The Domain of Attraction of the Desired Path in Vector-Field Guided Path Following

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in [1]. In this paper, we strengthen the result in [1, Theorem 2] by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an n-dimensional ambient manifold {\mathcal{M}}, is homeomorphic to \mathbb {R}^{n-1} \times \mathbb {S}^{1}, and not just homotopy equivalent to \mathbb {S}^{1} as shown in [1, Theorem 2]. This result is extended for a k-dimensional compact manifold for k \ge 2.