Piecewise Linear Price Function of a Differential Game with Simple Dynamics and Integral Terminal Price Functional

Abstract

In this paper, we consider an antagonistic differential game of two persons with dynamics described by a differential equation with simple motions and an integral terminal payment functional. In this game, there exists a price function, which is a generalized (minimax or viscous) solution of the corresponding Hamilton–Jacobi equation. For the case where the terminal function and the Hamiltonian are piecewise linear and the dimension of the phase space is equal to 2, we propose a finite algorithm for the exact construction of the price function. The algorithm consists of the sequential solution of elementary problems arising in a certain order. The piecewise linear price function of a differential game is constructed by gluing piecewise linear solutions of elementary problems. Structural matrices are a convenient tool of representing such functions. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.