Exponential and adaptive synchronization of inertial complex-valued neural networks: A non-reduced order and non-separation approach
Introduction
During the recent decades, artificial neural networks received plenty of attention in virtue of their increasing wide applications in various fields such as image processing (Kotropoulos et al., 1994, Song and Chen, 2018, Zeng et al., 2018), combinatorial optimization (Kwok & Smith, 1999) and deep learning (Wen et al., 2019, Xiao et al., 2018). Particularly, as an important topic, synchronization of chaotic neural networks has been extensively addressed since the chaotic behavior of neurons was proposed in Aihara, Takabe, and Toyoda (1990). Actually, synchronization can be utilized not only to secure communication but also to explain self-organization feature in ecological system and the brain (Milanović & Zaghloul, 1996). Nowadays, different types of synchronization, including asymptotical synchronization, exponential synchronization and adaptive synchronization, have been widely investigated for cellular neural models, Cohen–Grossberg neural networks, memristor-based neural networks and other neural systems described by the first order differential equations (Cao et al., 2019, Hu et al., 2010, Hu and Zeng, 2017, Sader et al., 2018, Wang et al., 2015).
In 1986, a new type of neural networks, called inertial neural networks, was proposed by introducing inductor into neural circuit to show the inertial feature and the dynamical models for such networks were depicted by second-order differential equations (Babcock & Westervelt, 1986). It was found that the inertial neural networks not only can exhibit more complicated dynamics in contrast to the standard resistor–capacitor first-order model (Babcock and Westervelt, 1986, Babcock and Westervelt, 1987) but also possess wide biological backgrounds. For example, the squid axon and the quasi-active membrane behavior of neurons can be simulated by circuits involving inductance (Angelaki and Correia, 1991, Mauro et al., 1970). Hence, it is of vital significance to examine the dynamics and control of inertial neural networks (Wang, Zeng, Ge, & Hu, 2018).
At present, the synchronization of inertial neural networks has been vastly explored and numerous results have been reported. Based on matrix measure and Halanay inequality, the exponential stability and synchronization were studied in Cao and Wan (2014) for inertial BAM neural networks. The synchronization for inertial neural networks with time-varying delays was discussed respectively by means of linear feedback control (Lakshmanan et al., 2016, Prakash et al., 2016), quantized sampled-data control (Zhang, Zeng, Park, Liu, Zhong, 2018), nonlinear output control (Abdurahman, Jiang, & Sader, 2019), periodically intermittent control (Tang & Jian, 2019). In Alimi, Aouiti, and Assali (2019) and Jian and Duan (2019), proportional delays were introduced to inertial neural networks and finite-time synchronization was investigated. Recently, inertial memristor-based neural networks were closely addressed for exponential synchronization and finite-time or fixed-time synchronization (Chen, Li et al., 2019, Gong et al., 2019, Huang et al., 2017, Lu et al., 2019). In Gu, Wang, and Yu (2019), a kind of inertial fractional-order neural networks was proposed and global synchronization was analyzed by means of the theory of fractional-order equations. The authors in Xiao, Huang, and Zeng (2019) investigated exponential stability and synchronization under the timescale framework for a type of discrete-time inertial neural networks. In addition, the synchronization for various coupled inertial neural networks was also discussed by using graph theory and Lyapunov theory (Dharani et al., 2017, Feng et al., 2018, Guo et al., 2018, Li and Zheng, 2018, Rakkiyappan et al., 2016). Nevertheless, in the above results, the original inertial systems were transformed into the first-order models under some variable substitutions, which may increase the dimensions of models and make the theoretical analysis more complicated. In Li, Li and Hu (2017), without applying the common reduced order idea, a new Lyapunov functional was constructed to directly analyze the asymptotical stability and synchronization of time-delayed inertial neural networks. Now, the non-reduced order method has been extended to dispose stability, stabilization and periodicity of various inertial neural models (Huang and Liu, 2019, Huang et al., 2019, Zhang, Hu et al., 2018, Zhang and Zeng, 2019). Unfortunately, these results mainly focus on inertial neural networks with real variables but neglect the complex-valued feature in practical applications.
Actually, complex variables are frequently used in numerous applications including image reconstruction (Tanaka & Aihara, 2009), nonlinear filtering (Aizenberg, 2017) and pattern recognition and classification (Cha & Kassam, 1995). Compared with real-valued ones, complex-valued neural networks can solve a wider range of practical problems. A famous example is XOR problem, which is unsolvable for real-valued neural networks but can be easily solved by means of complex-valued ones (Song et al., 2018a, Tripathi and Kalra, 2011). Moreover, complex-valued neurons have stronger processing ability and faster computing power since it is revealed in Amin and Murase (2009) that a single layer in complex-valued neural networks can present equivalent or higher performance than multilayered real-valued counterparts. Currently, the synchronization for the first-order complex-valued neural networks has been widely studied by separating the complex-valued models into two real-valued subsystems (Hu and Zeng, 2017, Kan et al., 2019, Li, Fang et al., 2017, Liu et al., 2018, Xie et al., 2019, Zhang and Wang, 2017). Meanwhile, without using the separation technique, based on complex function theory and constructing Lyapunov functionals in the complex domain, some synchronization results were also reported for the first-order complex-valued neural networks (Chen, Chen et al., 2019, Li et al., 2019, Wang et al., 2019, Yang et al., 2018, Yuan et al., 2019). Evidently, the non-separation technique is more effective and convenient. In contrast to numerous synchronization results on complex-valued neural networks described by the first-order equations, there seems to be no published work to discuss the synchronization of complex-valued inertial neural networks to our knowledge. Hence, a natural and challenging problem is how to develop some effective methods in complex domain to analyze the synchronization of complex-valued inertial neural networks without utilizing reduced order and separation means.
Enlightened by the above discussion, the main aim of this paper is to investigate exponential and adaptive synchronization of inertial complex-valued neural networks with time delays by constructing some innovative Lyapunov functionals instead of the traditional reduced order and separation approach. In comparison to the existing results, the innovative contents of this paper mainly lie in the following respects.
In contrast to plentiful synchronization efforts on the first-order complex-valued neural networks (Chen, Chen et al., 2019, Li et al., 2019, Wang et al., 2019, Yang et al., 2018, Yuan et al., 2019), the synchronization for the second-order inertial complex-valued neural networks is first considered in this paper and the earlier results are further complemented.
Unlike the traditional separation method used in the most of existing results (Hu and Zeng, 2017, Kan et al., 2019, Liu et al., 2018, Xie et al., 2019, Zhang and Wang, 2017), in which the complex-valued models are firstly divided into two real-valued subsystems and then two real-valued controllers are respectively designed onto them, some complex-valued control schemes are proposed in this paper and the exponential and adaptive synchronization are directly discussed by using the theory of complex-variable functions.
Without applying the standard reduced-order variable transform, some new Lyapunov functionals, including the state variables and the derivatives of them, are constructed to directly analyze the synchronization of inertial models, the theoretical analysis is essentially simplified compared with the previous reduced order technique utilized in Abdurahman et al., 2019, Alimi et al., 2019, Feng et al., 2018, Jian and Duan, 2019, Lakshmanan et al., 2016, Prakash et al., 2016 and Tang and Jian (2019).
Compared with the work Li, Li et al. (2017), a type of more general systems, complex-valued inertial neural models, is considered in this paper and a faster convergent synchronization namely exponential synchronization is investigated.
The remaining of this paper is organized as follows. The driving and response complex-valued inertial neural networks are provided in Section 2 along with some definitions and lemmas. In Section 3, exponential and adaptive synchronization are discussed as the main results. Several numerical simulations are provided in Section 4 to verify the derived theoretical results. In the end, a brief summary of this paper is given.
Notations: Throughout this paper, , , , , stand for the sets of all real and complex numbers, the sets of all -dimensional real and complex-valued vectors, respectively. denotes the imaginary unit, that is, . denotes a set composed of continuous mapping from to . For any , and respectively denote the real and imaginary parts of it, the norm is defined as , where is the conjugate of . For each , the norm is denoted by . Without loss of generality, the notation in this paper represents the norm if and denotes the absolute value if .
Section snippets
Preliminaries
A type of complex-valued inertial neural networks is considered in this paper and depicted as where corresponds to the number of neurons, represents the state of the th neuron at time , the second derivative is called an inertial term of (1), and , and are connection weights related to the neurons without delays and with delays, respectively, and are the
Exponential and adaptive synchronization
This section focuses on the analysis of exponential and adaptive synchronization for the neural models (1), (3) by developing some new Lyapunov functionals instead of the common reduced order and separation technique.
For , let be the synchronization error, and the error system can be easily described by where , for .
Firstly, the
Numerical simulations
This section provides a numerical example and some simulations to verify and support the theoretical results.
Consider the following driving inertial neural network containing two complex-valued neurons and the controlled response neural network is given by where , , , , for ,
Conclusions
The synchronization problem of complex-valued inertial neural networks has been investigated in this paper. Totally different from the previous reduced-order method for inertial neural networks and separation approach for complex-valued systems, some innovative Lyapunov functionals have been constructed to directly discuss exponential and adaptive synchronization of complex-valued inertial neural systems. Particularly, some conditions including multi-parameters have been obtained to ensure
Acknowledgments
This work was supported partially by the National Natural Science Foundation of China under Grant 61866036, Grant 61963033 and Grant U1703262, partially by Tianshan Youth Program, China under Grant 2018Q001, partially by the Natural Science Foundation of Xinjiang, China under Grant 2018D01C057 and partially by the Innovation Team Program of Universities in Xinjiang Uyghur Autonomous Region, China under Grant XJEDU2017T001.
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