  We consider a general class of mean field control problems described by
stochastic delayed differential equations of McKean-Vlasov type. Two numerical
algorithms are provided based on deep learning techniques, one is to directly
parameterize the optimal control using neural networks, the other is based on
numerically solving the McKean-Vlasov forward anticipated backward stochastic
differential equation (MV-FABSDE) system. In addition, we establish a necessary
and sufficient stochastic maximum principle for this class of mean field
control problems with delay based on the differential calculus on function of
measures, as well as existence and uniqueness results for the associated
MV-FABSDE system.

Optimization and Control