Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems
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253 Vadim Komkov Texas Tech University, Lubbock, Tx/USA
Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems
Springer-Verlag Berlin· Heidelberg· NewYork 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and 8. Eckmann.Zurich
253 Vadim Komkov Texas Tech University, Lubbock, Tx/USA
Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems
Springer-Verlag Berlin· Heidelberg· NewYork 1972
AMS Subject Classifications (1970): Primary: 49B25 Secondary: 73C99, 73K05, 73K 10
ISBN 3-540-05734-X Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05734-X Springer-Verlag New York· Heidelberg· Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 73-188624. Printed in Germany. Offsetdruck:Julius Beltz, HemsbachlBergstr.
FOREWORD
This
intends to fill the existing gap in the
applications of optimal control theory to problems of damping (or excitation) of simple elastic systems.
Some of the
material follows closely the contents of artioles oonoerning the control of hyperbolic systems of D. Russell and of artioles of the author oonoerning the control of beams and plates. the material has never appeared in print before.
Some of
Some obvious
generalizations have been omitted, but some more diffioult generalizations, such as the control of a vibrating arbitrary three dimensional elastic body, have not been solved yet. This monograph is intended to be a self-oontained exposition of the basic principles of optimal damping of vibrations of simple elastio systems.
The reader is assumed to be familiar
with advanced calculus, some elementary concepts of funotional analysis and some concepts of partial differential equations. For the sake of convenience the author includes a basic discussion of admissible distributional controls in Appendix 1, and an expository discussion of the olassical form of Pontryagin's principle is offered in an appendix.
List of Contents
........
1
CHAPTER I. A Summary of Some Results on Controls of Hyperbolic Partial Differential Equations • • • • ••
4
Appendix to Chapter I. Remarks Concerning Concepts from the Theory of Generalized Functions Used in Chapter I • • • • • • • • • • • • • • ••
58
Introductory Remarks
CHAPTER II. The Optimal Control of Vibrating Beams
61
Appendix 2.1. Formulas for Torsion Constant C and Warping Constant Cw for Some Cross-Sections • • • 118 CHAPTER III. Optimal Control Theory for Thin Plates
• 119
CHAPTER IV. Classification of the Bou
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