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Originally published in the series: Systems & Control: Foundations & Applications
1st ed. 1997. 2nd printing, 1997, XX, 570 p. 17 illus., Softcover
ISBN: 978-0-8176-4754-4
A Birkh?user Boston product
Contents
Preface xi
Basic notations xv
Chapter I. Outline of the main ideas on a model problem 1
1. The infinite horizon discounted regulator 1
2. The Dynamic Programming Principle 2
3. The Hamilton-Jacobi-Bellman equation in the viscosity sense . . . . 3
4. Comparison, uniqueness and stability of viscosity solutions 6
5. Synthesis of optimal controls and verification theorems 10
6. Pontryagin Maximum Principle as a necessary and sufficient
condition of optimality 13
7. Discrete time Dynamic Programming and convergence of
approximations 17
8. The viscosity approximation and stochastic control 20
9. Bibliographical notes 21
Chapter II. Continuous viscosity solutions of Hamilton-Jacobi
equations 25
1. Definitions and basic properties , 25
2. Some calculus and further properties of viscosity solutions 34
3. Some comparison and uniqueness results 50
4. Lipschitz continuity and semiconcavity 61
4.1. Lipschitz continuity 62
4.2. Semiconcavity 65
5. Special results for convex Hamiltonians 77
5.1. Semiconcave generalized solutions and bilateral supersolutions 77
5.2. Differentiability of solutions . . ._ 80
5.3. A comparison theorem 82
5.4. Solutions in the extended sense 84
vi CONTENTS
5.5. Differential inequalities in the viscosity sense 86
5.6. Monotonicity of value functions along trajectories 90
6. Bibliographical notes 95
Chapter III. Optimal control problems with continuous value
functions: unrestricted state space 97
1. The controlled dynamical system 97
2. The infinite horizon problem 99
2.1. Dynamic Programming and the Hamilton-Jacobi-Bellman
equation 99
2.2. Some simple applications: verification theorems, relaxation,
stability 110
2.3. Backward Dynamic Programming, sub- and
superoptimality principles, bilateral solutions 119
2.4. Generalized directional derivatives and equivalent notions
of solution 125
2.5. Necessary and sufficient conditions of optimality, minimum
principles, and multivalued optimal feedbacks 133
3. The finite horizon problem 147
3.1. The HJB equation 147
3.2. Local comparison and unbounded value functions 156
3.3. Equivalent notions of solution 160
3.4. Necessary and sufficient conditions of optimality and
the Pontryagin Maximum Principle 170
4. Problems whose HJB equation is a variational or
quasivariational inequality 183
4.1. The monotone control problem 184
4.2. Optimal stopping 193
4.3. Impulse control 200
4.4. Optimal switching 210
5. Appendix: Some results on ordinary differential equations 218
6. Bibliographical notes 223
Chapter IV. Optimal control problems with continuous
value functions: restricted state space 227
1. Small-time controllability and minimal time functions 227
2. HJB equations and boundary value problems for the minimal
time function: basic theory 239
3. Problems with exit times and non-zero terminal cost 247
3.1. Compatible terminal cost and continuity of the value function 248
3.2. The HJB equation and a superoptimality principle 254
4. Free boundaries and local comparison results for undiscounted
problems with exit times 261
5. Problems with state constraints 271
6. Bibliographical notes 282
CONTENTS vii
Chapter V. Discontinuous viscosity solutions and applications 285
1. Semicontinuous sub- and supersolutions, weak limits, and stability . 286
2. Non-continuous solutions 295
2.1. Definitions, basic properties, and examples 295
2.2. Existence of solutions by Perron's method 302
3. Envelope solutions of Dirichlet problems 308
3.1. Existence and uniqueness of e-solutions 309
3.2. Time-optimal problems lacking controllability 313
4. Boundary conditions in the viscosity sense 316
4.1. Motivations and basic properties 317
4.2. Comparison results and applications to exit-time problems
and stability 326
4.3. Uniqueness and complete solution for time-optimal control . . 333
5. Bilateral supersolutions 342
5.1. Problems with exit times and general targets 342
5.2. Finite horizon problems with constraints on the endpoint
of the trajectories ?'. 348
6. Bibliographical notes 357
Chapter VI. Approximation and perturbation problems 359
1. Semidiscrete approximation and e-optimal feedbacks 359
1.1. Approximation of the value function and construction
of optimal controls 360
1.2. A first result on the rate of convergence 369
1.3. Improving the rate of convergence 372
2. Regular perturbations 376
3. Stochastic control with small noise and vanishing viscosity 383
4. Appendix: Dynamic Programming for Discrete Time Systems . . . 388
5. Bibliographical notes 395
Chapter VII. Asymptotic problems 397
1. Ergodic problems 397
1.1. Vanishing discount in the state constrained problem 397
1.2. Vanishing discount in the unconstrained case:
optimal stopping 402
2. Vanishing switching costs 411
3. Penalization 414
3.1. Penalization of stopping costs 414
3.2. Penalization of state constraints 417
4. Singular perturbation problems 420
4.1. The infinite horizon problem for systems with fast
components 420
4.2. Asymptotics for the monotone control problem 426
5. Bibliographical notes 429
viii CONTENTS
Chapter VIII. Differential Games 431
1. Dynamic Programming for lower and upper values 433
2. Existence of a value, relaxation, verification theorems 442
3. Comparison with other information patterns and other
notions of value 448
3.1. Feedback strategies 448
3.2. Approximation by discrete time games 457
4. Bibliographical notes 468
Appendix A. Numerical Solution of Dynamic Programming
Equations by Maurizio Falcone 471
1. The infinite horizon problem 472
1.1. The Dynamic Programming equation 473
1.2. Synthesis of feedback controls 478
1.3. Numerical tests 481
2. Problems with state constraints 484
3. Minimum time problems and pursuit-evasion games . . .y ' . . . . . 488
3.1. Time-optimal control 488
3.2. Pursuit-evasion games 494
3.3. Numerical tests 496
4. Some hints for the construction of the algorithms 499
5. Bibliographical notes 502
Appendix B. Nonlinear Hoo control by Pierpaolo Soravia 505
1. Definitions 506
2. Linear systems 510
3. Woo control and differential games 512
4. Dynamic Programming equation 515
5. On the partial information problem 521
6. Solving the problem 527
7. Exercises 530
8r Bibliographical notes 531
Bibliography 533
Index 565
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