Approximation of minimax solutions to Hamilton-Jacobi functional equations for delay systems

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 questions of the correct transition from the infinite-dimensional functional argument of the desired solution to the finite-dimensional one. 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 ordinary Hamilton-Jacobi equations with partial derivatives 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.