On the Stability by the Nonlinear NonstationaryHybrid Approximation

The paper investigates the effect of nonstationary perturbations on the stability of nonlinear nonautonomous systems with switching and impulsive effects. Sufficient conditions guaranteeing the asymptotic stability of a given equilibrium position of the original system are obtained, and restrictions under which the asymptotic stability is preserved under perturbations acting on the system are established. Note that the nonstationarities both in the system itself and in the perturbations can be described by functions unbounded with respect to time as well as by functions arbitrarily close to zero. It is assumed that the unperturbed system is homogeneous in the state vector. To find the desired results, Lyapunov’s second method is used in conjunction with the theory of differential inequalities.