In this paper we consider a control problem for a Partially Observable
Piecewise Deterministic Markov Process of the following type: After the jump of
the process the controller receives a noisy signal about the state and the aim
is to control the process continuously in time in such a way that the expected
discounted cost of the system is minimized. We solve this optimization problem
by reducing it to a discrete-time Markov Decision Process. This includes the
derivation of a filter for the unobservable state. Imposing sufficient
continuity and compactness assumptions we are able to prove the existence of
optimal policies and show that the value function satisfies a fixed point
equation. A generic application is given to illustrate the results
