We consider a class of ${ℝ}^d$-valued stochastic control systems, with possibly unbounded costs. The systems evolve according to a discrete-time equation $x_{t+1}=Gₙ(x_t,a_t)+{\xi }_t$ (t = 0,1,... ), for each fixed n = 0,1,..., where the ${\xi }_t$ are i.i.d. random vectors, and the Gₙ are given functions converging pointwise to some function $G_{\infty }$ as n → ∞. Under suitable hypotheses, our main results state the existence of stationary control policies that are expected average cost (EAC) optimal and sample path average cost (SPAC) optimal for...