Unbalanced Optimal Transport and Density Control for Discrete-Time Linear Systems

TL;DR

This paper introduces unbalanced optimal transport and density control methods for discrete-time linear systems, providing convex formulations and demonstrating their effectiveness through a numerical example.

Contribution

It extends unbalanced optimal transport to include system dynamics and constraints, with a focus on Gaussian references and convex optimization.

Findings

Both UOT and UDC admit globally optimal convex formulations.

The methods are applicable to Gaussian reference measures in linear systems.

Numerical experiments illustrate the practical implementation of the approach.

Abstract

This article studies unbalanced optimal transport (UOT) and its dynamical extension, unbalanced density control (UDC), for a class of constrained discrete-time linear systems. UOT compares measures with unequal total mass by balancing transport cost and fidelity to reference measures, while UDC incorporates system dynamics and constraints into this framework. Focusing on Gaussian references and discrete-time linear systems, we show that both problems admit globally optimal convex formulations, analogous to covariance steering. A numerical experiment is provided to illustrate our approach.