Solving the Hamilton-Jacobi Equation

This chapter reviews some major approaches for solving the Hamilton-Jacobi equation (HJEs), and present one algorithm that may yield global solutions. It discusses some popular polynomial and Taylor-series approximation methods for solving the HJEs and examines a factorization approach which may yield exact and global solutions. It explores the extension of this approach to Hamiltonian mechanical systems. The chapter proposes some major approaches for solving the Hamilton-Jacobi-Bellman equation (HJBE) and Hamilton-Jacobi-Isaacs equation (HJIE) that have been proposed in the literature. In a recursive procedure for the HJBE for a class of nonlinear systems. It presents a factorization approach that may yield exact global solutions of the HJIE for the class of affine nonlinear systems. The chapter deals with a discussion of sufficiency conditions for the existence of exact solutions to the HJIE which are provided by the Implicit-function Theorem.