Intralayer Synchronization of Multiplex Dynamical Networks via Pinning Impulsive Control

These days, the synchronization of multiplex networks is an emerging and important research topic. Grounded framework and theory about synchronization and control on multiplex networks are yet to come. This article studies the intralayer synchronization on a multiplex network (i.e., a set of networks connected through interlayer edges), via the pinning impulsive control method. The topologies of different layers are independent of each other, and the individual dynamics of nodes in different layers are different as well. Supra-Laplacian matrices are adopted to represent the topological structures of multiplex networks. Two cases are considered according to impulsive sequences of multiplex networks: 1) pinning controllers are applied to all the layers simultaneously at the instants of a common impulse sequence and 2) pinning controllers are applied to each layer at the instants of distinct impulse sequences. Using the Lyapunov stability theory and the impulsive control theory, several intralayer synchronization criteria for multiplex networks are obtained, in terms of the supra-Laplacian matrix of network topology, self-dynamics of nodes, impulsive intervals, and the pinning control effect. Furthermore, the algorithms for implementing pinning schemes at every impulsive instant are proposed to support the obtained criteria. Finally, numerical examples are presented to demonstrate the effectiveness and correctness of the proposed schemes.