Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses

This paper studies the synchronization problem of complex dynamical networks (CDNs) on time scales. A pinning impulsive control scheme that takes into account of time-delay effects is designed to achieve synchronization of CDNs on time scales with the state of an isolated node. Based on the theory of time scales and the direct Lyapunov method, a synchronization criterion is established for linear CDNs on general time scales. Our result shows that, by impulsive control a small portion of nodes, the consensus of CDNs on time scales can be achieved. According to our pinning impulsive control scheme, different numbers of nodes will be selected at each impulsive instant and time delay is considered in the pinning impulses. The modeling framework developed in this paper is a unification and generalization of many existing continuous-time and discrete-time CDN models, while the pinning impulsive control scheme is an extension of the existing control scheme for synchronization of continuous-time CDNs. Moreover, the idea of studying dynamical systems on time scales provide a unified approach to investigate continuous-time system and its discrete-time counterpart simultaneously. Numerical simulations are given to illustrate the effectiveness of the theoretical analysis.