## Abstract

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.

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.

Original language | English |
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Title of host publication | European Control Conference (ECC) |

Publication status | Accepted/In press - 2021 |