Asymptotic Stability Investigation of the Zero Solution for a Class of Nonlinear Nonstationary Systems by the Averaging Method

Abstract

A system of nonlinear differential equations is considered that describes the interaction of two coupled subsystems, one of these subsystems is linear, and the other is nonlinear and homogeneous with an order of homogeneity greater than one. It is assumed that this system is affected by nonstationary perturbations with zero mean values. Using the averaging method, sufficient conditions are determined under which perturbations do not disturb the asymptotic stability of the zero solution. The derivation of these conditions is based on the use of a special construction of the nonstationary Lyapunov function which takes into account the structure of the acting perturbations. In addition, we consider the case where there is a constant delay in the right-hand sides of the system. An original approach to the construction of the Lyapunov-Krasovskii functional for such a system is proposed. Using this functional, conditions are found that guarantee the preservation of the asymptotic stability for any positive delay.