# Optimal transport over a linear dynamical system

arXiv: https://arxiv.org/abs/1502.01265 · HTML: https://mlsanity.com/paper/1502.01265/

## Key facts

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

## Built on by (most cited first, 14 of 14)

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- [Optimal Mass Transport over the Euler Equation](https://mlsanity.com/paper/2304.00595/) (2023)
- [Sliced Wasserstein Steering between Gaussian Measures](https://mlsanity.com/paper/2604.22807/) (2026)
- [WTF?! Simulation-Free Reinforcement Learning with Wasserstein-Tilted Flow Maps](https://mlsanity.com/paper/2609.27033/) (2026)
- [Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps](https://mlsanity.com/paper/2412.13372/) (2024)
- [The Ground Cost for Optimal Transport of Angular Velocity](https://mlsanity.com/paper/2504.03190/) (2025)
- [Neural solver for Wasserstein Geodesics and optimal transport dynamics](https://mlsanity.com/paper/2602.22003/) (2026)
- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)

## Abstract (arXiv)

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator (dot x(t) = u(t)) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schr"odinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.