  The main goal of this paper is to derive sufficient conditions for the
existence of an optimal control strategy for 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. In order to do that we apply the so-called vanishing discount
approach to obtain a solution to an average cost optimality inequality
associated to the long run average cost problem. Our main assumptions are
written in terms of some integro-differential inequalities related to the
so-called expected growth condition, and geometric convergence of the post-jump
location kernel associated to the PDMP.
