On the equivalence between static and dynamic optimal transport governed by linear control systems
Amit Einav, Yue Jiang, Alpár R. Mészáros
TL;DR
This work extends optimal transport to a setting with non-autonomous linear control dynamics by introducing a $p$-energy cost $c_p(x,y)$ and establishing a constructive equivalence between static and dynamic transport formulations: $\min_{\pi\in\Pi(\mu,\nu)} \int c_p\,d\pi = \mathcal{C}_p(\mu,\nu) = \mathcal{D}_p(\mu,\nu)$. The authors develop a rigorous framework around the end-point map $E_{s,t}^x$ and the state-transition map $\Phi$, prove controllability under a Kalman-type condition, and derive regularity properties for the optimal control $\alpha_p^*(\cdot;x,y)$, including a Lagrange-multiplier characterization that yields explicit forms in the quadratic case $p=2$. They further connect trajectory-level optimal controls to density evolutions via a generalized continuity equation and a superposition principle, culminating in a generalized Benamou--Brenier formula that links static and dynamic OT in this controlled setting. The results provide a constructive methodology to translate minimizers between static and dynamic OT and extend kinetic-type OT perspectives to linear-control frameworks with potential applications to multi-agent and mean-field problems.
Abstract
In this paper we revisit a class of optimal transport problems associated to non-autonomous linear control systems. Building on properties of the cost functions on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ derived from suitable variational problems, we show the equivalence between the static and dynamic versions of the corresponding transport problems. Our analysis is constructive in nature and relies on functional analytic properties of the end-point map and the fine properties of the optimal control functions. These lead to some new quantitative estimates which play a crucial role in our investigation.