🤖 AI Summary
This work addresses the problem of designing feedback control laws to steer the state distribution of a dynamical system from a given initial distribution to a prescribed terminal distribution. Building upon sliced optimal transport, the authors propose a finite-horizon sliced feedback control framework that defines directional terminal conditions via one-dimensional projections and combines minimum-energy control with directional averaging to construct deterministic feedback laws, thereby circumventing the need for high-dimensional optimal transport maps. The approach preserves the affine structure for Gaussian distributions, enabling exact regulation of both mean and covariance, and incorporates distribution-dependent gains that guarantee linear decay of the sliced Wasserstein distance. Theoretical analysis shows that the proposed stochastic controller converges to the mean sliced flow as the sampling period tends to zero, and the framework naturally extends to general linear systems, with numerical experiments confirming its efficacy.
📝 Abstract
Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.