  In this paper, the fixed-time cluster synchronization problem for complex
networks via pinning control is discussed. Fixed-time synchronization has been
a hot topic in recent years, which means that the network can achieve
synchronization in finite-time and the settling time is bounded by a constant
for any initial values. To realize the fixed-time cluster synchronization, a
simple distributed protocol by pinning control technique is designed, whose
validity is rigorously proved, and some sufficient criteria for fixed-time
cluster synchronization are also obtained. Especially, when the cluster number
is one, the cluster synchronization becomes the complete synchronization
problem; when the intrinsic dynamics for each node is missed, the fixed-time
cluster synchronization becomes the fixed-time cluster (or complete) consensus
problem; when the network has only one node, the coupling term between nodes
will disappear, and the synchronization problem becomes the simplest
master-slave case, which also includes the stability problem for nonlinear
systems like neural networks. All these cases are also discussed. Finally,
numerical simulations are presented to demonstrate the correctness of obtained
theoretical results.
