On Piecewise Linear Minimax Solution of Hamilton–Jacobi Equation with Nonhomogeneous Hamiltonian

Abstract

Terminal Cauchy problem for nonhomogeneous Hamilton–Jacobi equation is considered in the case when state space is Euclidean plane. The Hamiltonian and the terminal function are piecewise linear. This problem reduces to a problem with a homogeneous Hamiltonian in three-dimensional state space. A finite algorithm for the exact construction of the minimax and/or viscosity solution is developed. The algorithm consists of a finite number of consecutive stages, at each of which elementary problems of several types are solved and the continuous gluing of these solutions are carried out. The solution built by the algorithm is a piecewise linear function. Cases are also indicated when the original problem with a nonhomogeneous Hamiltonian can be solved on a plane without moving to the problem in three-dimensional space.