Finite-time synchronization of Markovian neural networks with proportional delays and discontinuous activations

In this paper, finite-time synchronization of neural networks (NNs) with discontinuous activation functions (DAFs), Markovian switching, and proportional delays is studied in the framework of Filippov solution. Since proportional delay is unbounded and different from infinite-time distributed delay and classical finite-time analytical techniques are not applicable anymore, new 1-norm analytical techniques are developed. Controllers with and without the sign function are designed to overcome the effects of the uncertainties induced by Filippov solutions and further synchronize the considered NNs in a finite time. By designing new Lyapunov functionals and using M-matrix method, sufficient conditions are derived to guarantee that the considered NNs realize synchronization in a settling time without introducing any free parameters. It is shown that, though the proportional delay can be unbounded, complete synchronization can still be realized, and the settling time can be explicitly estimated. Moreover, it is discovered that controllers with sign function can reduce the control gains, while controllers without the sign function can overcome chattering phenomenon. Finally, numerical simulations are given to show the effectiveness of theoretical results.