Stochastic Neutral Evolution Equations on Hilbert Spaces and Their Partially Observed Optimal Relaxed Control
In this paper we consider the question of existence of mild solutions for a class of Neutral stochastic evolution equations on Hilbert space driven by relaxed controls. The controls are based on partial information. We prove continuous dependence of solutions on controls in appropriate topologies. Then we consider an optimal control problem of Bolza type. Further on, we construct the measure valued functions corresponding to the mild solutions and study the weak compactness properties of attainable sets of probability measures in two different topologies. Using these results we solve several interesting standard and nonstandard control problems involving evolution of probability measure valued functions and relaxed controls.