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详细说明：经典控制理论教程Analysis and Control of Nonlinear Infinite Dimensional SystemsThis 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 Jovgnovich, Publishers
Boston San Diego New York
London sydney Tokyo Toronto
This book is printed on acid-free paper. E.
Copyright 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 bt
ACADEMIC PRESS LIMITED
24-28 Oval Road, London nw] 7DX
Library of Congress Cataloging-in-Publication Data
Barbu,Ⅴ morel.
Analysis and control of nonlinear infinite dimensional systems
Viorel Barbu
P. cm.(Mathematics in science and engineering; v 189)
Includes bibliographical references and index
ISBN0-12-078145-X
1. Control theory. 2. Mathematical optimization. 3. Nonlinear
perators. I. Title. Il. Series
QA4023B3431993
003′.5-dc20
92-2851
CIP
PRINTED IN THE UNITED STATES OF AMERICA
9293949EB987654321
Contents
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
17
Chapter 2 Nonlinear Operators of Monotone Type
35
2. 1 Maximal monotone operators
35
2.2 Generalized Gradients( Subpotential Operators)
57
2.3 Accretive Operators in Banach Spaces
Bibliographical Notes and Remarks
123
Chapter 3 Controlled Elliptic Variational Inequalities
125
3.1 Elliptic Variational Inequalities. Existence Theory 125
3.2 Optimal Control of Elliptic Variational Inequalities 148
Bibliographical Notes and Remarks
196
Chapter 4 nonlinear accretive Differential equations
199
4.1 The Basic Existence results
199
4.2 Approximation and Convergence of Nonlinear
Evolutions and Semigroups
240
4.3 Applications to Partial Differential Equations
255
Bibliographical Notes and remarks
311
Chapter 5 Optimal Control of Parabolic Variational Inequalities 315
5.1 Distributed Optimal Control Problems
315
5.2 Boundary Control of Parabolic
Variational Inequalities
342
5.3 The Time-Optimal Control Problem
364
Contents
5.4 Approximating Optimal Control Problems via the
Fractional Steps Method
Bibliographical Notes and Remarks
404
Chapter 6 Optimal Control in Real Time
407
6.1 Optimal Feedback Controllers
407
6.2 A Semigroup Approach to the Dynamic
Programming Equation
433
Bibliographical Notes and remarks
456
References
459
Subject Index
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(mono
tone)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 nonlin
ear 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 euclides
e 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 optimiza
tion problem, which requires a specific treatment. In concrete situations
such a problem reduces to a nonlinear distributed optimal control prob
lem, 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 mathemat-
ical 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 differ
ential 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
vI
P
Peace
Part of the material presented in Chapter Ill 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 lasi and University of Cincinnati
Part of the manuscript has been read by my colleagues and former stu
dents, Professor I. Rabie, Dr. D. Tataru, and Dr. S. Anita, who contri
buted 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
Iasi, November 1991
Notation and symbols
R
the N-dimensional Euclidean space
R
th
e real line
R
+∞),R=(-∞,0,R=(-∞,+
open subset of RN
the boundary of n2
Q=9×(0,T),Σ=09×(0,T) where0
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