# Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

Springer Science & Business Media, Sep 14, 2004 - Mathematics - 304 pages

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 Copyright

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