Stability of Time-Delay Systems with Delayed Impulses: Average Impulsive Estimation Approach

This paper addresses the exponential stability of impulsive systems with both pointwise and distributed delays, where delayed impulses are considered. Based on impulsive control theory, some Lyapunov-based sufficient conditions for exponential stability involving both impulsive perturbation and impulsive control are derived, respectively. Especially, the derived conditions do not impose any restriction on the magnitude relationship between the delay in continuous flow and impulsive delay in the case of impulsive perturbation. It also shows that the delay in continuous flow might have a potential effect on system stability. It may not be reasonable for existing results to assume a common threshold of impulsive strength at every impulse point, such as with in the case of impulsive perturbation. Here, based on the proposed concepts of average impulsive estimation and average positive impulsive estimation, impulsive estimation can be time-varying, and the information of impulsive delay can be integrated into it to guarantee the effect of impulse. The results of stability analysis are applied to the synchronization of complex networks with mixed delays and impulses. Numerical examples illustrate the efficiency of the derived results.