Synchronization of Discontinuous Neural Networks with Delays via Adaptive Control

The drive-response synchronization of delayed neural networks with discontinuous activation functions is investigated via adaptive control. The synchronization of this paper means that the synchronization error approaches to zero for almost all time as time goes to infinity. The discontinuous activation functions are assumed to be monotone increasing which can be unbounded. Due to the mild condition on the discontinuous activations, adaptive control technique is utilized to control the response system. Under the framework of Filippov solution, by using Lyapunov function and chain rule of differential inclusion, rigorous proofs are given to show that adaptive control can realize complete synchronization of the considered model. The results of this paper are also applicable to continuous neural networks, since continuous function is a special case of discontinuous function. Numerical simulations verify the effectiveness of the theoretical results. Moreover, when there are parameter mismatches between drive and response neural networks with discontinuous activations, numerical example is also presented to demonstrate the complete synchronization by using discontinuous adaptive control.