Stochastic Systems and Control

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0204669DuncanThe major focus of this research is the application of a stochastic calculus for a fractional Brownian motion to stochastic systems, especially for the problems of identification and control. The importance of a fractional Brownian motion in stochastic models has been exhibited in a wide variety of applications, such as, hydrology, economics and telecommunications. One goal of the research is to develop further a stochastic calculus for a fractional Brownian motion similar to the way that the stochastic calculus for Brownian motion was developed to provide the tools for solving problems of stochastic systems with a fractional Brownian motion. Some tools of stochastic calculus that are planned for development are necessary for solving the problems of identification, filtering and control of a stochastic system with a fractional Brownian motion. The fractional Brownian motions for study include processes with values in both finite and infinite dimensional spaces. The study includes the existence and the uniqueness of solutions of stochastic differential equations and stochastic partial differential equations with a fractional Brownian motion and martingales that are formed from a fractional Brownian motion. Another family of stochastic models that are planned for study is hidden Markov models. An investigation of adaptive control for systems described by both fractional Brownian motions and hidden Markov models is planned.This research focuses on a stochastic calculus for a fractional Brownian motion. A fractional Brownian motion is a stochastic process that has been important in modeling physical systems such as those arising in hydrology, economics and telecommunications. Stochastic calculus provides the basic tool for solving problems of stochastic models that contain a fractional Brownian motion. Fractional Brownian motions have a self similarity property and many have a long range dependence property that often occur in physical phenomena. It is planned to study the control of partially known systems that contain fractional Brownian motion and hidden Markov models. Such control problems naturally arise in applications.