Information Transfer of Control Strategies: Dualities of Stochastic Optimal Control Theory and Feedback Capacity of Information Theory

The control-coding capacity of stochastic control systems is introduced, and its operational meaning is established using randomized control strategies, which simultaneously control output processes encode information, and communicate information from control processes to output processes. The control-coding capacity is the analog Shannon's coding-capacity of noisy channels. Furthermore, duality relations to stochastic optimal control problems with deterministic and randomized control strategies are identified including the following. First, extremum problems of stochastic optimal control with directed information payoff are equivalent to feedback capacity problems of information theory, in which the control system act as a communication channel. Second, for Gaussian linear decision models with average quadratic constraints, it is shown that optimal randomized strategies are Gaussian, and decompose into a deterministic part and a random part. The deterministic part is precisely the optimal strategy of the linear quadratic Gaussian stochastic optimal control problem, whereas the random part is the solution of an water-filling information transmission problem that encodes information, which is estimated by a decoder.