Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

by Cannarsa, Piermarco; Sinestrari, Carlo

• ISBN13: 9780817643362
• ISBN10: 0817643362
• Format: Paperback
• Copyright: 2004-09-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	vii
A Model Problem	1	(28)
Semiconcave functions	2	(2)
A problem in the calculus of variations	4	(2)
The Hopf formula	6	(3)
Hamilton-Jacobi equations	9	(2)
Method of characteristics	11	(7)
Semiconcavity of Hopf's solution	18	(7)
Semiconcavity and entropy solutions	25	(4)
Semiconcave Functions	29	(20)
Definition and basic properties	29	(9)
Examples	38	(3)
Special properties of SCL (A)	41	(2)
A differential Harnack inequality	43	(2)
A generalized semiconcavity estimate	45	(4)
Generalized Gradients and Semiconcavity	49	(28)
Generalized differentials	50	(5)
Directional derivatives	55	(1)
Superdifferential of a semiconcave function	56	(9)
Marginal functions	65	(3)
Inf-convolutions	68	(5)
Proximal analysis and semiconcavity	73	(4)
Singularities of Semiconcave Functions	77	(20)
Rectifiability of the singular sets	77	(7)
Propagation along Lipschitz arcs	84	(4)
Singular sets of higher dimension	88	(6)
Application to the distance function	94	(3)
Hamilton--Jacobi Equations	97	(44)
Method of characteristics	98	(7)
Viscosity solutions	105	(7)
Semiconcavity and viscosity	112	(9)
Propagation of singularities	121	(3)
Generalized characteristics	124	(11)
Examples	135	(6)
Calculus of Variations	141	(44)
Existence of minimizers	142	(5)
Necessary conditions and regularity	147	(5)
The problem with one free endpoint	152	(9)
The value function	161	(9)
The singular set of u	170	(4)
Rectifiability of Σ	174	(11)
Optimal Control Problems	185	(44)
The Mayer problem	186	(5)
The value function	191	(12)
Optimality conditions	203	(10)
The Bolza problem	213	(16)
Control Problems with Exit Time	229	(44)
Optimal control problems with exit time	230	(7)
Lipschitz continuity and semiconcavity	237	(16)
Semiconvexity results in the linear case	253	(5)
Optimality conditions	258	(15)
Appendix	273	(22)
A.1 Convex sets and convex functions	273	(9)
A.2 The Legendre transform	282	(6)
A.3 Hausdorff measure and rectifiable sets	288	(1)
A.4 Ordinary differential equations	289	(3)
A.5 Set-valued analysis	292	(1)
A.6 BV functions	293	(2)
References	295	(8)
Index	303