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Foundations of Optimal Control Theory E. B. Lee L. Markus Center for Control Sciences Institute of Technology University of Minnesota 0 g ROBERT E. KRIEGER PUBLISHING COMPANY MALABAR, FLORIDA To Judy and Lois Original Edition 1967 Reprint Edition 1986with corrections Printed and Published by ROBERT E. KRIEGER PUBLISHING COMPANY, INC. KRIEGER DRIVE MALABAR, FL 32950 Copyright 0 1967 by John Wiley and Sons, Inc. Reprint by Arrangement All rights reserved. No part of this book may be reproduced in any form or by any elec- tronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher. No liability is assumed with respect to the use cf the information contained herein. Printed in the United States of America Library of Congress Cataloging in Publication Data Lee, E. B. (Ernest Bruce), 1932- Foundations of optimal control theory. Reprint. Originally published: New York : Wiley, 1967. Bibliography: p. Includes index. 1. Control theory. I. Markus, L. (Lawrence), 1922- 11, Title. QA402.3.L38 1986 629.8’312 84-21772 ISBN 0-89874-807-O 109876543 Preface to Reprint Edition Almost twenty years have passed since the first printing of this treatise, Foun- dations of Optimal Control 7’heov; yet, the demand for this text still remains at a high level. In order to satisfy the demand and to take the opportunity to correct and amend the text at several points, the authors welcome the new edition by Krieger Publishers. Nevertheless, some explanations should be offered for the reprinting of a treatise on control theory, a theory that has been in a rapid state of explora- tion, development, and modification for over a generation. In the volume by Professors E .B . Lee and L . Markus the theory of controllability, observabili- ty, stabilizability, and identifiability-in other words, systems theory-is for- mulated and presented in a coherent and accessible mathematical style. As stated in the first edition: “Many of the basic concepts of control theory still demand an approach through the qualitative theory of differential systems, and it is from this viewpoint that this text has been prepared . . . As interpreted by the theory of nonlinear ordinary differential equations, the fundamental problems of control theory have been mathematically posed and answered”. These opinions remain those of the authors and they explain the long- duration validity and usefulness of this text-and its renewed importance in the most recent resurgence of systems theory. In the modem applications to nonlinear problems, for instance in robotic engineering, in the practical im- plementation by high speed large computers, in the current emphasis on robust theories of engineering and computation numerical schemes, the “approach through the qualitative theory of differential systems, and . . . the theory of nonlinear ordinary differential equations,” appears more and more basic and central to the future research of systems science. During the past two decades, many other important volumes have been published dealing with the algebraic techniques of linear.and bilinear control systems. Because of the special properties of linearity, the methods of algebra have been highly successful. But as large nonlinearities and large-scale phenomena have been seen as more significant physically, and also more V vi Preface amenable to computational powers, the fundamental mathematical analyses of global nonlinear differential equations have been required as a replacement for the algebra of linear theory. The treatise of Lee and Markus fills this fundamental engineering and mathematical demand now, just as previously-and it is now seen as a valuable foundation for future research and development of systems science theory and applications. E.B. Lee Professor of Electrical Engineering (Center for Control Science and Dynamical Systems) L. Markus Regents’ Professor of Mathematics University of Minnesota