The mean-field linear quadratic optimal control problem for stochastic systems controlled by impulses

In the present note, we state and solve the linear quadratic (LQ) control problem for a class of McKean-Vlasov stochastic differential equations for which the control actions are of impulsive nature. After reformulating the original optimization problem using an orthogonal decomposition of the state variables, we introduce an adequately defined system of two coupled matrix Lyapunov-type differential equations with jumps. Such equations are central for the definition of the optimal feedback gains corresponding to the closed-loop representation of the optimal LQ control.