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 $\mathsf{SO}(3)$. The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .

翻译：本文研究了紧致连通李群上运动学方程的薛定谔桥问题。其目标是在给定初始和终端密度支撑于李群的条件下，引导受控扩散过程，同时最小化控制代价。我们发展了一种无坐标形式的随机最优控制问题表述，该表述尊重李群的底层几何结构，从而避免了局部参数化或嵌入欧几里得空间所带来的局限性。我们证明了相应薛定谔系统解的存在唯一性。我们的结果是构造性的，导出了在支撑于李群的概率密度之间实现最优插值的几何控制器。为说明结果，我们给出了在$\mathsf{SO}(2)$和$\mathsf{SO}(3)$上的数值示例。代码和动画公开于https://github.com/gradslab/SbpLieGroups.git。