On the existence of local smooth repulsive stabilizing feedbacks in dimension three

Given an affine control system in subject to the Hörmanderʹs condition at the origin, we prove the existence of a local smooth repulsive stabilizing feedback at the origin. Our construction is based on the classical homogenization procedure, on the existence of a semiconcave control-Lyapunov function, and on the classification of singularities of semiconcave functions in dimension two