  This paper studies a systemic risk control problem by the central bank, which
dynamically plans monetary supply to stabilize the interbank system with
borrowing and lending activities. Facing both heterogeneity among banks and the
common noise, the central bank aims to find an optimal strategy to minimize the
average distance between log-monetary reserves of all banks and the benchmark
of some target steady levels. A weak formulation is adopted, and an optimal
randomized control can be obtained in the system with finite banks by applying
Ekeland's variational principle. As the number of banks grows large, we prove
the convergence of optimal strategies using the Gamma-convergence argument,
which yields an optimal weak control in the mean field model. It is shown that
this mean field optimal control is associated to the solution of a stochastic
Fokker-Planck-Kolmogorov (FPK) equation, for which the uniqueness of the
solution is established under some mild conditions.
