基于切片最优传输控制的分布引导
Distribution Steering via Sliced Optimal Transport Control
- The University of Tokyo（东京大学）
- KTH Royal Institute of Technology（瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对动力系统分布引导问题，提出基于切片最优传输的有限时间控制框架，通过投影最优传输映射构造反馈律，实现分布引导并验证其收敛性，扩展至线性系统并给出数值示例。
AI中文摘要：
分布引导旨在寻找反馈律，以驱动动力系统的状态分布在规定的初始分布和终端分布之间。最优传输提供了一种自然的几何方法，但其实现通常需要全状态空间中的传输映射或耦合。切片最优传输通过一维投影避免了这种全维构造，但得到的投影映射仅指定方向位移，本身并不能规定可实现的反馈律。为此，我们开发了一种基于切片最优传输的有限时间控制框架：在每个采样时刻，投影最优传输映射定义了一个方向终端条件，其最小能量实现产生了一个随机单向控制器；对投影方向取平均则得到确定性切片反馈。对于单积分器动力学，该平均反馈使到目标的切片 Wasserstein 距离非增；对于高斯端点分布，该反馈是仿射的，保持高斯性，并将均值和协方差引导至规定的终端值。我们进一步确定了依赖于分布的增益，该增益可使切片 Wasserstein 距离线性衰减，并明确刻画了控制能量；还证明了随着采样周期趋于零，随机控制器收敛到平均切片流。最后，我们将该构造扩展到线性动力系统：可达性归一化坐标允许均匀全驱动系统瞬时实现切片速度，而局部可控性格拉姆矩阵为一般可控系统提供精确的有限步实现。数值示例说明了所得的分布流。
英文摘要：
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.
