Optimal encoder and control strategies in stochastic control subject to rate constraints for channels with memory and feedback

This paper is concerned with the joint design of encoder/control strategies which minimize a control payoff subject to a rate constraint on the feedback information provided by the system. The control system is described by a conditional distribution, while the rate is described by the directed information of a noisy communication channel. By generalizing the Posterior Matching Scheme (PMS), a scheme that achieves operational capacity, to channels with memory and feedback, the joint optimality of encoder/control strategies is analyzed. A principle of optimality is derived and a dynamic programming equation is obtained, which together describe how the optimal encoder/control strategies should be selected. It is shown that joint optimality separates into an optimal encoder strategy based on the PMS, and a control strategy obtained via the dynamic programming equation.