Pinning a Complex Delayed Dynamical Network to a Homogenous Trajectory

Time delay exists in many fields, such as biology, neural network, mechanics, ecology, and so on. It is important for us to investigate the control of interconnected time-delay systems. In this brief, we propose two schemes to pin a complex delayed dynamical network to a homogenous trajectory. In particular, we prove that the delayed dynamical network is asymptotically synchronized with linear feedback control, and it is exponentially asymptotically synchronized with adaptive feedback controllers. We also find that the number of pinned nodes satisfies an inequality for synchronization. Additionally, the coupling matrix is not necessarily symmetric, and the pinned nodes can randomly be selected. Moreover, the linear feedback gain need not be large. Finally, two well-known network models are provided as illustrative examples to confirm the effectiveness of the technique.