  This paper investigates the stabilization and control problems for linear
continuous-time mean-field systems (MFS). Under standard assumptions, necessary
and sufficient conditions to stabilize the mean-field systems in the mean
square sense are explored for the first time. It is shown that, under the
assumption of exact detectability (exact observability), the mean-field system
is stabilizable if and only if a coupled algebraic Riccati equation (ARE)
admits a unique positive semi-definite solution (positive definite solution),
which coincides with the classical stabilization results for standard
deterministic systems and stochastic systems.
  One of the key techniques in the paper is the obtained solution to the
forward and backward stochastic differential equation (FBSDE) associated with
the maximum principle for an optimal control problem. Actually, with the
analytical FBSDE solution, a necessary and sufficient solvability condition of
the optimal control, under mild conditions, is derived. Accordingly, the
stabilization condition is presented by defining an Lyaponuv functional via the
solution to the FBSDE and the optimal cost function.
  It is worth of pointing out that the presented results are different from the
previous works for stabilization and also different from the works on optimal
control.
