Flow Matching for Measure Transport and Feedback Stabilization of Control-Affine Systems

TL;DR

This paper extends flow matching from generative modeling to control-affine systems by formulating measure interpolation under admissible trajectories and controls, and by recasting stabilization as a time-reversed denoising task. It develops exact, approximate, and output-based flow matching to transport distributions via a continuity equation, and introduces two stabilization schemes—PMP-based noising and randomized-control noising—with existence and regularity guarantees. The analysis relies on the superposition principle, disintegration, and controllability/Hörmander-type conditions to ensure well-defined measure flows, even when controls are non-smooth. Numerical experiments on linear and nonlinear driftless systems, as well as classic controllers and stochastic-noise inspired setups, illustrate the framework’s ability to realize measure transport and stabilization in practice. Overall, the work connects flow matching with Pontryagin principles and diffusion-model intuitions to provide scalable, regression-based tools for control-oriented measure transport and stabilization with theoretical guarantees.

Abstract

We develop a \emph{flow-matching framework} for transporting probability measures under control-affine dynamics and for stabilizing systems to points or target sets. Starting from the continuity equation associated with the control affine system dx/dt = f_0(x) + \sum_{i=1}^m u_i f_i(x), we construct measure interpolations through exact and approximate flow matching, and extend the approach to \emph{output flow matching} when only output distributions must align. These constructions allow to directly import standard control tools, such as feedback design, oscillatory inputs, and trajectory steering, and yield sample-efficient, regression-based controllers for measure-to-measure transport. We also introduce a complementary ``noising + time-reversal'' perspective for classical state or set stabilization, inspired by denoising diffusion models. Here stabilization is interpreted as a denoising problem. We propose two methods for constructing the noising process: (i) PMP-based noising, which leverages the Hamiltonian system from Pontryagin's Maximum Principle and recovers the optimal controller for linear systems with convex costs, while providing feasible feedback laws in the nonlinear case; and (ii) randomized-control noising, which employs regular (non-white noise) controls through the endpoint map and naturally accommodates control constraints. Both approaches avoid the score blow-up seen in stochastic differential equation-based denoising methods. We establish existence of solutions to the corresponding ODEs and regularity of the induced flows on measures, even when control laws are nonsmooth. Finally, we illustrate the framework on linear and nonlinear systems, demonstrating its effectiveness for both measure transport and stabilization problems.

Figures (5)

• Figure 1: Comparison of final output positions projections of target distribution and trained six-state linear system.
• Figure 2: Comparison of final positions projections of target distribution and trained driftless system \ref{['eq:drsys1']}.
• Figure 3: Trajectories of the time-reverse system visualized in 3D, stabilized to the origin.
• Figure 4: Trajectories of the time-reverse system visualized in 3D, stabilized to the unit sphere.
• Figure 5: Time-reversed trajectories of the Martinet system visualized in 3D. The learned policy drives a wide range of initial conditions close to the origin.

Theorems & Definitions (31)

• Theorem 4.1: Flow Matching with Control System Constraints
• proof
• Corollary 4.2: Controlled Flow Matching with Initial and Final Measure Constraints
• Theorem 4.3: Superposition Principle ambrosio2014continuity
• Theorem 4.4: Realization of Process
• proof
• Theorem 4.5: Approximate Flow Matching
• proof
• Theorem 4.6: Output Flow Matching
• proof