Flow matching for stochastic linear control systems

TL;DR

This paper introduces a flow matching approach for steering probability distributions in stochastic linear control systems, with applications to robotic swarms and thermodynamics, providing explicit solutions and numerical methods.

Contribution

It develops a novel flow matching framework tailored for control channels in stochastic linear systems, including explicit formulas and approximation techniques.

Findings

Explicit control laws derived for special cases

Numerical procedure demonstrated with examples

Applicable to robotic swarms and thermodynamics

Abstract

This paper addresses the problem of steering an initial probability distribution to a target probability distribution through a deterministic or stochastic linear control system. Our proposed approach is inspired by the flow matching methodology, with the difference that we can only affect the flow through the given control channels. The motivation comes from applications such as robotic swarms and stochastic thermodynamics, where agents or particles can only be manipulated through control actions. The feedback control law that achieves the task is characterized as the conditional expectation of the control inputs for the stochastic bridges that respect the given control system dynamics. Explicit forms are derived for special cases, and a numerical procedure is presented to approximate the control law, illustrated with examples.