Average cost Markov control processes: stability with respect to the Kantorovich metric

We study perturbations of a discrete-time Markov control process on a general state space. The amount of perturbation is measured by means of the Kantorovich distance. We assume that an average (per unit of time on the infinite horizon) optimal control policy can be found for the perturbed (supposedly known) process, and that it is used to control the original (unperturbed) process. The one-stage cost is not assumed to be bounded. Under Lyapunov-like conditions we find upper bounds for the average cost excess when such an approximation is used in place of the optimal (unknown) control policy. As an application of the found inequalities we consider the approximation by relevant empirical distributions. We illustrate our results by estimating the stability of a simple autoregressive control process. Also examples of unstable processes are provided.