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

TL;DR

This paper develops a rigorous framework for constrained mean-field control with singular controls by introducing a customized relaxed-control approach and a compactification argument to handle dynamic state-control-law constraints. It recasts the forward McKean–Vlasov dynamics as an infinite-dimensional equality constraint and derives a stochastic maximum principle with a constrained BSDE via Lagrange multipliers, resulting in a nonstandard FBSDE with an adjoint pair $(Y,Z)$. The authors prove the existence of optimal relaxed controls and, under mild regularity and monotonicity conditions, the existence, uniqueness, and stability of solutions to the resulting constrained FBSDE, including a KKT-type condition for optimality. The results extend mean-field control theory to settings with both singular controls and joint state-law constraints, providing a solid foundation for constrained policies in risk regulation and liquidity management contexts.

Abstract

This paper studies a class of mean-field control (MFC) problems with singular control 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 the singular control. To 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 constrained BSDE using a novel Lagrange multipliers method. Additionally, we investigate the uniqueness and the stability result of the solution to the constrained FBSDE associated to the constrained MFC problem with singular control.

Theorems & Definitions (10)

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• proof : Proof of \ref{['strictcontrol']}
• proof : Proof of \ref{['closedset']}