Can neural networks with arbitrary delays be finite-timely synchronized?

Finite-time synchronization means the optimality in convergence time, thus many contributions have been made to it in the literature. However, to the best of our knowledge, most of the existing results on finite-time synchronization do not include time-delay. Considering the fact that time-delays especially infinite-time distributed delays are inevitably existing in neural networks, this paper aims to study global synchronization in finite time of neural networks with both time-varying discrete delay and infinite-time distributed delay (mixed delays). The techniques that we apply in this paper are not only different from the techniques employed in existing papers, but also applicable to differential systems with or without delay. Based on new Lyapunov–Krasovskii functional candidate and the new analysis techniques, sufficient conditions guaranteeing the finite-time synchronization of the addressed neural networks are derived by using a class of simple discontinuous state feedback controller. Conditions for realizing finite-time synchronization of neural networks with finite-time distributed delay and without delay are also given. Moreover, estimation of the upper bound of synchronization-time is also provided for neural networks with finite-time distributed delay and without delay. It is shown that the synchronization-time depends on both the initial values and the time-delays of the drive-response systems. Numerical examples demonstrate the effectiveness of the theoretical results.