Convergence of the optimal values of constrained Markov control processes

We consider a sequence of discounted cost, constrained Markov control processes (CCPs) with countable state space, metric action set and possibly unbounded cost functions. We give conditions under which the sequence of optimal values of the CCPs converges to the optimal value of a limiting CCP, and, furthermore, the accumulation points of sequences of optimal policies for the CCPs are optimal policies for the limiting CCP. These results are obtained via an approximation theorem for general minimization problems.