Relaxed long run average continuous control of piecewise deterministic Markov processes

In this paper we consider the long run average continuous control problem of piecewise-deterministic Markov processes (PDP's for short). The control variable acts on the jump rate λ and transition measure Q of the PDP. We consider relaxed open loop policies which choose, at each jump time, randomized (rather than deterministic) control actions. The advantage of allowing randomized actions is that the optimality equation for the continuous-time problem can be re-written as a discrete-time Markov decision process with compact action space. The main goal of this paper is to show the compactness proprieties of the action space for discrete-time problem as well as to prove the equivalence between the optimality equations of the continuous and discrete-time problems.