**1. Introduction**

Fractional calculus is a mathematical theory that has been studied and applied in different fields for the past 300 years. Compared with traditional integer-order systems, fractional-order (FO) derivatives provide an excellent tool for the description of memory and inherent properties of various materials and processes, with applications in many areas, such as heat conduction, electronics, and abnormal diffusion [1,2]. As a result, fractional calculus has attracted increasing attention from physicists and engineers [3–7]. Moreover, fractional calculus has been applied to numerous neural network models [8,9]. Hence, the research on fractional neural networks (NNs) is important for practical applications, and many important results on chaotic dynamics, stability analysis, stabilization, synchronization, dissipativity, and passivity have been reported [10–16]. This popularity is due to the fact that fractional calculus has the ability to include memory when describing complex systems and gives a more precise characterization than the standard integer-order approach. A key characteristic is that the FO derivatives require an infinite number of terms, whereas the integer-order derivatives only indicate a finite series. Consequently, the integer derivatives are local operators, whereas the FO derivative has the memory of all past events.

In the real world, there are different types of uncertainty that can attenuate the performance of the system and affect its stability. These uncertainties may result from parameter variations and external disturbances. If a structural process is observed experimentally, it is not possible to assign precise values to the observed events. This means data uncertainty occurs, which may result from scale-dependent impacts that are not considered, which create inaccuracies in the estimations and incomplete sets of observations. In this manner, the estimated results are more or less described by the data uncertainty that begins with

**Citation:** Hymavathi, M.; Syed Ali, M.; Ibrahim, T.F.; Younis, B.A.; Osman, K.I.; Mukdasai, K. Synchronization of Fractional-Order Uncertain Delayed Neural Networks with an Event-Triggered Communication Scheme. *Fractal Fract.* **2022**, *6*, 641. https://doi.org/ 10.3390/fractalfract6110641

Academic Editors: Thach Ngoc Dinh, Shyam Kamal, Rajesh Kumar Pandey and Riccardo Caponetto

Received: 12 August 2022 Accepted: 27 October 2022 Published: 2 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

imprecision. In addition, the parameter uncertainties are unavoidable while displaying a neural network, which creates unstable results. It is known that a precise physical model of an engineering plant is difficult to build because of the uncertainties and noises. In actual operation, due to the existence of some external or internal uncertain disturbances, system states sometimes are not always fully accessible [17–26].

Generally speaking, an event-triggered control strategy is more appealing than the traditional time-triggered one from an economic perspective, since the control input is updated only when the predetermined triggering condition is reached. Since the eventtriggered control approach can reduce information exchange in systems, event-triggered synchronization or consensus for fractional-order systems has received increasing attention in recent years. Recently, there has been significant research on the event-triggered control (ETC) strategy [27–29]. Compared to the time-driven consensus, the event-triggered consensus is more realistic. The event-triggered controller introduced in the field of networked control systems has the advantage of using limited communication network resources efficiently. Recently, an event-triggered scheme (ETS) provided an effective way of determining when the sampling action should be carried out and when the packet should be transmitted. A number of researchers have recommended event-triggered control. To deal with network congestion, the ETS has been proposed to improve data transmission efficiency. In the past few years, event-triggered control has proved to be an efficient way to reduce the transmitted data in the networks, which can relieve the burden of network bandwidth. Thus event-triggered control strategies have been employed to study networked systems [30–32].

In addition, in many practical applications, the system is expected to reach synchronization as quickly as possible. Synchronization is an important phenomenon in the real world, which exists widely in practical systems, as well as in nature. The problem of achieving synchronization in a neural network is another research hotspot. Different kinds of synchronization, such as pinning synchronization [33], local synchronization [34,35], lag synchronization [36], and impulsive synchronization [37] have been considered in the literature. Recently synchronization has also attracted attention in the field of complex networks systems [38,39]. Synchronization techniques require communication among nodes, which creates network congestion and wastes network resources. Moreover, the treatment of the synchronization problem of fractional-order systems with input quantization is quite limited in the literature. Numerous consequence have been described for the synchronization-based event-triggered problem [40–42]. As collective behaviors, consensus and synchronization are important in nature.

There is no doubt that the Lyapunov functional method provides an effective approach to analyze the stability of integer-order nonlinear systems. The synchronization and stabilization of fractional Caputo neural network (FCNNs) were proved by constructing a simple quadratic Lyapunov function and calculating its fractional derivative. The contributions of this article are listed below:

1. The synchronization of fractional-order uncertain delayed neural networks with an event-triggered communication scheme is investigated.

2. A fractional integral, which is suitable for the considered fractional-order error system, is proposed.

3. A Lyapunov–Krasovskii (L–K) functional is established, and the conditions corresponding to asymptotic stability are derived for the design of an event-triggered controller based on linear matrix inequalities (LMIs).

4. The derived conditions are expressed in terms of linear matrix inequalities (LMIs), which can be checked numerically via the LMI toolbox very efficiently.

5. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed stability results.

The following notations are used in this paper. R and R<sup>n</sup> denote the set of real numbers and the <sup>n</sup>-dimensional real spaces, respectively; <sup>R</sup>n×<sup>n</sup> denotes the set of <sup>n</sup> <sup>×</sup> <sup>n</sup> matrices. I denotes the identity matrix of appropriate dimension. The super script "T " denotes the matrix transposition. "(−1)" represents the matrix inverse. X > 0 (X < 0) means that X is positive definite (negative definite). I represents the identity matrix and zero matrix with compatible dimensions. In symmetric block matrices or a long matrix expression, we use an asterisk (\*) to represent a term that is induced by symmetry. L2[0, ∞) denotes the space of square-integrable vector functions over [0, ∞).
