Mathematical Theory of Optimal Processes.pdf

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L. S. PONTRYAGIN SELECTED WORKS Volume 4 The Mathematical Theory of Optimal Processes

Classics o f Soviet Mathematics L. S. PONTRYAGIN SELECTED WORKS Edited by R . V. Gamkrelidze Volume 1: Selected Research Papers Volume 2: Topological Groups Volume 3: Algebraic and Differential Topology Volume 4: The Mathematical Theory o f Optimal Processes ISSN 0743-9199 This book is part of a series. The publishers will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

L. S. PONTRYAGIN SELECTED WORKS Volume 4 The Mathematical Theory o f Optimal Processes L. S. Pontryagin, V. G. Boltyanskii, R . V. G am krelidze, and E. F. Mishchenko Translated from the Russian by K. N. TrirogofF Aerospace Corporation, El SegunJo, California English Edition Edited by L. W. Neustadt Aerospace Corporation El SegunJo, California CRC Press \Cf^ J Taylor Si Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an inform a business

0 1986 by Gordon and Bmch S c i m Publishtrs BA.. P.O. Box 161.1820 Montrew 2, Swittetland. All rights mmd. Gordon and Breach SEieaa Publishers P.O. Box 784 Cooper Station New York, NY 10276 United Starts d Americp P.O. Box 197 London WC2E 9PX Englafid 58, rue Lhomond 75005 M Ftrac~ 1C9 Okubo 3 4 o m t Shiajuku-ku, Tokyo la J a m s Originally plblishcd m Russh as Mamrmwcnmn r.rrpnn omwvlnrnhlx n ~ e o m U k l ' g v n Nulltr. M-. Firsl pblirlmd in EngliP by Inkmcitaee R 1 M i . divirion of John Wiky & W. Inc. @ t%2 by John Wiey & Soar. Inc. Thad M h g . -bet Repainlad. by F . by Gwdon md Bmch Scicaoc Publihhen S.A.. 1986. flepnntcd from I copy m the c a l m of Ihe Bmddya PuMic I d m y . I%l, 1%5. by tikvf Of c w c8-- P m t r y e ~ . L. 8. (Lcv Semenovich), 1- IhU of Soviet mathmaties ISSN 0743-9199) The mathematical themy of optimal 7. (L. S. P o n m n sekcted works ; v. 4) (W i Tranlrion ok yalermtiebcska& hnik Reppint Originally published: Ncw York : Intedenct PuMirhcrs. 1962. Wilh new introd. optimal 'nykh p r m v . Bibliography: p. lncludea index. I. Mathematical optimimtiw. 1. MeurtPdt, L i e n W. 11. Titk. 111. Title OplimPl p-. IV. Setics: Pontq.pin, L S. (Lev Stmeawich), 1 4 0 8 . !kkthu. Poly@. IHS;v.4. V. Srk Clasrics of Swia mathematics. QA3.P76 1985 vol. 4 ISBN 2-88124477-1 (Switzrrbnd) IQA402.51 510 1 [519] 864732 Volume 4: ISBN 24812M7-1; Qvolume m: ISBN 248laC1344. No pad ofthis or utilimd in any form or by any means, tkaronie or mcchani- bDok m y be @wed crr by l a y idormalion stow or reprkval 4. k l w l i ~ pboloCoPYjng a d d systm, without p m h a h n in mitig fmm Iht publirhm. R i n d m G m t Britain by Bell and Bain Ltd.. (3-. g , Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence b m those they have been unable to contact.

Lev Semenovich Pontryagin

N í ilU I

Contents Editor’s Preface................................................................................ xi Preface to the English Translation................................................ xxiii Introduction...................................................................................... 1 Chapter I. The Maximum Principle.............................................. 1. Admissible Controls.............................................................. 2. Statement of the Fundamental Problem............................ 3. The Maximum Principle...................................................... 4. Discussion of the Maximum Principle.............................. 5. Examples. The Synthesis Problem...................................... 6. The Problem with Variable Endpoints and the Trans- versality Conditions...................................................... 7. The Maximum Principle for Non-Autonomous Systems. 8. Fixed Time Problems............................................................ 9. The Relation of the Maximum Principle to the Method of Dynamic Programming................................................ 9 9 11 17 21 22 45 58 66 69 Chapter II. The Proof of the Maximum Principle...................... 75 75 10. Admissible Controls.............................................................. 11. The Formulation of the Maximum Principle for an Arbi trary Class of Admissible Controls............................ 12. The System of Variational Equations and its Adjoint 83 System............................................................................ 86 13. Variations of Controls and Trajectories............................ 92 14. Fundamental Lemmas.......................................................... 15. The Proof of the Maximum Principle................................ 99 16. The Derivation of the Transversality Conditions............ 108 79 Chapter III. Linear Time-Optimal Processes.............................. 115 17. Theorems on the Number of Switchings............................ 115 18. Uniqueness Theorems.......................................................... 123 vu

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