Fractional calculus has experienced more than 300 years of history, but the research on fractional calculus has become popular and achieved good results in the past 20 years. Such as fluid mechanics [1], financial investment [2], image processing [3], neural networks and so on have achieved gratifying results. Due to its memory and genetic properties, some researchers have introduced it into neural networks and used it to represent the dynamic behavior of neurons, and because fractional calculus can more accurately describe objects with more dynamic and algebraic characteristics. Thus, forming fractional neural networks [4, 5]. In real life, there is an inevitable time delay in the study of artificial neural networks and it cannot be eliminated. In neural networks, discrete-time delays are unavoidable due to the limited speed of switching between neurons and amplifiers [6, 7]. Neural networks often have spatial characteristics due to the existence of a large number of average paths with different ayon sizes and lengths, and the propagation speed along the path is different, resulting in propagation distribution delay [8–10]. Obviously, time delay is disadvantageous to the stability of neural network system, so it is valuable to consider delay in neural network. In 1971, Zeili used symmetry prediction to propose a fourth circuit module based on resistance, inductance, and capacitance, and recorded it as a memristor. Thus, the concept of memristor was born [11]. Then, in 2008, HP LABS demonstrated the memristor for the first time [12]. Since then, the research on memristors has entered a stage of rapid development. Memristors are often used because of their advantages such as small size, fast propagation speed, low energy consumption and scalability, which are different from resistors [13, 14]. When the memristor is turned off, it can record the charge of the historical switch, which can also be used to simulate the synapses of biological neurons after stimulation. Because of these properties mentioned above, the research of memristor has received extensive attention [15, 16]. Synchronization, as an important dynamic behavior in dynamics, is of great significance in biological systems, secure communication, finance and image processing. At the same time, it can also be used in complex neural networks, such as cryptography, sensor network synchronization and brain modeling. The methods of synchronization are also very rich, such as anti-synchronization, complete synchronization, finite time synchronization and exponential synchronization, e.g., see [17–21]. Among the above synchronization methods, global synchronization is undoubtedly an important processing scheme. Jin Zhou and Lan Yiang [22] used the stability theory of time-delay dynamic systems to analyze and derive the delay-independent and delay-dependent criteria for global synchronization of networks. ZhenyuanGuo and ShaofuYang [23] researched the theoretical results of global exponential synchronization of multiple statistical networks and the existence of eternal noise by two types of distributed pin control. Ali Kazemy [24] studied the global synchronization of coupled neural networks with mixed coupling. The fractional-order global synchronization problem of uncertainty in neural networks with time-delay is studied in [25]. In [26], the global exponential synchronization of multi-memory reaction-diffusion neural networks with time delay is studied.
Inspired by the above discussion, a fractional-order neural network with mixed time-varying delay is studied in this paper. The main contributions of this paper can be summarized as follows:
• In this paper, the limitations on the assumption of bounded activation functions in [27-32] are weakened to make them more general to activation functions, while making the results less conservative.
• In order to avoid chattering in existing results, a new unsigned effective state feedback controller and an adaptive controller are proposed. The designed controller can also reduce the control cost. It also appears to be the first time such two controllers have been mentioned in the system.
• By utilizing the knowledge of norm and fractional calculus, a suitable candidate for the Lyapunov Krasovsky function was established, and sufficient conditions for global synchronization of the system were obtained.