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

TL;DR

This paper introduces a novel Sinkhorn recursion with memory to solve the control-affine Schrödinger bridge problem when input and noise channels mismatch, extending existing methods to nonlinear PDEs.

Contribution

It develops a new algorithm leveraging PDE structure to handle channel mismatch in nonlinear Schrödinger bridge problems, with proven local stability.

Findings

The proposed Sinkhorn recursion with memory effectively solves mismatched channel problems.

The algorithm demonstrates local stability in the nonlinear PDE setting.

It extends the applicability of density steering solutions to more general control scenarios.

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.