Research in Differential Games

首席研究员打算发展的理论， 所谓的微分博弈。 这个术语指的是描述的问题 通过微分方程，其中有两个或多个不同的 其中存在两个或更多个不同控制变量的方程， 一个用于每个决策者或“参与者”。 例如，竞争 股票市场上的游戏或追捕-逃避游戏（在空中遇到） 战斗）可以通过使用以下理论进行数学分析： 微分对策 后者代表了更多的混合物 传统的博弈论、控制和各种最近的数学 关于偏微分方程的理论 微分对策理论是由校长 研究者将被用来开发近似的方法， 微分对策的解到任意的精确度。 第一种方法是近似连续 一个多步无限博弈序列的时间博弈， 时间间隔的离散化。 它将表明， 这些无限多步博弈的解收敛于 原始的游戏。 的收敛速度的估计 第n阶段博弈的值与微分博弈的值之比 将获得。 如果Isaacs条件不成立，则混合 战略将被使用。 多步无限对策的求解方法 也将被调查，因为这种游戏的解决方案是一个 该计划的重要组成部分。 对于多步游戏，基于 在必要条件和直接向后递归将是 研究了 将开发的另一种方法是基于 艾萨克方程的数值解 由于值 函数是这个方程的粘性解， 在文献中提出的数值技术的问题将是 研究了 前面的程序将首先应用于以下游戏： 固定持续时间，然后扩展到广义追求的游戏， 逃避和生存游戏。 在追踪和逃避问题中出现的一个问题是 与两人零和博弈不同的是， 可以确保捕获的初始点以及 可以确保规避的初始点。 我们将 研究这个问题时，双方追求者和逃避者都被允许， 选择他们的行动在每一个瞬间的时间，知道前 双方球员的行动。 这与前苏联和 其他只允许选择其中一个对手的工作 他的行为随着游戏的发展而发展。 应用数学项目主任建议24 与空军科学办公室联合资助的一个月的奖励 Research.

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.