Semiconcave Functions, Hamilton—Jacobi Equations, and Optimal Control

  • Piermarco Cannarsa
  • Carlo Sinestrari

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


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.


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

  • Piermarco Cannarsa
    • 1
  • Carlo Sinestrari
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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