Mathematical Sciences: Stochastic Control and Nonlinear Estimation

本课题涉及最优随机控制和非线性估计等几个课题。主要研究的是风险敏感最优控制。风险敏感最优控制提供了随机和确定性（鲁棒控制）方法之间的联系，以模拟控制系统中的干扰。两个参与者，零和微分博弈在分析中自然出现。风险敏感控制还涉及随机过程的大偏差理论。所使用的方法包括非线性偏微分方程的粘度解技术。在金融理论的应用中提出的其他类型的随机控制问题将被研究。在非线性估计中，将研究图像分析应用和隐马尔可夫模型中出现的部分观测随机场问题。数学控制理论和控制应用中的一个主要问题是控制策略的设计对控制系统中可能出现的不利干扰不敏感。对于可以建模为线性控制系统的植物，有几种有效的策略可以在系统扰动存在的情况下获得良好的系统性能，并清楚地了解主要方法之间的联系。然而，许多（如果不是大多数的话）实际的控制系统，例如高性能飞机所需的控制系统，只是很差，或者根本没有建模为线性控制系统。本项目旨在开发非线性随机控制系统鲁棒控制的方法及其相互联系，类似于线性随机控制系统。

This project involves several topics in optimal stochastic control and nonlinear estimation. The main topic is risk sensitive optimal control. Risk sensitive optimal control provides a link between stochastic and deterministic (robust control) approaches to modelling disturbances in control systems. Two player, zero-sum differential games arise naturally in the analysis. Risk sensitive control also involves large deviations theory for stochastic processes. The methods to be utilized include viscosity solution techniques for nonlinear partial differential equations. Other classes of stochastic control problems suggested by application in finance theory will be studied. In nonlinear estimation, some problems of partially observed random fields arising in image analysis applications, and hidden Markov models, will be investigated. One of the main issues in mathematical control theory and in control applications is the design of control strategies which are insensitive to unfavorable disturbances which may occur within the control system. For plants which can be modelled as linear control systems, there are several effective strategies for obtaining good system performance in the presence of system perturbations and a clear understanding of the connections between the main methods. However many, if not most, practical control systems, such as those required on high performance aircraft, are only poorly, or not at all, modelled as linear control systems. This project seeks to develop methodologies and their interconnections for robust control of nonlinear stochastic control systems analogous to those for linear stochastic control systems.