This article studied the locally exponential stability (LES) of time-delay systems subject to delayed impulses. Some Lyapunov-Razumikhin (L-R) theorems were presented, in which the information about the delays within the impulses was fully incorporated and then integrated into the stability analysis of the concerned systems. Our results highlight a critical finding: the delays in impulses can have dual effects on the stability of the systems, i.e., they may either destabilize the systems or contribute to the stability of the systems. Moreover, the effects of the nonlinear rate in discrete dynamics were fully considered, where a new relationship between the discrete dynamics, the continuous dynamics, and the initial region was established. As applications, several sufficient conditions that formulated in terms of linear matrix inequalities (LMIs) were obtained to ensure the stability of certain time-delay systems with highly nonlinear delayed impulses. To illustrate the applicability and effectiveness of the proposed results, two numerical examples were provided.

This paper investigates the lag synchronization problem of complex-valued interval neural networks with both discrete and distributed time-varying delays under delayed impulsive control. A distributed delayed impulsive controller that depends on the accumulation of the states over a history time period is designed to guarantee the exponential lag synchronization between the drive and the response systems. By employing the complex Lyapunov method and a novel impulsive differential inequality technique, some delay-dependent synchronization criteria are established in terms of complex-valued linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.