The Pontryagin Maximum Principle in the Wasserstein Space

TL;DR

This paper establishes a Pontryagin Maximum Principle for optimal control problems in Wasserstein spaces, using subdifferential calculus and geometric needle variations to handle non-local transport dynamics.

Contribution

It introduces a novel formulation of the Pontryagin Maximum Principle in Wasserstein spaces, integrating geometric and variational methods for non-local transport equations.

Findings

Proves a first-order optimality condition in Wasserstein space.

Connects geometric needle variations with subdifferential calculus.

Provides a framework for control problems with non-local dynamics.

Abstract

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.