Partially Observed Control of Markov Processes. III

Let $\nu$ denote the value function of a partially observed control problem. If $\nu$ is once differentiable in a certain direction $\hat{B}$, then optimal controls are characterized by a feedback involving the directional derivative $\hat{B}\nu$. It is also shown that $\nu$ satisfies the corresponding Bellman equation, an infinite-dimensional PDE on the space of measures, in the viscosity sense of Crandall and Lions.