The finite-time synchronization (FNTS) problem for a class of delayed fractional-order fully complex-valued dynamic networks (FFCDNs) with internal delay and non-delayed and delayed couplings is studied by directly constructing Lyapunov functions instead of decomposing the original complex-valued networks into two real-valued networks. Firstly, a mixed delay fractional-order mathematical model is established for the first time as fully complex-valued, where the outer coupling matrices of the model are not restricted to be identical, symmetric, or irreducible. Secondly, to overcome the limitation of the use range of a single controller, two delay-dependent controllers are designed based on the complex-valued quadratic norm and the norm composed of its real and imaginary parts’ absolute values, respectively, to improve the synchronization control efficiency. Besides, the relationships between the fractional order of the system, the fractional-order power law, and the settling time (ST) are analyzed. Finally, the feasibility and effectiveness of the control method designed in this paper are verified by numerical simulation.

This paper focuses on the finite-time (FNT) cluster synchronization issues for a class of delayed fractional-order fully complex-valued community networks (FFCVCNs). A new mathematical expression of the complex networks is developed with internal delay, non-delayed and delayed couplings, complex-valued state variables, system function, coupling strengths, inner coupling matrices and outer coupling matrices. Instead of transforming the complex-valued (CV) networks into two independent realvalued (RV) systems, the delay-dependent controllers are designed based on the quadratic norm and a novel norm composed of the absolute-valued norm to realize the cluster synchronization for the proposed complex networks in FNT, respectively. In addition, the upper bounds of the settling time (ST) when the system could reach finite-time cluster synchronization are estimated. The obtained results are less conservative than some of the existing studies due to the characteristics of fully fractional-order complex-valued (FOCV) dynamical networks. The feasibility and effectiveness of the main results are demonstrated by simulation examples.INDEX TERMS Finite-time synchronization; cluster synchronization; fully complex-valued community networks; delay.