Mathematical Sciences: Stochastic Control and Nonlinear Estimation

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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.