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

9404990赫尔姆斯/斯托克布里奇该项目由应用数学项目和统计与概率项目共同支持。它的目的是为自适应和奇异控制问题的马尔可夫控制的存在和辨识提供一种建设性的方法。所发展的数学工具扩展并提供了一种解决随机控制问题的新方法。重点将放在线性规划(LP)方法上，当目标是优化长期平均标准时，这些方法自然会出现。将这些线性规划方法推广到有限期限和无限期限贴现准则。对平稳分布的直接刻画可以识别马尔可夫控制，并允许将该值作为无限维线性规划的解而不是作为Hamilton-Jacobi-Bellman偏微分方程解来获得。约束的形式导致了一种自然的近似过程，该过程允许对接近最优的控制进行数值计算。该项目由应用数学计划和统计与概率计划联合资助。它的目标是发展一种新的、建设性的方法来解决自适应和奇异随机控制问题，以及用于近最优控制的数值计算的近似过程。这项研究有望有许多重要的应用。两个这样的例子是管理股票投资组合以优化某些财务目标的任务，以及期权的估值问题。当不存在交易成本时，期权定价理论是众所周知的，但当成本是交易的一部分时，现有的方法不再适用。自适应控制应用的其他例子包括从机器人机械手到高性能飞机的飞行控制系统。所有这些系统都必须在广泛工作条件下运行，导致系统参数Rs变化很大。这些系统的复杂性要求实现自适应控制器。控制的实际实现还需要复杂的数值方法来寻找接近最优的自适应控制器。

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.