Stabilization of singularly perturbed systems with a polynomial right-hand side

The article considers the problem of stabilization of singularly perturbed systems of ordinary differential equations with homogeneous right-hand side in the form of polynomials of odd degree. The values of the perturbing parameter are supposed to be small. Sufficient conditions are obtained for stabilizing the zero solution of these systems to asymptotic stability by feedback control in the form of polynomials of the same degree as the right-hand side of the original system. It is assumed that only components of the vector of slow variables are subject for measurement and that control can only enter into the slow subsystem. For various cases, methods for constructing stabilizing controls are described. As a method of investigation, decomposition of a singularly perturbed system into a fast and slow subsystems of smaller dimension is applied. For stability analysis, the Lyapunov function method is used.