Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

In this review of Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control the reviewer finds a rigorous, self-contained treatment aimed at graduate students and researchers who need a deep theoretical foundation. The book's single biggest reason to buy is its comprehensive exposition of semiconcavity theory and its clear linkage to both optimal control and viscosity solutions, making it a unique reference for those developing or applying analytical methods to Hamilton-Jacobi problems.

Key Features

• Comprehensive theory: The book presents the general theory of semiconcave functions with full proofs, so readers gain a complete understanding rather than a collection of isolated results.
• Applications-focused: Later chapters apply the theory to the Bolza problem and optimal exit time problems, illustrating how abstract results inform concrete control problems.
• Self-contained prerequisites: Included material from convex analysis, nonsmooth analysis, and viscosity solutions reduces the need for additional textbooks when approaching the text.
• Illustrative examples: Significant examples accompany theory to clarify subtle points and demonstrate typical behaviors encountered in applied problems.
• Suitable for research: The level and scope make it a practical reference for mathematicians working on nonlinear differential equations and control theory.

Who It's For

The primary audience is graduate students in applied mathematics and researchers in optimal control, partial differential equations, or calculus of variations who require a rigorous source on semiconcave functions. It also suits teachers preparing advanced courses on Hamilton-Jacobi equations who want a single-volume reference for both theory and applications.

Practitioners seeking quick computational recipes or an elementary introduction should look elsewhere, since the book emphasizes rigorous development and proofs rather than algorithmic implementation or introductory exposition.

Pros & Cons

Pros

• Thorough presentation of semiconcavity that consolidates scattered literature into a unified account.
• Clear treatment of applications to the Bolza problem and optimal exit time, linking theory to classical control problems.
• Self-contained appendices on convex and nonsmooth analysis reduce external prerequisites for motivated readers.

Cons

• The text is demanding and assumes mathematical maturity, so it is not a casual or introductory read for non-specialists.

Specifications

Title	Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control
Series	Progress in Nonlinear Differential Equations and Their Applications, 58
Authors	Piermarco Cannarsa, Carlo Sinistrari
Coverage	Theory of semiconcave functions and applications to optimal control
Applications	Bolza problem; optimal exit time problems for nonlinear control systems
Includes	Prerequisites from convex analysis, nonsmooth analysis, and viscosity solutions

Our Verdict

For serious students and researchers in differential equations and control theory this book is an authoritative, good-value reference that ties abstract semiconcavity to concrete optimal control problems. Its self-contained style and breadth make it a durable addition to an academic library, though casual readers should expect a steep learning curve.

Frequently Asked Questions

Does the book require prior knowledge of convex analysis?
The book includes the necessary material on convex analysis and nonsmooth analysis, so it is largely self-contained for readers with sufficient mathematical maturity.

Are practical control algorithms covered?
The focus is theoretical: applications to control problems are analytical rather than algorithmic, so readers looking for numerical methods should consult complementary sources.

Is this suitable for a semester course?
Yes, motivated instructors can base an advanced graduate course on this text, especially for topics connecting Hamilton-Jacobi equations and optimal control.