  This paper is concerned with a mean-field linear quadratic (LQ, for short)
optimal control problem with deterministic coefficients. It is shown that
convexity of the cost functional is necessary for the finiteness of the
mean-field LQ problem, whereas uniform convexity of the cost functional is
sufficient for the open-loop solvability of the problem. By considering a
family of uniformly convex cost functionals, a characterization of the
finiteness of the problem is derived and a minimizing sequence, whose
convergence is equivalent to the open-loop solvability of the problem, is
constructed. Then, it is proved that the uniform convexity of the cost
functional is equivalent to the solvability of two coupled differential Riccati
equations and the unique open-loop optimal control admits a state feedback
representation in the case that the cost functional is uniformly convex.
Finally, some examples are presented to illustrate the theory developed.
