Exponential stability of delayed discrete-time systems with averaged delayed impulses

This article investigates the exponential stability of nonlinear discrete-time systems with time-varying state delay and delayed impulses, where the delays in impulses are not fixed. Specifically, the study is separated into two cases: (1) stability of delayed discrete-time systems with destabilizing delayed impulses, where the time delays in impulses can be flexible and even larger than the length of the impulsive interval; (2) stability of delayed discrete-time systems with mixed delayed impulses (means stabilizing and destabilizing delayed impulses exist simultaneously), where the time delays in impulses are unfixed between two adjacent impulsive instants. First, the concept of average impulsive delay (AID) is extended to discrete-time systems with delayed impulses. Then, some Lyapunov-based exponential stability criteria are provided for delayed discrete-time systems with delayed impulses, where the impulses satisfy the average impulsive interval (AII) condition and the delays in impulses satisfy the AID condition. For the delayed discrete-time systems with mixed delayed impulses, the exponential stability criterion is provided by limiting the geometric average of the impulses' strengths. The obtained results can be used to study the delayed discrete-time systems with some large impulses which may even be unbounded. It's also shown that the delays in impulses can have a stabilizing effect on the stability of delayed discrete-time systems. Some numerical examples are also provided to illustrate the effectiveness and advantage of the theoretical results.