Adaptively Synchronize the Derivative Coupled Complex Networks With Proportional Delay

This article investigates adaptive control and exponential synchronization of a kind of derivative coupled complex dynamical networks (CDNs) with proportional delay. Based on impulsive control and adaptive pinning control protocols, sufficient criteria for achieving of the exponential synchronization on CDNs are obtained by jointly applying the proportional delayed impulsive comparison principle, the extended parameters variation formula and the definition of average impulsive interval. Meanwhile, suitable control gains for achieving adaptive synchronization are acquired according to the efficiently designed adaptive updating laws. In addition, the convergence velocity of the exponential synchronization is precisely estimated. Finally, one numerical simulation is presented to illustrate the validity of the adaptive pinning control protocols and theoretical results. By introducing the concept of impulsive distance for the first time, the example further explains the dynamic balance between the impulsive effects and the feedback control gains, which provides a method in controller designing.