Modified Fixed-Time Synchronization Criteria of Complex Networks with Time-Varying Delays via Continuous or Discontinuous Control

This paper investigates modified fixed-time synchronization (FxTS) of complex networks (CNs) with time-varying delays based on continuous and discontinuous controllers. First, for the sake of making the settling time (ST) of FxTS is independent of the initial values and parameters of the CNs, a modified fixed-time (FxT) stability theorem is proposed, where the ST is determined by an arbitrary positive number given in advance. Then, continuous controller and discontinuous controller are designed to realize the modified FxTS target of CNs. In addition, based on the designed controllers, CNs can achieve synchronization at any given time, or even earlier. And control strategies effectively solve the problem of ST related to the parameters of CNs. Finally, an appropriate simulation example is conducted to examine the effectiveness of the designed control strategies.