Turnpike Properties in the Calculus of Variations and Optimal Control
This book is devoted to the recent progress on the turnpike theory. The turnpike property was discovered by Paul A. Samuelson, who applied it to problems in mathematical economics in 1949. These properties were studied for optimal trajectories of models o
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Nonconvex Optimization and Its Applications VOLUME 80 Managing Editor: Panos Pardalos University of Florida, U.S.A.
Advisory Board: J. R. Birge University of Chicago, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A. J. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany H. Tuy National Centre for Natural Science and Technology, Vietnam
______________________________________ TURNPIKE PROPERTIES IN THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
By ALEXANDER J. ZASLAVSKI The Technion—Israel Institute of Technology, Haifa, Israel
13
Library of Congress Cataloging-in-Publication Data Zaslavski, Alexander J. Turnpike properties in the calculus of variations and optimal control / by Alexander J. Zaslavski. p. cm. — (Nonconvex optimization and its applications ; v. 80) Includes bibliographical references and index. ISBN-13: 978-0-387-28155-1 (alk. paper) ISBN-10: 0-387-28155-X (alk. paper) ISBN-13: 978-0-387-28154-4 (ebook) ISBN-10: 0-387-28154-1 (ebook) 1. Calculus of variations. 2. Mathematical optimization. I. Title. II. Series QA316.Z37 2005 515´.64—dc22 2005050039 AMS Subject Classifications: 49–02
ISBN-10: 0-387-28155-X e-ISBN-10: 0-387-28154-1
ISBN-13: 978-0387-28155-1 e-ISBN-13: 978-0387-28154-4
© 2006 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 know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the 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 11405689
Contents
Preface Introduction
ix xiii
1. INFINITE HORIZON VARIATIONAL PROBLEMS 1.1 Preliminaries 1.2 Main results 1.3 Auxiliary results 1.4 Discrete-time control systems 1.5 Proofs of Theorems 1.1-1.3
1 1 3 7 17 20
2. EXTREMALS OF NONAUTONOMOUS PROBLEMS 2.1 Main results 2.2 Preliminary lemmas 2.3 Proofs of Theorems 2.1.1-2.1.4 2.4 Periodic variational problems 2.5 Spaces of smooth integrands 2.6 Examples
33 33 37 54 59 62 69
3. EXTREMALS OF AUTONOMOUS PROBLEMS 3.1 Main results 3.2 Proof of Proposition 3.1.1 3.3 Weakened version of Theorem 3.1.3 3.4 Continuity of the function U f (T1 , T2 , x, y) 3.5 Discrete-time control systems 3.6 Proof of Theorem 3.1.2 3.7 Preliminary lemmas for Theorem 3.1.1
71 71 76 79 83 88 90 94
vi
TURNPIKE PROPERTIES
3.8 3.9 3.10 3.11 3.12
Preliminary le
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