Constrained mean-field control with singular controls: Existence, stochastic maximum principle and constrained FBSDE

This paper studies a class of mean-field control (MFC) problems with singular controls under general dynamic state-control-law constraints. We first propose a customized relaxed control formulation to cope with the dynamic mixed constraints and establish the existence of an optimal control using compactification argument in the proper canonical spaces to accommodate singular controls. To further characterize the optimal pair of regular and singular controls, we treat the controlled McKean-Vlasov process as an infinite-dimensional equality constraint and recast the MFC problem as an optimization problem on canonical spaces with constraints on Banach space, allowing us to derive the stochastic maximum principle (SMP) and a class of constrained BSDE using a new Lagrange multipliers method. Additionally, we investigate the uniqueness and the stability result of the solution to the constrained FBSDE associated with the constrained MFC with singular controls.