In this study, we elucidated the exponential synchronization of a complex network system with time-varying delay. Then the exponential synchronization control of several types of complex network systems with time-varying delay under no requirements of delay derivable were explored. The dynamic behavior of a system node shows time-varying delays. Thus, to derive suitable conditions for the exponential synchronization of different complex network systems, we designed a linear feedback controller for linear coupling functions, using the Lyapunov stability theory, Razumikhin theorem, and Newton-Leibniz formula. The exponential damping rates for the exponential synchronization of different complex network systems were then estimated. Finally, we validated our conclusions through a numerical simulation.

This paper investigates the finite-time synchronization problems of complex spatiotemporal networks with time delays and diffusion terms. First, a boundary controller based on Lyapunov stability theory, Wirtinger's inequality, and finite-time analysis is designed. Subsequently, sufficient conditions for finite-time synchronization are obtained, and the setting time of finite-time synchronization is estimated. Finally, a simulation example is given to demonstrate the effectiveness of the obtained result.INDEX TERMS Finite-time synchronization, complex network, time delay, diffusion term, boundary control.