PART I. CALCULUS OF VARIATIONS
Section 1. Introduction 3
Section 2. Example Solved 12
Section 3. Simplest Problem—Euler Equation 14
Section 4. Examples and Interpretations 21
Section 5. Solving the Euler Equation in Special Cases 30
Section 6. Second Order Conditions 41
Section 7. Isoperimetric Problem 47
Section 8. Free End Value 52
Section 9. Free Horizon—Transversality Conditions 57
Section 10. Equality Constrained Endpoint 65
Section 11. Salvage Value 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
Section 17. Diagrammatic Analysis
Section 18. Several Functions and Double Integrals
PART II: OPTIMAL CONTROL
Section 1. 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 164
Section 9. Equilibria in Infinite Horizon Autonomous Problems 174
Section 10. Bounded Controls 185
Section 11. Further Control Constraint 195
Section 12. Discontinuous and Bang-Bang Control 202
Section 13. Singular Solutions and Most Rapid Approach Paths 209
Section 14. The Pontryagin Maximum Principle, Existence 218
Section 15. Further Sufficiency Theorems 221
Section 16. Alternative Formulations 227
Section 17. State Variable Inequality Constraints 230
Section 18. Jumps in the State Variable, Switches in State Equations 240
Section 19. Delayed Response 248
Section 20. Optimal Control with Integral State Equations 253
Section 21. Dynamic Programming 259
Section 22. Stochastic Optimal Control 264
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 313
Section 7. Line Integrals and Green's Theorem 320
APPENDIX B. DIFFERENTIAL EQUATIONS
Section 1. Introduction 325
Section 2. Linear First Order Differential Equations 328
Section 3. Linear Second Order Differential Equations 332
Section 4. Linear nth Order Differential Equations 339
Section 5. A Pair of Linear Equations 344
Section 6. Existence and Uniqueness of Solutions 350
References 353
Author Index 367
Subject Index 371
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