Fixed-Time Cluster Synchronization of Discontinuous Directed Community Networks via Periodically or Aperiodically Switching Control

The fixed-time cluster synchronization problem for a class of directed community networks with linear couplings and discontinuous nodes is investigated in this paper. First, by means of reduction to absurdity and mathematical induction, two new and vital differential equalities are proposed and proved to study the fixed-time stability. Besides, by designing periodically switching controllers and aperiodically adaptive switching controllers, as well as using the theory of differential inclusions, some fixed-time cluster synchronization criteria are derived in which the settling time function is bounded for any initial values. Finally, the numerical simulations are performed to show the feasibility and effectiveness of the control methodology by comparing with the corresponding finite-time synchronization problem.