Mathematical Sciences: Existence and Computation of Optimal Markov Controls for Adaptive Control Problems

数学科学：自适应控制问题的最优马尔可夫控制的存在性和计算

• 批准号: 9404990
• 负责人: Kurt Helmes
• 金额: $ 6万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1997-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404990&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Existence; Computation; Optimal

9404990 Helmes/Stockbridge This project is supported jointly by the Applied Mathematics Program and the Statistics and Probability Program. It goal is to provide a constructive approach to the existence and identification of Markov controls for adaptive and singular control problems. The mathematical tools developed extend and provide a new approach to the solution of stochastic control problems. The focus will be on linear programming (LP) methods which arise naturally when the objective is to optimize a long-term average criterion. These LP methods will be extended to finite horizon and infinite horizon discounted criteria. A direct characterization of the stationary distributions identifies Markov controls and allows the value to be obtained as the solution of an infinite-dimensional LP rather than as a solution to the Hamilton-Jacobi-Bellman partial differential equation. The form of the constraints leads to a natural approximation procedure which allows numerical computation of nearly optimal controls. This project is supported jointly by the Applied Mathematics Program and the Statistics and Probability Program. Its goal is the development of a new, constructive approach to the solution of adaptive and singular stochastic control problems and of approximation procedures for the numerical computation of nearly optimal controls. This research is expected to have many important applications. Two such example are the task of managing a portfolio of stocks so as to optimize some financial goal, and the problem of valuing options. The theory of option pricing is well-understood when there are no transaction costs, but existing methods are no longer appropriate when costs are part of the transaction. Other examples of adaptive control applications range from robotic manipulators to flight contpol systems of high-performance aircraft. All these systems have to operate over a wide range of working conditions, leading to great variability in the system paramete rs. The complexity of these systems requires the implementation of adaptive controllers. The practical implementation of the controls further requires sophisticated numerical methods to find nearly optimal adaptive controllers.

9404990 Helmes/Stockbridge这个项目由应用数学计划和统计和概率计划共同支持。 IT目标是为Markov控制措施的存在和识别提供一种建设性的方法，以解决适应性和奇异控制问题。 数学工具开发了扩展并为解决随机控制问题解决方案提供了新的方法。重点将放在线性编程（LP）方法上，当目标是优化长期平均标准时，自然会出现。这些LP方法将扩展到有限的地平线和无限的地平线折扣标准。固定分布的直接表征标识了马尔可夫控件，并允许获得该值作为无限二维LP的解决方案，而不是作为汉密尔顿 - 雅各布斯 - 贝尔曼部分部分微分方程的解决方案。约束的形式导致自然近似程序，该过程允许对几乎最佳控制的数值计算。 该项目由应用数学计划以及统计和概率计划共同支持。它的目标是开发一种新的，建设性的方法来解决自适应和奇异随机控制问题的解决方案以及几乎最佳控制的数值计算的近似程序。这项研究预计将有许多重要的应用。 这样的两个例子是管理股票投资组合以优化某些财务目标以及评估选项的问题。如果没有交易成本，则可以将期权定价理论理解，但是当成本是交易的一部分时，现有方法不再适合。自适应控制应用程序的其他示例范围从机器人操纵器到高性能飞机的飞行contpol系统。所有这些系统都必须在各种工作条件下运行，从而导致系统参数Rs的差异很大。 这些系统的复杂性需要实施自适应控制器。控件的实际实现进一步需要复杂的数值方法，以找到几乎最佳的自适应控制器。