This paper studies input-to-state stability (ISS) of general nonlinear time-delay systems subject to delay-dependent impulse effects. Sufficient conditions for ISS are constructed by using the method of Lyapunov functionals. It is shown that, when the continuous dynamics are ISS but the discrete dynamics governing the delay-dependent impulses are not, the impulsive system as a whole is ISS provided the destabilizing impulses do not occur too frequently. On the contrary, when the discrete dynamics are ISS but the continuous dynamics are not, the delayed impulses must occur frequently enough to overcome the destabilizing effects of the continuous dynamics so that the ISS can be achieved for the impulsive system. Particularly, when the discrete dynamics are ISS and the continuous dynamics are also ISS or just stable for the zero input, the impulsive system is ISS for arbitrary impulse time sequences. Compared with the existing results on impulsive time-delay systems, the obtained ISS criteria are more general in the sense that these results can be applied to systems with delay dependent impulses while the existing ones can not. Moreover, when consider time-delay syste

This article investigates the problem of exponential bipartite synchronization of fractional-order multilayer signed networks via hybrid impulsive control. First, the mathematical model of the networks is established by integrating the fractional dynamics of nodes, the multilayer network edges, and the positive and negative weights (antagonistic relationship), which is more pluralistic and practical. Second, a hybrid impulsive controller is designed, which is composed of the feedback control part and the impulsive control part to realize the exponential bipartite synchronization objective. Both positive and negative impulsive effects are considered and the ranges of the impulsive control gain related to the order of Caputo fractional derivative are discussed. In addition, some sufficient conditions for exponential bipartite synchronization of fractional-order multilayer signed networks are obtained by using the signed graph theory, Lyapunov method, and average impulsive interval method. Furthermore, in order to illustrate the practicability of the theoretical results, fractional-order coupled Chua’s circuits model and fractional-order power systems built on multilayer signed networks are established and the exponential bipartite synchronization issues are analyzed. Numerical examples and simulations are provided to show the effectiveness of the obtained results.

In this article, the sampled-data-based event-triggered synchronization control for fractional and impulsive complex networks (CNs) with time-varying delay is investigated and a class of more general network structure based on the switching topologies at impulsive instants is considered. First, a class of novel fractional-order integral inequalities is produced to obtain depend-delay synchronization criteria and estimate Lyapunov–Krasovskii functions. Then, a sampled-data-based event-triggered control is designed, which can ensure synchronization of fractional and impulsive CNs (FICNs) with time-varying delay. Next, by using the Lyapunov direct method, some criteria are obtained to guarantee the synchronization of FICNs. Numerical simulations are given to demonstrate that the designed sampled-data-based event-triggered synchronization strategy can effectively not only achieve synchronization of FICNs but reduce the frequency of controller update compared to the previous related works.