This was part of

**Distributed Solutions to Complex Societal Problems Reunion Workshop**## Mean-field Control under State Constraint: Optimality Conditions and Mean-field Limit

**
Samuel Daudin, Université Paris Dauphine
**

**Tuesday, February 21, 2023**

**Abstract**: We present a stochastic control problem where the probability distribution of the state is constrained to remain in some region of the Wasserstein space of probability measures. Reformulating the problem as an optimal control problem for a Fokker-Planck equation, we derive optimality conditions in the form of a mean-field game system of partial differential equations. The effect of the constraint is captured by the presence, in this system, of a Lagrange multiplier which is a non-negative Radon measure over the time interval. Our main result is to exhibit geometric conditions on the constraint, under which this multiplier is bounded and optimal controls are Lipschitz continuous in time. As a consequence we prove, in a second time, that the stochastic control problem with constraints in law, arises as limit of control problems for large number of interacting agents subject to almost-sure constraints.