Schrödinger Bridge Over A Compact Connected Lie Group
This work addresses the Schrödinger bridge problem for kinematic equations on Lie groups, which is incremental as it extends existing methods to a geometric setting with applications in domains like robotics or physics.
The paper tackles the problem of steering a controlled diffusion between given initial and terminal densities on a compact connected Lie group while minimizing control effort, by developing a coordinate-free formulation that respects the geometric structure and establishing existence and uniqueness of the solution, with numerical examples on SO(2) and SO(3).
This work studies the Schrödinger 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 Schrödinger 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 $\mathsf{SO}(3)$.