# Maximum Principle of Optimal Probability Density Control

arXiv: https://arxiv.org/abs/2505.18362 · HTML: https://mlsanity.com/paper/2505.18362/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 37 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Mean-Field Optimal Control](https://mlsanity.com/paper/1306.5913/) (2013)
- [A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems](https://mlsanity.com/paper/1810.13117/) (2018)
- [Optimal Transport-based Coverage Control for Swarm Robot Systems: Generalization of the Voronoi Tessellation-based Method](https://mlsanity.com/paper/2011.08337/) (2020)
- [Density control of large-scale particles swarm through PDE-constrained optimization](https://mlsanity.com/paper/2104.06373/) (2021)
- [Velocity Field Generation for Density Control of Swarms using Heat Equation and Smoothing Kernels](https://mlsanity.com/paper/1710.00299/) (2017)
- [Approximate and exact controllability of the continuity equation with a localized vector field](https://mlsanity.com/paper/1710.09287/) (2017)
- [Mean-field optimal control as Gamma-limit of finite agent controls](https://mlsanity.com/paper/1803.04689/) (2018)
- [Mean-Field Pontryagin Maximum Principle](https://mlsanity.com/paper/1504.02236/) (2015)
- [The Pontryagin Maximum Principle in the Wasserstein Space](https://mlsanity.com/paper/1711.07667/) (2017)
- [Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces](https://mlsanity.com/paper/2101.10668/) (2021)
- [High-dimensional Optimal Density Control with Wasserstein Metric Matching](https://mlsanity.com/paper/2307.13135/) (2023)
- [Numerical Algorithm for Optimal Control of Continuity Equations](https://mlsanity.com/paper/1708.05516/) (2017)

## Abstract (arXiv)

We develop a general theoretical framework for optimal probability density control on standard measure spaces, aimed at addressing large-scale multi-agent control problems. In particular, we establish a maximum principle (MP) for control problems posed on infinite-dimensional spaces of probability distributions and control vector fields. We further derive the Hamilton--Jacobi--Bellman equation for the associated value functional defined on the space of probability distributions. Both results are presented in a concise form and supported by rigorous mathematical analysis, enabling efficient numerical treatment of these problems. Building on the proposed MP, we introduce a scalable numerical algorithm that leverages deep neural networks to handle high-dimensional settings. The effectiveness of the approach is demonstrated through several multi-agent control examples involving domain obstacles and inter-agent interactions.