Approximation of Minimax Solutions of Hamilton—Jacobi Functional Equations for Time-Delay Systems

Abstract

A minimax solution of the Cauchy problem for a functional Hamilton-Jacobi equation with coinvariant derivatives and a condition at the right end is considered. Hamilton-Jacobi equations of this type arise in dynamical optimization problems for time-delay systems. Their approximation is associated with additional issues of a valid transition from an infinite-dimensional functional argument of the desired solution to a finite-dimensional argument. Earlier, the schemes based on the piecewise linear approximation of the functional argument and the correctness properties of minimax solutions were studied. In this paper, a scheme for the approximation of Hamilton-Jacobi functional equations with coinvariant derivatives by usual Hamilton-Jacobi partial differential equations is proposed and justified. The scheme is based on the approximation of the characteristic functional-differential inclusions used in the definition of the desired minimax solution by ordinary differential inclusions.