解决控制与噪声通道不匹配时的非线性密度调控问题
Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schrödinger Bridge Problem
- 提出带记忆的动态Sinkhorn算法，应对非线性偏微分方程
- 在通道不匹配条件下实现最优密度路径的精确计算
- 适合研究随机控制与生成建模的科研人员
Schrödinger桥问题及其推广可导出受控扩散过程中的最优反馈控制策略。数值求解通常采用动态Sinkhorn递推方法，其数学基础是通过Hopf-Cole变换将最优性条件转化为边界耦合的线性偏微分方程组。近期研究指出，对于控制仿射型Schrödinger桥问题，该线性化仅在控制与噪声通道成比例时成立；当两者不匹配时，经Hopf-Cole变换后的方程仍为非线性，现有方法无法求解。本文提出一种带记忆的Sinkhorn递推算法，利用此类非线性PDE的结构，成功求解控制与噪声通道不匹配下的控制仿射型Schrödinger桥问题，并证明了所提算法的局部稳定性。
原文摘要 · Abstract (English)
Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.
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