Homogeneous state feedback stabilization of homogenous systems

We show that for any asymptotically controllable homogeneous system in eucl idean space (not necessarily Lipschitzat the origin) there exists a homogen eous control Lyapunov function and a homogeneous, possibly discontinuous st ate feedback law stabilizing the corresponding sampled closed loop system. If the system satis es the usual local Lipschitz condition on the whole spa ce we obtain semiglobal stability of the sampled closed loop system for eac h sufficiently small fixed sampling rate. If the system satis es a global L ipschitz condition we obtain global exponential stability for each sufficie ntly small fixed sampling rate. The control Lyapunov function and the feedb ack are based on the Lyapunov exponents of a suitable auxiliary system and admit a numerical approximation.