Mathematical Sciences: Research in Differential Games

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8901462 Berkovitz This research will be concerned primarily with the determination of optimal strategies - and numerical approximations to them - in cases where optimal strategies exist and with the determination of near-optimal strategies in other cases. In previous work it was shown that in games with Lipschitz continuous value, the directional derivates satisfy certain inequalities. In the proposed research it will be shown that given a Lipschitz continuous function that satisfies these inequalities and the same boundary conditions as the value, one can obtain optimal strategies, in cases where they exist, by considering the values of the control variable at which the inequalities are achieved. In cases where optimal strategies do not exist, near-optimal strategies will be obtained. A method for constructing such a function in many cases will be given. A method for determining the value function by a backward recursive solution of the inequalities will also be developed. Computational feasibility, proof of convergence and estimates of the rate of convergence will also be considered. Since the value function is a viscosity solution of the Isaacs equation, the adaptability to the problem at hand of numerical techniques for finding approximate viscosity solutions of first order partial differential equations will be investigated. Another method of approximating solutions that will be investigated is that of approximating the continuous time game by a sequence of infinite multi-move games corresponding to discretization of the time interval. The convergence of the solutions of these multi-move games to the solutions of the continous time game will be established and estimates of the rate of convergence will be obtained. A theory of differential games in which a time lag in 1nformation is present will also be developed.