Mathematical Sciences: Research in Differential Games

8901462 贝尔科维茨 这项研究将主要关注 确定最佳策略-和数值 在最优策略 存在，并确定接近最佳的战略， 其他案件。 在以前的工作表明，在游戏中， Lipschitz连续值，方向导数满足 某些不平等。 在拟议的研究中， 证明了给定满足以下条件的Lipschitz连续函数 这些不等式和相同的边界条件， 值，人们可以获得最佳策略，在这种情况下， 通过考虑控制变量的值， 这些不平等的实现。 在最佳情况下， 策略不存在，接近最优的策略将 得到了 在许多情况下，构造这种函数的方法 案件将被提交。 一种用于确定值的方法， 函数由不等式的向后递归解 也将得到发展。 计算可行性，证明 收敛性和收敛速度的估计也将 被考虑。 因为价值函数是粘性的 Isaacs方程的解， 问题在手头的数值技术找到近似 一阶偏微分粘性解 方程将被研究。 的另一种方法 近似的解决方案，将被调查的是， 用一个序列来近似连续时间博弈， 无限多步对策对应的离散化 时间间隔。 这些解的收敛性 连续时间对策解的多步对策 将建立和估计的收敛速度 将获得。 一种微分对策理论，其中时间滞后 1信息也将得到发展。

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.