The paper derives the stochastic maximum principle for mean-field control under dynamic constraints by embedding the problem in Banach-space optimization and applying generalized Fritz-John conditions to obtain a BSDE as the Lagrange multiplier.
Clarke (1976): A new approach of Lagrange multipliers
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Extended mean-field control under constraints: The generalized Fritz-John conditions and Lagrangian method
The paper derives the stochastic maximum principle for mean-field control under dynamic constraints by embedding the problem in Banach-space optimization and applying generalized Fritz-John conditions to obtain a BSDE as the Lagrange multiplier.