  We study mean-field control problems in discrete-time under the infinite
horizon average cost optimality criteria. We focus on both the finite
population and the infinite population setups. We show the existence of the
average cost optimality equation (ACOE) and the existence of optimal stationary
Markov policies for finite population problems under (i) a minorization
condition that provides geometric ergodicity on the collective state process of
the agents, and (ii) under standard Lipschitz continuity assumptions on the
stage-wise cost and transition function of the agents when the Lipschitz
constant of the transition function satisfies a certain bound. Furthermore, we
provide a relative value iteration algorithm that approximates the optimal
solution of the problem. For the infinite population problem, we establish the
existence of the ACOE, and the existence of optimal policies under the
continuity assumptions on the cost and the transition functions, and we provide
a convergent relative value iteration method for the infinite population
problem as well. Finally, we prove that the optimal value of the finite
population problem converges to the optimal value of the infinite population
problem as the number of agents grows to infinity. This result has been proven
for finite horizon and infinite horizon discounted cost problems, however, it
is the first result for the average cost optimality criteria to our knowledge.

Optimization and Control;Systems and Control;Systems and Control