Stochastic Optimal Control with Markovian Lossy State Observations

We consider optimal control of a diffusion process where the state is either observed exactly or completely lost at the controller, as described by a binary Markov chain. The random observation loss, coupled with the nonlinear dynamics, makes the conventional optimal control techniques difficult to apply. We introduce an abstract state space from the point of view of a hybrid system and next apply dynamic programming. For this purpose, we apply tools of differentiation of functions defined on a space of probability measures, which has no linear structure. Our approach is explicitly illustrated by a linear quadratic (LQ) control problem.