# Density control of interacting agent systems

arXiv: https://arxiv.org/abs/2108.07342 · HTML: https://mlsanity.com/paper/2108.07342/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2021.

## Built on by (most cited first, 13 of 13)

- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithms](https://mlsanity.com/paper/2206.03792/) (2022)
- [Go With the Flow: Fast Diffusion for Gaussian Mixture Models](https://mlsanity.com/paper/2412.09059/) (2024)
- [SwarmDiff: Swarm Robotic Trajectory Planning in Cluttered Environments via Diffusion Transformer](https://mlsanity.com/paper/2505.15679/) (2025)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Deep Generalized Schr"odinger Bridges: From Image Generation to Solving Mean-Field Games](https://mlsanity.com/paper/2412.20279/) (2024)
- [Exact Solutions to the Quantum Schr"odinger Bridge Problem](https://mlsanity.com/paper/2509.25980/) (2025)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)
- [Slack More, Predict Better: Proximal Relaxation for Probabilistic Latent Variable Model-based Soft Sensors](https://mlsanity.com/paper/2603.11473/) (2026)
- [Learning Sampled-data Control for Swarms via MeanFlow](https://mlsanity.com/paper/2603.20189/) (2026)
- [Mutual Information Optimal Density Control of Linear Systems and Generalized Schr"odinger Bridges with Reference Refinement](https://mlsanity.com/paper/2605.09349/) (2026)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (2026)

## Abstract (arXiv)

We consider the problem of controlling the group behavior of a large number of dynamic systems that are constantly interacting with each other. These systems are assumed to have identical dynamics (e.g., birds flock, robot swarm) and their group behavior can be modeled by a distribution. Thus, this problem can be viewed as an optimal control problem over the space of distributions. We propose a novel algorithm to compute a feedback control strategy so that, when adopted by the agents, the distribution of them would be transformed from an initial one to a target one over a finite time window. Our method is built on optimal transport theory but differs significantly from existing work in this area in that our method models the interactions among agents explicitly. From an algorithmic point of view, our algorithm is based on a generalized version of the proximal gradient descent algorithm and has a convergence guarantee with a sublinear rate. We further extend our framework to account for the scenarios where the agents are from multiple species. In the linear quadratic setting, the solution is characterized by coupled Riccati equations which can be solved in closed-form. Finally, several numerical examples are presented to illustrate our framework.