The Well-Posedness and Attainability of an LQ Optimal Control Problem for Mean-Field Linear Stochastic Systems Controlled by Impulses

In this work, the well-posedness and attainability of a finite horizon time linear quadratic (LQ) optimal control problem for a class of McKean–Vlasov stochastic linear differential equations controlled by impulses are addressed. No assumption regarding the sign of the weighting matrices involved in the performance criterion is made. First, employing an orthogonal decomposition of the state variables and of the control parameters, we obtain an equivalent version of the original problem of the optimal control. Then, the new optimal control problem is embedded into a so-called extended LQ optimal control problem. The well-posedness and the closed-loop attainability properties initially defined for the original LQ optimal control problem are then adapted to the extended optimal control problem. The main tool involved in the analysis of the well-posedness and the closed-loop attainability properties of the extended LQ optimal control problem is the solution of a system of two coupled backward jump matrix Lyapunov-type differential equations with a Riccati-type jumping operator. This solution is subject to a set of constraints due to the absence of information about the rank and the sign of the weighting matrices from the objective functions. We show that the existence of such a solution is a necessary and sufficient condition for the closed-loop attainability of the extended LQ optimal control problem. Explicit formulae of the gain matrices of the optimal feedback are provided.