Cauchy Problems for Certain Isaacs-Bellman Equations and Games of Survival

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Abstract
Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set , at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of then they are continuous everywhere. To discuss continuity of the values on the boundary of certain comparison theorems for the values of the game are established. In particular if there are sub- and super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussing value functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.

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