APPROXIMATION SCHEMES FOR CONSTRUCTING MINIMAX SOLUTIONS OF HAMILTON-JACOBI EQUATIONS

A grid algorithm is proposed for constructing the optimal guaranteed r esult function (which need not be differentiable) in control problems. Wherever it is differentiable, this function satisfies the Isaacs-Bel lman equation, which is a first-order partial differential equation of Hamilton-Jacobi type. A convergent finite-difference method is propos ed for Hamilton-Jacobi equations. Unlike the classical grid method, in which one approximates the gradients of the unknown function, which n eed not necessarily exist, this method requires the computation of sub differentials of locally convex bulls. Underlying the method is the co ncept of a generalized minimax (viscosity) solution [1-4] of the Hamil ton-Jacobi equation, with the corresponding infinitesimal construction s-directional differentials and subdifferentials-replacing the classic al derivative.