Dynamic Optimization

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ADVANCED TEXTBOOKS IN ECONOMICS VOLUME 31 editors C.J. BLISS M.D. INTRILIGATOR Advisory editors W.A. BROCK D, W JORGENSON A P KIRMAN 1. J LAFFONT J -F. RICHARD ELSEVIER Amsterdan· Lausanne… New York· Oxford· Shannon Singapore . Tokyo DYNAMIC OPTIMIZATION The calculus of variations and Optimal control in economics and management Morton I KAMIEN an d lancy L. SCHWARTZ+ J. L. Kellogg graduate School of management Northwestern University Evanston. lllinois U.S.A Second edition ELSEVJER Amsterdam. Lausanne New York· Oxford· Shannon· Singapore· Tokyo ELSEVIER SCIENCE B V Sara Burgcrhartstraat 25 P.O. Box 211. 1000 AE Amsterdam. The Netherlands @1991 Elsevier Science B V. All rights reserved This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. 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Because of rapid advances in the medical sciences, in particular, independenr or ideas products liability, negligence or otherwise, or from any use Dr operation of any methods, products, instructions verification of diagnoses and drug dosages should be made. first edition 199 1 Fourth impression: 1998 pression: 1993 Fifth impression: 2000 ird impression: 1995 Sixth impression: 200)1 Library of Congress Cataloging in Publication Data A calalog record fruin the Library of Congress is on lile ISBN:0444,01609.0 E The paper used in this publication meets the requirements of ANSINISO Z3948-1992 (Permanence f Paper) Printed in The Netherlands INTRODUCTION TO THE SERIES The aim of the series is to cover topics in economics, mathematical economics and econometrics, at a level suitable for graduate students or final year undergraduates specializing in economics. There is at any time much material that has become well established in journal papers and discussion series which still awaits a clear, self-contained treatment that can easily be mastered by students without considerable preparation or extra reading. Leading specialists will be invited to contribute volumes to fill such gaps. Primary emphasis will placed on clarity, comprehensive coverage of sensibly defined areas, and insight into fundamentals, but original ideas will not be excluded. Certain volumes will therefore add to existing knowledge, while others will serve as a means of communicating both known and new ideas in a way that will inspire ind attract students not already familiar with the subject matter concerned Christopher 3. Bliss Michael D. Intriligato This page intentionally left blank Contents Preface to the Fourth Printing Preface to the Second edition Preface to the first edition PART I CALCULUS OF VARIATIONS Section 1. introduction Section 2. Example solved Section 3. Simplest Problem-Euler Equation Section 4. Examples and Interpretations 21 Section 5. Solving the Euler Equation in Special Cases Section 6. Second Order Conditions Section 7. Isoperimetric Problem 47 Section Free end value 52 Section 9. Free Horizon- Transversality Conditions 57 Section 10. Equality Constrained Endpoint 65 Valu 71 Section 12. Inequality constraint Endpoints and sensitivity analysis 77 Section 13, Corners 86 Section 14. Inequality Constraints in(t, x) 90 Section 15. Infinite Horizon Autonomous Problems 95 Section 16. Most Rapid Approach Paths 97 Section 17. Diagrammatic Analysis 102 Section 18. Several Functions and Double Integrals 112 Contents PART I: OPTIMAL CONTROL Section Introduction 121 Section 2. Simplest Problem-Necessary Conditions 124 Section 3. Sufficiency 133 Section 4. Interpretations 136 Section 5. Several variables 142 Section 6. Fixed Endpoint Problems 147 Section 7. Various Endpoint Conditions 155 Section 8. Discounting, Current Values, Comparative Dynamics Section 9. Equilibria in Infinite Horizon Autonomous Problems 174 Section 10. Bounded Controls 185 Section 11. further Control constrai 195 Section 12. Discontinuous and Bang-Bang Control 202 Section 13. Singular Solutions and Most rapid approach paths Section 14. The Pontryagin Maximum Principle, Existence 218 Section 15. Further Sufficiency Theorems 21 Section 16. Alternative formulations 227 Section 17. State Variable Inequality Constraints 230 Section 18. Jumps in the State Variable, Switches in State Equations Section 19. Delayed Response 248 Section 20. Optimal Control with Integral State Equations Section 21. Dynamic Programming 259 Section 22, Stochastic Optimal Control Section 23. Differential Games 272 APPENDIX A CALCULUS AND NONLINEAR PROGRAMMING Section 1. Calculus Techniques 291 Section 2. Mean-Value Theorems 294 Section 3. Concave and Convex Functions 298 Section 4. Maxima and minima 303 Section 5. Equality Constrained Optimization 307 Section 6. Inequality Constrained Optimization Section 7. Line Integrals and greens Theorem 320 Contents APPENDIX B DIFFERENTIAL EQUATIONS Section 1. Introduction 325 Section 2. Linear First Order Differential Fquations 328 Section 3. Linear Second Order Differential equations 332 Section 4. Linear nth Order Differential Equations 339 Section 5. A Pair of Linear Equations Section 6. Existence and Uniqueness of Solutions 350 R exercices 353 Author index 367 Subject Index 371

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