Non-Smooth Stabilizers for Nonlinear Systems with Uncontrollable Unstable Linearization

We prove that every chain of odd power integrators perturbed by a C triangular vector eld can be stabilized in the large via continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback. The proof is constructive and accomplished by developing a machinery|a continuous type of adding a power integrator|that enables one to explicitly design a C globally stabilizing feedback law as well as a C control Lyapunov function which is positive de nite and proper.