Stabilization of Nonlinear Planer Systems using Homogeneous Eigenvalues

Homogeneous systems play important roles in nonlinear control theory. We proposed homogeneous eigenvalues, and provided sufficient conditions for asymptotic stability using homogeneous eigenvalue analysis in the previous papers. In this paper, we prove that there exists a homogeneous system which has uncountable infinite homogeneous eigenvalues. Moreover, we introduce the definition of the oscillation in nonlinear systems and prove the condition for oscillation. Finally, we construct the controller which does not oscillate each solution of the controlled system