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An optimal control problem is studied for a linear mean-field stochasticdifferential equation with a quadratic cost functional. The coefficients andthe weighting matrices in the cost functional are all assumed to bedeterministic. Closed-loop strategies are introduced, which require to beindependent of initial states; and such a nature makes it very useful andconvenient in applications. In this paper, the existence of an optimalclosed-loop strategy for the system (also called the closed-loop solvability ofthe problem) is characterized by the existence of a regular solution to thecoupled two (generalized) Riccati equations, together with some constraints onthe adapted solution to a linear backward stochastic differential equation anda linear terminal value problem of an ordinary differential equation.

Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Closed-Loop Solvability