Lyapunov criteria for local and global finite-time stability of impulsive systems with proportional delayed impulses

This article focuses on the issue of finite-time stability (FTS) of nonlinear systems with stabilizing and destabi-lizing impulses exhibiting proportional delays. Some Lyapunov-based local/global FTS criteria with settling-time estimation are established under a relaxed Lyapunov inequality condition. Specifically, two local FTS criteria are obtained by constructing an appropriate set of impulsive instant sequences that can be used to estimate the upper bound of the number of impulses experienced by the state before reaching zero. Furthermore, by utilizing the average impulsive interval (AII) approach, we expand the domain of attraction to a global scope in the case of stabilizing delayed impulses, while only local FTS can be achieved in the case of destabilizing delayed impulses. Our results indicate that under the relaxed Lyapunov condition, the time delay in stabilizing impulses can either promote or compromise the system stability, depending on the continuous dynamics of the system. In either case, sufficient conditions are established to guarantee the FTS of the system, regardless of the impact of the delayed impulses. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.