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© 2004

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)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Pages 1-28
  3. Pages 29-47
  4. Pages 141-183
  5. Pages 185-228
  6. Back Matter
    Pages 273-304

About this book

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 Springer Book Archive
  • 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 Partial Differential Equations
    Measure and Integration
    Optimization
  • Buy this book on publisher's site
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Reviews

"The main purpose of this book is to provide a systematic study of the notion of semiconcave functions, as well as a presentation of mathematical fields in which this notion plays a fundamental role. Many results are extracted from articles by the authors and their collaborators, with simplified—and often new—presentation and proofs.... One of the most attractive features of this book is the interplay between several fields of mathematical analysis.... Despite the many topics addressed in the book, the required mathematical background for reading it is limited because all the necessary notions are not only recalled, but also carefully explained, and the main results proved.

The book will be found very useful by experts in nonsmooth analysis, nonlinear control theory and PDEs, in particular, as well as by advanced graduate students in this field. They will appreciate the many detailed examples, the clear proofs and the elegant style of presentation, the fairly comprehensive and up-to-date bibliography and the very pertinent historical and bibliographical comments at the end of each chapter."

—Mathematical Reviews