A Pontryagin Maximum Principle for State-Constrained Control Problems in Wasserstein Spaces

Abstract. We develop necessary optimality conditions in the form of a Pontryagin maximum principle for a Bolza optimal control problem, where the dynamics are governed by a nonlocal continuity equation, and the problem includes pointwise state constraints. To derive the main result, we combine diffusive perturbations of the control with classical variational techniques, such as Ekeland’s variational principle. Particular attention is given to the interplay between the nonlocal nature of the dynamics and the structure of the state constraints, requiring a careful adaptation of traditional methods.