Error Estimation for Quasi-Synchronization of Multilayer Dynamical Networks: A Pinning Delayed Impulsive Control Scheme.

In this article, we address the error estimation problem of quasi-synchronization for a class of multilayer dynamical networks. The proposed network model simultaneously accounts for interlayer and intralayer time-varying coupling structures, network directionality, and interlayer communication delays. To achieve synchronization in a cost-effective manner, we design a novel pinning impulsive control strategy that leverages large-scale impulse delay information together with the number of pinned nodes. By employing an iterative algorithm, we establish a new delay-dependent impulsive differential inequality, which precisely characterizes the convergence domain and provides flexibility in the choice of impulse delays. Then, some quasi-synchronization criteria are derived to guarantee convergence of multilayer networks within a prescribed error level, and explicit analytical expressions for the synchronization error bounds are obtained. Finally, to demonstrate the practical applicability, the proposed criteria are applied to the synchronization of multilayer single-link robot arm networks under error bounds, with numerical examples validating the effectiveness of the method.