Mathematical Sciences: Research in Differential Games

8901462伯克维茨这项研究将主要关注最优策略的确定--以及它们的数值近似--在存在最优策略的情况下，以及在其他情况下确定近乎最优的策略。前人的工作证明了在具有Lipschitz连续值的对策中，方向导数满足一定的不等式。在所提出的研究中，将证明给定一个满足这些不等式的Lipschitz连续函数并且与该值具有相同的边界条件，在存在的情况下，可以通过考虑达到这些不等式的控制变量的值来获得最优策略。在不存在最优策略的情况下，将得到接近最优的策略。文中将给出一种在多种情况下构造这种函数的方法。还将开发一种通过向后递归解这些不等式来确定值函数的方法。还将考虑计算的可行性、收敛的证明和收敛速度的估计。由于值函数是Isaacs方程的粘性解，因此我们将研究求一阶偏微分方程粘性近似解的数值技术对手头问题的适应性。另一种将被研究的近似解的方法是通过对应于时间间隔的离散化的无限多步博弈序列来逼近连续时间博弈。建立了这些多步对策解对连续时间对策解的收敛性质，并给出了收敛速度的估计。还将发展一种存在信息时滞的微分对策理论。

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.