Schrödinger Bridges over Compact Lie Groups
Generalizing coordinate-free density steering from SO(2) to any compact connected Lie group, with numerical examples on SO(2) and SO(3).
This work lifts our SO(2) result to its natural level of generality: the Schrödinger bridge problem on any compact connected Lie group. The goal is the same — steer a controlled diffusion between given initial and terminal densities supported on the group while minimizing control effort — but the formulation is fully coordinate-free, respecting the group’s geometry and avoiding local parameterizations or Euclidean embeddings.
Working in the Banach space of continuous functions on the group and using Hilbert’s projective metric, we prove existence and uniqueness of the solution to the corresponding Schrödinger system, and the result is constructive: it yields the geometric controller that optimally interpolates the densities. This geometric perspective matters for groups like SO(3), which appear throughout attitude control, DNA statistical mechanics, flexible-needle steering, and mobile robotics.
Optimally interpolated densities steered over the group — illustrated on the rotation groups SO(2) and SO(3).
Contributions
- A coordinate-free formulation of the SBP for any compact connected Lie group.
- Existence–uniqueness of the Schrödinger system solution via a contraction in Hilbert’s projective metric.
- A constructive geometric controller, with numerical examples on SO(2) and SO(3).
Schrödinger Bridge Over a Compact Connected Lie Group — Hamza Mahmood, Abhishek Halder, Adeel Akhtar (IEEE Control Systems Letters).