A Geometric Solution of the Schr\"odinger Bridge Problem on $\mathsf{SO}(2)$ via Stochastic Optimal Control
We present a geometric coordinate-free solution to the isotropic Schr\"odinger bridge problem (SBP) for the kinematic equation on the Lie group $\mathsf{SO}(2)$. We consider the angular velocity of the system as the control input and assume that the given initial and terminal state probability density functions defined on $\mathsf{SO}(2)$ in our SBP are continuous and strictly positive. We solve the SBP by proving the existence and uniqueness of a solution to the so-called Schr\"odinger system of equations on $\mathsf{SO}(2)$, by showing that a fixed-point recursion is contractive in a complete metric space with respect to the Hilbert's projective metric. The geometric controller thus designed only uses the intrinsic geometric structure of $\mathsf{SO}(2)$ and does not embed it in the Euclidean plane to achieve the optimal density control. The numerical simulation verifies the validity of the theoretical construction of the Schr\"odinger bridge. The code and animations are publicly available at \texttt{\href{https://gitlab.com/a5akhtar/sbp}{https://gitlab.com/a5akhtar/sbp}}.