Pinning lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay

In this paper, a linear and adaptive feedback pinning strategy is used to study the lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay. Firstly, based on the Lyapunov stability theory, a novel Lyapunov functional, which involves the estimated error \(\hat{e}_{i}(t)\) rather than the general synchronization error \(e_{i}(t)\), is constructed. Secondly, in view of the unknown information of the channel time-delay, two available pinning controllers are designed such that the considered networks achieve lag synchronization. Furthermore, by a proper adaptation mechanism, we estimate the unknown channel time-delay successfully under the case that the initial value of the estimated channel time-delay is larger than true channel time-delay, i.e., \(\hat{\tau }(0)>\tau \) and the another case \(\hat{\tau }(0)<\tau \). Finally, the effectiveness and correctness of the lag synchronization criteria are verified through two simulation experiments.