# The Pontryagin Maximum Principle in the Wasserstein Space

arXiv: https://arxiv.org/abs/1711.07667 · HTML: https://mlsanity.com/paper/1711.07667/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2017.

## Built on by (most cited first, 3 of 3)

- [Policy Gradient for Continuous-Time Mean-Field Control](https://mlsanity.com/paper/2605.20718/) (2026)
- [Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints](https://mlsanity.com/paper/2411.19193/) (2024)
- [Maximum Principle of Optimal Probability Density Control](https://mlsanity.com/paper/2505.18362/) (2025)

## Abstract (arXiv)

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.