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arXiv · 2604.23370

Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schr\"{o}dinger Bridge Problem

Georgiy A. Bondar·Asmaa Eldesoukey·Yongxin Chen·Abhishek Halder
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Abstract

Solutions to the Schr\"{o}dinger 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 Schr\"{o}dinger 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 Schr\"{o}dinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

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Georgiy A. Bondar, Asmaa Eldesoukey, Yongxin Chen, Abhishek Halder. 2026-04-25. Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schr\"{o}dinger Bridge Problem. https://arxiv.org/abs/2604.23370

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