Extended mean field control problems with constraints: The generalized Fritz-John conditions and Lagrangian method

This paper studies the extended mean field control problems under dynamic expectation constraints and/or dynamic state-control constraints. We aim to pioneer the establishment of the stochastic maximum principle (SMP) and the derivation of the backward SDE (BSDE) from the perspective of the constrained optimization using the method of Lagrangian multipliers. To this end, we first propose to embed the constrained extended mean-field control (C-(E)MFC) problems into some abstract optimization problems with constraints on Banach spaces, for which we develop the generalized Fritz-John (FJ) optimality conditions. We then prove the stochastic maximum principle (SMP) for C-(E)MFC problems by transforming the FJ type conditions into an equivalent stochastic first-order condition associated with a general type of constrained forward-backward SDEs (FBSDEs). Contrary to the existing literature, we recast the controlled Mckean-Vlasov SDE as an infinite-dimensional equality constraint such that the BSDE induced by the FJ first-order optimality condition can be interpreted as the generalized Lagrange multiplier to cope with the SDE constraint. Finally, we also present the SMP for constrained stochastic control problems and constrained mean field game problems as consequences of our main result for C-(E)MFC problems.