Semiconcave functions, Hamilton-Jacobi equations, and optimal control.

*(English)*Zbl 1095.49003
Progress in Nonlinear Differential Equations and their Applications 58. Boston, MA: Birkhäuser (ISBN 0-8176-4084-3/hbk; 0-8176-4336-2/pbk). xii, 304 p. (2004).

This book is the first systematic and comprehensive study of the theory of semiconcave functions. The book consists of 6 Chapters and an Appendix. The Chapter 1 is introductory to the whole text. There is studied a model problem from the calculus variations: this allows to present, in a simple situation, some of the main ideas, that will be developed in the rest of the book. In the Chapters 2, 3 and 4 the theory of semiconcave functions without aiming at specific application is developed. In the Chapters 5,6,7 and 8 the authors discuss contexts in which semiconcavity plays an important role such as Hamilton-Jacobi equation, calculus of variations, optimal control theory. In the Appendix all the needed definitions and most proofs of the basic results are presented. 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.

Reviewer: A. A. Tolstonogov (Irkutsk)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49K20 | Optimality conditions for problems involving partial differential equations |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

26B25 | Convexity of real functions of several variables, generalizations |

35F20 | Nonlinear first-order PDEs |

49L20 | Dynamic programming in optimal control and differential games |