Homogeneous Lyapunov function for homogeneous continuous vector field

The goal of this article is to provide a construction of a homogeneous Lyapunov function V associated with a system of differential equations xdotf(x), x ϵRn (n≥1), under the hypotheses: (1) f ϵ C(Rn, Rn) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function V(x) tends to infinity with ‖x‖, and belongs to C∞(Rn/{0}, R)∩Cp(Rn, R), with pϵN∗ as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.