Research in Differential Games

首席研究员打算发展所谓的微分对策理论。这个术语指的是由微分方程描述的问题，其中有两个或两个以上不同的方程，其中有两个或两个以上不同的控制变量，每个决策者或“玩家”一个。例如，股票市场上的竞争博弈或追逐逃避博弈（在空战中遇到）可以使用微分博弈理论进行数学分析。后者代表了更传统的博弈论，控制和各种关于偏微分方程的最新数学理论的混合。由首席研究员开发的微分博弈理论将用于开发近似微分博弈解的方法，以达到任意程度的精度。第一种方法是用与时间间隔离散化相对应的多步无限对策序列来逼近连续时间对策。将证明这些无限多步对策的解收敛于原对策的解。将得到第n阶段对策的值收敛到微分对策值的速率的估计。如果艾萨克条件不成立，则使用混合策略。解决多步无限游戏的方法也将被调查，因为这些游戏的解决方案是程序的重要组成部分。对于多步棋，基于必要条件和直接反向递归的方法将被研究。另一种将被开发的方法是基于艾萨克斯方程的数值解。由于值函数是该方程的黏性解，因此将研究文献中提出的数值技术对我们问题的适应性。前面的程序将首先应用于固定时间的游戏，然后扩展到一般的追逐和逃避游戏和生存游戏。在追逐和逃避问题中出现的一个问题与两人零和博弈不同，即确定哪些初始点可以确保捕获，哪些初始点可以确保逃避。我们将研究当追逐者和逃避者都被允许在每个瞬间选择他们的行动，并且知道双方之前的行动。这与之前的苏联和其他作品形成鲜明对比，在这些作品中，随着游戏的发展，只有一个对手可以选择自己的行动。应用数学项目主任建议与空军科学研究办公室联合资助为期24个月的奖项。

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.