Homogeneous Lyapunov functions and necessary conditions for stabilization

We provide necessary conditions for the stabilization of nonlinear control systems with the additional requirement that a time-invarianthomogeneous Lyapunov function exists for the closed-loop system.The authors gratefully acknowledge research support from the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology, and Culture, and from the EC-Science Project SC1-0433-C(A). The first author is Charge de recherches F.N.R.S, on leave from CESAME, Université Catholique de Louvain, Belgium. He acknowledges partial support from the following organizations: National Science Foundation under Grant ECS-9203491, Air Force Office of Scientific Research under Grant F-49620-92-J-0495, Belgian American Educational Foundation, and North Atlantic Treaty Organization. The scientific responsibility rests with the authors.