The vanishing approach for the average continuous control of piecewise deterministic Markov processes

This paper deals with the long run average continuous control problem of piecewise deterministic Markov processes (PDMP¿s) taking values in a general Borel space and with compact action space depending on the state variable. The control variable acts on the jump rate and transition measure of the PDMP, and the running and boundary costs are assumed to be positive but superiorly unbounded. Our main result is to obtain the existence and characterization of an ordinary optimal feedback control for the long run average cost problem using the so-called vanishing discount approach.