Global Mittag-Leffler synchronization of fractional-order neural networks with discontinuous activations

Highlights

• Discontinuous activations are taken into account in fractional-order neural networks.

• Filippov solution is introduced to study the dynamics behavior of fractional-order neural networks with discontinuous activations (FNNDAs).

• Prove the existence of global solution in the sense of Filippov for FNNDAs based on a singular Gronwall inequality.

• Some sufficient conditions on global Mittag-Leffler synchronization of FNNDAs are obtained.

Abstract

This paper is concerned with the global Mittag-Leffler synchronization for a class of fractional-order neural networks with discontinuous activations (FNNDAs). We give the concept of Filippov solution for FNNDAs in the sense of Caputo’s fractional derivation. By using a singular Gronwall inequality and the properties of fractional calculus, the existence of global solution under the framework of Filippov for FNNDAs is proved. Based on the nonsmooth analysis and control theory, some sufficient criteria for the global Mittag-Leffler synchronization of FNNDAs are derived by designing a suitable controller. The proposed results enrich and enhance the previous reports. Finally, one numerical example is given to demonstrate the effectiveness of the theoretical results.

Introduction

Fractional calculus dates from the 17th century, and it is a name for the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and integration (Kilbas et al., 2006, Podlubny, 1999). It is well known that physical phenomena in nature can be described more accurately by fractional-order models than integer-order ones (Hilfer, 2000). Moreover, fractional-order model is an excellent instrument for the description of memory and hereditary properties of various materials and processes (Chen et al., 2010, Isfer et al., 2010, Kilbas and Marzan, 2005, Szabo and Wu, 2000). Therefore, fractional calculus and fractional-order model play important roles in different fields, such as dynamics of complex materials or porous media (Carpinteri, Cornetti, & Kolwankar, 2004), fluid mechanics (Tripathi, Pandey, & Das, 2010), bioengineering (Magin, 2004, Magin, 2010, Magin and Ovadia, 2008), viscoelasticity (Soczkiewicz, 2002), etc.

Fractional-order neural networks (FNNs) can be used to simulate neurons in the human brain. In Arena et al., 1998, Arena et al., 2000, the authors built the model of fractional-order cellular neural networks by replacing the traditional first-order cell with a noninteger-order one, and revealed the existence of chaotic behavior of fractional-order cellular neural networks. Subsequently, FNNs have become a promising research topic, and various dynamical behaviors of FNNs have been widely investigated, such as stability (Chen et al., 2013, Wang et al., 2013, Zou et al., 2014), synchronization (Yu et al., 2012, Zhou et al., 2008), chaos (Kaslik & Sivasundaram, 2012), etc. However, all the activations of these networks were Lipschitz-continuous (Chen et al., 2013, Wang et al., 2013, Yu et al., 2012, Zou et al., 2014).

Compared with continuous activations, discontinuous activations have been proved really useful as ideal models of activations with very-high gain, and such models have been frequently applied to solve constrained optimization problems via a sliding mode approach (Chong et al., 1999, Forti et al., 2004). In fact, Forti and Nistri (2003) have performed a thorough analysis of global convergence for a large class of neural networks with discontinuous activations (NNDAs) in 2003. Meanwhile, this paper pointed out that NNDAs were frequently encountered in the applications, such as impacting machines, systems oscillating under the effect of an earthquake and dry friction (Cortes, 2008, Danca, 2004, Popp et al., 1995). Inspired by the ground-breaking work (Forti & Nistri, 2003), more and more researchers pay more attention to studying NNDAs, such as Cortes (2008), Danca (2002), Forti, Nistri, and Papini (2005), Liu and Cao (2009) and Lu and Chen, 2006, Lu and Chen, 2008, etc. However, these results were built in the case of integer-order neural networks. To the best of our knowledge, there are few results on the dynamics analysis of fractional-order neural networks with discontinuous activations (FNNDAs). Generally, FNNDAs can be better used to simulate neurons in the human brain, and their dynamics behaviors have brilliant application prospect. Therefore, to investigate the dynamics behaviors of FNNDAs is necessary.

Motivated by the above discussion, we investigate the global Mittag-Leffler synchronization for a class of FNNDAs. The main advantages of this paper lie in the following aspects. Firstly, the classical solutions cannot be applied to FNNDAs due to the discontinuities of activations. To cope with the problem of solution for FNNDAs, we introduce the concept of Filippov solution in the sense of Caputo’s fractional derivation. Secondly, the peculiarities of fractional calculus, namely, non-locality and weak singularity, are unmanageable to investigate the existence of global Filippov solution for FNNDAs. Here, the existence of global solution in the sense of Filippov is given based on a singular Gronwall inequality. Thirdly, by using the nonsmooth analysis and control theory, some sufficient criteria for the global Mittag-Leffler synchronization of FNNDAs are presented. It is noted that our results still hold for integer-order neural networks with discontinuous activations, and we extend the results of FNNs with continuous activations, such as Zhang, Yu, and Wang (2015).

The organization of this paper is as follows. The systems and some preliminaries are introduced in Section  2. The existence of global Filippov solutions is provided in Section  3.1, and some sufficient criteria for the global Mittag-Leffler synchronization of FNNDAs are proposed in Section  3.2. Numerical simulations are given to demonstrate the effectiveness of the obtained results in Section  4. Finally, conclusions are drawn in Section  5.