## Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal ControlSemiconcavity 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 |

302 | |

### Other editions - View all

Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control Piermarco Cannarsa,Carlo Sinestrari No preview available - 2004 |