# Approximate and exact controllability of the continuity equation with a localized vector field

arXiv: https://arxiv.org/abs/1710.09287 · HTML: https://mlsanity.com/paper/1710.09287/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2017.

## Built on by (most cited first, 4 of 4)

- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Neural ODE Control for Trajectory Approximation of Continuity Equation](https://mlsanity.com/paper/2205.09241/) (2022)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Maximum Principle of Optimal Probability Density Control](https://mlsanity.com/paper/2505.18362/) (2025)

## Abstract (arXiv)

We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately one to another with Lipschitz controls when the uncontrolled dynamics allows to cross the control set. We also show that the exact controllability only holds for controls with less regularity, for which one may lose uniqueness of the associated solution.