Homogeneous State Feedback Stabilization of Homogenous Systems

We show that for any asymptotically controllable homogeneous system in euclidean space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system. If the system satisfies the usual local Lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate. If the system satisfies a global Lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate. The control Lyapunov function and the feedback are based on the Lyapunov exponents of a suitable auxiliary system and admit a numerical approximation.