The proposed research is part of an ongoing program of work which revolves around problems of controlling nonlinear systems with imperfect or partial observations. The main problems investigated are: smoothness of the optimal value function, behaviour of the optimal control, the asymptotic behavior of the system as the time or noise parameters approach zero or infinity, and establishing a class of explicitly solvable case examples. A special case of the partially observed problem with a finite parameter set has been solved in the investigator's previous work. It is known under the name of the so-called Adaptive Control problem formulated as a stochastic control Bayesian problem. Of interest is now the extent to which these results can be established for the continuous parameter case. The general partially observed problem is another topic of interest for this investigation. Recently, the investigator has established the general equivalence of the partially observed problem and the so-called separated problem, which is a special infinite dimensional completely observed problem. The next step in this research is the establishing of the Bellman equation. The investigator intends to rely here on the recent results on viscosity solutions. Finally, he intends to establish a general continuity result, i.e. to prove that the value function depends continuously on the generators. Another topic of interest that the investigator wants to pursue is the exit problem. Stochastic control and filtering is an area of active research with potential applications to stock management, control of noisy dynamical systems and filtering of signals.
