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.
