Finite-time adaptive cluster synchronization of complex-variable fractional-order dynamic networks: A new finite-time fractional-order differential inequality approach

This paper is devoted to the problem of finite-time adaptive cluster synchronization (CS) for complex-variable fractional-order dynamic networks (CVFODNs). Initially, based on the idea of variable substitution and the extreme value theorem, a new finite-time fractional-order (FO) differential inequality is developed, which provides a very useful tool to explore finite-time adaptive synchronization or stability in FO systems. Subsequently, a practical and cost-effective adaptive control approach is proposed, where the signum function is excluded and the adaptive law has a concise form. Furthermore, by the aid of the newly developed inequality and the proposed adaptive controller, some new criteria on finite-time CS are derived for the addressed CVFODNs and the corresponding cluster-synchronized settling times are reliably estimated. Our findings indicate the resultant estimates are closely related to the control parameters and the order of the fractional derivative. At last, the validity of the derived results is confirmed by a numerical example.