Logarithmic Connections on Principal Bundles and Their Applications to Geometric Control Theory

In this research, we establish a precise correspondence between the theory of logarithmic connections on principal G-bundles over compact Riemann surfaces and the geometric formulation of control systems on curved manifolds, providing a novel differential–geometric framework for analyzing optimal control problems with non-holonomic constraints. By characterizing control systems through the geometric structure of flat connections with logarithmic singularities at marked points, we demonstrate that optimal trajectories correspond precisely to horizontal lifts with respect to the connection. These horizontal lifts project onto geodesics on the punctured surface, which is equipped with a Riemannian metric uniquely determined by the monodromy representation around the singularities. The main geometric result proves that the isomonodromic deformation condition translates into a compatibility condition for the control system. This condition preserves the conjugacy classes of monodromy transformations under variations of the marked points, and ensures the existence and uniqueness of optimal trajectories satisfying prescribed boundary conditions. Furthermore, we analyze systems with non-holonomic constraints by relating the constraint distribution to the kernel of the connection form, showing how the degree of non-holonomy can be measured through the failure of integrability of the associated horizontal distribution on the principal bundle. As an application, we provide computational implementations for SL(2,C) connections over hyperbolic Riemann surfaces with genus g≥2, explicitly constructing the monodromy-induced metric via the Poincaré uniformization theorem and deriving closed-form expressions for optimal control strategies that exhibit robust performance characteristics under perturbations of initial conditions and system parameters.