From Artstein-Sontag theorem to the min-projection strategy

Abstract

Given a finite or countable family of continuous vector fields {fi(x)}i∈ I , we show that there exists a feedback k : Rn → I such that the switched system x = fk(x)(x) is globally asymptotically stable whenever there exists a smooth control Liapunov function V such that for all x ≠ 0, ∇V(x)fi(x)<0, for some i ∈ I.