Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization

Without imposing any growth condition, we prove that every chain of odd power integrators perturbed by a C¹ triangular vector field is globally stabilizable via non-Lipschitz continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback because the Jacobian linearization may have uncontrollable modes whose eigenvalues are on the right half-plane. The proof is constructive and accomplished by developing a machinery – adding a power integrator – that enables one to explicitly design a C⁰ globally stabilizing feedback law as well as a C¹ control Lyapunov function which is positive definite and proper.