ON REGULAR DIFFERENTIAL GAMES OF PURSUIT WITH FIXED DURATION

In any differential game the programmed maxmin is a guaranteed payoff of first player. For a long time, due to the simplicity of geometric interpretation of programmed maxmin and difficulties of implementation for Isaacs' method, programmed maxmin was extensively studied. Researchers were interested in finding conditions under which programmed maxmin is the value of differential game. These conditions are called regular conditions. Differential games satisfying these conditions are called regular games. The programmed iteration method could be considered a non-smooth version of the dynamic programming method. Initially the programmed iteration method was aimed at studying non-regular differential games. Later it became obvious that the scope of application of programmed iteration method is wider. For example based on results of the programmed iteration method the theory of differential games could be built. One more example is provided in this article. Based on results of programmed iteration method, theorem on convex-concave functions and the theorem on measurable selector of multi-valued map we provide simple proof of well-known regular condition for linear differential game of approach with fixed duration.