On the Stability by the Nonlinear Nonstationary Hybrid Approximation

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