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.