The Pontryagin Maximum Principle in the Wasserstein Space

Benoît Bonnet (1) , Francesco Rossi (2)

1 LSIS - Laboratoire des Sciences de l'Information et des Systèmes
2 Dipartimento di Matematica "Tullio Levi-Civita"

We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on needle variations and on the evolution of the covector (here replaced by the evolution of a mesure on the dual space) can be translated into this formalism.

Mots clés: Optimal Control; Mean Field; Non-local PDEs; Subdifferential calculus; Wasserstein Space; Pontryagin Maximum Principle; Multi-Agent

Domaines: Optimisation et contrôle [math.OC]

Benoît Bonnet, Francesco Rossi. The Pontryagin Maximum Principle in the Wasserstein Space. 2017. ⟨hal-01637050v3⟩