PARTIALLY OBSERVED STOCHASTIC EVOLUTION EQUATIONS ON BANACH SPACES AND THEIR OPTIMAL LIPSCHITZ FEEDBACK CONTROL LAW.

In this paper we consider optimal feedback control problems for a general class of nonlinear partially observed stochastic evolution equations on Banach spaces. The system is governed by a pair of (coupled) stochastic evolution equations, one representing the main system and the other representing the observer. Both are governed by stochastic evolution equations on unconditional Martingale difference Banach spaces. The state of the second system, which is observable, is used to provide the input to control the main system. We present the existence of mild solutions of the system equations and then introduce a class of admissible nonlinear feedback operators. The space of feedback operators is endowed with the topology of pointwise convergence on the domain space with respect to the weak topology in the range space giving a compact Hausdorff space. This is then used to prove the existence of an optimal output feedback control law for the Bolza problem. Also we prove the weak compactness of the attainable set of induced measures and prove the existence of optimal feedback control laws for several nontypical control problems involving measures. [ABSTRACT FROM AUTHOR]