Abstract
<abstract><p>This paper addresses the problem of exponential synchronization in continuous-time complex dynamical networks with both time-delayed and non-delayed interactions. We employ a proportional integral derivative (PID) control strategy and a dynamic event-triggered approach to investigate this synchronization problem. Our approach begins with constructing a general model for complex dynamical networks that incorporate delays. We then derive synchronization criteria based on the PID control parameters, utilizing linear matrix inequality techniques in conjunction with a dynamic event-trigger mechanism. The application of Lyapunov stability theory and inequality techniques allows us to establish these criteria, considering the presence of hybrid delays. To illustrate the effectiveness of our proposed model, we provide two numerical examples showcasing synchronization dynamics. These examples demonstrate the successful theoretical results of a novel PID controller and dynamic event-trigger mechanism.</p></abstract>
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