Studies in Adaptive and Optimal Control of Stochastic Systems

科学和工程中的大量过程和物理现象必须与数据测量中的不确定性或驱动系统的力相抗衡。这种情况由随机系统描述，其中对系统中未知参数和强迫的随机性质作出适当的假设。通常，人们对通过控制参数或以优化给定“成本”函数为目标的强制来控制系统感兴趣。例如，在金融的背景下，净资产是用一个随机微分方程来描述的，控制是放置在各种资产中的份额，目标是优化净资产的函数。本研究的目的是研究随机系统的最优控制和自适应控制。这类问题的解决需要参数辨识和确定显式最优控制的算法。在这个项目中使用的噪声过程是经验确定的，属于一类分数布朗运动。除了研究随机最优和自适应控制问题外，还将研究一些二人零和随机微分对策问题。这些游戏可以模拟两个相互竞争的玩家或利益集团的许多情况，所以它们经常出现。这些随机问题在控制和微分对策将研究连续和离散时间系统。该提案还将吸引本科生和研究生以及高中生参与随机问题的研究。各种系统、成本函数和噪声过程的随机最优控制问题将被研究，特别强调获得明确的最优控制。提议者已经开发了一种不需要求解Hamilton-Jacobi-Bellman方程或使用随机极大原理的方法，并且可以应用于具有非马尔可夫或半鞅的一般噪声过程的系统。系统包括离散和连续时间系统，线性和非线性系统。二人零和随机微分对策也将研究线性和非线性方程，以确定两个参与者的显式最优控制策略。由于随机系统通常包含未知参数，自适应控制的问题，包含参数的同时识别和系统的控制，将研究线性系统的遍历（长期平均）二次和指数二次成本函数和噪声过程是分数布朗运动或其他过程的平稳增量。所研究的随机控制系统既有有限维的，也有无限维的。在希尔伯特空间中演化的无限维系统可以模拟具有分数阶布朗运动噪声的抛物型和双曲型随机偏微分方程。控制和噪声可以限制在域的边界上，也可以限制在域中的离散点上。

A large number of processes and physical phenomena in science and engineering have to contend with uncertainties in the measurements of the data or the forces that drive the system. Such situations are described by stochastic systems, where suitable assumptions are made about the stochastic nature of the unknown parameters and forcing in the system. Frequently, one is interested in controlling the system through control parameters or forcing with the goal of optimizing a given "cost" functional. For example, in the context of finance, the net worth is described by an stochastic differential equation, the controls are the shares placed in the various assets, and the goal is to optimize a functional of the net worth. The goal of this research is the study of optimal control and adaptive control of stochastic systems. The solutions of such problems require algorithms for parameter identification and the determination of explicit optimal controls. The noise processes used in this project are empirically determined and belong to a class of fractional Brownian motions. In addition to the investigation of problems of stochastic optimal and adaptive control some problems of two-person zero-sum stochastic differential games will be studied. These games can model many situations of two competing players or interest groups so they often arise. These stochastic problems in control and differential games will be studied for both continuous and discrete time systems. The proposal will also engage undergraduate and graduate students, as well as high school students, in research on stochastic problems.Stochastic optimal control problems for a variety of systems, cost functionals, and noise processes will be studied with the particular emphasis on obtaining explicit optimal controls. The proposers have developed a method that does not require solving Hamilton-Jacobi-Bellman equations or using a stochastic maximum principle and can be applied to systems with general noise processes that are not Markov or semimartingales. The systems include both discrete and continuous time and linear and nonlinear systems. Two person zero sum stochastic differential games will also be investigated for both linear and nonlinear equations to determine explicit optimal control strategies for the two players by a direct method. Since stochastic systems often contain unknown parameters, the problems of adaptive control, which connote the simultaneous identification of parameters and the control of the system will be investigated for linear systems with ergodic (long run average) quadratic and exponential quadratic cost functionals and noise processes that are fractional Brownian motions or other processes with stationary increments. The stochastic systems for control to be studied are both finite dimensional and infinite dimensional. The infinite dimensional systems that evolve in Hilbert spaces can model both parabolic and hyperbolic stochastic partial differential equations with fractional Brownian motion noise. The control and the noise can be restricted to the boundary of the domain or to discrete points in the domain.