Asymptotic stabilization of three dimensional homogeneous quadratic systems

Necessary and sufficient conditions are given for the asymptotic stabilizability of a generic class of single-input, three-dimensional, homogeneous quadratic systems. Three key lemmas related to the stabilization of homogeneous systems are described. The first one gives a sufficient condition, the second one gives a necessary condition for the stabilizability, and the third one describes a class of deformation on the space of homogeneous systems, which will preserve the hypothesis of the first two lemmas. These lemmas are sufficient to determine the stabilizability of a generic class of homogeneous quadratic systems