An optimal control problem is studied for a linear mean-field stochastic
differential equation with a quadratic cost functional. The coefficients and
the weighting matrices in the cost functional are all assumed to be
deterministic. Closed-loop strategies are introduced, which require to be
independent of initial states; and such a nature makes it very useful and
convenient in applications. In this paper, the existence of an optimal
closed-loop strategy for the system (also called the closed-loop solvability of
the problem) is characterized by the existence of a regular solution to the
coupled two (generalized) Riccati equations, together with some constraints on
the adapted solution to a linear backward stochastic differential equation and
a linear terminal value problem of an ordinary differential equation.Comment: 23 page
