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

• First comprehensive and systematic exposition of the properties of semiconcave functions and their various applications, particularly to optimal control problems, by leading experts in the field

• A central role in the present work is reserved for the study of singularities

• Graduate students and researchers in optimal control, the calculus of variations, and PDEs will find this book useful as a reference work on modern dynamic programming for nonlinear control systems

Textbook

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 58)

1. Front Matter
Pages i-xiii
2. Pages 1-28
3. Pages 29-47
4. Pages 49-76
5. Pages 77-96
6. Pages 97-139
7. Pages 141-183
8. Pages 185-228
9. Pages 229-271
10. Back Matter
Pages 273-304

### Introduction

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.

### Keywords

cal. variation geometric measure theory optimal control calculus convex analysis dynamic programming equation function functions Jacobi measure theory model Natural programming time

#### Authors and affiliations

1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

### Bibliographic information

• Book Title Semiconcave Functions, Hamilton—Jacobi Equations, and Optimal Control
• Authors Piermarco Cannarsa
Carlo Sinestrari
• Series Title Progress in Nonlinear Differential Equations and Their Applications
• DOI https://doi.org/10.1007/b138356
• Copyright Information Birkhäuser Boston 2004
• Publisher Name Birkhäuser Boston
• eBook Packages
• Hardcover ISBN 978-0-8176-4084-2
• Softcover ISBN 978-0-8176-4336-2
• eBook ISBN 978-0-8176-4413-0
• Edition Number 1
• Number of Pages XIV, 304
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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