Analysis and control of nonlinear infinite dimensional systems, Volume 190 (Mathematics in Science and Engineering)

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Analysis and Control of Nonlinear Infinite Dimensional Systems

This is volume 190 in MATHEMATICS IN SCIENCE AND ENGINEERING Edited by William F. Ames, Georgia Institute of Technology A list of recent titles in this series appears at the end of this volume.

ANALYSIS AND CONTROL OF NONLINEAR INFINITE DIMENSIONAL SYSTEMS Viorel Barbu SCHOOL OF MATHEMATICS

UNIVERSITY OF IASI IASI, ROMANIA

ACADEMIC PRESS, INC. Harcourt Brace Jooanooich, Publishers

Boston San Diego New York London Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright 0 1993 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC.

1250 Sixth Avenue, San Diego, CA 92101-4311

United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX

Library of Congress Cataloging-in-PublicationData

Barbu, Viorel. Analysis and control of nonlinear infinite dimensional systems Viorel Barbu. p. cm.-(Mathematics in science and engineering; v. 189) Includes bibliographical references and index. ISBN 0-12-078145-X 1. Control theory. 2. Mathematical optimization. 3. Nonlinear operators. I. Title. 11. Series. QA402.3.B343 1993 003'.5-d~20

PRINTED 92939495

IN THE UNITED STATES OF AMERICA

EB

9 8 7 6 5 4 3 2 1

92-2851 CIP

Contents

vii ix

Preface Notation and Symbols Chapter 1 Preliminaries 1.1 The Duality Mapping 1.2 Compact Mappings in Banach Spaces 1.3 Absolutely Continuous Functions with Values in Banach Spaces 1.4 Linear Differential Equations in Banach Spaces

9 17

Chapter 2 Nonlinear Operators of Monotone Type 2.1 Maximal Monotone Operators 2.2 Generalized Gradients (Subpotential Operators) 2.3 Accretive Operators in Banach Spaces Bibliographical Notes and Remarks

35 35 57 99 123

Chapter 3 Controlled Elliptic Variational Inequalities 3.1 Elliptic Variational Inequalities. Existence Theory 3.2 Optimal Control of Elliptic Variational Inequalities Bibliographical Notes and Remarks

125 125 148 196

Chapter 4 Nonlinear Accretive Differential Equations 4.1 The Basic Existence Results 4.2 Approximation and Convergence of Nonlinear Evolutions and Semigroups 4.3 Applications to Partial Differential Equations Bibliographical Notes and Remarks

199 199

Chapter 5 Optimal Control of Parabolic Variational Inequalities 5.1 Distributed Optimal Control Problems 5.2 Boundary Control of Parabolic Variational Inequalities 5.3 The Time-Optimal Control Problem V

1 1 5

240 255 311 315 315 342 364

vi

Contents

Approximating Optimal Control Problems via the Fractional Steps Method Bibliographical Notes and Remarks 5.4

Chapter 6 Optimal Control in Real Time 6.1 Optimal Feedback Controllers 6.2 A Semigroup Approach to the Dynamic Programming Equation Bibliographical Notes and Remarks References Subject Index

389 404 407 407 433 456 459 475

Preface In contemporary mathematics control theory is complementarily related to analysis of differential systems, which concerns existence, uniqueness, regularity, and stability of solutions. In fact, as remarked by Lawrence Markus, control theory concerns the synthesis of systems starting from certain prescribed goals and the desired behavior of solutions. We tried to write this book in this dual perspective: analysis-synthesis having as its subject the class of nonlinear accretive control systems in Banach space. Since its inception in the 1960s the theory of nonlinear accretive (monotone) operators and of nonlinear differential equations of accretive type has occupied an important place among functional methods in the theory of nonlinear partial differential equations, along with the Leray-Schauder degree theory. Its areas of application include existence theory for nonlinear elliptic and parabolic boundary value problems and problems with free boundary. The optimal control problems studied in this book are governed by state equations of the form Ay = Bu + f and y ’ + Ay = Bu + f , where A is a nonlinear accretive (multivalued) operator in a Banach space X , B is a linear continuous operator from a controller space U to X , and u is a control parameter. Very often in applications A is an elliptic operator on an open domain of the Euclidean space with suitable boundary conditions. The cost functional is in general not differentiable, and since the state equation is nonlinear this leads to a nonsmooth and nonconvex optimization problem, which requires a specific treatment. In concrete situations such a problem reduces to a nonlinear distributed optimal control problem, and a large class of industrial optimization processes can be put into this form. In fact, the optimal control theory of nonlinear distributed parameter systems has grown in the last decade into an applied mathematical discipline with its own interest and a large spectrum of applications. However, here we shall confine ourselves to the treatment of a limited number of problems with the main emphasis on optimal control problems with free and moving boundary. Nor is this book comprehensive in any way as far as concerns theory of monotone operators and of nonlinear differential equations of accretive type in Banach spaces. The exposition was restricted to a certain body of basic results and methods along with certain significant examples in partial differential equations. vii

viii

Preface

Part of the material presented in Chapter I11 and V appeared in a preliminary form in my 1984 Pitman Lectures Notes, Optimal Control of Variational Inequalities. This book was completed while the author was Otto Szasz Visiting Professor at University of Cincinnati during the academic year 1990-1991 and the material has been used by the author for a one year graduate-level given course at University of Ia$ and University of Cincinnati. Part of the manuscript has been read by my colleagues and former students, Professor I. Vrabie, Dr. D. Tiitaru, and Dr. S. Anifa, who contributed valuable criticism and suggestions. To Professor Gh. MoroSanu I also owe special thanks for his careful reading of the original manuscript and his constructive comments, which have led to a much better presentation. V. Barbu Ia$, November 1991

Notation and Symbols

the N-dimensional Euclidean space the real line ( - 0 0 , 00) +m), R-= (-a, 03, R = (-03, +m] n open subset of R N JR the boundary of R Q = R x (0, T ) , 2 = J R x (0, T ) where 0 < T < w the norm of a linear normed space X the dual of the space X the space of linear continuous operators from X to Y the gradient of the mapf: X + Y the subdifferential (or the Clarke gradient) of f: X -+ R the adjoint of the operator B the closure of the set C the interior of C the convex hull of C X the signum function on X : sign x = - i f x # O llxllx sign 0 = {x; IIxII I 11 the space of real valued functions on R that are continuously differentiable up to order k, 0 I k I 03 the subspace of functions in Ck(R)with compact support in R the space Ct(R)

RN

R R + = [0,

the derivative of order k of u : [ a , b ] + x the dual of S ( R ) , i.e., the space of distributions on R the space of continuous functions on the space of p-summable functions u : R + R endowed with the norm llullp =

Ilullo;= ess sup lu(x)l for p X€f2

l/P

lu(x)lp dx) =

, 1 I P < w,

00

the Sobolev space { u E LP(R); D"u E L P ( R ) ,IaI _< rn), l < p < m the closure of C;(R) in the norm of W m S P ( R )

x

Notation and Symbols

1 1 the dual of WT*P(fl); - + - = 1. P 4 H"(fl), H i ( fl) the spaces of W k T 2 ( fand l ) W,k,'(fl), respectively. LP(0, T ; X) the space of p-summable functions from (a, b ) to X (Banach space) 1 < p I a,- a < a < b I~0 AC([a, 61; X ) the space of absolutely continuous functions from [ a , b ] to x BV([a, 61; X ) the space of functions with bounded variation on [ a , b ] du E LP(a, b; X I } WkJ'([a,b]; X ) the space { u E AC([a, bl; X I ;

w-rn '4( fl)

Chapter 1

Preliminaries

The aim of this chapter is to provide some basic results pertaining to geometric properties of normed spaces, semigroups of class C,, and infinite dimensional vectorial functions defined on real intervals. Some of these results, which can be easily found in textbooks or monographs, are given without proof or with a sketch of proof only. 1.1. The Duality Mapping

Throughout this section X will be a real normed space and X * will denote its dual. The value of a functional x * E X * at x E X will be denoted by either ( x , x*) or x * ( x ) , as is convenient. The norm of X will be denoted by II 1 , and the norm of X * will be denoted by I1 * II* . If there is no danger of confusion we omit the asterisk from the notation II I[* and denote both the norms of X and X * by the symbol II 11. We shall use the symbol lim or -+ to indicate the strong convergence in X and w-lim or for the weak convergence in X . By w*-lim or we shall indicate the weak-star convergence in X * . The space X * endowed with the weak star topology will be denoted by X,* . Define on X the mapping J : X + 2x':

-

-

-

J(x)

=

{x*

-

€

x*;( x , x * ) = llxIl2 = lIX*1l2}

vx

E

x.

(1.1)

By the Hahn-Banach theorem we know that for every x , E X there is some x,* E X * such that (x,, x,*) = llxoll and Ilx,*ll I 1. Indeed, by the Hahn-Banach theorem the linear functional f: Y + R defined by f ( x ) = a(IxolIfor x = a x , , where Y = { a x , ; a E R}, has a linear continuous 1

2

1. Preliminaries

extension x: E X * on X such that Kx:, x)l I llxll Vx E X . Hence, ( x : , x , ) = llxoll and Ilx;ll I 1 (in fact, Ilx,*ll = 1). Clearly, x~llxollE J(x,) and so J ( x ) # 8’ for every x E X .

The mapping J : X .+ X * is called the duality mapping of the space X . In general, the duality mapping J is multivalued. For instance, if X = L’(fl), where fl is a measurable subset of R N then it is readily seen that every u E L”(fl) such that u ( x ) E sign u ( x ) . IlullLl(n, a.e. x E fl belongs to J(u). (Here, we have denoted by sign the function sign u = 1 if u > 0, sign u = - 1 if u < 0 and sign 0 = [ - 1,1].) It turns out that the properties of the duality mapping are closely related to the nature of the spaces X and X * , more precisely to the convexity and smoothing properties of the closed balls in X and X * . Recall that the space X is called strictly convex if the unity ball B of X is strictly convex, i.e., the boundary dB contains no line segments. The space X is said to be uniformly convex if for each E > 0,O < < 2, there is ti(&) > 0 such that if llxll = 1, llyll = 1, and IIx -yll 2 E , then IIX +yll I 2 0 - ti(&)). Obviously, every uniformly convex space X is strictly convex. Hilbert spaces as well as the spaces L P ( f l ) ,1 < p < m, are uniformly convex spaces (see, e.g., G. Kothe [l]). Recall also that in virtue of the Milman theorem (see, e.g., K. Yosida [l]) every uniformly convex Banach space X is reflexive (i.e., X = X * * ) . Conversely, it turns out that every reflexive Banach space X can be renormed such that X and X * become strictly convex. More precisely, one has the following important result due to E. Asplund [l]. Theorem 1.1. Let X be a reflexive Banach space with the norm II .I[. Then there is an equivalent norm )I * [lo on X such that X is strictly convex in this norm and X * is strictly convex in the dual norm II * 115. Regarding the properties of the duality mapping associated with strictly or uniformly convex Banach spaces, we have Theorem 1.2. Let X be a Banach space. If the dual space X * is strictly convex then the duality mapping J : X + X * is single valued and demicontinuous, i.e., it is continuous from X to X,* . If the space X * is uniformly convex then J is uniformly continuous on every bounded subset of X .

ProoJ

Clearly, for every x E X , J ( x ) is a closed convex subset of X * .

1.1. The Duality Mapping

3

Since J(x) c dB, where B is the open ball of radius llxll and center 0, we infer that if X * is strictly convex then J(x) consists of a single point. Now let {x, 1 c X be strongly convergent to xo and let x: be any weak star limit point of {Jx,}. (Since the unit ball of the dual space is w*-compact (Yosida [l],p. 137) such a x: exists.) We have (xo xg*) 9

2

llxoll 2 llx:l12

=

because the closed ball of radius llxoll in X * is weak star closed. Hence 2 llxoll = llx~1I2= ( xo ,x;). In other words, x i = J(xo), and so

-

J(x,)

J(XO),

as claimed. To prove the second part of the theorem let us establish first the following lemma. Lemma 1.1. Let X be a uniformly convex Banach space. If x, limsup,,, Ilx,ll IIlxll, then x, -+ x as n + CQ.

Proof

By hypothesis, (x,, x*)

+

-

x and

(x, x*) for all x E X, and so

llxll I liminf Ilx,ll I Ilxll. n+m

Hence limn+mIlx,ll

-

= Ilxll.

Y,

Now, we set x,

=

Ilx,ll ’

y

X

=

-.

llxll

Clearly, y, y as n -, m. Let us assume that y, + y and argue from this to a contradiction. Indeed, in this case we have a subsequence y,,, ly,, -yll 2 E , and so there is S > 0 such that lly,, +yll I2(1 - 6). Letting nk + 00 and recalling that the norm y + llyll is weakly lower semicontinuous, we infer that llyll I 1 - 8. The contradiction we have arrived at completes the proof of the lemma. Proof of Theorem 1.2 (continued). Assume now that X * is uniformly convex. We suppose that there exist sequences {u,},{u,) in X such that llu,ll, Ilu,ll I M , IIu, - unll-, 0 for n + CQ, M u , ) - J(u,)ll 2 E > 0 for all n , and argue from this to a contradiction. We set x, = u,llu,ll~ 1 , y, = ~,llu,ll-~. Clearly, we may assume without any loss of generality that Ilu,ll 2 a > 0 and that IIu,lI 2 a > 0 for all n. Then, we can easily see that IIx, - y J

-, 0

as n -, 00

1. Preliminaries

4

and ( x , J(x,) 9

+J(Y,))

=

IIxnI12 + IIynI12 + ( x n

-Yn

9

J(Yn))

2 2 - Ilx, -y,ll.

Hence

+ J(y,)lI

$IlJ(x,)

2 1 - $llxn -Ynll

Vn.

Inasmuch as IIJ(x,)ll = llJ(y,)ll = 1 and the space X * is uniformly convex, this implies that limn~m ( J ( x , ) - J ( y , ) ) = 0. On the other hand, we have

J(u,)

- J(v,) =

ll~,Il(J(X,)

so that limn~m (J(u,) - J(u,)) proof.

-J(y,)) =

+ (llu,ll0

-

llU,Il)J(Y,),

0 strongly in X*.This completes the

Now let us give some examples of duality mappings. 1. X = H is a Hilbert space identified with its own dual. Then J = Z the identity operator in H. If H is not identified with its dual H', then the duality mapping J : H H' is the canonical isomorphism A of H onto H'. For instance, if H = H,'(R) and H' = H-'(R), then J = A is defined bY

( A u , ~=) / - V ~ . V u d r

VU,LJ€H,'(R).

( 1 -2)

In other words, J = A is the Laplace operator - A under Dirichlet boundary conditions in R c RN. 2. X = LP(R), where 1 < p < m and R is a measurable subset of RN. Then, the duality mapping of X is given by

J(u)(x)= I

U(X)I~-~U(X)IIU~~~~~~

a.e. x

E

a,

V u E Lp(R). (1.3)

Indeed, it is readily seen that

where l/p

J(u)

+ l/q =

=

1. If X

{ u E L"(R);

= L'(R),

U(X)

E

then as we will see later

-

sign u ( x ) llull~l(a,a.e. x

E

a } . (1.4)

1.2. Compact Mappings in Banach Spaces

3. Let X = W,,'pP(R), subset of RN.Then,

5

where 1 < p < co and R is a bounded and open

In other words, J : Wb.P(R)

+

W-',q(fl), l / p

+ l / q = 1, is defined by

a

4. Let X = C(fi), where is a compact subset of R N .Then, the duality mapping J: C ( a ) + M ( a ) (the space of all bounded Radon measures on is given by (E. Sinestrari [l])

a)

J(Y)

=

(CLllYllccn); POI) 5 m={y(xo) sign Y(XO), xo CL

VY

E M(.R))

E

E

My},

C(fi),

where

M y= ( x o

E

fi; lY(X0)l = IlYllc(n)}.

We shall see later that the duality mapping J of the space X can be equivalently defined as the subdifferential (Ghteaux differential if X * is strictly convex) of the function x + ~ 1 1 ~ 1 1 ~ . 1.2. Compact Mappings in Banach Spaces Let X, Y be linear normed spaces. An operator T: X + Y is called compact if it maps every bounded subset of X into a relatively compact subset of Y. If D is a bounded and open subset of X and T: D + X is continuous, then for any p E X such that p G F ( d D ) , F = Z - T ( J D is the boundary of D ) , we may define the topological degree of F in p relative to D, denoted d(F, D, p ) having the following properties (the Leruy-Schauder degree): (i)

d(Z,D,p)= 1

if p

E

D,

6

1. Preliminaries

if p 6C D; d(Z,D,p)=0 (ii) If d ( Z , D , p ) # 0, then the equation Fx = p has at least one solution x E D ; (iii) If H ( t ) is a compact homotopy in X such that p 6C H ( t X d D ) V t E [0,TI, then d ( l - H ( t ) , D , p ) = constant V t E [O,T]. (2.1) The construction of the Leray-Schauder topological degree is a classical result, which can be found in most textbooks devoted to nonlinear equations in infinite dimensional spaces (see, for instance, J. Schwartz [ll). Here, we shall confine ourselves to deriving as a simple consequence of Leray-Schauder degree theory an important result of nonlinear analysis, the Schauder fixed point theorem.

Theorem 2.1 (Schauder). Let K be a closed, bounded and convex subset of a Banach space X . Let T : K -+ K be a continuous and compact operator that maps K into itself. Then the equation Tx = x has at least one solution x E K . Proo$ Consider first the particular case where K = (u E X ; llull I R). Consider the homotopy H ( t ) = tT, 0 I tI 1, and note that 0 P (I H(t))dK V t E [O, 0, i.e., u - tTu # 0

for llull = R and t

E

[0,1).

Hence d(Z - H ( t ) , K,O) = constant, and therefore d(Z - T , K,O) f 0, as claimed. In the general case, consider a ball B centered in origin such that K c and take a compact extension f of T on B such that R ( f ) c K ( R ( f )is the range of f).(The existence of a such a map f is a well-known result due to Djugundi, which extends the classical theorem of Tietze.) Hence, f maps B into itself and according to the first part of the proof it has at least one fixed point x E B. Since R(f)c K , we infer that x E K and Tx = x , as claimed. rn It is clear that the following version of Schauder theorem is also true: Let K be a compact convex subset of a Banach space X and let T : K + K be a continuous operator. Then T has at least one fixed point in K. The Schauder fixed point theorem along with the Leray-Schauder degree theory represent powerful instruments in the study of infinite dimensional equations with compact nonlinearities. Their applications include a large spectrum of nonlinear problems in theory of partial

1.2. Compact Mappings in Banach Spaces

7

differential equations, ordinary and integral equations, game theory, and other fields. Here, we shall indicate only one application, to the existence of a saddle point for convex-concave functions (J. von Neumann theorem). First, we shall indicate an extension of Schauder theorem to multivalued operators (Kakutani theorem). Recall that a multivalued mapping T : X + 2' ( X ,Y are metric spaces) is said to be upper semicontinuous (u.s.c.) in x if for every E > 0 there is 6 = 6 ( x , E ) > 0 such that

Here, B ( x , 6 ) is the ball of radius 6 and center x in X, whilst

The mapping T is said to be upper semicontinuous on X if it is upper semicontinuous at every x E X. It is readily seen that the graph of an U.S.C. mapping is closed. Conversely, if R ( T ) is compact and the graph of T is closed, then T is U.S.C. Theorem 2.2 (Kakutani). Let K be a compact convex subset of a Banach space and let T : K + X be an upper semicontinuous mapping with convex values T ( x ) such that T ( x ) c K , Vx E K . Then there is at least one x E K such that x E T ( x ) .

Proo$ We will prove first that for every function f,: K + K such that

E

> 0 there is a continuous

( B( x , E ) X B( f,( x ) , E )) n T Z 0

Vx E K .

(2.3)

In other words, for every x E K there is ( x s , y , ) E K X K such that E Tx,, IIx, - xII I E , Ily, -f,(x)II I E. Indeed, consider a finite cover of K with subsets of the form B ( x i , 6 ( x i , E ) ) = U ; , x i E K , i = 1 , . ..,m , and take a finite subordinated partition of unity { a i } E l ,i.e., 0 I ai I 1, supp ai c U ; , Cy=n=l ai = 1, and the ai are continuous. For instance, we may take

y,

8

1. Preliminaries

where p i ( x ) = max(0, 6 ( x i ,E ) - IIx fine the function

-

xill). Then, pick yi E T ( Q ) and de-

m

L ( x )= C

i= 1

Vx

"i(x)yj

E

K,

which satisfies the condition (2.3). Moreover, f , ( K ) c K. Then, by the Schauder theorem, there is x, E K such that f,(x,) = x,. In other words, for every E > 0 there are ( i eye) , E K X K such that Ilf, - x,II I E , Itye - f&(x,)Il I E and 9, E lX,. Since the set K is compact, we may assume without any loss of generality that x, + x and 9, + x for E -+ 0. By the upper semicontinuity property we infer that x E Tx, thereby completing the proof. Let U be a convex subset of X.We recall that the function f: U said to be convex if

AX + ( 1 - A)y)

I Af(x)

+ (1 - A)f(y)

+

R is

V X ,E ~ U,O I A I 1.

If U C X and V c Y are two convex subsets of linear spaces X and Y, respectively, the function H: U x V + R is called convex-concave if H ( u , u ) is convex as function of u and concave as function of u. (A function g is said to be concave if -g is convex.) Theorem 2.3 (J. von Neumann). Let X and Y be real Banach spaces and let U c X , V c Y be compact convex subsets of X and Y, respectively. Let H : U X V R be a continuous, convex-concave function. Then there is (uo,uo)E U X Vsuch that -+

H ( uo ,u ) I H ( uo ,u o ) I H ( u , u o )

V u E U , Vu E V .

(2.4)

Such a point ( u o ,u,) is called a saddle point of the function H and it is readily seen that (2.4) implies min-max equality:

H ( u o , u o )= min m a x H ( u , u ) U € U

V€V

=

max m i n H ( u , u ) , V€V U € U

which plays an important role in game theory. Proof of Theorem 2.3. Define the mappings Tlu =

{ U E

T~u= [U

E

V ;H ( u , u ) 2 H ( u , w ) V w

E

V},

U;H ( u , u ) IH(w,u) VWE U } ,

1.3. Absolutely Continuous Functions with Values in Banach Spaces

9

Clearly T : U x V + U x V is upper semicontinuous and T ( u , u ) is a convex set for every (u,u ) E U X V.Then, by Kakutani's theorem there is at least one ( u o ,u,) E U x V such that ( u o ,uo) E T ( u o ,uo). Clearly, this point satisfies (2.41, i.e., it is a saddle point of H. This complete the proof.

If the spaces X and Y are reflexive, Theorem 2.3 remains valid if U and

V are merely bounded, closed, and convex subsets. This follows by the

same proof using the Schauder theorem (Tichonov theorem) in locally convex spaces (see Edwards [l]).

1.3. Absolutely Continuous Functions with Values in Banach Spaces

Let X be a real (or complex) Banach space and let [ a , b ] be a fixed interval on the real axis. A function x : [ a , b ] -+ X is said to be finite& valued if it is constant on each of a finite number of disjoint measurable sets A , c [ a , b ] and equal to zero on [ a , b ] \ U k A , . The function x is said to be strongly measurable on [ a , b] if there is a sequence { x , } of finite valued functions that converges strongly in X and almost everywhere on [ a , b ] to x . The function x is said to be Bochner integrable if there exists a sequence { x , } of finitely valued functions on [ a , b ] to X that converges almost everywhere to x and such that

A necessary and sufficient condition that x : [ a , b ] -+ X is Bochner integrable is that x is strongly measurable and that 1," Ilx(t)ll dt < CQ. The space of all Bochner integrable functions x : [a, bl + X is a Banach space with the norm

and is denoted L'(a, b; X).

10

1. Preliminaries

More generally, the space of all (classes) of strongly measurable functions x on [ a ,b ] to X such that

for 1 Ip < 03 and llxllm= ess S U ~ ~Ilx(t)ll ~ [ <~w, , will ~ ~ be denoted LP(a, b; XI. This is a Banach space in the norm II * ( I p . If X is reflexive, then the dual of LP(a, b; X ) is the space Lq(a, b; X*), where l / p l / q = 1 (see Edward's book [l]). Recall also that a function x : [ a , b ]+ X is said to be weakly measurable if for any x* E X * , the function t + ( x ( t ) ,x * ) is measurable. According to the Pettis theorem, if X is separable then every weakly measurable function is strongly measurable, and so these two notions coincide. A n X-valued function x defined on [ a ,b ] is said to be absolutely continuous on [ a ,bl if for each E > 0 there exists S ( E ) such that N C,= Ilx(t,> - x(s,>ll I E , whenever C,"= It, - s,I 5 S ( E ) and (t,, s,) n ( t , ,s,) = 0 for m # n. Here, ( t , ,s,) is an arbitrary subinterval of ( a , b). A classical result in real analysis says that any real valued absolutely continuous function is almost everywhere differentiable and it is expressed as the indefinite integral of its derivative. It should be mentioned that this result fails for X-valued absolutely continuous functions if X is a general Banach space. Indeed, if X = L'(a, b ) and x : [ a , bl -+X is defined by

+

(i

x(t) =

ifassst, ift<s
then it is readily seen that x is absolutely continuous on [ a , bl but it is not differentiable in any point of the interval. However, if the space X is reflexive we have (Komura [l]):

Theorem 3.1. Let X be a reflexive Banach space. Then evey X-valued absolutely continuous function x on [ a ,b ] is almost e v e y h e r e differentiable on [ a ,b ] and ~ ( t =) x ( a ) +

1t -dsdx ( s ) a

ds

V t E [ a ,b ] ,

where dx/dt: [ a ,b ] + X is derivative of x , i.e., d -x(t) dt

=

lim

E'O

x(t

+ &) &

-x(t)

13. Absolutely Continuous Functions with Values in Banach Spaces

Proof

11

Since x is absolutely continuous, it is continuous and so the set

{ ~ ( t )t ;E [ a , b ] }is compact. Replacing X by the strong closure of the subspace generated by ( x ( t ) ; t E [ a , b]},we may assume without loss of generality that X is separable. Let us denote by V ( x ,t ) the total variation of x on the interval [ a , t ] , i.e.,

c n

V ( x , t ) = sup

Ilx(t1=1)

- X(ti)Il,

i= 1

where the supremum is taken over all partitions a = to < t , < ... < t, = b of [a, bl. Clearly, t + V ( x ,t ) is bounded and monotonically increasing on [ a , b]. Note also the inequality Ilx(t

+E)

-x(t)ll IV(x,t

+ E ) - V(x,t),

a It

+

E

< b,

which yields

l

b--E

Ilx(t

+ &)

-x(t)ll

&

dt I V ( x ,b ) < 00.

Hence, lim sup

Ilx( t

+ E)

- x ( t)ll

&

E’O

<

a.e. t

00

E

(a,b).

Let {x,* -1; be a sequence in X * that is dense in X*.For every n , there exists a subset I,, of [a, b ] such that t + ( x ( t ) , x,*) is differentiable at all t E [ a , b ]\I, and rn(Z,) = 0 (rn is the Lebesgue measure). We set t

and N E)

= U

:=,

- x(t)ll/&

E

[ a , b ] ;limsup E-0

E

Z,. Clearly, m(Z) = 0, and for every t is bounded for E + 0. Hence,

E

[a,b l \

N, Ilx(t

+

exists for all t E [ a , b ]\ N and all x * E X*. In other words, x is weakly differentiable a.e. on [a, b]. The function x (the weak derivative of x ) is weakly measurable and so by Pettis’ theorem it is strongly measurable. By

1. Preliminaries

12

(3.2) we infer that

and so by the Fatou lemma llxll On the other hand, we have (x(t) -x(a),x*) =

E

L'(a, b).

/f(x(s),x*)ds a

Vt

E

[ a , b ] , x *E X * ,

and therefore x(t) =x(a)

+ /'i(s)ds

Vt E [a,b].

a

This yields lim

'E

0

x(t

+E)

-x(t)

strongly in X ,a.e. t

=i ( t )

&

E

( a , b).

Hence x = d r / d t , and this completes the proof. Let us denote by N a , b ) the space of all infinitely differentiable real valued functions on [ a , b] with compact support in (a, b). The space 9 ( u , b ) is topologized as an inductive limit of LBK(a,b ) where K ranges over all compact subsets of (a, b ) and g K ( a ,b ) = { cp E @a, b); supp cp c K}. We shall denote by B Y a , b; X ) the space of all continuous operators from H a , b ) to X. An element u of g ' ( a , b; X ) is called a X-valued distribution on (a, b). If u E ~ ' ( u b; , X) and j is a natural number, then the equation u"'(cp) = (-l)Ju(cp"')

defines another distribution

vcp E L a ( U , b )

which is called the derivative of order j of

U.

We note that every element u bution (again denoted u )

E

u ( q ) = /"(t)cp(t) a

L'(a, b; X ) defines uniquely the distri-

dt

vcp E q a , b ) ,

(3.3)

and so L'(a, b; X ) can be regarded as a subspace of 9 Y a , b; XI. In all that follows, we shall identify a function u E L'(a, b; X ) with the distribution u defined by (3.3).

13. Absolutely Continuous Functions with Values in Banach Spaces

13

Let k be a natural number and 1 I P Iw. We shall denote by Wk'P([a, b]; X ) the space of all X-valued distributions u EL@'(a,b; X ) such that

u ( j ) ~ L P ( a , b ; X ) f o r j = 0 , 1 , ..., k.

(3.4)

Here, u(j)is the derivative of order j of u in the sense of distributions. We shall denote by A'.P([a, b];X ) , 1 Ip I03, the space of all absolutely continuous functions u from [a,bl to X having the property that they are a.e. differentiable on ( a , b ) and du/dt E LP(a, b; X I . If the space X is reflexive, it follows by Theorem 3.1 that u E A'*P([a,b];X ) if and only if u is absolutely continuous on [a ,b ] and du/dt E LP(a, b; X ) . It turns out that the space W'gP can be identified with A'*P. More precisely, we have:

Theorem 3.2. Let X be a Banach space and let u w. Then the following conditions are equivalent:

E

LP(a, b; X ) , 1 I p I

(i) u E W'*P([a, bl; X I ; (ii) There is uo E A ' * P ( [ a b, ] ;X ) such that u ( t ) = uo(t),a.e. t Moreover, u' = duo/dt a.e. in ( a , b).

E

( a , b).

Proof For simplicity, we shall assume that [a,b ] = [O, TI. TI; X ) , i.e., u E LP(0, T ; X ) and u' E LP(0, T ; X ) , Let u E W'*P([O, and define the regularization u, of u , u,(t)

=

/'u(s)p((t 0

- s ) n ) ds

V t E [O,T],

(3.5)

where p EN R ) is such that j p ( s ) ds = 1, p(t ) = p( - t ), supp p c [ - 1,ll. It is well-known that u, + u in LP(0, T ; X ) for n -+ 03. Note also that u, is infinitely differentiable. Let q E N O , T ) be arbitrary but fixed. Then, by (3.9, we see that

ifsuppcpc

if,.-

f).

14

1. Preliminaries

Hence, du,/dt = a.e. in ( l / n , T - l / n ) . On the other hand, letting n tend to a in the equation

we get u(t) - u(s)

=

a.e. t , s

/‘urdr

E

(O,T),

S

because ( u r I n+ u r in LP(0, T; X I . The latter equation implies that u admits an extension to an absolutely continuous function uo on [O, TI that satisfies the equation Vt

u o ( t ) - u o ( 0 ) = / ‘ u r ( r )d r 0

u’(q)

= -

1 u ( t ) q ’ ( t )dt T

0

+ &lim +O

1

E

L

= -

& u ( t ) q ( t-

1 lim -

&+O

E )

E

/

dt

T

0

E

u(t)

[O,T].

q ( t ) - q(t -

dt

E

Vq E g ( O , T )

Hence

This shows that u’ E LP(0, T ; X ) and u r = du/dt. This completes the proof of Theorem 3.2. In particular, it follows by the proof of Theorem 3.2 that we may choose + ji du,/dsds V t E [ a , b].

u1 E A’.P([a,b];X ) such that u,(t) = u,(a)

Theorem 3.3. Let X be reflexive and let u E LP(a, b; X I , 1 0 such that

lb-hllu(f+ Proof

h ) - u ( t ) l l P dt I ClhlP

V h E [O,b - a ] .

(i) * (ii). By Theorem 3.1, we know that

where u1 E A’sP([a, b];X ) , i.e., du,/dt [b-hllu(t

+ h ) - u ( t ) l l Pdt

E

LP(a, 6; X I . This yields

I lhlP

lbPh dt [+’

U

1 211

ds.

Since

-

we find the estimate (ii). (ii) (9. Let u, be the regularization of u. A little calculation involving formula (3.5) reveals that ( u ; ) is bounded in LP(a, b; X ) . Since u, + u in L p ( a , b ; X), u; + u’ in g ’ ( u , b ;X I , and { u ; } is weakly compact in LP(a, b; X I , we infer that u’ E LP(a, b; X I , as claimed. Remark 3.1. If u that

E

lb-hllu(f +

W’*’([a, b ] ;X I , then it follows as in the preceding

h ) - u(t)ll dt I Clhl

V h E [O,b - a ] .

However, this inequality does not characterize the functions u in W ’ .‘ ( [ a ,b];X ) , but the functions u with bounded variation on [ a , 61 (see H. BrCzis [4]). Let V and H be a pair of real Hilbert spaces such that V c H c V’ in the algebraic and topological senses. Here, V’ is the dual space of V and H is identified with its own dual. Denote by I * I and II * II the norms of H and V , respectively, and by the duality between V and V’. If u l , u2 E H, then (u, , u,) is the scalar product in H of u1 and u 2 . ( - , a )

16

1. Preliminaries

Denote by W([a,b];V ) the space W ( [ a , b ] ; V= ) {u

E

L 2 ( a , b ; V ) ; u ’E L 2 ( a , b ; V ’ ) ) ,

(3.6)

where U ’ is the derivative of u in the sense of g ’ ( u , b; V’).By Theorem 3.2, we know that every u E W ( [ a , b ] ; Vcan ) be identified with an absolutely continuous function u o : [ a , b ]+ V’. However, we have a more precise result.

Theorem 3.4. Let u E W([a,b]; V). Then there is a continuous function u o :[a, b ] + H such that u ( t ) = u o ( t ) a.e. t E ( a , b). Moreouer, if u , u E W([a,b]; V ) then the function t + ( u ( t ) ,d t ) ) is absolutely continuous on [ a , bl and d -(u(t),v(t)) dt

=

( u ’ ( t ) , u ( t ) )+ ( u ( t ) , v ’ ( t ) )

a.e. t

E

(a,b).

(3-7) ProOJ: Let u, u E W ( [ a ,b]; V )and cC/(t)= (u(t), u(t)).As we have seen in Theorem 3.2, we may assume that u, LJ E AC([a,bl; V ’ )and

Then, we have

Hence cC/

E

W’, ‘ ( [ a ,bl; R) and

as claimed. Now, in Eq. (3.7) we take u = u and integrate from s to t. We get

1.4. Linear Differential Equations in Banach Spaces

17

Hence, the function t lu(t)l is continuous. On the other hand, for every u E V the function t + ( u ( t ) ,u ) is continuous. Since lu(t)l is bounded on [ a ,b ] , this implies that for every u E H the function t + (u(t), u ) is continuous, i.e., u ( t ) is H-weakly continuous. Then, from the obvious equation --f

lu(t) - u(s)I2 = lu(t)I2 + lu(s>I2- 2 ( u ( t ) , u ( s ) ) ,

Vt,s

E

[a,b]

it follows that lims+,lu(t) - u(s)l = 0, as claimed. The spaces W'*P([a,b ] ;X ) , as well as W ( [ a ,bl; V ) , play an important role in theory of differential equations in infinite dimensional spaces. The following compacity result, which is a sharpening of Arzelh-Ascoli theorem, is in particular useful in this context. Theorem 3.5 (Aubin). Let X , , X , ,X , be Banach spaces such that X , cX , cX , ,

Xireflexive for i

=

O,l,

and the injection of X , into X , is compact. Let 1 < pi < w, i the space

=

0 , l . Then

W = L P o ( ab, ; X,,) n W 1 , P l ( [ ab, ] ;X , ) is compactly imbedded in LPU(a,b; X , ) .

The proof relies on the following property of the spaces Xi(see J. L. Lions [ l ] ,p. 58). For every E > 0 there exists C, > 0 such that

Ilullx, I

EIIUIIX,

+ c&llullx,

vu E

x.

1.4. Linear Differential Equations in Banach Spaces In this section, we shall review some classical results on semigroups of linear operators and the linear Cauchy problem in Banach space. For a detailed presentation of the subject we refer the reader to the books of K. Yosida [ l ] ,E. Hille and R. S. Phillips [ l ] ,and A. Pazy [2].

18

1. Preliminaries

1.4.1. Semigroups of Class C ,

Let X be a Banach space with the norm 11.11. A one parameter family S t ) , 0 I t < 03, of continuous linear operators from X to X is a semigroup of class C , on X i f (i) S(0) = I ; (ii) S ( t ) S ( s ) = S(t + s) for all t , s 2 0 (the semigroup property); S(t)x = x for every x E X. (ii) lim, ~

,

(Here, Z is the unity operator in X . ) If S ( t ) is a semigroup of class C , then it follows that there exist constants o E R and M 2 1 such that IIS(t)llL(X,X)

s Me"'

V t 2 0.

(4.1)

(Here, L ( X , X ) is the algebra of linear continuous operators on X and II * IIL(X, X ) is the norm in L ( X , X I . ) In particular, if M = 1 and w = 0, then S ( t ) is called a semigroup of contractions on X. The linear operator A: D ( A ) c X -+ X defined by t Ax

=

lim 11.0

S(t)x -x t

Vx

E

D(A),

(44

is called the infinitesimalgenerator of the semigroup S ( t ) . The set D ( A ) is the domain of A. Proposition 4.1. The infinitesimal generator A is a closed, linear, and densely defined operator in X . Zf x E D ( A ) ,then S ( t ) x E D ( A ) for all t 2 0, S ( t ) E C'([O,co);X ) , and

d -S(t)x =AS(t)x dt

=

S(t)Ax

V t 2 0.

(4.3)

Proof We shall prove first that D ( A ) is a dense subset of X. Let x be arbitrary but fixed. For every t > 0, define x,

=

t-' j b S ( s ) x d s .

EX

1.4. Linear Differential Equations in Banach Spaces

It is readily seen that x , precisely, one has

+x

as t

+

0 and x ,

E D ( A ) for

S(t)x -x lim S( & ) X I - x , -

810

Now let x

all t > 0. More

> 0.

Vt

t

&

19

D ( A ) . By the semigroup property, it follows that

E

S(&) - I &

S(t)x

=

S(t)

S( & ) X - x

V t 2 0, &

&

> 0.

This yields lim

8

S( &) - I

10

Hence, S ( t ) x

&

S(t)x

=

lim

S( & ) X - x

810

&

=

S(t)Ax.

E D ( A ) and

df -S(t)x dt

=AS(t)x

=

S(t)Ax

V t 2 0.

To conclude the proof of (4.31, it remains to be shown that the left derivative of S(t>x exists and equals S(t)Ax. This follows from lim 8

10

(

S(t)x - S(t &

&)X

- S(t)Ax

1

=

lim S ( t - &)

810

(

S(&): -

-4

+ 8lim S(t - &)Ax - S(t)Ax 10 =

0.

Let us prove now that A is closed. For this purpose, consider a sequence { x , 1 c D ( A ) such that x , -+ x , and Ax, + y o in X for n -+ m. Letting n tend to m in the equation

we get

which implies that y o = Ax,, as claimed.

W

20

1. Preliminaries

The characterization of the infinitesimal generators of C,-semigroups is given by the famous theorem of Hille and Yosida (Theorem 4 . 0 , which represents the core of the whole theory of C,-semigroups. Theorem 4.1. A linear operator A in a Banach space X is the infinitesimal generator of a semigroup of class C, if and only if

(i) A is closed and densely defined; (ii) There exist M > 0 and w E R such that

I I ( A I -A)-"IIL(x,x) I

M (A -

0)"

forall A > w , n

=

1 , 2 ... .

Proof: Suppose that A is the infinitesimal generator of a C,-semigroup S ( t ) satisfying condition (4.1). Define, for A E C, Re A > w,

R( A ) x

=

~me-"S(t)xdt

Vx

E

X

We have d" dA"

e - A f (- t ) " S ( t ) x d t

Vn

=

1 , 2 , .. .,

and this yields

On the other hand, it is readily seen that AR(A)x = AR(A)x - x for all A E C, Re A > w . Hence, R(A) = ( A Z - A)-' for Re A > w . Taking in account the well-known equation

we obtain for ( A Z - A ) - " the estimate (4.4). We shall prove now that if A satisfies condition (4.41, then A generates a C,-semigroup satisfying condition (4.1). For simplicity, we shall assume that w = 0, the general case being obtained by appropriately translating the particular case we are considering.

1.4. Linear Differential Equations in Banach Spaces

21

For n natural, consider the operators J,

=

(I

-

n-'A)-',

A,

=

n(Jn - I ) ,

and note that AJ,x = J , A x = A,x for all x E D ( A ) . Moreover, A,, and J , belong to L ( X , X ) for all n and IIJ,xll I Mllxll by virtue of condition (4.1). Hence, IIAnxll I MIIAxlI

Vx E W A ) ,

lim J,x = x

Vx

n-rm

Since A,

E

E

D(A )

(4.5)

=X

.

(4.6)

L ( X , X I , the approximating equation dun dt

t 2 0,

- =A,,u,, Un(0) =x,

has for each x

E

(4.7)

X a unique solution u, S,( t ) x

=

un(t )

E

C'([O, 03); X). We set

Vt 2 0.

By uniqueness of the Cauchy problem (4.7) we see that S,(t) is a C,-semigroup on X and A , is the infinitesimal generator of S,(t). As a matter of fact, S,(t) is given by the exponential formula

and this yields (by virtue of condition (4.4)) IIS,(t)llL(X,X) I M

Vt

> 0, n

=

1 , 2 ... .

(4.8)

On the other hand, by (4.7) we have d

-(S,(t ds

-s)S,(s)x)

=S,(t - s ) ( A , - A , ) S , ( s ) x

forOIsIt.

This yields IlS,(t)x - S,(t)xll 5 /bIIS,,(t - ~

) I ~ L ~ x , x ~ I I ~ ~ ( ~ ) I -I LA,xl( Ix , ~ds~ ~ ~ ~ , ~ V m , n , and t 2 0,

22

1. Preliminaries

and by (4.5) and (4.8) we infer that S,(t)x converges (uniformly) in t (on compacta) for every x E X . We set S ( t ) x = nlim S,(t)x +m

V t 2 0, x E X .

(4.9)

It is readily seen that S ( t ) is a semigroup of class C , on X and S ( t ) x - x = [S(s)Rrds

Vx~D(A),t20.

Hence, if A' is the infinitesimal generator of S ( t ) , we have D ( A ) c D ( A ) and A = A' on D(A). On the other hand, by first part of theorem we know that (I - n-'A')-' E L ( X , X ) . Hence, ( I - n-'A)-' = ( I - n - 2 I - l and D ( A ) = D ( k ) , as desired. rn We shall denote by eAT the semigroup generated by A. In particular, we derive from Theorem 4.1 the following characterization of infinitesimal generators of C,-semigroups of contractions. Corollary 4.1 (Hille-Yosida-Philips). The linear operator A is the infinitesimal generator of a C,-semigroup of contractions if and only if:

(i) A is closed and densely defined; and (ii) II(AI - A)-'IIL(X,X) I 1 / A VA > 0. We shall see later that condition (ii) characterizes the m-dissipative operators in X. 1.4.2. Analytic Semigroups

Let S(t):[O,w) -+ L ( X , X ) be a semigroup of class C,, and let A be its infinitesimal generator. The semigroup S ( t ) is said to be differentiable if S ( t ) x E D ( A ) for all t > 0. By Proposition 4.1, it follows that if S ( t ) is differentiable then for any x E X the function t -+ S ( t ) x is infinitely differentiable and

Moreover, by the closed graph theorem A S 0 1

E

L ( X , X ) for all t > 0.

1.4. Linear Differential Equations in Banach Spaces

23

The semigroup S ( t ) is said to be analytic if for every a > 0 (equivalently, for some a > 01,

sup tllAS(t)ll < a.

(4.11)

O
It follows by (4.10) that if the semigroup S ( t ) is analytic, then the function t + S ( t ) x is analytic on (0, m). Conversely, if for every x E X , S ( t ) x is analytic on (0,m) the semigroup S ( t ) is differentiable and (4.11) follows from (4.10).

Theorem 4.2. Let A be a closed and densely defined linear operator on X . Then A is the infinitesimal generator of an analytic semigroup S(t), IlS(t)llL(x,x ) _< M , for all t > 0 if and only if there exist 0 E (7r/2, 7r) and M > 0 such that (4.12) where C

=

{A

E

C ; A # 0, larg A1 < 6).

Proof Assume first that condition (4.12) is satisfied. Then, define SO) E L ( X , X ) by the Dunford integral s(t)

E)

(27ri)-'

eA'(AZ - A ) - ' d A

Vt

2

0,

(4.13)

is a piecewise smooth curve in 2 defined by r = I?, u rl u r2, -- { r e - i ( ~ - ~3 ) 1 . -< r < a ),rl = 1 5 r < a),r2= (e'"; - ( e -

where

r0

=

r

By condition (4.121, S ( t ) is well-defined and belongs to L ( X , X ) for all t 2 0. We have A S ( t ) x = (27ri)-' LAe"(AZ - A ) - ' x d A -(21ri)-'xLeA'dh =

(27ri)-' LAe"'(AZ - A ) - ' x d A

Vx E X

24

1. Preliminaries

Hence, by (4.12), we have IIAS(t)xll IC ( A m I e r e - ' ( ' ~dr &)'l

+ /Imlere'('-c)rl dr + 1)llxll IC,t-'llxIl

Vt

> 0.

On the other hand, d -S(t)x dt

=

(27ri)-' LAe"'( A Z - A ) - ' x d h

=AS(t)x,

vt

> 0.

Hence, u ( t ) = S(t)x is the solution to the Cauchy problem du

in

-=AM

dt u(0) = x .

[O,w),

(4.14)

Let us prove that problem (4.14) has at most one solution Ilu(t)ll IM V t 2 0. Indeed, if u and u are two such solutions, then w = u - u satisfies the equation dw/dt = Aw, w ( 0 ) = 0, and its Laplace transform $(A), A > 0, is the solution to homogeneous equation ( A Z - A)$(A) = 0. Hence, $(A) = 0 for A > 0, which clearly implies that w = 0. Then, by the uniqueness of the Cauchy problem (4.14) it follows that S(t + s) = S ( t ) S ( s ) and that A is the infinitesimal generator of S ( t ) . Conversely, if A is the infinitesimal generator of an analytic semigroups S(t), then by (4.10) and (4.11) it follows that

Then, the semigroup S ( t ) has an extension S ( z ) defined by S(z) =S(t)

+ nc =l

d"S(t) ( z - t)" ~

dt"

n!

in a sector { z E C; Iz - tl 0,

1.4. Linear Differential Equations in Banach Spaces

25

where r is a piecewise smooth curve in A of the form encountered before. W By using this formula we find estimate (4.12). 1.4.3. The Cauchy Problem If A is the infinitesimal generator of a C,-semigroup S ( t ) = eA', then, as seen before, the homogeneous initial value problem (the Cauchy problem) du

dt = A u ,

tLO,

(4.15)

u(0) = x ,

has for every x E D ( A ) a unique solution u E C'([O,w); X ) given by u(t)= e A r x Consider now the nonhomogeneous Cauchy problem

u(0) = x ,

(4.16)

where f E L'(0, T ; X )and x E X. The continuous function u: [0, TI stants formula u ( t ) = eA'x

+

t

+

X defined by the variation of con-

eA('-s)f(s) ds

V t E [0,TI

(4.17)

is called mild solution to Cauchy problem (4.16). X is said to be a classical solution to problem A function u: [0, TI (4.15) if it is continuously differentiable on [0, TI, u ( t ) E D ( A ) for all t E [O, TI, and it satisfies Eq. (4.16) on [0, TI. It is readily seen that if A generates a C,-semigroup, then problem (4.16) has at most one classical solution u ( t) . Indeed, if y is a solution to dY dt

- = Ay

in

then, for 0 I s I t < m, we have

Hence e A ( ' - s ) y ( s= ) 0.

[O,m),

y ( 0 ) = 0,

1. Preliminaries

26

Regarding the existence of classical solutions to problem (4.161, we have the following well-known theorem whose proof will be omitted.

Theorem 4.3. Assume that A is the infinitesimal generator of a C,-semigroup e A ron X and x E D ( A ) . Assume either (i) f (ii) f

C'([O, TI; X I , or E N O ,TI; X ) , A f E N O , TI; X ) . E

Then the function u given by formula (4.17) is the unique classical solution to the Cauchy problem (4.16). W A function u E W'*'([O, TI; X ) that satisfies almost everywhere Eq. (4.16) and the initial condition is called strong solution to problem (4.16).

Theorem 4.4. Assume that A is the infinitesimalgenerator of a semigroup of class C,. Let x E D ( A ) . I f f E W'*'([O, TI; X ) , then the mild solution u to problem (4.16) is a strong solution. Proof

We have

u(t) =

] ' e A ( r - s ) f ( sds ) 0

=

Vt

e A s f ( t- s) ds,

E

[ O , T ] , (4.18)

and this implies that u E W'.'([O,TI; X ) . Hence, u is a.e. differentiable on (0, T ) and by a little calculation we see that

Hence, u ' ( t ) = A u ( t ) + f ( t >a.e. t

E

(0, T ) , as claimed.

H

If the semigroup e A t is analytic, then the mild solution is a strong solution under weaker assumptions on f. More precisely, we have (see A. Pazy [2]):

Theorem 4.5. Let A be the infinitesimal generator of an analytical semigroup e A r .If x E X and f is Holder continuous, i.e., IIf(t) - f ( S ) I I

ICIt - sI'

Vt,s

E

[O,T],

(4.19)

1.4. Linear Differential Equations in Banach Spaces

27

where 0 < 8 I 1, then the mild solution u to problem (4.16) is continuously differentiable on (0, TI, Au E C([6, TI; H ) for every 6 > 0, and u satisfies Eq. (4.16) on the interval (0, TI. Moreover, if x E D ( A ) , then Au E C([O, TI; X ) and u is a classical solution to (4.16). Proot

We have u ( t ) = eA'x

+ u(t),

Vt

E

[O,T],

where u is defined by formula (4.18). Since eA'x E C'([O, TI; X ) and satisfies the homogeneous equation (4.151, it suffices to prove the theorem for u = u, i.e., in the case x = 0. We may write u as u(t) = =

0

f ( s ) - f ( t ) ) ds

+ [ e A ( ' - ' ) f ( t ) ds

+ u2(t).

u*(t)

It is readily seen that u,(t) E D ( A ) and A u 2 ( t )= ( e A f- I ) f ( t ) , V t E [0,TI. Hence, Au, E C([O, TI; X I . Now, by condition (4.19) and analyticity condition (4.11) it follows that A e 4 - S ) ( f ( s ) )- f ( t ) ) E L'(0, T ; X ) , and since A is closed this implies that u l ( t ) E D ( A ) for all t and A u , ( t ) = / ' A e A ( f - Sf)(( s ) - f ( t ) ) ds. 0

Since

d -dt4 t )

=Au(t)

vt

E

[O,T],

as claimed.

Theorem 4.6. Let A be the infinitesimal generator of an analytic semigroup in a Hilbert space H . If f E L2(0,T ; H ) and x E D ( A ) , then the mild

28

1. Preliminaries

(Here,

II IID(A) is the graph norm IlAxll + Ilxll.)

Proof

Let u be the function (4.18). We have

G( A)

j0

m

=

e-%( t ) dt

=

( AZ - A )

-'f( A)

for A

=

a

+ it,

where a 2 p > 0. This yields (we extend u and f by 0 outside [O, TI) AG(A) =A(AZ-A)-'f(A)

-f(A)

VA,ReA= a ,

and by the Parseval formula, m

[ [ A ; (A)1I2 d (

=

2 ~ / ~ e - ~ ~ ~ l I Adut .( t ) l l ~ 0

Hence

and estimate (4.20) follows.

rn

1.4.4. Examples A large class of infinite dimensional problems, including linear parabolic equations, the wave equation, symmetric hyperbolic systems, and linear functional-delay equations, can be studied in the framework of Co-semigroups. Here, we shall present only a few standard examples of linear operators that generate Co-semigroups.

Proposition 4.2. Let A be linear, self-adjoint, positive operator on a real Hilbert space H . Then - A generates an analytic semigroup of contractions on H.

Proof Let H be endowed with the norm 1 * I and scalar product Denote by R(AZ + A ) the range of AZ A, A 2 0. Since

+

( A u ,u ) 2 0

Vu

ED(A),

(a,

- 1.

1.4. Linear Differential Equations in Banach Spaces

29

it is readily seen that R(AZ + A ) is a closed subset of H . On the other hand, R(AZ + A ) is dense in H. Indeed, otherwise there would exist y E H , y # 0, such that (Ax+Ru,y)

=o

VXED(A).

This implies that x -+ ( A x , y ) is continuous on H and so y E D(A).Then, for x = y the previous equation gives y = 0. Hence R(AZ + A ) = H VA > 0. On the other hand, we have

Hence, -A generates a C,-semigroup of contractions S(t) = C A I . For x E D ( A ) the function u ( t ) = C A ' x is the (classical) solution to the Cauchy problem du

V t 2 0,

u(0) = x .

z)

Vt>0.

- +Au = 0

dt

We have du

t -

ldtl

+ t

Hence,

(

Au,-

=o

=

0

V t 2 0.

(4.21)

(4.22)

On the other hand, we have (multiplying Eq. (4.21) by u ) 1 d

- -lu( t ) I 2

2 dt

+ ( A u ( t ) , u( t ) ) = 0

Vt 2 0

(4.23)

and I d

- -lu(t 2 dt

+ h ) - u(t)l2+ (Au(t + h ) -Au(t),u(t + h ) - u ( t ) ) = 0 V t ,h 2 0.

Hence, the function t + I(du/dt)(t)l is decreasing on (0, m). Then, integrating (4.22) on the interval (0, t ) , we get 2

~ s l ~ ( s ds ) l I( ( A u ( s ) , u ( s ) ) d s

V t 2 0,

30

1. Preliminaries

whilst by (4.23) 1 , 1 ( ( A u ( s ) , u ( s ) ) d s + -lu(t)l 2 I -21 ~ 1 ' .

Hence t2

du -(t) ldt

i.e.,

I

V t > 0,

IlXI2

as claimed. Let fl be a bounded and open subset of RN with a smooth boundary

3 0 . Then, it follows by Proposition 4.2 that the operator A,: D ( A , )

C

L2(fl) -+ L2(fl) defined by A,u

=

-Au,

D(A,)

=

H,'(fl) n H 2 ( f l ) ,

(4.25)

generates an analytic semigroup of contractions on ~ ' ( f l ) . More generally, consider for 1 I p < 00 the operator A, = - A with D ( A , ) = W;-p 1 and D ( A , ) = { u E Wd*';u E ~ ' ( f l )for ) p = 1. Proposition 4.3. For 1 I P < 03, A,, generates an analytic semigroup of contractions in X = L P ( f l ) .

Proof. It follows that A , is m-accretive in L P ( f l ) , i.e., R(AZ + A , ) = LP(fl) for all A > 0 and II(Al + A,)-' I I L ( x , x ) 5 A-' VA > 0 (see Proposition 3.9 in Chapter 2). Moreover,

D(A , ) c Wd,q(fl),

N where 1 < q < N-1'

(4.26)

Hence, A , is the infinitesimal generator of a C,-semigroup of contractions on LP(fl). The analyticity of the semigroup e-Apf follows in the case 1 I P 5 2 from Proposition 4.4 following and the analyticity of e-Azr.The case p > 2 follows from verifying directly conditions of Theorem 4.2 (see Pazy [21).

31

1.4. Linear Differential Equations in Banach Spaces

Proposition 4.4. Let R be a bounded and open subset of R N and let S ( t ) be the semigroup generated on L’(R) by A with zero Dirichlet conditions. Then S ( t ) ( L ’ ( R ) )c L”(R) for all t > 0 and

llS(t)UollL”(n) 5 Ct-N/211UOllL~(~)v u ,

E

L ’ ( W t > 0, (4.27)

where C is independent of uo. Interpolating between L ’ ( R ) and L”(R), it follows by inequality (4.27) that IIS(t)u,llLqn,

Icf-(N/2x1-1/~)llu IIL l ( n )

‘it > 0.

(4.28)

Prooj For N = 1, inequality (4.27) follows by representing S(t)u, as Fourier series with respect to the Laplace eigenfunctions. We set u = S ( t ) u , and recall that dU

in R

--AUTO dt

u(0,x)

=

uo(x),

X

R’

x

E

R.

(4.29)

Without loss of generality, we may assume that u is a classical solution to (4.29). Let k be a positive integer. We multiply Eq. (4.29) by tk+21ylp-2y, where p > 2, and integrate on R X R+.We get

/--lVu( x, t ) h ( x,

d zllu( t)ll,” + ( p - l ) t k + 2

tk+2p - 1

q v - 2

ak

I0

v t > 0, i.e.,

d tk+’ - - ~ ~ u ( +t 2) t~k +~ 2~~ ~ l V l u ( x , t ) I p / ’ak I 25 0 . dt Then, by the Sobolev imbedding theorem,

1-

IVIU(x,t)lp/212ak 2

if N > 2. If N

=

2,

c

(N-2)/N

lu(x,t)lpN/‘N-2)dX

(4.30)

32

1. Preliminaries

Then, substituting this in (4.30), we get

d

+ ~ t ~ + ~ I l u ( t )Il lo, P t l t > 0 , 2 ) and 11 - [Ip is the L p ( 0 ) norm. Now, we integrate

t k + 2-llu(t)Il,P dt

where a = N / ( N from 0 to t and get

(4.31)

Now we have, by the Holder inequality,

where 0 0,

1.4. Linear Differential Equations in Banach Spaces

33

Then, using the well-known inequality a p

a61 P

1 1 -+-=1,

bq

+ -, 4

P

4

we get

where k is arbitrary and ( p - pX1 - p ) - ' p + ( a - l ) / a = 2 / N . Hence, IIu(t)IIL"(n) I C ( k

=

a p . For p

+ 2)N/2t-N/2

IIUOIIL'(R)

+

vt

03,

we see that

> 0,

(4.33)

as claimed. Remark 4.1. In particular, it follows by Proposition 4.4 that for yo E L'(R) and f E L'((0, TI; L ' ( R ) ) the boundary value problem

-dY_ dt

Ay

y(0) =yo

has a unique mild solution y y(x,t)

=

(S(t)yo

and for q < ( N

in R x ( O , T ) ,

=f

in R, E

+ (S(t

+ 2)/N

y

=

0

in d R ,

C([O, TI; L'(R)): - s)f(s) d s ) ( x )

vt 2 0 , x E

a,

we have

IlYllLqn X(0, T ) ) I C ( IluollLl(n, +

IlfllLl(nx(0, T ) ) ) .

(4.34)

34

1. Preliminaries

The Wave Equation. Consider the second order differential equation d2Y + Ay

=f

dt2

Y ( 0 ) =Yo

in (0, T ) , dY z(0) =Y ,

7

(4.35)

9

in a real Hilbert space H, where A is a self-adjoint positive definite operator on H and f E L'((0, T ) ;H ) . Equation (4.35) can be written as a first order differential equation dY = 2,

dt dz _ -- -AY dt

+f, (4.36)

on the product space X = V x H , where V = D(A'/'). (The space X is endowed with the scalar product ( ( Y ,Z ) , ( Y l ,

21))

=

( A ' i 2 y ,A1'2Yl) +

( 2 ,z , ) . )

It is readily seen that the operator

is m-dissipative on X , i.e., R(Z + d )= X and M y , z ) ,( y , 2)) 2 0 for all (y, z ) E X . Hence, d generates a Co-semigroup on X and so for y o E D ( A ) , y 1 E V , and f E C'([O,TI; H ) , problem (4.35) has a unique solution Y E C'([O, TI; V ) n C([O,TI; D ( A ) ) with dy/dt E C'([O, TI; H ) r~ N O , TI; V ) . In particular, for H = L2(fl) and A = - A , D ( A ) = H,'(fl) n H 2 ( n ) , and (4.35) reduces to the wave equation.

Nonlinear Operators of Monotone Type

Chapter 2

In this chapter we introduce the concept of maximal monotone operator, along with the basic results of this theory and its relationship with convex analysis and the theory of generalized gradients. A separate section is devoted to nonlinear m-accretive operators. We present several examples and applications to nonlinear partial differential equations, such as nonlinear elliptic boundary value problems, the nonlinear diffusion equation, and first order quasi-linear partial differential equations. 2.1. Maximal Monotone Operators

2.1.1. Definitions and Basic Results

If X and Y are two linear spaces, we will denote by X X Y their Cartesian product. The elements of X X Y will be written as [ x , y ] where x E X and y E Y. If A is a multivalued operator from X to Y , we may identify it with its graph in X X Y :

{[x,yIE X X y ; y E

W.

(1.1)

Conversely, if A c X X Y , then we define Ax

=

R(A) =

D ( A ) = { x E X ; Ax

{ y E Y ;[ x , y ] € A } ,

u

X€D(A)

Ax,

A-'

=

( [ y , ~ [] x;, Y l € A } . 35

z 0 } ,(1.2) (1.3)

36

2. Nonlinear Operators of Monotone Type

In this way here and in the following, we shall identify the operators from X to Y with their graphs in X X Y and so we shall equivalently speak of subsets of X X Y instead of operators from X to Y. If A, B c X x Y and A is a real number, we set: A

AA

=

+B

=

{ [ x , A y l ;b , Y l

€4;

(1.4)

{ [ X , Y + 21; [ X , Y l E A , [ x , z l E B ) ; (1.5) . A B = { [ x , z ] ; [ x , y ] ~ B , [ y ,~z A ] f o r s o m e y ~ Y )(1.6)

Throughout this chapter, X will be a real Banach space with dual X*. Notations for norms, convergence, and duality pairings will be that introduced in Chapter 1, Section 1. In particular, the value of functional x * E X * at x E X will be denoted by either (x, x * ) or (x*, x). If X is a Hilbert space unless otherwise stated, and we shall implicitly assume that it is identified with its own dual. Definition 2.2. The set A c X X * ) is said to be monotone if

X * (equivalently the operator A: X

X

(xl - x 2 , y l - y 2 ) 2 0

V [ x i , y i ]E A , =~ 1,2.

+

(1.7)

A monotone set A c X X X * is said to be maximal monotone if it is not properly contained in any other monotone subset of X X X*.

Note that if A is a single valued operator from X to X * then A is monotone if (x1 - x 2 , A X 1

2 0,

-AX2)

(1.8)

Vx,,x, ED(A).

A simple example of a monotone subset of X X X * is the duality mapping J of X (see Section 1.1). Indeed, by definition of J , (x1 - X 2 , Y 1

-Y2)

2

=

lklll + llx21I2 - ( X l , Y 2 )

2

(IlXlll

- llx211)2

- (X2,Yl)

V[Xi,Y,I E J .

As a matter of fact, it turns out that J is maximal monotone in X Now let us show that the map 4: L'(R) + La(.) defined by

4 ( u ) = { u E La(R); u ( x ) E sign u ( x ) .IlullLl(n, a.e. x

E

X

X*.

R)

is the duality mapping of L'(R). (Here, sign u = ~ 1 for u~ # 0,1 sign0 ~ = [ - 1,111. Indeed, since 4 c J , as easily seen, it suffices to show that 4 is

~

2.1. Maximal Monotone Operators

maximal in L 1 ( R )X LYR). Let ( u , , u,) /n(U,

- u)(u, - u)dx 2

0

Note that the equation sign u

+ sign u

*

IlullLl(n)3 uo

E L1(R) x

37

L " ( R ) be such that

vu E L'(R),uE 4(u).

+ sign u,

a.e. in R

has at least one solution u l . Substituting in the preceding inequality yields

Hence, u,

=

L

( u , - ul)(sign u , - sign u l ) dr I0.

u1 and u,

E

sign u1 * llulll~yn),i.e., [u,,u,l

E

4, as claimed.

Definition 1.2. Let A be a single valued operator from X to X * with D(A) = X . The operator A is said to be hemicontinuous if, for all X,Y E X , w - limA(x + h y ) = A x . A-rO

A is said to be demicontinuous if it is continuous from X to X,* , i.e.,

w

-

x,

limAx, = A x .

' X

A is said to be coercive if Iim ( y , , x , - xo)llx,ll-

1

=

( 1*9)

03

for some x o E X and all [ x , , y,] E A such that limn+" Ilx,ll Proposition 1.1. Let A c X

X

= m.

X * be maximal monotone. Then:

A is weakly-strongly closed in X x X * ; i.e., if [ x , ,y,] E A, x , inXandy, +yinX*,then[x,yl EA; (ii) A - ' is maximal monotone in X * X X ; (iii) For each x E D(A), Ax is a closed convex subset o f X * .

(i)

Proo$

-

x

(i) From the obvious inequality (x, - u,y, - u) 2 0

V [ U , U ]=

A,

we see that ( x - u , y - u ) 2 0 V [ u ,u ] E A , and since A is maximal this implies [ x , y ] E A, as claimed. (ii) is obvious. (iii) By (i) it is obvious that Ax is a closed subset of X * for each x E D ( A ) . Now let y o ,y , E Ax and let yh = hy, + (1 - h ) y l , where

2. Nonlinear Operators of Monotone Type

38

0 < A < 1. From the inequalities ( X - U,Y, - U )

2 0,

(X

-

U , Y ~- V ) 2

0

V [ U , V ]€ A ,

we deduce that (x - u , y, - u ) 2 0 V [ u ,u ] E A , which implies that [x,y h ] E A because A is maximal. The proof is complete. W It has been shown by G. Minty in the early 1960s that coercive maximal monotone operators are surjective. This important result, which implies a characterization of a maximal monotone operator A in terms of the surjectivity of A + J ( J is the duality mapping) is a consequence of the following existence theorem. Theorem 1.1. Let X be a reflexive Banach space and let A and B be two monotone sets of X X X * such that 0 E D ( A ) , B is single valued, hemicontinuous, and coercive, i .e.,

lim ( B x ,x)/llxll

Ilxll-

Then there exists x

E

(U

K

=

= +m.

(1.lo)

conv D( A ) such that

-x,Bx+u) 2 0

V [ U , U ]€ A .

(1.11)

It should be noted that if A is maximal monotone, it follows from (1.11) that 0 E Ax Bx. We will prove Theorem 1.1 in several steps. We shall prove first the following lemma.

+

Lemma 1.1. Let X be a finite dimensional Banach space and let B be a hemicontinuous monotone operatorfrom X to X * . Then B is continuous.

Proof: Let us show first that B is bounded on bounded subsets. Indeed, otherwise there exists a sequence { x , } c X such that IIBx,II -, 00 and x, xo as n + 03. We have --f

(x, -x, Bx, and therefore

- Bx) 2

0

VX E X ,

2.1. Maximal Monotone Operators

Without loss of generality, we may assume that Bx,IIBx,II-' n -+ w. This yields (xo-x,y,)

20

39

- ) y o as

VXEX,

and therefore y o = 0. The contradiction we have arrived at shows that B is indeed bounded. Now, let { x , } be convergent to xo and let y o be a cluster point of {Bx,}. Again by the monotonicity of B , we have ( x 0 - x , y 0 - Bx)

If in this inequality we take x X , we get

=

tu

>0

VX E X .

+ (1 - t h o , 0 I t I 1, u arbitrary in

(~,-~,yo-B(tu+(l-t)~,,))>O tlt€[O,l],~EX. Then, letting t tend to zero and using the hemicontinuity of B , we get (x0 - u,y0 - B x ~2 ) 0

which clearly implies that y o

= Bx,,

VUE X ,

as claimed.

Now we shall prove Theorem 1.1 in the case where X is finite dimensional. Lemma 1.2. Let X be a finite dimensional Banach space and let A and B be two monotone subsets of X X X * such that 0 E D(A ) , and B is single valued, continuous, and satisfies (1.10). Then there exists x E conv D(A ) such that

(u-x,Bx+u)>O

V[U,U]EA.

(1.12)

Proof Redefining A if necessary, we may assume that K = conv D(A ) is bounded. Indeed, if Lemma 1.2 is true in this case, then replacing A by A , = ( [ x , y ] E A ; llxll I n} we infer that for every n there exists x , E K , = K fl ( x ; llxll I n} such that (U

-x,,Bx,

+U) >0

V [ U , U ]€ A , .

(1.13)

This yields (x,

, ~ ~ , ) l l ~ , I l ~ I' II 5 11,

for some

5 E AO,

and by the coercivity condition (1.10) we see that there is M > 0 such that Ilx,ll I M for all n. Now, on a subsequence, for simplicity again denoted n ,

we have x , + x . By (1.13) and the continuity of B, it is clear that x is a solution to (1.12), as claimed.

2. Nonlinear Operators of Monotone Type

40

Let T: K + K be the rnultivalued operator defined by

T x = ( ~ E K ; ( u - ~ , B x + ~u O) V [ U , U€ A ] }. Let us show first that Tx

K,,

=

#

0, Vx E K . To this end, define the sets

( y E K ; ( U - y , BX +

and notice that

n

TX =

2 01,

U)

K,,.

[u,vIEA

Inasmuch as the K,, are closed subsets (if nonempty) of the compact set K , to show that fl[ U , U I E A K , , # 0 it suffices to prove that every finite collection ( K,, ,", ; i = 1, . . . ,rn} has a nonempty intersection. Equivalently, it suffices to show that the system (u; - y , B x

+ u;) 2 0,

i = 1,..., m ,

has a solution y E K for any set of pairs [ u ; , ui] E A , i Consider the function H: U X U + R,

H(A,p)

=

where U=

" (

C p;

i= 1

1

+ u ; , c hjuj - U ; m

BX

j=l

i

=

(1.14) 1,. . . , m.

VA, p

A E R m ; A = ( A l,..., A , ) , A j 2 0 ,

E

U , (1.15)

cAj=l m

i= 1

1

.

The function H is continuous, convex in A, and concave in p. Then, according to the J. Von Neumann theorem (see Theorem 2.3 in Chapter 1) it has a saddle point ( A o , pol E U X U, i.e.,

H ( A0 P ) 9

I

H ( A0

9

PO)

5 H ( A,

VA, P

PO)

On the other hand, we have

j= 1

m

m

.

.

m

m

E

u.

(1.16)

2.1. Maximal Monotone Operators

41

Then, by (1.161, we see that H(Ao,p) 5 0

VPE

i.e., pi( B x i= 1

+ ui,

In particular, it follows that.

i

m

Bx+ui,

1

m

( A j ) o ~ -j u i 5 0

1

( A j ) o ~ j - ~ 5i 0 j= 1

u, Vp

E

U.

V i = 1 , ..., m .

Hence, y = C;! ( A j ) o ~ E j K is a solution to (1.14). We have therefore proved that T is well-defined on K and that T ( K ) c K. It is also clear that for every x E K, Tx is a closed convex subset of X and T is upper semicontinuous on K. Indeed, since the range of T belongs to a compact set, to verify that T is upper-semicontinuous it suffices to show that T is closed in K x K, i.e., if [ x , , y,] E T , x , + x and y, + y, then y E Tx. But the last property is obvious if one takes in account the definition of T. Then, applying Kakutani’s fixed point theorem (Theorem 2.2 in Chapter 1) we conclude that there exists x E K such that x E Tx, thereby completing the proof of Lemma 1.2. Proof of Theorem 1.1. Let A be the family of all finite dimensional subspaces X , of X ordered by the inclusion relation. For every X , E A, denote by j,: X , + X the injection mapping of X , into X and by j : : X * --+ X,* the dual mapping, i.e., the projection of X * onto X,* . The operators A, = j : Aj, and B, = j,* Bj, map X , into X: and are monotone in X , X X,*. Since B is hemicontinuous from X to X * and the j,* are continuous from X * to X: it follows by Lemma 1.1 that B, is continuous from X -, to X,*. We may therefore apply Lemma 1.2, where X = X,, A = A , , B = B,, and K = K, = conv D( A,). Hence, for each X , E A, there exists x , E K, such that ( U -x,,

B,x,

+ U) 2 0

V [ U , U ]E A ,

or equivalently, (U

-x,,Bx,

+ u) 2 0

V [ u , u ]€ A , .

(1.17)

42

2. Nonlinear Operators of Monotone Type

By using the coercivity condition (l.lO), we deduce from (1.17) that {x,} remains in a bounded subset of X. Since the space X is reflexive, every bounded subset of X is sequentially compact and so there exists a sequence { x , ) c {x,} such that in X

x," - x

as n

-+

(1.18)

03.

Moreover, since the operator B is bounded on bounded subsets, we may assume that in X *

Bxan- y

as n

(1.19)

+ w.

Since the closed convex subsets are weakly closed, we infer that x By (1.171, we see that limsup (xu",BX,") I (u -x,u) n-m

+ (u,y)

E

K.

V [ u , u ] E A . (1.20)

Without loss of generality, we may assume that A is maximal in the class of all monotone subsets A' c X X X * such that D(A') c K = conv D(A). (If not, we may extend A by Zorn's lemma to a maximal element of this class.) To complete the proof, let us show first that (1.21)

lim sup ( xan - x , B x , ) I 0. n-m

Indeed, if this is not the case, it follows from (1.20) that (u - x , u + y ) 2 0

V[u,u] € A ,

and since x E K and A is maximal in the class of monotone operators with domain in K it follows that [x,- y ] E A . Then, putting u = x in (1.20), we obtain (1.211, which contradicts the working hypothesis. Now, for u arbitrary but fixed in D ( A ) consider uA = Ax + (1 - A h , 0I A I 1, and notice that by virtue of the monotonicity of B we have ( xan - UA

9

Bxan)

2 (xan -

uA

9

BuA)*

This yields (1

-

A)(Xmn -

2

u , BX,)

( 1 - A)(

+ A(-%"

X a n - U ,Bu,)

- x , BX,")

+ A( X a n

- X , Bu,)

and, by (1.21) and (1.20),

( x - u , B u A ) I1imsup(xan - u , B x a n ) I n-m

( U - X , U )

V[u,u] € A .

2.1. Maximal Monotone Operators

43

Since B is hemicontinuous, the latter inequality yields (U

V [ U , U €] A ,

-x,u+Bx) 2 0

rn

thereby completing the proof of Theorem 1.1.

We shall use now Theorem 1.1 to prove a fundamental result in theory of maximal monotone operators due to G. Minty and F. Browder. Theorem 1.2. Let X and X * be refexive and strictly convex. Let A c X X X * be a monotone subset of X X X * and let J : X + X * be the duality mapping of X . Then A is maximal monotone if and only if, for any A > 0 (equivalently, for some A > O), R( A + A J ) = X * . Proof: “If” part. Assume that R ( A + A J ) = X * for some A > 0. We suppose that A is not maximal monotone, and argue from this to a contradiction. If A is not maximal monotone, there exists [ x o ,y o ] E X X X * such that [ x o , y o ]G A and ( x -xo,y -yo) 2 0

( 1.22)

V[X,Yl € A .

On the other hand, by hypothesis, there exists [xl, y , ] E A such that AJ(x1) + Y 1

=

AJ(x0) +YO.

Substituting [xl, y l ] in place of [ x , y ] in (1.22) yields (x1

- xo 7 J ( x 1 ) - J ( x 0 ) ) I0.

Taking in account definition of J , we get llx1112 + llxo112 I( x 1 , J(X0))

+ (xo

(x1 , J ( x o ) ) =

llx1ll

9

J(Xl)),

and therefore (XO,J(Xl))

=

2

=

2

llxoll

’

Hence J(x0) = J(Xl),

and since the duality mapping J-’ of X * is single valued (because X is strictly convex), we infer that x o = xl. Hence [ x o ,y o ] = [xl, y l ] E A , which contradicts the hypothesis.

44

2. Nonlinear Operators of Monotone Type

“Only if” part. The space X * being strictly convex, J is single valued and demicontinuous on X (Theorem 1.2 in Chapter 1). Let y o be an arbitrary element of X * and let A > 0. Applying Theorem 1.1, where BU = A J ( u ) - yo

we conclude that there is x (U

EX

- x , A J ( x ) -yo

VU

E

X,

such that

+U) 2 0

V [ U , V ]E A

Since A is maximal monotone, this implies that [ x , AJ(x) - y o ] E A, i.e., yo E AJ(x) Ax. Applying Theorem 1.1, we have implicitly assumed that 0 E D ( A ) . If not, we apply this theorem to Bu = AJ(u + uo>- yo and def Au = A(u + uo),where uo E D ( A ) .Thus, the proof is complete.

+

We shall see later that the assumption that X * is strictly convex can be dropped in Theorem 1.2. Now we shall use Theorem 1.1 to derive a maximality criteria for the sum A B .

+

Corollary 1.1. Let X be rejlexiue and let B be a hemicontinuous and bounded operator from X to X * . Let A c X X X * be maximal monotone. Then A + B is maximal monotone.

Proof By Asplund’s theorem (Theorem 1.1in Chapter l), we may take an equivalent norm in X such that X and X * are strictly convex. It is clear that after this operation the monotonicity properties of A, B , A + B as well as maximality do not change. Also, without loss of generality, we may assume that 0 E D ( A ) ;otherwise, we replace A by u -+ A(u uo),where uo E D ( A ) and B by u + B(u + u,). Let yo be arbitrary but fixed in X*. Now, applying Theorem 1.1, where B is the operator u -+ Bu J(u) - y o , we infer that there is an x E conv D( A ) such that

+ +

(U

-x,Jx

+ BX - y o + U ) 2 0

Since A is maximal monotone, this yields

y o E Ax as claimed.

+ Bx + Jx,

V[U,U]€ A .

2.1. Maximal Monotone Operators

45

In particular, it follows by Corollary 1.1 that every monotone, hemicontinuous, and bounded operator from X to X * is maximal monotone. We shall prove now that the boundness assumption is redundant. Theorem 1.3. Let X be a reflexive Banach space and let B : X -+ X * be a monotone hemicontinuous operator. Then B is maximal monotone in X X X *.

Suppose that B is not maximal monotone. Then there exists [x, , y o ] E X x X * such that y o # Bx, and

Proof

(x, - U,Y, - Bu) 2 0

For any x E X, we set u, (1.23). We get (x, - X , y o

=

-

Ax,

(1.23)

VU EX.

+ (1 - A)x, 0 I A I 1, and put u = u,,in VA

Bu,) 2 0

E

[0,1], u

E

X,

and, letting A tend to 1, (xo-x,y,-Bx,)

VXEX.

20

Hence y o = Bx,, which contradicts the hypothesis. Corollary 1.2. Let X be a reflexive Banach space and let A be a coercive maximal monotone subset of X x X * . Then A is surjective, i.e., R ( A ) = X * . Proof Let y o E X * be arbitrary but fixed. Without loss of generality, we may assume that X,X * are strictly convex, so that by Theorem 1.2 for every A > 0 the equation

+

AJ(xA)

(1.24)

YO

has a (unique) solution x, E D ( A ) . Multiply Eq. (1.24) by x,, - xo, where xo is the element arising in the coercivity condition (1.9). We have

AII~,II~+ (xA

- x 0 , A,) = (xA - x o , y o )

+~(x,,~x,).

By (1.91, we deduce that (x,) is bounded in X and so we may assume (taking a subsequence if necessary) that 3 x , E X such that

w

-

lim x, A10

=x,.

Letting A tend to zero in (1.24), we see that lim Ax = y o . A10

2. Nonlinear Operators of Monotone Type

46

Since, as seen earlier, maximal monotone operators are weakly-strongly closed in X X X * , we conclude that y o E Ax,. Hence R ( A ) = X * , as claimed. W In particular, it follows by Corollary 1.2 and Theorem 1.3 that: Corollary 1.3. A monotone, hemicontinuous, and coercive operator B from a reflexive Banach space X to its dual X * is surjective.

Let us now pause briefly to give an immediate application of this surjectivity result to existence theory for nonlinear elliptic boundary value problems. Let R be an open and bounded subset of the Euclidean space R"'. Consider the boundary value problem

c

,..., Dmu)= f

(-l)"D* A,(x ,u

inR,

(1.25)

lalsm

in d o ,

1,

where f E V r n V q ( R ) , l/p + l/q satisfy the following conditions: (i) A ,

=

1, is given and A , : R

6 ) are measurable in x , continuous in such that

IA,(x,

()I

IC(161p-'

+g(x))

Vx

E

Rk + R

6, and 3 g

= A,(x,

Lq(R)

X

E

R,6 E R k ; (1.26)

(ii)

C

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

2 0

lalsm

for all Let V

=

6, r]

E

Rk and a.e.

x

WrgP(R)and let a: V

a(u,u)=

E

X

R.

V

-+

R be the Dirichlet functional

c 1D % . A , ( x , u ,...,

Dmu)Ctx,

u , u E V . (1.27)

laism

By (i) and (ii), it is readily seen that a is well-defined on all of V x V and Ia(u, .)I

I~ ( I I ~ l l m , p ) l l ~ l l m , p V U , U €

V.

(1.28)

2.1. Maximal Monotone Operators

47

(Here, )I I l m , p is the norm in W;,p(R).) Moreover, a(u,u - u ) -a(u,u - u ) 2 0

The function u

E

VU,U€

(1.29)

V.

V is called weak solution to problem (1.25) if a(u,u) = ( u , f )

VUE

( 1.30)

V.

(Here, (., . ) denotes the usual pairing between V and V'.) We will assume further that (iii) lim

lIull,,p-+m

a(u, ~

) / l l ~ l l m= , p 03.

Proposition 1.2. Under assumptions (i)-(iii), for every f problem (1.25) has at least one weak solution u E W ; . P ( R ) .

Proof

Define the operator A: V

+

E

W-m9q(R)

V' by

(u,Au)=a(u,u)

VU,V€

V.

The operator A is monotone and coercive. To apply Corollary 1.3, it suffices to show that A is hemicontinuous. As a matter of fact, we shall prove that A is demicontinuous. To this end, let (u,} be strongly convergent to u in V as n + m. Extracting further subsequences if necessary, we may assume that

Dau,

+ D"u

a.e. in R for la1 I m.

This implies that A a ( x , u n,...,D m u , ) - + A , ( x , u,..., D m u )

a.e. x

E

R , la1 I m .

Taking in account estimate (1.261, this implies that

,llm j a A a ( x , u ,,..., =

D m u , )D b d v

/-A,(x,u ,..., D m u )D%dx

Vu E V , la1 I m . (1.31)

Indeed, by the Egorov theorem, for every E > 0 there exists 0, c R such that m ( n \ RE)I E and A,(x, u,, . . . , D m u , ) + A,(x, u,. . . ,D m u ) uni-

48

2. Nonlinear Operators of Monotone Type

formly in

.,/I

a,. On the other hand, A,( x , u, ,.. . ,DrnU,)D"u dw

uniformly in n. Now by (1.31) we conclude that w - lim,

2.1.2.

~~

Au,

= Au

in I/' as desired.

rn

The Sum of Two Maximal Monotone Operators

A problem of great interest because of its implications for existence theory for partial differential equations is to know whether the sum of two maximal monotone operators is again maximal monotone. Before answering to this question, let us first establish some facts related to Yosida approximation of the maximal monotone operators. Let us assume that X is a reflexive strictly convex Banach space with strictly convex dual X * , and let A be maximal monotone in X x X * . According to Corollary 1.1 and Corollary 1.2, for every x E X the equation 0

E J ( x , - X)

+A h ,

( 1.32)

has a solution x,. Since ( X

- U ,JX

-

Ju) 2

(IIxII

-

IIuII)~

V X , UE X ,

and J-I is single valued (because X is strictly convex), it is readily seen that x, is unique. Define J,x A,X

= x,, =

P J (-X x,).

(1.33)

for any x E X and A > 0. The operator A,: X -+ X * is called the Yosida approximation of A and plays an important role in the smooth approximation of A. We collect in

49

2.1. Maximal Monotone Operators

Proposition 1.3 several basic properties of the operators A , and J,

.

Proposition 1.3. Let X and X * be strictly convex and reflexive. Then:

(i)

A , is single valued, monotone, bounded and demicontinuous from X to (ii) IIAAxII I [Ax[= inf {llyll; y E Ax) for every x E D ( A ) , A > 0; (iii) J,: X + X is bounded on bounded subsets and

x*;

lim JAx

A-0

(iv) If A,

-+ 0,

x,

--+

x , A,?,

=x

-

V x E conv D(A ) ;

(1.34)

y and

(1.35)

lim sup ( x , - x , ,A,,x, - A,,x,) I 0, n,m-m

(v)

-

then [ x , y ] E A and lim,,,,+m(x,, - x m , Ahox,- A,,x,) = 0; For A + 0, A,x Aox Vx E D ( A ) , where Aox is the element of minimum norm in Ax. If X * is uniformly convex, then A,+x+ Aox V x E D(A ) .

The main ingredient of the proof is the following lemma.

-

-

Lemma 1.3. Let X be a reflexive Banach space and let A be a maximal monotone subset of X X X * . Let [u,, u,] E A be such that u, u, u, u, and either

lim sup ( u , - u, ,u, - urn)I 0

(1.36)

n,m-rm

or

limsup ( u , - u,u, - u ) I 0. n-m

Then [ u , u ] E A and (u,, u,)

+

(u, u ) as n

+

00.

Prooj Assume first that condition (1.36) holds. Since A is monotone, we have lim

n,m-m

( u , - u, ,u, - urn)= 0.

Let nk + 00 be such that (unk,unk) .+ p. Then, clearly, we have p I ( u , u ) = 0.

2. Nonlinear Operators of Monotone Type

50

-

-~

Hence limn

(u, ,u,) I ( u , u), while by monotonicity of A we have ( u , - x , v, - y ) 2 0

WX,

yl

EA,

and therefore 20

(u -x,u-y)

V[x,y]€A,

which implies [ u , u ] E A because A is maximal monotone. The second part of the lemma follows by the same argument. rn

Proof of Proposition 1.3. (i) We have ( x - y , A,x - AAy)

=

(JAx -

+((’

and since A,x

E

A,x - A , y ) -

-JAx)

(Y

-AAy)7

M,x, we infer that ( X

- y , A,x - A , y ) 2 0

because A and J are monotone. Let [ u , u ] E A be arbitrary but fixed. If multiply Eq. (1.32) by J,x - u and use the monotonicity of A , we get (J,x - u 7 J ( J , x - x ) ) I h(u - J ~ x , u ) ,

which yields

IIJ,x

I Ilx - uII lIJ,x

-

+ Allx - uII llVll + AllUll IIJ,X

-

- XI1.

This implies that J, and A, are bounded on bounded subsets. Now, let x , + x o in X. We set u, = J,x, and u, = A,x,. By the equation

J(u,

-

x,)

+ Au,

=

0,

it follows that ((un

- x n)

-

(urn

+ A( U, - U,

-xrn),J(un

,U,

- u,)

- x n )

+ A( X ,

-J(urn - X,

,U,

-Xrn))

-

u,)

=

0.

Since as seen previously J, is bounded, this yields lim

n,m-rm

and

lim

m,n+m

( ( u , - x n ) - (urn- x , , J ( u ,

( u , - u m , u n- urn)I 0

-xn)

-J(u, - x m ) )

=

0.

2.1. Maximal Monotone Operators

-

-

-

51

Now, let n k + be such that unk u , unk u and J(unk- x n k ) w. By Lemma 1.3, it follows that [ u , u ] E A , [ u - x , , w ] E J , and therefore J( u

+ AU = 0.

x,)

-

-

-

We have therefore proven that u = J A x o ,u = A , x , , and by the uniqueness of the limit we infer that JAx, JAxo and A A x n A A x Oas , claimed. (ii) Let [ x , x * ] E A. Again, by the monotonicity of A we have 0I ( x - JAx,X * - A , x ) I Ilx*ll IIx - xA\II- A-'Ilx - xAll

2

Hence, AIIAAxII = IIx - x A I I I AIIx*II

which implies (ii). (iii) Let x E convD( A ) and [ u , u * ] (JAX -

E

VX*

E Ax,

A. We have

u , A,x - u * ) 2 0 ,

and therefore IIJAX - X I [

Let A,

-+

2

< A(JAX - U , U * ) -k

0 be such that

J(JAp- x )

-

-

2

lim IIJAx- xII

(U - X , J ( J A X

-X)).

y in X*. This yields

I (u -x,y).

A-0

Since u is arbitrary in D ( A ) , the preceding inequality extends to all u E conv D( A), and in particular we may take u = x . (iv) We have ('n

-' =

m

- AAmxm)

7

(x n - x m

= (JA,,xn

9

-

AJAnxx,

- JA,,,xm

- ( x m - JA,,,xm

9

AJAnxn

- AJAmxm)

-

(xm

- JAmxm))

( ( x ~- J A n x n )

-

(xm

-JAmxm),

A,'J(x,

-

(('n

- JAnxn)

- AAmXm)

7

- JAnxx,)

(<xn

+

JAp,)- A i 1 J ( x m - JA,xm)).

-AA,,,xm)

2. Nonlinear Operators of Monotone Type

52

Since A,?, = A;'J(J,?, - x,) and x, remain in bounded subsets of X * and X, respectively, we infer that lim

m,n-rw

(x, - x, , Ah?,

- A,,xm) =

0

and lim

( J,?x, - J,,xm , A,,x, - A,,xm)

m,n+m

=

0.

Then, by Lemma 1.3 we conclude that [x,y ] E A because lim (Jh,x, - x,)

n+m

= -

lim A,J-'( AAnxAn) = 0.

n-rm

(v) Since Ax is a closed convex subset of X*, and X * is reflexive and strictly convex, the projection Aox of 0 into Ax is well-defined and unique. Now, let x E D ( A ) and let A,, + 0 be such that AA.X y in X * . As seen in the proof of (iv), y E Ax, and since llA,?Il IIIAoxll we infer that y = A'x. Hence, A,x Aox for A + 0. If X * is uniformly convex, then by Lemma 1.1 in Chapter 1 we conclude that A,x + Ax (strongly) in X * as A + 0.

-

-

Proposition 1.4.

If X

=

H is a Hilbert space, then:

J, = ( I + AA)-' is nonexpanswe in H (i.e., Lipschitz with Lipschitz constant 1); (ii) IIA,x - A,yll IA-'llx - yll VX, y E D ( A ) , A > 0; (iii) Iim,+,, A,x = Aox Vx E D ( A ) .

(i)

Prooj (i) We set x, = (I + A A ) - ' x , y , operator in H I . We have

=

XA-yA+A(AYA-AYA)

(I

+ AA)-'y

( I is the unity

3x-Y'

(1.37)

Multiplying by x, - y , and using monotonicity of A, we get

1Ix~- y ~ l l IIIx -yll

VA

> 0-

Now, multiplying Eq. (1.37) by Ax, - A y , we get (ii). Regarding (iii), it follows by Proposition 1.3(v). Corollary 1.4. Let X be a reflexive Banach space and let A be maximal monotone in X X X * . Then both D( A ) and R( A ) are convex.

2.1. Maximal Monotone Operators

53

Prooj Without any loss of generality, we may assume that X and X * are strictly convex. Then, as seen in Proposition 1.2, JAx + x for every x E conv D(A ) . Since JAx E D ( A ) for all A > 0 and x E X , we conclude that c o n v D ( A ) = D ( A ) , as claimed. Since R ( A ) = D ( A - ' ) and A - ' is maximal monotone X * X X , we conclude that R( A ) is also convex. W We shall establish now an important property of monotone operators with nonempty interior of the domain. Theorem 1.4. Let A be a monotone subset of X bounded in any interior point of D(A).

X

X * . Then A is locally

Let us first prove the following technical lemma. Lemma 1.4. Let ( x , ) c X a n d {y,} c X * be such that x , -+ 0 and lly,ll + as n -+ m. Let B(0,r) be the closed ball { x ; llxll I r}. Then there exist x o E B(0, r ) and { x , ) c {x,}, (y,) c {y,) such that (1.38)

Prooj Suppose that the lemma is false. Then there exists r > 0 such that for every u E B(0, r ) there exists C , > -m such that V n E N.

( x , - u,y,) 2 C,

We may write B(0, r ) = U k ( E ~ B(0, r); ( x , - u , y,) 2 - k V n } .Then, by the Hausdorff-Baire theorem (see e.g., K. Yosida [l],p. 111, we infer that there is k , such that int(u

E

B ( 0 , r ) ;( x ,

In other words, there are ( u ; IIu - uoll I E}

E

> 0, k ,

c (u

E

> - k , V n } z 0.

- u,y,)

N, and

u,

B(0,r);( x ,

-

E

E B(0, r )

such that

u,y,) > -k, V n } .

Now, we have ( x , - u,yn) 2 -k,

and

(x,

+ u o , y n ) 2 C-,,,.

Summing up, we get (2x,

+ u, - u,yn) 2

-k,

+c

Vu E B(u,,

E),

54

2. Nonlinear Operators of Monotone Type

where C = Cue. Now, we take u = uo + 2x, n sufficiently large, we therefore have

(w,y,) 2

-c

+ w ,where llwll = ~ / 2 .For

Vw,llwll=

+y,

which clearly contradicts the fact that llyJ

--f

E/2,

a as

n

a.

Proof of Theorem 1.4. Let xo E int D ( A ) be arbitrary. Without loss of generality, we may assume that xo = 0. (This can be achieved by shifting the domain of A . ) Let us assume that A is not locally bounded at 0. Then there exist sequences {x,} c X , {y,J c X * such that [x,, y,] E A , Ilx,ll + 0, and lly,ll + 03. According to Lemma 1.4 there exists, for every ball B(0, r ) , x o E B(0, r ) and { x , ) c {x,), {y,) c { y J such that

Let r be sufficiently small so that B(0, r ) c D ( A ) . Then, x o by the monotonicity of A it follows that (xnk- x o , A x o ) -+ -a

as k

+

E

D ( A ) and

00.

The contradiction we have arrived at completes the proof.

W

Now we are ready to prove the main result of this section, due to R. T. Rockafellar [4]. Theorem 1.5. Let X be a reflexive Banach space and let A and B be maximal monotone subsets of X x X * such that

(int D( A ) ) Then A

n D( B ) ) # 0.

+ B is maximal monotone in X

X

(1.39)

X*.

Prooj As in the previous cases, we may assume without loss of generality that X and X * are strictly convex. Moreover, shifting the domains and ranges of A and B, if necessary, we may assume that 0 E (int D ( A ) ) n D ( B ) , 0 E AO, 0 E BO. We shall prove that R(J + A + B ) = X * . To this aim, consider an arbitrary element y in X * . Since the operator BA is demicontinuous, bounded, and monotone, and so is J : X + X * , it follows by Corollaries 1.1 and 1.2 that for every A > 0 the equation J(xJ +

+ BAxA 3 Y

(1.40)

55

2.1. Maximal Monotone Operators

has a solution x, E D ( A ) . (Since J and J - ’ are single valued and X , X * are strictly convex, it follows by standard arguments involving the monotonicity of A and B that x, is unique.) Multiplying Eq. (1.40) by x, and using the obvious inequalities ( XA BAx,) 2 0,

( XA A,) 2 0 ,

9

7

we infer that

vA > 0.

llx~llIllyll

Moreover, since 0 E int D ( A ) , it follows by Theorem 1.4 that there exist the constants p > 0 and M > 0 such that

Ilx*ll IM

(1.41)

v x * € A X , llxll 5 p .

Multiplying Eq. (1.40) by x, - pw and using the monotonicity of A , we get ( X A - p W , J X , 4-

+ ( X A - p W , A( p w ) ) I 0

BAXA- y)

vllwll

=

1.

By (1.411, we get

+ IIxAlI) + IIxAII( P + IIyII).

lIxAII2 - p ( w , BAxA) I M ( p

Hence, IIx,J12 + pllBAxAll 5

IIx,lI(

p

vA > 0-

+ M + IIyII) + M p

We may conclude, therefore, that {B,x,) and (yA= y - Jx, - B,x,,) are bounded in X * as A -, 0. Since X is reflexive, we may assume that on a subsequence, again denoted A,

BAxA-Y~,

~A-xxg,

Inasmuch as A

+J ( X,

JX,-YO.

Y A E A A - Y ~ ?

is monotone, we have -

VA, p > 0.

x,,, BAxA- B,,x,) 5 0

Then, by Proposition 1.36~1, lim ( x,

A,cL-O

-

x,, ,BAxA- B,,x,)

=

0

and [ x ~yl, ] E B. Then, by Eq. (1.40), we see that lim ( X, - X, , Jx,

A,CL-+O

+ yA- Jx,

-

y,,)

=

0

YA

A,

7

yp

7

and since J + A is maximal monotone it follows by Lemma 1.3 that [xo,yo + y2] E A + J . Thus, letting A tend to zero in (1.40) we see that

2. Nonlinear Operators of Monotone Type

56

thereby completing the proof.

W

In particular, Theorem 1.5 leads to: Corollary 1.5. Let X be a reflexive Banach space, A c X X X * a maximal monotone operator and let B: X + X * be a demicontinuous monotone operator. Then A B is maximal monotone. W

+

More generally, it follows from Theorem 1.5 that if A, B are two maximal monotone sets of X x X*, and D ( B ) = X*, then A + B is maximal monotone. We conclude this section with a result of the same type in Hilbert spaces.

Theorem 1.6. Let X = H be a Hilbert space and let A , B be maximal monotone sets in H X H such that D( A ) n D( B ) # 0 and ( u , A , u ) 2 -C(llul12

Then A

+ AllA,ull* + llA,ull + 1)

V [ U , U ]E B . (1.42)

+ B is maximal monotone.

Proof We have denoted by A, = A-'(Z - ( I + A ) - ' ) the Yosida approximation of A. For any y E H and A > 0, consider the equation

x,

+ Bx, + A,x,

3y

,

( 1.43)

which by Corollaries 1.4 and 1.5 has a solution (clearly unique) x, E D(B). Let xo E D ( A ) n D ( B ) . Taking the scalar product of (1.43) with x, - x o and using the monotonicity of B and A, yields (xA,xA -xo)

+ YO,^, - x o ) + ( A , x o , x , - x o )

5 (Y,x, -xo).

Since, as seen in Proposition 1.3, IIAAxoIII IAxoIVA > 0, this yields

Ilx,Il I M

VA > 0.

Next, we multiply Eq. (1.43) by A,x, and use inequality (1.42) to get, after some calculations, IIAAxAII I C

VA

> 0.

2.2. Generalized Gradients (Subpotential Operators)

Now, for a sequence A,

+

57

0, we have

where y , = y - x, - A,x, E Bx,. Then, arguing as in the proof of Theorem 1.5 it follows by Proposition 1.3 that [ x , y l ] E A , [ x , y 2 ] E B , and this implies that y E x + Ax + Bx, as claimed. W 2.2. Generalized Gradients (Subpotential Operators)

2.2.1. Subdifferential of a Convex Function

Let X be a real Banach space with dual X * . A proper convexfirnction on X is a function cp: X + ] - 00, + 4 = % that is not identically + 03 and that satisfies the inequality cp((1 - A ) x + AY) I ( 1 - A)cp(x) + M Y )

(2.1)

for all x , y E X and all A E [O, 11. The function cp: X + ] - 00, + m ] is said to be lower semicontinuous (1.s.c.) on X if lim inf cp(u) 2 cp( x) u+x

Vx E X ,

or equivalently, every level subset {x E X; ~ ( x I) A) is closed. Since every level set of a convex function is convex and every closed convex set is weakly closed (this is an immediate consequence of Mazur's theorem, (K. Yosida [ l ] ,p. 1091, we may therefore conclude that a proper convex function is lower semicontinuous if it is weakly lower semicontinuous. Given a lower semicontinuous convex function cp: X + ] - 00, +..I = R, cp f 00, we shall use the following notations: (the effective domain of cp),

D( cp)

=

( x E X ; cp( x )

< 00)

Epi ( cp)

=

(( x , A)

R; cp( x ) I A)

E

X

X

(2.2) (the epigraph of cp). (2.3)

It is readily seen that Epi(cp) is a closed convex subset of X properties are closely related to those of the function cp.

X

R, and its

2. Nonlinear Operators of Monotone Type

58

Now, let us briefly describe some elementary properties of l.s.c., convex functions. Proposition 2.1. Let cp: X + R be a proper, 1.s.c. and convex function. Then cp is bounded from below by an afine function, i.e., there are x: E X * and a E R such that cp(x) 2 ( x , x , * )

+a

vx EX.

(2-4)

Let E(cp) = EpKcp) and let x o E X and r E R be such that cp(xo) > r. By the classical separation theorem, there is a closed hyperplane H = { ( x , A) E X X R; ( x ; , x ) A = a} that separates E(cp) and ( x o , r ) . This means that

Proof

+

( x : ,x )

and

+A2a

(x,*,xo)

Hence, for A

=

vx

E

E ( cp),

+ r < a.

cp(x), we have

(x:,x)

+ cp(x)

which implies (2.4).

2 (x,*,x,)

+r

VXEX,

W

Proposition 2.2. Let cp: X continuous on int D(cp).

+

R

be proper, convex, and 1.s.c. Then cp is

Proof Let x o E int D(cp). We shall prove that cp is continuous at x o . Without loss of generality, we will assume that x o = 0 and that q(0) = 0. Since the set ( x : cp(x) > - E } is open it suffices to show that { x : cp(x) < E } is a neighborhood of the origin. We set C = { x E X ; cp(x) I E } n ( x E X ; cp(-x) s E ) . Clearly, C is a closed, balanced set of X (i.e., a x E C for la1 I 1 and x E C). Moreover, C is absorbing, i.e., for every x E X there exists a > 0 such that a x E C (because the function t + cp(tx) is convex and finite in a neighborhood of the origin and therefore it is continuous). Since X is a Banach space, the preceding properties of C imply that C is a neighborhood of the origin, as claimed. W

The function cp*: X *

+

@ defined by

cp*(p) = sup{(x,P )

is called the conjugate of cp.

-

cp(x); x E XI

(2.5)

59

2.2. Generalized Gradients (Subpotential Operators)

Proposition 2.3. Let cp: X --+ R be l.s.c., convex, and proper. Then I.s.c., convex, and proper on the space X*.

cp* is

ProoJ: As supremum of family of affine functions, cp* is convex and I.S.C. W Moreover, by Proposition 2.1 we see that cp* f m. Proposition 2.4. Let X be reflexive and let cp be a I.s.c. function on X. Further assume that

lim cp(x)

Ilxll+

=

proper convex

00.

m

Then there exists x o E X such that cp(xo) = i n f ( c p ( x ) ; x E X } .

ProoJ: Let d = inf{cp(x); x E X) and let { x , } c X such that d Icp(x,) I d + l / n . By (2.6),we see that { x , } is bounded in X and since the space X is reflexive it is sequentially weakly compact. Hence, there is { x n k }c { x , } such that x n k x as nk + 00. Since cp is weakly semicontinuous, this W implies that cp(x) Id . Hence cp(x> = d , as desired.

-

Given a function f from a Banach space X to R, the mapping f’: X X + R defined by

X

(if it exists) is called the directional derivative of f at x in direction y . The function f : X + R is said to be Gfiteaux differentiable at x E X if there exists Vf(x) E X* (the Giiteaux differential) such that f’(X?Y) =

(Y,Vf(X))

VY E X .

(2.8)

If the convergence in (2.7) is uniform in y on bounded subsets, then f is said to be Frbchet differentiable and Vf is called the Frichet differential (derivative) of f. Given a l.s.c., convex, proper function cp: X + R, the mapping dcp: X + X* defined by d c p ( ~ )= { x * E X * ; c p ( ~ ) Icp(y)

is called the subdifferential of cp.

+(X

- y , x * ) , V y E X ) (2.9)

2. Nonlinear Operators of Monotone Type

60

In general, dcp is a multivalued operator from X to X*, and in accord with our convention we shall regard it as a subset of X X X*. An element x * E dcp(x) (if any) is called a subgrudient of cp in x. We shall denote as usual by D(dcp) the set of all x E X for which dcp(x) # 0. Let us pause briefly to give some simple examples. 1. cp(x) = +1Ix1l2.Then, dcp = J (the duality mapping of the space X). Indeed, if x * E Jx, then (x - y, x*) = 1 1 ~ 1 1 ~ ( y , x*) 2 $(11x112 - lly112>V y E X . Hence x * E dcp(x). Now, let x * E dcp(x), i.e., ~(11x112- Ilyl12) I ( x - y , x*)

We take y

=

Ax, 0

vy

E

x.

(2.10)

< A < 1, in (2.10), getting (x,x*) 2

+I1X1l2(1

+ A).

Hence (x, x*> 2 llx112. 2 If y = Ax where A > 1, we get that (x, x*) I Ilx1l2.Hence, (x, x*) = llxll and Ilx*ll 2 IlxIl. On the other hand, taking y = x + A u in (2.101, where A > 0 and u is arbitrary in X, we get

+

( u , x*)

s

( u , x*)

s llxll Ilull.

+(IIx

- IIxlI2),

which yields Hence Ilx*Il I Ilxll. We have therefore proven that (x*, x) = Ilx1l2 = llx*1I2, as claimed. 2. Let K be a closed convex subset of X. The function I,: X + R defined by

0

ifxEK, if x @ K,

= {+a

(2.11)

is called the indicatorfunction of K, whilst its dual function H,

&(PI

=

sup((p,u); u

E

KJ

is called the supportfunction of K. It is readily seen that D(dZ,) = K, dZ,(x) nonempty) and that dZ,(x)

=N,(x)

=

VP E X * , =

0 for x

{ x * E X * ; ( X - u,x*) 2 OVU E K)

E

int K (if

VX E K. (2.12)

2.2. Generalized Gradients (Subpotential Operators)

61

For every x E K, N K ( x ) is the normal cone at K in x . Parenthetically, note that if K , and K, are two closed convex subsets, then K , c K, if and only if

3. Let p be Gsteaux differentiable at x . Then d p ( x ) since p is convex, we have

and letting A tend to zero we see that Vp(x) arbitrary element of d p ( x ) . We have d x ) - P(Y) 5 ( x

E

=

Vp(x). Indeed,

d p ( x ) . Now, let w be an

VY E X .

-Y,W)

Equivalently,

and this implies that ( y ,Vcp(x) - w ) 2 0 for all y

E X.

Hence, w

= Vp(x).

By the definition of d p it is obvious that p(x) = i n f ( p ( u ) ; u ~ X }

iffOEdp(x).

There is a close relationship between d p and d p * . More precisely, we have: Proposition 2.5. Let X be a refexiue Banach space and let p: X .+ R be a conuex, proper function. Then the following conditions are equiualent:

L.s.c.,

(i) x * E dp(x); (ii) p(x> + cp*(x*> (iii) x E dp*(x*>.

=

In particular, dp*

( d p ) - ' and (p*>*= p.

=

( x , x*);

62

2. Nonlinear Operators of Monotone Type

Pro05

By definition of cp*, we see that cp*(x*) 2 ( x , x * )

-

cp(x)

vx

E

x,

with equality if and only if 0 E d,((x, x * ) - cp(x)). Hence, (i) and (ii) are equivalent. Now, if (ii) holds, then x* is a minimum point for the function cp*(p) - ( x , p ) and so x E dcp*(x*). Hence, (ii) 3 (iii). Since conditions (i) and (ii) are equivalent for p*, we may equivalently express (iii) as

Thus, to prove (ii) it suffices to show that (cp*)* = cp. It is readily seen that (cp*)* = cp** I cp. We suppose now that there exists xo E X such that cp**(x,) > cp(x,), and we will argue from this to a contradiction. We have, therefore, (x,,, cp**(x,)) 6E Epi(cp) and so by the separation theorem it follows that there are x i E X * and a E R such that ( x : , x , ) + acp**(xo) > sup{(x;, x ) + ah; ( x , A) E Epi(cp)}. After some calculation involving this inequality, it follows that a < 0. Then dividing this inequality by - a , we get

which clearly contradicts the definition of cp**. Theorem 2.1. Let X be a real Banach space and let cp: X -+ R be a 1.s.c. proper convex function. Then dcp is a maximal monotone subset of X X X * . Pro05 It is readily seen that dcp is monotone in X X X * . To prove that dcp is maximal monotone, we shall assume for simplicity that X is reflexive and refer the reader to Rockafellar's work [2] for the general case. Continuing, we fix y E X * and consider the equation

Jx

+ dcp(x) 3 y .

(2.13)

2.2. Generalized Gradients (Subpotential Operators)

Let f: X

4

R be the convex, I.s.c. f(x)

= +11X1l2

63

function defined by

+ cp(x)

- (X,Y).

By Proposition 2.1, we see that l i m l , x l , + m f ( x=) + m , and so by Proposition 2.4 we conclude that there exists x o E X such that

f( x o ) = inf{f( x ) ; x

EX

}.

This yields +llx01l2 +

cp(X0)

- ( X 0 , Y ) 5 +11x112+ cp(x) - ( X , Y )

vx E

x 7

i.e., cp( xo) - q ( x ) 5 ( x o - x , y

) +

+ ( x -xo,Jx)

I (xo-x,y)

In the latter inequality we take x = tx, an arbitrary element of X. We get cp(X0)

+(11x112- 11xol12) vx E X .

+ (1 - t ) u , 0 < t < 1, where

- cp(u) 5 ( x o - U , Y >

u is

+ ( u -xo,w,),

where w, E J ( t x , + (1 - flu). For t 4 1, w, w E J ( x o ) because J is maximal monotone and so it is strongly-weakly closed in X X X*. Hence,

-

q(x0)

-

q(u) I (x,

-

u,y

-

vu E X ,

w)

and this inequality shows that y - w E dcp(xo), i.e., x o is a solution to Eq. (2.13). We have therefore proven that R(J + d q ) = X*, thereby completW ing the proof. Proposition 2.6. Let cp: X + R be a 1.s.c. convex and proper finction. Then D(dcp) is a dense subset ofD(cp1.

Prooj Let x be any element of D(q)and let x A = JAx be the solution to the equation (see (1.32))

J(x,

-X )

+ hdcp(x,)

Multiplying this equation by x A - x , we get

3

0.

2. Nonlinear Operators of Monotone Type

64

Since by Proposition 2.1 cp is bounded from below by an affine function and cp(x) < CQ, this yields lim x,,

A-0

As x,, E D(dcp) and D( cp)

=

x

D( dcp), as claimed.

=x.

is arbitrary in D(cp), we conclude that

rn

Proposition 2.7. Let cp be a l.s.c., proper, convex function on X . Then int D(cp) c D(dcp).

ProoJ: Let x o E int D(cp) and let V = B ( x , , r ) = { x ; IIx - xoll < rJ be such that V c D(cp). We know by Proposition 2.2 that cp is continuous on V and this implies that the set D = { ( x , A) E I/ X R; cp(x) < A} is an open convex set of X x R.Thus, there is a closed hyperplane, H = { ( x , A) E X x R; ( x , x , * ) A = a ) , which separates ( x o , cp(xo)) from 0. Hence, ( x o ,x : ) + cp(xo) < a and

+

(x,x,*)+A>a

V(x,A)EB.

This yields cp(xo) - cp(x)

<

-(xo -x,x;>

vx

E I/.

But, for every u E X , there exists 0 < A < 1 such that x = Ax, + (1 A)u E V. Substituting this x in the preceding inequality and using the convexity of cp, we obtain cp(xo) I cp(u)

Hence, x o E D(dcp) and x:

E

+ ( x o - u,x,*)

vu EX.

dcp(x,).

For every A > 0, define the function

where cp: X + R is a 1.s.c. proper convex function. By Propositions 2.1 and 2.4 it follows that cp,,(x) is well-defined for all x E X and the infimum defining it is attained (if the space X is reflexive). This implies by a straightforward argument that cp, is convex and 1.s.c. on X . (Since cp,, is everywhere defined, we conclude by Proposition 2.2. That cp, is continuous.)

65

2.2. Generalized Gradients (Subpotential Operators)

The function cp, is called the regularization of cp, for reasons that will become clear in the following theorem. Theorem 2.2. Let X be a reflexive and strictly convex Banach space with strictly convex dual. Let cp: X -,R be a 1.s.c. convex, properfunction and let A = dcp c X X X * . Then the function cp, is convex, continuous, Giteaux differentiable, and Vcp, = A , for all A > 0. Moreover:

lim cp,(x)

A+O

=

(2.16)

~ ( x ) Vx EX ;

If X is a Hilbert space (not necessarily identijied with its dual) then cp, is Frkchet differentiable on X . IIx Proo$ We observe that the subdifferential of the function u ~11~/2+ A cp(u) is just the operator u -, A - ' J ( u - x ) + dcp(u) (see Theorem 2.3 following). This implies that every solution x, of the equation -+

A- ' J( u

+ dcp(u) 3 0

- X)

is a minimum point of the function u 1/(2A)llx - ul12+ cp(u). Recalling that x A = JAx, we obtain (2.15). Regarding inequality (2.171, it is an immediate consequence of (2.14). To prove (2.16), assume first that x E D(cp). Then as seen in Proposition 1.3, lim,+ J,x = x , and by (2.17) and the lower semicontinuity of cp we infer that -+

cp( x ) I lim inf cp( A-0

J , x ) I lim inf cp,( x ) I cp( x ) . A-0

If x E D(cp), i.e., cp(x) = +m, then limA+ocp,(x) wise would exist (A,,} -, 0 and C > 0 such that cp,,(x) I C

= +a

because other-

Vn.

Then, by (2.13, we see that limn J,,x = x , and again by (2.17) and the lower semicontinuity of cp we conclude that cp(x) I C, which is absurd. ~

3

c

66

2. Nonlinear Operators of Monotone Type

because A , is demicontinuous. Hence, cp, is Giiteaw differentiable and VqA = (avo),= A , . Now, assume that X is a Hilbert space. Then by the monotonicity of A we see that lIJA(x) - JA( y)ll

s IIx

- yll

vx, y

x7

> O*

This implies that A , : X + X is Lipschitz with Lipschitz constant 2/h. (As a matter of fact, it follows that it is l / A J Then, by the inequality (2.181, we see that

Let us consider the particular case where cp subset of X , and X is a Hilbert space. Then IIX =

- PKX1I2

2A

tlx

=

ZK,K a closed convex

EX

, A > 0,

(2.19)

2.2. Generalized Gradients (Subpotential Operators)

67

where PKx is the projection of x on K. (Since K is closed and convex, P K x is uniquely defined.) Moreover, as seen in the preceding, we have PK = Jh

=

( I + AA)-'

VA

> 0.

(2.20)

A problem of great interest in convex optimization as well as for calculus with convex functions is to determine whether given two l.s.c., convex, proper functions f and g on X, d ( f + g ) = df + dg. The following theorem due to R. T. Rockafellar [3] gives a general answer to this question. Theorem 2.3. Let X be a Banach space and let f:X + R and g : X + R be two I.s.c., convex, properfunctions such that D( f) n int D ( g ) # 0. Then d(f+g)

=

(2.21)

df+ dg.

Proof If the space X is reflexive, (2.21) is an immediate consequence of Theorem 1.5. Indeed, as seen in Proposition 2.7, int D ( d g ) = int D ( g ) and so D ( d f ) n int D ( d g ) # 0. Then, by Theorem 1.5, df + dg is maximal monotone in X x X*. On the other hand, it is readily seen that df + dg c d ( f + g). Hence, df d g = d ( f + g). In the general case, Theorem 2.3 follows by a separation argument we will present subsequently. Since the relation df + dg c d ( f g ) is obvious, let us prove that d ( f g ) c df dg. To this end, consider xo E D(df)n D ( d g ) and w E d ( f gxx,), arbitrary but fixed. We shall prove that w = w 1 + w,,where w1 E d f ( x o ) and w, E d g ( x o ) . Replacing the functions f and g by x + f ( x + x , ) - f ( x , > - ( x , zl) and x + g ( x + x o ) - g ( x o ) - ( x , z,), respectively, where w = z1 z 2 ,we may assume that x , = 0, w = 0, and f(0) = g(0) = 0. Hence, we should prove that 0 E d f ( 0 ) + dg(0). Consider the sets Ei, i = 1,2, defined by El = { ( x , A) E X X R;f ( x ) I A), E , = { ( x , A) E X X R; g ( x ) I -A). Inasmuch as 0 E d ( f + gXO), we have

+

+ +

+

+

+

0

=

(f + g)(O)

=

inf{(f

+g)(x);x

E XI,

and therefore El n intE, = 0. Then, by the separation theorem there exists a closed hyperplane that separates the sets E, and E,. In other words, there are w E X * and a E R such that ( w , x ) + a h ~ O V(x,A)€El, (w,~+ ) ah 2 0

V(x,A)

E

E,.

(2.22)

68

2. Nonlinear Operators of Monotone Type

Let us observe that the hyperplane is not vertical, i.e., a f 0. Indeed, if a = 0, then this would imply that the hyperplane ( w , x ) = 0 separates the sets D ( f ) and D ( g ) in the space X, which is not possible because D( f ) n int D ( g ) # 0. Hence a # 0, and to specify the situation we shall assume that a > 0. Then, by (2.22) we see that VXE X ,

g(x) I -A I (w,x) I -af(X)

and therefore ( l / a ) w E d f ( O ) , - ( l / a ) w as claimed.

E dg(O),

i.e., 0

E

df(0)

+ dg(O),

Theorem 2.4. Let X = H be a real Hilbert space and let A be a maximal monotone subset of H X H . Let q : H -+ R be a l.s.c., convex, proper function such that D( A ) n D( d q ) # 0 and, for some h E H , q((Z

+ A A ) - ' ( x + Ah)) Iq ( x ) + CA(1 + q ( x ) ) , Vx

Then A

E

D( q ) , A > 0.

(2.23)

+ d q is maximal monotone and - D ( A + dq) = D ( A ) n ~ ( p ) .

Proof We shall proceed as in the proof of Theorem 1.6. Let y be arbitrary but fixed in H . Then, for every A > 0, the equation x,+ +A,ix,

+

d(~(x,i) 3~

has a unique solution x A E D(dq).We multiply the preceding equation by x - J,(x, + Ah) and use condition (2.23). This yields IlA,xA1l2+ ( AAx,

+ Ah)) s CA(IIyII + llhll + IIxAII + q(xA) + I), JA( x A )

- JA(

XA

where JA = (I + AA)-'. We get IIAAxAll2 ICA(IIyll

+ llhll + IIxAII + q ( x A ) + 1).

On the other hand, multiplying the preceding equation by x, - x o , where E D ( A ) n D(dq), we get

xo

IIxAII~

+ P(xA)

+

+ 1).

I C ( I I A , X , , I I ' c~(x,)

Hence, {A,x,) and ( x , } are bounded in H. Then, as seen in the proofs of Theorems 1.5 and 1.6, this implies that x, x , where x is the solution to

-

69

2.2. Generalized Gradients (Subpotential Operators)

the equation x

+ dcp(x) + A x

3

y.

Now, let us prove that

c D ( A ) n ~ ( c p )c D ( A ) n ~ ( d c p ) . - Let u E D ( A ) n D( cp) be arbitrary but fixed and let h be as in D(A) n

condition (2.23). Clearly, there is a sequence {uA}C D(cp) such that uA + Ah E D(cp) and uA -+ u as A -+ 0. Let % = JA(uA+ Ah) E D ( A ) n D(cp) (by condition (2.23)). We have

I l y - uII because u

E

+

5 IIJA(uA Ah) - JAull

+ Ilu - JAulI

-+

0

as A

-+

0,

D( A ) (see Proposition 1.3). Hence,

c D ( A ) n~ ( 9 ) .

D(A) n

Now, let u be arbitrary in D ( A ) n D(q)and let x A E D ( A ) n D(dcp) be the solution to xA

+ A ( h , + dc p( x , ) )

3

u.

By the definition of dcp, we have V ( xA) - V ( u ) I ( u - xA -

9

xA - u )

5 -1IU - XA1I2 4-hllAoUll IIU - XAll

Vh > 0.

Hence x A -+ u for A -+ 0, and so D ( A ) n D(cp) c D ( A ) n D(dcp), as claimed. This completes the proof. Remark 2.2. In particular, condition (2.23) holds if (AA(x

for some h

E

+ Ah),y) 2

-C(1

+~

( x ) )

VA

> 0,

H , and all [ x , y ] E dcp.

We conclude this section with an explicit formula for dcp in term of the directional derivative. Proposition 2.8. Let X be a Banach space and let cp: X convex, proper function on X . Then for all x,, E D(dcp),

-+

R

be a l.s.c.,

dcp(xo) = { x : E X * ; c p ' ( x 0 , u ) 2 ( U , X ; ) V UE X } .

(2.24)

70

2. Nonlinear Operators of Monotone Type

Prooj

Let x:

E dcp(x,).

Then, by the definition of dcp,

cp(xo) - c p ( x , + t u )

I- t ( u , x , * )

VUEX,t>O,

which yields c p f ( x o , u )2 ( u , x , * )

Vu E X .

Assume not that ( u , x:) Icp’(xo,u ) V u E X . Since cp is convex, the function t + (cp(xo + tu) - cp(x,))/t is monotonically increasing and so we have ( u , xi) 5 Hence x:

E

t-l(

cp( x o

+ tu) - cp(xo))

Vu

€

x,t > 0.

dcp(xo), and the proof is complete.

Formula (2.24) can be taken as definition of the subdifferential dcp, and we shall see later that it may be used to define generalized gradients of certain nonconvex functions. It turns out that if cp is continuous at x, then c p f ( x 0 , u )= SUP{(U,X:); X:

E

dcp(x,)},

u EX.

(2.25)

2.2.2. Examples of Subdifferential Mappings

There is a general characterization of maximal monotone operators that are subdifferentials of 1.s.c. convex functions due to R. T. Rockafellar [l]. A set A c X X X * is said to be cyclically monotone if

( x 0 - x ~ , x , * )+ *.. + ( x , - ~ - x , , x , * - ~ ) + (x, -x0,xg*) 2 0 (2.26) for all [ x i xi*] , E A , i = 0,1,. . . , n. A is said to be maximal cyclically monotone if it is cyclically monotone and has no cyclically monotone extensions in X X X * . It turns out that the class of subdifferential mappings coincides with that of maximal cyclically monotone operators. More precisely, one has: Theorem 2.5. Let X be a real Banach space and let A c X X X * . The set A is the subdifferential of a I.s.c. convex, proper function from X to R if and only $ A is maximal cyclically monotone.

We leave to the reader the proof of this theorem and we shall concentrate on some significant examples of subdifferential mappings.

2.2. Generalized Gradients (Subpotential Operators)

71

1. Maximal monotone sets (graphs) in R X R. Every maximal monotone set (graph) of R X R is the subdifferential of a 1.s.c. convex proper function on R. Indeed, let p be a maximal monotone set in R X R and let P o :R R be the function defined by

P o ( r ) = { y E p ( r ) ; lyl

=

inf{lzl; z

E

We know that D( p ) = [ a , b ] ,where -a Ia monotonically increasing and so the integral j(r)

=

j r p 0 ( u )du

p(r)}}

V r E R.

Ib I 03.

The function P o is

V r E R,

(2.27)

ro

where ro E D( p 1, is well-defined (unambiguously a real number or + a). Clearly, the function j is continuous on (a, b ) and convex on R. Moreover, lim inf, j ( r ) 2 j ( b ) and lim inf, a j ( r ) 2 j(a). Finally,

Hence p = d j , where j is the 1.s.c. convex function defined by (2.27). 2. Self-adjoint operators. Let H be a real Hilbert space with scalar product and norm 1.1, and let A be a linear self-adjoint positive operator on H . Then, A = d q where ( a , . )

(2.28) Conversely, any linear, densely defined operator that is the subdifferential of a 1.s.c. convex function on H is self-adjoint. To prove these assertions, we note first that any self-adjoint positive operator A in a Hilbert space is maximal monotone. Indeed, it is readily seen that the range of the operator I + A is simultaneously closed and dense in H . On the other hand, if cp: H + R is the function defined by (2.28), then clearly it is convex, l.s.c., and q ( x ) - q(u) =

+(IA'/2XI2 - I A ' / 2 U I 2 )

vx

I ( h , x E

-

u)

D ( A ) ,u E D ( A ' / 2 ) .

Hence A c d q , and since A is maximal monotone we conclude that A = dq. NOW,let A be a linear, densely defined operator on H of the form A = dJI, where JI: H + R is a 1.s.c. convex function. By Theorem 2.2, we

72

2. Nonlinear Operators of Monotone Type

know that A,

=

V$,, where A,

d -t,b,(tu) dt

=

A-'(I

=

-

and therefore $,(u) = $ ( A , u , u ) for all u FrCchet derivative of I),, we see that =

+ h A ) - ' ) . This yields

VUE H , t

t ( A,u ,U )

V$, = A ,

(I

E

E

[0,1],

H and A > 0. Calculating the

+(A,+A;).

Hence A, = A : , and letting A + 0 this implies that A = A*, as claimed. More generally, if A is a linear continuous, symmetric operator from a Hilbert space V to its dual V * (not identified with V ) , then A = dcp, where cp: V + R is the function cp(u)

= +(AU,U)

vu

E

V.

Conversely, every linear continuous operator A: V + V' of the form dcp is symmetric. In particular, if R is a bounded and open domain of R N with a sufficiently smooth boundary (of class C2, for instance), then the operator A: D ( A ) c L 2 ( R )+ L2(R) defined by

A y = -Ay is self-adjoint and A

Vy ED(A), =

dcp where cp:

-

D ( A ) =H,'(R) n H 2 ( R ) ,

L2(R)

R is given by

otherwise. This result remains true for a nonsmooth bounded open domain if it is convex (see Grisvard [l]). 3. Convex integunds. Let R be a measurable subset of the Euclidean space R N and let LP(R), 1 s p < M, be the space of all p-summable functions on R.We set LL(R) = (LP(R))". The function g: R X R" + R is said to be a normal convex integund if the following conditions hold: (i) For almost all x E R,the function g ( x , * ): R" + R is convex, I.s.c., and not identically + w ; (ii) g is 9 x 9 measurable on R X Rm;that is, it is measurable with respect to the u-algebra of subsets of R X R" generated by products of Lebesgue measurable subsets of R and Bore1 subsets of R".

2.2. Generalized Gradients (Subpotential Operators)

73

We note that if g is convex in y and int D ( g ( x , )) # 0 for every x E R,then condition (ii) holds if and only if g = g ( x , y ) is measurable in x for every y E R" (Rockafellar [51). A special case of a 9 X B measurable integrand is the Caratheodory integrand. Namely, one has:

Lemma 2.1. Let g = g ( x , y): R X R"' + R be continuous in y for every x E R and measurable in x for every y. Then g is measurable.

Proof Let {zi}T=,be a dense subset of R N and let A E R arbitrary but fixed. Inasmuch as g is continuous in y , it is clear that g ( x , y ) I A if and only if for every n there exists zi such that llzi - yll 4 l / n and g ( x , , z i ) I A l / n . Denote by Rin the set { x E R;g(x,zi) I A + l / n } and put y n = { y E R"; IIy - zill 5 l / n } . Since

+

C

C

m

{ ( ~ , ~ ) ~ R x R " ' ; g ( x , y ) ~0h j U = RjnXYn, n=l i=l

we infer that g is 9x 9 measurable, as desired. Let assume in addition to conditions (i), (ii) the following: (iii) There are a

E L4,(R),

l/p

g(x,y) 2 (a(x),y)

+ l / q = 1, and

+ p(x)

ax. x

p

E

E

L'(R) such that

R 7 yE

R", (2.29)

where ) is the usual scalar product in R"; There is yo E LL such that g ( x , y o ) E L ' ( R ) . (a,

(iv)

Let us remark that if g is independent of x , then conditions (iii) and (iv) automatically hold by virtue of Proposition 2.1. Define on the space X = L$(R) the function Zg: X -+ R,

Proposition 2.9. Let g satisfi assumptions (i)-(iv). Then the function Zg is convex, lower semicontinuous, and proper. Moreover, dZg(y) = { w

E

L4,(R); w ( x ) E d g ( x , y ( x ) ) a.e. x

Here, dg is the subdifferential of the function y

-+

E

g ( x , y).

a}. (2.31)

74

2. Nonlinear Operators of Monotone Type

Proof. Let us show that Zg is well-defined (unambiguously a real number or +m) for every y E L4,(R). Note first that for every Lebesgue measurable function y: R + R" the function x + g(x, y ( x ) ) is Lebesgue measurable on R. For a fixed A E R,we set

Let us denote by 9the class of all sets ?,? c R X Rm having the property that the set ( x E R;( x , y ( x ) ) E i}is Lebesgue measurable. Obviously, i contains every set of the form T X D,where T is a measurable subset of R and D is an open subset of R". Since 9is a a-algebra, it follows that it contains the a-algebra generated by the products of Lebesgue measurable subsets of R and Bore1 subsets of Rm. Hence E €9, and therefore g ( x , y ( x ) ) is Lebesgue measurable and so Zg is well-defined. By assumption (i), it follows that Zg is convex, whilst by (iv) we see that Zg f + m . Let ( y , ) c L$(R) be strongly convergent to y. Then there is { y n k }c ( y , } such that ynk(x ) -+ y ( x )

a.e. x

E

R

for nk + w .

Then, by assumption (iii) and Fatou's lemma, it follows that

and therefore lim inf I,( y a k ) 2 Zg( y ) . nk+x

Clearly, this implies that lim inf, Z&y,) 2 Zg(y), i.e., Zg is 1.s.c. on X. Let us now prove (2.31). It is easily seen that every w E L4,(R) such that w ( x ) E d g ( x , y ( x ) ) belongs to dZ&y). Now let w E dZ&y), i.e., ~~

75

2.2. Generalized Gradients (Subpotential Operators)

Let D be an arbitrary measurable subset of R and let u defined by u(x) =

y(x)

for x

E

D,

for x

E

R \O,

E

LL(R) be

where y o is arbitrary in R". Substituting in the previous inequality, we get

The case p = co is more difficult since the elements of d I g ( y )C (L',(R))* are measures on R. However, we refer the reader to Rockafellar [5]for the complete description of d I g in this case. Now, let us consider the special case where

K being a closed, convex subset of R". Then, Ig is the indicator function of the closed convex subset Z of LL(R) defined by

Z ={ y E LL(R); y ( x ) E K a.e. x

E

R},

and so by formula (2.31) we see that the normal cone 'N c LQ,(R) to Z is defined by N'(y)

=

( w E L4,(R); w ( x ) E N , ( y ( x ) ) a.e. x

E

R ) , (2.32)

where N , ( y ) = ( 2 E R"; (2,y - u ) 2 0 V u E K } is the normal cone at K in y E K. In particular, if m = 1 and K = [ a , b ] ,then N,(y) = { w ~ L q ( R ) ; w ( x=)O a . e . i n [ x ~ R ; u< y ( x ) < b ] , w ( x ) 2 0 a.e. in [ x E R ; y ( x ) in [ x E R ; y ( x ) = u ] } .

=

b ] ,w ( x ) I0 a.e.

(2.33)

76

2. Nonlinear Operators of Monotone Type

Let us take now K

(y

=

E

R"; llyll I p ) . Then,

NK(Y)= and so Nz is given by Nz(y)

=

(I>,,

if llyll < p , if I I Y I I= p ,

u AY

{w E L4,(R); w ( x ) = 0a.e. in [ x E R ; Ily(x)ll < p ] , w ( x ) = A(x)y(x) a.e. in [ x E R ; Ily(x)ll = p ] , where A E L4,( R ) , A ( x ) 2 0 a.e. x E 0).

We propose that the reader calculate the normal cone Ne to the set

i

27 = y

E

L2(R); a s y ( x ) Ib a.e. [ x

E

R ; Jny(x) dx

=

I])

and prove that N,(y)={z=w+A;AER,wEL2(R),w(x) = O a . e . i n [ a < y ( x ) O a . e . i n [ x ; y ( x ) = b ] , w ( x ) I0 a.e. in [ x ; y( x ) = a ] } .

4. Semilinear elliptic operators in L2(R). Let R be an open, bounded subset of RN, and let g: R + R be a lower semicontinuous, convex, proper function such that 0 E D ( d g ) . Define the function cp: L2(R) + R by + g ( y ) ) dx

if y

E

H,'(R) and g ( y )

E

L'(R),

otherwise.

(2.34) Proposition 2.10.

if the boundaly dR

The function cp is convex, 1.s.c. and f + m. Moreover, is suficiently smooth (forinstance, of class C 2 )or if R is

convex, then dq(y)

~ ( R ) y; w ( x ) + Ay(x)

= {W E L

Proot It is readily seen that strongly convergent to y as n

H , ' ( ~ In ) W(R), E dg(y(x)) a.e. x E R ) .

E

(2.35)

is convex and f +m. Let (yn}c L2(R) be As seen earlier,

+ m.

2.2. Generalized Gradients (Subpotential Operators)

77

and it is also clear that

Hence, liminf,,, cp(y,) 2 cp(y). Let us denote by r c L2(fl) x L’(fl) the operator defined by the second part of (2.33, i.e.,

r = ( [ y , ~E] ( & ( a )n ~ ~ ( xn~ )~ ()f l ) ; w( x ) E - A y ( x ) + d g ( y ( x ) ) a.e. x E 0). Since the inclusion r c dcp is obvious, it suffices to show that r is maximal monotone in L’(fl). To this end, observe that r = A + B, where Ay =

V y E D ( A ) = Hd(R) n H2(R), and By = ( u E L2(fl); 4 x 1 E d g ( y ( x ) ) a.e. x E fl}. As seen earlier, the operators A and B are maximal monotone in L’(fl) X L’(fl). Replacing B by y + By - y o , where yo E BO, we may assume without loss of generality that 0 E B(0). On the other hand, it is readily seen that ( B A u X x )= p,(u(x)) a.e. x E R for all u E L2(fl), where p = dg and PA = K’(1 - (1 + AP)-’), is the Yosida approximation of p. We have -by

because pi 2 0. Then applying Theorem 1.6 (or 2.4) we may conclude that r = A B is maximal monotone.

+

Remark 2.2. Since A + B is coercive, it follows from Corollary 1.2 that R ( A + B ) = L’(R). Hence, for every f E L2(R), the Dirichlet problem -Ay

+ p ( y ) 3f y=O

a.e. in R , in d f l ,

has a unique solution y E Hd(fl) n H’(fl). In the special case where p c R X R is given by

0

p ( r ) = (R-

if r > 0, if r = 0,

(2.36)

78

2. Nonlinear Operators of Monotone Type

problem (2.36) reduces to the obstacle problem a.e. in [ y > 01, -Ay = f - Ay 2 f , y 2 0 a.e. in R,

y =O

in dR.

(2.36)'

This is an elliptic variational inequality, which will be discussed in some detail later. We note that the solution y to (2.36) is the limit in H,'(R) of the solutions y, to the approximating problem

-Ay + P&(Y)= f y=o

in a, in dR.

(2.37)

Indeed, multiplying (2.37) by y, and Aye, respectively, we get

lly,112H;(n) + llAy,11~2(n,Ic

YE

> 0,

and therefore {yJ is bounded in H,'(R) n H2(R). This yields

and therefore

/

/Rlv(Y& -YA)I2 dX + R ( p & ( Y & ) & 5 09 -&(YA))(E@,(Y&)- ~PA(YA))

because p,(y) E p((1 + ~ p ) - ' y ) and p is monotone. Hence, {y,) is Cauchy in H,'(R), and so y = lim, ,y, exists in Hd(R). This clearly also implies that ~

Ay, Y&

-+

-+

Ay

weakly in L2(R),

Y

weakly in H 2 ( R ) ,

P&(Y&> -+g

weakly in L ~a). (

Now, by Proposition 1.3(iv), we see that g ( x ) y is the solution to problem (2.36).

E

p(y(x)) a.e. x

E

R,and so

5. Nonlinear boundaly Neumann conditions. Let R be a bounded and open subset of RN with the boundary dR of class C2. Let j : R R be a l.s.c., proper, convex function and let p = d j . Define the function -+

2.2. Generalized Gradients (Subpotential Operators)

79

otherwise. Since for every u E H'(R) the trace of u on dR is well-defined and belongs to H ' l 2 ( d R ) c L2(dR),formula (2.38) makes sense. Moreover, arguing as in the previous example it follows that q is convex and 1.s.c. on L2(R). Regarding its subdifferential d q c L2(R) x L2(R), it is completely described in Proposition 2.11, due to H. BrCzis [2, 31. Proposition 2.11.

We have

(2.39)

and d / d v is the outward normal derivative to dR. Moreover, there are some positive constants C , , C , such that

I I u I I H ~ ( ~I ) CIIIU - A u I I L ~ ( + ~ )C , Pro05

Let A: L2(R)

--f

i

E

D( d q ) .

(2.40)

L2(R) be the operator defined by u ED(A),

AU = - A u ,

D(A) = u

VU

1

dU

E

H 2 ( R ) ; - - E ~ ( ua.e. ) in dR dv

.

Note that A is well-defined since for every u E H2(R), d u / d v H ' / 2 ( d R > .It is easily seen that A c d q . Indeed, by Green's formula, Au(u - u ) dx

In

=

/ Vu(Vu

- V U ) dx

+

dx

R

2

jnlVUI2

+/

dfl

+ /a nj ( u ) dx

P(U)(U -

u ) dx

E

80

2. Nonlinear Operators of Monotone Type

for all u

E

D ( A ) and u E H'(R). Hence, vu €D(A),UEL2(R).

(Au,u - u) 2 q(u) - q(u)

(Here, is the usual scalar product in L2(R).) Thus, to show that A = d q , it suffices to prove that A is maximal monotone in L2(R) X L2(R), i.e., R(Z + A ) = L2(R). Toward this aim, we fix f E L2(R) and consider the equation u + Au = f, i.e., (.,a

dU

dV

u-Au=f

inR,

+ p(u) 3 0

in dR.

u-Au=f

inR,

(2.41)

We approximate (2.41) by dU dV

+ p,(u)

=

in d R ,

0

(2.41)'

where PA= K ' ( 1 - (1 + A @ ) - ' ) , A > 0. Recall that PA is Lipschitz with Lipschitz constant l / h and p,(u) + p o ( u )t l u E D( p), for A + 0. Let us show first that Eq. (2.41)' has a unique solution uA E H2(R). T Indeed, consider the operator u + uldn from L 2 ( d R )to L 2 ( d f i ) ,where u E H'(R) is the solution to linear boundary value problem in R ,

u - Au=f

u + A-

dU

dU

=

(1

+ hp)-'u

in dR. (2.42)

(The existence of u is an immediate consequence of the Lax-Millgram lemma.) Moreover, by Green's formula we see that

1 <- A

1 ((1 + A ~ ) - ' u

-

dn

(1

+ h p ) - ' i i ) ( ~- 5)d u x ,

where {u, u } and (5,ii} satisfy (2.42). Since l(1

+ Ap)-'x

we infer that

-

(1

+ hp)-'yl

IIx

-yl

Vx,y E R, h > 0,

81

2.2. Generalized Gradients (Subpotential Operators)

Since by the trace theorem the map u -+ uldn is continuous from H'(R) into H'/2(dR) c L 2 (d R ),we have

and so the map T is a contraction of L2(dR). Applying the Banach fixed point theorem, we therefore conclude that there exists u E L2(dR) such that Tu = u, and so problem (2.41)' has a unique solution u, E H'(R). We have u, - Au,

=f

in R,

dUA - - -p,(u,) dV

in dR.

(2.43)

Since p,(u,\) E H'(R) (because p, is Lipschitz) and so its trace to dR belongs to H'/2(dR), we conclude by the classical regularity theory for the linear Neumann problem (see, e.g., Lions and Magenes [ll, BrCzis [71) that u, E H2(R). Let us postpone for the time being the proof of the following estimate:

Ilf I l ~ ~ ( n ) ) VA > 0 ,

(2.44)

where C is independent of A and f. Now, to obtain existence in problem (2.411, we pass to (2.43). Inasmuch as the mapping u -+ ( u l a n , d u / d v l , n ) from H2(R) to H3l2(dR) x H'l2(dR) and the injection H'(R) c L2(R) is compact, we may assume, selecting a necessary, that, for A + 0,

limit A 0 in in continuous of H2(R) into subsequence if

IIUA\IIH~5 ( ~ )c ( 1 -t

-+

(2.45)

2. Nonlinear Operators of Monotone Type

82

It is clear by (2.431, (2.43, and (2.46) that u - Au

=f

dU

in R, a.e. in dR.

-+g30 dv

Let us show that g ( x ) E p ( u ( x ) ) a.e. x L2(dR) X L2(dR) defined by

E

R. Indeed, the operator

p = { [ u , u ] E L ~ ( ~ xR L) ~ ( ~ R ) ;

U(X) E

p ( u ( x ) ) a.e. x E

fi c

an)

is obviously maximal monotone, and 6A(u)(x)

((I + A 6 ) - ' u ) ( x )

= pA(u(x)),

=

(l +

Ap)-lu(x) a.e. x

-

E

0.

Since, by (2.46), p,(u,) g, (I + A p ) - ' u , + u , and p,(u,) E &(I + Ap)-'u,), we conclude that g E p ( u ) (because is strongly-weakly closed). We have therefore proved that u is a solution to Eq. (2.40, and since f is arbitrary in L2(R) we infer that A = d q . Finally, letting A tend to zero in the estimate (2.44), we obtain (2.40), as claimed.

p

Proof of estimate (2.44). Multiplying Eq. (2.43) by u, - u o , where uo E

D( p ) is a constant, we get after some calculation involving Green's lemma that

(We shall denote by C several positive constants independent of A and f . ) Hence, I l u ~ l l ~ l (5n )c(I

~ ~ I I L ~ +( ~ 1) )

VA

> 0.

If R' is an open subset of R such that R'c R,then we choose p such that p = 1 in F . We set v = pu, and note that u - Av

=

pf - u,Ap - 2Vp. Vu,

in

a.

(2.47) E

C;(R) (2.48)

Since u has compact support in R we may assume that v E H 2 ( R N ) ,and Eq. (2.48) extends to all of RN. Then, taking the Fourier transform and using Parseval's formula, we get IIUlIH2(RN)

5

C ( IlfllLZ(n) + lIuAJIH'(n)),

2.2. Generalized Gradients (Subpotential Operators)

83

and therefore, by (2.471,

where C is independent of R’ c c 0. To obtain H2-estimates near the boundary dR, let xo E dR, U be a neighborhood of x,, and cp: U -+ Q be such that cp E C 2 ( U ) , cp-’ E C2(Q),cp-’(Q+)= R n U,and cp-’(Qo) = dR n U,where Q = { y E R N ; Ily‘ll < 1, ly,l < 11, Q+= { y E Q; 0 < Y , < 11, Q o = {Y E Q; Y , = 01, and y = ( y ’ , y N )E RN. (Since dR is of class C2 such a pair (U,cp) always exists.) Now we will “transport” Eq. (2.48) from U n R on Q , using the local coordinate cp. We set

and observe that w satisfies on Q , the boundary value problem

+ C ( Y ) W= g ( y ) dW

-

dn

+ p A ( w )= 0

in in

Q+,

Q,,

(2.50)

and dW

- --

an

c

i,j=1

d w dcpj

- -cos(v,xi)

d y j dx,

( v is the outward normal derivative to dR). Since cpN(x) = 0 is the equation of the surface dR n U we may assume that d c p N / d x j = -cod u , xi), and so dW

N

dn

j=1

_.=-

dw -ajN dYj

in Q,.

84

2. Nonlinear Operators of Monotone Type

Assuming for a while that f - 1, satisfies the equation N

E

C'(n), we see that x

d

=

d w / d y i , 1 < i IN

+ C ( y ) z + C'(y)w

dZ

-=

dn

-p;(uA)z

+

d w dajN c-

j=1

dYj

dYi

in Q,.

(2.51)

Now, let cp E C ; @ + ) be such that p ( y ) = 0 for lly'll 2 2/3, 2/3 < y , < 1, and p ( y ) = 1 for lly'll < 1/2 and 0 s y N I1/2. Multiplying Eq. (2.51) by p2z and integrating on Q , , we get

- jQ+(CZ

+ C ' w ) z p 2 dy.

Taking into account that

k,j=l

we find after some calculation that

Hence,

for i

=

1 , 2,..., N - 1 , j = 1,..., N.

2.2. Generalized Gradients (Subpotential Operators)

Since a N N ( y )2 w o > 0 for all y estimate that

E

85

Q,, we see by Eq. (2.50) and the last

Hence,

II PWllH2(Q,)

I C(llfJILZ(R)

+ 1).

Equivalently,

II(

p * cp)uAllH~(u n n) IC(IlfIlL2(n)

+ 1)

VA

> 0.

Hence, there is a neighborhood U' c U such that IIuAIIHZ(u'nn)I

C ( I I ~ I I L ~+( ~1))

VA > 0.

(2.52)

Now taking a finite partition of unity subordinated to a such a cover {U')of dR and using the local estimates (2.49) and (2.52), we get (2.44). This completes the proof of Proposition 2.11. We have incidentally proved that for every f E L2(R), the boundary value problem (2.41) has a unique solution u E H2(R). If p c R X R is the graph p(0)

=

RN

p ( r ) # 0 for r

#

0,

then (2.41) reduces to the classical Dirichlet problem. If (2.53) then problem (2.41) can be equivalently written as y - Ay

=f

in R,

This is the celebrated Signorini's problem, which arises in elasticity in connection with the mathematical description of friction problems. This is a problem of unilateral type and the subset r,, that separates { x E dR; y > 0) from { x E dR; d y / d v > 0) is a free boundmy and it is one of the unknowns of the problem. For other unilateral problems of physical significance that can be written in the form (2.411, we refer to the book of Duvaut and Lions [l].

86

2. Nonlinear Operators of Monotone Type

Remark 2.3. Proposition 2.11 and its corollaries remain valid if R is un open, bounded, and convex subset of RN. The idea, developed by P. Grisvard in his book [l], is to approximate such a domain R by smooth domain R,, to use the estimate (2.44) (which is valid on every R, with a constant C independent of E ) , and to pass to limit. It is useful to note that the constant C in estimate (2.44) is independent of p. 6. The nonlinear diffusion operator. Let R be a bounded and open subset of RN with a sufficiently smooth boundary dR. Denote as usual by H,'(R) the Sobolev space of all u E H'(R) having null trace on dR and by H-'(R) the dual of Hd(R). Note that H-'(R) is a Hilbert space with the scalar product (u,u) = (J-'u,u)

V U , U E

H-'(R),

where J = - A is the canonical isomorphism (duality mapping) of H,'(R) onto H-'(R) and (., . ) is the pairing between H,'(R) and H-'(R). Let j : R + R be a l.s.c., convex, proper function and let p = d j . Define the function cp: H-'(R) + R by

if u

j ( u( x ) ) du

E

L'( R) and j ( u )

E

L'( a), (2.55)

otherwise. Proposition 2.12.

Let us assume further that lim j ( r ) / l r l

lrI+m

(2.56)

= +a.

Then the function cp is convex and lower semicontinuous on H-'(R). Moreover, dcp = { [ u , w ]E

(H-'(R)

u(x) E p(u(x))

Proof

n L'(R)) x H - ' ( R ) ; w

a.e. x

E

0).

=

- A U , U E Hd(R), (2.57)

Obviously, cp is convex. To prove that cp l.s.c., consider a sequence

{u,} c H-'(R) n L'(R) such that u, + u in H-'(R) and cp(u,) IA, i.e., j(u,)dx 5 A V n . We must prove that Jn j ( u ) a k I A. We have already seen in the proof of Proposition 2.9 that the function u + Jn j(u)du is

la

lower semicontinuous on L'(R). Since this function is convex, it is weakly lower semicontinuous in L'(R) and so it suffices to show that {u,} is weakly compact in L'(R). According to Dunford-Pettis criterion (see e.g.,

2.2. Generalized Gradients (Subpotential Operators)

87

Edwards [l],p. 2701, we must prove that {u,} is bounded in L'(R) and the integrals jlu,l du are uniformly absolutely continuous, i.e., for every E > 0 there is N E )such that j E lu,(x)l du I E if m ( E ) 5 S ( E ) ( E is a measurable set of a). By condition (2.561, for every p > 0 there exists R ( p ) > 0 such that j ( r ) 2 plrl if Irl 2 R ( p ) . This clearly implies that jnlu,(x)l dx 5 C. Moreover, for every measurable subset E of R, we have

if we choose p > ( 2 ~ I - sup l jnlun(x)l du and (u,) is weakly compact in L'(R).

m(E)I ~ / 2 R ( p ) Hence, .

To prove (2.571, consider the operator A c H-'(R)

X

H-'(R) defined

bY Au

=

{ -Au; u E H,'(R),

U(X) E

p ( u ( x ) ) a.e. x E

a},

where D ( A ) = { u E H-'(R) n L'(R); 3u E H,'(R), u ( x ) E p(u(x)) a.e. E a). To prove that A = d q , we will show separately that A c d q and that A is maximal monotone. Let us show first that A is maximal monotone in H-'(R) X H-'(R). Let f be arbitrary but fixed in H-'(R). We must show that there exist u E H-'(R) n L'(R) and u E H,'(R) such that

x

u

-

Au= f

in 0,

U(X) E y ( u ( x ) )

a.e. x

E

R;

equivalently, u - Au=f

a.e. x

E

R, u

E

in R , u(x) E p(u(x)) H - ' ( R ) n L'(R), U E H,'(R),

(2.58)

where y = p-'. Consider the approximate equation yh(u) - A u = f

in R ,

u=

0 in d R ,

(2.59)

where y, = K ' ( 1 - (1 + hy)-'), A > 0. It is readily seen that (2.59) has a unique solution u, E H,'(R). Indeed, since - A is maximal monotone from H,'(R) to H-'(R) and u + y,(u) is monotone and continuous from

88

2. Nonlinear Operators of Monotone Type

H,'(R) to H-'(R) (in fact, from L2(R) to itself) we infer by Corollary 1.1 that u + y,(u) - Au is maximal monotone in H,'(R) X H-'(R), and by Corollary 1.2 that it is surjective. Let uo E D(y).Multiplying Eq. (2.59) by u, - uo, we get

/nlv%lz

dr +

/n r('O)(u!

-

uO)

dr

I(vh

UO?f)*

-

Hence, {u,} is bounded in Hd(R>.Then, on a subsequence, again denoted A, we have u,

-

u

in H,'(R),

in L 2 ( R ) .

u, + u

Thus, extracting further subsequences, we may assume that u,(x)

(1

+

+ A~)-'u,(x)

+

U(X)

a.e. x

E

R,

u(x)

a.e. x

E

a,

(2.60)

because by condition (2.56) it follows that D(y)= R( p ) = R ( p is coercive) and so limA+o(l+ A y ) - ' r = r for all r E R (Proposition 1.3). Then, letting A tend to zero in (2.591, we see that We set g, = g, + u in H - ' ( Q ) and inR,

u-Au=f

UEH,'(R).

It remains to be shown that u E L'(R) and 4 x 1 E y(u(x)) a.e. x Multiplying Eq. (2.59) by u,, we see that g,u,

dr

IC

VA

dr

IC

R.

> 0.

On the other hand, for some uo E D ( j ) we have j ( g , ( x ) ) - uo)u Vu E p(g,(x)). This yields

i( gA( X))

E

VA

Ij ( u o )

+ (g,(x)

> 0,

because (1 + Ay)-'u, E @(g,). As seen before, this implies that (g,) is weakly compact in L'(R). Hence, u E L'(R) and g,

-

u

in L'(R)

for A

-+ 0.

(2.61)

On the other hand, by (2.60) it follows by virtue of the Egorov theorem that for every E > 0 there exists a measurable subset E, c R such that

2.2. Generalized Gradients (Subpotential Operators)

m ( R \ E,) 5

E,

+ Ay)-'q}

is bounded in LYE,), and

{(l

(1 + hy)-'u,

+

89

u

uniformlyin E,

as A

+

0.

(2.62)

Recalling that g,(x) E y((1 + Ay)-'q(x)) and that the operator ? = ( [ u ,u ] E LYE,) x LYE,); 4 x 1 E y ( u ( x ) ) a.e. x E E,} is maximal monotone in L'(E,) x LYE,), we infer, by (2.61) and (2.621, that [ u , u ] € ?, i.e., u ( x ) E y ( u ( x ) ) a.e. x E E,. Since E is arbitrary, we infer that u ( x ) E y ( u ( x ) ) a.e. x E R, as desired. To prove that A c d q , we shall use the following lemma, which is a special case to a general result due to BrCzis and Browder [l]. Lemma 2.2. Let R be an open subset of RN. lf w and u E H ; ( R ) are such that

a.e. x

w ( x ) u ( x ) 2 -Ih(x)l

for some h

E

L'(R), then wu

E

E

E

H-'(R> n L'(R)

a,

(2.63)

L'(R> and

w(u) =

j n w ( x ) u ( x ) du.

(Here, w ( u ) is the value of functional w

E

(2.64)

H-'(R) at u

E H;(fl).)

Prooj The proof relies on an approximation result for the functions of H,'(R) due to Hedberg [l]. Let u E H;(R). Then there exists a sequence {u,) C H,'(R) n L"(Rn) such that supp u, is a compact subset of 0, u , + u in H,'(R), and lu,(x)l I inf(n,lu(x)I),

a.e. x

u,(x)u(x) 2 0

E

R . (2.65)

Then,

w(u,)

=

jn w ( x ) u , ( x ) d u

vn.

On the other hand, by (2.63) we have wu,

+ Ihl-unU

=

u,

(wu + \hi)-

and so by the Fatou lemma wu

U

+ Ihl

because, on a subsequence, u , ( x )

+

E

a.e. in R ,

2 0

L'(R) and

u ( x ) a.e. x

E

fl.

(2.66)

2. Nonlinear Operators of Monotone Type

90

We have therefore proved that wu lim inf n-m

E

L'(R) and

jn wu, d\: 2 jn wu h.

On the other hand, wu, .+ wu a.e. in R and, by (2.651, Iwu,( I IwuI a.e. in R. Then, by the Lebesgue dominated convergence theorem, wu, -+ wu in in (2.66) we get (2.641, as desired. W L'(R), and letting n --f

Now, to conclude the proof of Proposition 2.12, consider an arbitrary element [ u , -Au] E A , i.e., u E H-'(R) n L'(R), u E H$R), u ( x ) E p ( u ( x ) ) a.e. x E R. We have ( A U , ~ u)

= (U,U

-

Vii

ii)

E

H-*(R) n ~ ' ( 0 ) .

Since u ( x X u ( x ) - i i ( x ) ) 2 j ( u ( x ) ) - j ( i i ( x ) ) a.e. x Lemma 2.2 that ( A u , u - ii)

=

(u,u - ii) =

jn u ( x ) ( u ( x )

E

R, it follows by

-

i i ( x ) ) du

E

H-'(R),

Hence, (Au,u - u) 2 q(u) - p(U)

Vii

thereby completing the proof. Remark 2.4. Condition (2.56) is equivalent to R( p ) = R and p-' bounded. Indeed, by the definition of d j = p, we have j ( r ) 5j(r,) + y ( r

-

r,,)

Vr E

is

R,

and (2.56) implies that p is coercive and so p-' is everywhere defined and bounded on bounded subsets. Conversely, if p-' is bounded on bounded subsets, then for every z E R, Izl I p there is u E D( p ) and M > 0 such that [ u , 21 E p and IuI < M . Then, by the inequality j(r) -j(u) 2 z(r - u)

Vr

E

R,

we get plrl 5 j(r1 + M, V r E R. Hence lim,r,+m j(r)/lrI = 00, as claimed. It should be noted that this result remains true for a general 1.s.c. convex function j on a Banach space X .

2.2. Generalized Gradients (Subpotential Operators)

91

2.2.3. Generalized Gradients Here we shall present the Clarke generalized gradient along with some of its most important properties. Throughout this subsection, X is a real Banach space with the norm denoted II * 1 , the dual X*, and the duality pairing (., . 1. A function f : X + R is called locally Lipschitz if for every p > 0 there exists L, > 0 such that

If(x)

- f(y)l I L,llx - yll

Given a locally Lipschitz function f: X defined by

fo(x , u )

=

lim sup Y' 1

f(Y +

Vllxll,llyll I p.

+

,

R,the function fo: X x X + R

-f(Y)

Vx, u E X,

A

(2.67)

A10

is called the directional den'uatiue off. It is easily seen that f o is finite, positively homogeneous, and subadditive in u, i.e.,

f o ( x , Au)

=

Afo(x,u),

f0(x,u1+ u2)I f o ( x , v l ) +f0(x,u2).

Then, by the Hahn-Banach theorem, there exists a linear functional on X, denoted 77, such that v ( u ) IfO( x, u )

Vu E

Since Ifo(x,u)l I Lllull Vu E X , it follows that Hence, 7 E X * and (u,7)) < f O ( x , u )

x.

(2.68)

7)

is continuous on X.

VvEX.

(2.69)

By definition, the generalized gradient (Clarke's gradient) of f at x, denoted df(x) is the set of all v E X * satisfying (2.69). In other words, df(x)

=

{v E X * ; ( u , v ) I f O ( x , u )

VUEX).

( 2.70)

Clearly, this definition extends to all functions f that are Lipschitz in a neighborhood of x . Note that if x is a local minimum point or maximum point of f, then 0 E df(x). Let us observe that if f is locally Lipschitz and Giiteaux differentiable at x, then Vf(x> E df(x). Indeed, we have (u,Vf(x))=f'(x,u)
VUEX.

92

2. Nonlinear Operators of Monotone Type

If f is continuously differentiable in a neighborhood of x , then Vf(x) = d f ( x ) because in this case f r ( x ,u ) = f o ( x ,u ) Vu E X . On the other hand, if f is convex (and locally Lipschitz), then df reduces to the subdifferential of f defined in Section 2.2 (see (2.9)). Indeed we have, by (2.671, f 0 ( x , u ) = lim

sup

& l oIly-xII<eS

sup A - ' ( f ( y O
+ Au) - f ( y ) ) ,

where 6 is positive and arbitrary. Since the function A + A - ' ( f ( y - f ( y ) ) is decreasing (because f is convex), we infer that f O ( x , u )= lim

sup Ily-xlt<se

~-'(f(+ y

EU)

+ Au)

-f(y))

Hence f o < x ,u ) I f r ( x ,u ) . Since the opposite inequality is obvious, we have that f 0 ( x , u ) = f ' ( x , u ) , and by Proposition 2.8 we conclude that d f ( x ) coincides with the subdifferential of f at x . Proposition 2.13. in X * . Moreover,

For every x

EX

f O ( x , u )= max{(u,v);

, d f ( x ) is convex and weak star compact E

df(x)}

(2.71)

VX,UEX,

and the map df: X + zx' is weak star upper semicontinuous, i.e., i f x , and y, E d f ( x , , ) is weak star convergent t o y in X * , then y E d f ( x ) .

+

x

Prooj By definition it is obvious that d f ( x ) is convex, closed, and bounded in X*. Hence, by the Alaoglu theorem, it is weak star compact. To prove (2.70, we will assume that f o ( x ,u,) > max{(u,, 7);77 E d f ( x ) ) for some u, E X and will argue from this to a contradiction. Indeed, by the Hahn-Banach theorem there is f E X * such that

( % f )5 f 0 ( x , u ) V U E X ,

and

(u,,l)

=f0(x,v0).

Hence, f E d f ( x ) and f o ( x ,u,) > (u,, f 1, which leads to a contradiction. We shall prove now that f o is upper semicontinuous. Let x , + x and u, + u. For every n , there exists h , E X and A,, E (0,l) such that Ilh,ll + A, Il / n and fo(x,,v,) I(f(x,

+ h,, + A,u,)

-f(x,

1

+ h , ) ) A i ' + -. n

2.2. Generalized Gradients (Subpotential Operators)

93

This yields limsupfo(x, ,u,) s f o ( x , u ) , n-rm

as claimed. Now, let x , (V,

+ x,

7,

-

77,) i f"(x,

7 be such that rl,

,v )

E

d f ( x , ) . We have

VtJE X ,

and letting n + m we get ( u , 77) I f '(x, u ) Vu E X. Hence, 7 E d f ( x ) . The proof is complete. In the case X = RN,d f ( x ) can be equivalently defined as the convex hull of all limit points of Vf(y) for y + x . The function f being locally Lipschitz it is almost everywhere differentiable (Rademacher's theorem). Let us denote by rf the set of all points x E R N where f is not differentiable. Denote by rn the Lebesgue measure in RN.

Theorem 2.6. Let x be arbitraiy in R N and let S c RN be such that m ( S ) = 0. Then i f f is locally Lipschitz in a neighborhood of x, we have

Proof

Since Vf(x,)

E

d f ( x , ) , it follows by Proposition 2.13 that

lim{Vf(x,); x , + x , x , P

s u rf)c df(x),

and since d f h ) is convex, we see that E

=

conv{limVf(x,); x , + x , x , P S U rf)c df(x).

Let us denote by H the support function of E, i.e., H ( p ) = sup{(p,u); E E } . To prove that d f ( x ) c E, it suffices, according to Proposition 2.13, to show that

u

H(u) 2fO(x,u)

Let u # 0 and

E

VUEX.

> 0. We will prove the inequality

f 0 ( x , u ) - ~ ~ l i m s u p { ( V f ( y ) , u ) ; y + x , y P S ~ (2.73) ~~]. We denote by p the right hand side of (2.73). There is 6 > 0 such that (Vf(y),u) s p + E for all y P S u rf,Ily - xII I6. We have

94

2. Nonlinear Operators of Monotone Type

and since by the Fubini theorem the Lebesgue measure of the set {y + tu; 0 < t < 6/2llull) n (S U rf) is zero, we infer that f(y + tu) - f(y) 5 ( p + &It for 0 < t < 6/21(ull and Ilx - yll I 6. This implies (2.731, as desired. Let g : R -+ R be a measurable and locally bounded function. Then, its indefinite integral

is locally Lipschitz on R, and by Proposition 2.13 it follows that m y )

=

n n

-

S>O m ( N ) = O

convg(B,(y)\w

VY

E

R,

(2.74)

where B,(y) = ( x E R Ix - yl < 6) and m is the Lebesgue measure. The multivalued map f arising in the right hand side of (2.74) is well-known from the theory of Filipov solutions for differential equations with discontinuous right hand side. It can be equivalently represented as f(Y)

=

E(g(y))

=

[=i(g(Y))mg(Y))l

VY

E

R7

(2.75)

where lim

8-0

essinf

u€[y-G,y+S]

g(u),

If C is a closed subset of X, denote by d , ( x ) the distance from x to C , i.e., d,( x )

=

inf{I1x

-

ull; u

E

C}.

Obviously, d, is Lipschitz with Lipschitz constant 1 (i.e., d, is nonexpansive). By definition, the vector u E X is said to be tangent at C in x if d ; ( x , u ) = 0. The set of all tangent vectors at C in x will be denoted by T,(x), and it is called the tangent cone at C in x . (By properties of d o , it is readily seen that T,(x) is a closed cone of X . ) The normal cone at C in x is by definition the set

N,(x)= ( ~ E X * ; ( UI ,O~V )U E T ~ ( X ) } .

(2.76)

2.2. Generalized Gradients (Subpotential Operators)

95

Let f and g be locally Lipschitz functions on the Banach

Proposition 2.14. spaceX. Then

+ dg(x)

d ( f + g ) ( x ) c df(x)

(2.77)

vx E X .

If C is a closed subset of X and iff attains its infimum on C in x , then 0

Prooj

E

+ N,(x).

df(x)

(2.78)

By definition of the directional derivative, we have ( f + g ) " x , ). I f'( x , v )

+ g O G ,u )

vu E

x,

and since, by Proposition 2.13, f ' ( x , . ) is the support function of f(x), i.e., f ' ( x , * ) = H d f ( x )we , have 5

Hd(f+gXx)(4

Hdf(x)(V)

+ Hdlg(X)(V) I

E

Hdf(x)+dg(x)(4

x,

and this implies (2.77). Now, let x E C be such that f(x) = inf{f(u); u E C } , and let S be a closed ball centered at x such that f is Lipschitz with constant L on S. Replacing if necessary C by C n S (the sets C and C n S have the same normal cone at x > we may assume that C c S. Consider the function g ( y ) = f(y) + pd,(y), where p 2 L. We have g(x> 5 g ( Y )

and therefore 0

E dg(x).

0

because ad&)

E

VY

E

s,

Then, by (2.771,

df(x) + p . d d c ( x ) C df(x) + N c ( x ) ,

c N,(x).The proof is complete.

rn

Let us show now that if C is convex, then N J x ) coincides with the normal cone at C in x in the sense of convex analysis, i.e., N,(x)

=

(7E X * ; ( x - u , q ) 2 0

vu

E

C}.

Indeed, if ( x - u, 7)2 0 for all u E C , then x is a minimum point on C for the function u + ( x - u , 7)and so, by formula (2.781,

0

E

-77

+ N,(x).

Conversely, if 77 E N c ( x ) , then (7, u ) I 0 for all u E X such that d:(x, u ) = 0, On the other hand, since the function d , is convex, we have

2. Nonlinear Operators of Monotone Type

96

d o = d ' , and so 1 d o ( x , u - x ) = d ' ( x , u - x ) = lim - ( d ( l - h ) x + A u ) - d ( x ) ) s O AJO

Vu

E

c.

Hence, (v,x - u ) 2 0 V u E C , as claimed. Now, let V be a real separable Hilbert space and let V * be its dual (not identified with V ) .Let (en);= be an orthonormal basis in V and let X,, be the finite dimensional space generated by ( e l ,. . .,en}.Denote by P,, : V + X,, the projection of X into X,, , i.e., n

P,,U = C , ( u , e i ) e , . i= 1

Let A,, :R" + X,, be the operator n

A,(T) =

Tie,,

T =

(T,,...,T,,).

i= 1

If f: V + R is a locally Lipschitz function, denote by f": V + R the function f"(u)

=

lRnf( P,,u & A , , T ) ~ , , (dT T)

V u E I/,

-

(2.79)

where n = [ E - ' ] and p,, E C;(R") is a mollifier in R",i.e., p,,(B) = 0 for llell,, > 1 , p,, 2 0, jRn p,,(O)dO = 1 , and p,,(e) = p J - 8 ) Ve E R". Proposition 2.15.

The function f " is Frichet differentiable and

lim f"( u )

&+

If

0

=

f( u )

(uJ is strong& convergent to u in V as

Vf"(u,)

-6

then 6 E d f ( u ) . Prooj

Clearly, we may rewrite f" as

Vu E +

E

V.

0 and

in V * ,

(2.80)

2.2. Generalized Gradients (Subpotential Operators)

97

from which we see that f " is FrCchet differentiable and Vf" is continuous (i.e., f" E C ' ( V ) ) .On the other hand, we have

which yields

If"(

u ) - f( u)l 5 L( IIU

-

Pnull+

VE

E )

> 0,

and this implies (2.80). Now, if {u,} is as before, by the mean value theorem we have A - ' ( f"( U, =

+ Az)

- f"( u,)

+

A - ' j R n ( f ( P n u e P,,z - E A , ~ -)f ( P n u E - & A , T ) ) ~ , ( T d7. )

Hence, by the Beppo-Levi theorem (or by the Fatou lemma),

Since the function f o is upper semicontinuous in V + u, P,z -, z for E + 0 ( n -+ 031, we have limsup n+m

Hence,

X

V and Pnu, - EA,T

/- f O ( P n u ,- E A , r , P n z ) ~ , ( ~ ) d 7 1 f o ( u , z ) . R"

( 6 , z ) If o ( u ,z ) for all

x

E

V and so

5 E d f ( u ) , as

claimed.

It must be said that this approximating result can be extended to separable Banach spaces with Schauder basis. Corollary 2.1. If u and u are two distinct points of V , then there is a point z on the segment [ u , ul and q E df(z) such that

f(u)

=

(77,u - u ) .

Boot Let f" constructed as before. By the classical mean value theorem, there exists z , such that z, = A,u + (1 - A,)u, 0 IA, I1, and f"(u) -f"(v)

= (Vf"(Z,),U

-

.).

98

2. Nonlinear Operators of Monotone Type

Extracting a subsequence, we may assume that z,

and Vf"(z,)

+z

-7

as

E +

0.

Then, by Proposition 2.15, it follows that

f(u)

-f(u)

(7,u

- V),

rn

as desired.

Consider now on the space X tion

where g: R

=

X

R

-+

=

Lf'(R;z),1 s p < m, f2 c R N ,the func-

R is measurable in x and Lipschitz in y , i.e.,

Ig(x,Y) -g(x,z)IS a ( x ) I y - z I

Vy,zER,xER,

where a E L q ( a ) ,l/p + l / q = 1, and g(x,O) Clearly, f is Lipschitz in LP(R) and df(y)

C

{w

E

E

L'(R).

Lq(R); w ( x ) E d g ( x , y ( x ) ) a.e. x

E

a). (2.81)

Indeed, we have

fo( y , u )

=

lim sup Z-+Y

ALO

5

L

g(x7 4 x 1 + W X ) ) - g ( x 3 z ( x ) ) du

/-gO(x,y,u)dx

A VUELP(a).

Hence, if 7) E d f ( y ) , we have

and this yields, by standard arguments, (q(x),u) sgO(x,y(x),u)

VUE R

a.e. x

E

a,

as claimed. This concept of generalized gradient extends to functions f: X I - 00, +..I. If E(f)is the epigraph of such a function, i.e.,

E(f) =

{ ( x , A) E X X R ; f ( x ) 5 A},

+

99

23. Accretive Operators in Banach Spaces

+ w, then, by definition, J f ( x ) = (77 Ex*; (77, -1) E & ( / ) ( X J ( X ) ) } .

and if -a < f ( x ) <

(2.82)

The set d f ( x ) c X * is weak star closed in X * (if nonempty). In particular, if f = I , is the indicator function of a closed subset of X , then dZ,(x)

Indeed, 77

E

=

N,(x)

vx E

c.

dZ,(x) if and only if

(77, -1)

E

&(,,)(X70)

=

~ c x [ o , m ] ( x , o= )

XI -

?Ol.

Hence, 77 E dZ,(x) if and only if 77 E N J x ) . This definition agrees with that given for locally Lipschitz functions. Indeed, it can be proved that if f is locally Lipschitz on X , then 77 E d f ( x ) if and only if (77, - 1 ) E N&X, f ( x > >(see, e.g., F. Clarke [2]). We shall not pursue further the study of df in this general context and refer the reader to F. Clarke [2] and R. T. Rockafellar [7, 81. We mention, however, the following extension of Proposition 2.14. Proposition 2.16. Let fi:X 4 R, i = 1,2, be finite in x and let Lipschitz in a neighborhood of x . Then one has d(f1 + f 2 ) ( x )

= dfdx)

+ df*(x)*

f2

be

(2.83)

2.3. Accretive Operators in Banach Spaces

2.3. I . Definition and Basic Properties Throughout this subsection, X will be a real Banach space with the norm II * 1 , X * will be its dual space and (., the pairing between X and X * . We will denote as usual by J : X -,X * the duality mapping of the space X . Definition 3.1. A subset A of X x X (equivalently, a multivalued operator from X to X ) is called accretive if for every pair [ x , ,y,], [ x 2 ,y21 € A, there is w E J ( x , - x , ) such that

(Y, -Y29W) 2 0.

(3.1)

100

2. Nonlinear Operators of Monotone Type

An accretive set is said to be maximal accretive if it is not properly

contained in any accretive subset of X X X. An accretive set A is said to be m-uccretiue if R(I+A)

=x.

(3.2)

Here, we have denoted by I the unity operator in X; when there will be no danger of confusion we shall simply write 1 instead of I. A subset A is called dissipative (resp., maximal dissipatwe, m-dissipatwe) is - A is accretive (resp., maximal accretive, m-accretive). Finally, A is said to be *accretive (w-m-accretiue), where o E R, if A + W I is accretive (resp., m-accretive). The accretiveness of A can be equivalently expressed as 11x1

- x2II I

11x1

+ A(y, - y2)II VA > 0, [ x j , y i ] E A , i = 1 , 2 , (3.3)

- x2

using the following lemma (Kato’s lemma).

Lemma 3.1. Let x , y

if and only if

E X.

Then there exists w

llxll I

IIx

+ Ayll

VA

EJ(x)

such that (y, w ) 2 0

> 0.

(3.4)

Proof: Let x and y in X be such that (y, w ) 2 0 for some w Then, by definition of J, we have

11~1= 1 ~ (x, w )

I (x

+ Ay, w ) I

IIx

+ Ayll. llwll = IIx + AylI

*

EJ(x).

llwll

VA > 0 ,

and (3.4) follows. Suppose now that (3.4) holds. For A > 0, let w, be an arbitrary element of J(x + Ay). Without loss of generality, we may assume that x # 0. Then, w, # 0 for A small. We set fA = wAllwAII-I. Since {fA)A,o is weak star compact in X*,there exists a generalized sequence, again denoted A, such that fA -f in X*. On the other hand, it follows from the inequality IlxII I IIx

+ Ay II = ( x + Ay, fA)

I

IIxII + A(Y,

that (y,f~)2 0

Vh > 0.

f,)

2.3. Accretive Operators in Banach Spaces

101

Hence, ( y , f ) 2 0 and llxll I ( x , f ) . Since llfll s 1, this implies that llxll = ( x , f ) , llfll = 1, and therefore w =fllxll E J ( x ) , (y,w) 2 0, as claimed. Proposition 3.1. A subset A of X X X is accretive if and only (3.3) holds for all A > 0 and all [xi,yi] E A, i = 1,2.

if inequality

Proposition 3.1 is an immediate consequence of Lemma 3.1. In particular, it follows that A is o-accretive iff IIx1 - x z

+ A(yl -yz)II

2 ( 1 - Aw)IIxI -xzII forO
Hence, if A is accretive then the operator ( I R(Z + AA), i.e.,

ll(Z

+ AA)-'

(3.5)

is nonexpansive on

+ A A ) - ' x - ( I + AA)-'yll

I IIx -yII VA > 0, x , y E R(Z

+ AA).

If A is w-accretive, then it follows by (3.5) that ( I + AA)-' is Lipschitz with Lipschitz constant 1/(1 - Am) on R(Z + AA), 0 < A < l / w . Let us define the operators J, and A,:

J,x

=

(I + AA)-'x,

A,x

=

A - ' ( x - J,x),

R(Z + AA); x E R(Z + A A ) . x E

(3.6) (3.7)

The operator A, is called the Yosida approximation ofA, and in the special case when X = H is a Hilbert space it has been studied in Section 2.2 (see Proposition 2.4). In Proposition 3.2 following, we collect some elementary properties of J, and A,. Proposition 3.2.

(a) (b)

Let A be o-accretive in X

X

X . Then:

+

IIJAx - JAyll I (1 - Aw)-'Ilx - yll VA E (0, l/w), x , y E R(Z AA); A, is w-accretive and Lipschitz with Lipschitz constant 2/(1 - A w ) in R(Z AA), 0 < A < l/w;

+

+

(c) A,x E AJ,x V x E R(Z AA), 0 < A < l/w; (d) (1 - Aw)llA,xIl I lAxl = inf{llyll; y E Ax); (el lim,+ J,x = x V x E D( A ) fl < <, I , w R(Z + AA).

102

2. Nonlinear Operators of Monotone Type

Proof: (a) and (b) are immediate consequences of inequality (3.5). (c) Let x E R(Z + AA). Then, A,x E A-'((Z + AA)J,x - J,x) E M , x . (d) For x E D ( A ) n R(Z + AA), we have A,x = Au1(J,(Z + AA)x J A x ) and, therefore, llA,xll I Ihl(1 - Awl-' tlx E D ( A ) . (el For every x E D ( A ) n R(Z + AA), we have

,

Hence lim,+ ,J,x = x . Clearly, this extends to all of D(A ) fl < ,< + AA), as claimed.

R(Z

In the following we shall confine ourselves to the study of accretive subsets, the extensions to the *accretive sets being immediate. Proposition 3.3.

An accretive set A c X

X

X is m-accretive if and only

R(Z + AA) = X for all (equivalently, for some) A > 0. Proof. Let A be m-accretive and let y Then, the equation

E

X, A > 0 be arbitrary but fixed.

+A

h3 y

*=(Z+A)-'(;

+ (1

x

if

(3.8)

may be written as -

+).

Then, by the contraction principle, we infer that the equation has solution for 1/2 < A < + w . Now, fix A, > 1/2 and write the preceding equation as

Since (Z + A,A)-' is nonexpansive, this equation has a solution for A E (A,/2, a).Repeating this argument, we conclude that R(Z AA) = X for all A > A,/2", n = 1,. . ., and so R(Z + AA) = X for A > 0. Assume now that R(Z + A,A) = X for some A, > 0. Then, if we set Eq. (3.8) into the form (3.91, we conclude as before that R(Z + AA) = X for all A E (A,/2, m) and so R(Z + A) = X for all A > 0, as claimed.

+

2.3. Accretive Operators in Banach Spaces

103

Combining Proposition 3.2 and 3.3 we conclude that A c X x X is m-accretive if and only if for all A > 0 the operator (I AA)-' is nonexpansive on all of X. Similarly, A is w-m-accretive if and only if, for all 0 < A < l/w,

+

By Theorem 1.2, if X = H is a Hilbert space, then A is m-accretive if and only if it is maximal accretive. A subset A c X X X is said to be demiclosed if it is in closed X X X,, i.e., if x, + x , y, y , and [ x , , y,] E A, then [ x , y ] E A (recall that denotes weak convergence). A is said to be closed if x , -+ x , y , y , and [x,, y,] E A for all n imply that [ x , y ] E A .

-

-

-+

Proposition 3.4. Let A be an m-accretive set of X and if A, E R, x , E X are such that A, + 0 and X,

for n

AAnx,,+ y

+x ,

X

+

X . Then A is closed

m,

then [ x , y ] E A . If X * is uniformly convex then A is demiclosed, and X,

Let x ,

Pro06

IIx,

-+

x, y,

+x,

+

for n

AA.X - y

y , [ x , , y,]

-

uII I I I x , + Ay,

-

uII I I I x + Ay

-

(U

E

+

m,

(3.11)

if (3.12)

A . Since A is accretive, we have

+ Au)\~

V [ U , UE ] A , A > 0.

Hence,

IIx

- (U

+ Au)Il

V [ U , U ]€ A , A > 0.

Now, A being m-accretive, there is [ u , u ] E A such that u + Au = x + Ay. Substituting in the latter inequality, we see that x = u and y = u E Ax, as claimed. Now, if A,, x , satisfy condition (3.10, then (AAnx,}is bounded and so J,+x, - x , + 0. Since Ah>,, E MA>,,, JA,x, + x , and A is closed, we have that [ x , y ] E A . We shall assume now that X * is uniformly

104

2. Nonlinear Operators of Monotone Type

convex. Let x , ,y, be such that x,, -+ x , y , accretive, we have

(y,

- u,J(x,

-

-y ,

[ x n,y , J

E

V[u,u]EA, n E

u)) 2 0

A. Since A is

N*.

On the other hand, recalling that J is continuous on X (Theorem 1.2, Chapter 1) we may pass to limit n .+ m to obtain

( y - u,J(x

- u))

V[u,u] € A .

10

Now, if we take [ u , u ] E A such that u + u = x + y , we see that y = u and = u. Hence [ x , y ] E A , and so A is demiclosed. The final part of Proposition 3.4 is an immediate consequence of this property, rememberrn ing that A,,x, E RIAnx,. Note that an m-accretive set of X X X is maximal accretive. Indeed, if [ x , y ] E X X X is such that

x

IIx

-

uII

II I x

+ Ay - ( U + Au)II

V [ U , U ]€ A , A > 0,

then choosing [ u , u ] E A such that u + Au = x + Ay, we see that x = u and so u = y € A x . In particular, if X * is uniformly convex, then for every x E D ( A ) we have Ax

=

( y EX*;( y - u , J ( x - u ) ) 2 O V [ u , u ] E A } .

Hence, Ax is a closed convex subset of X. Denote by A o x the element of minimum norm on Ax, i.e., the projection of origin into Ax. Since the space X is reflexive, A o x # 0 for every x E D ( A ) . The set A' c A is called the minimal section of A . If the space X is strictly convex, then A' is single valued. Proposition 3.5. Let X and X * be uniformly convex and let A be an m-accretive set of X X X . Then:

(i> A,x .+ A o x V x E D ( A ) for A (ii) D(A ) is a convex set of X .

+

0;

-

Proof: (i) Let x E D ( A ) . As seen in Proposition 3.2, lIA,xIl I(Ax1 = llAoxll VA > 0. Now, let A,, .+ 0 be such that AAnx y . By Proposition 3.4 we know that y E Ax, and therefore

23. Accretive Operators in Banach Spaces

105

Since the space X is uniformly convex, this implies that A,,x -,y = Aox (Lemma 1.1, Chapter 1). Hence, A,x + Aox for A + 0. (ii) Let xl, x 2 E D ( A ) , and 0 I (Y I 1. We set x , = a x l + (1 - (Y)x,. Then, as is easily verified,

and since the space X is uniformly convex, these imply by a standard device that IIJA(X,)

where Iim,+, a ( A )

=

- X,II

0. Hence, x ,

I a( A) E

D(A).

VA

> 0, W

We know that a linear operator A: X --+ X is m-dissipative if and only if it is the infinitesimal generator of a C,-contraction semigroup in X (Theorem 4.2, Chapter 1). Regarding linear m-accretive (equivalently, m-dissipative operators), we note the following density result. Proposition 3.6. Let X be a Banach space. Then any m-accretive linear operator A : X -,X is densely defined, i.e., D( A ) = X .

Proo$ Let y E X be arbitrary but fixed. For every A > 0, the equation x A + A h , = y has a unique solution X, E D ( A ) .We know that llxAllIllyll for all A > 0 and so, on a subsequence A, + 0, x,, - x ,

A,hAn - y - x

in X.

-

Since A is closed, its graph in X x X is weakly closed (because it is a linear subspace of X X X ) and so h,xA, -, 0, A(h,xAn) y - x imply that y - x = 0. Hence, (1

+ A,A)-*y - y .

We have therefore proven that y E D( A ) (recall that the weak closure of D ( A ) coincides with the strong closure). This completes the proof. We conclude this section by introducing another convenient way to define the accretiveness. Toward this aim, denote by [*, I, the directional

-

2. Nonlinear Operators of Monotone Type

106

derivative of the function x [x,y],

((XI(, i.e. (see (2.711,

IIx + AyII - IIxII

lim

=

Since the function A I-,* I, as

4

A

A10

-+

IIx

X,Y

7

+ Ayll is convex, we may define, equivalently,

Let us now briefly list some properties of the bracket [-, *

[a,

1, : X x X

-+

R is upper semicontinuous;

[ a x , P ~ l , = I P l [ x , ~ l i, f a , P > O , x , y E X ; [x, a x + yl, = allxll + [ x , yl, if a E R, x E X; I[x, yI,I I IIyII, [ x , y + 21, I [x, YI, + yIs Vx, Y, [x, y],

=

I,.

Let X be a Banach space. We have:

Proposition 3.7. (i) (ii) (iii) (iv) (v)

(3.13)

EX.

max{(y, x*); x *

E

W x ) ) Vx, Y

EX

2 E X;

,

where @(x) = ( x * E X * ; (x, x * ) = llxll,llx*ll @(O)

=

{x*

=

1)

if x

+ 0,

Ex*;Ilx*ll 5 1).

Pro08 (i) has been proven in Proposition 2.13, whilst (ii), (iii), and (iv) are immediate consequences of the definition. To prove (v), we note first that q x ) =

a( Ilxll)

Vx

EX

,

and apply Proposition 2.13. Now coming back to the definition of accretiveness, we see that condition (3.3) can be equivalently written as [xl -x2,yl

-yZ], 2 0

V[xi,y,] E A , =~ 1,2.

(3.15)

Similarly, the condition (3.5) is equivalent to [XI

-x2,Y1 -y2], 2 --wIIxl -x211

V[xi,yi] E A , =~ 1,2. (3.16)

2.3. Accretive Operators in Banach Spaces

Summarizing, we may see that a subset A of X one of the following equivalent conditions hold: (i) If [ x l , yl], [ x 2 ,y2] E A , then there is w (y,

-

X

E J(x,

107

X is w-accretive if - x,)

such that

y2 , w ) 2 - wllx1 - x211;

(ii) lIxl - x2 + A(y, - y,)ll 2 (1 - Aw)llx, - x211 for 0 < A < l / w and all [xi,yi] E A , i = 1,2; (iii) [xl - x 2 , y 1 -y21s 2 -wIIxl -x211V[xi,yil € A , i = 1,2. In applications it is more convenient to use condition (iii) to verify the accretiveness. We know that if X is a Hilbert space, then a continuous accretive operator is m-accretive (Theorem 1.3). This result was extended by R. Martin [l] to general Banach spaces. More generally, we have the following result established by the author in [l] (see also G. F. Webb [ll). Theorem 3.1. Let X be a real Banach space, A an m-accretive set of X X X and let B: X -,X be a continuous, m-accretive operator with D ( B ) = X . Then A + B is m-accretive.

For the proof of this theorem, we refer to the author’s book [21. A direct proof making use of sharp estimates and convergence results for a discrete scheme of the form + B x n + ~ )3 e n + , , along with several generalizations of this theorem can be found in the work of Crandall and Pazy [41. Other m-accretivity criteria for the sum A + B of two m-accretive operators A , B c X X X can be obtained approximating the equation x +Ax Bx 3 y by x + A x + BAx 3 y , Xn+1 - x n

+ hn+l(xn+l + & , + I

+

where BA is the Yosida approximation of B. We shall illustrate the method on the following example. Proposition 3.8. Let X be a Banach space with uniformly convex dual X * and let A and B two m-accretive sets in X X X such that D( A ) n D( B) # 0 and (Au,J(B,u)) 2 0 VA>O,u€D(A). (3.17)

Then A

+ B is m-accretive.

108

2. Nonlinear Operators of Monotone Type

Proof Let f E X and A > 0 be arbitrary but fixed. We shall approximate the equation u +Au

bY u

+ Bu 3 f

(3.18)

+ Au + BAu 3 f ,

A > 0,

(3.19)

where BA is the Yosida approximation B, i.e., BA = A-'(Z - ( I We may write Eq. (3.19) as u=

(

1+-

l + h

l + A

+

+ AB)-').

+ AB)-'U

(I

which by the Banach fixed point theorem has a unique solution uAE D ( A ) (because (I + AB)-' and ( I + AA)-' are nonexpansive). Now, we multiply the equation UA

+ AuA + BAuA 3 f

(3.20)

by J(BAuA)and use condition (3.17) to get

IIBAuAII I IIfII + IIuAII

VA

> 0.

On the other hand, multiplying Eq. (3.20) by J(u, - uo), where uo E D ( A ) n D ( B ) , we get llUA -

+ l l f l l + 116011 + ~ ~ B A u o ~ ~ VA > Illu~ll+ llfll + 115011 + IBuoI

uoll 5 lluoll

0 7

where toE A u , . Hence, IluAlI + IIBAuAII I C

Vh > 0 .

(3.21)

Now we multiply the equation (in the sense of duality between X and X*) UA

- U,

+ AuA - Au, + BAuA - B,u,

3

0

by J(u, - up). Since A is accretive, we have IIuA

-

u,I* + (B,u,

-

B,u,, J ( U A

-

u,))

I

0

VA,

> 0. (3.22)

109

2.3. Accretive Operators in Banach Spaces

On the other hand,

+

because B is accretive and BAu E B(Z AB)-'u). Since J is uniformly continuous on bounded subsets and by (3.21)

this implies that {uA}is a Cauchy sequence and so u Extracting further subsequences, we may assume that

Then, by Proposition 3.4, we see that y solution (obviously unique) to Eq. (3.18).

E

=

limA+ uA exists.

Bu, z E Au, and so u is a W

2.3.2. Some Examples

Throughout this section, R is a bounded and open subset of R N with a smooth boundary, denoted d o . 1. Sublinear elliptic operators in LP(R). Let p be a maximal monotone graph in R X R such that 0 E D( p). Let C LP(R) X L P ( R ) ,1 I p < 00, be the operator defined by

b

b(u) D(

=

{ u E L P ( R ) ;u ( x )

p ) = ( u E LP(R); 3u

It is easily seen that

E

E

p ( u ( x ) ) a.e. x E

a),

LP(R) such that u ( x )

E

p ( u ( x ) ) , a.e. x

6 is m-accretive in LP(R) X LP(R) and

E

(3.23)

0).

110

2. Nonlinear Operators of Monotone Type

Let B:L P ( 0 )

Proposition 3.9.

+ P(u)

BU = - A M

+ L P ( 0 ) be

VU E D ( B ) ,

D ( B ) = W , ’ > P (n ~ )w ~ * P ( o n,D( ) D(B) = {u

E

the opemtor defined by

W,’*’(0);Au

6)

if 1 < p < a,

L 1 ( 0 ) } n D(

E

6))

if p

=

1. (3.24)

Then B is m-accretive and surjective. We note that for p

=

2, this result has been proved in Proposition 2.6.

Proof Let us show first that B is accretive. If u1 ,u 2 E D ( B ) and u1 E Bu, , u2 E Bu,, 1 < p < a,we have, by Green’s formula, llu1

- u,Ile,’,)
- V2 9

-jnA(ul

-

+ jn( Nu,)

J ( u , - u2))

u 2 )Iul

-

U~I”-’(U~

-

- P ( u , ) ) ( u , - u211u1

h )

~

2

-

u2Ip-,

2 0

because p is monotone (recall that J( uX x) = l u ( ~ > l ~ - ~ u ( x ) Iisl the ull~~~~~ duality mapping of the space LP(fl)). In the case p = 1, consider the function yE:R + R defined by yE(r) =

for r > E , for - E < r IE , for r < --E.

E-lr i‘1

(3.25)

The function y, is a smooth approximation of the signum function, signr= If [ui, ui]E B, i

j-u, -

V2)

=

[-1,1] i‘1

for r > 0, forr=O, for r < 0.

1,2, then we have

Y€(ul -

u 2 ) dx =

j p ,- u2>12Yr,l
20

j-( P ( u 1 )

-

u2) h

P(u2))Y&(ul - u2) dx

V E > 0.

For E + 0, y,(u, - u,) + g in Lm(0), where g E J(u)llullL$n), u u 2 , i.e., g ( x ) E sign u ( x ) a.e. x E a. Hence, B is accretive.

=

ul

-

2.3. Accretive Operators in Banach Spaces

111

We shall prove that B is m-accretive, considering separately the cases l
vu

E

D(A,).

(3.26)

Hence, A, is m-accretive. Let us prove now that R(Z + A, + $1 = L p ( f l ) . Replacing, if necessary, the graph p by u + p ( u ) - u o , where uo E p(O), we may assume that 0 E p ( 0 ) and so &(O) = 0. Then, by Green’s formula,

VA

and by Proposition 3.7 we conclude that R(Z + A , + claimed. To prove surjectivity of A, + consider the equation

> 0, (3.27)

p > = Lp(flR), as

6,

&U

+A,u

+ p(u) 3 f ,

& >O

,fELP(il),

(3.28)

which as seen before has a unique solution u,, and u,

=

lim uf

in L P ( f l ) ,

A-0

where uf is the solution to approximating equation By (3.27)’ it follows that

where C is independent of

E

EU

+ A, u + &i,
and A. Hence,

IIApu,llLP(n) I

c

v.5 > 0 ,

which by the estimate (3.26) implies that (u,} is bounded in W ~ ~ f W2*P(fl). Selecting a subsequence, for simplicity again denoted u,, we

112

2. Nonlinear Operators of Monotone Type

may assume that u,

weakly in W 2 , P (R),strongly in L P (R),

+u

Apu, +APu

bh,) EM,

weakly in LP( R), weakly in L P ( R),

g + 0 +

strongly in LP( R).

Since by Proposition 3.4 we know that g E p(u), we infer that u is the solution to the equation APu + p(u) 3 f , i.e., u E W2*P(R)and -Au

+ p ( u ) 3f

in dR.

u =O

Case 2. p

=

a.e.in R, (3.29)

+ p) = L'(R), i.e., for

1. We shall prove directly that R ( A , E D ( A , ) = {u

E

Wb*'(R); A M E

If,} c L2(R) such

that f, + f in

f~ L'(R), Eq. (3.29) has a solution u L'( R )I. We fix f in L'(R) and consider L'(R). As seen before, the problem

- A M , , + p(u,) 3 f n u,=O

in R, indR,

(3.30)

has a unique solution u, E H,'(R) n H2(R). Let u,(x) = f n ( x ) + A u , ( x ) E p ( u , ( x ) ) . By (3.30) we see that

because p is monotone and

for all u EH,'(R) n H2(R) and some 8 E Lm(R) such that N x ) sign u ( x ) a.e. x E R. Hence, strongly in L'( R),

v, + v

Au,,

+

6

strongly in L'( R).

E

(3.31)

Now, let hi E LP(R), i = O , l , ..., N , p > N. Then, by a well-known result due to G. Stampacchia [l] (see also Dautray and Lions [11, p. 4621,

2.3. Accretive Operators in Banach Spaces

113

the boundary value problem

cp=O

in d R ,

(3.32)

has a unique weak solution cp E Hd(R) n L"(R) and (3.33) This means that (3.34) Substituting JI

=

u, in (3.341, we get

and therefore, by (3.331,

'

Since { h i } E oc (Lp(R))" are arbitrary, we conclude that {(u,, d u n / a x 1 , .. ., d u , / d x , , ) } ~ , is bounded in (Lq(R))" l/q l/p = 1. Hence,

',

IIu, IIwd qn) IC IIA u, II

+

~l(n),

(3.35)

where 1 < q = p / ( p - 1) < N / ( N - 1). Hence, {u,,) is bounded in W,'.q(R) and compact in L'(R). Then, extracting a further subsequence if necessary, we may assume that u,

u

weakly in

W,.q(

a),strongly in L'(a).

(3.36)

Then, by (3.311, it follows that 6 = Au, and since the operator fi is closed in L'(R) X L'(R), we see by (3.31) and (3.36) that d x ) E p ( u ( x ) ) a.e. x E R and u E W/*q(R). Hence R ( B ) = L'(R), and in particular B is m-accretive.

114

2. Nonlinear Operators of Monotone Type

We have proved, therefore, the following existence result for the semilinear elliptic problem in L'(R). Corollary 3.1. For every f

-Au

E

L1(R), the boundary value problem

+ p(u) 3f u =O

a.e. in R , in d R ,

(3.37)

has a unique solution u E W ~ ~ q ( R with ) Au E L'(R), where 1 Iq < N / ( N - 1). Moreover, the following estimate holds: Ilullw;.qn,

ICllf 1ILyn)

V f E L'(R).

(3.38)

Inparticular, D ( A , ) c Wb.q( and Ilullwd,q(n) I cllAuII~~(n)

V U E D(A1).

Remark 3.1. It is clear from the previous proof that Proposition 2.8 and Corollary 3.1 remain true for linear second order elliptic operators A on

R.

2. The nonlinear di@sion operator in L'(R). In the space X define the operator Au

=

D(A)

=

-AP(u) VU ED(A), ( U E L'(R); P(u) E Wd.'(R), AP(u)

E

= L'(R),

L'(R)), (3.39)

where P is a maximal monotone graph in R x R such that 0 is an open bounded subset of RN with smooth boundary.

E

P(0) and R

Proposition 3.10. The operator A is m-accretiue in L'(R) X L'(R).

Prooj Let u , u E D(A ) and let y be a smooth monotone approximation of sign (see (3.25)). Then, we have

Letting y

-+

sign, we get

/-(AM

-AlJ)(dW 2 0,

2.3. Accretive Operators in Banach Spaces

115

where [ ( x ) E sign( p ( u ( x ) ) - p(u(x))> = sign(u(x) - u ( x ) ) a.e. x E R. Hence, A is accretive. Let us prove now that R(Z + A ) = L'(R). For f E L'(R), the equation

u+Au=f can be equivalently written as p - ' ( u ) - Au = f

in R,

u E Wb,'(R), Au

E

L'(R). (3.40)

But according to Proposition 3.9, (Corollary 3.11, Eq. (3.40) has a solution u E Wd,q(
L'(RN).We shall use the notation a = ( a , , a 2 , .. .,a N ) , d u ) , = CE1(d/d)xi)a,(u(x)) = divdu). The function a: R + RN is assumed to be continuous. We will define A as the closure of the operator A , C L'(R) X L'(R) defined in the following.

in the space X cpx = ( c p x , ,

=

. . ., cpx,),

It is readily seen that if a E C'(R) and u E Ct(RN) then u E D ( A , ) and A,u = a(u>,. Indeed, if p is a smooth approximation of r + Irl, then we have

116

2. Nonlinear Operators of Monotone Type

where a' = ( a ; ,a ; , . ..,a h ) is the derivative of a. Now, letting p' tend to sign,, we get

for all cp E C,(RN). Hence, u E D ( A , ) and A,u = ( U ( U ) ) ~ . Conversely, if u E D ( A , ) n Lm(RN)and u E A o u , then using the inequality (3.42) with k = IIuIIL-(RN) + 1 and k = - ( l l ~ l l ~ - ( + ~ ~11,) we get

RN be continuous and lim sup, Proposition 3.11. Let a: R Irl < w. Then A is m-accretive. -+

~

,Ila(r)ll/

We shall prove this important result in several steps.

Lemma 3.2. A is accretive in L'(RN) X L'(RN). ProoJ: Let [ u , U ] and [U,V] be two arbitrary elements of A,. By Definition 3.2 we have, for k = U(y), f ( x ) = 9(x, y ) ( 9 E C;(RN X RN), 9 2 01,

+.(x)+(x,y))

h d y 2 0.

(3.43)

Now, it is clear that we can interchange u and ti, u and 5, x and y to obtain, by adding to (3.43) the resulting inequality,

2.3. Accretive Operators in Banach Spaces

117

Substituting in (3.441, we get

(3.45)

where N y ) E sign(u(y) - ii(y)) a.e. y E R N . Hence, for every cp E C,"(RN),cp 2 0, there exists 8 E J(u - U) such that (3.46) holds ( J is the duality mapping of the space L'(R)). If in (3.46) we take cp = a ( ~ I I y l l ~ ) , where a E Cr(R), a 2 0, and a ( r ) = 1 for Irl I1, and let E + 0, we get

for some 8

E

J(u

-

U). Hence, A, is accretive and hence so is A.

W

In order to prove that A is m-accretive, we will show that the range of I + A o is dense in L'(RN), i.e., that the equation u + a(u), = f has a solution (in the generalized sense) for a sufficiently large class of functions f. We shall approximate this equation by the family of elliptic equations u

+ a(u), - EAU =f

in R N .

(3.47)

Lemma 3 3 . Let a E C ' , a' bounded, and let E > 0. Then for each f L 2 ( R N ) ,Eq. (3.47) has a solution u E H 2 ( R N ) .

Pro05

E

Denote by A the operator A

=

-A,

D(A) =H2(RN)

and Bu = -a(u), V u E D ( B ) = H ' ( R N ) .The operator T = ( I + EA)-'B is obviously continuous and bounded from H ' ( R N ) to H 2 ( R N ) ,and therefore it is compact in H ' ( R N ) .

2. Nonlinear Operators of Monotone Type

118

For a given f

E

L 2 ( R N )Eq. , (3.47) is equivalent to

(3.48) + ( I + &A)-'f. = {u E H'(RN); ( I u ( ~ ~ ~ ( R +N E) I I V U ( I ~ Z ( R N<) R 2 } , where R =

u

Let D

=

TU

+ 1. We note that ( I + ~ A ) - ' f e( I

llfI(~z(R~)

-

Indeed, we suppose that there is u u - EAU

E

dD and t

+ t a ( ~=)f~

(3.49)

0 st I 1.

tT)(dD),

E

[O, 11 such that

in R N ,

and we will argue from this to a contradiction. Multiplying the last equation by u and integrating on R N ,we get

On the other hand, we have

/ R N a ( u ) x u d x= where b ( u ) =

-1

RN

(a(u),ux)h=

-1 divb(u)dx RN

=

0,

/{ a(s)ds. Hence, +

I I ~ I I ~ L ~ ( R ENI )I V U I I ~ L ~ ( I R NI)I ~ I I L z ( R N ~ I I u I I L I ~ ( R( NR)

- 1 ) <~~~7

and so u CZ dD. Let us denote by d(Z - t T , D,(Z+ &A)-'f) the topological degree of the map Z - tT relative to D at the point ( I + &A)-'f. By (3.49), it follows that

d ( Z - t T , D , ( Z + &A)-'f) = d ( Z , D , ( Z + &A)-'f) for all 0 I t I 1. Hence,

d(Z

-

T,D,(Z

+ &A)-'f)

=

d(Z,D,(Z + &A)-'f)

=

1

because ( I + ~ A ) - l f E D. Hence, Eq. (3.48) has at least one solution W u E D ( A ) = H 2 ( R N )The . proof of Lemma 3.3 is complete. Lemma 3.4. Under assumptions of Lemma 3.3, iff then u E L p ( R N )and IIUIILP(RN)

IlfllLP(RN)-

E

Lp(RN),1 Ip I 03, (3 S O )

2.3. Accretive Operators in Banach Spaces

Prooj We shall treat first the case 1 n, if r < - n .

lrlp-2r

If multiply Eq. (3.47) by a,(u)

3

119

L 2 ( R N )and integrate on R N ,we get

because, as previously seen,

R'

whilst

because a, is monotonically increasing. Note the relation a,(r)r 2 Ia,(r)I4

Vr

E

R

l/p

+ l/q

=

1.

Then, using the Holder inequality in (3.511, we get

Hence,

which clearly implies that u E L p ( R N )and (3.50) holds. In the case p we multiply Eq. (3.47) by S,(u), where

if Irl I n - ' , if r > n - ' , if r < - n - '

=

1,

120

2. Nonlinear Operators of Monotone Type

Note that n211u11i2(RN).

6,(u) E L 2 ( R N ) because m{x Then, arguing as before, we get

E

RN; lu(x)l > n-'1 I

Then, letting n + ~0we get (3.501,as desired. In the case p = 03, put M = l l f l ( L = ( R ~ ) . Then, we have u - M + a ( u ) , - ~ A ( u-M) = f - M 1 0

a.e.inRN.

Multiplying this by ( u - M I + (which, as is well-known, belongs to H ' ( R N ) ) , we get IR4(u - MI+)' du I0 because

Hence, u ( x ) s M a.e. x u

E

R N .Now, we multiply the equation

+ M + (a(u)), - &A(u + M ) =f + M 2 0 and get as before that ( u + M I - = 0 a.e. in

by ( u + M ) u E L m(RN) and

R N . Hence,

as desired. Lemma 3.5. Underassumptions of Lemma 3.3, let f, g E L2(RN n L'(RN and let u, u E H 2 ( R N )n L'( RN )be the corresponding solutions to Eq. (3.47). Then we have

Il(f IIu - VIIL1(RN) IIl(f

Il(u -

U ) + I I L ' ( R N )I

-g)+llL1(RN))

(3.52)

- g ) l l L ' ( R N )*

(3.53)

121

2.3. Accretive Operators in Banach Spaces

Prooj Since (3.53) is an immediate consequence of (3.52) we shall confine ourselves to the latter estimate. If multiply the equation

u by t ( x )

E

-U+

(u(u) -u(u)),

-

E A ( U- U) = f - g

sign(u - u)' and integrate on R"',we get

Now,

by the divergence theorem because in d ( x ; u ( x ) > u(x)}.

a(u) = a(u)

Hence, IKu - u ) + I I L ~ ( R N )

I Il(f

as claimed.

- g)+llL1(RN),

W

Proof of Proposition 3.ZZ. Let us show first that L ' ( R N )n Lw(RN)c R(Z +A,). To this end, consider a sequence {a,} of C' functions such that &+ 0 a,(O) = 0, a , -+ a uniformly on compacta. For f E L'(RN)n Lw(RN),let u, E H 2 ( R N )n L ' ( R N ) n Lw(RN)be the solution to Eq. (3.47). Note the estimates II~,IIL~(RN) I Ilfll~1(R~)

II~,IIL=(RN)

I II~IIL"(RN).

Moreover, applying Lemma 3.5 to the functions u y ) , we get the estimate

=

u , ( x ) and u = u,(x

Vy

E

+

R"'.

According to a well-known compactness criterion these estimates imply that {u,) is compact in L:,,(R"'). Hence, there is a subsequence, which for simplicity will again be denoted u,, such that u, u,( x )

-+

u

-+

u( x )

in every L ' ( B , ) , VR > 0 , a.e. x

E

R"',

where BR = { x ; llxll I R}. We shall show that u

(3.54)

+ A,u

=

f.

122

2. Nonlinear Operators of Monotone Type

Let cp E C ; ( R N ) , cp 2 0, and let a E C2(R) be such that a” 2 0. Multiply the equation satisfied by u, by a’(u,)cp, and integrate on R N . Integration by parts yields

Letting

E

tend to zero and using (3.541,we get

a ” ( s ) a ( s ) ds -

+ cpx[

for all cp E C t ( R N ) ,cp O,(s - k ) , where

2

0, k

E

and y, is given by (3.25). Letting

jR

R, and a

E

dx E

I IR

N f a ’ ( u ) dx

C2(R), a” 2 0. Let a ( s ) =

tend to zero, this yields

sign,( u - k ) [ ucp - (a( u ) - a( k ) ) cpx - fcp] dx 4 0.

,

On the other hand, since lim sup,,, a(r)/lrl < 00, a(u> E L’(RN).We have therefore shown that f E u + A,u. Now, let f E L’(RN),and let f,, E L ’ ( R N )n Lm(RN)be such that f,,+ f in L ’ ( R N )for n + 00. Let u,, E D ( A , ) be the solution to the equation u + A , u 3 f,, . Since A , is accretive in L ’ ( R N )X L’(RN),we see that {u,,} is convergent in L’(RN). Hence, there is u E L ’ ( R N )such that ~

This implies that f

E u

+ Au, thereby completing the proof.

W

Bibliographical Notes and Remarks

In particular, we have proved that for every f partial differential equation

E

123

L ' ( R N )the first order

(3 3 ) has a unique generalized solution u Lipschitz in L'(RN).

E L'(RN),

and the map f

+

u is

Bibliographical Notes and Remarks

Section 1. Theorem 1.1 is due to G. Minty [2], Theorems 1.2 and 1.3 were originally given by G. Minty [l] in Hilbert space and extended to reflexive Banach spaces by F. Browder [l, 21. Theorems 1.4 and 1.5 are due to R. T. Rockafellar [3,4], but the proofs given here are due to P. M. Fitzpatrick [l] and to H. BrCzis et al. [l], respectively. For other significant results in the theory of monotone operators, we refer the reader to the survey of F. Browder [2]. Section 2. The results of Section 2.2.1 are essentially due to J. Moreau [2] and to R. T. Rockafellar [l]. Theorem 2.2 has been established in Hilbert space by H. BrCzis [2, 41 (see also J. Moreau [l]) and in this form arises in the author's book [2]. Theorem 2.4 is due to H. BrCzis [4]. For other results on convex functions and their subdifferentials, we refer to the monographs of R. T. Rockafellar [l], H. BrCzis [4], and V. Barbu and T. Precupanu [ll. The concept of generalized gradient developed in Section 2.2.3 along with its basic properties are due to F. Clarke [l, 21 (see also R. T. Rockafellar [7, 81). Proposition 2.15 has been established in the author's book [7]. Section 3. The general theory of m-accretive operators has been developed in the works of T. Kato [3] and M. G. Crandall and A. Pazy [l, 21 in connection with theory of semigroups of nonlinear contractions. Proposition 3.9 is due to H. BrCzis and W. Strauss [l] and Proposition 3.11 to M. G. Crandall [ 13. Other examples and applications to partial differential equations can be found in author's book [2] as well as in the monographs of R. Martin [21 and D. Zeidler [l].

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Chapter 3

Controlled Elliptic Variational Inequalities

This chapter is concerned with optimal control problems governed by variational inequalities of elliptic type and semilinear elliptic equations. The main emphasis is put on first order necessary conditions of optimality obtained by an approximating regularizing process. Since the optimal control problems governed by nonlinear elliptic equations, and in particular by variational inequalities, are nonconvex and nonsmooth the standard methods to derive first order necessary conditions of optimality are usually inapplicable in this situation. The method we shall use here is to approximate the given problem by a family of smooth optimization problems containing an adapted penalty term and to pass to limit in the corresponding optimality conditions. We shall discuss in detail several controlled free boundary problems to which the general theory is applied, such as the obstacle problem and the Signorini problem. 3.1. Elliptic Variational Inequalities. Existence Theory 3.1.1. Abstract Elliptic Variational Inequalities

Let X be a reflexive Banach space with the dual X * and let A: X + X * be a monotone operator (linear or nonlinear). Let cp: X R be a lower semicontinuous convex function on X, cp f +m. If f is a given element of X, consider the following problem. Find y E X such that --f

(AY,Y

-2)

+4 Y )

- d z ) 5

125

(Y - z , f )

vz E X .

(1.1)

126

3. Controlled Elliptic Variational Inequalities

This is an abstract elliptic variational inequality associated with the operator A and convex function cp, and can be equivalently expressed as AY + dcpo(Y)

3

f

( 1*2)

9

where dp c X x X * is the subdifferential of cp. In the special case where cp = I , is the indicator function of a closed convex subset K of X , i.e., zK(x)

=

{0

+oo

ifxEK, otherwise,

problem (1.1) becomes: Find y

E

K such that

It is useful to notice that if the operator A is itself a subdifferential d+ of X + R, then the variational inequality a continuous convex function (1.1) is equivalent to the minimization problem (the Dirichlet principle)

+:

min{$(z) + cp(z) - ( z , f ) ; z E X }

(1.4)

or, in the case of problem (1.31, min{+(z) - ( z , f);z E K}.

(1.5)

As far as concerns existence in problem (l.l), we note first the following result.

Theorem 1.1. Let A : X + X * be a monotone, demicontinuous operator and let cp: X + R be a lower semicontinuous, proper, convex function. Assume that there exists y o E D(cp) such that

lim ((AY,Y - y o ) + ~ ( Y ) ) / l l Y l l=

llyll-+ 30

+w.

( 1.6)

Then problem (1.1) has at least one solution. Moreover, the set of solutions is bounded, convex, and closed in X . If the operatorA is strictly monotone, i.e., ( A M- Av, u - v) = 0 u = v, then the solution is unique.

+

Proof By Theorem 1.5 in Chapter 2, the operator A dp is maximal monotone in X X X * . Since by condition (1.6) it is also coercive, we conclude (see Corollary 1.2 in Chapter 2) that is surjective. Hence, Eq. (1.2) (equivalently, (1.1)) has at least one solution.

3.1. Elliptic Variational Inequalities. Existence Theory

127

Since the set of all solutions y to (1.1) is ( A + d c p ) - ’ ( f ) , we infer that this set is closed and convex (Proposition 1.1 in Chapter 2). By the coercivity condition (1.61, it is also bounded. Finally, if A (or more generally, if A + d q ) is strictly monotone, then ( A + dcp)-’f consists of a single element. In the special case cp

=

ZK, we have:

Corollary 1.1. Let A: X -+ X * be a monotone demicontinuous operator and let K be a closed convex subset of X . Assume either that there is y o E K such that

lim ( A y ,y - yo)/llyll =

II yll-

(1.7)

+w,

m

or that K is bounded. Then problem (1.3) has at least one solution. The set of all solutions is bounded, convex, and closed. If A is strictb monotone, then the solution to (1.3) is unique.

To be more specific we shall assume in the following that X Hilbert space, X * = V ’ , and VcHcV’

=

V is a (1.8)

algebraically and topologically, where H is a real Hilbert space identified with its own dual. The norms of V and H will be denoted by II * II and I . I, respectively. For u E V and u’ E V’ we denote by (u, u ’ ) the value of u‘ in u; if u, u’ E H , this is the scalar product in H of u and u ‘ . The norm in V‘ will be denoted by II II*. Let A E L ( V , V ’ ) be a linear continuous operator from V to V’ such that, for some w > 0,

-

( A u ,u )

2 wllu112

vu E V .

Very often the operator A is defined by the equation ( u , Au) = a ( u , u )

V U , U EV ,

(1-9)

where a: V x V -+ R is a bilinear continuous functional on V X V such that a( u, u ) 2 wlluIl2

vu E V .

( 1 .lo)

128

3. Controlled Elliptic Variational Inequalities

In terms of a, the variational inequality (1.1) on V becomes

a(y,y

-2)

y

K,

+ V(Y) - cp0)

(Y - z , f )

Vz

E

E

K.

v,

(1.11)

and, E

a(y,y

-

z ) I ( y - z ,f )

Vz

(1.12)

As we shall see later, in applications V is usually a Sobolev space on an open subset R of R"', H = L 2 ( R ) , and A is an elliptic differential operator on R with appropriate homogeneous boundary value conditions. The set K incorporates various unilateral conditions on the domain R or on its boundary dR. By Theorem 2.1 of Chapter 2 we have the following existence result for problem (1.11). Corollary 1.2. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let cp: V + R be a l.s.c., convex proper function. Then, for every f E V' , problem (1.11) has a unique solution y E V. The map f + y is Lipschitzfrom V' to V.

Similarly for problem (1.12): Corollary 1.3. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let K be a closed convex subset of V. Then, for every f E V ' , problem (1.12) has a unique solution y . The map f + y is Lipschitzfrom V ' to V.

A problem of great interest when studying Eq. (1.11) is whether Ay To answer this problem, we define the operator A,: H + H ,

A,y=Ay

fory€D(A,)={uEV;AuEH).

E

H.

(1.13)

The operator A, is positive definite on H and R(Z + A,) = H ( I is the unit operator in H ). (Indeed, by Theorem 1.3 in Chapter 2 the operator Z + A is surjective from V to V'.) Hence, A, is maximal monotone in H X H.

Theorem 1.2. Under the assumptions of Corollary 1.2, suppose in addition that there exists h E H and C E R such that

3.1. Elliptic Variational Inequalities. Existence Theory

Then, i f f

E

129

H , the solution y to (1.11) belongs to D ( A H )and lAyl IC(I +

Proof: Let A ,

E

(1.15)

Ifl).

L ( H , H 1 be the Yosida approximation of A,, i.e.,

A,

=

A-'(Z - (I + AA,)-'),

A

> 0.

Let y E V be the solution to (1.11). If in (1.11) we set z + Ah) and use condition (1.141, we get

=

(I + A A H ) - ' ( y

(Here we have assumed that A is symmetric; the general case follows by Theorem 2.4 in Chap. 2.) We get the estimate IA,+ylIc(1 + I f

I)

VA

> 0,

where C is independent of A and f . This implies that y estimate (1.15) holds. Corollary 1.4.

Zn Corollaty 1.3, assume in addition that f

(I + AA,)-'(y

+ Ah) E K

for some h

E

E

E

D(AH) and

H and

H and all A > 0. (1.16)

Then the solution y to variational inequality (1.3) belongs to D(A,), and the following estimate holds: lAyl IC(I + I f l )

Vf

E

H.

(1.17)

3.1.2. The Obstacle Problem Throughout this section, R is an open and bounded subset of the Euclidean space RN with a smooth boundary dR. In fact, we shall assume that dR is of class C 2 .However, if R is convex this regularity condition on dR is no longer necessary.

130

3. Controlled Elliptic Variational Inequalities

Let V = H'(R), H

=

L2(R), and A : V + V' be defined by N

a2 > 0. If

by

Here, ao,a,, E LYR) for all i, j

c

=

1,. .., N , a,,

N

a , ( x ) 2 0,

2

t,1=1

a,,(x)6,6, 2 w11611~

=

a,,, and

V t E R N ,X

E

a,

(1.20)

where w is some positive constant and 11. IIN is the Euclidean norm in R N . If a 1 = 0, we shall assume that a o ( x ) 2 p > 0 a.e. x E R. The reader will recognize of course in the operator defined by (1.18) the second order elliptic operator

c N

AOY

=

-

( a l l y x , ) , , + aoy

(1.21)

with the boundary value conditions aIy

dY + a2 =0 dU

in d R ,

(1.22)

where d / d u is the outward normal derivative, d

-y

dv

c N

=

i,j=1

a i j y x cos( , u, X i ) .

(1.23)

Similarly, the operator A defined by (1.19) is the differential operator (1.21) with Dirichlet homogeneous conditions: y = 0 in dR.

3.1. Elliptic Variational Inequalities. Existence Theory

131

Let I) E H2(R) be a given function and let K be the closed convex subset of V = H’(R) defined by K

=

{ y E V ;y ( x ) 2 $ ( x ) a.e. x

E

a}.

(1.24)

Note that K # 0 because $+= max($,O) E K . If V = Hd(fl), we shall assume that $(x) 4 0 a.e. x E dR, which will imply as before that K # 0. Let f E V’. Then, by Corollary 1.3, the variational inequality ( 1.25) VZ E K a(y,y -z ) I ( Y - z7f) has a unique solution y E K . Formally, y is the solution to the following boundary value problem known in the literature as the obsfacfeprobfem,

in R + = { x E R ; y ( x ) > $ ( x ) } , A,y2f, y > $ inR,

A,y = f

y = $

inR\R+

dY

- = -

’ du

a* du

in d R + \ dR,

(1.26) (1.27)

Indeed, if $ E C ( a ) and y is a sufficiently smooth solution, then R + is an open subset of R and so for every a E C;(R+) there is p > 0 such that y f p a 2 $ on R, i.e., y f p a E K . Then if take z = y f p a in (1.251, we see that

Hence, A,y = f in 9 ’ ( f l + ) . Now, if we take z = y a , where a

+

E

H’(R) and a

2

0 in R, we get

and therefore A,y 2 f in S’(Sz). The boundary conditions (1.27) are obviously incorporated into the definition of the operator A if a2 = 0. If a2 > 0, then the boundary 0 conditions (1.27) follow from the inequality (1.25) if a,$ + a2 d $ / d u I a.e. in dR (see Theorem 1.3 following). As for the equation d y / d u = d $ / d u in dR+, this is a transmission property that is implied by the conditions y 2 I) in R and y = $ in d R +, if y is smooth enough.

132

3. Controlled Elliptic Variational Inequalities

In problem (1.261, (1.27), the surface d o + \ dQ = S, which separates the domains R + and R \ fi' is not known a priori and is called the free boundary. In classical terms, this problem can be reformulated as follows: Find the free boundary S and the function y that satisfy the system A,y =f y = $ a,y

dY + a2 = 0 dV

inR+, dY a* _ --

inR\R+,

dv

dv

in S,

in dR.

In the variational formulation (1.29, the free boundary S does not appear explicitly but the unknown function y satisfies a nonlinear equation. Once y is known, the free boundary S can be found as the boundary of the coincidence set { x E a; y ( x ) = + ( x ) } . There exists an extensive literature on regularity properties of the solution to the obstacle problem and of the free boundary. We mention in this context the earlier work of Br6zis and Stampacchia [ll and the books of Kinderlehrer and Stampacchia [l] and A. Friedman [l], which contain complete references on the subject. Here, we shall present only a partial result. Proposition 1.1. Assume that aij E C'(fi), a , E Lm(R), and that conditions (1.20) hold. Further, assume that I!,I E H2(R) and

a*

a,@ + a2 - I0 dV

a.e. in dR.

(1.28)

Then for every f E L2(SZ) the solution y to variational inequality (1.25) belongs to H ' ( a )and satisfies the complementarity system

ax. x E R,Y(X) 2 (A,Y(X) -f(X))(Y(X) - *(XI) = 0 A,y(x) 2f(x) a.e. x E SZ,

*(XI,

(1.29)

along with boundary value conditions

(1.30) Moreover, there exists a positive constant C independent off such that IlyIlfP(n) IC(llfIlL2(n)

+ 1).

(1.31)

133

3.1. Elliptic Variational Inequalities. Existence Theory

Proof. We shall apply Corollary 1.4, where H = L 2 ( 0 ) ,V = H ' ( 0 ) (respectively, V = H d ( 0 ) if a2 = O), A is defined by (1.18) (respectively, (1.19)), and K is given by (1.24). Clearly, the operator A,: L 2 ( 0 )-+ L 2 ( 0 )is defined in this case by (AHY)(X)

D(A , ) A

=

=

(A,y)(x)

i y

E H'(

ax.

X E

0 ) ;a l y

We shall verify condition (1.16) with h

> 0 the boundary value problem

0, Y

ED(A,),

dY + a2 = 0 a.e. in d 0 . dV } = A,$.

+ AA,w = y + AA,$ dw alw + a2 - = 0 in d 0 , dV w

To this end, consider for in R ,

which has a unique solution w E D ( A , ) . Multiplying this equation by (w - $)-E H'(R) and integrating on 0, we get, via Green's formula,

( y - $)(w - $)- du 5 0 . Hence, (w - $I-= 0 a.e. in 0 and so w E K as claimed. Then, by Corollary 1.4, we infer that y E D ( A , ) and II&yIlL2(n) 5 C(llfllLqn)

+ l),

and since dR is sufficiently smooth (or R convex) this implies (1.31). Now, if y E D ( A , ) , we have

and so by (1.25) we see that jR

( A , y ( X ) - f( x ) ) ( y ( X ) - Z( x ) ) du

I0

VZ E K .

(1.32)

134

3. Controlled Elliptic Variational Inequalities

The last inequality clearly can be extended by density to all z where

K O= { u If in (1.32) we take z get

E

=

L 2 ( R ) ; u ( x ) 2 + ( x ) a.e. x

=

0).

KO,

(1.33)

+ + a , where a is any positive L 2 ( 0 )function, we

(A,y)(x) -f(x) 2 0

Then, for z

E

E

a.e. x

E

R.

+, (1.32) yields

which completes the proof. We note that under assumptions of Theorem 1.3 the obstacle problem can be equivalently written as ( 1.34)

dZ,o(y)

=

i

u E L2(

I

a ) ;1u( x ) ( y ( x ) - z( x ) ) dr 2 0 v z E KO R

or, equivalently,

where p : R

+

2R is the maximal monotone graph, if r > 0, if r = 0, if r < 0.

(1.35)

Hence, under the conditions of Theorem 1.3, we may equivalently write the variational inequality (1.25) as

a,y

dY + ff2 =0 dV

a.e. in d R ,

(1.36)

3.1. Elliptic Variational Inequalities. Existence Theory

135

arid as seen in Section 2.2, Chapter 2, it is equivalent to the minimization problem

(1.36)' where j : R

is defined by

+

j(r)

=

ifr20 otherwise.

(0

+m

( 1.37)

As seen elsewhere, Eq. (1.36) can be approximated by the smooth boundary value problem A o y + p,(y - $) = f

a.e. in fl, ( 1.38)

where P,(r) = -(l/&)r- ( p, is the Yosida approximation of p). In this context, we have a more general result. Let p be a maximal monotone graph in R X R, and let $ E H'(fl). Let p, = ~ - ' (-l (1 + ~ p ) - 'be ) the Yosida approximation of p. Then, for each f E L'(fl), the boundary value problem AOY + PAY a,y

+ a'

$1 = f

in

JY

in d f l

-= 0 dV

f l 7

( 1.39)

has a unique solution y,f E H'(fl). (Problem (1.39) can be written as A,y p,(y - #) = f , where y -+ p,(y - t,h) is monotone and continuous in ~ ' ( f l ) . )

+

Proposition 1.2.

Assume thaf

Then y,f

+

yf

strongly in ~ ' ( f l )weak& , in ~ ' ( f l ) ,

(1.41)

136

3. Controlled Elliptic Variational Inequalities

where y,f is the solution to boundary value problem (1.39).Moreover, i f f , + f weakly in L’(R), then y,f. + y f

w e a k l y i n H 2 ( R ) ,strong&inH’(R).

ProoJ Let y, = y,f.. Multiplying Eq. (1.39) by p,(y, on R,we get, by Green’s formula,

(1.42)

9 ) and integrating

Hence

Inasmuch as ( d / d u X y , -

9 ) - (a$/&

9)P,(Y,

-

9 ) = -(a1/a2XY,

+ ( a 1 / a 2 ) 9Be() 9 ) s 0 in

-

d f l , we infer that

9 ) P,(y,

-

is bounded in L2(R)

( P,( y, - @)}

and {A,y,) is bounded in L2(R>.We may conclude, therefore, that {y,) is bounded in H2(R) and on a subsequence, again denoted (ye},we have Y,

+

weakly in H 2 ( R ) ,strongly in H ’ ( R ) , (1.43)

Y

P,(Y, - 9 ) 5 A,y,=A,y, -A,y +

weakly in L2(R),

( 1.44)

weaklyin L2(R).

(1.45)

Clearly, we have Aoy+ dY

5=f

a’ - + a , y dU

=

0

a.e. in R a.e. in J R .

On the other hand, if denote by B c L2(R) X L’(fl) the operator By = ( w E L2(fl); w ( x ) E P ( y ( x ) - $ ( x ) ) a.e. in R},then B is maximal monotone and its Yosida approximation B, is given by B,(y)

=

P,(y -

9)

a.e. in R.

3.1. Elliptic Variational Inequalities. Existence Theory

137

Then, by (1.44) and Proposition 1.3 in Chapter 2, we deduce that 6 E By, i.e., ( ( x ) E p ( y ( x ) - @(XI). Hence y = y f , thereby completing the proof of Proposition 1.2. In particular, Condition (1.40) is satisfied if p is given by (1.35) and so Proposition 1.1 can be viewed as a particular case of Proposition 1.2. A simple physical model for the obstacle problem is that of an elastic membrane that occupies a plane domain R and is limited from below by a rigid obstacle J/ whilst it is under the pressure of a vertical force field of density f. We assume that the membrane is clamped along the boundary dR (see Fig. 1.1). It is well-known from linear elasticity that when there is no obstacle the vertical displacement y = y ( x ) , x E R, of the membrane satisfies the Laplace-Poisson equation. In the presence of the obstacle y = @ ( x ) , the deflection y = y ( x ) of the membrane satisfies the system (1.26). More precisely, we have -Ay =f

in { x

-Ayzf,

y z @

y=O

E

R; y ( x ) > @ ( x ) ) , inn,

in dR.

( 1.46)

The contact region { x E R; y ( x ) = @(XI) is not known a priori and its boundary is the free boundary of the problem. Let us consider now the case of two parallel elastic membranes loaded by forces f,, i = 1,2, that act from opposite directions (see Fig. 1.21.The variational inequality characterizing the equilibrium solution y is (see, e.g.,

Figure 1.1.

138

3. Controlled Elliptic Variational Inequalities

Figure 1.2.

Kikuchi and Oden [ll)

VY, . V ( Y , - 21) dU + Pz 5

/-fl(YI

+

- 21)

/ VY,

/n

R

fz(Y2

*

V Y ,

- 22) d X

dU

V(2,

- 22)

7

22)

K7 (1.47)

where f, 2 0, f, 5 0, and

K = { ( y , , y , ) €H,'(R) x H , ' ( R ) ; y , - y , s f a . e . i n R } .

(1.48)

Here, pl, p2 are positive constants, f is the distance between the initial positions of the unloaded membranes, and y , ( x , , x , ) 2 0, y,(x,, x , ) 5 0 are the deflection of the membranes 1 and 2 in ( x , ,x , ) = x . This problem is of the form (1.251, where H = L 2 ( f l )X L2(R), I/ = H,'(R) x H,'(R), K is defined by (1.48), f = ( f l , f,), and

4 Y , 2)

= PI

/ VY, R

v z , dU + Pz for Y

Formally, the solution y free boundary problem

AYI =f,, y, -y, 5 1 - P , A Y , Sf17 y, = 0 , -PI

=

=

/ VY, R

( Y , ,y2),

*

v z , dU

=

(zl, 2,) E I/ x I/.

( y , ,y,) to problem (1.47) is the solution to the

in { x ; Y d X ) -Pz AY, = f 2 in R , -P2 AY, 2f2 in R , y,=O in dR.

-Yz(X)

< f } 7

139

3.1. Elliptic Variational Inequalities. Existence Theory

The free boundary of this problem is the boundary of the contact set { x ; y , ( x ) - y,(x> = I ) . An important success of the theory of elliptic variational inequalities has been the discovery made by C. Baiocchi [l]that the mathematical model of

the water flow through an isotropic homogeneous rectangular dam can be described as an obstacle problem of the type just presented. Let us now briefly described this problem. Denote by D = (0, a ) X (0, b ) the dam and by Do the wetted region (see Fig. 1.3).The boundary S that separates the wetted region Do from the dry region D, = D \ Do is unknown and it is a free boundary. Let z be the piezometric head and let p(xI,x2)be the unknown pressure at the point (x1,x2)E D . We have z = p + x , in D and, by the D'Arcy law (we normalize the coefficients), Az=O

inD,.

(1.49)

Note also that z satisfies the obvious boundary conditions z dz

-=0 dx2

=

h , in AF,

inAB,

z =x2 dZ

- -dU

in S

U

0 in S ,

where d / d u is the normal derivative to S.

GC,

z

=

h , in BC,

(1 S O )

140

3. Controlled Elliptic Variational Inequalities

Introduce the function

and consider the Baiocchi transformation

Lemma 1.1. The function y satisfies the equation Ay

(1.51)

in g'( D)

= XD,

and the conditions

y > 0 in Do

y=O

in B \ D o ,

y =g

in d D , (1.52)

where

in FH

U HL U L C ,

g

=

0

g

=

1 2 -(x2 - h2) 2

We have denoted by

inCB,

g

=

1 2 -(x2 - h,) 2

xD, the characteristic function of

inAF.

Do.

Prooj We shall assume that y E H ' ( D ) and that the free boundary S is sufficiently smooth. Then, if x 2 = a ( x , ) is the equation of S, we have, for every test function cp E C,"(D),

141

3.1. Elliptic Variational Inequalities. Existence Theory

J-YA+=J

Dn

c p ( ~ I ~ ~ 2 ) ~ = x D , ( 4 0 )VcpECXD),

and Eq. (1.51) follows. Conditions (1.52) follow by the definition of y and by (1.50).

W

By Lemma 1.1, we may view y as the solution to the obstacle problem

- A y 2 -1, y 2 0 in D, Ay=1 in(xED;y(x)>O), y=g in d D ,

( 1.53)

or, in the variational form,

Vy * V(y

- u ) clx

+ JD(y

- u ) clx I0,

Vu E K,

(1.54)

where K = { u E H ’ ( D ) ; u = g in dD, u 2 0 in D). By Corollary 1.3, we conclude that problem (1.54) (and consequently the dam problem (1.53)) has a unique solution y E K. The free boundary S can be found solving the equation y(x,, xz) = 0. For sharp regularity properties of y and S, we refer to the book of A. Friedman [l].

142

3. Controlled Elliptic Variational Inequalities

3.1.3. An Elasto-Plastic Problem

Let R be an open domain of RN and let a: H,'(R) defined by

X

Hd(R)

-+

R be

( 1.55)

Introduce the set K

=

( y E Hd(R); IIVy(x)llN I1 a.e. x

E

R},

(1.56)

where II . I I N is the Euclidean norm in RN, and consider the variational inequality y

E

a ( y , y - 2)

K,

I(

y -z , f )

Vz

E

K,

(1.57)

where f E H-'(R). By Corollary 1.3, this problem has a unique solution y . If y is sufficiently smooth, then it follows as for the obstacle problem that -Ay

in { x E R ; I l V y ( x ) l l ~< I} in R, = R \ R e ,

=f

llVyllN = 1

in dR.

y=O

=

Re, (1.58)

The interpretation of problem (1.58) is as follows: The domain can be decomposed in two parts Re (the elastic zone) and R, (the plastic zone). In R e , y satisfies the classical equation of elasticity whilst in R,, I l V y ( x ) l l ~ = 1; the surface S that separates the elastic and plastic zones is a free boundary, which is not known a priori and is one of the unknowns of the problem. This models the elasto-plastic torsion of a cylindrical bar of cross-section R that is subject to an increasing torque. The state y represents in this case the stress potential in R. As noted earlier, (1.57) is equivalent to the minimization problem min{ f a ( z , z ) - ( z ,f ) ; z

E

K).

(1.59)

If f E L2(R) and dR is sufficiently smooth, then the solution y to (1.57) belongs to H2(R)(BrCzis and Stampacchia [l]). It is useful to point out that y

=

lim ye

E'O

in L'(R),

3.1. Elliptic Variational Inequalities. Existence Theory

where y ,

E

143

Hd(fl) is the solution to the boundary value problem -

Ay - div dh,( Vy)

=

in 0,

f

indfl,

y=O where h , : R N .+ R N is defined by

he(u)

=

I"

if l l u l l ~< 1,

(IlullN -

if l l u l l ~2 1,

2E

and ah,: R N -+ R N is its differential, i.e., dh,(u)

=

i"

if IIuIIN < 1,

( u l l u l l ~- 1)

if IlullN 2 1.

EIIUIIN

Let us calculate starting from (1.59) the solution to problem (1.57) in the case where fl = (0,l). If make the substitution

w( x )

=

l X Z ( s) ds,

x E (0, l ) ,

0

then problem (1.59) becomes i n f ( ~ / o l W ' ( x ) d X - / '0w ( x ) l I fX ( s ) d s d X ; w where U = { u E L2(0,1); lu(x)l to (1.60) satisfies the equation

I1

a.e. x

E

where Nu is the normal cone to U. Hence,

y'(x)

=

w(x) =

1

1

E

(0, l)}.Hence, the solution w

144

3. Controlled Elliptic Variational Inequalities

3.1.4. Elliptic Problems with Unilateral Conditions at the Boundary

Consider in R c R N the boundary value problem cy

-

Ay

dY

=f

+ P(Y) dV

3

g

y

=

0

in R,

(1.61)

r, , in r,, in

rl and r2 are two open, smooth, and disjoint parts of dR, rl u r, = dR, f E L2(R), g E L2(rl), c is a positive constant and p is a

where -

maximal monotone graph in R X R. Let j : R -+ R be a lower semicontinuous, convex function such that dj = p. We set V={y~H~(fi);y=oinr,}

and define the operator A: V + V‘ by

p(z)

=

1j (

z ) dz

Vz

E

V,

rl

and let fo

E

V’ be defined by

(fo 7 z > =

1nf ( x ) z ( x ) dx + 1g ( x ) z ( x W

Vz

E

V.

Vz

E

‘v,

rl

By Corollary 1.2, the variational inequality a ( y , y -2)

+ V(Y) - V(Z) 5

( f 0 , Y -2)

(1.63)

has a unique solution y E V. Problem (1.63) can be equivalently written as min{+a(z,z)

+ p(z)

- ( f o , z ) ;z E

v}.

(1.64)

The solution y to (1.63) (equivalently, (1.64)) can be viewed as generalized solution to problem (1.61). Indeed, if in (1.63) we take z = y - a, where a E C t ( R ) , we get (cya

In

+ v y . V a ) dx = 1nfa dx.

3.1. Elliptic Variational Inequalities. Existence Theory

Hence, cy - b y

=

f in g ' ( R ) . Now, multiplying this by y R,we get

z E V , and integrating on

- z,

145

where

More precisely, we have

where ( ;) is the pairing between H112(rl) and H-'I2(r1)(if y E H'(R) and A y E L2(R), then y E H112(rl)and d y / d v E W1I2(rl) (see Lions and Magenes [l]). Then, by (1.631, we see that

( 2 )

4

( j ( y ) -j(z))dx I g - -,y

-z

vz

E

v.

Hence, if g - d y / d v E L2(rl), we may conclude that g - d y / d v a.e. in r l .Otherwise, this simply means that

E

p(y)

where a+: H1'2(rl)-, H-'Iz(Fl) is the subdifferential of the function 6: H112(rl)+ R defined by

In the special case where rl = dR and g = 0, then as seen in Section 2.2, Chapter 2 (Proposition 2.10, y E H2(R) and llylIH2(n) IC(I

+ IlfllL2(n))

Vf E L2(R),

(1.65)

where C is independent of f. Moreover, we have y

=

lim0 y ,

&+

weakly in H 2 ( R ) ,strongly in H ' ( R ) ,

(1.66)

146

3. Controlled Elliptic Variational Inequalities

where y , E H2(R) is the solution to the approximating problem cy, - Ay, dY,

+ p,(y,) dU

=

f

in R,

=

0

in d R ,

=

1 -(r

(1.67)

and p,(r)

E

-

(1

+ E@)-'r)

Vr E

R, E > 0.

In this case, we have a more precise result. Namely: Proposition 1.3. Let f, + f weakly in L2(R). Then the solutiony, E H2(R) to problem (1.67) is weakly convergent in H2(R) and strongly convergent in H'(R) to the solution y f to problem (1.61).

Proof: As seen in the proof of Proposition 2.11, Chapter 2, we have for y , an estimate of the form (1.63, i.e.,

where C is independent of p, (i.e., of denoted E , we have Y, 4 Y

E).

Hence on a subsequence, again

weakly in H ~ ( R )strongly , in H ' ( Q ) ,

dY

weakly in H1I2(an),strongly in L ~ ( R ) ,

P&(Y&)

5

strongly in L~( d R ) ,

Y&

Y

strongly in L ~an). (

dY,

-4 du du +

+

(1.68)

Arguing as in the proof of Proposition 1.2, we see that 6 E p ( y > a.e. in dR. Then, letting E tend to zero in (1.681, we see that y is the solution to W (1.61). We shall consider now some particular cases. If j ( r ) = golrI, r E R, where go is some positive constant, then

3.1. Elliptic Variational Inequalities. Existence Theory

147

and so problem (1.61) becomes

cy dY

- +go

dv

Ay

-

sign y

=f

0

3

in R , in dR.

( 1.69)

Equivalently, cy-Ay=f

inR, y

JY

+ golyl = 0

a.e. in ail.

(1.70)

The boundary conditions can be rewritten as

=o y's o LO

if

1-1

dY dV

< go ,

dY dv

if - = g o ,

a.e. in dR.

dY

if - = - g o , dV

Hence, there are apriori two regions r' and r2on dR where Idy/dvl < go and I d y / d v l = g o , respectively. However, r' and r2 are not known, so problem (1.69) is in fact a free boundary problem and as seen before it has a unique solution y E H2(R). Problem (1.70) models the equilibrium configuration of an elastic body C! that is in unilateral contact with friction on dR (see Duvaut and Lions [l], Chapter IV).

The Signon'ni Problem. Consider now problem (1.61) in the special case g = 0, and where rl = dR, r2= 0, r > 0, r=0, r < 0,

V r ER.

(1.71)

i.e. cy-Ay=f

dY

a.e.inR,yLO, - 2 0 , dV

dY

y-=O dv

a.e.indR. (1.72)

148

3. Controlled Elliptic Variational Inequalities

This is the famous problem of Signorini, which describes the conceptual model of an elastic body R that is in contact with a rigid support body and is subject to volume forces f. These forces produce a deformation of R and a displacement on dR, with the normal component negative or zero. Other unilateral problems of the form (1.61) arise in fluid mechanics with semipermeable boundaries, climatization problems or in thermostat control of heat flow, and we refer the reader to the previously cited book of Duvaut and Lions [l]. In mechanics, one often meets problems in which constitutive laws are given by nonmonotone multivalued mappings that lead to problems of the following type (see P. D. Panagiotopoulos [l]):

where A: V .+ V' is defined by (1.18) and y is the generalized gradient (in the sense of Clarke) of a locally Lipschitz function 4: V + R. In particular, we may take

where j E L";b,(R) is such that lj(01IC(1 + l&Ip-'), 5 E R, and SO (see . s))), R(j(y(x>))I a.e. x E d+(y) c {w E L'(a); W ( X > E [fi(J(Y( (2.75) in Chapter 2). This is a hemivariational inequality, and the existence theory developed in the preceding partially extends to this class of nonlinear problems (see the book [l] by Panagiotopoulos, and the references given there).

3.2. Optimal Control of Elliptic Variational Inequalities

In this section, we shall discuss several optimal control problems governed by semilinear elliptic equations and in particular that governed by elliptic variational inequalities and problems with free boundary. The most important objective of such a treatment is to derive a set of first order conditions for optimality (maximum principle) that is able to give complete information on the optimal control. Since the optimal control problems governed by nonlinear equations are nonsmooth and nonconvex, the standard methods of deriving necessary conditions of

3.2. Optimal Control of Elliptic Variational Inequalities

149

optimality are inapplicable here. The method is in brief the following: One approximates the given problem by a family of smooth optimization problems and afterwards tends to the limit in the corresponding optimality equations. An attractive feature of this approach, which we shall illustrate on some model problems, is that it allows the treatment of optimal control problems governed by a large class of nonlinear equations, even of nonmonotone type and with general cost criteria.

3.2.1. General Formulation of Optimal Control Problems

Let V and H be a pair of real Hilbert spaces such that V is a dense subset of H and

VcHcV’ algebraically and topologically. We have denoted by V’ the dual of V and the notation is that of Section 1.1. Thus (.; ) is the pairing between V and V’ (and the scalar product of H ) and II 1 , I * I are the norms in V and H, respectively. Consider the equation

where A: V + V’ is a linear continuous operator satisfying the coercivity condition ( A u , u ) 2 wllull cp: V

+

2

Vu E V

for some w > 0,

R is a lower semicontinuous, convex function,

(2.2)

dcp: V + V’ is the

subdifferential of cp, B E L(U, V’),and f is a given element of V’. Here, U is another real Hilbert space with the scalar product denoted ( ,. ) and norm I * 10 (the controller space). As seen in Section 3.1 a large class of nonlinear elliptic problems, including problems with free boundary and unilateral conditions at the boundary, can be written in this form. The parameter u is called the control and the corresponding solution y is the state of the system. Equation (2.1) itself will be referred to as the state system or control system.

3. Controlled Elliptic Variational Inequalities

150

The optimal control problem we shall study in this chapter can be put in the following general form:

(PI Minimize the function

on ally

E

V and u

E

U satisfving the state system (2.1).

Here, g: H + R and h: U following conditions: (i)

+

R

are given functions that satisfy the

g is Lipschitz on bounded subsets of H (i.e., g is locally Lipschitz) and

bounded from below by an afine function, i.e.,

where (Y E H and C is a real constraint; (ii) h is convex, lower semicontinuous, and

(iii) B is completely continuousfrom U to V ' ,

The last hypothesis is in particular satisfied if the injection of V into H is compact. Roughly speaking, the object of control theory for system (2.1) is the adjustment of the control parameter u, subject to certain restrictions, such that the corresponding state y has some specified properties or to achieve some goals, which very often are expressed as minimization problems of the form (P). For instance, we might pick one known state y o and seek to find u in a certain closed convex subset Uo c U so that y = y o .Then the least square approach leads us to a problem of the type (PI, where

g(y)

=

31y

-

y0l2

and

h(u) =

if u E U,, elsewhere.

Other control problems such as that of finding the control u such that the free boundary of Eq. (2.1) (if this equation is a problem with free boundary) is as close as possible to a given surface S, though more

3.2. Optimal Control of Elliptic Variational Inequalities

151

complicated, also admit an adequate formulation in terms of (PI. This will be discussed further in a later section. Now let us briefly discuss existence in problem (PI. A pair (y*, u * ) E V x U for which the infimum in (P) is attained is called optimalpair and the control u* is referred as optimal control. Proposition 2.1. optimal pair.

Under assumptions (i)-(iii) problem (PI has at least one

Pro08 For every u E U we shall denote by y" E U the solution to Eq. (2.1). Note first that the map u -+ y" is weakly-strongly continuous from U to V. Indeed, if {u,} c U is weakly convergent to u in U, by assumption (iii) we see that

Bu,

+ Bu

strongly in V'

whilst by (2.1) and (2.2) we have wIIyUn- yumI12I IIBu, because d q is monotone in V lim y'n

n+m

=y

Now letting n tend to

X

-

B~,II*IIyUn- yUmII

V ' . Hence,

exists in the strong topology of V .

+m

in

Ay"n

+ d q ( y".)

E Bu,

+f ,

we conclude that y = y", as claimed. Now let d = inf{g(y") + h(u); u E U ) . By assumptions (i), (ii), it follows that d > - w . Now, let (u,} c U be such that d I g(y,)

+ h ( u , ) I d + n-'

Vn,

where y, = y".. By (i) and (ii) we see that {u,) is bounded in U, and so on a subsequence (for simplicity again denoted (u,}) we have u, + u*

y,

-+

y*

weakly in U , strongly in V ,

because B is completely continuous. Since as seen in the preceding y* = y"' and g(y*) h(u*) = d (because g is continuous and h is weakly lower semicontinuous), we infer that (y*, u * ) is an optimal pair for problem (PI.

+

152

3. Controlled Elliptic Variational Inequalities

From the previous proof it is clear that as far as existence is concerned, hypotheses (i)-(iii) are too strong. For instance, it suffices to assume that g is merely continuous from V to R.Also, if g is convex, assumption (iii) is no longer necessary because g is, in this case, weakly lower semicontinuous on V. On the other hand, it is clear that the convexity assumption on h cannot be dispensed with since it assures the weak lower semicontinuity of the function u + g ( y " ) + h(u), a property that for infinite dimensional controllers space U is absolutely necessary to attain the infimum in problem (PI.

3.2.2. A General Approach to the Maximum Principle Here we shall discuss a general approach to obtain first order necessary conditions of optimality for optimal control problems and in particular for problem (PI. Most of the optimal control problems that arise in applications can be represented in the following abstract form:

(6) Minimize the functional on all ( y , u ) E X

X

U, subject to state equation F ( y ,U)

=

0.

(2.7)

Here, L : X X U + R is a continuous function, h: U + R = I - m, +..I is convex and lower semicontinuous, and F : X X U + Y is a given operator, where X, Y,U are Banach spaces. We shall assume of course that for every u E U, Eq. (2.7) has a unique solution y " E X. Let us first assume that the map u + y" is Giteaux differentiable on U and that its differential in u E U , z = D,y"(u), is the solution to the equation

where Fy and F, are the differentials of y -+ F ( y , u ) and u + F ( y , u ) , respectively. (This happens always if the operator F is differentiable.)

3.2. Optimal Control of Elliptic Variational Inequalities

153

Let ( y * ,u * ) be an optimal pair for problem (PI. We have A - l ( L ( y U * + A " ,+~ *A u ) - L ( Y * , u * ) ) + A-'(h(u* Vu E U, A

+ Au)

-

h ( u * ) )2 0.

> 0, and if L is (GPteaux) differentiable on X X U this yields

(L,(Y*,U*),z)

+ (L,(y*,u*),u) + h'(u*,u)2 0

VUE U , (2.9)

where h ' is the directional derivative of h and (.,. 1, ( * ,. ) are the dualities between X, X * and U, U*, respectively. Denote by F; the adjoint of F y , and further assume that the linear equation (2.10)

q y * , u * ) p = L,(Y*, u * ) (the dual state equation) has a solution p we see that

E

(Ly(y*,u*),z)= (p,F,(y*,u*)z)

Y*. Then, by (2.8) and (2.10)

=

-(F;L(y*,u*)p,v),

and substituting in (2.9) yields VUE

(F;L(Y*>U*)P - L,(y*,u*),u) Ih'(u*,u)

u.

By Proposition 2.8 in Chapter 2, this implies

F,*( y * , u * ) p - L,( y * , u*) E dh( u * ) .

(2.11)

Equations (2.71, (2.101, and (2.11) taken together represent the first order optimality system for problem (P), and if F is linear and L is convex these are also sufficient for optimality. If L and F are nonsmooth, one might hope to obtain a maximum principle type result in terms of generalized derivatives of L and F by using the following method: Assume that the control space U is reflexive and that there are a family (F,),,, of smooth operators from X X U to Y and a set of differentiable functions L,: X X U + R such that

(i)'

For every E > 0 and u E U the equation F,(y, u ) = 0 has a unique solution y =y," and there exists p 2 1 such that lly:llx

I

C(1

+ lull;)

Vu E

u.

(2.12)

'

154

3. Controlled Elliptic Variational Inequalities

(ii)' If {u,} is weakly convergent to u in U then { y , " ~ is } strongly convergent to y" in X and liminf L,(y,Ue,u,) &+

0

2

(2.13)

L(y",u).

Moreover, lim, L,(y,", u> = L ( y " , u ) V u E U. (iiiY L,(y, u> 2 -C(lyl + lul) V ( y ,u> E X X U , where C is independent of E . ~

Let ( y * , u * ) be any optimal pair in problem penalized problem

i

1

(P,) inf ~ , ( y , u )+ h ( u ) +

-IU

2P

- u*IZP;

(el. We consider the

F,(Y,u)

= O,U E

1

u

,

and assume that this problem has a solution (y,,u,) (this happens, for instance, if the function u -+ L,(y,U, u ) is weakly lower semicontinuous). Then, we have Lemma 2.1. For

E

-+

0, we have u, + u*

y, Pro05

For all

E

-+

y*

strongly in U , strongly in X .

> 0 we have

and by condition (2.12) and assumption (iiiY, {u,) is bounded in U and so, on a subsequence E, 0. -+

weakly in U ,

u&n-+ ii y,,

-+

y

=y z

strongly in X .

Since h is weakly lower semicontinuous, we have lim inf h( u,,) 2 h( U ) E" +

and so, by assumption (ii)',

0

3.2. Optimal Control of Elliptic Variational Inequalities

155

Hence, J = y * , U = u * , and u,, -+ u* strongly in U on some subsequence {qJ.Since the limit u* is independent of the subsequence, we conclude that u, + u* strongly in U, as claimed. Now, since problem (6,) is a smooth optimization problem of the form (61, (y,,u,) satisfy an optimality system of the type (2.10), (2.11). More precisely, there is p , E Y * that satisfies along with (y,, u,) the system

F&(Y&, U & > = 07 ( F & ) ; ( Y &U, & ) P &= ( L & ) J Y CU"), , (F,),*(y,,u,)p,

-

( L & ) u ( Y & , uE &W ) u , ) + J ( u , - u*)Iu,

- u*I:p-27

(2.14)

where J : U + U* is the duality mapping of the space U. By virtue of Lemma 2.1, we may view (2.14) as an approximating optimality system for problem (P). If one could obtain from (2.14) sufficiently sharp a priori estimates on p , , one might pass to limit in (2.14) to obtain a system of first order optimality conditions for problem (6). We shall see that this is possible in most of the important cases, but let us see first how this scheme looks for problem (P) considered in the previous section. We shall assume that g and h satisfy assumptions (9, (ii) (with the eventual exception of condition (2.511, B E L ( U , V r ) , f~ V r , and the injection of V into H is compact. Let ( y * ,u * ) be an optimal pair in problem (PI. Then, we associate with this pair the penalized optimal control problem (adapted penalty):

Here, g" is defined by (see formula (2.79) in Chapter 2)

where P,u = Cy,, u,e,, u = Cy=, u,e,, A,,T = Cy=,u,e,, {e;} an orthonorma1 basis in H , and cp": V + R is a family of convex functions of class C2 on V such that cp"(u) 2 - C ( llull

lim cp"(u) = cp(u)

E-0

+ 1) Vu E V

Vu E V , E

> 0,

156

3. Controlled Elliptic Variational Inequalities

whilst, for any weakly convergent sequence u,

+

u

in V ,

lim inf cp"( u,) 2 cp( u ) . E'O

By Proposition 2.1, we know that problem (P,) has at least one solution ( y , , u,). Since our conditions on cpE clearly imply assumptions (i)', (ii)', (iii)' of Lemma 2.1, where X = H, Y = V ' , and F,(Y,U) =AY L & ( Y ,u ) we conclude that, for

E +

u,

+

y,

+ y*

+ Vcp"(Y> - B u -f,

=g"(y)

+ h(u),

0,

strongly in U , strongly in H, weakly in V .

u*

On the other hand, the optimality system for problem (P,) is (see (2.14)) AY, + Vcp"(Y , ) -

=

Bu, + f,

A*P&- V*cp"(Y&)P& = Vg"(y,), B*p, E d h ( u , ) + U, - u*.

(2.16) (2.17) (2.18)

Note that since V*cp"(y,) E L(V,V ' ) is a positive operator, Eq. (2.17) has a unique solution p , E V. Moreover, since {Vg"(y,)}is bounded on H (because g is locally Lipschitz), we have the estimate wllp,ll IIVg"(y,)l Hence, on a subsequence

E, +

P&+P A*p, + A*p Vg"(y,)

-3

5

c

v.5 > 0.

0, we have weakly in V , strongly in H, weakly in V', weakly in H.

By Proposition 2.15 in Chapter 2 we know that 6 E dg(y*),where dg is the generalized gradient of g whilst by (2.18) it follows that B*p E dh(u*) or, equivalently, u* E dh*(B*p). Summarizing, we have proved: Proposition 2.2. Let ( y * ,u * ) be an optimalpair orproblem (PI. Then there is p E V such that

-A*p - 77 E d g ( y * ) , u* E dh*( B * p ) ,

(2.19)

3.2. Optimal Control of Elliptic Variational Inequalities

where in V.

=

w - lim,+ oV2cp"(y,)p, in V ' , p ,

+

p , and y,

+

157

y * weakly

We may view (2.19) as the optimality system of problem (P). There is a large variety of possibilities in choosing the approximating family {cp"). One of these is to take cpE of the form (2.151, i.e.,

where n

=

[ E - ' I , P,, : V -,X,,An: R"

+

V and

(2.21)

or

if cp is lower semicontinuous on H (this happens, for instance, if cp is coercive on V ) . One of the interesting features of the adapted penalty problem (P,) is that every sequence u, of corresponding optimal controllers is strongly convergent to u*. If instead of (P,) we consider the problem

(P")

min{g"(y)

+ h( u ) ; Ay + Vcp"(y )

= Bu

+f

u E U},

then under assumptions (i)-(iii), (P") admits at least one optimal pair ( y E ,u E )and, on a sequence E, + 0, uE* + u: yen

+yf

weakly in H, strongly in V ,

where (yf ,u f ) is an optimal pair of (PI. The result remains true if replace h by a smooth approximation, for instance, h, as in (2.22), and one might try to calculate u s by a gradient type algorithm taking in account that the differential &$ of the function 4: u + R,

4(u) =g"((A

+ Vcp")-'(Bu +f)) + h , ( u ) ,

u E

U,

158

3. Controlled Elliptic Variational Inequalities

is given by d + ( u ) = dh,(u) - B*p,

where -A*p - V'cps( y ) p Ay

=

Vg"( u),

+ V p E (y ) = Bu + f .

Now, we shall present another approach to problem (PI, apparently different from the scheme developed in the preceding, but keeping similar features. To this end, we shall assume that, beside the assumptions (i) and (ii), the injection of V into H is compact, cp is lower semicontinuous on H, B E L(U, H I , f E H, and hypothesis (1.14) holds. Then, as seen earlier, the solution y to variational inequality (2.1) belongs to D ( A , ) , i.e., Ay E H. Let (y*, u * ) E D ( A , ) X U be an optimal pair for problem (P). We associate with (y*, u * ) the family of optimization problems

(Q,)

inf(g(y) + h ( u ) + e-l(cp(y) + cp*(u)

+ +lu - U * l 2 + 31.

-

(Y,.))

- u*Ih},

where the infimum is taken on the set of all (y, u , u ) E V subject to

X

U

X

H,

Ay=Bu-u+f. Here, u* = Bu* gate of cp.

+ f - Ay* E dcp(y*) c H

and cp*: H

+

R

is the conju-

Lemma 2.2. Problem Q, has at least one solution (y,, u,, u,) H.

E

V

X

U

X

(2.23) Since {u,} is bounded in H and {y,) is bounded in V,we may pass to limit in the previous inequality to get the existence of a minimum point. W

3.2. Optimal Control of Elliptic Variational Inequalities

Lemma 2.3. For

E +

159

0, u,

-,u*

strongly in U ,

u,

-+

u*

strongly in H ,

y,

+

y*

strongly in I/.

(2.24)

Proof: We have

+ E-'(cp(Y,> + cp*(V,)

g(Y€) + N u , )

+~(IU,

-

u * Iu2

-

(Y&lU&))

+ Iu, - u*I2) _< g ( y * ) + ' h ( u * )

VE > 0, (2.25)

because (see Proposition 2.5 in Chapter 2) cp(Y*) + 'p*(v*)= ( Y * , . * ) . Note also that

cpo(Y,)

+ cp*(.,)

Hence, on a subsequence

E, +

V

weakly in H,

-+

v.2 > 0.

2 0

(2.26)

0, we have

weakly in U ,

yen + j j E,,

(Y,,.,)

uEn+ ii u,.

NOW,letting

-

strongly in H, weakly in I/.

(2.27)

tend to zero in (2.25) and using (2.26) and (2.27) we get

Ig(y*)

+ h ( u * ) = inf P,

(2.28)

respectively, cp(Y)

+ cp*(V)

- (J,V) =

0,

because cp and cp* are weakly lower semicontinuous and

2 ( f , A J ) - (jj,Bii

+f)

=

-(J,V).

Then, by Proposition 2.5, Chapter 2, we infer that V E dcp(jj), and so by (2.28) we get (2.24).

160

3. Controlled Elliptic Variational Inequalities

Lemma 2.4. Let ( y e ,u,, u,) be optimal for problem p,. Then there is p , E V that satisfies the system

Here, dcp: V

+

-A*P, v,

E

B*p,

E

M Y & )+ &-‘(dcp(Y€) M Y , - EP, + E(V* -

(2.29)

dh(u,)

(2.31)

+ U, - u*.

(2.30)

7I),.

V’ is the subdifferential of cp: V + R.

Proof: Let us denote by L,(y, u , u ) the function L A ( y ? u , u )= g ^ ( y ) + h ( u ) + & - ‘ ( P A ( y )

+ cp*(v)

+ +(Iu - U * l 2 + Iu - U * I h ) + +(Iu - u,IU2 + IU - L J , ~ ~ ) ,

where g ” is defined by (2.15) and problem

(ph

(y,.))

A > 0,

by (2.22). Clearly, for every A > 0 the

inf(L,(y,u,u);u~U,u~H,Ay=Bu-u+f}

(2.32)

has a solution ( y A u, ” , u ” ) .We have Lh(YA7UA,Uh)

< g ” ( Y c ) + h ( u ~ +) YE) + q * ( u . )-

+ +(I.,

-

U*l2

+ Iu,

- U*lh)

(Y~7’e))

VA > 0.

Since {u”}),{u”}are bounded in U and H , respectively, we may assume that U” +

ii

weakly in U

u”

+

V

weakly in H

yA

+

7

weakly in V , strongly in H.

Then we have g ” ( y ” )+ g ( y ) , liminf cpA(yA)2 c p ( j j ) , A-0

liminf cp*(u”) 2 c p * ( ~ ) , A-rO

liminf h ( u ” ) 2 h ( E ) A+O

161

3.2. Optimal Control of Elliptic Variational Inequalities

This yields

+ &-I( <

Hence,

+ cp*(cv)

cp( J )

-

( 1 , V ) ) + +(I.

- U*l2

+ l i i - .*I;)

inf Q,.

1=y,, ii = u,, V = uE and for A

+

0

U”

-+

u,

strongly in U ,

u”

-+

u,

strongly in H,

Y ” -+YE

strongly in I/.

(2.33)

Now, since problem (2.32)is smooth one can easily find the corresponding optimality conditions. Indeed, we have

(vgA(yA),z)+ h r ( U h , W )

+ &-‘((vVA(YA),z) + ( c p * ) r ( u Ae, ) - ( 2 , ~ ” ) ( y A ,e ) ) + (d- u * , e )

+ ( u ” - u * , w ) + ( u A- U & , W ) 2 0

for all w

E

(2.34)

U, 8 E H and z satisfying the equation A Z = B W - ~ .

Now, let p ”

E I/

be the solution to

-A*p”

=

VgA(yA)+ &-‘(VpA(y”)

- u”).

(2.35)

Substituting (2.35)in (2.34), we get

+

+

- ( B * ~ ~ , w ) @ , P A ) h r ( u A , w )+ d ( ( c p * ) ’ ( d , e ) - (y^,e))

e)

+(dfor all w

E

U, 0

E

+ (u^

-

u*, e )

+ (u” - u * , w ) + (u” - u , , w ) 2 0

H. This yields

B*p” E dh(u”) + 2 ~ -” U, - u*,

(2.36)

-PA

(2.37)

E

& - * ( d c p * ( u ” )- y A )

+ 2uA - u* - u,.

We may equivalently write (2.37)as lJA E

acp(yA - &PA

+ & ( U * + U& - 2u”)).

(2.38)

162

3. Controlled Elliptic Variational inequalities

Then, substituting in (2.35) and multiplying the resulting equation by ( I + A dp)F'yA - yA+ & p A- E(U* + u, - 2q), we get the estimate VA

IlpAll IC

> 0,

because d p is monotone and (Vg*} is uniformly bounded on bounded subsets. Hence on a subsequence, A + 0, we have weakly in V , weakly in V ' ,

P A+P& A* p A+ A*p,

whilst by Propositions 2.15 and 1.3, in Chapter 2, VgA(yA)+ 6 E dg(y,)

weakly in H ,

7 E dp( y,)

weakly in V'

and VpA(y ^)

+

because {VpA(yA)) is bounded in V' and y A (2.36) it follows that B*p, E d h ( u , )

+ y,

strongly in V. Finally, by

+ U, - u*,

and V& E dcp(Y, - E P ,

+

E(U*

-

.&>>

thereby completing the proof. By virtue of Lemma 2.4 we may view (2.29H2.31) as an approximating optimality system for problem (P). Let 6, = u* - us. Then, we may rewrite (2.29) as -A*P,

E

MY,)

+ E-'(dv(Y&)

-

MY,

- E P , + &O&)>* (2.39)

If multiply this equation by p , - 0, and use the monotonicity of dcp along with the coercivity condition (2.21, we get 11p,11 5

because (Vg(yJ is bounded. Hence, there is a sequence

E, +

+

U," + u* +

u*

E

VE

> 0,

0 such that

weakly in V , strongly in H , weakly in V ' , strongly in U ,

P&, + P A*P&" A*P

%

c

dcp(y*)

strongly in H .

3.2. Optimal Control of Elliptic Variational Inequalities

Then, letting

E = E,

163

tend to zero in (2.29)-(2.31), we get -A*p B*p

E E

d g ( y * ) + w*, dh(u*),

(2.40)

where

in the weak topology of V ' . These conditions can be made more explicit in the specific problems that will be studied in what follows. 3.2.3. Optimal Control of Semilinear Elliptic Equations In this section, we shall study the following particular case of problem (P): Minimizeg(y) + h ( u ) o n a l l y subject to A,y

E H,'(R)

n H 2 ( R )andu

+ p ( y ) = f + Bu

E

U , (2.42)

in R , in dR.

y = o

(2.43)

Here, A , is the elliptic differential operator (1.211, i.e., N

AOY

=

-

C

i,j = 1

where a i j ,a,, E L"(R), a i j = aji for all i , j N

C

i,j=l

(2.44)

(aij(x)yx,)x,+ a , ( x ) ~ ,

a i j ( x ) t i t j2 W I I ~ I I ~

=

1,. . ., N , a , 2 0 a.e. in R,

~tE R

~ , a.e. x

E

a,

for some w > 0, p is a monotonically increasing continuous function on R, f E L2(R), and B E L(U, L2(R)). We shall assume that the functions g and h satisfy assumptions (i) and (ii) of problem (P). More precisely, we shall assume: (a) The function g : L2(R) + R is Lipschitz on bounded subsets and there are C E R and f,, E L2(R) such that

d Y )

2 (f0,Y) +

c

VY E L 2 ( R ) *

(2.45)

164

3. Controlled Elliptic Variational Inequalities

(b) The function h: U -+ R is convex, lower semicontinuous, and satisfies condition (2.5). Problem (2.421, (2.43) is a particular case of problem (P), where H L2(R), V = H,'(R), A : V -+ V' is defined by (1.191, i.e.,

=

( 2.46)

and ~ ( y =)

1j ( y ( x ) ) du, n

y

E

H,'(R);

dj

=

p, j : R

-+

R . (2.47)

The function g : L2(fl) -+ R may arise as an integral functional of the form (2.48) where g o : R x R -+ R is measurable in x and Lipschitz in y . We shall assume that R is a bounded and open subset of R N , either with smooth boundary (of class C2 for instance) or convex. As seen earlier, this implies that for every u E U, problem (2.43) has a unique solution y E H,'(R) n H2(R). Moreover, if h satisfies condition (2.9, then problem (2.42) has at least one optimal pair ( y , u). Theorem 2.1 following is a maximum principle type result for this problem.

Theorem 2.1. Let ( y * ,u * ) be any optimal pair in problem (2.42), (2.431, where p is a monotonically increasing locally Lipschitz function on R, and g , h satisjj assumptions (a), ( b ) . Then there exist functions p E H,'(R), q E L'(R), and 6 E L 2 ( R )such that A p E (L"(R))* and

-(Ap), T(X) EP(X)

If either 1 I N

-

q

=

6

a.e. in R ,

W y * ( x > > , 6(x) E d g ( y ( x ) ) B*p E dh(u*). I 3 or

(2.49) a*e*x

E

a,

(2.50) (2.51)

p satisfies the condition

P ' ( r ) IC ( l p ( r ) l + Irl

+ 1)

a.e. r

E

R,

(2.52)

3.2. Optimal Control of Elliptic Variational Inequalities

then Ap

E L’(R)

165

and Eq. (2.49) becomes -Ap - 77

=

6

a.e. in 0.

(2.53)

Here, d p and dg are the Clarke generalized gradients of p and g, respectively (see Section 2.3 in Chapter 21, (L”(R))* is the dual space of Lm(R)and (Ap), is the absolutely continuous part of Ap E (L”(R))*. We can apply the Lebesgue decomposition theorem to the elements p of the space (,!,“(a))*: p = pa+ ps,where pa E L’(R) is the absolutely continuous part of p and p, the singular part (see, e.g., Ioffe and Levin [l]). This means that there exists an increasing sequence of measurable sets R, C R such that m(R \ R , ) k 2 m 0 and p,(cp) = 0 for all cp E LYR) having support in R,. Thus, (2.49) should be understood in the following sense: There exists a singular measure vs E (L”(R))* such that Ap = v, - 77 - 6 , where 77 E L’(R) and 6 E L2(R) satisfy Eq. (2.50).

Proof of Theorem 2.1. We shall use the approach described in Section 2.2 by approximating problem (2.41) by a family of problems of the form (Qe). Namely, we shall consider the approximating problem: Minimize

Ay

=

BU

-

v

+f.

(2.55)

Here, u* = Bu* + f - Ay* E dcp(y*) = p(y*) a.e. in R and dcp c L2(R) X L2(R) is the subdifferential of cp given by (2.471, as a 1.s.c. convex function from L ~ ( R to ) R. Similarly, cp*(u) =sup{(u,p) - cp(u); UEL2(R))

=

jR j * ( u ( x ) ) dx,

UEL*(R).

(Throughout this section [ - I 2 denotes L2 norm and the L2 scalar product.) Without any loss of generality, we may assume that p(0) = 0. ( - , a )

166

3. Controlled Elliptic Variational Inequalities

As seen in Lemmas 2.2 and 2.3, problem (2.54) has at least one solution ( y , , u,, u,) E (H,'(R) n H2(R)) x U x L2(R) and, for E -+ 0, u,

+

u*

strongly in U ,

u,

+

u*

strongly in L ~a), ( stronglyin H,'(R) n H 2 ( a ) .

y, + y *

(2.56)

Now, by Lemma 2.4 it follows that there are p, E H,'(Rn) 17 H2(R) and

& E L2(R) such that -AP,

=

u,

=

(&(XI

B*p,

E

5,

+

&-I(

P ( Y & ) - v,)

+

P ( y , - ~ p , E ( U * - u,)) dg(y,(x)) dh(u,) + U, - u*.

a.e. in R, a.e.in R,

(2.57) (2.58)

a.e. in R ,

(2.59) (2.60)

To pass to limit in system (2.57)-(2.60), we need some a priori estimates on p,. These come down to multiplying Eq. (2.57) by p, and integrating on Q. We have

2 -115,1121IP,112

-

(I1 P(y,)ll2 + Ilu,ll2)llu*

-

v,l12

V€ > 0.

Since { &} is bounded in L2(R>(because g is locally Lipschitz), we have Ilp,llH:(n, Ic

VE

> 0,

(2.61)

and so we may suppose that p,

Ap,

+p

weakly in H,' ( R),strongly in L2(0),

+ Ap

weakly in H-'

Extracting a further subsequence P,,(X)

.,Ax)

(a).

E,, +

P(X) - u * ( x ) -+ 0 +

y,,(x) + y * ( x )

(2.62)

0, we can assume by (2.56) that ax. x

E

a,

a.e. x

E

R,

a.e. x E R.

(2.63)

3.2. Optimal Control of Elliptic Variational Inequalities

167

Now, by the Egorov theorem, for every S > 0 there is a measurable subset E, c R such that rn(E,) I 6 and the sequences p,,, u,. - u*, y,, are bounded in Lm(R\ E,) and uniformly convergent on R \ E,. Now multiply Eq. (2.57) by LA(p , ) where 5, is a smooth approximation of signum function, i.e., for A > 0, for

ldr)

r 2 A,

for - A < r < A,

=

-1

for

r

I -A.

We have

and, therefore,

:1 P(Y.s) < :/

~5-l

-

&-'

-

u ~ ) l , ( -p ~E ( U *

P(Y,)

+115,11L~(n,

-

- .&>)

u . ) ( lA(p&- &(.*

- u . ) ) - lA(pe))

V& > 0.

Now, let S > 0 be arbitrary and let E8 c R be such that rn(E,) I 6 and the convergences in (2.63) are uniform on R \ E,. Then, letting E = E, tend to zero in the previous inequality, we see that

<

c + 2 limsup E ; ~ E" +

0

IP(y,,)

- v,,l

h,

and this yields

IP(y,,) - v,,lh,

(2.64)

3. Controlled Elliptic Variational Inequalities

168

where C is independent of 6 (for 6 sufficiently small). On the other hand, since ( / 3 ( y E n )- uEn}is bounded in L2(R), we have

L

I P ( Y & , ) - U&,I du

Ic6"2

where C is independent of n and 6. Then, for 6 =

E,)?,

(2.64) yields (2.65)

from which (2.66) Since

is bounded in L 2 ( f l ) ,we may assume that weakly in L2(R)

ten+ 6

and, by Proposition 2.13 in Chapter 2, 6 E d g ( y * ) . Similarly, letting E, tend to zero in (2.60) we get (2.51). By the estimates (2.65) and (2.66), it follows that there exists a generalized subsequence of &, say &, such that weak star in ( Lw(a))*,

ApEA+ p,

1

-(

P ( y e A )- u,)

-+

7

weakstar in (L"(R))*,

(2.67)

&A

where p

= Ap

on LYR) n H,'(R) and in particular on C,"(R). We have -p, = 7

+ 4.

(2.68)

On the other hand, by the mean formula (Corollary 2.1, Chapter 21, we have 1 -(P(Y&,)

-

&A

=

e,(P~A - ( .*

- ueA))

R,

E @ ( z , ) , ycAI zA 5 yEA- &,(pEA - u* + uJ). Now since 0, and P(yEA)- uE,>are uniformly bounded on R \ E,, we may assume that

where 0, &;I(

OEA+ 0

weak star in Lw(R \ E,)

(2.69)

and 1

-( &A

P(yJ

- u,)

+

Op

weak star in Lm(R\ E , ) .

(2.70)

169

3.2. Optimal Control of Elliptic Variational Inequalities

By (2.69) we have

This yields ( P o is the directional derivative of p )

Hence, V w E R, a.e. x E

O(x)w I po(y*(x),w)

R

E,,

and therefore O ( x ) E d p ( y * ( x ) ) a.e. x E R \ E,. We have therefore proved that 77 E Lm(R\ E,) and ~ ( x E) d p ( y * ( x ) ) p ( x ) a.e. x E R \ E,. Then, by (2.68), we see that the restriction of p to R \ E, belongs to LYR \ E , ) and a.e. x

pa(x ) = p( x ) E - d p ( y * ( x ) ) p ( x ) - (( x )

E

R

\

E, .

Since 6 is arbitrary, we conclude that -&Ax)

E

dP(Y*(X))P(X)

a*e*xE

+ ((x)

a,

as claimed. Suppose now that 1 I N I 3. Then, by the Sobolev imbedding theorem, H2(R) c C ( a ) a n d so y, are uniformly bounded on On the other hand, we have, by (2.57),

a.

4EP,)

=

-&,

-

P(Y&) +

LL

and, since { p ( y , ) - u,) is bounded in L2(R), we infer that { ~ p , ), , is bounded in H2(R)and therefore in C(a). Finally, we have y, -

EP&

+ E(U*

-

U&)

E p-l(v,),

y* E p - ' ( u * ) .

Subtracting and multiplying by u, - u*, we get EIU*

- U&l I ly,l

+ elp,l

a.e. in R

170

3. Controlled Elliptic Variational Inequalities

and, since p is locally Lipschitz, this implies that IAp,l

I15,l

+ L(lp,l + Iv,

-

.*I)

a.e. x

E

a.

Hence, {Ap,} is bounded in L2(R) and we conclude, therefore, that p E H,'(R) n H2(R), 7 E L2(R), and Ap = (Ap*),, as claimed. Assume now that condition (2.52) holds. We shall prove that {q, = ( 1 / ~ )p(y,) ( - u,)} is weakly compact in L'(R). We have

3.2. Optimal Control of Elliptic Variational Inequalities

171

One might obtain the same result if one uses the approach described in Proposition 2.2. Namely, we approximate problem (2.42), (2.43) by the following family of optimal control problems: Minirnizeg"(y)

+ h ( u ) + $ I U - U*I;

o n d ( y , u ) E (H,'(R) nH2(a))x U ,

(2.72)

subject to Ay where A y fined by

= A,y

+ p " ( y ) = Bu + f

in R ,

(2.73)

with D ( A ) = H,'(R) n H2(R), and p"

E

CYR) is de-

Here, p,(r) = ~ - ' ( r (1 + & p ) - ' r ) , r E R,and p is a C: mollifier in R, i.e., p E C"(R), p ( r ) = 0 for Irl > 1, p ( r ) = p ( - r ) V r E R, / p ( t ) dt = 1. Note that p is monotonically increasing, Lipschitz, and

I P E ( r ) - p,(r)l

I2 8

Vr

E

R.

(2.75)

We are in the situation described in Section 2.2, where

Throughout the sequel, we set B E = ( BE)'. Let ( y &u,) , be optimal for problem (2.72). Then we have u,

-+

u*

strongly in H i ( R ) , weakly in H 2 ( R ) ,

Y, +Y* p"(y,)

+

strongly in U,

p(y*)

weaklyin L 2 ( R ) .

(2.76)

Indeed, from the inequality g"( y,)

+ h( u,) + +IU" - u*l;

Ig"(

y"')

+ h( u * )

172

3. Controlled Elliptic Variational Inequalities

( y " is the solution to Eq. (2.7311, we see that {u,) is bounded in U. Hence, on a subsequence E,, + 0, we have

u,"

+

weakly in U .

U

On the other hand, we may write Eq. (2.73) as

AY, + P & ( Y & )= Bu, + f + P & ( Y & ) - P E ( Y & ) and so, by Proposition 1.2, y,, + j j

strongly in H ' ( 0 ) and weakly in

H2(a), where j j is the solution to (2.43), where u = U. This yields g ( j j ) + h ( ~+) limsup +1uEn- u*Ic I g ( y * ) + h ( u * ) = inf(P). En -+

0

Hence, us" -+ u* strongly in U and j = y * , U = u*. The sequence being arbitrary, we have (2.76). Moreover, there is p , E H,'(R) n H2(a) such that

-4, - B"(Y,>P,

=

Vg"(y,)

B*p,

E

dh(u,)

in

a,

E,,

(2.77)

+ U , - u*.

(2.78)

Now, multiplying Eq. (2.77) by p , and sign p,, we get the estimate

Ilp,ll2Hd(n, +

/-I B"(Y,)P,l dx I c

Hence on a subsequence, again denoted

p, Vg"(y,)

+p +

5

we have

weakly in H; ( a),strongly in

( a),

weakly in L 2 ( a ) ,

and, on a generalized subsequence

be^( y,,)p,,

E,

v.5 > 0.

+

v

{EJ,

weak star in ( L"( a))*.

Hence, -Ap - q = 5 E d g ( y * ) in a. Now, by the Egorov theorem, for each S > 0 there exists a measurable such that m ( E , ) I 6 and y * , p E Lm(a \ E,) and subset E, of

y,( x )

+ y*( x ) ,

p,( x )

Since {( BE(y,)) is bounded in subsequence

b " ( y , ) +f,

+p ( x

)

uniformly on

Lm(a \ E,), weak star in

\

E,.

we may assume that, on a

Lm(a \ E6).

3.2. Optimal Control of Elliptic Variational Inequalities

173

Then, by Lemma 2.5 following, we infer that f s ( x ) E d p ( y * ( x ) ) a.e. x E Cl \ E, and so 77,(x) = f , ( x ) p ( x > E

ap(Y*(x)>P(x>

a*e*x

E

\E,*

The last part of Theorem 2.1 follows as in the previous proof since { b E ( y E ) }is bounded in Lm(CL)if 1 I N < 3 (because I b”(y,)I I 1 p’(y,)I < C in a ) and is weakly compact in L’(R) if p satisfies condition (2.52). The main ingredient to pass to limit in the approximating optimality system is Lemma 2.5 following, which has intrinsic interest.

Lemma 2.5. Let X be a locally compact space and let u be a positive measure on X such that u( X ) < 03. Let yE E L’( X ; u ) be such that y, + y strong& in L ’ ( X ; u ) and @“(y,) + f o

weaklyin L’(x; u ) .

Then fo(x)E dp(y(x))

Prooj

a.e. x E X .

On a subsequence, again denoted ye, we have y,(x) + y ( x )

v x E X \A,

u ( A ) = 0.

On the other hand, by Mazur’s theorem there is a sequence o, of convex combinations of { b E ( y E )such ) that on+ fo

strongly in L’( X ; u ) ,

where w,(x) = X i a,!, pEc(yi(x)).Here, I, is a finite set of positive n = 1. Hence, there is a integers in [n,+ M[, yi =y,, and a: 2 0, C i E I a,!, subsequence, again denoted o,,such that o,(x) -+ f o ( x ) V x E Cl \ B , u ( B ) = 0. In formula (2.74) we make the substitution t = r - c28 to get ps(r)= - & - z / p , ( t ) p ( y ) d t - /pE(-Eze)p(e)d0

and this yields b E ( r )= . ~ - ‘ / ~ + “ ~ P r, -( tt ) p ~ ( ~ ) d t r- c2 = --E-~

/ b E ( r - & ) p f ( 0 ) do.

174

3. Controlled Elliptic Variational Inequalities

where 8; 2 0, C ~ 8;L= ~1. Hence,

B

Y;( x 1)

(2.79) where

On the other hand, we have

Hence, l(e;e;)-'((i

+ Eip)-'Yi

- (1

+ Eip)-l(yi

-

&;e;)> - I ( I

cEi

Because be, uniformly bounded on every bounded subset ( p is locally Lipschitz). Then, by (2.791, it follows that

bet(y i ( x ) )

c 6;q; m,

=

k= 1

+'-yj,

where

'y; +

0 as i

+ 03.

(2.80)

3.2. Optimal Control of Elliptic Variational Inequalities

175

On the other hand, 8; - yi + 0 uniformly with respect to k, and by (2.80) we have bEi(yi(x))h _<

m,

k= 1

8 ; P 0 ( ~ ; , h+) -yih

Vh

E

R.

Hence, limsupbEi(yi(x))h _< P o ( y i ( x ) , h )

V h E R.

i-rm

Finally,

It should be said that the methods of Section 2.2 are applicable to more general problems of the form (2.42), for instance, for the optimal control problem: Minirnizeg(y)

+ h ( u ) on all ( y , u ) E ( w ~ , R) P (n W2vP(R)) x U , 1 IP < 03,

(2.81)

subject to -Ay

+ P ( y ) = f + Bu y=o

a.e. in 0, in a R ,

(2.82)

where f E LP(R), B E L(U, LP(R)), g locally Lipschitz on LP(R), h is convex and 1.s.c. on U, and P is monotone and locally Lipschitz. One gets a result of the type of Theorem 2.1 by using the approximating control process (2.721, i.e.:

-Ay

+ p"(y)

= f + Bu

y=o

in R, in dR

(2.83)

176

3. Controlled Elliptic Variational Inequalities

Then one writes the optimality system and passes to limit as in the proof of Theorem 2.1. 3.2.4. Optimal Control of the Obstacle Problem We shall study here the following problem: Minimizeg(y) + h ( u ) o n a l l ( y , u )E ( H ~ ( R n ) H2(R))X U , y E K , ( 2.84) subject to a ( y , y - z ) I( f where a: H,'(R) set (1.24).

X

H,'(R)

+

+ Bu,y

-z)

Vz

E

(2.85)

K,

R is defined by (1.19) and K is the convex

As seen in the preceding, (2.85) is equivalent to the obstacle problem (A,y -f

-

B u ) ( y - @)

=

a.e. in R ,

0

A,y - f - Bu 2 0 , y 2 @

a.e. in

a,

in dR.

y=O

(2.86)

Here, B E L(U, L2(fl)), f E L 2 ( f l ) ,and g : L 2 ( R )+ R, h: U assumptions (a) and (b) in Section 2.3.

+

R satisfy

Theorem 2.2. Let ( y * ,u * ) be an optimal pair for problem (2.84). Then there exist p E H,'(fl) with A p E (L"(R))*and 6 E L2(R) such that $, E dg(y*) and

+

a.e. in [ x; y*( x ) > @( x ) ] , (Ap), 6 = 0 p ( A y * - Bu* - f ) = 0 a.e. in R , a ( p , X(Y* -

@)I + ( 6 7 ( Y * - @ ) x )= 0 B*p

E

dh(u*),

a ( P , P ) + ( 6 , P ) I 0. Zf 1 I N

I3,

vx E C ' ( h ) ,

(2.87) (2.88) (2.89) (2.90) (2.91)

then Eq. (2.87) reduces to ( A p + 6 ) ( y * - @)

=

Oin R .

(2.92)

177

3.2. Optimal Control of Elliptic Variational Inequalities

We have denoted by A the operator A, with the domain D ( A ) = H,'(R) n H 2 ( C l ) and by ( A p ) , the absolutely continuous part of A*p. If N I3, then y* E H2(R) c C(n) and so A p ( y * - #) is well-defined as an element of (L"(R))*. In particular, (2.92) implies that Ap* = in [ x ; y * ( x ) > +(x)l. The system (2.871, (2.89) represents a quasivariational inequality of elliptic type.

Proof of Theorem 2.2. Consider the penalized problem (2.93) subject to Ay

+ P E ( Y - S)

=

Bu

(2.94)

+f,

where

In other'words, p" is defined by formula (2.741, where p is the graph (1.35). By Proposition 2.1, problem (2.93) has at least one optimal pair (u,,y , ) E ux ( ~ ~ n ( H~(R)). 0 ) Arguing as in the proof of Theorem 2.1 (see problem (2.72)), it follows by Proposition 1.2 that

u,

+

u*

stronglyin H,'(R),weaklyin H 2 ( R ) ,

y, + y *

p " ( y , - 9)

strongly in U ,

+ Bu* - Ay*

weakly in L 2 ( n ) .

+f

Moreover, we have for (2.93) the optimality system (see (2.771, (2.78)) -AP, - P ( Y & - S I P , B*p,

E

=

W(Y,)

+

d h ( ~ , ) U, - u*.

in

(2.96) (2.97)

3. Controlled Elliptic Variational Inequalities

178

Then multiplying Eq. (2.96) first by p, and then by sign p , , we get the estimate

c.

IcIIvg"(y,)lL2(n) I

Hence, there is a subsequence, again denoted P, Vg"(y,) Letting

E

+

-+

such that

P

weakly in H i ( R),strongly in L2(R ) ,

6 E dg(y*)

weakly in L 2 ( R ) .

tend to zero in (2.97), we get B*p

Now, let bY

E,

(2.98)

&: R

.+

R and 7,:R

On the other hand, we have

--j

E

dh(u*).

R be the measurable functions defined

179

3.2. Optimal Control of Elliptic Variational Inequalities

This yields IP&P"(Y&-

$11 I EIP&PE(Y&- $ W - ' l Y & - $16, +E-'ly& -

+ 2clp,l

$177,)

a.e. in R . (2.101)

We note that p " ( y , - $)q& = &-'(ye - $177, + C q , remain in a bounded subset of L2(R), whilst by the definition of 6, wee see that ~-'ly~(x )$(x)l&.(x)

a.e. x

IE

E

R.

Since ( p , P " ( y , - #)) is bounded in L'(R), it follows by (2.99) and (2.101) that, for some subsequence E,, + 0, p & , ( x ) p E f l ( Y & , ( x-) $ ( x ) )

+

0

ax. x

E

a,

(2.102)

whilst p,, p " n ( y E n- $)

+p ( f

+ Bu* - A y * )

weakly in L'( R).

Together with (2.102) and the Egorov theorem, the latter yields a.e.inR,

p(f+Bu* - A y * ) = O and therefore

stronglyin L'(S1). (2.103)

p E a p e n ( y e n- $) -+p(f+ Bu* - A y * )

Then, by (2.1001, we see that ( y E n- $ ) P " n ( y , , - $ ) p , ,

Inasmuch as I p " ( y , - $) (2.103), and (2.104) that

+

0

+ &-'(y8 - $)-I

(Y&n- $)+ P Y Y , , - $ ) P & n

+

Since (yen +&JY&"

-

$)+E

0

stronglyin L'(fl). (2.104) I C E , it follows by (2.99),

strongly in

Jw).

Hi(R), applying Green's formula in (2.96) yields

$I+ x ) + ( v g Y Y , , ) , ( Y & " - @ I +x )

-+

0

vx E c l m

Since p,, + p weakly in Hd(R) and (yen - $ ) + + y * - $ strongly in H'(R), we get (2.89). Regarding inequality (2.91), it is an immediate consequence of Eq. (2.96) because we have

(4+, V g & ( Y & ) , P &I) 0

V& > 0-

180

3. Controlled Elliptic Variational Inequalities

Now, selecting a further subsequence, if necessary, we may assume that y,,(x)

+

a.e. x

y*(x)

E

R.

On the other hand, by estimate (2.98) it follows that there is p of E, such that and a generalized subsequence p"~(y,,- $ ) p ,

weakstar in

-+ p

E

(L"(R))*

(,!,"(a))*.

This implies that Ap admits an extension as element of (L"(R))*, and we have -A*p - p

=

5 E dg(y*).

Now, by Egorov's theorem, for every 6 > 0 there is a measurable subset E, of R such that rn(E,) I 6, y* - $ is bounded on R \ E, = R, and

y,"

- $ -+

y*

- $

uniformly on R,.

Then, by (2.104), it follows that p(y* - $)

=

0 in R,, i.e.,

On the other hand, there is an increasing sequence and ps = 0 on L"(Rk). Hence,

rn(n \ Rk)I k-'

(ak)such

that

Thus, ( y * - $ ) p a = 0 a.e. in R,, and letting 6 tend to zero we infer that ( y * - $ ) p a = 0. Hence,

-(Ap), If 1 s N

=

5E

a.e. in [ y * >

dg(y*)

$1.

s 3, then H2(R) c C ( n ) and so y,(x)

-+ y

*(x)

uniformly on

a.

Since $ E H2(R) c C(n), it follows by (2.104) that ( y * - $ ) p

(Y*

-

$)(AP + 5 )

This completes the proof of Theorem 2.2.

=

0.

=

0, i.e.,

181

3.2. Optimal Control of Elliptic Variational Inequalities

Remark 2.1. Theorems 2.1 and 2.2 remain valid if one assumes that ( y * , u * ) is merely local optimal in problem (2.42) (respectively, (2.841, i.e., g(y*) + h(u*)I g ( Y ) + h(u)

for all ( y , u ) satisfying (2.43)) (respectively, (2.85)) and such that lu - u*Iu < r. Indeed, in problem (2.72) (respectively, (2.94)) replace the cost functional by g"(y)

+ h ( u ) + aIu - u*1;,

where a is sufficiently large that Iim sup g"( y,"') &+

0

+ h( u * ) I ar'.

Then Iu, - u*Iu I r for all E > 0 and this implies as before that u, strongly in U.The rest of the proof remains unchanged.

+

u*

Problems of the form (2.84) arise in a large variety of situations, and we now pause briefly to present on such an example. Consider the model, already described in Section 2.2, of an elastic plane membrane clamped along the boundary an, inflated from above by a vertical field of forces with density u and limited from below by a rigid obstacle y = $ ( x ) < 0, Vx E n (see Fig. 1.1). We have a desired shape of the membrane, given by the distribution y = yo(x> of the deflection, and we look for a control parameter u subject to constraints. lu(x)l I p

a.e. x

E

n,

(2.105)

such that the system response y" has a minimum deviation of y o . For instance, we may consider the problem of minimizing the functional / , ( y ( x ) - y0(x))' dw on all ( y , u ) E (H,'(R) n H 2 ( n ) )X L'(fl), subject to control constraint (2.105) and to state equation (2.861, where f = u. This is a problem of the form (2.84), where A, = - A , B = I , U = L'(n), f = 0, and

h(u) =

if lu(x)l I p a.e. x otherwise.

E

n,

182

3. Controlled Elliptic Variational Inequalities

By Proposition 2.1, this problem has at least one solution ( y * , u * ) whilst by Theorem 2.2 such a solution must satisfy the optimality system Ay*

+ u*

=

y*

=

0

+,

y* = O

in R + = ( x

R; y*(x) > + ( x ) } , a.e. in R, = R \ R', A + + u* I0 indR,

Ap =y* -yo

p(u*

+ Ay*) = 0

E

a.e. in R + ,

a.e. in R ,

u* = p sign p

(2.106) p

(2.107)

=

in d R ,

0

(2.108)

a.e. in R .

(2.109)

Assume that IA+I # p a.e. in 0. Then, by Eq. (2.108), we see that p R,. Hence, p is the solution to boundary value problem in R + , A p = y* - y o p=O inR,, p=O

indR.

=

0 in

(2.110)

This system could be solved numerically using an algorithm of the following type: yi 2

+,

( A y , + u i ) ( y i - +) = 0 Ay, + u , I0 a.e. in R

Ap, = y i - y o

ui+

a.e. in 0, y,

=

0

in d R ,

in Ri = ( x E R ; y i ( x ) > + ( x ) } , pi = 0 in d R i , =

p sign p i

a.e. in R i .

Let us assume now that y o E HdCCl) n H2(R> and A y o ( x ) 2 max(A+(x), p ) a.e. x E 0. Then, by Eqs. (2.106), we see that A(y* - y o ) I0 a.e. in R and, therefore, by the maximum principle y* - y o 2 0 in a. Then, assuming that R + is smooth enough, we deduce by (2.110) and by virtue of the maximum principle that p < 0 a.e. in a+,and therefore, by (2.1091, u* = - p

inR+

Hence, y* E H,'(R) n H2(R) satisfies the variational inequality Ay* = p

inat,

Ay* 

y*

=

in R,

y* = O

+

inR, inR+, indR,

(2.111)

183

3.2. Optimal Control of Elliptic Variational Inequalities

from which we may determine R+. For instance, if R = ( 0 , l ) and = - 1, then clearly the solution to (2.111) is convex and so it is of the following form: y * ( x ) = -1

forO
y*(O) = y * ( l ) = 0

-1
<0

forx

E

(0,a) u ( b , l )

=

R+.

Hence for x

E

[a, b),

for x

E

[a,b],

and one determines constants a, b, c, and C , from the continuity conditions y*(a) =y*,(b)

=

-1

(y*)’(a)

=

(y*)\(b)

=

0.

A problem of great interest in the study of a physical system modeled by the obstacle problem is that of controlling the incidence set R, = ( x E R, y ( x ) = $ ( X I } or its boundary dR,. For instance, in the case of the contact problem (1.46) recalled in the preceding a problem of interest would be to find the force field f (viewed as a control parameter) such that the set of all points where the membrane is in contact with the obstacle be as close as possible (in a certain acceptable sense) to a given measurable subset D of 0. The least squares approach to this problem leads us to consider the optimal control problem: Minimize (2.112)

on all ( y , u ) E H,’(R) X U subject to (2.85). Here, ,yo is the characteristic function of D and xy is that of the set R; y ( x ) = $ ( X I } . However, since the function y + xy is not locally Lipschitz on L2(R) we shall replace it by g,(y) = A / ( ( y - $I++ A), A > 0, and so we shall consider the problem:

{x E

184

3. Controlled Elliptic Variational Inequalities

To be more specific, we shall assume that A, is given by h: L2(a> + h(u)

=

{L

=

-A, B

if lu(x)l I p a.e. in otherwise.

= I,

a,

This problem has at least one solution (y, , u,) and since, for A g,(y)

+

x,

f = 0, and

+

0,

strongly in L2(a )

for every y E L 2 ( a ) ,it is readily seen that for A + 0, u, u* weak star in Lm(a), y, + y* weakly in H2(a), where (y*, u * ) is an optimal pair for problem (2.112). On the other hand, the optimality system of (2.87H2.90) is in this case Ay,

+ u, = 0

y, 2 $, Ay,

pIp,I +p, A $

+ u,

I0

=

at = [ y , = $1, a.e. in a.

a.e. in

0

=

$1

a:, a.e. in a ,

a.e. in [y, >

uA = p sign p,

This problem can be treated as in the previous examples. Now we shall study problem (2.84) in the case where g is a continuous for instance, convex function on

~(a),

g ( y ) = IlY -Yollccn,, The subdifferential d g : C ( a ) Chapter 1, Section 1)

a;

+

Y

E

cm.

(2.113)

M ( a ) of g is given by (see Example 4 in

where M , = (xo E Iz(x,)l = Ilzll~~n,}. is a bounded, open, and smooth For simplicity, we shall assume that domain of R3, and A, = -A.

185

3.2. Optimal Control of Elliptic Variational Inequalities

Theorem 2.3. Let ( y * ,u * ) be optimal in problem (2.84) where g is given by and (2.113). Then there existsp E Wdvq(R),1 < q < 3/2, with A p E p E dg(y*) such that

Ap

=

p ( x)( Ay*( x)

p in { x E

R;y * ( x ) > @ ( x ) ) ,

+ Bu*( x ) + f( x)) B*p

E

=

dh(u*).

0

a.e. x

E

R,

Proog Since the proof is essentially the same as that of Theorem 2.2, it will be sketched only. We consider the approximating control problem:

Minimizeg,(y) + h ( u ) + n~ ~ ( x 0 Usatisfy ) ) -Ay

31.

- u*It,

+ p " ( y - @)

subject to having all ( y , u ) E (H,'(R) = Bu

+f

in R .

Here, g , is the usual convex regularization of g ( z ) = llz - y o l l ~ - ( ni.e., ),

Let (y,, u,) be optimal in the preceding problem. Then we have

p"(y,

-

u,

+

Y,

+

Y* @) + Bu*

and there are p,

Ap,

-

E

strongly in U ,

u*

+ by* +f

strongly in H,'(R) n ~

'(n),

weakly in L2(R),

H,'(R) n H'(R) such that

B E ( y , - @ ) p ,= Vg,(y,)

in ~ , B * P E , dh(u,) + u,

-

u*.

Since sup{ll,$ IIL~(n);5 E d g ( y ) } I1, we have

l l v g & ( y & ) l l LI ~ (c~ )

V& > 0.

Then, multiplying the latter equation by sign p , and integrating on R,we get

186

3. Controlled Elliptic Variational Inequalities

Now, let hi E La(R),i = 0,1,2 (see Section 3.21, the problem

has a unique solution 8

E

CY

> 3. Then, as mentioned in Chapter 2

H,'(R) n LYR) and

and, therefore,

Ilp&llw;4(n)Ic where l/q quence,

+ l/a

=

V,F

> 0,

1, i.e., 1 < q < 3/2. Hence, on a generalized subsep,

+

Q&(Y&) j 6 ( y &- $ ) p ,

p

weakly in W , . q (R ) ,

CL

vaguely in M ( Q ,

v

vaguely in

+

-+

~(fi).

We have, therefore, in g'(fl),

Ap

=

v+ p

B*p

E

dh(u*).

Since y, + y * uniformly on R (because H2(R) c C ( 1 ) compactly), we have v

=

0

in { x

E

R; y*(x) > $ ( x ) } .

Then, arguing as in the proof of Theorem 2.2, we get

p(x)(Bu*(x) Finally, letting

E

+ Ay*(x) + f ( x ) ) = 0

a.e. x

tend to zero in the obvious inequality

E

R.

187

3.2. Optimal Control of Elliptic Variational Inequalities

we infer that

Remark 2.2. In applications, the function g : L2(R) + R that occurs in the payoff of problem (P) and subsequent optimal control problems considered here is usually an integral functional of the form

where g o :R x R -+ R is measurable in x and locally Lipschitz in y , whilst condition (i) or (a) requires that go be global Lipschitz in y. However, it turns out that most of the optimality results established here remain valid if instead of (i), one merely assumes that (i)' go 2 0 on R X R, go(.,O) E L'(R), and for each r > 0 there is h, E L ' ( 0 ) such that l g o ( x , y ) - g o ( x , z)l I h,(x)ly - zI

a.e. x

E

a,

for all y , z E R such that lyl, IzI 5 r. (ii)' There exists some positive constants a, C , , C , and p such that

a.e. x

E

E

L'(R)

R,y

E

R.

For the form and the proof of the optimality conditions under these general assumptions on g we refer to author's work [3].

3.2.5. Elliptic Control Problems with Nonlinear Boundary Conditions We will study here the following problem:

(2.114)

188

3. Controlled Elliptic Variational Inequalities

on all ( y , u ) E H2(R) X U, subject to

y - Ay

=

f + Bu

dY

+p(y)3 0 dU

a.e. in R, (2.1 15)

a.e.in d R ,

where B E L(U, L2(R)), f E L2(R), and p c R X R is a maximal monotone graph. The functions g: L2(R) + R and h: U + R satisfy assumptions (a), (b) of Section 2.3. We know by Proposition 2.1 that if h satisfies the coercivity condition (2.5) then this problem admits at least one solution. Let ( y * , u * ) be any optimal pair for problem (2.114). Then, using the standard approach, we associate with (2.114) the adapted penalized problem: Minimize

g"( y )

+ h( u ) + $1.

(2.116)

- u*I21,

on all ( y , u ) E H 2 ( n ) x U, subject to

y - Ay

=

f + Bu

dY

+ P"(Y> 3 0 dU

a.e. in R, (2.117)

a.e. in d R ,

where p" is defined by (2.74). Let (y,*,u,*)be an optimal pair for problem (2.116). Then, by the inequality

g"( y , )

+ h( U & ) + ;Iu& - u*I2 Ig"( y,"') + h( u * )

(y," is the solution to (2.117)), we deduce as in the previous cases that u,

+

u*

y, + y *

strongly in U weaklyin H2(R),stronglyin H ' ( R ) .

Indeed, by Proposition 1.3 if u, + ii weakly in U then y , H2(R), where y" is the solution to (2.115). This yields limsup &+

0

IU, -

u*12 = o

-+

(2.118)

y" weakly in

3.2. Optimal Control of Elliptic Variational Inequalities

189

because lim inf, h(u,) 2 h(E) and g " ( y , ) + g ( y " ) . Now the optimal pair ( y , , u,) satisfies, along with some p , E H2(R),the first order optimality system ~

+ Ap, = Vg"(y,)

-p,

dP&

-+ dV

PYYJP, B*p,

E

=

a.e. in R , a.e. in d R ,

0

+

(2.1 19)

d h ( ~ , ) U, - u*.

Now, multiplying Eq. (2.119) first by p , and then by sign p , , we get the estimate (2.120) Hence, on a subsequence, again denoted

-+

we have

weakly in H'(

P, + P Vg"(y,)

E,

a),

weaklyin L ' ( S Z ) ,

6 E dg(y*)

strongly in L ~ ( R )weakly , in H ' ( R ) , (2.121)

P,+P and, by (2.118), P"(Y,)

+

weakly in L'(

770

where q o ( x ) E p ( y * ( x ) ) a.e. x Note also that, by (2.120),

E

an),

dSZ.

weak star in (L"( all))*,

P e ~ ( y S , ) p s+ , p

(2.122)

where ( E J is a generalized subsequence (directed subset) of ( E } . Now, letting E tend to zero in Eq. (2.1191, we see that p satisfies the system -p A p E dg(y*) a.e. in R , (2.123) dP - + p = o in d R ,

+

dV

B*p

E

(2.124)

dh(u*).

We note that since p E H'(SZ) and A p well-defined by the formula

E

L2(R), d p / d v

E

H-'/'(dR) is

190

3. Controlled Elliptic Variational Inequalities

where yocp is the trace of cp to dR. The boundary condition in (2.123) means of course that dP

-(

dV

cp)

+ p( cp)

4

vcp E L"( an) n ~ l / ~ ( d R ) ,

o

and in particular it makes sense in 9 ' ( d R ) . These equations can be made more explicit in some specific situations.

Theorem 2.4. Let ( y * , u * ) be optimal for problem (2.114), where p is monotonically increasing and locally Lipschitz. Then there are p E H'(R), q E L'(dR), and 6 E L 2 ( R ) such that p - A p E L2(R>, dp/dv E (L"(dR))*and inR, -p+Ap=( ((x) E dg( y*( x)) a.e. x E 0, (2.125)

(

+q =0

T(X) E

a.e. in d a ,

P(Y*(X>)P(X) ax. x B*p E dh(u*).

E

(2.126) (2.127)

dR,

If either 1 I N I 3 or p satisfies condition (2.52) then d p / d v and Eq. (2.126) becomes dP

dV

+ d p ( y * )p

3

0

E L'(dR)

a.e. in d R .

(2.128)

Here, ( d p / d v ) , is the absolutely continuous part of dp/dv. &oo$ By the Egorov theorem, for every 6 > 0 there is E, c dR such that m(E,) I 6, p,, y , are uniformly bounded on dR \ E,, and

y, p,

y* +p +

and, on a subsequence

{E,)

PYY,") +fo

uniformly on dR uniformly on dR

\

\

E6, E,

c {E}, weak star in L'( dR

\

Eo).

Then, by Lemma 2.5, we infer that f o ( x ) E dp(y*(x>>a.e. x and so, by (2.122) (see the proof of Theorem 2.1), pa(x)

E

d p ( y * ( x)) p ( x)

a.e. x

E

dR

\

E, .

E

dR

\

E,

3.2. Optimal Control of Elliptic Variational Inequalities

191

Since 6 is arbitrary, Eq. (2.126) follows. Now, if 1 I N I 3 then H2(R) c C(n), and so ( y J is bounded in C(fi). We may conclude, therefore, that

l p " ( y & ) lI c

vx

E

dR.

This implies that { b"(y,)) is weak star compact in L Y R ) and so 77 = = - d p / d u E LYdR). If condition (2.52) holds then we derive as in the ] proof of Theorem 2.1, via the Dunford-Pettis criterion, that ( b E ( y E ) is weakly compact in L ' ( R ) . Hence p E L'(dR), and the proof of Theorem 2.4 is complete. Now we shall consider the case where p is defined by (1.71). Then, Eq. (2.13) reduces to the Signorini problem

y

-

Ay

=

Bu

+f

a.e. in R , y dY

dY -20,

0,

a.e. in d R , (2.129)

y-=o

dU

2

dU

which models the equilibrium of an elastic body in contact with a rigid supporting body. The control of displacement y is achieved through a distributed field of forces with density Bu. Theorem 2.5. Let ( y * , u * ) E H2(R) X U be an optimal pair for problem (2.114) governed by Signorini system (2.129). Then there exist functions p E H'(R) and 6 E L 2 ( R ) such that d p / d u E (L"(dR))*, 6 E dg(y*), and

-p

+ Ap = 6 B*p

E

a.e. in R,

(2.130)

(2.132)

h( u * ) .

If 1 I N I 3, then y * -JP dU

=o

a.e. in d R .

(2.133)

Prooj The proof is almost identical with that of Theorem 2.2. However, we sketch it for reader's convenience.

192

3. Controlled Elliptic Variational Inequalities

Note that in this case P" is given by formula (2.95). Let A,: dR + R be the measurable functions

A&(X) =

We have

i

0

if y , ( x ) >

- E ~ ,

1

if y , ( x ) 4

--E

2

6,: dR

and

.

I P&( x 1P "( Y , ( x 11I I E I P&( x 1B ( Y , ( x 11I( E - ' I Y&( x ) I 6, ( x 1

+ 2~lp,(x)l

+~-'Iy,(x)A,(x)l)

+R

a.e. x

E

dR.

Since { P"(y,)) is bounded in L2(dR) and P " ( Y & ) A & ( x )= E - ' Y & ( x ) A & ( x ) +

EA&(X) /l

0

we infer that { ~ - l y , A , ) is bounded in L2(dR).Note also that ~-'Iy,l 6, I E a.e. in dR and, since ( b " ( y , ) p , ) is bounded in L'(dR), there is a sequence E,, + 0 such that p,,(x)pEn(yE,(x))

+

0

a.e. x

E

dR.

Then, by (2.120, we conclude that

P&,P Y Y & , )

+

-P

dY *

-= O dV

strongly in L'( an).

Finally, by (2.134) we see that yEnBCn(yEn)pEn +o

strongly in L ' ( ~ R ) .

(2.135)

Now, by the Egorov theorem, for every 6 > 0 there exists E, c dR such that rn(E,) I 8, ye" are uniformly bounded on dR \ E, and yCn+ y*

uniformly on dR

\

E,.

3.2. Optimal Control of Elliptic Variational Inequalities

193

By virtue of (2.fZZl this impfies that y*p

=

in dR

0

\

Es

and, arguing as in the proof of Theorem 2.2, we see that y * p , dR, which along with (2.123) yields (2.131). If 1 I N I 3, then H2(R) c C(n) and y,

+ y*

uniformly on

=

0 a.e. in

a.

Then, by (2.122) and (2.13.9, we deduce that Y*P

=

0,

where y * p is the product of the measure p y * E C(dR). H

E

(L”(dR))* with the function

3.2.6. Control and Observation on the Boundary We consider here the following problem: Minimize

(2.136)

in R ,

Y-AY=f

-+p(y) dY

dv

3

B,u

in

rl,

y

=

0

in

r,,

(2.137)

where p c R X R is maximal monotone, B, E L(U, L2(rl)), f E L2(R), and aR = rl u r,, where rl and I’, are smooth disjoint parts of dR. The functions g, : L2(R) + R and h : U + R satisfy conditions (a), (b) of Section 2.3, whilst g o : rl x R + R is measurable in x , differentiable in y , and

194

3. Controlled Elliptic Variational Inequalities

As seen in Section 1.4, for each u E U, Eq. (2.137) has a unique solution y E H'(R) (if n F2 = 0, then y E H2(fl)).As a matter of fact (2.136) is a problem of the form (PI considered in Section 2.1, where

r,

g(Y)

= gdx)

so(x,Y(x))

+

VY

E

I

and V ,A are defined as in Section 1.4 (see (1.62)). Let ( y * , u * ) be an arbitrary optimal pair for problem (2.136). Then, following the general approach developed in Section 2.3, consider the penalized problem:

subject to in 0

Y-AY=f dY

dU

+ p & ( y )= Bou

in

rl,

y

=

in

0

r2.

Here, g ; is defined by (2.15) and p" by (2.74).

By a standard argument, it follows that, for

E

-+

u,

+

u*

strongly in U ,

y,

-+

y*

weakly in

y*

strongly in L ~d(a ) ,

y, p"(y,)

--f

-+

fo

HI(

0,

a),strongly in L'( a),

weakly in L2(an),

(2.140)

where f o ( x ) E P ( y * ( x ) ) a.e. x E dR. On the other hand, the optimality principle for problem (2.138) has the form dP,

-+

dv

P, - AP€ = v g ; ( Y & )

in

a,

B"(Y,>P&= - v g o ( x Y Y € )

in

rl,

0

in

r2,

P, B*p,

E

=

dh(u,)

+ U, - u*.

( 2.141a)

(2.14lb)

3.2. Optimal Control of Elliptic Variational Inequalities

195

Now, multiplying Eq. (2.141a) by p , and sign p , , (more precisely, with [ ( p e l , where 5 is a smooth approximation of sign), we find the estimates

and so on a subsequence, again denoted

Letting

E

dP

we have

tend to zero in Eqs. (2.141a), we get

dU

E,

+ p = -Vgo(x,y*) B*p

E

in

rl,

p

=

0

in

r,,

(2.142) (2.143)

dh(u*),

where p E (L"(T,))*. One can give explicit forms for these equations if p is locally Lipschitz or if p is the graph of the form (1.71). Since the proofs are identical with that of Theorem 2.2 and 2.4 respectively, we only mention the results. Theorem 2.6. Let ( y * ,u * ) be optimal in problem (2.136), (2.137), where /3 is monotonically increasing and satisfies condition (2.52). Then there exists p E H'(R) such that A p E L 2 ( R ) ,d p / d u E L2(rl), -p

+ Ap E dgl(y*) B,*p E d h ( u * ) .

a.e. in R ,

(2.144)

196

3. Controlled Elliptic Variational Inequalities

Consider now the case where /3 is given by (1.71). In this case, the state equation (2.137) reduces to the unilateral problem inR,

y-Ay=f

dY

LO, - - B o u > O , dV

y

=

0

in I‘,.

(2.145)

Theorem 2.7. Let ( y * ,u * ) be optimal forproblem (2.136) governed by state equations (2.145). Then there is p E H’(R) such that A p E L2(R), d p / d u E (L”(I‘,))*and -p

+ Ap E

-)

p ( B o u * - dY*

dgl(y*)

a.e. in R , a.e. in ( x E

=

0

a.e. in

rl ; y * ( x ) > 0},

rl,

B,*p E d h ( u * ) . Similar results can be obtained if one considers problems of the form (2.135) governed by variational inequalities on R with Dirichlet boundary control (see V. Barbu [71, p. 107). Bibliographical Notes and Remarks Section 1. The results of this section are classical and can be found in standard texts and monographs devoted to variational inequalities (see, for instance, J. L. Lions [l], Kinderlehrer and G. Stampacchia [l], A. Friedman [l], and C. M. Elliott and J. R. Ockendon [l]). For other recent results on analysis and shape of free boundary in elliptic variational inequalities and nonlinear elliptic boundary value problems, we refer to J. Diaz [l]. Section 2. Most of the results presented in this section rely on author’s work [3,7]. For other related results, we refer the reader to the works of A. Friedman [2], V. Barbu and D. Tiba [l],V. Barbu and Ph. Korman [l],D. Tiba [l], G. MoroSanu and Zheng-Xu He [l], and L. Nicolaescu [l]. An attractive feature of the approach used here is that it allows the treatment of more general problems such as optimal control problems governed by

Bibliographical Notes and Remarks

197

not well-posed systems (J. L. Lions [2]) or by hemivariational inequalities (Haslinger and Panagiotopoulos [ 11, Panagiotopoulos [2]). A different approach to first order necessary conditions for optimal control problem governed by elliptic variational inequalities is due to F. Mignot [l] (see also Mignot and Puel [l]), and relies on the concept of conical derivative of the map u + y". Let us briefly describe such a result, for the optimal control problem (2.84), where J!,I = 0, U = L2(R), B = I , and

= {rp E H,'(R), rp 2 0 in 2,; uf) = O}, where y" is the solution to (2.85), then ( y * , u * ) is optimal if and only if there is p E -Sue such that u* = p and

If Z,

=

( x E R, y " ( x ) = 0} and S,

(rp, Aoy" -

(Mignot and Puel [l]). A different approach related to the method developed in Section 2.2 (see problem Q,) has been used by Bermudez and Saguez ([1-4]). In a few words, the idea is to transform the original problem in a linear optimal control problem with nonconvex state constraints and to apply to this problem the abstract Lagrange multiplier rule in infinite dimensional spaces. A different approach involving Eckeland variational principle was used by J. Yong [4]. Optimal controllers for the dam problem were studied by A. Friedman et' a f . [ 11. Optimality conditions for problems governed by general variational inequalities in infinite dimensional spaces were obtained in the work of Shuzong Shi [l] and Barbu and Tiba [l]. For some earlier results on optimal control problems governed by variational inequalities and nonlinear partial differential equations, we refer to J. L. Lions [4] (see also [2]). There is an extensive literature on optimal control of free boundary in elliptic variational inequalities containing control parameters on variable domains (shape optimization). We mention in this direction the works of Ch. Saguez [l], J. Haslinger and P. Neittaanmaki [l], P. Neittaanmaki, et a f . [l], J. P. Zolesio [l], V. Barbu and A. Friedman [l], W. B. Lui and J. E. Rubio [l], Hlavacek, et al. [l], and Hoffman and Haslinger [l]. A standard shape optimization problem involving free boundaries is the following: Let R, be a domain in R N that depends upon a control variable u E U and let its boundary dR, = ro u r,, To n T, = 0, where r0 is prescribed inde-

198

3. Controlled Elliptic Variational Inequalities

pendently of u. The problem is to find u E U such that robecomes the free boundary and R, the noncoincidence set of a given obstacle problem in a domain R 3 R,, for instance: A y = f in [ y > 01, A y s f in R,, y > 0 in R, , y = 1 in r, . Such a problem is studied by the methods of this chapter in the work of Barbu and Friedman [l] and an explicit form of the solution is found in a particular case. A different approach has been developed in Barbu and Tiba [2] and Barbu and Stojanovic [l]. The idea is to reduce the problem to a linear control problem of the following type: Find u E U such that d y / d v = 0 in r0and y > 0 in R,, where y is the solution to Dirichlet problem A z = f in r, , z = 0 in r0,z = 1 in r, .

First order necessary conditions for state constraint optimal control problems governed by semilinear elliptic problems have been obtained by Bonnans and Casas [l] using methods of convex analysis (see also Bonnans and Tiba [l]).

Chapter 4

Nonlinear Accretive Differential Equations

This chapter is devoted to the Cauchy problem associated with nonlinear accretive operators in Banach space. The main result is related to the Crandall-Liggett exponential formula for autonomous equations, from which practically all existence results for the nonlinear accretive Cauchy problem follow in a more or less straightforward way. A large part deals with applications to nonlinear partial differential equations, which include nonlinear parabolic equations and variational inequalities of parabolic type, first order quasilinear equations, the nonlinear diffusion equations, and nonlinear hyperbolic equations.

4.1. The Basic Existence Results

4.1.1. Mild Solutions Let X be a real Banach space with the norm (I* (I and dual X * and let A c X x X be an w-accretive set of X x X; i.e., A + W Z is accretive for some w E R (Section 3 in Chapter 2). Consider the Cauchy problem

Y(0)

= Y o 9

(1.1)

where f E L'(0, T; X ) and y o E D ( A ) . Frequently, we shall write Eq. (1.1) in the form y ' + A y 3 f, y ( 0 ) = y o . 199

200

4. Nonlinear Accretive Differential Equations

Definition 1.1. A strong solution of (1.1) is a function y E W','((O,TI; X) n C([O, TI; X) such that

'

Here, W'.'((O,T I ; X ) = { y E L (0, T ; X I ; y ' E L ' ( 6 , T ; X ) V6 E (0, T I ) It is readily seen that any strong solution to (1.1) is unique and is a continuous function of f and y o . More precisely, we have: Proposition 1.1. Let A be w-accretive, f, E L'(0, T ; X ) , yd E D(A ) , i = 1 , 2 and let yi E W','((O,TI; X I , i = 1,2, be corresponding strong solutions to problem (1.1). Then, IlYl(f)

-

Y*(t)ll

IeofllyA - yi11

+

<

/d

e 4 - s )[ Y l ( S )

ewfllyA-yo211

-Y*(S),fdS)

-f*(S)l,

+ /d.~(f-s)llfl(s)-f2(s)ll

ds

ds.

( 1.2)

The main ingredient of the proof is Lemma 1.1 following. Lemma 1.1. Let y = y ( t ) be an X-valued fitnction on [O, TI. Assume that y ( t ) and Ily(t)ll are differentiable at t = s. Then

Here, J : X Pro08

+X

Let

E

* is the duality mapping of X .

> 0. We have

4.1. The Basic Existence Results

201

Similarly, from the inequality (Y(S - ).

-y(s),w)

(Ily(s -

.)I1

-

lly(~)ll)llwll,

we get

as claimed. In particular, it follows by (1.3) that

where (see Proposition 3.7 in Chapter 2) [x,Y], = inf A-’(Ilx A>O

+ Ayll - Ilxll) = max((y,x*); x*

E

B’(x)} (1.5)

Proof of Proposition 1.1. We have d -ds ( Y ~ ( s ) - Y ~ ( s ) )+ A Y ~ s ) A Y ~ S3) f d ~ -fz(s) ) a. e. s E ( 0 , T ) . On the other hand, since A is *accretive, [Yl(S) - YZ(S), AYdS) - AYZ(S)I, 2 -WllYdS) - Yz(S)ll and so, by (1.41, we see that

[Yl(S) -Yz(S),fdS) -fz(S)l, + WllYl(S) - Y z ( S ) l l a. e. s E ( 0 , T ) . Then integrating on [O, t] we get (1.21, as claimed. Proposition 1.1 shows that, as far as concerns uniqueness and continuous dependence of solution of data, the class of *accretive operators A offers a suitable framework for the Cauchy problem. However, for existence we must extend the notion of solution for Cauchy problem (1.1).

4. Nonlinear Accretive Differential Equations

202

We now define this extended notion. Let f E L'(0, T ; X ) and E > 0 be given. A n &-discretization on [0, TI of I the equation y ' + A y 3 f consists of a partition 0 = to I t , 5 t 2 I t , of the interval [0, t,] and a finite sequence (A};"=,c x such that t , - t,-' < E

for i

f 1''

=

Ilf(s>

r=l

l , . . . ,N , T -

E

< t, I T ,

( 1*6)

-All ds < 8.

(1.7)

'1-1

We shall denote by D i ( 0 = t o , t,, . . . , t,; f,,.. .,f,) this cdiscretization. A solution to the &-discretization Di(0 = t o ,t,, . ..,t,; f,,. . . ,f,) is a piecewise constant function z : [O, t,] -+ X whose values 2, on (t,- ,t,l satisfy the equation

,

zi - z i - ,

ti - ti-

I

+ A Z i 3 f, ,

i

=

1,..., N .

(1*8)

Such a function z = {zi},"= is called &-approximate solution to the Cauchy problem (1.1) if it further satisfies IlZ(0) - Yoll I

(1.9)

E.

Definition 1.2. A mild solution of the Cauchy problem (1.1) is a function y E C([O, T I ; X ) with the property that for each E > 0 there is an capproximate solution z of y ' + Ay 3 f on [ 0, TI such that Ily(t) - z(t)ll < E for all t E [O, T I and y ( 0 ) = x . Let us notice that every strong solution y E C([O, T I ; X ) n W','((O,T I ; X ) to (1.1) is a mild solution. Indeed, let 0 = to I t , I ... I t , be an &-discretization of [O, TI such that

,

and 1 Ilf(t) - f(t,)ll dt 5 &(t, - t,_ ,I. Then z = y ( t , ) on (t,- , t,] is a solution to the cdiscretization D i ( 0 = t o ,t , , . . . ,t,; f,,. . .,f,), and if we choose the discretization {t,) so that Ily(t) - y(s)ll I E for t , s E ( t , - , t , ) we have Ily(t) - z(t)ll 5 E for all t E [O, TI, as claimed. Theorem 1.1 following is the main result of this section. !,'I I -

,

Theorem 1.1. Let A be w-accretiue, y o E D(A ) , a n d f E L'(0, T ; X ) . For each E > 0, let problem (1.1) have an capproximate solution. Then the Cauchy problem (1.1) has a unique mild solution y . Moreover, there is a

4.1. The Basic Existence Results

continuous function 6 = mate solution of (1.1) then

such that S(0)

Ily(t) - z(t)ll I6( E )

=

for t

0 and

E

[0, T

203

if z is an &-approxi-

&I.

(1 .lo)

Let f , g E L'(0, T; X ) andy, 7 be mild solutions to (1.1)corresponding to f and g, respectively. Then Ily(t) - j ( t ) l l

4

em('-')Ily(s) - J(s)ll JS

f o r O I s < t I T . (1.11) This important result, which represents the core of existence theory of evolution processes governed by accretive operators will be proved in several steps. Let us first present some of its immediate consequences. Theorem 1.2. Let C be a closed convex cone of X and let A be w-accretive in X X X such that D(A)c C c

n

R(I

O
+ AA)

forsome A, > 0.

(1.12)

Let y o E D(A ) and f E L'(0, T; X ) be such that f ( t ) E C a.e. t E (0, T). Then problem (1.1) has a unique mild solution y . If y and 7 are two mild solutions to (1.1) corresponding to f and g , respectively, then Ily(t) - J(t)ll Iem('-')Ily( s) - 7(s)ll

forOIs
E

(1.13)

L'(0, T; X ) and

where (ti}/!,, t, = 0, is a partition of the interval [O, t N ]such that ti - ti< E , T - E < t, < T. By assumption (1.121, it follows that for E small enough the function z = zi on ( t i - ] , t i ] ,z , = y o , defined by (1.8) is an &-approximatesolution to (1.1). (It is readily seen by assumption (1.2) and the w-accretivity of A that Eq. (1.8) has a unique solution {zi},?,.) Thus,

204

4. Nonlinear Accretive Differential Equations

Theorem 1.1 is applicable and so problem (1.1) has a unique mild solution satisfying (1.13). rn In the sequel, we shall frequently refer to the map ( y o ,f ) + y from D( A ) X L'(0, T ; X ) to C([O, TI; X ) as the evolution associated to A . It should be noted that in particular the range condition (1.12) holds if C = X and A is wm-accretive in X x X . In the particular case when f = 0, if A is waccretive and

R ( I + A A ) 3 D(A)

for all small A > 0,

(1.14)

, E D ( A ) , if E is sufficiently then Eq. (1.8) has a unique solution { z i ) E o zo small. Hence, for all E > 0 sufficiently small, problem (1.1) with f = 0 has an &-approximatesolution, and so by Theorem 1.1 we have:

Theorem 1.3 (Crandall and Liggett). Let A be waccretiue, satisfiing the range condition (1.14) and y o E D( A ) . Then the Cauchyproblem dY dt

-+Ay30,

t>0,

(1.15) has a unique mild solution y . Moreover

(1.16)

n+m

uniformly in t on compact intervals.

rn

Indeed, in this case if to = 0, ti = is, i = l , . . ., N , then the solution z, to the &-discretization D,(O = t o , t , , . . . ,t , ) is given by z,

=

(I

+ &A)-'yo

on ((i - I ) & ,~ E J ] .

Hence, by (l.lO), we have

which implies the exponential formula (1.16). We note that in particular the range conditions (1.12), (1.14) are automatically satisfied if A is wm-accretive, i.e., if ol + A is m-accretive for some real o.

4.1. The Basic Existence Results

y

205

We shall apply now Theorem 1.2 to the mild solutions y = y ( t ) and to the equations

=x

and y'

in ( O , T ) ,

3f

y' + A y

+ Ay 3 u

in (0, T ) . ,

u E Ax,

respectively. We have, by (1.131, I ew('-s)lly(s)- x 1 1 + l r [ y ( T )- y , f ( T ) - u],e"('-')dT

Ily(t)

S

(1.17)

V O s s < t s T , [ x , u ]€ A .

Such a function y E C([O, TI; X ) is called (BCnilan [l]) an integral solution to Eq. (1.1). We may conclude, therefore, that under assumptions of Theorem 1.2 the Cauchy problem (1.1) has an integral solution, which coincides with the mild solution of this problem. On the other hand, it turns out that the integral solution is unique (BCnilan [l]) and under the assumptions of Theorem 1.2 (in particular, if A is w-m-accretive) these two notions coincide. Let us now come back to the proof of Theorem 1.1. Let z be a solution to an &-discretization D:(O = t o , t , , ...,t,; f , , .. . ,f , ) and let w be a solution to D i ( 0 = s o , sl, . . . ,s M ;g , , . . . , g M ) with the nodal values zi and w j , respectively. We set a j j = llzi - wjll,

si = ( t i - ti-

yj =

(Sj

- sj-

1).

For all 1 I i I N , 1 I j I M , we have

Lemma 1.2.

(1.18) Moreover, for all [ x , ul ai,0

s

ai,,llz0

E

A we have

-XI1

+ llwo - XI1 +

c 1

k= 1

'Yi,kSk(IlfkII

+ Ilvll),

0 I i I N , (1.19)

4. Nonlinear Accretive Differential Equations

206

and a o , j IP,,~IIW~ - xII

i

+ IIzo - xII + C

k= 1

Pj,kYk(IIgkII+

Ilvll),

0 I j IM , (1.20) where

n (1 i

ai.k

=

m=k

-

pj,k =

OSm)-l,

i

n

m=k

(1 - w Y m ) - I .

(1.21)

We have

Prooj

f,

+ 8;1(zL-l - z L ) E A Z , ,

g,

+ Y;I(w,-l

-

w,)E A W , , (1.22)

and since A is w-accretive this yields

[ z ,- w,,f, +

8;'(Z1-l

-21)

-g, -

-w,)ls

Y;l(W,-l

2 -wIIz, - w,II.

Hence,

--wIIz,

- W/II I12, -

w/, f ,

+ r,-"z, I[ z , -

- S,l, +

-

w,,w, - w,-lls

w,,f, - g , ] ,

- Y,-l(llzL

8:l-"fl - w/, 2 , - I - ZLlS

- 8:1(llz,

- w,ll - llz,

- w,II -

11~,-1 -

yll)

- W,-lIl),

and rearranging gives (1.18). To get estimates (1.19), (1.20), we note that since A is w-accretive we have respectively l l W j -XI1

1 ( 1 - rjW)-'IIwj --x

+ r j ( g j + X:'(wj-,

- Wj) -

U)II'

for all [ x , u ] E A . Hence, llz;

-XI1

I ( 1 - 8;w)-111zj-l

-XI1

llwj

--I1

I(1 - Yjw)-'llwj-l

-XI1

+ (1 - 8;w)-18;(llfjll + Ilvll), + (1 - YjO)-lYj(llgjll + IIUII),

and (1.19), (1.20) follow by a simple calculation.

H

4.1. The Basic Existence Results

Now, consider the functions equation

a*

-(f, dt

s)

+

I )

and cp on [O,Tl that satisfy the linear

s)

-(f, ds

207

w*(f, s)

-

for 0

=

d fs),

It IT , 0

IS

IT

, (1.23)

and the boundary conditions $ ( t , s)

=

for t

b ( t - s)

=

0 or s

=

(1.24)

0,

where b E C ( [- T , TI). There is a close relationship between Eq. (1.23) and inequality (1.18). Indeed, if define the grid It,,,
A=((tj,sj);O=to~I t, O=S,<S,I

<s,
and approximate (1.23) by the difference equations

for i where 8; = t i - t i - , yj rearrangement

=

sj

- sj-

=

, and

i

1,..., N , j

+;,

=

=

=

1 ,..., M . (1.25)

cp(ti, sj), we get after some

1,..., N , j

=

1,..., M. (1.26)

Define the functions q ( t , s)

=

Ilf(r)

- g(s)ll,

+;,j

=

Ilf,

- gjll,

i = l ,..., N , j = l , ..., M ,

where fi and gj are the nodal approximations of f and g respectively.

E

L'(0, T ; XI,

208

4. Nonlinear Accretive Differential Equations

if 0

-

e"'b(t - s)

0 Is <

t IT ,

+ /'e"('')cp(T,s 0

-t

+ T)dr

(1.27)

has a unique solution {t,bij}i = 1,. . ., N , j = 1,.. . ,M. Denote by '4' = HA(B,4 ) the piecewise constant function on R defined by '4'

=

+bij

on (ti-1,til x

(Sj-1,

sj1,

(1.30)

i.e., the solution to (1.26), (1.29). Lemma 1.3 following gives the convergence of the finite difference scheme (1.251, (1.29) as m ( A ) 4 0. Lemma 1.3.

Let b

E

C([- T,TI)and

cp E L1(R) be given. Then

IlG( b , cp) - HA( B , 4)IIL"(R(A))

0

(1.31)

4.1. The Basic Existence Results

209

as

m(A)

+ Ilb - B l I ~ " ( - s , , t ~ )IICP - 4lln(A)

+

0.

Proof In order to avoid a tedious calculus, we shall prove (1.31) in the accretive case (i.e., w = 0). Let us prove first the estimate

I\HA(B,4)IlL"(n(A))5 \ I B \ l L " ( - s , , f , )

+ 1\4Iln(A)*

(1.32)

Indeed, we have HA(B,4) = HA(B,O)+ HA(0, 4), and by (1.2% (1.30) we see that the values of HA(B,0) are convex combinations of the values of B. Hence,

II HA( B , 0) IIL " ( ~ ( A ) )5 II B IIL"( - sy, I N It remains to show that IIHA(O,4)1IL"(O(A)) 5 Il+lln(A).

-

By the definition of the II IlncA,-norm,we have

Now, let g i , j = ai + Pj 2

14i,jland i

d ' .. I . =

ffkSk

k= 1

+

i

C

PkYk.

k=l

It is readily seen that +i, = di, satisfy the system (1.26) where c#+, = g;,j . Hence, d = HA(B,g ) provided d i , = bi, for i = 0 or j = 0, where d = Id;,j ) , B = { h i , j ) and g = {g;,j ) . Inasmuch as g;, 2 I+;, jl, we have

d if

= HA(B, g ) 2

HA(o, 4) 2 IHA(O,

411

bi, 2 0. Hence IIHA(O,4)IIL"(n(A)) 5 lldllL"(n(A))5 Ildln(A)

as claimed. JSs Now, let = G 6 , 4) and assume first that (1.23), we see that &, = $ ( t i , s j ) satisfy the system

4

tjj,o

=

b(ti),

=

b(-sj),

i

=

0,1, ..., ~

E

3

L"(R). Then, by

, = 0,1, j ...,M ,

210

4. Nonlinear Accretive Differential Equations

where

Proof of Theorem 1.1. (continued). We shall apply Lemma 1.3, where ~p(t,s)= IIf(t) - g(s)II, 4 = I+J, +j,, = If, - gill, 1 5 j IM , 1 I iI N, where f, and gj are the nodal values of f and g , respectively, and B ( t ) = Bi,o

for

B ( s ) = Bo,,

for - s j

=

1, ..., N ,

~ - ~ ,j =

1,..., M .

i

Here, Bi,o is the right hand side of (1.19) and B,,, is the right hand side of (1.20). It is easily seen that, for E + 0, B ( t ) + b ( t ) = ew‘llzo - xII

+/be-(‘-T)

(llf(T)11

and

B( s)

+

b( -s)

=

+ llwo - xII i- Ilull) d7

ewSllwO- xII

+ llzo - xII

Vt

E

[O,T],

4.1. The Basic Existence Results

211

By (1.7) and (1.31), we have IIq -

4 l l f i ( A ) I2 8

whilst, by Lemma 1.2, ai,,= llt; - w,ll IHA( B , +)i,

Vi, j .

j

Then, by Lemma 1.3, we see that for every 77 > 0, we have Ilz(t) - w(s)ll 5 G ( b , q ) ( t , S )

+ rl

Vs,t

as soon as 0 < E < ~ ( 7 ) . If f = g and zo = w o ,then G(b, q ) ( t ,t ) = e"'b(0) so, by (1.351,

Ilz(t) - w(t)ll

5 77

+ 2ew'llzo - xlI

E

=

[O,Tl,

(1.35)

2ew'llzo - xII and

Vx E D ( A ) , t E [ & T I ,

for all 0 < E I47).Since JJzo- soJJIE , yo E D(A ) , and x is arbitrary in D ( A ) , it follows that the sequence 2, of &-approximate solutions satisfies the Cauchy criterion and so y(t) = lim, z,(t) exists uniformly on [O, TI. Now we take the limit as E + 0 in (1.35) with s = t + h , g =- f, and zo = w o = y o . We get -t

Ily(t

+ h ) -y(t)ll

IG ( b , q ) ( t

+ h , t ) = ewt(ewh+ l ) l l y o --I1 +

+/ t

0

7 )

c

w ( h - T'

(llf(~)II + Ilull) dT

I l f ( ~ + h ) - f ( ~ ) l l dT

V[x,uI € A ,

and therefore y is continuous on [0, TI. Now by (1.35) we have, for f = g, t = s,

Ilz(t)

- y(t)ll

I S(

8)

vt

E

[O,TI,

where t is an &-approximate solution and 8 ( & ) -, 0 as take t = s in (1.35) and let E tend to zero. We get

E -+

0. Finally, we

212

4. Nonlinear Accretive Differential Equations

Then

G ( h , qo)(t,t )

=

e"'lly(0)

- Y(0)ll

+p('-'"Y(s)

- Y(s>,f(s>- g(s)ls,

and so (1.11) follows for s = 0 and consequently for all s E (0, t ) . Thus, the proof of Theorem 1.1 is complete. The convergence theorem can be made more precise in the autonomous case of (1.15) i.e., for f = 0. Corollary 1.1. Let A be o-accretive and satisjj condition (1.14), and let y o E D( A ) . Let y be the mild solution to problem (1.15) and let y, be an &-approximatesolution to (1.15) with y,(O) = y o . Then

Ily,(t) -y(t)ll for all x

E

CT(lly0

+ IhI(& + t'/2&'/2 1)

V t E [O,TI, (1.36)

D( A). In particular, we have

yoII I CT(lIy0 - X I+ tn-'/21Axl) (1.36)' for all t E [O, TI and x of x and y o .

E

D( A ) . Here, C , is a positive constant independent

Prooj Since the mappings y o + y and y o + y e are Lipschitz with Lipschitz constant ewT,it suffices to prove estimate (1.36) for yo E D ( A ) . By the estimate (1.34) we have, for all T > 0, IIG(b,O)

- HA(B,O)IIL=(~(A)) I Ilb -

~IIL=(-T,T)

+ IIB - ~ ~ I L = ( - T , T )

+ CE(II$rrIIL=(n)+ I I $ ~ ~ I I L = ( ~ ) ) ~ where 6 = G(b,O), 6 is a sufficiently smooth function on [ - T , TI, K! = ( 0 , T ) x (0, T ) , and C is independent of E , b, and B. We shall apply this inequality for B and b as in the proof of Theorem 1.1, i.e.,

b ( t ) = w-'(e"l'l

- 1)Ihl

Vt € [-T,T].

Then we have

b ' ( t ) = e"l'llh1sign t .

4.1. The Basic Existence Results

213

We shall approximate the signum function by

6 such that

and so we construct a smooth function P(t)

=

sign tlAxleWl'l

w

&O)

=

0 and

+ ~'(t)(RuIeWlfl.

Hence, S U P { I ~ " ( So) II; s I t }

+ A-I)

IeWIIIIAxI(w

and, therefore, Ilb - 6 I b - f J )

+ CEt(ll~IIIIL"((O,I)X(O,f))

+ II ksIIL"((0,I ) x ( 0 , I N ) <

Ct&IAxl(l

+ A - I ) + CAlAxl

V t E [O,T],

where C depends on T only. Similarly, we have

-

IIB - bII~=(-t,i) 5 C(E

+ A)IAI.

Finally, IIC(b,O) - HA(B,O)IILG(R,(L\)) I C ( E+ A

+~EA-')IAI,

where R f = (0, t) x (0, t). This implies that (see the proof of Theorem 1.1) IIy,(t) - y ( t ) I I IG ( b , O ) ( t , t )

for all

t E

[0, TI and all A > 0. For A

Ily,(t) - y(t)ll I ClAxl( E

which completes the proof.

+ C I A ~ I ( E+ A + t c h - ' )

=

(tE)'/' this yields

+ t'/2E'/2)

Vt

E

[O,TI,

W

4.1.2. Regularity of Mild Solutions

A question of great interest is that of circumstances under which the mild solutions are strong solutions. One may construct simple examples that

214

4. Nonlinear Accretive Differential Equations

show that in a general Banach space this might be false. However, if the space is reflexive then under natural assumptions on A, f , and y o the answer is positive. Theorem 1.4. Let X be reflexive and let A be closed, w-accretive and let it satisfL assumption (1.12). Let y o E D ( A ) and let f E W','([O,TI; X ) be such that f(t) E C Qt E [0,TI. Then problem (1.1) has a unique strong solution y E W',"([O,TI; X ) . Moreover, y satisfies the estimate

(1.37)

Proof Let y be the mild solution to problem (1.1) provided by Theorem 1.2. We shall apply estimate (1.13) where y ( t ) : = y ( t h ) and g ( t ) : = f ( t + h). We get

+

Ily(t + h ) -y(t)II

+tlIf(s +

IIly(h)

I

Ch

-y(O)IIe"' h ) -f(s)lleo('-s)ds

+ Ily(h) - y(O)lle"',

because f E W " ' ( [ O , TI; X ) (see Theorem 3.3 and Remark 3.1 in Chapter 1). Now, applying the same estimate (1.13) to y and y o , we get

We may conclude, therefore, that the mild solution y is Lipschitz on [O, TI. Then, by Theorem 3.1 of Chapter 1, it is a.e. differentiable and belongs to W'*"([O, TI; X). Moreover, we have

Ily( t + h ) h

- y ( t)Il

4.1. The Basic Existence Results

Now, let t

E

215

[O, 7'1 be such that

dY dt

-(t)

=

1 lim (y(t h-0 h

+h) -y(t))

exists. By inequality (1.171, we have

Since the bracket [ u , ul, is upper semicontinuous in ( u , u), and positively homogeneous and continuous in u, this yields -

wlly(t> - XI1 I [ Y ( t ) - x,f(t) - wls V [ x , w ]€ A .

This implies that there is mapping)

6 E J ( y ( t > - x > such that ( J is the duality

( 2 ( t ) - o ( y ( t ) - x ) -f(t)

Now we have

-w,t

(1.38)

216

4. Nonlinear Accretive Differential Equations

where g ( h ) + 0 as h + 0. On the other hand, by condition (1.121, for every h sufficiently small and positive, there are [x,, ,wh]E A such that Y(t - h )

+ h f ( t ) = x,, + hw, .

Substituting successively in (1.38) and in (1.39), we get

Hence, x,, + y ( t ) and w,, we conclude that

as claimed.

f ( t ) - (dy(t))/dt as h

4

+

0. Since A is closed

rn

Remark 1.1. If A is w-m-accretive, then as seen earlier it is closed in X X X and so assumptions (1.12) hold automatically. In particular, we have: Theorem 1.5. Let A be w-accretive, closed and satisfi, the range condition (1.14). Let X be reflexive and y o E D ( A ) . Then problem (1.15) has a unique strong solution y E w'*"([O, 03); X I . Moreover,

More can be said about the regularity of strong solution to problem (1.1) if the space X is uniformly convex. TI; X ) , y o E D ( A ) , Theorem 1.6. Let A be w-m-accretive, f E W'?'([O, and X be uniformly convex along with the dual X * . Then the strong solution y to problem (1.1) is everywhere differentiable from the right, ( d + / d t ) yis right

217

4.1. The Basic Existence Results

continuous and d+ -y(t) dt

+ ( A y ( t )-f(t))'

=

0

Vt

E

Vt

(1.41)

[ O,T),

E

[0, T). (1.42a)

Here, ( A y - f>' is the element of minimum norm in the set A y - f.

ProoJ Since X and X * are uniformly convex, A y is a closed convex subset of X for every x E D ( A ) (see Section 3 in Chapter 2) and so ( A y ( t ) - f(t))' is well-defined. Let y

E

W'.m([O,TI; X > be the strong solution to (1.1). We have

d -dh (Y(t

+ h ) - Y ( t ) ) +AY(t + h ) 3 f ( t + h ) a.e. h > 0, t

E

(O,T),

and since A is w-accretive, this yields d ($Y(t

+ h) -Y(l))-

5 4Y(t

5

)

+ h ) -y(t)1I2 + ( f ( t + h ) - 7 7 ( f ) , 5)

V77( t 1 E AY ( t 1

7

where 5 E J ( y ( t + h ) - y ( t ) ) . Then, by Lemma 1.1, we get Ily(t

+ h ) -y(t)ll

I/ h e w ( h - s ) l l q ( t )-f(t 0

+s)llds,

(1.42b)

which yields (($(t)IlI

In other words,

~ ~ f (-t > V(~)II

~v(tE > A y ( t ) , a.e. t

E

(0,T).

218

4. Nonlinear Accretive Differential Equations

and since dy(t)/dr dY dt

-

+ A y ( t ) 3 f ( t >a.e. t E (0, TI, we conclude that

+ ( A y ( t ) -f(t>)' = o

a.e. t

E

(0,~).

(1.43)

Observe also that y satisfies the equation d -(Y(t dt

+h)

-

Y ( t > ) + AY(t + h ) - A Y ( t )

3f(t + h ) - f ( t ) a x . in (0, T ) .

Multiplying this by J ( y ( t see by Lemma 1.1 that d -lly(t dt

+ h ) -y(t)ll

+ h)

-

y ( t ) ) and using the w-accretivity of A , we

I wlly(t

+ h ) -y(t)ll + Ilf(t + h ) - f ( t ) l l a.e. t , t

and therefore Ily(t

+ h ) -y(t)ll

+ h E (0, T ) ,

+ h ) -y(s)ll

I ew('-s)lly(s

Ilf(T

+ h)

-f(T)IldT.

(1.44)

Finally,

a.e. 0 < s < t < T . (1.45) Similarly, multiplying the equation

by J ( y ( t ) - y o ) and integrating on (0, t ) , we get the estimate I l y ( t ) -yell I / ' e w ( t - s ) l l ( Ay, - f(s))'Il 0

and substituting in (1.44) with s

=

ds

Vt

E

[0, T I , (1.46)

0, we get

a.e. t

E

( 0 , T ) . (1.47)

4.1. The Basic Existence Results

219

Since A is demiclosed (Proposition 3.4 Chapter 2) and X is reflexive, it follows by (1.43) and (1.47) that y(t) E D ( A ) V t E [O, TI and II(Ay(t) -f(t))oll

For t arbitrary but fixed, consider h, ~ ( +t hn) - ~ ( t )

hn

vt E [ O , T ] .

Ic +

(1.48)

0 such that h , > 0 for all n and

- 6 i n X

asn-w.

By estimate (1.42), we see that

11511 2 II(AY(t1 -f(t))oll,

whilst

5 E f ( t ) - Ay(t) because

(1.49)

A is demiclosed. Indeed,

where q E LYO, T ) and q ( t ) E Ay(t) V t E [O, TI. We set q,(s) = q ( f + sh,) and y,(s) = y(t + sh,). If we denote again by A the realization of A in L2(0,1; X ) x L2(0,1; X),we have yn + y(t) in L2(0,1; X I , q, + f ( t ) - 5 weakly in L2(0,1; X I . Since A is demiclosed in L2(0,1; X ) x L2(0,1; X ) we have that f ( t > - 5 E Ay(t), as claimed. Then, by (1.49), we conclude that 5 = (Ay(t) - f(t))' and, therefore, d+ Y(t + h ) -Y(t) -y(t) = lim h dt h 1.0 Next, we see by (1.44) that

=

-(Ay(t)

-f(tp

Vt E[O,T).

Let t , + t be such that t, > t for all n. Then, on a subsequence, again denoted by t,,

where - 5 E Ay(t) - f(t) (because A is demiclosed). On the other hand, it follows by (1.50) that

11511 2

limsupII(Ay(t,) n+m

-f(t,)>OII

2

II(AY(t) -f(t))oll.

220

4. Nonlinear Accretive Differential Equations

Hence, 6 = - ( A y ( t ) - f(t))' and ( d + / d t ) y ( t , ) -+ 6 strongly in X (because X is uniformly convex). We have therefore proved that ( d + / d t )y ( t ) is right continuous on [0, T ) , thereby completing the proof. In particular, it follows by Theorem 1.6 that if A is w-m-accretive, y o E D ( A ) , and X,X * are uniformly convex, then the solution y to the autonomous problem (1.15) is everywhere differentiable from the right and d+ -y(t) dt

+ AOy(t) = 0

(1.51)

V t 2 0,

where A' is the minimal section of A. Moreover, the function t -+ A o y ( t ) is continuous from the right on R+. It turns out that this result remains true under weaker conditions on A. Namely, one has: Theorem 1.7.

Let A be w-accretive, closed, and satisb the condition

n

conv D(A ) c

R( I

O
+ AA)

for some A, > 0.

(1.52)

Let X and X * be uniformly convex. Then for every x E D(A ) the set Ax has a unique element of minimum norm A'x, and for every y o E D(A ) the Cauchy problem (1.1) has a unique strong solution y E W'3"([0,m);X I , which is eveywhere differentiablefrom the right and d+ -y(t) dt Moreover, the function t

-+

+ AOy(t) = 0

V t 2 0.

(1.53)

A o y ( t ) is continuous from the right and (1.54)

Proot We shall assume first that A is demiclosed in X set B c X x X by Bx

=

convAx,

x

ED

(B)

X

X. Define the

=D(A).

It is readily seen that B is w-accretive. Moreover, by (1.52) it follows that D(A) c

n

R(I+AB).

O
Let x E D ( A ) . Then, x A = ( I + AA)-'x and y, = A,x are well-defined x for 0 < A < A,. Moreover llAnxll IIAxl = inf{llwll; w E Ax} and x , -+

4.1. The Basic Existence Results

221

-

and A + 0 (see Proposition 3.2 in Chapter 2). Let A, + 0 be such that AAnx y. Since A,: E AxAnand A is demiclosed, it follows that y E Ru. On the other hand, we have llAAxll

=

IIBAxll 5 lBxl

=

IIBoxll*

( B o x exists and is unique because Bx is convex.) This implies that = Box E Ru. Hence, Ax has a unique element of minimum norm A'x. Then we may apply Theorem 1.6 to deduce that the strong solution y to problem (1.15) (which exists and is unique by Theorem 1.5) satisfies (1.53) and (1.54). (In the proof of Theorem 1.6, the w-m-accretivity has been used only to assure the existence of a strong solution, the demiclosedness of A , and the existence of A'.) To complete the proof, we turn now to the case where A is only closed. Let A be the closure of A in X X X , , i.e., the smallest demiclosed extension of A. Clearly, D ( A ) c D ( A ) c D( A ) and A' satisfies condition (1.52). Moreover, since the duality mapping J is continuous we easily see that A is w-accretive. Then, applying the first part of the proof, we conclude that problem

y

d'u

-

dt

+Aou= o 40)

in [o,co),

=Yo7

has a unique solution u satisfying all conditions of the theorem. To conclude the proof it suffices to show that D ( k ) = D ( A ) and A ' = A'. Let x E D ( k ) . Then, there is [xA,y,] E A c A' such that x = x, - Ay, for 0 < A < A,. We have x, = ( I + AA)-'x and y , = A,x = 2,x.Since x E D ( i ) , A-rO

+ x and llyAll Il hl= IIA'xll. As A' is demiclosed and X is uniformly convex this implies by a standard device that y, + Aox as A 4 0. Since A is closed, this yields Aox E Ax and x E D ( A ) . Hence, D ( 2 ) = D ( A ) and A'Ox = Aox Vx E D ( A ) . The proof of Theorem 1.7 is complete.

X,

Remark 1.2. If the space X * is uniformly convex, A is w-m-accretive, E W','([O,TI; X), and yo E D ( A ) , then the strong solution y E W'sm([O,T I ; X ) to problem (1.1) (Theorem 1.4) can be obtained as

f

y(t)

=m !

-0

y,(t)

in X , uniformly on [0, T I ,

(1.55)

222

4. Nonlinear Accretive Differential Equations

where y, system

E

C'([O, TI; X ) are the solutions to the Yosida approximating

On the other hand, multiplying the equation d2YA dt2

-

+ -dtdA , y , ( f )

=

df dt

-

a.e. t

E

(0, T ) ,

by J(dy,/dt) yields

because A, is w-accretive. This implies

Hence, IIA,y,(t>ll IC VA E (0, AJ, and IlyA(t) - ( 1 + AA)-'y,(t)ll ICA. Since J is uniformly continuous on bounded sets, it follows by (1.57) that

223

4.1. The Basic Existence Results

{y,} is a Cauchy sequence in the space C([O, TI; X )and y ( t ) = lim, yA(t> exists in X uniformly on [O, TI. Let [ x , w ] be arbitrary in A and let x, = x + Aw. Multiplying Eq. (1.56) by J(y,(t) - x,) and integrating on [s, tl, we get ~

because J is continuous. This yields

By estimate (1.57), we see that y is absolutely continuous on [O,T] and dy/dt E L"(0, T ; X ) . Hence, y is a.e. differentiable on (0, T ) . If s = t o is a point where y is differentiable, by (1.58b) we see that dY

( f ( t o )+ Z(t0)- w + o ( y ( t 0 ) - x > , J ( y ( t o )

-XI

1

2 0

V[x,w]€ A .

Since A

+ oZ is m-accretive, this implies that

Hence, y is the strong solution in problem (1.1).

W

224

4. Nonlinear Accretive Differential Equations

4.1.3. The Cauchy Problem Associated with Demicontinuous Monotone Operators We are given a Hilbert space H and a reflexive Banach space V such that V c H continuously and densely. Denote by V’ the dual space. Then, identifying H with its own dual, we may write VCHCV’ algebraically and topologically. The norms of V and H will be denoted 11. II and I I, respectively. We shall denote by ( u I , u 2 )the pairing between u1 E V’ and u2 E V; if u1,u2 E H, this is the ordinary inner product in H. Finally, we shall denote by 11 * It* the norm of V’ (which is the dual norm). Besides these spaces we are given a single valued, monotone operator A: V -+ V’. We shall assume that A is demicontinuous and coercive from V to V‘. We begin with the following simple application of Theorem 1.6.

Theorem 1.8. Let f E W’.‘([0,TI; H ) and y o E V be such that A y , Then there exists one and only one function y : [0,TI + V that satisfies y

E

Wl,m([O,T I ; H ) ,

dY -dt( t )

Ay

E L”(0,T

a.e. t

+Ay(t) = f ( t )

E

;H ) ,

E

H.

(1.59)

(O,T), (1.60)

Y ( 0 ) =Yo* Moreover, y is evelywhere differentiablefrom the right (in H ) and d+ -dt y(t)

Proof: Define the operator A,: H A,u = A M

V t E [O,T).

+Ay(t)= f ( t )

VU

E

+

H,

D(A,)

= {U E

V ; AME H } .

(1.61)

By hypothesis, the operator u + u + A M is monotone, demicontinuous, and coercive from V to V’. Hence, it is surjective (see, e.g., Corollary 1.3 in Chapter 2) and so A, is m-accretive (maximal monotone) in H X H. W Then we may apply Theorem 1.6 to conclude the proof.

4.1. The Basic Existence Results

225

Now, we shall use Theorem 1.8 to deduce a classical result due to J. L. Lions [ l ] .

Theorem 1.9. Let A: V satisfies the conditions

+

V ' be a demicontinuous monotone operator that

( A u , u ) 2 wllullP + c, IIAull* I C,(1

+ llullp-l)

vu E

v,

vu

E

(1.62)

v,

(1.63)

where w > 0 andp > 2. Given y o E H andf E L4(0,T ; V ' ) ,l / p + l / q = 1, there exists a unique absolutely continuousfunction y:[O,TI + V' that satisfies

y

E

dY -(t) dt

C ( [ O , T ] ;H ) n L P ( O , T ; V )n W ' 7 4 ( [ 0 , T ] ; V ' ) , (1.64) +Ay(t)= f ( t )

a.e. t E ( O , T ) ,

y ( 0 ) = y o , (1.65)

where d/dt is considered in the strong topology of V ' . Prooj Assume that y o E D ( A , ) and f E W'*'([O, TI; H I . Then, by Theorem 1.8, there is y E W'*m([O,TI; H ) with A y € LYO, T ; H ) satisfying (1.65). Then, by assumption (1.62), we have

and therefore

V t E [O,T]. (1.66)

Then, by (1.631, we get

(We shall denote by C several positive constants independent of y o and f.) Let us show now that D ( A , ) is a dense subset of H. Indeed, if x is any element of H we set x, = ( I + &A,)-'x ( I is the unity operator in H I . Multiplying the equation x, &Ax, = x by x,, it follows by (1.62) and

+

226

4. Nonlinear Accretive Differential Equations

(1.63) that lX&I2

and

+ OEIIX,IIP < Ix,I Ilx,

-

1x1 + CE

YE

> 0,

xII* I EIIAx&II* 5 CE(llx,ll”-’

+ 1)

WE

> 0.

x in Hence, {x&}is bounded in H and x , + x in I/’ as E + 0. Hence, x, H as E + 0, which implies that D ( A , ) is dense in H . Now, let y o E H and f E Lq(0, T ; V’). Then there are the sequences {yo”} c D ( A , ) , { f n ) c W’,’([O,TI; H ) such that

in H ,

y,” + y o

f, + f

in L q ( O , T ; V ’ ) ,

as n + m. Let y, E W’*”([O, TI; H ) be the solution to problem (1.65), where yo = yo” and f = f, . Since A is monotone, we have

a.e. Integrating from 0 to

t,

we get

+ 2(/$(s)

ly,(t) - y r n ( r ) l Z I ly; -Y:?

x ([Ily,(.)

rp

-f,(s)ll%ds

- y r n ( s ) l l p ds

.

t E

(0, T ) .

Y

(1.68)

On the other hand, it follows by estimates (1.66) and (1.67) that { y n ) is bounded in Lp(O,T ; I/) and (dy,/dt} is bounded in Lq(O,T ; I/’). Then, it follows by (1.68) that y ( r ) = limn+my,(r) exists in H uniformly in r on [O, TI. Moreover, extracting a further subsequence if necessary, we have Yn

weakly in

+Y

dYn dr

d~ dr

weakly in Lq(0, T ; I/’),

-+ -

where dy/dr is considered in the sense of I/’-valued distributions on (0,T). In particular, we have proved that y E CUO, TI; H ) n Lp(O,T ; I/> n W’-q([6,TI; V’).It remains-to prove that y satisfies a.e. on (0, T ) Eq. (1.65). Let x E I/ be arbitrary but fixed. Multiplying the equation dYn dr

-

+ Ay,

=

f,

a.e.

t E

(0, T )

4.1. The Basic Existence Results

by Y ,

-

227

x and integrating on (s, t ) , we get

Hence.

( 1.69)

We know that y is a.e. differentiable from (0, T ) into V’ and

Let to be such a point where y is differentiable. By (1.691, it follows that

and since x is arbitrary in V and A is maximal monotone in V implies that

X

V’ this

thereby completing the proof. Remark 1.3. Theorm 1.9 applies neatly to the parabolic boundary value problem

DPy=o

indRx(O,T),

IPlim-1,

y(x,o)

where f E Lq(0,T ; W m . q ( n )y)o, E L 2 ( n )and the A , satisfy conditions (i)-(iii) of Proposition 1.2 in Chapter 2.

228

4. Nonlinear Accretive Differential Equations

Theorem 1.9 remains true for time-dependent operator A ( t ) :V + V’ satisfying assumptions (1.62) and (1.63). More precisely, we have: Theorem 1.9‘. Let ( A ( t ) ;t E [0,T I ) be a family of nonlinear, monotone, and demicontinuosu operators from V to V’ satisfying the assumptions:

The function t + A ( t ) u ( t ) is measurablefrom [O, TI to V’ for euery measurable function u: [0,TI + V ; (ii) ( A ( t ) u , u )2 wllullp + C , V u E V , t E [O,T]; (iii) IIA(t)uII, I C,(I + I I U I I P - ’ ) vu E V , t E [o, T I , where w > 0, p 2 2. (i)

Then for every y o E H and f E Lq(0,T ; V ’ ) l / p + l / q = 1, there is a unique absolutely continuousfunction y E W ‘ - q ( [ O , TI; V ’ ) that satisfies

Y dY -dt( t )

E

C([O, T I ; H ) n LP(0,T ; V ) ,

+ A ( t ) y ( t )= f ( t )

a.e. t E ( O , T ) ,

Y ( 0 ) =Yo.

(1.70)

Proof: Consider the spaces 7= Lp(O,T ; V ) ,

2=L2(0, T ;H ) ,

7’= Lq(0, T ; V ’ ) .

Clearly, 7 and Y ’ are dual pairs and

7 c 2 c 7’ algebraically and topologically. Let y o E H be arbitrary and fixed and let B: 7+7’be the operator du BU = -, dt

u

E

D(B)

du

E

T ’ ,u ( 0 )

where d/dt is considered in the sense of vectorial distributions on (0, T ) . We note that D ( B ) c W’,q(O,T ; V’) n LP(0, T ; V ) c C([O,TI; H ) , so that y ( 0 ) = y o makes sense. Since B is clearly Let us check that B is maximal monotone in 7 X 7’. monotone, it suffices to show that R ( B + 4) = 7’, where 4 ( u ( t ) )= F(u(t))lIu(t)ll”-’,

u

E

7,

and F : V + V’ is the duality mapping of V. Indeed, for every f equation Bu

+ 4 ( u ) =f,

E

7’the

229

4.1. The Basic Existence Results

or equivalently du

- + F ( ~ ) l l u l l =~ f- ~ dt

in [O,T],

u(0) = y o ,

has by virtue of Theorem 1.9 a unique solution u

E

du dt

- E Lq(0, T ; V ' ) .

C ( [ O ,T I ; H ) n LY(0, T ; V ) ,

(Renorming the spaces V and V f ,we may assume that F is demicontinuous and that so is the operator u -+ F ( ~ ) l l u l l ~ -Hence, ~.) B is maximal monotone in 7~7 ' . Define the operator A,: F'-+ 7 'by a.e. t E ( 0 , T ) .

(A,u)(t) =A(t)u(t)

Clearly, A, is monotone, demicontinuous, and coercive from T to V f . Then, by Corollaries 1.2 and 1.5 of Chapter 2, A, + B is maximal , completes the monotone and surjective. Hence, R ( A , + B ) = 7 ' which proof. rn 4.1.4. Continuous Semigroups of Contractions Definition 1.3. Let C be a closed subset of a Banach space X . A continuous semigroup of contractions on C is a family of mappings M t ) ; t 2 0) that maps C into itself with the properties:

(i) (ii) (iii) (iv)

S(t

+ SIX = S(t)S(s)x V x E C , t , s 2 0;

S(0)x = x V x E C ; For every x E C, the function t

S ( t ) x is continuous on IlS(t)x - s(t)yll IIIx -yll V t 2 0; x , y E C . -+

[O,w).

More generally, if instead of (iv) we have (iv) IlS(t)x - S(t)yll Ieo'llx - yll V t 2 0; x , y E C , we say that S ( t ) is a continuous w-quasicontractive semigroup on C. The operator A,: D ( A ) c C + X defined by

where D ( A , ) is the set of all x E C for which the limit (1.71) exists, is called the infinitesimal generator of the semigroup S(t).

230

4. Nonlinear Accretive Differential Equations

There is a close relationship between the continuous semigroups of contractions and accretive operators. Indeed, it is easily seen that - A , is accretive in X x X. More generally, if S ( t ) is quasicontractive, then - A , is w-accretive. Keeping in mind the theory of C,-semigroups of contractions, one might suspect that there is an one-to-one correspondence between the class of continuous semigroups of contractions and that of m-accretive subsets. As seen in Theorem 1.3, if X is a Banach space and A is an w-accretive mapping satisfying the range condition (1.141, then for every y o E D(A ) the Cauchy problem (1.15) has a unique mild solution y ( t ) = S,(t)y, given by the exponential formula (1.161, i.e., ( 1.72)

S A ( t ) y O= lim n+m

For this reason, we shall also denote S,(t) by e-,'. We have: Proposition 1.2. c = D(A ) .

S,
Proot It is obvious that conditions (ii)-(iv) are satisfied as a consequence of the existence theorem, Theorem 1.3. To prove (i), we note that, for a fixed s > 0, y l ( t ) = SA(t + s ) x and y 2 ( t ) = S,(t)S,(s)x are both mild solutions to problem dY

dt

+Ay=O,

t 20,

and so by uniqueness y , = y , . Let us assume now that X, X * are uniformly convex and that A is an *accretive set that is closed and satisfies the condition (1.521, i.e., convD(A) c

n

R(I

0 < A < A,,

Then, by Theorem 1.7, for every x the right on [0, +m) and -A'X

=

lim f

10

+ AA) E

(1.73)

D ( A ) , S,(t)x is differentiable from

S,(t)X - x t

forsome A, > 0.

vx E D(A ) .

Hence -Ao c A,, where A, is the infinitesimal generator of S,(t).

W

4.1. The Basic Existence Results

231

As a matter of fact we may prove in this case the following partial extension of Hille-Yosida-Philips theorem to continuous semigroups of contractions. Proposition 1.3. Let X and X * be uniformly convex and let A be an o-accretive and closed set of X X Xsatisbing condition (1.73). Then there is a continuous o-quasicontractwe semigroup S ( t ) on D(A ) whose generator A , coincides with -Ao.

Prooj For simplicity, we shall assume that w = 0. We have already seen that A' (the minimal section of A ) is single valued, everywhere defined on D ( A ) ,and -A,x = Aox Vx E D ( A ) . Here, A , is the infinitesimal generator of the semigroup S,(t) defined on D(A ) by the exponential formula (1.72). We shall prove that D ( A , ) = D ( A ) . Let x E D ( A , ) . Then h

h10

and by the semigroup property (i) it follows that t + S,(t)x is Lipschitz on every compact interval [0, TI. Hence, t + S,(t)x is a.e. differentiable on (0,a) and d -SA(t)x dt

=AOSA(t)x

a.e. t > 0.

Now, since y ( t ) = S,(t)x is a mild solution to (1.15) that is a.e. differentiable and ( d / d t ) y ( O )= A,x it follows by Theorem 1.5 that S,(t)x is a strong solution to (1.159, i.e., d -S,(t)x dt

+ AoS,(t)x 3 o

a.e. t > 0.

Now, -A,x

=

lim

h10

JhA0S,(t)xdt, h o

and this implies as in the proof of Theorem 1.6 that x E D ( A ) and -A,x E Ax (as seen in the proof of Theorem 1.7, we may assume that A is demiclosed). This completes the proof. If X is a Hilbert space it has been proved by Y. Komura [2] that every continuous semigroup of contractions S ( t ) on a closed convex set C c X is

232

4. Nonlinear Accretive Differential Equations

generated by an m-accretive set A , i.e., there is an m-accretive set A c X X X such that -Ao is the infinitesimal generator of S ( t ) . This result has been extended by S. Reich [l] to uniformly convex Banach spaces X with uniformly convex dual X*. Remark 1.4. There is a simple way due to Dafermos and Slemrod [ll to transform the nonhomogeneous Cauchy problem (1.1) into a homogeneous problem. Let us assume that f E L'(0, w; X ) and denote by Y the product space Y = X X L'(0, CQ; X ) endowed with the norm Il(x,f)llv

Let d:Y

-+

=

llxll + ~ m l l f ( t ) l l d t , ( x , f )

E

Y.

Y be the (multivalued) operator

= q x , f ) = {Rr - f ( O > , - f ' ) , D(d)

=D

(A)

X

(x,f) E D ( 4 , W ' . ' ( [ O , w ) ;X ) ,

where f ' = df/dt. It is readily seen that if y is a solution to problem (1.1) then Y ( t ) = ( y ( t ) ,f,),f,(s) = f ( t s), is the solution to the homogeneous Cauchy problem

+

d -Y(t) dt

+A Y ( t ) 3 0, Y(0)=

t 2 0,

(Y07.f).

On the other hand, if A is w-m-accretive in X

X

X, so is d in Y

X

Y.

4.1.5. Nonlinear Evolution Associated with Subgradient Operators

We shall study here problem (1.1) in the case in which A is the subdifferential dcp of a lower semicontinuous convex function cp from a Hilbert space H to R = ] - a,+m]. In other words, consider the problem dY

dt + M y ) 3 f ( t ) Y ( 0 ) = Yo

7

in (O,T),

(1.74)

in a real Hilbert space H with the scalar product (.,. ) and norm I * I. It turns out that the nonlinear semigroup generated by A = dcp on D(A ) has regularity properties that in the linear case are characteristic to analytic semigroups.

4.1. The Basic Existence Results

233

If cp: H + R is a lower semicontinuous, convex function, then its subdifferential A = dcp is maximal monotone (equivalently, m-accretive) in H X H and D ( A ) = D(cp) (see Section 2.1 in Chapter 2). Then, by Theorem 1.2, for every y o E D(A ) and f E L'(0, T ; H ) the Cauchy problem (1.74) has a unique mild solution y E C([O,TI; H ) , which is a strong solution if y o E D ( A ) and f E W','([O,TI; H ) (Theorem 1.4). Theorem 1.10 following amounts to saying that y remains a strong solution to (1.74) on every interval [ 6, TI even if y o 6Z D ( A ) and f is not absolutely continuous. In other words, the evolution generated by dcp has a smoothing effect on initial data and on the right hand side f of (1.74). Theorem 1.10. Let f E L2(0,T;H ) and y o E D ( A ) . Then the mild solution y to problem (1.1) belongs to W ' T ~S (, TI; [ H ) for evey 0 < S < T , and

y(t) ED(A) t'/2 dy dt

E~

a.e. t E ( O , T ) ,

( 0T ,; H )

c p ( u )E ~ ( 0T ), ,

dY -(t) dt

+ d p ( y ( t ) )3 f ( t )

dY dt

L2(0,T;H),

-E

(1.75)

a.e. t E ( 0 , T ) .

cp(y) E W ' * ' ( [ O , T ] ) .

( 1.76)

(1.77)

( 1.78)

The main ingredient of the proof is the following chain rule differentiation lemma. Lemma 1.4. Let u E W',2([0, TI; H ) and g E L2(0,T ; H ) be such that g ( t ) E dcp(u(t)) a.e. t E (0,T). Then the function t -, cp(u(t))is absolutely continuous on [0,TI and

Prooj

Let cpA be the regularization of cp, i.e., 2

Iu - vl

+cp(u);utH),

uEH, A>O.

234

4. Nonlinear Accretive Differential Equations

We recall (see Theorem 2.2 in Chapter 2) that cpA is Frechet differentiable on H and

vcpA= (dcp),

=

A-1([

- (1

+ Adcp)-'),

> 0.

Obviously, the function t -+ cpA(u(t))is absolutely continuous (in fact, it belongs to W','([O,TI; H ) ) and

Hence

Now, by taking u = u(to + follows.

E)

we get the opposite inequality, and so (1.79)

Proofof Theorem 1.10. Let x o be an element of D(dcp) and yo E dcp(xo). If we replace the function cp by + ( y ) = cp(y) - cp(xo) - ( Y O ,u - xo), Eq. (1.74) reads

Hence, without any loss of generality, we may assume that min{ cp( u ) ; u

E

H}

= cp(

x o ) = 0.

4.1. The Basic Existence Results

235

Let us assume first that y o E N d q ) and f E W'g2([0,TI; H I , i.e., df/dt E L2(0,T;H ) . Then, by Theorem 1.2, the Cauchy problem (1.74) has a unique strong solution y E W'*m([O, TI; H). The idea of the proof is to obtain a priori estimates in W'**([ 6,TI; H ) for y , and after this to pass to the limit together with the initial values and forcing term f. To this end, we multiply Eq. (1.74) by t d y / d t . By Lemma 1.4, we have a.e. t Hence

and, therefore,

E

(0,T).

4. Nonlinear Accretive Differential Equations

236

Now, combining estimates (1.80) and (1.81), we get

Multiplying Eq. (1.74) by dy/dt and integrating on (0, t ) , we get

Hence,

Now, let us assume that y o E D(dp) and f E L2(0,T ; H I . Then there exist sequences (yo"}c D(dp) and {fJ c W's2([0,TI; H ) such that yo" + y o in H and f, + f in L2(0,T ; H ) as n + a.Denote by y , E W'."([O,TI; H ) the corresponding solutions to (1.74). Since d p is monotone, we have (see Proposition 1.1)

dY, + dY dt

dt

in g ' ( 0 , T ; H),

and by estimate (1.82) it follows that t ' I 2 dy/dt E L2(0,T ; H ) . Hence, y is absolutely continuous on every interval [ 6 , TI and y E W'p2([6,TI; H ) for all 0 < S < T . Moreover, by estimate (1.81) we deduce by virtue of Fatou's lemma that p ( y ) E L'(0, T ) and

We may infer, therefore, that y satisfies estimates (1.81) and (1.82). Moreover, y satisfies Eq. (1.77). Indeed, we have $lY,(t) - XI2 I $lY,(

S)

-XI2

+ /'(f,(T) S

- W ,Y , ( T ) - X

)

d7

4.1. The Basic Existence Results

for all 0

Is

IT

237

and all [ x , w ] E d q . This yields

and, therefore,

Letting s

+

t we get, a.e. t E (0, T ) ,

for all [ x , w ] E A , and since A = d q is maximal monotone this implies that y ( t ) E D ( A ) and ( d / d t ) y ( t ) E f ( t ) - A y ( t ) a.e. t E (0, TI, as desired. Assume now that y o E D(q).We choose in this case y," = ( I + n-l d q ) - ' y o and note that y," + y o as n + m, and

and, letting n

-+

01,

we find the estimate

(1.84) because q is lower semicontinuous. This completes the proof of Theorem 1.10.

W

In the sequel, we shall denote by W'*P((O,T];H ) , 1 IP I01, the space of all y E LP(0, T; H ) such that d y / d t E L p ( 6 ,T; H ) for every 6 E (0, T ) .

4. Nonlinear Accretive Differential Equations

238

Theorem 1.11. Assume that y o E D(A ) and f solution y to problem (1.74) satisfies dY tdt

E

L”(0,m; H ) ,

d+ -y(r) dt

y(t)

E

+ ( A y ( t ) -f(t))’

E

W’x ‘(10, TI; H ) . Then the

Vt

D(A), =

0

Vr

€

E (07

TI,

(O,T].

(1.85)

( 1.86)

By Eq. (1.74), we have

Proofi

d -ly(t dt

+ h ) -y(t)l

IIf(t

+ h ) -f(t)l

a.e. t , t

+ h E (0,T).

Hence,

This yields -s

2

’dI

-(t)

dt

(* I

: 1 ’1

S

-(s)

+s

([IgldT)’

a.e.O<s < t < T .

Then integrating from 0 to t and using estimate (1.821, we get

In particular, it follows by (1.88) that lim sup h-0

( y t+

-y(t)

1

< 03

Vt

E

[O,T].

h>O

Hence, the weak closure E of ( ( y ( t + h ) - y ( t ) ) / h } for h + 0 is nonempty for every t E [O, TI. Let 77 be an element of E. We have proved earlier the inequality

4.1. The Basic Existence Results

for all [x, w ]E d p and t , t (777

239

+ h E (0, TI. This yields

Y ( t ) - x ) I( f ( t ) - w ,u ( t ) - x)

Vt

E

(07 T I ,

and since [ x , w ]is arbitrary in d q we conclude that y ( t ) E D ( A ) and f ( t ) - 77 E A y ( t ) . Hence, y ( t ) E D ( A ) for every t E (0, T ) . Then, by Theorem 1.6, it follows that

because for every E > 0 sufficiently small y ( ~E) D ( A ) and so (1.89) holds for all t > E . This completes the proof of Theorem 1.11. In particular, it follows by Theorem 1.11 that the semigroup S ( t ) generated by A = d p on D( A ) maps D( A ) into D ( A ) for all t > 0 and d+ t -S(t)yo Idt

1

IC

Vt

> 0.

More precisely, we have: Corollary 1.2. Let S ( t ) be the continuous semigroup of contractions generated by A = dp. Then S(t)D(A ) c D ( A ) for all t > 0, and =

IAoS(t)yolIlA0xl +

for all y o E D( A ) and x

1

-Ixt

-yol

Vt

> 0, (1.90)

E D(A).

Prooj Multiplying Eq. (1.74) (where f = 0) by d y / d t and integrating on (0, t ) , we get

Next, we multiply the same equation by y ( t ) - x and integrate on (0, t). We get

240

4. Nonlinear Accretive Differential Equations

Combining these two inequalities, we obtain 1 2

- ( I y ( 0 ) - X I 2 - ly( t )

- XI')

+t(4X)-dY(t))

<

1 - ( l y ( O ) - XI2 - l y ( t ) - XI2) 2 + t ( AOx, x - y ( t ) )

1 < ?IY(O) - - I 2

-

+

t21A0xl2

V t > 0.

Since by (1.87) the function t + Kdy/dtXt)l (and consequently t 4 K d + / d t ) y(t)l) is monotonically decreasing, this implies (1.90), thereby completing the proof. rn Remark 1.5. Theorems 1.10 and 1.11clearly remain true for equations of the form

Y ( 0 ) = Yo 7 where w E R. The proof is exactly the same.

4.2. Approximation and Convergence of Nonlinear Evolutions and Semigroups 4.2.1. The Trotter-Kato Theorem for Nonlinear Evolutions

Consider in a general Banach space X a sequence A,, of subsets of X x X. The set lim inf A, is defined as [x, y ] E lim inf A,, if there are sequences x,, y , such that y , E A,x,,, x,, 4x and y , + y as n 4 00. If the A,, are wm-accretive there is a simple resolvent characterization of lim inf A, . Proposition 2.1. Let A,, lim inf A,, if and only if

for 0 < A < w - ' .

+ w l be m-accretive for n = 1,2,. .. . Then A C

4.2. Approximation and Convergence of Nonlinear Evolutions

241

Proof: Assume that (2.1) holds and let [ x , y ] E A be arbitrary but fixed. Then we have

+ hy) = x

( I + AA)-'(x

Vh E ( 0 , u-')

and, by (2.0,

(I + AA,)-'(x

+ hy)

-+

(I

+ h A ) - ' ( x + hy) = x .

In other words, x , = (I + AA,)-'(x + Ay) x as n CQ and x, + hy, = x + hy, y , E Ax,. Hence, y, -+ y as n -+ CQ, and so [ x , y] E lim inf A,. Conversely, let us assume now that A c lim inf A , . Let x be arbitrary in X and let x , = (I + AA)-'x, i.e., -+

x,

+ hy,

Then, there are [ x , , y,] We have x,

+ hy,

=x,

E A, = z,

where yo E Ax,.

such that x ,

-+

x,

-+

+ Ay,

-+

=x

x , and y,

as n

-+

--+

yo as n

-+

w.

w.

Hence

(I+ AA,)-'x

-+

x,

=

( I + AA)-'Y,

for 0 < A <

W-l,

as claimed. Theorem 2.1 following is the nonlinear version of the Trotter-Kato theorem from the theory of C,-semigroups (see T. Kato [l]). Theorem 2.1. Let A, be u-accretive in X X X , f n 1,2, . .. . Let y, be the mild solution to

dYfl

dt + A n y ,

3fn

in [ O , T I ,

E

L'(0; T ; X ) for n

Y,(O) =yo".

=

(2.2)

Let A c lim inf A, and

Then y , ( t ) (1.1).

-+

y(t) uniformly on [0, TI, wherey is the mild solution to problem

Proof: Let Did0 = t o ,t,, .. .,t,; f,", .. . ,f ; ) be an &-discretization of problem (2.2) and let Di(0 = t o , t , , ...,t,; f , , . . . ,f , ) be the correspond-

4. Nonlinear Accretive Differential Equations

242

ing &-discretizationfor (1.1). We shall take ti = iE for all i. Let y,, be the corresponding capproximate solutions, i.e., y,,,(t)

where y,:

= y,"

=y:,,,

, y,"

y,(t)

=y:,

qti-1,41,

i

I,. ..,N ,

= yo, and

+ EA,~:,, 3;;y: + ~f:, y: + &Ay: 3y:-1 + &fi,

y:,

By the definition of lim inf A,, for every such that

Ilj ;,,

fort

-yill

+ Ilwi,,

EW)

+~

=

(2 *4)

1)...)N .

(2.5)

for n

2 ~ ( T ,EI ), .

(2.6)

Ay:. Then, using the o-accretivity of

E

-1

=

> 0 there is [ j j ; , , , w;,,]E A,

T,I

i

+

I(1 -

i

- well IT,I

Here, w: = (l/&Xy:-' ef, - y:) A,, by (2.4142.6) it follows that lljj:,n -Y;J

and y,

-i-l

IIY,,,

-y:l;

( -1~ ~ ) - l l l f i -ni l l

+ CET,I

Vi,

for n 2 S(T,I, E ) . This yields ~Ijj:,,

+

-~:,,II

ICT,I CE

-Y:II

ICT,I CE

i

C (1 -

-~"II,

i

=

1,..., N .

C (1 - E W ) - " I I ~ / - ~ " I I ,

i

=

1,..., N ,

EW)-"II~/

k= 1

Hence,

II~;,,

+

1

k= 1

for n 2 ~ ( E , T , I ) . We have shown, therefore, that, for n 2 Ilye,n(t> - Y ~ ( ~ ) I 5 I

c ( .+ jbTllfn(t)

where C is independent of n and Now, we have

E.

a(&,71,

- f ( t ) d~ t ~)

~t

E

[ O ~ T I (2.7) ,

243

4.2. Approximation and Convergence of Nonlinear Evolutions

Let 7 be arbitrary but fixed. Then, by Theorem 1.1,

Ily,(t) -y(t)ll I 77 V t

if0 < E < ~ ~ ( 7 7 ) .

[(),TI,

E

Also, by estimate (1.35) in the proof of Theorem 1.1, we have llYE,,(f)

- Y,(t)ll

for all 0 < E < ~ ~ ( 7 where 7) (2.81, we have

IIy,(t) -y(t)II

IC

I 77

cl(r))does

Vt E [O, TI,

not depend of n. Thus, by (2.7) and

i + kT 77

IIf"(t) -f(t)ll dt

1

Vt

E

[(),TI

for all n sufficiently large. This completes the proof of Theorem 2.1. Corollary 2.1. Let A be w-m-accretive, f E L'(0, T ; X I , and y o E D( A ) . Let y, E C'([O, TI; X ) be the solution to the approximating Cauchy problem

dY +A,y = f dt

-

in [ O , T ] ,

y(0) = y o ,

where A , = A - ' ( Z - ( I + A A ) - ' ) . Then lim,+ y,(t) on [0,TI, where y is the mild solution to problem (1.1).

1 0 < A < -, 0

= y ( t ) uniformly in

Prooj It is easily seen that A c lim inf,, o A , . Indeed, for a we set

x,

=

(I

+ aAJ1x,

u =(I

(2.9)

E

t

(O,l/w)

+ aA)-'x.

After some calculation, we see that

Subtracting this equation from u of A, we get

+ aAu 3 x

and using the w-accretivity

Hence, limA,o x, = u = (I + a A ) - ' x for 0 < a < l/A, and so we may apply Theorem 2.1. W

Remark 2.1. If X is a Hilbert space and S, is the semigroup generated by A,, then condition (2.1) is equivalent to the following one (H. BrCzis

244

4. Nonlinear Accretive Differential Equations

[61): For every x E D(A ) , 3(x,,} c D ( A , ) such that x,, -+ x and S,,(t)x,, S(t)x V t > 0, where S O ) is the semigroup generated by A. -+

Theorem 2.1 is useful in proving the convergence of many approximation schemes for problem (1.1). If A is a nonlinear partial differential operator on a certain space of functions defined on a domain R c R", then very often the A,, arise as finite element approximations of A on a subspace X,, of X (see e.g., H . T. Banks and K. Kunisch [l], and F. Kappel and W. Schappacher [ll. 4.2.2. The Nonlinear Chemoff Theorem and Lie-Trotter Products We shall prove here the nonlinear version of the famous Chernoff theorem (Chernoff [l]), along with some implications for the convergence of the Lie-Trotter product formula for nonlinear semigroups of contractions.

Theorem 2.2. Let X be a real Banach space, A be an accretive operator satisfying the range condition (1.14), and let C = D( A ) be convex. For each t > 0, let F(t):C -+ C satisfy: (i)

IIF(t)x - F(t)ull IIIx

Then, for each x

E

-

uII

Vx,u

E

C and t

E

[O,T];

C and t > 0, (2.10)

uniformly in t on compact intervals. Here, SA(t)is the semigroup generated by A on C = D(A ) . The main ingredient of the proof is the following convergence result.

Proposition 2.2. Let C c X b e nonempty, closed, and convex, let F : C nonexpanswe, and let h > 0. Then the Cauchy problem du dt

- + h-'(I has a unique solution u

E

-F)u =

0

~ ( 0 =) X

E

C,

C'([O,a);X I , u ( t ) E C , for all t 2 0.

-+

C

(2.11)

4.2. Approximation and Convergence of Nonlinear Evolutions

245

Moreover, the following estimate holds:

for all n 2 0. In particular, for t

=

nh we have

IIF"x - u(nh)ll In'/211x - Fxll,

n

=

1 , 2 , . . ., t 2 0.

(2.13)

Proofi The initial value problem (2.11) can be written equivalently as

u( t )

e-'Ihx

=

+

t

e-(r-s)/hFu( s ) ds

V t 2 0,

and it has a unique solution u ( t ) E C V t 2 0, by the Banach fixed point theorem. Making the substitution t -, t / h , we can reduce the problem to the case h = 1. Multiplying Eq. (2.11) by w E J ( u ( t ) - x ) , we get d -llu(t) - xII IIlFx - xII dt

a.e. t > 0,

because I - F is accretive. Hence, Ilu(t)

-XI1

I tllFx

-41

(2.14)

V t 2 0.

On the other hand, we have u ( t ) - F"x

=

and

e-'(x

-

F"x)

+ /'e"'(Fu(s)

- F " x ) ds

0

n

IIx - F"xll I

C I I F ~ - ' ~- F ~ ~ I II nllx - FXII

Vn.

k= 1

Hence, Ilu(t) - F"xll Ine-'llx - Fxll

+ /'e'-'Ilu(s) 0

- F"-'xll

ds.

We set cp,(t)= Ilu(t) - F'xlI IIx - Fxll-'e'. Then, we have cp,(t) In

+ / 0' ~ " - ~ ( xds)

Vt 2 0, n

=

1 , 2,...,

(2.15)

and, by (2.14), rp,,(t) I te'

V t 2 0.

(2.16)

246

4. Nonlinear Accretive Differential Equations

Solving (2.19, (2.161, we get

Since tn+j+l

(j

+ l)!(n

(n+ j

-

+ l)!

l)! 7

we get

= C ~ t Ik n - k l s

C

( n -k k! ) 2 t k ] " 2 e f / 2

[k=O

k=O

Hence

as claimed.

rn

Proof of Theorem 2.2. We set A h = h - ' ( I - F ( h ) ) and denote by s h ( t ) the semigroup generated by Ah on C = D ( A ) . We shall also use the notation Jh

=

(I

+ AA)-',

Since J,"x + J,x V x E C, as h every x E C, sh(t)x

+

SA(t)x

+ 0,

J!

=

(I

+ AA,)-'.

it follows by Theorem 2.1 that for

uniformly in t on compact intervals. (2.17)

247

4.2. Approximation and Convergence of Nonlinear Evolutions

Next, by Proposition 2.2 we have S,(nh)x

-

F"(h)xll s IIS,(nh)J,hx IIIx - J,hxll(2

Now we fix x

E

D ( A ) and h

=

- F"(h)J,hxIl

+ A-'hn'/').

+ 211x

-

J,hxll

n - ' t . Then the previous inequality yields

+ A - ' t n - ' / * ) ( l l x - J , x \ ~+ I I J ~ / " x- J,xII) V t > 0 , A > 0. < (2 + A - ' t n - ' / ' ) ( A ~ A x+ IIJL/"X - J,xll) <

(2

Finally, IlS,,,(t)x

-

F" ( i ) x l l < 2A(Ax(+ t n - ' / ' ( h (

+(2

+ A-'tn-'/')11J~/"x- J,xI~ V t > 0 , A > 0.

Now fix A > 0 such that 2 A b 1 (2

I~

+ A-'tnp'/2)11Jhf/"x

-

/ 3 Then, . by (ii), we have J,xll I~ / for 3 n > N( E ) ,

and so by (2.17) and (2.18) we conclude that, for n

i: 1

+

llS,(t)x

-

F" - x Now, since

SA(t)x

(2.18)

+ 03,

uniformly in t on every [ 0 ,T I .

S,(t)yll IIx -yl

v t 2 0, x , y

E

(2.19)

c,

and

(2.19) extends to all x E D(A ) = C. The proof of Theorem 2.2 is complete.

a

Remark 2.2. The conclusion of Theorem 2.2 remains unchanged if A is *accretive, satisfies the range condition (1.14), and F ( t ) : C + C are Lipschitz with Lipschitz constant L ( t ) = 1 + o t + o ( t ) as t + 0. The proof is essentially the same and relies on an appropriate estimate of the form (2.13) for Lipschitz mappings on C.

248

4. Nonlinear Accretive Differential Equations

Given two m-accretive operators A, B c X x X such that A m-accretive, one might expect that S,+,(t)x

=

lim

x

n-m

V t 2 0,

+B

is

(2.20)

for all x E D( A ) n D( B ) . This is the Lie-Trotter product formula and one knows that it is true for C,-semigroups of contractions and in other situations (see A. Pazy [ l ]p. , 92). It is readily seen that (2.20) is equivalent to the convergence of the fractional step method scheme for the Cauchy problem dY dt

- + Ay + B y

3

Y ( 0 ) = Yo 1

in [ O , T ] ,

0

(2.21)

7

.e ., dY dt

- + Ay

3

0

in [i., ( i + l ) ~ ]i ,= 0,1,. . . , N - 1 , T i

y + ( i & )= z ( E ) ,

=

0,1, ..., N - 1, (2.22)

Y+(O) =Yo, dz

-

dt

+ Bz 3 0

= NE,

in [0, E ] , (2.23)

z(0) = y - ( i E ) .

In a general Banach space, the Lie-Trotter formula (2.20) is not convergent even for regular operators B unless S,(t) admits a graph infinitesimal generator A, i.e., for all [ x , y ] E A there is xh .+ x as h + 0 such that h - ' ( x h - S,(h)x) + y (Benilan and Ismail [l]).However, there are known several situations in which formula (2.20) is true and one is described in Theorem 2.3 following.

Theorem 2.3. Let X and X * be uniformly convex and let A , B be m-accrefive single valued operators on X such that A + B is m-accretive and S,(t), S,(t) map D( A ) n D( B ) into itself. Then S , + , ( ~ ) X = lim n-m

x

V x € D ( A ) n D ( B ) , (2.24)

and the limit is uniform in t on compact intervals.

4.2. Approximation and Convergence of Nonlinear Evolutions

Proof

249

We shall verify the hypotheses of Theorem 2.2, where F ( t ) = and C = D ( A ) n D ( B ) . To prove (ii), it suffices to show that

SA(t)sB(t)

lim t

x - F(t)x

io

X

=AX

+ BX

vx E D ( A ) n D ( B ) .

(2.25)

Indeed, if t

and x, =

( I + A(A + B ) ) - ' x

then we have x,

A

+ -t( X I

and, respectively, x,

(2.26)

-F(t)x,) = x

+ A h , + ABx,

(2.27)

=x.

Subtracting (2.26) from (2.271, we may write x, - x o

A

+ -((I t

- F(t)x, -

(I - F(t)x,))

Multiplying this by J ( x , - x,), where J is the duality mapping of X,and using (2.25) and the accretiveness of I - F ( t ) , it follows that

,

Hence, lim, x , = x , , which implies (ii). To prove (2.23, we write t - ' ( x - F ( t ) x ) as t-'(x - F(t)x) = t-'(x - SA(t)x)

Since t - ' ( x - S A ( t ) x ) + A x as t that =

-+

+ t-'(SA(t)x

- SA(t)SB(t)x).

0 (Theorem 1.71, it remains to prove

t-' ( S A ( t ) x - S , ( t ) S , ( t ) x )

+ Bx

as t

+

0.

(2.28)

250

4. Nonlinear Accretive Differential Equations

Since S,(t) is nonexpansive, we have

llzlll

llBxll

I t-'IISB(t)x - XI1

V t > 0.

(2.29)

On the other hand, inasmuch as I - S,(t) is accretive, we have

Vu

Let t,

--f

E

C , t > 0. (2.30)

- z. Then, by (2.301, we have

0 be such that z,

( A u + Bx - A x - z , J ( u -x)) 2 0

Vu E D(A),

and since A is m-accretive, this implies that Ax + z - Bx = A x , i.e., = Bx. By (2.29), recalling that X is uniformly convex, it follows that 2," + Bx (strongly). Then (2.28) follows, and the proof of Theorem 2.3 is rn complete. z

Remark 2.3. Theorem 2.3, which is essentially due to H. BrCzis and A. Pazy [ll was extended by Y . Kobayashi [2] to multivalued operators A and B in a Hilbert space H . More precisely, if A , B and A B are maximal monotone and if there is a nonempty closed convex set C c D(A ) n D(B ) such that ( I + A A ) - ' C c C and ( I + AB)-'C c C VA > 0, then

+

S A + B ( t ) x = lim n-m

uniformly in t on compact intervals.

4.2.3. Null Controllability of Nonlinear Accretive Equations Let A be an m-accretive subset of X Consider the controlled system dY dt

-

+ Ay 3 u

X

X , where X is a Banach space.

in R + = [0, +m)

Y(0) = Y o ?

where y o E D(A ) and u ZP= { u

E

E

(2.31)

Lt,,(R+; X ) . Consider the set

L"(R+;X ) ; Ilu(t)ll I p a.e. t > 0).

(2.32)

4.2. Approximation and Convergence of Nonlinear Evolutions

251

The parameter function u is called the control and the corresponding mild solution y = y " ( t >to problem (2.31) is called the state of the system. A problem of great interest is to find u E 2YP that steers y o into the origin in a finite time T . If this happens, we say that the system (2.31) is gPnull controllable. Let us define the mapping sgn: X + X , (2.33) It is readily seen that this mapping is m-accretive in X x X . Indeed, for every f E X the equation has the unique solution (2.34)

and after some computation we see that

Proposition 2.3. Assume that 0 E A 0 and that the operator A + p sgn is m-accretive in X x X . Then the feedback law u = - p sgn y steers y o E D(A ) into the origin in a finite time T I p-'llyoll, i.e., the solution y to the system dY +Ay dt

-

+ psgn y 3 0

in R ', (2.35)

Y(0) =Yo, has the support in [O, T I , i.e., y ( t ) = 0 for t 2 T .

Proof If A + p sgn is m-accretive then the Cauchy problem (2.35) has a unique mild solution y E C(R+;X I , given by y ( t ) = lim y,(t) &+

0

(2.36)

V t 2 0,

uniformly on compact intervals, where ye is the solution to the difference equation

y,( t )

+ &Aye(t ) + ~p sgn ye( t ) 3 y,( ye( t )

= yo

t

-

for t 2 E) Vt I 0.

E,

(2.37)

252

4. Nonlinear Accretive Differential Equations

Multiplying this by O,(t)

E

sgn y,(t), we get

Ily,(t)ll + Ep

Vt 2

IIly,(t - s)ll

8,

because A is accretive. This yields Y,(~E)

+~

E

k

I P Ilyoll,

=

0,1,. . .,

and so y , ( k s ) = 0 for k s 2 p-'lly0ll. Now, letting E tend to zero, it follows by (2.36) that y ( t ) = 0 for t 2 p-'lly0ll, as claimed. Proposition 2.3, which can be deduced from a more general result regarding the finite extinction time property for the solution to nonlinear m-accretive equations y ' By 3 0 with int BO # 0, provides a simple feedback law that steers all initial states yo E D ( A ) into the origin in finite time. We do not know whether A p sgn is m-accretive in a general Banach space X. However, this happens if X * is uniformly convex (see Proposition 2.4 following) and in several other significant situations that will be discussed later.

+

+

Proposition 2.4. Let X * be uniformly convex and A be m-accretive. Then A + p sgn is m-accretive.

boo$ Without loss of generality, we may assume that 0 E AO. Put B = psgn. By (2.341, we see that the Yosida approximation BA of B is given by

Hence,

and so by Proposition 3.8 in Chapter 2 we infer that A as claimed. rn

+ B is m-accretive,

By Proposition 2.3 we may conclude, therefore, that if X * is uniformly convex then the system (2.31) is 2ZP null controllable.

4.2. Approximation and Convergence of Nonlinear Evolutions

253

4.2.4. Compact Evolutions

Let A be anw-m-accretive mapping in a Banach space X and let N t ) :D(A ) + D(A ) , t 2 0, be the semigroup generated by on D(A ) . The semigroup S ( t ) is said to be compact if S ( t ) is a compact mapping in X for each t > 0.

Theorem 2.4. The semigroup S ( t ) is compact two conditions are satisfied:

if and only if

two following

(i) JA = ( I + AA)-' is compact for all 0 < A < w - ' ; (ii) For each bounded subset M c D(A ) and to > 0, lim, ,oS(t)x= S(to)xuniformly in x E M (i.e., S ( t ) is equicontinuous). ~

Prooj Let us assume first that (i) and (ii) are satisfied. By inequality (1.11) in Theorem 1.1, we have

for all [ x , y ] E A and t > 0. Since

we get

IIS(t)yo - xII I eW1llyo- xII

If we take x

= JAyo and

y

= AAyo,we

get

254

4. Nonlinear Accretive Differential Equations

and this yields

for all A,t,s > 0. If in the latter inequality that zero, it follows by assumption (ii) that Vs

lim JAS(s ) y o = S( s ) y o

A-0

=

A and let A tend to

> 0,

uniformly on every bounded subset of D ( A ) . Since JA is compact we conclude that S(s) is compact for every s > 0, as claimed. Assume now that S ( t ) is compact. Again by inequality (1.10, we have

This yields lim S ( t ) J A x = JAx

1-0

Vt

> 0,

uniformly in x on bounded subsets. Hence, JA is compact for every A > 0. Regarding (ii), it follows by a standard argument we do not reproduce w here.

4.3. Applications to Partial Differential Equations

255

An example of compact semigroup is that generated by A = dcp on

cp: H -+ R is a lower semicontinuous convex function on a Hilbert space H such that for all A, R > 0 the set { x E H; llxll IR , cp(x) IA) is compact. Indeed, it is readily seen that in this case JA = (Z + hA)-' is compact because

D(A ) , where

where y o E D(cp). On the other hand, by Corollary 1.2 we have

IS(t)yo- S ( t

+ &)yoIIE l A O X l + -lIx t &

-yo11

V& > 0, t

'0,

for all yo E D(A ) and x E D ( A ) . This clearly implies that S ( t ) is equicontinuous. Regarding the compactness of evolutions generated by w-m-accretive operators, we mention the following result due to Baras [l]. Theorem 2.5. Let A be w-m-accretive in a Banach space X and let Q:D(A ) X L'(0, T ; X ) + L'(0, T ; X ) be the evolution generated by A , i.e., Q ( y o , f ) = y where y is the mild solution to (1.1). Zf the semigroup S ( t ) generated by A on D(A ) is compact then Q is compact, too, from D(A ) X L'(0, T ; X ) to L'(0, T ; X ) .

4.3. Applications to Partial Differential Equations In this section we present a variety of applications to nonlinear partial differential equations illustrating the ideas and general existence theory developed in the previous sections.

4.3.1 Semilinear Parabolic Equations

Let p c R X R be a maximal monotone graph such that 0 E D( p), and let R be an open and bounded subset of RN with a sufficiently smooth boundary dR (for instance, of class C2). Consider the boundary value

256

4. Nonlinear Accretive Differential Equations

problem in R x ( 0 , T ) = Q ,

y(x,O) =yo(x) y=o

vx

E

a,

in dR x ( 0 , T ) = 2

(3.1)

where yo E L2(R) and f E L2(Q). We may represent (3.1) as a nonlinear differential equation in the space H = L2(R):

dY +Ay 3 f in[O,T], dt Y(0) = Yo where A: L 2 ( 0 )+ L 2 ( 0 )is the operator defined by -

(3 4

9

Ay D(A)

= =

-Ay + w ,w ( x ) E p ( y ( x ) ) a.e. x E R}, {y E H , ' ( ~ In ) H ~ ( R ) 3; w E L ~ ( o )W, ( X ) E p ( y ( ~ ) ) a.e. x E O}. (3.3) { z E L2(R); z

=

Recall (see Proposition 2.10 in Chapter 2) that A is maximal monotone (i.e., m-accretive) in L2(R) X L 2 ( 0 )and A = drp, where

rp(Y) and dg

=

=

3 /n lVYI2 dX + /R g ( y ) dX,

Y

E L2(R),

p. Moreover, we have

IlYIlH2(n) + IlYllH;(n) IC(IIAoyllL2(n)+ 1)

vy

ED(A).

(3.4)

Writing Eq. (3.1) in the form (3.2), we view its solution y as a function of c from [O,Tl to L2(R). The boundary conditions that appear in (3.1) are implicitly incorporated into problem (3.2) through the condition y(t) E D ( A ) vt E [O, TI. The function y: R X [O, TI + R is called a strong solution to problem (3.1) if y: [O, TI + L2(R) is continuous on [O, TI, absolutely continuous on (0, T ) , and satisfies d -dty ( x , t )

-

Ay(x,t) + P(y(x,t)) 3 f ( x , t )

a.e. t a.e. x a.e. x

E

( O , T ) , x E R,

E

R,

E

dR, t E ( 0 , T ) . (3.5)

4.3. Applications to Partial Differential Equations

257

Here, ( d / d t ) y is the strong derivative of y : [0, TI + L2(sZ)and b y is considered in the sense of distributions on a. As a matter of fact, it is readily seen that if y is absolutely continuous from [ a ,b ] to L'(R), then dy/dt = d y / d t in 9 ' ( ( a , b); L'(R)), and so a strong solution to Eq. (3.1) satisfies this equation in the sense of distributions in (0, T ) X a. For this reason, whenever there will not be any danger of confusion we shall write d y / d t instead of dy/dt.

Proposition 3.1. Let y o E L 2 ( f l ) and f such that

E

L2(0,T ; L 2 ( a ) )= L 2 ( Q ) be

y o ( x ) E o(p) a.e. x E a. Then problem (3.1) has a unique strong solution y E C([O, TI; L 2 ( a ) )fl W'*'((O,TI; L2(a)) that satisfies t 1 I 2dy E L2(0, T ; L2( a ) ) . dt (3.6)

t'/'y E L2(0,T ; H J ( a ) n H 2 ( a))

If, in addition, f E Wl9'([O,TI; L 2 ( a ) )then y ( t ) E H,'(R) n H2(a)for every t E (0,Tl and dY dt

t - E L"(0, T ; L2(

a)).

(3.7)

I f y o E H,'(a),g ( y o ) E L'(R), a n d f E L2(0,T ; L2(sZ>>, then dY dt

- E ~ ~ (T ;0~ , ~

( a ) y) E, ~ " ( 0T ,; H;( a ) )n ~

~ (T ;0H , ~a() ) . (3.8)

Finally, i f y o E D ( A ) a n d f E Wl9'([O, TI; L 2 ( f l ) ) then , dY dt

- E ~ " ( 0T,; L ~a( ) ) ,

y

E

~ " ( 0T ,; H ~ ( n H;( a)) (3.9)

a)

and

d+ -dt y(t)

+ ( - A y ( t ) + P(y(t)) -f(t))'

=

0

Vt

E

[O,T). (3.10)

Proof: This is direct consequence of Theorems 1.10, 1.11, and 1.6. Here, we have used the fact that D ( A ) = { u E L ~ ( R )u; ( x ) E D( p ) a.e. x E

a},

4. Nonlinear Accretive Differential Equations

258

which follows by Theorem 2.4 in Chapter 2 because

where A , = - A , D ( A , ) = H,'(R) n H 2 ( 0 ) , and p w sume that 0 E p(0)).

=

dg (we may as-

'

In particular, it follows that for y o E H d ( 0 ) , g ( y , ) E L ( 01, and f~ L2(R x (O,T)), the solution y to problem (3.1) belongs to the space H 2 , ' ( Q )= { y E L2(0,T ; H 2 ( 0 ) ) ,d y / d t E L 2 ( Q ) ) ,Q = 0 X (0, T ) . Problem (3.1) can be studied in the LP setting, 1 5 p < to, if one defines A : L P ( 0 ) + L P ( 0 ) as Ay

=

{ z E L P ( 0 ) ;z

D(A )

=

{y

E

=

-Ay

+ w , w ( x ) E p ( y ( x ) ) a.e. x E a),

(3.11)

Wi*P(0) n W2,P(0 ) ;w

E

LP( 0) such that

if p > 1, w( x ) E p ( y ( x ) ) a.e. x E 0) D ( A ) = { y E W d , ' ( 0 ) ;A y E L ' ( 0 ) ,3w E L ' ( 0 ) such that if p = 1 w ( x ) E p ( y ( x ) ) a.e. x E 0)

(3.12) (3.13)

As seen earlier (Proposition 3.9 in Chapter 21, the operator A is m-accretive in L P ( 0 ) x L P ( 0 ) and so the general existence theory is applicable. Proposition 3.2. Let y o E D ( A ) a n d f E W ' 3 ' ( [ 0TI; , L P ( 0 ) )1, < P < O0. Then problem (3.1) has a unique strong solution y E C([O, TI; L p ( 0 ) ) which , satisfies

d

zy

E

d' -y(t) dt

L ~ O , TL P; ( . ~ ) ) ,

y

E

~ " ( 0T, ; w;,P(~I) nw'.P(~)),

(3.14)

+ ( - A y ( t ) + p ( y ( t ) )-f(t))'

=

0

Vt

E

[O,T). (3.15)

Pro05 Proposition 3.2 follows by Theorem 1.6 (recall that X rn uniformly convex for 1 < p < w ) .

= L P ( 0 )is

If y o E D(A ) and f E L'(0, T ; L P ( 0 ) ) , then according to the general theorem, 1.1, the Cauchy problem (3.2), and implicitly problem (3.11, has a unique mild solution y E C([O, TI; LP(0)).

4.3. Applications to Partial Differential Equations

259

Since the space X = L ' ( R ) is not reflexive, by the general theory, the mild solution to the Cauchy problem (3.2) in L'(R) is only continuous, even if yo and f are regular. However, also in this case we have a differentiability property of mild solutions comparable with the situation encountered in the linear case (see Proposition 4.4 in Chapter 1). Proposition 3.3. Let p : R + R be a maximal monotone graph, 0 E D( /3 1, and p = dg. Let f E L2(0,T ; L"(R)) and yo E L ' ( R ) be such that y0(x) E o(p)a.e. x E R. Then the mild solution y E C([O,TI; L ' ( R ) ) to problem (3.1) satisfies

(3.17) Proof Without loss of generality, we may assume that 0 E p(0). Also, let us assume first that yo E H,'(R) n H2(R). Then, as seen in Proposition 3.1, the problem (3.1)has a unique strong solution such that t'/'y, E L2(Q), t ' l 2 y E L2(0,T ; H,'(R) n H2(R)), i.e., dY -(x, dt

t > - Ay(x, t )

+ P ( y ( x , t ) ) 3 f ( x , t ) a.e. ( x , t ) E Q ,

Y(X,O) =Yo(x), y=o

x E R, in dR X ( 0 , T ) .

(3.18)

Consider the linear problem di!

Az

=

Ilf(t)ll~=(n) in Q ,

z(x,O)

=

Iyo(x>l,

-

dt

-

z=o

R, in dR X ( 0 , T ) .

x E

(3.19)

4. Nonlinear Accretive Differential Equations

260

Subtracting these two equations and multiplying the resulting equation by

( y - z)+, after integration on R we get

I d

+

I V ( Y - z ) + 1' o!.x I

- - ~ ( y - z ) + 112Lqn)

2 dt

o a.e. t E ( o , T ) ,

( y - z ) + (0) I o

in R,

because z 2 0 and p is monotonically increasing. Hence, y ( x , t ) s A x , t ) a.e. in Q and so l y ( x , t)l I z ( x , t ) a.e. ( x , t ) E Q. On the other hand, the solution z to problem (3.19) can be represented as z(~,t= ) S(t)(IyoI)(x)

+ l r S ( t - S)(IIf(S)IILyn))

dS

0

a.e. ( x , t )

E

Q,

where S ( t ) is the semigroup generated on L'(R) by - A with Dirichlet homogeneous conditions on an. We know by Proposition 4.4 in Chapter 1 that IIS(t)u,llL=(n) I Ct-"211UOIIL'(n)

VU, E

L'(R),

> 0.

t

Hence, I y ( x , t ) l ICt-"'IIyolILl(n)

+

l

Ilf(S)lIL=(n)

dS,

(t,X)

E

Q. (3.20)

Now, for an arbitrary y o E L'(R) such that yo E D( p ) a.e. in R we choose a sequence {y,") c H,'(R) n H2(R), y," E D( p ) a.e. in Q, such that y," + y o in L'(R) as n + 03. (We may take, for instance, y," = S(n-'X(l + n - ' p ) - ' y o ) . ) If y, is the corresponding solution to the problem (3.11, then we know that y, + y strongly in C([O, TI; L'(R)), where y is the solution with the initial value y o . By (3.201, it follows that y satisfies the estimate (3.16). Since y ( t ) E L"(R) c L2(R) for all t > 0, it follows by Proposition 3.1 that y E W1*'([6, TI; L2(R)) n L2(S,T ; H,'(R) n H2(R))for all 0 < S < T and it satisfies Eq. (3.18) a. e. in Q = R X (0, TI. (Arguing as before, we may assume that y o E H,'(R) n H'(R) and so y,, y E L'(0, T; L2(R)).) To get the desired estimate (3.171, we multiply Eq. (3.18) by y,tk+' and integrate on Q to get

joT n/ tk+'y? dxdt + 51 o =

LTjnt

k+

' y ,f d x d t ,

n

t k + ' J V y l : dxdt

d

+ j T / tk+' - - g ( y ) o n

dt

dxdt

4.3. Applications to Partial Differential Equations

where y, = d y / d t and dg This yields

k+2

=

p.

tk+'lVy12dxdt

y/Q

I

+

1

261

jDTfk+'y,! dxdt

+ ( k + 2 ) / t k + ' g ( y )dxdt

+ -/

Q

1

2 Q

'

tk+'f dxdt.

Hence,

I(

k

+ 2 ) / tk+'lVy12dxdt + 2 ( k + 2 ) Q

+ T k + 2lQf

Q

tk+@(y)ydx

dxdt.

Finally, writing p ( y ) y as ( f get

+ A y -y,)y

and using Green's formula, we

+&tk+'lVylZ dxdt I( k

+ 2 ) ( k + 1) / y 2 t k dxdt + Tk+' lQfz dxdt Q

+ 2 ( k + 2 ) / tk+'lfllyldxdt Q

+ Tk+'/af2dxdt).

IC ( j Q t k y 2 d x d t

(3.21)

262

4. Nonlinear Accretive Differential Equations

Next, we have, by the Holder inequality

for p

=

2 N ( N - 2)-'. Then, by the Sobolev imbedding theorem,

On the other hand, multiplying Eq. (3.18) by signy and integrating on R x (0, t ) , we get

because, as seen earlier j a A y sign ydx I0.

Then, by estimates (3.21) and (3.221, we get

On the other hand we have, for k

=

N/2,

4.3. Applications to Partial Differential Equations

263

Substituting in the latter inequality, we get after some calculation t(Nt4)/2y:dxdt

+ 1t ( N + 2 ) / 2 1 V y (t)I2 x , dxdt Q

+ T ( N + 4 ) /,Ivy( /2 c,

(

I l Y O l4l L/ (~N( f+12))

x , T)12dx

+

41

)

(N+2)/2

f ( x , t)l dxdt

(3.23)

+ C 2 T ( N + 4 )JQf2(x,t) /2 dxdt, as claimed.

In particular, it follows by Proposition 3.3 that the semigroup S ( t ) generated by A (defined by (3.10, (3.13)) on L'(fl) has a smoothing effect on initial data, i.e., for all t > 0 it maps L'(R) into D ( A ) and is differentiable on (0,031. In the special case where if r > 0, if r = 0,

problem (3.1) reduces to the parabolic variational inequality (the obstacle problem)

y(x,O) =yo(x)

in 0,

y

=

0

in dR

X

( 0 , T ) = Z. (3.24)

More will be said about this in the next section. We also point out that Proposition 3.1 and partially Proposition 3.2 remain true for equations of the form dY

- AY + P ( X , Y ) dt

3 f

y(x70) =yo(x) y=o

in

Q,

in fl, in C,

264

4. Nonlinear Accretive Differential Equations

where p : R x R + 2R is of the form p ( x , y ) = d,g(x,y) and g : R X R + R is a normal convex integrand on R X R (see Section 2.2 in Chapter 2). The details are left to the reader. Now, we consider the equation dY

-

dt

d

-YdU

in R x ( 0 , T ) = Q,

-Ay=f

+ P(Y)

3

in 2 ,

0

y(x,O) = y o ( x )

in

a,

(3.25)

where p c R X R is a maximal monotone graph, 0 E D( p ) , y o E L2(R), and f E L2(Q).As seen earlier (Proposition 2.11 in Chapter 2) we may write (3.25) as

dY dt

- +Ay

where A y

an).

=

=f

in(O,T),

-Ay, V y E D ( A ) = { y E H2(R); 0 E d y / d v

More precisely, A

= dq,

+ p ( y ) a.e. in

where q : L2(R) + R is defined by

and d j = p. Then, applying Theorems 1.10 and 1.11, we get:

Proposition 3.4. Let y o E D(A ) and f E L2(Q).Then problem (3.25) has a unique strong solution y E C([O,TI; L2(R)) such that t’”

JY dt

-E

L2 ( 0 ,T ; L2(a)),

If y o E H’(R) and j ( y o ) E L’(R), then

t”’y

E L2(0,T

; H 2 ( R))

4.3. Applications to Partial Differential Equations

265

Finally, i f y o E D ( A ) a n d f , d f / d t E L'(0, T ; L 2 ( a ) ) ,then dY dt

- E Lye, T ; L2(

a)),

y

E L"(0, T

; H 2 (a ) )

and d+ -y(t) dt

-

V t E [0, T ) .

Ay(t) = f ( t )

It should be mentioned that one also uses here the estimate

An important special case is

Then, problem (3.25) reads 8Y

-

dt

inQ,

-Ay=f

JY y-==O, dV

y( x,O)

= y o (x

Y

20, dY dV

)

in

rO

inZ,

a.

(3.26)

A problem of this type arises in the control of a heat field. More generally, the thermostat control process is modeled by Eq. (3.29, where

ai 2 0, 8 , E R, i = 1,2. In the limit case, we obtain (3.26). The black body radiation heat emission on d a is described by Eq. (3.25) where p is given by (the Stefan-Boltzman law) a ( y 4 - y,")

for y r 0, for y < 0,

266

4. Nonlinear Accretive Differential Equations

whilst in the case of natural convection

P(Y)

=

j0

ay5I4

for y 2 0, for y < 0.

Note also that the Michaelis-Menten dynamic model of enzyme diffusion reaction is described by Eq. (3.0, where f

V

\@

for y < 0,

where A, k are positive constants. We note that more general boundary values problems of the form

JY

--

dt

AY + Y ( Y ) 3.f

where P and y are maximal monotone graphs in R X R such that 0 E D( P I , 0 E D(y),can be written in the form (3.2) where A = drp and rp: ,!,'(a)+ R is defined by

\ +w

otherwise,

and dg = y , d j = P. We may conclude, therefore, that for f E L 2 ( a )and y o E H'(R) such that g ( y o ) E L'(R), j ( y J E L ' ( d S 1 ) the preceding problem has a unique solution y E W','([O,TI; L'(Ln)) n L2(0,T ; H ' ( a ) ) . On the other hand, semilinear parabolic problems of the form (3.1) or (3.25) arise very often as feedback systems associated with the linear heat equation. For instance, the feedback control u

=

- p sign y ,

(3.27)

4.3. Applications to Partial Differential Equations

267

where sign r

=

if r # 0,

[ii

applied to the controlled heat equation in R x R', in dR

y=o y(x,O) = y , ( x )

in

X

R',

a,

(3.28)

transforms it into a nonlinear equation of the form (3.0, i.e., dY dt

in R

--Ay+psignysO

X

R',

in dR x R',

y=o

in R .

y(x,O) = y , ( x )

(3.29)

This is the closed loop system associated with the feedback law (3.27) and, according to Proposition 3.2, for every yo E L'(R) it has a unique strong solution y E C(R+; L2(R)) satisfying y ( t ) E L"(R) t(N'4)/4yI , t(N+4)/2yE L:,,,(R';

vt > 0, L2(a)).

Let us observe that the feedback control (3.27) belongs to constraint set { u E L"(a X R'); I I u I I ~ = ( ~ I ~ ~p +) and ) steers the initial state yo into the origin in a finite time T . Here is the argument. We shall assume first that yo E L"(R2) and consider the function w ( x , t ) = lly,ll~=(n,- pt. On the domain iR X (0, p-lIIyoll~-(n,>= Q,, we have dW

--

dt

Aw

+ psignw 3 0

w ( 0 ) = IlYollL=(n) w 2 0

in Q , , in R ,

(3.30)

in dR x (0, P-lllyollL-(n,).

Then subtracting equations (3.29) and (3.30) and multiplying by (y - w)' (or simply applying the maximum principle), we get (y - w ) + s0

in Q,.

268

4. Nonlinear Accretive Differential Equations

Hence, y I w in Q,. Similarly, it follows that y 2 -w in Q, and, therefore, E Qo.

Hence, y(t) = 0 for all t 2 T = p-'I1y0ll~=(n).Now if yo E L'(R), then applying in the system (3.28) the control u(t)

=

i"

t IE, for 0 I for t > E ,

-psigny(t)

we get a trajectory y(t) that steers yo into the origin in the time

where C is independent of E and yo (see estimate (3.16)). If we choose E > 0 that minimizes the right hand side of the latter inequality, we get

We have therefore proved the following null controllability result for the system (3.28): For any yo E L'(R) and p > 0 there is u E LYC! X R '), I I U I I L = ( ~ ~ R +< ) p, that steers yo into the origin in a finite time T(y,).

Proposition 3.5.

This result extends to controlled semilinear equations of the form (3.1) (see Barbu [ l o ] ) . Remark 3.2. Consider the nonlinear parabolic equation dY

--by at

+ IyIp-'y=O

Y(X,O) = Y,(X), y=o

a x R+, x E a, in

in d f l

X

R ' ,

(3.31)

where 0 < p < ( N + 2 ) / N and yo E L'(a).By Proposition 3.3, we know that the solution y satisfies the estimates Ily(t)llL=(n, I Ct-N/211yollL'(n) 9

Ily(t)llLl(n) I CllyOllLl(n)

7

4.3. Applications to Partial Differential Equations

269

for all t > 0. Now, if yo is a bounded Radon measure on R, i.e., y o E M(R) = (Co(fi))* (CO(a)is the space of continuous functions on fi

that vanish on an),there is a sequence {yi) c Co(R) such that I l y i l l ~ l ( ~ ) IC and y i + y o weak star in M(R). Then if y' is the corresponding solution to Eq. (3.31) it follows from the previous estimates that (see BrCzis and Friedman [ll) in Lq(Q), 1 < q < ( N

YJ + Y lyjlP-lyj

y

+1~1 p- 1

+ 2)/N,

in L'(Q).

This implies that y is a generalized (mild) solution to Eq. (3.31). If p > ( N + 2)/N, then there is no solution to (3.31). Remark 3.2. Consider the semilinear parabolic equation (3.1), where /3 is a continuous monotonically increasing function, f E Lp(Q), p > 1, and Yo E Wg,2-2/p(il), j(yo) E L'(R) where j(r> = 10'I p ( s ) l p - $ ( s ) d s . Then the solution y to problem (3.1) belongs to W(:'Q) and

c(

I I ~ I I L ; . ~ ( QI )

+ IIyOllt;g.2-2/qn) + j,j(Yo)

I I ~ I I ~ P ( ~ )

h ) . (3.32)

Here, Wp'.'(Q) is the space (y E Lp(Q); d r + s / d t ' dx?y E Lp(Q), 2r + s I2). For p = 2, W$ '(Q) = H2''(Q). Indeed, if we multiply Eq. (3.1) by I p ( ~ ) l ~ - ~ pwe ( y get ) the estimate (for f and yo smooth enough this problem has a unique solution y E W'*"([O,TI; LP(R)), y E LYO, T; W f ( i 2 ) )

where l/p

+ l/q

=

1. In particular, this implies that

II P(Y)IILP~Q) I C ( I l f l l ~ q ~+) Ilj(yo)llLl(~)) and by the LP estimates for linear parabolic equations (see, e.g., 0. A. Ladyzenskaya et al. [l] and A. Friedman [l]) we find the estimate (3.32), which clearly extends to all f~ LP(Q) and yo E Wg*2-2/p(Rz), j(yo) E L'(R).

270

4. Nonlinear Accretive Differential Equations

4.3.2. Parabolic Variational Inequalities

Here and throughout the sequel, V and H will be real Hilbert spaces such that V is dense in H and V c H c V’ algebraically and topologically. We shall denote by I * I and 11. II the norms of H and V , respectively, and by (-,. the scalar product in H and the pairing between V and its dual V’. The norm of V’ will be denoted II * ]I*. We are given a linear continuous and symmetric operator A from V to V’ satisfying the coercivity condition 2

( A y , y ) + ‘YIYI 2 wlly1I2

vy

E

(3.33)

V,

for some w > 0 and a E R. We are also given a lower semicontinuous convex function cp: v + i? = ] - m, +a], cp + m . For y o E V and f E L2(0,T ; V’),consider the following problem:

+

Findy

E

L2(0,T ; V ) n C([O, TI; H ) n W’+2([0, TI;V ’ ) such that

Here, y‘ = dy/dt is the strong derivative of the function y : [O, TI + V’. In terms of the subgradient mapping dcp: V + V’ problem (3.34) can be written as

This is an abstract variational inequality of parabolic type. In applications to partial differential equations, V is a Sobolev subspace of H = L2(R) (R is an open subset of RN), A is an elliptic operator on R,and the unknown function y : R x [0,TI + R is viewed as a function of f from [0,TI to L 2 ( f l ) .Finally, the derivative y’ is the partial derivative of y = y ( x , t ) in t in the sense of distributions. In the special case where cp = Z , is the indicator function of a closed convex subset K of V , i.e. cp(y)

=

0 if y

E

K,

cp(y) =

+m

if y

K,

(3.36)

4.3. Applications to Partial Differential Equations

271

the variational inequality (3.34) reduces to

Regarding existence for problem (3.341, we have: Theorem 3.1.

Let f

E

W','([O,TI; V ' ) and y o E V b e such that (3.38)

{AYO + dCO(Y0) -f(O)} n H + 0.

Then problem (3.34) has a unique solution y E W'92([0, TI; V ) n W1s"([O, TI; H ) and the map ( y o ,f 1 + y is Lipschitzfrom H X L'(0, T ; V ' ) to C([O, TI; H ) n L2(0,T ; V ) . Iff E W'*'([O,TI; V ' ) and dye) < 03, then theproblem (3.34) has a uniquesolutiony E W','([O,TI; H ) fl C,([O, TI; V ) . Iff E L2(0,T ; H ) and q ( y o ) < 00, then the problem (3.34) has a unique solution y E W','([O,TI; H ) n C,([O, TI; V ) ,which satisfies y'(t) = ( f ( t ) -Ay(t) - dq(y(t)))O Proof: Consider the operator L : D ( L ) c H

4

a.e. t

E

(0, T ) .

H,

VY E D ( L ) , Ly = { A y + d q ( y ) ) n H D ( L ) = { y E V ; { A y+ d q ( y ) } n H # 01. Note that al + L is maximal monotone in H X H ( I is the identity operator in H). Indeed, by hypothesis (3.33) the operator al + A is continuous and positive definite from V to V'. Since d q : V + V' is maximal monotone we infer by Theorem 1.5 (or Corollary 1.5) in Chapter 2 that al + L is maximal monotone from V to V' and, consequently, in H X H. Then, by Theorem 1.4, for every y o E D ( L ) and g E W'.'([O,TI; H ) the Cauchy problem dY dt

- + Ly

3

g

a.e. in (0, T ) ,

4. Nonlinear Accretive Differential Equations

272

has a unique strong solution y E W'*?'([O, TI; H ) . Let us observe that &pa = al + L, where qa:H + R is given by RAY)

=

W Y + aY,Y) + d Y )

YY E H .

(3.39)

Indeed, cp, is convex and lower semicontinuous in H because

and cp, is lower semicontinuous on V. On the other hand, it is readily seen that al L c &pa, and since al + L is maximal monotone,we inferthat a l + L = &pa, as claimed. In particular, this implies that D(L) = D(cp,) = D(q ) (in the topology of H). Now, let y o E V and f E W ' , 2 ( [ 0TI; , V ' ) ,satisfying condition (3.38). Let (yo"}c D ( L ) and {f,} c W'92([0,TI; H ) be such that

+

yo" + y o

strongly in H, weakly in V ,

f,,+ f

strongly in ~ ~ (T ;0v'), ,

f,' + f'

strongly in L 2 ( 0 ,T ; v')

Let y , E W'~m([O,TI; H ) be the corresponding solution to the Cauchy problem dYn

dt

+~

y 3fn ,

Y,(O) =yo" *

a.e. in ( O , T ) , (3.40)

If we multiply Eq. (3.40) by y , - y o and use condition (3.331, we get I d

- --ly,(t) 2 dt

- Y,I2

+ d y , , ( t )-Y J 2

where 6 E A y , + d p ( y o ) c V'. After some calculation involving Gronwall's lemma, this yields

273

4.3. Applications to Partial Differential Equations

Now we use the monotonicity of dcp along with condition (3.33) to get by (3.40) that

1 d 2 dt

- -ly,(t)

-ym(t)I2

+ wlly,(t)

-yrn(t)1I2

Ia I ~ n ( t )-Yrn(t)l2 + I l f n ( t )

- ~ r n ( ~ ) ~ ~ * ~ -~y Ym n( t() ~ )

a.e. t

E

(0,T).

Integrating on (0, t ) and using Gronwall's lemma, we obtain the inequality

(

IC ly,"

Thus, there is y

E

y,

j 0T l l ~ n ( r )- y m ( t ) 1 1 2 dt

+

lyn(t) -ym(t)I2

+ /"IIfn(t)

-y;I2

0

-frn(t)1I2

dt

C([O, TI; H ) n L2(0,T; V ) such that +

in

y

c([o, T I ;

1

*

H ) n ~ ~ (T ;0v ), .

(3.43)

Now, again using Eq. (3.40), we get I d --ly,(t 2 dt

+ h ) -y,(t)I2

-

+ wIly,(t + h ) -y,(t)l12

+ h ) -y,(t)I2 + Ilf,(t + h ) -f,(~)ll*IlY,(~

Ialy,(t

+ h)

-y,(t)ll

a.e. t , h

E

( 0 ,T ) .

This yields

IY,(~ + h ) - y,(t)I2 + / T - h 0

IC(ly,(h)

and, letting n tend to

Il~,(t + h ) - y,(t)ll2 dt

+ kT-hllfn(t + h ) -f,(t)Il:dt

-y$

+w,

ly(t + h ) - y(t)12 +

/T-h 0

IC(ly(h) - y J 2

Ily(t

+ h)

+ /"-'Ilf(t 0

- y(f)1I2

1

dt

+ h ) -f(t)ll:dt) Vt

E

[O,T - h ] . (3.44)

274

4. Nonlinear Accretive Differential Equations

Next, by (3.41) we see that if then we have I d

- -ly,(t)

2 dt

+ olly,(t)

-YJ2

Integrating and letting n

5 E Ay, + dq(y,)

-+ co

is such that f(0) -

5E H,

-Y,1I2

we get, by the Gronwall inequality,

Vt

E

[O,T] .

This yields Iy(t) -y,l

ct

I

Vt

E

[O,T].

Along with (3.44) the latter inequality implies that y is H-valued, absolutely continuous on [0, TI, and

a.e. t

E

(0, T ) .

Hence, y E W'7"([0, TI; H ) n W'92([0,TI; V ) . Let us show now that y satisfies equation (3.34) (equivalently, (3.35)). By (3.40) we have

a.e. t where z

E

D ( L ) and 77

E

Lz. This yields

E

(O,T),

275

4.3. Applications to Partial Differential Equations

and, letting n

+

00,

i(ly(t

+E)

- Zl2 -

5 f+'(f(s)

ly(t) - Z l 2 )

+ a y ( s ) - 77, y ( s )

- zds.

Finally, this yields (Y(t

l t + ? f ( s ) + UY(S)

+ E ) -Y(t),Y(t)

-z) I

-

%Y(S) - 2)

Since y is a.e. H-differentiable on (0, T ) , we get a.e. t E ( O , T ) , ( y ' ( t ) - a y ( t ) + 77 - f ( t ) , y ( t ) - z ) I0 for all [z, 771 E L. Now, since L is maximal monotone in H X H, we conclude that a.e. t E ( O , T ) , f(t) ~ y ' ( t )+ Ly(t)

as desired. Now, if (y6,fi), i = 1,2, satisfy condition (3.38) and the yi are the corresponding solution to Eq. (3.351, by assumption (3.33) it follows that lYl(t) - Y 2 W I 2 +

/ =0 l l Y , W

ly; -yi12

- Y2(t)ll2 dt

+ /Tllfl(t) 0

-f2(t)ll:dt)

V t E [O,T].

Now assume that f E W1y2([O,TI; V ' ) and yo E N q ) . Then, as seen earlier, we may rewrite Eq. (3.35) as y'

+ dqa(y) - a y 3 f Y(0)

a.e. t

-

(O,T), (3.45)

= Y o 7

where qa:H -+ R is defined by (3.39). For f have the estimate

d lY,'(t)12 + - p ( Y , ( t ) )

E

=

f, and yo = y," , y

,we

a d

5 ZIYfl(t)l2 I(ffl(t),YXt)) a.e. t

This yields

= y,

E

(0,T).

4. Nonlinear Accretive Differential Equations

276

Finally,

Then, arguing as before, we see that the function y given by (3.43) belongs to W'*2([0, TI; H ) n L"(0, T ; V ) and is a solution to Eq. (3.34). Since y E C([O,TI; H ) n L"(0, T ; V ) ,it is readily seen that y is weakly continuous from [0, TI to V. If f E LYO, T ; H ) and y o E D(qc,) we may apply Theorem 1.10 to Eq. (3.45) to arrive at the same result. Theorem 3.2.

Let y o E K and f

E

W ' T ~ ( [TI; O , V ' ) be given such that

(f(0) - A y , - & , y o - u ) 2 0

(3.46)

VUEK,

for some toE H . Then problem (3.37) has a unique solution y E W'."([O,TI; H ) n W ' * 2 ( [ 0TI; , V). If y o E K and f E W'v2([0,TI; V ' ) , then problem (3.37) has a unique solutiony E W',2([0,TI; H ) n C,([O, TI; V).Iff E L2(0,T ; H ) andyo E K , then problem (3.37) has a unique solution y E W'72([0, TI; H ) n C,([O, TI; V ) . Assume in addition that

( A y , y ) 2 ~llY1I2 for some o > 0, and that there is h

(I Then Ay

E

EH

VY

E

(3.47)

V,

such that

+ & A , ) - ' ( y + Eh) E K

V&> 0 , V y E K .

(3.48)

L2(0,T ; H ) .

Pro05 The first part of the theorem is an immediate consequence of Theorem 3.1. Now assume that f E L2(0,T ; H I , y o E K , and conditions (3.47), (3.48) hold. Let y E W',2([0, TI; H ) n C,([O, TI; V )be the solution to (3.37). If in (3.37) we take z = (I + & A H ) - ' ( y+ Eh) (we recall that A,y = Ay n H ) , we get ( Y ' W + N t ) , A & Y ( t )- ( I I (f(t),A,y(t)

where A, = A ( I

+ &A,)-'

+ cA,)-'h)

- ( I + &A,)-'h) =

&-'(I

-

a.e. t

E

(I + & A H ) - ' ) .Since

(O,T),

4.3. Applications to Partial Differential Equations

277

and

we get

a.e. t

E

(0,T).

Hence,

and by Proposition 1.3 in Chapter 2 we conclude that A y claimed. rn

E L'(0,

T ; H I , as

Now, we shall prove a variant of Theorem 3.1 in the case where

cp: V + R is lower semicontinuous on H. (It is easily seen that this

happens, for instance, if cp(u)/llull

-+ + w

as llull + 00.

Proposition 3.6. Let A: V + V f be a linear, continuous, symmem'c operator satisfying condition (3.33) and let cp: H + R be a lower semicontinuous convex function. Further, assume that there is C independent of E such that either ( A y , v p & ( y ) )2 -c(1 + IvV,(Y)l)(l + lyl)

vY

ED(AH),

(3*49)

or q((Z

+ & A , ) - ' ( y + Eh)) Icp(y) + C

VE > O V y

E

H , (3.50)

for some h E H , where A , = (YI+ A H . Then for every y o E D(cp) n V and every f E L'(0, T ; H I , problem (3.37) has a unique solution y E W','((O,TI; H ) n C([O, TI; H ) such that t'/'y' E L2(o,T ; H I , t'/'Ay E L'(0, T ; H ) . If y o E D(cp) n V , then y E W'/'([O,TI; H ) n C([O, TI; V ) . Finally, if y o E D ( A H )n D(dcp) and f E W'*'([O, TI; H I , then y E W'sm([O, TI; H ) .

278

Here,

4. Nonlinear Accretive Differential Equations 'p,

Proof

is the regularization of cp.

As seen previously, the operator

A,y

=

ay

+Ay

vy

E

D(A , )

=

D(A H ) ,

is maximal monotone in H x H . Then, by Theorems 1.6 (if condition (3.49) holds) and, respectively, 2.4 in Chapter 2 (under assumption (3.50)), A , + dcp is maximal monotone in H x H and M,YI

C(I(A,

I

Moreover, A ,

+~

V ) ~ ( Y+ )I YI I+ 1)

VY

E

D(A,) n~ ( d c p ) .

+ dcp = dcp", where (see (3.39))

and writing Eq. (3.35) as y'

+ dcpa(y) - a y 3 f

a.e. in ( O , T ) ,

Y(0) =Yo, it follows by Theorem 1.10 that there is a strong solution y to Eq. (3.39) satisfying the conditions of the theorem. Note, for instance, that if y o E D(cp) n V then y E W'.2([0,TI; H ) and cp"(y) E W'*'([O,TI). Since y is continuous from [0, TI to H this implies that y is weakly continuous from [0, TI to V. Now, since t cp"(y(t)) is continuous and cp:H -+ R is lower semicontinuous, we have -+

lim ( A Y ( t , ) , Y ( t , ) ) -,r

t,

and this implies that y

E

I

(AY(t),Y(t))

C([O, TI; V ) ,as claimed.

Vt

E

[O,TI,

W

Corollary 3.1. Let A : V -+ V' be a linear, continuous, and symmetric operator satisfying condition (3.33) and let K be a closed convex subset of H with (Z+cA,)-'(y+ch)~K

VE>O,VYEK,

(3.51)

for some h E H . Then, for evely y o E K and f E L2(0,T ; H ) , the variational inequality (3.37) has a unique solution y E W','([O, TI; H ) n C([O, TI; V ) n

279

4.3. Applications to Partial Differential Equations

L2(0,T ; D(A H1). Moreover, one has

where N K ( y ) c L2(R) is the normal cone at K in y .

As an example, consider the obstacle parabolic problem

aly

dY + a2 =0 dU

in C

=

dR x ( O , T ) ,

where R is an open bounded subset of RN with smooth boundary (of class C ' , ' , for instance), $ E H2(R), and a l , a2 2 0, a 1 + a2 > 0. This is a problem of the form (3.37), where H = L2(R), V = H'(R), and A E L(V, V ' ) is defined by

if a2 # 0, or (AY,Z)

=

/-n V y . V z h

if a2 = 0. (In this case, V = H,'(R), V' The set K c V is given by

K

=

{y

E

v y , z EHgl(R), =

(3.54)

H-'(R).)

H'(R); y ( x ) 2 $ ( x ) a.e. x

E

a),

(3.55)

and condition (3.51) is satisfied if (see Proposition 1.1 in Chapter 3) a,$

a* + a2 I0 dU

a.e. in dR.

(3.56)

280

4. Nonlinear Accretive Differential Equations

Note also that A,: D ( A , ) c L2(fl) + L2(fl) is defined by A,y

D(A , ) and IlYllH2(R)

a.e. in fl, V y E D ( A , ) ,

-Ay

=

i

i

dY

y E H'( fl); a I y + a2- = 0 a.e. in afl ,

=

dv

C(IIA,YllL2(n, + IlyllL2(n,)

v y E D(A,).

Then we may apply Corollary 3.1 to get: Corollary 3.2. Let f E L2(Q), y o E H'(fl) ( y o E Hd(fl) if a2 = 0) be such that y o 2 a.e. in fl. Assume also that E H 2 ( f l )satisfies condition (3.56). Then problem (3.52) has a unique solution y E W192([0,TI; L2(fl)) n L2(0,T ; H2(fl>)n C([O, TI; Hi(fl)).

+

+

Noticing that (see Section 1.1 in Chapter 3), N,(y) = {u E L2(fl); u ( x ) E P ( y ( x >- 1,!4x)) a.e. x E fl}, where P : R + 2R is given by P(r)

=

(i R-

r > 0, r = 0, r
it follows by Corollary 3.1 that the solution y satisfies the equation

d -dty ( t )

+ ( - A y ( t ) + P ( y ( t ) - +) - f ( t ) ) '

=

a.e. t E ( 0 , T ) .

0

Hence, the solution y to problem (3.52) given by Corollary 3.2 satisfies the system d

-Y(x,t) dt

-

AY(X,t) = f ( x , t ) a.e. in ( ( x ,t ) d

-y(x,t) dt

=

E

Q;Y ( X , t ) > + ( x ) } ,

m a x ( f ( x , t ) + A+(X),O}

a.e. in ( ( x , t ) ;y ( x , t )

=

+ ( x ) } , (3.57)

because y(., t) E H2(fl) and so A y ( x , t ) = A+(x>a.e. in ( y ( x ,t> = +(x>}. Since the previous problem can be equivalently written as

4.3. Applications to Partial Differential Equations

281

we may recover Corollary 3.2 as a particular case of Theorem 1.10 (see Proposition 3.1). It follows, therefore, that the solution y to the obstacle problem (3.52) is given by y(t)

=

limy,(t)

in C ( [ O , T ] ; L ~ ( ~ ) ) ,

E’O

where y, is the solution to penalized problem

a,y

+

cr2

dY

-=0 dv

in 2 .

(3.58)

Now let us consider the obstacle problem (3.52) with nonhomogeneous boundary conditions, i.e.,

dY dt

--Ay>f,

y 2 0

cry+ dY = g

in 2 ,

dv

=

= YO(X)

in

and fo: [O, TI

+

=

(0,T)

a,

where d a = rl U r2,r, n r, = 0, and g H ’ ( a ) ; y = 0 in r2},define A : V -+ I/’ by (Ay,z)

rl x

in C2 = r2x ( O , T ) ,

y=o Y(X,O)

inQ,

1nvy

V’ by

*

vzdr

(3.59) E L2(C,).If

+ a jr,yzdcT

we take V = (y E

vy, z

E

v,

282

4. Nonlinear Accretive Differential Equations

we may write problem (3.59) as

Vz E K, a.e. t

E

(3.60)

Y(0) =Yo, where F = f + fo Equivalently,

E

(O,T),

L2(0,T ; V ' )and K is defined by (3.55).

(3.61)

Vz E K , t E [ O , T ] .

Applying Theorem 3.2, we get: Corollary 3.3. Let f E w ' , ~ ( T[I o ; ,~ ~ ( ng)E) ,w ' * ~ (T [I ;oL2(rJ), , and y o E K . Then the problem (3.61) has a unique solution y E W1,2([0,TI; V ) r l C,([O, TI; V ) . If, in addition, dY 0

dV

+ ay,

=g(x,O)

a.e.in ( x

E

r l ;y o ( x ) > + ( x ) ) ,

(We note that condition (3.62) implies (3.46).) It is readily seen that the solution y to (3.61) satisfies (3.59) in a certain generalized sense. Indeed, assuming that the set E = { ( x , t ) ; y ( x , t ) > + ( x ) } is open and taking z = y ( x , t ) & prp in (3.611, where rp E C;(E) and

4.3. Applications to Partial Differential Equations

283

p is sufficiently small, we see that

dY

--

Ay = f

in g ’ ( E ) .

(3.63)

dY dt

Ay 2 f

in g ’ ( Q ) .

(3.64)

dt

It is also obvious that --

Regarding the boundary conditions, by (3.611, (3.631, and (3.64) it follows that dY

dV

+ ay =g

in g ’ ( E n X I ) ,

respectively

-+ ay 2 g dY

dV

in g ’ ( 2 , ) .

In other words,

a9

-

dv

+a@>g

in{(x,t)EC1;y(x,t)=9(x)}

Hence, if g satisfies the compatibility condition d* + a+< dV

gin&,

then the solution y to problem (3.61) satisfies l..e require1 boundary conditions on 2 , . Also in this case, the solution y given by Corollary 3.3 can be obtained as limit as E + 0 of the solution ye to the equation Y E

--

dt

AYE + P&(Y&-

9)=f

in R x ( O , T ) ,

Y & ( X , O ) = Y o ( x ) in

dyE dv

-

+ ay,=g

in Z,,

ye = 0

in C z , (3.65)

284

4. Nonlinear Accretive Differential Equations

where p J r ) = --(l/&)r- Vr E R. If Q + = { ( x , t ) E Q; y(x, t ) > $ ( x ) } , we may view y as the solution to free boundary problem dY

-

dt

-Ay=f

y(x,O) = y , ( x ) dY

a,y+a2-=0 dV

inZ,

y=$,

inQ+, in

a,

JY dv

- = -

dv

in dQ'(t), (3.66)

where dQ+(t) is the boundary of the set Q + ( t ) = { x E a;y ( x , t ) > $(XI}. We call dQ+(t) the moving boundary and dQ' the free boundary of the problem (3.66). In problem (3.66), the noncoincidence set Q + as well as the free boundary dQ' are not known a pn'on' and represent unknowns of the problem. In problems (3.37) or (3.61) the free boundary does not appear explicitly, but in this formulation the problem is nonlinear and multivalued. Perhaps the best-known example of a parabolic free boundary problem is the classical Stefan problem, which we will briefly describe in what follows and which has provided one of the principle motivations of the theory of parabolic variational inequalities. The Stefan Problem. This problem describes the conduction of heat in a medium involving a phase charge. To be more specific, consider a unit volume of ice R at temperature 8 < 0. If a uniform heat source of intensity F is applied, then the temperature increases at rate E / C , until it reaches the melting point 8 = 0. Then the temperature remains at zero until p units of heat have been supplied to transform the ice into water ( p is the latent heat). After all the ice has melted the temperature begins to increase at the rate h / C , ( C , and C , are specific heats of ice and water, respectively). During this process the variation of the internal energy e ( t ) is therefore given by where

e ( t ) = c(e(t)) + PH(O(t)),

4.3. Applications to Partial Differential Equations

285

and H is the Heaviside graph,

In other words, e=y(e)=

[1::

[O,p]

if 8 < 0, if8=0, if 8 > 0.

(3.67)

The function y is called the enthalpy of the system. Now, let Q = R X (0, and denote by Q - ,Q+ ,Q, the regions of Q where 8 < 0, 8 > 0, and 8 = 0, respectively. We set S + = dQ+ , S- = dQ- , and S = S + u S - . If 8 = 8 ( x , t ) is the temperature distribution in Q and q = q ( x , t> the heat flux, then, according to the Fourier law,

(3.68)

q ( x , t ) = -kVO(x,t),

where k is the thermal conductivity. Consider the function K(8)

=

k18 k,8

if 8 s 0, if 8 I0,

where k , , k , are the thermal conductivity of the ice and water, respectively. If f is the external heat source, then the conservation law yields

d

-

dt

/,.e ( x , t ) d x = -1

dR'

for any subdomain R* X ( t l , t , ) are smooth. Equivalently,

= -

(q(x,t),u)da+ C

+ / R- F ( x , t ) d x

Q ( v is the normal to d o * ) if e and q

/,,div q( x , t ) dx

where V ( t )= --N,/llNxIl is the true velocity of the interface S ( N ( N , , N,) is the unit normal to S) and E.1 is the jump along S.

=

286

4. Nonlinear Accretive Differential Equations

The previous inequality yields

(3.69) Taking into account Eqs. (3.67H3.691, we get the system de

C, - - k, A 8 = f

in Q - ,

dt

de

C, - - k, A0 dt

( k , VB+- k , V K , N,)

in Q + ,

=f =

in S,

pN,

e+= 8 - = o If we represent the interface S by the equation t read

( k ,V V - k, Ve-,Vu)

=

(3.70)

ins. =

(3.71)

u ( x ) , then Eqs. (3.71)

in S ,

-p

e+= e-= 0.

(3.72)

To Eqs. (3.70), (3.721, usual boundary and initial value conditions can be associated, for instance,

e=o e(x,O) = e,(x)

in dR x ( O , T ) ,

(3.73)

in R ,

(3.74)

or Neumann boundary conditions on dR. This is the classical two phase Stefan problem. Here we shall study with the methods of variational inequalities a simplified model described by the one phase Stefan problem de

- - A ~ = o dt

e=o

v,O(x,t).Vu(x)

=

-p

e=o e2o

in Q + = { ( x , t )

E

Q; u ( x ) < t < T } ,

in Q - = { ( x , t )

E

Q ;0

in S in S

= {(x,t);t =

U

in Q , .

( x ) ) ,

~ ( x ) } ,

Q-,

(3.75)

287

4.3. Applications to Partial Differential Equations

These equations model the melting of a body of ice R c R3 maintained at 0°C in contact with a region of water along a portion r2 of the boundary dR. So assume that dR = r, u r,, where rl and r2 have no common boundary and r, is in contact with a heating medium with temperature 8, ; t = a h ) is the equation of the interface (moving boundary) S,, which separates the liquid phase (water) and solid phase (ice) (see Fig. 4.1).Thus, to Eqs. (3.75) we must add the boundary conditions

8=0

in C2 = r2x (0, T )

(3.76)

and the initial value conditions

Instead of (3.761, we may impose the Dirichlet boundary conditions 8=u

in&,

We set R' = {x E R;0 < t < Ro = R, (see Fig. 4.2).

8=0

in&.

(3.76)'

a ( x ) } . Then, dR' = S , , and we assume that

There is a simple device due to G. Duvaut [l] that permits us to reduce problem (3.7W3.77) to a parabolic variational inequality via a Baiocchi

Figure 4.1.

288

4. Nonlinear Accretive Differential Equations

X

Figure 4.2.

transformation (see Section 1.2, Chapter 3). To this end, consider the function

\o

if ( x , t )

E

Q-,

and let

Lemma 3.1.

Let 8

E

H ' ( Q ) and u E H ' ( f l ) . Then (3.80)

where

x

is the characteristic function of Q . +

4.3. Applications to Partial Differential Equations

proof. By (3.78), we have

289

290

4. Nonlinear Accretive Differential Equations

-

because V,O(x, d x ) ) V d x ) = - p Vx we see that

E

s1 \ a,. Then, by Eqs. (3.73,

-

as claimed. By Lemma 3.1 we see that the function y satisfies the obstacle problem

in { ( x , t ) y =0 and the boundary value conditions dY

-=

dv

where i , ( x , t ) have:

=

-a(y -

il)

/; Ol(x,s ) d s

in X I ,

V(x,t)E

E

Q; ~ ( x >) t } ,

y

=

0

in Z 2 ,

(3.81)

(3.82)

X I . Then, by Corollary 3.2, we

Corollary 3.4. Let O l E L2(Xl) be giuen. Then the problem (3.81), (3.82) has a unique (generalized) solution y E W'v"([O,TI; L 2 ( f l ) )n W'.*([O,TI; H'(s1)).

Keeping in mind that S, = d ( ( x , t ) ; y ( x , t ) = O}, we can derive from Corollary 3.4 an existence result for the one phase Stefan problem (3.7343.77). Other mathematical models for physical problems involving free boundary such as the oxygen diffusion in an absorbing tissue (E. Magenes [l], C. M. Elliott and J. R. Ockendon [l]) or electrochemical machining processes lead by similar devices to parabolic variational inequalities of the same type.

4.3. Applications to Partial Differential Equations

291

4.3.3. The Nonlinear Diffusion Equation

We shall study here the nonlinear Cauchy problem dY

--

dt

ANY)

3f

y = o Y(X,O) = y o ( x )

in R

in dR in

( 0 , T )= Q,

X

X

a,

( 0 , T ) = C,

(3.83)

where s1 is a bounded and open subset of R"' with smooth boundary and p: R + 2R is a maximal monotone graph in R X R such that 0 E D( p). The function y E C([O, TI; L'(s1))is called a generalized solution to Eq. (3.83) if

for all cp E C 2 " ( e ) such that cp(x,T) = 0 in Cl and cp = 0 in 2. Let us first briefly describe some physical problems that lead to equation of this type. 1 . The flow of gases in porous media. Let y be the density of a gas that flows through a porous medium that occupies a domain R c R3. If p denotes the pressure, we have p = p o y a for some a 2 1. Then, the conservation law equation dY k , - + divyv = 0 dt

combined with Darcy's law

( k , is the porosity of the medium, k , the permeability, and v viscosity) yields the porous medium equation (3.85) where 6

=

k 2 ( k , ( a + l)-ypo)-'.

292

4. Nonlinear Accretive Differential Equations

2. Two phase Stefan problem. Consider the problem (3.70), (3.70, (3.73), (3.74), i.e.,

Clef - k, A8 = f

in Q - = { ( x , t ) ;8 ( x , t ) < 0 } ,

C,8, - k, A8 = f

in Q + = { ( x , t ) ; 8 ( x , t ) > 0},

(k2V8+-k, VO-,Va(x)) =

(3.86)

-p,

where t = a ( x ) is the equation of the interface S. We may write system (3.86) as d --y(8)

-

dt

where y : R

AK(8)

in (2,

3f

(3.87)

-, 2R is y(r) =

C1r

for r < 0,

[O, PI C,r + p

for r = 0, for r > 0,

(3.88a)

and K(8)

=

for 8 < 0,

Indeed, for every test function cp E Cr(Q) we have

+

/S((k,V8+- k, ve-,Vu) +

p ) ds

=

0.

(3.88b)

4.3. Applications to Partial Differential Equations

293

If we denote by p the function y-lK,i.e., for r < 0, for0 I r < ki'p,

Ci'(k,r

- p)

(3.89)

for r 2 k i ' p ,

we may write (3.87) in the form (3.83). Equation (3.83) is also relevent in the study of other mathematical models, such as population dynamics (Gurtin and MacCamy [l]). Problem (3.83) can be treated as a nonlinear accretive Cauchy problem in two functional spaces: H - ' ( R ) and L'(R). 3. The Hilbert Space Approach. In the space H-'(R), consider the operator

We shall assume that

p-' is everywhere defined and bounded on the bounded subsets of R. (3.90) Then, by Proposition 2.12 in Chapter 2, A is maximal monotone in H - ' ( R ) X H - ' ( R ) . More precisely, A = dcp where cp: H - ' ( R ) + R is defined by

where d j = p. We may write problem (3.83) as

dY dt

-+Ay3f

in(O,T), (3.91)

and so by Theorem 1.10, we obtain:

4. Nonlinear Accretive Differential Equations

294

Theorem 3.3. Let p be a maximal monotone graph in R X R satisfying condition (3.90). Let f E L'(0, T ; H-'(R)) and lety, E H-'(R) n L'(R) be such thaty,(x) E D( p ) a.e. x E 0. Then there is a uniquepairoffinctions y E C([O, TI; H-'(R)) n W ' 9 2 ( ( 0TI; , H-'(R)) and u: Q + R satisfying dY - A u = ~ dt

a.e. in Q

-

u(x,t) E

Y(X,O) t'/2

=Y,(X)

dY dt

E

Q,

u=0

in 2 , (3.92)

1 / 2 L~~ ~( o , T ; H , ' ( R ) ) .(3.93)

L'(R) then

E

u E L2 (0 ,T ; H i ( a)).

L2 ( 0 7T ; H-'(R)),

D ( A ) and f

R x (O,T),

a.e.in R.

dt

-E E

a.e. ( x , t )

dY E L ~ ( o , T ; H - ~ ( R ) ) ,t

Moreover, i f j ( y , )

Zfy,

P(y(x,t))

=

E

(3.94)

W','([O,TI; H-'(R)I7 then

dY dt

- E L"(O,T;H-'(R)),

U E

L"(O,T;Hd(R)).

H

(3.95)

We note that the derivative d y / d t in (3.92) is the strong derivative of the function t + y ( - ,t ) from [O, TI into H-'(R), and it coincides with the derivative dy/dt in the sense of distributions on Q. It is readily seen that the solution y provided by Theorem 3.3 is a generalized solution to (3.83) in the sense of definition (3.84).

The L'-Approach. In the space X A

=

=

L ' ( R ) ,consider the operator

{ [ y , ~E] L ' ( R ) X L'(S1); w U E

=

-Au,

W d . ' ( R ) , u ( x )E p ( y ( x ) )a.e. x

E

a}.

(3.96)

We have seen earlier (Proposition 3.10 in Chapter 2) that A is m-accretive in L ' ( R ) x L ' ( 0 ) . Then, applying the general existence Theorem 1.2, we obtain: Proposition 3.7. Let p be a maximal monotone graph in R x R such that 0 E p(0). Then for every f E L'(0, T ; L ' ( R ) ) and every y o E L ' ( R ) , y , ( x )

4.3. Applications to Partial Differential Equations E

D( p ) a.e. x

E

295

a, the Cauchyproblem dY +Ay dt

-

in(O,T),

3f

(3.97)

Y ( 0 ) =Yo7 has a unique mild solution y

E

C([O, TI; L'(R).

We note that D(A ) = { y o E L'(R); y o ( x )E D( p ) a.e. x E a).Indeed, ( 1 + ~ p ) - ' y+ ~ y o in L'(R) as E + 0, if y o ( x )E D( p ) a.e. x E R, and (1 + E A ) ~ ' ~-+ , y o if j ( y o ) E L'(R). Proposition 3.7 amounts to saying that in L'(R), uniformly on

y ( t ) = &lim y,(t) + 0

[o,~],

where y, is the solution to difference equations

u,(x,t)

E

P(y,(x,t))

y,(t) = y o

a.e.in

f o r t 5 0, x

E

R.

x (o,T), (3.98)

The function t + u,(t) E W,,','(R) is piecewise constant on [O,T] and f,(t) = f , V t E [ie,(i+ OE] is a piecewise constant approximation of f:[O, TI -+ L'(R). By (3.98) it is readily seen that y is a generalized solutiion to problem (3.83). In particular, it follows by proposition 3.7 that the operator A defined by (3.96) generates a semigroup of nonlinear contractions S(t): D(A ) + D(A ) . This semigroup is not differentiable in L'(R), but in some special situations it has regularity properties comparable with those of the semigroup generated by the Laplace operator on L'(R) under Dirichlet boundary conditions (see Proposition 4.4 in Chapter 1).

Theorem 3.4. Let p E C'(R ing the conditions

\

(0) n C(R) be a monotonefunction satisfy-

4. Nonlinear Accretive Differential Equations

296

where a > 0 if N I 2 and L"(R) for every t > 0,

(Y

> ( N - 2)/N i f N

Ils(t)y 0 11 L Y n ) < ct-N/(Na+2-N) and S(tXLP(R)) c LP(R) for all t

Proof

2 3. Then S(tXL'(R))c

I l y o l l2L/ ~ ( 2( + n ,N ( u - ' ) )

>0

Vt

> 0, (3.100)

and 1 I p < a.

We shall establish first the estimates

[ [ ( I+ h A ) - ' f l l , 4

Ilfll,P

+ Ch(/nl(I + h A ) - ' ~ ~ ' p " - ' ' N ~ ' N - 2 ) Vf E L P ( R ) , > 0,

(3.101)

for N > 2, and

(3.102)

-

if N = 2. Here, II [ I p is the LP norm in R, C is independent of p 2 1, and A is the operator defined by (3.96). We set u = (I + h A ) - ' f , i.e., in R , in dR.

u - hAp(u) =f p(u) = O

(3.103)

We recall that p ( u ) E Wb*q(R>,where 1 < q < N/(N - 2) (see Corollary 3.1 in Chapter 2). Multiplying Eq. (3.103) by (uIp-' sign u and integrating on R, we get

Now using the identity

and condition (3.991, we get jnIulP

dx +

4 W P - 1) (p+a-1)2

I jnlflPdx.

q

n

IVlul(p+"-')/2

I2 & (3.104)

297

4.3. Applications to Partial Differential Equations

On the other hand, by Sobolev imbedding theorem

c

( v l y J ( P + l - 4 / 2 1 2&

lyl(P+1-W/(N-2)

&

(N-2)/N

if N > 2,

and

for N = 2. Then, substituting these inequalities into (3.104), we get (3.101) and (3.1021, respectively. We set JA = (I + A A ) - * and denote q ( u ) = IIull;, J / ( u ) = p + a- 1 cllull(p+ a- l ) N / ( N - 2 ) .

Then inequality (3.101) can be written as V(JAf) +

AJ/(JAf)

'fE

I q(f)

LP(R).

This yields (P(Jhkf)

+ AJ/(J/f)

Summing these equations from k

Recalling that J:/,f

+ S(t)f

qp(S(t)f)

=

=

for n

44-Y)

1 to k

+ m,

=

Vk.

n and taking A = t / n yields

the last equation implies that

+ / '0J / ( S ( ~ ) f d~ ) = p(f)

(3.105)

V t 2 0.

In particular, it follows that the function t + cp(S(t)f) is decreasing and so is t + J / ( S ( t ) f ) . Then, by (3.105), we see that

V(S(t)f> + t J / ( S ( t ) f ) 5

i.e., IlS(t)fll,P

p + a- 1

df) vt >

+ c t l l ~ ( t ) f l l ( p + a -I ) N / ( N - ~ 5 ) Ilfll,P

Vt

0 9

> 0, (3.106)

where C is independent of p and f. Let P , , + ~= ( p , a - 1)N/(N - 2). Then, by (3.1061, we see that

+

4. Nonlinear Accretive Differential Equations

298

where t , = 0 and t, some calculation

+

> t, . Choosing t ,

+

- I,, =

t/2"+

:)

', we get after

P

limsup

I I S ( t ) f I l b ~ ~ ~ / N ) n p n +_
C I I ~ I I ~ ~ ( Vt

n+x

where p

=

> 0,

N/2. Since N

"=

"

(m) "

+

Nff

2(N-2)

N

((m)nP1)

(here, we have used the fact that a > ( N - 2)/N, we get 2 P o / ( 2 P o + N ( a - I ) ) t - N / ( 2 p o + N ( u - 1))

IlS(t)fll?: ICllfIlP"

vpo2 1,

as claimed. The case N = 2 follows similarly. Moreover, by by inequality (3.101) and the exponential formula defining S(t), it follows that IlS(t)fllp 5

Vf E L P ( f l )t, 2 0.

Ilfllp

This complete the proof of Theorem 3.4. 4.3.4. First Order Quasilinear Partial Differential Equa tions

We shall consider here the Cauchy problem dy

-+ dt

C i=l N

d

-ai(y) dxi

=

O

Y(X,O) =yo(x),

in RN X R + ,

x

E

RN.

(3.107)

in the space X = L1(RN),where a = ( a l , . . .,a , ) is a continuous map from R to RN satisfying the condition lim sup Ila( r)ll/lrl < CQ, lr1+0

and y o E L1(RN). We have seen earlier (see Proposition 3.11 in Chapter 2) that the first order differential operator y -, Cfl ( d / d x i ) a , ( y ) admits an m-accretive extension A c L ' ( R N )X L1(RN) defined as the closure in L ' ( R N )X L1(RN)of the operator A , given by Definition 3.2 (Chapter 2).

4.3. Applications to Partial Differential Equations

299

Then, by the generation theorem, 1.3, the Cauchy problem dY +Ay 3 0 dt Y ( 0 ) = Yo

in [0, + m ) ,

-

3

has for every y o E D(A ) a unique mild solution y ( t ) = S ( t ) y o given by the exponential formula (1.16) or, equivalently, y(t )

=

uniformly on compact intervals,

lim y,( t )

&+

0

where y, is the solution to difference equation E - l ( y & ( t ) -y,(t

-

0 y,(t) = y o

c)) + A y , ( t )

=

fort > E , for t < 0.

(3.108)

We call such a function y ( t ) = S ( t ) y o a semigroup solution to the Cauchy problem (3.107).

Theorem 3.5. Let S ( t ) be the semigroup of contractions generated by A on D ( A ) and let y = S ( t ) y , be the mild solution to problem (3.107). Then (i) S ( t ) L p ( R N )c L p ( R N )for all 1 r p Im and IIS(t)YolILp(RN)5 (ii) If y o E

IIYOIILp(RN)

VyO

D(A)

L p ( R N ) .(3.109)

D(A) n Lm(RN), then

jOTjR,(iy(X,

t ) - klcp,(x, t ) + signo(y(x9 t ) - k ) ( a ( y ( x ,t ) ) (3.110)

t 0 - a ( k ) , cpx(x, t ) ) ) d ~ d 2 for every cp T > 0.

E

C,"(RN x (0,T ) ) such that cp 2 0, and all k

E

RN and

Inequality (3.110) is Kruikov's [l] definition of generalized solution to the Cauchy problem (3.107).

Proof of Theorem 3.5. Since, as seen in the proof of Proposition 3.11 (Chapter 2), ( I + AA)-' maps L p ( R N )into itself and

II(Z + A A ) - ' u I I L ~ ( R N ) r I I ~ I I L ~ ( P ) VA > 0 , u E LP(R") for 1 r p r m, we deduce (i>by the exponential formula (1.16).

4. Nonlinear Accretive Differential Equations

300

On the other hand, we have (Y,(X, t - 8 ) - Y&(X,t ) ) Signo(Y,(x, t ) - k ) =

(y,(x,t -

8)

-

k ) sign,(y,(x,t)

-

k)

-(Y,(X, t ) - k ) Signo(Y,(x, t ) - k ) IZ , ( X , t

-

E )

- z,(x,t)

where z,(x, t ) = ly,(x, t ) - kl. Substituting this into (3.111) and integrating on RN X [0, TI, we get t ) - k ) (a(y,(x, t ) - 4 k ) , 43(x, t ) ) /6T/RNsigno(Y&(x, +&-1(Z,(X,t

-

E)

- z , ( x , t ) ) q ( x , t ) ) d x d t 2 0.

Now letting E tend to zero, we get (3.110) because y,(t) on [0, TI in L’(RN) and

/&/ly(

=&-I

0 RN

+ &-I &fO/‘/

0

RN

X,t)

N

/

-klq( x,t)dxdt-

ZE( x ,

t ) ( q( x , 1

+

+ y(t)

uniformly

ze(x , t ) q ( x,t)dudt

T--E R N E )

- q( x , t ) ) dxdt

ly(x,t) - k l q o , ( x , t dxdt. )

This complete the proof of Theorem 3.5. Equation (3.107) is known in literature as the law consemation equation and has a large spectrum of applications in fluid mechanics and applied

4.3. Applications to Partial Differential Equations

301

sciences. Controlled systems of the form in RN x Y(X,O) =yo(x),

x

E

(O,m),

RN.

arise in the mathematical description of traffic control (M. Slemrod [l]) and in other problems of practical interest. For such a system we may prove a controllability result similar to Propositon 2.3. More precisely, it turns out that (Barbu 1171) the feedback control u(t) = -psignoy(t) steers every initial data yo E D(A ) n Lm(RN)into the origin in a finite Here, sign'y = y / l l ~ l I ~ if~ (y ~#~0, and signO0 time T I p-lIIyollL=(R~). = 0. 4.3.5. Nonlinear Hyperbolic Equations

We are given two real Hilbert spaces V and H such that V c H c V' and the inclusion mapping of V into H is continuous and densely defined. We have denoted by V' the dual of V and H is identified with its own dual. As usual, we denote by II II and I * I the norms of V and H, respectively, and by ) the duality pairing between V and V' and the scalar product of H. We shall consider here the second order Cauchy problem

-

(.,a

3f(t)

in(O,T), (3.112)

where A is a linear continuous and symmetric operator from V to V' and B c V X V' is maximal monotone. We assume further that ( 4 , y ) + alyI2 2 wlly1I2

vy

E

V,

(3.113)

where w > 0 and a E R. One principal motivation and model for Eq. (3.112) is the nonlinear hyperbolic boundary value problem

4. Nonlinear Accretive Differential Equations

302

where /3 is a maximal monotone graph in R X R and C! is a bounded open subset of R"' with a smooth boundary. Regarding the general problem (3.1121, we have the following existence result: Let f

Theorem 3.6. such that

E

W'?'([O, TI; H ) and y o E V , y ,

E

D ( B ) be given

{ AY, + BY,} n H z 0.

Then there is a unique function y satisfies

E

(3.115)

W1,lr([0,TI; V ) n W2'm([0, TI; H ) that

(3.116)

where ( d + / d t ) d y / d t is considered in the topology of H and ( d + / d t ) y in V. Proof

Let X

V X H be the Hilbert space with the scalar product

=

w,

9

U2)

=

(Au,

7

u2)

+ a ( 4 ,u 2 ) + ( u ,

9

v2),

where U, = [ u , , u , ] , U2 = [u2,u21.In the space X, define the operator d : D ( d ) c X - X by

D ( d ) = { [ u , ~E] v x H ; { A u + Bu} n H z 0}, d [ u , u ] = [ - u ; ( A u + BU) n H ] (+[u,u], [ u , u ]E ~ ( d ) , (3.117)

+

where (+ = sup{a(u, u ) / ( ( A u ,u ) + aluI2 + 1 1 ~ 1 ~ ) ; u We may write Eq. (112) as

E

V , u E HI.

dz dt

- +Ay+Bz=f. Equivalently, d -U(t) dt

+dU(t)

-

( + U ( t )= F ( t ) ,

t

E

(O,T), (3.118)

4.3. Applications to Partial Differential Equations

303

where

u(t)= [ Y ( t ) , Z ( t ) l ,

F(t)

=

[O,f(t)l,

uo = [ Y , , Y l l .

It is easily seen that d is monotone in X X X. Let us show that it is maximal monotone, i.e., R ( I +d)= V X H where I is the unity operator in V X H . To this end, let [ g , h ] E V X H be arbitrarily given. Then the equation U d U = [ g , hl can be written as

+

y Z

Substituting y = (1 system, we obtain

Ay

-2

+ay =g,

+ BZ + a2 3 h .

+ o ) - ' ( z + g ) in the second equation of the previous

+ a ) z + (1 + a ) - ' A z + Bz 3 h - (1 + a ) - ' & . Under our assumptions, the operator z 3 (1 + a ) z + (1 + a)-'Az is continuous, positive, and coercive from V to V ' . Then R(T + B ) = V' (see (1

Corollary 1.5 in Chapter 2), and so the previous equation has a solution z E D(B)and a fortiori [ g , h ] E R ( I +d). Then the conclusions of Theorem 3.6 follow by Theorem 1.2 because there is a unique solution U E W',m([O,TI; V X H ) to problem (3.118) satisfying d+ -U(t) dt

+dU(t) - uU(t) 3 F(t)

vt E

[O,T),

i.e.,

where ( d + / d t ) y ( t ) is in the topology of V whilst ( d + / d t ) z ( t )is in the topology of H . rn The operator B that arises in Eq. (3.112) might be multivalued. For instance, if B = dcp where cp: V + R is a lower semicontinuous convex function, problem (3.1 12) reduces to a variational inequality of hyperbolic type. In order to apply Theorem 3.6 to the hyperbolic problem (3.1141, we take V = H,'(fl), H = L 2 ( 0 ) ,V' = H - ' ( O ) , A = - A , and B : H,'(fl) -,

4. Nonlinear Accretive Differential Equations

304

H-'(R) defined by B

=

H,'(R)

dcp where cp:

+

R is the function

The operator B is an extension of the operator ( B o y X x ) = (w E L2(R); w ( x ) E p ( y ( x ) ) a.e. x E R} from H,'(R) to H-'(R). More precisely (see H. BrCzis [5]),if p E d q ( y ) E H-'(R), then p is a bounded measure on R and p = p, dr + p, where the absolutely continuous part pa E L'(R) has the property that p a ( x ) E P ( y ( x ) ) a.e. x E R. If D( /3) = R, then p E L'(R) and p ( x ) E P ( y ( x ) ) a.e. x E R. On the other hand, it follows by Lemma 2.2 in Chapter 2 that if p E H-'(R) n L'(R) is such that p ( x ) E P ( y ( x ) ) a.e. x E R,then p E By. Then, applying Theorem 3.6, we get: Corollary 3.5. Let /3 be a maximal monotone graph in R X R and let B = d q , where rp is defined by (3.119). Let y o E H,'(R) n H 2 ( a ) , y1 E H,'(R), a n d f E L2(Q)be such that d f / d t E L2(Q)and

a.e. x

p o ( x )E p ( y , ( x ) )

Then there is a unique ficnction y

E

E

for some p o E L 2 ( R ) . (3.120)

R

N O , TI; H,'(R)) such that

*Y

(3.121)

- ELm(0,T;L2(R));

dt2

d+ dy

-- ( t )

dt

dt

-

Ay(t)

+B

3f(t)

y Assume further that D( P )

d + dy

--(t)

dt dt

where p ( x , t )

=

=

0

Vt

in dR

E

X

[O,T)

(0,T).

in 0, (3.122)

R. Then A y ( t ) E L'(R) for all t

E

[0, T ) and

x E R,t

E

[O,T),

- Ay(x,t) + p(x,t) =f(x,t),

(3.123) E

P((dy/dtXx,t))ax. x

E

R.

305

4.3. Applications to Partial Differential Equations

(We note that condition (3.120) implies (3.1151.) Problems of the form (3.114) arise frequently in mechanics. For instance, if P ( r ) = rlrl this equation models the behavior of elastic membrane with the resistance proportional to the velocity. As another example, consider the unilateral hyperbolic problem

d 2Y

dt2 2

y

=

d

A ~ + f inQ,

0

~

y

>

+ in^,

in dR x [ O , T ) , (3.124)

+

+

where E H2(R) is such that I 0 a.e. in dR. Clearly, we may write this variational inequality in the form (3.1121, where V = H,'(R), H = L2(R), A = - A , and B c H,'(R) X H-'(R) is defined by

for all u E D ( B ) = K = {u E H,'(R); u 2 Then, applying Theorem 3.6, we get:

a.e. in R}.

Corollary 3.6. Let f,f, E L2(Q)and y o E Hdn H2(R), y , E Hd
I/ , f ( x , t ) ( : ( x , t )

- u(x)

1

ak

Vt

E

[O,T),

Vu

E

K,

306

4. Nonlinear Accretive Differential Equations

The Nonlinear Wave Equation. We shall study now the problem

dY

y

=

in

- j p ' O ) =Yl(x)

Y(X,O) = y , ( x ) ,

in d f l x ( 0 , T ) = C,

0

a, (3.126)

where fl is a bounded and open subset of R"', with a sufficiently smooth boundary (of class C2, for instance), and g E W'."(R)satisfies the following conditions: (i)

Ig'(r)l _< L ( l

+ Irl')

a.e. r

E

R,

where 0 I p I 2/(N - 2 ) if N > 2, and p is any positive number if 15Ns2; (ii)

rg(r) 2 0

V r E R.

In the special case where g ( y ) = p I y I p y , assumptions (i), (ii) are satisfied for 0 2, and for p 2 0 if N I 2. For p = 2 this is the Klein-Gordon equation, which arises in the quantum field theory (see Reed and Simon [l]). In the sequel, we shall denote by I) the primitive of g, i.e., I+!&-) = /; g(t> dt. Let f, d f / d t

Theorem 3.7.

E

L 2 ( Q ) and y o E H,'(fl) flH2(fl), y l E Then under assumptions ( i ) , (ii) there is

H i ( n ) be such that $ ( y o ) E L'(R). a unique function y that satisfies

y

E

L"(o,T;H,'(R) n H ~ ( R ) n ) C'([O,T]~ ; ,'(fl)),

and d 2Y -

dt2

Ay

+ g ( y ) =f

a.e. in Q,

y(x,O) = y , ( x ) ,

dY

z ( x , O ) =y,(x)

a.e. x

E

fl. (3.128)

4.3. Applications to Partial Differential Equations

ProoJ

307

We shall write Eq. (3.126) as a first order differential equation in x L ~ ( R )i.e., ,

x = H,'(R)

dY dt

- -z =

0,

dz dt

- - Ay

+ g(y) =f

in [O,T].

(3.129)

308

4. Nonlinear Accretive Differential Equations

where 1/p + 1/S + 1/2 = 1. Now, we take in the latter inequality S = N and /3 = 2 N / ( N - 2). We get

(We have denoted by

II Ilp the LP norm.) This yields

and, therefore, IlG(Y1, 21) - G(Y2 ,z2)llx

I CIIy, - y,IIH;(n)(l

+ max(llylllEg(n), IIy211E;(n))) VY, ,Y,

as claimed.

E H,'(fq,

(3.131)

rn

Let r > 0 be arbitrary but fixed. Define the operator

6:X + X ,

By (3.131) we see that the operator 6 is Lipschitz on X. Hence, A , + G is w-m-accretive on X and by the standard existence theorem (Theorem 1.2) we conclude that the Cauchy problem

(3.132)

43. Applications to Partial Differential Equations

309

has a unique solution U E W',"([O,TI; X).This implies that there is a unique y E W',"([O,TI; H,'(R)) with dy/dt E W's"([O,TI; L2(R)) such that

(3.133)

Choose r such that IlyollH;(n) < r. Then there is an interval [0, T,] such that Ily(t)llH;(n) Ir for t E [O,T,] and Ily(t)ll > r for t > T,. Then we have

and multiplying this by y, and integrating on R equality

Since +(y) 2 0 and +(yo) lly,(t)Ilz I(llYlll:

E

X

(0, t ) , we get the energy

L'(R), by Gronwall's lemma we see that

+ Ilyoll2H;(n) + 2ll+(Yo)llLl(n))

'/2

310

4. Nonlinear Accretive Differential Equations

and, therefore,

The latter estimate shows that given yo E H,’(R), yI E L ( a),T > 0, and f~ L 2(QT ),there is a sufficiently large r such that Ily(t)ll~;(n)Ir for

t E [0,TI, i.e., T = T,. We may infer, therefore, that for r large enough the function y found as the solution to (3.133) is in fact a solution to Eq. (3.128) satisfying all the requirements of Theorem 3.7. The uniqueness of y satisfying (3.1271, (3.128) is the consequence of the fact that such a function is the solution (along with z = dy/dt) to the o-accretive differential equation (3.132). By the previous proof, it follows that if one merely assumes that

then there is a unique function y,

that satisfies Eq. (3.126) in a mild sense. However, if $(yo) 6L L’(R) or if one drops assumption (ii) then the solution to (3.126) exists only in a neighborhood of the origin. The same method can be used to obtain existence for the nonlinear wave equation with nonhomogeneous Dirichlet conditions,

y=u

in C.

(3.134)

Bibliographical Notes and Remarks

311

Bibliographical Notes and Remarks Section 1. The main result of Section 1.1 is due to M. G. Crandall and L. C. Evans [ l ] (see also M. G. Crandall [2]), whilst Theorem 1.3 has been previously proved by M. G. Crandall and T. Liggett [l]. The existence and uniqueness of integral solutions for problem (1.1) (see Theorem 1.2) is due to Ph. BCnilan [l]. Theorems 1.4-1.7 were established in a particular case in Hilbert space by Y. Komura [l] (see also T. Kato [2]) and later extended to Banach spaces with uniformly convex duals by M. G. Crandall and A. Pazy [l], 121. Note that the generation theorem, 1.3, remains true for @accretive operators satisfying the extended range condition (Kobayashi [Ill

1 lim inf -d( x , R( I h10 h

+ AA)) = 0

vx E

D(A ) ,

where d ( x , K ) is the distance from x to K (see also R. H. Martin [l]). These results have been partially extended to time-dependent accretive operators A ( t ) in the works of T. Kato [2], M. G. Crandall and A. Pazy [3], N. Pave1 [l], and K. Kobayashi et al. 111. The basic properties of continuous semigroups of contractions have been established by Y. Komura [2], T. Kato [4], and M. G. Crandall and A. Pazy [ l , 21. For other significant results of this theory, we refer the reader to author’s book [2]. The results of Section 1.5 are due to H. BrCzis [l, 41. These results were partially extended to the time-dependent case ( d / d t ) y + dcp(t,y) 3 0 in the works of J. C. Peralba [l], J. Moreau [3], H. Attouch and A. Damlamian [l, 21, J. Watanabe [l], N. Kenmochi [l], and Otani [l]. For some extensions of Theorem 1.10 to the subgradient differential equation (1.74) with a nonconvex function cp we refer to E. DeGiorgi et al. [l] and M. Degiovanni, et al. [l]. Section 2. Theorem 2.1 is essentially due to Kobayashi and Miyadera [ll. A number of related results have been obtained previously by Miyadera and Oharu [l], BrCzis and Pazy [l], J. Goldstein [l], and H. BrCzis [61 (see Remark 2.1). In the case A, = dp, , the convergence of (2.1) is equivalent to the convergence of cp, in the sense of Mosco, and the previous approximation results become more specific (see H. Attouch [ l , 21). Theorem 2.2 is due to BrCzis and Pazy [2] (see also Miyadera and Oharu [l]). Theorem 2.3 along with other results of this type have been established in the previously cited paper of BrCzis and Pazy [l]. For other related results we refer the reader to S. Reich [ l , 21 and E. Schecter [ll.

312

4. Nonlinear Accretive Differential Equations

There is a large literature on numerical approximation of solutions to Eq. (1.1) but we simply refer the reader to the book [ l ] of J. W. Jerome and the bibliography given there. Approximation schemes derived from the nonlinear Chernoff theorem or Lie-Trotter product formula were obtained in the works Y. Kobayashi [2], E. Magenes and C. Verdi [l], and E. Magenes [2]. Theorem 2.3 is due to H. BrCzis [6]. For other significant results in this direction and recent references, we refer to Vrabie's book [l]. The general theory of nonlinear semigroups of contractions and evolutions generated by accretive mappings is treated in the books of V. Barbu [2], R. H. Martin [2], K. Deimling [l], and in the survey of M. G. Crandall [2]. For other results such as asymptotic behavior and existence of periodic and almost periodic solutions to problem (l.l), we refer the reader to the monographs of A. Haraux [ l ] and Gh. MoroSanu [l]. Section 3. The theory of nonlinear accretive differential equations was first applied to semilinear parabolic equations and parabolic variational inequalities by H. BrCzis [2, 31. In the L' setting, these problems were studied by Ph. BCnilan [l], Y. Konishi [l], Massey [l], and L. C. Evans 111 (Proposition 3.3). There is an extensive literature on parabolic variational inequalities and the Stefan problem (see J. L. Lions [l], G. Duvaut and J. L. Lions [l], A. Friedman [l], C. M. Elliott and J. R. Ockendon [l], and A. M. Meirmanov [l] for significant results and complete references on this subject). Here we were primarily interested in the existence results that arise as direct consequences of the general theory developed previously, and we tried to put in perspective those models of free boundary problems which can be formulated as nonlinear differential equations of accretive type. The L'-space semigroup approach to the nonlinear diffusion equation (3.79) was initiated by Ph. BCnilan [l] (see also Y. Konishi [l]), whilst the H-' approach is due to H. BrCzis [2] (Theorem 3.3). The smoothing effect of semigroup generated by the nonlinear diffusion operator in L ' ( f l ) (Theorem 3.4) was discovered by Ph. BCnilan [2] and L. VCron [ l ] but the proof given here is essentially due to A. Pazy [l]. For other related contributions to existence and regularity of solutions to the porous medium equation, we refer to BCnilan, et al. [l], M. G. Crandall and M. Pierre [l], H. BrCzis and A. Friedman [l]. Other properties of the semigroup generated by - A/3 in L'(R) such as the existence of a finite extinction time were established by I. Diaz [2]. The null controllability of the system y, - Ap(y) 3 u is proved

Bibliographical Notes and Remarks

313

in V. Barbu [181. We refer to 0. Car@ [l]for other controllability results of nonlinear systems of accretive type. The semigroup approach to the conservation law equation (3.10) (Theorem 3.5) is due to M. G. Crandall [l].Theorem 3.6 along with other existence results for abstract hyperbolic equations have been established by H. BrCzis [3] (see also the author’s book [2] for other results of this type).

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Chapter 5

Optimal Control of Parabolic Variational Inequalities

In this chapter we will consider optimal control problems governed by semilinear parabolic equations and variational inequalities of parabolic type studied in the previous chapter. The main emphasis is put on first order necessary conditions of optimality. We will illustrate the breadth of applicability of these results on several problems involving the control of moving boundary in some physical problems. 5.1. Distributed Optimal Control Problems 5.1.1. Formulation of the Problem

Consider a general control process in R c RN governed by the abstract parabolic equation y’+Ay+p(y-$)

3Bu+f

inQ=Rx(O,T), in R,

Y(0) =Yo

(1.1)

and with the pay-off G ( y , u ) = J T ( d t 7 Y ( t ) )+ h ( u ( t ) ) )dt + cpo(Y(T)). 0

Here, y ’ ( t ) = ( d y / d t X t ) is the strong derivative of the function y : Q -, R viewed as a function of t from [0, TI to L2(R), and f E L2(Q), y o E L2(R) are given functions. Throughout in the following, R c R N is a bounded open subset with C2-boundary. As for the operators A: D ( A ) c L2(R) -, 315

316

5. Optimal Control of Parabolic Variational Inequalities

L2(fl), B : U + L2(fl) and the - functions p : R --+ 2R, g : [0, TI X L2(fl) + R, p0:L2(fl) -+ R, h: U -+ R = ] - m, m], their properties are specified in hypotheses (i)-(iv) in the following. Throughout this chapter, H will be the space L2(fl) endowed with the usual scalar product (., ) and norm I 1 2 . We are given also a real Hilbert space V that is dense in H , and

VcHcV' algebraically and topologically. We shall denote by II )I the norm of V and by (.,.) the pairing between V and V' that coincides with the scalar product of H on H X H . The norm of V' will be denoted by )I II*. The following hypotheses will be in effect throughout this chapter: (i) The injection of V into H is compact. (ii) A is a linear, continuous, and symmetric operator from V to V' such that

where w > 0 and a E R. (iii) p is a maximal monotone graph in R J/ E H 2 ( f l ) .

There exists C independent of

E

X

R such that 0

E D(

p ) and

such that

Moreover,

for every monotonically increasingfunction 6 E C'(R), S(0) ~ - ' ( l- (1 + E P ) - ' ) and D ( A , ) = (y E V; Ay E H } .

=

0. Here, p,

=

(iv) B is a linear continuous operatorfrom a real Hilbert space U to H . The norm and the scalar product of U will be denoted by ( , >,respectively.

-

1.1"

and

317

5.1. Distributed Optimal Control Problems

(v) The function h: U -+ R is convex and lower semicontinuous (1.s.c.). There exist c1 > 0 and c2 E R such that

h(u) 2

2 C ~ I U ~U ~2

V U E U.

(vi) The function g : [0,TI X H + R is measurable in t , g ( - , 0) E L'(0, TI, and for evely r > 0 there exists L , > 0 independent o f t such that

Instead of hi), in most of the situations encountered subsequently one might alternatively assume that g is a normal continuous convex integrand and qo is continuous and convex, i.e.: (vi)' The function g : [0,TI X H + R is measurable in t , continuous and convex in y , and qo: H + R is continuous and convex. There are the functions y E L2(0,T ; H ) , S E L'(0, T ) such that

As seen earlier in Chapter 4 (Section 3.2), we may equivalently write Eq. (1.1) as

y'

+ A y + d q ( y ) 3 f + Bu

in (0, T),

Y ( 0 ) =Yo where q ( y ) = Jn j ( y ( x ) - J / ( x > ) d r V y t+b) E L'(R)} and

(1.4) E

L2(R) = H , d j

=

P. Then,

D(q)= { y E H ; j ( y -

aq(y)

=

{ w E L ~ ( R )w; ( x ) E p ( y ( x ) - + ( x ) ) a.e. x

E

a).

By Proposition 3.6, Chapter 4, for every f E L2(Q),u E L2(0,T ; U),and every yo E D ( p ) n V = { y E j ( y - $1 E L'(R)} the Cauchy problem (1.4) has a unique solution y = y ( t , y o , u )E W'*2([0,Tl;H ) fC([O,Tl; l V),A y E L2(0,T ; H ) . If y o E D( q ) n V = D( q), then

y

E

C([O,T I ; H ) n L2(0,T ; V ) ,

and

t1I2AyE L2(0,T ; H ) .

t"2y'

E L2(0,T

;H ) ,

318

5. Optimal Control of Parabolic Variational Inequalities

Examples of equations of the form (1.1) satisfying the preceding hypotheses have been discussed in Chapters 3 and 4. For instance, if I/ = Hd(Q), A = - A , $ = 0, and P is a monotonically increasing continuous function, then the control system (1.1) reduces to a semilinear parabolic equation y , ( x , t ) - A y ( x , t ) - P ( y ( x , t ) ) = Bu =

in

Q,

in Q , in C = dR x (0, T ) .

y(x,O) =y,(x) y(x,t)

+f

0

(1.5)

If P is the multivalued graph, P(r)

=

L,

1 - W,Ol,

r > 0, r = 0, r < 0,

( 1.6)

Vr ER,

and $ I 0 in dQ then assumptions (i)-(iii) are satisfied and problem (1.1) reduces to the obstacle parabolic problem a.e. in { ( x , t ) ;y ( x , t ) > $ ( x ) ) ,

y, - A y

=

Bu + f

t)

=

max{f(x, t > + ( B u ) ( x ,t ) + A $ ( X ) , O }

Y,(X,

y ( x ,t ) 2 $ ( x ) y = o

a.e. in { ( x ,t ) ; y ( x , t ) a.e. in Q , in C,

y(x,O) =y,(x)

in Q .

=

$(XI}, ( 1*7)

Instead of - A we may consider of course a general second order elliptic differential operator N

A,Y

=

-

C (aijyx,)xj + ~ O Y , i, j = 1

+

with the boundary conditions a2 d y / d v a , y = 0 in dR (see Section 1.2 in Chapter 3, and Section 3.2 in Chapter 4). Recalling that A , dcp is maximal monotone in H X H , we may write Eq. (1.4) as

+

319

5.1. Distributed Optimal Control Problems

The optimal control problem we shall study here is the following: (PI Minimize ( k T ( g ( t , y ( tYo , u)) E

where Y

4,

9

+ h ( u ( t ) ) )dt + cpo(Y(T9Yo

9

u>>;

L~(o,T;

= y ( t ,y o ,u )

is the solution (1.1) (equivalently (1.8)).

Proposition 1.1. For every y o E D( 4 ) = D( cp) n V, problem (P) has at least one solution u.

Proof The proof is standard. However, we outline it for reader’s convenience. Let d be the infimum in (PI. It is readily seen by virtue of assumption (vi) or (viY that d > -m. Hence, there is a sequence {u,) c L2(0,T ; U ) such that

where y , = y(t, y o , u,) and G ( y ,u ) = /:(g(t, y(t, y o , u)) + h(u(t)))dt + (~o(y(T Y O, u)). By assumption (v), (vi) or (v$, we see that the u, remain in a bounded subset of L2(0,T ; U ) . Hence, there is u* E L2(0,T ; U ) such that on a subsequence, again denoted u, , 9

u,

+

u*

weakly in L*(o,T ; U )

Now if take the scalar product of (1.8) (where u = u, ,y = y,) with y , and ty; , we get the estimates (see the proof of Theorem 1.10 in Chapter 4)

Hence, (y,) remains in a bounded subset of L2(0,T; V ) n W’.’([q,Tl; H ) n C ( [ q ,TI; V ) for every q E (0, T ) . Then, by the Arzeli-Ascoli theo-

320

5. Optimal Control of Parabolic Variational Inequalities

rem we infer that, on a subsequence,

(y*)’

weakly in L2(0,T; V ) , strongly in every C([r),T]; H), weaklyineveryL2(r),T;H),

5

weakly in every L2(r ) , T; H).

Yn

+Y*

yi

+

d+(Yn)

+

In particular, this implies that yn + y* strongly in L2(0,T; H ) and a.e. in (0, T ) , d+(y*(t)) because the operator y + { w E L2(a,b; H ) ; w ( t ) E d+(y(t)) a.e. t E (a, b)) is maximal monotone. Hence, y* = y(t, yo, u*). Next, we have, for n -+ 03, 5(t) E

4oo(Yn(T))

+

4oO(Y*(T)),

if (vi) holds, and

if (vi)’ holds. Finally, since the function u + 10’ h(u)dt is weakly lower semicontinuous (because it is convex and l.s.c.), we have liminf i T h ( u n ( t ) )dt 2 j T h ( u * ( t ) )dt. 0

n+m

Then by (1.9) we see that G(y*, u * ) = d, as desired. To prove existence in problem (P) we have used the fact that every level subset (y: +(y) I A} is compact in H (as a consequence of hypothesis 6)). Thus, Proposition 1.1 remains true for every control system of the form (1.8) with this property. In particular, this applies to the following optimal control problem:

(PI) Minimize G(y, u ) on all y u E L2(0,T; U),subject to y, - Ay y( x , 0)

E

=

W1.2([0, TI; H ) n L2(0,T; H 2 ( Q ) ) and Bu

+f

= yo(x

)

a.e. in Q, a.e. x E Q , a.e. in C..

(1.10)

5.1. Distributed Optimal Control Problems

321

Here, f E L2(Q),and p is a maximal monotone graph in R p = d j where j : R + R is a 1.s.c. convex function. As seen earlier, problem (1.10) can be written as (1.8) where

X

R, i.e.,

and d + ( y ) = - A y , D(d+)= ( y E H2(fl); d y / d v + p ( y ) 3 0 a.e. in dfl}. Hence, if y o E D(+)under assumptions (iv)-(vi) (or (vi)') problem (P,) has at least one solution.

5.1.2. The Approximating Control Process Let ( y * , u * ) E (W'.2([0,TI; H ) n L2(0,T ; D ( A , ) ) ) x L2(0,T ; U )be any optimal pair for problem (P), where y o E D(cp) n V , i.e.,

For every E > 0, consider the adapted optimal control problem:

(PJ Minimize G " ( y , u )= / T ( g E ( t , y ( t ) + ) h ( u ( t ) ) + ;lu(t) 0

- u * ( t ) ( : ) dt

+ cp,"(Y(T))

on ally

E

W'.2([0,TI; H ) n L2(0,T ; D(A,)), u

y'

+ Ay + p " ( y - @)

= Bu

Y(0) = Y o . Here, g": [O, TI X H (2.79) in Chapter 2)

+

+f

E

a.e.

L2(0,T ; H ) subject to f E

(O,T),

(1.12)

R and cp,": H + R are defined by (see formula

(1.14)

322

5. Optimal Control of Parabolic Variational Inequalities

where n = [&-'I, p, is a mollifier in R" and P,: H + X n , A,:R" + X , are given by

c u,ei, n

P,u

=

c uie,,

i= 1

c riei n

co

u =

A,,r =

i= 1

i= 1

((ei)yx1is an orthonormal basis in H ) . The functions p": R -+ R are continuously differentiable, monotonically increasing, p"(0) = 0 and they satisfy the following conditions: (k) (kk)

lim inf j E (r,) &+

0

if lim r,

2j( r )

lim j & ( r )= j ( r )

Vr

8'0

=r;

E-0

(kkk) IP"(r) - p,(r)l 5 C

E

R;

V-E> 0, r

E

R;

- (1 + & @ ) - I ) and p = J j . For where j " ( r ) = 1,' p"(s)ds, p, = instance, the functions p" defined by (see formula (2.71) in Chapter 3)

P"(r) =

j-lm(P,(r - -E2(8) - P , ( - E 2 e ) ) P ( e >

dB +

PAO), (1.15)

where p is a C;-mollifier, satisfy these conditions. If /3 is defined by (1.16) we may take p" of the following form: f o r r s -E, for --E < r I0, for r > 0.

(1.17)

As noted earlier, we may equivalently write Eq. (1.12) as y ' ( t ) + A y ( t ) + V q " ( y ( t ) ) = B u ( t ) +f(t) Y(0) =Yo, where cp"(Y)

=

/Y(Y n

-

VY

a.e. t

E

(O,T), (1.18)

EJ%W.

It is readily seen that for E + 0 the solution y," E W 1 3 2 ( [TI; 0, H ) n L2(0,T; D ( A , ) ) to problem (1.18) approximates the solution y" to (1.1). Moreover, we have:

5.1. Distributed Optimal Control Problems

Lemma 1.1. Let u,

+

u weakly in L2(0,T ; U )for

323 E

-+

0. Then

weakly in W','([O,T I ; H) n L 2 ( 0 ,T ; D(A H ) ) , stronglyin L 2 ( 0 , T ;V ) n C ( [ O , T ] ;H ) .

y,"~ y" -+

ProoJ: We write Eq. (1.12) as Y ' + A Y + VCp,(Y)

=

Bu

where Vcp,(y) = ( d q ) , ( y ) = & ( y + and condition (kkk) it follows that

+ f + VCp,(Y) - VCp"(Y), $1 a.e. in R. Then, by hypothesis (1.2)

IIVP&(Y,")IILZ(O,T;H ) + II~Y,UIlL~(O,T; H ) + IIY,"IlW'.2([O,Tj;H ) 5

C(llUllL2(0,T;O)

+ 11,

where C is independent of u. Then by the Ascoli-Arzelh theorem and Aubin's compactness lemma we conclude that there is a subsequence, again denoted E , such that Y,". Ay,"e

(Y,"")' vP"(Y,"c)

+

-+

+

+

Y

strongly in C([O, T I ; H ) n L 2 ( 0 ,T ; V ) ,

AHy

weakly in L 2 ( o ,T ; H ) ,

Y'

weakly in L 2 ( 0 ,T ; H ) ,

5

weakly in L'(0, T; H ) .

By condition (kkk),

Inasmuch as for 2), we have

E -+

0, j,(z)

lim l Q j & ( z ( x , t )&+

0

-+ j ( z )

+(XI)

in L'( Q ) (Theorem 2.2) in Chapter

hdt

=

1j ( z ( x , t ) 8

- +(x))

vz

Similarly, by condition (kk) and the Fatou lemma,

hdt E

L~(Q).

5. Optimal Control of Parabolic Variational Inequalities

324

Then, letting

E

tend to zero in the inequality

By Proposition 1.1, for every optimal pair (y&,u s ) .

Lemma 1.2.

For

E +

> 0, problem (PJ has at least one

0,

u,

+

u*

strongly in L2(0,T; V ) ,

y,

-+

y*

strongly in L2(0,T ; V ) n C([O, TI; H ) , weakly in

P"(y,

E

- $)

6

-+

where

5 = f + Bu*

Proof

For any

E

G"(Y,, uE)

w ' . ~ ( [To],; H )n ~

~ (T ;0D(, A ~ ) ) , (1.19)

weaklyin L 2 ( 0 , T ;H), -

(y*)'

-Ay* E

(1.20)

dP(y*) a.e. in Q.

> 0, we have cp,"(~,"'(T)) + / = ( g & ( t ,y,"*(t)> + h ( u * ( t ) ) )dt. 0

(We recall that y," is the solution to Eq. (1.12).) By Lemma 1.1, y:* in C([O,TI; H ) and so, by Proposition 2.15 in Chapter 2, g " ( t , Y,"*(f)) g ( t , Y*(t>) +

Vt E 10, TI.

+

y*

325

5.1. Distributed Optimal Control Problems

Then, by the Lebesgue dominated convergence theorem,

l i ii g " ( t , Y , ( t ) > dt T

=

/ T g ( t 9Y * ( t ) ) 0

Similarly lim Vo,"(Y,U*(T))= V , ( Y * ( T ) ) ,

€+

0

whence limsup G 6 ( y , , u , ) €+

(1.21)

IG ( y * , u * ) .

0

On the other hand, since by assumption (v), (us)is bounded in L2(0,T; U ) , there exists u1 E L2(0,T; U ) such that, on some subsequence E + 0, weakly in ~ ~ (T ;0U ,) ,

u, -+ u1

and so, by Lemma 1.1, ye

+

y,

Since the function u L2(0,T; U ) , we have

strongly in C ( [ O , T I ; H ) .

= y"'

+

10' h(u(t))dt

is weakly lower semicontinuous on

lim inf G"(y e , u,) 2 G ( y , ,u l ) 2 G( y * , u * ) E+O

and, by (1.21),

lii Hence y ,

= y * , u1 =

i

T

Iu,

- u*I;

dt

=

0.

u * , and (1.191, (1.20) follow by Lemma 1.1.

Consider the Cauchy problem

P:

- AP,

- P ( Y &- 9 ) P ,

=

Vg"(t,ye)

PATI

=

-VVo"(Y,(T)),

in ( 0 ,T ) , ( 1.22)

which has a unique solution p, E L2(0,T ; V ) n C([O,TI; H ) with p: E L2(0,T; V ' )(see e.g., Theorem 1.9' in Chapter 4). (Here, 4' = ( fl")'.) On the other hand, since ( y , , u,) is optimal for problem (PJ, we have G E ( y u ~ + h u+, Au) ~ , 2 G E ( y E , u E ) VA > 0 , u E L 2 ( 0 , T ;V ) .

326

5. Optimal Control of Parabolic Variational Inequalities

This yields joT(hYu,,u) + ( V g " ( t , y , ) , z , ) + (u, - u * , v > ) d t L 2 ( 0 , T ;U ) , (1.23)

+ ( v ( ~ o " ( y , ( T ) ) , z , ( T2) )0

where z, E W ' * 2 ( [ 0 , T ]H; ) n L2(0,T ; D ( A , ) ) is the solution to the linear equation 2'

+ A Z+ b c ( y , - +)z

=

~u

z(0)

=

0

a.e. t

E

(o,T),

and h' is the directional derivative of h (see formula (2.7) in Chapter 2). If multiply Eq. (1.22) by z , and integrate on (0, T ) we get, by (1.23), LT(h'(u,,u) + (u,-u*,u)

-

(B*p,,u))dt~O VUEL~(O,T;V),

and this yields (Proposition 2.8 in Chapter 2) B*p,(t)

E

dh(u,(t)) + u,(t) - u*(t)

a.e. t

E

( 0 , T ) . (1.24)

Equations (1.221, (1.24) taken all together represent the Euler-Lagrange optimality conditions for problem (P,). Lemma 1.3. There is C > 0 independent of

E

such that

Proof: We take the scalar product of (1.22) with p J t ) and integrate over [ t ,TI. Since j E2 0, we get 1

d p , ( t ) l 22 5 31p,(T)I:

-

~ ~ T l l p E ( ~ds) 1+1alTIp,(s)I: 2 ds

+ p & ( s Y&(s))121p,(s)12 , ds,

t

E

[O, TI.

On the other hand, by hypothesis (vi), Section 1.1, we have IIVgB(t,YE)IIL"(O,T;H) + IV(P;(y&(T))12I c

VE

> 0,

5.1. Distributed Optimal Control Problems

327

and so by Gronwall's lemma we arrive at estimate (1.25). To get (1.26), we multiply Eq. (1.22) by f ( p , ) and integrate on Q = X (0, TI, where 5 is a smooth monotonically increasing approximation of the sign function such that f ( 0 ) = 0. For instance,

f

m

= SA(r)

= /-,(YA(r

-

h e ) - %(-Ae))P(e) de,

where yA(r)= rlrl-' for Irl 2 A, yA(r)= A-'r for Irl < A, and p is a C;-mollifier. Then ( A f ( p , ( t ) ) ,[ ( p , ( t N 2 0 and, therefore,

+ llP,(T)lILl(Q)

V& > 0.

Then, letting f tend to the sign function, we get estimate (1.26). Now, since {Ap,} is bounded in L2(0,T ; V ' ) and { P E ( y E- + ) p , } is bounded in L'(0, T ; L ' ( f l ) ) ,we may infer that { p : } is bounded in L'(0, T ; L'(R) + V ' )and so, by the Sobolev imbedding theorem, {PI.} is bounded in L'(0, T ; Y *), where Y * = (HYR))' + V ' (s > N / 2 ) is the dual of Y = HYR) n V. Since the injection of H into Y * is compact and the set { p , ( t ) ) is bounded in H for every t E [O, TI, by the Helly theorem we conclude that there is a function p E BV([O,TI; Y * ) such that, on a subsequence En 0, p,,(t) + p ( t ) stronglyin Y * , Vt E [O,T].

-

(Here, BV([O,TI; Y *) is the space of all Y *-valued functions p : [O, TI + Y * with bounded variation on [O, TI.) On the other hand, by estimate (1.25) we see that p,, + p

weak star in L"(0, T ; H ) , weakly in L 2 ( 0 ,T ; V ) . (1.27)

Now since the injection of V into H is compact, for every A > 0 there is S(h) > 0 such that (see J. L. Lions [ l ] ,Chapter 1, Lemma 5.0,

IP,p)

- P(t)12 5

IlP,,(t) - P(t>ll + W ) I I P , , ( t ) - p ( t ) l l v * V t E [O, T I , V n E N*.

This yields P,,

+

P

strongly in L2(0,T ; H ) ,

( 1.28)

328

5. Optimal Control of Parabolic Variational Inequalities

and

p,,(t)

weakly in H , V t

+p ( t )

E

(1.29)

[O, T I .

Moreover, by estimate (1.26) we infer that there is p that, on a generalized subsequence A of E, ,

E

(L"(Q))* such

weakstar in ( L " ( Q ) ) * .

p h ( y h- + ) p A+ p

(1.30)

(We may also view p as a bounded Radon measure on e.1 Now, since {Vcp,E.(y,$T))} is bounded in H , we may assume by Proposition 2.15, Chapter 2, that VcpI?(Y&,(T))

+

-P(T)

and V,,g"n(t , y e n ) -+ ( Lemma 1.4.

E

dcpO(Y*(T))

weakly in H ,

weak star in L"(0, T ; H).

We have ((t)

E

ag(t,y*(t))

a.e. t

E

(0,T).

5.1. Distributed Optimal Control Problems

329

Now, letting E = E, + 0 in Eq. (1.22), we conclude that there is p BV([O,TI; Y * ) n L2(0,T ; V ) n L"(0, T ; H ) that satisfies the equation

p' - A p

-p E

P(T) E

dg(t,y*) in ( O , T ) , -dvo(y*(T)),

E

(1.31)

where p' is the derivative of p in the sense of vectorial distributions on (0, T ) . Moreover, since the map dh: U + U is closed we see by (1.19), (1.24), and (1.27) that

B*p(t) E dh(u*(t))

a.e. t

E

(0,T).

(1.32)

Modifying p on a set of measure zero, we may also assume that the lateral limits p ( t + 0) and p ( t - 0) exist in L i ( R ) everywhere on ( 0 , T ) (respectively, at t = 0 and t = TI. Summarizing, we have proved the following weak form of the maximum principle for problem (P):

Proposition 1.2. Let ( y * , u*) be an arbitrary optimal pair for problem (PI. Then there exist the function p E L"(0, T ; H ) n L2(0,T ; V ) n BV([O,TI; Y * )and a measure p E (L"(Q))*satisfiing Eqs. (1.31) and (1.32). We shall call such a function p the dual extremal arc of problem (P). The properties of p as well as the optimality system (1.31) will be explicated for some particular cases. 5.1.3. Optimal Control of Semilinear Parabolic Equations

We shall study here problem (P) in the special case where A = - A , V = H,'(R), V' = H-'(R), (1, = 0, and f l : R + R is a locally Lipschitz, monotonically increasing function. In other words, we will consider the case where the state equation is given by dY

--

dt

Ay

+ p ( y ) = Bu + f

in Q

y=o y(x,O) = y o ( x )

in C = dR x ( O , T ) , in R .

=

(R X ( O , T ) , ( 1.33)

5. Optimal Control of Parabolic Variational Inequalities

330

Theorem 1.1. Let ( y * , u * ) E W ' 3 2 ( [ 0TI; , H ) n L2(0,T ; (H,'(R) n H2(R))) X L2(0,T ; U )be any optimalpair forproblem (P) having (1.33) as state vstem, where p is locally Lipschitz and monotonically increasing. Then there are p and p

E

E

BV([O,T I ; Y * ) n L2(0, T ; H,'(R)) n L"(0, T ; H )

(L"(Q))* such that p' - A p - p

E Lm(O,T ;

p' - A p - EL E d g ( t , y * ) pa(x,t ) E p ( x , t ) W ( Y * (t ~ ) ), P ( T ) + dVo,(Y*(T))3 0 B*p(t) E dh(u*(t))

H ) and

a.e. in Q , a x . ( x ,t ) E Q, in fl, a.e. t E ( 0 , T ) .

(1.34) (1.35)

a.e. r

(1.38)

(1.36) (1.37)

Further assume that 0 Ip ' ( r ) IC(l P(r)l

+ Irl + 1)

E

R.

Then p E AC([O,TI: Y * )n C,([O, TI; H ) n CUO, TI; L'(R)) and pa = p E L'(Q). Here, p a is the absolutely continuous part of the measure p and d p is the generalized gradient of p. Proof: Let p E L"(0, T ; H ) n L2(0,T ; H,'(R)) n BV([O, TI; Y * ) be the function arising in Proposition 1.2 and let p E (L"(Q))* be the measure defined by (1.30). It remains to prove (1.35). By the Egorov theorem, for each 7 > 0 there is a measurable subset Q, c Q such that y*, p E LYQ,), m
b E n (

-+

P

strongly in L"(Q,),

weak star in L"( Q , ) .

y E n )-+ g ,

By Lemma 2.4, Chapter 3, we have g,(x,t)

E

Wy*(x7t))

W , t )E

Q,.

This clearly implies that pa(x,t ) E p ( x , t ) dp(y*(x,t ) ) a.e. (x,t ) E Q because bA(yA)pA p -+

weakstarin(L"(Q))*.

Now if p satisfies condition (1.381, then we have

o5

I~

( P1 E ( r ) I+ JrI + 1)

a.e. r

E

R,

331

5.1. Distributed Optimal Control Problems

and so for any measurable subset E of Q we have

Since { p")y,)} is bounded in L2(Q)whilst p , and ye are strongly convergent, the last inequality implies that, for every S > 0, 3 d S ) such that

if m ( E ) < w ( S ) . Then, by the Dunford-Pettis criterion, ( B E ( y , ) p E }is weakly compact in L'(Q) and therefore p E L'(Q), thereby completing the proof. Remark 2.1. Assume now that f E Lq(Q), B E L(U, Lq(n)),and y o E wo2 - 2 / q * q ( 1 R ) , j ( y o ) E L'(R) where q > max(N, 2) and j ( y ) = /J I p(s)Iq-*p(s) ds. Then the solution y, to system dY,

Aye dt y,(O) = y o --

+ p " ( y , ) = Bu, + f in a , y, = 0

in Q, in d o ,

satisfies II p " ( y , ) l l L 4 ( ~I ) C (see Remark 3.1 in Chapter 4). Hence, { y e } is bounded in W$ ' ( Q )and therefore it is compact in C ( D ) .This implies that ( B " ( y , ) )is bounded in LYQ) and so p E L2(Q).Therefore, the optimality system (1.34)-(1.36) becomes, in this case, dP dt

-

+ Ap - p d p ( y * )

p ( x , T ) E -dqo,(y*x,T))

3

a.e.in Q

0

a.e. in

a,

p

=

0

in C.

5.1.4 The Optimal Control of the Obstacle Problem We shall consider now the case where p is defined by (1.16), i.e.,

p(r)

=

0 if r > 0,

p(0) = ] - 03,0],

p(r)

=

0 if r < 0.

As seen earlier, in this case Eq. ( 1 . 1 ) reduces to the obstacle problem (1.7).

332

5. Optimal Control of Parabolic Variational Inequalities

Theorem 1.2. Let ( y * , u*) be optimal in problem (P), where A = - A , V = H;(Q). Then there is a firnetion p E L2(0,T ; H ; ( Q ) ) n Lm(O,T ; L2(Q))n BV([O,TI; Y *) with p' - Ap E (Lm(Q))*that satisfies the equations a * e * i n[ Y * > 919 (1.39) ( p ' + Ap)a E d g ( t , y * ) (1.40) a.e. in [ y * = 91, p ( f + Bu* + A y * ) = 0 P ( T ) + d%(Y*(T))3 0 in Q , (1.41) B*p(t) E dh(u*(t)) a.e. t E ( 0 , T ) . (1.42) Here, p' is the derivative of p:[O,T]+ L 2 ( Q ) in the sense of distributions. Equation (1.39) should be understood of course in the following sense: There is an increasing family {Q,};=, of measurable subsets of Q such that m(Q\Q,> I k-' and

IQp ( x , -

t ) y , ( x , t ) dxdt -

jQ V p ( x ,t )

*

V y ( x ,t ) dxdf

j-p(x, t ) y ( x , T ) dx + j8 y ( x , t ) d g ( t , y * ( x ,t ) ) dxdt

=

0

for all y E L2(0,T ; H,'(R)) n C(rz> n C([O, TI; Y ) such that yt E L2(0,T ; H - ' ( f l ) ) , y ( x , 0 = 0 and supp y c K x , t ) E Q; y*(x,t ) > 9 ( x ) ) n Q k . Equivalently, Pt

+ AP

=

5, + 5 s

in

[(X,t>;

y * ( x , t ) > Icl(x)l,

E L'(Q), S , ( x , t ) E dg(t,y*(x,t)) a.e. ( x , t > E Q, and tS is a where singular measure with respect to the Lebesgue measure on Q.

Proof of Theorem 1.2. The proof closely follows that of Theorem 2.2 in Chapter 3. However, for the sake of simplicity we will take here, in the approximating problem (Pel,P" of the form (1.17). Let E = E, be the sequence that occurs in the proof of Proposition 1.2. We have

P,P"(Y, =

L

9 ) - P&(Y€- 9 ) B " ( Y &- 9 ) in [ y , - 9 1 2-'P&

-

-&I,

2 - ' ( y , - ~ 1 ) ~ p , in [ - E < y e - 9 I 01, in [ y , - 9 > 01.

Since {p,B"(y, - @ ) } is bounded in L'(Q), this yields IlP,P"(Y, -

9 ) -P,(Y,

-

$ ) b " ( Y , - 9 ) I I L ' t Q ) I CE

vE>

O. (1-43)

5.1. Distributed Optimal Control Problems

333

On the other hand, since { b " ( y , - +)pel is bounded in L'(Q), we have

whilst

Since

(because ( p " ( y , - +)) is bounded in L2(Q)),it follows by (1.44) and (1.45) that

p,p"(y,

-

$)

+

strongly in L ' ( Q )

0

( 1.46)

and, by (1.431,

P,P"(Y,

-

$)(Y,

-

$1

+

0

strongbin L ' (Q).

(1.47)

On the other hand, as seen earlier in the proof of Proposition 1.2,

p,b"(y,

-

+) -+p(f+ Bu* - A y * )

weaklyin L ' ( Q ) .

Hence,

p(f

+ Bu* - A y * ) = 0

a.e. in Q.

Next, by the Egorov theorem, for every q > 0 there is a measurable subset Q, c Q such that m
y,

+

y*

unformly on Q, .

334

5. Optimal Control of Parabolic Variational Inequalities

This implies that (see (1.30)) in

p ( y * - $) = O

Q,,

i.e.,

for all cp

E

L"(Q) such that supp cp

( y * ( x , t )- $(x))pa(x,t)

C

Q,.This yields

=

0

a.e. ( x , t )

E

Q, n Qk,

where {Q,}Y= is an increasing family of measurable sets Qk c Q such that m(Q \ Q k ) Ik-I and ps = 0 on Q k . Hence, a.e. in Q, (y* - $)pa= 0 and this clearly implies Eq. (1.39). Remark 1.2. If f E Lq(R), B E L(U, L q ( R ) ) ,and y o E W:-z'q7q(RR), y o 2 $ a.e. in Q, where q > max(N, 2), then as seen earlier (see Remark 1.1) {y,} is bounded in W:('Q) and compact in C(lz>. Then, by (1.30) and (1.471, we conclude that P(Y* -

$1

=

0

and so Eq. (1.39) becomes, in this case,

Pr + AP

E

dg(t,y*)

in [ Y * >

$1.

The previous methods applies mutatis-mutandis to the optimal control problem (PI). Theorem 1.3. Let ( y * , u * ) be optimal in problem (PI), where p is locally Lipschitz and satisfies condition (1.38). Then there is p E AC([O,TI; (H'(R))') n CJO, TI; L2(R)) n L2(0,T ; H'(R)), s > N / 2 such that dp/dv E L Y Z ) and

pr + A p

E

dg(t,y*)

p ( x 9 T ) + dcpo(y*(T))(x)3 0

B*p

E

dh(u*)

a.e. in Q,

(1.48)

a.e. in R,

(1.49)

a.e. in C,

( 1.50)

a.e. in ( 0 , T ) . (1.51)

335

5.1. Distributed Optimal Control Problems

In the case of Signorini problem, i.e., where the state equation is y, - A y

7

BU

y(x,O) = y o ( x ) y20,

dY

in

Q,

in

a,

dY

-20,

y-=O

dV

dV

inC,

( 1.52)

we have: Theorem 1.4. Let ( y * , u * ) be optimal. Then there is p E BV([O,T]; (H'((.n))')n L2(0,T ; H'(R)) n LYO, T ; L2(SZ)) with dp/dv E (L"(C))* such that

y*(s) a

=o

P ( T ) + dcpO(Y*(T))3 0 B*p

E

dh(u*).

in C,

(1.55)

a.e. in SZ,

(1.56) ( 1.57)

Equations (1.52)-(1.57) taken all together represent a quasivariational inequality (see, e.g., J. L. Lions [4]).Equations (1.53), (1.551, and (1.56) should be interpreted in the following sense:

+L

d g ( t , y * )c p ( x , t ) h d t = 0

for all cp E L2(0,T; H'(SZ)) n W'.*([O,T];L2(SZ))such that cp(x,O) = 0 and cp = 0 in Ck n [ y * = 01; Ck is a sequence of measurable subsets of C such that m(C,) + 0 as k -, 03. Proof of Theorem 1.3. Since the proof is essentially the same as that of Theorem 1.1, it will be sketched only.

336

5. Optimal Control of Parabolic Variational Inequalities

Consider the approximating problem

dY

-

dv

+ p"(y)

=

0

a.e. in 2,

(1.58)

where y o E H'(R), j(yo) E L'(R), p" satisfies (k)-(kkk) in Section 1.2, and (y*, u * ) is an arbitrary optimal control for problem (PI). Arguing as in the proof of Lemma 1.1, it follows that if u, LZ(O,T;U )then

y , " ~+ y"

-+

u weakly in

strongly in L2(0,T ; H I ( a)) n C([ 0 ,T I ; L2(a)), weakly in W','( [0, T I ; L2(R)) n L2(0,T ; H 2 ( a)). (1.59)

(Here, y" is the solution to (1.58)J Moreover, p"(y,) + - dy"/dv E p(y") weakly in L'(2). Indeed, multiplying Eq. (1.58) by y, =y,"a and -Ay,, we get the estimates

Hence, {y,} is bounded in C([O,TI; LZ(R))n LTO, T ; H'(R)) n L2(0,T ; H2(R)) and {dy,/dt} is bounded in L2(0,T ; L2(R)). Thus, {y,} is compact in C([O,T];L2(R)) n L2(0,T; H'(R)) and weakly compact in L2(0,T; H2(R)) n W ' T ~ ( [TI; O , L2(R)). Then (1.59) follows by standard arguments. Now, let (y*, u * ) be optimal for problem (PI). (The existence of such a pair follows by Proposition 1.1.) Then, arguing as in the proof of Lemma

5.1. Distributed Optimal Control Problems

337

1.2, it follows that u,

+

strongly in L'(0, T; U ) ,

u*

strongly in C ( [0, T I ; L'(

Y, + Y *

P"(Y&)

JY *

+

a ) )n ~ ' ( 0T ,; H ' ( a ) ) ,

weakly in L' ( Z) ,

--

dV

weakly in W','( [0, T I ; L2(a ) )n L'(0, T ; H 2 ( a ) )

Y, + Y *

Now, let p , E L2(0,T ; H ' ( f 2 ) ) n W',2([0,TI; ( H ' ( i 2 ) ) ' ) n C([O,TI; L 2 ( f l ) be ) the solution to boundary value problem

in Z. (The existence of p , follows, for instance, by Theorem 1.9' in Chapter 4, where V = H ' ( n ) and A t ) : V + V' is defined by ( A ( t ) u , u )= j n V u * V u d t

+ Jdnbc(y,)uudcr

Vu,uEH'(R).

Then, reasoning as in the proof of Proposition 1.2, it follows that B*p,

E

dh(u,)

+ u, - u*

a.e. in (O,T),

and Ip,(t)lz

+ /UTIlp,(t)llkl(n, dt + L l 6 " ( y , ) p , l

drdt IC

VE > 0.

+

Hence, {dp,/dt) is bounded in L'(0, T; L'(a)) L'(0, T ; ( H ' ( a ) ) ' ) c L'(0,T; (H'((n))'), where s > N/2. Then, by the Helly theorem, there is p E BV([O,TI; (H'((n))') such that, on some subsequence, again denoted E,

p,(t) + p ( t )

P&+P

stronglyin ( H ' ( ( R ) ) ' ,

Vt

E

[ O,T],

weakly in L2(0,T ; H ' ( a ) ) , weak star in Lm(O,T; L2(a ) ) .

( 1.60)

338

5. Optimal Control of Parabolic Variational Inequalities

This implies as in the proof of Proposition 1.2 that and

p,

+

B"(y,)p,

+

p

strongly in L2(Q)

6

weak star in ( L " ( C ) ) * .

Clearly, p satisfies the system dP

-

dt

+ Ap E dg(t, y * )

dP -+6=0

in 2 ,

dU

B*p

E

in Q,

dh(u*).

Now, if p satisfies condition (3.18) it follows as in the proof of Theorem 1.1 that { B " ( y , ) p , } is weakly compact in L ' ( 0 ) and so 6 E L'(C), whilst by Lemma 2.4 in chapter 3 we see that 6 ( x , t ) E d P ( y * ( x ,t ) ) p ( x , t )

a.e. ( x , t )

E

2,

because y , + y * strongly in L2(C) and p , + p weakly in L2(C). This completes the proof of Theorem 1.3. Proof of Theorem 1.4. Arguing as in the proof of Theorem 1.2, we find (see (1.43)-(1.45)) that

0

strongly in L'( C ) ,

p,fiE(y,)y, + o

strongly in L ' ( C ) ,

p , p "( y , )

+

The latter implies, as in the proof of Theorem 1.2, that Say* = 0 in 2. On the other hand, by (1.60) it follows that on a subsequence, again denoted E , we have a.e. in C l , p, + 0 p"(y,)

-+

0

a.e. in C, ,

where C = Cl u 2 , . Since @ " ( y e )+ - d y * / d v p , + p weakly in L2(C),we conclude that p as claimed.

dY *

=

0

a.e. in C,

weakly in L2(C) and

5.1. Distributed Optimal Control Problems

339

5.1.5. Optimal Control Problems with Infinite Horizon

Consider the problem inf ( C ( P ( Y ( S ) ) + h ( u ( s ) ) )d s ; u

E

L Y R + ; U ) ) = rcrm(Yo), (1.61)

subject to dY dt

-

+ Ay + d p ( y )

3

a.e. t > 0,

Bu

Y(0) = y o ,

( 1.62)

where A is a linear continuous symmetric operator from V to V' satisfying the coercivity condition

-

(Ay,y)

B E L(U, H ) , and and

p:H

2

~lIY1l2,

y

E

V,

R is a l.s.c., convex function such that 0 E dp(0)

(AY, dP&(Y))2 -c(1 + ldP&(Y)12)(1+ lYl)

VY

E

D(A,),

where D ( A , ) = (y E C Ay E HI, dqE(y) = ~ - l ( y- (1 + ~ d p ) - l y ) . Here, U, V , and H are real Hilbert spaces such that V c H c V ' algebraically and topologically. We shall assume that the injection of V into H is compact and we shall use the standard notations for the norms in V , H , and U (see Section 1.1). The functions h: U + R and g : H + R satisfy the conditions: (j)

h is convex, lower semicontinuous, h(0) = 0, and h ( u ) 2 rlulZl

VY

E

u,

for some y > 0. (jj) g is locally Lipschitz, g 2 0, and g(0) We denote by 4 the function i(Ay, y)

+ dp.

=

0.

+ p(y) and recall that

&$

=A,

Proposition 1.3. Assuming the preceding conditions, the function & :D( 4 ) -+ R is locally Lipschitz, and for euey yo E D(4 ) the infimum

defining &,(yo)is attained.

5. Optimal Control of Parabolic Variational Inequalities

340

Proof Let y o E D( 4) be arbitrary but fixed. If y ( t , y o , u ) is the solution to the Cauchy problem (1.62), then we have ly(t,yo,O)l I Ce-"'lyol and so g ( y ( t , y o , 0)) be such that

E

L'(R). Hence, qm(y0)< to. Now let u,

/ (g(Yn(s))+ m

YO)

V t > 0,

I

0

E

h ( u n ( s ) ) ) ds I qm(Yo> +

L2(R+;U )

1

Lj

where y,(t) = y ( t , y o , u,). ) so, by Eq. (1.621, we By (j),we see that (u,} is bounded in L 2 ( R + ; Uand have ~ y , ( t )I ~ e-O'lyol

+ I I B I I / ~ I ~ , ( ~ ) I ~ ~-c-c" ( ' - ~t~ )2~0,~

L T t l y ; ( t ) l Zdt

0

+ /Tllyn(t)l12 dt I C, 0

V T > 0.

(1.63)

Hence, on a subsequence, again denoted {n}, u,

+

Yn(t)

+

weakly in L2(R+,U )

u*

~ ( tY O, u*> 3

strongly in C([ 6, T I ; H ) ,

VO < 6 < T ,

and by the Fatou lemma, V T > 0,

J O T ( g ( Y ( t 9 Y ou*>> + h ( u * ( t ) ) )dt I&(Yo) 7

and this implies that [(g(Y(t,Yo Now, for y o E B,

=

9

u*))

+ h ( u * ( t ) ) )dt

=

+m(YO>.

{ y E H ; lyl Ir } , we have

/

I C ~ ~ ( Y O )I

m

0

g ( y ( t , y O , O ) ) dt

ICr

and so we may confine the infimum in (1.61) to those u which m

y / , lu(t)lc dt Ij m h ( u ( t ) dt ) I C,. 0

E

L2(R+;U)for

341

5.1. Distributed Optimal Control Problems

Then, by estimate (1.631, we have ly(t,yo,u)l I re-@' + IIBll(Cr + C)

IC,!

Vt 2 0, yo E B,

Now, if y o , zo E Br, we have

where u*

E

L2(R+;U ) is such that

In particular, it follows by Proposition 1.3 that for every yo E D( 4) problem (1.65) has at least one optimal pair (y*, u * ) E C(R+; H ) X L2(R+; U). We notice that, for every f > 0,

Proposition 1.4. In addition to ( j ) , ( j j ) , let us suppose that F = d q E C ' ( H ) , N ( B * ) = {0},and h is GBteaux differentiable on U.Then if (y*, u * ) is an optimal pair for problem (1.61) there is p E C([O,m); H ) n L2(R+; V ) n L"(R+; H ) such that dP -(t) dt

-Ap(t) -

VF( y*( t ) ) p (t )

E

dg( y*( t ) )

a.e. t > 0,

~ ( t E) -d+m(Y*(t))

V t 2 0,

(1.65)

B*p(t) = dh(u*(t))

Vt 2 0.

(1.66)

342

5. Optimal Control of Parabolic Variational Inequalities

Proof As seen in the preceding, for every t > 0, ( y * , u * )is optimal for problem (1.64). This implies by a standard device that there is p' E C[O,t ] ; H ) n W',2((0,tl; H ) n L2(0,t ; V )such that dP' -(s) ds

-Ap'(s)

-

a.e. s

V F ( y * ( s ) ) p ' ( s )E d g ( y * ( s ) )

E

(O,t),

P ' ( t > E -d+m(y*(t)), a.e. s

B*p'(s) = dh(u*(s))

E

(1.67)

(0, t ) .

Since N ( B * ) = (0) and dh is single valued it follows that p' for t If. Now, let p : [0, m) + H be the function p(s) =p'(s) Since y*

E

for0

IS

= p'

on [O, t ]

It.

L"(O,m:H), we see by (1.67) that lP'(S)I2 + /fIlp'(s)112ds 0 Ic

where C is independent of t. Hence, p rn Eqs. (1.65), (1.66), as claimed.

E

vs E

[O,t],

L"(R+; H ) n L2(R+;V )satisfies

Remark 1.3. In order to avoid a tedious argument, we did not put the preceding result in its most general form. However, it is readily seen that it remains true for semilinear parabolic equations of the form (1.1) with p locally Lipschitz as well as for the obstacle problem.

5.2. Boundary Control of Parabolic Variational Inequalities 5.2.1. The Obstacle Problem with Dirichlet Boundary Conditions

We will consider here the controlled obstacle problem dY dt

--

AY + P ( Y -

+I

3f

Y(X,O) =Yo(X)

y = u

in Q in

=

R

X

(O,T),

a,

in Z

=

dR x ( O , T ) ,

(2.1)

343

5.2. Boundary Control of Parabolic Variational Inequalities

where P : R

+

2R is is the graph r > 0, r = 0, r < 0,

I) E C2(n), $ 5 0 in dR, and

R is a bounded open subset of RN with a

smooth boundary dR (of class C2,for instance). Regarding the functions f , y o , and u , we assume that

f

E

u

E

Wo2 - 2 / P , ~ ( f i ) , wp 2 - l / P , 1 - 1 / 2 P ( C ) , u 2 $ in C

Lp(Q),

YO E

yo 2 I) in 0,

u(x,O) = y o ( x ) ,

(2.2) x

E

R,

(2.3) where p 2 2. Here, W;','(R) is the usual Sobolev space on C (see, e.g., Ladyzhenskaya et a f . [l],p. 96). As seen earlier, problem (2.1) can be written equivalently as

y(x,O) = y o ( x )

in R,

y

=

u in C .

(2.4)

Regarding existence, we have:

where C is independent of y o , f , and u . f f p > ( N 2)/2 then y is Holder continuous in ( x , t ) and the map ( y o , f , u ) + y is compact from w ~ ~ - ' / P , P ( ~ Ix) L P ( R ) x 2-'/P91-1/2P(C) to c@>.

+

wp

Proof The last part of the proposition is an immediate consequence of regularity properties of the elements of W : '(Q) (Ladyzhenskaya et a f . [ll, p. 98).

344

5. Optimal Control of Parabolic Variational Inequalities

For existence in (2.0, consider the approximating equation

JY

at - AY + P " ( Y -

$1 = f

Y ( X , O ) =yo(x)

in

a,

in C,

y=o

(2.6)

where P" is defined by (1.17). The existence for problem (2.6) follows from the following linear result (Ladyzhenskaya et al. [l],p. 388).

Lemma 2.1. The boundary value problem Yt-AY=f Y ( X , O ) =yo(x)

in

Q,

in

a,

in C,

y=u

(2.7)

has for every f E L p ( Q ) and y o ,u satis-ing (2.21, (2.3) a unique solution y E Wp'.' ( Q ) ,which satisfies the estimate

II~IIw:.~(Q) Now, let

IC ( IIyoIIw~-2/~.p(n,+

5E

+ I I f IILP(Q)).

II~IIw2-l/p.1-1/2p(n)

(2.8)

Wp'.'(Q) be the solution to (2.7). Then the problem

- Az, + P E ( z , +

(z,),

z,(x,O)

=

5 - $1 = f

0

in

z, = 0

in

a,

Q,

in 2 ,

(2.9)

has a unique solution z, E W: ' ( Q ) . Indeed, by the standard contraction principle it follows that (2.9) has a unique mild solution in C([O, TI; LP(a)).Hence, p"(z, + 6 - $) E Lp(Q) and by Lemma 2.1 we infer that z, E Wp2". If we set y, = z , + 6 , we get d

-Y, at

-

AY, +

P"(Y,

-

$1 = f

in

Q, (2.10)

345

5.2. Boundary Control of Parabolic Variational Inequalities

We may write Eq. (2.10) as d

-y, df

where

.,(X7f)

=

-

I

Y,(X,O) =Yo(X) Y, = u 1 2

(Y,

=

4v2 2E2 -

-(l/EXy,

1 + ;(ye-

9)

in Q,

in a , in C.,

ify,-

-

0

We set 8,

Ay, - E-'(Y, - * ) - = f + v,

(2.11)

@ <- E ,

if

-*so,

if y, - JI > 0. (2.12)

-

+>-.We have

and

Multiplying Eq. (2.11) by IS,lp-2S, and integrating on Q, we get, after some calculation involving the Holder inequality,

and so, by Gronwall's lemma,

5. Optimal Control of Parabolic Variational Inequalities

346

Finally, by estimate (2.13) we see that, on a subsequence, y,

--f

weakly in

y

w,','(Q)

and, in particular,

Y& + Y

dyE dt

P " ( Y & - JI)

--f

+

in L 2 ( 0 , T ;Hi(fl)),

2 dt

weaklyin L 2 ( Q ) ,

77

weakstar in L " ( 0 , T ; L P ( f l ) ) .

(2.15)

We have, therefore, dY --Ay+7=0 dt y(0) = y o , y = u

inQ, (2.16)

in 2

This implies that dY jQ(z(y-z) +Vy.V(y-z)

for all z

E

K

=

{z

=

E

dxdt+

jQ7 7 ( y - z ) d x d t 2 0

(2.17)

W?'(Q); z 2 0). Since

j Q 7 ( x , t ) ( y ( x , t )- z ) d x d t > O

VZEK,

We infer by (2.15) and (2.16) that y is the solution to Eq. (2.11, i.e., ~ ( xt ), E P ( y ( x , t ) - + ( x ) ) a.e. ( x , t ) E Q. Estimate (2.5) is implied by (2.13) and (2.141, whilst the uniqueness is obvious. As seen earlier in Chapter 4 (Eq. (3.13)), perhaps the most important physical model for problem (2.1) is the one phase Stefan problem describing the melting process of a solid. 5.2.2. The Optimal Control Problem

We shall study here the optimal control problem: g ( t , y ) ) d t + 4 ( u ) + cpo(y(T))

(2.18)

5.2. Boundary Control of Parabolic Variational Inequalities

on ally E WpZ"(Q) and u E W2-1/P*1-1/2P (I;),p > ( N the state system (2.1) and to the constraints u 2 0

in I;,

u(x,O) = y , ( x )

+ 2)/2,

in R.

347

subject to (2.19)

(For simplicity, we take I(I = 0.1 Here, g : [O, TI x L2(R) + R ' , p,: L2(R) + R + are locally Lipschitz and 6:W2-1/p*1-1'2p(2) + R is a lower semicontinuous convex function such that for all u E D(+),u 2 0 a.e. in 2, u(x,O) = y , ( x ) Vx E dR. The latter assumption allows us to incorporate the control constraints (2.19) into the cost functional of problem (2.18). We set XP = W2-1/P,1-1/2P(I;) and denote by Ill . Illp the natural norm of X p . Theorem 2.1. Let ( y * ,u * ) E W:('Q) X X p be an optimal pair for problem (2.18). Then there is 6 E L2(Q), p E L"(0, T ; L2(R)) n L2(0,T ; H,'(R)) n BV([O,TI; H-'(R)), s > N/2 such that

(2.20) y*

=

0,

6 E dg(t,y*)

Ay* - f ) p

=

o

dP

- - E a+(u*) dU

We note that the product y * ( p , + A p 'c C@>. w

p

'

a x . in Q , (2.21)

a.e. in Q, in I;.

(2.22) (2.23)

6 ) makes sense because y*

E

g

Prooj We shall use the standard method. Consider the approximating control problem:

Minimize

subject to (2.1), (2.19).

348

5. Optimal Control of Parabolic Variational Inequalities

Here, g" and ppo"are defined as in the previous cases and, since the map

u - + y is compact from X p to C(is>, it follows by a standard device that

problem (2.24) has for every E > 0 at least one solution ( y , , u s ) E Wp'.'(Q) x X p . Moreover, using Lemma 2.1 and arguing as in the proof of Proposition 1.2, it follows that, for E 0, -+

u,

-+

y,

+y*

p"( y , )

u*

+

strongly in X p , strongly in

f + by*

T2v1( Q) c C ( e), (2.25)

weakly in LP( Q).

- y:

Now let p , E H 2 -'(Q) n L2(0,T; H,'(R)) be the solution to the boundary value problem

JP,

+ AP, dt

- P , ~ " ( Y , )=

in C,

p,= 0 P,(V

in Q,

Vg"(t,y,)

+ Vqo,"(Y&(T))= 0

in

a.

(2.26)

Then, by a little calculation, we find that

-+ dP, dv

d+(u,)

+ F ( u , - u*) 3 0

(2.27)

in 2 ,

where F: X p -+ X; is the duality mapping of X p and d+: X p -+ X; is the subdifferential of 4. Now multiplying Eq. (2.26) by p , and sign p , , we get the estimate

kT

l l ~ , ( t ) l I & (+~ )

Ilp,(t)llk:(n) dt +

/n I

b"(y,)p,l h d t

IC .

(2.28)

Hence, on a subsequence, we have (see the proof of Proposition 1.2)

P , -+P

stronglyin L2(Q),weaklyin L 2 ( 0 , T ;H , ' ( Q ) ) , weak star in L"(0, T; L 2 ( a ) ) ,

p,(t)

-+

p(t)

strongly in

a ) for every t E [0, T I ,

where p E BV([O,TI; H-'(R)), s > N/2. Moreover, there is p such that, on a generalized subsequence of { E } ,

BE(y , ) p ,

+

p

E

weak star (vaguely) in ( Lm( Q))* .

(2.29) (L"(Q))*

349

5.2. Boundary Control of Parabolic Variational Inequalities

Then, letting

E

tend to zero in Eq. (2.26), we get dP

+Ap dt

-p E

in Q ,

dg(t,y*)

in R ,

P ( T ) + dcp,(Y*(T)) 3 0 p = o

In other words, 35 E L2(Q),t ( x , t ) that

E

in

X.

(2.30)

d g ( t , y * X x , t ) a.e. ( x , t )

E

Q, such

for all cp E L2(0,T ; H,'(R)) n L'(Q) such that d q / & E L2(0,T ; H-'(R)) and cp(x,O) = 0. Moreover, by (1.46) and (1.47), we have p,P"(y,) + p ( f p , b E ( y E ) y E+ o

JY *

7+ A Y * ) = 0

stronglyin L ' ( Q ) ,

strongly in L ' ( Q ) .

Then, by (2.29, we infer that PY*

=

0,

and Eqs. (2.20, (2.22) follow. Now, let x E Wp2"(Q)be the solution to the boundary value problem JX

--Ax=O dt x=w x(x,O) = xo(x)

in C, in a,

where w E X p , xo E W;-2/p(R) and x0(x) = w ( x , 0) a.e. x by Lemma 2.1,

II xIIw2.1p( Q ) IC( 111 w III p + II

(2.31) E

R. Then,

xOIIW,~-~/P(~~)).

On the other hand, by the trace theorem (see, e.g., Ladyzhenskaya et al. [l]) we may choose xo E W;-2/p(R) such that xo = w(*,O)in dR and

II

XJIw;-2/P(n)

ICllw(0, *)llW,2-%(R)

c 111 w 111 p .

I

350

5. Optimal Control of Parabolic Variational Inequalities

With such a choice, we have

II XIlW,2.'(Q,

I Cllwll,

vw E

xp.

(2.32)

Now multiplying Eq. (2.30) by y, and integrating on Q, after some calculation we get

/ 151Ixldxdl

IIIL(X)I+

Q

(I xol IP,(X,O)I

+ l P & ( X , T)I I x ( x , T)I) h,

and by estimate (2.32) and the trace theorem, we get

Hence, { d p , / d u ) is bounded in X * p (the dual of X p >and, letting zero in Eq. (2.29, it follows by (2.27) that

E

tend to

as claimed. We will consider now a variant of problem (2.18). Minimize on ally system

E

W,'.'(Q>,u dY

E

i T g ( t , y ( t ) ) dt

+ po(y ( T ) )

(2.33)

LYO, T ) , and u E W'*l0([O, TI), subject to the state

dt - AY = f o

in { ( x , t ) E

Q ;y ( x , t ) > 01,

Y =go(x)u(t) in c, in x E a, y(x,O) = 0, u' + f ( u ) = u a.e. t E ( O , T ) , u ( 0 ) = 0,

(2.34a) (2.34b)

5.2. Boundary Control of Parabolic Variational Inequalities

351

and to the control constraints a.e. t

0 5 u ( t ) IM

k T u ( t )dt Here, f o

E

LP(Q), go

E

W:-'/P(dil),

=

E

(2.35)

L.

p >(N

g o 2 0 indil,

(O,T),

+ 2)/2,

and (2.36)

O<MT
The function f : R -+ R is Lipschitz and continuously differentiable. Denote by Uo c LYO, T ) the class of functions u satisfying the constraints (2.35). To be more specific, we shall assume that g(t,y) =

1g ' ( y ( x ) )dx,

4oo(Y)

n

where g': R + R, i ing the conditions

=

=

1S 2 ( Y W ) n

Y

d x 7

E

L2(W

1,2 are continuously differentiable functions satisfy-

+ lyl)

g ' ( y ) 2 -C(1

Vy E R,i

=

(2.37)

1,2.

Theorem 2.2. Let ( y * , u * ) be optimal for problem (2.33). Then there are p E L2(0,T ; H,'(Q)) n BV([O,TI; H-'((n)) n L"(0, T ; L2(il)>and q E AC([O,TI) such that dP

dt + Ap

=

p

=

0

in [ y *

P

=

0

in 2 ,

q ' ( t ) - f ' ( u * ( t ) ) q ( t )=

-g:(y)

=

in [ y * > 01,

0, f o z 01,

p ( x , T ) = -g,2(y*(x,T)),

1'1 go zdPd x d s I

q ( T ) = 0,

an

a.e. t

E

(2.38) (O,T), (2.39)

where h is some real number. Proo& Since the proof is essentially the same as that of Theorem 2.1, it will be sketched only.

352

5. Optimal Control of Parabolic Variational Inequalities

Consider the approximating penalized problem: Minimize

{ / Q g ' ( y ( x , t ) ) dxdt

1

LT(

+

+ / g ' ( y ( x , T ) ) dx n

(2.41)

u( t ) - u*( t ) ) 2dt

on all ( y , u, u ) , subject to dY

-at

AY + P e ( y ) = f o

in Z,

y =g,u

in

Q, inn,

y(x,O) = O

and to constraints (2.34b), (2.35). If ( y * , u * , u * ) is optimal for problem (2.411, it follows as before that u, -,u*

strongly in ~ ' ( 0T,) ,

ye -,y *

strongly in W,'.'(Q),

u, -, u*

strongly in w','( [0, T I ) .

On the other hand, problem (2.41) has the following optimality system: dP&

-+

dt

Pe

=

0

in Z,

p,(x,T)

=

-g,2(ye(x,T)),

x E

q:(t) -f'(ve(t))qe(t)

qe(T)

q,(t)

in

Ape - P"(Ye)Pe = g i ( Y e )

+ u*(t) - u,(t)

E

=

lTd'

=

0,

Q,

a,

(2.42)

/ang o ( x ) 'Pe du, (2.43)

dh(u,)(t)

a.e. t

E

( O , T ) , (2.44)

where dh: L2(0,T ) -,L2(0,T ) is the subdifferential of the indicator function h to the subset U,,i.e., h(u)

=

if u E U,, otherwise.

Now, we may represent dh as (see, e.g., Section 2 in Chapter 2) dh(u)

=

dh,(u)

+ dh,(u),

u E L*(O,T),

5.2. Boundary Control of Parabolic Variational Inequalities

353

where h , is the indicator function of {u E L2(0,T ) ; 0 Iu IM a.e. in (0, T)) and h , is the indicator function of {u E L2(0,T ) ; :/ u ( t ) dt = L). As shown earlier, we have dh,(u)

= {w E

L~(O,T); w(t)

=

o a.e. in [ t ; o < u ( t ) < M I ,

w 2 0 a.e. in [ t ; u ( t ) and dh,(u) = {A

E

w 2 0 a.e. in [ t ; u ( t )

=MI,

=

O]}

R). Then, by (2.441, we see that

+ u*(t) - u,(t)

a.e. in [ t ; q , ( t )

=

< A,],

(2.44)‘

where A, E R. Now using the standard estimate for the solution p , to Eq. (2.42), it follows that, on a subsequence, stronglyin H - ” ( R ) , V t E [ O , T ] ,

p,(t) + p ( t )

strongly in L 2 ( Q ) weakly , in L 2 ( 0 , T ;Hj(R)),

PE + P

where s > N/2 and p

E

B”(Y&)P,

BV([O,TI; H - ” ( R ) ) Moreover, . we have +

weak star in (L”( Q ) ) * ,

P

aPE aP + dV

weakly in Xp* .

dV

Then, letting E tend to zero in the system (2.42H2.441, we see that p satisfies the system JP

+ Ap dt

-p

=g:(y*)

in Q,

P ( X , T ) = -g,’(y*(x, T ) ) q r ( t ) - f ’ ( u * ( t ) ) q ( t )=

Y,, in a, in

p = o

1‘1

d n

t

go

JP

dads

a.e. f

q ( T ) = 0. Similarly, letting

E

(2.45) E

(o,T), (2.46)

tend to zero in Eq. (2.44)’, we get

u*(t) =

i“

M

a.e. [ t a.e. [t

( 0 , T ) ;q ( t ) < A], E ( 0 , T ) ;q ( t ) > A], E

(2.47)

354

5. Optimal Control of Parabolic Variational Inequalities

where A

E R. Moreover, as seen in the previous proof we have p = 0 in Q; y * ( x , t ) > 0) and p = 0 in { ( x , t ) E Q; y * ( x , t ) = 0, fo + 01, thereby completing the proof.

{(x,t) E

In particular, it follows by Theorem 2.2 that if d p / d u 2 0 in dR (this happens, for instance, if gi s 0, g; 2 0, by virtue of the maximum principle for parabolic equations), then q is monotonically increasing and so every optimal control u* for problem (2.32) has at most one switching point t , . More will be said about this in Example 2 following. We note also that Theorem 2.2 extends to control systems (2.33) with boundary conditions of the following form (see, e.g., Barbu and Barron [ll): m

y

=

+f ( u ) = u

U’

where u = ( u , , .. . , u r n > , u W2-l/p(dR), i = 1,. . . ,m .

C gi(x)ui(t)

in 2

i= 1

a.e. in ( O , T ) , =

( u l , . . . , un),

f

u( 0 ) = 0, = (fl,.

. . , f,,,),

and gi

E

5.2.3. Examples 1. Control of oxygen difision in an absorbing tissue. The oxygen diffusion in absorbing tissue R c R3 is governed by the obstacle problem (see, e.g., E. Magenes [l], and Elliott and Ockendon [l], p. 127)

dY dt

- - Ay

+ 1= 0

y(x,O) = y o ( x ) , y=u

in ( ( x , t ) x

E

E

Q;y ( x , t )

> O},

a,

in C

=

dR x ( O , T ) ,

(2.48)

where y o is the initial distribution of oxygen concentration in the tissue and u ( t ) is a prescribed concentration on dR at moment t , which satisfies the constraints u ( t ) E U, V t E [0,TI,

u ~ = { ~ E x ~ ; u ( x , o ) = ~ ~ ( x ) v ~ E ~ R , ~ ~ o ~

5.2. Boundary Control of Parabolic Variational Inequalities

355

We associate with control system (2.48) the cost functional

1141; + J ( y ( x , T ) ) - Y o ( x ) ) 2h,

(2.49a)

R

where y o E ,!,'(a). This is a problem of the form (2.181, where g

cp,(y)

=

y

Ily -y0112L2(n,,

= 0 and

EL2(Q).

Then, &$(u) = Fu + Nu@) V u E X, where F: X p + X; is the duality mapping of X p = W;-'/P,1-1/2P(.C) and Nu, c X; is the normal cone to U,, i.e., Nu,(u)

= (r]

EX;; (r],u - u ) 2 OVUE

u,}.

If we take u = u + cp, there cp 2 0, cp E C;(Z), we get ( r ] , ~ )I0. Hence, r] is a nonpositive Radon measure on 2, i.e., r] E M ( 2 ) . Moreover, it is readily seen that r] = 0 in ((x,t ) E 2; u ( x , t ) > 0). Then, by Theorem 2.1, the optimality system for this problem is JP at

-

+ Ap = 0

in [ y * > 01, in [ y * in 2,

p =0 p =O

=

01,

(2.49b)

whilst the optimal control u* is given by

- + Fu* dP dU

=

0

u*=o

because (77, u * ) u* =

0,

r] I0

-F-1(2) dV

in

=

for all

["

dV

in [ u * > 01, in[S>o],

r] E NuI,(u*).Equivalently,

< 01,

2. Optimal control of the one phase Stefan problem. Consider the melting process of a body of ice Q c R3 maintained at 0°C in contact with a region

5. Optimal Control of Parabolic Variational Inequalities

356

of water on T, and at controlled temperature u on r,. The boundary d f l is composed of two disjoint parts rland r, , and n = 0. Let N x , t 1 be the water temperature of point x E fl at time t. Initially, the water occupies the domain fl, c fl at temperature 8, (see Fig. 4.1). If t = u ( x ) is the equation of the water-ice interface, then the temperature distribution 8 satisfies the one phase Stefan problem

r1 r2

8=0 e(x,o)

=

in { ( x , t ) vx E

e,(x)

E

Q; ' + ( x ) 2 t } ,

a,,

e(x,o) =

o

vx

E

R\R,,

(2.51)

along with the Dirichlet boundary conditions 8( x , t ) = go( x ) u ( t ) e(x,t)

=

o

rl x (0, T ) , in 2, = r, x ( O , T ) ,

in 2 ,

=

(2.52)

where go E

w:-l/p(rl),

8,

c(fio),

E

go 2 0

p >(N

in

rl,

+ 2)/2,

eo(x)

>o

vx

E

no. (2.53)

We will consider here the following model optimization problem associated with the controlled melting process (2.51), (2.52): (2.54) on all 8 and u subject to (2.50, (2.52) and to the control constraints u where u

E L"(0, T

) ;0 I u ( t ) I M ,

We will assume of course that MT > L.

1 T

0

u ( t ) dt =

E

Uo,

5.2. Boundary Control of Parabolic Variational Inequalities

357

As seen in Chapter 4, (Section 3.21, by the transformation y(x, t )

/ k x , s ) e ( x , $1ds,

=

0

where x is the characteristic function of Q + = { ( x , t ) ; u ( x ) < t system (2.50, (2.52) reduces to the controlled obstacle problem

0

in R,

y(x,O)

=

y(x,t) y(x,t)

=g,(x)v(t)

u’(t) = u(t)

=

0

V(x,t)

a.e. t

E

I

T ) , the

(2.55) V(x,t)

E c1;

(2.56)

E 22,

(O,T),

u(0) = 0,

(2.57)

where fo = 8, in R, and fo = - p in R \ R,. In terms of y, problem (2.54) becomes: (2.58)

Maximize jay( x , T) dx on all (y,u ) E W$ ‘(Q)X Uo satkfiing (2.55142.57).

Applying Theorem 2.2, where f = 0, g ’ = 0, g2(y) = -y, we find that every optimal control u* is of the form (2.401, where

(2.59) and

dP dt

-

+ Ap = 0 p =0 p = O

in

{(x,t) E

Q ; y*(x,t) > O } ,

in { ( x , t ) E Q ; y*(x,t) in2, p(x,T) = 1

=

o , f o ( x , t ) + 01, V ~ E R . (2.60)

Inasmuch as dy*/dt - A y * > 0 in Ro X (O,T), we infer that y* > 0 in 0, x (0,T).

358

5. Optimal Control of Parabolic Variational Inequalities

Hence, p satisfies Eq. (2.60) in { ( x , t ) E Q;x E R,, t E (0, T ) ) and, by standard regularity results for parabolic Dirichlet problems (see for instance Chapter 4, Section 3.11, we know that p E C2’ X [O, T ) ) and p E C([O, TI; L2(R)). Then, by virtue of the maximum principle for linear parabolic operators (see, e.g., Porter and Weinberger [l], p. 170) we have

‘(a,

p > 0 in R, x ( 0 , T )

JP

dV

< 0 in

rl x ( 0 , T ) .

Then, by Eqs. (2.47) and (2.59), it follows that q is increasing and the optimal control u* has one switch point t o , i.e., (2.61) where Mt, = L. We have therefore proved: Corollary 2.1. Under assumptions (2.53) the optimal control problem (2.54) W has a unique solution u * , given by (2.61).

We will consider now the following problem: Given the surface So = ( ( t ,x ) ; t = K x ) } , find u E U, such that So is as “close as possible” to the free boundary S = dQ, of problem (2.51). This is an inverse Stefan problem in which the melting surface is known and the temperature on the surface rl has to be determined. Let y o be a given smooth function on Q such that y O ( x , t )= 0 for 0 I t I l ( x ) and d YO -

yo I 0

Ayo
in R ,

(2.62)

where fo is defined as before. Then the least square approach to this inverse problem (which in general is not well-posed) leads us to a problem of the form (2.33), i.e.: Minimize

/n

(y ( x , t )

-

y o ( x , t))’ dxdt

on all ( y , u ) E W: ‘(Q) X U, satisbins (2.55)-(2.57).

(2.63)

5.2. Boundary Control of Parabolic Variational Inequalities

359

Then, by Theorem 2.2, every optimal control u* of problem (2.63) is given by Eq. (2.401, where

and JP dt

- + Ap

=

in [ y * > 01,

2(y* - y o )

in [ y * = 0; fo z 01, in C, p(x,T) =0

p = o p = o

in R . (2.64)

By (2.551, (2.62), and the maximum principle, it follows that y * 2 y o in Q . Then, by (2.64), we conclude (again by virtue of the maximum principle) that d p / d u > 0 in Cl and so, by (2.401, we see that u* = M on [O, t o ] , u* = 0 on [ t o ,TI, where Mt, = L. 5.2.4. The Obstacle Problem with Neumann Boundaiy Control

The previous results remain true for optimal control problems with payoff (2.65) and governed by the variational inequality (2.11, i.e., dY

- A Y =fo dt dY

- Ay dt

in{(x,t) E Q ; y ( x , t ) > O ) ,

in Q

2fo, y 2 0

Y(X,O) =yo(x),

x

E

=

R

X

(O,T),

a,

(2.66)

with the boundary conditions dY +cry dU

du

-

dt

=u

+ AU = BU

in C , , Vt

E

y =O (O,T),

inC2,

(2.67)

~ ( 0= ) 0.

(2.68)

Here, R is a bounded, open subset of R" with a sufficiently smooth boundary dR = rl u r2,TI n T2 = 0, C i= ri x (0, T ) , i = 1,2; A is a

5. Optimal Control of Parabolic Variational Inequalities

360

linear continuous operator from L 2 ( 2 , )to itself, B is a linear continuous operator from a Hilbert space of controllers U to L2(Cl), a > 0, and fo

E

(2.69)

W'72([0,T1; L2(fl)),

y o ~ ~ ~ ( f ly o) =, 0 in

r,,

Regarding the functions g : [0, TI + R we will assume that:

dY0 dv

-

X

+ "yo = 0

in I?,,

yo 2 0.

(2.70)

L2(fl) + R, (po: L 2 ( f l )+ R,and h: U

(i) h is convex, lower semicontinuous and

h ( u ) 2 yllullZl +

c

Vu

E

u,

(2.71)

for some y > 0 and C E R, (ii) g is measurable in r , g ( t , 0) E L"(0,T ) , and there exists C that g ( t , y ) + %(Y)

2 C(llYIlL2(n) + 1)

VY

E

E

R such

L2(fl). (2.72)

For every r > 0, there exists L , > 0 such that Ig(t,y) - g ( t , z)l + IVO(Y) - cpo(z)l 5 L,llY - Z l l L 2 W

for all t E [O,Tl and IlyII~qn)+ I I z I I L ~ R ) Ir. Under assumptions (2.69), (2.70) the boundary value problem (2.66)-(2.68) has for every u E U a unique solution y E W ' 3 2 ( [ 0 , T ] ; V ) n W'9"([0,T];H ) (see Corollary 3.3, Chapter 4). Here, V = {y E H'(fl); y = 0 in r,}, H = L2(fl). Moreover, y = lim,+o y, strongly in C([O,TI; H ) and weakly in W'."([O,TI; H ) n W ' T ~ ( [TI; O , V ) ,where y is the solution to the approximating equation a "

-y dt

-

dv

-

dt

Ay

+ p"(y)

+ Av = Bu

where p" is defined by (1.17).

=fo

in Q

in ( O , T ) ,

=

fl x ( O , T ) ,

u ( 0 ) = 0,

(2.73)

5.2. Boundary Control of Parabolic Variational Inequalities

361

The following estimate holds: J l y , I l w ~ ~ ~ (H[)on,W~I]. ;~ ( [ O , Tv]); I C(1 + Ilullu). Then, by a standard device (see Proposition 1.11, it follows that optimal control problem (2.65) admits at least one optimal control u*. Regarding the characterization of optimal controllers, we have: Theorem 23. Let ( y * ,u * ) be an optimal pair for problem (2.65), (2.66). Then there exists p E L2(0,T ; V ) n L“(0,T ; L2(fl)) n BV([O,TI; (V n Hs(fl))‘), s > N/2, such that d p / d t + A p E (L“(Q))*and

($+ A p ) .

E

d g ( t ,y * )

a.e. in { ( x , t )

P ( T ) + dcp,(Y*(T)) dP

dV

p

=

0

+ ap=O

inZ,,

3

E

Q ; y*( x , t )

in fl,

0

in&,

p=O

a.e. in { ( x , t ) E Q ; y*( x , t )

=

> 0 } , (2.74) (2.75) (2.76)

0 , f o ( x , t ) Z O}, (2.77)

B* l T e - A * ( s - t ) p ( s ) ds E d h ( u * ) .

(2.78)

(A* is the adjoint of A J If N = 1, then y* E C ( 8 ) and Eq. (2.74) becomes

where 6 E L2(Q). Here, BV([O,TI; (V n Hs(fl))’) is the space of functions with bounded variation from [O, TI to (V n Hs(fl))’.

Proof Since the proof is essentially the same as that of Theorem 2.1, it will be sketched only. Also, for the sake of simplicity we will assume that g and cpo are differentiable on L2(fl). For every E > 0, consider the approximating control problem: Minimize

5. Optimal Control of Parabolic Variational Inequalities

362

Let ( y & u,) , be a solution to problem (2.78). By assumptions (9, (ii) we see that, for E + 0, u,

+

u*

strongly in U ,

y,

+

y*

strongly in C([O,T I ; H ) , weakly in W ' T [0, ~ ( T I ; H ) n W ' , 2 [0, ( TI;V ) ,

u,

u*

strongly in

dv, dt

+ Av& = Bp&

-+

[o, T I ;

L ~q)), (

where -

a.e. in ( O , T ) ,

u,(O)

=

0.

On the other hand, we have j u T ( V y g ( t , y & ( f ) ) , Z ( t ) ) d+t h f ( u , , w )

+ (u,

-

u*,w) 2 0

Vw

E

U,

where h' is the directional derivative of h , (., and ( * , ) are the scalar products in L2(R) and U, respectively, whilst z is the solution to Z, -

A Z + P"(Y,)z

in Q, in R ,

0 z(x,O) = 0

dv dt

-

=

+ Ru = Bw

in ( O , T ) ,

v(0) = 0.

Let p , E W',2([0, TI; L2(R)) n L2(0,T ; V ) be the solution to boundary value problem d

-P, dt + AP,

-

P,P&(Y&)= V,g(t, Y,)

x

P , ( X , T ) + V % ( Y & ( T ) ) ( x )= 0 ,

dP& + ap, dV

=

0

in Z , ,

After some calculation, we get that h'(u,,w)

+ (u, - u*,w)

in

p,

=

0

in

Q9 E

a,

Z2.

(2.80)

5.2. Boundary Control of Parabolic Variational Inequalities

This yields B * ( [ T e - A * ( ~ - r % ( gds )

1+

u,

- u* E

363

dh(u,).

Next, we multiply Eq. (2.80) by p , and sign p , and integrate on Q. We obtain the estimate

Hence, {(P,)~} is bounded in L'(0, T ; L'(s1)) + L2(0,T ; V ' ) c L'(0, T ; ( H s ( ( R )n V ) ' )for s > N / 2 (by Sobolev's imbedding theorem). Thus, on a subsequence, we have p,

+

p

weakly in L 2 ( 0 ,T ; V ) ,

weak star in Lm(O,T ; L2(a ) ) ,

and by the Helly theorem, stronglyin ( H ' ( ( R ) n V ) ' ,

p,(t) -+p(t)

Vt

E

[O,T].

Now, since the injection of V into L2(a)is compact, for every A > 0 we have

Finally, arguing as in the previous proofs we see that (on a subsequence)

P , P " ( Y & ) + ~ ( f o -Y: + A Y * ) PJ"(Y,)

+

strongbin L ' ( Q ) , strongly in L'( Q) .

0

Combining the preceding relations, we conclude that p satisfies Eqs. (2.74H2.78). If N = 1, then it follows that y, + y* in C(&) and so we infer that PY*

=

( p I+ A p

-

6 ) =~0

in

Q,

5. Optimal Control of Parabolic Variational Inequalities

364

where 6 = lim, p with y * .

--t

,V,,g(t,y,) (in L2(Q))and p y * stands for the product of

Theorem 2.3 can be applied as in the previous example to optimal control of the one phase Stefan problem with boundary value conditions de

-++(8-u)=O

dv

in&,

8 = 0 in2,.

5.3. The Time-Optimal Control Problem 5.3.1. The Formulation of the Problem

Consider the control process described by the nonlinear Cauchy problem

where M is a maximal monotone mapping in a Hilbert space H with the norm I I and scalar product (., * ). Then, as seen earlier, for every y o E D ( M ) and u E L:,,(O,m; H ) problem (3.1) has a unique mild solution y = y ( t , y o , u ) E C([O, m); H ) . Denote by %! the class of control functions u ,

-

W = { u E L"(0,m; H ) ; u ( t ) E K a.e. t > 0),

(3.2)

where K is a closed bounded and convex subset of H. Let y o ,y , E D(M ) be fixed. A control u E %! is called admissible if it steers y o to y , in a finite time T, i.e., y ( T , y o ,u ) = y l . If K = ( u E H ; IuI Ip ) , then we have: Lemma 3.1. Assume that y o E D(M ) and y , there is at least one admissible control u E W.

E D(M),

IMoy,l < p. Then

Proof We shall argue as in the proof of Proposition 2.3 in Chapter 4. Consider the feedback law

5 3 . The Time-Optimal Control Problem

Since the operator My problem

365

+ sign(y - y,) is monotone in H X H, the Cauchy

Y(0) =yo3 has a unique mild solution y E C([O,m); H ) . If yo E D ( M ) , then y is a.e. differentiable on (0, m) and we have, therefore, I d -ly(t) 2 dt

-y,I2

-

+ ply(t)

-y,l

IIM0YlllY(t) - Y 1 l

a.e. t > 0,

because M is monotone. (Here, M o t is the minimal section of M . ) This yields ly(t) -y,l

I(IMOy,l- P ) t

+ ly,

-y,l

V t 2 0.

Hence, y(t) = y, for t 2 ( p - IMoy,l)-'lyo - yll. This clearly extends to all yo E D( M). The smallest time t for which y(t, yo, u ) = y, is called the transition time of the control u, and the infimum T(yo,y,) of the transition times of all admissible controls u E % is called minimal time, i.e., T ( y o ,y,)

=

inf{T ; 3 u

E % such

that y( T, y o , u )

= y,).

(3.3)

A control u E % for which y(T(yo, y,), yo, u ) = 0 (if any) is called a time-optimal control of system (3.1) and the pair (y(t, yo, u), u ) is called a time-optimal pair of system (3.1)). Proposition 3.1. Let M be maximal monotone and let S ( t ) = e -M ' , the semigroup generated by M on D( M ) , be compact for every t > 0. Then under conditions of Lemma 3.1 there exists at least one time-optimal control for system (3.1).

Proof Let yo E D ( M ) be arbitrary but fixed. We know that T o = T(y,,y,) < m. Hence, there is a sequence T, + T o and u, such that y(T,, yo, u,) = y, . Let y, = y(t, yo, u,) be the corresponding solution to Eq. (3.1). Without loss of generality, we may assume that u, + u* weak star in Lm(O,T; H ) , where T o < To < 03 (we extend u, be zero outside the

366

5. Optimal Control of Parabolic Variational Inequalities

-

interval [O,T,]) and by Theorem 2.4, Chapter 4, we have, on a subsequence, Y,(t)

Y(t9

Yo u * ) * 9

This clearly implies that y ( T o ,y o ,u ) = y , , and so u* is a time-optimal control for system (3.1). Recall that if M = d 4 where 4: H R is a lower semicontinuous convex function, then the assumptions of Proposition 3.1 hold if, for every A E R, the level sets {x E H ; 4(x) s A) are compact in H. -+

In the linear case, every time-optimal control is a bang-bang control and satisfies a maximum principle type result (Fattorini [l]). More precisely, if M is the generator of an analytic semigroup then every time-optimal control u* for system (3.0, where K = { u E H ; IuI I p } , can be represented as a.e. t > 0, u * ( t ) = p sgn p ( t ) where p is the solution to adjoint equation p ' - M*p

=

0

a.e. t > 0,

and sgn p = pIp1-l if p # 0, sgn 0 = { w ; IwI I1). (For other results of this type, we refer to Balakrishnan [l], J. L. Lions [2], and H. 0. Fattorini [21.) Next we shall prove a similar result for some classes of nonlinear accretive systems of the form (3.1). 5.3.2. The Time-Optimal Control Problem for Smooth Systems We shall consider here the time-optimal problem for system (3.1) in the case where M=A+F,

and: (i)

(3 -4)

-A is the infinitesimal generator of a Co-semigroup of contractions that is analytic and compact; (ii) F: H -+ H is continuously differentiable, monotone, and its FrCchet derivative F' is bounded on bounded subsets; (iii) K is a closed, convex, and bounded subset of H and { p E H ; Ipl I y ) c K for some y > 0. e-At

367

53. The Time-Optimal Control Problem

H

In particular, it follows by (i), (ii) that A X H.

+F

is maximal monotone in

Theorem 3.1. Assume that y o ,y 1 E D ( A ) and that hypotheses (i)-(iii) are satisfied. Let ( y * , u * ) be any time-optimalpair corresponding to y o ,y , , where IAy, + Fy,l < y . Then u*(t)

dHK(

P ( t 1)

a.e. t

E

( O , T * ) (3.5)

in [O,T*], (3.6) a.e. t E (O,T*). (3.7)

p ’ ( t ) - A * p ( t ) - ( F ’ ( y * ) ) * p= 0 HK(p(t)) - (Ay*(t)+ Fy*(t),p(t)) = 1

Here, T* = T ( y , , y l ) is the minimal time and HK is the support function of K, i.e., HK(P)

=

Since, by (3.71, p ( t ) # 0 V t bang-bang control, i.e., u*(t)

K,

sup{(p,u);

E

E

Vp

[0,T*l it follows by (3.5) that u* is a

FrK

a.e. t

E

(O,T*).

(dHKis the subdifferential of HK.)

The solution p to Eq. (3.6) is considered of course in the mild sense and A* denotes the dual of A. The idea of proof is to approximate the time-optimal problem by the free time-optimal control problem min{T

+ lT( h( u( t ) ) + -lu( 0 2 &

t ) I 2 ) dt

1

+ -le-’&( 2E

y ( T ) - y1)I2

where the minimum is taken over all T > 0 and u E L2(0,T ; U),y C([O, TI; H ) satisfying Eq. (3.1) with M = A + F. Here, h: H + R is the indicator function of K, i.e.,

h(u) =

E

if u E K , otherwise.

It is readily seen that problem (3.8) has at least one solution (yE,uE,T,). Lemma 3.2.

Let (y,, u,, T,) be optimal in problem (3.8). Then for

E +

0,

5. Optimal Control of Parabolic Variational Inequalities

368

we have T,

T*

+

l(u,

=

-

T ( y o ,y l ) and

- u*> cis

o

strongly in

~2(0,m;

HI,

(3.9)

u,

+

u*

weak star in L"(0, T * ; H),

y,

+

y*

strongly in C([O, T * ] ;H ) , weakly in W ' , 2 ( [ 0T, * ] ;H).

Proof

(3.10) (3.11)

We have T,

+

i'(5lu,l2 + T

E

1

h ( u , ) ) dt

+- [dt1lr(uE 2

0

+ ( 2 ~ ) - ' l e - ~ " ( y , ( T-,y) l ) 1 2

- u * ) ds

Iu*I2d t .

(We extend u, and u* by 0 on [T,, +m) and [ T * ,+..I, T, IT * and Hence, lim sup,

(3.12)

respectively.)

~

ly,(T,) -ylI

+

0

as

E +

0.

Now, let E,, 4 0 be such that T,, + To and us, + uo weak star in LYO, m; H ) . Since - A generates an analytic semigroup and y o E D ( A ) , we have (see Theorem 4.6 in Chapter l), Ily;IIL2(0,TO;ff, Ic

VE

> 0.

Note also that ly,(t)l Ic

Vt

E

(0,m).

Now, since the semigroup e-A' is compact we deduce by the Arzelh-Ascoli theorem that {y,] is compact in C([O,Tol;H ) . Hence, on a subsequence, again denoted E,, , we have y,"

3

j7

strongly in C([O, T o ] ;H); weakly in W'*'([O,T o ] ;H ) ,

where j is the solution to (3.1) with u = 6. Clearly, j ( T o ) = y , and so uo is admissible. Hence To = T * and by (3.12) we have also that

as claimed.

rn

5.3. The Time-Optimal Control Problem

369

Let (y,, u,, T,) be optimal in problem (3.8). Then there is W’.2([0, T,k H ) n C([O,T,]; H ) such that

Lemma 3.3.

p,

E

y:+Ay,+Fy,=u,

a.e.

p: -A*p, - (F’(y,))*p, = 0

y,(O) p,(t)

E

=y

,,

p,(T,)

(3.13)

a.e.tE(O,T,),

r

E

(O,T,),

1

= - -e-A’Ee-AE & (Y&(T&) -Yl),

(3.14) (3.15)

dh(u,(t)) + &U,(t) + j T , d s / ‘ ( u , ( T ) - u * ( T ) ) dT 0

r

V t E [O,T,], (3.16) ((u,(s)

- u * ( s ) ) ds =

0,

u,(t)

= u*(t)

V t 2 T,, (3.17)

-(AY,(T,) + FY&(T&), P,(T,)) + HK(P,(T,) - &U&(T,>) &

2 + -Iu,(T&)I = 1. 2

(3.18)

Here, d h h ) = {u E H ; (w,u - u ) 2 0 Vu, llull Ip ) and H , ( p ) sup{(p,u); u E K ) is the support function of K. Proof

=

Since ( y e ,u,, T,) is optimal, we have

IkTs(h(u,(t)

+ -21 ( d t

+ A u ( t ) ) + -2l u , ( t ) + Av(t)I2)dt

l((u,

&

+ Au - u * ) ds VA > 0 , u E Lm(O,CQ; H ) ,

370

5. Optimal Control of Parabolic Variational Inequalities

where y ( t , u , y o ) is the solution to system (3.1). Subtracting, dividing by A, and letting A tend to zero, we get joT"(h'(u,(r). 4 t ) ) + +(dt

dt - p ( P & ( % u ( t ) ) dt

&(U,(t),V(t)))

0

(/'(u,(s) - u*(s)) d s , / ' u ( ~ d )r 0

0

1

20

Vu E L"(0, m; H ) ,

where p , is the solution to (3.14), (3.15) and h' is the directional derivative of h. This yields

+((u(7),/mdt~'(u& -u*)ds 7

0

i

d72 0

V u E L " ( 0 , a J ;H ) ,

which implies (3.16) and (3.17). It remains to prove (3.18). We note first that

T,

&

+ ( 2 & ) - ' 1 e - ~ " ( y , ( ~ -y1)12 ,) + 5/~"IU,(~)I' I T, - A

+ (2&)-'le-A"(y,(T,

+ 5 k'-Alu,(t)12 &

VO

dt

- A)

dt

-y1)12

< A < T,.

(3.19)

Since ( & I + ah)-' is Lipschitz on H and p, E W',2([0,T,];H ) (because p,(T,) E D ( A * ) and the semigroup e-A*' is analytic), we see by Eq. (3.16) that u, is Holder continuous on [O,T,]. Hence, y, E C'([O,T,]; H ) (see Theorem 4.5 in Chapter 1) and so we may pass to limit in (3.19), getting &

- ( Y ; ( T & ) , P € ( T & ) )+ ?lU&(T&)I2 5 -1. Equivalently, &

( A Y A T , ) + FY,(T&) - k ( T & ) , P & ( T &+) ),lU&(T&)125 -1.

(3.20)

On the other hand, it follows by (3.16) that - ( & ( T E ) 9 P € ( 7 , € ) )+ &IU,(T&)12+ ( J h ( u , ( T , ) ) ,U & ( T & ) ) = 0 ,

5.3. The Time-Optimal Control Problem

371

and this yields the opposite inequality

Proof of Theorem 3.1. Since 0

E

int K, we have

HK(P) 2 YIP1

VP E H ,

where y > 0. Then, by Eq. (3.181, it follows that &

-lU,(T)I2 + YlP&(T&) - &U&(T&>I 2 5 1 + IFY&(T,) + Ay,l lP,(T&)l

because the operator e-A*Ee-AEAis positive. Since, as seen in Lemma 3.2, y,(T') + y , as implies that &

yIU&(T,)12+ lp,(T,)l

c

I

+ CE E +

V.9 > 0.

0 the preceding

372

5. Optimal Control of Parabolic Variational Inequalities

Then, by the variation of constants formula

and the compactness of the semigroup e - A * t , we conclude that on a subsequence, again denoted E , we have (3.21) strongly in H, V t E [0, T * ] , p , ( t ) -+ p ( t ) Where p is the solution to Eq. (3.6). Then, letting E tend to zero in (3.16), we see that a.e. t E (0, T * ) , p ( t ) E d h ( u*( t ) ) which is equivalent to (3.5). It remains to prove (3.7). To this end, we note first that by Eqs. (3.131, (3.14) we have

5 3 . The Time-Optimal Control Problem

Then, letting that

E

373

tend to zero, it follows by Lemma 3.2, (3.18), and (3.21)

thereby completing the proof.

5.3.3. The Time-Optimal Control Problem for Semilinear Parabolic Equa tions We shall study here the time-optimal control problem in the case where H = L2(R), K = {u E L2(R); lu(x)l Ip a.e. x E R), My = - A y + p ( y ) V y E D ( M ) = { y E H2(R) n H,'(R); 3w E L2(R) such that w ( x ) E p ( y ( x ) ) a.e. x E R}.Here, R is a bounded and open subset of R N with a sufficiently smooth boundary (of class C',',for instance) and /3 is a maximal monotone graph in R X R such that 0 E p(0). In other words, we shall study the problem:

where y ( t , y o , u) is the solution to semilinear parabolic boundary value problem dY dt

--

Ay

+p(y)3u

in R

X (O,m),

and

Regarding y o and y l , we shall assume that y,,y,

E

D ( M ) n L"(R)

and

IIMoylllr(n)< P ,

(3.24)

374

5. Optimal Control of Parabolic Variational Inequalities

It turns out that under the preceding assumptions problem (P) has at least one solution (T*, y*, u*). This follows as in the proof of Lemma 3.1, using Lemma 3.4 following. Lemma 3.4. Under assumptions (3.241, there is at least one admissible control u E Zpforproblem (P).

Proofi We note that Lemma 3.1 is inapplicable here since int K = 0. However, we shall use the same method to prove the existence of an admissible control (see also Proposition 3.5 in Chapter 4). Namely, consider the feedback control 4x9t)

where sign r problem

JY dt

= =

- p sign(y(x, t ) - Yl(X))

r/lrl if r

#

0, sign 0

=

x

t)E

(O,W),

[ - 1,1]. Then the boundary value

AY + P ( Y ) + psign(y - y l ) y(x,O) =yo(x), y=o

V(X,

X E

in

30

x

(o,~),

a,

in dR x (O,m),

has a unique solution y E W'>*([O,T]; L2(R)) n L2(0,T; H,'(R) n H2(R)) for every T > 0 because by Theorem 2.4, in Chapter 2, the operator

MY = MY + psign(y

-Y1),

y

E

WM),

is maximal monotone in L2(R). As a matter of fact,

=

d+ where

5.3. The Time-Optimal Control Problem

375

and by the maximum principle we see that Y ( X , t ) -Y d X )

4x7t)

V ( x , t ) E R x ( O , ( p - P)-lIIYo -YlllL=(nJ. Hence, y ( x , t ) - y , ( x ) I llyo - y,lI~=(n) - ( p - p ) t , and by a ~ymmetric argument it follows that y(x,t) -y,(x)

2

-IlYo -Y,llL-(n) + ( P - P ) f

for x E R, 0 5 t 5 ( p - ~ ) - ' l l y- Y ~ ~ I I L = ~ ~ ) . Hence, y ( x , t ) = y , ( x ) for x E R and t 2 ( p - p)-'llyo - ylIIL=(n), as claimed. Now we shall formulate a maximum principle type result for problem (P). We shall assume that y o ,y , satisfy (3.24) and

Yo

E w;-2/q,q

(01,

4 > max(N,2).

(3.25)

Theorem 3.2. Let ( y * ,u * , T * ) be optimal in problem (P). Then

u*(x,t) E psignp(x,t)

a.e. ( x , t ) E R

X

( O , T * ) , (3.26)

where p E L2(0,T * ; W ~ ~ q " < n R )BV([O, ) T*l; W S ( R + ) W - l . q ( R ) ) ,s > N / 2 , satisfies the system d

-p at

+ Ap - u = 0

+ p / nI p ( x , t ) l d~ = 1

in R x ( O , T * ) ,

a.e. t

E

(o,T*).

(3.27)

(3.28)

Here 5 E L2(R x (0, T * ) ) is such that ( ( x , t ) E p ( y * ( x ,t ) ) a.e. ( x , t ) E R X (0, T * ) and v E (L"(R X (0, T*)))*. In particular, it follows by Eq. (3.28) that p ( t , - ) f 0 a.e. t E (0, T * ) and so for almost every t E (0, T * ) there is R, c R, m(R,) > 0 such that lu*(x,t)l = p a.e. x E a,. Proof of Theorem 3.2. Let ( y * ,u * , T * ) be any optimal pair for problem (P). Proceeding as in the proof of Theorem 3.1, consider the approximating

376

5. Optimal Control of Parabolic Variational Inequalities

control problem

where y E W ' * 2 ( [ 0TI; , L2(R)) n L2(0,T ; H,'(R) n HZ(R))V T > 0 is the solution to system dY

-

dt

+ Ay + p " ( y )

=

Y(X,O) =Yo(x> y=o

u in

(o,~),

in R x

a,

in dR x ( 0 , ~ ) .

(3.30)

v(&)&z0~,

Here, T ( E ) > 0, I * l2 is the L2-norm on R and p" is a smooth approximation of p satisfying conditions (k)-(kkk) in Section 1.2. The function h: L2(R) -+ R is the indicator function of K O = (u E L2(R); lu(x)l Ip a.e. x E a}. Let (y,, u,, 7")be optimal in problem (3.29). Then, by Lemma 3.3, there is p, E C([O, T,]; L2(R)) n L2(0,T,; H,'(R) n H2(R)) such that

dye

- Ay, dt y,(x,O) =yo(x)

+ p"(y,)

=

u,

in R,

y,

=

in Q, 0

=

in C,

R

=

X

dR

(O,T,), X

(O,T,), (3.31)

5.3. The Time-Optimal Control Problem

Now, arguing as in the proof of Lemma 3.2, we see that T, T ( y o ,yl), y,(T,) y , in L 2 ( n ) ,and

377

-, T* =

-+

u,

c ( u , - u * ) ds

-+

-+

u*

weak star in L"(0, T*; L2(a ) ) ,

o

strongly in P ( O , ~ ; ~ 2 ( a ) ) ,

strongly in C( [0, T * ] ;L2(a ) ) , y, y* weakly in L2(0,T*; H i ( a ) n H 2 ( a ) )n W l v 2[0, ( T * ] ;L2(a ) ) . -+

(3.35) Now, multiplying Eq. (3.32) by sign p , and integrating on Q,, we get

because by Eq. (3.34) and the monotonicity of p" it follows that (p,(T,)) is bounded in L'(R). To obtain further estimates on p,, we consider the boundary value problem

where hi E L2(0,T*; L Q ( f l ) )i, = 1,. .., N, q > 2. Problem (3.36) has a unique solution u E L"(0, T*; H,'(R)) with d u / d t E L2(0,T*; H-'(R)) (see, e.g., Theorem 1.9 in Chapter 4). Moreover, if q > N then u E L"(Q*) and N

llvllL-(Q*, 5

c C IlhillL2(0,T'; L y l ) ) i= 1

(see Ladyzhenskaya et al. [l],p. 213). Now, if we multiply Eq. (3.32) by u and integrate on Q*, we get the inequality I N

and, since h

=

I

N

( h , , . . .,h,) is arbitrary in L2(0,T; Lq(a)),

5. Optimal Control of Parabolic Variational Inequalities

378

+ l / q ' = 1. Hence, { d p , / d t } is bounded in L'(O,T*; WS(fl) + W-',q(fl)), where s > N/2. (If extend u, by 0 on [T,, +m) we may

where l/q

assume that p , are defined on [O,T*].) Then, according to the Helly theorem, there is p E BV([O,T*]; W S ( R ) + W-'*9(fl)) n L2(0,T*; W,,'*q"
p,,(t) + p ( t ) P&,

+

Vt

[O,T*],

E

weakly in L 2 ( 0 ,T*; W,,'*q'(fl)).

P

On the other hand, since the injection of W ~ ~ q " 0 there is ~ ( 6 >) 0 such that (see J. L. Lions [l], p. 71) IIP&,(t)- p ( t ) l l L q n ) I 6llP&,(t)- p ( t ) l l w ; . q n ,

+ T ( s ) l l p & , ( t )- p(t)llH-r(n)+w-'.4(n) Vt

E

[O,T*].

This implies that p,,

Now, letting

E,

+

strongly in ~ ' ( 0 T, * ; ~

p

a)).

q ' (

(3.38)

tend to zero in Eq. (3.321, we see that dP

-

dt

+ Ap - v = 0

in Q * ,

where v = w - lim p " ( y , ) p , on some generalized sequence { E } . Moreover, by (3.33) we get (3.26). If we multiply Eq. (3.31) by P"(y,)I P"(y,)1q-2 and integrate on Q , = fl X (0, T'), we see that { P"(y,)} is bounded in Lq(Q*). This implies that ( y , ) is bounded in Wq2"(Q*) and so on a subsequence, again denoted E, , AY&"-

PYY&")

+

AY* -

5

weakly in

Lq( Q * ),

in c(e.1, Y&" Y * ((x,t) E P(y*(x,t)) a.e. ( x , t ) E Q * .

(3.39)

+

(3.40)

Now, multiplying Eq. (3.31) by p i , (3.32) by y$ , and subtracting the results, we get, as in the proof of Theorem 3.1,

53. The Time-Optimal Control Problem

379

where - u * ( T ) )dT

U,(X,f) = /?'dS/'(U&(T) f 0

and h ; ( p ) = sup((p,u), - (~/2)llull~;u and (3.39), it follows that

as claimed.

E

KO).Then, by (3.341, (3.351,

W

Now we shall consider some particular cases. The first one is that where for r 2 0, for r = 0. As seen earlier, in this case the control system (3.33) reduces to the obstacle controlled problem dY dt

- - Ay dY

--

dt

=

u

in { ( x , t ) ;y ( x , t ) > 0), in CR x

Ay 2 u, y 2 0

y(x,O) = y o ( x )

in C R ,

y

=

0

We take P" in the following form (see (1.17)):

and set

(O,m),

in dR

X

( 0 , ~ ) . (3.41)

5. Optimal Control of Parabolic Variational Inequalities

380

Then we have P,P"(Y,)

=

( P & b " ( Y & ) Y+&2-'p,)x,' + 2-'P&b"(Y&)Y&x,2 a.e. ( x , t ) E Q,, (3.43)

where x:, i = 1,2, is the characteristic function of Q:. Since {p,b"(y,)} is bounded in L'(Q,), {y,) in C@>, and {P"(y,>} is bounded in Lq(Q,), it follows by (3.43) that, for some E, 0, -+

p,, P"n(y,,)

0

+

a.e. in Q*,

(3.44)

whilst by (3.391, (3.40) we have weakly in L'(Q*)-

P,, P " ~ ( Y , , ) P ( A Y * - Y:) E P P ( Y * ) +

Hence, p,, p"n(y,,)

--f

strongly in L ' ( Q * ) ,

0

and so p(u*

+ dt JY* + A y * ) = 0

a.e. in Q*.

Now, using (3.42) once again, we see that

P,( P"(Y,)

-

because m ( Q i ) ,m(Qf)

P,, ~

~"(Y,)Y,) 0 +

-+

0 as

E +

strongly in L'(Q*>,

0. Hence, on a subsequence,

Y Y , ~ )0 Y ,strongly ~ in L'(Q*). +

Since as seen earlier ySn+ y * in C@>, this implies that vy* i.e.,

:(

-

+ Ap

)y* = O

=

0 in Q*,

inQ*.

We have therefore proved the following theorem:

Theorem 33. Let y o E Wo2 - 2 / q , q ( f i ) n H ~ ( R )n L"(R), q > max(N,2) be such thaty, 2 0 in R, and f e t y , E H , ' ( f l ) n H2(R) n LYR) be such that y, 2 0 in R, IlAy,llL-~n,< P . Let ( y * , u * ) be any optimal pair for the time-optimal problem associated with system (3.41). Then there is p E L2(0,T * ; WdVq'(R)) n BV(/([O, T I ; H-S(RZ)

53. The Time-Optimal Control Problem

381

+ W-1.9(R)) such that ( d / d t ) p + A p E M ( p ) and dP dt

-+Ap

0

in { ( x , t ) E Q*;y * ( x , t ) > 0},

(3.45)

in { ( x , t ) E Q*; y * ( x , t ) = 0}, p =0 u * ( x , t ) E psignp(x,t) a.e. ( x , t ) E Q * ,

(3.46)

=

A y * ( x , t ) p ( x , t )dr

=

a.e. t

1

E

(3.47)

(O,T*). (3.48)

p.

Here, M ( p ) is the space of bounded Radon measures on This theorem clearly extends to the time-optimal problem for the variational inequality dY -Ay=u dt

-

i n { y > @},

dY - Ay

in R x

2 u, y 2 @ at y(x,O) = y o ( x ) , y = 0

(O,m),

in dR x

(O,m),

where @ E C2(fi) is a given function such that @ I0 in dR. Now we shall consider the special case where y 1 = 0. If we take, in the approximating problem (3.291, V ( E ) = E - ~ / ' , multiply Eq. (3.31) by d y , / d t , and integrate with respect to x , we get j " ( y , ( x , t ) ) dr

c

I

VE

> 0, t

E

[O,T&],

where C is independent of E and t . We recall that j " ( r ) = 10' p"(s) ds and so, by (3.42), it follows that

j f ( x , T , ) dr

ICE

VE > 0.

If multiply Eq. (3.32) by p,' and integrate on R x ( t , T'), we get

382

5. Optimal Control of Parabolic Variational Inequalities

Hence

in Q*,

P I 0 dP dt

-

+ Ap = 0

in { ( x , t ) E Q*; y * ( x , t ) > O),

in { ( x , t ) E Q*; y*(x,t) = 0), 0 in Q*. (3.49) u*( x , t ) E p sign p ( x , t ) If the open set E = { ( x , t )E Q*; y * ( x , t ) > 0) is connected, then by the maximum principle we conclude that p < 0 in E, and so

p

=

u * ( x , t ) = -1

(3.50) in { ( x , t ) E Q*; y * ( x , t ) > 0 ) . We have obtained, therefore, a feedback representation for the time-optima1 control u*. In general, it follows by (3.50) that u* = - 1 in at least one component of the noncoincidence set { ( x ,t ) ; y*(x,t ) > 0). We shall consider now the case where p is a monotonically increasing locally Lipschitz function on R. Then we may take p" defined by the formula (2.74) in Chapter 3. By (3.40) we see that { b"(y,)} is bounded in L"(Q*), and so extracting a further subsequence if necessary we may assume that pen(

yen)+ g

weak star in L"( Q * ) ,

where g ( x , t ) E d p ( y * ( x , t ) )a.e. ( x , t ) E Q* (see Lemma 2.5 in Chapter 3). Then, by (3.38), we infer that v E Lq'(Q*) and v ( x , t ) E dp(y*(x,t ) )p ( x , t ) a.e. ( x ,t ) E Q*, where d p is the generalized gradient of p. Then, by Theorem 3.2, we have:

Theorem 3.4. Let y o , y , satisfv (3.24), (3.25) and let p be monotonically increasing and locally Lipschitz on R. Then if ( y * , u * , T * )is optimal for problem (PI there are p E L2(0,T * ; Wb.q"cfl>> n BV([O,T * ] ; H - ' ( f l ) + W-'*q(fl)), s > N/2, and 77 E L"(Q*) such that u * ( x , t ) E psignp(x,t)

- + Ap dP dt

-

qp

=

0

a.e. ( x , t ) E Q*,

(3.51)

in Q*,

T ( x , ~E ) dp(y*(x,t))

a.e. ( x , t ) E Q*,

(3.52)

53. The Time-Optimal Control Problem

383

Remark 3.1. If in problem (P) we replace the set 2Zp by

then Theorems 3.2-3.4 remain true except that Eqs. (3.281, (3.481, and (3.53) are replaced by -

la(VY*(X,t )

*

V p ( x ,t ) + 6 ( x , t ) p ( x ,t ) )lfx

+ pIIp(t)ll~2(n) = 1

a.e. t

E

(O,T*),

- v p ( x , t ) d~ + pllp(t)llLzcn)= 1

ax. t

respectively -javy*(x,t>

E

( 0 ,T * ) ,

in the case of the obstacle problem. 5.3.4. Approximating Time-Optimal Control by Infinite Horizon Controllers

Though the results of this section remain true for more general time-optima1 problems, we confine ourselves to parabolic systems of the form (3.23). More precisely, we shall consider the time optimal control problem:

where =

{u

E L"(O,W;~~(fl));u ( t ) E K

a.e.

r > O}.

(3.54)

K is either the set { u E Lm(fl); lu(x)l Ip a.e. x E fl) or { u E L2(fl); Ilull2 5 p ) , and y = y ( t , y o , u ) is the solution to system (3.23). Here, p is a maximal monotone graph in R X R such that 0 E p(0). We shall assume throughout this section that yo E

~ , ' ( a ) , i ( y o ) E L'(fl) ( ~ =i p )

(3 3 )

if K = ( u ; llul12 Ip), and y o E Lm(fl)n Hj(fl), p ( y o ) E L'(a) if K = { u ; llullL=(n,5 p}. As seen earlier (Lemmas 3.1 and 3.4), problem (P,) admits

at least one optimal pair ( y * , u*). We shall approximate problem (PI) by the following family of infinite

384

5. Optimal Control of Parabolic Variational Inequalities

horizon optimal control problems: Minimize

(P")

iw(+

h,( u( t ) ) ) dt

g " (y ( t ) )

on all u E LToc(R+;L2(R)) a n d y n H2(R)), subject to

E

W,i:([O,m); L2(R)>n L:,,(R';

dY - Ay + p " ( y ) = u

in R

drt

R',

X

in a, in drR x R'.

y(x,O) =y,(x) y = o

H,'(R)

(3.56)

Here, p" is a smooth approximation of p, i.e., p" E C2(R), bE2 0, p"(0) = 0, BE E L"(R), and these satisfy assumptions (k)-(kkk) in Section 1.2. The functions g": L2(R) + R and h,: L2(R) + R are defined by h,(u)

=

inf

Iu

- vI22

UEL2(R),

(3.57)

and (3.58) where

TE

C'(R+) is such that

T' 2

1 Lemma 3.5. For all solution ( y 6 ,ue).

0,O I

TI

1 in R', and

for y 2 2,

sufficiently small, problem (P") admits at least one

E

Proo$ It is readily seen that there exists at least one admissible pair ( y , u ) in problem (P"). For instance, we may take u as in the proof of Lemma 3.1 and 3.4. Hence, there are the sequences u , , y , satisfying system (3.56) and such that

/ ( g " ( y , ) + h , ( u , ) ) dt W

d I

0

I d

+ n-',

where d is the infimum in (P"). Then, by the definition of h", we see that the u, remain in a bounded subset of LToc(R+;L2(R)). Hence, on a subsequence, u,

+

u

weakly in LT,,(R'; L 2 ( R ) ) ,

5.3. The Time-Optimal Control Problem

385

and by Eq. (3.56) we see that the y,, remain in a bounded subset of W,i:([O, m); L2(R)) n L:,,(R+; H,'(R) n H2(R)). Hence, we may assume that, for every T > 0, y,, + y = y ( t , y , , u )

stronglyin L ~ ( o , T~ ; ~ (n 0L)~)( o , TH; , ' ( ~ I ) ) ,

weakly in ~ ~ (T 0; H, 2 ( a)), and by the Fatou lemma,

liminf i m g " ( y , , dt ) 2 l m g F ( yd) t , 0

n+m

and

lim inf n-m

because the function u L2(R)). Hence,

as desired.

.m

jo h,( u,,) dt 2 --+

10"h,(u) dt

.m

is convex and 1.s.c. on L:,,(R+;

rn

Theorem 3.5. Let ( y , , u,) be optimal in problem (P"). Then, on a subsequence, E -+ 0, u,

--+

u*

weak star in Lm(O, T*; L 2 ( R ) ) ,

y,

+

y*

weakly in W ' . 2 ( [ 0T, * ] ;L2(a)) n L2(0,T * ; H 2 ( a)), strongly in C([O,T * ] ;L 2 ( R ) ) n L2(0,T * ; H;(R)), (3.59)

where T * is the minimal time and ( y * ,u * ) is an optimalpairforproblem (PI).

Pro05 Let ( y : , u:) E W',2([0,T * ] ; L2(R)) n L2(0,T*; L2(R)) be any optimal pair in problem (PI). (We have already noted that such a pair exists.) We extend u: and y : by 0 on [ T * , +m) and note that yf ,u: is a solution to (3.23) on R X (0,m). Now, let 9, be the solution to Eq. (3.56) for u = u:. Since h,(u:(t)) = 0 a.e. t > 0, we have i r n W ( Y & ( t )+) h , ( u , ( t ) ) ) dt

J0r n g E ( 9 , ( t )dt )

whereas, by Lemma 1.1, I j , ( t ) -yT(t)l2

I CE"~

Vt

E

[O,T*].

(3.60)

5. Optimal Control of Parabolic Variational Inequalities

386

On the other hand, by Eq. (3.56) we have the estimate Ie

IY,(t)l2

(because u:

=

-u(I-T*)

Vt 2 T*

Iyc(T*)12

0 on [T*,m)),and along with (3.60) this yields Iys(t)12

Vt 2 T*.

ICE"~

Then, by the definition (3.58) of g", it follows that, for all small, i m g & ( y , ( t ) ) dt = i T * g " ( y , ( t ) ) dt Ii ' * g " ( y f ( t ) ) dt

E

sufficiently

+CE'/~,

and so limsup j " ( g E ( y & ( t )+) h,(u,(t))) dt &+O

0

IT*

(3.61)

On the other hand, since {uJ is bounded in L:,,(R+; L2(SZ>)it follows by Lemma 1.1 that there exists u* E LYo,(R+; L2(SZ))such that, for every T > 0, uEn-+ u*

ySn -,y*

weakly in L2(0,T ; L2(a ) ) ,

weakly in W ' s 2 ( [ 0T, I ; L2(a ) )n L 2 (0 ,T ; H 2 ( a ) ) , strongly in C ( [0, T I ; L2(a ) )n L2(0,T ; H ' ( S Z ) ) ,

where y* = y ( t , y o , u*). We shall prove that u* is a time-optimal control. We note first that, by (3.58) and (3.611, it follows that the Lebesgue measure of the set {t > 0; ly,(t)li 2 2 ~ ' is~ smaller ~ ) than T * . Thus, there are E, -+ 0 and t, E [O, 2T*] such that 1y,jt,)12I2&,'i4

(3.62)

Vn.

Extracting a further subsequence, we may assume that t, -+ To. On the other hand, since {dyEn/dt)is bounded in every L2(0,T ; L 2 ( f l ) ) , we have ICIt - t,I'/2

Iy,jt) - y,jt,)I2

Vt

E

[O, To].

Then, by (3.621, we conclude that y*(To)= 0. Let f = inf { T ; y * ( T ) = 0). We will prove that = T*. To this end, for every E > 0 consider the set E, = { t E [O, f ] ; ly,(t)l; 2 2 ~ ' / ~By ) . (3.611, we see that limsup m(E,) &+

-

IT* IT ,

0

where m denotes the Lebesgue measure. On the other hand,

5.3. The Time-Optimal Control Problem

387

lim SUP, rn(E,) = f,for otherwise there would exist S > 0 and E, + 0 such that rn(Een)If - S V n . In other words, there would exist a sequence of measurable sets A, c [ O , f ] such that rn(A,) 2 6 and lys$t)l; I 2 ~ ; "V ~t E A,. Clearly, this would imply that ~

+

vt E A , ,

i y * ( t ) i 2 I( 2 ~ , 1 / 4 ) ' / ~ v,

where v, + 0 as n + 03. On the other hand, since y * ( t ) lim rn(t

E

n-rm

#

0 Vt

E

[O,

f],we have

[0,f1;ly*(t)12 I ( 2 & ; / 4 ) ' / 2 + v,)

=

0.

The contradiction we have arrived at shows that indeed lim sup, rn(E,) = f and therefore f = T*, as claimed. This completes the proof. ~

To be more specific, let us assume that K = { u E L2(fl); lul2 s PI. We may pass to limit in system (3.63), (3.64) to get that the optimal pair ( y * , u * ) given by Theorem 3.5 satisfies a maximum principle-type system. Indeed, by (3.63) and (3.56) we get d

dt ( ( P , , -AYE

&

+ P " ( Y & ) > - PlP,(t)l2 - ~ I P , " ) l : + g " ( y , ) )

and, therefore, ( P , ( f ) , - AY&(t> + P & ( Y & ( t ) ) - PlP&(t>l2 &

-

-IP,(t)l; 2

+ g " ( y , ( t ) ) = c.

=

0

388

5. Optimal Control of Parabolic Variational Inequalities

Since p , E L2(R+;L2(fl)), g"(y,) E L'(R+),and ( P e ( t n ) , - A ~ s ( t n )+ P " ( y e ( t n ) ) >

+

for some

0

tn

+

03,

we find that &

~ l ~ & ( t )+I 2T I p e ( t ) I : =

Vt 2 0. (3.65)

( P&('), - A Y e ( t ) + P " ( Y & ( ~ ) )+) g " ( y & ( t ) )

On the other hand, a little calculation reveals that

d -( ) P " ( Y & ( ~ ) )2> 0, dt P e ( t ) , - A Y & ( ~ + and so ( ~ & ( t -) A, Y , ( ~ ) + P " ( Y e ( t ) > ) I0

Vt 2 0.

This implies that plp,(t)lz

I1

V t 2 0.

(3.66)

Noticing that G E ( y , ( t ) ) = 0 for lye(t)l: 2 Z ? E ' / ~ ,it follows by estimate (3.66) and Eq. (3.63) that { p , } is bounded in L"(O,T* - 6; L2(fl)) n L2(0,T* - 6; H,'(R)) for every 6 > 0. Then, by using a standard device we find as in previous proofs that, on a subsequence en + 0, P,, pe,(t)

+

strongly in ~ ~ (T *0 ; L , ~a)), (

P

+p

(t)

strongly in ~ - ' ( f l ) weakly , in .L2( fl) for t E [O, T * ) ,

where p E L'(0, T * ; L2(fl)) n L2(0,T * ; H,'(R)) n BV([O,T * ] ; H-'(fl)) satisfies the equations dP

-+Ap dt

- u=

0

u * ( t ) = p sgn p ( t )

plp(t)l2 - ( p ( t ) , - A y * ( t )

in R x ( O , T * ) ,

(3.67)

a.e. t

(3.68)

+ P(y*(t)))

E

( 0 ,T * ) ,

=

1

a.e. t

E

(O,T*), (3.69)

where u E (L"(R x (0, T*))*. In particular, it follows by (3.68) and (3.69) that u* is a bang-bang control, i.e., lu*(t)l2 = p a.e. f E (O,T*). For special choices of P (for instance, P locally Lipschitz or a maximal monotone graph of the form (1.6)), we may deduce Theorems 3.3 and 3.4

5.4. Approximating Optimal Control Problems

389

from the preceding optimality system (see Remark 3.1). We refer the reader to author's book [7] for other results in this direction. 5.4. Approximating Optimal Control Problems via the Fractional Steps Method 5.4.1. The Description of the Approximating Scheme

We will return now to the optimal control problem (PI in Section 1.1, i.e.,

on all ( y , u ) E C([O,TI; H ) n L2(0,T ; U),subject to the state system

y'(t)

+Ay(t) + Fy(t) 3 (Bu)(t) + f ( t )

a.e. t

E

(O,T), (44

Y(0) = y o ,

in a real Hilbert space H. Here, B E L(L2(0,T ; U),Lz(O,T ; H ) ) , g : [0,TI x H + R, cpo: H + R, h: U + R satisfy assumptions (v), (vi) in Section 1.1, and U is a real Hilbert space. The operator A : V + V' is linear, continuous, symmetric, and coercive, i.e., Vy E V

( A Y , ~2) ~lly1I2 whilst F = dcp: H that

+ H,

where cp: H

Vy

(Ay,FAy) 2

FA = K 1 ( Z

+

-

R

is a l.s.c., convex function such A > O,

D ( A , ) 9

(I + hF)-')

= dqA.

(4.3)

We will assume further that the projection operator P of H onto K = D( F ) maps V into itself and (APy,Py)

(AY,Y)

VY

E

(4.4)

Here, V is as usually a real Hilbert space compactly, continuously, and densely imbedded in H, with the norm denoted )I * I(. We will assume, finally, that

390

5. Optimal Control of Parabolic Variational Inequalities

As seen earlier, assumptions (4.3), (4.5) imply that A monotone. More precisely, A + F = d+, where

+(Y)

=

+(AY,Y) + 4 Y )

+ F is maximal

VY E V *

Then, the Cauchy problem has for every u E L2(0,T ; U ) a unique solution y" E W'g2([0,TI; H ) n L2(0,T ; D ( A , ) ) . Since the map u + y" is compact from L2(0,T ; U ) to C([O, TI; H ) , problem (4.1) has a solution. Here, we will approximate problem (4.1) by the following one: Minimize

on ally: [O, TI

+

j b T ( g ( t ,y ( t ) ) + h ( u ( t ) ) )dt + v o ( y ( ~ ) ) (4.6)

H, u

E

L2(0,T ; V ) , subject to

a.e. t E ( i e , ( i + l ) ~ ) , y ' ( t ) + A y ( t ) = ( B u ) ( t ) +f(t) y + ( i ~=) w ; ( E ) for i = 1,..., n - 1, y + ( O ) = y o , (4.7) wf + Fwi 3 0 E = T / n , in ( 0 , E ) , (4.8) ~ ~ (=0P y)- ( i ~ ) for i = 1 , 2 , . . ., n - 1.

Here, y - ( i ~ and ) y + ( i ~are ) respectively the left and right limits of y at iE. Since, by assumption (4.31, e - F ' V c V for all t > 0, it is readily seen that problem (4.7), (4.8) has a unique solution y : [O, TI + H, which is piecewise continuous and belongs to W's2([i&,(i+ 0 . ~ 1 ;H ) n L 2 ( i s , ( i+ 1 ) ~V ; ) on every interval [ i ~ , (+i l ) ~ ] Then, . by a standard device, it follows that the optimal control problem (4.6) has for every E > 0 at least one solution u f . We set W U ) =

j b T M Y " ( t ) ) + h ( u ( t ) ) )dt + % ( Y " ( T ) )

and ?&(U)

=

/T(s(Y,"(w + h ( u ( t ) ) )dt + cpo(Y,"(T)), 0

where y," is the solution to system (4.71, (4.8). Then, in terms of ? and ?=, we may rewrite problem (4.1) and (4.6) as min{?(u); u

E L 2 ( 0 , T ;U

min{?&(u); u

E

)},

(4.9)

L 2 ( 0 , T ;U ) } .

(4.10)

respectively, The main result of this section is the following convergence theorem.

5.4. Approximating Optimal Control Problems

391

Theorem 4.1.

Assume that beside the preceding hypotheses at least one of the following assumptions holds: (i) B is compact from L2(0,T ; U ) to L2(0,T ; H I ; (ii) F = dI,, where C is a closed convex subset of H .

Then lim (inf{qc(u); u

E+

0

E

L 2 ( 0 , T ;U ) } ) = inf{*(u); u

E

L2(0,T;U ) } , (4.11)

and if {u:} is a sequence of optimal controls for problem (4.6) then q(uz)

+

inf{*(u); u

E L 2 ( 0 , T ;U

Moreover, every weak limit point of {u:} for problem (4.1).

E +

)}.

(4.12)

0 is an optimal control of

It is apparently clear that conceptually and practically the decoupled problem (4.6) is simpler than the original problem (4.1). Now we shall briefly present some typical situations to which Theorem 4.1 is applicable. 1. Consider the distributed control system

du

-

dt

+ Du = B,u 4 0 ) = uo,

a.e. t u=

E

(O,T),

(q,...,%),

(4.14)

in an open domain R c R"' with a sufficiently smooth boundav. Here, (Y 2 0, p: R + R is a maximal monotone graph (eventually, multivalued) such that D( p ) = R and 0 E p(O), D is a Lipschitz mapping from R" to itself, Bo E L(RP,R"), ai E L"(R) for i = 1,. . . ,m , and

f where p

=

E

dj.

L2(Q),

yo E H ' ( W

Aye)

E

L'W,

392

5. Optimal Control of Parabolic Variational Inequalities

We may apply Theorem 4.1, where U A : V -+ V' is defined by

(respectively, A y

=

- A y and V

=

H,'(R) if a

( F y ) ( x ) = (w E L2(R); w ( x )

c

= RP,

E

=

V = H'(R), H

= L2(R),

O), and

p ( y ( x ) ) a.e.x

E

a},

m

( B u ) ( t , x )=

i= 1

u

a,(x)u;(t),

ELZ(0,T;U).

Assumption (i) is obviously satisfied by virtue of the Arzelii theorem. We leave it to the reader to write the iterative scheme and to formulate the approximating problem (4.6) in the present situation. 2. Consider the optimal control problem (4.1) governed by the free boundary problem (the obstacle problem) dY dt

inQ

--Ay>u

Y(X,O) = y , ( x ) y = o

in a, in 2 ,

(4.15)

where y o E H,'(R), y o 2 0 a.e. in R. As seen earlier, this control system is of the form (4.21, where H = L2(R), V = H,'(R), A = - A , and F = dZ, where C = { y E Hi(R); y ( x ) 2 0 a.e. x E R).We note that in this case ( P y X x ) = y + ( x ) = max(y(x),O}, a.e. x E R,and assumptions (4.3) (4.4) are clearly satisfied since lIvPylILz(n) IIIvyIILz(n) Since e-F'y in this case,

=y

+ V t 2 0, and all y

dY --AY=u dt

in

E

Q6

V y E H,'(R).

L2(R), system (4.7), (4.8) becomes,

=

R x ( i ~ , ( i+ l ) ~ ) ,

+

in 2; = dR x ( i ~ , ( i I ) & ) , y = o Y(X,O) =y,(x) in a, y + ( x , i ~=) m a x { y - ( x , i E ) , O } a.e. x E R.

(4.16)

5.4. Approximating Optimal Control Problems

393

Arguing as in the proof of Theorem 1.2, we get for the corresponding problem (4.6) the following optimality system (assume that g is Giiteaux differentiable and 'p,, = 0):

p - ( x , ( i + 1 ) ~ =) p + ( x , ( i+ 1 ) ~ ) p-(x,(i + 1 ) ~= ) 0 p-(x,T) =0 p

E

> 01, in { x ; (Y,*)+ ( x , ( i + in ( x ; ( y , * )+ ( x , ( i + 1 ) ~ = ) 0}, in 0, (4.17)

dh(u*)

a.e. t

E

(4.18)

(0,T).

This system can be solved numerically by a gradient type algorithm and the numerical tests performed by V. Arniiutu (see Barbu [ll]) show that a large amount of computing time is saved using this scheme. 5.4.2. The Convergence of the Scheme

We will prove Theorem 4.1 here. The main ingredient of the proof is Proposition 4.1, which also has an interest in itself. Proposition 4.1. Under the assumptions of Theorem 4.1, if (u,) is weakly convergent to u as E,, -+ 0, then

y : ~+ y " ( t )

strongly in H , V t E [ O , T I .

(4.19)

We recall that y," is the solution to ystem (4.7), (4.8). Let us postpone for the time being the proof of Proposition 4.1 and derive now Theorem 4.1. Let u,* be an optimal controller for problem (4.6) and let y,* be the corresponding solution to system (4.7), (4.8). By assumption (v) in Section 1.1, (u,} is bounded in L2(0,T;U )and so, on a subsequence E,, + 0 as n + 03, u,"

+

u*

weakly in L2(0,T ; U ) ,

(4.20)

whilst by Lemma 4.1, y,*Jt) + y " ' ( t )

strongly in H , V t

E

[O,T].

(4.21)

5. Optimal Control of Parabolic Variational Inequalities

394

(We set yzn = y,"cn.) This clearly implies that

g(Yz"> and since u

+

+

in L'(0, T ) ,

g(y"')

(4.22)

:/ h ( u ) is weakly lower semicontinuous, we have

On the other hand, we have *&,(U&")

I *&,(G*)

Vn,

where G* is optimal in problem (4.1). Letting n tend to (4.21144.23) that

a, we

get by (4.24)

* ( u * ) 5 liminf *&,(uEn)I *(G*) n-m

and, therefore, * ( u * ) = lim V&,(u&,) = n+m

*(fi*)

=

inf{*(u); u

E

L 2 ( 0 , T ;U)},

i.e., u* is an optimal control for problem (4.1). To prove (4.121, we set = y " : and note that, by (4.20) and the ArzelL-Ascoli theorem,

Y&" + y *

=yu*

strongly in

c([o, TI;

H).

Hence,

whilst by (4.24) we see that

Therefore, *(uZn) + *(u*), as claimed. This completes the proof.

W

To prove Proposition 4.1 under hypothesis (i) we shall establish first a Lie-Trotter product formula for the nonhomogeneous Cauchy problem

y*

+ Ay + Fy = 4 , y(0) =x,

f 2 0,

(4.25)

5.4. Approximating Optimal Control Problems

395

where q E L'(R+; H I , x E D( A ) n D( F ) = D(F ) = K , and A, F satisfy assumptions (4.31, (4.4). As mentioned earlier (Remark 1.4 in Chapter 4) we may write (4.25) as an autonomous differential equation

d -S(t)(x,q) +dS(t)(x,q) dt

and w

=

=

0

t

2

0,

(4.26)

e - F r xis the solution to

w'

+ Fw = 0

inR+,

w(0) = x .

It is easily seen that S,(t) and S 2 ( t ) are generated by the operators d,and d2: d 1 ( x , q ) = [ A- q(O), -4'1,

Now let P: H + K be the projection on K K x L'(R+; H ) be the operator

Lemma 4.1.

For all ( x , q ) €3, we have

uniformb on compact intervals.

d 2 ( x 3 q ) = [FX,Ol. =

D( F ) , and let Q: X -+Z=

396

5. Optimal Control of Parabolic Variational Inequalities

Prooj We will use the nonlinear Chernoff theorem (Theorem 2.2 in of nonexpansive Chapter 4). To this end, consider the family operators on 3, t 2 0, r ( t )= Q s , ( t ) s z ( t ) , and set

X,

=

(I

+ At-'(I

-

t 2 0.

r(r)))-'(x,q),

(4.30)

Then, according to Chernoff theorem, to prove (4.29) it suffices to show that lim X I = ( I + A A ) - ' ( x , q ) ,

VA > 0.

1-0

(4.31)

According to (4.271, (4.281, we may rewrite (4.30) as

(t

+ A)y'(s)

-

Ay'(s

+ t ) = tq(s)

VS 2 0,

(4.33)

where X I = ( x ' , y ' ) E X and e-A' is the semigroup generated on H by -A". Inasmuch as the operators (I ,W-' and (I + A t - ' ( I - r(t)))-'are nonexpansive, without loss of generality we may assume that x E D ( A ) n D ( F ) and q , q' E L2(R+; V ) n L'(R+; V ) (the general case follows by density). Then, by Eq. (4.33), we see that y' E L'(R+; V ) n L2(R+;V ) ,y' is V-absolutely continuous on compact intervals, and

+

IlyfllL1(R+;V)

llqllLi(R+;V),

i

=

1,2,

Hence, { Y ' ) ~ , is compact in L'(R+; H ) n C(R+; H ) and, therefore,

y'

strongly in H, uniformly in s on compacta.

+w

Since, by (4.33), ly'(s)l ds Ijrnlq(s)lds jPrn P

Vp

> 0,

we may conclude, therefore, that y'

+w

stronglyin L'(R+; H).

(4.34)

5.4. Approximating Optimal Control Problems

+ E W',2([0,

On the other hand, for each

and, therefore, for t 1 -(y'(s f

H ) , @(O)

=

CQ);

397

0, we have

0,

+

+ t ) -y'(s))

+

weaklyin L2(R+; H).

w'

Then, letting t tend to zero in (4.331, we see that a.e. in R + .

w - w' = q Next, by (4.34), it follows that 1

-

t

/'e-A('-s)y'(s) ds o

+

strongly in H as t

w(0)

To complete the proof it remains to be shown that, for t x'

+

+

0.

0,

strongly in H ,

+ x:

(4.35)

where x: is the solution to the equation x:

+ A( Ax: + Fx:)

=x

+ Aw(0).

(4.36)

To this aim, we set q' = t - ' / i e-A('-s)y '( s ) d s . Noticing that P = (I dZ,)-', we may equivalently write (4.32) as t-'(xt -

e-A'X')

+ At-'

+

+ At-le-A'(x' - e-FIX')

dZK(A-'(t

+ A)x'

- tA-'x) 3x

+ Aq'

-XI.

(4.37)

Let z ' ( s ) = e-Asx' and let u be arbitrary but fixed in V. Multiplying the equation z' + Az = 0 by z' - u and integrating over [O,t l , we get (e-AtX'

+

- XI, x'

t(

- u)

+

ile-A'x' - x'12

cPdz'(s)> - cP,(u))

where q l ( y ) = i ( A y , y ) , y Similarly,

E

5 0,

(4.38)

V.

(e-FIXf - X I , XI - u ) + +le-"'x' - x'12

+

t(

cp( e - F s x ' ) - cp( u ) ) ds I

0.

(4.39)

398

5. Optimal Control of Parabolic Variational Inequalities

Hence, t-'(e-Frx' - x',e-Ar(x' - u ) )

+ t-1 /b(cp(e-F'x') <

t-'(e-F'x'

+ ( 2 t ) - ' 1 e - ~ ' x ' - X'I'

- cp(u) e-A'X'

-

+ cpl(e-ASx') x')

-

- cpl(u)) ds

+ t-'le-F'x'

- x'I le-A'u - uI.

(4.40)

+

Now multiply (4.37) by x' - e-F'x' t h - ' ( x ' - x ) , and use the accretivity of ePA'along with the definition of JZ, to get t-l(xr

<

e - ~ ~ X -~ , - FXr X~' )

- ( X I

+(x

-

e-A'x',x' - x ) - ( x ' - e-F'x',e-Ar(x' - x ) )

+ Aq'

- x ' , x ' - e-F'x')

+ t h - ' ( x + hq' - x ' , x '

-x).

Combining this with (4.38) and (4.40) yields h(t-'(e-A'x' - x ' ) , x ' - u )

+ At-'

/b(

I A(cp(u)

q( eCFsx')

+ h(t-'e-A'(e-F'x'

-x'),x'

- u)

+ cpl( e-Asx')) ds

+ q , ( u ) ) + (2t)-'hle-A'u

- uI2

+ Ix' - e-F'~'1(21x' - XI + Ix + hq' - x'l) + t h - ' l x + hq' - x ' I Ix' - X I .

(4.41)

Next, by (4.321, we have XI

=

+

t ( ~t ) - ' x

+ A ( A + t)-lP(e-A'e-F'x' + t q ' ) ,

whilst by assumption (4.3) it follows that cpl(e-F'x) I cpl(x)

V t 2 0, Vx E V .

This yields

Ilx'll

4 llxll

+ Allq'll

IC

Vt

> 0,

because as previously seen ( q ' } is bounded in LYO, 1; V ) . We may conclude, therefore, that { x ' } is a compact subset of H and so on a subsequence, again denoted ( t } ,we have x'

+ x,"

strongly in H .

(4.42)

399

5.4. Approximating Optimal Control Problems

Since the functions cp and cpl are lower semicontinuous, by the Fatou lemma we have

+ cpl(e-ASx'))

liminf t-1 l ( c p ( e - F ' x ' ) 1-0

+ cpo,<x,">. (4.43)

cis 2 cp(x;>

Next, from (4.42), we see that (see also Theorem 1.10 in Chapter 4)

+ le-A'x'

t - ' l u - e-A1u12

- X'I

+ le-F'x' - X ' I + o

as t

+

0. (4.44)

Now, coming back to Eq. (4.371, it follows by the definition of dZ, that

+ h)x'-

ht-'(dZ,(h-'(t

+ h ) ~-' t h - ' x ) , x ' - x )

2 -(dZK(h-'(f = -h-l(e-A'Xl

- th-'(x

th-'x),x' - U)

-xf,xf-x ) -

+ hq'

-x',x'

(e-Ftx'

-x))

-X',e-AI(x'

-x),

and by (4.44) we infer that liminf t-'(dZ,(h-'(t t+O

+ A)x'

- th-'x),x' - u ) 2

0.

Along with (4.371, (4.40, and (4.43), this yields -

where

=

cp

lim ( x

t+o

+ cpl.

+ hq'

- x ' , x' - u ) 5 A(

=

dcp

-

C#J(

x')),

Hence,

(x," - hw(0) - - x , x '

Since &$

C#J( u )

- U ) Ih ( + ( u ) - C # J ( X ' ) ) .

+ d p l and u is arbitrary in H , this implies that x + h ( O ) - x," E A( A + F ) x , " ,

i.e., x," is the solution to Eq. (4.36), as desired. Proof of Proposition 4.1. We shall assume first that hypothesis (i) holds. Let (u,) be weakly convergent to u in L2(0,T; U ) and let y, = y , " ~be the corresponding solution to (4.71, (4.8). Then {Bu,) is strongly convergent to Bu in L2(0,T; H I , and if we extend Bu, and Bu by 0 on ( T ,CQ),then by Lemma 4.1 the sequence 9, defined by [9&(t)9ii~(t)]

= (Qs1(E)S2(E))i(y07f+

Bu~)?

'

+ ')'I,

(4.45)

400

5. Optimal Control of Parabolic Variational Inequalities

or, equivalently Vt

Y & ( t ) = P ( y , ) - ).i(

E

[i&,(i

+ l).],

(4.46)

is strongly convergent to y " ( t ) for every t E [O, TI. On the other hand we have, by (4.7) and (4.81, I(y,")- ( i ~ )- ( y , " ) , (i.)l=

l(y,")- ( i . ) - e - F " P ( y , " ) - (i.)l

(4.47)

+f l

(4 S O )

and

Note also that

IBu,

ds I CE"'

Vi.

5.4. Approximating Optimal Control Problems

401

Along with (4.471, this yields l 0 lim l(yE)- ( i ~ )- ( y & ) +( i ~ ) =

&+

0

because e-Ff is continuous in t and { ( y & ) - ( i ~ )is} compact in H. Now, using once again Eq. (4.71, we see that lY&(t) - (YE>+( i d 2 I

+ IfI2)h)

C(El(Y&)+( i E ) I 2 + jr(lBu,12 is

Vt

E

[i&,(i

+l)~],

and along with (4.50) this implies that ye( t )

y"( t )

strongly in H , V t

E

[0,T I ,

as claimed. Now we shall assume that hypothesis (ii) holds, i.e., F = dZ,. Then, ePfFP= P = (I + A dZ,)-l for all t > 0, K = C, and the system (4.71, (4.8) becomes y ' ( t ) + A y ( t ) = ( B u ) ( t ) +f(t) y+(i.) = P y - ( i E ) .

a.e. t

E

(i~,(i+ l ) ~ ) ,

(4.51)

Let uEn-+ u be weakly convergent in L2(0,T; U).For simplicity, we set E, = E and: :y = y, . By the estimate (4.491, we have

uEn- u,,

c1

N- 1 i=o

(i+ 1 ) s

IE

(lYl(t)l'

+ IlY.(t)l12) (4.52)

and, therefore,

(4.53) On the other hand, we have

(4.54)

5. Optimal Control of Parabolic Variational Inequalities

402

Since e-A'C c C V t

2

0, we have

Substituting this in (4.54),we get the estimate

Along with (4.521, this yields T

v y, + Ily,(t)ll

4

0

c

Vn,t

E

[O,T],

where V y, stands for the variation of y,: [O, TI H. Since the injection of V into H is compact we conclude, by virtue of the infinite dimensional Helly theorem, that on a subsequence, again denoted y, , -+

stronglyin H , V t

y,(t) - + y ( t )

E

[O,T].

(4.55)

By estimate (4.53), it follows that A y E L2(0,T; H ) . Now, let z E C be arbitrary but futed, and let t E [ k ~ , (+k l ) ~ ] , s E [ i E , ( i + l ) ~ ] i, < k, be two points on the interval [O, TI. By Eq. (4.51) we get 1

n(IYn(t)

--I2 k

- I(yn)+

+3 C1 ( K Y n ) j=

( k ~ -z12) )

( j ~ - - I)2

- I(Yn)+ ( ( j -

1 ) ~ -21') )

+ + ( I ( y , ) _ ( ( i + 118) - z12 - ~ y , ( s ) - z12) f

=

This yields

[(Bun +f-AY,,Y,

-z)dT.

5.4. Approximating Optimal Control Problems

403

On the other hand, we have

and letting n tend to

+a

we get

T1 ( I y ( t ) - 21’ - l y ( s ) - 21’)

I/‘(Bu

+ f - A y , y - z ) d~ (4.56)

S

for all 0 Is It I T . Before proceeding further, let us observe that y ( t ) E C a.e. t Indeed, we have

ly,(t) - Py,(t)l I

E

(0, T ) .

lI:

e - A ( f - S ) ( B u+, f ) ds I CE”’,

because e-AfCc C V t 2 0. Hence, y ( t ) = Py(t), as claimed. Now, in (4.56) take z = y ( s ) . By Gronwall’s lemma, we get ~ y ( t -) y ( s ) ~I/ ‘ I B ~+ f + ~ d7 y ~

for 0

Is

t

T,

S

and therefore the function y : [O, TI + H is absolutely continuous and almost everywhere differentiable. On the other hand, by (4.56) we have (Y(t)-Y(S),Y(S)

- 2)

404

5. Optimal Control of Parabolic Variational Inequalities

Then, dividing by t - s and letting s tend to t , we see that ( y ' ( t ) + A y ( t ) - ( B u ) ( t )- f ( t ) , y ( t ) for all z

E

= y",

I0

a.e. t

E

(O,T),

C. Hence,

Y ' ( t ) +AY(t)

i.e., y

-2)

+ dZ,(Y(t>)

3

( B u ) ( t )+ f ( t )

ax. t

E

(O,ET),

as claimed. This completes the proof of Proposition 4.1.

Bibliographical Notes and Remarks

Section 1. Theorems 1.1-1.4 along with other related results were established in the author's work [4, 5 , 71. In a particular case, Theorem 1.2 has been previously given by Ch. Saguez [2]. D. Tiba [ll (see also [31 and [41) has obtained similar results for optimal control problems governed by hyperbolic equations of the form y,, - Ay + y ( y , ) 3 Bu, nonlinear parabolic equations in divergent form, and the nonlinear diffusion equation y, - A p ( y ) 3 Bu (see also D. Tiba and Zhou Meike ill). In this context, we also mention the work of S. Anifa [l] on optimal control of a free boundary problem that models the dynamics of population. By similar methods and a sharp analysis of optimality system, A. Friedman [3] has obtained the exact description of the optimal controller for the obstacle problem with constraints of the form { u E L"(Q); 0 Iu IM , u h d t = L}. Periodic optimal control problems for the two phase Stefan problem were studied by Friedman et al. [2]. Theorems 1.1 and 1.2, were extended by Zheng-Xu He [l] to state constraints problems of the form (PI (see also D. Tiba [31). Numerical schemes for problems of this type were studied by V. ArnZutu [l]. We mention in this context the works of I. Pawlow [l], and M. Niezgodka and I. Pawlow [l]. Section 2. In a slightly different form, Theorems 2.1-2.3 were established first in the author's work [5, 7, 91 (see Friedman [3, 41 for the one phase Stefan problem; see also Ch. Moreno and Ch. Saguez [l]). There is an extensive literature on the inverse Stefan problem and optimal control of moving surfaces, and we refer the reader to the survey of K. H. Hoffmann and M. Niezgodka [l] for references and significant results. A different approach to the inverse Stefan problem that consists of reducing it to a linear optimal control problem in a noncylindrical domain was used in the works of V. Barbu [16], and V. Barbu, G. DaPrato, and J. P. Zolesio 111 (see also V. ArnZutu [21). The control of the moving boundary of the

Bibliographic Notes and Remarks

405

+ ay = u

inZ,,

two phase Stefan problem, y, - AP(y)

3f

in Q,

dY

-= dV

0 in Z,,

dY

dV

where /3 is the enthalpy function (see Section 3.3 in Chapter 4) has important industrial applications and was studied by the methods developed here by several authors, including Ch. Saguez [31, D. Tiba [4], D. Tiba and Zhou Meike [l], V. Arngutu and V. Barbu [l], and D. Tiba and P. Neittaanamaki [l]. The optimal control of the moving boundary of a process modeling growth of a crystal was discussed in the work of Th. Seidman [l,21. Section 3. The main results of this section (Theorems 3.1 and 3.4) were established in the author’s work [lo, 121. The approach presented in Section 3.3 was first used in the author’s work [7, 91 to get first order necessary conditions of optimality for the time-optimal control problem. A different approach involving the Eckeland variational principle was developed by H. 0. Fattorini [3]. Section 4. The contents of this section closely follows the author’s work [14]. For other related results, we refer to the author’s work [15, 191.

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Chapter 6

Optimal Control in Real Time

In this chapter we will be concerned with the feedback representation of optimal controllers to problems studied in the previous chapter. We will see that under quite general conditions such a control is a feedback control of the form u = d h * ( - B * d , J l ( t , y ) ) , where Jl is a generalized solution to a certain Hamilton-Jacobi equation associated with the given problem (the dynamic programming equation). 6.1. Optimal Feedback Controllers 6.1.1. Closed Loop Systems

Consider a general control process E [O,TI, Y ’ ( t ) + MY(t) 3 W t ) + f ( t ) , Y ( 0 ) = Yo (1.1) in a real Hilbert space H , where M = d 4 , 4: H -+ R is a lower semicontinuous convex function, d 4 : H + 2 H is the subdifferential of 4, B E L(U, H I , and U is another real Hilbert space. Here, y o E D(4) and f E L2(0,T: H ) . The control function u : [0, TI -+ U is said to be a feedback control if it can be represented as a function of the present state of the system (1.11, i.e., 9

-

u(t) E A(t,y(t)) a.e. t E ( O , T ) , ( 14 where A: [O, TI X H U is a multivalued mapping. Of course, some continuity and measurability assumptions on A are in order. A map A: H + U is said to be upper semicontinuous at y from H to U, if for 407

6. Optimal Control in Real Time

408

every weakly open subset D of U satisfying M y ) c D there exists a neighborhood B ( y , 8 ) of y such that A ( B ( y , 8 ) ) c D. The multivalued map N t ) : [0, TI -+ U is said to be measurable if for each closed subset C of U the set {t E [O,Tl; A ( t ) n C # 01 is Lebesgue measurable. It is easily seen that if R ( A ) is bounded in U then A is upper semicontinuous from H to U,, and with weakly closed values if and only if A is closed in H X U,. (Here, U, is the space U endowed with the weak topology.) If in the state system (1.1) we replace the control u by the feedback control (1.2), we obtain the closed loop system Y' +MY

-

t E (O,T),

BA(t,Y) 3 f ( t ) , Y ( 0 ) = Yo *

(1.3)

We say that the feedback control A is compatible with system (1.1) if (1.2) has at least one local solution. In general, the Cauchy problem (1.2) is not well-posed unless we impose further conditions on the feedback law A. We mention in this direction the following result due to Attouch and Damlamian [ l ] More . general results of this type can be found in Vrabie's book [ll (see also the monographs of Aubin and Cellina [ll and Filipov [l] for a complete treatment of nonmonotone differential inclusions in R N).

Proposition 1.1. Let H be a separable Hilbert space and let A, = B A : [0,TI x D(M ) + H , be upper semicontinuous in y , measurable in t , and with compact convex values. Assume further that (a) For each y o E D(M ) there exist r > 0 and h, that SUP{

IIvIILI;

vE

Ao(f7

Y ) ) 5 ho(t)

and every level set { y

E

H ; +( y )

a-e. t IA}

E

E

L2(0,T ; R+) such

(0, T)7 IIy

Ir

is compact in H . (1.4)

Then for each y o E D ( M ) there is 0 < To < T such that the Cauchy problem (1.3) has at least one strong solution y on [0,To]that satisfies y dY t'/2 dt

E

E

C([O,ToI; H ) ,

L 2 ( 0 ,T o ; H ) ,

Y(t)E D ( M ) +(y)

E

L'(0, T ) ,

as.t

E

t'l2My

(O,T,), E

L 2 ( 0 ,T ; H ) .

(1.5) I f y o E D(+),then y

E

W',*([O,T o ] ;H ) and + ( y ) E AC([O,To];H I .

6.1. Optimal Feedback Controllers

409

This means that there exists one measurable selection A ( t ) of B N t , y ( t ) ) such that A E L2(0,T; H ) and dY -(t) dt

+ My(t) 3 A(t) +f ( t )

hoof: Denote by q :L'(0, T o ; H )

a.e. t

(0,T).

) the operator

+ L2(0,To; H

( q z ) ( t )= y ( t )

E

a.e. t E (O,To),

where y is the solution to the Cauchy problem y ' ( t ) + M y ( t ) = z ( t ) +f ( t ) Y(0) =Yo Let D c L2(0,T o ;H )

D

=

X

a.e.t

E

(O,T,), ( 1 -6)

*

L2(0,T o ; H ) be the multivalued mapping

{ [ y , u ]E L 2 ( 0 , T o H ; ) x L 2 ( 0 , T oH ; ) ;y ( t )

E

D(M),

l y ( t ) - yol Ir a.e. t E (0, T o ) , u ( t ) E B A ( t , y ( t ) ) a.e. 1 E (O,To)]. We have: Lemma 1.1. D is upper semicontinuous and with compact convex values from L2(0,To; H ) into L;(O, To; H ) . Moreover, Dy # 0for ally E L2(0,To; H I , y ( t ) E D(M ) , Iy(t) - yol Ir a.e. t E (0, To). (Here, Lt(0, T o ; H ) is the space L2(0,T o ; H ) endowed with the weak topology.) Proof: Let y , + y strongly in L2(0,To; H ) and u, E Dy, , u, in L2(0,T,,;H ) . We have, therefore, u,(t) E B h ( t , y , ( t ) )

a.e. t

E

+

u weakly

(0,T).

By Mazur's theorem, there is {wm},a finite combination of the u,, n 2 m, such that w, + u strongly in L2(0,T o ; H ) as m + w. Thus, there is a measurable subset I c (0, T ) such that m(Z) = T and on a subsequence, again denoted n , we have y,(t) Y,(t)

+y ( t ED

)

strongly in H, V t E I ,

( M ) , %(t) E BA(t,Y,(t)), strongly in H , V t E Z. wm(t) + u ( t )

V t E I,

410

6. Optimal Control in Real Time

Since BR(t, * ) is upper semicontinuous to U,,, , for every weakly neighborhood 7 of B A ( t , y ( t ) ) there is a neighborhood % of y ( t ) such that B M t , x) c 7 for all x E %. This clearly implies that u ( t ) E B M t , y ( t ) ) V t E I. Now let y E L2(0,T o ;H ) be such that y ( t ) E D( A ) , l y ( t ) - y o ( I r a.e. t E (0, To).Then, the multivalued mapping t + B N t , y ( t ) ) is clearly measurable and so, according to a well-known selection result due to C. Castaign (see, e.g., C. Castaign and M. Valadier [11) it has a measurable selection, which by condition (1.4) is in L2(0,To ; H I . Now we come back to operator ‘4’ previously defined. For any 6 E L2(0,T o ; H I , denote by xg the set ( z E L2(0,T o ; H I ; Iz(t)l I 6 0 ) a.e. t E (0,T)). Lemma 1.2. The operator 1I’ defined on X , is continuous from L i ( 0 ,To ; H ) to LYO, To; H ) .

Proo$ Let 2, E X be weakly convergent to z in L2(0,To; H ) and denote by y, the corresponding solutions to (1.6). We have the estimates (see Section 1.5, Chapter 4)

and by the Arzelh-Ascoli theorem we infer that (y,) is compact in E > 0. Hence, on a subsequence, we have

C ( [ E To]; , H ) for every

y,(t) + y ( t )

stronglyin H , V t

E

[O,TO],

uniformly on every interval [ E , To], E > 0. By the Lebesgue dominated convergence theorem, it follows that y E L2(0,T o ;H ) and y, + y strongly in L2(0,To;H ). By standard arguments, this implies that y = 1I’z is the solution to (1.61, as claimed. Proof of Proposition 1.1 (continued). We may write problem (1.3) as y

E

*D(y),

l y ( t ) -yol 5 r

a.e. t

E

@,To)

or, equivalently, WED’4’W

WEXg,

where 6 = l l B l l ~ H , ~)~h O ,(see assumption (3.3)).

(1.7)

6.1. Optimal Feedback Controllers

411

By Lemmas 1.1 and 1.2, the operator DlIr is upper semicontinuous on Lt(0,T o ; H I , has compact convex values and X , is a compact subset of Lt(0,T o ;H ) . Moreover, DlIr maps X , into itself if To is sufficiently small. Indeed, for z E X , we have, by Eq. (1.61,

+ 4(Y(t)>- 4(Yo)

3(lY(t) -Yo12)’

a.e. t

5 ( w ( t ) +f(t),Y(t) - Y o )

E

(O,TO),

and this yields l Y ( t ) -Yo1 5

c(t + @s)

+ If(sN2ds)

Vt

E

[O,TOl.

(We assume first that y o E D(+).) Hence, for To sufficiently small we have l*w(t)

- y,l

sr

Vt

E

[O, To],

and so by assumption (1.4) it follows that D q w E X , , as claimed. - If y o E D(4) = D(M ) , the same conclusion follows by density. Then, by the Kakutani theorem (see Theorem 2.2 in Chapter 1) in the space Lt(0, T o ; H I , we infer that the operator D q has at least one fixed point w E X,. Equivalently, there is y = q w such that

w

=@

*

in L 2 ( 0 , T o ;H ) .

By the definitions of D and we see that y is a solution to the Cauchy problem (1.3). We note that the conditions (1.5) follow by Theorem 1.10 in Chapter 4. Consider now the optimal control problem:

subject to

y’

=

d + ( y ) 3 Bu + f

a.e. t

E

(O,T),

(1.9)

Y(0) =Yo9 where the functions g : [0, TI X H -+ R, q o :H + R and h: U conditions (v), (vi) from Section 1.1 in Chapter 5.

+

R satisfy

412

6. Optimal Control in Real Time

An optimal control u for problem (120, (1.9) that is represented in the feedback form (1.2) is called optimal feedback control. We shall see in the sequel that for a quite general class of optimal control problems of the form (1.8) every optimal control can be represented as a feedback control, and the synthesis function A can be easily described in terms of the solutions to the optimality system associated with (1.81, (1.9). To be more specific, we will consider the optimal control problem (1.8) governed by the semilinear parabolic equation

JY dt

Ay

--

+ P ( y ) = Bu + f

in Q

=

R x (O,T),

in 2 = dR x ( O , T ) , in a,.

y = o y(x,O) =yo(x)

(1 .lo)

where is a locally Lipschitz real valued function that satisfies the condition 0 IP ' ( r ) IC(IP(r)I

+ Irl + 1)

a.e. r

E

(1.11)

R.

As seen earlier this is a problem of the form (1.91, where H = L 2 ( n )and d 4 ( y ) = - A y + P ( y > V y E D ( 4 ) = {Z E H,'(R) n H2(R>; P ( z ) E L2(R)}. It is readily seen that D ( d 4 ) is dense in L2(R). Indeed, for any z E H,'(Q) we have (1 E P ) - ' z E (we may assume P(0) = 0) and

+

(1 + E P ) - ' z ( x )

+z ( x )

a.e. x

E

for

1(1+ E P ) - ' Z ( X ) I

IIZ(X)I

a.e. x E

(a).

+

Hence, (1 E P ) - ' z + z strongly in L 2 ( n )as Define the map r: [O, TI x H + H by

E +

E +

0,

0.

where ( y ' , u', p ' ) satisfy the system d

-y'

dS

d

-p' dS

-

+ Ap'

y'(x,t) y'

Ay' -

+ P ( y ' ) = Bu' dp(y')p'

= z,(x), = p' =

0

E

in Q' = R

+f

dg(s,y')

=

dC!

B*p'(s) E dh(u'(s))

(t,T),

in Q',

p ' ( x , T ) 3 -drp,(y(x,T)) in C'

X

x ( t ,T ) , a.e. s E ( t , T ) ,

in

a, (1.13) (1.14)

6.1. Optimal Feedback Controllers

and ( y ' , u ' ) problem

E

413

C ( [ t ,TI; H ) x L2(t,T; U ) is a solution to optimization

{ l T ( g ( s , y ( s ) )+ h ( u ( s ) ) )ds + cp,(y(T)), (1.15)

Minimize

subject to u E L 2 ( t , T ;U )and dY

-dS

Ay

+ p ( y ) = Bu + f in R,

y(x,t) =zo(x)

in Q',

y

=

in Z'.

0

(1.16)

As seen in Proposition 1.1 and Theorem 1.1 in Chapter 5, for every ( t , z , ) E [O,T]x L2(R) there are y',u', and p' € A C ( [ t , T ] ; Y * )n C,([t, TI; L2(R)) n L2(t,T; H,'(R)) satisfying Eqs. (1.13), (1.14). Hence, r is well-defined on [O, TI X L2(R).

Theorem 1.1. Every optimal control u* for problem (1.81, (1.10) has the feedback representation a.e. t

u*(t) E dh*(B*r(t,y*(t)))

E

(1.17)

(O,T),

where y* is the corresponding optimal state. Theorem 1.1 amounts to saying that every optimal control of problem (1.8), (1.10) is an optimal feedback control with the synthesis function V ( t , y ) E [O,T] X L 2 ( n ) . (1.18)

A ( t , y ) = dh*(B*r(t,y))

Pro05 Let ( y * , u * ) be any optimal pair for problem (1.8), (1.10). Then, obviously, ( y * ,u * ) is also optimal for the problem: Minimize

(lT(

g( s, y ( s))

d

-y

dS

-

Ay

+ h( u( s)))

ds

+ cpo( y ( T ) ) ,

+ p ( y ) = Bu + f

y(x,t) =y*(x,t)

in R,

y

subject to

in R', =

0

in Z',

(1.19)

and so by Theorem 1.1 in Chapter 5 we infer that there are p' AC([t,T l ; Y * )n CJt, TI; L2(R>)n L2(t,T; H,'(R)) such that d

-pf dS

+ Ap'

-

dp(y*)p'

E

dg(s,y*)

in

E

'p',

p f ( x , t )E -d'p,(y*(x,T)), x E R, p' = 0 in u*(s) E dh*(B*p'(s)) a.e. s E ( t , T ) .

X',

(1.20) (1.21)

414

6. Optimal Control in Real Time

The function u* being measurable, it is a.e. approximately continuous on [0, TI. This means that for almost all t E (0, T ) there is a measurable set El c (0, T ) such that t is a density point for El and u*IE, is continuous at t. Let us denote by E' the set of all s E [ t ,TI for which (1.21) holds. Since, for almost all t E (0, TI, t is a density point for E' n El there is a sequence s, + t such that u*( s,)

+

and

u*(t)

u*( s,) E dh*( B*p'( s n ) ) .

Since p' is weakly continuous on [ t ,TI, we have

B*p'(s,)

--$

B*p'(t)

weakly in L'(fl),

and dh* being weakly-strongly closed in U X U we conclude that u * ( t ) E dh*(B*p'(t))a.e. t E (0, TI. In other words, we have shown that u*(t)E dh*(B*r(t,y*(t)))

a.e. t

E

(O,T),

thereby completing the proof. Regarding the properties of the synthesis function A, we have: Proposition 1.2. For each t E [0,TI, the map r(t,* 1: L'(fl) + Lk(fl) is uppersemicontinuous and bounded on bounded subsets. For each yo E L'(fl), r(-,yo):[O, TI -+ L'(fl) is measurable.

Proog It is easily seen that r(t, ) is bounded on every bounded subset. We assume that r is not upper semicontinuous from L'(fl) to L;(fl), and argue from this to a contradiction. This would imply that there are y o E L'(fl), y," -+ y o strongly in L2(fl), and .I," E r(t,y,") such that 7; E D for all n. Here, D is a weakly open subset of L'(fl) and

T ( t , Y , ) c D.

(1.22)

By the definition of r there are ( y : , u f , p i ) satisfying the system (1.13), (1.14), and p ; ( t ) = $, y:(t) = y," . By (1.15, (1.16) it is easily seen that { y i } is bounded in C ( [ t ,TI; ,!,'(a)) and It"' dyi/ds} is bounded in L2(t,T; L2(fl)). Thus, on a subsequence, again denoted (n),we have u:, --$ u' y:(s) - + y ' ( s )

weakly in L'(t, T; U ) , stronglyin H , V s E [ t , T ] ,

weaklyin w ' . ' ( [ s , T ] ;H ) VS E ( O , T ) , Y,: - j Y ' where y' is the solution to (1.13) with the Cauchy condition y ' ( t ) = y o . Then, by the estimates proved in Proposition 1.2, Chapter 5, we know that

6.1. Optimal Feedback Controllers

415

(see also Theorem 1.1 in Chapter 5 ) strongly in ~ ~ T( ; tL*( , a)),

P:,+PI

weak star in L"(t, T; L 2 ( a ) ) , and stronglyinY*,Vs

p:,(s)+ p ' ( s )

E

[t,T],

where p' is the solution to (1.13), (1.14). and p:,(t) - + p ' ( t ) strongly in Y* Since ( p i ( t ) }is bounded in L2(Cl),we conclude that

3

weakly in L ~a( ) ,V t E [0, TI.

p : , ( t ) --+ p ' ( t )

Hence, .I," + p ' ( t ) E r ( t , y , ) weakly in L 2 ( f l ) ,which by virtue of (1.22) implies that .I," E D for all n sufficiently large. The contradiction at which we have arrived completes the proof. Let y o E L 2 ( f l )be arbitrary but fixed. Then by a similar argument it follows that the multivalued function t -+ r(t,y o ) = p ' ( t ) is weakly upper semicontinuous from [0, TI to L 2 ( a ) , which clearly implies that (U*,y,))Y'(C) is a closed subset of [O,Tl for each closed subset C of L 2 ( f l ) ,i.e., IX-,y o ) is measurable, as claimed. Theorem 1.1 provides a simple method to compute the synthesis function A associated with problem (1.8), (1.9) by decoupling the corresponding optimality system. However, since in general the multivalued mapping (1.18) is neither upper semicontinuous nor has convex values, the existence result given by Proposition 1.1 is not applicable to closed loop system (1.2) and so it is not clear if the corresponding feedback control (1.17) is compatible with system (1.10). However, if h ( u ) = ~ I u I we ~ may replace u by the relaxed feedback control

u

=

B*conv T ( t , y ) ,

which by virtue of Propositions 1.1 and 1.2 can be implemented into system (1.10). Let us observe that if, in the problem (1.8, (1.101, g ( t , . 1, po,and p are continuously differentiable with Lipschitz derivatives and if ah* is Lipschitz, then by standard fixed point arguments it follows that for It - TI I6 sufficiently small, the system (1.13), (1.14) has a unique solution ( y ' , p ' ) . Hence, for T sufficiently small, r(t,* ) is single valued and continuous on a

416

6. Optimal Control in Real Time

given bounded subset of L 2 ( n ) .Then the feedback control (1.17) is continuous, and so by Proposition 1.1 it is compatible with the system (1.10). Now, if we approximate g, q o , and h by g", (qo>",p", and h,(u) = h(u) + E I U I ; , respectively, we may construct an approximating feedback control compatible with the system (1.10) on a sufficiently small interval of time. Remark 1.1. If p is merely locally Lipschitz or in the case of the obstacle problem, we may define the map r as U t ,2 0 ) = p ' ( t

+ 01,

where ( y ' , p ' ) satisfy the corresponding optimality system on [ t ,TI, and p'(t + 0) = w - lims+t,s, I p'(s> in ~ ~ ( n ) . 6.1.2. The Optimal Value Function

We shall consider here problem (PI studied in Chapter 5, i.e.,

on all ( y , u ) E C([O,TI; H ) X L2(0,T ; U),subject to dY

-dt( t )

+Ay(t)

+ d q ( y ( t ) ) 3 Bu(t)

a.e. t

E

(O,T), ( 1.23)

Y(0) =Yo7

where A is a linear, continuous, and symmetric operator from V to V' satisfying the coercivity condition ( A y , y ) 2 ~llY1I2 and q : H

+

vy

E

v,

(1.24)

R is a lower semicontinuous convex function such that

( A y , d q & ( y ) )2

-c(l + ldq&(y)12)(1+ Iyl)

where D ( A H )= { y & > 0.

E

V; Ay E H ) and dqc

vy = &-l(Z

D(AH),

-(I

(1.25)

+ &dq)-',

Here, H and V are real Hilbert spaces such that V c H c V' algebraically and topologically, and the injection of V into H is compact.

6.1. Optimal Feedback Controllers

-

417

We shall denote as usual by 1 the scalar product of H, and also the pairing between V and V’, and by I I (respectively, II * 11) the norm of H (respectively, V ) . As seen earlier, the operator (a,

-

+

A,u = A u n H ,

dc#~=A, dq,

is maximal monotone in H

X

H , and

Regarding the functions g: H + R, q o :H + R, and the operator B: U + H, we shall assume that hypotheses (iv), (v), and (vi) or (vi)’ of Section 1.1 in Chapter 5 are satisfied. As noted earlier, for every y o E D( 4) the Cauchy problem (1.23) has a unique solution y E C([O, TI; H ) n L’(0, T; V )with t’/’ dy/ds E L’(0, T ; H ) , t*/2A,y E L’(0, T; H ) . For 0 < t < s < T and y o E D( 4), denote by y ( s , t, y o ) the solution to the Cauchy problem

dY

-(s)

dr

+ d ~ $ ( y ( s ) 3) Bu(s)

The function

a.e. s

E

(t,T),

y ( t ) = y o . (1.26)

9:[O. TI x D(4) + R,

is called the optimal value function of problem (P).

Proof: By Proposition 1.1 in Chapter 5 we know that the infimum defining $ ( t , y o ) is attained. Since d+ is monotone, we have ly(s, t , y o u ) - Y ( S , t , 2 0 u)l 5 I Y O - ZOI 9

9

Vs E [ t , T ] . (1.28)

418

6. Optimal Control in Real Time

Now if multiply Eq. (1.23) by y(s) - yo, where yo E D ( d 4 ) , and integrate on [ t ,sl, we get ly(s,t,yo,u)l I lyol + C ( [ ' l u ( ~ ) l ~d ~ 1) +

Vs

E

[ t , T l , (1.29)

where C is independent of t and s. Now let yo E D(4 ) be such that lyol 5 r. We have @(t?YO)5 /?'(g(Y(sJ7Yo,0)) + 4 0 ) ) ds + %(Y(T,t,YO,O))

c,.

<

( 1.30)

(We may assume that 0 E D(h).) By virtue of assumption (v) we may therefore restrict the optimization problem (1.24) to the class A, of u E L2(t,T; U ) satisfying the condition dT

LTIU(T)l:

c:.

Then, by estimates (1.281, (1.291, it follows that for u EA,the function yo .+ JtT g(y(s, t , yo, u ) ds + (po(y,T, t , yo, u)) is Lipschitz on B, = { y E D( d 4 ) ; lyl Ir ) with the Lipschitz constant independent of u €Ar. Since ~

for yo, zo E B, and some u $(t, 20)

=

E&,

such that

[=(g(Y(s, t , 2 0 u ) ) + 4 u ) ) ds + cPo(Y(T7 9

20

7

u)),

we obtain that l$(t,yo)

-

$ ( t , 2,)l

I L,lYo - 201

VYOJO

E

B,,

as claimed. On the other hand, for every yo E D ( d 4 ) we have Iy(s, & Y o u ) - Y(S, t , Yo ,u)l 9

I ly(t,f,yo,u) -yo

+

t t

lBu(~)ld~ ld4(yo)lIt + - fl.

(1.31)

Now let us observe that without any loss of generality we may assume that, for yo E B,, A, c { u E LTt, T; U ) ; l u ( ~ )I l~ C,), where C , is independent of t. Indeed, if u' is a minimum point for the functional (1.27) then by

6.1. Optimal Feedback Controllers

419

(1.29) we see that the corresponding optimal state y' has an independent bound in C ( [ t ,TI; H ) . On the other hand, a.e. s

u'(s) E dh*(B*p'(s))

where p'

E LYt, T ;

E

(t,T),

H ) and

Ip'(s)l

c;

vs E [ t , T ] .

I

This follows as in the proof of Proposition 1.2 in Chapter 5, approximating problem (1.27) by a family of smooth control problems and taking the limit of the corresponding optimality system, dY& + Ay ds

-

+ VCp"(y,) = Bu,

in ( t , T ) ,

dP& = Vg"(y,) ds -4,- V2Cp"(Y&)P&

in ( t , T ) ,

-

Y & ( t )= Y o ,

P & ( T )= -V(cpO)&(Y&(T)).

Since, by assumption (v), dh* is bounded on bounded subsets, we conclude that l u ' ( s ) I ~I C j

V y o E Br, s

[t,T],

E

and so by (1.31) we have Iy(s,f,yo,u) -y(s,t,yo,u)l for all t If such that

Is IT .

IC;It

-

fl

Now let u , E L2(t,T ; H ) and y,

vu

€4,(1.32)

= y ( s , t , y o ,u,)

be

+(t,YO) = / T ( g ( Y f ( s ) )+ h ( u , ( s ) ) )ds + cpO(Yl(T)), and let 4 s ) = u o for s Muo) < a.We have

E

[f,t], and u ( s )

=

u I ( s ) for s E [ t , T l , where

+(f, Y o ) - + ( t , Y o ) 5 [ ' M Y ( S , f, Yo u ) ) + h ( u 0 ) ) ds 9

+ l T M Y ( s .f, Y o u ) ) - g ( Y ( s ,t , Yo u ) ) ) ds 9

9

+ (Po(Y(T,f, Yo u ) ) - CpO(Y(T3t , Yo , u ) > , 9

which along with the estimate (1.32) yields l + ( f , y o ) - +(t,yo)l 5 Llt - fl

V t , f E [%TI.

(We note that L depends on r ( y o E Br n D(d+)).)

420

6. Optimal Control in Real Time

Now we shall prove the dynamic programmingprinciple for problem (P).

Prooj Let ( y , u ) y ( s , t , y o ,u ) and

E

C([t,TI; H ) x L 2 ( t , T ; U ) be such that y

=

This yields

+ @ ( s , y ( s , t , y o , u ) ) ;24 On the other hand, for all u have

E

E

1

L 2 ( t , T ;U )

*

L2 (t,T ; U ) and y ( s ) = y ( s , I , y o ,u), we

We may choose y and u in a such a way that

6.1. Optimal Feedback Controllers

421

and so

thereby completing the proof. (The continuity of $ is obvious.)

W

6.1.3. The Dynamic Programming Equation As in the classical theory of calculus of variations, we associate with the problem (PI the Hamilton-Jacobi equation $ f ( t , Y ) - h * ( - B * I C ; ( t , y ) ) - ( d 4 ( Y ) , c G ; ( t , Y ) +) d Y ) = 0 in [ O , T ] x H $(T,Y) = Vo(Y) VY = H . (1.34) where I/Jf and i,hy represent the partial derivative of $ with respect to t and y , whilst h* is the conjugate of h . Equation (1.34) is called the dynamic programming equation corresponding to problem (PI. In general, this equation does not have a solution in the classical sense even if the space H is finite dimensional and d 4 , g , and qo are smooth. However, under supplementary conditions on problem (P) the optimal value function I) satisfies Eq. (1.34) in some generalized sense. Let us denote by 0,’ $ ( t , y ) the superdifferential of $(t, * at y , i.e., the set of all 7 E H such that limsup ( ( + ( t , z) - + ( t , y ) - (7,z - y ) ) l z - y t - ’ }

I0.

(1.35)

Z+Y

Similarly, Dy- $(t, y ) , the subdifferential of of all w E H such that

$0, ) at y , is defined as the set

We will assume now that: (a) (b)

A is a linear, continuous and symmetric operator from V to V’ satisfying condition (1.24); V c H c V’ algebraically and topologically; the injection of V into H is compact;

6. Optimal Control in Real Time

The operator F = dcp: H -+H is FrCchet differentiable on H with locally Lipschitz FrCchet differential VF; The functions g and cp, are continuously differentiable on H with locally Lipschitz derivatives Vg and Vcp, , respectively. The function h: U + R is convex, lower semicontinuous, and

h ( u ) 2 alult

+y

Vu

E

U,

where a > 0. Moreover, the conjugate function h* is differentiable with locally Lipschitz derivative Vh*. Theorem 1.2. Let assumptions (a)-(e) be satisfied. Then the optimal value [0,TI X H + H is continuous, locally Lipschitz in y for every function t E [0,TI, Lipschitz in t for every y E D ( A , ) , 0,’ $ ( t , y ) # 0 V ( t , y ) E [O, TI X H , and

+:

More precisely, Eq. (1.36) holds for all ( t , y ) for which @ ( . , y ) is differentiable at t. satisfies the Roughly speaking, Theorem 1.2 amounts to saying that Hamilton-Jacobi equation (1.34) in the following weak sense: * f ( l 9 Y ) - h*( - B * D , + @ ( t , Y )

( A H Y + F Y , ~ , + * ( t , Y )+) d Y ) a.e. t

E

( O , T ) ,y

ED(A,).

=

0

(1.36)‘

Regarding the optimal feedback controllers, we have: Theorem 1 3 . Under assumptions (a)-(e), every optimal control u* of problem (PI is expressed as a function of the optimal state y* by the feedback law

u * ( t ) E Vh*( - B * D , + J I ( t , y * ( t ) )

V t E [O,T].

(1.37)

Proof of Theorem 1.2. Let ( t , y o ) E [0,TI X D ( A , ) be arbitrary but fixed. Then, as seen in Chapter 5 (Proposition 1.11, there are y‘ E C ( [ t ,TI; H )

6.1. Optimal Feedback Controllers

and u'

E L2(t,T ; lJ) such

423

that

and d -y'(s) ds

+ A , y ' ( s ) + F y ' ( s ) = Bu'(s) Y'(t> =Yo

u'(s)

=

=

(1.39)

*

Then, by the maximum principle, there is p'

P'(T)

a.e. s E ( t ,T ) ,

E

C ( [ t ,TI; H ) such that

(1.40)

-Vvo(y'(T)),

Vh*(B*p'(s))

Vs

E

(1.41)

(t,T).

Since y' E W 1 , 2 ( [TI; t , H ) , the functions s -+ V g ( y ' ( s ) )and s -+ V F ( y ' ( s ) ) are Holder continuous. Moreover, inasmuch as p' E W 1 , 2 ( [61; t , H ) for every t < S < T , we conclude that s + V g ( y ' ( s ) ) + V F ( y ' ( s ) ) p ' ( s ) is Holder continuous on every interval [ t , 61 c [ t ,TI, and so p' is a classical solution to Eq. (1.40) on every [ t , 61 (see Theorem 4.5 in Chapter 1). Finally, since by (1.41) u' is Holder continuous on every [ t , 61, we infer that y' is a classical solution (Le., a C'-solution) to Eq. (1.39) on [ t ,TI. Now, if multiply Eq. (1.39) by dp'/ds, Eq. (1.40) by dy'/ds, and subtract the results, we get d - [ ( A Y ' ( s ) , P ' ( s ) ) + ( F Y ' ( S ) , P ' ( S ) ) + g ( y ' ( s ) > - h*(B*p'(s))l ds =0 Vs E ( t , T ) , because, by virtue of Eq. (1.41), d -h*(B*p'(s)) ds

=

Vs

E

(t,T).

424

6. Optimal Control in Real Time

We set

r(t,yo)

=

{ - p ' ( t ) ; p' satisfies Eqs. (1.40), (1.41) along with some optimal pair (y', u')}.

We have: For euery ( t ,yo) E [0,TI

Lemma 1.4.

r(t,Yo) = Proof

X

H , we have

qw,Yo).

( 1.43)

For any x E H we have

@ ( t ,x ) - @ ( t Yo) ,

I lT(g(y,(sN

-

d Y W ) ) ds + Vo(Y,(T))

-

Vo(Y'(T)),

(1.44)

where

y;(s) +Ay,(s)

+ Fy,(s)

s E

= Bu'(s),

(t,T),

YAt) = x * Clearly, we have

ly,(s) -y'(s)l I Clx -Yo1

vs

E

( 1.45)

[t,T).

Now let w be the solution to the equation

w' + A w

+ VF(y')w

0 w ( t ) = x -yo.

It is easily seen that if lyo - XI I

ly,(s) -y'(s)

=

in [ t , T ) , (1.46)

then

- w(s)l I ElYO

--I

vs E [ t , T ) ,

(1.47)

and so by (1.44) we have @(t,X)

-

@(f,Yo)

I j l T ( W Y ' ( S ) ) , w 0 ) ds + (VVo(Y'(T)),w(T))

+

for Ix -yol I

EIX

a(&).

-Yo1 (1.48)

Now we take the scalar product of Eq. (1.40) with w and integrate on [ t ,TI. We get @ ( t , x ) - @(t,YO) I -(p'(t),w(t))

+

EIX

-Yo1

for Ix - yol I 6( E).

6.1. Optimal Feedback Controllers

425

Remark 1.2. It is easily seen that for T - t sufficiently small, the system (1.39)-(1.41) has a unique smooth solution ( y ' , p ' ) and so r(t,. ) is single valued. Moreover, arguing as in the previous proof, it follows that for every R > 0 there is v ( R ) > 0 such that @(r, ) E C ' ( B , ) and

-

V,W,Yo) for all y o

E B, =

=

r(t,Yo)

{ y E H ; lyl

Vt

E

[T - v(R),TI,

IR}.

Proof of Theorem 1.2 (continued). We shall assume first that V q o ( y ) E D ( A ) for all y E H. Then, by Eq. (1.40), we see that p' E C'([t,TI; H ) and y' E C ' ( [ t ,TI; H I . Now let y o E D ( A ) be arbitrary but fixed and let t E [0, TI be such that y o ) is differentiable at t. By the definition of $, we have #(a,

*(t,YO)

=

inf j l o T - t g ( y ( s ) ) + 4 4 s ) ) ) Cis + rpo(Y(T - t ) ) ;

y'

i +

+ Ay + Fy = Bu in [0, T - t ] ,y ( 0 ) = y o .

We set z'(s) = y'(t + s), d ( s ) = u'(t + s), q ' ( s ) = p'(t - t], where ( y ' , p ' ) is defined by (1.38)-(1.41). We have

s) for s E [O, T

426

6. Optimal Control in Real Time

427

6.1. Optimal Feedback Controllers

By assumption (e) it follows that (u:} is bounded in L2(t,T; U ) and so on a subsequence, again denoted { E } , we have

where

weaklyin L 2 ( t , T ;U ) ,

6'

u:

+

y:

+yl

stronglyin C ( [ t , T ] ;H),

p:

+pl

stronglyin C ( [ t , T ] ;H),

(y', g', 2 ) satisfy Eqs. (1.39)-(1.41). *"(t,Yo)

Then, letting W

E

E

lim ,,(t,YO)

&+

0

+

It is also clear that as

*(t,Yo)

E

+

0.

tend to zero in (1.491, we see that - h*(B*lj'(t)) + (AYO

Since, as seen in Lemma 1.4, f i ' ( t )

E

-D,'

+ FYO,fi'(t))+ g ( Y o )

=

0.

$0,y o ) we infer that a.e. t E ( O , T ) ,

for some E D,'@(t, yo). This completes the proof of Theorem 1.2.

W

Remark 1.3. Under the assumptions of Theorem 1.2, for every R > 0 there is T = T ( R ) such that I) is a classical solution to Eq. (1.36) on the domain ( B R n D ( A , ) ) X (0, T ) (see Remark 1.2). Proof of Theorem 1.3. Let ( y * , u*> be any optimal pair in problem (P). Then, by Lemma 1.3, we see that for every t E (O,T), ( y * ,u * ) is also optimal for the problem

This means that u*(s)

=

Vh*(B*p'(s))

Vs E [ t , T ] ,

where p 1 is a solution to the system (1.40) where y' = y*. Then, by Lemma 1.4, we conclude that u * ( t ) E Vh*( -B* D,'+(t, y*(t))) V t E [O, TI, as claimed. rn

6. Optimal Control in Real Time

428

Now we shall recall the concept of viscosity solution for the Hamilton-Jacobi equation (1.34) (M. G. Crandall and P. L. Lions [31). Let rp E C([O, TI x H). Then, y is a viscosity solution to (1.34) on [0, T ] x H if for every continuously differentiable function x : [0, TI X H + R and 8: H -, R satisfying:

x

is weakly sequentially lower semicontinuous and V x , A V X are continuous; (ii) 8 is radial, nondecreasing, and continuously differentiable on H .

(i)

if ( t o ,y o ) E (0, T ) x H is a local maximum (respectively, minimum) of rp - x - 8 (respectively, rp + x + O), we have x , ( t o Y o ) - h*( -B*( X y G o Y o ) + W Y O ) ) ) - ( X x y ( t 0 Y o ) , Y o ) 7

+YO

9

7

9

XJtO

9

(1 S O )

Yo) + VO(Y0)) + d Y 0 ) 2 0

(respectively, X L t o Y o ) + h*( - B * ( x y ( t o ,Yo 9

-( xy@o Y o ) + V 8 ( Y O ) ?FYO) 9

+ V 8 ( Y O ) ) ) - (A x y ( t 0 Y o ) ,Y o ) 7

-g(Yo) 2

0.)

(1.51)

Proposition 1.4. Under assumptions (a)-(e), the optimal value jhction @ is a uiscosity solution to Eq. (1.34).

6.1. Optimal Feedback Controllers

429

For every uo E D(h), we may choose u E W ' * * ( [ t 0 , T ]U; ) such that h(u) E C ( [ t oTI). , For instance, we may take u to be the solution to the Cauchy problem ur

+ dh(u) 3 0

a.e. in ( t o , T ) ,

u ( t o )= u o .

Then, by (1.52), it follows that X r ( t o , Y o ) - ( A ~ y ( t O , y o ) , y o) ( F Y O , X ~ ( ~ O , Y+ O )V O ( y 0 ) ) + g ( y O ) 2

h*(

-

B*( xy@o, Y o ) + V8(YO)),

because 8 is radial nonincreasing, i.e., 8 ( x ) = dlxl), w r 2 0, and so ( A y o , v e ( y o ) )2 0. Assume now that cp + y, 8 has a local minimum at ( t o ,yo). We have, therefore, by Lemma 1.4

+

X(t0,YO) - X ( t , Y ( t ) > + 8(Yo) - O(Y(t))

as claimed. It follows by the uniqueness results in Crandall and Lions [3] that J/ is the unique viscosity solution of Eq. (1.34). In the general case of unbounded operators dqb, the Cauchy problem (1.34) is still well-posed in a certain class of generalized solutions introduced by D. Tgtaru [l] (see also [3]).

430

6 Optimal Control in Real Time

As a matter of fact such a problem can be approximated by a family of smooth optimal control problems of the form

(P,)

Minimize

subject to dY dt

- + Ay

+ dcp"(y) Y(0)

= Bu

in [0, T I ,

=Yo9

where g", cp,", and cp" are smooth approximations of g , cpo, and cp, respectively. Moreover, the optimal value function $ " of problem (P,) satisfies Eq. (1.34) in the sense of Theorem 1.2 and $ &-+ $ on [0, TI X D(+) as E -+ 0. In this context, we may view $ as a generalized solution to Eq. (1.34). More will be said about this equation in Section 2. Now let us come back to the case H = L2(R), A = - A , V = Hi(R), and dcp(y)(x)

=

p(y(x))

tly

E

L2(R),a.e. x

E

R,

(1.53)

where p: R -+ R is a locally Lipschitz function. Though Theorems 1.2 and 1.3 are not applicable to the present situation we have, however: Theorem 1.4. If p satisfies condition (1.11) then every optimal control u* is expressed as function of the corresponding optimal state y * by the feedback law

u*(t)

E

ah*( - B * d $ ( t , y * ( t ) ) )

a.e. t

E

(O,T),

(1.54)

where $: [0,TI X L2(R) + R is the optimal value function and dtC, is the Clarke generalized gradient of $ ( t , * 1. Proo$ According to Lemma 1.4, for every t E [O, TI the optimal pair (y*, u * ) is on the interval [0, t ] optimal for the problem i n f ( ~ ( g ( Y ( s , o , Y o , u )+ ) h ( u ( s ) ) )ds + $ ( t , y ( t , O , y o , u ) ) ; u E L2(0,t;

a)).

431

6.1. Optimal Feedback Controllers

Then, by Theorem 1.1 in Chapter 5, for every t E [O,T] there is p' AC([O,t ] ; Y * ) n C,([O, tl; H ) that satisfies the equations

E

(1.55) (1.56)

B*p'(s) E dh(u*(s)), P ' ( t > E -dtCl(t,y*(t)).

Since the function u* is approximately continuous on [O, T I , arguing as in the proof of Theorem 1.1 it follows by Eq. (1.55) that u * ( t ) E dh*(B*p'(t))

a.e. t

E

(O,T),

which implies (1.541, as desired. Remark 1.4. If, under assumptions of Theorem 1.4, h is G2teaux differentiable and R ( B ) (the range of B ) is dense in H then any dual extremal arc associated with ( y * , u * ) satisfies the equation p ( t ) E -d+(t,y*(t))

a.e. t

E

(0,T).

(1.57)

Indeed, by Eq. (1.55) we have p ( s ) = p ' ( s ) Vs E [O, t ] , which by virtue of (1.56) implies (1.57). (See C. Popa [2]for a similar result in the case of the obstacle control problem and Clarke and Vinter [ l ]for a related finite dimensional result.) 6.1.4. Feedback Controllersfor the Optimal Time Control Problem

We shall consider here the time-optimal control problem (PI studied in Chapter 5 (Section 3.3) with the state system (3.23) and the control constraints u E 2'& = ( u E L"(0,m; L 2 ( f l ) )Iu(t)12 ; I p a.e. t > O}. We shall assume here that p: R + R is locally Lipschitz and 0I P'(r) s C(l

+ I p ( r ) l + Irl)

a.e. r

E

R.

Denote by cp the minimal time function cp(

y o ) = inf ( T ; 3 u

E

gpsuch that y ( T , y o ,u ) = 0) .

( 1.58)

6. Optimal Control in Real Time

432

Also, it is readily seen that (the dynamic programming principle) cp(yo) = i n f k

e}

+ q ( y ( t , y o , u > >u; E

vt

> 0, Y O

E~ ~ ( f i > .

(1.59)

Here, y ( t , y o , u ) is the solution to system (3.23), i.e., dY -by dt

-

+ p ( y ) = Bu

Y(X,O) =y,(x)

in

in Cl x R ' ,

a,

in dR

y = o

X

(1.60)

R ' .

This implies that every time-optimal pair ( y * , u * ) is also optimal in problem (1.58), and so by Theorem 1.1 in Chapter 5 for every f E [0, T * ) (T* is the minimal time) there is p t E AC([O,t ] ; Y*)n C,([O,t]; L2(R>> n C([O, tl; L'(R)) such that dP' ds

- + Apt

-p'ap(y) 3

0

in R x ( O , t ) ,

a,

p'(t)

E

-8cp(y*(t))

in

u*(s>

E

P sgn P ' ( S >

a.e. s

E

( O , t ) , (1.61)

where sgn p' = pf/lp'12 if p' # 0, sgnO = (w E L2(R); lwlq I 1). Then, arguing as in the proof of Theorem 1.1, we conclude that u * ( t ) E -psgn ( ( t ) ,

t ( t )E

d q ( y * ( t ) ) a.e. t E ( O , T * ) , (1.62)

where d q is the generalized gradient (in the sense of Clarke) of cp: L2(R) +

R.

We have proved, therefore: Theorem 1.5. Under the assumption (1.58), any time-optimal control u* of system (1.60) admits the feedback representation (1.62).

It turns out that, at least formally, the minimal time function cp is the solution to the Bellman equation associated with the given problem, i.e.,

( W Y ) , MY) + PIDV(Y)l2 = 1, d o ) = 0,

Y

E L2(W,

(1.63)

433

6.2. A Semigroup Approach to the Dynamic Programming Equation

where My = - A y + p ( y ) V y E D ( M ) = ( y E Hd(fl>n H2(fl);p ( y ) E L2(fl)). Indeed, coming back to approximating problem (P,) in Section 3.3, Chapter 5, we see by Eqs. (3.641, (3.65) there that

+ ep,(t)

u , ( t ) = psgnp,(t)

a.e. t > 0,

and therefore u,(t>

E

- P sgn d V & ( Y & ( t )-) E J 9 & ( Y & ( t ) )

Then, by (3.651, it follows that Hamilton-Jacobi equation

'p,

a x . t > 0. (1.64)

is the solution to the stationary

&

( d 9 & ( Y )MY) , + PldV&(Y)lZ + p J & ( Y ) 12 2- g " ( y )

VY

E

L2(fl)9

v(0) = 0 , i.e., 3v,(y)

E

( 1.65)

d ~ o , ( ysuch ) that

Since by Theorem 3.5, Chapter 5, 'p, + q~as E + 0, Idcp,(y)l2 I p - ' , and g" + 1 as E + 0, we may view 'p, as an approximating solution to Eq. (1.63) and so cp itself is a generalized solution to Eq. (1.63). As a matter of fact, it turns out that cp is a viscosity solution to Eq. (1.63) and that under a supplementary growth condition on p, it is unique in the class of weakly continuous functions on L2(fl) (see Barbu [20], Tgtaru [2]). 6.2. A Semigroup Approach to the Dynamic Programming Equation

6.2.1. Variational and Mild Solutions to the Dynamic Programming Equation We shall study here the Hamilton-Jacobi equation (1.34) in a more general context, namely in the case when d 4 is replaced by a general maximal monotone operator M c H X H. By the substitution q(t, y ) = +(T - t, y ) we reduce this problem to the forward Cauchy problem cp,(t,Y) + h*( --B*cp,(t,y)) + ( M Y , q ( t , y ) ) = d Y ) ,

cp(0,Y)

=

%(Y).

(2.1)

434

6. Optimal Control in Real Time

As seen earlier, there is a close connection between Eq. (2.1) and the optimal control problem

where y = y ( s , x , u ) E C([O,t];H ) is the mild solution to the Cauchy problem

dY ds

-

+ My 3 Bu

in ( O , t ) ,

y(0)

=x E

D(M).

(2.3)

The following hypotheses will be in effect throughout this section:

H and U are real Hilbert spaces with the norms denoted I * I and I . I u , respectively. B E L(U, H ) and M is a maximal monotone subset of H X H with the domain D ( M ) . (ii) h: U R is a lower semicontinuous convex function such that (i)

-+

Denote by h* the conjugate function of h , i.e., h * ( p ) = sup{(p,u) h(u); u E U ) . We have denoted by (., * ) the scalar product of H and by ( * , ) the scalar product of U. B* is the dual operator of B. Given a metric space X we shall denote by B U C ( X ) the space of all bounded and uniformly continuous real valued functions on X endowed with the usual sup norm: llfllb

=

sup{If( x)l; x

E

XI

9

f E BUC( X ) .

By Lip(X), we shall denote the space of all Lipschitz functions f: X -+ R. In the following, the space X will be the closure D( M ) of D ( M ) in H . If g, cpo E B U C ( D ( M ) ) , then the function cp is well-defined on [O,m) x D(M). We set (S(t)cpo)(x)

=

cp(t,x),

t 2 0, x

E

D(M),

(2.5)

and call it variational solution to Hamilton-Jacobi equation (2.1). We have

6.2. A Semigroup Approach to the Dynamic Programming Equation

435

In other words, S(t) is a semigroup of contractions on the Banach space

Y = BUC(D( M ) ) . Proof Obviously, S(t)cpo is bounded on D ( M ) for every t 2 0. For x, xo E D(M ) we have, for any E > 0, S(t)cPo(x) - S ( t ) c p o ( x o )

and y ( s , x, u ) is the mild solution to (2.3). Recalling that (see Theorem 1.1, Chapter 4)

l y ( t , x o ,u > - y ( t , x, (v)t 5 lx0 - XI

+/b(u 0

- v ) l u ds,

we conclude that S(t)cpo is uniformly continuous on D(M ) . The semigroup property (2.6) is an immediate consequence of the optimality principle, whilst (2.6)' follows by the obvious inequality ( S ( t ) c p o ) ( x ) - ( S ( t ) S o ) ( x ) 5 cpO(Y(t9 x, u ) ) - $ o ( t , x, u ) +

where

E

is arbitrary and suitable chosen.

E?

(2.7)

rn

The main result of this section (Theorem 2.1 following) amounts to saying that the semigroup S ( t ) is generated on a certain subset of Y by an m-accretive operator.

436

6. Optimal Control in Real Time

The generator A? of S ( t ) is constructed as follows. For any f E BUC(D( M ) ) and A > 0, define the function (R(A)f)(x)

=

inf{ke*'(f(y(W u

E L:,,(R+;

U ) , y'

+ h ( u ( t ) ) )d t ;

+ My 3 Bu in R f , y ( 0 ) =

It turns out that R(A) is a pseudo-resolvent in Y = BUC(D( M ) ) . More precisely, we have:

Lemma 2.2.

For every A > 0, R(A) maps Y into itself and

R(A)f=R(p)((p-A)R(A)f+f)

Moreover, i f f

E

O < A I p < w .

(2.9)

Lip(D(M)) then R ( A ) E ~ Lip(D( M I ) .

Pro06 Let uo E U be such that h(u,) < co and let y o ( t ) = y ( t , x, u,,) be the corresponding solution to (2.3). We have the obvious inequality

This would imply that 10"e-A'h(u,(t))dt -, - 03 as n + 00, which by virtue of the convexity of h and assumption (2.4) leads to a contradiction. Hence, R(A)f is bounded on D(M). To prove that R(A)f is uniformly continuous consider xl, x2 arbitrary but fixed in D(M). For every E > 0, there are u, and iie such that

6.2. A Semigroup Approach to the Dynamic Programming Equation

for all (y, u ) satisfying Eq. (2.3). This yields

437

438

6. Optimal Control in Real Time

+ k m C h ’ ( f ( r * ( s- t ) ) + h ( u * ( s - t ) ) )ds (2.10) where ( y * , u * ) and ( z * , u * ) satisfy Eq. (2.3) with the initial conditions y*(O) = x , z*(O) = y * ( t ) and are chosen in a such a way that

R( El.)(( El. - A ) R ( A ) f + f ) ( x ) 2

k m e - * l ( f ( y * ( t ) )+ h ( u * ( t ) ) )dt

+kme-”(f(z*(s))

+(

+ h ( u * ( s ) ) )ds -

-

A) j m e - P ‘ d t 0

(2.11)

E.

Now, if we multiply (2.10) by e - ( P - A ) r and integrate on (0,m), we find after some calculation involving (2.11) that

( R ( A ) f ) ( x ) 2 R( El.)(( El.

-

A)R(A)f+f)(x) +

E

> 0,

which completes the proof. Now let d:D ( d ) c Y -+ Y be the operator (eventually, multivalued) defined by

dR(1)f = f - R( 1)f

Vf

E

Y = BUC( D(M)),

D ( d ) ={cp=R(l)f;f€Y).

(2.12)

If the operator A is continuous on H and f,h* are smooth, then it is readily seen that cp = R(l)f satisfies the stationary Hamilton-Jacobi equation

M Y ) + (MY777(Y)) + h*(-B*77(y)) = f ( Y ) y VY E W M ) . 77(Y) E d c p ( Y ) , (This follows easily by the optimality system associated with the infinite horizon optimal control problem (2.81.) Thus, we may view the operator d as an extension on Y of the operator y + (My, cpy(y))+ h*(-B*cpy(y)). If

6.2. A Semigroup Approach to the Dynamic Programming Equation

439

M is linear, then we have (see Barbu and DaPrato [l]) dp=

Y ;W ( X ) = ( ~ ( x ) , M x )+ h*( - B * ~ ( x ) ) V X E D ( M ) , bE V(X) E W x ) }

{W E

w-4.

Lemma 2.3. The operator d is m-accretive in Y ( A Z +&')-'

=

VA

R(A)

X

E

Y and

(0,1].

(2.13)

(I is the unity operator in Y.) Proof: By Eqs. (2.91, (2.12) we see that ( AZ

+ d)

=

for 0 < A

R( A)

I1,

(2.14)

whilst by definition of R(A) it is readily seen that IIR( A)f - R( A)gIIb I A-'

IIf

- gIIb

VA > 0, f,g

E

Y.

Hence, &' is m-accretive (see Section 3.1 in Chapter 2). Moreover, by the resolvent equation ( A Z + d ) - ' f = ( p Z +d)-'(( p - A)(AZ + d ) - ' f + f ) , we infer that (2.13) holds for all A > 0. According to the Crandall-Liggett generation theorem (Theorem 1.3 in Chapter 41,for every 'po E D ( d ) and g E Y the Cauchy problem dP dt

-+dq 38 Cp(0) =

has a unique mild solution 40 formula

E

in

[O,w),

(2.15)

4009

C([O,w); Y ) , given by the exponential

(2.16)

q & ( t )= 40,

for

-8

I t I 0.

(2.17)

440

6. Optimal Control in Real Time

- Moreover, the map T ( t ) :D ( d ) -+ D ( d ) defined by Vcpo

T(t)cpo,= cp(t)

E

(2.18)

D ( d ) ,t 2 0,

is a continuous semigroup of nonlinear contractions of D ( d ) . The function t -+ T(t)cpo is called mild solution to Eq. (2.1). 6.2.2. The Equivalence of Variational and Mild Solutions Coming back to the semigroup S ( t ) defined by formula (2.51, one might suspect that S(t)

=

onD(d).

T(t)

Indeed, we have:

Theorem 2.1. Assume that hypotheses (i), (ii) hold and that g L i p { ( m ) . Then S( t ) 9 0

=

T ( t )~

o ,

E

(2.19)

V t 2 0 , Vpo, E D ( d ) .

Moreover, the operator d is single valued and for every cpo E D ( d ) one has 1 lim - ( ( S ( t ) c p o ) ( x ) - cpo(x)> t10 t

=

-(dvo)(x) +g(x)

Vx E D(M)* (2.20)

Before proving Theorem 2.1 we shall give a precise description of the closure D ( d ) of D ( d ) in Y. We shall denote by Z the set of all cp E Y having the property that the function t -+ SM(t)cp= cp(e-M'x) defined from [O,m) to Y is continuous into the origin. In other words, 2 is the domain of the Co-semigroup S M ( t ) : [O,m)

Proposition 2.1.

-+

Y.

Under the preceding assumptions 2

=

D(d).

441

6.2. A Semigroup Approach to the Dynamic Programming Equation

or, equivalently,

i

8 = cp

E

5 L,(f

BUC(D(M))n Lip(D(M));

Icp(y(t, x , u ) ) - c p ( x ) ~

+ l I u ( s ) I u d s ) V t 2 0 , u E L'(R+; U ) , x

E

D(M)), (2.21)'

where y ( t , x , u) is the solution to (2.3). It is readily seen that Z =8(the closure of 8 in B U C ( D ( M ) ) ) . Indeed, the space B U C ( D ( M ) )n Lip@( M)) is dense in B u C(D( M ) ) (see, e.g., Lasry and Lions [l]) and by the same argument it follows that 2 n Lip(D( M ) ) is dense in 2. Now it is readily seen that for every cp E Z n Lip@( M ) ) ,

and cp& E A *

This implies that 2 =8, as claimed. Hence, to prove Proposition 2.1 it suffices to show that 5 = D ( d ) .Toward this aim, we shall prove first that

(2.23)

0, (2.24)

because cp is Lipschitz (Lemma 2.3). (Here and everywhere in the following we shall denote by C several positive constants independent of E , u,

442

6. Optimal Control in Real Time

and t.) On the other hand, by the optimality principle we have

y’

+ My 3 Bu in ( 0 , t),y(O) = x)

Vt

> 0.

In view of (2.231, this yields 1

+ h(s)) d s ~_< E

- ~ c p ( x )- e-‘/”cp(y,(t)) - / ‘ e - ’ / / ” ( - f ( y , ( s ) )

CL

0

and, therefore, ~ c p ( x )- e-‘/”cp(y,(t)>l<

c(t + ( j b h ( u , ( s ) )d s l ) + E

V t 2 0.

(2.25) On the other hand, by (2.24) we have 1 jbe-sl” ( -El.f ( Y , ( s ) ) + h ( u A s ) ) ) 1 -f(yo) CL

I te-s/”(

1

+ e-””cp(y,(t))

+ h ( u o ) cis + e - ‘ / ” q ( y o ( t > )+ E ,

where uo E U is such that M uo) < w and y o = y ( t , y , uo). This yields

j b h ( u , ( s ) ) ds

I C

(+ a t

Iu,(s)Iu

ds

1+

E

Vt

> 0,

because h is bounded from below by an affine function. Then, by assumption (2.41, we see that

Substituting this into (2.24) and using (2.25), we get I q ( y ( t ) ) - q(x)l

c(r + jblu(s)lu

I

ds)

+E

VE

> 0, t > 0,

where C is independent of E , u, and t. Hence, cp €9. Since the space BUC(D(M)) n Lip@( M ) ) is dense in BUC(D( M ) ) and the operator (I p d - ’ is nonexpansive on Y = BUC(D( M ) ) , we conclude that

+

(I + @)-lf€G

Vf€ Y.

6.2. A Semigroup Approach to the Dynamic Programming Equation

On the other hand, we have (see Proposition 3.2 in Chapter 2), for p

+p~)-'f+f

(I

443 -+

0,

in Y ,

for every f E D ( d ) . Along with (2.22), this implies that D ( d ) C g . To prove that %L c D ( d ) consider f, an arbitrary element of 9, and set

f,

=

(1+

R(

@)-lf=

p-lf),

P

> 0.

We shall prove that f, + f in Y as I*. + 0. Since f, E D ( 4 for all p > 0 this clearly will imply that f E D ( M ) , thereby completing the proof. For every p > 0 and all x E D ( M ) , there are (y,,u,) E C([O,a); H ) n L:,,(O,m); U),y, = y ( t , x , u,) such that P

and

for all u E L:,,(O,m; y ( t , x , u o ) we have

f,G>-f(x)

U.In particular, for 1 . .

j

P O

u

e-'/"(f(Yo(t))

= uo = constant and y o =

-f(X)

+ P.h(uo)) dt

Vp

because f EL^. Similarly,

f ( x > -f,<x>

> 0 , (2.26)

444

6. Optimal Control in Real Time

On the other hand, by (2.26) we see that

Then, by assumption (2.4), we see that for every 6 > 0 there is N ( 6 ) such that

Hence,

This implies that :/ e-'/'h(u,(t))dt + 0 as p -+ 0, and so by (2.261, W (2.27) we conclude that f, + f in Y as p + 0, as claimed.

=s

Proofof Theorem 2.1. We shall prove first that S ( t ) maps D ( d ) into By the optimality principle we have, for itself. To this end, we fix p0 in 8. all 0 I S < t < 03,

6.2. A Semigroup Approach to the Dynamic Programming Equation

445

where uo = constant, h(uo)< m, and y o = Y O , x , uo). Moreover, there exist y,, u, satisfying Eq. (2.3) and such that

and yields

Then, using once again assumption (2.4), we get

Now, coming back to inequalities (2.281, (2.291, we get

where (y, u ) is an arbitrary pair of functions satisfying (2.3). On the other hand, by the definition of S ( t ) it follows by similar argument that

which along with (2.30) implies that S ( t ) q o ~ gas claimed. ,

446

6. Optimal Control in Real Time

Now we shall apply the nonlinear Chernoff theorem (Theorem 2.2 in Chapter 4) on the space Y where C =%, F ( t ) = S(t), and A = d o , docp = dcp + g. To this end, we shall prove that for every p > 0,

=

(I

+ pA?o)-'cpo = ( I + @ ) - ' ( ( p a

+ pg)

Vcpo € 3 . (2.31)

Let us postpone for the time being the proof of (2.31). If (2.31) holds, then we have T(t)cpo =

,'Fm(S(:))"cpo

=

S(t)cpo

Vcpo E % , t 2 0 , (2.32)

i.e., T ( t ) = S ( t ) on 3 = D ( d ) . Now, by definition of S ( t ) , we have ( S ( t ) c p o ) ( x ) - cpo(x)

for all ( y , u ) satisfying Eq. (2.3). If that

(pa E

D ( d , then there is

f E Y )such

6.2. A Semigroup Approach to the Dynamic Programming Equation

i.e.,

[e-”h(u,(s)) ds + j 0f e - ” f ( Y & ( s )ds ) + e-bo(y,(t)) Icpo(x)

+E

This yields

because by Eq. (2.3) we have lye( t ) - XI

ItlMxl

+1 1 ‘Bu,( 0

V t 2 0.

s)l ds

We may conclude, therefore, that jb(lh(~,(s))l

+ lu,(s)lu)

ds IC ( t +

E )

Vt

where C is independent of E and t. We take E = t2. Then, by (2.341, we have

where z:

+ Mz, 3 Bu,

in ( O , . ? ) ,

z,(O)

=x ,

2

0,

447

448

6. Optimal Control in Real Time

and

and so by (2.36) we have

tlx

ED(M),

which along with (2.35) implies that

for all x E D ( M ) and all cpo E D ( d ) , as desired. To complete the proof, it remains to verify (2.31). Since (I @I-' and (I + ( ~ / E X -I s ( E ) ) ) - ' are nonexpansive on 3 = ~ ( d it)suffices , to prove (2.31) for cpo ES3. We set

+

We have

6.2. A Semigroup Approach to the Dynamic Programming Equation

449

Equivalently,

y'

+ My 3 Bu, y ( 0 ) = X , u E L ' ( 0 , c ;

(2.38)

Recall that by the definition of S ( t ) we have

yi

+ My,

3

Bu,

and

Icp,(x)

+ E2

This yields

where y o = y ( t , x, uo), h(u,) < 00,

in

(o,~),

y,(O)

= x,

450

6. Optimal Control in Real Time

Then, by the estimate (2.391, we have

and, using once again assumption (2.41, we get ((lh(u,(t))l

where C is independent of

+ lu,(t)lo) E.

=

(I

+ @)-‘(cp0

VE > 0 ,

(2.41)

Since cpo €9, this implies that

Icp,(y(t)) - cp,(x)l I CE We set cp

dt I CE

Vx

E

W M ) ,t E ( 0 , .).

(2.42)

+ p g ) . As noted earlier, we have

and so there exists a pair ( z , , u,) satisfying the Cauchy problem (2.3) on (0, E ) and such that

((lh(v,(t))l

+ Iu,(t)lu)

dt I CE

VE > 0

(2.44)

6.2. A Semigroup Approach to the Dynamic Programming Equation

451

Now, by (2.421, (2.431, it follows that cp&(X)

- cp(x)

(2.47)

Using the estimates (2.391, (2.411, (2.421, (2.44), and (2.451, in inequalities (2.461, (2.471, we see that

which completes the proof of (2.31).

Remark 2.2. Theorem 2.1 is related to existence of Lie generators for nonlinear semigroups of contractions (J. W. Neuberger [l, 211. Indeed, if g = 0, B = 0, and =

(S(t)cp)(x)

=

(:

if u = 0, otherwise,

cp(e-M'x)

Vx E D ( M ) ,

for all cp E Y = BUC(D( M ) ) . Then, by Theorem 2.1, it follows in particular that this semigroup is generated on Z by a single valued m-accretive operator.

452

6. Optimal Control in Real Time

6.2.3. Approximation of the Dynamic Programming Equation We shall consider here the forward Hamilton-Jacobi equation (2.0, where M = A + dcp and A E L(V,V ' ) , cp: H + R, g , h , and cpo satisfy the assumptions of Section 1.2. As mentioned earlier, the function

+ Ay + d c p ( y ) 3 Bu a.e. in ( O , t ) ,

y'

y(0)

(2.48)

= x , u E L'(0, t ; U ) ) ,

called a variational solution to Eq. (2.11, i.e., +At, x ) + h*( -B*+At,

XI)

+ ( A+ dcp(t, X I , + A t , XI) ( t ,x )

+(O,X)

=

E

cpdx),

=g(x),

(0, T ) x H , (2.49)

is under certain regularity assumptions on cp, a strong solution in some generalized sense (Theorem 1.2). It should be said that due to its complexity Eq. (2.49) is hard to solve or approximate by standard methods. Here, we will briefly discuss the Lie-Trotter scheme (the method of fractional steps) for a such a equation. Formally, a such a scheme for Eq. (2.49) is defined by + f E ( tX, ) + h*( - B * + / ( t , x ) ) + ( A ,+ E ( t , x ) )

+

=

cpo(x)

Vx

E

0

in [ i E , ( i + l ) ~ x] D(cp),

t+b,"(is,x) = + , " ( i E , e - E d C x ) E g ( e - E d p x ) , +"(O,X)

=

i

=

1,..., N - 1, (2.50)

D(cp),

where NE = T. In general, we do not know whether this scheme is convergent for E + 0. However, this happens in some significant situations and in particular if cp = Z,, where C is a closed subset of H such that

(A4,,PV) 2 ( A Y , Y )

(z + AA,)-'c

c

c

VY

E

V,

VA > 0.

(Here, P is the projection operator on C.)We recall that e-' V t 2 0.

(2.51) dQ

=P

453

6.2. A Semigroup Approach to the Dynamic Programming Equation

As seen earlier, this corresponds to an optimal control problem governed by the variational inequality (y’(t)

+ A y ( t ) - B u ( t ) ,y ( t ) - z ) 5 0

a.e. t

E

( O , T ) ,Vz

E

C,

Y(0) =Yo. Theorem 2.2.

Under the preceding assumptions +“(t,X)6G0+(t,X)

V ( t , x ) E [O,TI

where JI is the variational solution to (2.48).

x

c,

(2.52)

W

Theorem 2.2 is a direct consequence of convergence Theorem, 4.1, in Chapter 5 but we omit the details (see the author’s paper [151). It should be observed that Eq. (2.50) is structurally simpler than the original equation and its solution cp can be explicitly written. Indeed, we have

y’

+ Ay = Bu in ( i c , t ) , y ( 0 ) = x, u E ~ ‘ ( 0t ;, u)), t

E

[ i E , ( i + l ) ~ ] ,(2.53)

and in view of the maximum principle (optimality system) we have

where

+ Ayz

=

Bu,*

in ( i c , t ) ,

p: - Ap,

=

0

in ( i s , t ) ,

(y:)’

y,*(ic) = x , u,*(s ) E ah*( B*p,( s ) )

a.e. s

E

( i c ,t ) .

6. Optimal Control in Real Time

454

Hence, uf( s )

for some p

E

+"( t , x )

E

a.e. s

ah*( B * e - ( f - S p) A)

E

(iE,t),

H . Substituting this into (2.531, we get =

inf

PGH

is:

h( ah*(B * e - ( ' - S ) pA ) ) ds

for t

[i&,(i

E

+ Eg( e - ( r - i s ) A x )

+ l ) ~ ]x, E C .

(2.54)

Moreover, V t E [O, T I ,

u = dh*( -B*rG;"(T - t , y ) )

is a suboptimal feedback control for problem (P) in Section 1.2. We will illustrate this on the following example: H = L2(fl), V = Hi(fl), A = - A , C = { y E Hi(fl); y ( x ) 2 0 a.e. x E fl), B = I, h ( u ) = +/, u2 dx, g = 0, and cp,(y) = i / J y - y o ) 2dx. This corresponds to following optimal control problem: Minimize on all u E L2(Q), Q y , E L2(Q),subject to

/Qu2dxdt + =

R

X

( 0 , T ) and y

dY

- - by = u

in ( ( x , t )

at

dY dt

--Ay>u,

i / n ( y ( ~ , T )- y ' ( ~ ) )dx~

y 2 0

y(x,O) =yo(x)

E

E

Q;y ( x , t ) > 0),

inQ, in fl.

L2(0,T ; H,'(fl) n H2(R)),

6.2. A Semigroup Approach to the Dynamic Programming Equation

455

In this case, Eq. (2.49) becomes

where &(y)

=

{w E L 2 ( R ) ; w(x) = 0 a.e. in [ x

w(x) I 0

a.e. in [ x

E

E

R; y ( x )

a; y ( x ) > 01, =

O]}.

By Theorem 2.2, we have

Vt

E

[i&,(i

+ l ) ~ ]y, E L 2 ( R ) ,

(2.56)

where w + = max(w,O) and dz dS

+Az=O

z=o z(t,x) = p ( x ) dW

--Aw=z dS

w=o w(is,x)=y(x)

i n ( i E , t ) x R , i = O , l , ... , N - l , in ( i E , t ) x 3 0 , x

E

a,

in (i&,t) x R , in ( i ~ , t )x d R , in R .

(2.57)

Thus, at every step the calculation of $" reduces to a minimization problem on the space L2(R), which can be solved numerically by standard methods.

6. Bibliographical Notes and Remarks

456

As a second example, we shall consider the dynamic programming equation associated with the optimal control problem

y’ + P ( y )

=

u , lu(t)l

1, Y ( 0 ) = y o

i.e.,

I$xl + p ( x ) q X = g ( x ) ,

$( +

$ ( O , x ) = cpo(x),

x E R, t x

E

E

)

(2.58)

9

(0,779

(2.59)

R,

where p is a locally Lipschitz function. In this case, we have lim, $ “ ( t , x ) W t , x ) E [0, TI X R, where

$(t,x) =

~

$ :

+ $lI:

=

$,E(is,x) =

0

in [ i ~ , ( i + l ) ~ X] R,

+ $ ~ ( ~ E , z ( E ) ) , i = 0,1, ..., N z’ + p ( z ) = 0 in (0, E ) ,

Eg(X)

z(0) = x ,

-

1, (2.60) (2.61)

and arguing as in the proof of (2.54) we get $&(t,x) =

u

=

+ ( t - i ~ ) p +) $ f ( i ~ , w ( ~ ) ) ; w ’ + p (w ) = 0 in (0, E ) , w( 0) = x + ( t - i ~ ) p } , inf { E g ( x

Ipls 1

-sign $,,?(T - t , y )

(2.62)

as an approximating (suboptimal) feedback control of problem (2.58). Bibliographical Notes and Remarks

Section 1. Most of the results of this section have been previously established in a related form in the works of Barbu and DaPrato [l] and Barbu [6, 7, 101. For a recent treatment of dynamic programming for finite dimensional optimal control problems, we refer to the book of St. Mirica [l]. For other related results we refer to the recent works of Cannarsa and Frankowska [11, Cannarsa and DaPrato [11, Barbu, Barron, and Jensen [l], and Fattorini and Sritharan [l] (the last concerned with optimal feedback controllers for the Navier-Stokes systems).

6.2. A Semigroup Approach to the Dynamic Programming Equation

457

The theory of viscosity solutions to Hamilton-Jacobi equations was developed in the works of M. G. Crandall and P. L. Lions [l-31, and M. G. Crandall et al. [ll. In particular, the results of D. Tgtaru [l] cover the existence and uniqueness theory of viscosity solution for the dynamic programming equation associated with problem (PI or the time-optimal problem (in this context, we mention also the work of M. Bardi [l]). Section 2. The results of Section 2.1 and 2.2 have been previously established in the author’s work [13]. The semigroup approach to Hamilton-Jacobi equations was also used by Barbu and DaPrato [2] and Th. Havirneanu [l] (The latter work extends the results of Section 2.1 to Hamilton-Jacobi equations with nonconvex Hamiltonian.) Theorem 2.2 along with other results of this type were given in the author’s work [14, 151. Related results were obtained in a more general context by C. Popa [l] and Th. Havimeanu [2]. We mention also the work of L. C. Evans [2] for a min-max type approximation formula for the solutions to Hamilton-Jacobi equations.

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Index

Closed loop system, 408 Cone tangent, 94 normal, 61, 94 Control feedback, optimal feedback, 407,412 optimal, 151 problem, 149 time optimal, 365 Convergence strong, 1 weak, 1 weak star, 1 Convex function, 57 integrand, 72

Generalized gradient, 91 Leray-Schauder degree, 5 Mapping, compact, 5 duality, 2 measurable, 408 upper-semicontinuous, 7 Operator coercive, 37 demicontinuous, 37 elliptic, 130 hemicontinuous, 37 Problem Cauchy, 25 elasto-plastic, 142 free boundary, 132 obstacle, 131 Signorini, 85, 147 Stefan, 284, 292

Directional derivative, 31, 59 Domain effective, 87 Dynamic programming principle, 420 Epigraph, 57 Equation dynamic programming, 421 Hamilton-Jacobi, 421 law conservation, 300 Free boundary, 85, 130 Function absolutely continuous, 10 Bochner integrable, 9 conjugate, 58 finitely valued, 9 FrCchet differentiable, 59 G2teaux differentiable, 59 indicator, 60 indicator of A , 60 lower semicontinuous, 57 optimal value, 417 proper convex, 57 strongly measurable, 10 support, 60 weakly measurable, 10

Regularization of a function, 65 Saddle point, 8 Schauder fixed point theorem, 6 Semigroup analytic, 23 of class C,, 18 of contractions, 18, 22 differentiable, 22 oquasi contractive, 229 Set accretive, 100 closed, 103 demiclosed, 103 maximal monotone, 36 monotone, 36 m-accretive, 100 w-accretive, 100 om-accretive, 100

475

476

Index

Solution classical to Cauchy problem, 25, 200 mild to Cauchy problem, 25, 202 mild to Hamilton-Jacobi equation, 440 strong to Cauchy problem, 25, 202 variational to Hamilton-Jacobi equation, 434 viscosity to Hamilton-Jacobi equation, 428 Space strictly convex, 2 uniformly convex, 2

Subdifferential, 59, 421 Subgradient, 60 Superdifferential, 421

Variational inequality elliptic, 126 parabolic, 270

Yosida approximation, 48

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