This work considers two problems: In one, it seeks to find when convergence of a sequence of channels imply convergence of the optimal cost values for a stochastic control or estimation problem; where setwise convergence and weak convergence are particularly studied and the inadequacy of weak convergence is exhibited. The second problem is with regard to a concavity result on the optimal cost with regard to the channels. These findings have applications in optimal vector quantization design, and robust control. Some connections with design of experiments will also be discussed.