Synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights and discontinuous nodes based on sliding mode control strategy.

This paper is concerned with the global synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights, where the dynamic nodes are modeled to be discontinuous, and subject to the local Hölder nonlinear growth in a neighborhood of continuous points. Firstly, an inequality with respect to variable-order fractional derivative for convex functions is proposed. On the basis of the proposed inequality, a global convergence principle in finite time for absolutely continuous functions is developed. Secondly, based on proposed convergence principle in finite time, a new sliding mode surface is presented, and an appropriate sliding mode control law is designed to drive the trajectory of the error system to the prescribed sliding mode surface in finite time and remain on it forever. In addition, on the basis of differential inclusions theory and Lur'e Postnikov-type convex Lyapunov function approach, the sufficient conditions with respect to the global stability in finite time are established in terms of linear matrix inequalities for the error system on designed sliding mode surface. Moreover, the upper bound of the settling time is explicitly evaluated. Finally, the effectiveness and correction of synchronization strategies are illustrated through two simulation experiments.