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This paper deals with discrete-time Markov control processes in Borel spaces and unbounded rewards. Under suitable hypotheses, we show that a randomized stationary policy is optimal for a certain expected constrained problem (ECP) if and only if it is optimal for the corresponding pathwise constrained problem (pathwise CP). Moreover, we show that a parametric family of unconstrained optimality equations posseses compactness and convergence properties that lead to an approximation scheme which allows us to obtain constrained optimal policies as the limit of unconstrained deterministic optimal policies. In addition, we give sufficient conditions for the existence ofdeterministic policies that solve these constrained problems.

Deterministic optimal policies for Markov control processes with pathwise constraints