L. S. PONTRYAGIN SELECTED WORKS Volume 4 The Mathematical Theory of Optimal Processes Classics of 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 of 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 of Optimal Processes L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, 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 informa business 0 1986 by Gordon and Bmch ScimP ublishtrs 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 s Ftrac~ 1C9 Okubo 34omt Shiajuku-ku, Tokyo la Jam Originally plblishcd m Russh as Mamrmwcnmn r.rrpnn omwvlnrnhlx n ~ e o bmy Ukl'gvn Nulltr. M-. I%l, Firsl pblirlmd in EngliP by Inkmcitaee R1Mi. divirion of John Wiky & W.In c. @ t%2 by JohFn W. iey & Soar. Inc. Thad Mhg. -bet 1%5. Repainlad. by by Gwdon md Bmch Scicaoc Publihhen S.A.. 1986. flepnntcd from Ic opy m the c a l mo f Ihe Bmddya PuMic Idmy. tikvf c w Of --8c IhU Pmtrye~L.. 8. (Lcv Semenovich), -1 The mathematical themy of optimal 7. (L. S. Ponmns ekcted works ;v . 4) (Wi of Soviet mathmaties ISSN 0743-9199) Tranlrion ok yalermtiebcska& hnik optimal 'nykh p r m v . Reppint Originally published: Ncw York : Intedenct PuMirhcrs. 1962. Wilh new introd. Bibliography: p. lncludea index. I. Mathematical optimimtiw. 1. MeurtPdt, Lien W. 11. Titk. 111. Title OplimPl p-. IV. Setic.s : Pontq.pin, L S. (Lev Stmeawich), 1408 !kkthu. Poly@. IHS;v.4. V. Srk Clasrics of Swia mathematics. QA3.P76 1985 vol. 4 IQA402.51 510 1 [519] 864732 ISBN 2-88124477-1 (Switzrrbnd) Volume 4: ISBN 24812M7-1; Qvolume m: ISBN 248laC1344. No pad ofthis bDok m ybe @wed or utilimd in any form or by any means, tkaronie or mcchani- 4.k l w l ip~bo loCoPYjng a dd g ,crr by lay idormalion stow or reprkval systm, without pmhahn in mitig fmm Iht publirhm. Rind m Gmt Britain by Bell and Bain Ltd.. (3-. Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence b mt hose they have been unable to contact. Lev Semenovich Pontryagin N í I il U Contents Editor’s Preface................................................................................ xi Preface to the English Translation................................................ xxiii Introduction...................................................................................... 1 Chapter I. The Maximum Principle.............................................. 9 1. Admissible Controls.............................................................. 9 2. Statement of the Fundamental Problem............................ 11 3. The Maximum Principle...................................................... 17 4. Discussion of the Maximum Principle.............................. 21 5. Examples. The Synthesis Problem...................................... 22 6. The Problem with Variable Endpoints and the Trans- versality Conditions...................................................... 45 7. The Maximum Principle for Non-Autonomous Systems. 58 8. Fixed Time Problems............................................................ 66 9. The Relation of the Maximum Principle to the Method of Dynamic Programming................................................ 69 Chapter II. The Proof of the Maximum Principle...................... 75 10. Admissible Controls.............................................................. 75 11. The Formulation of the Maximum Principle for an Arbi­ trary Class of Admissible Controls............................ 79 12. The System of Variational Equations and its Adjoint System............................................................................ 83 13. Variations of Controls and Trajectories............................ 86 14. Fundamental Lemmas.......................................................... 92 15. The Proof of the Maximum Principle................................ 99 16. The Derivation of the Transversality Conditions............ 108 Chapter III. Linear Time-Optimal Processes.............................. 115 17. Theorems on the Number of Switchings............................ 115 18. Uniqueness Theorems.......................................................... 123 vu viii CONTENTS 19. Existence Theorems.............................................................. 127 20. The Synthesis of the Optimal Control................................ 135 21. Examples................................................................................ 140 22. A Simulation of Linear Time-Optimal Processes by Means of Relay Circuits............................................................ 172 23. Linear Equations with Variable Coefficients.................... 181 Chapter IV. Miscellaneous Problems............................................ 189 24. The Case Where the Functional is Given by an Improper Integral............................................................................ 189 25. Optimal Processes with Parameters.................................... 191 26. An Application of the Theory of Optimal Processes to Problems in the Approximation of Functions.......... 197 27. Optimal Processes with a Delay.......................................... 213 28. A Pursuit Problem................................................................ 226 Chapter V. The Maximum Principle and the Calculus of Variations 239 29. The Fundamental Problem of the Calculus of Variations 240 30. The Problem of Lagrange.................................................... 248 Chapter VI. Optimal Processes with Restricted Phase Coordinates 257 31. Statement of the Problem.................................................... 258 32. Optimal Trajectories Which Lie on the Boundary of the Region............................................................................ 264 33. The Proof of Theorem 22 (Fundamental Constructions). 270 34. The Proof of Theorem 22 (Conclusion)............................ 291 35. Some Generalizations.......................................................... 298 36. The Jump Condition............................................................ 300 37. Statement of the Fundamental Result. Examples............ 311 Chapter VII. A Statistical Optimal Control Problem................ 317 38. The Concept of a Markov Process. The Kolmogorov Differential Equation.................................................... 318 CONTENTS ix 39. The Precise Statement of the Statistical Problem.............. 322 40. The Reduction of the Evaluation of the Functional J to the Solution of a Boundary Value Problem for the Kolmogorov Equation.................................................. 324 41. The Evaluation of the Functional J in the Case Where the Kolmogorov Equation has Constant Coefficients ... 327 42. The Evaluation of the Functional J in the General Case 348 References.......................................................................................... 354 Index.................................................................................................. 357