The Minimal Solution of Matrix Riccati Differential Equations Arising in Connection with the Mean-Field Stochastic Control: Application to the Mean-Field LQ Optimal Control with Free End Points

In this article, a system of two coupled matrix Riccati differential equations (CMRDEs) arising in connection with mean-field linear quadratic (LQ) control problems is considered. Our aim is to provide conditions that guarantee the existence and uniqueness of the minimal and positive semidefinite solution of such kind of matrix Riccati differential equations. In our approach, the proofs of the existence, as well as the properties of the minimal and positive semidefinite solutions, are done involving the solutions’ properties of the differential equations with positive evolution on the ordered Hilbert space of the symmetric matrices. Then, as an application of the minimal and positive semidefinite solution of the CMRDE, a mean-field linear quadratic optimal control problem with free end value is considered. We show that under the considered assumptions this optimal control problem has a unique optimal control and this control is in a state feedback form. The gain matrices of the optimal feedback are computed based on the minimal and positive semidefinite solution of a CMRDE associated with the problem under consideration.