The principal investigator intends to develop the theory of the so-called differential games. This term refers to problems described by differential equations in which there are two or more different equations in which there are two or more different control variables, one for each decision maker or "player". For instance, competition games on stock markets or pursuit-evasion games (encountered in aerial combat) can be mathematically analyzed by using the theory of differential games. The latter represents a mixture of more traditional game theory, control and various recent mathematical theories about partial differential equations. The theory of differential games developed by the principal investigator will be used to develop methods for approximating the solutions of differential games to an arbitrary degree of accuracy. The first method to be used is that of approximating the continuous time game by a sequence of multimove infinite games corresponding to the discretization of the time interval. It will be shown that the solutions of these infinite multimove games converge to the solutions of the original game. Estimates of the rates of convergence of the values of the n-th stage games to the value of the differential game will be obtained. If the Isaacs condition does not hold then mixed strategies will be used. Methods for solving multimove infinite games will also be investigated, as the solutions of such games is an essential part of the program. For the multimove games, methods based on necessary conditions and direct backward recursions will be investigated. Another method that will be developed is one based on the numerical solution of the Isaacs equation. Since the value function is a viscosity solution of this equation, the adaptability to out problem of numerical techniques proposed in the literature will be investigated. The preceding program will first be applied to games of fixed duration and then extended to games of generalized pursuit and evasion and to games of survival. A question that arises in pursuit and evasion problems that is different from the two person zero-sum game is that of determining those initial points from which capture can be assured and those initial points from which evasion can be assured. We shall investigate this problem when both pursuer and evader are permitted to choose their actions at each instant time, knowing the previous actions of both players. This is in contrast to previous Soviet and other work in which only one of the antagonists is allowed to choose his action as play evolves. Program Director for Applied Mathematics recommends a twenty-four month award funded jointly with the Air Force Office of Scientific Research.
