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

We develop a \emph{flow-matching framework} for transporting probability measures under control-affine dynamics and for steering systems to points or target sets. Starting from the continuity equation associated with the control affine system of the form dx/dt = f_0(x) + \sum_{i=1}^m u_i f_i(x), we construct measure interpolations through exact, approximate flow matching, and extend the approach to 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 feedback 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: noising corresponds to destabilizing the system through excitations, while denoising corresponds to stabilization via time reversal. We propose two methods for constructing the noising process: (i) Randomized-control noising, which employs regular (non-white noise) controls through the endpoint map and naturally accommodates control constraints. (iI) PMP-based noising, which leverages the Hamiltonian system from Pontryagin's Maximum Principle, corresponding to fixed or variable end-point optimal control problems, to explore the configuration space by randomizing the adjoint vectors and recovers the optimal controller for linear systems with convex costs, while providing feasible feedback laws in the nonlinear case. Finally, we numerically illustrate the framework on linear and nonlinear systems, demonstrating its effectiveness for measure transport, steering systems to target sets and path planning in a domain with obstacles.