On the equivalence between static and dynamic optimal transport governed by linear control systems

TL;DR

This paper demonstrates the equivalence between static and dynamic optimal transport problems linked to linear control systems, using functional analysis and variational methods to establish new quantitative estimates.

Contribution

It establishes a constructive proof of the equivalence between static and dynamic optimal transport in linear control systems, introducing new quantitative estimates.

Findings

Proves the equivalence between static and dynamic transport problems.

Provides new quantitative estimates for control-based transport.

Uses functional analytic techniques to analyze the end-point map.

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.