Stability analysis and applications of time-delay systems subject to delayed impulses

The present study explores the stability problem of nonlinear time-delay systems subject to delayed impulses. Specifically, this paper addresses the dual aspects of both delayed impulsive control and perturbation issues on nonlinear systems with time-varying delays. In particular, by employing Halanay-type inequality techniques, we establish some sufficient criteria to ensure the exponential stability of the aforementioned time-delay systems. Moreover, we remove previous restrictions on the relative sizes of time delays in both continuous dynamics and impulses. It is shown that, under certain conditions, the delays in impulses may be arbitrarily finite and even exceed the duration of the impulsive intervals, but such delayed impulses solely affect the convergence rate of the addressed system without undermining its stability. Furthermore, this study's theoretical results are applied to the stabilization of sewage treatment systems and the impulsive synchronization of Chua's circuits. Two illustrative examples are given to demonstrate the effectiveness and validity of the derived results.