Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

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Springer Science & Business Media, Sep 14, 2004 - Mathematics - 304 pages
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Semiconcavity is a natural generalization of concavity that retains most of the good properties known in convex analysis, but arises in a wider range of applications. This text is the first comprehensive exposition of the theory of semiconcave functions, and of the role they play in optimal control and Hamilton–Jacobi equations.

The first part covers the general theory, encompassing all key results and illustrating them with significant examples. The latter part is devoted to applications concerning the Bolza problem in the calculus of variations and optimal exit time problems for nonlinear control systems. The exposition is essentially self-contained since the book includes all prerequisites from convex analysis, nonsmooth analysis, and viscosity solutions.

A central role in the present work is reserved for the study of singularities. Singularities are first investigated for general semiconcave functions, then sharply estimated for solutions of Hamilton–Jacobi equations, and finally analyzed in connection with optimal trajectories of control systems.

Researchers in optimal control, the calculus of variations, and partial differential equations will find this book useful as a state-of-the-art reference for semiconcave functions. Graduate students will profit from this text as it provides a handy—yet rigorous—introduction to modern dynamic programming for nonlinear control systems.

 

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Contents

A Model Problem
1
11 Semiconcave functions
2
12 A problem in the calculus of variations
4
13 The Hopf formula
6
14 HamiltonJacobi equations
9
15 Method of characteristics
11
16 Semiconcavity of Hopfs solution
18
17 Semiconcavity and entropy solutions
25
55 Generalized characteristics
124
56 Examples
135
Calculus of Variations
141
61 Existence of minimizers
142
62 Necessary conditions and regularity
147
63 The problem with one free endpoint
152
64 The value function
161
65 The singular set of u
170

Semiconcave Functions
29
22 Examples
38
23 Special properties of SCLA
41
24 A differential Harnack inequality
43
25 A generalized semiconcavity estimate
45
Generalized Gradients and Semiconcavity
48
31 Generalized differentials
50
32 Directional derivatives
55
33 Superdifferential of a semiconcave function
56
34 Marginal functions
65
35 Infconvolutions
68
36 Proximal analysis and semiconcavity
73
Singularities of Semiconcave Functions
77
42 Propagation along Lipschitz arcs
84
43 Singular sets of higher dimension
88
44 Application to the distance function
94
HamiltonJacobi Equations
97
51 Method of characteristics
98
52 Viscosity solutions
105
53 Semiconcavity and viscosity
112
54 Propagation of singularities
121
66 Rectifiability of
174
Optimal Control Problems
184
71 The Mayer problem
186
72 The value function
191
73 Optimality conditions
203
74 The Bolza problem
213
Control Problems with Exit Time
229
81 Optimal control problems with exit time
230
82 Lipschitz continuity and semiconcavity
237
83 Semiconvexity results in the linear case
253
84 Optimality conditions
258
A 1 Convex sets and convex functions
273
A 2 The Legendre transform
282
A 3 Hausdorff measure and rectifiable sets
288
A 4 Ordinary differential equations
289
A 5 Setvalued analysis
292
A 6 BV functions
293
References
295
Index
302
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Page 301 - WESTPHAL U., FRERKING J., On a property of metric projections onto closed subsets of Hilbert spaces, Proc. Amer. Math. Soc. 105 (1989), 644-651.
Page 297 - On the regularity of semipermeable surfaces in control theory with application to the optimal exit-time problem, II, SIAM J. Control Optim. 35 (1997), 1653-1671. 51. CAROFF N., Caracteristiques de 1'equation d' Hamilton-Jacobi et conditions d'optimalite en controle optimal non lineaire, Ph.D.
Page 297 - CLARKE FH, LEDYAEV YS, STERN RJ, WOLENSKI PR, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Springer- Verlag, New York, 1998.
Page 297 - On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations, Indiana Univ. Math. J. 36 (1987), 501-524.

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