Fixed-time synchronization in probability of drive-response networks with discontinuous nodes and noise disturbances

This paper proposes a novel lemma to study the fixed-time synchronization in probability, which is less conservative and its estimated time is independent of the system's initial values. Based on the new lemma, the fixed-time synchronization in probability of stochastic drive-response system with discontinuous nodes is investigated by designing a universal and simple controller. As two special cases, two kinds of convenient controllers are designed to realize the fixed-time synchronization in probability of the continuous dynamical system with noise disturbances and the fixed-time synchronization of discontinuous networks. Finally, numerical simulations illustrate the correctness of our main results. Particular, it is found that the gap between the estimated settling time and the real synchronization time can be bridged if the parameters and the index in controllers increase.