Separation Principle for Partially Observed Linear-Quadratic Optimal Control for Mean-Field Type Stochastic Systems

We consider the partially observed linear-quadratic (LQ) optimal control problem for mean-field type stochastic systems driven by Brownian motion. The control does not have access to complete state information, but only to noisy state information from the (stochastic) observation model. The dynamics and observation model as well as the objective functional include the expected values of state and control variables, known as the mean-field variables. The main result is the separation between optimal control and state estimation. Specifically, we show that the classical separation principle can be extended to the LQ mean-field type problem, where the optimal solution can be obtained by a simple replacement of the state in the complete information case with the state of the optimal filtering process. The main result is proved by decomposing the original problem into stochastic and mean-field parts leading to an equivalent lifted problem, constructing the optimal filtering process for the lifted problem using the innovation approach, and employing the completion of squares method through the orthogonal projection property of the filtering process. Numerical examples are provided to illustrate the theoretical result of the article.