Singularities and Metric Structures in Sub-Riemannian Geometries with Applications to Control Theory

Sub-Riemannian geometry extends classical Riemannian frameworks by defining metrics only on constrained directions within manifolds, naturally modeling systems with nonholonomic constraints. This paper investigates the nature and impact of singularities—points where the geometric structure or metric degenerates—on the local and global properties of sub-Riemannian manifolds. We analyze metric behavior near singularities through nilpotent approximations and study their influence on geodesic existence, uniqueness, and stability, with particular emphasis on abnormal geodesics. Further, we explore the crucial role these singularities play in optimal control problems for constrained dynamical systems, illustrating how they affect controllability, accessibility, and trajectory synthesis. Through canonical examples like the Heisenberg group and Martinet distribution, this work highlights the intricate interplay between geometry and control, laying groundwork for future advances in both theoretical understanding and practical applications.