The quasi-synchronization problems of the delayed multiplex networks with stochastic perturbations are investigated in this article. Different from previous studies on complete synchronization for the delayed multiplex networks, quasi-synchronization, which is more practical and more challenging than complete synchronization, is discussed. To adopt a better cost-saving control, an impulsive pinning scheme is designed. That is, the key nodes in the multiplex networks are pinned and controlled at impulsive instants. At the same time, a virtual leader is introduced. Based on the theory of stochastic impulsive differential equations and the Lyapunov–Krasovskii functional method, the sufficient conditions are derived for the error systems to converge into a bounded region in the sense of the mean square. Therefore, stochastic quasi-synchronization is guaranteed. Finally, the two-layer Newman–Watts (NW) small-world networks and two-layer Barabási–Albert (BA) scale-free networks are provided to illustrate the effectiveness of the theoretical results.

This article investigates the synchronization of stochastic delayed neural networks under pinning impulsive control, where a small fraction of nodes are selected as the pinned nodes at each impulsive moment. By proposing a uniformly stable function as a new tool, some novel mean square decay results are presented to analyze the error system obtained from the leader and the considered neural networks. For the divergent error system without impulsive effects, the impulsive gains of pinning impulsive controller can admit destabilizing impulse and the number of destabilizing impulse may be infinite. However, if the error system without impulsive effects is convergent, to achieve the synchronization of the stochastic neural networks, the growth exponent of the product of impulsive gains can not exceed some positive constant. It is shown that the obtained results increase the flexibility of the impulsive gains compared with the existing results. Finally, a numerical example is given to illustrate the practicality of synchronization criteria.

In this article, by using Razumikhin-type technique, we investigate pth moment exponential stability of stochastic functional differential equations with Markovian switching and delayed impulses. Several stability theorems of impulsive hybrid stochastic functional differential equations are derived. It is assumed that the state variables on the impulses can relate to the finite delay. These new results are employed to a class of n-dimensional linear impulsive hybrid stochastic systems with bounded time-varying delay. Moreover, an effective M-matrix method is introduced to study the exponential stability of these hybrid systems. Meanwhile, some examples and simulations are given to show our results.