Optimal ControlOptimal Control brings together many of the important advances in 'nonsmooth' optimal control over the last several decades concerning necessary conditions, minimizer regularity, and global optimality conditions associated with the Hamilton–Jacobi equation. The book is largely self-contained and incorporates numerous simplifications and unifying features for the subject’s key concepts and foundations. Features and Topics: * a comprehensive overview is provided for specialists and nonspecialists * authoritative, coherent, and accessible coverage of the role of nonsmooth analysis in investigating minimizing curves for optimal control * chapter coverage of dynamic programming and the regularity of minimizers * explains the necessary conditions for nonconvex problems This book is an excellent presentation of the foundations and applications of nonsmooth optimal control for postgraduates, researchers, and professionals in systems, control, optimization, and applied mathematics. ---------------------------------Each chapter contains a well-written introduction and notes. They include the author's deep insights on the subject matter and provide historical comments and guidance to related literature. This book may well become an important milestone in the literature of optimal control. —Mathematical Reviews This remarkable book presents Optimal Control seen as a natural development of Calculus of Variations so as to deal with the control of engineering devices...Thanks to a great effort to be self-contained, [this book] renders accessibly the subject to a wide audience. Therefore, it is recommended to all researchers and professionals interested in Optimal Control and its engineering and economic applications. It can serve as an excellent textbook for graduate courses in Optimal Control (with special emphasis on Nonsmooth Analysis). —Automatica. The book may be an essential resource for potential readers, experts in control and optimization, as well as postgraduates and applied mathematicians, and it will be valued for its accessibility and clear exposition. —Applications of Mathematics |
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Contents
Overview | 1 |
12 The Calculus of Variations | 4 |
13 Existence of Minimizers and Tonellis Direct Method | 18 |
14 Sufficient Conditions and the HamiltonJacobi Equation | 21 |
15 The Maximum Principle | 25 |
16 Dynamic Programming | 30 |
17 Nonsmoothness | 35 |
18 Nonsmooth Analysis | 39 |
The Extended EulerLagrange and Hamilton Conditions | 233 |
72 Properties of the Distance Function | 237 |
73 Necessary Conditions for a Finite Lagrangian Problem | 242 |
Nonconvex Velocity Sets | 252 |
Convex Velocity Sets | 259 |
76 Dualization of the Extended EulerLagrange Condition | 264 |
77 The Extended Hamilton Condition | 277 |
78 Notes for Chapter 7 | 280 |
19 Nonsmooth Optimal Control | 52 |
110 Epilogue | 56 |
111 Notes for Chapter 1 | 60 |
Measurable Multifunctions and Differential Inclusions | 61 |
22 Convergence of Sets | 62 |
23 Measurable Multifunctions | 64 |
24 Existence and Estimation of FTrajectories | 75 |
25 Perturbed Differential Inclusions | 86 |
26 Existence of Minimizing FTrajectories | 91 |
27 Relaxation | 94 |
28 The Generalized Bolza Problem | 100 |
29 Notes for Chapter 2 | 108 |
Variational Principles | 109 |
32 Exact Penalization | 110 |
33 Ekelands Theorem | 111 |
34 MiniMax Theorems | 115 |
35 Notes for Chapter 3 | 125 |
Nonsmooth Analysis | 127 |
42 Normal Cones | 128 |
43 Subdifferentials | 133 |
44 Difference Quotient Representations | 139 |
45 Nonsmooth Mean Value Inequalities | 144 |
46 Characterization of Limiting Subgradients | 149 |
47 Subgradients of Lipschitz Continuous Functions | 154 |
48 The Distance Function | 161 |
49 Criteria for Lipschitz Continuity | 166 |
410 Relationships Between Normal and Tangent Cones | 170 |
Subdifferential Calculus | 179 |
52 A Marginal Function Principle | 181 |
53 Partial Limiting Subgradients | 185 |
54 A Sum Rule | 187 |
55 A Nonsmooth Chain Rule | 190 |
56 Lagrange Multiplier Rules | 193 |
57 Notes for Chapters 4 and 5 | 197 |
The Maximum Principle | 201 |
62 The Maximum Principle | 203 |
63 Derivation of the Maximum Principle from the Extended Euler Condition | 208 |
64 A Smooth Maximum Principle | 214 |
65 Notes for Chapter 6 | 228 |
Necessary Conditions for Free EndTime Problems | 285 |
82 Lipschitz Time Dependence | 288 |
83 Essential Values | 295 |
84 Measurable Time Dependence | 297 |
85 Proof of Theorem 841 | 301 |
86 Proof of Theorem 842 | 310 |
87 A Free EndTime Maximum Principle | 313 |
88 Notes for Chapter 8 | 318 |
The Maximum Principle for State Constrained Problems | 321 |
92 Convergence of Measures | 324 |
93 The Maximum Principle for Problems with State Constraints | 329 |
94 Derivation of the Maximum Principle for State Constrained Problems from the EulerLagrange Condition | 334 |
95 A Smooth Maximum Principle for State Constrained Problems | 339 |
96 Notes for Chapter 9 | 359 |
Necessary Conditions for Differential Inclusion Problems with State Constraints | 361 |
102 A Finite Lagrangian Problem | 362 |
Nonconvex Velocity Sets | 368 |
Convex Velocity Sets | 375 |
105 Free Time Problems with State Constraints | 382 |
106 Nondegenerate Necessary Conditions | 387 |
107 Notes for Chapter 10 | 396 |
Regularity of Minimizers | 397 |
112 Tonelli Regularity | 403 |
113 Proof of The Generalized Tonelli Regularity Theorem | 408 |
114 Lipschitz Continuous Minimizers | 417 |
115 Autonomous Variational Problems with State Constraints | 422 |
116 Bounded Controls | 425 |
117 Lipschitz Continuous Controls | 428 |
118 Notes for Chapter 11 | 432 |
Dynamic Programming | 435 |
122 Invariance Theorems | 442 |
123 The Value Function and Generalized Solutions of the HamiltonJacobi Equation | 452 |
124 Local Verification Theorems | 465 |
125 Adjoint Arcs and Gradients of the Value Function | 474 |
126 State Constrained Problems | 483 |
127 Notes for Chapter 12 | 487 |
References | 493 |
505 | |