Schr\"odinger Bridge Over A Compact Connected Lie Group

TL;DR

This paper develops a geometric, coordinate-free approach to the Schr"odinger bridge problem on compact Lie groups, providing existence, uniqueness, and constructive solutions with numerical examples on SO(2) and SO(3).

Contribution

It introduces a geometric, coordinate-free formulation for the Schr"odinger bridge problem on Lie groups, ensuring solutions respect the group's structure and deriving an optimal interpolating controller.

Findings

Proved existence and uniqueness of solutions to the Schr"odinger system on Lie groups.

Developed a geometric controller for optimal density interpolation.

Provided numerical examples on SO(2) and SO(3).

Abstract

This work studies the Schr\"odinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schr\"odinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$. The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .