A unified Razumikhin-type criterion on input-to-state stability of time-varying impulsive delayed systems

This paper discusses the problem of input-to-state stability (ISS) of time-varying impulsive delayed systems. By introducing a switching parameter and using the notion of average impulsive interval, a unified Razumikhin-type criterion on ISS which is simultaneously effective for stabilizing impulses and destabilizing impulses is derived. The condition which requires the coefficient of the estimated upper bound of the derivative of a Lyapunov function to be constant in the existing results on ISS of impulsive systems is weakened. The results in this paper allow the coefficient of the derivative of a Lyapunov function to be time-varying function which can be both positive and negative and may even be unbounded. Furthermore, the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and large enough upper bound simultaneously. As a by-product, a unified criterion on ISS for time-varying impulsive delay-free systems is also presented. Two examples are presented to illustrate the effectiveness of our results.