Research in Differential Games

微分博弈研究

• 批准号: 8700813
• 负责人: Leonard Berkovitz
• 金额: $ 8.39万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8700813&HistoricalAwards=false
• 关键词: Research; Differential; Games

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.

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，其中有两个或更多不同的方程，其中有两个或更多不同的控制变量，每个决策者或参与者一个控制变量。例如，股票市场上的竞争博弈或空战中遇到的追逃博弈都可以用微分对策理论进行数学分析。后者代表了更传统的博弈论、控制论和各种关于偏微分方程的最新数学理论的混合。由首席研究员发展的微分对策理论将被用来开发将微分对策的解逼近到任意精度的方法。第一种方法是将连续时间博弈近似为对应于时间间隔离散化的多步无限博弈序列。证明了这些无限多步对策的解收敛于原对策的解。我们将得到第n阶段对策的值与微分对策的值的收敛速度的估计。如果艾萨克斯条件不成立，那么将使用混合策略。求解多步无限对策的方法也将被研究，因为这类对策的解是程序的基本部分。对于多步对策，将研究基于必要条件和直接向后递归的方法。另一种将被开发的方法是基于Isaacs方程的数值解的方法。由于值函数是该方程的粘性解，因此将研究文献中提出的数值技术对OUT问题的适应性。前面的程序将首先应用于固定持续时间的博弈，然后扩展到广义追逃博弈和生存博弈。与二人零和博弈不同，追逃问题中的一个问题是确定哪些初始点可以保证抓获，哪些初始点可以保证逃避。当追赶者和逃避者都被允许在每个瞬间选择他们的行动时，我们将调查这个问题，知道双方之前的行为。这与以前的苏联和其他作品形成了鲜明对比，在这些作品中，随着游戏的发展，只有一个对手被允许选择自己的动作。应用数学项目主任建议与空军科学研究办公室联合资助一项为期24个月的奖励。