A brief survey on stability and stabilization of impulsive systems with delayed impulses

This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.