Nonlinear Optimal Control Theory

Chapter 12

Hamilton-Jacobi Theory

12.1 Introduction

In Section 6.2 we used dynamic programming to derive the nonlinear par-

tial differential e quation (12.1.1) for the value function associated with an op-

timal control problem. This partial differential equation is called a Hamilton-

Jacobi-Bellman (HJB) equation, als o Be llman’s equatio n. Typically, the value

function W is not smooth, and (12.1.1) must be understood to hold in some

weaker sense. In particular, under suitable assumptions W satisfies (12.1.1) in

the Crandall-Lions viscosity solution sense (Section 12.5). Section 12.6 gives

an alternate characterization (12.6.2) of the value function