Exponential stability of time-delay systems with highly nonlinear impulses involving delays

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.