Refining dichotomy convergence in vector-field guided path-following control

In the vector-field guided path-following problem, the desired path is described by the zero-level set of a sufficiently smooth real-valued function and to follow this path, a (guiding) vector field is designed, which is not the gradient of any potential function. The value of the aforementioned real-valued function at any point in the ambient space is called the level value at this point. Under some broad conditions, a dichotomy convergence property has been proved in the literature: the integral curves of the vector field converge either to the desired path or the singular set, where the vector field attains a zero vector. In this paper, the property is further developed in two respects. We first show that the vanishing of the level value does not necessarily imply the convergence of a trajectory to the zero-level set, while additional conditions or assumptions identified in the paper are needed to make this implication hold. The second contribution is to show that under the condition of real-analyticity of the function whose zero-level set defines the desired path, the convergence to the singular set (assuming it is compact) implies the convergence to a single point of the set, dependent on the initial condition, i.e. limit cycles are precluded. These results, although obtained in the context of the vector-field guided path-following problem, are widely applicable in many control problems, where the desired sets to converge to (particularly, a singleton constituting a desired equilibrium point) form a zero-level set of a Lyapunov(-like) function, and the system is not necessarily a gradient system.