Control of Stochastic Systems
Project Award Date: 09-01-2011
Since most controlled stochastic systems have been described with a Brownian motion, it is important to study control problems for stochastic systems with other fractional Brownian motions. These control problems require significantly different analysis methods than the well-known methods for the control of systems with Brownian motions. These major differences have resulted in few available results for the optimal control of linear systems driven by an arbitrary fractional Brownian motion.
ITTC researchers have initiated a major study of these control problems for linear systems. This study not only obtains results for an arbitrary fractional Brownian motion but also provides results for the control of linear systems driven by a wide spectrum of other continuous stochastic processes. Researchers will expand the study of control problems to other cost functionals and to other types of systems, both finite and infinite dimensional. Specifically they will determine explicit expressions for the optimal control and the optimal cost for an ergodic (or long run average) quadratic cost functional. Researchers will study the control of linear stochastic partial differential equations for both finite and infinite time horizon control problems with a quadratic cost functional. The noise stochastic process and the control are allowed to be restricted to the boundary of the domain for the partial differential equation. Typically the stochastic systems have unknown parameters so the control problems are adaptive control problems.
The research will expand the initial ITTC work on adaptive control for a scalar linear system to multidimensional linear systems with an arbitrary fractional Brownian motion. Researchers will examine the control of bilinear systems driven by a fractional Brownian motion and having a quadratic cost functional. Since some physical models require discontinuous processes, it is proposed to study a controlled linear system with a quadratic cost functional and a discontinuous stochastic process, such as a Levy process. Some computationally related aspects of these control problems are also proposed such as the discretization of continuous time algorithms and the control of discrete time linear systems.
Primary Sponsor(s): National Science Foundation (NSF)