A piecewise linear value function for a differential game with simple motions

This article investigates the value function of a differential game. It is known that in a domain where the value function is differentiable, it satisfies a first-order partial differential equation called the fundamental equation of the theory of differential games, or the Bellman-Isaacs equation. The investigation of the value function is closely connected with the development of the general theory of Hamilton-Jacobi equations, since under sufficiently general assumptions the Hamilton-Jacobi equation coincides with the Bellman-Isaacs equation for a suitably chosen differential game. Interest in these equations has intensified in recent years, due to the development of a new approach which essentially amounts to replacement of the original equation by a pair of differential inequalities. It is known that the Hamilton-Jacobi equation and, in particular, the Bellman-Isaacs equation do not, as a rule, have classical solutions. In the framework of the new approach a definition is proposed for a solution that exists, is unique, and coincides with the value function of the corresponding differential game.