Bifurcation of Extremals in Optimal Control
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1216 Jacob Kogan
Bifurcation of Extremals in Optimal Control
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Jacob Kogan Department of Mathematics The Weizmann Institute of Science Rehovot, Israel
Mathematics Subject Classification (1980): 49 ISBN 3-540-16818-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16818-4 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-In-Publication Data. Kogan, Jacob, 1954- Bifurcation of extremals in optimal control. (Lecture notes in mathematics; 1216) Bibliography: p. Includes index. 1. Control theory. 2. Bifurcation theory. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1216. QA3.L28 no. 1216510 s 86-24878 [QA402.3] [629.8'312] ISBN 0-387-16818-4 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage In data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To my uncle Boris
Preview The study concerns bifurcations of extremals in optimal control problems. The roots of this topic descends to the classical theory of the calculus of variations. The question of intersection of extremals has been considered in the calculus of variations under a sufficiency criterion, which enables to derive the famous Jacobi necessary condition. This condition guarantees the absence of conjugate points, namely the points of intersection of neighboring extremals. We show in this work that in .optimal control conjugate points exist even under a natural generalization of the sufficiency criterion of the calculus of variations. However, we discover that the set of the conjugate points has a simple and elegant structure. The set of the conjugate points is described in the study for three different types of optimal control problems: an optimal control system with a scalar cost, an optimal control system with a vector cost functional, and an optimal control problem with constraints. In the case of a linear control equation we find out that the conjugate points are the points where the dimension of the attainable set increases; in particular these points do not depend on a cost functional. In a nonlinear case the conjugate points of an extremal x(t) are determined by the attainable set of the system linearized about the extremal x(t). The Jacobi necessary condition is recovered as a special particular case. The first chapter of the study is an overview of the concepts, definitions, methods and results, the last, however, without proofs. The remainder of the work contains full proofs of the results.
In conclusion I wish to thank the numerous people for their assistance in
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