  Different from most of the previous works, this paper provides a thorough
solution to the fundamental problems of linear-quadratic (LQ) control and
stabilization for discrete-time mean-field systems under basic assumptions.
Firstly, the sufficient and necessary condition for the solvability of
mean-field LQ control problem is firstly presented in analytic expression based
on the maximum principle developed in this paper, which is compared with the
results obtained in literatures where only operator type solvability conditions
were given. The optimal controller is given in terms of a coupled Riccati
equation which is derived from the solution to forward and backward stochastic
difference equation (FBSDE). Secondly, the sufficient and necessary
stabilization conditions are explored. It is shown that, under exactly
observability assumption, the mean-field system is stabilizable in mean square
sense if and only if a coupled algebraic Riccati equation (ARE) has a unique
solution $P$ and $\bar{P}$ satisfying $P>0$ and $P+\bar{P}>0$. Furthermore,
under the exactly detectability assumption, which is a weaker assumption than
exactly observability, we show that the mean-field system is stabilizable in
mean square sense if and only if the coupled ARE has a unique solution $P$ and
$\bar{P}$ satisfying $P\geq 0$ and $P+\bar{P}\geq 0$. The key techniques
adopted in this paper are the maximum principle and the solution to the FBSDE
obtained in this paper. The derived results in this paper forms the basis to
solve the mean-field control problem for continuous-time systems and other
related problems.
