A bounded cost condition for the existence of average optimal stationary policies of linear stochastic systems

This note considers the long-run average cost control problem of a discrete-time stochastic system. The stochastic system is assumed to be linear with respect to the state but the controls possess a general structure, possibly nonlinear. The main contribution of this paper is to show the existence of an optimal stationary policy for the long-run average cost problem provided that (i) there exists an ε-optimal policy for the discounted problem and this policy generates uniform bounded costs by stage, and that (ii) a persistency of excitation condition is satisfied. A numerical example illustrates the derived result.