Book Review: Generalized solutions of Hamilton-Jacobi equations

Abstract

Introduction. The Hamilton-Jacobi equation is probably known to most engineers and physicists as a partial differential equation which pops up in the study of (Lagrangian or Hamiltonian) mechanics, yielding solutions of a system of ordinary differential equations, as its characteristics, after a varia-tional procedure is used. It is also known, again through its relation to the calculus of variations, to people studying control theory, differential games, or other optimization problems, although it is sometimes referred to as the " Bellman equation" in these contexts. The last thirty years has seen the rise of a new interest in the Hamilton-Jacobi equation. With the rise of computers and new numerical techniques, the failure of classical smooth solutions to describe physical situations except in limited (local) domains, and the needs of mathematical modeling, aerospace engineering , and other applications to have solutions described everywhere, many mathematicians have become interested in global solutions (whatever that means). As nearly the most general first order partial differential equation, and as an equation for which global results were possible, the Hamilton-Jacobi equation became a natural target for mathematicians studying global solutions. In order to clarify the object of interest a little better, let us define the Hamilton-Jacobi equation. In its most familiar classical form, the Hamilton-Jacobi equation is