Average control of Markov decision processes with Feller transition probabilities and general action spaces

This paper studies the average control problem of discrete-time Markov Decision Processes (MDPs for short) with general state space, Feller transition probabilities, and possibly non-compact control constraint sets A(x)A(x). Two hypotheses are considered: either the cost function cc is strictly unbounded or the multifunctions Ar(x)={a∈A(x):c(x,a)≤r}Ar(x)={a∈A(x):c(x,a)≤r} are upper-semicontinuous and compact-valued for each real rr. For these two cases we provide new results for the existence of a solution to the average-cost optimality equality and inequality using the vanishing discount approach. We also study the convergence of the policy iteration approach under these conditions. It should be pointed out that we do not make any assumptions regarding the convergence and the continuity of the limit function generated by the sequence of relative difference of the αα-discounted value functions and the Poisson equations as often encountered in the literature.