Optimal Control Theory Applications to Management Science and Econom
Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the authors have applied to business management problems developed from their research an
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OPTIMAL CONTROL THEORY Applications to Management Science and Economics Second Edition
Suresh P. Sethi The University of Texas at Dallas
Gerald L. Thompson Carnegie Mellon University
Springer
Library of Congress Cataloging-in-Publication Data Sethi, Suresh P. Optimal control theory : applications to management science and economics / Suresh P. Sethi, Gerald L. Thompson.—2"^* ed. p. cm. Includes bibliographical references and index. ISBN: 0-7923-8608-6 (hardcover) ISBN: 0-387-28092-8 (paperback) 1. Management—Mathematical models. 2. Control theory. 3. Operations research. I. Thompson, Gerald Luther, 1923- II. Title. HD30.25 .S47 2000 658.4'033—dc21
00-040540
First paperback printing, 2006 © 2000 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com
SPIN 11532040
This book is dedicated to the raemory of our parents: Manak Bai and Gulab Chand Sethi Sylva and Luther Thompson
Contents Preface t o First Edition
xiii
Preface t o Second Edition
xv
What is Optimal Control Theory? 1.1 Basic Concepts and Definitions 1.2 Formulation of Simple Control Models 1.3 History of Optimal Control Theory 1.4 Notation and Concepts Used
1 2 4 7 10
The Maximum Principle: Continuous Time 2.1 Statement of the Problem 2.1.1 The Mathematical Model 2.1.2 Constraints 2.1.3 The Objective Function 2.1.4 The Optimal Control Problem 2.2 Dynamic Programming and the Maximum Principle . . . 2.2.1 The Hamilton-Jacobi-Bellman Equation 2.2.2 Derivation of the Adjoint Equation 2.2.3 The Maximum Principle 2.2.4 Economic Interpretations of the Maximimi Principle 2.3 Elementary Examples 2.4 Sufficiency Conditions 2.5 Solving a TPBVP by Using Spreadsheet Software
23 23 24 24 25 25 27 27 31 33
The Maximum Principle: Mixed Inequality Constraints 3.1 A Maximum Principle for Problems with Mixed Inequality Constraints 3.2 Sufficiency Conditions
57
34 36 44 48
58 64
viii
Contents 3.3 3.4 3.5 3.6
Current-Value Formulation Terminal Conditions 3.4.1 Examples Illustrating Terminal Conditions Infinite Horizon and Stationarity Model Types
....
65 69 74 80 83
4
The Maximum Principle: General Inequality C!onstraints 97 4.1 Pure State Variable Inequality Constraints: Indirect Method 98 4.1.1 Jimip Conditions 103 4.2 A Maximimi Principle: Indirect Method 104 4.3 Curren
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