Stability of time-varying systems with delayed impulsive effects

In this article, we investigate the global uniform exponential stability problem of linear time-varying systems with delayed impulsive effects. Two cases are considered: in the first case, the impulse-free system is uniformly asymptotically stable and in the second case, the impulse-free system may not be uniformly asymptotically stable. When the impulse-free system is uniformly asymptotically stable, by the virtue of a uniformly asymptotically stable function, Lyapunov-based sufficient conditions are derived to ensure the uniform exponential stability of linear time-varying systems, where destabilizing and stabilizing impulses are considered, respectively. Different from existing results, the Lyapunov function is not assumed to decrease all the time and it is shown that under some conditions, the delayed impulsive effects are stabilizing. For the case that the impulse-free system is not uniformly asymptotically stable, a time-varying function is used to characterize the upper bound on the derivative of the Lyapunov function, and it is illustrated that the linear time-varying systems can be stabilized by the delayed impulsive effects. Moreover, it is also shown that the impulsive input delay-dependent criteria are less conservative than the delay-independent ones when the impulsive input delay is small. Some examples are presented to demonstrate the effectiveness of our theoretical results.