A Feedback-Type Optimal Solution for Partially–Observed Linear–Quadratic Risk-Sensitive Optimal Control Problem of Mean-Field Type Stochastic Systems

We study the linear–quadratic (LQ) risk-sensitive optimal control problem for mean-field type stochastic differential equations (MF-SDEs) driven by Brownian motion. The expected values of state and control variables are included in the MF-SDE as well as the objective functional, and the objective functional is of the risk-sensitive type. The control has access to the noisy state information from the mean-field type stochastic observation model. Under this setting, we obtain the practically implementable explicit feedback-type linear optimal solution to the problem. In particular, we decompose the original problem into the (control-constrained) partially observed LQ risk-sensitive control problem and the LQ risk-neutral problem for the mean-field dynamics. While the optimal solution of the former is characterized by the risk-sensitive state estimator and satisfies the associated control constraint, the optimal solution of the latter is represented by the state-feedback mean-field type process. Then, by combining the optimal solutions of these two problems, we obtain the explicit feedback-type linear optimal solution to the original problem. We provide the simulation results of the modified national income problem to demonstrate that our feedback-type optimal solution is practically implementable.