Stabilization of homogeneous polynomial systems in the plane

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In this paper, we study the problem of stabilization via homogeneous feedback of single-input homogeneous polynomial systems in the plane. We give a complete classification of systems for which there exists a homogeneous stabilizing feedback that is smooth on $\mathbb{R}^2 \setminus\{ (0,0)\}$ and preserve the homogeneity of the closed loop system. Our results are essentially based on Theorem of Hahn in which the author gives necessary and sufficient conditions of stability of homogeneous systems in the plane.