Limiting discounted-cost control of partially observable stochastic systems

This paper presents two main results on partially observable (PO) stochastic systems. In the first case, we consider a general PO system x_t+1=F(x_t, a_t, ξ_t), y_t=G(x_t, η_t), on Borel spaces, with possible unbounded cost-per-stage functions, and give conditions for the existence of α-discount optimal control policies (0<α<1). In the second result we specialize (*) to additive-noise systems x_t+1=F_n(x_t, a_t)+ξ_t, y_t=G_n(x_t)+η_t, t ∈ N, in Euclidean spaces, with F_n(x, a) and G_n(x) converging pointwise to functions F_∞(x,a) and G_∞(x), respectively, and give conditions for the limiting PO model x_t+1=F_∞(x_t, a_t)+ξ_t, y_t=G_∞(x_t)+η_t, t ∈ N, to have an α-discount optimal policy.