  This papers deals with the constrained discounted control of piecewise
deterministic Markov process (PDMPs) in general Borel spaces. The control
variable acts on the jump rate and transition measure, and the goal is to
minimize the total expected discounted cost, composed of positive running and
boundary costs, while satisfying some constraints also in this form. The basic
idea is, by using the special features of the PDMPs, to re-write the problem
via an embedded discrete-time Markov chain associated to the PDMP and
re-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 in
the same framework of a general constrained discrete-time Markov Decision
Process and, due to that, some conditions are required to get the equivalence
between the continuous-time problem and the LP formulation. We provide in the
sequel 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 the
proof of this generalization which, we believe, is of interest on its own. The
paper is concluded with some examples to illustrate the obtained results.
