Asymptotic bp-stabilization of a given closed invariant set

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.