  The mean field control problem for a multi-dimensional diffusion-aggregation
system with Coulomb interaction is considered. The existence of optimal control
is proved through the Gamma convergence of the control problem of a regularized
particle control problem. The optimal control problem on the particle level is
studied by using the corresponding Liouville equation. Because of strong
aggregation effect, additional difficulties arises from control function in the
well-posedness theory, so that the known method for multi-dimensional
Keller-Segel equation can not be directly applied. We use a combination of
local existence result and boot-strap argument to obtain the global solution
with small initial data. Since we obtain a strong propagation of chaos result
by combining the convergence in probability and relative entropy method, the
compact support requirement for control functions, which has been often used in
the literature, is not need.
