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This papers deals with the constrained discounted control of piecewisedeterministic Markov process (PDMPs) in general Borel spaces. The controlvariable acts on the jump rate and transition measure, and the goal is tominimize the total expected discounted cost, composed of positive running andboundary costs, while satisfying some constraints also in this form. The basicidea is, by using the special features of the PDMPs, to re-write the problemvia an embedded discrete-time Markov chain associated to the PDMP andre-formulate the problem as an infinite dimensional linear programming (LP)problem, via the occupation measures associated to the discrete-time process.It is important to stress however that our new discrete-time problem is not inthe same framework of a general constrained discrete-time Markov DecisionProcess and, due to that, some conditions are required to get the equivalencebetween the continuous-time problem and the LP formulation. We provide in thesequel sufficient conditions for the solvability of the associated LP problem,based on a generalization of Theorem 4.1 in [8]. In the Appendix we present theproof of this generalization which, we believe, is of interest on its own. Thepaper is concluded with some examples to illustrate the obtained results.

A Linear Programming Formulation for Constrained Discounted Continuous Control for Piecewise Deterministic Markov Processes