Differential games with Lipschitz control functions and fixed initial control positions

AbstractDifferential games in which one or both players are restricted to choosing control functions which are uniformly Lipschitz continuous and which start at fixed initial conditions always have a value. We derive the Hamilton-Jacobi equation which this value satisfies a.e. as a function of the initial time t, the initial state x, and the initial control positions. We also show that a “Lipschitz Game” has an approximate saddle point in pure strategies. The approach of Friedman to differential games is used