Properties of discontinuous value functions in time-optimal differential games

Properties of discontinuous value functions in time optimal difference games, in which the payoff functional is the time spent to reach a given terminal space, are discussed. Sufficient conditions for a given discontinuous function to coincide with the value function of the game are obtained that are formulated in terms of the classical notions of u- and v- stable functions. The classical approach to solve differential games consists of finding the value function that assigns an optimal guaranteed result to each initial position of the game. If the optimal result is differentiable then the problem of finding it reduces to solving the corresponding boundary value problem for a first order partial differential equation (PDE). If the value function is differentiable, then searching it reduces to solving a boundary value problem for a first-order PDE. Discontinuous value function arises in time-optimal problems and finding it requires considering semicontinuous u- and v- stable functions.