**6. Conclusions**

In this paper, the synchronization problem was investigated for neural networks. It is well known that the Lyapunov direct method is the most effective method to analyze the stability of neural networks; the authors gave an important inequality on the Caputo derivative of quadratic functions, which plays an important role in analyzing the stability of fractional-order systems. By using Lyapunov functionals and analytical techniques, we obtained some sufficient conditions, and we derived event triggering to guarantee the synchronization of the delayed neural networks. We appled the Lyapunov functional method and the LMI approach to establish the synchronization criteria for the fractionalorder nerual network matrix. A linear matrix inequality approach was developed to solve the problem. Numerical examples were given to demonstrate the effectiveness of the proposed schemes. Future work will focus on event-triggered control for fractional-order systems with time-delay and measurement noises. In addition, more effective eventtriggered schemes such as an adaptive one, a dynamic one, and a hybrid one will also be considered for the stability analysis of fractional-order systems.

**Author Contributions:** Conceptualization, M.H., M.S.A., T.F.I. and B.A.Y.; methodology, K.I.O., M.H., M.S.A., T.F.I. and B.A.Y.; software, M.H., M.S.A. and K.I.O.; validation, M.H., M.S.A., T.F.I. and K.M.; formal analysis, K.I.O., M.H., M.S.A., T.F.I. and K.M.; investigation, M.H., T.F.I., B.A.Y. and K.I.O. resources, M.H., M.S.A., T.F.I. and K.M.; writing—review and editing, M.H., M.S.A., T.F.I., B.A.Y. and K.M.; visualization, M.H., M.S.A., T.F.I. and B.A.Y.; supervision, K.I.O., M.H., M.S.A., T.F.I. and B.A.Y.; project administration, M.H., M.S.A., T.F.I. and K.M.; funding acquisition, M.S.A. and K.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large groups (project under grant number RGP.2/47/43/1443). Moreover, this research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [grant number B05F650018].

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
