stable. Proof. Let y ( t ) and L 2 0 be as in the first part of the proof of Theorem 6.1. Conditions (iii), (iv) imply that
Y(%)
= V((P(Ti),Ui) L V((P(Ti+l),%) L V((P(7i+l),Ui+l) = Y(Ti+l) (15)
for each i = 0 , 1 , 2 , . ... This yields limi--t+ooV(p(Ti+l),ui) = limi,+, V ( ( P ( T ~ +ui+l) ~ ) , = L. Now, using (iv’) we obtain
It follows limi++, p ( N ( ( ~ ( ~ i + l ) , u i ) ) = 0 and hence limi++, N ( ( P ( T ~ + I ) = , u 0. ~ )Using again (15) and the continuity of V, we
99
conclude t h a t also limi.++m V ( ~ ( T ~ + I ) ,= Ulim++m ~ + I ) y(q+l) = L = 0. Recalling that y ( t ) is nonincreasing, we finally infer limt--t+my(t) = 0. It is now easy to conclude t h e proof. 0
References 1. V. Andriano, A. Bacciotti and G. Beccari, Nonlinear Anal. 28,1167 (1997). 2. Z. Artstein, Examples of stabilization with hybrid feedback, in Hybrid Systems 111: Verification and Control, eds. R. Alur, T.A. Henzinger and E.D. Sontag, Lecture Notes in Computer Science, Vol. 1066 (Springer Verlag, Berlin, 1996): 173-185. 3. A. Bacciotti and L. Rosier, Liapunov Functions and Stability i n Control Theory, Communications and Control Engineering Series, second edition (Springer Verlag, London, 2005). 4. A. Bacciotti, Analisi della stabiliti, Quaderni dell’unione Matematica Italiana, N. 49 (Pitagora Editrice, Bologna, 2006). In Italian. 5. A. Bacciotti and L. Mazzi, Nonlinear Anal. 67,2167 (2007). 6. M. Branicky, IEEE Trans. Automat. Control 43,475 (1998). 7. K. Cooke and J. Wiener, A survey of differential equations with piecewise continuous arguments, in Delay Differential Equations and Dynamical Systems, eds. S . Busemberg and M. Martelli, Lecture Notes in Mathematics, Vol. 1475 (Springer Verlag, Berlin, 1991): 1-15. 8. E. De Santis, M. Di Benedetto and L. Berardi, IEEE Trans. Automat. Control 49,184 (2004). 9. B. Francis and T. Georgiou, IEEE Trans. Automat. Control 33,820 (1988). 10. R. Goebel, J. Hespanha, A. Teel, C. Cai and R. Sanfelice, Hybrid systems: Generalized solutions and robust stability, in Proceedings IFA C-NOLCOS (Stuttgart, 2004): 1-12. 11. D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications (Birkhauser, Boston, 2003). 12. D. Liberzon and A. Morse, IEEE Control Syst. Mag. 19,59 (1999). 13. J. Lygeros, K . Johansson, S. SimiC, J. Zhang and S.S. Sastry, IEEE Trans. Automat. Control 48,2 (2003). 14. A. Morse, IEEE Trans. Automat. Control 41,1413 (1996). 15. H. Schumacher and A. van der Schaft, A n Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Inform. Sci., Vol. 251 (Springer Verlag, 2000). 16. H. Ye, A. Michel and L. Hou, Stability analysis of switched systems, in Proceedings of the 35th Conference on Decision and Control (Kobe, Japan, 1996): 1208-1212. 17. H. Ye, A. Michel and L. Hou, IEEE Trans. Automat. Control 43,461 (1998).
100
A REVIEW ON STABILITY OF SWITCHED SYSTEMS FOR ARBITRARY SWITCHINGS U. BOSCAIN’ SISSA-ISAS, via Beirut 2-4, 34014 Theste, Italy and Le24 CNRS, UniversitC de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France, E-mail: boscain4sissa.it In this paper we review some recent results on stability of multilinear switched systems, under arbitrary switchings. An open problems is stated Keywords: Switched systems, Asymptotic stability, Lyapunov functions
1. Introduction
By a switched system, we mean a family of continuous-time dynamical systems and a rule that determines at each time which dynamical system is responsible of the time evolution. More precisely, let {fu : u E U } (where U is a subset of Rm) be a finite or infinite set of sufficiently regular vector fields on a manifold M , and consider the family of dynamical systems,
The rule is given by assigning the so-called switching function, i.e., a function u(.): [O,oo[--, U c R”. Here, we consider the situation in which the switching function is not known a priori and represents some phenomenon (e.g., a disturbance) that is not possible to control. Therefore, the dynamics defined in (1) also fits into the framework of uncertain systems (cf. for instance Ref. 10). In the sequel, we use the notations u E U t o label a fixed individual system and u(.) to indicate the switching function. These kind of systems are sometimes called “n-modal systems”, “dynamical polysystems”, “input systems”. The term “switched system” is often reserved to situations in which the switching function u(.) is piecewise continuous *This research has been financed by “Rkgion Bourgogne” , via a project FABER.
101
or the set U is finite. For the purpose of this paper, we only require u(.) to be a measurable function. When all the vector fields f,, u E U are linear, the switched system is called multilinear. For a discussion of various issues related to switched systems, we refer the reader to Refs. 6,12,14,16. A typical problem for switched systems goes as follows. Assume that, for every fixed u E U ,the dynamical system j: = .f,(x) satisfies a given property (P). Then one can investigate conditions under which property (P) still holds for j: = f,(t)(x), where u(.)is an arbitrary switching function. In this paper, we focus on multilinear systems and (P) is the asymptotic stability property. The structure of the paper is the following. In Section 2 we give the definitions of stability we need, we state the stability problem and we recall some sufficient conditions for stability in dimension n due to Agrachev, Hespanha, Liberzon and M ~ r s eIn. Section ~ ~ ~ 3 ~we~discuss ~ the problem of existence of Lyapunov functions in certain functional classes (in particular in the class of polynomials). This problem was first studied by Molchanov and P y a t n i t ~ k i i . ' ~ -More ~ l recently new results were obtained by Dayawansa, Martin, Blanchini, Mianil5@~l2 and in Ref. 18 in collaboration with Chitour and Mason. In Section 4, we discuss the necessary and sufficient conditions for asymptotic stability for bidimensional single input systems (bilinear systems) that were found in Ref. 7 (see also Ref. 18) and, in collaboration with Balde, in Ref. 4. In Section 5 we state an open problem. 2. General properties of multilinear systems
By a multilinear switched systems (that more often are simply called switched linear systems) we mean a system of the form,
x E R", {A,},ELI C (2) where U c Rm is a compact set, u(.) : [0, m[-+U is a (measurable) switching function, and the map u A, is continuous (so that {A,},Eu is a j : ( t )= A,(t)z(t),
-
compact set of matrices). For these systems, the problem of asymptotic stability of the origin, uniformly with respect to switching functions was investigated, in Refsw. 1,4,7,12,15. A particular interesting class is the one of bilinear (or single-input) systems,
+
j:(t)= u(t)Ax(t) (1 - u ( t ) ) B x ( t ) ,z E R",
A, B
E R"'".
Here the set U is equivalently [ O , 1 ] or { O , 1 } (see Remark 2.2 below). Let us state the notions of stability that are used in the following.
(3)
102
Definition 2.1. For 6 > 0 let Bg be the unit ball of radius 6, centered in the origin. Denote by U the set of measurable functions defined on [0,00[ and taking values on the compact set U . Given xo E R", we denote by T ~ ~ , ~ ( , the ) ( . )trajectory of (2) based in zo and corresponding to the control u(.). The accessible set from 20, denoted by d(zo),is 4x0) =
U,(.)ELISUPP(Y,,,~(.)(.)).
We say that the system (2) is 0
0
0
unbounded if there exist zo E Rn and u(.)E U such that ~ ~ ~ , ~ ( , ) ( goes to infinity as t -+ 00; uniformly stable if, for every E > 0, there exists 6 > 0 such that d(x0) c B, for every zoE Bg; globally uniformly asymptotically stable (GUAS, for short) if it is uniformly stable and globally uniformly attractive] i.e., for every 61,62 > 0, there exists T > 0 such that yzo,,(,)(T) E Bbl for every u(.) E U and every $0 E B62;
In this paper we focus on the following problem.
Problem 1 For the system (2) (resp. for the system (3)), find under which conditions on the compact set { A U } u L E(resp. ~ on the pair ( A , B ) )the system is GUAS. Next we always assume the following hypothesis,
(HO) for the system (2) (resp. for the system (3)) all the matrices of the compact set { A u } u E ~(resp. A, and B ) have eigenvalues with strictly negative real part (Hurwitz in the following), otherwise Problem 1 is trivial.
Remark 2.1. Under our hypotheses (multilinearity and compactness) there are many notions of stability equivalent t o the ones of Definition 2.1. More precisely since the system is multilinear] local and global notions of stability are equivalent. Moreover, since { A u } u E ~is compact, all notions of stability are automatically uniform with respect to switching functions (see for instance Ref. 3). Finally, thanks to the multilinearity, the GUAS property is equivalent t o the more often quoted property of GUES (global exponential stability, uniform with respect to switching) see for example Ref. 2 and references therein.
103
Remark 2.2. Whether systems of type (3) are GUAS or not is independent on the specific choice U = [0, 11 or U = (0, l}. In fact, this is a particular instance of a more general result stating that the stability properties of systems (2) depend only on the convex hull of the set { A u } u Esee ~ ; for instance Ref. 18. Example One can find many examples of systems such that each element of the family { A u } u Eis~GUAS, while the switched system is not. Consider for instance a bidimensional system of type (3) where,
Notice that both A and B are Hurwitz. However it is easy to build trajectories of the switched systems that are unbounded, as shown in Figure 1.
Integral curve of Bx
Integral curve of Ax
rajectory of the switched system going to infinity Fig. 1. For a system of type (3), where A and B are given by formula (4),this picture shows an integral curve of Ax, an integral curve of Bx,and a trajectory of the switched system, that is unbounded.
104
Let us recall some results about stability of systems of type (2), subject to (HO). In Refs. 1,15 it is shown that the structure of the Lie algebra generated by the matrices A,, g = {A, : u E U}L.A.,
is crucial for the stability of the system (2). For instance one can easily
prove that if all the matrices of the family { A , } , E ~ commute, then the switched system is GUAS. The main result of Ref. 15 is the following. Theorem 2.1. (Hespanha, Morse, Liberzon)
If g is a solvable Lie
algebra, then the switched system (2) is GUAS. In Ref. 1 a generalization was given. Let g = r K s be the Levi decomposition of g in its radical (i.e., the maximal solvable ideal of g ) and a semi-simple sub-algebra, where the symbol K indicates the semidirect sum. Theorem 2.2. (Agrachev, Liberzon) the switched system (2) is GUAS.
If s is a compact Lie algebra then
Theorem 2.2 contains Theorem 2.1 as a special case. Anyway the converse of Theorem 2.2 is not true in general: if s is non compact, the system can be stable or unstable. This case was also investigated. In particular, if g has dimension at most 4 as Lie algebra, Agrachev and Liberzon were able to reduce the problem of the asymptotic stability of the system (2) to the problem of the asymptotic stability of an auxiliary bidimensional system. We refer the reader to Ref. 1 for details. For this reason the bidimensional problem assumes particularly interest and, for the single input case, was solved in Ref. 7 (see also Ref. 18) and, in collaboration with Balde, in Ref. 4. Before stating the stability conditions found in Refs. 4,7,18, let us recall the advantages and disadvantage of the most used method to check stability, i.e. the method of common Lyapunov functions. 3. Common Lyapunov functions
For a system (2), it is well known that the GUAS property is a consequence of the existence of a common Lyapunov function.
-
Definition 3.1. A common Lyapunov function (LF, for short) V : Rn Itf,for a switched system ( S )of the type (2), is a continuous function such
105
that V is positive definite (i.e. V ( x ) > 0, Vx # 0, V ( 0 ) = 0) and V is strictly decreasing along nonconstant trajectories of (S). Vice-versa, it is known that, given a GUAS system of the type (2) subject to (HO), it is always possible to build a C““ common Lyapunov function (see for instance Refs. 12,19-21 and the bibliographical note in Refs. 14). Clearly is much more natural to use LFs to prove that a given system is GUAS (one has just t o find one LF), than to prove that a system is unstable (i.e. proving that a LF does not exist). Indeed this is the reason why, usually, is much more easy t o find (nontrivial) sufficient conditions for GUAS than necessary conditions. (Notice that all stability results given in the previous section in terms of the Lie algebra g, are, in fact, sufficient conditions for GUAS.) Indeed the concept of LF is useful for practical purposes when one can prove that, for a certain class of systems, if a LF exists, then it is possible to find one of a certain type and possibly as simple as possible (e.g. polynomial with a bound on the degree, piecewise quadratic etc.). Typically one would like to work with a class of functions identified by a finite number of parameters. Once such a class of functions is identified, then in order to verify GUAS, one could use numerical algorithms to check (by varying the parameters) whether a LF exists (in which case the system is GUAS) or not (meaning that the system is not GUAS). The idea of identifying a class of function where to look for a LF (i.e. sufficient t o check GUAS), was first formalized by Blanchini and Miani in Ref. 6. They called such a class a “universal class of Lyapunov functions.” For instance, a remarkable result for a given class C of systems of type (2) (for instance the class of single input system (3), in dimension n ) could be the following.
Claim: there exists a positive integer m (depending on n) such that, whenever a system of C admits a LF, then it admits one that is polynomial of degree less than or equal to m. In other words, the class of polynomials of degree a t most m is sufficient to check GUAS (i.e. the class of polynomials of degree a t most m is universal, in the language of Blanchini and Miani). The problem of proving if this claim is true or false attracted some attention from the community. For single input bidimensional systems of type (3), with n = 2, Shorten and K. Narendra provided in Ref. 23 a necessary and sufficient condition
106
on the pair (A,B ) for the existence of a quadratic LF, but it is known (see for instance Ref. 11,12,22,24) that there exist GUAS bilinear bidimensional systems not admitting a quadratic LF. See for instance the paper by Dayawansa and Martin12 for a nice example. In Ref. 12, for systems of type (3), Dayawansa and Martin assumed that the Claim above is true and posed the problem of finding the minimum m. More precisely, the problem posed by Dayawansa and Martin is the following:
Problem 2 (Dayawansa and Martin): Let E be the set of systems of type (3) in dimension n satisfying (HO). Define EGUASc E as the set of GUAS systems. Find the minimal integer m such that every system of EGUASadmits a polynomial LF of degree less or equal than m. Remark 3.1. In the problem posed by Dayawansa and Martin, it is implicitly assumed that a GUAS system always admits a polynomial common Lyapunov function. This fact was first proved by Molchanov and Pyatnitskii in Refs. 19,20, for systems of type (2), under the assumption that the set { A u } u Eis~of the form { ( a i j ) i , j = ~ , , ,:, ~a; 5 aij 5 a t } . In Ref. 21 Molchanov and Pyatnitskii state the result, with no further details, under the more general hypothesis { A u } u Ejust ~ compact. In the case in which the convex hull of { A u } u E is ~finitely generated, the existence of a polynomial common Lyapunov function for GUAS systems was proved by Blanchini and Miani,5>6in the context of uncertain systems. In Ref. 18, in collaboration with Chitour and Mason, a simple proof for a set { A u } u Esatisfying ~ the weaker hypothesis that its convex hull is compact, (without necessarily requiring U to be compact) is provided. The core of the paper18 consists of showing that the Problem of Dayawansa and Martin does not have a solution, i.e. the minimum degree of a polynomial LF cannot be uniformly bounded over the set of all GUAS systems of the form (3). More precisely, we have the following:
Theorem 3.1. If (A,B ) is a pair of n x n real matrices giving rise to a system of EGUAS, let m(A,B ) be the minimum value of the degree of any polynomial LF associated to that system. Then m(A,B ) cannot be bounded uniformly over - G U A S . In other words the set of polynomials of fixed degree is not a universal class of Lyapunov functions. Finding which is the right functional class where t o look for LFs is indeed a very difficult task. Sometimes, it is even easier to
107
prove directly that a system is GUAS or unstable. Indeed this is the case for bidimensional single-input switched systems. 4. Two-dimensional bilinear systems
In Ref. 7 (see also Ref. 18) and Ref. 4, we studied conditions on A and B for the following property to be true:
( P ) The switched system given by k(t) = u(t)AZ(t)
+ (1- u(t))Bx(t),
where x E R2, A, B E R2”,
u(.) : [0, oo[+ (0, l},
(5)
is GUES at the origin. The idea (coming from optimal control, see for instance Ref. 9) is that many information on the stability of (5) are contained in the set 2 where the two vector fields Ax and Bx are linearly dependent. This set is the set of zeros of the function Q ( x ) := det(Ax, Bx).Since Q is a quadratic form, we have the following cases (depicted in Figure 2):
A. 2 = (0) (i.e., Q is positive or negative definite). In this case the vector fields preserve always the same orientation and the system is GUAS. This fact can be proved in several way (for instance building a common quadratic Lyapunov function) and it is true in much more generality (even for nonlinear systems, see the paper Ref. 8, in collaboration with Charlot and Sigalotti). B. 2 is the union of two noncoinciding straight lines passing through the origin (i.e., Q is sign indefinite). Take a point x E 2 \ (0). We say that 2 is direct (respectively, inverse) if Ax and Bx have the same (respectively, opposite) versus. One can prove that this definition is independent of the choice of x on 2. Then we have the two subcases: B1. 2 is inverse. In this case one can prove that there exists uo E]O,1[ such that the matrix uoAx (1 - uo)Bx has an eigenvalue with positive real part. In this case the system is unbounded since it is possible to build a trajectory of the convexified system going to infinity with constant control. (This type of instability is called static instability.) B2. 2 is direct. In this case one can reduce the problem of the stability of (5) to the problem of the stability of a single trajectory called worst-trajectory. Fixed xo E R2 \ {0}, the worst-trajectory yzois
+
108
Case B1
Case A
7t
Z
(inverse)
1
unbounded
GUAS
Case B2
Z (direct)
GUAS
unbounded
Case C l
Z-
GUAS
p -Y
Case C2
Z-
7P
(inverse)
(direct)
uniformly stable but not GUAS
GUAS
Fig. 2.
the trajectory of (5), based at ZO, and having the following property. At each time t , qzo(t)forms the smallest angle (in absolute value) with the (exiting) radial direction (see Figure 3). Clearly the worst-trajectory switches among the two vector fields on the set 2. If it does not rotate around the origin (i.e., if it crosses the set 2 a finite number of times) then the system is GUAS. On the other side, if it rotates around the origin, the system is GUAS if and only if after one turn the distance from the origin is decreased. (see Figure 2, Case B2). If after one turn the distance from the origin is increased then the system is unbounded (in this case, since there are no trajectories of the convexified system going
109
to infinity with constant control, we call this instability dynamic instability).If yzois periodic then the system is uniformly stable, but not GUAS.
smallest angle
z
worst-trajectory
Fig. 3.
C . In the degenerate case in which the two straight lines of Z coincide (i.e., when Q is sign semi-definite), one see that the system is GUAS (resp. uniformly stable, but not GUAS) if and only if 2 is direct (resp. inverse). We call these cases respectively C2 and C1. A consequence of these ideas is that the stability properties of the system
( 5 ) , depend only on the shape of the integral curves of Az and B x and not on the way in which they are parameterized. More precisely we have:
+
Lemma 4.1. I f the switched system i = u(t)Aa: (1 - u ( t ) ) B x , u(.): [0,co[+ (0, l}, has one of the stability properties given in Definition 2.1, then the same stability property holds for the system i = u(t)(A/aA)x+ (1 - u ( t ) ) ( B / a ~ ) afor : , every C X A , ~ > B 0. The main point to get stability conditions is t o translate the ideas above in terms of coordinate invariant parameters. To this purpose one have first t o find good normal forms for the two matrices A and B , in which these parameters appear explicitly. We treat separately the case in which the two matrices are diagonalizable (called diagonalizable case) and the case in which one or both are not (called nondiagonalizable case).
110
4.1. The diagonalisable case In the case in which both A and B are diagonalizable, we assume HI: Let X 1 , & (resp., X 3 , X 4 ) be the eigenvalues of A (resp., B). Then Re(X1), Re(X2), Re(X3), Re(X4) < 0. H2: [A,B ] # 0 (that implies that neither A nor B is proportional to the identity). H3: A and B are diagonalizable in @. (Notice that if (H2) and (H3) hold, then XI # h , A3 # X4.) H4: Let V1,Va E @P1(resp., V3,V4 E CP') be the eigenvectors of A (resp., B). Then Vi # Vj for i E {1,2}, j E {3,4}. (Notice that, from (H2) and (H3), the are uniquely defined, V1 # Vz and V3 # V4, and (H4) can be violated only when both A and B have real eigenvalues.) Condition ( H l ) is just the condition that A and B are Hurwitz (cf. condition (HO) in Section 2). Condition (H2) is required otherwise the system is GUAS as a consequence of Theorem 2.1. The case in which ( H l ) and (H2) hold but (H3) does not is treated in the next section. The case in which ( H l ) , (H2) and (H3) hold but (H4) does not can be treated with arguments similar to those of Ref. 7, and it possible to conclude that ( P ) is true. Theorem 4.1 below, gives necessary and sufficient conditions for the stability of the system (5) in terms of three (coordinates invariant) parameters given in Definition 4.1 below. The first two parameters, P A and p ~ depend , on the eigenvalues of A and B , respectively, and the third parameter K depends on T r ( A B ) ,which is a Killing-type pseudoscalar product in the space of 2 x 2 matrices. As explained in Ref. 7, the parameter K contains the interrelation between the two systems x = Ax and x = Bx, and it has a precise geometric meaning. It is in 1 : 1 correspondence with the cross ratio of the four points in the projective line @P1 that corresponds to the four eigenvectors of A and B . For more details, see Ref. 7.
Definition 4.1. Let A and B be two 2 x 2 real matrices and suppose that ( H l ) , (H2), (H3), and (H4) hold. Moreover, choose the labels (1) and ( 2 ) (resp., (3) and (4)) so that IX21 > 1x11 (resp., IX41 > IX31) if they are real or Im(X2) < 0 (resp., Im(X4) < 0) if they are complex. Define
111
. A3 + A4
P B := - a p .
A3
K
:= 2
- Aq ’
Tr(AB)- $ T r ( A ) T r ( B )
(A1 - A2)(A3 - A 4 ) Moreover, define the following function of P A , P B , K:
D := K 2
+ 2 P A P B K - (1 + p i + p i ) .
Notice that P A is a positive real number if and only if A has nonreal eigenvalues and P A E iR, p A / i > 1 if and only if A has real eigenvalues. The same holds for B . Moreover, 2) E R.
4.1.1. Normal forms in the diagonalizable case Under hypotheses ( H l ) to (H4), using a suitable 3-parameter changes of coordinates, it is always possible to put the matrices A and B , up the their norm, in the normal forms given in the following Proposition, where PA,PB,Kappear explicitly. (See Ref. 7 and Ref. 18 for the proof.) We will call, respectively, (CC) the case where both matrices have nonreal eigenvalues, (RR) the case where both matrices have real eigenvalues, and (RC) the case where one matrix has real eigenvalues and the other nonreal eigenvalues.
Proposition 4.1. Let A , B be two 2 x 2 real matrices satisfying conditions ( H I ) , (HZ), (HS), and (H4) . I n the case in which one of the two matrices has real and the other nonreal eigenvalues (i.e., the (RC) case), assume that A is the one having real eigenvalues. Then there exists a 3parameter change of coordinates and two constant Q A , C Y B > 0 such that the matrices A/aA and B/aB (still denoted below by A and B ) are an the following normal forms. Case in which A and B have both nonreal eigenvalues ((CC) case):
+
where P A , p~ > 0, IEl > 1. I n this case, K = ; ( E &). Moreover, the eigenvalues of A and B are, respectively, - P A fi and -PB f i.
112
Case in which A has real and B nonreal eigenvalues ((RC) case):
0
-pA/i-
1)
’
B=
( - dp ~ - Km/ i -JCF +K/i) ’ -PB
where P B > 0 , p A / i > 1, K E ilw. In this case, the eigenvalues of A and B are, respectively, - p A / i f 1 and -PB f i. Case in which A and I3 have both real eigenvalues ((RR) case):
where p A / i , P B / i > 1 and K E IW \ { f l } . In this case, the eigenvalues of A and B are, respectively, -PA12 f 1 and - P B / i f 1 . Using these normal forms, following the ideas presented at the beginning of this section, one gets the following stability conditions (see Ref. 7’18 for the proof).
4.1.2. Stability conditions in the diagonalizable case Theorem 4.1. Let A and B be two real matrices such that ( H l ) , (H2), (H3 ) , and (H4), given in section 4, hold and define , o A , ~ BK, , V as in Definition 4.1. W e have the following stability conditions. Case (CC) If A and B have both complex eigenvalues, then
V < 0 , then ( P ) i s true; V > 0, then Case (CC.2.1). if K < -1, then ( P ) is false; Case (CC.2.2). if K > 1, then ( P ) is true if and only if it holds
Case (CC.l). if Case (CC.2). if
the following condition:
(7) -PB
arctan
(P A - P B K ) (PAPB (PAPB
7r
- ?(PA +pB)]
+ K ) + d5 < 1. + Ic) - d5
113
Case (CC.3). If 2) = 0, then ( P ) holds true or false whether K: > 1 or K: < -1. Case (RC). If A and B have one of them complex and the other real eigenvalues, define x := ,OAK: - p ~ where , P A and p~ are chosen in such a way P A E iR, p~ E R. Then Case (RC.l). if V > 0 , then ( P ) is true; Case (RC.2). if V < 0 , then x # 0 and we have: Case (RC.2.1). if case K / i < 0; Case (RC.2.2). if
x > 0, then
( P ) is false. Moreover, in this
x < 0, then
Case (RC2.2.A). if K:/i 5 0 , then ( P ) is true; Case (RC2.2.B). if K:/i > 0 , then ( P ) is true if and only if it holds the following condition:
x
(d-rn-
sinf - c o s t - g s i n t ) ) (
< 1,
t
where
Case (RC.3). If V = 0 , then ( P ) holds true whether x < 0 or x > 0. Case (RR). If A and B have both real eigenvalues, then Case (RR.l). if V < 0, then ( P ) is true; moreover we have 1x1 > 1; Case (RR.2). if V > 0 , then K: # - p ~ p(notice ~ that - p ~ p> ~1) and Case (RR.2.1). if K: > - p ~ p ~then , ( P ) is false; Case (RR.2.2). if K < - p ~ p ~then , Case (RR.2.2.A). if K: > -1, then ( P ) is true; Case (RR.2.2.B). if K: < -1, then ( P ) is true if and only if the following condition holds:
PRR := -fSym(PA, x
P B , K)fasym(pA,P B I X)
fasYm
( P B ,P A , K : )
< 1,
(9)
114
where
Case (RR.3). If D = 0, then ( P ) holds true or false whether K: < - P A P B or K: > - P A P B . Finally, if (P)is not true, then in case CC.2.2 with pCc = 1, case (RC.2,2.B), with PRC = 1, case (RR.2.2.B), with /IRR = 1, case (CC.3) with K < -1, case (BC.3) with x > 0 and case (RR.3) with K: > - p ~ p ~ , the origin is just stable. In the other cases, the system is unstable.
Remark 4.1. Notice that cases (CC.2.1), (RC.2.1) and (RR.2.1) correspond to 2 inverse, while cases (CC.2.2), (RC.2.2) and (RR.2.2) correspond to 2 direct. 4.2. The nondiagonalizable case In the case in which one or both the matrices are nondiagonalizable, we assume
(H5) A and B are two 2 x 2 real Hurwitz matrices. Moreover A is nondiagonalizable and [A,B] # 0. In this case new difficulties arises. The first is due to the fact that eigenvectors of A and B are at most 3 noncoinciding points on CP'. As a consequence the cross ratio is not anymore the right parameter describing the interrelation among the systems. It is either not defined or completely fixed. For this reason new coordinate-invariant parameters should be identified and new normal forms for A and B should be constructed. These coordinate invariant parameters are the three real parameters defined in Definition 4.2 below. One (q) is, up to time reparametrization, the (only) eigenvalue of A, the second ( p ) depends on the eigenvalues of B and the third (k) plays the role of the cross ratio of the diagonalizable case. For 2 E R define
+1 if z > 0 -1 if z < 0.
115
Definition 4.2. Assume (H5) and let 6 be the discriminant of the equation d e t ( B - XId) = 0. Define the following invariant parameters:
k=
[&
( T r ( A q ) - -Tr(A)Tr(B) 1 2
T r ( A B )- i T r ( A ) T r ( B )if 6 = 0. 2
Remark 4.2. Notice that 6 = (XI - XZ)’ E R, where XI and XZ are the eigenvalues of B . Notice moreover that B has non real eigenvalues if and only if 6 < 0. Finally observe that 77, p < 0 and k E R. Definition 4.3. In the following, under the assumption (H5), we call regular case (R-case for short), the case in which k # 0 and singular case (S-for short), the case in which k = 0. 4.2.1. Normal forms in the nondiagonalizable case
We have the following (see Ref. 4 for the proof):
Lemma 4.2. (R-case) Assume (H5)and k # 0 . Then it is always possible to find a linear change of coordinates and a constant r > 0 such that A/. and B / r (that we still call A and B ) have the following form:
Moreover in this case [A,B ] # 0 is automatically satisfied. Lemma 4.3. (5’-case) Assume (H5) and k = 0. Then S > 0 and it is always possible to find a linear change of coordinates and a constant r > 0 such that A / r and B / r (that we still call A and B ) have the following form,
A= or the form, A=
(;?) , B = ( (: i), B (’+
, called &-case,
(12)
) , called S-1-case.
(13)
p - 1 p +0 l )
=
0
p-1
116
4.2.2. Stability conditions in the nondiagonalizable case First we need to define some functions of the invariants r], p, k. Set A = k2 - 4r]pk sign(6)4v2. By direct computation one gets that if k = 2qp then A = -4det Adet B < 0. It follows,
+
Lemma 4.4. Assume (H5).Then A 2 0 implies k
# 2qp.
> 0 and k < 0, define
Moreover, when A
I e x pa ( T + pe8ig7L(6))l (14) -2
sign(6) = -1), where
e-l
=
{
+ + r] # o + + = 0,
if I C ( ~ 2) if k ( p f )
arctan
O1 = arctanh
a
a
r]
k(P - f >- r]’
eo = -.
kP
Notice that when k 00 are well defined.
< 0, then k ( p - f )
- r]
> 0 and
kp
> 0. Hence 01 and
The following Theorem states the stability conditions in the case in which A is nondiagonalizable. For the proof see Ref. 4. The letters A., B., and C. refer to the cases described at the beginning of this section and in Figure 2. Recall Lemma 4.4. Theorem 4.2. Assume (H5). We have the following stability conditions for the system (5).
A. If A < 0 , then the system is GUAS. B. If A > 0 , then: B1. if k B2. if k
> 2r]p, then the system is unbounded, < 2r]p, then
117 0
in the regular case (k # 0), the system is GUAS, uniformly stable (but not GUAS) or unbounded respectively if
R < l , R = l , R > 1. 0
I n the singular case (k = 0), the system is GUAS.
C. If A = 0 , then: C1 If k > 27p, then the system is uniformly stable (but not GUAS), C2 if k < 27p, then the system is GUAS. 5. An open problem Many problems connected to the ideas presented in this paper as still open. In the following we present a problem that, in our opinion, is of great interest.
Open Problem Find necessary and sufficient conditions for the stability of a bilinear switched system in dimension 3. This problem seems to be quite difficult, and the techniques presented in this paper cannot be applyed. (The reason is that these techniques use implicitly the Jordan separation lemma.) Even if studying all possible cases is probably too complicated, one would like to see if the set of all pairs of 3 x 3 matrices giving rise to a a GUAS system can be defined with a finite number of inequalities involving analytic functions, exponentials and logarithm. This was the case for bilinear planar systems, and it would be already very interesting to see if the same holds in dimension 3. Another formulation of this problem is to extend to bilinear systems in dimension 3 the following corollary of the results given in the previous section. Corollary 5.1. The set of all pairs of 2 x 2 real matrices giving rise to a GUAS system is a set in the log-exp category. For the precise definition of the log-exp category, see for instance Refs. 13,17.
Remark 5.1. Notice that the topological boundary of the set described by Corollary 5.1 is related to the problem of finding the right functional class where to look for LFs. Another interesting problem is to clarify this relation. This problem is already interesting in dimension
118
Acknowledgments
I would like to thank Andrei A. Agrachev. Indeed t h e open problem above comes essentially from him.
References 1. A. A. Agrachev and D. Liberzon, SIAM J. Control Optim. 40,253 (2001). 2. D. Angeli, A note on stability of arbitrarily switched homogeneous systems, Systems Control Lett. (to appear). 3. D. Angeli, B. Ingalls, E. D. Sontag, and Y . Wang, J. Dynam. Control Systems 10,391 (2004). 4. M Balde, U. Boscain, Commun. Pure Appl. Anal. 7,1 (2008). 5. F. Blanchini, Automatica J . IFAC 31, 451 (1995). 6. F. Blanchini and S. Miani, IEEE Transactions on Automatic Control 44,641 (1999). 7. U. Boscain, SIAM J . Control Optim. 41,89 (2002). 8. U. Boscain, G. Charlot, M. Sigalotti, Discrete Contin. Dyn. Syst. 15,415 (2006). 9. U. Boscain, B. Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds, SMAI, Vol. 43, (Springer-Verlag, Berlin, 2004). 10. Z. Bubnicki, Analysis and Decision Making in Uncertain Systems, Communications and Control Engineering Series (Springer-Verlag, London, 2004). 11. G. Chesi, A. Garulli, A. Tesi, A. Vicino, Automatica J. I F A C 3 9 , 1027 (2003). 12. W. P. Dayawansa and C. F. Martin, IEEE Trans. Automat. Control 44,751 (1999). 13. L. van den Dries, A. Macintyre, D. Marker, Ann. of Math. (2) 140, 183 (1994). 14. D. Liberzon, Switching in systems and control, Systems & Control: Foundations & Applications (Birkhauser, Boston, 2003). 15. D. Liberzon, J. P. Hespanha, and A. S. Morse, Systems Control Lett. 37,117 (1999). 16. D. Liberzon and A. S. Morse, IEEE Control Syst. Magaz., 1 9 59 (1999). 17. J.M. Lion, J. P. Rolin, Ann. Inst. Fourier 47,859 (1997). 18. P. Mason, U. Boscain, and Y . Chitour, SIAM J . Control Optim. 45, 226 (2006). 19. A. P. Molchanov, E. S. Pyatnitskii, Awtomat. i Telemekh. 3 , 6 3 (1986) [in Russian]. English translation: Automat. Remote Control 47,344 (1986). 20. A. P. Molchanov, E. S. Pyatnitskii, Automat. i Telemekh. 4,5 (1986) [in Russian]. English translation: Automat. Remote Control 47,443 (1986). 21. A. P. Molchanov, E. S. Pyatnitskii, Systems Control Lett. 13, 59 (1989). 22. A. Olas, On robustness of systems with structured uncertainties, in Mechanics and control (Proc. 4th Workshop on Control Mechanics, Los Angeles, CA, 1991), eds. J. M. Skowronski, H. Flashner and R. S. Guttalu, Lecture Notes in Control and Inform. Sci. Vol. 170 (Springer, Berlin, 1992): 156-169.
119
23. R. Shorten and K. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems, in Proceeding 1999 American Control Conf. (Sun Diego, CA, 1999): 1410-1414. 24. A. L. Zelentsovsky, IEEE Trans. Automat. Control 39,135 (1994).
120
REGULARITY PROPERTIES OF ATTAINABLE SETS UNDER STATE CONSTRAINTS P. CANNARSAt and M. CASTELPIETRAt
Dipartimento d i Matematica, Universita d i Roma Tor Vergata, 00133 Roma, Italia t E-mail: cannarsaQmat.uniroma2.it *E-mail: castelpiQmat.uniroma2.it http://www.mat.uniroma2.it P. CARDALIAGUET
Laboratoire de Mathdmatiques, UniversitC de Bretagne Occidentale 29285 Brest, FYance E-mail: Pierre. Cardaliaguet Quniv- brest.fr http://maths2. univ- brest.f r The Maximum principle in control theory provides necessary optimality conditions for a given trajectory in terms of the co-state, which is the solution of a suitable adjoint system. For constrained problems the adjoint system contains a measure supported at the boundary of the constraint set. In this paper we give a representation formula for such a measure for smooth constraint sets and nice Hamiltonians. As an application, we obtain a perimeter estimate for constrained attainable sets.
Keywords: Maximum principle; Interior sphere property; Perimeter.
1. Introduction
The Maximum principle is a fundamental result in optimal control theory. Not only does it provide necessary conditions for candidate solutions of an optimal control problem, but it can also be used to obtain regularity results for optimal trajectories. To fix ideas, consider a control system of the form
i
d ( t ) = f ( t , z ( t ) , u ( t ) ) ,u ( t ) E U
a.e. t 2 0
4 0 ) = $0,
where u(.)is a control, and x(.) denotes the corresponding trajectory.
(1)
121
Consider an open set R C R”.For a point zo E R we say that a control u(.) is admissible in zo on [O,T]if
x(t) E
n
vt E
[O,T],
a
where x(.) is the corresponding (then, admissible) trajectory. is called the constraint set. Denote by d(t)the attainable set in time t from a closed set K: C R. We call optimal an admissible trajectory x(.) on the interval [O,T]if T = inf{t 2 0 : z ( T ) E d ( t ) } ,that is, if the trajectory x(.) minimizes the time to reach the point x(T). Every optimal trajectory satisfies the Maximum Principle: this result was obtained by several authors in the smooth case18~10~12 and, as a natural evolution, also in the nonsmooth case.2~7~9~11~13 In constrained problems there are singularity effects for the trajectories that touch the boundary of R. These effects are due to an additional term (containing a measure) that appears in the Maximum Principle. More precisely, introducing the Hamiltonian
we have that, if x(.) is an optimal trajectory, then there exist an arc p ( . ) and a Radon measure p such that for a.e. t >. 0
( W -p‘(t),z’(t)) , E a H ( t , 4 t ) , P ( t )+ +(t)),
(2)
where
h ( t )= H ( t , z ( t ) , p ( t + ) +(t)). Here, VQ(Z) belongs to the normal cone to R at 2, and the measure p is supported only by the set of times t for which x ( t ) E dR (see Ref. 7). If we have a “nice” Hamiltonian (in Sec. 2 we clarify the meaning of nice), then the generalized gradient of H turns to be the triplet (DtH I D, H , D,H), and inclusion ( 2 ) becomes an equality. The main result of this paper, developed in the next section, gives a representation of the additional term in the Maximum Principle. That is, we show that, if aR is smooth, then z’(t) = D,H(t, x ( t ) , p ( t ) )and
+
-PW
= a H ( t , 4 t ) , P ( t ) )- X(t).~(Z(t))ldsl(Z(t)),
where X is a measurable function, depending on dR and H I that we explicitly compute.
122
In the last section of this paper, we apply the above formulation of the Maximum Principle to extend a result by Cardaliaguet and Marchi' for perimeters of attainable sets. 2. Maximum principle under state constraints
C Rn be an open set with uniformly C2 boundary. We define the signed distance from dR by
Let R
d ( z ) = dn(x)- dnc(x)
II: E
R" ,
where we have denoted by ds(x) = inf{lx-yl : y E S } the distance function from a set S & R". The boundary of R is of class C2 if and only if there exists some r] > 0 such that
d(.)
E Cz
on dR
+ B, = {y E B(x,r]): z E d R } ,
(3)
where C t is the set of functions of class C2 with bounded derivatives of first and second order. We consider the following controlled system, subject to state constraints,
z ' ( t ) = f ( t ,x ( t ) , u ( t ) ) , a.e. t 2 0 x(t)E t20 (4) x(0) = xo, where U C R" is a compact set, and u : R+ + U is measurable function, in short an admissible control. We assume that f : R+ x R" x U is a continuous function such that, for some positive constants L and k ,
{
{
a,
If(t,~,U)-f(t,Y,UU.)I I L l x - y l v ~ , y E R " , v ( t , UE) R + x V ( t ,IC, U ) E R+ x R" x U. I f ( t ,z, ).I I k(1 )1.1
+
u,
(5)
Let K: R be a given closed set. The set of admissible trajectories from K: (at time t ) is I,d(lC,t)= {trajectories z(.) that solve (4) in [O,t],such that zo E Ic}. The attainable set from K: at time t is defined by
d(K:,t)= {Z(t): z(.) E z d ( X , t ) } . We introduce the minimum time function
~ x ( x=) inf{t : x
E d(K:,t)},
that is the time needed to reach a point x, starting from K:. For state constrained systems the definition of "extremal solutions" differs, somewhat, from the unconstrained case. In fact, it is not sufficient to
123
request that trajectories be on the boundary of d(K:,t),because a trajectory can stay on the boundary of R without being “optimal”. Example 2.1. Take 52 = {x E lR2 : (z,ez) < l}, where {el,eZ} is an orthonormal basis of R2. Let K: = {0}, and f ( t ,x,u)= u with U = B,and T = 3. Consider the trajectories x1 (.) and x2(.) associated, respectively, to the controls
~ ( t= )el,
t E [O, 31
{
t E [O, 11 t E (1,2] t E (2,3].
and
uz(t)=
e2, $el,
0,
Then both 2 1 and z2 are in Bd(IC,t) for every t E [0,3],but “morally” only x1 is really extremal. Indeed, the point y = 2 2 ( 2 ) = x z ( 3 ) is reached in time d/fiby the trajectory x3(.) associated to the control 1 u3(t)= -el 2
6 + -e2. 2
On the other hand, the point z = xl(3) cannot be reached in any time t < 3. Definition 2.1. A solution x(.) of control system (4)is an optimal traject o y (or eztremal solution) on [O,T]if TK(x(t))= t for all t E [O,T].We denote the set of optimal trajectories on [O,T]by X ( T ) .
Note that the relation T K ( z ( T )=) T suffices to guarantee that, for all t in [O,T],we also have TK(x(t))= t. Now, we can focus our attention on the Maximum Principle. Define the Hamiltonian function as
H ( t , x , p )= s u p ( f ( t , z , u ) , p )
V(t,Z,P)E R+ x
Exn x R”.
UEU
Proposition 2.1. If z(.) is an optimal trajectory on [0,TI, then there exist an arc p ( . ) , with Ip(0)l > 0 , and a Radon measure p such that for a.e.
t E P,TI (h’(t),-P’(t), x’(t))E d H ( t 7 4 t ) , p ( t+) INt)),
+
(6)
where +(t) = J[o,t)Dd(x(s))p(ds)and h(t) = H ( t , x ( t ) , p ( t ) + ( t ) ) .Here, measure p is supported only by the set of times t for which x ( t ) E a R .
124
For a proof, see Ref. 7. In general, we don't know if (hence h) is continuous. To handle this boundary term, we need some regularity of the Hamiltonian. We will assume that $J
{
H is of class C2 on R+ x R" x (R"\{O}), MIPl
L H ( t , z , p ) 2 alp17
(7)
rim2 D ; ~ H ~5,(P~) 2,
for some constants a , y, I?, M
> 0, and for all (t,z , p ) E R+ x R"
x (Rn\{O}).
Remark 2.1. Note that the last assumption of (7) is made for H 2 , and not for H . In fact, such an assumption, if imposed on H , would be too restrictive (see example 2.2). Moreover, we will impose the following growth conditions for the derivatives of H : for all ( t , z , p ) E R+ x R" x (Rn\{O}),
Remark 2.2. We can give sufficient conditions for f to satisfy some of the assumptions in (8). If we assume that, for all ( t , z , u )E R+ x R" x U , we have I f ( t , z , u ) l I M and l D t f ( t , z , u ) I 5 M and IIDzf(t,z,u)ll I M , then the first three bounds in (8) are satisfied. Moreover, if { f ( t , z ,U ) } ( t , z ) E ~ +isx ~a nfamily of uniformly convex sets of class C2, then H E C2(R+ x R" x (R"\{O})), see, e.g., Ref. 5. Under the above regularity assumptions for the Hamiltonian, the Maximum Principle takes the more precise form described below.
Theorem 2.1. Assume that (3), (5), (7) and (8) hold true. Let x(.) be an optimal trajectory on the time interval [0,TI. Then there exists a Lipschitz continuous arc p : [O,T]-+R", with Ip(t)l > 0 , and a bounded measurable function X(t) 2 0 such that
z'(t) = D , H ( t , z ( t ) , p ( t ) ) -p'(t)
for all t E [O,T],
= D , H ( t , z ( t ) , p ( t ) )- X ( t ) D d ( z ( t ) )
for a.e. t E [O,T].
125
Moreover, X is the function given below:
Proof. The idea of the proof is to approximate system (4)by a penalized control system without state constraints. Then we will apply the nonsmooth Maximum Principle to such a system. Finally, we will retrieve useful information for the original system (4). Let E < q / 2 be a positive fixed constant (where q is defined in (3)), let f & ( t , Z , U ) := f(t,x,.)
(
:
1 - -dn(z)
>+
,
and consider the unconstrained system
{ z'(t)
= f,(t,z ( t ) ,u ( t ) ) ,
a.e. t
20
z(0) = 2 0 .
(9)
The associated perturbed Hamiltonian and the set of optimal trajectories are denoted as follows
Hdt, z, P ) = F$f&(4 2,U > ,P ) X,(T)
= {optimal
V ( t ,Z, P ) E R+ x R" x Rn,
trajectories of system (9) on [0,TI}.
We note that f, is Lipschitz continuous, but nonsmooth on the boundary of 52, and that
(
H,(t,z,p) = H ( t , z , p ) 1 - ,dn(z)
>+
We shall prove that, as soon as E is small enough, any trajectory of X ( T ) is actually a trajectory of X,(T), i.e. X ( T ) 5 X,(T). This is not an obvious fact. In fact, it is clear that any constrained solution of (4)is still a solution of system (9), but an optimal trajectory x(.) E X ( T ) may fail to be optimal for (9). Indeed, system (9) can have more trajectories that arrive at the point z ( T )E
n.
126
We split the proof in 3 steps. The first and second step are devoted to show that X ( T ) C X,(T) (so that we can use the nonsmooth Maximum Principle). In the third step we use the nonsmooth Maximum Principle to recover the conclusion in the constrained case. Step 1 Let us check that, for a suitable choice of E , any optimal trajectory of system (9) that stays in at time T > 0, remains in for all t < T . Equivalently, we want to prove that
n
a
X,'(T) := {x(.) E X,(T) : z ( T ) E
n} C z d ( x , T ) .
Take z,(.) E X,"i(T). By the definition of f,, dn(z,(t))
< E < 712
Vt E [O,T].
(10)
Since z,(.) is optimal for an unconstrained problem, we can use the Maximum Principle of proposition 2.1 for nonsmooth hamiltonians without the boundary term +(t). So, we find that there is some adjoint map p , : [O,T]4 R" such that, for a.e. t E [O,T],
{
1
d ( t )= q J H ( t , z & ( t ) , P & ( t )-) (,dn(4)))+ l
(11)
-PL(t) E azH€(t,Z E ( t ) , P E ( t ) ) .
We observe that, for each t such that x,(t) 4 852, the generalized gradient dzH,(t, z,(t),p,(t)) reduces to a singleton and
where
Note that Ipe(t)l > 0 for all t E [O,T].In fact we can normalize p E so that Ip,(O)l = 1. Moreover, setting h,(t) = H,(t,z,(t),p&(t)), we have that
thanks to ( 6 ) and (8). We recall that, in view of (7), H , ( t , z , p ) 2 Crlpl(1 dn (z)/ E ) + . Applying Gronwall's lemma we get h, ( t ) 2 e-(M/a)thE (0), and, since H ( t ,X & ( t ) , P E ( t ) ) L H & ( tz,(t),p,(t)), , M
H(t,z,(t),p,(t)) 2 e - ~ t H ( O , z , ( 0 ) , p e ( O )2) e-eta > 0.
(14)
127
In particular, this implies that IpE(t)l> 0 on [O,T]. Now, in order t o conclude the proof of step 1, suppose, by contradiction, that there exists an interval (a, b) on which z E ( t )$! E, and z,(a), z,(b) E do. Then, using right and left derivative o f t H d ( z E ( t ) in, ) respectively, a and b, we obtain
i
&d(ze(t))lt=a
= ( ~ p ~ ( a , z , ( a ) , p . z (~ad)()z, c ( a ) )2) 0 and
+(z&))lt=b d-
= ( D , ~ ( b , ~ € ( b ) , P , ( bDd(z,(b))) )), 5 0*
Since H is nonnegative, these inequalities can also be rewritten as
i
( D p H 2 ( a , 2 , ( a ) , p , ( a ) Dd(z,(a))) ), L 0 and (DpH2(b,% ( b ) , P € ( b ) ) ,Dd(z,(b))) 5 0.
Observe that there exists a constant C (not depending on E ) such that, for all t E [a,b], 11D2~(.€(t))llI
c,
(15)
since, thanks t o ( l o ) , z e ( t )is in a set in which d(.) is of class Ct. Hence
where y is defined by (7) and M' = 2M3(2 Finally, we get a contradiction setting &
:=
aye-
+ C ) + 4M2 + r M
%T
4M'
'
128
+
Indeed H(t,z,,p,)y/E - M‘(1 Ip,l) > 0 if, for lp,l 5 1, we choose E < a y e ~ ( ~ / ~ ) ~ / (and, 2 M for ’ ) lpEl , > 1, we choose E < a y / ( 2 M ‘ ) . Step 2 We want to show that an extremal solution of the constrained problem (4) is also an extremal solution of (9), i.e., that X ( T ) C X E ( T ) . Let z(.) E X ( T ) , and let T, the minimum time to reach the point z ( T ) for the perturbed problem (9). Since any solution of the constrained problem (4) is also a solution of the unconstrained problem (because fEIn = fin), we have that T, 5 T . Now, let z,(.) be a trajectory of the perturbed problem (9) such that z,(T,) = z ( T ) .Then z,(.) is in XEE(TE), and, by step 1, we know that it is also a solution of the original problem (4), as it ends in Then T, 2 T , and we have that T, = T and z(.) E X E ( T ) . Step 3 Finally, let z(.) E X ( T ) and p ( . ) be an associated adjoint map such that ( z ( + ) , p ( .satisfies )) (11).We want to find an explicit expression for p’(t). For this we define the set of times t for which the trajectory z(.) stays on the boundary, that is
n.
Ex= {t E [O,T]: z ( t ) E 80). Observe that, by standard properties of the generalized gradient,
for all z E dR (see Ref. 7). Moreover, for all z E dR,
dZH(t,Z , P ) = D x H ( t ,ZIP), Recalling (11) and (12), we have that there is a measurable function A, : [O,T]-+ [0,1] such that, for a.e. t E [O,TI,
-PW
= D x H ( t ,z ( t > , p ( t )-)
where A,@) is given by (13) for t $!
x e ( W ( t , z ( t ) , P ( t )D)( j ( z ( t ) ) ,
p7)
&
Ex.Now define the function
cp(t) := d ( z ( t ) )
t
E
[O,T].
Since R is of class C2 and z’(t) exists for a.e. t , we have that cp is differentiable for a.e. t E Ex.Moreover, for a.e. t E Ex,
(Dd(X(t)), D,H(t, z ( t ) , p ( t ) )= ) ( D d ( z ( t ) )z’(t)) , = cp’(t) = 0.
(18)
In fact, let t E Ex.If cp’(t) > 0 then, by continuity, there exists some q > 0 such that cp(s) > 0 for all s E (t,; q), in contrast to z ( t ) E
+
129
We claim that, for Ip(t)l > 0, ( D d ( z ( t ) )DEpH(t, , z(t),p(t))Dd(z(t)> ) ) 0. Indeed, for a unit vector C E JR”, we have that
thanks to (7). Then,
As a consequence of (18), there exists a constant 6 > 0 such that
Finally, recalling the expression of p’ in (17), we find a representation of X by setting
(4( t , z ( t ) , p ( t ) ) . X ( t ) := -X -EH &
(where E is defined in (16)) we have that X is bounded and p ( . ) is Lipschitz continuous. Observing that the right-hand side of the equality
130
= D,H(t, z , p ) is continuous, we conclude that this equality holds for all t in [O, TI. 0 2'
Example 2.2. We discuss a class of control systems that satisfy the assumptions of this section for the maximum principle. Consider the control system
{
z'(t)= f ( t , z ( t ) ) u ( t ) , u ( t )E B z(0) = 2 0 E
Ic,
where f : R+ x R" + Mn(W) is of class C2, and the matrix f ( t , z ) is bounded and invertible, with bounded inverse. Moreover, the Hamiltonian is
H ( t , Z , P ) = m a _ . ( f ( t , z ) w ) = max('LL,f*(t,z)p)= If*(t,z)PI. UEBl
UEBl
So, assumptions (7) are satisfied, since f and
f-l
are bounded, and
D;pH2(t,z,P)= 2f(t,z)f*(t,z). Since we can consider ( t , ~in) a compact set, then D t f ( t , z ) and D z f ( t , z ) are bounded, and, thanks t o remark 2.2, D t H ( t , z , p ) and D,H(t, z , p ) satisfy assumptions (8). Moreover,
so that it would be impossible to satisfy T I , 2 D & H ( t , x , p ) for this simple control system.
2 y l n , even
131
3. Perimeter estimates for the attainable set In this section we will study the special case of control systems in R2 of the form x'(t) = f ( t , x ( t ) ) u ( t ) , a.e. t 2 0 x(t)E t20 (20) ~ ( 0= ) zo E K ,
{
where u : R+ such that 0 0 0
-+
a,
B is a measurable function, and f : R+ x R2
f is of class C2 in R+ x
n
---f
M2(R)is
f ( t , z ) is invertible for any ( t , x ) E lR+ x f ( t , x ) and f - ' ( t , x ) are bounded by M > 0 for any ( t , z )E R+ x
a.
The Hamiltonian for this dynamic is wt,2 , P ) = max(f ( t ,x)u,P> = If * ( t Z)Pl. , uEB
The aim of this section is to estimate the perimeter of the attainable set
d ( t ):= d ( K , t ) . Cardaliaguet and Marchi6 proved such an estimate for f ( t , z ) = c ( t , z ) l n , where cis a scalar function. Applying theorem 2.1 we can use their technique to extend this analysis to system (20).
Theorem 3.1. Let T > 0 be fixed. For any t E (O,T] the attainable set d ( t ) is of finite perimeter, and there exist C1, C2 > 0 such that
In particular, if set K has the interior sphere property, then the attainable set d ( t ) has finite perimeter for any t E [0,TI, and the perimeter of d(t) is bounded on [0,TI. In order to prove that d ( t ) has finite perimeter, we cover the boundary of the attainable set with the following sets
n
Bbnd(t):= {x E : 3x(.) E K ( t ) , 3s E [ O , t ] , with x ( t ) = z, z(s) E a O } , Bint(t):= {x E s1 : 3x(.)E X ( t ) , with x ( t ) = z and x([O,t])n a R = 8). Note that Bint(t)is the set of points that can be reached by extremal trajectories contained in the interior of R , while Bbnd(t)is the set of points reached by extremal trajectories that touch the boundary 80. Then
c
a d ( t ) Bint(t)u Bbnd(t)u dR.
132
We point out that, in this case, the adjoint system of theorem 2.1 is given by
-p’(t)
=
IP(t)l ( D , H ( t , x ( t ) , IP(t)I - X ( t ) D d ( x ( t ) ) )I
(21)
where X is a bounded positive function. Consequently, for all T > 0, we have that the set X ( T ) of the extremal trajectories for system (20) is compact with respect to the C 1 norm. With an easy adaptation we can recover from Ref. 6 a Lipschitz estimate for the velocities of extremal trajectories on the boundary of 0.
Lemma 3.1 (Ref. 6, lemma 4.1 ). Let T and 6 be given positive numbers, let t l , t 2 E (O,T],and let yi : [O, ti
+ 61
4
R2
(i = 1 , 2 )
be optimal trajectories for (20). There is a positive number u such that, if y i ( t i ) E dR (i = 1 , 2 ) and Iyl(t1) - yz(t2)l I u, then
where Cn is a positive constant depending only o n the regularity of 6’0. In order to estimate N1(Bbnd(t))the main idea is to use the perimeter of dR. At this aim we define the set
Bbnd(t,6) := { X E Bbnd(t): x ( ( t - 6, t ] )n XI= 0). Lemma 3.2. Let T > 0 be fixed, let t E (O,T],and let y1,y2 E Bbnd(t,6). For i = 1 , 2 , let y i ( . ) E X ( t ) and let si E [0,t - 6 ) such that y i ( t ) = Y { , yi(si) E 80,
y i ( ( s i , t l ) n 80 = 0.
There is a constant CO> 0 such that, for any 6 > 0, a constant exists, so that, i f lYl(S1)
-
Y2(S2)I
u6
>0
5 U6,
then IY1 - Y2l
I COIYl(S1) - YZ(S2)l.
Proof. Without loss of generality, we can suppose s1 I s2. There exist adjoint states pi(.) such that, for s 2 si the extremal trajectories yi(.) solve
133
We normalize arcs p i ( . ) so that H(si,Yi(si),Pi(si))= If*(si,Yi(si))pi(si)~ = 1,
i = 1,2.
Applying Gronwall's lemma we have
Now, we focus our attention on the right-hand side of this inequality. For the first term, we have that IYl(S2) - Y2(S2)1
+ IYl(S2) - Yl(S1)l I IYl(S1) - Y2(S2)1 + MIS1 - s21 I (1 + MLo)lYl(sl) - Y2(S2)JI I IYl(S1) - Y2(S2)1
invoking the Lipschitz continuity (of rank Lo) of the minimum time function. Similarly, we have that IPl(S2.1 -P2(S2)1
I
IPl(S1) -P2(S2)1
+ IPl(S2) -Pl(sl)l
I IPl(S1) -Pz(Sz)l + C'IYl(S1) - Y2(S2)1. Recalling the expression of D,H(t,z,p) in (19), we have that IPl(S1) -Pz(SdI
1
=1
P
= I f * ( ~ l , Y l ( ~ l ) ) P l ([f(sl,Yl(sl))f*(sl,yl(sl))]-ly:(sl) ~l)I =1
P
Moreover, we also have that
Since, for i = 1 , 2 , lyl(si)lyyb(si)/lyb(si)l aB, we obtain that
= f(si,yi(si))ui(si), and ui(si) E
As a consequence of lemma 3.1 and of Lipschitz continuity o f f we have the conclusion. 0
134
Thanks to lemma 3.2 we have a bound for the perimeter of Bbnd(t).We give a proof for the reader's convenience.
Proposition 3.1. Let T > 0 be fixed. There is a constant CO> 0 such that, for all t E [0,TI,
IFI1(Bb"d(t))i CoIFI1(aR). Proof. For any fixed 6 > 0, let Co and C T ~the constants of lemma 3.2. Let c > 0, and let {&},EN be a family of sets such that
aR C
u
+
B,,
'Hl(d0)
00
E
2
n€N
diam(B,),
diam(B,) 6 inf{ag,E}.
n= 1
Consider the covering of Bbnd(t,6), given by
Kn = { X E Bbnd(t,6 ) : In view of lemma 3.2, for all n E
X(S)
E B,, z ( ( s , ~n] dR ) = 0}.
N,we have that
diam(K,) 5 COE, diam(K,) 6 Codiam(B,). Thus, 00
IFI&oE(Bb"d(t, 6))i
diam(K,)
i Co(IFI'(dR) + E).
n=l
Now, letting c independent of
---f
E
0 and then 6 -+ 0, we have the desired result, since Co is and 6.
Now, we turn our attention to set Bint(t).Following the main ideas of Ref. 3, we can see that this set has the interior ball property, and then it has finite perimeter for any t > 0.
Remark 3.1. Let x ( . ) be an extremal solution on the interval [0,TI, such that x([O,TI) R. Then there exists some r] = r](z) > 0 such that, for all s E [0,TI, we have that B V ( x ( s ) )C 0. Proposition 3.2. Let T > 0 be fixed. Then there exists a constant CT > 0 such that for any x E Bint(T),with z(.)extremal solution so that x ( T ) = x , then there exists some q = q ( x ) > 0 such that
135
From Ref. 1 and proposition 3.2, we can derive the following bound for t h e perimeter.
Proposition 3.3. Fix T > 0. There exists C > 0 such that c/'
7l1(Bint(t))I -.
CTt
T h e perimeter of Bint(t)is in inverse ratio t o the time t , since the interior sphere property is proportional t o t he time. This means t hat , if we fix a time 6 > 0, for any t 2 6 we have a uniform estimate (i.e. C / c ~ 6 )In . addition, if set K has t he interior sphere property of radius T , then for all
t E P,TI 'FI'(B'"t(t)) 5
-.Cr
References 1. 0. Alvarez, P. Cardaliaguet, R. Monneau, Interf. B e e Bound. 7,415 (2005). 2. A.V. Arutyunov and S.M. Aseev, SIAM J. Control Optim. 35,930 (1997). 3. P. Cannarsa and P. Cardaliaguet, J. Convex Anal. 13,253 (2006). 4. P. Cannarsa and H. Frankowska, ESAIM Control Optim. Calc. Var. 12,350 (2006). 5. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton- Jacobi equations and optimal control (Birkhauser, Boston, 2004). 6. P. Cardaliaguet and C. Marchi, SIAM J. Control Optim. 45,1017 (2006). 7. F.H. Clarke, Optimization and Nonsmooth analysis (Wiley-Interscience, New York, 1983). 8. A.Y. Dubovitskii and A.A. Milyutin, USSR Comput. Math. and Math. Physics 5,1 (1965). 9. H. Frankowska, J. Convex Anal. 13,299 (2006). 10. R.V. Gamkrelidze, Izk. Akad. Nauk., USSR Sec. Mat. 24,315 (1960). 11. P. Loewen and R.T. Rockafellar, Truns. Amer. Math. SOC.325,39 (1991). 12. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical theory of otpimal processes (Interscience Publishers John Wiley & Sons Inc., New York-London, 1962). 13. R.B. Vinter, Optimal Control (Birkhauser, Boston, Basel, Berlin, 2000).
136
A GENERALIZED HOPF-LAX FORMULA: ANALYTICAL AND APPROXIMATIONS ASPECTS I. CAPUZZO DOLCETTA
Dipartimento di Matematica, Universita d i Roma ”La Sapienza”, 00185 Roma, Italia E-mail:
[email protected] In this paper we discuss the validity of the Hopf-Lax representation formula for solutions of evolution Hamilton-Jacobi equations governed by statedependent and perhaps non coercive Hamiltonians. Some applications to PDE’s on the Heisenberg group and finite differences approximations are also presented.
Keywords: Hamilton-Jacobi, Hopf-Lax formula, Heisenberg group, finite differences approximations
1. Introduction
The purpose of this Note is to report some recent research on scalar evolution Hamilton-Jacobi equations of the form
t + (Ho(II:,D,u)) { uu(0,z) , @
=0, = g(z) II: E RN,
(t,II:)E (0, +m) x RN
(1)
where p -+ H o ( z , p ) is convex and 1-positively homogeneous and @ is a convex, nondecreasing scalar function. The focus is on Hamiltonians HO which are state-dependent and perhaps non coercive with respect to the gradient variable. The main issue addressed here is the validity of a representation formula of Hopf-Lax type (often referred also as Hopf-Lax-Oleinik formula) for the solution of the above initial value problem. In the language of optimization theory, these type of formulas represent the solution of problem (1) as a marginal function, parametrized by the time variable t. In the model state-independent example where H0(Dzu) = lDul and @ ( r )= ;r2, is well-known that the Hopf-Lax solution is given by
137
Representations formula as the above are of course very useful in the analysis of Hamilton-Jacobi equations. We just mention here applications in optimal control theory3yz3, large time behaviour of solutions5 , weak KAM theory16 , transport theoryz9 , geometric front propagation26 . The validity of such represefitation formulas has been widely investigated in the state-independent coercive case starting from the seminal paper Ref. 20, see also Refs. 2,28 for recent general results. Few results are available in the state-dependent case, see however Refs. 24,27 in this respect. Our formula
where Q,* is the convex conjugate of Q, and d is a distance function suitably associated with the 1-positively homogeneous function Ho, covers in particular the case of Hamilton-Jacobi equations associated with sub-Riemannian metrics; in this case Ho(x,p)= Iu(x)pI where u(x) is a M x N matrix satisfying the Chow-Hormander rank condition, see for example Ref. 8 . Let us recall for convenience that this condition amounts to the requirement that there exists an integer k 5 N such that the Lie algebra spanned by the columns of u(x) by means of commutators of length 5 k has rank N at any x E I W ~ . An interesting special case discussed in greater detail in Sections 3 and 4 is that of the Heisenberg Hamiltonian; in that case the matrix u(x) is given, for x = ( X I ,x 2 , x 3 ) E EX3, by 1 0 2x2
4 . ) = ( 0 1 -2xJ In that case, our Hopf-Lax type formula can be regarded as a simplified expression for the value function of the Mayer optimal control problem for the well-known Brockett’sg system where * denotes transposition:
Some application of our representation formula to the asymptotic analysis of the heat operator on the Heisenberg group and to the error analysis of an approximation scheme for the numerical computation of the solution of problem (1) are outlined in Section 4. Most of the results reported here are taken from the recent papers or preprints1~10~11~15
138
2. A generalized eikonal equation
Consider a set-valued mapping x
+C(x)
C RN such that
C ( x ) is convex and compact and 0 E C(z) ,Vx E RN ,
(2)
where dist(a, C ( y ) ) ; max dist(a, C ( x ) )
d E ( C ( z ) ,C ( y ) ) = max
bEC(9)
is the usual Hausdorff metric. We assume also that C ( x ) is "fat" enough in the following sense:
there exist 6 > 0 , a natural number M 5 N and for each x E RN a n M x N smooth matrix u(x)with bounded Jacobian such that the columns of u(x) satisfy the Chow-Hormander rank condition of order k at any x E RNand C * ( x ) B W M (0,s)
c ( x ) , vx E RN.
The above condition, labelled ( C C ) from now on, is trivially satisfied if C ( x )contains an N-dimensional ball; observe however that ( C C ) may be fulfilled even when C ( x ) has empty interior at some x,see examples below. Denoting by Bl(0) the Euclidean unit ball, it is easy to check that condition ( C C ) holds e.g. in the following cases:
C ( x ) = a(.) BWM (0,6), a(.) C ( x )= m
B
2
a0
>0
(4)
W(0,6), ~ A ( x ) positive definite
(5)
C ( x )= u*(x)B W M ( O , 6 ) , u ( x ) satisfying ( C C ) .
(6)
let us consider now the support function of the set C(z), namely
Ho(x,p) = SUP q . P
(7)
PEC(X)
It is not hard to check that under the assumptions made, HO satisfies p
H
Ho(x,p)convex , Ho(z, Xp) = XHo(z,p) for all X > 0,
(8)
139
Furthermore, it is well-known from convex analysis that
where apH denotes the partial subdifferential with respect to the p variable. The Hamiltonians corresponding to the cases (4),(5),(6)are given respectively, by HO(ZIP) = 4Z)lPI
7
HO(ZIP) = lU(Z)Pl.
HO(ZIP) = J 1
This last example includes, in particular, the case of the Grushin Hamiltonian given on R by Ho(z,p) = as well as that of Heisenberg Hamiltonian given on R3 by
Jm,
Ho(Z,P) = J(P1
+ 2 C ? P 3 ) 2 -k
(P2
- 221p3)~.
(10)
See Ref. 8 for a discussion of the role of these Hamiltonians in subRiemannian geometry. Further examples are provided by Hamiltonians of the form H ~ ( z , p = ) max [HO((z,p),. . . , H ~ ( z , p ) ]where , the H i are of the types above. Condition ( C C ) and the classical Chow Connectivity Theorems imply that the set Fz,yof all trajectories X ( . ) of the differential inclusion
X ( t ) E d p H o ( X ( t ) , O ), X ( 0 ) = Z, X ( T ) = y for some T = T ( X ( . ) ) > 0 , is non empty for all z and y in consequently the minimum connection time d ( z ; y ) :=
RN
and,
T(X(.))
inf
x E & a, is well defined and finite on RN.Moreover, d(x,y) is a sub-Riemannian metric on RN such that I
and therefore, z + d ( z , y ) is a $-Holder continuous function, see Ref. 8 for a nice presentation of these topics. By Dynamic Programming techniques one can verify that z -+ d(z, y) solves for each fixed y E RN the generalized eikonal problem
in the viscosity ~ e n s e. ~ ' ~ ~
140
3. The generalized Hopf-Lax formula Let us consider now the evolution problem
{ u ( +0 , z )( H o ( z ,, ut
G U ) ) =0
@
= g(z)
zE
, (t,z) E (0, +oo) x
RN
IWN
(12)
where @ : lRN -+ [0, +m), @(O) = 0, is convex, increasing and HO as in (7). In this setting, an Hopf-Lax type formula holds for the solution of (12) under the assumption for Ho made in the previous section.
Theorem 3.1. Assume H o ( z , p ) = supqEc(z)q . p with C(z) satisfying conditions (2),(3) as well as condition (CC). If g E C(IWN)is linearly bounded below, i.e. g(z)
2 -C(1+
)1.1
for some C > 0 ,
then the function
where @* is the convex conjugate of @, is the unique continuous linearly bounded below viscosity solution of the Cauchy problem (12). This representation formula generalizes to the present framework the classical Hopf-Lax for the case of uniformly convex state independent Hamiltonians H = H ( p ) as well as the one established in Ref. 24 for the special case of the Heisenberg Hamiltonian (10). Comparison and uniqueness results for problem (12) suited to the present setting can be found in Refs. 11,13 . The proof of the fact that the Hopf-Lax function (13) is a solution of problem (12) is based on viscosity calculus and makes essential use of stability properties of viscosity solutions with respect to inf operations. We briefly sketch it here and refer to Ref. 11 for full details. The first step is to show that the functions vY defined by V Y ( t , z)
=g(y)
+ t@* -
(“7
y’>
’
where y E EXN plays the role of a parameter, do satisfy equation(l2) for each fixed y . A formal argument is as follows: assume that (11) has smooth solutions d ( z ) = d ( z ,y ) for each fixed y and suppose that vY is smooth.
141
Then, denoting by v; the partial derivative of vY with respect to the t variable,
v r ( t , z )= a*(.)
-.(a*)'(.)
, D,vY(t,z) = ( @ * ) ' ( T ) D , d ( z , y )
where r = > 0. It is not hard to check, using the positive homogeneity of Ho, see assumption (8), that d solves the eikonal equation (11).Assuming for simplicity that @* is smooth, the reciprocity formula of convex analysis
+ @*(s) = s ((a*)+)
@ ((@*)'(s))
,
yields
v!(t, z)
+ @ (HO(2,D , V Y ( t ,
z))) = 0 ,
( t ,z)
E (0, +w) x
RN .
The procedure outlined above can be made rigorous by viscosity calcuIus. The second step of the proof is to show, using the stability properties of viscosity solutions with respect to inf operations, that the lower envelope
u ( t , z )= inf vY(t,z) Y ERN
of this family of special solutions, is lower semicontinuous and solves (12) in the viscosity sense. Observe that, since p -+ @ ( H o ( z , p ) )is convex, it is enough at this purpose to check that
x + H ( z ,77) = 0
Y(v, A) E D-u(z, t )
at any (5, t ) ,where D-u(z, t ) is the subdifferential of u at (z, t ) ,see Ref. 6. The third step is to check the initial condition u ( 0 , z )= g(z) and the fact that u is bounded below by a function of linear growth. This can be done much in the same way as in Ref. 2. Let us conclude this section by observing that in the model example
t + $fY(z)D,u12 { uu(0,z) g(z) , =
with
f~
=0
2 E
, (t,z) E (0, +m) x RN
RN
(14)
as in (6), the Hopf-Lax solution given by (13) is
YEW*
which can be regarded as a generalized form of the classical Yosida-Moreau inf-convolution of function 9, see Ref. 14 for more informations on this issue.
142
4. The Hopf-Lax formula for the Heisenberg Hamiltonian
The Heisenberg Hamiltonian Ho(x,P) = J(P1
+ 2?2P3)2+ ( P 2 - 2511)3)~= la(x)1)l
where x = (21,x2,23) and
41.
1 0 222 = (0 1 - k l )
is strictly related to the dynamics of the famous Brockett’s exampleg in control theory k ( t ) = a*(x(t))c(t) ,
(15)
2(0) = 2
and to the analysis of some asymptotic properties of the heath kernel on the Heisenberg group, see Ref. 7. Thanks to the Lie bracket relation [(I,0,2ZZ), (0,1, - 2 4 = -4 (O,O, 1) , the matrix a(.) satisfies the Chow-Hormander rank condition with k = 2. Then, as a particular case of the results of the previous section, the minimum time for connecting x to y by system (15) with controls c : [O, +m) --$
R2 , Ic(t)l 5 1,
is the appropriate distance d(s;y) occurring in the Hopf-Lax function
solving in the viscosity sense (note that x -+ d(x;y) is in the present case Holder continuous) the Cauchy problem: just
4-
{
+
ut @ (Ia(x)DuI)= 0 , u(0,2) = g(x) , 2 E R3
( t , z )E (0, +m) x R3
(17)
The Hopf-Lax formula defines the nonlinear Lax-Oleinik semigroup16 S ( t ) on the space B C ( R N )of bounded continuous function:
Using this, it is easy to deduce, see Ref. 1, some useful properties of the solution of problem (17), namely
S(t
+ s)g 5 S(t)g , v s > 0
,
i$g
5 S(t)g 5 supg w3
143
and
II (S(t)g - s(t)a)+
1103
I Ilk -i9+1103*
Moreover, if E M -+ +oo as s + +m , then the expansion of support property holds: if suppg 5 {z E
R3 : d(z;O) 5 R}
then supp S(t)g G {z E R3 : d(z;0 ) 5 R
+ R(t)}
where R(.)is a nondecreasing function depending on I1g-llo0 and
a*.
These and other properties of Hopf-Lax function are useful in the error analysis of finite difference schemes for the numerical computation of the solution of (17) which will be briefly described later on.
4.1. A singular perturbation problem on the Heisenberg group Next we present a result from Ref. 15 , see Ref. 22 for previous result of this type. The proof here exhibits the role played by the Hopf-Lax function in a singular perturbation problem for the heat equation on the Heisenberg group H1 = (It3,@). The Heisenberg group is a stratified Lie group whose group operation @ is defined as
+
y @ z = (z1 y11zz
+ yz, 2 3 + y3 + 2(21yz - zzy1))
Note that y @ z - z @ y = 2 ( J z . y) e3
where J is the matrix
J=
(iig)
It is immediate to check that the trajectory z ( t ; z , y ) of the Brockett’s control system k ( t ) = a*(z(t))y
z(0) = z
144
with control y = (y1, y2,O) is given by 4 t ;2,Y) = (21
+ ty1, + tyz, + 2t(z1yz 22
23
22Yl)) = 2
@ ty
In particular, z(1; IC, y) = 2 @ y. The horizontal gradient D H I Uof a smooth function u is defined as D H l U ( 2 ) = a ( Z ) D U ( Z )= ( U z l
- 2Z1Uz3)
222Uz37Uz2
while its Heisenberg Laplacian'l A H u I is given by the second order expression
AHIU = Tr ( a * ( z ) a ( z ) D 2 U ). Consider for
{
E
> 0, the heat diffusion problem
w;(t,Z)- E
' ( 2) t ,= 0 , (t,2) E (0,+CO) x R3 , 2 E R3 .
AH1~
w'(0,z) = e-+
(18)
The question we address is to identify the limit of functions wE as the diffusion coefficient E is sent to Of. It is natural in this context to look at the log transform17 of function w', namely
u'
= -2E log W E.
Function uEsatisfies the quasilinear equation
{ UZ
+ + ~ C J ( ~ ) D , U '=~ ~0 , ( t , ~E )(0,
-E AHIu' =g(2)
U'(0,Z)
+CO)
, 2 E w3 .
x R3
(19)
The formal guess is that u ' = - 2 ~log wE should converge to the viscosity solution u(t,2) of the Hamilton-Jacobi equation
{
+
ut ! ~ I D H ~=u 0~ ,~ ( t , z )E (o,+co) x R3 u'(0,z) = g ( 2 ) , 2 E R3 ,
(20)
that is to the Hopf-Lax function
2t where, by the result of the previous section, d is the minimum time needed to connect 2 to y through the trajectories of the controlled Brockett's system
k ( t ) = a*(z(t)) c ( t ) , Ic(t)l L 1 .
145
Let us sketch briefly the elements of the proof of the formal guess; it makes essential use of the Hopf-Lax's formula and the Large Deviations Principlelg . The functions wE have the integral representation wE(t,z) =
e-*p(Et, L
where p is the fundamental solution of wt-6 see Ref. 22, by
for some M
z, y) d y
3
AH1 w.
Function p is estimated,
> 1 and small t , where d solves the eikonal equation ldx)D,d)I = 1 I z E
\ {Y}
(21)
Hence, the family of probability measures
satisfy the conditions needed to apply Large Deviation Principle with rate function ItJ(y) = -. This gives the Laplace-Varadhan type result: lim clog E+O+
(A3
e
a 2E
d p E ( y ) ) = sup [g(y) - I ( ~ U ) I yER3
and, as a consequence, lim -26 logw' = inf E'O+
YER3
2t
The above result confirms that even in the present sub-riemannian context, the inf-convolution can be regarded as a singular limit of integral convolutions, see Ref. 10 for the standard euclidean heat equation and also Ref. 18 for a general remark in this direction. Similar asymptotic results for more general matrices satisfying the ( C C ) condition can be found in Ref. 14 . The way of deriving the Hopf-Lax function via the logarithmic HopfCole transform and the Large Deviation Principle is also closely related to the Maslov's approach25to Hamilton-Jacobi equations based on idempotent analysis. In that approach, the basefield R of ordinary calculus is replaced by the semiring R* = Ru(00) with operations a n b = min{a, b } , a u b = a+b . A more detailed description of this relationship goes beyond the scopes of the present paper; just observe that the nonsmooth operation an b has the smooth approximation a n b =
146
4.2. Convergence rate of finite diflerences approximation
We address in this section some issues related to the computation of approximate solutions of
+ a (IcT(2)DuI) (t, { u(0,z) R3 ut
=0, =g(2), 2 E
2) E
(0, +m) x Ex3
(22)
where cT(2)Du(5)= J(u,,
+
2z2uz3)2
+ (uz2-
221uz3)2
is the Heisenberg Hamiltonian considered in the previous section. Since o(2) has rank 2 , ( 2 2 ) is a non coercive equation. For Hamilton-Jacobi equations ut H ( z ,D,u)= 0 which are coercive, that is limlpl-t+m H ( z , p ) = +m, finite differences schemes with uniform grid size h > 0 have been investigated in Ref. 12 and the following optimal error estimate
+
IIu - ~ a p p r l l m< CJTE (EE) was established. The validity of this estimate in the coercive case depends, in particular, on the Lipschitz continuity of the viscosity solution. The purpose in Ref. 1 is to design an approximation scheme for equation ( 2 2 ) (as previously mentioned the solution is just Hdlder continuous in this case) in such a way as to obtain the same rate of convergence proved in Ref. 12 for the coercive case. The idea is to construct a grid compatible with the Heisenberg group translations
a-
9 @ 2 = ( z 1 -k y 1 , 2 2 -k y 2 , z3 -k 9 3
+ 2(21?/2 - 2 2 y 1 ) )
7
(note that @ is non commutative) and dilations a * 5 = ( a 2 1 , a z 2 , a 2 z3)
a>O.
The grid nodes of discrete lattice are chosen to be ti,j,k
= (ih,jh,(4k
+ 2ij)h2)
where h > 0 and i , j , k are integers. Observe that
@ fh
*
el = t i * l , j , k ,
ti,j,k
@ fh
*
e2
= ti,j&l,kFi.
tl,m,n@ t i , j , k =
Formulas above show compatibility between the discrete lattice and the group operations @ and .
147
where (Aiu)i,j,k
= ui+l,j,k - Ui,j,k,
(A:u)i,j,k
= Ui,j+l,k-i - ui,j,k.
The numerical Hamiltonian Hnumused in the scheme is chosen to be monotone and consistent with (22). For example, one can use the Osher-Sethian26 upwind scheme is Hnurn(u1,~2,vl,v2) =a
(( min(ul,0>2+ max(u2,0>2+ min(vl,0)2 + max(w2,0)2)’)
which is monotone on the interval [-A, A] if 1- y W ( 2 A ) 2 0. Under the assumption that @ : RN 4 [ O , + o o ) , @ ( O ) = 0 is convex, increasing and
a*(s)
-+
+oo
S
as
s -+ +oo
and some extra conditions on the initial datum g, including 0 0
g has compact support, g is Lipschitz continuous with respect to left translations, S ~ P Z E W 3Ig(z a3 6e3) - 4 z ) I I L l S l ,
in Ref. 1 it is proved that, for some positive constant C,
@:=
(EE)
sup
SUP
i,j,kEZ n=1,
for all 0 < h
...,%
Iu.j,k
- u(pAt,c z , j , k ) l
5C A
< 1.
An important role in the proof is played by the fact, which can be checked using the Hopf-Lax formula (16), that the exact solution of problem (22) is Lipschitz continuous with respect to right translations with some explicitly computable constant L ( t ) .Proof makes use in a crucial way of the typical viscosity comparison technique and is based on ”doubling variables’’ and careful analysis of the auxiliary function
8 : [O,T]x R3 x
(Ji,j,k,nAt),i,j,Ic
E Z,n = 0, ...,
At
148
+ ( D H A ) I ( - E o@ V O ,t o - SO)
5 Chi ,
+ (DH&)I(-
I Chi,
SO)
+ (D~k4)2(-<0@ q o , t o - so) I C h i ,
+ ( D H P ~ ) ~ ( -@ETO, o to -
SO)
I Chi .
We refer to Ref. 1 for full details of the proof and numerical experiments. References 1. Y. Achdou, I. Capuzzo Dolcetta, Approximation of solutions of HarniltonJacobi equations on the Heisenberg group, cpde-preprint yar30073 (June 2006), available at http://cpde.iac.rrn. cnr.it/ricerca.php.
149
2. 0. Alvarez, E.N. Barron, H. Ishii, Indiana University Mathematical Journal, 48,993 (1999). 3. M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton - Jacobi - Bellman Equations, Systems & Control: Foundations and Applications (Birkhauser Verlag, Boston, MA, 1997). 4. M. Bardi L.C. Evans, Nonlinear Anal., 8 , 1373 (1984). 5. G. Barles, P.E. Souganidis, S I A M Journal Math. Anal., 31,925 (2000). 6. E.N Barron, R. Jensen, Comm. Partial Differential Equations, 15, 1713 (1990). 7. R. Beak, B. Gaveau, P.C. Greiner, J. Math. Pures Appl. (9), 79,633 (2000). 8. A. Bellaikhe, The tangent space in sub-Riemannian geometry, in SubRiemannian Geometry, eds. A. Bellache and J.-J. Rider, Progr. Math., Vol. 144 (Birkhauser Verlag, Basel, 1996): 1-78. 9. R. W. Brockett, Control theory and singular Riemannian geometry, in New directions in applied mathematics, eds. P.J. Hilton and G.J. Young (SpringerVerlag, New York, 1982): 11-27. 10. I. Capuzzo Dolcetta, Representations of solutions of Hamilton-Jacobi equations, in Nonlinear equations: methods, models and applications (Bergamo, 2001), Progress in Nonlinear Differential Equations and Applications, Vol. 54 (Birkhauser Verlag, Basel, 2003): 79-90. 11. I. Capuzzo Dolcetta, H. Ishii, Hopf-Lax formulas for state dependent Hamilton-Jacobi equations (in preparation, 2007). 12. M.G. Crandall, P.L. Lions, Math.Comp., 43,1 (1984). 13. A. Cutri, F. Da Lio, Comparison and existence results for evolutive noncoercive first-order Hamilton-Jacobi equations, E S A IM Control Optim. Calc. (SeptemVar. (to appear), available at http://cpde.iac.rm.cnr.it/ricerca.php ber 2005). 14. F. Dragoni, Discrete Contin. Dyn. Syst.-A, 17,713 (2007). 15. F. Dragoni, NoDEA Nonlinear Diflerential Equations Appl., 14,429 (2007). 16. A. Fathi, Weak K A M theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2007). 17. W. H. Fleming, M. H. Soner, Controlled Markou Processes and Viscosity Solutions, Applications of Mathematics, Vol. 25 (Springer-Verlag, New York, 1993). 18. J.B. Hiriart-Urruty, C.LemarBcha1 Fundamentals of Conuea: Analysis, Grundlehren Text Edition (Springer-Verlag, Berlin, 2001). 19. F. den Hollander Large Deviations, Fields Institute Monographs, Vol. 14 (Amer. Math. SOC., Providence, 2000). 20. E. Hopf, J. Math. Mech., 14,951 (1965). 21. D. Jerison, A. Sanchez Calle, Subelliptic second order differential operators, in Complex analysis, I11 (College Park, Md., 1985-86), Lecture Notes in Mathematics, Vol. 1277 (Springer, Berlin, 1987): 46-77. 22. R. LBandre, J. f i n c t . Anal., 74,399 (1987). 23. P.-L. Lions, Generalized solutions of Hamilton- Jacobi equations, Research Notes in Mathematics, Vol. 69 (Pitman, Boston, Mass., 1982).
150
24. J. J. Manfredi, B. Stroffolini, Comm. Partial Differential Equations, 27,1139 (2002). 25. V.P. Maslov, On a new principle of superposition for optimization problems, Uspekhi Mat. Nauk (Russian Math. Surveys), 42,39, 255 (1987). 26. S. Osher, J.A. Sethian, J. Comput. Phys., 79,12 (1988). 27. R.T. Rockafellar, P.R.Wolenski, SIAM J . Control Optim., 39,1351 (2000). 28. A. Siconolfi, Trans. Amer. Math. SOC,355, 1987 (2003). 29. C. Villani, Optimal transport, old and new (Springer, Berlin, 2007).
151
REGULARITY OF SOLUTIONS TO ONE-DIMENSIONAL AND MULTI-DIMENSIONAL PROBLEMS IN THE CALCULUS OF VARIATIONS F. H. CLARKE Institut universitaire de fiance et UniversitC de Lyon Institut Camille Jordan U M R 5208 Universitt! Claude Bernard Lyon 1 La Doua, 69622 Villeurbanne, France E-mail:
[email protected] We review the long-standing issue of regularity of solutions to the basic problem in the calculus of variations, in both the onedimensional and the multidimensional settings. It is shown how certain recent results fit in with the classical ones, in particular the theories of De Giorgi and Hilbert-Haar. Keywords: Calculus of variations; regularity; necessary conditions; existence.
1. Introduction We begin in the middle, with two of the celebrated problems proposed by Hilbert in Paris in 1900:
The 20th problem: Is it not the case that every regular variational problem has a solution, provided certain assumptions on the boundary conditions are satisfied, and provided also, if need be, that the concept of solution is suitably extended? The 19th problem: Are the solutions of regular problems in the calculus of variations always analytic? These questions bear upon the following basic problem in the calculus of variations: to minimize the functional
J ( u ) := over the functions u : SZ
---f
F ( z ,u ( z ) ,Du(x))dx
lR assuming prescribed values on U(x) = ~
( X C )2,
E
r
:= aR:
152
Here R is a domain in Rn:an open bounded connected set, and Du denotes the gradient of u.In fact, Hilbert was referring to the case n = 2 in his problems, but we shall assume only n 2 2 for now; the case n = 1, which is the context in which the subject began (in 1696, arguably, but certainly no later than 1744), has a markedly different character, and will be considered in the final section. Hilbert also took the function F ( x , u , z ) (the Lagrangian) to be analytic; the problem is regular if F,, is positive definite everywhere. We do not discuss in this article the so-called vector case of the problem in which u is vector-valued (that is, in which there are several unknown functions). The decade preceding the formulation of Hilbert’s problems had been marked by a controversy over the Dirichlet principle, which affirms the equivalence between functions u minimizing the Dirichlet functional ( n = 2)
+
and solutions u of Laplace’s equation uzz uyy = 0. As Weierstrass and Hilbert pointed out in response to (notably) Riemann’s assertions, the existence of a minimum here (and the very class in which t o seek one) is problematic. Hilbert went on to give the first rigorous treatment of the issue in 1904, in a context which succeeded in limiting the class of functions u involved to Lipschitz ones. But it became clear that a more general type of function space was needed, and eventually the work of Levi, Tonelli, Morrey, Sobolev and others, led to the theory of Sobolev spaces, which provides a suitable context in which to assert the existence of a solution to the basic problem. In the meantime, however, significant progress on Hilbert’s regularity question was made: (1) If u is C 3 ,then u is analytic [Bernstein 19041; (2) If u is C 2 ,then u is C3 [Lichtenstein 19121; (3) If u is C1ta (that is, has a gradient which is Holder continuous of order (u E (0, l]),then u is C2 [Hopf 19291.
These results lowered the regularity threshold to C1ia;once this level of regularity is present in the solution, then it is as regular as the Lagrangian permits: C‘ ( r 2 2), C”, or analytic. Letting W1>’(s2)denote the usual Sobolev space, we now consider the following reduced basic problem (P): minimize J ( u ) :=
s,
F ( D u ( x ) )dx : u - 4 E Wt”(R)
153
under the following Standing Hypotheses:
> 0 everywhere; > 1: for certain constants
(1) F is of class C2 and F,, (2) F is coercive of order p
F ( z )2 (3)
4 : R"
-+ R
(TIZIP
(T
> 0 and p ,
+ p , z E R";
is Lipschitz.
It is a standard exercise in the theory of Sobolev spaces to invoke the direct method introduced by Tonelli, in which one exploits the weak sequential compactness of a minimizing sequence and the weak lower semicontinuity of the convex functional J , to deduce the existence of a solution u t o problem (P) (u is the unique solution, since J is strictly convex). Note that this is an answer of sorts to Hilbert's 20th problem. The issue now becomes the regularity of the solution u,especially since functions in the Sobolev space W1>l (0)are not even continuous necessarily. The reasons for wanting regularity of the solution u are manifold. For example, continuity of u (or more precisely, of one of its representaives) on the closure of R would mean that the boundary conditions are assumed in the conventional pointwise manner (rather than in the sense of trace). Differentiability of u would mean that Du can be interpreted as the true gradient, and not just the weak distributional derivative. If u is locally Lipschitz in R (or equivalently, Lipschitz on compact subsets of R), then the Euler equation in weak form can be asserted to hold:
V F ( D u ( z ) )* D l c , ( ~dx ) = 0 , 1c, E Cc(R). And finally, in view of the results cited above, local C12" regularity would imply that the full regularity of the Lagrangian F is inherited by u,up t o and including analyticity (as in Hilbert's 19th problem). There are two major 20th century developments on the regularity issue to report, and both were first obtained in the context of the reduced problem (P). We now examine these in turn. 2. The theorem of De Giorgi
The Lagrangian F is said to uniformly elliptic provided that for some E > 0 we have F,,(z)2 €1 for all z E R". It is said to be almost quadratic if for certain constants cg, c1, do > 0 , d l > 0 we have co
+ dolz12
5 F ( z ) 5 c1 + d l ) Z ) 2 ' z
E IWn.
154
The Dirichlet integrand, which is precisely quadratic, is the canonical example of such a Lagrangian. As we know from harmonic analysis, its minimizers are analytic in R. In 1957, De Giorgi15 proved the following celebrated result:
Theorem 2.1. Let F be C2, uniformly elliptic and almost quadratic. Then the solution u t o (P) is locally C1@in R. Note that the boundary function q5 plays no role here, and that the theorem is strictly one of ‘interior regularity’. De Giorgi’s proof proceeded by obtaining a linearized Euler equation for u,to which a new regularity result on elliptic pde’s was then applied to get the required conclusion. This difficult result was introduced in the same article, and other proofs of it were later given by Nash (1958), and by Moser (1960). The main effect of the theorem, from the point of view of the present discussion, is to reduce the regularity threshold to ‘locally Lipschitz’. Let us make this explicit by recording the following simple consequence of the theorem which, surprisingly, is not stated in De Giorgi’s article:
Corollary 2.1. Under merely the hypotheses that F is C2 and regular, i f the solution u t o the problem (P) is locally Lipschitz, then it is locally C’@. This is proved by replacing the original Lagrangian by one which agrees with F on a bounded set containing the values of Du(z) and which is uniformly elliptic and almost quadratic. Then u is still a solution of the problem (by convexity), and De Giorgi’s theorem applies to yield the conclusion. The theorem above has been extended in a variety of ways. Certain evident limits t o such extensions, however, as well as possible grounds for pessimism, arise from certain examples due t o Giusti and Giaquinta. We refer here to special cases of (P) in which the Lagrangian is uniformly elliptic and satisfies
and yet the solution fails t o be continuous in R. In such examples, though, the boundary function q5 is itself discontinuous. This motivates the thought that if some regularity properties were imposed on q5, then perhaps this would induce regularity of the solution u. This idea, which harkens back to Hilbert’s successful analysis of the Dirichlet principle, is precisely the one that underlies the other significant approach t o the regularity issue; we turn to it now.
155
3. Hilbert-Haar theory
The classical Hilbert-Haar approach, which in contrast to Theorem 2.1 makes no additional structural assumptions on F , requires instead that q!~ satisfy the bounded slope condition (BSC) defined below. The ingredients of the theory come from several sources. Hilbert is responsible for the first version of the comparison (or maximum) principle (in the Dirichlet context), which Haar extended to other Lagrangians. Rado introduced the ‘three point condition’, a forerunner of the BSC below. The idea of applying comparison to a translated solution is due to von Neumann. The BSC was formulated and studied in its present form by Hartman and Nirenberg, while Stampacchia21 coined the term BSC and applied it to variational problems. The bounded slope condition of rank K is the requirement that given any point y on the boundary, there exist two afine functions
(c;
(cyf,Y - 7)+ 4(Y) agreeing with 4 at y whose ‘slopes’ satisfy
7
Y - 7)+ 4(YL Y
and such
that
(c;,y’-y)+4(d
M Y l ) 5 (c;,y/-T)+q!J(y)
V’y’ E r .
Let Lip(R) denote the class of globally Lipschitz functions on R. The Hilbert-Haar theorem (see Chapter 1 of Giusti“) is the following:
Theorem 3.1. Let 4 satisfy the BSG of rank K . Then there is a solution u of problem ( P ) when it is restricted to Lip(R), and the solution u is Lipschitz on R of rank K . It is possible to show3~14~18 that the solution of (P) relative to Lip(R) is in fact the solution relative to W1yl(R), so we now deduce
Corollary 3.1. If 4 satisfies the BSC of rank K , then the solution u of problem ( P ) is Lipschitz on 52 of rank K . It is clear that Theorems 2.1 and 3.1, or rather their corollaries, work in tandem to assert the higher regularity of the solution u to (P), as follows: if q!J satisfies the BSC, then u is locally C13” and inherits the full regularity of the Lagrangian up to analyticity. It is natural to ask now how restrictive the BSC is. It is certainly a serious limitation of the allowable boundary conditions on ‘flat parts’ of I?, since it forces q!J to be affine. But the BSC becomes more interesting when
156
R is sufficiently curved. R is said to be uniformly convex if, for some E > 0, every point y on the boundary admits a hyperplane H through y such that dH(y’)
2E
J ~’ yI2
Vy’ E
r.
Miranda’s Theorem20 states that when R is uniformly convex, then any 4 of class C2 satisfies the BSC. Later, Hartman17 showed that when R is uniformly convex and is C’~’, then 4 satisfies the BSC if and only if 4 is itself C’,’. We can say therefore that as a hypothesis, the BSC essentially restricts the boundary data to be smooth. When 4 is not affine, the BSC also forces R to be convex, a hypothesis that we will add to our Standing Hypotheses as we turn now to some new results that center around a weakening of the BSC. 4. New boundary hypotheses
We now turn our attention to the present, or at least the very recent past. We assume throughout this section that R is convex. Clarkeg has introduced a new hypothesis on 4, the lower bounded slope condition (lower BSC) of rank K : given any point y on the boundary, there exists an affine function Y H (&,Y - y) 4(y) with l&yl I K such that
+
Being one-sided, the lower BSC naturally admits a counterpart: an upper BSC that is satisfied by 4 exactly when -4 satisfies the lower BSC.
4.1. Interior regularity The significance of the ‘partial’ BSC hypothesis stems from the following result:9
Theorem 4.1. If 4 satisfies the lower bounded slope condition, then the solution u of ( P ) is locally Lipschitz in R . I n fact, there is a constant we with the property that for any subdomain 0‘ of distance 6 > 0 from have
r,
Thus the one-sided BSC gives the crucial regularity property: u is locally Lipschitz in R. This allows us to assert that u is a weak solution of the Euler equation, in the absence of any restrictive growth conditions on F , and of course allows the application of Theorem 2.1 to deduce higher-order
157
regularity. Informally, it appears that in return for 'half the BSC hypothesis', we obtain considerably more than half the conclusion. Of course the principal thing that has been sacrificed is the continuity of u up to the boundary, but as we shall see below, this can be recovered under a variety of additional hypotheses. As in the case of the BSC, it behooves us to examine the conditions under which the one-sided BSC can be asserted to hold. In this new context, the property that R be curved has less importance than before; flat parts of the boundary do not force q5 to be affine. Nonetheless, curvature can still serve a purpose: Bousquet' has shown that when R is uniformly convex, then q5 satisfies the lower (upper) BSC if and only if it is the restriction to I' of a function which is semiconvex (semiconcave), a familiar and useful property in pde's (see for example Ref. 5). In the uniformly convex case, therefore, Theorem 4.1 extends Hilbert-Haar theory to boundary data that is semiconvex or semiconcave rather than C2 (or C1il). We remark that the proofs of Theorems 3.1 and 4.1 have something in common: both of them construct a new minimizer from u itself with which to compare u. In the classical case, this is done by translation. The principal new idea in the proof of Theorem 4.1 is to construct a new minimizer through dilation rather than translation. 4.2. Continuity at the boundary
In a variety of situations, it turns out that the lower or upper BSC does imply continuity at the boundary, and even a global Holder condition in some cases. A counterexample due to Bousquet and Cannarsa (in the Dirichlet context) shows, however, that the gradient of u can become unbounded.
Theorem 4.2. Suppose that in addition t o the hypotheses of Theorem 4.1, one of the following holds:
(a) I' is a polyhedron, or (b) I' is Cl?' and p > ( n 1)/2, or (c) R is uniformly convex.
+
T h e n u is continuous o n dition o n
n.
a. I n cases (a) and (b), u satisfies a Holder con-
We remark that we know of no example in which (under the hypotheses of Theorem 4.1) the solution fails to be continuous o n n . Indeed, we know of no example in which q5 is Lipschitz and u fails to be continuous.
158
4.3. More general Lagrangians
It turns out t o be challenging to extend the new results given above to Lagrangians depending on x and u as well as Du, principally because the comparison principle does not hold in that case. However, using the technique of barrier functions, Bousquet and Clarke2 have obtained a result for Lagrangians of the form F ( D u ) G(x,u ) ;we describe it now. We assume that F is uniformly elliptic, that G ( x , u )is measurable in x and differentiable in u, and that for every bounded interval U in R there is a constant LU such that for almost all x E R ,
+
IG(x,U ) - G ( x ,u ’ ) I
5 L U ~-Uu‘I
V U ,U‘
E
U.
We also postulate that for some bounded function b, the integral G(z, b(x))dx is well-defined and finite.
s,
Theorem 4.3. Under these hypotheses, and when q5 satisfies the Lower or Upper BSC, any bounded solution u of the basic problem is locally Lipschitz in R. We remark in connection with this result that it is possible to formulate additional hypotheses on the data which imply a priori that any solution to the basic problem is bounded, and that additional structural hypotheses lead as before to continuity a t the boundary.
5. The one-dimensional case Finally, let us address the beginning of the subject: the one-dimensional case, for which there is always a different notation. The basic problem (P) now corresponds to the minimization of the functional
J ( x ) :=
Jd
b
F ( t ,x ( t ) ,x’(t))dt
over the functions x : [a,b] 4 RN in some given class X , and subject to prescribed endpoint conditions: .(a) = A , x(b) = B . Note that we now allow N > 1, the case of several unknown functions; the generic name for the variables becomes ( t ,x , v) rather than (z, u, 2 ) . For Euler and his contemporaries, all functions were smooth, so the issue of the regularity of the solutions did not arise; implicitly, X was a space of very smooth functions. On a more rigorous level, when the degree of smoothness becomes a consideration, we can deduce that if x is C1to start with, and if F is regular (which continues to mean F,, > 0) and at
159
least C2, then z inherits the full degree of regularity of F , up to analyticity. This is known as the Hilbert-Weierstrass theorem (circa 1890), and is a consequence of the Euler equation together with the implicit function theorem. By the 19th century, however, the possible nonsmoothness of solutions began to be recognized as an important point, in view of such concrete evidence as nonsmooth soap (minimal) surfaces. Results were obtained for the class PWS of piecewise smooth functions, notably. An important breakthrough was duBois-Reymond’s proof of the integral form of the Euler equation: if x solves (P) relative to PWS, then there exists a piecewise smooth function p such that
( P ’ ( t ) , P ( t > ) = VfYt,xc(t>,x‘(t)) at all non-corner points; here, V F refers to the gradient in the ( x , v )variables. This condition subsumes the earlier Erdmann condition, and like it, can sometimes be used to deduce the smoothness of x (in this setting, the absence of corners). Note however that x has to be assumed piecewise smooth a priori. This is unfortunate, since, although suitable necessary conditions can be asserted in PWS, the class of piecewise smooth functions is of no help in regard to the existence issue. It was Tonelli’s major contribution to show that existence theory can be developed successfully in the class AC of absolutely continuous functions. But within AC, the ability to derive the Euler equation fails in general, so our steps forward (on existence) are accompanied by one step back (on the necessary conditions). A way out of this quandary is to find reasonable supplementary hypotheses on the Lagrangian which will imply that x is Lipschitz. The reason for this is that all the classical results for the class PWS carry over to this class (now that Lebesgue has given us his integral). In other words, just as in the multi-dimensional case treated in the previous sections, the regularity threshold is situated at Lipschitz continuity of the solution. Of course, in the one-dimensional case there is little help to be found from examining the boundary conditions (as in the Hilbert-Haar theory). In fact, the methodology used to obtain regularity theorems has been overwhelmingly based upon analyzing the necessary conditions. This is in stark contrast to the multi-dimensional case, where the necessary conditions don’t seem to yield very much directly. We conclude therefore that our best hope to extend regularity theory lies in the possibility of deriving stronger necessary conditions in more general circumstances. We proceed to report on just such a recent development. For this pur-
160
pose, we examine a Lagrangian F ( t ,x , w) which is merely measurable in t and (z,w) (see" for the precise meaning of this) and lower semicontinuous in ( x ,w). No hypotheses of smoothness or convexity are made. Instead, we assume the following generalized Tonelli-Morrey condition [GTM]: for every bounded subset S of Rn there exist a constant c and a summable function d such that for almost every t , for every ( x ,V) E S x R", for every (C,1cI) E dpF(t,x,w), one has
Here, dpF refers t o the proximal subgradient of F with respect t o the ( x ,w) variables, a basic construct in nonsmooth analysis. When F is C2 (or somewhat less), the growth condition of [GTM] is equivalent to
ID81
I c1 {IF1 + 14 + IDWFI) + d l ( t ) + {c2 (IF1 + )1.
+ d2(t)}I D W F I .
The special case c2 = d2 = 0 corresponds to a class of Lagrangians that has been considered by Clarke and Vinter13 in connection with regularity. In this case, and when in addition the term involving ID,FI is placed on the left side of the inequality, as follows:
(thereby making the condition a more stringent hypothesis), we obtain a growth condition first postulated by Tonelli in order to be able t o derive the necessary conditions in the class AC. In the much more general setting now being considered, the new [GTM] still has the same effect: it allows one to deduce necessary conditions that must be satisfied by any solution to (P). These conditions have the nature of the classical ones, but expressed in such a way as t o take account of the nonsmoothness of F . Here is just one such result taken from Clarke:lo
Theorem 5.1. Let x be a solution to (P) relative to the class AC, where F satisfies the generalized Tonelli-Morrey growth condition [GTM]. Then there exists an arc p satisfying the Euler inclusion
p ' ( t ) E co {w : ( w , p ( t ) )E d L F ( t , x ( t ) , x ' ( t ) ) }a.e.,
and the Weierstrass condition: for almost all t E [a,b] we have F ( t ,z ( t ) ,V) - F ( t ,~ ( t x)' (, t ) ) 2 ( p ( t )v, - x'(t)) V v E R".
161
The Euler inclusion in this statement involves the limiting subdifferential &F; it reduces to the integral form of the Euler equation when F is smooth. The Weierstrass condition is the familiar one of the classical theory. This theorem admits a more general form in which the cost depends on the endpoint values of the arc x , which need be only a local minimum in a specified sense. Beyond this, and most significantly, the theorem can be stated for extended-valued Lagrangians F , so that problems in optimal control are subsumed by it. But these are chapters in a different story, so let us proceed instead with our quest for regularity consequences. We say that the Lagrangian F is coercive if for any bounded subset S of Rn there exists a function 6 : [0, co)--f R satisfying 8 ( r ) - +co lim -
and such that
q t ,x ,
2 e(lvl)
v (t,x , v ) E [a,b1 x s x I W ~ .
We remark that coercivity is a familiar ingredient in the theory of existence of solutions. The symbiosis between necessary conditions and regularity is well illustrated by the following new result.
Corollary 5.1. If x solves (P) relative to the class AC, where F is coercive, bounded above on bounded sets, and satisfies the generalized Tonelli-Morrey growth condition [GTM], then x is Lipschitz on [a,b]. Proof. In view of Theorem 5.1, we know that an arcp exists which satisfies the Weierstrass condition. Let M be an upper bound on
for t E [a,b ] , and let 0 be a coercivity function for F when the variable x is restricted to the bounded set consisting of the values of z ( t ) on [a,b]. Then, taking v := x’(t)/(l /x’(t)l)in the Weierstrass inequality leads to (almost everywhere)
+
e(lx’(t)l)5 F ( t , x ( t ) , x ’ ( t ) )5 M
+ IP(t)l Ix’(t)l.
Since Ip(t)l is bounded and limT+mO(r)/r = +a, it follows from this 0 inequality that x’(t) is essentially bounded on [a,b]. This corollary even extends (relative to previous results) the class of smooth Lagrangians for which the necessary conditions can be asserted and
162
regularity inferred. A simple exampleof this is provided (for N = 1) by
+ +t2)w2} .
F ( t , x,w) = exp { (1 x 2
This Lagrangian satisfies the hypotheses of the classical Tonelli existence theorem as well as those of the corollary. Thus a solution to (P) over AC exists, and any solution x is Lipschitz. Because F is strictly convex in w, it then f01lows'~that x is C1, and finally we derive all the higher regularity of x from the Hilbert-Weierstrass theorem. There are other structural hypotheses yielding regularity results that serve to highlight the extremely general Lagrangians that can be treated in the one-dimensional case, as compared to the very special structure that seems to be required in the multi-dimensional case. We end the discussion with one further example," which asserts the regularity of the solution for one-dimensional problems having autonomous Lagrangians ( F is called autonomous when it has no explicit dependence on the variable t ) .
Theorem 5.2. Let x solve (P) relative to AC, where the Lagrangian F is coercive, bounded above on bounded sets, and autonomous. Then x is Lipschitz o n [a,b ] . This result (among others that we do not touch upon here) was first proved by Clarke and Vinter13 under the added requirement that F be locally Lipschitz and convex in w. A more direct and simplified proof in that setting can be given.ll The literature on the venerable subject of regularity in the calculus of variations, and on the inextricably linked issues of existence and necessary conditions, is huge and still growing. We make no attempt here to be complete, but rather we refer the interested reader to the classic books of Giaquinta, Giusti and Morrey, and to the representative (but not exhaustive) more recent references appearing below, in which detailed bibliographic information appears.
References 1. P. Bousquet, J . Convex Analysis 14, 119 (2007). 2. P. Bousquet and F. Clarke, J. Differential Equations 243, 489 (2007). 3. G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon, in Recent Developments in Well-Posed Variational Problems, eds. R. Lucchetti and J. Revalski (Kluwer, Dordrecht, 1995) pp. 1-27. 4. G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, Oxford Lecture Series in Mathematics and its Applications, Vol. 15 (Clarendon Press, 1998).
163
5. P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton- Jacobi Equations, and Optimal Control (Birkhauser, Boston, 2004). 6. A. Cellina, SIAM J. Control Optim. 40, 1270 (2001). 7. F. H. Clarke, Regularity, existence and necessary conditions f o r the basic problem in the calculus of variations, in Contributions to the modern calculus of variations, ed. L. Cesari (Longman, London, 1987) pp. 80-90. 8. F. H. Clarke, Trans. Amer. Math. SOC.336,655 (1993). 9. F. Clarke, Ann. Scuola Norm. Sup. Pisa C1. Sci. (5) 4, 511 (2005). 10. F. Clarke, Necessary Conditions in Dynamic Optimization, Memoirs of the Amer. Math. SOC.,No. 816, Vol. 173, 2005. 11. F. Clarke, Ergodic Theory Dynam. Systems 27,1 (2007). 12. F. H. Clarke and R. B. Vinter, J. Differential Equations 59,336 (1985). 13. F. H. Clarke and R. B. Vinter, 'Pans. Amer. Math. SOC.289,73 (1985). 14. R. DeArcangelis, Ann. Univ. Ferrara 35,135 (1989). 15. E. D. Giorgi, Mem. Accad. Sci. Torino 3,25 (1957). 16. E. Giusti, Direct Methods in the Calculus of Variations (World Scientific, Singapore, 2003). 17. P. Hartman, Pacific J. Math. 18,495 (1966). 18. C. Mariconda and G. Treu, Proc. Amer. Math. SOC.130,395 (2001). 19. C. Mariconda and G. Treu, J. Optim. Theory Appl. 112,167 (2002). 20. M. Miranda, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19,233 (1965). 21. G. Stampacchia, Comm. Pure Appl. Math. 16,383 (1963).
164
STABILITY ANALYSIS OF SLIDING MODE CONTOLLERS F. H. CLARKE Institut universitaire de h n c e et UniversitC de Lyon Institut Camille Jordan UMR 5208 UniversitC Claude Bernard Lyon 1 La Doua, 69622 Valleurbanne, France E-mail: clarkeQmath.univ-lyonl.fr http://math. univ-lyonl .fr R. B. VINTER Department of Electrical and Electronic Engineering Imperial College, London hibition Road, London SW7 2BT, UK E-mail: r.vinterQimperia1. ac. uk http://www3. imperial. ac.uk In this paper we announce a new framework for a rigorous stability analysis of sliding-mode controllers. We give unrestrictive conditions under which such feedback controllers are robustly stabilizing. These conditions make allowance for large disturbance signals, for modeling, actuator and observation measurement errors, and also for the effects of digital implementation of the control. The proposed stability analysis techniques involve two Lyapunov-type functions. The first is associated with passage to the sliding surface in finite time; the second, with convergence to the state associated with the desired equilibrium point. Application of the techniques is illustrated with reference to higher-order linear systems in control canonical form. Keywords: Sliding Mode Control, Discontinuous Control, Lyapunov Functions, Feedback, Stabilization
1. Introduction
Sliding-mode control is a presence of disturbances. geon and Edwards3 and a subset C of the state
technique for feedback stabilization, in the We refer to the books of Utkiq7 SpurSlotine and LL6 The state is driven to space, the sliding surface, in finite time.
165
Subsequently, the state trajectory remains in C and moves asymptotically to a value consistent with the desired equilibrium point: See Figure 1.
Fig. 1. A Closed-Loop State Trajectory under Sliding-Mode Control
A notable feature of sliding-mode controls is their stabilizing properties for dynamic systems, in the presence of large disturbances. Sliding-mode controllers, relating the control u to the current state z, commonly take the form
in which g(z) and x(z) are respectively smooth and discontinuous terms. The purpose of the discontinuous term ~ ( z is) to force the state to approach C at a uniformly positive rate (controllers containing such terms are said to satisfy the ‘sliding condition’). The continuous term g(z) is preliminary feedback, configuring the system so that the sliding condition can be achieved. Since, if ever the state trajectory departs from C, the controller drives it back towards C, we expect that the state trajectory remains in C in some sense.
166
If sliding-mode control is implemented digitally, with a high sample rate, the control values generated by the control law are typically observed to switch rapidly, after the state trajectory first crosses C, in such a manner that the state trajectory remains close to C, and lies in C in the limit, as the sample period tends to zero. Traditional approaches to analysing the stabilizing properties of sliding-mode controllers, expounded for example by Utkiq7 are along the following lines. Since rapid switching keeps the state close to C,we might expect that the state trajectory will be similar to that generated by an ‘equivalent’ control law,
in which xequiv(x) is a smooth function, chosen to ensure that the time derivative of the evolving state vector is tangent to C. Classical techniques (Lyapunov theory or eigenvalue analysis in the linear case) can be used to study the stability properties of the equivalent control law and, by implication (perhaps) those of discrete implementations of the original control, in the limit as the sample period tends to zero. We present analytic tools for a rigorous stability analysis of sliding-mode controllers. Broad, unrestrictive conditions are given under which these controllers are stabilizing, in the presence not only of large disturbances, but also of modeling, actuator and observation errors. These conditions invoke the existence of two Lyapunov type functions Vl and V2,the first associated with passage to the sliding surface in finite time, and the second with convergence to the state associated with the desired set point. The approach takes account, from the beginning, of implementational constraints. The main stability theorem concerns the stabilizing properties of a ‘sampleand-hold’ implementation of the feedback controller (l),in the presence of modeling, actuator and measurement errors, when the bounds on these errors and the sample period are suitably small. The stability theorem is accompanied by a description of the ‘dynamics’ of the state trajectory, in the form of a differential inclusion, in the limit as the actuator and measurement errors, and also the upper bound on sample period, tend to zero. The advantages of our approach over the classical one are as follows. First, by examining the behaviour of the sample-and-hold implementation of the controller directly we avoid the additional hypotheses and the analytical apparatus in Utkin7 associated with defining Filippov solutions to the continuous time dynamic and controller equations (via equivalent controls), and also the difficulties of choosing the ‘correct’ equivalent control from among the candidate equivalent controls, when they are not unique.
167
Second, our approach tells us that the controller is robust: it retains its stabilizing properties when we take account of implementational constraints and small modeling, actuator and measurement errors. The sliding-mode stability tools introduced in this paper are used to demonstrate the stabilizing properties of a class of sliding-mode controllers applied to a general nth-order linear system in control canonical form, with large disturbances. The controllers employed here are well-known; see, e.g., the books of Spurgeon and Edwards and of Slotine and Li,3>6where emphasis is on the second-order case and the ‘disturbances’ describe, in part, departures from linearity of a nonlinear model such as the rigid pendulum. Our analysis brings to these old controllers new insights, however, regarding their robustness to modeling errors and the effects of digital implementation. The systematic use of a pair of Lyapunov functions appears to be a novel approach to analysing the stability properties of digitally-implemented sliding-mode control systems, in the presence of modeling and measurement errors. But this paper includes many ingredients from earlier research. In particular, the sliding-mode controller is discontinuous; the use of a single control Lyapunov function to establish the stabilizing properties of sampleand-hold implementations of discontinuous feedbacks has been systematically studied, and is the subject of numerous papers, those of Clarke et a1.lI2 for example. The link between the existence of a single C2 control Lyapunov function and robustness of the corresponding control system to measurement errors is also very well understood; see for example Refs. 4,5. A future paper, besides providing full details of the proofs of the stability results announced here, will report on their application to the analysis of new sliding-mode controllers for nonlinear systems, such as the nonholonomic integrator. It will show that, with small changes, the stability analysis presented here can be applied when the ‘delay-free sample-and-hold digital control’ implementation scheme considered in this paper is replaced by a wide range of alternative implementation schemes (involving, for example, time delays, pre-filtering of the control system, and various forms of ‘regularization’ of the discontinuous sliding-mode control law). In Euclidean space, the length of a vector z is denoted by 1x1,and the closed unit ball {z : 15 - a1 5 R} by B(a,R). &(z) denotes the Euclidean distance of the point z from the set C, namely min{ Iz - 2’1 : z’ E C}.
168
2. System Description
Let { t o = 0, tl . . . , } be an increasing sequence of numbers such that ti -+ 03 as i + 03. (We refer to such a sequence as a 'partition of [0,00)'.) Consider the dynamic system, governing the evolution of the closed-loop state trajectory x(.),described as follows: a.e. t E [0,03) f (4%4%4 t ) ) u ( t ) = ui + g ( x ( t i )+ mi) + vi a.e. t E [ti,ti+l), i = 0 , 1 , 2 , . . . 2 = 0,1,2,. . * wi = x(z(t2)+mi) d(t) E D a.e. t E [O, 00) , k(t) =
in which f : Rn x R" x Rk given functions satisfying
-+
R",
g : R" -+
x(x) C V for all x
E
R" and R"
x : R"
+
]
(2)
R" are
,
and D c Rk and V c R" are given sets. In the above equations, d(.) : [0,00)-+ Rk is a measurable, essentially bounded function describing the disturbance signal. The sequences { ui} and {mi}describe the n-vector actuator errors and m-vector measurement errors at successive sample instants, respectively. The following hypotheses will be imposed: ( H l ) f and g are continuous and of linear growth: there exist c f , c g > 0 such that
for all (x,u,d) E R" x R" x D I c f ( l + 1x1 +)1.1 Ig(x)l I cg(l + 1x1) for all x E Rn. For any bounded sets X c R" and U c R", there exists K
If(x,u,d)I (H2)
=
K ( X ,U ) > 0 such that the following Lipschitz condition holds:
If
(x,u,d ) - f (x',u,d)l 5 Klx - x'1
for all x,x' E XI (u, d) E U x
D
(H3) V and D are closed bounded sets. 3. Lyapunov Functions for Sliding Mode Control
We assume that the control feedback design
u = g(x) + x ( x ) has been carried out on the basis of a Lyapunov stability analysis, in ignorance of the measurement and actuator errors and on the basis of a possibly
169
inaccurate dynamic model: j: =
f o ( x , u , d ).
To study the effects of sliding-mode control via Lyapunov stability analysis, it is helpful to introduce not one, but two Lyapunov-like functions V1 : R" -+ [ O , o o ) and & : Rn -+ [0, co),a decrease function W : R" -+ [0, co) associated with V2, and a subset C c R" of the state space (the 'sliding set'). V1 will be used to capture the property that the sliding-mode control drives the state into a 'boundary layer' about C, in finite time. V2 is associated with the subsequent motion of the state to a neighbourhood of the origin. Vl, V2 and W will be required to satisfy the following conditions. (LF1): V1 is continuous. & is of class C1 on Rn\C. Furthermore, there exists w1 > 0 such that
+
VVl(z). fO(x,g(z) x ( x ) , d ) < -w1 Vl(z) = 0 for z E C
for all z E R"\C,
dED
&(x) > 0 for z $ C. Now write
n
Fo(z)':=
{ f O ( x ' , g ( z '+) v ' , d ) : z1 E B ( x , p ) ,v'
E
~ ( x ' )d ,E D } .
P>O
We may think of Fo(x)as consisting of all possible velocity values j: (in limiting terms, and for the nominal dynamics given by fo) when the state is at x. (LF2): & and W are continuous. Vz is of class C' on R"\{O}. Furthermore sup
(VV2(x),w) 5 -W(z) for all IC E C\{O}.
wEFo(x)
V2(0) = 0 and V2(z) > 0 for z E C\{O} W ( 0 )= 0 and W ( z )> 0 for x E C\{O}. (LF3): V1 + V2 is a proper function; i.e., for any a {z : Vl(z) Vz(2) 5 a } is bounded.
+
2 0, the
sub-level set
4. Sufficient Conditions for Stability
In this section we give sufficient conditions for stabilization to the origin, via sliding-mode control. Since the control system description we have adopted
170
takes account of digital implementation and allows for disturbances, we cannot expect state trajectories to converge to the zero state, as t tends to a. Instead we give conditions for practical semiglobal stability, i.e. conditions under which, for any two balls B(0,R) (the set of initial states) and B ( 0 , r ) (the target set) in R", R > r > 0, a state trajectory that issues from the set of initial states is driven to the target set in bounded time, and remains in the target set thereafter.
Theorem 4.1 (Conditions for Practical Semiglobal Stabilization). Assume (Hl)-(H3). Suppose there exist functions V1, VZ and W ,and a set C satisfying hypotheses (LFl)-(LF3). Choose any numbers R > 0, r > 0 (R > r), ij E (0, w1) and 5 > 0. Then there exist positive numbers R, > R (R, depends only on R), em, e,, 6, e l , e2 and T > 0 , with the following properties: Take any sequences { m i } and {ui} in Rm and R" respectively, partition { t i } , measurable function d : [0,00) -+ D and zo E B(0,R ) satisfying /mil I em,
[ail5 e,
and
Iti+l - ti1
I6
f o r all i.
(3)
Suppose in addition that we have I ( V v l ( . ) , f ( ~ , g ( z ) + ~ , d-)f 0 ( ~ 1 9 ( ~ ) + v , d ) )I l el for all w E ~ ( z )IL:, E B(0,R,)\C,
dED
(4)
and I ( V V 2 ( ~ ) , f ( ~ , g ( ~ ) + ~ , d ) - f 0 ( I L : , g ( ~ ) +I~e2 ,d))I for all w E ~ ( z )z, E B(0,R,)\{O}, d E D. (5)
Let z(.) : [0,00) 4 R" be any solution to eqn. (2) with z(0) = zo. (One such solution exists). Then
z ( t ) E B(0,R,) for all t 2 0 and
z ( t ) E B(0,r) for all t 2 T . Furthermore, d c ( z ( t ) ) I5 for all t
E
[V~(z(O)/ij, 00).
A proof of the theorem will be given in a future publication. We remark that the way in which the modeling error f - f a is constrained in (4)and (5) (relative to inner products with the gradients of V1 and Vz) leads to a new interpretation of 'matched' errors.
171
A standard 'sequential compactness' analysis, combined with an application of the results of the preceding theorem, provides information, summarized as Prop. 4.1 below, about the set of solutions to eqn.(2), in the limit as the sample period tends to zero and modeling, actuator and measurement errors vanish. Full details will be reported elsewhere. Let Q : R"
@ ( x ):=
us
R" be the set-valued function
n
+
co {f( x ,g(x) v,d ) I v = x ( x ' ) ,x' E B ( x ,P ) , d E D} . (6)
0 0
Here, c o S denotes 'closed convex hull of S', and T s ( x )is the tangent cone to the closed set S at a point x E S:
T s ( x ) := {v E Rn : 3vi
-+ v
and
~i
1 0 s.t. x + ~ j v Ei S for all i } .
Proposition 4.1. Assume (H1)- (H3). Suppose that there exist functions V1, Vz and W satisfying (LF1)-(LF3) (when f replaces fo). Take a measurable function d ( . ) : [0,03) + D and families of sequences of n-vectors {mjo,mj1,.. and of m-vectors {ajo,a j l , . . .}plsatisfying
.}zl
lim sup ImjiI
j+w
a
+ 0,
lim sup lajil + 0 .
j-+w
i
Let {tjo,tjl,.. .}F,be a family of partitions satisfying
Fix xo E Rn . For each j , let x j ( . ) be a solution to eqn.(Z) (with initial value xo), when {tjo,t j l , . . .}, {ajo, a j l , . . .} and {mjo,mjl, . . .} replace {to,t l , . . .}, {ao,a1,... } and {mo,m l , . . .} respectively. Then there exists a locally Lipschitz continuous arc x(.) : [0,co) + R" and T I 2 0 such that ~,'Vl(XO) Tl I and, for some subsequence (we do not relabel), xj(.) + x(.) as j + 03 uniformly on bounded subsets of [0,co). We have
Rn\C for t < T I fort2T1,
x(t)E{C
x(t)
---f
0
as t
--+ 03
172
and, for a.e. t E [ O , c o ) ,
{
,qt)
Q(4t)) ift
TI
I
This proposition describes the behaviour of state trajectories under ‘continuous time’ control feedback, when the response to continuous time control feedback is interpreted as the limit of a sample-and-hold implementation, as the sample period goes to zero. It asserts that, if there exist a pair of Lyapunov functions with the stated properties, the state trajectories reach the switching surface in finite time. They then move asymptotically towards the origin, while remaining in C. The proposition also provides details of a differential inclusion that the state trajectory satisfies, after it has entered C.
5 . An Example
Consider the linear system in control canonical form, relating the n-vector state z ( t ) to the scalar control u(t): d -z(t) dt
=
+
Az(t) b(d(t) + u(t))
(7)
in which
0 0
1
o..o
0 0
0 10.0
. ....
A = 0 -a0
and
b =
oo..o 1 -a1 . . . -un-l *
Here ao, . . . , a,-l are known parameters, d is an unmeasured scalar disturbance signal, assumed to satisfy, for some given dmax(> 0), the condition dmax Id(t>l I We seek state feedback u
=
4b),
to achieve closed loop asymptotic stabilization, for arbitrary disturbance signals d(.). This design problem is addressed in the sliding-mode literature as follows (see, e.g., Ref. 3). Fix coefficients XO, . . . ,X,-1 of a stable polynomial, of degree n - 1, X(a) = A0
+ X1a + . . . + Xn_2an--2 + an-l
and define the scalar-valued function of the state s(z) = Aoz1
+
A1z2
+ ... +
+ z,
An-2zn-1
*
Define also the k vector
k Fix K
= col{ao, a1 - Ao,
. . . ,an-1
- An-2}
.
> dmax. The following control law has been proposed: =
1 'L
$(x),
where $(z) = k T z - Ksgn{s(z)}.
Here, sgn(.) is the 'signum function' sgn(s) =
+1 if s > 0 -1 if s 5 0 .
The rationale here is that, if we substitute control law (9) into (7) and take account of the fact that $xi = zi+l, for i = 1,.. . ,n - 1, there results (in the case s ( z ( t ) )# 0 ) d
d
+ An-zzn + . - + X O ~ Z ] = sgn{s(z(t))}[-aozl+ . . . - ~ , - ~ z + , aoz1
zls(z(t))l = sgn{s(z(t))}[zzn
+
+ .. .+ -Ksgn{s(z(t)} + d(t) + An-zzn + . . . + AOZZ] +(a1
- AO)XZ
(an-1
An-2)~n
+ Id(t)l
I -Ksgn{s(zc(t)}sgn{s(zo) I -w,
where w = (K-dmax) ( > 0 ) . These calculations suggest that, if s(z(0)) 0, then z ( t ) arrives at the set
c
= {z : s(z) = O}
#
(11)
at a positive time T (T I Is(z(O))I/(K- dmax)), where it remains thereafter. Furthermore, for t > T we have that dn- 1 dn-2 s(z(t)) = -X I + An-221 Aoz1 = 0 . (12) dtn-l dtn-2 From this we can surmise that z l ( t )(and hence also z z ( t ) ,. . . , , z), converge to zero, as t 4 0. The methods of this paper can be used to justify these conclusions, taking account of implementational effects and the presence of modeling and measurement errors. We show:
+...+
174
Proposition 5.1. Take R > 0 and r > 0 (R > r). Take also dmaa: > 0 and K > dmax. T h e n there exist positive numbers 8 > 0, m > 0, $ > 0 and T > 0 with the following properties:
Take a n y partition { t i } , measurable function d ( . ) quences {ai} and {mi} satisfying Iti+l - ti1
:
[O,m)-i R, and se-
< 8, lail 5 a, Imil 5 % f o r all i, lld(.)llp= 5 dmaa:.
Then, for a n y zoE B(0,R ) , the solution x : [0,m) 4 R" t o
+
+
k ( t ) = A x ( t ) b ( d ( t ) u ( t ) ) ) a.e. t E [0,m) u ( t )= ai +(%(ti) +mi) a.e. t E [ti,ti+l) x ( 0 ) = 20
+
satisfies
x ( t ) E B ( 0 , r ) for all t 2 T . Here, +(.) is the mapping (10).
Proof. We apply Thm. 4.1, making the following identifications: f (x,u, d ) = Ax
D
+ b ( u + d ) , g(x) = k T x , x ( x ) = Ksgn{s(x)}
= [-dmax, Sdmax] and
V = [-K, +K] .
The hypotheses (Hl)-H(3) on the dynamics are satisfied for this choice of data. The assertions of the propositon will have been proved, if we can construct functions &, fi and W satisfying the conditions (LFl)-(LFS). Take Vl : R" 4 R+ to be the function
Vl
= Is(x)I,
where s(-) is the function (8). The sliding set is
c = {x 1 &(x)
=0) =
{x 1 s(z)
= 0).
V1 is continuous, of class C1 on Rn\C. It can, furthermore, be shown that, for any x E Rn\C:
OK).( . f (x,g ( x ) + X b ) , d ) 5 -w1 where w1 is the positive number w
= K - dmax.
Thus,
7
V1
satisfies (LFl).
175
With a view t o constructing
in which
V2
and W , we introduce the matrices
is the vector
x
= Col(x0,.
. . , A n-2} .
Note that J has full column rank and that C = range{ J } . It follows that J T J is invertible and, given any x E Rn, there exists a unique E Rn-' such that a: = JE. Since A is a 'stable' matrix, there exists a symmetric matrix P and Y > 0 and c > 0 such that
(See Ref. 8.) Define
Po
=
in which
[Po]
0 is the ( n- 1) vector col{O, . . . ,0). Finally we set ' T Vz(a:)= -x Pa: for all z
2-
E
Rn
The function W is taken t o be
W ( Z )= y l [ J T J ] - 1 J T ~ 1, 2 The functions V2 and W constructed in this way are continuous, vanish a t the origin, and are of class C2. Note also that, if x E C\{O}, then z = JE for some 5 # 0. It follows that
W ( z ) = -y1<12 > 0 and &(x) = ETPE > 0
,
i.e. Vz and W are positive on C\{O}. It can also be shown that, for any x E C\{O}, the decrease property of (LF2) holds. It is a straightforward matter to show that V1 +Vz is proper (condition (LF3)). All the hypotheses of Thm. 4.1 are satisfied. The assertions of the proposition follow.
176
References 1. F. Clarke, Y . S. Ledyaev, L. Rifford and R. Stern, SIAM J . Control Optim. 39,25 (2000). 2. F. H. Clarke, Y . S. Ledyaev, E. D. Sontag and A. I. Subbotin, IEEE Duns. Automat. Control 42, 1394 (1997). 3. C. Edwards and S. K. Spurgeon, Sliding Mode Control (Taylor and Francis, London, 1998). 4. Y . S. Ledyaev and E. D. Sontag, Nonlinear Analysis 37,813 (1999). 5. Y . Lin, E. D. Sontag and Y . Wang, SIAM J. Control Optim. 34, 124 (1996).
6. J. J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, Englewood Cliffs NJ, 1991). 7. V. I. Utkin, Sliding Modes in Control Optimization (Springer Verlag, Berlin, 1992). 8. J. L. Willems, Stability Theory of Dynamical Systems (Thomas Nelson and Sons, London, 1970).
177
GENERALIZED DIFFERENTIATION OF PARAMETERIZED FAMILIES OF TRAJECTORIES M. GARAVELLO
Dipartimento d i Matematica e Applicazioni, Universitb d i Milano-Bicocca, 20125 Milano, Italy, Present address: Di.S. T.A . , Universitb del Piemonte Orientale, 15100 Alessandria, Italy E-mail: mauro.garavel1oQmfn.unipmn.it
E. GIREJKO Bialystok Technical University, ul. Wiejska 45a, 15-351, Bialystok, Poland. E-mail: [email protected] B. PICCOLI
I.A.C., C.N.R., 00161 Roma, Italy E-mail: b.piccoliQiac.cnr.it We consider the problem of weak differentiability for a parameterized family of trajectories. The study is motivated by various problems in ordinary differential equations and control theory. Some results are presented and their applicability is shown by means of few examples. The methods are based on generalized differentiation of flows or, more generally, of set-valued maps.
Keywords: Parameterized families of trajectories; generalized differentials; necessary and sufficient conditions for optimality.
1. Introduction
This paper deals with weak differentiability of parameterized families of trajectories of vector fields. This topic is extensively studied in the literature, since various problems in mathematical analysis can be formulated in such a way. First let us describe precisely the setting. Let P be a normed space and consider the following family of Cauchy
178
problems
where p E P is the parameter, f p : R" x [0, +m[+ Rn is a parameterized family of vector fields and z p E Rn a parameterized family of initial conditions. We indicate by xp a solution to (l),thus we get a parameterized family of trajectories (in case of non uniqueness of solutions to (1) one has to take a selection). We list some examples of problems which can be stated in terms of differentiability of the family xp, at some time in the domain of definition, with respect to the parameters. Differentiability with respect to initial data. The most classical example is the differentiability of a trajectory of a C1 vector field with respect to the initial data. In this case x,p E R", and for fixed po E R" we set
{ o:v:fL
xp0(O)
+
21,
where v E R". It is well known that the family is differentiable at every time (at which solutions are defined) and is expressed by the solution of the adjoint matrix equation:
where fx is the Jacobian matrix of f , in the sense that d / d p xP(t)Ip=po=
V(t). When the vector field f is not smooth, especially when f is not continuous or it has not the uniqueness property for trajectories, then this result does not hold in general. Hence the necessity to introduce weaker concepts of differentiability. For this, various results on generalized differentials of maps may be used, see Refs. 2,3,11,13,19,22-25,28. Differentiability with respect to the vector field. The second example is the investigation of how the solution to a Cauchy problem for the same initial data depends on the vector field. Let us consider directly discontinuous vector fields. For instance, fix a time dependent family of discontinuous vector fields fE : R x R 4 R+ ( E 2 0) with exactly one discontinuity at z > 0, so that each fE can be written in the form
f;(x,t), if x 6 3, f,S(x,t),if x > 2,
179
with f; and f$ C' vector fields. This means that we take parameter p = E E R. For every E 2 0, consider the Cauchy problem
and solutions x,(.) on [O,T]crossing Z at a certain time 0 means that each x,(.) solves
{
< t, < T . It
& ( t )= f;(x,(t), t ) , if 0 < t < t,, k E ( t )= f,+(x,(t),t), if t, < t 5 T ,
d o ) = 0,
(4)
Z E ( t E )= z.
Our aim is to compute the differential of x E ( t )for t > to with respect to E. Variations of trajectories of control systems. The main ingredient to prove necessary conditions for optimality in optimal control problems is that of considering variations of trajectories. The most known are "needle" variations used in the proof of Pontryagin Maximum Principle; see Ref. 21. Consider for example the optimal control problem in Bolza form:
x
R", u E u,
(5)
x ( 0 ) = 2, x(T)E s,
(6)
E
the controlled dynamics, U is the set of controls, L is the Lagrangian cost, 3 E R" is the initial condition and S c Rn is the target. Assume f regular enough and U compact so to have a solution to (5) for every control u(.)defined on [0,TI. G'iven a candidate optimal control u*,a control variation is a parameterized family of controls u,, u o = u*, so that the corresponding trajectories x, (satisfying the initial condition) give a variation of the trajectory ZO. Needle variations consist in fixing T E [0,TI (Lebesgue point for both f(.,ZO(.), u g ( . ) ) and L ( . , x ~ ( . ) , u o ( . ) )w) ,E U , and setting u, = w if t E [T - & , T I and u, = uo otherwise. Computing differentials of trajectory variations amounts to consider the above problem with p = E E R, f,(z,t)= f ( t , x , u , ) and
z p = z. Needle variations give rise to trajectory variations differentiable in classical sense only after time T . A setting for which generalized differentiability holds, for every time, was introduced in Refs. 12,20. Other possible variations can be stated in terms of switching times; see Example 5.2 below. Other works in this direction are Refs. 1,4,5,7,10,15,16,26,27.
180
Optimal Syntheses. A technique to provide solutions to problems as (5)(6) is to look for an optimal synthesis. Roughly speaking an optimal synthesis consists of a family of trajectories z,(.), one for every initial condition z ( 0 ) = x, thus embedding (5)-(6) in a family of problems. To construct an optimal synthesis, one usually finds a synthesis formed by extremal trajectories, i.e. satisfying Pontryagin Maximum Principle. Then, if the synthesis is regular enough, it follows optimality, in the sense that each z, solves the corresponding optimal control problem. Here regularity can be expressed in terms of weak differentiability of zz with respect to initial conditions (plus some other conditions); see Refs. 6,9,17,18,20. The aim of this paper is thus to provide general results on weak differentiability of x p with respect to p , so to address all examples above. The approach we use is quite natural. We start from the incremental ratio, then we split it in two different ways and consider two different methods depending on how the incremental ratio was split. The first approach is inspired by the results of weak differentiability for the flow obtained in Refs. 12,20 and on the use of additional families of trajectories, to apply some generalizations of the Theorem 2.9 of Ref. 8. In the second approach, we consider the (possibly multivalued) flow of an ordinary differential equation and we apply generalized differentiation theories for multivalued maps. The paper is organized as follows. In Section 2 we describe the problem and the methods used. Sections 3 and 4 develop respectively the first and the second approach and contain the main results with the proofs. Finally Section 5 presents some examples of applicability of the theory developed. 2. Basic definitions
Let P be a normed space and fix po, qo E P. Consider the family of Cauchy problems (l),fix T > 0 and assume
(C-1) for every p E P the vector field fp is measurable w.r.t. t and continuous w.r.t. x; (C-2) for p , q in a neighborhood of po there exists a solution defined on P,TI to
Remark 2.1. Notice that the assumption (C-1) with an additional growth condition on f, is sufficient to ensure local forward existence of
181
a Caratheodory solution to the Cauchy problem (1). By a Caratheodory solution to (1) we mean an absolutely continuous function z : [0, h] ---t R", ( h > 0), such that
for every t E [O, h].
Notation 2.1. We denote by zp(.)a solution to the Cauchy problem (1) and by z:(.) a solution to the Cauchy problem (7). If the Cauchy problem (7) admits a unique solution, then the function z:(.) is clearly fixed. In the other case we will select xi(.) from the family of solutions so to satisfy some additional conditions. We denote by ~ : ~ , ~ ( yp>"(t) t), and respectively, the directional derivative in the direction Y E P of z:(t) w.r.t. p at p = PO,the directional derivative in the direction E P of z:(t) w.r.t. q at q = qo and the directional derivative in the direction 21 E P of z p ( t ) w.r.t. p at p = P O . Our aim is to compute ypo,w(t)under suitable assumptions. There are at least two natural ways to evaluate it: (a) the first one consists in writing
(b) the second one consists in writing
Remark 2.2. Note that the first addendum of the equation (9) and the second one of the equation (8) look like generalized differentials of the flow with respect to initial data (see for example Ref. 2,3,11,13,19,24-26). More precisely, the first addendum of (9) is similar to lim,,o y::;", i.e. the limit
182
on E of the differential of the flow, which can be granted by uniformity assumptions on such differentials. On the other hand, the remaining addenda in (9) and (8) look like weak differentiations with respect to the dynamics. Indeed, the second term of (9) is exactly ZJ~:,~, i.e. a weak differential with respect t o the dynamics a t xpo along the direction w,whilst the first term in (8) is similar to limE+o ygE:Eu, i.e. the limit as E -+0 of a weak differentiation with respect t o the dynamics at X,O+&V. Before considering the general case, we propose an example where the dynamics f, are smooth vector fields.
Example 2.1. Consider the following Cauchy problem
where p , q E P and the vector fields f, are of class C2. In particular (10) admits a unique local solution for every initial point. Let w E P and E > 0. Consider the trajectory x::, solution to (10) on [0,TI where p = po EW and q = po E W , and Tt : [t,TI --f R" the solution to
+
+
Using Taylor's expansion and classical differential with respect to the initial point, we have
Yt+h(T) - Y t P ) = [Yt+h(t
+ h ) - rt(t + h)l
P T
+ h) - Yt(t + h ) + h 1 I h fpo,z(Yt(S), s ) M ( s , t + h)v(t + h)ds + o(h) = Y t + h ( t + h ) - yt(t + h) + h M (T , t + h)w(t + h ) - hw(t + h ) + o(h), = Yt+h(t
where f,,,x denotes the Jacobian matrix with respect to x of the flow f,,, M ( s ,t ) is the fundamental matrix solution to the linear system
183
and
v(t + h ) = lim yt+,(t ,-Po+
+ h ) - %(t+ h) 17
Since the vector fields are of class C2,then
+
~ (+ th ) = Yt+h(t + h ) - rt(t h ) + 4 h ) h
and so
Y t + h ( T ) - y t ( T ) = M ( T ,t
+ h) [ ~ t + h (+t h) - yt(t + h ) ]+ o(h).
Define the function $ ( t ) := y t ( T ) . Clearly $ is differentiable and
T
= lim h-O+
1h 1 M ( T ,t + h ) [Yt+h(t + h ) - ~ t (+t h ) ]d t . 0
Observing that
Notice that the final equation of the example expresses the numerator of the first addendum of equation (8). We consider now weaker assumptions on the vector fields in order to have similar expressions.
3. Approach (a) This section deals with the estimate of the directional derivative ypo,w( t ) using the decomposition given in (8). The first part of the Section is dedicated to the statement of the main results (Theorems 3.1 and 3.2), while the second one to the proofs. In order to do this, we prove Proposition 3.1 and some technical lemmas.
184
Theorem 3.1. Fix T > 0. Suppose that the function P
go( T )
is Gateaux differentiable at PO E P . (1) Fix v E P . For every E
> 0 and t E [0,T ] assume that the Cauchy problem
admits a unique solution Tt : [t,TI
4
R”. Define
f = s u p { t E [ O , T ] : V S E[ O , ~ ] V T[Es l t ] - y s (=~z) ; : : ~ : ( T ) } Assume that f p o is a Lipschitz continuous function and f o r p in a neighborhood of PO, f p is bounded on 7 := { ( r t ( s ) , s :) t E [ t , T ]s, € [ t , T ] } . Then, there exists a constant C > 0 such that
Moreover, if
then IfPo+&w(Z;::::(t),t) - f?Jo(~;::z:(t),t)pt +ly;;w
1.
(2) Fix v E P . Assume that, for every p E P , the vector field C2 and the limit f o r E O+ of
f p
is of class
--f
exists, where M ( s ,t ) is the fundamental matrix solution to the system
and rt(s) is the solution to
185
Then T
Y P o , v ( T ) = Elim 'O&
1s0
M ( T , t ) [fPo+EV(Z;::::(t)4 - f P O ( X ; : ; : : ( t ) , t ) ]
dt
+YF:'~(T). Remark 3.1. Notice that the fundamental matrix solution M ( t , s ) depends also on E . Theorem 3.2. Fix T t E [O,T],consider )(.:z:;
> 0 , PO
E P and v E P . For every E > 0 and and assume that the Cauchy problem
{ y%s)( t )
= f p o ( Y ( S ) r s), = x;::::w,
admits a unique solution yt : [t,T ]-+ Rn. Suppose that the function P
$o(T>
is Gateaux differentiable at PO. (1) Assume that the functions w ( t ) = z;zz::(t) and y t ( s ) satisfy for some
c>o
I tl 5 t 2 I T lytz(T) - T t l ( T ) [ I cl%(t2)
( A l ) for every 0
-Ttl(t2)I
+o(lt2 - t l \ ) ;
(13)
(A2) there exists h > 0 s.t. for every t E [O,T[ Iw(t Then
Moreover, if
then
+ h) - yt(t + h)l I C h .
(14)
186
(2) Assume that, for e v e y satisfy
E
> 0, the functions w ( t ) = xE:Tz",t) and
yt(s)
( B l ) there exists 71 > 0 and a positive constant C such that
(B2) there exists a Radon measure pe o n [0,T ] such that the function
tI+
w(t
+ h ) - rt(t + h )
h weakly* converges, as h -+ O+, to p e an the space of Radon measures, seen as the dual of CO; (B3) for every h there exists a continuous function ah : [0,TI -+ R n X n such that uniformly o n [O,T]as h -+ 0 and
+
+
Yt+h(T) - Y t ( T ) = Q h ( t ) (Yt+h(t h) - rt(t + h ) ) o ( h ) . If the limit for E Of of Q E ( t ) d p E ( texists, ) then -+
(3) Assume that, for e v e y satisfy
E
> 0 , the functions w ( t ) = x ~ ~ ~and~y t (~s ) ( t )
( B l ' ) the function $ ( t ) := y t ( T ) is approximately continuous at 0 , i.e.
(Be') the function
t-
w(t
+ h) - rt(t + h )
h is continuous in [O,T]for h suficiently small and converges uniformly to a continuous function mE on [O,T]as h -+ O+; (B3') for e v e y h > 0 there exists in L1(O,T ) a function Oh : [0,T ] -+ W n X nsuch that ah
-*
Q&,
ash+O
where a€ is a Radon measure on [0,TI and the convergence is in the weak* topology. Moreover, Yt+h(T) - Y t ( T ) = ah(t)(Yt+h(t
+ h) - Y t ( t + h ) )+ o ( h ) .
187
If the limit for E -+ O+ of
m,(t)da,(t) exists, then
3.1. Technical proofs This subsection deals with the proofs of Theorems 3.1 and 3.2. We now proceed with technical lemmas giving boundedness and estimates of trajectories produced by variations.
Lemma 3.1. Fix T > 0 and a continuous function w : [0,T ] -+ Rn. Assume that, for every t E [O,T],there exists a continuous curve yt : [t,T]-+ R” such that y t ( t ) = w ( t ) and the function ( s , t ) H yt(s) is measurable. Moreover, suppose that there exists a positive constant C > 0 such that ( A l ) and (A2) of Theorem 3.2 hold. Then
Proof. The assumptions on w and y imply that the function
is measurable. Define
$4t) := l-/t(T) - T o ( T ) I . Clearly, $(t) is a measurable function. Moreover, for every t E [t,T ]
I$‘(t+h)-?l(t)) 5 I%+h(T)-?’t(T)( 5 CIYt+h(t h ) - Yt(t + h)l + o(h) = C ( w ( t h ) - yt(t h)l o(h) 5 C 2 h o(h).
+
+
+ +
+
Therefore II, is a Lipschitz continuous function and consequently $ is differentiable almost everywhere. Moreover,
5W t ) for almost every t E [0, TI. Define the Lipschitz continuous function
z ( t ) := +(t)- J 0
$(s)ds.
188
It is clear that k ( t ) I (C - l ) 4 ( t )for almost every t E [O,T]and ~ ( 0 = ) 0. Thus T
and so
which is the thesis.
0
Lemma 3.2. Fix T > 0 and a measurable function w : [O,T].+ Rn. Assume that, for every t E [O,T]there exists a curve -yt : [t,T]+ Rn such that rt(t)= w ( t ) and the function ( s , t ) H yt(s) is measurable. Moreover, suppose that, for the functions w and Yt, hypotheses (Bl), (B2) and (B3) of Theorem 3.2 hold, with Q and p in the place, respectively, of a, and p E . Then T
Proof. We prolong r s ( t ) and w ( t ) in the following way. We assume that
(1) w ( t ) = w ( T ) for every t 2 T ; (2) rs(t)= Y T ( T )for every s,t 2 T ; (3) rs(t)= y,(T) for every s E [O,T]and t 2 T . Define the function
@(t> =rtw
- ro(T).
We have
dJ(t + h) - @ ( t = ) h
(w(t
+ h ) - rt(t + h ) h
I t thus holds
In particular, @ is a BV function. Indeed, for some L
>0
189
Passing to the limit when h
I'
-+
O+ in (16) we get
d$(t) = I ' 4 t ) d P '
where we use the BV property of $, and the hypotheses (B2) and (B3). Hence
The proof is completed.
0
The next lemma is dual with respect to the previous one, since the convergence required in hypotheses (B2') and (B3') are in the dual topologies with respect to the convergences required in hypotheses (B2) and (B3) of the previous lemma.
Lemma 3.3. Fix T > 0 and a measurable function w : [O,T]4 Rn. Assume that, for every t E [O,T]there exists a curve ^It : [t,T]-+ Rn such that r t ( t ) = w ( t ) and the function ( s , t ) H yt(s) is measurable. Moreover, suppose that, for the functions w and yt, hypotheses (B1'), (B2') and (B3') of Theorem 3.2 hold, with CY and m in the place, respectively, of a, and m,. Then
w ( T ) - -yo(T)= I'mda. Proof. We follow the same proof of Lemma 3.2. We prolong the functions w ( t ) and rs(t)as before. One can write
By assumptions we have that
and the limit as h
-+
O+ of
w ( t + h ) - rt(t h
+ h)dt
exists and is finite. So the conclusion easily follows. The next corollary is an easy consequence of the previous lemma.
0
190
Corollary 3.1. Assume the same hypotheses as in Lemma 3.2. If for every t E [o,TI there exists AT,t, a collection of linear maps LT,t : Rn 4 R", such that Q h ( S ) E hT,s+h for s E [O,T]and h sufficiently small, then T
where
and
A is the closure of the set U t E [ o , q h ~ , t .
Proposition 3.1. Fix v E P and E > 0. Consider a trajectory x;:$E:. For every t E [O,T]assume that the Cauchy problem
admits a unique solution Tt : [t,TI
-+ Rn.
Define
t = s u p { t E [O,T]: V s E [ O , t ] V T E[ S , t ] Y S ( 7 )=x;;:::(T)}.
Assume that f p , is a Lipschitz continuous function and for p in a neighborhood of pol f p is bounded on
7 := { ( y t ( s ) ,s ) : t E [ f , T ]s, E [ t , T ] } . Then there exists C > 0 such that
Proof. We apply Lemma 3.1 in the case where the function w is equal to We have
2Po+EV PO+Ew.
T
rT
where we use the Lipschitzianity of f p o . I t follows from Gronwall's Lemma that ( A l ) is fulfilled. Moreover we get
Iw(t + h) - rt(t
+ h)\ I
t+h
IfPo+6w(w(s),S) - fpo(Yt(s),s))d 5s ZCh,
191
where C is defined by
c = SUP
{fPO+EW(7),
fPO(7)I
7
which is finite from the assumptions. So also (A2) is satisfied. Therefore
and the proof is completed.
0
Remark 3.2. If the Cauchy problem (17) has not a unique solution for every t E [O,T],then Proposition 3.1 can also be applied with yo(T) instead of zF:+""(T).So, in order to evaluate 1xi:TE:(T)- z$:+~~(T)~ one has to consider also the additional term lyo(T) - ZF:+""(T)\. We can now proof the main results about the estimate of ~ ~ ~ , ~ ( t ) .
and so, since E
> 0,
On the other hand, by assumption of Gateaux differentiability, we get
So the first part of the theorem clearly holds.
192
Consider now point 2. Example 2.1 shows that
zPo+Ev
po+ev(T)
- z;:+&T)
The assumption of Gateaux differentiability permits to conclude.
0
Proof of Theorem 3.2. Consider first point 1. By Lemma 3.1 we have T
Iz;:::E(T)
- z;:+Ew(T)I
IfPo+sv(z;:T::(t),t) - fPo(z;:+f:E(t),t)Idt.
The assumption of Gateaux differentiability permits to conclude in the same way as in the proof of the previous theorem. Consider now point 2. By Lemma 3.2 there exist a continuous function a&and a Radon measure p, such that
1 T
%PO +&'U
po+ev(T)
-
z;:+Ew(T) =
0
a&(t)dP&(t).
The proof of the point 3 is similar to that of the point 2.
0
Remark 3.3. If in point 2 of Theorem 3.2 we use Lemma 3.3 we come to the conclusion T
Yp0,v =
mda(t)+Y;:'W.
Remark 3.4. The convergence as E
O+ of the integrals
is assured for example by the following condition. There exists a real number ,LI E [0,1] such that a,&-@converges uniformly in Co(O,T)and p & ~ @ - l weakly* converges t o a Radon measure. 4. Approach (b)
To deal with approach (b) we need to consider generalized differentiation theories: we indicate by D a generalized differentiation theory and by D-differential the corresponding generalized differential. We write A E V ( F ; z, y; C) which means that A is a V-differential of a set-valued map F : Rn --H R" at the point (z, y) of the graph of F in the direction C , where C is a closed convex cone in R". The object D ( F ;z, y; C) is required to
193
be a set of nonempty compact subsets of Lin(Rn,Rn), the space of linear maps from Rn to R". For examples of generalized differentiation theories see Refs. 11,14,23,28. For every p E PI define the possibly multivalued map = @,(tO,tl,.) as the flow map of the vector field f,, where t o 5 tl are respectively the initial and final time.
@2it1(.)
Definition 4.1. Fix p E P. We say that A, is a minimal generalized differential of the flow map at the point ( ~ 1 ~ x where 2) xz E @?ytl(xl), if for every y solution to
@2>"(.)
?(t)= f,(Y(t),t)I
{ -/(to)
= 5,
there exists L E A, such that Y ( t 1 )- 2 2 = L ( x - 2 1 )
.+I(
- XI[).
We indicate by min2)(@2it1(.); X I ,XZ) the set of all minimal 2)-differentials.
Theorem 4.1. Fix po E P , u E P , and u1 E Rn. Assume that, for every E > 0 , there exists a trajectory x;:+~, such that the followings hold. (1)
Zpo+sv =
z,,
+ + EUl
O(E).
(2) There exists APO+EV E minD(@;br+,,; ~ ' p o X r ;;+€~(T)). (3) There exists a compact set A of linear functions from Rn to Rn such that U E > ~ A p o + ECvA. (4) For every t E [0,TI, the Cauchy problem ?(s) = f,o(Y(S)r = x;:+,,(t)l
{ y(t) admits a unique solution (a) there exist K > 0 and
-/t :
[t,TI
--+
S)I
Rn such that:
6 > 0 such that, for every t E [0,T ]
(b) there exists a Radon measure pE on [0,T ] such that the function t H
X;:+,,(t
+ h ) - rt(t + h) h
weakly* converges to pE in the sense of measures as h
---f
O+;
194
(c) for every h there exists a continuous function ah : [O,T]-+ R n X n such that Qh
uniformly on [O,T]as h
+0
+ QE
and
+
Yt+h(T) - Y t ( T ) = Q h ( t )(-Yt+h(t h) - rt(t
(5) The real number
$5,
T
cxE(t)dpEconverges as
Then there exists L E A and a sequence E,
E --+
+0
+ h ) )+ 4 h ) .
O+.
as n + +oo such that
Proof. By the definition of a minimal generalized differential we have that, for every e > 0, xPo f e w
p o + e m
-X;:+Ewm =L€(W
Dividing by c
= L€(~PO+EW -zpo)
+ O(l%IO+EW -zpoI)
+ 4 E ) ) + o(l~po+&w - zpol).
> 0 we get
By assumptions, L, belongs to the compact set A and hence there exists a sequence E~ and L E A such that LEn+ L. Therefore
we obtain that
Applying Lemma 3.2 with w = X;:+Enw
(TI - x:: ( T )=
and so
This concludes the proof.
I'
(t)dpEn (t)
0
Remark 4.1. Notice that the previous theorem holds even if we substitute assumption 4 according to the hypotheses of Theorems 3.1 and 3.2. Moreover we can substitute assumption 5 of the theorem with a hypothesis on the strong convergence of ae in a topological space and on the weak convergence of measures pE in the dual space.
195
5. Applications of the main results In this section we present some scalar examples that clarify how to apply the results of previous sections. This choice follows from the fact that generalizations to the vector-valued case are trivial, but would render this part more involved and hard to be followed.
Example 5.1. (Differentiability with respect to the vector field). Fix a time dependent family of discontinuous vector fields f& : R x JR -+ IW+ ( E 2 0) with exactly one discontinuity at 4 > 0 as in (2) and lim,,,+ f€(z,t) > 0 for every E 2 0 and t L 0. Define z, as in (3), (4). Fix E > 0, set w(.) = z,(.) and, for every t E [0, T ]consider a solution rt(-)
which converges pointwise to
I f & ( W , t-) fo(w(t),t)I as h -+ Of. More precisely we have that the convergence is uniform on the intervals [0,fl and [f,TI, where f satisfies w(f) = 3. Moreover the function a h ( t ) , defined in Theorem 3.2, is given by 1+
*
[IT
max(t2 ,t+h}
1
+Tt+h (t + h)1- rt (t + h) fo+(rt+h(s),5)ds -
s'
max{tz,t+h}
where tk and tz are respectively the times satisfying y t + h ( t k ) = 4 and rt(tz)= 3. Therefore, the boundness of vector fields implies the boundness of ah(t) in the space of Radon measures and so we have compactness of ah(t)with respect to the weak* convergence. Hence point 3 of Theorem 3.2 can be applied.
196
Example 5.2. (Switching times). Fix fl and fi, two bounded C’ vector fields on R, and 0 < f < T . Let x ( t ) be defined as follows. On [O,f],x ( t ) satisfies
{
k(t)= fl(dt),t), x ( 0 ) = xo,
where zo E R, while on [ f , T ]x,( t ) satisfies k ( t ) = f Z ( Z ( t ) ,t )
o
We perform a variation in this way. For 5 t 5 f. On [f,f+ E ] , x e ( t )solves
E
> 0, x C E ( is t ) equal to z ( t ) if
(4,t ),
{ &( t )
= fl(2, zCEm = 4%
while on
[f+ E , TI, x e ( t ) satisfies kCE ( t )= f2 (%& ( t ), t ) .
Define the functions yt(s) in the following way. If t 5 f,then yt(s) = Z(S) for every s E [ t , T ] If . t 2 E + E , then yt(s) = x,(s) for every s E [ t , T ] . Finally, if f < t < f+ E , then yt(s) is the solution to
?dS)
= f2(7t(S), s),
rt(t)= 4 t h see Figure 1.
I
I
I
-7: I
I/ V -
I
I 4
I
I
I
X
0
T
Fig. 1. The trajectory zE and the functions yt for a variation on the switching time.
It is easy to check that (Bl’) and (B2’) of Theorem 3.2 hold. More precisely, the term
+ h) - yt(t + h)
Z&(t
h converges uniformly to fc(xE(t),t ) - fo(z&(t), t).
197
In the same way as in Example 2.1, we deduce also (B3'). Thus we can apply Theorem 3.2.
Example 5.3. (Control theory) Consider the following Cauchy problem
where u E [0,1] is the control, t E [0,TI and q 2 0 is a C1 function. Let uo = 0 and xo 0 be a corresponding solution. We perform a variation of the trajectory xo and we apply Lemma 3.3. Notice that in this case the differential of the final point of the trajectory with respect to the initial condition is zero, since we take the initial condition always equal to 0. Fix a function a l ( ~E]O,T[ ) and a function ~ z ( E €10, ) T - a 1 ( ~ )Define ]. the family of controls
--
U & ( t ):=
{ 0,
(P(E),
if 0 5 t 5 T if T - W ( E ) < t 5 T ,
with p(e) E [0,1] and a corresponding family of trajectories
it 5 T -W(E) -~ z ( E ) , , if ~ - a ! l ( e ) - a z ( E ) i t i ~ - a l ( ~ ) , if 0
w(t)=z,(t) =
{
2 (t-T+a1f)+a2(~))
O' 4 4
if T - ~
1
5 t 5 T,
I ( E )
(20)
where the function z satisfies
i
.i(t>=
m+ r l ( t ) c p ( E ) , (
)
2
z ( T - a l ( E ) ) = az(.)* 2 1
see Figure 2. Further, for every t E [0, T - C Y ~ ( E ) ]define , "It : [t,T] + R by
while, for every t E [T - a 1 ( ~TI, ) , define the Cauchy problem %(SO
=
m,
rt(t)= 4% see Figure 3.
: [t,T]--f
B by the solution to
198
Fig. 2.
The trajectory z E .
Fig. 3. The curves y.
We check the hypotheses (Bl’), (B2’) and (B3’). In order to simplify the notation we omit the dependence on E . Note that (Bl’) is granted by definition. Let us focus on (B2’). Notice that
Iw(t for t
+ h) - y t ( t + h)l = 0
< T - a1 and h sufficiently small. If t 2 T - a1, then
199
Passing to the limit with h -+ 0, the first term of the right hand side of the above equation tends to 0, while the second one tends to prl(t) pointwise. Moreover,
t+h
1
which, for every t 2 T - a l l is bounded when h is sufficiently small. By Ascoli-ArzelB theorem. we conclude that
h converges when h -+ O+ uniformly, and hence also in the sense of measures, to cpr](t).Thus we get (B2’). Let us pass to (B3’). We have
h
T-t-h
Thus
where x denotes the characteristic function of a set. Passing to the limit with h --f 0 we conclude that Qh
in L1(O,T ) ,where
-+
QO,
200
Hence (B3') is satisfied. Lemma 3.3 permits to conclude that
We have
a2
Consider first the special case when q = 1, cp = E ~ / ~ + Pwith ~ , P1 2 0 and ,& 2 0. We get
Il
= 1, a1 = E
~
/ and ~ ~
Z E P 1 - 3 ,
I2 5 &ZPl-Pz - 3 I3 =
(&PI
+ 4
&P2)2
.
Choosing PI = $ and p2 €10, +[, we conclude that I1 = 1, and I z ,I3 tend to 0 as E -+ Of; thus we have produced a first order non zero variation. Let us take now q(t) = (T - t ) P 3 , cp = $ 4 , a1 = E ~ / ~ +and P ~ a2 = E ~ / with ~ ,& ~ 2 ~0, P~2 2 ,0, ,& 2 0 and ,& 2 0. In this case
P
~
201
In order to get
I1
constant and
12, I 3
going to 0 we need
which is equivalent to
i.
i,
If = then the first equation of the The first equation implies P 1 5 system implies that p 3 = P 4 = 0, thus cp E 1 and r] = 1 as before. Finally, choosing p 1 < we get infinitely many solutions to (22).
Remark 5.1. If we consider the control system (18) and we want to maximize the cost z(T), then the previous example shows that the trajectory x E 0 is not optimal, since the first order variation xE(t) of Example 5.3 satisfies
Remark 5.2. Notice that the classical needle variations are not useful in the above example. Indeed, consider a C1 function r] such that r](t) > 0 for every t E [O,T[and r](T) = 0 (for example r](t) = (T - t)P3 with P 3 > 0 ) . Two cases happen. (1) A needle variation is originated at t = T . It means that controls u, are defined by
In this case the corresponding trajectories take on the form xE(T)= cp~(T - E)E
+ o(E).
Thus Z E ( T ) - zo(0) &
as E
+ O+.
+4E)
VV(T &
0,
202
(2) A needle variation is originated at t = r u, are defined by
< T . It means that controls
0, i f 0 5 t I r - c E , cp, if r - E
< t 5 T,
0, i f r - E < t t T . As before we have
while
so
is unbounded for
E -+
O+.
Therefore the needle variation in the first case is not a first order variation, while in the second case is not an admissible variation.
Example 5.4. (Control theory) Consider the following Cauchy problem
{ 4x (t0)) -m+ 4% = = 0,
where u > 0 is the control, t E [O,T].Let u g = 0 and x g = 0 be a corresponding solution. We perform a variation of this trajectory using the controls
u,(t)
=
&.
Thus the trajectories x,(.) satisfy
{ while the functions
&(t)=
-&m+
rt(.)satisfy ?t(s) =
-v5ml1
rt (4 = 2, ( t ) ; see Figure 4.
Ji,
4 0 ) = 01
203
0
T Fig. 4.
The trajectory zc and the functions
We want to apply Lemma 3.3. As in the previous example the differential of the trajectory with respect to the initial condition is zero. The hypothesis (Bl’) is trivially satisfied. We have
and so
+ h) - rt(t + h) + J z
Z&(t
h
as h 4 O+. This implies (B2’). We have x e ( t )5 E for every t 2 0 and setting
rt(t if rt(t)E
f = 2&
+ i)= 0
[ O , A Then it is clear that =0
for every t E [O,T - 2 4 and hence ah(t) is equal to 0 for every t E [0, T - 2&[ and h sufficiently small. Moreover, if T is sufficiently big, then
204
Therefore we get ah@)=
{
g h + 4466- 2- (hT - t )
,
if t E [T - 2&,T], if t E [O,T- 2@[,
and so ah(t)-+ ao(t)when h -+Of in L1(O,T), where ao(t) =
, if t E [T - 2&,T], if t E [O,T- 2&[.
Thus (B3’) is satisfied. Lemma 3.3 permits to conclude that XE(T) =
4
dt = E .
Remark 5.3. One can deal classical needle variations as before (possibly adding a function q). Acknowledgments The second author was partially supported through a European Community Marie Curie Fellowship and in the framework of the CTS, and by Bialystok Technical University under the grant W/WI/2/07.
References 1. A. A. Agrachev, G. Stefani, P. Zezza, S I A M J. Control Optim. 41,991 (2002). 2. J. P. Aubin, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions, in Mathematical analysis and applications, Part A , Adv. in Math. Suppl. Stud., Vol. 7, eds. L. Nachbin (Academic Press, Orlando, 1981) pp. 160-232. 3. J. P. Aubin, H. Frankowsh, Set-Valued Analysis, Systems & Control: Foundations & Applications, Vol. 2 (Birkhauser, Boston, MA, 1990). 4. R. M. Bianchini, M. Kawski, S I A M J . Control Optim. 42,218 (2003). 5. R. M. Bianchini, G. Stefani, S I A M J . Control Optim. 31,900 (1993). 6. V. G. Boltyanskii, S I A M J . Control 4, 326 (1966). 7. A. Bressan, S I A M J . Control Optim. 23,38 (1985). 8. A. Bressan, Hyperbolic systems of conservation laws (Oxford University Press, Oxford, 2000). 9. P. Brunovski, J . Differential Equations 38, 317 (1980). 10. F. H. Clarke, S I A M J . Control Optim. 14, 1078 (1976). 11. F. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Vol. 5 , second edition (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990). 12. M. Garavello, B. Piccoli, S I A M J . Control Optim. 43, 1867 (2005). 13. E. Girejko, Rend. Semin. Mat. Uniu. Politec. Torino 63, 357 (2005).
205
14. E. Girejko, B. Piccoli, Set-Valued Anal. 15,163 (2007). 15. H. Halkin, Necessary conditions for optimal control problems with differentiable or nondifferentiable data, in Mathematical control theory (Proc. Conf., Australian Nut. Univ., Canberra, 1977), Lect. Notes in Math. 680 (SpringerVerlar, Berlin , 1978), pp. 77-118. 16. B. Kdkosz, S. Lojasiewicz Jr., Nonlinear Anal. 9, 109 (1985). 17. M. Kiefer, H. Schatler, SIAM J. Control Optim. 37, 1346 (1999). 18. U. Ledzewicz, A. Nowakowski, H. Schatler, J . Optim. Theory Appl. 122,345 (2004). 19. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I & 11, Grundlehren der mathematischen Wissenschaften, Vol. 330 and 331 (Springer-Verlag, Berlin, 2006). 20. B. Piccoli, H. J. Sussmann, SIAM J . Control Optim. 39, 359 (2000). 21. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Translated by D. E. Brown (A Pergamon Press Book. The Macmillan Co., New York 1964). 22. R. T. Rockafellar, R. J. B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften, Vol. 317 (Springer-Verlag, Berlin, 1998). 23. H. J. Sussmann, New theories of set-valued diflerentials and new version of the maximum principle of optimal control theory, in Nonlinear Control in the Year 2000, Vol. 2, Lecture Notes in Control and Inform. Sci., 259, eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek (Springer-Verlag, London, ZOOO), pp. 487-526. 24. H. J. Sussmann, Path-integral generalized differentials, in Proceedings of the 4 l s t IEEE 2002 Conference on Decision and Control, Las Vegas, Nevada (Las Vegas, Nevada, December 10-13, 2002): 1101-1106. 25. H. J. Sussmann, Warga derivate containers and other generalized differentials, in Proceedings of the 41st IEEE 2002 Conference on Decision and Control (Las Vegas, Nevada, December 10-13, 2002): 4728-4732. 26. H. J. Sussmann, Generalized differentials, variational generators, and the maximum principle with state contraints, in Nonlinear and Optimal Control Theory (Lectures given at the C.I.M.E. Summer School held in Cetraro (Cosenza), June 21-29, 2004), Lecture Notes in Mathematics, eds. G. Stefani and P. Nistri (Springer-Verlag (Fondazione C.I.M.E.), to appear). 27. R. Vinter, Optimal control, Systems & Control: Foundations & Applications (Birkhauser Boston, Inc., Boston, MA, 2000). 28. J. Warga, SIAM J. Control Optim. 21, 837 (1983).
206
SAMPLED-DATA REDESIGN FOR NONLINEAR MULTI-INPUT SYSTEMS L. GRUNEt and K. WORTHMANNt
Mathematisches Znstitut, Fakultat fur Mathematik, Physik und Informatik Universitat Bayreuth, 95440 Bayreuth, Germany t E-mail: 1ars.grueneQuni-bayreuth. de t E-mail: karLworthmannQuni. bayreuth.de We investigate the sampled-data redesign problem for nonlinear control affine multi-input systems and consider sampled-data feedback laws for which the trajectories of the sampled-data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as sampling time vanishes. We analyze geometric conditions for the existence of such sampled-data feedback laws and give formulae and algorithms for their computation.
Keywords: sampled-data system, controller design, Lie brackets, Fliess expansion, Taylor expansion, nonlinear control affine system, convergence rate
1. Introduction
Feedback controllers are nowadays typically implemented using digital devices. In contrast to analog implementations, these devices are not able to evaluate the feedback law continuously in time but only at discrete Sampling time instances. Thus, the controller must be designed as a sampleddata controller, whose simplest (and most widely used) implementation is a zero order hold, i.e., the feedback law is evaluated at each sampling time and the resulting control value is kept constant and applied on the sampling interval until the next sampling time. A popular design method for sampled-data controllers is the design of a controller based on the continuous-time plant model, followed by a discretization of the controller. In other words, the continuous control function generated by the continuous-time controller is replaced by a piecewise constant and thus nonsmooth control function generated by the sampled-data controller. If the sampling interval is sufficiently small, then the choice of an appropriate sampled-data controller can be done in a very straightforward way, however, hardware or communication constraints may prohibit
207
the use of small sampling intervals, in which case more sophisticated techniques have to be used. A good introduction to this subject in the nonlinear setting considered here can be found in the survey paper Ref. 11. An important class of such techniques is the sampled-data redesign, in which a sampled-data controller is constructed which inherits certain properties of a previously designed feedback law for the continuous-time system. The survey papers Refs. 5,12 summarize a couple of such redesign techniques. The analytical approaches in these papers are restricted to single-input systems, i.e., for systems with a one dimensional control variable, a condition which we relax in this paper. More precisely, in this paper we extend the redesign technique presented in Ref. 4 to multi-input control affine nonlinear systems. This technique solves the redesign problem by designing a controller which is asymptotically optimal in the sense that we maximize the order at which the difference between the trajectories of the continuous and the sampled-data system converges to zero as the sampling time tends to zero. This amounts to investigating whether a sampled-data feedback law for a desired order exists and, in case the answer is positive, how it can be computed. Concerning the conditions for the existence of higher order sampled-data feedback laws, it turns out that like in the single-input case the answer lies in the geometry of the system, expressed via the possible directions of the solution trajectories, which in turn are determined by the Lie brackets of the vector fields. Compared to the single-input case, the main difference of our multi-input results lies in the fact that the presence of more control variables typically facilitates the construction of a higher order sampleddata feedback law, an effect we illustrate in our numerical examples. In particular, it turns out that the design of sampled-data feedback laws of arbitrary order is always feasible if the control dimension equals the state dimension and the matrix composed of the control vector fields has full rank. Since for higher orders the existence conditions and formulae for the sampled-data feedback laws become fairly complicated, we restrict our analytical results to low orders in order to illustrate the geometric nature of the conditions. For general orders we provide a Maple code which checks the respective conditions and computes the resulting sampled-data feedback, if possible. Here the second main difference to the single-input case appears: while in the single input case this computation was based on the successive solution of several one dimensional linear equations, the multi-input case can be tackled algorithmically via the solution of a suitable least squares
208
problem. 2. Problem formulation
We consider a nonlinear plant model k(t)= fb(t),u(t))
(1)
with vector field f : R" x U -+ R" which is continuous and locally Lipschitz in z, state z ( t ) E R" and control u ( t ) E U c R". Throughout the paper we assume that a smooth static state feedback uo : R" -+ IRm has been designed which solves some given control problem for the continuous-time closed-loop system k ( t ) = f ( x ( t ) uo(x(t))) ,
4 0 ) = xo.
(2)
Our goal is now to design u ~ ( zsuch ) that the corresponding sampled-data solution 4 ~ ( XO, t , U T ) of the closed-loop system using a sampler and zero order hold
i ( t )= f ( . ( t ) , u r ( z ( W ) ) , t E [ k T ,(k + 1)T)
(3) k = 0 , 1 , . . ., reproduces the behavior of the continuous-time system and thus improves the performance of the sampled-data closed loop system. Our approach uses an asymptotic analysis in order to study the difference between the continuous-time model (2) and the sampled-data model (3). To this end, for a function a : R x IR" -+ IR we write a(T,x) = O(Tq), if for any compact set K c R" there exists a constant C > 0 (which may depend on K ) such that the inequality a(T,x) 5 CTq holds for all elements z E K . If we consider a specific set K we explicitly write a ( T , z )= O(Tq) on K . In order to obtain asymptotic estimates, we consider an "output" function h : R" -+ IR and derive series expansions for the difference
Ah(T,zO,uT):= Ih($(T,xO))- h($T(T,xO,uT))I,
(4)
where $(T,xo) denotes the solution of the continuous-time system (2). Note that h here is not a physical output of the system but rather a scalar auxiliary function which can be chosen arbitrarily. In particular, we will use hi(z) = xi, i = 1,.. . , n, in order to establish A h i ( T , z o , u ~=) C3(Tq) which then implies
A'$(T, z0, u T ) := /14(T,x0) - $'T(T,20, U T > l l c c = O ( T q )
(5) measured in the maximum norm 11 . From this estimate it follows by a standard induction argument that on each compact interval [ O , t * ] we
[Ic .
209
obtain A$(t,zo, U T ) 5 O(T4-l) for all times t = k T , k E N with t E [0,t*] which in particular allows to carry over stability properties from $ to &, see Refs. 17,18. In order to facilitate this analysis we restrict ourselves to control affine systems where the ordinary differential equations in (1)-(3) take the form m
i ( t )= 9 o ( z ( t ) )
+C9i(z(t)).o,i(w),
(6)
i= 1
with smooth vector fields g o , g 1 , . . . ,gm : R" 4 R" and controls U O , ~. ., . ,~ 0 : Rn , + ~ R. Note that the continuous-time feedback uo(z) = (uo,l(z), . . . , U O , ~ ( X ) is) ~represented in a vectorial form. In Refs. 4,14 we investigated single input systems, i.e., u ( t ) E R. In this paper we extend these results to the multi-input case, i.e., m > 1. We look at sampled-data feedback laws meeting the following definition.
Definition 2.1. An admissible sampled-data feedback law U T is a family of maps UT : Rn + Rm, parameterized by the sampling period T E (0, T*]for some maximal sampling period T*,such that for each compact set K c R" the inequality SUP z E K , TE(O,T*]
11 uT(z) lloo <
holds. Note that for existence and uniqueness of the solutions of (3), we do not need any continuity assumptions on U T . Local boundedness is, however, imposed, because unbounded feedback laws are physically impossible to implement and often lead to closed-loop systems which are very sensitive to modelling or approximation errors, cfr., e.g., the examples in Refs. 3,15,17. A special class of sampled-data feedback laws is given by M 'llT(z)= c
Tjuj(z),
(7)
j=O
where the uj(z)are vectors ( u j , l ( z ).,. . , ~j,,(z))~. We will see later that this is exactly the form needed for our purpose. Inserting the sampled-data feedback (7) into our affine control system (6) leads to m
210
Here the second index of uj,i(x)denotes the i-th component of the vector u j ( x ) .Another way of writing (8) is
(
)
. gm,1(x) 2 = go(x) + G ( x ) u ~ ( x with ) G ( x )= .: ... ; . (9) 91,n(x)* . . grn,n(x) In the sequel we use the following notation: for subsets D c EXn we write cl D , int D for the closure and the interior of D. The notation I . I stands for Sl,l(Z)
* *
the Euclidean norm while llxlloo = maxi=l,...,n/ x i /denotes the maximum norm in Rn.Furthermore, cfr. Ref. 6, we denote the directional derivative of a function h : Rn + Iw in the direction of g : Rn -+ R" by
d L,h(x) := -h(x) dx
'
g(x)
and the Lie bracket of vector fields g i , g j : Rn -+ R" by d [gi,g j ] = -gj dx
*
d dx
gi - -gi
*
gj.
3. Fliess series expansion
In this section we provide the basic series expansion used for the redesign of U T . Although the admissible sampled-data feedback UT according to Definition 2.1 may in principle be completely unrelated to uo = ( U O J , . . . ,~ 0 ,T ,~
)
in the sequel it will turn out that a certain relation between uo and UT must hold. More precisely, we will see that the resulting sampled-data feedback (if existing) will be of the form (7) with U O , J ( Z ) ., . . , UO,,(Z) from (2) and u1,1(x),. . . , u ~ , ~ :( Rn x )-+ R being locally bounded functions. This structure appears to be rather natural and was also obtained as the outcome of the design procedure in several other papers, cfr. Refs. 1,10,16 and also for our problem in the single input case.4 Thus, we develop our series expansion for these feedback laws. In order to formulate our results we define multinomial coefficients n n! (no ... n,> := no!n1!...n M ! as well as multi-indices v := (no,121,. . . ,n ~ ) M . and use the notations IvI := no n1 . . . n M and llvll = Ci=ozni. Our analytical considerations are based on the following theorem which is a generalization of [14, Theorem 3.11 to the multi-input case.
+ + +
Theorem 3.1. Consider the control afine system (6), a smooth function h : Rn + R, the continuous-time closed-loop system (2) and the sampled-
211
data closed-loop system (3) with controller ficiently small T , we can write:
UT
given by ( 7 ) . Then, f o r suf-
M
where po(z) = Lgoh(z)and p s ( z , u o , . . . , us-l), s = 1 , . . . , M , is given by
k= 1 io =O,. ., i k = O
' 5(
,
C 6 , vi=s-k with ui = ( u i , ~ . .,. ,~
i
,
~Here ) ~cj,
)fiu;y)
ci (nO,inl,j ' ' ' n M , i
VENF:
1=o
llvj ll=vi
denotes #{il
I 1 = 1 , . . . ,k : il = j } .
For the proof, we need the following result, which can be found, e.g., in 18, Theorem 4.21
Proposition 3.1. For ai E R, i = 0 , 1 , 2 , . . . ,M and n E equality
N
we have the
Proof. (Theorem 3.1) Using the Fliess series expansion, see [6, Theorem 3.1.51, we can write 00
h($(t,.)) - h ( z ) =
2
k=O io=o, ...,i k = o
s,"
L,io
' ' *
Lgik
h(z)
Jd
t d
' '
"
The expressions d
212
Using Equation (12) we can write
Like in the single-input case14 we use Lemma 3.1 in order to transform the components of (7). This leads to the expression
with
I
Ivj =
M
nlj
and
I Ivj I I =
+
M
2nl,j. Hence
it follows
+ zj”=, +
denotes an O ( M 1) term. Define s := k llvjll = k CY==, C,”=, ln1,j and sum over all terms of order 5 M . Collecting all terms of order strictly greater than M in H1 we can rewrite the last equation as
where
H1
Observe that for each s > 0 the sum for k = 1 . . . , s is exactly (11). Thus, in order to complete the proof it remains to show that the summands for k = 0 equals the remaining terms in (10). To this end first consider s = 0 and k = 0. This leads to s - k = 0 and the only vector v E W p satisfying C,”=, vi = s - Ic is the zero vector. Consequently we obtain
213
For s = 1,..., M and k = 0 one computes Czlu.i = s and differs between the cases io = 0 and io > 0. For io = 0 this provides L,,h(z) multiplied with a sum with respect t o the empty set because it holds cj = 0 for all j = 1, . . . ,m and thus lvjl = 0 for all j = 1,..., m. This causes llvj 11 = 0 for all j = 1,.. . m and leads to a contradiction due to Czl llvjll = ui = s > 0. Accordingly io = 0 doesn't provide any additional terms. Continuing we consider io = i E { 1,. . . , m } . It holds ci = 1 and cj = 0 for all j = 1,.. . , m with j # i. This leads to Ivjl = 0 and )lvj11 = 0 for j = 1,.. . m with j # a. Consequently we have to consider u = sei where ei denotes the i-th unit vector. Hence we multiply Lgih(z) with u8,idue to
xzl
cj =
1 for j = i , 0 otherwise.
finishes the proof.
0
Remark 3.1. While (10) is a straightforward extension of the single-input case, the p , differ in our multi-input case in terms of an additional combinatorial condition. One has to choose all vectors u E W2;" whose components add up to s - k. Remark 3.2. For later reference we will explicitly compute the components of p l and p z according t o (11).For pl(z,uo) we have s - k = 0 and thus the combinatorial condition u E N2;" : Czl ui = s - k is only satisfied for u = ON^. Now we have to distinguish three cases, the first one is io = il = 0. Here cj = 0 for all j E (1,.. . ,m} and it results ~L,,L,,h(z). The second case is io = 0, il # 0 and respectively io # 0, i l = 0.Here it exists exactly one c j > 0 and due to I Ivj I I = 0 it follows
The last case, s = 1, io > 0,i l > 0,provides
For p2(zlU O ,u1) we need to distinguish between k = 1 and k = 2. For k = 1 we have s- k = 1 and thus u = e l , . , . , em. Furthermore, we have to discern
214
three cases. The first is i o = il = 0 which leads to the empty set. The second, i o = 0, il > 0 respectively io > 0, il = 0, provides
i= 1
Finally, in the case io > 0 and il > 0 we obtain
i=
2
l,j=1
c m
1 ~ ~ g a L , j h ( ~ ) u l , i ( z ) u o ,+ j ( z ) L;ah(z)ul,i(z)uo,i(s). i= 1
i#j
For k = 2 it holds s - k = 0 and thus we have only to regard v = On;;.. The computations take course analogously to the case s = k = 1. Again we obtain terms dependent on components of U O . Hence, altogether p z ( z , U O ,u1) provides
f 2 (Pg0Lg,+ L,iL,o1 i=l . m
m
h ( z )+ C L S i L g j h ( z ) u 0 , j ( 4 j=1
Remark 3.3. Computer algebra systems, such as Maple, can be used to compute expansions of the difference (4) for particular examples, cfr. Remark 4.5. 4. Necessary and sufficient conditions
In this section we investigate necessary and sufficient conditions for the existence of an admissible feedback law U T which achieves
Ah(T,z, U T ) = O(T4)
(15)
A($(T,2, U T ) = o ( T q )
(16)
or
and provide formulae for these feedback laws. Since the computations with respect to the sufficient condition turn out to be fairly involved we restrict
215
our analytical computations to the case q 5 4 and provide a Maple procedure for the general case. As we will see, q = 4 is the first nontrivial case in the sense that that (15) and (16) for q 5 3 can always be satisfied without any further conditions. For the case q = 4 it turns out that the cases (15) and (16) require different conditions. In particular, for (16) we obtain a much stronger necessary condition than for (15). Thus, we state them in two separate theorems starting with (15). The next theorem is a consequence from Theorem 3.1 performing a careful evaluation of the pi-terms. It generalizes corresponding results for the single input-case rn = 1, see [14, Theorem 4.111 for the cases (i) and (ii), [4, Theorem 3.11 for case (iii), and also [5, Theorem 3.21. For the formulation of the theorem we use the notation
Note that this definition coincides with the continuous-time controller for i = 0. Furthermore we abbreviate
Theorem 4.1. Consider the vector field (6), the continuous-time closedloop system (2), the sampled-data closed-loop system (3), a smooth function h : R" .+ R and a compact set K c R". Then the following assertions hold for from (17): (i) Ah(T,zo, U T ) = O ( T 2 )holds on K for
u+)
= u(O)(z).
(ii) Ah(T,zo, U T ) = O ( T 3 )holds on K for UT(2)
+
= u(O)(z) T u ( l ) ( z ) .
(aaa) If there exists a bounded function cq, : K m
c
Lgt h ( z ) a h , a ( z )
a=1
m a=
1
Rm satisfying
m
[L[g,,gt]h(z) -k
=
-+
c
L[g,,g,lh(z)'%(z)]iLt(z),(19)
3=1
32%
216
then there exists U T such that A h ( T ,X O , U T ) = O(T4) holds on K with
where k := {z E K 13i : Lgih(x)# 0). Conversely, if an admissible sampled-data feedback law i i = ~UT+O(T~) for U T from (21) satisfies (20) on a set k k , then there exists a bounded function Q satisfying (19) on elk. Proof. Our smoothness assumptions enable us to use the Taylor series expansion for the solution of the ordinary differential equation (6). To this end we define the differential operator 2 := Lg,+-pl L , , ~ , , and ~ apply the Taylor series expansion to our output-function h, i.e.,
Hence, from the Taylor expansion of h ( $ ( t , x ) )in t = 0 we obtain the identity
using Theorem 3.1 in the last step. Thus (i) holds. For the proof of (ii) we apply twice to the output-function h and
217
exploit the shape of pl(z, U O ) outlined in Remark 3.2:
T2 h($(t,z)) = h ( z )+ TLA(z)+ -L2h(z) 2
+0(T3)
s=o
+
where we used Theorem 3.1 and U T = U(’) Tu(l)in the last step. This shows (ii). In order t o prove (iii) we have to examine the Taylor series expansion up to order four. To this aim we consider the threefold application of 2 and use the identity iLj(z) = Luo,j(z):
m
m
m
m
A comparison between (27) and the terms resulting from (14) using
UT
218
L
c m
-
LgaLg0h(x)-
+ h(x) + 0(T4).
LgaL,,h(Z)'ILO,j(x)
j=1 j#i
With LgoLg,h(x)- LgiLgoh(x) = L[go,galh(x) one obtains
+ h ( x )+ 0(T4)=
h((bT(T,5,U T ) ) + 0(T4), (28) where we used Theorem 3.1 and the definition of UT and a h in the last step. Thus, the choice (21) ensures a forth order approximation. Note that the function a h is bounded on K by assumption, which in particular implies that the control law (21) is admissible in the sense of Definition 2.1 on k. For the converse statement, if UT from (21) satisfies (20) on G k and Definition 2.1, then the function a in (21) must be bounded. Hence, for each boundary point x E ail we can find a sequence Xk --+ x such that a(Xk) is convergent and define a ( x ) = limk,, Q(Zk). Then, since all coefficients in (19) are continuous, we obtain that (19) also holds for x E i.e., on cl 2 and the boundedness follows immediately. 0
z
az,
Remark 4.1. Note that the converse part of statement (iii) is rather weak, as it only provides a necessary condition for the existence of feedback laws of the specific form (21) but not for arbitrary admissible sampled-data feedback laws satisfying (20). It is however, an important building block for the much stronger necessary condition for A(b(T,x,U T ) = 0(T4) given in Theorem 4.2, below.
219
Remark 4.2. In (21) we distinguish between z E c l k and x $ c l k . This case differentiation can be interpreted in terms of relative degree (see Ref. 7 for a definition and Ref. 12 for the role of the relative degree in sampled-data feedback design). System (6) with output function h has relative degree one on c l k while the relative degree is strictly larger on K \ c l k . This explains why the feedback law (21) has different structure inside and outside cl k. Remark 4.3. For driftless systems, i.e., go(.) E 0, the lie-bracket [go,gi] in (19) is equal to zero for all i = 1, . , . , m. Hence, condition (19) is always satisfied for m = 1 and easier to evaluate otherwise.
Now we turn to (16) and deduce assertions for the full state trajectory from Theorem 4.1 by choosing hi(.) = xi, i = 1 , . . . ,n. Theorem 4.2. Consider the control afine system (6), the continuous-time closed-loop system (2), the sampled-data closed-loop system (3) and a compact set K c Rn satisfying K = clint K . Then the following assertions hold for u ( ~from ) (17): (i) A ~ ( T , ~ O ,=UOT( T) 2 )holds on K for UT(Z)
= U(O)(X).
(ii) A ~ ( T , Z O , U =TO)( T 3 )holds on K for U T ( Z ) = U(O)(Z)
+Tu(l)(z).
(iii) If there exists a bounded function a : K m
1
-+ Rm
satisfying
m
then
holds on K for
with k := { x E K I3i : g i ( x ) # 0). Furthermore, on
K* = {z
E
K I G(z) from (9) has full column rank},
any feedback law GT satisfying A4(T,2 0 ,GT) = O(Tq),q = 2 , 3 , 4 , is of the form GT(z) = U T ( Z ) O(Tq-l) for UT from (i), (ii) or (iii), respectively,
+
220
and the function Q in (29) is unique if it exists. On c l K * the suficient condition (29) is also necessary for the existence of UT in (iii). Proof. Note that (16) is equivalent t o (15) for hi(x) = xi, i = 1,.. . , n . Hence, assertions (i) and (ii) follow immediately from Theorem 4.1 applied to hi(x) = xi,i = 1,.. . ,n. For the proof of (iii), we first show that under condition (29) any feedback of the form (30) satisfies the assertion. First note that for x $ cl k the feedback value U T ( ~ is) indeed arbitrary. This follows since on K \ cl k the control system is given by j. = go(x). Thus, on the open set int ( K \ cl k)the Taylor series expansions of $(t,x) and $T(t,x,U T ) coincide for any order, regardless of the values of uo and U T , i.e., we obtain (16) for any 1M > 0 for arbitrary U T . By continuity of the expressions in the Taylor series expansion this property carries over to clint ( K \ c l k ) which contains K \ cl k because we have assumed K = cl int K. It is, hence, sufficient to show that UT satisfies the assertion for x E cl k. Assume that the function Q exists and is bounded. Fix i E (1,. . . , n } and consider the function hi(x)= xi. A simple computation using the identities
L[gk,gjlhi(z)= [ g k , g j ] ( z ) i
Lgjhi(x) = gj,i(x) and
shows that the function a from (29) satisfies m
m
+
c Q i ( z ) g i ( x )=
[gO,gi](x) C [ ~ j , g i ] ( ~ ) u o , j ( x ) i= 1
i= 1
j=1
j#i
j=1
i= 1
j#i
Thus, the feedback is of the form (21) for h = hi and we can use Theorem 4.1 to conclude (15) for q = 4 and i = 1,.. . , n and thus (16). Now we show the claimed form of the GT on K * : €+om Theorem 3.1 for M = 0 it follows that any i i satisfying ~ A$(T,ZO,G T ) = O ( T 2 )must fulfill m
m
i=l
i= 1
for k = 1,. . . , n in order to get the equality “(23) = (24)” (for i i instead ~ of U T ) for all hk. Using again L g j h i ( z )= gj,i(x)one sees that this is equivalent
221
to
+
G(z)uo(z)= G ( z ) i i ~ ( z )O(T)
+
and since G(z) has full column rank this implies U T ( X ) = uo O(T). The statements for (ii) and (iii) now follow analogously by induction using the equalities “(25) = (26)” and (28). The uniqueness of a on K* follows again from the full column rank of G(z) because the right hand side of (29) equals
G(x)a(x). Finally, using the uniqueness of UT in (iii) up to higher order terms, the necessity of (29) on cl K* follows from the converse statement in Theorem 4.1(iii) for ?j = K*. 0
Remark 4.4. Theorem 4.2 has a nice geometric interpretation if we consider the possible directions of the system trajectories. To this end, consider the expansions
in which the vectors ui and wi determine the directions of the respective solution trajectories. While the control value in $ may vary in time, the control value in $T is constant on the sampling interval [O,T). Thus, for each i = 0,1,. . . the set of possible directions vi which can be generated by different choices of uo is larger or equal than the corresponding set of possible directions Wi generated by different U T . The cases (i) and (ii) now show that the sets of possible directions vi and wi are indeed identical for i = 0 , l and 2, because (i) and (ii) are unconditionally feasible provided UT is chosen appropriately. Note that the T-dependence of UT is crucial in (ii) because it gives us the additional flexibility needed for achieving w2 = 212. This is no longer possible for the directions 213 and w3 which affect the trajectories with order O(T3).Indeed, our analysis shows that the direction 213 can be decomposed as 213 = ui 2132, such that w3 = 213’ can always be achieved via the u(’)term in UT while cannot in general be reproduced by w3. This direction 2132 is exactly the expression appearing on the left hand side of (29) which depends on the Lie brackets of the vector fields and on the continuous-time feedback law U O . Condition (29) now demands that 2132 lies in span(g1,. . . ,gm) such that it can be compensated by the a-term of the sampled-data feedback law U T .
+
222
Remark 4.5. While the formulation of condition (29) is suitable for the geometric interpretation, it is difficult to generalize it to orders O(T'J),q 2 5. However, using Theorem 3.1 directly we can obtain a simple recursive procedure for computing U T for arbitrary orders: Assuming that U O ,. . . , U M - ~ in (7) are already determined and realize the order O(TM+').Then, comparing the summand for s = M in (10) with the summand for s = M in the Taylor expansion of $ ( T , x ) leads to a (in general overdetermined) linear system
G ( x ) u M ( x= ) b(~).
(31)
If (31) admits a solution, then this defines the M-th component of UT in (7) which then realizes the order O ( T M t 2 )If . (31) does not admit a solution, then the order O(TM+')cannot be achieved by a sampled data feedback law. This procedure can be efficiently implemented in MAPLE using the least squares solver in order to solve (31) and checking the residual in order to decide whether (31) is solvable. The MAPLE implementation is available on www.math.uni-bayreuth.de/~lgruene/publ/redesign_multiinput.html.
Furhermore, this procedure shows that we can always achieve any desired order if the matrix G is square, i.e., the control dimension m equals the space dimension n, and invertible.
Remark 4.6. In Ref. 13 it was shown for single-input systems, i.e., m = 1, that the condition [go,g1] E span(g1) is necessary and sufficient for the existence of sampled-data feedback laws UT realising A $ ( T , x ) = O ( T Q ) for all q 2 2 and all continuous-time feedback laws U O . We conjecture that the generalization of this condition to the multi-input case is [gi,gj] E span(g1,. . . ,gm) for all i , j = 0 , . . . , m. Note that the sampleddata feedback laws considered in Ref. 13 are not necessarily locally bounded and thus may not fulfill our Definition 2.1. 5. Examples We illustrate our results by two examples. We first consider the second order version of the Moore-Greitzer jet engine model
with the continuous-time stabilising backstepping feedback law U O ,(x) ~ = -721 5x2 derived in [9, Section 2.4.31. Here the condition (29) shows that
+
223
no admissible sampled-data feedback u~ satisfies A+(T,IC, u ~ 5)0(T4), cfr., [4, Section 41. Now we examine this system with an additional control U O ,E ~ 0, i.e.,
Note that this is now an academical example because the vector field g 2 = ( 9 2 , 1 , 9 2 , ~ and ) ~ its control u0,2 do not have any physical meaning. Nevertheless, the additional input allows for the design of higher order sampled-data feedback laws. Indeed, while u0,2 = 0 implies that the left hand side of our condition ( 2 9 ) coincides for ( 3 2 ) and (33) and evaluates to
with Ul(x) from (18),on the right hand side of condition ( 2 9 ) the coefficients of g 2 yield additional degrees of freedom and ( 2 9 ) becomes
It is easily seen that this equality is satisfied, e.g., for g ~ ( x= ) ( l , O ) T and a(.) = ( O , - U ~ ( Z ) ) ~The . performance for this choice of 9 2 , a , and the resulting feedback law UT(X)
+
+
= ~ ( " ( 2 ) T u ( l ) ( ~ ) T2u(')(x)
T + -a(.) 12
is shown in Figure 1. By means of our Maple-procedure, we may compute feedbacks of even higher order. We took this approach to compute the trajectory for q = 6 in Figure 2 . Remarkable is that neither the sampled-data feedback for q = 2 (i.e., u~ = uo)nor the feedback for q = 3 preserve the asymptotic stability of the continuous-time system. In contrast to that the fourth order feedback preserves asymptotic stability and the feedback for q = 6 provides an even better performance despite the large sampling period. For further investigation of our analytically constructed control laws we analyze the three dimensional Moore-Greitzer modeI. Adding an additional control input u0,2= 0 analogously to (33) we obtain the system
224 xl(t),T= 0.15,~(0)=[-0.65,2.5] I
-0.4 I
A
!.O--0.6
-1.4
0
0.5
1
1.5
xZ(t),T= 0.15,~(0)=[-0.65,2.5]
05
1
1.5
Fig. 1. z ( t ) for example (33), T = 0.15. Continuous-time solution (--); sampledclata solution for order q = 2 ( o ) , q = 3 (x) and q = 4 (0).
with
CT = 2
U0,1(2)
and continuous-time controller (cfr. [9, Section 2.4.21)
= -(C1 - 321)
u0,2(2)= 0
using the parameters c1 = 1 and c2 = 50. Again, the Maple-routine provides
225 x i (t), T= 0.5, ~(0)=[-0.65,2.5] 2
'
-5' 0
0.2
0.4
25,
,
,
0.6
0.8
1
1.2
1.4
T(t),TI 0.5,~!0)=[-065,2.5]
1.6
1.6
,
,
201
-201
0
I 2
/t
"
0.2
0.4
"
0.6
0.8
"
1
1.2
"
1.4
1.6
'
1.8
I 2
Fig. 2. z ( t ) for example (33), T = 0.5. Continuous-time solution (--); sampledclata solution for order q = 2 ( o ) ,q = 3 (x), q = 4 (0) and q = 6 (*).
sampled-data feedbacks for q = 4,5, but reveals that there does not exist a control law for order q = 6. However, the order q = 6 becomes feasible if one adds g 3 ( 2 ) = (O,O, l ) T ~ 0 , 3 ( ~with ) U O ,E ~ 0 as a second additional control term. Figure 3 presents the numerical simulations of this design procedure. The sampled continuous-time feedback UT = uo does not retain the asymptotic stability of the continuous-time solution. Instead, it exhibits an
226 Xl(t), T=O.O4, X(0)=[-1.85,18.5,10.5]
-2'
0
"
0.02 0.04
"
0.06 0.08
"
0.1
0.12
"
0.14
0.16
0.18
Fig. 3 . ~ ( tfor ) example ( 3 6 ) , T = 0.04. Continuous-time solution (--); sampled-data solution for orders q = 2 (o), q = 4 (0) and q = 6 (*).
asymptotically stable periodic trajectory and even divergence for sampling periods T L 0.052. In contrast to that UT for q = 4 and q = 6 preserve the asymptotic stability for T 5 0.05 for q = 4 and T 5 0.064 for q = 6, respectively, while for larger sampling intervals the solutions become first periodic and eventually divergent, too.
227
References 1. A. Arapostathis, B. Jakubczyk, H.-G. Lee, S. Marcus and E. D. Sontag, Syst. Contr. Lett. 13,373 (1989). 2. L. Griine and P. E. Kloeden, Numer. Math. 89,669 (2001). 3. L. Griine and D. NeSiC, SIAM J . Control Optim. 42, 98 (2003). 4. L. Griine, D. NeSid and K. Worthmann, Automatica (to appear). 5. L. Griine, D. NeSid, J. Pannek and K. Worthmann, Redesign techniques for nonlinear sapled-data systems, at-Automatisiemngstechnilc (to appear). 6. A. Isidori, Nonlinear control systems, 3rd edn. (Prentice Hall, Upper Saddle River, New Jersey, 2002). 7. H. K. Khalil, Nonlinear systems, 3rd edition (Prentice Hall, Upper Saddle River, New Jersey, 2002). 8. V. Krishnamurthy, Combinatorics: theory and applications (Affiliated EastWest Press, Madras, 1985). 9. M. KrstiC, I. Kanellakopoulos and P. V. KokotoviC, Nonlinear and adaptive control design (John Wiley & Sons, New York, 1995). 10. D. S. Laila and D. NeSiC, Automatica 39,821 (2003). 11. D. S. Laila, D. NeSiC and A. Astolfi, Sampled-data control of nonlinear systems, in Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005, eds. A. Loria, F. Lamnabhi-Lagarrigue and E. Panteley, Lecture Notes in Control and Information Sciences, Vol. 328 (Springer-Verlag, Berlin, Heidelberg, 2006) pp. 91-137. 12. S. Monaco and D. Normand-Cyrot, Europ. J . Control 7 ,160 (2001). 13. S. Monaco and D. Normand-Cyrot, Input-state matching under digital control, in Proceedings of the 45th Conference on Decision and Control (San Diego, CA, 2006): 1806-1811. 14. D. NeSiC and L. Grune, Automatica 41,1143 (2005). 15. D. NeSiC and A. R. Teel, IEEE Trans, Automat. Control 49 (7), 1103 (2004). 16. D. NeSiC and A. R. Teel, Automatica 42, 1801 (2006). 17. D. NeSiC, A. R. Teel and P. V. KokotoviC, Syst. Contr. Lett. 38,259 (1999). 18. D. NeSiC, A. R. Teel and E. D. Sontag, Syst. Contr. Lett. 38,49 (1999).
228
ON THE DEFINITION OF TRAJECTORIES CORRESPONDING TO GENERALIZED CONTROLS ON THE HEISENBERG GROUP P. MASON Institut Elie Cartan U M R 7502, Nancy- U n i v e r s i t ~ / C N R S / I N R I A , POB 239, Vandoeuwre-12s-Nancy 54506, France E-mail: Paolo. Mas0nQiecn.u-nancy.fr In this paper we consider control-affine systems driven by controls that are not absolutely continuous, so that a solution in the classical sense is not defined. We discuss the possibility of defining a generalized notion of solution. Focusing on the Heisenberg example, we state some conjectures that would justify an extension of the definition of solution, and we analyze them by means of examples and counterexamples.
Keywords: Generalized Controls; Motion Planning; Stochastic Differential Equations.
1. Introduction
The aim of this paper is to discuss the possibility of generalizing the notion of solution of an affine control system where the control is supposed to belong to a functional space such that the classical results about local existence and uniqueness do not apply. This problem has been studied extensively by control theorists for its applications to impulsive and it is also interesting for probabilists since, as we will precise later, it is connected to the field of stochastic processe~.~>~?~ The control systems under consideration here have the form m a= 1
where x E R" and the vector fields fa are smooth for every i , and satisfy some suitable growth condition (for instance we can assume that they are sublinear or bounded). We will not insist on the regularity properties of the
229
fields f i , which actually could be sharpened, since this is not the target of this paper. In fact our aim is to discuss the minimal requirements on w(.) that enable us to give a meaningful definition of solution for (1). Notice that the equation (1) is affine with respect to the derivative of the control and therefore, under the previous assumptions on fi, it is quite natural to expect that the possible solutions have the same regularity as w(.). We now define the main objects used throughout this paper. An admissible control for (1) is an absolutely continuous function w(.) = (v1(.), ...,urn(. )) : [O,T]--t Rm. Notice that, if w is an admissible control, then, setting u = 6 in equation (l),we obtain the affine control system
+
j. = fo(.)
c m
Wfi(.)
7
4 . 1 E L1([0,TI, Rm)
*
i=l
An admissible trajectory is an absolutely continuous curve x(.) satisfying (1) a.e. for some admissible control w. Observe that in general, if m is smaller than the dimension n of the state space, the set of the admissible trajectories for (1) is a proper subset of the class of absolutely continuous curves on R". Let 3 =span{fo ELl uifi, u E Rm} and consider the vector space generated by the Lie brackets of elements of 3
+
It is well-known that the condition Lie,(?=) = T,R" 11 R" for every x E Rn, i.e. .?= is a bracket-generating family of vector fields, is a genericity condition (with respect to the C2 topology) for families of vector fields generated by at least two vector fields. If it is the case, then, by Krener's Theorem, the attainable set for (l),starting from any given point xo A ,,
:= {x(T) E R" :
x(.) admissible trajectory for (l),T > 0) 0
0
is "full dimensional" in the sense that clos(d,,)> A,, (here d,, denotes the set of interior points of d,,).In particular, in the driftless case fo = 0, the attainable set coincides with R",as a consequence of Chow's Theorem. Moreover every smooth trajectory can be uniformly approximated by admissible trajectories, and therefore the Motion Planning Problem is meaningful in this case (see for instance Refs. 4,9) and it is clearly strictly connected to the problem of generalizing the notion of solution for (1).
230
1.1. Previous approaches and results In order to give a meaning to (1) in the case in which v is non-admissible one can introduce the following equation m.
whose solutions are defined by means of the Stieltjes integration. It is clear that this new formulation allows one to give a meaning to the solution in the more general case in which v(.) has bounded variation. This solution can be seen as a generalized solution to the equation (1). Nevertheless, motivated by mechanical models (see for instance Refs. 1,2), a weaker notion of generalized solution, that applies to the case of discontinuous controls with bounded variation, has been proposed in Ref. 3. In this framework, a generalized solution can be simply regarded as the limit of a converging sequence of admissible solutions with uniformly bounded variation (see also Ref. 7). In particular, with this new definition and in the case in which the vector fields fi, i = 1,.. . , m do not commute, the uniqueness of the solution is no more guaranteed. Now we call Vadm, X a d m the spaces of the admissible controls and the admissible trajectories and we consider the input-output map associated to (l),i.e.
A natural way to define a notion of generalized solution of the equation (1) consists in fixing some topological spaces V 3 Vadm, X 3 X a d m in such a way that the input-output map can be extended to a continuous map Po: V 4 X. Then a generalized solution associated to some v(.) belonging to the closure of U a d m with respect to the topology of V can be simply defined as (w)(.). For instance it is well-knowns that in the case in which m = 1 the input-output map is continuous in the topology of uniform convergence and therefore in this case, associated to each continuous control v(.), one can define a generalized solution. This is no more true if m > 1 and, as one could expect, the vector fields fi, i > 0 do not commute. In this case one needs to consider more appropriate topologies. The notion of generalized solution is also useful in order to exploit the relation between stochastic differential equations (SDE) and ordinary differential equations of the form (1).In this case v(.) and z(.) represent paths of the stochastic processes that correspond respectively to the input and
23 1
the output of the SDE. Notice that a path of a stochastic process is in general (with probability 1) a continuous path, so that the case of continuous (but not regular) paths turns out to be of particular importance in the literature. One important result in this direction has been obtained by T . Lyons in Ref. 5, where the spaces V and X are identified with the space of continuous functions with bounded pvariation, with p > 2 (for precise definitions, see the next section). More precisely in Ref. 5 the Picard iteration method has been used to prove the existence and uniqueness of the solutions of the equation (3). Such results are inspired by the work of L.C. Young," who essentially proved that the Stieltjes integrals of the form f dg, where f , g have finite p and q variation and > 1, are well-defined. In Ref. 6 the problem of generalizing the notion of solution to the case in which u(.)has finite p-variation, with p 2 2, has been treated. Even if the results obtained are powerful, they are very difficult to apply for several reasons. Indeed such approach does not guarantee the existence of a generalized solution for any u in the class of functions with bounded pvariation. Moreover, if a solution exists, in general it is not unique. Finally the very theoretical approach precludes the possibility of direct applications of the results. Our aim is to discuss some alternative ways of defining generalized solutions which are easier to handle. In particular we analyze some conjectures related to the Heisenberg system with the help of some examples and counterexamples.
+
2. Functional spaces and topologies
In this section we introduce some functional spaces that can be seen as generalizations of the classical spaces of Lipschitz, BV and absolutely continuous functions. The first interesting functional space is the space of the Holder-a functions Co@([O, TI),with a E (0, l ) ,i.e. the space of real-valued (or vector-valued) functions satisfying S U ~ ~ ~ , ~ ~ ~< [+m. ~ It , is easy to verify that it is a Banach space with norm
Jf/i:L%:2)L
It is interesting to investigate the properties of Coya([O, TI) as a varies. For this purpose the following simple result is useful.
Proposition 2.1. I f f E Co)a'([O,T]) with a' > a , limn--tmf n = f uniformly and 11 is uniformly bounded, then lim f n = f in the topology fnllat
n+m
~
~
232
Proof. Assume by contradiction that fn does not converge t o f in the norm 11 . / I o 1. Then there must exist a subsequence fnk and two sequences Z k , Yk E [o, T ] such that
for some C > 0. Therefore we can suppose that, up to subsequences, limk,, xk = Z, limk,, Yk = g, and we deduce, from the uniform convergence of fn to f,that the inequality (6) may hold only if Z = g. This clearly contradicts our hypotheses, since the right-hand side of (6) is bounded by Ilfnk - f l l a ' 1 Z k - yklol'-a, which tends to 0. 0 Since for every f E CO+'([O,T]) it is easy to find a sequence of smooth functions approximating f as in the hypotheses of the previous result, we get the following corollary.
Corollary 2.1. The closure ofC"([O,T]) with respect to ( [ 0 TI). , the set
u
11 . ]Ia
contains
Cola'
a'>a
Notice that the closure of C-([O,T])with respect t o 11 . [la doesn't coincide with Co>a([O,T]) since, for instance, the function f ( t ) = ta cannot be approximated by a smooth function in this norm. Observe also that Co@([0,TI) contains properly the space of Lipschitz functions, but it does not contain the space of absolutely continuous functions, since to is not Holder-a for every p < a. This means that each possible definition of solution to the equation (1) that applies to the case in which v(.) is Holder cannot be considered as a generalization of the classical notion of Carathkodory solution. Therefore we introduce new spaces that are more general. The space BV, of functions with finite p-variation, introduced by Wiener, is the set of functions f satisfying the following condition
In particular by the Minkowski inequality it follows trivially that the map
233
defines a norm on BV, and the space BV, turns out to be a Banach space. Moreover it is clear that Co9'lP c SV,. We give a particular importance t o the BV, functions that are also continuous. In particular we mention two functional spaces that are contained in the class of continuous and BV, functions. The first one is the set of regular BV, functions5 (that we denote by BT/preg),i.e. the functions f satisfying the following condition
C If(ti+l)-
limsup ...< t N = T
O=to
rnax Iti+l
f(ti)lP =0 .
(8)
-til+O
I t is not difficult to see that the closure of C" with respect to the norm 11 l l ~ v ,is contained inside the space of BVp'"g functions. The other functional space that we introduce is a natural generalization of the space of absolutely continuous functions: we say that f E AC, if it satisfies the following property: V& > 0 36 s.t.
c
/ti- Sil
< 6*
cIf
i
(ti)- f (%)I"
<E.
(9)
a
Notice that for every p 2 1 we have C 0 ) l / P c AC,. Moreover it is easy to see that we have the following inclusions for every 1 5 p < p':
BV, n C
c BT/pegC AC,#
C BV,,
nC
(10)
Indeed these inclusion can be easily derived from the definitions and the uniform continuity (we are working with continuous functions defined on an interval). In some sense this series of inclusions shows that the spaces that we have introduced are "almost" equivalent, since infinitesimal changes of the parameter p can reverse the inclusions.
3. The Heisenberg example We consider now the simplest driftless completely non holonomic system, namely a system of the form X
= SlFi(z)
+S ~ F Z ( Z ) ,
where z E R3 and F1 and Fz generates the Heisenberg algebra i.e. F1 =
(
1 lFZ
=
(
z:,2)
(11)
234
Notice that [PI,J”] = (O,O, l ) T and that the system is nilpotent, in the sense that the brackets of order greater than two annihilate. For every admissible control z ~ ( . ) the first two components of the corresponding admissible trajectory coincide with (z11(-), z12(.)) up to additive constants. Moreover the third component is characterized by the equation 1 2
5 3 = -(XI&
- &22)
(13)
that means that, if we set
that is the area computed in counterclockwise sense of the region spanned
t
v2
positive area
Fig. 1. The area spanned by a curve
by the planar curve ( L C ~ ( . ) , X ~ ( - )(see ) Figure l),then z ~ ( t=) A[tc(.)](t). Consider now a non admissible control v(.) in equation (ll),and suppose that it is possible to define a generalized solution for z~ by means of a continuous extension of the input-output map. Then the first two components must coincide with the corresponding controls up to constants. Therefore the problem of generalizing the notion of solution reduces to the following:
235
Problem: Given a continuous, but not absolutely continuous, curve x ( . ) = (XI(.),2 2 (.)), is it possible to give a definition of the area generalizing (14) ? We focus in particular on curves belonging to the spaces Coia([O,T]).We start with a simple example after which we will discuss some possible conjectures. Fix a ~ ] 0 , 1and [ consider the following sequence of admissible controls:
1 na
1
w g ( t ) = (-cos(nt),-sin(nt)).
(15)
na
WE(.)
It is clear that Ilwz(.)llw -+ 0 and therefore is an approximating sequence of the zero function in the Co topology. The corresponding area is the following:
d[wE(.)](t) = n(1-2a)t.
(16)
Therefore we can distinguish three cases: If a > 1/2 then d[w,"(.)](t) converges uniformly to zero as n goes to 00, If a = 1/2 then d [ v E ( . ) ] (converges t) uniformly to the function w ( t ) = t , If a < 1/2 then d[wE(.)](t) d'iverges. This example with a 5 1/2 is enough to conclude that the Co norm does not render the input-output map continuous. Therefore we should look for a stronger norm. For this purpose, it is interesting to notice that the functions vE(.) are such that the sequence IIwEllCy is uniformly bounded. More precisely lim 7LW '
IlwEllat
=
{
O if a' > a 2 if a'= a 00 TQ if a' < a
This suggests some natural conjectures. Let V:;: Then:
= {w E Coo,
1 1 ~ 1 51 ~ K } .
C1 If a > 1/2 then the input-output map can be extended continuously t o the set {w E IIvlla 5 K } , seen as the closure of : ; U with respect to the norm ll.llw. Cola,
In5 it was proved that, if we consider the topology given by 11 . [ ~ B vwith , p < 2, the input-output map can be extended continuously to a map @lo defined on SV, n C. Combining this result with Proposition 2.1 we obtain
236
easily that the conjecture C1 is true:
As a consequence, we obtain that it is possible t o generalize the definition of the area d[v(.)](t) for w E Cot" and a > 1/2.For a 5 1/2 the previous example shows that the analogous conjecture is not true. However for a = 1/2 one could still expect the possibility of defining the area on subsequences. C2 Given w E Co)6, for every sequence of smooth functions w, converging uniformly to w and with 11wnl11/2 uniformly bounded, there exists a subsequence such that the corresponding admissible solutions of (11) (or, more in general, of (1)) form a Cauchy sequence in the uniform topology. If this conjecture was false than one could still try to consider a stronger norm on the space of controls, getting the following conjecture.
C3 If the space of the admissible controls is endowed with the topology given by 11 . then it is possible to extend continuously the inputoutput map. In the previous example the functions were constructed in order to converge to 0, but it is clear that this is far from being a natural way of approximating the function w = 0. One can also try to overcome this problem by defining directly a notion of convergence of sequences of functions. For instance, given a function w and a sequence of times 0 = to < tl < . . . < tk-1 < tk = T , one can consider the piecewise affine approximations obtained joining successively by segments the points w(th) and w(th+l),for h = 0 , . . . , k. Therefore another possible conjecture is the following
C4 Let w E Coi6 and d k )be a sequence of piecewise affine approximating functions defined as before and characterized by the times t r ) , and assume that lim sup ItfJl - tr'l = 0. Then the sequence d [ d k ) ] ( T is ) k-w
h
a Cauchy sequence. We discuss the above conjectures with the help of some examples.
Example A. Consider the following function
which represents a closed curve in the complex plane. We want to prove the following facts. 0 0
v(.) ECO'b.
9
Consider the sequence of times t h = h = 0,. . . , 4k - 1 and define the piecewise affine function obtained joining the point W ( t h ) to v(th+l) for every h = 0 , . . . , 4 k - 1 , i.e ?& as
lim d [ v ( ' ) ( . ) ] ( 2 n )= +a.
Then
k-++ca
For the first issue we first fix two real numbers x and y. Then we define an integer number Q depending on x and y (in the computations that follow, [.]denotes the integer part):
Q
:= [log,
-
( d-.1)]YI 1 5-
I
1
&q < 2Q+'
We have that
For the second term we have
I
C
+a
1
$(e4"iz
-e'"iy)I
+m
C
<2
k=Q+1
k=Q+1
1 2k
- =2-Q+1
while we can estimate the first one using the inequality
I
Q
-
k=O
-
1 2(e4k2r - e4"Y
<
)I-
Q
-
1
<4
m
leis - eiPl < Icu-pI: Q
C
T4klx - yI = 2'112 - yI k=O 2 k=O < 2Q+llz - yI 5 2 J M .
Finally we have found that the inequality Iv(x) - v(y)l I 6 J m holds for every x and y in the interval [ 0 , 2 n ]and therefore w(.) E C o l t .
238
Remark 3.1. Observe that each term in the Fourier series of v(.) is Holderwith the same optimal constant.
;
Now we want t o see that the function w(.) defined above, as a function from [0,2n] to R2, disproves the conjectures C2 and C4. Consider the 4N times th = h = 0 , . . . , 4N - 1. Then the area corresponding to the segment connecting 4 t h ) t o . ~ ( t h + iis) given by ( m ( t h ) v l ( t h + i ) - ~ l ( h ) v ~ ( t h + l ) ) , where v1 , v2 denote the first and second components of v. In particular we have
9
;
c
N-1
~ ( t h= )
k=O
c3 c N-1
00
1 2k
- C O S ( ~ " ~ ~ T ~ ) +
=
k=N
1 gc0~(4~-"2~h)+2-~+',
k=O
Therefore N-1 2 - N + 1
v1(th)w2(th+1)- w2(th)v1(th+1) = c k = O
2k
( sin(4kth+l)- sin(4kth))
+ Cn,m=O 2n+m ( c o ~ ( 4 ~ tsin(dmth+1) h) - c 0 s ( 4 ~ t h + lsin(4%)) ) N-1
1
.
If we consider the sum over h = 0, . . . 4N - 1 of these terms and we exchange the order of summation we obtain the area N-1
C
.4[JN'(*)1(27r) =
n,m=O
.
2nSm 1
c
4N-1
+
~ i n ( 4 ~ - ~ 2 7 r (1) h - 4n-N2nh) .(18)
h=O
Now we want to see that
c
4N- 1
+
~ i n ( 4 ~ - ~ 2 n (1) h - 4n-N2nh) = 0
h=O
+
") +4m = if n # m (n,rn < N ) . We rewrite 4m(h 1) - 4*h = h( hP + 4. N Let M be the greatest common divisor of p and 4N, and define D := If we prove that
&.
(h
+
4N + D)p+q = h p + q + 2
+
+ +
(mod d N ) ,
(20)
then $ ( ( h 0 ) p q) = $(hp q ) 7r and it is clear that the sum of 2 0 successive terms in (19) is 0. Therefore, since 2 0 divides 4N, the whole series (19) is 0. To prove (20) notice that 2Dp = 4 N $ = 0 (mod 4N), so that either D p = 0
239
%
(mod 4N) or Dp = (mod 4N). In the first case we would have 4N& = 0 (mod 4N) which means that 2M divides p , which is impossible, since otherwise 2M would be a common divisor of p and 4N. Therefore we obtain N Dp = (mod 4N) that immediately gives (20). So the only non-vanishing terms in (18) correspond to n = m N
%
c$
N-1
d[v")(.)](2n) =
n=O
_.
4N-1
~in(4~-~2rr) h=O
N-1
N
n=O
m=l
Since lim 4m ~ i n ( 4 - ~ 2 . r= r ) 1 we have that m-oo
lim d[vcN)(.)](2rr)= +co .
N-++oo
Remark 3.2. There is a easy geometric interpretation of this example. Indeed, in (17) each term in the summation corresponds to 4k turns on a circle centered a t the origin with radius The area corresponding to such path is exactly rr and it is easy to see that the area of a finite sum of 1 elements in (17) is exactly lrr. As suggested by Figure 2, where the image of v(.) is drawn, the fact that v(.) is given by the sum of an infinite number of circles with decreasing radius and increasing speed renders this function very irregular.
&.
Remark 3.3. A similar example has been discussed by L.C. Young," in order to prove that a particular inequality, that would justify the existence of the Stieltjes integral f dg, does not hold in general if f and g have bounded 2-variation. Remark 3.4. We proved that there is a sequence of piecewise affine approximations of v such that the corresponding area goes to infinity, but we don't know if this remains true for all possible converging sequences of piecewise affine approximations. However, the geometric interpretion given in Remark 3.2, suggests that this is true. On the other hand, in general, the area does not converge t o infinity for smooth approximations (also with uniformly bounded Holder constant), since
is such that d[v(N)](2rr)= 0 VN > 0.
240
Example B. We consider now a slightly modified example:
In this case, exactly as before, one can prove the following. 0 0
V(+) is Holder-$. h = O , . . .4' - 1 is such that the The sequence of times th = area of the corresponding piecewise affine approximation of a(.) does not converge as k goes to infinity even if it is uniformly bounded. More precisely the area of such approximation is k
d[V(k)(.)](2.rr) = (-I)k
C (-4)msin(4-m2.rr) m= 1
and it is easy to see that ~ t [ d ~ ~ ) ( . ) ] ( 2and . r rd[d2'++')(.)](2n) ) converge to two different values as k goes to infinity. The image of fi is depicted in Figure 2.
Fig. 2. The images of
2)
and 5 in the complex plane
24 1
Example C. Consider now the following function.
This function satisfies the same properties as the one of the first example, and such properties can be proved exactly in the same way. The most interesting feature of this example is that the approximating sequence d N )= C,”==, &e4“’ converges to 6 in the norm (1 . II1/2, while we still have lim~4+ood[dN)](27r)= +cm. Moreover, similarly t o Remark 3.4 it is easy to construct an approximating sequence converging to 6 in the norm 11 . 111/2 and such that the corresponding area is uniformly bounded. Therefore also conjecture C3 turns out to be false. 4. Conclusion
We have seen that it is not easy to find a definition of generalized solution that applies to the case of Holder-$ controls. However notice that, for instance, the first example we have constructed is very particular, since the components vl and 212 seem to “cooperate” in order to increase the area. Therefore it seems quite natural to look for some particular %onresonance” condition on the space of controls, that could also represent the case of two paths of independent stochastic processes. Then one could try t o define a generalized solution of (11) extending continuously the input-output map relatively to the class of controls satisfying this non-resonance condition. Our future research on this field will be based on this new approach.
Acknowledgements The author wishes to thank Prof. Andrei A. Agrachev, Dr. Ugo Boscain and Prof. Alberto Bressan for their suggestions and comments on the subject of this paper.
References 1. Aldo Bressan, Atti Accad. Naz. dei Lincei, Memorie classe di Scienze Mat. Fis. Nut. (9) 1, 149 (1989). 2. Aldo Bressan, Atti Accad. Naz. dei Lincei, Memorie classe di Scienze Mat. Fis. Nut. (8) 19,249 (1989). 3. Alberto Bressan, F. Rampazzo, Boll. Un. Mat. Ital. B (7) 2, 641 (1988). 4. W. Liu, SIAM J. Control Optim. 35,1989 (1997).
242
5. T. Lyons , Math. Res. Lett. 1, 451 (1994). 6. T. Lyons, Rev. Mat. Iberoamericana 14, 215 (1998). 7. P. Mason, Sistemi di Controllo Impulsivi su VarietA, Tesi di Laurea, Universit& degli studi di Padova, (Padova, 2002). 8. H.J. Sussmann, Ann. Probability 6, 19 (1978). 9. H. J. Sussmann and W. Liu, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, in Proceedings of the 30th IEEE 1991 Conference on Decision and Control, Brighton, UK (Brighton, UK, December 11-13, 1991): 437-442. 10. L.C. Young, Acta Math. 67, 251 (1936).
243
CHARACTERIZATION OF THE STATE CONSTRAINED BILATERAL MINIMAL TIME FUNCTION C. NOUR Computer Science and Mathematics Division Lebanese American University Byblos Campus, P.O. Box 36 Byblos, Lebanon E-mail: [email protected] We characterize the state constrained bilateral minimal time function as the unique proximal solution of a partial Hamilton-Jacobi equations with certain boundary conditions. This generalizes [13,Theorem 3.41 where Stern studied the unilateral case.
Keywords: Bilateral and unilateral minimal time function; State constraint, Hamilton-Jacobi equations; Proximal analysis; Nonsmooth analysis
1. Introduction
Let F be a multifunction mapping points z in Wn to subsets F ( z ) of Rn and let S c Rn a closed set. Associated with F is the differential inclusion k ( t ) E F ( z ( t ) ) a.e. t E [O,T],z(0) = TO.
(1)
A solution to (1) is an absolutely continuous function z(.) defined on the interval [0,TI with initial value z(0) = zo, in which case we say that z(.) is a trajectory of F that originates from 20. The notation k ( t ) refers to the derivative of z(.) at t and is the right derivative if t = 0. We assume throughout this paper that F satisfies the standing hypotheses; that is, F takes nonempty compact convex values, has closed graph, and satisfies a linear growth condition: for some positive constants y and c, and for all z E Rn, ,v E F ( z )
* ll4l I rll4l +c.
The multivalued function F is also taken to be locally Lipschitz: every z E Rn admits a neighborhood U = U ( x ) and a positive constant K = K ( z )
244
such that 51,x2 E
u ===+ F ( x 2 ) c F(z1) + Kllx1 - z211z.
We associate with F the following function h, the lower Hamiltonian: h ( x , p ) := min{(p, w) : w E F ( z ) } . Now let S c R" be a closed set. A trajectory x ( t ) of F over [O,T] which satisfies the state constraint x ( t ) E S for all t E [O,T]is called an S-trajectory of F . For C c S , a closed set, the S-constrained (unilateral) minimal time function (associated with C) Ts(.,C): S R" U {+m} is defined as follows: For a E S , Ts(a,C)is the minimum time taken by an S-trajectory to go from a! to C (when no such trajectory exists, Ts(a,C) is taken to be +m). We assume throughout this note that Ts(.,C) is extended to be +m on S", the complement of S. When S = R" (unconstrained case) then Ts(.,C) coincides with the well known minimal time function (denoted by T(.,C)) which has a large literature; see for example Refs. 4,10,11,14,15 for the regularity study of T ( . ,C), and Refs. 1,3,6,7,12,15for its HamiltonJacobi characterization. We also invite the reader to see [2, Chapter 41 for a thorough history of such results. The Hamilton- Jacobi characterization of the state constrained (unilateral) minimal time function is studied (apparently for the first time) in Stern13 where a characterization of this function is given as the unique proximal solution of Hamilton- Jacobi equations with certain boundary conditions. Let us recall the main result of this paper. We begin with some definitions. Our reference will be the book of Clarke et aL5 and the paper of Stern13 . For a lower semicontinuous function f : R" RU{+co} and a point x E domf := {x' : f(x') < +m}, we denote by d p f ( x ) the proximal subdifferential of f at x. We recall that C E a p f ( x ) if and only if there exists (T = ( ~ ( C) x , 2 0 such that
-
-
f(Y)
- f(x)
+ 4 I Y - 4122 (C,Y - 4,
for all y in a neighborhood of x. We say that S is wedged provided that at each boundary point x one has pointedness of N g ( x ) ;that is, Ng(x)n{-Ng(x)}= {0}, where N g ( x ) denotes the Clarke normal cone to S at x. We note that wedgedness implies S is the closure of its interior, and that any closed convex body is wedged. We also say that strict inwardness condition is satisfied if
h(x,C) < 0 VC E N g ( x ) , Vx E bdry S.
245
-
A function cp : W" -1 - 00, +00] is said to be (C, S)-continuous if there exist yV > 0 and a function wV : [0,7] [0, +00[ such that limw(s) = 0 and cp(z) 5 w V ( d c ( z ) ) 'dx E S U {C 310
+ yVB},
where d c ( . ) is the Euclidean distance function and B is the unit open ball. If Ts(.,C)is (C,S)-continuous then we say that F is S-constrained small time controllable. It is well known that the S-constrained small time controllability condition is equivalent to the continuity of T , ( . ,C) on S n {C vB} for some v > 0. Now we can state the main result of Stern13.
+
Theorem 1.1. Assume that S is compact and wedged, and that the strict inwardness condition holds. Assume also that F is S-constrained small time controllable. Then the function T s ( . ,C ) is the unique lower semicontinuous - 00, +00] which is ( C ,S)-continuous, bounded below function cp : Wn 1on Rn, identically 0 on C , identically +00 on S" and satisfies 0 0 0
+ + +
h ( z ,apcp(x)) 1 = 0 for all x h ( z ,apcp(z)) 1 2 0 for all z h ( z ,apcp(x)) 1 I:0 for all z
{int S } \ C, {int S } n C, and E {bdry S } \ C, E
E
-
The purpose of this note is to generalize the preceding result to the bilateral case, that is, find a Hamilton-Jacobi characterization for the Sconstrained bilateral minimal tame function Ts : S x S [0, +00] defined by the following: For ( a ,0)E S x S , Ts(a,p) is the minimum time taken by a 5"-trajectory to go from a to p (when no such trajectory exists, Ts(a, p) is taken to be +00). We note that the unconstrained bilateral case is studied in Nour' where we give a Hamilton-Jacobi characterization and some regularity results. In the next section we present our main result and then we sketch its proof. More detaiIs can be found in the forthcoming paper of Nour & Sterng. 2. Main result
We begin with the following definition. We say that a function 'p : R" x R" -1 - 00,+co] is bilateral S-continuous if there exist yV > 0 and a function wV : [0,yv] [0, +oo[ such that limw,(s) = 0 and
-
(P(z,g) 5 w,(l.
810
- YII) 'da E
s, 'dx,Y E ( a + 7pB) n s.
If the function Ts(.,.) is bilateral S-continuous then we say that F is bilateral S-constrained small time controllable.
246
Let D : { ( a ,a ) : a E R"} be the diagonal set. The following is our main result (here the function Ts(.,.) is also assumed to be +03 outside S x S ) .
Theorem 2.1. Assume that S is compact and wedged, and that the strict inwardness condition holds. Assume also that F is bilateral S-constrained small time controllable. T h e n the function Ts(.,.) is the unique lower semi- 03,+03] which is bilateral S continuous function cp : Rn x Rn -] continuous, bounded below o n R" x IR", identically O o n D n ( S x S ) , identically +03 o n ( S x S)" and satisjies 0
V a # ,Ll E int S and V(C, 0) E 6'py(a,p) we have: 1
0
V a E int S and V((, 0) 6 apcp(a,a ) we have: 1
0
+ h(a,C) = 0 , + qa,C) L 0,
V a E b d r y S and V,Ll E S such that a
#p
we have:
V(C,Q) E &4%,Ll), 1 + h(a,C) 5 0, and V(5,Q) E 6'PY(P, a ) , 1 + h(P, C) I 0. Sketch of the proof. We consider the multifunction F defined over C := D n ( S x S ) we can verify that the S x S-constrained (unilateral) minimal time function for the new dynamic F associated to C (denoted by Fsxs(.,C)) coincides with the S-constrained bilateral minimal time function Ts(.,.). Then the result of the theorem follows by applying Theorem 1.1 to Fsxs(.,C) (which is Ts(.,.)) after verifying that:
R" x Rn by F(z,y) := F ( z ) x (0). For
0
0
We have an S x S-constrained trajectory tracking result for F (this follows since we already have an S-constrained trajectory tracking result for F and since a trajectory of 8' is of the form ( z ( t )p) , where z ( t )is a trajectory of F ) .
A function y : R" x R" -1 - 03, +03] is bilateral S-continuous if and only if it is (c,S)-continuous (for F ) .
References 1. M. Bardi, SIAM J . Control Optim. 27,776 (1989). 2. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhauser, Boston, 1997).
247
3. M. Bardi and V. Staicu, Acta Appl. Math. 31,201 (1993). 4. P. Cannarsa and C. Sinestrari, Calc. Var. Partial Differential Equations 3, 273 (1995),. 5. F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, Vol. 178 (SpringerVerlag, New York, 1998). 6. F. H. Clarke and C. Nour, J . Convex Anal., 11,413 (2004). 7. C. Nour, ESAIM Control Optim. Calc. Var. 12,120 (2006). 8. C. Nour, J . Convex Anal. 3,61 (2006). 9. C. Nour and R. J. Stern, The state constrained bilateral minimal time function (in progress). 10. N. N. Petrov, J . Appl. Math. Mech. 34,785 (1970). 11. P. Soravia, SIAM J . Control Optim. 31,604 (1993). 12. P. Soravia, Comm. Partial Differential Equations 18,1493 (1993). 13. R. J. Stern, SIAM J . Control Optim. 43,697 (2004). 14. V. M. Veliov, Journal of Optimization Theory and Applications 94, 335 (1997). 15. P. Wolenski, Y. Zhuang, SIAM J . Control Optim. 36,1048 (1998).
248
EXISTENCE AND A DECOUPLING TECHNIQUE FOR THE NEUTRAL PROBLEM OF BOLZA WITH MULTIPLE VARYING DELAYS N. L. ORTIZ
Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23284, USA E-mail: nlortizOvcu. edu In this article we consider the generalized problem of Bolza with different time varying delays appearing in the state and velocity variables. We derive existence of solutions and prove a natural extension of a decoupling theorem, which was originally introduced by Clarke in the non delay casez and was previously extended7 to the neutral case when both of the delays appearing are given by one function. Keywords: Generalized problem of Bolza; Multiple delays; Non-smooth analysis; Neutral systems; Optimal control.
1. Introduction
In this article we study the generalized problem of Bolza involving two different time-varying delays that appear in the state and velocity variables. We will refer to this problem as the multiple delays neutral problem of Bolza, or multiple delays NPB. We seek to extend previous results' to guarantee existence of solutions and use a decoupling technique introduced by Clarke' in the case without delays as an important tool to derive necessary conditions analogous to Ref. 7. In this note we consider the functional
n ( ~:=C(z(T)) ) +
iT
L ( t , ~ ( t ) ,-~ A( ti ( t ) ) , k ( t ) , k (-t A'@))) d t . (1)
Here L : [0,TI x + (--00, co] and C : Rn -+ (-GO, m] are given and are allowed to reach 00 in order to incorporate constraints. The given delay functions A1 : [O,T]-, [O,Ao] and A2 : [O,T]-, [O,Ao], with A0 > 0 a constant, are assumed to be continuous. In addition, we assume that Al(0) = 0, A,(.) is Lipschitz with Lipschitz constant Kal = 1, and extend
249
Al(s) := 0 for each s E [-A,, 0)U [T,T + A,]. We are also given the initial data c : [-A,, 01 -+ R",which we assume to be in L2[-A0, 01. We now state the main problem involved: minimize H(J:)
(2)
over all absolutely continuous functions J: : [0, T ] -+ R" with initial conditions ~(0) = 2 0 ,z ( t ) = c ( t ) for each t 5 0. The generalized problem of Bolza was introduced by Rockafellar" as a means to study a broad class of optimal control problems. In Ref. 11 he proved existence of solutions introducing a method of proof now referred to as the "direct method" in the calculus of variations. We previously extended this method t o prove existence of solutions of NPB when the delay present was given by a single constant.* In this note we offer a natural extension to the multiple delay NPB ( 2 ) . A number of results involving necessary conditions for the generalized problem of Bolza can be obtained in the literature in the case without d e l a ~ l ? ' and ? ~ various cases involving d e l a y ~ . ~In> Ref. ~ ~ ' 2, Clarke introduced a decoupling technique and a decoupling theorem that served as a powerful tool to derive necessary conditions. This "decoupling" theorem was extended to NPB with varying delay given by the same function7 and here we provide a proof for the more general case with multiple time delay functions. Further, we show that our results subsume the ones previously obtained. 2. Main Assumptions and Preliminaries
Our assumptions are the following: (i) l! is lower semi-continuous and bounded below; (ii) L ( t , a , y , v , w ) is lower semi-continuous in ( ~ : , y , v , w ) ,is C x Bmeasurable on [0,T ]x EX4", and is (jointly) convex in (v,w). (iii) There exists a nondecreasing function 0 : [0,00) 4 R satisfying limr-+mO(r)/r= 00 so that L(t,s,y,v,w) 2
e(IvI) + ~ ( l w l )
for all v , w E Rn.
In what follows, we use the natural modification of the maximized Hamiltonian:
H : R x R4" 4 R
250
As in non delay problems (see Ref. ll), H is upper semi-continuous in (z, y , w,w), C x B-measurable on [0,TI x R4n, and is (jointly) convex in (P,4 ) .
3. Existence of Solutions In order to prove our main existence theorem for (2), we rely on the following preliminary lemmas and proposition.
Lemma 3.1. Suppose X
C dC[O, T ] and V 2 L1[O,T ] are
nonempty and
T h e n V is weakly sequentially precompact. Proof. By the Dunford-Pettis criterion4 it is enough to show that (a) SUPW(.)€V11~(.)111< 00, and (b) for all c > 0 there exists a 6 Iw(s>ld s < E for all w(.) E V .
sI
> 0 such that m ( I ) < 6 implies
Given hypothesis (iii) and our assumed estimate above (3) we have that for each w(.) E V ,
p
e(lw(t>l>+ w J ( t- Az(t)>l>dt < K .
Thus, J:O(lv(t)l)dt 5 K - TO(0) since O(0) is the minimum value of 0. Let I CC [O,T]be any measurable set, R > 0 so that 8(R) > 0, and define A := I n {t : Iv(t)l I R} and B := I f l {t : Iv(t)l > R } . Then
where Ic := ( K - T O ( 0 ) ) . Since r/O(r) -+ 0 as T -+ 00, (a) follows for large R by letting I = [O,T].If we let E > 0, choose R > 0 such that ~ u p ~ ~ ~ r /
25 1
Next we show the weak lower semicontinuity of the integral part of (1)
and rely on the following proposition, a proof of which can be found in Theorem 1 of Ref. 10.
taken over ( p ( . ) , q ( . ) ) in L” (= the bounded measurable functions defined from [o,T]into
Lemma 3.2. Suppose sequences {xi(.)} C L2[0,T ] and {vi(.)} G L1[O,T ] are such that zi(t) -i z ( t ) for almost all t in [O,T]and wi(t) 2 V ( t ) (weak convergence in L’) where each zi(s) := c(s) and vi(s) := 0 for each s E [-Ao, 0). Then,
252
Proof. It follows from Proposition 3.1 that T
l i m i n f l L ( t , x i ( t ) , x i ( t - Al(t)),vi(t),vi(t - A,(t)))dt a 4 0 0
We now proceed to our main theorem.
Theorem 3.1. If there exists x(.) E dC[O,T] that is feasible for (2), then there exists an arc Z ( . ) E dC[O,T]that solves (2).
Proof. Suppose that z(.) is feasible. Select a minimizing feasible sequence {xi(.)} c dC[O, T ]so that for each i , !(zi(T)) A(xi) 5 l ( x ( T ) ) A(.) < 03. Since t(.)is bounded below it follows that A(xi) is bounded above, and so by Lemma 3.1 there exists 5 ( . ) E L1[O,T]and a subsequence (which we do not relabel) satisfying ii(.) 2 a(.). Define %(-)E dC[O,T]by
+
+
E(t) = ~ ( 0 )
rt
@(s) ds
+
253
and set equal to c(t) for t E [-Ao, 01. Then zi(t)+ %(t) for all t E [-Ao,T]. It follows from Lemma 3.2 and our hypothesis (i) that
Since {xi(.)} is a minimizing sequence, it follows that 2 ( . ) solves (2).
0
4. The Decoupling Technique
Let X := Rn x L2[0,T] x L2[0,T] x L1[O,T] x L1[O,T]. We define a new .~ . . Bolza-type functional I? : X -+ (--00, 001, similar to (1) by ~~
:= e(y)
+
1
T
L(t,u(t),w(t),w1(t),'U:!(t))d t .
Our original problem (2) is equivalent to minimizing the following constraints
u ( t )= z(0)
+
r over X
(4)
subject to
t
w1(s) ds
where A(s) := A,(s) - A,(s), wl(t) := 0 for each t E [-A,,O), and E(t) is defined by,
t ( t ):=
1, if t - Al(t) 0, else.
< 0;
(9)
Indeed, ( 5 ) says that z ( t ) := u(t) is absolutely continuous, (6) implies that y is the endpoint z ( T ) , and (7)-(8) together say that w ( t ) = z(t A,(t)) for t E [O,T].We introduce the following function 2) : X + R to measure how far an element (y, u(+), w(.), wl(.), wz(.)> E X is from satisfying
254
+IT
( w ( t )- J ( t ) [ c ( t- Al(t)) - z o ] - zo
-
vz(s - A(s))[l - Al(s)]ds
It is clear that (5)-(8) hold if and only if ~ ( ~ , u ( . ) , w ( . ) , v l ( . ) , v 2 ( . )= ) 0. We now extend the decoupling technique to the multiple delays NPB.
Theorem 4.1 (Decoupling Theorem). Suppose the main assumptions (i)-(iii) hold and e > 0. Then there exist a constant IS > 0, an element (T,ii(.), a(.), fjl(.),' u 2 ( . ) ) E X , and absolutely continuous arcs p ( . ) and q ( . ) defined on [O,T]so that
(4 q 7 ,q . 1 , a(.), fjl(.),@ 2 ( 4 < 6 ;
(b) I'(7, ii(.), a(.)@ ,I ( . ) ,@ 2 ( . ) ) is within E of the minimum value in (2); (c) for almost all t E [O,T],the map
L ( t , u ,w, ' U l , vz) - (Ij(t),4 - ( 4 ( t ) w , ) - ( P ( t ) ,211)
(u, w , 211, 'uz)
- (4(t
+a
{
+ A(t))[l - A l ( t + A(t))],.2)
121 -
U(t)I2
+ ( w - a(t)I2+ Jv1 - v1(t)I2+ lvz -
a,(t)12
1
where A(t) := Al(t) - Az(t) is minimized at (u,w,v1,v2) = a(t>,@,(t),fjz(t)); and (d) the map
(w,
Y
l(Y) + (7,P(T))+ ISIY - TI2
is minimized at y = 7. Proof. We introduce a new problem P,,,a9pand we seek to minimize, over all arcs z(.), the functional Av,a(.),p(.) (z) :=
+v) +
l(z(T)
Jd'
+
L ( t ,z ( t ) ~ ( t z)( ,t - A1 ( t ) )+p(t),k ( t ) ,k ( t - A d t ) ) ) dt.
(10) Here z(0) = 50, z ( t ) is set equal to c(t) for t E [-Ao,O), where c(0) = x o , (q,a(.),p(.)) E R" x L2[0,T]x L2[0,T],and k ( t - Az(t)) := 0 when
255
- A2 < 0 and p(t) := 0 when t - Al(t) < 0. Define a value function V : Rn x L2[0,T] x L2[0,T] (-00,00] by setting V ( q , a ( . ) , P ( . )as) the minimum value in (10). Here V ( q , a ( . ) , P ( . )= ) 00 if there are no feasible arcs for Pq,a,p, V is lower semi-continuous and if V ( q ,a(.),P(.)) < 00 then a solution to Pv,or,pexist^.^ Fix E > 0. From the Density Theorem of proximal analysis3 we select ( f j , b ( . ) , P (E. )R” ) x L2[0,T] x L2[0,T] with lfjl 11bl12 IlPllz < E and with l V ( f j , b ( . ) , P ( .-)V)( O , O , O ) I < E . The theorem also guarantees that the proximal subgradient of V is not empty and so there exists (I,$(.), $(.)) E Rn x L2[0,T]x L2[0,T] with ((,4(.),$(.)) E a p V ( f j , b ( . ) , B ( . ) ) .We note also that $(t) = 0 for all t E [O,A,] satisfying t < 0. By the proximal subgradient inequality there exist 6‘ > 0 and u’ > 0 such that for
t
.--)
+
all
+
(v, P ) E B((7 6, , P), u, Q,
Let Z(.) be an optimal solution to Pq,&,p,which implies that V(fj,Ci,P)= A q , & , ~ ( Z )Since, . for any arc z(.) one has V ( q , a , P )5 R,,,,p(z), (11) may be rewritten t o obtain:
p,
Here ibid ( f j , 6 , 2 ) represents the left hand side of the inequality but with the variables ( q ,a,P, z) substituted by ( f j , b, 2 ) . We now apply the following change of notation:
P,
+
Now, using the new variables and separating square terms (since ( a b)2 5
256
2a2 + 2b2) in (12) we obtain,
where estimations for z(.) in terms of v1 and v2 have been made and where CJ := 4a’(T T2).Define arcs p ( . ) and q ( . ) in L2[0,T]by
+
257
We know that (16) holds as long as IIQ - 6112 , Iq - f j l , and are each less than 6'. But by definition and since x can be bound in terms of vl and v2, we simply need to require that 1/21- tiiiJJz,JJw - W112,))ul - t j l J J 1 , 1 1 ~ 2- tj2111, and I y - 71 be small in order that (16) holds. We now prove the statements of the theorem. Part (a) of the theorem is satisfied since, by our change of variables definitions
and
l'
/w(t)- t ( t )[c(t- Al(t)) - XO] - 20=
Ilw(t) - %(t- Al(t>>ll2= 11~11;.
I"
32(s - A(t))[l - Al(s)] ds
Since I l C i l l 2 + l f j l + l l f l l 2 < E statement (a) is satisfied. Part (b) ofthe theorem is satisfied since IV(fj,Ci, - V(O,O,0 )l < E , and by setting u = E , u1 = 2 1 1 , ~=~t j 2 , w = W in (16) we can see that statement (d) is satisfied as well. Finally, t o verify that part (c) holds we let f t denote the function
p)
ft(u,w,?J1,m) = L(t,u,w,v1,vz) - (?j(t),'U) - ( d t ) , w ) - (P(t),Vl)
+ A(t))[l - Al(t + A(t))l, v2) + O { ~ U- U ( t ) I 2 + IW - G(t)I2+ lvl - G1(t)l2+ I w ~ - (dt
-
tj2(t)I2}. (17)
Assuming hypotheses (ii) and (iii), ft attains a minimum for almost every t . We will prove part (c) by establishing that for each T > 0 the set
has measure zero. Suppose not, then there exists T > 0 with m(A(r))> 0. Let Bi be a decreasing sequence of subsets of A(r) such that for each i one has 2 3 - ~ ( A ( T )<) m(Bi) < TvL(A(T)),
i
2
258
Choose (u’(.), w’(.),v{(.),vh(-))measurable such that for almost all t , ( ~ ’ ( tw’(t),vi(t),vh(t)) ), minimizes ft.To complete our proof we must first show that u‘,w‘, vi - vl, and v; - 8 2 are in L2[0,T]. By our choice of (u’(.), w’(.), vi(.),vh(.)) the inequality ft(u’(t),W’(t),v:(t), vb(t))
i f t ( a ( t ) , a ( t ) ,U l ( t ) , a 2 ( t ) )
(19) holds for almost all t. Let b be the lower bound for the function L . After simplifying and rearranging we obtain from our definition of f t and (19),
a{Iu’(t)- U ( t ) I 2
+ Iw’(t) - @@)I2 + Iv:(t)
+ Iv;(t) -
- a1(t)I2
v2(t)I2}
I (@(t),u’(t) - fiL(t))+ (4(t),W’W- w(t)) + ( d t ) 4 , ( t ) - Ul(t)) +( 4(t+A (t)) [1-A 1(t+A ( t ))] V ; ( t )-62 (t)) +L (t c(t ) ( t ) 61( t ) 212 (t)) -b . I
1
1
1
(20) Set
W ( t )= (u’(t)- a(t),w’(t) - @ ( t ) , v i ( t) vl(t),v;(t) - v 2 ( t ) ) g(t) =
(@MIQ ( t ) , P ( t ) Q(t , + A(t))[l - Al(t + A(t))l)
]
I
k ( t ) = L(t,a(t),a(t),v1(t),v2(t)) -b. Then inequality (20) becomes a IWI2 - ( g ( t ) , W ( t )-) k ( t ) I 0 and by the quadratic formula (2a llW11)2 I 4 11g1I2 8allkll. Furthermore, since g(.) E L2 and k(.) E L1, it follows that the integral of (20 llW11)2is bounded and thus, W E L2, proving that u’, w’,vi - D1, and vh - 6 2 are in L2[0,T ] and that each vi and v; is in L1 [0,TI. To finish our proof of (c) we define a family of functions ( u i 1 w ~ , v 1 ~ , vE2L~~) [ o , Tx] L ~ [ o , T x ]L ~ [ o , Tx] L ~ [ o , Tas: ]
+
Then, Jim ~ ~ ( u ~ , w i ,~ (i~i, @ ,~ , ~~i li~~2 ) I=l 20,
2’00
-+ co,m(Bi) -+ 0. From the definition of (u’, w’, wi,vh) and since m(A(r))> 0, it follows that
since as i
T
ft(Ui,Wirvli1v2i)dt<
Jo
T
ft(~,a1~1,a2)dt.
But setting y = 7,u= ui,w = wi, vl = vli, and v2 = v2i, this contradicts (16) for i sufficiently large. Hence, it must be that m(A(r))= 0, which proves (c) and completes our proof. 0
259
5 . Conclusion We expect that this more general version of the decoupling theorem will prove t o be a precursor for necessary conditions to our problem (2). If we let @(t)= g(t+A(t))[l - A l ( t + A ( t ) ) ] we can see that for almost all t E [O,T], the inclusions
( P ( t ) , 4 . ( t ) , p ( t ) , @ (Et ) )
.,
*I '7
.) ( u ( t ) , w ( t ) , ~ l ( t ) , v z ( t ) ) , (22) - p P ) E aPqY), (23)
which follow immediately from statements (c) and (d) of the Decoupling Theorem 4.1 are forms of the Euler-Lagrange and transversality conditions. In addition, our results are consistent and subsume our previous result^^^^ which are obtained by setting A1 := A2 in Ref. 7 and A2 := 0 in Ref. 8.
References 1. F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conf. Series in Applied Mathematics, Vol. 57 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989). 2. F. H. Clarke, J. Math.Anal.App1. 172,92 (1993). 3. F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, Vol. 178 (SpringerVerlag, New York, 1998). 4. R. E. Edwards, Functional Analysis. Theory and applications, Corrected reprint of the 1965 original (Dover, New York, 1995). 5. P. D. Lowen, R. T. Rockafellar, SIAM J . Control Optim. 34, 1496 (1996). 6. B. S. Mordukhovich, R. Trubnik, Ann. Oper. Res. 101,149 (2001). 7. N. Ortiz, J. Math.Anal.App1. 305,513 (2005). 8. N. L. Ortiz and P.R. Wolenski, J . Math.Anal.App1. 289,260 (2004). 9. N. L. Ortiz and P. R. Wolenski, Set Valued Anal. 12,225 (2004). 10. R. T. Rockafellar, Pacific J. Math., 39, 439 (1971). 11. R. T. Rockafellar, Adv. in Math., 15, 312 (1975).
260
STABILIZATION PROBLEM FOR NONHOLONOMIC CONTROL SYSTEMS L. RIFFORD Laboratoire J . A . DieudonnC, UniversitC de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France E-mail: riffordOmath.unice.fr We present the stabilization problem in the context of nonholonomic control systems and nonholonomic distributions. Then, we introduce the notions of smooth repulsive stabilizing feedbacks and sections, and review recent results on the existence of such objects. Keywords: Stabilization problem, nonholonomic control systems
1. Introduction
Throughout this paper, M denotes a smooth connected manifold of dimension n and a: a point of M .
1.1. Stabilization of nonholonomic control systems Let f l , . ’ . , f m be a family of m smooth vector fields on M . We say that the control system defined as, m a= 1
is nonholonomic on M (also called totally nonholonomic in Ref. 2) if the following property is satisfied*: Lie{fl,... , f m } ( z ) = T , M ,
VZE M .
*We recall that Lie { f l , . . ,fm} denotes the Lie algebra of vector fields generated by the family {fl,. . . ,fm}. It is the smallest vector subspace S of X o 3 ( M ) (the set of smooth vector fields on M ) containing the fi’s and such that [fi, g ] E S for any i = 1 , . . . , m and any g E S.
261
Recall that, for every x E M and every control u ( +E) L1([O,00); Rm), there is a unique maximal solution x(.) = x ( . ; u , x ): [O,T,) -+ M (with T, > 0) to the Cauchy problem m
i ( t >= C u i ( t ) f a ( x ( t ) ) , for a.e. t E [o,T,), x ( 0 ) = x. a= 1
As the next result shows, any nonholonomic control system is controllable on M (see Refs. 4 or 10). Theorem 1.1 (Chow-Rashevsky Theorem). Let (1) be a nonholonomic control system on M . Then, for every pair x,y E M , there exists a smooth control u ( . ): [0,I] -+ R" such that x(1; u, x) = y. Before presenting the stabilization problem, we need to recall the notion of globally asymptotically stable dynamical system. Let X be a smooth vector field on M , the dynamical system k = X ( z ) is said to be globally asymptotically stable at the point 3 (abreviated GAS2 in the sequel), if the two following properties are satisfied: (i) Lyupunov stability: for every neighborhood V of 3, there exists a neighborhood W of 3 such that, for every x E W , the solution of i ( t )= X ( x ( t ) ) , x ( O= ) x, satisfies x ( t ) E V , for every t 2 0. (ii) Attractivity: for every x E M , the solution of k ( t ) = X ( x ( t ) ) ,x ( 0 ) = z, tends to 3 as t tends to +00. The purpose of the stabilization problem is the following: Let (1) be a given nonholonomic control system, does there exist a smooth+ mapping k : M + Rm (called stabilizing feedback) such that the dynamical system (called closed-loop system) defined as, m a=
I
is GAS2 ?
1.2. Stabilization problem for nonholonomic distributions Let A be a (totally) nonholonomic distribution of rank m 5 n on M . This means that for every x E M , there is a neighborhood V , of x in M and a +Werestrict here our attention to smooth stabilizing feedbacks. In fact, it can be shown that if the control system (1) admits a continuous stabilizing feedback then this feedback can be regularized into a smooth stabilizing feedback.
262
and moreover, Lie { f:, .. . ,f&} (y) = T,M,
Vy E V,.
We call horizontal path between x to y, any absolutely continuous curve y(.) : [0,1] -+ M with y(0) = IC, $1) = y which satisfies
?(t)E A ( y ( t ) ) , for a.e. t E [0,1]. In the context of nonholonomic distribution, the Chow-Rashevsky Theorem takes the following form.
Theorem 1.2. Let A be a nonholonomic distribution on M . Then, any two points of M can be joined by a smooth horizontal path. The stabilization problem for nonholonomic distributions consists in finding, if possible, a smooth stabilizing section of A at 53, that is, a smooth vector field X on M satisfying X ( z ) E A(x) for every IC E M and such that the dynamical system i = X ( x ) is GASZ. 1.3. Two obstructions Given a nonholonomic control system (resp. a nonholonomic distribution), there are two obstructions to the existence of stabilizing feedbacks (resp. stabilizing sections). The first one is global while the second one is purely local. (i) Global obstruction: If fact, if the manifold M admits a smooth dynamical system which is GASz, then it must be homeomorphic to R" (see Ref. 12 for further details). (ii) Local obstruction: Since this obstruction is local, we can assume that we work in R", that is, in an open neighborhood U c R" of 3. If there is a smooth vector field X on U which is locally asymptotically stable at 5, then for all E small enough,
36 > 0 such that 6B c X ( 3
+EB),
where B denotes the open unit ball in Rn (see Ref. 12 for further details). In consequence, if there is a smooth feedback k : U -+ Rm such that the closed-loop system (2) is locally asymptotically stable at 3 ,
263
then the result above applies to the dynamics X ( x ) = Czl k(x)ifi(x). Thus, for all E small enough,
6B c
36 > 0 such that
i"
cuifi(Z
a=
+ E B )I u E R-}
,
1
which is the Brockett necessary condition (see Ref. 5). In particular, we deduce that any distribution A of rank m < n cannot admit a smooth locally stabilizing section at Z. The obstructions above make it impossible to prove the existence of smooth stabilizing feedbacks (resp. sections) for nonholonomic control systems (resp. distributions). Actually, they motivate the design of new kinds of stabilizing feedbacks. The main contributions in that direction have been: Sussmann,17 Coron18Clarke, Ledyaev, Sontag and S ~ b b o t i nand , ~ Ancona and Bressan.' The aim of the present paper is to highlight the notion of smooth repulsive stabilizing feedbacks (or sections) and to show that it permits to stabilize most of the nonholonomic control systems (or distributions). 2 . Examples
2.1. The Nonholonomic integrator Define in R3 the two smooth vector fields f l , f 2 by, f1=
a
a
ax1
ax3
-+x2-
and
f2
a
a
ax2
ax3
= - -xl-.
Note that at any point x E R3,the three vectors fl(z),f i ( x ) ,[ f l ,f2](x) are linearly independent. Hence the associated so-called nonholonomic integrator (or Brockett integrator) j: = Ulfl(.)
+ uzf2(z),
is nonholonomic on EX3. As it is well-known, this control system can be stabilized at the origin by a feedback which is smooth outside the vertical line$. Denote by S the vertical line of equation z1 = x2 = 0 in R3.Using an adapted control-Lyapunov function (see Refs. 11 or 9 for further details), we can construct a mapping k = (kl, kz) : R3 H R2 in such a way that the following properties are satisfied: SWe refer the reader to Ref. 3 for a detailed study of the stabilization of the nonholonomic integrator by discontinuous feedbacks.
264
(i) The mapping Ic is locally bounded and smooth on M \ S. (ii) The set R3 \ S is invariant with respect to the dynamical system
i =Icl(X)fl
+
IcZ(X)fZ(~),
and steers asymptotically all its trajectories t o the origin. In fact, the feedback above can be extended to the whole space R3. In this way, we can construct a feedback which is smooth outside S, discontinuous a t the points of S, and for which the closed-loop system is GAS2 in the sense of Carathkodory (see below). Such a feedback is an example of what we call a smooth repulsive stabilizing feedback. 2.2. The Riemannian case
Assume through this paragraph that the nonholonomic distribution A is given by A ( x ) = T,M for every x E M , our aim is t o show how to construct a stabilizing section for A. For that, consider a smooth and complete Riemannian metric g on M and denote by dg the Riemannian distance associated with g. We recall that, for any x,y E M , the Riemannian distance between x and y is defined as
where the infimum is taken over all the C1 paths y(.) : [0,1] x to y, and where r1
---f
M joining
.
Let V, : M -+ E% be the function defined by
V g ( x )= d g ( Z , x ) 2 , V x E M . It is easy to show that V, is Lipschitz continuous on M and smooth outside the set S defined as the cut-locuss from the point 3. Define the vector field X on M by
X ( X )= -V,V,(x),
VXE M
\ S,
§The cut-locus from 2 is defined as the closure of the set where V, is not differentiable. We refer the reader to Ref. 16 for a detail study of the distance function and the cut-locus from Z.
265
where V,V,(x) denotes the gradient of V, at z with respect to the metric g . By construction, any trajectory of k ( t ) = X ( z ( t ) )tends to 3 as t -+ 00 and satisfies the following property:
vt 2 0 , z ( t )!$
s.
In fact, X can be extended into a global section of A on M which is smooth outside the cut-locus from 3,discontinuous at the points of S and whose the associated dynamics drives all its Carathkodory trajectories asymptotically to 3. Such a stabilizing section corresponds to what we call a smooth repulsive stabilizing section of A on M . 3. Smooth repulsive stabilization
Our aim is now to make precise the notions of smooth repulsive stabilizing feedbacks or sections, and to show what kind of results we are able to prove.
3.1. SRSz,s vector fields Let S be a closed subset of M and X be a smooth vector field on M . The dynamical system x = X ( x ) is said to be smooth repulsive globally asymptotically stable at 3 with respect to S (denoted in short SRS2,s) if the following properties are satisfied: (i) The vector field X is locally bounded on M and smooth on M \ S. (ii) The dynamical system 2 = X ( z ) is globally asymptotically stable at 3 in the sense of Carathbodory, namely, for every x E M , there exists a solution of
i ( t ) = X ( x ( t ) ) , for almost every t
E [O, a),x ( 0 ) = x ,
(3)
and, for any E > 0, every solution of i ( t )= X ( x ( t ) )almost everywhere in [0,E ) (called Carathkodory solution of f = X ( x ) ) can be extended to [0, 00) and tends to % as t tends to 00.Moreover, for every neighborhood V of %, there exists a neighborhood W of 3 such that, for x W , the solutions of (3) satisfy z ( t ) E V , for every t 2 0. (iii) For every z E M , the solutions of (3) satisfy z ( t )$! S, for every t > 0. Given the nonholonomic control system (l),we shall say that a mapping k : M 4 Rm is a smooth repulsive stabilizing feedback at f (denoted in short SRSz feedback) for (l),if it is locally bounded on M and if there is a closed set S c M such that k is smooth on M \ S and such that its associated closed-loop system is SRS2.s. In the same way, given a nonholonomic
266
distribution A and a vector field X on M , we shall say that X is a smooth repulsive stabilizing section at 5 (denoted in short SRSz section) for A, if X is a section of A on M and if the dynamical system x = X ( x ) is SRSz,s for some closed set S c M .
3.2. Existence results of SRSz feedbacks Using a technique of cancellation of the bifurcation points of a “discontinuous” stabilizing feedback, we proved in Ref. 13 the following result:
Theorem 3.1. If M has dimension two, any nonholonomic control system of the form (1) admits a SRSz feedback. Using the classical method of local approximation of a nonholonomic control systems by an homogeneous one together with the technique of cancellation developed on surfaces, we were able in Ref. 14 to demonstrate the following result:
Theorem 3.2. If M has dimension three, any nonholonomic control syst e m of the form (1) admits a SRS, feedback defined o n a neighborhood of -
X.
We refer the interested reader to Refs. 13 and 14 for more details on these results.
3.3. Existence results of SRSz sections The method presented above in the Riemannian case can also be developed in the sub-Riemannian setting; we need for that to introduce material of sub-Riemannian geometry. Let A be a nonholonomic distribution of rank rn 5 n on M , the set of horizontal paths y(.) : [0,1] -+ M such that y(0) = Z, denoted by RA(%),endowed with the W191-topology,inherits of a Banach manifold structure (see Ref. 15 for further details). The end-point mapping f r o m 5 is the mapping Ez : RA(Z) -+ M defined by
EZ(?’(’)) = ?‘(I), v’Y(‘) E RA(%); it is a smooth mapping on R A ( ~ ) A . path y(.) is said to be singular if it is horizontal and if it is a critical point of the end-point mapping E,, that is, if the differential of Ez at y(.) is not a submersion. Let g be a smooth Riemannian metric on M , the sub-Riemannian distance d S R ( x ,y ) between two points x , y of M is defined by ~ S R ( XY ,) = inf
{length,(?(.))
I Y(.) E RA(%)}
267
According to the Chow-Rashevsky Theorem, since A is nonholonomic on M , the sub-Riemannian distance is well-defined and continuous on M x M . Moreover, if the manifold M is a complete metric space7 for the subRiamannian distance d S R , then, since M is connected, for every pair x , y of points of M there exists an horizontal path y(.) : [0,1] -+ M joining z to y such that d s R ( X , Y) = length,('Y(.)).
Such a horizontal path is said to be minimizing. The following result has been obtained recently with Trdat (see Ref. 15).
Theorem 3.3. Let A be a smooth nonholonomic distribution of rank m 5 n on M . Assume that there exists a smooth Riemannian metric g on A f o r which M is complete and no nontrivial singular path is minimizing. Then, there exist a section X of A on M , and a closed nonempty subset S of M , of Hausdorff dimension lower than or equal to n- 1, such that X is SRS,,s. Note that if m = n, then obviously there exists no singular path (it is the Riemannian case). In fact, the main assumption of Theorem 3.3 (the absence of nontrivial singular minimizing paths) is automatically satisfied for a large class of sub-Riemannian structures such as the fat distributions or the medium-fat distributions associated with a generic metric (we refer the reader to Ref. 15 for further details). Here, we just want to emphasize the fact that the main assumption of Theorem 3.3 is satisfied generically for distributions with rank greater than two. Let m 2 3 be a positive integer and 4, be the set of pairs ( A , g ) where , A is a rank m distribution on M and g is a smooth Riemannian metric on A, endowed with the Whitney Coo topology. It is shown in Ref. 6 that there exists an open dense subset W, of , Q such that every element of W, does not admit nontrivial minimizing singular paths. This means that, for m 2 3, generically, the main assumption of Theorem 3.1 is satisfied. Therefore, as a by-product of the Chitour-Jean-TrBlat Theorem, we have the following result:
Corollary 3.1. A generic nonholonomic distribution of rank 2 3 admits a SRSE section. VNote that, since the distribution A is nonholonomic on M , the topology defined by the sub-Riemannian distance d S R coincides with the original topology of M (see Refs 4 or 10).
268
We notice that in Ref. 15, we are able t o remove, in the compact and orientable three-dimension case, the assumption on the absence of singular minimizing paths. We refer the interested reader t o Ref. 15 for further details on that result.
3.4. A Nonholonomic dream In view of the results presented here, one might hope t h a t the following conjecture is true.
Conjecture. Any nonholonomic control system admits a SRS feedback. References 1. F.Ancona and A. Bressan, E S A I M Control Optim. Calc. Var., 4, 445 (1999). 2. A. A. Agrachev and Y . L. Sachkov, Control theory from the geometric uiew-
3. 4.
5.
6. 7. 8. 9. 10.
11. 12.
13. 14. 15.
point, Encyclopaedia of Mathematical Sciences, Vol. 87. Control Theory and Optimization, 11. (Springer-Verlag, Berlin, 2004). A. Astolfi. Eur. J. Control, 4 ( l ) ,49 (1998). A. Bellai'che, The tangent space in sub-Riemannian geometry, in SubRiemannian Geometry, eds. A. Bellache and J.-J. Risler, Progr. Math., Vol. 144, (Birkhauser Verlag, Basel, 1996), pp. 1-78. R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential geometric control theory (Houghton, Mich., 1982), eds. R. W. Brockett, R. S. Millman and H. J. Sussmann, Progr. Math., Vol. 27 (Birkhauser Boston, Boston, MA, 1983), pp. 181-191. Y.Chitour, F. Jean and E. TrBlat, J. Diff. Geom. 73 (l),45 (2006). F. H.Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, I E E E Trans. Automat. Control 42, 1394 (1997). J.-M. Coron, Math. Control Signals Systems, 5 (3), 295 (1992). Y.S.Ledyaev and L. Rifford (unpublished, 2007). R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91 (American Mathematical Society, Providence, RI, 2002). L. Rifford, ProblBmes de stabilisation en thBorie du contr6le, PhD thesis, UniversitB Lyon I (Lyon, France, 2000). L. Rifford, The Stabilization Problem: AGAS and SRS Feedbacks, in Optimal control, stabilization and nonsmooth analysis, eds. M. M. M.S. de Queiroz and P. Wolenski, Lecture Notes in Control and Inform. Sci., Vol. 301 (Springer, Berlin, 2004), pp. 173-184. L. Rifford, Rend. Semin. Mat. Torino 64 (l),55 (2006). L. Rifford, J. Dzfferential Equations, 226 (2), 429 (2006). L. Rifford and E. TrBlat, On the stabilization problem for nonholonomic distributions, J. Eur. Math. SOC. (JEMS) (to appear), available at http://arxiv.org/abs/math/0610363v1 (October 2006).
269
16. T. Sakai, Riemannian geometry, Translations of Mathematical Monographs, Vol. 149 (American Mathematical Society, Providence, RI, 1996). Translated from the 1992 Japanese original by the author. 17. H. J. Sussmann, J. Differential Equations 31,31 (1979).
270
PROXIMAL CHARACTERIZATION OF THE REACHABLE SET FOR A DISCONTINUOUS DIFFERENTIAL INCLUSION V. R. RIOS Departamento de Matemdtica, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo, Edo. Zulia, Venezuela E-mail: vriosQ1uz.edu.ve http://www. fec.1u.z. edu.ve
P. R. WOLENSKI Mathematics Department, College of Arts and Sciences, Louisiana State University, Baton Rouge, L A 70803, U S A E-mail: [email protected]. edu http://www.math. lsu. edu The graph of the reachable set is characterized via limiting Hamilton-Jacobi inequalities when the system dynamics satisfy a discontinuous dissipative Lipschitz condition.
Keywords: Reachable Set; Dissipative Lipschitz Maps; Hamilton-Jacobi Inequality; Differential Inclusion.
1. Introduction
This paper considers a dynamical control system governed by the differential inclusion
i ( t )E F ( x ( t ) ) a.e. t
E
I := [ O , o o ) ,
PI)
where F is a multifunction mapping Rn into the subsets of R" that satisfies assumptions that will be stated below. A solution (or trajectory) t o (DI) is an absolutely continuous (AC) function x(.) : I -+ R" satisfying (DI). Given a compact set M c R", the reachable set of F at t i m e t 2 0 is defined as
R F ( t ) := { x ( t ) : x(.) solves (DI) on [ O , t ] , with ~ ( 0 E) M } ,
271
and the graph of R F ( . is ) given by
{
~(RF := ) ( t , x ): t L 0,and z
E
RF(~)}.
We are interested in characterising ~ ( R F in terms ) of Hamiltonian inequalities for possibly discontinuous F . Throughout this paper, we will assume the multifunction F is endowed with the following structural properties (SP): 0 0
0
F(x) is a nonempty, compact, and convex set, for all x E R"; There is a constant c > 0 so that sup{llvll : w E F ( x ) } 5 c(1
+
11x11) for all z E R"; F ( . ) is upper semicontinuous (US), which in presence of the previous assumptions, is equivalent t o B ( F ) := {(x,v) : 21 E F ( x ) } being closed.
The fact that R F ( ~ is )nonempty is well known, see for instance the existence theory exposed in Refs. 1,5,6. Moreover, using compactness of trajectories (see Theorem 4.1.11 of Ref. 5) it is easy to check that R F ( ~ and ) ~(RF are) compact and closed, respectively. The motif of this note arises from Clarke2, where the author considers a time-dependent multifunction G : I x R" 3 R" satisfying (SP) and a Lipschitz condition, both jointly in ( t , x ) .The main result in Ref. 2 states that ~ ( R G can) be characterized as the unique closed subset S of I x R" for which the following conditions hold, with ST := {x E R" : ( T ,x) E S } :
O + H G ( t , z , C ) = O , forall(O,C) ~ N ; ( t , z ) a, n d a l l ( t , x ) E S ;
(1)
limST = M . TI0
(2)
Equation (1) involves the upper Hamiltonian of G, HG : I x R" x Rn defined by
-+
R,
) inf{(v,p) : w E (analogously, the lower Hamiltonian of G is h ~ ( t , x , p := G ( t , x ) } ) For . autonomous G the t-variable will be dropped in the previous definitions. Another key ingredient in the above result is the proximal normal cone N l ( z ) of a closed set S c Rm at z E S , defined as the set of elements C E Rm for which there exists 0 = o(C,x) 2 0 such that
272
In equation (2) the limit is interpreted via the Hausdorff metric means precisely that, given E > 0, there exists 6 > 0 such that 0 5 T < 6 ==+ ~ H ( S M T ,) := max
dH.
This
d(a,M ) , supd(P, S T ) OEM
where d ( a , K ) := inf{ Ila - 011 : Y E K } is the distance from a point a E R" to a closed set K c R". The goal of this article is to extend the above characterization to an important class of discontinuous dynamics F that we discuss below (see section 2). Following Ref. 2, our main result (Theorem 3.1) shows that under milder assumptions than heretofore imposed, a characterization of ~ ( R F ) can be provided by replacing (1) with an appropriate pair of Hamiltonian inequalities, one of which must reflect the possible discontinuity of F . It is noteworthy that equation (1) is obtained in Ref. 2 as a consequence of the invariant properties satisfied by the auxiliary pairs (RG, *E), where
E ( t , x ) := ( ( 1 , ~ :) Y E G ( t , z ) } . These invariant properties translate into ~ ( R G being ) simultaneously a subsolution and a supersolution of the exact Hamilton-Jacobi (HJ) equation (1) in the following sense (c.f. Ref. 2): a closed set S I x R" is a subsolution of (1) if
e + f f G ( t , % ,C) 2 0, for all (e, C) E N l ( t , x ) ,and all ( t , x )E S,
(3)
and a supersolution of (1) if
0 + HG(t,x,C) 5 0 , for all (0, C) E N l ( t , x ) , and all ( t , x )E S.
(4)
Property (4)is not necessarily enjoyed by ~ ( R F under ) discontinuous (DI), for the Lipschitz invariant result that is used in Ref. 2 to obtain (4) does not cover, for example, the type of discontinuous dynamics considered in this paper. This situation is illustrated in the next section (see also Refs. 8,9). The rest of this note is organized as follows. Section 2 contains precise assumptions on the data, and an invariant prerequisite that holds under these assumptions. The main result and its proof are presented in Section 3. 2. DL dynamics and invariance
Invariance properties of differential equations have been studied extensively. More recently, weak and strong invariance properties of differential inclusions have also received considerable attention due mainly to their significant applications in Hamilton-Jacobi theory. We suggest Refs. 4,9 (and
273
references therein) for concise histories on flow invariance theory and its repercussions. Regarding the terminology, let us recall that for a given closed set S & R" and F as above, the pair ( S ,F ) is said weakly invariant if for each zo E S , there is a solution z(.) to (DI), with z(0) = 2 0 , and satisfying z ( t ) E S for all t E I . Similarly, ( S , F ) is strongly invariant if for each zo E S every solution z(.) to (DI), with z(t0) = 20,also satisfies z ( t ) E S for all t E I . Weak invariance characterizations hold under general hypotheses similar to (SP) (see t,he works of Aubin' , F'rankowska et al.loill and Veliov16), and they have contributed to establishing subsolutions of the HJ equation in different approaches (minimax, proximal, viscosity: the history is sketched in Ref. 4). In this same sense, we will benefit from the following time-dependent result, which appeared as Theorem 1 in Ref. 9.
Theorem 2.1. Let T g I be a subinterval. Suppose a nonautonomous multifunction G : T X R" =t R" satisfies (SP), with the upper semicontinui t y requirement replaced by the weakened Scorza-Dragoni property, and let S c R" be closed. Then ( S ,G ) is weakly invariant if and only i f there exists a null set A c 7such that hG(t,z,C) 5 0,
I\
(5)
c
for all t E A , x E S , and E N C ( x ) . I n this case, (5) holds at all points of density of a certain countable family of pairwise disjoint closed sets I k C I (k=1,2,. . . ,) for which G(.,.) is upper semicontinuous o n each
-
Ik x
R".
On the other hand, supersolutions to the HJ equation (1) have generally been obtained by invoking classical criteria for strong i n ~ a r i a n c e ~ ~.' ' ~ ' ~ These criteria have typically required a Lipschitz assumption on the multifunction G(., .) (see Theorems 4.3.8 in Ref 5 , 5.2 in Ref. 3 , 4.10 in Ref. 11, and 2.1 in Ref. 13), which can be stated as follows (we shall restrict ourselves to global definitions to simplify the exposition, but local versions hold as well): a convex-valued multifunction G is Lipschitz if there exists a constant X such that, for all ( t , z ) ,( s , y ) E I x R",and all C E R" we have
IHG(trY,C) - H G ( S , z , c ) I
5 xllcll I l ( t - S , Y - z ) I I .
(6)
Within a non-Lipschitz framework, and consequently beyond the scope of the standard theory, a class of Dissipative dynamics has been used in Refs. 6, 14 t o model dry friction forces in physical phenomena. An autonomous convex-valued multifunction D : Rn =t Rn is Dissipative if
H D ( y ,y - x)
+ H D ( ~z ,- y ) 5 0,
for all z, y E R".
(7)
274
The main result of this paper will hold under a generalization of the previous dissipativity concept. The aforementioned notion was introduced by T. Donchev7 and consists of a quasi-quadratic perturbation on the righthand side of (7). More precisely, a convex-valued multifunction F is called Dissipative Lipschitz (DL) if there is a constant K: such that
H ~ ( y , y- z) - H F ( z ,y - x) I Klly - z1I2, for all x,y E Rn.
(8)
Property (8) is essentially weaker than properties (7) and (6): taking the multifunction G autonomous, it is clear that (8) follows from ( 6 ) by setting = y - z, and obviously (7)*(8). Nevertheless, the converse of these implications may fail, as can be appreciated by considering the multifunctions D ( z ) := -dllzlI, and F ( z ) := D ( z ) {z},where d denotes the subdifferential of convex functions, and z E R. Notice that D and F satisfy (SP), D is dissipative, and F is dissipative Lipschitz (with K: = 1).On the other hand, D and F are discontinuous at x = 0, which implies they are not Lipschitz. Moreover, F does not satisfy the dissipativity condition (7). In case of discontinuous DL dynamics the exact Hamilton-Jacobi equation (1) may not hold. In fact, letting A4 := (0) and S := ( ~ ( R = DI) x (0) we have the strict inequality
+
0
+ H D ( 0 , C ) = HE((0,O ) , (0, C))
> 0, for all (0, C) E N,P(O,O)\{(O, 0)).
-
However, the pairs (S,-5) and (S,5) are weakly and strongly invariant respectively, since any trajectory z ( . ) of the dissipative multimap D ( z ) := { (1,u): u E D ( z ) } ,with t ( 0 ) E S, is necessarily of the form z ( t ) = (t+to, 0 ) for some to E I (here, by state augmentation, we view t as a component of the state z = ( t , z )E I x Rn). The previous example suggests that, under assumption (8), finding an appropriate complementary condition to (3) that helps characterize ( ~ ( R F ) in terms of Hamilton-Jacobi inequalities, can be accomplished with a more general criterion for strong invariance. The following result, which appeared in Ref. 9, meets such a requirement. We include here an autonomous version of the proof for completeness and since it is somewhat simpler than the one given there. Proposition 2.1. (Corollary 5 of Ref. 9 ) Let S C Rn be closed and suppose F satisfies (SP) and (8). Then the system ( S ,F ) is strongly invariant if and only if limsupHF(y,C) 5 0, Y+C"
for all
2
E S, and all
E
Nl(x).
(9)
275
Remark 2.1. Given a nonzero vector C E Rn, the notation y -+c x in (9) signifies the limit of y approaching x along the vector C; in other words, y -+c x if and only if y -+ x and 4
fi &.
Proof. First, we show that (9) is sufficient for strong invariance. Given any trajectory x(.)of (DI), with x(0) = xo E S , the multifunction
G ( t , x ) := {W
E F ( x ) : (li:(t)- W ,x ( t ) - x) 5
Kllx(t) - x1I2}
satisfies (SP), with the weakened Scorza-Dragoni property (see page 70 of Ref. 15) replacing upper semicontinuity. Let T > 0, and set := [0,TI c I . The last property means that for any E > 0 there is a closed set I, c 7 with Lebesgue measure p ( T \ I € ) < E , such that the restriction of G to I, x R" is US in ( t , x ) .Let J , denote the points of density of I E ,and define A := (US>oJE)which has null measure in 7 (see page 274 in Ref. 12). Let x E S , and C E NC(x). Given t E A , there is some E > 0 for which t E J,. By taking the lim inf all over the sequences ( p , z ) + ( t ,x) in I, x Rn, the lower semicontinuity of h ~ ( ., . ,C) yields
T
T\
T\
Inequalities (9) and (lo), and the fact that h~ 5 H F imply
hG(t,x,C) 5 limsupHF(y,C) I 0, YC'"
and this inequality actually holds for almost all t E I , as can be readily seen by repeating the previous argument over each member of the sequence of intervals [T,27'1, [2T,32'1, . . . This in turn implies the weak invariance of G according to Theorem 2.1. If y(.) is an invariant trajectory of G with y(0) = 2 0 , the very definition of G guarantees that
which implies x ( t ) = y(t) 6 S, for all t E I via Gronwall's lemma. This establishes the strong invariance property of ( S ,F ) . Conversely, assume that the system ( S , F ) is strongly invariant. Let x E S, C E N;(x), and a sequence yi -'c x be given. For each i, consider wi E F(yi) such that H ~ ( y iC), = ( w i , C), and define Gi : Rn =t Rn by
Gi(y) := { W E F ( y )
: (wi - w , Y i
-
Y) 5 KlIYi - YI121.
It is easy to check that Gi satisfies (SP), and since Gi F the strong invariance of ( S ,F ) yields, in particular, the weak invariance of ( S ,Gi),for i = 1 , 2 , ... If wi E G i ( z ) satisfies (wi,C) = hGi(x,C),then Theorem 2.1
276
implies (wi,C) 5 0 (see also Theorem 4.2.10 of Ref. 5). Furthermore, due to (SP) the sequences vi and wi are bounded. Rearranging terms from the definition of Gi(x), and using the properties of vi, and wi we have lim sup HF (yi, C) = lim SUP (vi, C) i+cc
i-cc
= limsup
(wi,<) 5 0.
i-ca
Hence condition (9) is satisfied with the limsup taken all over the sequences !J yi -+c x. Therefore, (9) holds as stated. 3. Main result
We now proceed to establish a proximal characterization of G(RF) in terms of Hamilton-Jacobi inequalities. The proof given below follows the lines of Theorem 1 of Ref. 2, but extends the same result to the US-DL framework by incorporating the novel Hamiltonian condition provided in Proposition 2.1. Remark 3.1. We must clarify a t this point that the limsup over y -+c x in (9) can be replaced by the a priori weaker condition of taking this limit over S O+, with y = x+6C, without changing the equivalence with strong invariance. This simplification will be in effect when estimating inequality (b) below. -+
Theorem 3.1. A s s u m e F satisfies (SP) and property (8). T h e n the graph of i t s reachable set G(RF), i s the unique closed subset S of I x Rn satisfying t h e following f o r all (0, C) E N g ( t ,x), and all (t,x) E S : (a) 0 H F ( X , C) 2 0, (b) @ + l i m s u p H ~ ( y , CL) 0,
+
Y+C”
(c) limTlo ST = M . Proof. We start by recalling how to establish property (a). Let F ( t ,x) := (1) x F ( x ) . By definition of reachable set, for any ( 7 ,a ) E ~ ( R F there ) is a trajectory y(.) of (DI) defined on [ o , ~ ]with , Y(T) = a and y(0) E M . The previous also implies that (t,y(t)) E ~ ( R Ffor ) ,all t E [ O , T ] . Therefore, the
277
augmented time-reverse arc z ( t ) := (T
-
t ,y
(~ t ) )satisfies
i ( t )E - F ( z ( t ) ) a.e., t E [ o , T ] , with z ( t ) E G(RF)for all t E [ O , T ] , and z ( 0 ) = ( ~ , a The ) . last argument guarantees the weak invariance of the pair (G(RF), -F), which in light of Theorem 2.1 yields to
e + H F ( x ,c) = -h-F(tl X , (el5)) 2 0, for all ( t , x )E G(RF)and all (Ole)E N J ( a F ) ( t , x )We . now focus on property (b). Let ( 7 , a )E ~ ( R F and) .(.) be a trajectory of
~ ( tE )F ( w ( t ) ) a.e., t E I , with z ( 0 ) = ( ~ , c x )We . claim that ~ ( tE)~ ( R F for) all t 2 0. In fact, we first notice that necessarily z ( t ) = (t + T , y(t + T ) ) , for some trajectory y(.) of (DI) defined on I . We distinguish two cases: Case 1: If T = 0, the use of z ( . ) leads to y(0) = a. Since ( 0 , a )E G(RF), by definition of reachable set we must have y(0) E MI which implies ~ ( t=) (t,y(t)) E G(RF)r for t 2 0.
Case 2: Let us assume T > 0. Using the definition of G(RF)we obtain a trajectory x(.) of (DI) defined on [ O , T ] , with x(7) = a , and ~ ( 0 E) M . Then the piecewise arc
is a trajectory of (DI) satisfying w(0) = x(0) E M . Again, from the definition of G(RF)we have (t,w(t)) E G(RF), for all t 2 0. In particular, z ( t ) = (t T , w(t T ) ) E G(RF)for all t 2 0.
+
+
The verified claim implies (G(RF), F ) is strongly invariant. According to Remark 3.1, we have that ( 7 , ~+(e,,-) ) ( t , x ) if and only if T -0 t and y -)c x. Applying Proposition 2.1 we obtain @ + l i m s u p H ~ ( y= , ~ ) limsup Y-CX
H F ( ( T , Y ) , ( ~ 5, ~0 ), )
(T,Y)+(e,<)( t , ~ )
for all ( O , < ) E N:(RF)(tlx)l and all ( t , ~E) G ( R p ) . The linear growth condition in (SP) implies boundedness on compact sets, and this is readily seen to be the key requirement for G(RF) to satisfy condition (c). The proof of the uniqueness is based on the establishment of the double inclusion S C G(RF) S , for any closed S C_ I x R" for which (a), (b), and (c) hold. We remark that besides (SP), no additional assumption
278
is required on F to prove the maximality of ~ ( R Fsince ), S ~(RF) follows, for instance, from the weak invariance Theorem 4.2.10 in Ref. 5, and the compactness of trajectories property given in Theorem 4.1.11 in Ref. 5. However, the complementary inclusion B(RF)C S is obtained by applying the replacement Hamiltonian condition for strong invariance given in Proposition 2.1 (instead of Theorem 4.3.8 in Ref. 5 for Lipschitz F ) , together with a slight modification of Theorem 4.3.11 of Ref. 5 (Lipschitz continuous dependence on initial conditions), which makes it count for USDL dynamics. 0 Theorem 3.1 asserts that upper semicontinuity and the dissipative Lipschitz property are not in general sufficient ingredients to characterize B(RF) as a generalized solution to the exact HJ equation (1). However, the last property can be recovered by adding continuity to F , as the next statement confirms. Recall that F is continuous if HF(., .) is continuous.
Corollary 3.1. In addition to the assumptions given in Theorem 3.1, suppose F is continuous. Then the graph of the reachable set B(RF), is the unique closed subset S of I x Rn satisfying the following for all (0, <) E " ( t , z), and all ( t ,x) E S : (a) 0 -k HF(z,<) = 0, (b) limTlo ST = M . Proof. The result follows from Theorem 3.l(b), since in virtue of the continuity of HF we have
References J.-P. Aubin, Viability Theory (Birkhauser, Boston, 1991). F.H. Clarke, Systems and control Letters 27, 195 (1996). Clarke F., Ledyaev Yu., Radulescu M., J . Dynam. Control Syst. 3 , 4 9 3 (1997). F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, P.R. Wolenski, J . Dynam. Control Systems 1, 1 (1995). 5. F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, Vol. 178 (SpringerVerlag, New York, 1998). 6. K. Deimling, Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, Vol. 1 (De Gruyter, Berlin, 1992). 7. T. Donchev, Nonlinear Analysis 16, 543 (1991). 1. 2. 3. 4.
279
8. T.D. Donchev, V.R. Rios., P.R. Wolenski, A characterization of strong invariance for perturbed dissipative systems, in Optimal control, stabilization and nonsmooth analysis, eds. M.S. de Queiroz, M. Malisoff and P. Wolenski, Lecture Notes in Control and Inform. Sci., Vol. 301 (Springer, Berlin, 2004), pp. 343-349. 9. T.D. Donchev, V.R. Rios., P.R. Wolenski, Nonlinear Anal. 60,849 (2005). 10. Frankowska H. and Plaskacz S., Nonlinear Analysis, Theory, Methods, and Applications 26,565 (1996). 11. H. F’rankowska, S. Plaskacz, T . Rzezuchowski, Journal of Differential Equations 116,265 (1995). 12. E. Hewitt, K. Stromberg, Real and Abstract Analysis. A modern treatment of the theory offunctions of a real variable (Springer, New York, 1965). 13. M. Krastanov, Forward invariant sets, homogeneity and small-time local controllability, in Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), eds. B. Jakubczyk, W. Respondek and T. Rzeiuchowski, Banach Center Publ. 32 (Polish Acad. Sci., Warsaw, 1995), pp. 287-300. 14. M. Kunze, Non-smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744 (Springer, Berlin, 2000). 15. A. Tolstonogov, Differential Inclusions in a Banach Space, Mathematics and its Applications, Vol. 524 (Kluwer Academic Publishers, Dordrecht, 2000). Translated from the 1986 Russian original and revised by the author. 16. V. M. Veliov, Set-Valued Anal 1,305 (1993). 17. P.R. Wolenski, Y. Zhuang, SIAM J. Control Optim. 36,1048 (1998).
280
LINEAR-CONVEX CONTROL AND DUALITY R.T. ROCKAFELLARt and R. GOEBELt
Department of Mathematics, University of Washington Seattle, WA 98195-4350, USA t E-mail: [email protected] E-mail: goebe1Qmath.washington. edu An optimal control problem with linear dynamics and convex but not necessarily quadratic and possibly infinitevalued or nonsmooth costs can be analyzed in an appropriately formulated duality framework. The paper presents key elements of such a framework, including a construction of a dual optimal control problem, optimality conditions in open loop and feedback forms, and a relationship between primal and dual optimal values. Some results on differentiability and local Lipschitz continuity of the gradient of the optimal value function associated with a convex optimal control problem are included.
Keywords: Optimal control, generalized problem of Bolza, convex value function, duality, Hamilton-Jacobi theory
1. Introduction
The classical Linear Quadratic Regulator problem (CQR) involves minimizing a quadratic cost subject to linear dynamics. See Ref. 1 for a detailed exposition. When the system is subject to constraints or actuator saturation, or when nonquadratic costs or barrier functions are involved, the linear and quadratic techniques applicable to CQR are no longer adequate. Often though, the optimal control problem one needs to solve is convex. Much like what is appreciated in convex optimization, convex structure of an optimal control problem makes available techniques based on an appropriately formulated duality framework. Existence results, optimality conditions, and sensitivity to perturbations can be analyzed in such a framework even if constraints or nonsmooth costs are involved. Following the contributions made over the years by the first author and his students and former students, we outline in this paper some key aspects of a duality framework for the study of general finite-horizon and infinitehorizon convex optimal control problems. For simplicity of presentation,
281
but also with control engineering applications in mind, we specialize the key results t o a Linear-Convex Regulator (LCR):a problem in which the quadratic costs as in the LQR are replaced by general convex functions. Besides expository purposes, the choice of the duality related results we state is motivated by the task of analyzing the regularity properties of optimal value functions associated with a control problem. The properties of our interest include finiteness, coercivity, differentiability, strict convexity, and finally, continuity of the gradient. In particular, based on some recent results of the authors on the duality between local strong convexity of a convex function and the local Lipschitz continuity of the gradient of its conjugate, we identify some mild assumptions on a convex optimal control problem that guarantee that both the primal value function and the dual value function have locally Lipschitz continuous gradients. 2. Duality in finite horizon optimal control
The full power of convex duality techniques was brought to the calculus of variations problems on finite time horizons by Rockafellar in Ref. 18, and led, in particular, to very general existence results in Ref. 20. We point the reader to Ref. 18 for an overview of other early developments relying, to an extent, on convex duality. Regarding linear-quadratic optimal control problems with constraints, a duality framework was suggested in Ref. 22 and led, among other things, to simple optimality conditions for problems where some state constraints are present; see Ref 23. The duality tools were specialized to the study of value functions, in the Hamilton-Jacobi framework, by Rockafellar and Wolenski in Refs. 25,26. Comparison t o related Hamilton-Jacobi developments can be found in Refs. 9,25. 2.1. The (finite horizon) Linear- Convex Regulator
We will focus on the following optimal control problem: rT
minimize
J
{q(y(t))
+
T ( u ( ~ ) ) }dt
+g(z(T))
(1)
7
over all integrable control functions u : [.,TI namics
+
R', subject to linear dy-
4
k ( t )= A z ( t ) B u ( t ) , y ( t ) = C z ( t ) ,
(2)
and the initial condition X(T)
= <.
(3)
282
The absolutely continuous arc x : [7,T]+ R" denotes the state and y : [.,TI --+ Rm denotes the output of the linear system (2). The matrices A , B , and C are any matrices of appropriate dimensions; we will explicitly make assumptions of observability or controllability when needed. When the functions q , r , and g are quadratic and given by symmetric and positive definite matrices, the problem described above is the classical Linear-Quadratic Regulator; see Ref. 1 for background and detailed analysis. Here, we will allow q , r , and g to be more general convex functions. Consequently, we will call the problem defined by (l),(2), (3) a Linear-Convex Regulator, or just LCR. When an explicit reference to initial conditions is needed, we will speak of the LCR(T,t) problem. Below, E = [-m,+m] and a function f : R" --+ E is called coercive if limlxl+m f (.)/I4 = m.
Standing Assumptions. The functions q : R" g : R" --f E are elements of the class C , where
+
E, r : Rk + E,
and
f is convex, lower semicontinuous, f ( 0 ) = 0, f (x)2 0 for all z E Rz
i= 1
Furthermore, q ( y ) is finite for all y E R" while r is coercive. In what follows, we will say that f : Rn + R is proper if it never equals -m and is finite somewhere. 2.2. Convex conjugate functions
For any convex, lower semicontinuous (lsc.), and proper function f : Rn --+ R, its conjugate function f * : R" -+ E is defined by
The function f * is itself a convex, lsc., and proper function, and the function conjugate to it is f. It is also immediate from the definition of f * that f E C if and only f* E C ; this will in particular imply that our assumptions on LCR and on the problem dual to C C R are symmetric. A basic example of a pair of conjugate functions is
1
f (x)= -x 2
*
Qx,
1
f *(P) = Z P
*
Q-'P,
where Q is a symmetric and positive definite matrix. Another is
i x .Q x if x +m
EX
ifsex'
(5)
283
for Q as above and a closed convex and nonempty set X c R". Requiring that f(0) = 0 implies that 0 E X . If 0 E intX, then f* is quadratic on some neighborhood of 0, and given there by i p . Q-'p. In general, this f * is differentiable and Of*is Lipschitz continuous; related properties of not necessarily quadratic convex functions will be discussed in Section 3. A fundamental property of the pair f and f * ,that is the basis for EulerLagrange and Hamiltonian optimality conditions for convex optimal control problems, is that for any x,p E R",
+
af
while f(x) f*(p) = x.p if and only if p E (x)if and only if p E af*(x). Here, 13f is the subdifferential of the convex function f:
af(x) = { p E R" I f(d) 2 f(x) + p . (x' - x) for all x'
E
R"}
.
A standard reference for the material just presented is Ref. 17 2.3. General duality framework
Many of the results we will state or use were developed not in an optimal control setting, but rather, in the framework of duality for calculus of variations problems. We briefly outline this framework. Given convex, lsc., and proper functions L, 1 : R2n + E,consider
-
&,w)
= L*(W,P),
1(pT > p T ) = (pT -pT) 1
9
and two generalized problems of Bolza type: the primal problem P of minimizing, over all absolutely continuous x : 17, T ]-+ R", the cost functional
lT-
L ( x ( t ) , i ( t )dt )
+ 1(.(.),x(T)),
(8)
and the dual problem P of minimizing, over all absolutely continuous p : [.,TI -+ R", the (dual) cost functional
I ' & 4 t ) , P ( t ) ) dt +RP(.)IP(T)).
(9)
Note that the problem dual to p is the original P. Directly from (7), it follows that it is always the case that inf(P) 2 -inf(p), and that any absolutely continuous p : [.,TI + R" provides a lower bound on inf(P) through (9), while any absolutely continuous x : [.,TI -+ R" provides a lower bound on inf(F) through (8). Under some
284
mild assumptions, given in Refs. 18,20 for a more general time-dependent case and specialized in Ref. 25 to the autonomous case, the following holds: -oa
< inf(p) = -inf(F) < +oa,
(10)
and moreover, optimal solutions for both the primal and the dual problem exist. Regarding L , Ref. 25 required that: (i) the set F ( z ) = {w I L ( z , v ) < oa} be nonempty for all z, and there exist a constant p such that dist(O,F(z)) 5 p(1 1x1) for all z, and (ii) there exist constants a! and ,L? and a coercive, proper, nondecreasing function 0 on [ O , c o ) such that L ( z ,w) 2 O(max(0, IvI - alzl})- ,L?lzlfor all II: and w. A particularly attractive feature of these assumptions, besides their generality, is that L satisfies them if and only if 2: does. The Linear-Convex Regulator can be reformulated in the generalized Bolza framework. (For this, and other equivalences between optimal control problem formulations, see Chapter 1.3 in Ref 7.) To this end, for each fixed T 5 T , (' E one considers
+
an,
+
L ( z ,v) = q(Cz) min { r ( u )I Az l(zT,zT)
U
= 6C(zT)
+ Bu = v} ,
(11)
+ g(zT),
where 6~is the indicator of E: 6~((') = 0 while 6~(z)= -00 if z # 5. If a solution z : [ T , TI -+ Rn to the resulting Bolza problem P(T,<)is found, an optimal control u : [T,T]-+ R' for LCR can be then recovered, at almost every t E [T, TI, as the minimizer of r ( u ) over all u such that A z ( t ) Bu = i ( t ) .Given such L and 1, the dual Bolza problem F(T,E) is defined by
+
+
x ( p , w)= r*(B*p) min { q * ( z ) I - A*p
-
l(P7,PT) =
<
*PT
+ C*z = w},
f g*(-PT)'
Our Standing Assumptions on q , r , and g guarantee that L and 1 as above (and equivalently, 2: and meet the growth conditions of Ref. 25 and consequently, that (10) holds. We note that in most cases, and even if q and r are quadratic functions, L in (11) does not satisfy the classical coercivity conditions: L ( z ,w) is not bounded below by a coercive function of w. This can easily be seen in the case of B being an identity: then L ( z ,w) = q(Cz) T ( W - A X ) .
+
2.4. The primal and the dual value functions
The (primal) optimal value function V : (-co,T] x R" -+ is defined, for each ( T , <) E (--00, T ]x R", as the infimum in the problem LCR(T,('). For the quadratic case (of q, T , and g quadratic), V ( T .) , is quadratic for each
285 T
5 T . In general, it is immediate that V ( T , is a convex function. More a)
precisely, Theorem 2.1 and Corollary 7.6. in Ref. 25 state the following:
Theorem 2.1 (value function - basic regularity). 0
0
For each T 5 T , V ( T .) , is a n element of C . If g i s finite-valued, t h e n so is V(T,.). V ( T .), depends epi-continuously o n T E (-w, TI.
Epi-continuity above means that the epigraphs of V ( T .), , i.e., the sets a ) E Rnfl I cx 2. V ( T<)}, , depend continuously on T . Such concept of continuity of V takes into account the fact that the effective domains of V ( T .), may depend on T . (See Chapters 4 and 5 in Ref. 24 for details on set convergence and continuity of set-valued mappings.) The value function V can be equivalently defined, at each ( T , < ) , as the infimum in the generalized problem of Bolza P ( T , ~related ) to L C R ( T , ~through ) (11). - F'rom (10) it follows that V ( r , J )= inf(P(T,<))= - i n f ( P ( ~ , e ) ) .Since l ( p , , p ~ ) = inf,Ep{pT 7r bpT(7r)} and dpT(7r) = d,(p,), one obtains that V ( T , <equals )
{(el
-
+
The first and the last infimum above are taken over all arcs p : [T, T ]--+ Rn. The presence of the term d , ( p ( ~ ) ) ensures that the last infimum can be taken only over those arcs p(.) for which p ( ~ =) 7 r . This suggests that the last infimum defines a dual value function, parameterized by r and IT. We now make the formal definitions, following the ideas of Ref. 22 regarding the structure of a dual optimal control problem, and of Ref. 25 regarding the dual value function. The optimal control problem dual to LCR is as follows: minimize
JI'
{ r * ( s ( t )+ ) q*(z(t))}dt +g*(-p(T))
over all integrable control functions z : [.,TI dynamics
4
(12)
R", subject to linear
@(t)= -A*p(t) + C*z(t), s ( t ) = B*p(t),
(13)
286
(here, A* denotes the transpose of A, etc.) and the initial condition
The absolutely continuous arc p : [r,T]+ R" denotes the state and s : [T,T]-+ Rk denotes the output of the linear system (13). We will denote this problem by CCR, and when a direct reference to initial conditions is needed, by CCR(T,T ) . Note that the functions r * , q*, and g * ( - . ) are in the class C (recall (4)). Furthermore, r* is finite-valued (since T is coercive) and q* is coercive --u (since q is finite-valued). Thus C C R has the same growth properties that LCR has, based on the Standing Assumptions. Note also that if C C R is in fact a linear quadratic regulator, with q , r , and g quadratic, then so is CCR, thanks to (5). If CCR is a linear quadratic regulator with control constraints u E U for some closed convex set U (usually with 0 E intU) then the dual problem is unconstrained, but the state cost r* is not quadratic (recall (6)). is defined, at The (dual) optimal value function P : ( - m A x R" --+ a given ( 7 ,T ) , as the infimum in the problem CCR(7,T ) . By the symmetry of the Standing Assumptions, the properties attributed to V in Theorem 2.1 are present also for V . The idea of the computation that followed Theorem 2.1 can be now summarized. For details, see Theorem 5.1 in Ref. 25.
-
h -
h -
Theorem 2.2 (value function duality). For each T 5 T , the value functions V ( T , . )and V(T,.) are convex conjugates of each other, up to a minus sign. That is,
This result allows for studying properties of V ( T .) , by analyzing corresponding properties of ?(T, .). For example, it is a general property of convex functions that a function is coercive if and only if its conjugate is finite-valued. Thus, if g is coercive, then g* is finite-valued, then by Theorem 2.1 the function V(T,.) is finite-valued for each T , and so V ( T .), is coercive for each T . Further correspondences, between differentiability and strict or strong convexity, will be explored in Section 3.
287
2.5. The Hamiltonian and open-loop optimality conditions
With the generalized problems of Bolza P as described in Subsection 2.3, one can associate the (maximized) Hamiltonian H : R2n -+ E:
H ( x ,p ) = sup { p ' v - L ( x ,v)} . 2)
Convexity of L implies that H ( x ,p ) is convex in p for a fixed x and concave in x for a fixed p . In presence of the growth properties of L mentioned following (lo), it is also finite-valued. (The said growth properties have equivalent characterizations in terms of HI see Theorem 2.3 in Ref. 25.) The finiteness of H in turn guarantees that
- H ( x , p ) = sup {x * w - Z ( p , w)} . W
In other words, the Hamiltonian associated with the dual problem F is exactly ( p ,x) H - H ( x , p ) . This further underlines the symmetry between P and ?. A Hamiltonian trajectory on [.,TI is a pair of arcs x,p : [ T , T ]+ R" such that
j.(t> E &JH(X(t),P(t)),
-fi(t) E & m $ ) , P ( t ) )
(15)
. (15), L&H(z,p) is the subdifferential, in the for almost all t E [ T , T ]In sense of convex analysis, of the convex function H ( x , . ) , while & H ( x , p ) = ( - H ( x , p ) ) where the second subdifferential is in the sense of convex analysis. Under suitable assumptions (which also guarantee (10)) the following are equivalent:
-a,
0 0
x(.) solves P,p ( . ) solves F ; x ( . ) , p ( . )form a Hamiltonian trajectory and @(TI,
-P(T)) E W 4 7 ) 1 z ( T ) )
In the convex setting, the key behind this equivalence is the inequality (7) and the relationship between subdifferentials of L , 2: and HI rather than any regularity properties of these functions. For details, see Ref. 18. When the problem P comes from reformulating C C R in the Bolza framework as done in (11), one has
+
H ( x , ~=) p * AX - q(Cz) T * ( B * ~ ) .
(16)
The Hamiltonian system (15) turns into
+
k ( t ) = A x ( t ) Bu(t), u ( t )E d r * ( s ( t ) ) s, ( t ) = B*p(t), P ( t ) = --A*p(t) C*Z(t),4 t ) E %(Y(t)), Y(t) = C x ( t ) ,
+
(17)
288
if one additionally requires that u(t)be a minimizer of r ( u ) over all u such that i ( t ) = Az(t) Bu and that z ( t ) be a minimizer of q * ( z ) over all z such that @(t)= - A * p ( t ) C*z. Details are given in Lemma 3.8 of Ref. 10; see also Theorem 4.7 in Ref. 23.
+
+
Theorem 2.3 (open-loop optimality). For a control u : [.,TI the following statements are equivalent: 0
-i
Rk,
u(.) is an optimal control for C C R ( T ,<) and -T E &V(T,<); there exists a n integrable control z : [r,T]+ Rm so that arcs x,p : [T, TI + Rn corresponding to controls u(.), z ( . ) through the initial condition ( x ( T ) , ~ ( T )=) ( < , T ) and (17) satisfy
-AT) E d g ( z ( T ) ) . It can be added that if z ( . ) corresponds to an optimal u(.)as described in the theorem above, then z ( . ) is optimal for the problem defining ~ ( T , T ) , and also E &~(T,T).
-<
2.6. Hamilton-Jacobi results
The Hamiltonian H and the associated Hamiltonian dynamical system (15) is involved in two different characterizations of the value function. One is the (generalized) Hamilton-Jacobi partial differential equation, as written in Theorem 2.5 of Ref. 25. It says that, for all T < T , we have (T H ( z , p ) = 0 for each subgradient (a,p) (in the sense of Definition 8.3 of Ref. 24) of V ( T , < )We . stress that the subgradient is taken with respect to both time and space variables, and thus it can not be understood in the sense of convex analysis. Subject to a boundary condition, V is in fact the unique solution of this generalized Hamilton-Jacobi equation, as shown by Galbraith in Ref. 9. The other characterization will involve the flow S, generated by the Hamiltonian dynamical system (15):
+
We have the following result, as stated in Theorem 2.4 of Ref. 25:
Theorem 2.4 (Hamiltonian flow of ~ , V ( T.)). , For any gph (-d,V(T,
=
s, (gph(-dg))
'
T
5T,
289
In other words, the graph of the subdifferential of V ( T .) , is essentially obtained from the graph of the subdifferential of g by flowing backwards along Hamiltonian trajectories. This result can also be thought of as saying that the (generalized) method of characteristics works globally in this convex setting. Thanks to Theorem 2.2 and the fact that subdifferential mappings of a pair of convex conjugate functions are inverses of one another, Theorem 2.4 also gives a description of dnV(r,.). Since the subdifferential of a convex function is a monotone mapping, the graph of -8g is an “anti-monotone set”, that is, for any ( a , n )( ,~ 2 ~ E~ gph(-ag), 2 ) we have ( 2 1 - X Z ) . ( P I - PZ) I 0. Thanks to the concavity in x, convexity in p , of H ( x , p ) , the flow S, has a certain monotonicity, or rather anti-monotonicity preserving property:
(x1(7) - .2(.))
. (Pl(7) - P 2 ( 7 - ) ) I (Zl(T)- Zz(T)). (Pl(T) - P z ( T ) ) (18)
for any pair of Hamiltonian trajectories (xi(.),pi(.)),i = 1 , 2 , on [.,TI. This was noted in Theorem 4 of Ref. 19. In what follows, we will need this result specified to the case of CCR.
Lemma 2.1 (preservation of monotonicity). Let (xi(.),pi(.)),i = 1 , 2 , be Hamiltonian trajectories o n [r,T ]for the Hamiltonian (16). Then
E ( W C Z l ( t ) )- &I(Cxz(t))). (CZl(t) - Cx2(t))
+ (ar*(B*Pl(t))- 8 r * ( B * p z ( t ) ).)(B*Pl(t)- B * p z ( t ) ) 2 0 The inequality above can be easily shown directly from the definition of a Hamiltonian trajectory. It also obviously implies (18).
2.7. Feedback optimality conditions Open-loop optimal solutions of optimal control problems are not well-suited to cope with uncertainty or disturbance. This, and the classical control engineering problem of feedback stabilization, motivate the pursuit of an optimal feedback: a mapping that at each state produces the set of optimal controls to be applied. Theorem 2.3, Theorem 2.4, and the Hamiltonian system (15) suggest that optimal arcs z(.) satisfy the inclusion k ( t ) E d,H(x(t), -aEV(t,x(t))). Indeed, if a Hamiltonian trajectory ( x ( . ) , p ( . )on ) [r,TI is such that p ( T ) E -ag(z(T)) then, by Theorem 2.4, p ( t ) E - a < V ( t , ~ ( tfor ) ) almost all t [.,TI. A more difficult - and more important - issue is to what extent the
290
stated inclusion is sufficient for optimality. To an extent, this was addressed in Ref. 10. We state the relevant result in Theorem 2.5. The optimal feedback mapping CP : (-..,TI x R" for the problem C C R is defined, at each t 5 T , x E R", by
~ ( x) t ,= dr*(-B*dcV(t, x)) =
U
ar*(-B*p).
P€a,v(t,z)
Theorem 2.5 (optimal feedback). Fix (T,(') E (-..,TI x Rn. If u : [T, TI -+Rk is an optimal control for LCR(T,E ) and dcV(r, # 0, then
e)
u(t)E CP(t,z(t))for almost all t E [T,T].
(19)
O n the other hand, if a n integrable control u : [.,TI -+ Rk satisfies (19), and the absolutely continuous x : [r,TI + R" satisfying (2) and x(r) = is such that x ( t ) E intdomV(7, .) for almost all t E [.,TI, then u(.)is optimal for CCR(T,
<
c).
The interior intdomV of the effective domain of V, i.e., of the set domV of all points where V is finite, is nonempty if intdomg is nonempty, and (7, E ) E intdomV if and only if T < T and E E intdomV(.r, .); see Proposition 7.2 and Corollary 7.5 in Ref. 25. In general, the feedback mapping CP it is outer semicontinuous and locally bounded on intdomV, but need not have convex values; see Example 3.7 in Ref. 10. 3. Regularity of the value function
We now give some results on how regularity of the data of C C R may be reflected in the properties of V, and thus, of the optimal feedback mapping CP. The central role played by the Hamiltonian dynamical system (15) in the analysis of convex optimal control problems is reflected below in the fact that the only tools from the previous chapter we use are Theorem 2.4, which describes how V ~ V ( T.), evolves in the Hamiltonian system, and Lemma 2.1, about the monotonicity-preserving properties of the Hamiltonian flow. We note that similar regularity properties of the value function in a nonconvex setting require stronger assumptions than the ones posed here; see for example Refs. 4,6. To an extent though, regularity properties (or singularities) of the value function are propagated via the Hamiltonian system; see Ref. 8. 3.1. Convex-valued and single-valued optimal feedback
A natural condition to guarantee that the optimal feedback CP has convex values is that d c V ( ~<) , be single-valued, which is equivalent to essential
29 1
differentiability of V(T, .). Theorem 3.1 will show that this property is in a sense inherited by V from g . We need some preliminary material. Let f : Rn -+ be a proper, lsc., and convex function. The function f is essentially differentiable if D = intdomf is nonempty, f is differentiable on D ,and limi-+mlVf(xi)l= 0;) for any sequence { ~ i converging } ~ ~ to a boundary point of D.The function f is essentially strictly convex if f is convex on every convex subset of domd f = {x E R" I d f (x)# 8); that is, if for each convex S c domd f , we have (1 - A) f ( 2 1 ) Xf(z2) > f ((1- X)xl Xx2) for all X E (0, I), all 2 1 , x2 E S. The subdifferential mapping is a monotone mapping: for all 21~xE 2 d0md.f all 91 E af ( x i ) ,Y Z E ( ~ 2 () 2~1 - x2) * ( y i - Y Z ) L 0. It is strictly monotone if (21 - 2 2 ) . (y1 - y2) > 0 as long as 2 1 # 2 2 . The following statements are equivalent: (i) f is essentially differentiable; (ii) d f is single-valued on domd f ; (iii) d f * is strictly monotone; and (iv) f* is essentially strictly convex. The result below is a special case of Theorem 4.4 in Ref. 10. Note that no regularity is assumed about T .
a
+
+
af
Theorem 3.1 (value function - essential differentiability). Suppose that q and g are essentially differentiable. Then, f o r all V(T, .) is essentially differentiable.
T
5 T,
Proof. By Theorem 2.4, any two points in gph(-dcV(T,.)) can be expressed as ( z ~ ( T ) , P ~ ( T )i) = , 1 , 2 , where (xi(.),pi(.))are Hamiltonian trajectories on [T, T ]with pi(T) = - V g ( x i ( T ) ) , i = 1,2. If q ( 7 ) = 2 2 ( 7 ) , then by Lemma 2.1 one obtains 0 = (21(7)--2(7)).(p1(7)--P2(7)) 5 (21(T)-xZ(T)).(pl(T)-pz(T)) 50, with the last inequality above coming from monotonicity of V g ( . ) . Hence
h ( T )-.2(T))
*
( P l ( T ) -P2(T))= 0
which by essential strict convexity of g* (a property dual to essential differentiability of g ) implies that p l ( T ) = p 2 ( T ) . The fact that ( z l ( t )- 2 2 ( t ) ) . ( p l ( t ) - p z ( t ) ) is constant on [T, T] and Theorem 4 of Ref. 19 implies that ( x ~ ( . ) , p z ( . )is) a Hamiltonian trajectory. In particular,
+
@l(t)- @ 2 ( t ) = - A * ( p i ( t ) - ~ 2 ( t ) ) C* ( V q ( C x l ( t ) )- V q ( C X l ( t ) ) ) = -A*(m(t)- P 2 ( t ) ) , and since p l ( T ) = p 2 ( T ) , it must be the case that p 1 ( ~ = ) p2(7).
292
We have shown that, for any two points (xi(~),pi(~)), i = 1 , 2 , in gph(-dcV(.r, .)), if x ~ ( T = ) 2 2 ( 7 ) then also p l ( 7 ) = p2(7). This amounts to single-valuedness of dtV(7, .) which in turn is equivalent to essential differentiability of V(T, .). 0 Under the assumptions of Theorem 3.1, the feedback Q has convex values, since dr* has convex values. If furthermore dr* is single-valued, and so if r* is differentiable (as r* is finite, its essential differentiability is just differentiability), then Q is single-valued and in fact continuous. Thus, if q and g are essentially differentiable and r is strictly convex, then the optimal feedback @ is continuous on intdomV. (C.f. Proposition 3.5 in Ref. 10.) A different set of assumptions leading to essential differentiability of V(T, .) is strict convexity of both q and r* and observability of (A,C) and (-A*, B*).This combines arguments that apply to the case of strictly concave, strictly convex Hamiltonian, as in Ref. 21 and basic properties of observable linear systems (c.f. Theorem 3.1 in Ref. 13). We note though that strict convexity of r* may be absent in simple cases, like when r represents a quadratic cost and a convex control constraint set (recall (6)). The next natural property of the optimal feedback to pursue, with the uniqueness of solutions to the closed loop equation k ( t ) = Ax@)
+ B Q ( t ~, ( t ) )
(20)
in mind, is the local Lipschitz continuity of @(t, .). This motivates the developments in the next section. 3 . 2 . Convex functions with locally Lipschitz gradients
Let f : Rn -+ B be a proper, lsc., and convex function. The property of f being differentiable with Of Lipschitz continuous on R" with constant p > 0 is equivalent to f * being strongly convex with constant u = p - l :
for all X E (0, l ) , all p1,pz E R". A manipulation of (21) shows that strong convexity of f * with constant u is equivalent to f * - fa1 . l2 being convex. Furthermore, strong convexity of f * is equivalent to its subdifferential being strongly monotone with constant u: (p1 - p2) . ( 2 1 - 2 2 ) 2 alp1 - p2I2 for all ( P l , Z l ) , ( P 2 , ~ 2E) gphdf*. We now report some recent results from Ref. 14 regarding localization of the properties just mentioned. In what follows, we will say that a proper,
293
lsc., and convex function g : Rn -+ is essentially locally strongly convex if for any compact and convex K c domag, g is strongly convex on K , in the sense that there exists u > 0 such that (21) is satisfied for all p1,p2 E K and all X E ( 0 , l ) .
Theorem 3.2 (strong convexity and gradient continuity). For a proper, lsc., and convex function f : Rn -+ E, the following conditions are equivalent:
(a) f is essentially differentiable and V f i s locally Lipschitz continuous o n intdom f ; (b) the mapping d f is single-valued o n domaf and locally Lipschitz continuous relative t o gphaf in the sense that for each compact subset S c gphaf there exists p > 0 such that for any ( Q , Y I ) , ( z z , y z ) E S one has IY1
- YZl 5 P I X 1 - 221;
(c) the mapping af * is locally strongly monotone relative t o gpha f * in the sense that for each compact subset S c gphd f * there exists u > 0 such that for any ( y l , ~ (~y)z , z z ) E S one has (Y1
- YZ) . ( E l
-32)
L4Yl
- YZ?.
Iff is such that domdf" is convex and open, then each of the conditions above is equivalent to: (d) f * is essentially locally strongly convex. This in particular says that for a coercive (and proper and lsc) convex function f , essential differentiability of f and local Lipschitz continuity of V f is equivalent to (naturally understood) local strong convexity of f *. For convenience, we define the following classes of convex functions:
f is essentially differentiable and
V f is locally Lipschitz continuous on intdom f
C++ = {f E C+ I f is essentially locally strongly convex } . Since for any essentially differentiable f, domaf = intdomf is open and convex, for such f the condition that f is essentially locally strongly convex is equivalent, by Theorem 3.2, to f * E C+.Thus, f E C++ if and only if f,f * E C+.This immediately implies that f E C++ if and only if f * E C++.
294
3.3. Locally Lipschitz continuous optimal feedback
The result below is a local version of Theorem 3.6 of Ref. 11, where global Lipschitz continuity of V H and Vg was assumed. Here though, we make stronger symmetric assumptions on g. This simplifies the proof.
Theorem 3.3 (value function - strong local regularity). Suppose that q,r* E C+ while g E C++. Then, for all T 5 T , V ( r , . ) is an element of C++.
Proof. By Theorem 2.4, any point in gpha
M2
ll4~)IIIIP(T)II IIpo112' Lipschitz continuity of V H yields *
*This bound comes from reversing time, i.e., considering the backward Hamiltonian flow, with time 0 corresponding t o T and T - T corresponding to T . By Lipschitz continuity
295
Squaring both sides and multiplying by the previous displayed inequality yields
This shows that VcV(r,.) is Lipschitz continuous relative to X , x P,. Strong monotonicity of V ~ V ( T ,can . ) be shown similarly, by using a bound on I I ~ ( T ) I I in place of the bound on I I / ~ ( T ) I I in the arguments above. 0 Minor modifications of the proof above do show that the Lipschitz constant for V ~ V ( T , is . ) locally bounded in both T and <. This, and local Lipschitz continuity of Vr*,implies that the solutions to (20) are unique. 4. Infinite horizon problems
The original control engineering motivation for considering convex optimal control problems - construction of stabilizing feedbacks - calls for posing problems on an infinite time horizon. (Finite time horizon problems can then be used to approximate the infinite horizon ones, or, as seen in Receding Horizon control (also referred to as Model Predictive Control), to obtain stabilizing feedbacks without necessarily having them optimal; see Ref. 16 for a survey.) Besides control engineering, infinite-horizon problems have seen treatment in theoretical economics; see the book by Carslon, Haurie, and Leizarowitz, Ref. 5, for an overview, Ref. 2 for an example of duality techniques, and Ref. 21 for an illustration of the role played by the Hamiltonian dynamical system. For an infinite-horizon LCR, some convexity tools were used in Ref. 3. A detailed, and largely self-contained analysis of a Linear-Quadratic Regulator with control constraints was carried out in Ref 15. General calculus of variations problems on the infinite-time horizon were
296
treated in Ref. 12, based to an extent on the duality and Hamilton-Jacobi developments of Ref. 25. The (primal) infinite horizon optimal value function W : R" -i is defined as the infimum of
la
{ Q ( Y ( t ) )+ M t ) ) }dt7
taken over all locally integrable controls u : [0,00) ---t Rk, subject to dynamics (2) and the initial condition
x(0) = 6. The (dual) infinite horizon optimal value function as the infimum of
@ : R" + a is defined
la
{ r * ( s ( t )+ ) q * ( z ( t ) ) }d t ,
taken over all locally integrable controls z : [0,00) -+ R", subject to dynamics (13) and the initial condition
p ( 0 ) = 7T. Addition to the Standing Assumptions. The functions q, q*, T , r* are positive definite (0 at 0 , positive elsewhere). The pairs of matrices ( A , C ) and ( - A T , B T ) are observable. Of course, the value functions W and are convex. Results regarding their lower semicontinuity and the existence of optimal arcs in the problems defining them are in Ref. 12. The functions W and are both positive definite, and the optimal arcs, for both W and tend to 0 as t -i 00. Indeed, for W , note that W ( < )= 0 only if the optimal arc z satisfies C x ( t ) = 0 while the optimal control is u(t) = 0 for all t 2 0. But then k ( t ) = A z ( t ) ,and observability of ( A , C )gives x ( t ) = 0 for all t 2 0, in particular = x ( 0 ) = 0. Given an optimal arc x(.) with W ( z ( 0 ) )< 00, we have W ( x ( T ) = ) &?{q(Cx(t)) r ( u ( t ) ) dt } and by finiteness of W ( < )the , integral tends to 0 as T -i 00. Positive definiteness of W ( x ( T ) now ) implies that z ( T )-+ 0. Arguments for W are symmetric. Now, [12, Corollary 3.51 says that:
w
w w,
<
+ -
0
w(<)= W*(-<), equivalently, F(r)= W * ( - T ) ; f = W is the unique f E C such that H ( x , -af (x))= 0 for all z E Rn;
0
f =
0
w is the unique f
EC
such that H ( - a f ( p ) , p ) = 0 for all p E Rn.
297
In other words, the infinite horizon value functions are conjugate to each other, and are the unique solutions to the generalized stationary HamiltonJacobi equations. (The equation H ( x , - d f ( x ) ) = 0 is to be understood as H ( x , - p ) = 0 for all p E d f ( x ) . ) It is interesting to note that with such a concept of a solution, it is no longer true that under standard assumptions of ( A ,B ) being stabilizable and ( A ,C) being detectable, and with q and r quadratic, the (quadratic) value function W is the unique solution to the Hamilton-Jacobi equation. (This occurs even though the matrix defining W is the unique positive definite solution to the matrix Riccati equation, to which the Hamilton- Jacobi equation simplifies if one only looks for quadratic solutions.) See Section 4.1 in Ref. 12 for details. Another consequence of the fact that optimal arcs for both W and % converge to 0 is a result relating these two value functions to the finite horizon value functions discussed earlier. It turns out that both W and % can be simultaneously approximated by a pair of conjugate finite horizon value functions V ( T .) , and V(T,.). Indeed, as the proof of Theorem 3.4 in Ref. 12 suggests, we have the following:
Theorem 4.1 (finite-horizon approximation). Consider the functions WT,zT : R" + defined at points S,T E R" as the optimal values in LCR(0,C), CCR(0,T ) . Suppose that g and g* are finite-valued o n some neighborhood of 0. Then W T converges epi-graphically t o W while WT converges epi-graphically t o % as T + 00. h_
For the infinite horizon case, the Hamiltonian dynamical system (15) again yields a description of the subdifferential of W (and of %). Indeed, gph - dW consists of all points (5,a) for which there exists a Hamiltonian trajectory x , p on [0, co) with x ( 0 ) = 5, p ( 0 ) = a, and z ( t )+ 0, p ( t ) -+ 0 as t --+ 00. See Proposition 3.7 in Ref. 12 which generalizes the results given for strictly concave / strictly convex Hamiltonians in Ref. 21. We add that for such a Hamiltonian trajectory, x is optimal for W(5)while p is optimal for %(a). The said description of dW leads to the following result.
Proposition 4.1. Suppose that there exists a neighborhood of 0 o n which both q and r* are strictly convex. Then W is both essentially differentiable and essentially strictly convex. The assumption is equivalent to strict convexity of q and essential differentiability of r. By duality, the same assumptions imply that @ is also essentially differentiable and essentially strictly convex. The result can be
298
shown via arguments combining Lemma 2.1 and controllability assumptions we have. (Similar results, shown via slightly different methods, are in Theorem 4.1 of Ref. 21 and Theorem 3.1 of Ref. 13.) An interesting application of Proposition 4.1 lies in showing that for a wide class of linear systems with saturation nonlinearities there does exist a continuous and stabilizing feedback. (Such an issue was partially addressed in Refs. 27,28 via tools not related to optimality.) Consider a system
k ( t )= A x ( t )
+ Ba(u(t))
(22)
where a : Rk 4 Rk is a saturation nonlinearity. Often, for single input systems, a ( u ) = arctan(u) or a(u) = u for u E [-1,1], a(.) = -1 for u < -1, a(.) = 1 for u > 1. For these, and for many other commonly encountered in the control literature saturation functions, a = V s for a convex function s E C which is strictly convex near 0. Taking r = s* and, for simplicity, any positive definite quadratic q, leads to an essentially differentiable W . The optimal feedback @ ( x )= V s ( - B * V W ( x ) ) leads to asymptotic stability for the closed loop system k ( t ) = A x ( t ) B @ ( x ( t ) ) . This asymptotic stability can in fact be verified with W as a Lyapunov function. But since V s = a , this means that choosing u ( x ) = - B * V W ( x ) in (22) also leads to asymptotic stability. As VW is continuous, there does exist a continuous stabilizing feedback for the nonlinear system (22). For details, see Ref. 13. We conclude by stating a result on local Lipschitz continuity of V W . Since optimal trajectories for problems defining both W and converge to 0, local properties around 0 of these functions to an extent determine global properties. Another way to say this is that, on each compact subset of domW, W equals to a finite horizon value function defined by LCR with g = W in a neighborhood of 0, as long as T is large enough. This and Theorem 3.3 lead to the following.
+
w
Proposition 4.2. Suppose that there exists a neighborhood of 0 on which W is differentiable and strongly convex and VW is Lipschitz continuous. Then W E C++. An important example of problems where W is differentiable and strongly convex around 0 is provided by a C Q R problem with control constraints: u is restricted to some closed convex set U and 0 E intU. Indeed, then on some neighborhood of 0, W is quadratic and equals to the value function of the unconstrained LQR problem.
299
Acknowledgments Research by R.T. Rockafellar was partially supported by the NSF Grant
DMS-0104055. References 1. B. Anderson and J. Moore, Optimal Control: Linear Quadratic Methods (Prentice-Hall, Upper Saddle River, NJ, 1990). 2. L. Benveniste and J. Scheinkman, J. Economic Theory 27,1 (1982). 3. G. D. Blasio, SIAM J. Control Optim. 29,909 (1991). 4. C. Byrnes and H. Frankowska, C.R. Acad. Sci. Paris 315,427 (1992). 5. D. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems, 2nd edition (Springer-Verlag, New York, 1991). 6. N. Caroff and H. Frankowska, Trans. Amer. Math. SOC.348,3133 (1996). 7. F. Clarke, Optimization and Nonsmooth Analysis, Canadian Math. SOC.Series of Monogr. and Advanc. Texts (Wiley & Sons Inc., New York, 1983). 8. H. Frankowska and A. Ochal, J . Math. Anal. Appl. 306,714 (2005). 9. G. Galbraith, SIAM J . Control Optim. 39,281 (2000). 10. R. Goebel, Set-Valued Analysis 12,127 (2004). 11. R. Goebel, SIAM J. Control Optim. 43,1781 (2005). 12. R. Goebel, Trans. Amer. Math. SOC. 357,2187 (2005). 13. R. Goebel, IEEE Trans. Automat. Contr. 50,650 (2005). 14. R. Goebel and R. Rockafellar, Local strong convexity and local Lipschitz continuity of the gradient of convex functions, J . Convex Anal. (to appear). 15. R. Goebel and M. Subbotin, IEEE Trans. Automat. Contr. 52,886 (2007). 16. D. Mayne, J. Rawlings, C. R m and P. Scokaert, Automatica 36,789 (2000). 17. R. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28 (Princeton University Press, Princeton, N.J, 1970). 18. R. Rockafellar, J . Math. Anal. Appl. 32, 174 (1970). 19. R. Rockafellar, Pacific J. Math. 33,411 (1970). 20. R. Rockafellar, Trans. Amer. Math. SOC. 159,1 (1971). 21. R. Rockafellar, J . Optim. Theory Appl. 12 (1973). 22. R. Rockafellar, SIAM J. Control Optim. 25,781 (1987). 23. R. Rockafellar, SIAM J . Control Optim. 27, 1007 (1989). 24. R. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundl. der Mathemat. Wissensch. 317 (Springer, Berlin, 1998). 25. R. Rockafellar and P. Wolenski, SIAM J. Control Optim. 39,1323 (2000). 26. R. Rockafellar and P. Wolenski, SIAM J . Control Optim. 39,1351 (2000). 27. E. Sontag and H. Sussmann, Nonlinear output feedback design for linear systems with saturating controls, in Proc. 29 IEEE Conf. Decision and Control (Honolulu, Hawaii, 1990): 3414-3416. 28. E. Sontag, H. Sussmann and Y. Yang, IEEE Trans. Automat. Control 39, 2411 (1994).
300
STRONG OPTIMALITY OF SINGULAR TRAJECTORIES G . STEFAN1 Dipartimento d i Matematica Applacata, Universitb d i Firenze, via d i S.Marta 3, 50139 Farenze, Italia E-mail: gianna.stefaniQuniji.it The paper proves second order sufficient conditions for the strong optimality of singular extremals of the first kind. The conditions are given both in Hamiltonian formulation and by means of the coercivity of a suitable coordinatefree second variation. The conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where the necessary conditions have mild inequalities.
Keywords: sufficient conditions, singular control, second variation, Hamiltonian methods.
1. Introduction
In this paper we use a Hamiltonian approach to prove sufficient second order conditions for the strong optimality of a singular extremal of the following Mayer problem on the given interval [0,TI:
+
minimize co(S(0)) CT(S(T)) subject to
The state space is a smooth manifold M , NO and NT are smooth submanifolds, possibly reduced to a point, % and c1 are smooth functions, and fo, f l : M T M are smooth vector fields. By smooth we mean C", although the proof of the results requires only C2 data. Since the dynamics is linear with respect to the control, by Pontryagin Maximum Principle, generically optimal controls take values on the extremal points of the interval, nevertheless Filippov's theorem implies that ---f
301
interior points cannot be avoided. Here we study the strong local optimality of a reference trajectory relative to a singular control, i.e. to a control which takes values in the interior of the interval. Namely we study the strong local optimality of a reference couple (EG)satisfying (l),(2) and such that
.^(t) E (uo,u1) , vt E [O, 7.7. A
The trajectory is a strong local minimizer if it is a local minimizer in the Co topology of the trajectories on [0, TI, independently on the values of the associated controls. Since we deal with a minimizer corresponding to an interior control we require that the reference couple satisfies the first order optimality conditions in the form of Lagrange multipliers rule (see Lemma 2.1), which defines a lift of to the cotangent bundle, say : 10,TI --+ T * M , called adjoint covector. Remark that the adjoint covector given by Pontryagin Maximum Principle is opposite to the one we consider, which instead is linked to a minimization principle. A trajectory ?of the control system (1) associated to a singular control, satisfying first order conditions is called singular state-extremal. The Hamiltonian approach to sufficient conditions in optimal control corresponds to the classical construction in Calculus of Variations of a field of state extremals covering a neighborhood of the reference trajectory, see for example Ref. 10. The idea is to lift admissible trajectories to the cotangent bundle and to compare their costs, independently on the associated controls. When the minimized Hamiltonian Hman is sufficiently smooth the field of extremals can be obtained by projecting on the state manifold its flow emanating from a suitable Lagrangian sub-manifold and the coercivity of the second variation allows to invert the projection and lift the admissible trajectories, see Refs. 4,5 and the references therein. An important feature is that the comparison of the costs can be also obtained with a Hamiltonian which is smaller than or equal to Hmtn. In the present case we have to face two difficulties: from one hand Hmzn is not sufficiently smooth near the adjoint covector, on the other hand the classical second variation is completely degenerate. To overcome the problem we add a negative term to the minimized Hamiltonian which keeps the trajectories of the associated Hamiltonian system on the hyper-surface where Hmin is smooth, for more details see Section 3. It turns out that this new Hamiltonian is naturally linked to the extended second variation, defined in Section 5 . The construction is possible by assuming the strengthened
r^
302
generalized Legendre condition (9). For references on singular trajectories see Refs. 2,7-9,11,21, and the references therein. We underline that we study sufficient conditions for strong local optimality which to our knowledge do not appear in the literature on singular trajectories. We consider also the abnormal case, that is when the multiplier associated to the cost is zero. As the second variation is concerned, we start from the coordinate-free second variation defined in Ref. 3, we apply the Goh transformation and we end up with an extended second variation which is coordinate-free too and which is proved t o be the standard second variation of a new control system where the control appears up t o the second order. In Section 5 we give both the definition of the extended second variation and the necessary and sufficient conditions for its coercivity in terms of a suitable Hamiltonian system. In the particular case when the singular state extremals are integral curves of the same vector field fs,the conditions are given through the behavior of fs and the controlled vector field fl, see Subsection 6.2. The main theorem of the paper is the following: Theorem 1.1 (Main Theorem). Suppose that is a singular stateextremal and that the associated extended second variation J g , as defined in Section 5, is coercive, then in the normal case (PO= l), is a strict strong local minimizer for the original problem, in the abnormal case (PO = 0), is locally isolated between the solutions of (1) and (2)
<
-
<
We remark that the coercivity of J g implies the strengthened generalized Legendre condition, moreover the above stated sufficient conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where the necessary conditions have mild inequalities, see Theorem 5.1. The plan of the paper is as follows. In Section 2 we give the used notations and preliminary results including the first order conditions and the description of the geometrical picture of the cotangent space near the adjoint covector. In Section 3 we describe the Hamiltonian approach to strong optimality and we define the Hamiltonian x which allows the use of the method in the present case. In Section 4 we prove sufficient conditions from a Hamiltonian viewpoint (Theorem 4.1), describing how to compare the costs starting from the construction of a field of non intersecting almost extremals. In Section 5 we define the coordinate-free extended second variation J g
303
and we prove that its coercivity can be tested via a non singular problem. In Section 6 we describe the links between the extended second variation and the modified minimizing Hamiltonian and we prove the main theorem. In particular Lemma 6.3 gives necessary and sufficient conditions for the coercivity of J g in terms of almost extremals. In Section 7 we give some summarizing remarks on the paper and we describe future research perspectives. 2. Notations and preliminary results
2.1. Notations
In this paper we use some basic element of the theory of symplectic manifolds referred to the cotangent bundle T * M . For a general introduction see Ref. 6, for specific application to Control Theory we refer to Ref. 1 and Ref. 2. Let us recall some basic facts and let us introduce some specific notations. We identify any bilinear form Q on a vector space V with a linear form Q : V +. V* and we denote the associated quadratic form as
The kernel of Q as a quadratic form is the kernel of Q as a linear operator. We call nullity the dimension of kerQ and index the dimension of the maximal subspace on which Q is negative. For any subspace V c T x M , VL c T,M denotes its annihilator. If N is a sub-manifold of M , we write T k N for ( T X N ) l . We use two equivalent notations for the Lie derivative of a function q5 in the direction of the vector field f , namely we write
Lfq5 := f . 4 : z
(d4(z),f(.)
7
depending on typographic convenience. Denote by IT : T * M -+ M the canonical projection, the space T,*eM is canonically embedded in TeT*M as the space of tangent vectors to the fibers. The canonical Liouville one-form s on T * M and the associated canonical symplectic two-form u = ds allow to associate to any, possibly timedependent, smooth Hamiltonian Ht : T * M +. R, a Hamiltonian vector field &, by U ( W ,E i t ( e ) )
= (dHt(f!),w),
Y W E TeT*M
304
and we denote its flow from time 0 to time t by 'Ft : ( t , l )H 'Ft(t,l)= N t ( l ) .
We keep these notation throughout the paper, namely the overhead arrow denotes the vector field associated to a Hamiltonian and the script letter denotes its flow from time 0. Finally recall that any vector field f on the manifold M defines, by lifting to the cotangent bundle, a Hamiltonian
l E T * M H ( l , f ( n l )E) R. Coming back to our problem, we define the time-dependent reference vector field by
5 :M
-+
TM,
2
+ G(t)fi(z).
H fo(2)
The flow, from time 0 to time t , of neighborhood 0 of
6 is defined for all t E
[O,T]in a
h
Po := E(O), and it is denoted by ?t : 0 -+ M , t E [0,TI, while the time-dependent reference Hamiltonian defined by the vector field is denoted by gt. Following the notation of Ref. 2, we denote by Ho, HI the Hamiltonians associated to fo, fl respectively and by
5
HiliZ...ik
:=
j
{' * ' { H Z k - l
7
Hik}
*
"}
the Hamiltonian associated to filiz...ik := [fi,, [. . . [ f i k - , ,fi,] . . .I, where {, } denotes the Poisson parentheses between Hamiltonians and [,] denotes the Lie brackets between vector fields. 2 .2 . The Weak Maximum Principle
As we pointed out in Section 1 we assume the reference couple to satisfy the first order optimality conditions in the form of Lagrange multipliers rule, usually called the weak maximum principle (WMP). Recall that the adjoint covector given by Pontryagin Maximum Principle is the opposite to the one we consider, which instead is linked to the "minimization principle".
(c
Lemma 2.1 (WMP). If G) is a minimizing couple with G(t) E ( u o , u ~a.e.t ) , E [O,T],then there is po E {0,1} and an absolutely con-
305
and, by derivation, equation (6) leads to
so that we obtain
Hooi . q t ) = --(X(t)) Hi01
and
HOOl H i 10
-
= ---(X(t))
is called a singular extremal of the first kind, see for example Ref. 21.
Definition 2.1. Let u E Loo([O,TI,R) be such that u ( t ) E [uo, u l ] ,a.e. t E [0,TI. We call any non trivial trajectory X : [O,T]-+ T * M of the Hamiltonian vector field GO u ( t ) a l ,which satisfies (6), extremal of the control system, while we call its projection (' : [O,T]-+ M state extremal.
+
2.3. Geometry near singular extremals of the first kind
By (6), ( 7 ) and (8), any singular extremal of the first kind belongs to the (2n - 2)-dimensional symplectic manifold HOOl s= {t : H , ( t ) = HOl(t) = 0 , HllO(t) > 0, -(t) H110
which is contained in the hyper-surface C :=
{t E T * M : H i ( [ ) = 0 } ,
E (uo,u1)}
306
where the minimized Hamiltonian
Hmin : C H min{Ho(C)
+ uHl(C) : u E [uo,ul]}
coincides with Ho. By SGLC it is easy to prove the following result.
Lemma 2.2. I n a neighborhood U of S in T'M the following statements hold true. (1) C is an hyper-surface containing the symplectic manifold S (2) Eil is tangent to c (3) i i l is transversal to (4) 2 0 1 is transversal to c (5) The maps (s,l)H expsl?l(C) and ( T , s , C ) H e x p ~ f i o oexpsdl(C) 1 are local diffeomorphisms from lR x S to C and from R x R x S to T * M respectively. (6) C separates in U the regions defined by
s
Hmin = Ho + H1, Hmin = Ho - H I . Properties (5) in Lemma 2.2 yields to the possibility of defining the smooth function v : U --+ R by
v := Hool/Hllo on S
(10)
and extend it constant first on the integral lines I?1 and then on the ones of 201. In this way we get the Hamiltonian of singular extremal of the first kind by defining
Hs
:= Ho + v H 1 .
Indeed fis is tangent to S and any singular extremal of the first kind of our problem is an integral curve of 2,s contained in S : as a consequence A is C" and the same is u^ = v o A.
h
h
Remark 2.1. A particular case occurs when the function v depends only on nC E M , indeed in this case the Hamiltonian H s is the lift of a vector field fs = .fo vfl and any singular state extremal associated to the dynamics (1) is an integral line of fs. The vector field fs, after a reparametrization of the control, can be thought as the drift vector field f o and we can also think that the equality H O O = ~ ~0 Sholds. Remark that the above described case can be considered generic in dimensions 2 and 3, as it can be easily seen in any chart.
+
307
3. Hamiltonian approach to strong local optimality
The Hamiltonian approach to prove sufficient conditions to strong optimality consists in constructing a field of non-intersecting state almost-extremals covering a neighborhood of the given trajectory. We call almost extremal a solution of the Hamiltonian system associated to a Hamiltonian Ht (possibly time-dependent) with the following properties A
+
-
Ht 5 Hmin, H t o X = f i t o X , A = H t o A . (11) The field of state almost-extremals is obtained by projecting on the state manifold the flow associated to Ht emanating from a suitable horizontal Lagrangian sub-manifold A. If the map id x T'FI : [O,?] x A
4
[O,Q x M ,
( t , l )H (t,T'Ht(e))
is a diffeomorphism, then we can use symplectic arguments to compare the costs by lifting admissible trajectories to the cotangent bundle, independently on the associated controls. Remark that the third property in (11) is a consequence of the first two if Ht is differentiable along the adjoint covector and that the minimizing Hamiltonian Hmin satisfies the properties if it is sufficiently smooth. When the trajectory is singular we cannot satisfy (11)with Hmin unless G = 0 and in any case Hman does not guarantees that we can find the Lagrangian manifold A with the required properties, since generically the associated Hamiltonian system does not have the continuity property with respect to the initial conditions near the adjoint covector. Our idea for overcoming the problem is to use a Hamiltonian having the properties (11) and such that the associated vector field is tangent to C, so that Hmin = Ho along the chosen almost extremals, as explained in the following subsection.
3.1. The Hamiltonian
x
For the singular trajectories of the first kind, we pursue the above described strategy by adding to HO a negative Hamiltonian x.This possibility is given by the following Lemma 3.1. Remark that the result has been proved in a slight different form in Ref. 18.
Lemma 3.1. By possibly restricting U ,it is possible to define a smooth function p 1 U c T * M -+EX, with the following properties
x = $Hi, + 2 is tangent to C.
(1) The Hamiltonian 1?0
is such that the Hamiltonian vector field
308
P(4
= -1 for all C E S, hence, without loss of generality, we can suppose p < 0 (3) p can be chosen so that
(2)
(Eio
+ 217)~) = (exp(-6(C))Eil),Eio
0
(exps(e)Eil)(e) ,
vc E c
where d(C) is defined by Hol(expd(C)gl(C)) = 0
Proof.
and G ( l ) = 0 , VC E S.
f i o + 2 is tangent to C if and only if on C we have -.
(dH1, Ho
+ 2) = Hol(1-
1 -Hol(dp, 2
21)+ pHiio) = 0
and hence 1 1 - -Hoi(dp, 2
gi)+ pHiio = 0.
(12)
Since g1 is tangent to C, then we need to define p only on C. Any smooth extension of such a p has property (1). The previous equation reads p = l/Hlol on S, hence we can solve (12) by the characteristic method, obtaining for C E S, and It1 sufficiently small
This proves (1) and (2). From the above expression for p, taking into account that: + &Hlo o e x p t f i l = Hllo oexptH1 and ~ H oexptH1 o = Hlo oexptH1 we obtain in Cc = e x p t g l ( C s ) E C
-
x(&)
= Ho(Cs) - Ho(Cc).
-.
(13)
For C E C sufficiently close to S, it is possible to define d(C) by Hol(expd(C)fil(C))= 0 and d(C) = 0 , VC E S. With this notation, if we choose to extend lines of l ? o ~ , equation (13) reads
x constant along the integral
dx = dHo o (expdfil), and (3) is proved.
-
dHo 0
309
Remark 3.1. For each smooth function v : R x T * M -+ R,
(t,!) t-+ v ( t , e ) = vt(e) consider the Hamiltonian H[ := HO vtHl x. We have H[ 5 HO = Hmin on C and the associated vector field l?[ is tangent to C. Moreover if v ( t , x ( t ) )= Z ( t ) ,then d H [ ( x ( t ) )= dHt(X(t))therefore, in this case, is a
+
-
trajectory of
+
x
4
fir. In particular +
*
Ht := H t are tangent to C and
+2
+
and
gs := Hs + 2
x is a trajectory of both of them.
Lemma 3.2. If the function v is such that (14) v ( t ,x ( t ) )= Z ( t ) and (dvJ?l)(x(t)) = 0 , then 2 1 and do are invariant with respect to the flow of 'H," along the adjoint covector, i.e.
I?l(i(t>) = ('H:)*G~(X(O)) and & ( i ( t > >= ('H:>*GO(X(O)). Proof. at ( ( 7 f f ) ; 9 ? 1
= (7-lFl,v);1 (do1
0
'H,") (X(0)) = ('H,")*1
[&, 1'13 ( i ( t ) )=
+ [vt& + HlGt + H&P'/2 + H01piio1, I l l ] ) ( i ( t ) )=
= ('H,")?
(1 - P ( d H 0 1 , G l ) )fiOl(X(t))= 0,
the last equality follows by (12). In an analogous way we get at
(('H,");ll?o
0
'H,")
(m)('H,")2 ( W )+ =
p (dHOl,I?,)) GlO(X(t)),
which is identically zero by (lo), since p ( x ( t ) ) = l / H I o l ( x ( t ) ) .
0
4. Sufficient conditions from Hamiltonian viewpoint
To apply the Hamiltonian approach to prove sufficient conditions to strong optimality of singular extremals as explained in the previous Section 3, first we extend in a suitable way the cost functions, namely the smooth functions P O C O and p o c ~are defined on NO and NT respectively, but they can be extended t o the whole manifold M in such a way that the transversality conditions (4) and (5) hold on the whole tangent space. Therefore we denote by a , p : M -+ R functions such that
a = -POCO on NO , p = p o c ~ on NT i ( 0 ) = -da(&) on T ? , M , i ( T )= d p ( 2 ~ )on T z T M .
(15) (16)
310
In the normal case (PO = 1) a , p are cost functions equivalent to the original ones while in the abnormal case (PO = 0) they are extensions of the zero function. When po = 0 all the costs disappear, we study a problem with a zero cost and indeed we are studying the constraints. Proving that [ is a strict strong minimizer with po = 0 implies that it is isolated among the admissible trajectories. The sufficient conditions will be derived studying the following optimal control problem
+
Minimize J(E) := a([(O)) P(E(T))
(17)
subject to (l),(2) and (3). For any a with the properties (15), (16) and
LflL.fl4ZO) > 0 7
(18)
the set d = {z E 0 : L f l a ( z )= 0) is a hyper-surface of 0 and, by possibly restricting 0 , for 5 sufficiently small, we can state
o = {expsfl(z) : z E 6 , Is1 < E ) . We define & on 0 by setting &(expsfl(z)) = a ( z ) ,z E 6 , Is/ < 5, & is constant along the integral lines of manifold
fl,
therefore the Lagrangian sub-
A," := {l = - d & ( z ) , z E 0 ) is contained in C, so that can be also obtained as
l?1
= d&* fl is tangent to
A,* = {exp sl?,(l) : l
= -da(z), z E
A;. Remark that A;
6 , Is1 < C}.
Choose a function v with the properties (14) and consider the Hamiltonian H r as defined in Remark 3.1, with associated vector field I@ and flow 'Hf. Standard symplectic arguments show that the one-form w defined by
w = ( i d x 'H")*(s - HL d t ) is exact on [O,T]x A;, see Ref. 6. Setting w = d 0 we can choose 0 so that 0 ( O , l ) = -& o n ( l ) , l E A;, moreover whenever n o 'Hf is a local diffeomorphism from A," onto M we get
d (0, o (n o 7-t,")-1) = ('H"* s) (n o 'H,");'
= 'H," o ( n o'H,")-'.
Theorem 4.1. Suppose that x ( t ) E S f o r all t E [O,T].If there are functions a , p, v with the properties (15), (16), (18), (14) and such that:
311 -+ Tii,)M is an isomorphism for all t E [O,T], (1) .rr*'H;* : T-NO)Aff 0 (2) the quadratac map
6a: H u (d,&Gz,'FI&*(.rr'FI&)L16z) is positive definite on Tg(T,N T ,
s^
then is a strict strong local minimizer for the problem (17), therefore in the normal case (PO= 1) is a strict strong local minimizer for the original problem, in the abnormal case (PO = 0) E is locally isolated between the solutions of (1) and (2).
<
h
h
Proof. We consider the map i d x 'FI" : [O,T]x
(t,[)
A0
-+
[O,T]x C
(tl%(C)).
If assumption (1) is satisfied, then .rro'FIr:Ag-+M h
is a local diffeomorphism covering a neighborhood of ( ( t ) ,therefore, by possibly restricting 0 , there is a neighborhood V of the range of (id x in [O,T]x M such that the map
F)
id x .rroO' : A; x [O,T]+ V is a diffeomorphism. Consider an admissible couple ( u ,E ) of the problem such that
{ ( t ,E(t)>: t E [O, T I ) c v and denote A(t> = (.rr O P ( t ) = ( t ,A(t>),
'FI;>-W)) G(t)= ( t ,RO)) .
Since A(t) c C, then 'FI;(A(t))c C, therefore using that H r H I = 0 along 'FIr(A(t)), we compute
I Ho
and
312
Hence
a(c(o))- 4 2 0 ) + P ( W ) ) - P ( , m )
To))
L P(E(T))- O(T, Setting
'p :=
- (P(S^(T)) - Q(T,
P - Or o ( T ' H G ) - ~ , we have d'p = dP - 'HF o (x'HG)-'
d ' p ( F ( T ) )= dP(i-(T)) -
D2&T))(*, .)
m)
=0
(dP*(.)7 'HG* O (T'H$L'C))
=u
1
therefore $(T)is a local strict minimizer for $ on NT. By possibly restricting 0 we have a 6 by (18), so that (2) implies
J ( u , <)
J(Gl-)2 0.
To prove that the minimum is strict, remark that J(u,<)- J ( B , c ) = 0 w = 0. In particular the last implies L f l a ( c ( 0 ) )= 0, [ ( T )= g(T), inequality implies
sp
Hol
0
'H,V(X(t))
0
so that 'H," o X ( t ) E S . From Lemma 3.2 we obtain (x'H,")*I?1(X(t)) = and hence 'H,"*(T'H,");'fl(c(t)) = 0 ' H y ( X ( t ) ) for all t E [O,T]. Finally we compute fl(<(t))
=
(Go + u(t)G1)0 'H,V(X(t)).
In other words H ' ; o X is a singular extremal of the first kind, hence it is the integral curve of 2 s passing through x ( T ) so that = 0
c c. A
4.1. Remarks on Theorem 4.1 If f l ( ? o ) belongs to Te,No, then equation (18) is an assumption to be fulfilled by the function a. If fl(20) does not belong to T?,No, then a with the properties (15), (16) can be chosen so that (18) is fulfilled and 6 = NOn 0 , hence CY = 6 on NO.In any case a and 6 are determined on NO by the data, while we are free to chose the second order terms on any complement of NO. Remark that we can choose both
Hr
= HS = Hs
+x
A
and
H r = Ht
+x
313
in Theorem 4.1. The main theorem will be proved by using Theorem 4.1 with the Hamiltonian
Ht := @t
+X.
(19)
When the initial point of the original problem is free, we have the following sufficient conditions in terms of the data.
Corollary 4.1. Suppose that NO= M and i ( t ) E S for all t E [O,T].Let Ht be the Hamiltonian defined in (19) and suppose (1) LfiLfiCo(20) > 0
(2) .rr*?&* : T X ( ~ ) A-+Z Tr(,)M is an isomorphism for all t E [O,T]. (5') The quadratic form
6x H d
(dCT*bx,'FI$,TX$),~~Z)
is positive definite on TF(,NT then is a strict strong local minimizer for the original problem. Recall that in this case the extremal is normal, hence Q = Q. Remark 4.1. We have chosen to apply Hamiltonian methods starting from an initial Lagrangian sub-manifold and considering the forward flow of I?:. We could have chosen to start at any time, in particular some authors prefer to consider a final Lagrangian sub-manifold and integrate the Hamiltonian vector field backwards. This last strategy is obviously preferable when the final point is free. It is not difficult to reformulate Theorem 4.1 and Corollary 4.1 starting from the final time.
5. The extended second variation In this Section we use the second variation defined in Ref. 3, hence we need to define the pull back system. In the neighborhood 0 of 20 set gt = ?G1
fl
o ?t
p= p i& 0
;0
TM : 0 + R. --+
Choosing Q and ,8 with the properties (15), (16), we have
+ ,@(20)= - i ( O ) + ~ ( T ) ? T=, 0 , D 2 ( a + ,@(20)is a well defined bilinear d(m
so that y" = form on TzOM. The second variation, as defined in Ref. 3, is a linear quadratic form on
314
T2,No x L 2 ( [ 0 , T ] , Rrealized ) as a linear-quadratic (LQ) problem on the vector space T2,M. Set
y” = D2(a + @)(20) N; = S,-JTsTNT 6 e = ( 6 2 , ~ ET$,NO ) x L2([0,T],R), the second order approximation of the cost is given by 1 J”[6e]2= -y”[62I2 2
where
r]
+
I’
v(t)
L,, F(2o)dt
satisfies
Remark 5.1. The second variation is independent on the choice of the a,P with the properties (15) and (16), see Ref. 3.
functions
Introducing
and integrating by parts we obtain rT
and
rT
315
Notice that
9t
=
gE1[fO7 fll 0 gt
[St,gtI = %"[fO,flI,fll 0 gt [gt,gt]* P(2i-O)= (x(O), gE'[[fO,f l ] , f l ] 0 gt(2i.O))= Hllo 0 x(t) To simplify the notation set
R(t) = L[bt,gtjP(20) and Q(t)= L(.)Lg,P(2.o).
(21)
Simple computations show that rO[Sz,w0l2 can be equivalently written as 1 1 -yll[Sa: wof 1 ( 2 0 ) ] 2 - -w2 2 0 L2fl a ( 2 0 ) - wo L6,LflcX ( 2 0 ) . (22) 2 Since the map
+
v E L 2 ( [ 0 , T ] , RH ) (WO,W) E R x L 2 ( [ 0 , T ] , R )
(23)
is continuous and has a dense image, we extend J" to T?,No x R x L2([0,TI,R) by continuity and we call this extension the extended second variation. For all Se = (62, wo,w)we set
where c(t)is a solution of
t ( t )= w(t)9t(Po)
+
C(0) =-hwof1(20)
(25)
C(T)E N;
7
J Z is a coordinate free version of a classical one obtained by using the so called Goh transformation, Ref. 11, see also Ref. 2 where the same second variation is given in Hamiltonian form for a problem with fixed end-points. A
Theorem 5.1 (Necessary Conditions). If (Ti, <) is a singular optimizer f o r the problem (1 7), then (1) The generalized Legendre condition ( G L C ) holds true
R(t)= HllO
(m) (W, =
[fl, [ f l , f O l 1 ( 5 ^ ( t ) )
2 0.
(2) If the adjoint covector is unique u p t o a positive constant, then
JN 2 0 so that JZ 2 0 by the properties of the embedding ( 2 3 ) .
316
Proof. The proof from a geometric viewpoint can be found in Ref. 2: Corollary 20.18 pag.318 proves (l),while Proposition 21.5 proves (2). Remark that in Ref. 2 the sign of the adjoint covector is opposite to the one in this paper, hence all the inequalities are reversed. However both conditions are classical results: see for example the survey paper [9, eq. (47), p. 1411 for (l),and [13, Theorem 9.1, p. 2851 for (2). 0 Now we analyze the extended second variation more deeply. The case fl(20)$! T2,No. In this case Jg is realized as a LQ control problem on Tj, M with initial constraint given by N{ = T2,No @ Rfl(20). T h e case fl(20) E T2,No and L ; 1 a ( 2 ~=) 0. In this case, which includes the case fl(li.0) = 0, we have two possibilities: either L s , L f la ( f o ) = 0 for all 6e E T2,No x R x L 2 ( [ 0 , T ] , Rsuch ) that C(T) E N; or there is 6e such that L s z L f l a ( 2 0 )# 0 and ((7') E N;. In the first case, by (22), 6e = (fl, -1, w = 0) belongs to the kernel of Jg, so that Jg cannot be coercive and its index and nullity can be checked on the quadratic form obtained by setting wo = 0. In the second one the index of Jg is infinite and 2) cannot be a minimizer, see for example Ref. 2. T h e case fl(20)E T?,No and L ? 1 a ( 2 ~#) 0. In this case
(r,
is a linear subspace and we get
Substituting Sx = 6y+ wlfl(20) in (22), with 6y E Ng and performing the substitution (WO w1,wl) H ( w o , v ~ )we obtain
+
+ ~ o [ ~ +Y , ~ o I ~
J ~ [ ~ Y , v ~ , v=ov:L;la(fo) ,w]~ +
iLT
w(tI2R(t)+ 2w(t)(Q(t),C ( t ) ) d t
(27)
where
C(t)= w(t)jt@o) C(0) = SY + vofl(2.o)
7
C(T)E N ;
*
We underline that Jg is a direct sum of an LQ form on R and of an LQ form realized as a control problem on T?,M .
317
Remark 5.2. The coercivity of J g implies that SGLC must be satisfied. Moreover the sufficient condition implies fl(20)# 0, and if fl(20) E T?,No, then it splits in two conditions: Lfl L f l a ( 2 0 )> 0 and J g coercive on Ng x R x L 2 ( [ 0 , T ] , R When ). fl(20)# T?,No, without loss of generality, we can assume L ; 1 a ( 2 ~>) 0 and L ~ z L f l a ( 2=~0)for all 6x E T',No, see Remark 5.1. 5.1. Reduction to a non-singular problem In this Subsection we suppose that either f l ( 2 0 ) # T?,No, or that L ; 1 a ( 2 ~#) 0 and we show that index and nullity of the extended second variation can be calculated via the standard second variation associated to a suitable non necessarily singular problem. Before addressing the proof of the above statement we unify the notations denoting by J G : N[ x L 2 ( [ 0 , T ] , R-+ ) R the following quadratic forms realized as an LQ-problem on T2,M. (1) If f l ( 2 o )
# T?,No, then we set N[ = T?,No @ Rfi ( 2 0 ), Nc = No J ; [ ~ z vof1(20), w]' = J ~ [ ~ Z , w12 VO,
+
#O
= {expsfl(y) : Is1 < 3, y E
N O } and &(expsfl(y)) = a ( y ) .
(2) If fl(20) E T?,No and L ; 1 a ( 2 ~#) 0, then we set
N[ = Ng
@ Rfl(2.0)
and
N c = {y E NO: L f l a ( y )= 0)
J & [ ~ Y + v o ~ ~ ( ~= o )J ,~w[ ]~~Y , v o , w ] ~ #O
= {expsfl(y) : Is1 < 3, y E
N c } and &(expsfl(y)) = a ( y ) .
With the above notations consider the non-singular problem minimize & ( E ( O ) )
+ P(J(T)) subject to
i(t)= Z(t(t))+ w ( t ) [ f o fll(C(t)) , + + W 2 [ f l[,f l , f O ] l ( t ( t ) ) C(0) E f i o ,
C(T)E NT.
(28) (29)
(30)
Lemma 5.1. J G is the second variation of the above problem (28) subject to (29) and (30), relative to the reference couple (gw^(t)= 0 ) with associated adjoint covector A. h
Proof. It is easy to see that the couple (x,w^(t) = 0) is an extremal. Since ( d a ( b o ) ,fl(20))= 0 , then d & ( f o ) = d a ( & ) , hence, denoting ;U = 6 P,
+
h
318
dY(20) = 0. Set 7’’ = D2=j(20),the second variation as defined in Ref. 3 is given by 1 J”[6eI2 = -7”[6xl2 2
where
+
I’
+
~ ( t ) ~ R (2tw)( t ) Q(t)C(t) dt
(31)
Remark 5.3. Remark that SGLC is the strong Legendre condition for the above non-singular problem. Moreover we can always suppose that L ; 1 a ( 2 ~>) 0 and that NC = {y E fro : L f , a ( y ) = 0). Remark 5.4. If the initial point is free and the second variation is coercive, then a and ti are the same defined in Section 4. Remark 5.5. The extended second variation is a quadratic form on a Hilbert space, hence its index and nullity can be studied using the large literature on the subject. In particular if SGLC is satisfied, then we deal with a non singular LQ problem on a vector space so that it is possible to calculate the index and the nullity of J& through the idea of conjugate and semi-conjugate points, or by Riccati equation, see Ref. 20 and the references therein. In particular index and nullity can be calculated via the properties of the linear Hamiltonian flow 7-t: associated to the quadratic Hamiltonian H,“ : T;oM x Tj, M + R defined by -1 2 H,” : (p,dz) H -( ( P , B t ( 2 0 ) ) + LdZLBtP(W) (32) 2R(t) and starting from the Lagrangian subspace
Lb’ = {(-;u(62,.), 62) : 6J: E Nl} @ N:I.
(33)
319
6. Coercivity of J&
In this section we study the coercivity of J& from a Hamiltonian point of view, in particular we apply the results of Ref. 3 to problem (28) and we transfer the results to the Hamiltonian Ht defined in (19) by linking the associated Hamiltonian flows. The coercivity of J& implies SGLC, then we can define the "minimizing Hamiltonian" associated to the non-singular problem (28) and defined in a neighborhood of S, given by
and Ht coincide on S up to the third order, but the Hamiltonian vector -+ field Htis not tangent to C out from S, nevertheless the associated flows %t and 'Ht have the same tangent map at l o , i.e. we can prove the following lemma. fit
h
Lemma 6.1. Let %t and ' H t be respectively the flow of their tangent maps %t* and 'Ht* coincide on Tx(,)T*M .
Ht
and H t , then
Proof. The map %,'o'HFlt is the Hamiltonian flow associated to the Hamiltonian
Since d @ t ( i ( O ) )= 0, then it is easy to verify (see Ref. 14) that
(6;'
0
'Htlt), = '7?.t*'h!t* : Ti;(o)T*M-+ Tx(,)T*M
is a linear Hamiltonian flow associated to D z @ t ( i ( 0 ) )which , is easily seen 0 to be equal to zero.
The links between the quadratic Hamiltonian H," associated t o J& and defined in (32) and the Hamiltonian Ht can be described via the symplectic isomorphism described in Ref. 3
z : T;*,M x T$*, M (6p,6 2 ) H 6 p
-+
T X ( ~M ~T*
+ d(P)*62
.
Lemma 6.2. Let Gt be the pull-back Hamiltonian flow given by % ', = z-lGt*z holds through. then the equality
(35) o 'Ht,
320
Proof. Consider the pull-back Hamiltonian flow Gt = % ', o f i t , Corollary 3, pag. 702 in Ref. 3 reads in our case = z-'Gt,z, hence we get the statement by Lemma 6.1. 0
'Hr
Using the symplectic isomorphism defined in (35) and the results in Ref. 3 we state the conditions for J& to be coercive in terms of properties of the flow 'Ht, starting from the subspace LO of TzoT*M defined by
{
LO= zL; = -d&,dx
: 6 x E T * o f i ~@ } TAN.
Using the results in Ref. 20 and the techniques of Ref. 3 we can prove the following result.
Lemma 6.3. J& is coercive if and only if the following properties hold true
If n,'Ht,6e = 0 , 6e E L o , then n,'HS,6C = 0 and 'Hs,61 = %,,6e for all s E [O,t]. (2) 0 (d,f3+n,'HT,M, 'HFIT,6e)> 0 for all 6e E Lo such that 7r,'HT,6e is a nonzero element of TzTN T . (1)
Proof. Denote by V the space of the first order variations with fixed final point, i.e.
V = {6e = ( 6 x , w ) E N l x L 2 ( [ 0 , T ] , R:)[ ( T ) = 0). J& is coercive if and only if it is coercive on V and on the space V L orthogonal to V with respect to J&. Recall that V L is finite dimensional. Applying Corollary 5 in Ref. 3 to a problem with fixed final point we get that J G l V is coercive if and only if n,.Ft;e/l
= 0,
E L;
+
?-t;ell
=el/,
v s E [o,t].
(36)
Noting that %t+61 is vertical if and only if SC is and using Lemma 6.1, we get that J L l V is coercive if and only if the kernel of 7 r , f i t , is given by
{ s e E L~ : n,fisis*se = 0 , fi,,se
=
%,,se,
vs E
[o,ti),
hence J L l V is coercive if and only if (1) holds true. To prove (2) consider that 6e = ( S X , ~ belongs ) to V' there are po E N[' and p~ E T i o M such that
if and only if
1 1 J & ( d e , f e ) = - ( p o , & ~ ) + ~ ( p ~ , c ( T ) V) ,f e = (&x,W)E T 5 , M x L 2 ( [ 0 , T ] , R ) . 2
321
Denote l” = (Sp, 65) = (-T(6xl .) +pol 6x) E L:, (‘(t) the solution of (25) with initial condition C(0) = 6x and p ( t ) the solution of
i4t) = -w(t)Q(t)
7
~ ( 0=) 6 ~ .
Integrating by parts we obtain that 6e E V* if and only if for all 6e
; ( P T r ( T ) ) = -;(P(T)C(T))+
+;
s,’w(t) (w(t)R(t)+ &(t)C(t)+ d t ) j t ( f o ) ) d t
I
that is if and only if
so that ( p ( t ) ,(‘(t)) = Fly[‘’. Reasoning as in the proof of Lemmas 4 and 5 in Ref. 3 we can prove that 6e E VL if and only if there is P’ E L: such that 7r*i?” = 6 2 ,
and
7r,?i$l“
E N; ,
moreover in this case
On the other hand if (‘(T)= 7r*Fl$l” = 0, then by (36) we get (‘(t) = 0, so that 6x = 0 and w ( t ) g t ( & ) = 0, a.e. t E [O,T],hence w is the zero element of L2([0,TI,R), since gt cannot be zero in a subinterval of [0,TI. The proof of (2) follows easily applying the symplectic isomorphism, the invariance of cr with respect to Gt* and Lemma 6.1. For the free initial point case, the above conditions (1) and (2) coincide with conditions of Theorem 4.1, namely we get: Corollary 6.1. If NO= M , then JG is coercive if and only if (1) 7r*Flt* : Txco,Rg -+ Tg(,]M is an isomorphism for all t E [0,TI. (2) The quadratic m a p
6x H (T (dp*6x,XT‘*(T*h!T*)-16x) is positive definite on T F [NT ~,
Proof. (1) follows easily noting that LO = Txco,Rg. To obtain(,?) it is sufficient to substitute 6 x = 7r,‘FIT,6e. 0
322
6.1. Proof of the Main Theorem In this section we assume that the extended second variation is coercive, therefore we assume SGLC, J& coercive and L f l L f l a ( 2 0 )> 0, see Remark 5.2. We show that under the above stated assumptions we can find a function as which coincides with a on NO and satisfies the assumptions of Theorem 4.1 with the Hamiltonian Ht defined in equation (19). If NO= M we have nothing t o do, otherwise since f l ( 2 o ) # 0, then we can choose coordinates ( X I , . ' . , x,) a t ?O such that f l
a
= - and
8x1
NOis defined by xr+1 = . . . = x ,
= 0.
(37)
Since the second variation does not depend on a , p with the properties (15), (16), then without loss of generality we can assume
.
t i ( 2 ) = a(O,x2,.
*
,XT,
0,. .. , O )
+
c n
(38)
8@(20)Xi.
i=r+l
In these coordinates choose the non-negative quadratic form on .
T2, M
n
and extend it to T2,M x L2([0,TI,R) by r[6eI2 = r[6x:I2. On the other hand we consider J& as defined on the whole space T2,Mx L2([0,TI,R), and, since J& is coercive on the kernel of r, we can apply [12, Theorem 13.21 and conclude that there is a positive constant s such that
J&
+ sr
is coercive on W
=
(6e E T?,M x L2([0,TI, EX) : C(T)E N;}.
Defining
we get (1) L;ps(20) = L;p(Oo) (2) das(20) = -X(O) (3) tis = ti on No (4) ti^ =
>0
A
+ p)((ao) y + sr.
Corollary 6.1 proves that the assumptions of Theorem 4.1 are satisfied and hence the main theorem is proved.
323
6 . 2 . A particular case
A particular case occurs when the Hamiltonian of singular extremals H s as defined in Subsection 2.3 is the lifting of a vector field fs, see Remark 2.1. In this case we can give conditions for the extended second variation to be coercive in terms of the vector fields fs and f l . In particular the following Lemma gives necessary and sufficient conditions for the first property of Lemma 6.3 to be verified and describes r * ? i ~ * L oso, that the second property of the same lemma can be verified in any chart at E(T). h
Lemma 6.4. Let the Hamiltonian HS be the lift of the vector field fs. (1) If [fs,fll(2i.o)E
N:, then JhlVis coercive if and only if for all t
E (0, T]
fl(r^(t)) @ (exp t f s ) * V where
Moreover r * ? i ~ + L ois the linear space spanned by f l ( g ( T ) ) and (exp T f s ) * V . (2) [fs, f l ] ( f o ) @ N[, then JLlv is coercive if and only iffor all t E (0, T]
@ (exp tfs)*N;.
f1(5^(t))
Moreover T*'HT*Lo is the linear space spanned b y f l ( f ( T ) ) and (exp T f s ) * N [ . Proof. Without loss of generality we can assume that fs = fo, see Remark + 2.1, so that dt = HO x and Htls = I?ols is tangent to S. Consider the space V = {w E N : : L , L f o l & ( 2 ~=) 0) and observe that, N; = R fl(20)@ V , therefore we can choose coordinates ( X I ,. . . , z), at 20, with the properties (37), (38) and such that V is spanned by & , i = 2, ..,r. In these coordinates I ? l ( X ( O ) ) = d&&, d&ai E S , i = 2, ..,T , and
+
-4
LO= span(H1, d&&, dzj : i = 1,.., r , j = r + 1, ..,n}. 4
(1) If fol(20) E N:, then dzj E S, j and
=
r
+ 1,..,n, so that 7r*'Ht*dxj= 0
+
T*?it,Lo = Rfl(S^(t)) (exp t f o ) * v . Therefore Lemma 6.3 proves the statements for the first case since if 6e E span{d&& : i = 1, ..,r} belongs to ker7r,?it, and 7r,6C = 0, then
se = 0.
324
(2) If fol(20) $2 N:, then we can suppose without loss of generality, foi(2.0) = &+I SO that dxj E S for all j = T + 2,..,n and dxr+l = 6e yd&&, with 6e E S and y = l/H101(i(O)).In this case r*Xt*dxj = 0, for all j = T 2, ..,n, r*Xt+dx,+l = -y(exp tfo)*& and
+
+
Therefore Lemma 6.3 proves the statements for the second case since if 6 t E span{d&&, dx,+l : i = 1, ..,r } belongs to kern,Xt, and r,6t = 0, then be = 0. 0
7. Final remarks The paper proves second order sufficient optimality conditions for the strong local optimality of a singular trajectory [, that is the optimality is with respect to the admissible trajectories belonging to a neighborhood of F i n Co( [0,TI, M ) , independently on the values of the control. Remark that as a consequence we obtain that the trajectory is strong locally optimal even if the controls are unbounded. The proof is obtained in the framework of Hamiltonian approach to optimal control and points out the connections between the coordinate-free extended second variation and the modified minimizing Hamiltonian. The ideas of the paper have been applied t o prove strong local minimum time optimality of a singular trajectory,lg and of a bang-singular trajectory,1617 Strong local minimum time optimality may be of two kinds: either local with respect to both state and time (i.e. with respect to a neighborhood of the graph of the reference trajectory in R x M ) , or local only with respect to the state (i.e. with respect to a neighborhood of the range of the reference trajectory in M ) . The optimality considered in Refs. 16,17,19 is local with respect to both state and time, moreover the conditions for the bang-singular trajectories are not completely satisfactory. The work is in progress both t o extend the conditions t o the state local optimality and to improve the conditions for a reference extremal which contains bang arcs and singular arcs. A further research direction is to consider singular (or partially singular) trajectories of multi-input systems. The geometric picture is clear when only one control is singular and the others are bang: the result will appear elsewhere. It is the opinion of the author that the theory can be extended t o totally singular control if the controlled vector fields commute, while the general case seems to be much more difficult.
325
Finally it is t h e opinion of the author that the theory can give a deeper insight t o understand the possible sufficient conditions for the optimality of singular extremals of the second kind.
References 1. A. A. Agrachev and R. V. Gamkrelidze, Symplectic methods for optimization and control, in Geometry of Feedback and Optimal Control, eds. B. Jacubczyk and W. Respondek, Monogr. Textbooks Pure Appl. Math. Vol. 207 (Dekker, New York, 1998), pp. 19-77. 2. A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, Vol. 87. Control Theory and Optimization, 11. (Springer-Verlag, Berlin, 2004). 3. A. A. Agrachev, G. Stefani, and P. Zezza, Internat. J . Control 71,689 (1998). 4. A. A. Agrachev, G. Stefani, and P. Zezza, Proceedings of the Steklov Mathematical Institute 220, 4 (1998). 5. A. A. Agrachev, G. Stefani, and P. Zezza, SIAM J. Control Optimization 41, 991 (2002). 6. V. I. Arnold, Mathematical methods of classical mechanics. lkanslated from the Russian by K. Vogtmann and A. Weinstein. Graduate Texts in Mathematics, Vol. 60, second edition (Springer-Verlag, New York, 1989). 7. A.V. Dmitruk, English translation: Soviet Math. Dokl. 18,(1977). 8. A.V. Dmitruk, Dokl. Akad. Nauk SSSR 272, 285 (1983) [in Russian]. English translation: Soviet Math. Dokl. 28, 275 (1983). 9. V. Gabasov, F.M. Kirillova, SIAM J . Control Optimization 10, 127 (1972). 10. M. Giaquinta and S. Hildebrandt, Calculus of variations, I and 11, Grundl. der Mathemat. Wissensch. 310, 311 (Springer, Berlin, 1996). 11. B.S. Goh, SIAM J . Control Optimization 4, 309 (1966). 12. M. R. Hestenes, Pac. J. Math. 1,525 (1951). 13. M. R. Hestenes, Calculus of variations and optimal control theory (John Wiley & Sons, New York, 1966). 14. J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, A basic exposition of classical mechanical systems. Texts in Applied Mathematics, Vol. 17, (Springer-Verlag, New York, 1994). 15. Z. PBles and V. Zeidan, SIAM J. Control Optimization 32, 1476 (1994). 16. L. Poggiolini and G. Stefani, Minimum time optimality for a bang-singular arc: second order sufficient conditions, in Proc. 44th IEEE Conf. Decision and Control, and the European Control Conference ECC 2005 (Seville, Spain, 2005). 17. L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: second order sufficient conditions, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, eds. F. Bull0 and K. F'ujimoto, Lecture Notes in Control and Inform. Sci., Vol. 366 (Springer, Berlin, 2006)) pp. 281-291.
326
18. G. Stefani, On sufficient optimality conditions for a singular extremal, in Proc. 42nd IEEE Conf. Decision and Control (Maui, Hawaii, 2003): 27462749. 19. G. Stefani, Minimum-time optimality of a singular arc: second order sufficient conditions, in Proc. 43rd IEEE Conf. Decision and Control (Atlantis, Paradise Island, Bahamas, 2004): 450-454. 20. G. Stefani and P. Zezza, SIAM J. Control Optim. 35,876 (1997). 21. M. I. Zelikin and V. F. Borizov, Theory of Chattering Control, With applications to astronautics, robotics, economics, and engineering. Systems & Control: Foundations & Applications (Birkhauser Boston Inc., Boston, MA, 1994).
327
HIGH-ORDER POINT VARIATIONS AND GENERALIZED DIFFERENTIALS H. J. SUSSMANN’ Department of Mathematics Rutgers, the State University of New Jersey Hill Center-Busch Campus 110 Frelinghuysen Road Piscataway, NJ 0885.4-8019, USA E-mail: [email protected] http://www. math. rutgers. edu In a series of nonsmooth versions of the Pontryagin Maximum Principle, we used generalized differentials of set-valued maps, flows, and abstract variations. Bianchini and Stefani have introduced a notion of possibly high-order variational vector that has the summability property. We consider a slightly more general class of variational vectors than that defined by Bianchini and Stefani, and prove that the convex combinations of these vectors arise as “differentials” of variations that are differentiable in the sense of one of our generalized differentiation theories, namely, that of “approximate generalized differential quotients” (AGDQs).
1. Introduction In a series of papers (cf. Refs. 5-7,9,10), we showed how to derive general, nonsmooth versions of the Pontryagin Maximum Principle using generalized differentials of set-valued maps, flows, and abstract point variations. The use of general variations rather than the nedle variations used to prove the ordinary maximum principle makes it possible to obtain high-order versions of the maximum principle. The main technical difficulty with these general abstract variations is that they need not have the summability property, which is absolutely essential in order to derive the necessary conditions for optimality. R. M. Bianchini and G. Stefani (cf. Refs. 1-4) proposed a concept of high-order point variation that has good summability properties. The goal *Supported in part by NSF Grant DMS-05-09930
328
of this note is to relate this concept to a theory of generalized differentials, by describing a slightly more general version of the Bianchini-Stefani variations, and showing that they are differentiable in the precise sense of the theory of “Approximate Generalized Differential Quotients” (AGDQs). This makes it possible to use these variations in order to get additional necessary conditions for an optimum in situations such as the very general one described in Ref. 9, where the differentials involved are generalized differential quotients, and a fortiori AGDQs. 1.1. Preliminary remarks on notation We will use the notations and abbreviations of Ref. 9. In particular, “FDRLS” stands for “finite-dimensional real linear space,” “FDNRLS” for “normed FDRLS,” and “SVM” for “set-valued map.” If f is a SVM, then S o ( f ) , Ta(f), Gr(f), Do(f), Im(f) are, respectively, the source, target, graph, domain and image of a SVM f. (We recall that a SVM is a triple ( A ,B , G ) such that A, B are sets and G is a subset of A x B , in which case we say that f is a SVM from A to B , and define G - ’ e f { ( z ,y) : (y, z) E G } ,
f-’ef=
( B , A , G - ’ ) , S o ( f ) e f A , T a ( f ) e f B = So(f-’), Gr(f) = G , fb)= {Y : (Z,Y)E G r ( f ) l , Do(f) = {. : f(.) # 01, I m ( f ) = D 4 f - W We use S V M ( A , B ) to denote the set of all set-valued maps from A to B . The notation “f : A B” means “f is a set-valued map from A to B.” If f E S V M ( A , B ) then f is (i) single-valued if the set f ( z ) consists of a single member for every z E Do(f), (ii) one-to-one if f-’ is single-valued, (iii) surjective if Im(f) = Ta(f), (iv) everywhere defined if Do(f) = So(!), i.e., if f - l is surjective, (v) a ppd map (where H )
“ppd” stands for “possibly partially defined”) if it is single-valued, and (vi) an ordinary map if it is an everywhere defined ppd map. The notation “f : A ~t B” means “f is a ppd map from A to B.” If S is a set, then 1s is the identity map of S , i.e., the triple ( S ,S, As), where A s = {(z,x) : z E S } . The abbreviation “CCA” stands for for “Cellina continuously approximable.” (We recall that a CCA map from a metric space X to a metric space Y is a set-valued map F : X H) Y such that, for every compact subset K of X, (i) the set ( K x Y )n G r ( F ) is compact, and (ii) there exists a sequence { F j } Z , of single-valued continuous maps from K to Y such that the graphs Gr(Fj) converge to G r ( F ) , in the sense that lim sup{distx.y(q, G r ( F ) ) : q E Gr(F’)} = 0 . j-CC
329
(A detailed study of CCA maps appears in Ref. 9.) We use C C A ( X ,Y ) to denote the set of all CCA maps from X t o Y . If I is a totally ordered set, then we use 51to denote the order relation on I , and simply write 5 when the context makes I unambiguous. Also, “a < I b”-or, simply, “a < b”-means “a 51b and a # b.” A subinterval of I is a subset J of I such that c E J whenever a E J , b E J , c E I , and a I c 5 b. We use square bracket notation for subintervals of I that have an infimum and a supremum in I . (That is, if a, b E I and a 5 b, we write ] a ,b[Ief{t E I : a < t < b } , [a,b [ ~ e ~ { a } U b] [a ~, ],a ,b ] ~ e ~b][ laU,{ b } , def and [a,b ] =~ { a ,b } U ] a ,b [ I , and we omit the subscript when I is uniquely determined by the context.). Then every subinterval of I that has an infimum and a supremum in I is of one of the forms ] a ,b[, [a,b], [a,b[, ] a ,b]). When the totally ordered set is not specified, it is understood that it is the extended real line U {--00, +m}. We define R+ef{x E JR : x 1 0)
aefR
and JR+,>ef{zE JR : x > 0}, and let R+efR+ U {+a}. We use 8 to denote the class of all functions 6’ : R+,> H
0
R+
such that
6’ is monotonically nondecreasing (that is, O(s) 5 O ( t ) whenever s, t are such that 0 5 s 5 t < +m); lim,LoB(s) = 0.
E
R
If X is a FDNRLS, z* E X , T > 0 , then B x ( x * , T ) , B x ( ~ * , Tare, ) respectively, the open ball {x E X : 11x - x*11 < .} and the closed ball {x E X : 112 - z*11 5 r } . If X , Y are FDRLSs, then L i n ( X , Y ) , A f f ( X , Y ) will denote, respectively, the set of all linear maps and the set of all affine maps from X t o Y . By definition, the members of A f f ( X , Y ) are the maps a f f m L , h , for L E L i n ( X , Y ) , h E Y , where ~ f S ; n denotes ~ , ~ the a f i n e m a p with linear part L and constant part h, defined by affmL h(x)efL. x+h . We identify A f f ( X ,Y )with L i n ( X , Y )X Y by identifying kach map affmL,h E A f f ( X ,Y ) with the pair ( L ,h ) E L z n ( X , Y )x Y . If X and Y are FDNRLSs, then we endow L i n ( X ,Y ) with the operator norm 11 . IIop given by llLllop = sup{IILxll : 2 E X , IIxlI 5 l}, SO L i n ( X , Y ) is a FDNRLS as well. Also, we endow the linear space A f f ( X ,Y ) with the norm given by IbffmL,hll = IlLll + llhll. If A is subset of L i n ( X ,Y ) ,and 6 E R+,>, we define
As = {L E L i n ( X , Y ) : dist(L,A) I d } , where dist(L,A) = inf{llL - L’llop : L’ E A}. Also, if 6 ,E~R+,>, and we
330
still assume that A A(6,E)
C L i n ( X ,Y ) ,we let
- {affmL,h: L
E
L i n ( X , Y ) ,dist(L,A) 5 S, h E Y , llhll 5 S E } ,
Notice that if L E L i n ( X , Y ) ,then dist(L, 0) = +co. In particular, if A = 0 then R6 = 0. and = 0. Notice also that if A is compact (resp. convex) then A6 and A(s+) are compact (resp. convex). If X is a FDRLSs, then we use X t to denote the dual space of X , i.e., the space L i n ( X ,R). The word “manifold” will mean “finite-dimensional paracompact differentiable manifold without boundary.” If M is a manifold of class C1, and x E MI then T,M, T,*M denote, respectively, the tangent and cotangent space of M at x .
1.2. Approximate Generalized Dinerential Quotients
Definition 1.1. Assume that X , Y are FDNRLSs, F : X +++ Y is a set-valued map, A is a compact subset of L i n ( X , Y ) , 1% E X I jj* E Y , and S 2 X . We say that A is an approximate generalized dinerential quotient of F at (%*,y*) in the direction of S-and write A E AGDQ(F,%.,,jj,,S)-if there exists a function 6 E 8-called an AGDQ modulus f o r (A, F,&, jj*, S)-having the property that
(*) for every E E EX+,, such that e ( E ) < co there exists a set-valued map AE E CCA(Bx(%*, E)ns,Aff ( X IY ) ) ,with walues in A(e(E)ic), such that 0 jj., +A(x-%*) E F ( x ) whenever x E Bx (3*,E ) n S and A E A“( x ) . 1.2.1. Properties of AGDQs If A, B , C are sets, and S, 2 are sets of maps from A to B and from B to C , respectively, then the composite Z o Z is the set of maps from A to C given by 2 o E = {I o 6 : E 2, 6 E S } . The following statement, proved in Ref. 9, is the chain rule for AGDQs.
Theorem 1.1. For i = 1 , 2 , 3 , let X i be a FDNRLS, and let %*,i be a point Xi+l is a set-valued map, of X i . Assume that, for i = 1 , 2 , (i) Fi : X i (ii) Si is a subset of X i , and (iii) hi E AGDQ(Fil%c,,i7%*,i+17 Si).Assume, in addition, that (iv) Fl(S1) & S2, and either (v) 5’2 is a local quasiretract (cf. Remark 1.1) of X2 at %*,2 or (v’) there exists a neighborhood U of %*,I in X 1 such that the restriction F1 [ (UnS,) of F1 to U n S l is single-valued. Then h 2 o A 1 E A G D Q ( F ~ ~ F ~ , Z , , ~ , Z , , ~ , S ~ ) . 0 H )
331
Remark 1.1. The notion of a "local quasiretract" is defined in Ref. 9. The precise definition is as follows. First, if T is a topological space and S C: T I we say that S is a quasiretract of T if for every compact subset K of S there exist a neighborhood U of K and a continuous map p : U H S such that p(s) = s for every s E K . Then, if S C T and 3 E S , we say that S is a local quasiretract of T at 3 if there exists a neighborhood U of such that S n U is a quasiretract of U . An important example of a local quasiretract of a manifold M at a point s E M is a subset S of M such that, for some open neighborhood U of s, the set S n U is the image of a convex subset of an open neighborhood V of 0 in Rdim under a diffeomorphism @ of class C 1 from V onto U such that Q(0) = s. In particular, any set whose germ at s is, relative to some coordinate chart near s, the germ at s of a convex subset of Rdim , is a local quasiretract of M at s. 0 If M and N are manifolds of class C', Z, E MI y* E N, SC M I and F : M +.++ N, then it is possible to define a set AGDQ(F,3*,jj*, S ) of compact subsets of the space Lin(TE*Ml T g , N ) of linear maps from Tz*M to Tg*N as follows. We let m = dimM, n = dimN, and pick coordinate charts E : M ~f Rm, q : N c--) R",defined near z*, jj* and such that c(Z*) = 0 and q ( g * ) = 0, and declare that a subset A of Lin(TE,M,Tg*N) belongs to AGDQ(F,Z*, jj*, S ) if the composite set of maps Dq(jj,) o A o DE(Z*)-l is in AGDQ(q o F o E-', O,O, E(5'))). It then follows easily from the chain rule that, with this definition, the set AGDQ(F,Z*,j j * , S ) does not depend o n the choice of the charts E , q. In other words, the notion of a n AGDQ is invariant under C' diffeomorphisms and therefore makes sense intrinsically o n manifolds of class C 1 . Then the chain rule also holds on manifolds, as pointed out in Ref. 9.
Proposition 1.1. Assume that (I) for i = 1 , 2 , 3 , Mi is a manifold of class C' and Z*,,i E Mi, (11) Si C_ Mi,Fi : Mi M Mi+', and Ai E AGDQ(Fi,Z*,il Z*,i+l, Si)for i = 1,2, (111) either S2 is a local quasiretract of M2 or F1 is single-valued on UnSl for some neighborhood U of %*,I. Then the composite A2 o A1 belongs to AGDQ(F2 0 F 1 , Z ~ , l I % , 3SI). , 0 Furthermore, AGDQs have several natural properties. First, the following statement, proved in Ref. 9, says that classical differentials at one point of continuous maps and Clarke generalized Jacobians of Lipschitz maps are AGDQs.
332
-
Proposition 1.2. If M , N are manifolds of class C 1 , S C M , 3, E M , jj* E N , F : M N , U is an open neighborhood of% in M , and F ( & ) = {%}, then (1)
If (i) the restriction F r (U n S ) is a continuous everywhere defined
map, (ii) L is a differential of F at Z* in the direction of S (that is, L E Lin(T5*M,Tu,N)and llF( x) - F( 3 * ) - L . ( x --Z* )II = o ( ~ ~ x - Z * ~ ~ ) as x --t 2, via values in S , relative to some choice of coordinate charts about Z* and y*), then { L } belongs to AGDQ(F,Z*,y*, S ). (2) If (i) the restriction F U is a locally Lipschitz everywhere defined map, and (ii) A is the Clarke generalized Jacobian of F at 3,, then A belongs to AGDQ(F,z,,y,,M). 0 The following two propositions, also proved in Ref. 9, are the Cartesian product rule and the assertion that AGDQs are local, in the sense that the set AGDQ(F,Z,,jj,,S) is completely determined by the germ of the set S at Z* and the germ of the graph of F at (3*, y*). In Proposition 1.3, if A, B, C, D are sets and p : A t-+ C , u : B D , then p x u is the set-valued map from A x B to C x D that sends each point ( a ,b) E A x B to the subset p(a) x v(b) of C x D. (In particular, if p and u are ordinary single-valued maps, then p x u is an ordinary single-valued map, given by ( p x u)(a,b) = ( p ( a ) ,u(b))for a E A , b E B . ) If M , N are sets of SVMs from A to C and from B t o D ,respectively, then M x N is the set of all SVMs p x v,p E M , u E N .The spaces Tze,lM1x T z , j z MTu.,lN1 ~ x Tu*,,N2are identified with T(z*,l,z,,z)(M1 x M z ) and T(u*,l,u,,z)(N1 x Nz), respectively. H )
Proposition 1.3. (The product rule.) Assume that (1) for i = 1 , 2 , Mi and Ni are manifolds of class C 1 , Si G Mil 3*,i E Mi, g*,i E Ni, Fi : Mi t-+ Nil and hi E AGDQ(Fi,z*,i,g*,i, Si); (2) 3, = ( 3 * , 1 , 3 * , g* ~ ) ,= ( g * , l , & , z ) , S = S1 x SZ, and F = F1 x Fz.
Then A1 x
A2
E AGDQ(F,%,g,,S).
0
Proposition 1.4. (Locality.) Assume that (1) M , N , are manifolds of class C 1 , (2) Zt.,E M , (3) & E N , (4) Si G M and Fi : M t-+ N for i = 1 , 2 , and (5) the sets S1 and Sz have the same germ at Z,, and the graphs Gr(F1),Gr(Fz),have the same germ at (Z*,g*) (that is, there exist neighborhoods U , V of Z,, y*, in M , N , respectively, such that U n S l = U n S2 and(UxV)nGr(Fl)= ( U x V ) n G r ( F z ) )ThenAGDQ(Fl,%,y,,S1) . = AGDQ(F2,%,jL,Sz). 0
333
1.2.2. Uniform AGDQs
Assume that X and Y are FDNRLSs, and we are given a family {(F,,x,, y,, S,)},€A of 4-tuples, such that each F, is a set-valued map from X t o Y ,each x , is a point of X, each y , is a point of Y , and each S, is a subset of X . We say that a family {A,},,A of compact subsets of Lin(X, Y)is a u n i f o r m AGDQ of the maps F, at the points (x,, y,) in the direction of the S, if there exists a function 0 E 8 which is an AGDQ modulus for (A,, F,,x,, yalS,) for each a E A. The concept of a uniform AGDQ makes sense as well, in an intrinsic way, when X and Y are manifolds, provided that the family ( ( F , , X ~ , ~ ~ , S is~ )such } ~ that ~ A the set Q = { ( x a , y a ) : a E A } is precompact in X x Y.Indeed, let d x , dy be the dimensions of X and Y.If Q is precompact in X x Y ,then we can find a finite family C = {(<jlUj,77j,V,,Kj,Lj)}l<j~, such that (1) for each j , (i) <j is a coordinate chart of X with domain U j , (ii) qj is a coordinate chart of Y with domain V,, (iii) Kj is a compact subset of U j , and (iv) Lj is a compact subset of V,; ( 2 ) Q C U;F_l(Kjx Lj).
Then, if we let Aj = { a E A : (xa,y,) E Kj x L j } , it is clear that A = Uj”=,A,, and we can consider, for each j , the family @ j = { ( f l j , , , ~ , , y,, S j , , ) } a g ~ j , where Fj,,is the - set-valued map from Uj to V, whose graph is Gr(F,) n ( U j x V,), and Sj,, = S, n U j . If we identify U j , V, with open subsets oj, of R d X ,RdY, then { F j , a } a E ~isj a family of set-valued maps from RdX to RdY, the x , belong to RdX, the y , belong to R d Y ,and S, is a subset of R d X ,so we are in the situation of the previous paragraph, and it makes sense to talk about a “uniform AGDQ” {A,},EA~ of the family @ j . We then say that a family {A,},EA is a u n i f o r m AGDQ of the f a m i l y { ( F a ,x,, y,, S,)},€A if, for some choice of m and the family C = {([j, UjlVjlV,, Kj,Lj)}l<j<, - _ as above, it turns out that {A,},EA~ is a uniform AGDQ of @ j for each j. (It is easily seen that if this condition holds for one choice of m and C, then it holds for all such choices.) 1.2.3. A GDQ approximating multicones. A cone in a FDRLS X is a nonempty set C which is closed under multiplication by nonnegative real numbers, i.e., such that rc E C whenever c E C and r 2 0. The polar of a cone C in X is the subset Ct of X t defined by Ct = { p E X t : p ( c ) 5 0 whenever c E C}. Clearly, C t is always a closed
334
convex cone. If we identify X t t with X in the usual way, then C C C t t , and C = Ctt if and only if C is closed and convex. A multicone is a nonempty set of cones. A multicone M is convex if all the members of M are convex cones, The polar M t of a multicone M is the closure of the union of the polars Mt, M E M . Therefore M t is a always a closed cone in X t . Naturally, M t need not be convex in general.
Definition 1.2. Assume that M is a manifold of class C 1 , S is a subset of M , and 3* E S. An AGDQ approximating multicone t o S at 3* is a convex multicone C in T5* M such that there exist a nonnegative integer m, a set-valued map F : Rm M , a convex cone D in R”, and a A E AGDQ(F,O,?,,D), such that F ( D ) & Sand C = ( L D : L E R ) . 0 H )
1.3. l’ransversality of cones and multicones.
+
If 5‘1, S2 are subsets of a linear space X , we define the s u m S1 S2 and the diflerence S1 - S2 by letting S1 S2 = ( s 1 sz : s1 E S1, s2 E S2) ,
+
+
s2 : s1 E s1,s2 E S2). Recall that if S1, ,572 are linear subspaces of a FDRLS X, then 5’1 and S2 are transversal if S1 + S2 = X , or, equivalently, if S1 - S2 = X . If two submanifolds M I , M2 of class C 1 intersect a t a point x*,and S1, 5’2 are their tangent spaces a t x*, then it is well known that if 5’1 and 5’2 are transversal then M1 n M2 looks, near x*,like S1 n S2. In particular, if S1 n S2 # ( 0 ) (i.e., if dim S1 n S2 2 l),then M I n M2 contains a nontrivial curve going through x*.The following definitions generalize the concept of s 1-s 2 = (s1-
transversality and that of “transversality with a nontrivial intersection,” first to cones and then to multicones.
Definition 1.3. Let X be a FDRLS, and let C1, C2 be two convex cones in
X . We say that C1 and C2 are transversal, and write C16C2, if C1- C2 = X. We say that C1 and C2 are strongly transversal, and write C1dhC2, 0 if Cl6C2 and in addition C1 n C2 # ( 0 ) . In order to extend Definition 1.3 to multicones, it is convenient to start by reformulating the concept of strong transversality of cones, by making the trivial observation that CldhC2 if and only if the following two conditions hold: (i) C16C2 and (ii) there exists a linear functional p E X t such that p(w) > 0 for some w E C1 n C2.
In view of the above reformulation, we define a linear functional p : X H R to be intersection-positive on a pair ( C l , C 2 ) of multicones, if
335
the set {c E C 1 n C2 : p ( c ) > 0 } is nonempty for every C1 E C1 and every C2 E C2. Using this concept, the definitions of “transversality” and “strong transversality” of convex multicones are nearly identical to the definitions for cones.
Definition 1.4. Let X be a FDRLS. We say that two convex multicones C1 and C2
in X are transversal, and write C16C2, if ClhC2 for all C1 E C1, E C2. We say that C1 and C2 are strongly transversal, and write C2
if (i) C16C2, and (ii) there exists a linear functional p E X t which 0 is intersection-positive on (el,C2).
ClAC2,
Two convex cones C 1 C2 in a FDRLS X are linearly separated if there exists a nontrivial linear functional X E X t such that X(c) 5 0 whenever c E C1, and X(c) 2 0 whenever c E C2. (Equivalently, C 1 and C 2 are linearly separated if and only if C[ n (-Cz)t # {O}.) It is easy to see that C 1 and C2 are linearly separated if and only if they are not transversal. In view of this, we will call two convex multicones C1, C2, linearly separated if they are not transversal. Since strong transversality is a stronger property than transversality, its negation is weaker than the negation of transversality, i.e., than linear separation. So we will say that two convex multicones C1, C2, are weakly linearly separated if they are not strongly transversal. The following characterization of weak linear separation is proved in Ref. 10.
Proposition 1.5. Let C1, C2 be convex multicones in a FDRLS X. Then the following conditions are equivalent: 1. C1 and C2 are weakly linearly separated; 2. for every p E Xt\{O} there exist 7roI 7r1,7r2, C 1 , C2 such that
1.4. The nonseparation theorem.
The crucial fact about AGDQs that leads to the maximum principle is the transversal intersection property, that we now state (cf. Ref. 9 for the proof).
336
Theorem 1.2. Let M be a manifold of class C1, let S1, S2 be subsets of M , and let 3, E S1 n 5’2. Let C1, C2 be AGDQ-approximating multicones to S2 at S , such that C1AC2. Then S1 and S2 are not locally separated at 3,. (That is, the set S1 n S2 contains a sequence of points s j converging to 0 3, but not equal to g,.) 5’1,
Theorem 1.2 and Proposition 1.5 trivially imply the following result.
Corollary 1.1. Let M be a manifold of class C 1 , let S1, S2 be subsets of M , and let 3, E S1 n S2. Let C1, C2 be AGDQ-approximating multicones to S1, S2 at 3 , . Assume that S 1 and S2 are locally separated at 3,. (That is, there exists a neighborhood U of S , such that S1 n S2 n U = { 3 , } ) . Then Condition 2 of the statement of Proposition 1.5 holds. 0 The more familiar forms of the maximum principle for optimal control follow by applying Corollary 1.1 to suitable choices of M , S1, S2, C 1 , C 2 , 3,, and using the conclusion of the corollary with a suitable p. For example, consider a fixed time interval optimal control problem P whose data 9tuple D = ( M o ,U, a , b,U, f , L , i?,, S ) satisfies ( D l ) the state space MO is a smooth manifold, (D2) U is a set, (D3) a , b E IR and a < b, (D4) U (the class of “admissible controls”) is a set of U-valued functions on [a,b ] , (D5) ( L ( z , u , t , ) ,f (x,u, t ) ) E IR x T,Mo for each ( z , u , t ) E Mo x U x [a,b],(D6) i?* E Mo, and (D7) S Mo. Suppose that the objective of P is to minimize the integral L ( ( ( t ) ,q(t),t )d t , subject to the following conditions: ( C l ) ( : [a,b] H MO is absolutely continuous, (C2) q E U , (C3) ( ( t ) = f ( E ( t ) , q ( t ) , tfor ) almost all t E [a,b ] , (C4) ( ( a ) = Z,, and (C5) ( ( b ) E S. We then take M = IR x Mo. If a trajectory-control pair (E, q,) is a solution of P, we take S1 to be the set of all points (r,x ) E M such that x is reachable from 3, over [a,b] with cost r , and we take S2 to b e t h e s e t (]-m,r,[xS)U{q,},whereq, = (r,,(,(b)), a n d r , isthecostof ((,, q,). Then the optimality of ((, , q,) implies that S1 and S2 are locally separated at 4,. We then take C1 to be an AGDQ-approximating multicone to S 1 at q, obtained by constructing variations and propagating their effects to the terminal point of <,, and take C2 = { ] - m,O]x C : C E C}, where C is an AGDQ-approximating multicone to S at (,(b). We choose p to be the linear functional on T,,M IR x Tt*(b)Mogiven by p ( r , v ) = -r, so -p E C l for every C2 E C2. Corollary 1.1 then yields a decomposition 7r0p = m T Z , where T I E C!, T Z E Ci, T O 2 0, ( T O , T ~ , T#~ ()O , O , O ) , t T O 2 0, and C1 E C1, and C2 E C2. Then --TI = T Z - T O P . Since T Z E C,, - p E C;, it follows that -7r1 E Cl. If we write C2 = ]- m,O] x C , C E C,
s,”
N
+
337 7r1 = (-p,7i), then the fact that --TI E CJ. implies that p 2 0 and Ct. Then 7i and p are, respectively, the terminal adjoint vector (often called $ ( b ) or X ( b ) in the literature) and the additional multiplier (often
and let -7i E
called $0 or XO) conjugate to the cost T , and the familiar conclusions of the maximum principle follow.
2. Flows, trajectories, and generalized differentials of flows. 2.1. State space bundles and their sections
A t i me set is a nonempty totally ordered set. If I is a time set, we define 12>2= { ( t , s ) E I x I : t 2 s } , and 1392 = { ( t , s , ~ )E I x I x I : t 2 s 2 r } . A state-space bundle (abbr. SSB) over I is an indexed family X = { X t } t E I of sets. A state-space bundle is a pair X = (X,I ) such that I is a nonempty totally ordered set and X is an SSB over I . The set I is the time set of the SSB X . Remark 2.1. There are several reasons for using general totally ordered sets, rather than real intervals, as time sets for control systems. For a simple example, cf. Ref. 8, where an example is given of a problem for which the natural time set consists of a compact interval minus one interior point. 0
If C is a category whose objects are sets with some additional structure (for example, topological spaces, metric spaces, manifolds of class C" linear spaces, FDRLSs), then an SSB ( X , I ) = ( { X t } t E I , I is ) a bundle of Cobjects if each X t is a member of C. In particular, if k is a nonnegative integer, a Ck SSB is an SSB of manifolds of class Ck.Also, an FDRLS SSB is an SSB of finite-dimensional real linear spaces. Definition 2.1. Assume that X = ( X , I ) = ( { X t } t g ~ , Iis ) an SSB. A section of X is a single-valued everywhere defined map on I such that <(t)E X t for every t E I . We use S e c ( X ) to denote the set of all sections of x. 0
<
Definition 2.2. Let X = ( X , I ) = ({Xt}te=,I) be a C1 state-space bundle, and assume that E S e c ( X ) . The family T c X = { T c ( t ) X t } t g I is the 0 tangent bundle of X along E.
<
Clearly, the tangent bundle Tg X of a C1 SSB X along a section E E S e c ( X ) is an FDRLS SSB.
338
2.2. Flows and trajectories
Definition 2.3. Assume that C is a category whose objects are sets with some additional structure, and X = ( X , I ) = ({Xt}t,=I,I) is an SSB of C-objects. A C-flow on X is an indexed family f = { f t , , } ( t , s ) E I ~ , > such that (1) ft,, is a C-morphism from X, to Xt whenever (t,s) E 12yz; (2) f t , t is the identity morphism of Xt whenever t E I; (3) f t + 0 f,,,= ft,, whenever ( t ,s, T ) E 1332.
A C-flow is a pair 3 = ( X , f )such that X is an SSB of C-objects and f is a C-flow on X . 0
Example 2.1. If C is the category whose objects are all the sets, and whose morphisms are the set-valued maps, then a C-flow on an SSB X will just be called a flow on X . 0 Example 2.2. If C is the category whose objects are all FDRLSs, and whose morphisms are the linear maps, then a C-flow on an FDRLS SSB X will be called a linear FD flow. 0 Example 2.3. We use F D C L i n t o denote the category whose objects are all FDRLSs, and whose morphisms are defined as follows: if X, Y are FDRLSs, then the set of morphisms from X to Y is the set CLin(X, Y ) of all nonempty compact subsets of Lin(X, Y ) . (Composition of morphisms is defined in the obvious way: if A1 E CLin(X, Y) and A2 E CLin(Y, Z ) , then A2 o A I % ~ { Lo~L1 : Lz E A2, L1 E Al}.) An FDCLin-flow is a linear FD multiflow. 0
Remark 2.2. It is well known that every time set I can be regarded as a category c a t ( l ) , by taking the objects of cat(1) to be the members of I , and the set HomCat(I,(a,b)of morphisms from a E I to b E I to consist of a single object if a 51 b, and to be empty if b
I f f , f’ are SVMs, we write f 5 f’ if S o ( f ) = So(f’), Ta(f) = Ta(f’), and Gr(f) C Gr(f’). If, for i = 1 , 2 , 3i = ( X , f i ) are flows on the same SSB X , and f i = { f j , , } ( t , s ) E p , 2 ,we say that F 1is a subflow of F 2 ,or 32is a superflow of F’, and write F13 32,if f& 5 f& for all (t,s) E 12)->.
339
2.2.2. Dajec tories
Definition 2.4. Assume that X = ( X , I ) is a state-space bundle, 3 = ( X ,f ) is a flow, X = ( { X t } t E I~), , and f = { f t , s } ( t , s ) E 1 2 , t A . trajectory of 3 is a section [ of X such that [ ( t )belongs to f t , s ( [ ( s ) ) whenever (t,s) E I22.
We use T r u j ( 3 ) to denote the set of all trajectories of the flow 3. 0 2.3. AGDQs of ftows along trajectories
Definition 2.5. Assume that X = ( X , I ) = ( { X t } t E I , I )is a C1 SSB, 3 = ( X , f ) is a flow, f = { f t , s ) ( t , s ) E 1 2 , 2 , and E T r a j ( 3 ) . An AGDQ of 3 along [ is a linear FD multiflow g = { g t , s } ( t , s ) E z ~on , t the tangent bundle T E X such that gt,s E A G o Q ( f t , s ; [ ( s ) , [ ( t ) ; X ,whenever ) ( t , s ) E 12v1.
<
Remark 2.3. In view of our definitions, the condition that g is a linear FD multiflow on TcX means that (1) if (t,s) E 12i2,then gt,+ is a nonempty compact set of linear maps from T€(,)Xs to q ( t ) X t ; (2) gt,t = {he(t,xt} whenever t E I ; (3) gt+ o g8,,. = gt,,. whenever (r,s , t ) E 13>2. 0 2.3.1. Compatible selections
Definition 2.6. Assume that g = { g t , s } ( t , s ) E z ~is, >a linear FD multiflow on an FDRLS SSB ( Y ,I ) . A compatible selection of g is a linear FD flow 0 y = {yt,s}(t,s)E12,tsuch that ytY8E gt+ whenever ( t ,s) E 127>. We write C S e l ( g ) to denote the set of all compatible selections of g. Then C S e l ( g ) is a subset of the product space Pgdef = gt+. Since Pg is a compact space, by Tichonov’s theorem, and C S e l ( g ) is a closed subset of Pg-because C S e l ( g ) is the set of all y E Pg that satisfy a collection of equalities involving continuous functions on Pg-we can conclude that CSel(g) is compact.
n(t,s)E12,2
Remark 2.4. In view of our previous definitions, the condition that y is a linear FD flow means that yt,t = DT<(~)x for each t E I , and yt,sys,r= yt,,. whenever (t,s, r ) E 13 ? 2 . 0
340
2.3.2. Fields of variational vectors and adjoint covectors
-
Definition 2.7. Assume that g = {gt,s}(t,s)E12.tis a linear FD multiflow on an FDRLS SSB (Y, I ) = ({K}tEI,I ) . A field of variational vectors of g is a selection I 3 t v ( t ) E Yt such that vt E gt,sv, whenever ( t ,s ) E 1 2 1 2 . A field of adjoint covectors (also called, simply, an adjoint covector, or even an adjoint vector) of g is a selection I 3 t H w ( t ) E ytt of the dual bundle Yt = { y t t } t E l . such that ws E g;,,wt whenever ( t , s ) E 12>>, where g!.,
=
{rt : 7 E gt,s}.
0
The following result is an easy consequence of the compactness of CSel(g).
Proposition 2.1. Assume that g = { g t , s } ( t , s ) E I ~is, za linear FD multijlow on an F D R L S SSB (Y, I ) = ( { K } t E I , I ) Assume . that I 3 t H v ( t ) E K (resp. I 3 t H w ( t ) E ytt) is a selection of Y (resp. yt). Then v is a field of variational vectors (resp. w is a field of adjoint covectors) of g if and only if there exists a compatible selection y = {yt,s}(t,s)EI~,> of g such that vt = yt,svs (resp. ws = yJ,swt) whenever (t,s) E 12>2. 3. Variations, impulse variations, summability 3.1. Variations of set-valued maps Definition 3.1. Assume that F is a set-valued map and P is a FDRLS. A variation of F with ambient parameter space P is a family V = { V p } p E such ~ that (1) C is a closed convex cone in P with nonempty interior; (2) each V, is a SVM such that So(Vp) = S o ( F ) and Ta(V,) = Ta(F); (3) Gr(Vo) G G r ( F ) .
If F’ is another set-valued map such that So(F’) = So(F), Ta(F’) = Ta(F), and G r ( F ) G Gr(F’), we say that V is a variation in F’ if the inclusion Gr(V,) G Gr(F’) holds for every p E C , i.e., if V,(x) F‘(z) whenever p E C and x E So(F’). 0 If F, PI V are as in Definition 3.1, then the cone C is the parameter cone of V, and the dimension of C (or of P ) is the number of parameters of V. We will use to denote the SVM with source P x So(F) and target such that v(p,x) = V,(x) for all p E PI x E So(V0). (In particular, V(p, x) = 0 if p E P\C.)
F(F)
v
341
3.2. Infinitesimal impulse variations Definition 3.2. Assume that X = (X,I) = ( { X t } t E 1 , I )is a C1 statespace bundle, F = ( X , f ) is a flow, and E T r a j ( 3 ) . An infinitesimal is a triple (w,t,a) such that impulse variation (abbr, IIV) f o r (F,<) t E I , w E T E ( ~ ) and X ~ ,a is one of the symbols -. 0
<
+,
Remark 3.1. The purpose of including a in the above definition is to distinguish between "left" impulse variations, which will be labelled (w,t , -), and "right" impulse variations, labelled (w,t , +). Left and right impulse variations will differ in the way the concept of "carrier" of an IIV (w,t , a ) is defined, which will depend strongly on u. 0
3.3. Summability Definition 3.3. Assume that X = (X,I) = ( { X t } t E 1 , I )is a C1 statespace bundle, F = ( X , f ) is a flow, and € T r a j ( F ) . If (w,t,a) is an IIV for (F, E), we say that (w,t ,a ) is carried by a subinterval J of I if t E J and one of the following two conditions holds: (i) a = and there exists a t , E J such that t < t,, (ii) a = - and there exists a t , E J such that
<
+
t , < t. If V is a set of IIVs for (3, <), we say that V is carried by J if every member of V is carried by J . 0 If V is a finite set of IIVs for (.F,<),we let Rv, lRy denote, respectively, the set of all families p' = {pV}vEv of real numbers, and the set of all p" { p V } v ~ vE Rv such that pv 2 0 for all V E V. (Hence, if m is the cardinality of V, and V = { (wl,t l , al),. . . , (wm,tm,a m ) } ,the spaces EXv, R y , can be identified with Rm, Rl;., by identifying each family p" {pv},Ev . . . with the m-tuple ( f i l , . , . , f i m ) , where 151 = p("J~t3~u3) for j = 1,.. . ,m.) If g = { g t , s } ( t , s ) E ~ 2 , t is an AGDQ of F along <, y = { ~ t , ~ } ( t , ~ is) ~ a compatible selection of g , a , b E I , a 5 b, and V is carried by [a,b ] , we define a linear map LViyva,b: Rv x T,(,)X, H TE(b)Xbby letting
C
L ~ ~ ~ W)~ =~yb,,(w)+ ~ ~ ( J T p("'tiu)yb,t(21) , for J T E R ~W, E T ~ ( , ) X , . (",t,U)EV
We let hV,g,aib be the set of all the maps L V ~ y ~ afor ~ ball, y E C S e l ( g ) . Then R V i g ~ ais~the b image of C S e l ( g ) under the continuous map C S e l ( g ) 3 y H LViyiaib E Lin(RV X T,(,)x,,T,(b)xb), so hv,g3aib is a compact subset of Lin(Rv x Tc(a)XarT,(b)Xb)*
~ z ~ t
342
Definition 3.4. Assume that X = (X,I) = ( { x t } t E ~ , I ) is a C1 statespace bundle, 3 = (X,f) is a flow, E E T r a j ( 3 ) , g = {gt,s}(t,s)Ep,> is an AGDQ of 3 along E, and F' = (XIf') = (XI{fl,s}(t,s)EI~.>) is a superflow of 3. Let V be a set of IIVs for (3,E). We say that V is g-AGDQsummable within 3' if the following is true: 0
for every finite subset V of V , and every pair ( a ,b) E I x I such that a < b and V is carried b y the closed interval [a,b ] , there exists a variation W = { W F } F o~f f~b , ~, in f;,, such that the set h V ~ g i a ~isb an AGDQ
of the map
at ( ( O , ( ( a ) ) , < ( b )along ) :XE
0
x X,.
4. The A G D Q maximum principle
We now state and prove a general maximum principle in the setting of AGDQ theory. Instead of working with a control system k = f(z, u , t ) and a reference trajectory-control pair (& , q*), we consider the more general situation of a pair (3,3') of flows such that 3 is a subflow of 3'. We assume that 3 and 3' are defined on a common state space bundle X = ( & I ) = ( { X t } t g ~ , I )which , is of class C1, in the sense that the Xt are manifolds of class C1. In the control system case, (i) the time set I is a compact subinterval of R,(ii) all the state spaces Xt coincide, so there is a manifold X of class C1 such that Xt = X for all t E I , (iii) the domain of the reference control is I , (iv) 3 = ( X , f ) is the reference flow, i.e., the flow determined by the reference control v*,so that, i f f = { f t , s } ( t , s ) E I ~ , then >, f t , s ( z )for , zEXI ( t ,s ) E I Z > >is , the set given by
v*
f t , s ( z )= {E(t>: E E TW(rl*,f l s, t ) , E ( s ) = TI , where, if U is the class of admissible controls, then for any rl E U we use Traj(q,f,s,t) to denote the set of all E E W ' ~ ' ( [ s , t ] , Xsuch ) that ((7) = f(E(T),r](7),7) for a. e. 7 E [ s , t ] ,and W ' ~ ' ( [ s , t ] , Xis) the set of all absolutely continuous maps from [s,t] to X I (v) 3' = (XIf') is the flow of the full control system, so that, if f' = {fl,s}(t,s)EI~,>r then fl,s(z),for z E X and ( t ,s) E 12i2, is the reachable set from z over the interval [s,t ] , so f i , s ( z )is given by
fl,s(z)= {E(t): (37 E W ( E E Traj(rl, f ls, t ) )
((3)
=
.
(Notice that the maps ft,+ are single-valued-that is, each set ft+(x) is either empty or consists of a single member-if the ordinary differential equation j: = f ( z , q * ( t ) , t has ) uniqueness of trajectories, but for more
343
general reference vector fields ( x , t ) H f ( x , q , ( t ) , t )the f t + can be setvalued. On the other hand, the are never single-valued, except in trivial cases.) The flow formulation, together with the use of general totally ordered sets rather than real intervals (cf. also Remark 2.1), includes situations other than that of control systems, such as, for example, “hybrid systems” in which the state is allowed to jump at some time t from a state space X - to a state space X+. (This is achieved by treating t- and t+ as different times, with t- < t+, and having a family { J , } , E A of-possibly set-valued-jump maps from X - to X + , one of which is the reference jump map J,, . In that case, ft+,t- is the map Ja., and fl+,t- is the map such that f;+,t-(x) =
fl,,
UYEAJCY(X).)
Theorem 4.1. Assume that X = (X,I) = ( { X t } t E I , I )is a C’ statespace bundle, 3 = ( X , f ) is a flow, 3’ = (X,f’) is a superflow of 3, f = { f t , s } ( t , s ) E 1 2 2 1 f’ = { f l , s ) ( t , s ) € 1 2 2 E Traj(3)1 and g = {gt,s}(t,s)EIZ.> is an AGDQ of 3 along [. Let V be a set of infinitesimal impulse variations for (3, [) which is g-AGDQ-summable within 3’. Let a, b E I be such that a < b, and let S be a subset of x b such that [ ( b ) E S . Let C be an AGDQapproximating multicone of S at [ ( b ) . Assume that fL,a([(a))n S = { [ ( b ) } . Then for every nonzero h e a r functional p on T c ( b ) x b there exist (i) a compatible selection y = { y t , s } a < s < t < b of g, (ii) covectors ii, ii E T,(,)xb, and (iii) a nonnegative real number TO^ such that TOP = % ii,( T O , ii, ii)# (O,O, 0, ), iiE Ct, and T ( t ) . v 5 0 for every ( v ,t , u) E V which is carried by [a,b], where T ( t ) = ?i o yb,t for a 5 t 5 b. 1
c
+
Proof. Fix a p E T&b)Xb\{O}. Let VO be a finite subset of the summability of V, pick a variation { W C } ~ . ~ofVf b~, a in
w
+
v. Using fL,a such
at ( ( O , [ ( a ) ) , [ ( b )along ) that the set A V o ~ g ~ ais~ ban AGDQ of the map EX? x X,. For each compatible selection y of g, let 2 . be the linear map ~~0 3 @H L Vo,y,a,b(g,o), So that L(6) = &w,t,c)EVo P(w’t’“)yb,t(v). Let A = (2. : y E C S e l ( g ) } . Then A is an AGDQ of the set-valued map Rvo 3 fi w w ( f i , < ( a ) )C x b at ( O , [ ( b ) ) in the direction of EX?. Since w ( F , [ ( a ) ) C fL,,([(a)), the set M = { L y . : y E CSel(g)} is an AGDQ-approximating multicone of the set f L , a ( [ ( a ) )at [ ( b ) . Since fL,a([(a))n S = { [ ( b ) } , Corollary 1.1 implies that there exists a decomposition nop = ii+ ii,where ii E M f for some M E M , ii E Ct for some C E C, T O 2 0, and ( ~ o , i i , i i ) # (O,O,O). Since M E M , we can pick a y E CSel(g) such that M = 27 . E X Y O . Then the condition
Ry
344
that 7i E M t implies that ( 7 i , L Y ( p 3 ) 5 0 for every p’ E
a?.
Therefore
Ey. This implies that
( 5 ,~ ( w , t , a ) E V o p ( w ~ t l l T ) y b5, t (0wfor ) ) every p’ E
(7i,Y b , t ( V ) ) 5 0-i.e., that (?i o Yb,t, w) 5 0-for every (w,t , u ) E VO.Furthermore, the fact that iiE Ct implies that iiE Ct. It follows that the 4-tuple (Tr, ii, T O , y) satisfies all our conditions, except only for the fact that the inequality (7i o yb,$,w) 5 0 has only been shown to hold for (v,t , u ) in a finite subset VOof V. To prove the existence of a 4-tuple (Tr,F,ro,y) that satisfies (?i 0 ‘yb,t, w) 5 0 for all (w,t,u) E v, we use a familiar compactness argument. Fix a norm 11 . 11 on T & , ) X b . Let Q be the set of all 4-tUpleS (Tr,%,ro,y)such that ?i E T : * ( b ) x b , % E T : * ( b ) x b , T O E R, T O 2 0, T O 11Trll lliill = 1, and y E CSel(g). Then Q is a compact topological space, using on CSel(g) the topology induced by the product topology of gt,s. For each subset U of V, let Qu be the set of those (Tr,5,T O , y) E Q such that ii E Ct and (7i o y b , t , w) 5 0 for all (w,t,u)E U. Then every Qu is compact, and we have shown that Qu is nonempty if U is finite. Furthermore, it is clear that, if {Uj}jE(l,,,,,m) is a finite family of finite subsets of V, then Qul n n Qum = Q U I U . , . U U m , so Qul n . - . n Qum # 0. If U is the set of all finite subsets of V, we have shown that every finite intersection of members of the family { QU}uEu is nonempty. Therefore the set : U E U } is nonempty. But : U E U } = QV. So Qv is nonempty, concluding our proof. 0
+
+
n(t,s)EJZ,2
9
+
-
n{Qu
n{Qu
5. Generalized Bianchini-Stefani IIVs and the summability theorem We now present a class of IIVs that are infinitesimal generators of high-order variations in a sense that generalizes the definition proposed by Bianchini and Stefani. We assume that we are given ( D l ) a pair (3’,3) of flows, where
f ) and 3’= (X, f‘), (Dl. i) 3 = (X, (Dl. ii) X = (X, I ) ]= ( { X t } t E I , I ) is a C1 state-space bundle, (Dl.iii) f = { f t , s } ( t , s ) E p , and t f’ = {fl,s}(t,s)E12,2 are flows on the state=space bundle X, (Dl.iv) 3 is a subflow of F’,
c*
(D2) a “reference trajectory’’ E T r a j ( F ) , (D3) an AGDQ g = { g t , s } ( t , s ) E p , 2of 3 along
E*.
345
5.1. Times of right and left regularity Definition 5.1. Given 3’,3,&, g as above, a time of right (resp. left) ) a time f E I such that there exists a pair regularity of ( 3 ’ 1 3 1 & 1 g is (t,, X) for which (1) t , E I and f < r t , (resp. t ,
(3) if we let J~f[min(flt,),max(f,t,)]~, then
Xt
= X for all t E J , J ist a compact subinterval of R, the map J 3 t ++ & ( t ) E X is continuous, the family {gt,s}s,tEJ,s
(3.a) (3.b) (3.c) (3.d)
Remark 5.1. Condition (3.e) of the above definition is interpreted as follows: let n be a coordinate chart of X such that, for some f* E J\(Q, the interval J = [min(f,f*),rnax(f,~*)]~ is such that & ( t ) E Do(n) for every t 6 J (such a chart exists because of Condition (3.c)); we can then identify all the tangent spaces T,X. for 2 E Do(rc), with RdimX;then, if s,t E and s 5 t , all the members y of gt+ are linear maps from EdimXto EXdimX, and so is I T ~ * ( ~so X ,the difference y - I I T ~ * ( ~ ) xand its norm 117- T [ T ~ * ( ~ ) x I I make sense. 0 5.2. GBS IIVs Definition 5.2. Given F‘, 3,&, g as above, and a positive real number A, a triple (u,f,+) such that f E I and u E T c * ( q X fis a generalized Bianchini-Stefani (abbr. GBS) right infinitesimal impulse variation of order 1x. of (F’,3,<,, g ) at time f if (i)
f is a time of right regularity of (3’, 3,
g)
*We literally mean “is,” rather than just “can be identified with.” The rewon is that, when we consider several impulse variations with the same time f, we will not want the map identifying a right or left neighborhood of f with a real interval t o depend on the variation.
346
(ii) if t,, X , J are as in Definition 5.1, then there exists a 6-tuple ( a ,P, F,F,p, N ) (called a generator of (w,f,+)) such that (ii.1) 0 < a < P, F > 0, and E > 0, (ii.2) cp = { ( P ~ , ~ } ( ~x ,~ ~o ,) Eis~ ~a~ two-parameter ~ I family of set-valued maps from X to X , (ii.3) N is an open neighborhood of &(i?) in X , (ii.4) the set-valued map
N
x
[ o , ~3] ( x , ~ ) ( P ~ ( C , Z ) def = ( ~ c , e ( xC ) X H
is Cellina continuosuly approximable for each (ii.5) the maps p E satisfy (PE(0,
.>
G fr+pex,r++aax >.(
for
E
€10, E ] ,
E
N
2
7
(1)
as well as the asymptotic conditions u.w.r.t. c E [ O , F ] ,
(3)
p E ( c , E * ( f + a & ’ ) + h= ) ~,(f+P~~)+h+~~w+~(~+Ilhll) as ~ i O , h - , 0 ,
(4)
(cf. Remarks 5.2, 5.3), where “u.w.r.t.” stands for ”uniformly with respect to.” 0 The definition of what it means for a triple (v,f, -) t o be a GBS left IIV of order X of (F‘, 3,&, g ) at time f is similar, with obvious modifications.
Remark 5.2. Equation (3) is interpreted as follows: given any neighborhood U of & ( f )in X , there exist a positive number E* and a neighborhood U‘ of & ( f ) in X such that (pE(c,x)5 U whenever 0 < E 5 E * , x E U , and c E [O,C]. 0 Remark 5.3. In order to interpret Equation (4) precisely, we first agree, for each coordinate chart K. of X near &(l) such that Do(u;) C N,to write xn for the coordinate representation ~ ( x of ) a point z E Do(rc), and wn for the coordinate representation of a tangent vector w E T,X (so that wn = n,(w) = Du;(z) . w E WdimX). Then (3) implies-using Remark 5.2, with U = Do(K)-that there exists a positive number E* = E * ( K . , ( P ) having the following properties:
P1. 0 < &* 5 E,
347
P2. <(t)E Do(K) and @ ( & ( t ) " , E , ) C Im(r;) whenever f < t < BE?, P3. cpE(c,z) C Do(rc) whenever f 5 t I f+PE?, 0 < E I E,, c E [O, 4, and z E K - ' ( @ ( < , ( ~ ) " , E , ) ) . Wethenlet cp: be, for&E ] O , E * ] ,theset-valuedmapfrom [O,I?]X@(<,(~,)",E,) to Im(K) such that cp:(c,y) = (cpE(c,z))"--i.e., cp,"(c,y) = {z" : z E cp,(c,z)}-whenever E E ] O , E , ] , c E [O,E], z E Do(&), y E @ ( & ( f , ) & , ~ , ) are such that y = znand cp,(c,z) & Do(K).We then define the error En by
E"(c,E , h, 9) = y - <,(t+ PE')"
-
h - ECV"
,
for y E Im(K), E E ] O , E , ] , c E [O,E], and h E RdimX,and observe that E"(c,E,h , y ) belongs to RdimX. Then (4) is interpreted as asserting that
I
sup IIEn(c,~,h,y)ll: C E [O,C], rccp,"(c,C(i+.~').+h)} lim E~O,h-+O
E
=o.
+ llhll
It is easy to see that if this condition holds for some chart Do(&) C N,then it holds for every such chart.
K
such that
0
5.3. The summability theorem for GBS IIVs The following result is then our summability theorem.
Theorem 5.1. Let F ' , F , < , , g be data as in (Dl-2-3) above. Let V be the set of all generalized Bianchini-Stefani infinitesimal impulse variations of (3',3,<,, g ) . Then V is g-AGDQ-summable within 3'. 6. Proof of Theorem 5.1
We have to prove that, if V is a finite set of GBS IIVs of (F', F,[,, g ) , and a , b are such that V is carried by [a,b], then there exists a variation W = { W F }of ~ f b , a ~in ~ fL,a~such that the set A V ~ g ~ ais~an b AGDQ of W at ((0, <,(a)),J,(b)) along Ry x X,. It clearly suffices to consider the case when V is a nonempty finite set of GBS right IIVs at a point f E I , and to take a = t. Since V is nonempty, ?i is a time of right regularity for (F', .F, &, g ) , so we may pick t,, X such that the conditions of Definition 5.1 hold. Clearly, we may restrict t , further, and assume that t , 5 b. Furthermore, we may assume that all the points & ( t ) ,for t 5 t 5 t,, belong to the domain R of a coordinate chart K of X.
348
Let the members of V be listed as ( v l , f ,+), . . . , (vm,f,+), in such a way that the inverse orders XI,. . . ,A, satisfy A1 2 A 2 2 . . . 2 Am-1 2. ,A Then pick for each j a 6-tuple ( a j ,pj, Cj, E j ,cpj ,N j ) which is a generator of (vj,f,+) in the sense of Definition 5.2. I t is then easy t o see that
(*) without loss of generality, we m a y assume that A l . all the cj are equal to a positive number c such that 5 5 1, A2. all the cj are equal t o a positive number ?, A3. the inequalities pjcXj 5
aj+l~'j+l
for all j E { I , . . . , m - I},
E
E]o,c].
(5)
are satisfied, A4. the sets Nj all coincide; A5. f+ pmcAm5 t,. To see this, first replace each cj by min(cj, l),so all the pick a particular j . Then if p is small enough, p j p A 3. E x 3- I aj+l€xj+l
for all
E
8 are
1. Next,
E]O,C~],
(6)
because (i) &pxj 5 aj+l for small enough p (since X j , aj+l and pj are positive), and then (ii) the inequalities p j p ' j ~ ' j I aj+l& 5 aj+l~'j+l hold for 0 < E 5 Zj, because A j 2 Aj+l and E;. I 1. Then we may pick a p such that (6) holds, and replace the numbers a j , pj, and the family ' p j = { ( p z , E } c ~ p ~Ej l~, ~ ~ by , E the i ~ numbers ajnew, Drew and the family = { ~ , ~ e w } c ~ ~ ~ , E where ~ ~ ~ pyew ~ , E =~ p ~ j p '~j , , ajnew ~ ~ e=w% P ~ A,'3 , c -new j - pcj, pjinew
qew = min(l,p-lej), E
and
(p-$""
=
% ' jC , P E
[O,q""]
whenever c E it follows that
and
E ] O , q e w ] . Then, if we let ( p $ > n e w ( ~ , ~=) cp{;?'(z),
@new ( C , J * ( f + a j n e w E X j ) + h )
= E,(f+pj"ewEXj)+h+(pE)(p-lc)w+o(E+
llhll) ,
so that
&(f+ ajnew&)+ h ) = E*(f+ pYew~'j)+ h + ECV
(pj,new ( c ,
qew
+
O(E
+ llhll).
This means that the 6-tuple (ajnew, Drew, ' E-new j 7 N j ) is also a generator of (vj,f,+) and, after ( a j ,pj, C j , E j , ( p j , N j ) is replaced by (ajnew,Prew , c- nje w , E~ -new , ( ~ j ~ ~ ~ the~ desired , N j )inequality , (5) holds for our @lnew,
particular j . To get the inequality to hold for all j , we just carry out the replacements recursively, starting with j = m - 1 and moving backwards up to j = 1. Finally, when this is finished, we replace all the cj by their minimum, and do the same for the E j , thus obtaining a new family { ( a j ,pj, Zj, E j , @,Nj)}j=l,...,, of generators of the (vj,f,+) that satisfy
349
(A1,2,3). To get (A4) and (A5) to hold as well, we let N""" = Rn(nG,Nj), and replace each Nj by N""" and each family cpj by the family $ of the restrictions of the (pj, to [O,C] x N""". We then observe that the 6tuples ( a j ,pj , F j , E j , $, N""") are also generators of the (vj, f,+) that satisfy (A1,2,3,4). Finally, we make t, smaller, if necessary, to guarantee that the set {&(t) : f I t 5 t,} is contained in N""", and then make E smaller, if necessary, to satisfy (A5). n = We then use K to identify R with an open subset of R"-where dimX. Then all the tangent spaces T,X, for all z E 0, are identified with R".Since
{
lim sup 117- ~[w. 11 : y E gt,,, s E tlE
[t,tl } = o ,
(7)
we may assume, after making t, and E even smaller, that
llyll
I2
whenever y E gt,, and f I s
I t 5 t, .
(8)
We now use the fact that {gt,s}f5sltlt,is a uniform AGDQ of the maps at the points (&(s),<,(t))in the direction of X to choose a function 0 E 9 which is an AGDQ modulus for all the 4-tuples ( f t , , , & ( s ) , & ( t ) , X ) , for all s , t such that f 5 s 5 t 5 t,. We then fix a real number 8 such that (i) 0 < E 5 E , (ii) 0(E) 5 1, and (iii) the closed ball Bn(J,(s),8)is contained in R for all s E [f,t,]. We then choose, for each E E ] O , E ] and each pair ( s , t ) such that f 5 s 5 t 5 t,, a CCA map A:,, : B " ( ~ , ( S ) , E ) Aff(Rn,Rn), taking values in g$"""', such that & ( t ) + A ( h ) E ft,,(&(s) h). whenever h E @"(O,E) and A E A:,,(<,(s) h). We define set-valued maps &,, : a"(&(s),c) R",for E E I O , ~ ] ,f6 s 5 t 5 t,, by letting ft,s
H )
+
+
H)
$,,(<*W + h ) = E(t) + A4,s(h)(h)
+
(9)
+
+
(that is, A:,,(E*(s) h ) = { < ( t ) A ( h ) : A E AZ,,(J,(s) h ) } ) for h E Bn(O,&). It is then clear that A:,, E CCA(@"(<,(S),E),R~), and the estimate
I ~ Y - E*(t)ll I 4 l l x - t*(s)ll
whenever x E ~ " ( J * ( s ) , EY)E, A:,,(.) (10)
holds. In particular, &,,(i~(<,(s),p))
c ~ n ( t , ( t ) , 4 p )E R
-
E
o < p I E I -4.
(11)
B n ( < , ( s ) , ~ )and 4s 5 8.
(12)
if
In addition, it is clear that
AZ+(x)C ft,,(x)
whenever
IL: E
350
Next, we pick positive numbers ~ , , j= & , , j ( ~ , ( p j )that satisfy the properties of Remark 5.3 for the @, and are such that ~ , , j 6. We let E* = min{&,,j : j = 1,.. . , m}. It then follows that
s
,p{(c,z)
CR
if
E
I E + , 0 I c 5 E, f l t
I f+pj&?,z E @(&(t),&*). (13)
We then define the errors Ej by E j ( c , & , h , y )= y - E*(E+ @ex) - h - E C V ~ .
for y E R", E ~ ] O , e * , j ( r c , ( p j ) ] , c E [ O , E ] , and h E R", so E j ( c , ~ , h , y E) R". We then let C,(E), for 0 < E I E,, be the supremum of the numbers IIEj(c,p,h,y)II taken over all c E [ O , E ] , j E (1, . . . ,m } , h E R", p E ] O , E ] , such that llhll I E , and y E ,pj(c,&(E+ a j & ) h). We define &(&) = ~up{p-~C,(p): 0 < p I E } for 0 < E I E,, and 8,(&) = +m for E > E,. We then observe that the function 8, belongs to 8. Now, if E E ] O , E , ] , c E [ O , E ] , 0 < p I E , z E @(<,(E+ a j ~ ~ j ) ~and p ) , y E & ( c , z ) , we have I l y - J , ( t + p j e X j ) - h - & ~ j I I I &(E) I &8,(&), where h = z - &(t+ a j a x j ) . Since llhll 5 p, we conclude that
+
IIY - rdt+ P
5 P c+ E ( C I~I V ~ I I+~e,(&)). 5 E, and &(&#) 5 1, and let C
~
We fix E# such that 0 < E# Emax(llvlll,. . . , llwmll) &(&#). Then
+
(&
€10, &#] A c E [0,E] A z E @(J,(f+ IIy - < * ( t + p j E X j ) I I
aj&'j),p )
A y E ,pi(c, x))
=
+
I p+Ce
so (&
€lo,&#]A c E [ O , E ] A z E @ ( ~ . ( t + a j e X j ) , p ) ) ,p{(c,z)
c lE(E*(f+pj&.xj),p+CE).
(14)
In order to construct our variation W , we first define, for 0 < E 1.E # , ).(O-'E
= a;+alEA,,,(z),
QZ(c1,z) = , p ; ( c l 1 z 3 4 )
(that is, @ ( c l , z ) = U(pZ(c1,y) : y E Z:(x)})and then define, recursively,
(that is, 9 i + l ( c l , . . . , cj + l , z) is the union of the sets ,pZ+'(cl,. . . , c j, y ) for all y E ~ Z ( Q i ( c 1.,. . , cj, x))) for j = I , . . . , m - 1.
351
Next, we define ra(C1,...,Cm,X) =&,,f+,,,>,
(~~(cl,**.,cm,~)).
Successive applications of (11) and (14) show that a
a
if 0 < p and 4p I E , then Z:@(&(E), p ) ) if 0 < p and 4p Cc 5 E , then
+
Q:([o,E] x
C B(&(f+ QIE’~),
B(<.(Q,P)) G 1 ( 5 * ( f + P l E ” ) , 4 P + C E )
4p),
7
ifO<pand16p+4C~<~,then
E ~ @ ( E , ( f + p l ~ ’ ~ ) , 4 p + C ~C)B) ( & ( ~ + Q Z E ’1~6, p + 4 C ~ ) , a
ifO<pand16p+5C~<~,then *z([0,E]2 x B(&(Q,p))
a
CB(~*(f+Pi&”),16~+5C~),
ifO<pand16p+4C~5~ then ,
Ez(B(&(f+ PzE”), 4p
+ C E ) )G B(&(f+
64p
and so on, so that, for every j E (1,.. . , m } , if we let Gj then a
if 0 < p and 4jp
+~OCE),
=
3-l(4j - l),
+ G ~ C 5E E , then
Qi ( [ 0 , E ] j x B(&(Ej, p)) C B(&(f+
P j ~ ’ j ) ,4jp
+G ~ C E ) .
In particular, a
if 0 < p and 4mp x
9,“ ( [ O ,
+ GmCe 5 E , then B((* (E), p)) C B(E* (f + PmEAm),4mp + GmCE).
We now choose E# and E so that C 5 a
if 0
< 4mp 5
5 and E 5 E # ,
&-,
and conclude that
then
Q i ( [ O , E ] J x B(C*(E),P))
c B(C,(f+PjEAj),E).
jFrom now on, for each E €10, E # ] we fix p = P ( E ) = 2 - 1 - 2 m ~ so , 0 < 4mp 5 $, define Q{ = [O,$ x B ( & ( f ) , p ) , and let @?, be the restrictions of to QF. Then $F and ‘fEare set-valued maps from Qeto
QF, re,
f’F,
352
D ( & ( f + ,B,E'~),
$7
and
and
E)
B(&(f + ,O,E'~), 4 ~ )respectively. , Furthermore,
Y Eare composites of CCA maps, so GF E C C A ( Q , , ~ ( ~ , ( ~ + P , & X " ' ) , & ) ) Y EE CCA(QE,IfB(&(t,),4e)) .
In addition, it is easy to see that
p) . Tjrg(Ol x) & f,.,f(z)whenever 0 < E 5 E # , x E @(t*(f), We now define QZ = B(&(f), p ( ~ ) )a,0 = f. For j = 1,.. . ,m, we write Tj
=t+a.j&
,
+ , Zj = (c1,. . , C j ) , cj . w =c17J+
aj = t + a j E ' j
*
Lemma 6.1. There exists a family I? = { I ? ~ } j = O , ~ , . . . , , QZ ~f Aff(Rnl R") such that if (Zj, z) E QZ then
g ( z j 1 x )= ~ *
*
*
+cjwj .
of CCA maps I?: :
(a~j)+r~(~~,~ .w)). ) ( ~ - -(15) ~(f)+&
Furthermore, I? can be chosen so that
(#) there exists a family {Oj}j=o,l,.,.,m of members of 8 such that, for each j ,
q(+4 c Q o j , t
for all (Z',x) E QZ.
(ejF)'E)
Proof of Lemma 6.1.We define the set-valued maps I?; and the functions Oj, for j = 0 , 1 , . . . ,m, recursively. We first let I?: : Q: H Aff(Rn,Rn) be the map such that I'Z(x) = I I w ~for each x, and take 60 to be any member of 8 (for example, O o ( E ) = E ) . Next, we carry out the inductive step. We pick a j E (1,. . . , m } and has been defined. To construct I?:, we begin by letting assume that
P
= Aff(R",
R") x Aff(R", R") x Aff(R", R") x R" x R" ,
and defining M i be the set-valued map from QZ to P that sends each point (Z',x) E QZ to the subset MZ(Z',x) of P that corisists of all the 5-tuples (Ao, Al, Aa, u,w) for which
A0 E I?:-'(Z+1,x), u = &(gj-i) Ao(x - & ( f )
+
+
~ ( Z j - 1*
u))
(16) (17)
A1 E AfjrojJ4 w = C * ( T j ) + Al(U - [*(Cj-l)),
(18)
A E , j , T j (~t*(~j)).
(20)
1
442 E
(19)
353
We then define ri(Zj,x)to be the set of all A E A f f ( R n , R n )such that
A
= A2
0
A1 0 AO+ a h , , ,
[*(O + +
for some z E q i ( c j ,w ) - E*(oj)- (A2 o A1 o Ao)(zE ( Z ~. w)) and some ( A O , A ~ , A ~ , Z L E ,MZ(Zj,x). W) (Recall that if L E Lin(RP,RQ)and z E RQ then affmL,, is the affine map RP 3 z H L . z z E RQ.) If A E I'i(Zj,x), then there exist Ao, A1, A2, u,w , z , y, w, such that (Ao,A1,A2,u7w)E Mi(Zj,x), w = E * ( T ~ ) Ai(u - &(oj-i)), z = Y &(oj) - (A2 0 A1 0 A o ) (-~ S*(O ~ ( 3 '. v)), y E q i ( c j , w ) , and A = A2 o A1 o A0 affm,,,. It follows that
+
+
A(.
+ - c*(f) + E Z ~W) = Az(Ai(Ao(x- & ( f ) + E(Z' *
= (A2 0 A1 0 Ao)(x - c*(fl+
. w)))) + z E(Z' . v)) + y
- & ( g j ) - ( A z o A 1 o A ~ ) ( x - & ( +f )~ ( Z j * v ) ) =Y -E * b j )
7
+
so that A(. - S*(f) E ( Z ~. v)) E cpZ(c1,w ) - & ( a j ) . Furthermore, since w = E*(~j)+Al(u-E,(oj_l)) and A1 E A;j,uj-l(u), we see that w E E:(u). SO A(x-&(f)+~(C;..w)) E qi(cj,ZZ(u))-[*(oj). Since A0 E I':-'(Zj-l,x), and (15) holds for j - 1, so that E * ( ~ ~ +- r~; )- l ( z j + x ) ( x - t * ( f )
+ E ( Z ; -.~v ) ) = Q ~ - ' ( Z ~ + Z ) ,
it follows from (17) that u E Qi-'(E''-1,x). Therefore A(X-J,(O+E(Z~.W)) E
~ ( c j , Z i ( Q Z - l ( Z j - l , x )- <) *)( C j ) , so A ( X - < , ( O + E ( Z ' * WE) )@ ( Z j , ~ )- - & ( o j ) , and then E * ( o j ) + A ( z - ~ , ( O + ~ ( Z j . w ) ) E Qi(Zj,x).Since A is an arbitrary member of r:($,x), we conclude that r * ( o j )+r;(zj,x)(x + ( f )
+E@
.v)) c Q ~ ( z ~ , x ) .
(21)
To prove the opposite inclusion, we pick y E Qi(Zj,x), and find u E E Z i ( u ) such that y E q i ( c j , w ) . Since
Q i - l ( Z j - 1 , ~ ) and w
Z.j,(u)= A7j'6j-1(u)= C * ( T j )
+ A7j,uj-l(4(u- S * ( C j - l ) )
+
1
we can find A1 E A T j , u j - l ( usuch ) that w = & ( ~ j ) Al(u - &(oj--i)). Since u E Qi-l(Zj-l,z), the inductive hypothesis (i.e., that (15) holds for j - 1) implies that we can pick A0 E r:"(Zj-l,x) such that u = &(oj-1) + Ao(x - [*(Q E ( Z ~ - 1 . v)). Pick an arbitrary member A2 of A n j , T j ( w - & ( ~ jThen ) ) . the 5-tuple ( A o , A ~ , A ~ , ubelongs , w ) toMi(Z',z). Let z = y - &(oj) - (A2 o A1 o Ao)(x - & ( f ) E(C;. v)), and define. an
+
+
+
354
affine map A by letting A = A2 o A1 o A0 + afsm,,,. Then A belongs to rz(Zj,z), and y = & ( a j ) A(z - &(i)+ E ( Z ~. w)), so y is a member of [*(aj) +rj,(zj,z)(z- r * ( t ) +E(zj .I).
+
Since y was an arbitrary member of @(Zj,z), we have shown that Qz(c1,z) C &(aj) l?z(E",z)(z- &(i) E ( Z ~. w)). This fact, together with (21), implies that
+
+
~ j , ( ~ =~ C, * (~C T) ~+) rj,(zj,z)(z- &(i)+ &(zj. .)).
(22)
This completes the inductive construction of the , :'I and the proof that (15) holds. We now prove (#), also by induction. We assume that 0,-1 has been defined in such a way that 0,-1 E 8 and ( # j - 1 ) holds. Let A E r$(Z',z). Write A = A2 o A1 o A0 A O ,as~ before, and let A0 = afsmLo,zo,A1 = a f i L l , Z l , A2 = a f s m L z , , , . Then A = afsm,,,, where L = L ~ L I L o2, = L2Llzo Lzzl z2 z . On the other hand, we know from the inductive hypothesis that LO E g;I:/z' and llzoll I 8j-1((E)El
+
+
+ +
and we also know that L1 E gTj,gj-l, e(&) L2 E g!iE?J-l, IIz111 I O ( E ) E , and llz~llI 8 ( & ) E . Then (8) implies that llL0ll I 2 O j - l ( E ) , llLlll I 2 O(e), and llL2ll I 2 O(&), so
+
+
L2LlLO E
+
+
$ft", +
+
+
where 8, E 8. (Precisely, 8, = 88 4Oj-1 402 8OBj-1 3020+.1.) Also, llLzLlz0 L2z1+ z2ll I ~ , ( E ) E , where 8j belongs t o 8. (Precisely, ej = 38 40j-l O2 480j-1 828j-l.) As for z , we can estimate it as follows: we have
+
+ + +
+
355
Furthermore,
356
Hence, if we define ej(&)= max(8j(E),ej&(&)), it is clear that 0, E 8,and we have shown that ( # j ) holds, completing the proof of Lemma 6.1. We now define, for p" (pl, . . . ,p m ) E "I;", z E B(<*(f),P ( E ) )
~g(x= ) W ( p ' , x )= { T , ( ~ - ' p l , . . . ,&-'pm, x) : E
2 E-'
max(p1,. . . , p m ) ) ,
so each W g is a set-valued map, Gr(Wo) 2 Gr(ft*,F), and Gr(W,) C Gr(fl*,t). Then W = { W g } g E ~is~a variation of fT,t in f:,F. Also, given any positive E , the map [O, 4" x
B(C*(f),p(e)) 3 (pi x) H Z&(F,x ) ~ f T . , ( E - l g , x )
is a CCA map whose graph is contained in that of W . In addition, if we let 2, be the set-valued map that sends each point ( p i x ) E [O,&]m x @(S*(f), P ( E ) ) to the set
{(A,B): B E ~ & ( E - ' ~ , ~ ) , A E A ~ . , , ~ ( E * ( ( T ~ ) + B ( ~ - E * ( ~
+
(where urn = f as before), and use p to denote the map Aff(Rn,Rn) x Aff(Rn,Rn) 3 ( A , B ) H A o B E Aff(Rn,Rn), then the composite map 2, = p o 2, is a CCA map such that
+
+
ZE(3,x) = {E*(t*) M ( x - t * ( f ) p'. v) : M E
3)).
Let us now recall that, if y is a compatible selection of g, and V = {(q, f, +), . . . , (v,,f, +)}, then L v ~ y ~ f is ~ tthe * linear map ytYt,,t o i,where is the map (p',,) H h + j ? + v . We also recall that A V ~ g ~ Fis~ tthe * set of all maps L V ~ y ~ t for ~ t *all , yE CSel(g).The definition of 2, can be rewritten as
z&(p',x) = {t*(t*) + M ( i ( x- C*(O,PI) : M
E 2E(p',x))'
If M E 2,(pix), then M is the composite of a member B of I'~(E-lp',x) followed by a member A of At*,,,(y), where y = &(am)+B(x-&(f)+p',. v). If we write B = ajj'?nBl,bl,we know that B1 E g:",:' and Ilblll 5 em(&)&. Also, if A = ajj'?nAl,al,we know that A1 E gts,';;, and llalII I B ( E ) E . It follows that, if M i = ajj'?nK,k,then K = A l B l L and k = a1 Albl.
+
Therefore K E (gt,,t o i)'$(') and llkll I O $ ( E ) , where @(&) is an easily computable member of 8.(Precisely, we may take 8% = 2(1 Ili/l)(e
em + es,).)
+
+
~i E gt+ Now, the set gt,,t o i is precisely AV)g>t>t*. This shows that A V ~ g ~ Fis~ t * SO
(''(E),,)
an AGDQ of the map W at ( ( O , & ( f ) ) , & ( L ) ) in the direction of x X. If, for b 2 t,, we define W b= { W $ } F ~ Rby T ,letting W$(x) = Wb(@,x) =
357
Wp)(z)),then it is clear that W b is a variation of f b , f in f : and AV>gl'>bis an AGDQ of W bat ((0, &(f))l J,(b)) in the direction of R.;L x X . This completes our proof. 0 (fb,t, o
References 1. R. M. Bianchini, Variational approach to some optimization control problems, in Geometry in nonlinear control and differential inclusions (Warsaw, 1993), eds. B. Jacubczyk, W. Respondek and T. Rzeiuchowski, Banach Center Publ. Vol. 32 (Polish Acad. Sci. Inst. of Math., Warsaw, 1995), pp. 83-94. 2. R. M. Bianchini and G. Stefani, SIAM J. Control and Optim. 31,900 (1993) 3. R. M. Bianchini and G. Stefani, Good needle-like variations, in Differential geometry and control (Boulder, CO, 1997), Proc. Symposia Pure Math. 64 (American Mathematical Society, Providence, RI, 1999), pp. 91-101. 4. G. Stefani, Higher order variations: how can they be defined in order to have good properties?, in Non-smooth Analysis and geometric methods in deterministic optimal control (Minneapolis, MN, 1993), eds. B. s. Mordukhovich and H. J. Sussmann, IMA Vol. Math. Appl. Vol. 78 (Springer, New York, 1996), pp. 227-237. 5 . H.J. Sussmann, RBsultats recents sur les courbes optimales, in 15e Journe'e Annuelle de la Socie'te' Mathe'mathique de France (SMF) (Publications de la SMF, Paris, 2000), pp. 1-52. 6. H. J . Sussmann, New theories of set-valued differentials and new version of the maximum principle of optimal control theory, in Nonlinear Control in the Year 2000, Vol. 2, Lecture Notes in Control and Inform. Sci., 259, eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek (Springer-Verlag, London, ZOOO), pp. 487-526. 7. H. J. Sussmann, Combining high-order necessary conditions for optimality with nonsmoothness, in Proc. 43rd IEEE Conf. Decision and Control (Atlantis, Paradise Island, Bahamas, 2004). 8. H.J. Sussmann, Optimal control of nonsmooth systems with classically differentiable flow maps, in Proceedings of the Sixth IFAC Symposium on Nonlinear Control Systems (NOLCOS ZOO4), Vol. 2, (Stuttgart, Germany, September 13, 2004), pp. 609-704. 9. H.J. Sussmann, Generalized differentials, variational generators, and the maximum principle with state contraints, in Nonlinear and Optimal Control Theory (Lectures given at the C.I.M.E. Summer School held in Cetraro (Cosenza), June 21-29, 2004), Lecture Notes in Mathematics, eds. G. Stefani and P. Nistri (Springer-Verlag (Fondazione C.I.M.E.), to appear). 10. H.J. Sussmann, J. Differential Equations 243,448 (2007).
This page intentionally left blank
359
LIST OF PARTICIPANTS
1. ANDREIAGRACHEV[Trieste (IT)] .................. agrachevosissa.it 2. FABIO ANCONA[Bologna (IT)] . . . . . . . . . . . . . . . . . anconaQciram.unibo.it 3. ZVI ARTSTEIN [Rehovot (IL)] . . . . . . . . . . . zvi.artsteinQweizmann.ac.il 4. ANDREABACCIOTTI[Torino (IT)] . . . . . . . . andrea.bacciottiQpolito.it 5. MARTINOBARDI [Padova (IT)] . . .......... bardihath.unipd.it s)] . . rbarnardhath.lsu .edu 4. RICHARDBARNARD[Baton Rouge, 6. JERROLD BEBERNES[Boulder, CO (US)] ...... bebernesQcoloraodo.edu 7. UGO BOSCAIN[Trieste (IT)] ......................... boscainQsissa.it 8. ALBERTOBRESSAN[University Park, PA (US)] . . bressanOmath.psu.edu 9. ROGERBROCKETT[Cambridge, MA (US)] brockettQdeas.harvard.edu 10. FABIO CAMILLI[L’Aquila (IT)] . . . . . . . . . . camilliQing.univaq.it 11. PIERMARCO CANNARSA [Roma (IT)] .... arsahat .uniroma2.it 12. ITALO CAPUZZODOLCETTA[Roma (IT)] capuzzoQmat.uniromal.it 13. PIERRE CARDALIAGUET [Brest (FR)] Pierr iaguetQuniv-brest.fr 18. MARCOCASTELPIETRA[Roma (IT)] . . . castelpiQaxp.mat,uniroma2.it 15. ARRIGOCELLINA[Milano (IT)] . . . . . . . . . . . . arrigo.cel1inaQunimib.it 16. FRANCESCA CERAGIOLI[Torino (IT)] francesca.ceragioliQpolito.it 17. FRANCESCA CHITTARO[Firenze (IT)] ........ chittaroQmath,unifi.it 18. FRANCIS CLARKE[Lyon (FR)] . . . . . . . . . . . . clarkehath.univ-lyonl.fr 19. GIOVANNI COLOMBO[Padova (IT)] . . . . . . . . . . . . colombo0math.unipd.it 20. LUCACONSOLINI[Parma (IT)] ....... luca.consoliniQpo1irone.m.it 21. JEAN-MICHEL CORON [Paris (FR)] jean-michel.coronhath .u-psud.fr 22. MARCOCZARNECKI[Montpellier (FR)] ......... marcohniv-montp2.fr 23. WIJESURIYA DAYAWANSA [Lubbock, TX (US)] .......................... 24. ASEN DONTCHEV[Ann Arbor, MI (US)] . . . . . . . . . . . . . . . . . . aldQams.org 25. BRUNOFRANCHI [Bologna (IT)] ................ franchibQdm.unibo.it 26. HELENEFRANKOWSKA [Paris (FR)] . . . . frankoQshs.polytechnique.fr 27. DANIELGESSUTI [Trieste (IT)] ...................... gessutiQsissa.it 28. LARSGRUNE [Bayreuth (DE)] . . . . . . . . . . lars .grueneQuni-bayreuth.de 29. ALVAROGUEVARA[Baton Rouge, LA (US)] . . . aguevaraamath.lsu.edu 30. HENRYHERMES [Boulder, CO (US)] . . . . . . . . . . . . . . hankhermesQmsn.com 31. MATTHIASKAWSKI [Tempe, AZ (US)] . . . . . . . . . . . . . . . . . kawskiQasu.edu 32. EYUPKIZIL [Trieste (IT)] ............................. kizilQsissa.it 33. ARTHURJ . KRENER [Davis, CA (US)] .......... ajkreneroucdavis.edu 34. YURILEDYAEV[Kalamazoo, MI (US)] . . ....... 1edyaevQwmich.edu 35. SOFIALOPES [Guimarks (PT)] . . . . . . . . . . . . sofialopeshct .uminho.pt
360
36. CLAUDIO MARCHI [Cosenza (IT)] . . . . . . . . . . . . . . . marchiomat .unical.it 37. ANTONIOMARIGONDA[Pavia (IT)] . . . . . . antonio.marigondaQunipv.it 38. PAOLO MASON [Nancy (FR)] . . . . . . . . . . . Paolo .MasonQiecn.u-nancy.fr 39. FABIO MORBIDI[Siena (IT)] .................... morbidiQdii .unisi.it 40. JACQUELINE MORGAN[Napoli (IT)] ..................morganQunina.it 41. MONICAMOTTA [Padova (IT)] . . . . . . . . . . . . . . . . . . mottabath.unipd.it 42. CHADINOUR [Byblos (LB)] ......................... cnourQlau.edu.1b 43. NORMAORTIZ [Richmond, VA (US)] ................. nlortizQvcu.edu 44. BENEDETTO PICCOLI[Roma (IT)] .............. b.piccoli0iac.cnr.it 45. LAURAPOGGIOLINI [Firenze (IT)] ........ 1aura.poggiolinihnifi . it 46. FABIO PRIULI[Trondheim (NO)] ................. priulibath.ntnu.no 47. FRANCO RAMPAZZO [Padova (IT)] ........... rampazzoQmath.unipd.it 48. LUDOVICRIFFORD [Nice (FR)] . . . . . . . . . . . . . . . . riffordQmath.unice.fr 49. VINICIORIOS [Maracaibo (VE)] .................... riosQmath.lsu.edu 50. R. TYRRELL ROCKAFELLAR [Seattle, WA (US)] rtrbath .washington.edu 51. SERGIORODRIGUES [Trieste (IT)] .................. srodrigsQsissa.it 52. FRANCESCO ROSSI [Tkieste (IT)] .................... rossifrQsissa.it 53. REBECCASALMONI[Paris (FR)] .... rebecca.salmoniQlss.supelec.fr 54. CATERINA SARTORI[Padova (IT)] ........ caterina.sartoriQunipd.it 55. ANDREYSARYCHEV[Firenze (IT)] . . . . . . . . . . . . . . . . asarychevQunifi.it 56. ULYSSESERRES [Nancy (FR)] ... . . ulysse . serresQiecn.u-nancy .fr 57. CARLOSINESTRARI[Roma (IT)] . . . . . . . . . . . sinestrabat.uniroma2.it 58. MARCOSPADINI[Firenze (IT)] ...............marco.spadiniQunifi.it 59. GIANNA STEFANI [Firenze (IT)] ............. gianna.stefaniQunifi . it 60. HECTORSUSSMANN[Piscataway, NJ (US)] sussmannhath.rutgers.edu 61. GABRIELETERRONE [Padova (IT)] . . . . . . . . . . . . . gabterbath.unipd.it 62. MARIOTOSQUES [Parma (IT)] ............... mario.tosquesounipr.it 63. RICHARDVINTER [London (GB)] . . . . . . . . . . . r.vinterQimperia1.ac .uk 64. PETERWOLENSKI[Baton Rouge, LA (US)] . . . . wolenskiamath.lsu.edu
361
AUTHOR INDEX Alvarez, O., 1 Ancona, F., 28 Artstein, Z., 65 Bacciotti, A., 82 Bardi, M., 1 Boscain, U., 100 Bressan, A., 28 Cannarsa, P., 120 Capuzzo Dolcetta, I., 136 Cardaliaguet, P., 120 Castelpietra, M., 120 Clarke, F.H., 151, 164 Garavello, M., 177 Girejko, E., 177 Goebel, G., 280 Griine, L., 206 Marchi, C., 1 Mason, P., 228 Nour, C., 243 Ortiz, N.L., 248 Piccoli, B., 177 Rifford, L., 260 Rios, V.R., 270 Rockafellar, R.T., 280 Stefani, G., 300 Sussmann, H., 327 Vinter, R.B., 164
Wolenski, P.R., 270 Worthmann, K., 206
This page intentionally left blank
Series on Advances in Mathematics for Applied Sciences Editorial Board N. Bellomo Editor-in-Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy E-mail: [email protected]
F. Breui Editor-in-Charge lstituto Universitario di Studi Superiori Via Luino 4 1-27100 Pavia Italy E-mail: [email protected]
M. A. J. Chaplain Department of Mathematics University of Dundee Dundee DDI 4HN Scotland
S. Lenhart
C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence, RI 02912 USA
P. L. Lions University Paris XI-Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France
J. Felcman Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University in Prague Sokolovska 83 18675 Praha 8 The Czech Republic
0. Perthame Laboratoire J.-L. Lions Universitc!P. et M. Curie (Paris 6) BC 187 4, Place Jussieu F-75252 Paris cedex 05, France
M. A. Herrero Departamento de Matematica Aplicada Facultad de Matemdticas Universidad Complutense Ciudad Universitaria s/n 28040 Madrid Spain
S. Kawashima Department of Applied Sciences Engineering Faculty Kyushu University 36 Fukuoka 812 Japan M. Lachowicz
Department of Mathematics University of Warsaw UI. Banacha 2 PL-02097 Warsaw Poland
Mathematics Department University of Tennessee Knoxville, TN 37996-1 300 USA
K. R. Rajagopal Department of Mechanical Engrg. Texas A&M University College Station, TX 77843-3123 USA R. Russo Dipartimento di Matematica II University Napoli Via Vivaldi 43 81100 Caserta Italy
Series on Advances in Mathematics for Applied Sciences Aims and Scope This Series reports on new developments in mathematical research relating to methods, qualitative and numerical analysis, mathematical modeling in the applied and the technological sciences. Contributions related to constitutive theories, fluid dynamics, kinetic and transport theories, solid mechanics, system theory and mathematical methods for the applications are welcomed. This Series includes books, lecture notes, proceedings, collections of research papers. Monograph collections on specialized topics of current interest are particularly encouraged. Both the proceedings and monograph collections will generally be edited by a Guest editor. High quality, novelty of the content and potential for the applications to modern problems in applied science will be the guidelines for the selection of the content of this series.
Instructions for Authors
Submission of proposals should be addressed to the editors-in-charge or to any member of the editorial board. In the latter, the authors should also notify the proposal to one of the editors-in-charge. Acceptance of books and lecture notes will generally be based on the description of the general content and scope of the book or lecture notes as well as on sample of the parts judged to be more significantly by the authors. Acceptance of proceedings will be based on relevance of the topics and of the lecturers contributing to the volume. Acceptance of monograph collections will be based on relevance of the subject and of the authors contributing to the volume. Authors are urged, in order to avoid re-typing, not to begin the final preparation of the text until they received the publisher's guidelines. They will receive from World Scientific the instructions for preparing camera-ready manuscript.
SERIES ON ADVANCES IN MATHEMATICS FOR APPLIED SCIENCES
Published*: Vol. 62
Mathematical Models and Methods for Smart Materials eds. M. Fabrizio, B. LazzariandA. Morro
Vol. 63
Lecture Notes on the Discretization of the Boltzmann Equation eds. N. Bellomo and R. Gatignol
Vol. 64
Generalized Kinetic Models in Applied Sciences - Lecture Notes on Mathematical Problems by L. Arlotti et al.
Vol. 65
Mathematical Methods for the Natural and Engineering Sciences by R. E. Mickens
Vol. 66 Advanced Mathematical and Computational Tools in Metrology VI eds. P. Ciarlini et a/. Vol. 67
Computational Methods for PDE in Mechanics
byB. D'Acunto Vol. 68
Differential Equations, Bifurcations, and Chaos in Economics by W. B. Zhang
Vol. 69
Applied and Industrial Mathematics in Italy eds. M. Primicerio, R. Spigler and V. Valente
Vol. 70
Multigroup Equations for the Description of the Particle Transport in Semiconductors by M. Galler
Vol. 71
Dissipative Phase Transitions eds. P. Colli, N. Kenmochi and J. Sprekels
Vol. 72 Advanced Mathematical and Computational Tools in Metrology VII eds. P. Ciarlini et al. VOl. 73
Introduction to Computational Neurobiology and Clustering by B. Tirozzi, D.Bianchi and E. Ferraro
VOl. 74
Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure by C. Cattani and J. Rushchitsky
Vol. 75
Applied and Industrial Mathematics in Italy II eds. V. Cutello et a/.
Vol. 76
Geometric Control and Nonsmooth Analysis eds. F. Ancona et a/.
*To view the complete list of the published volumes in the series, please visit: http://www.worldscibooks.com/series/samas-series.shtml
