# Optimal Control of Continuity Equations

arXiv: https://arxiv.org/abs/1506.08932 · HTML: https://mlsanity.com/paper/1506.08932/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2015.

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

- [Neural ODE Control for Trajectory Approximation of Continuity Equation](https://mlsanity.com/paper/2205.09241/) (2022)
- [Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs](https://mlsanity.com/paper/2305.08849/) (2023)
- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)
- [PDE+: Enhancing Generalization via PDE with Adaptive Distributional Diffusion](https://mlsanity.com/paper/2305.15835/) (2023)

## Abstract (arXiv)

An optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.