Optimal Control

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Springer Science & Business Media, 2000 - Mathematics - 507 pages
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Optimal Controlbrings 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 application of nonsmooth optimal control for postgraduates, researchers, and professionals in systems, control, optimization, and applied mathematics.

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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
Index
505
Copyright

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About the author (2000)

Richard Vinter is Head of the Control and Power Research Group at Imperial College London.

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