Stochastic Control under Correlated Disturbances

Stochastic control is a fundamental concept in control theory. Traditionally, it is assumed that the system disturbances are independent. This assumption does not always hold. In this paper, we look into cases where the disturbances are correlated. We provide a modified version of the Dynamic Programming algorithm to find optimal policies without the independence assumption. Then, we look into the classic LQG problem but with correlated disturbances. We show that the optimal policy is closely related to the trajectory-following LQG problem. The trajectory in this problem is defined by estimating future disturbances and is time-varying. We provide numerical examples to illustrate the theoretical results.