Asymptotic bp-stabilization of a given closed invariant set

TL;DR

This paper presents a method to asymptotically stabilize a closed invariant set in a dynamical system using a family of control vector fields that ensure global attraction of the set.

Contribution

The authors develop a novel control design that guarantees asymptotic stabilization of any given invariant set in smooth dynamical systems.

Findings

Constructed control vector fields achieve global asymptotic stabilization.

The method applies to any closed invariant set in smooth systems.

Ensures attraction of all bounded positive orbits to the invariant set.

Abstract

Given a closed invariant set $\mathcal{C}$ of a dynamical system generated by a smooth vector field, $X$, for each $\lambda > 0$, we construct a control vector field, $X_{0}^{\lambda}$, such that the perturbed dynamics generated by the vector field $X+X_{0}^{\lambda}$, globally asymptotically bp-stabilizes the invariant set $\mathcal{C}$, that is, $\mathcal{C}$ attracts every bounded positive orbit of the perturbed dynamical system.