Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control

Abstract

This paper investigates pinning synchronization of coupled neural networks with both current-state coupling and distributed-delay coupling via impulsive control. A novel impulse pinning strategy involving pinning ratio is proposed and a general criterion is derived to ensure an array of neural networks with two different topologies synchronizes with the desired trajectory. In order to handle the difficulties of high-dimension criteria, some inequality techniques and matrix decomposition methods through simultaneous diagonalization of two matrices are introduced and low-dimensional criteria are obtained. Finally, an illustrative example is given to show the effectiveness of the proposed method.

Introduction

In the past three decades, there has been a substantial growth of research interest in the dynamics of neural networks including stability, bifurcation, periodic attractors, and chaotic attractors due to their wide applications in pattern recognition and associative memories, signal processing, dynamic optimization (Xiong, Ho, & Yu, 2016). On the other hand, synchronization, as a typical collective behavior, has been observed in biological systems such as synchronous fireflies, swarming of fishes, flocking of birds. It has been shown evidence that the presence or absence of synchrony in the brain is often linked to specific brain function or critical physiological state such as epilepsy (Freund, 1983) and Parkinson’s disease (Uhlhaas and Singer, 2006, Schnitzler and Gross, 2005) in neuroscience (Steur, Tyukin, & Nijmeijer, 2009). Therefore, it is important to investigate synchronous behaviors in an array of coupled neural networks.

The problem of complete synchronization of coupled neural networks has been well studied in the last decade. In He and Cao (2009), chaotic synchronization in master–slave delayed neural networks was investigated and a novel matrix measure method was applied with some less conservative criteria derived. Later, Cao and Wan extended the method to synchronization of master–slave inertial BAM neural networks (Cao & Wan, 2014). In Hu, Jiang, and Teng (2010), the synchronization problem was studied in a context of an array of coupled delayed neural network with reaction–diffusion effects. While fuzzy model of memristive neural networks (MNNs) was constructed in Yang, Li, and Huang (2016). Other types of synchronization including outer synchronization (Lu, Zheng, & Chen, 2016) with sampled-data control, cluster synchronization (Li, Ho, Cao, & Lu, 2016) were also studied in recent years.

On the other hand, controlled synchronization of coupled delayed neural networks has received increasing attentions, which is motivated by facts that in some cases coupled neural networks cannot reach synchronization via coupling forces or there are special requirements for the synchronous behavior. Among them, synchronizing a network to a desired trajectory is an important issue, and it has been well formulated as leader-following synchronization in which how to choose controlled nodes, known as pinning control, is one of key technical problems (Yu et al., 2009, Song et al., 2013, He et al., 2016, He et al., 2016). In Wen, Yu, Chen, Yu, and Chen (2014), a novel selective pinning strategy was proposed to study $H_{\infty }$ synchronization of directed networks with aperiodic sampled-data communications. While, in Su et al. (2013), a decentralized adaptive pinning controller was designed for cluster synchronization of dynamic networks. This paper mainly focuses on pinning synchronization of coupled neural networks via impulsive control. Impulsive control is practical in simulating the abrupt changes at certain instants in the fields such as medicine, biology and electronics. Compared with continuous control, impulsive control is an energy-saving control method, as it allows systems to possess discontinuous inputs. In Hu and Xu (2009), Lu, Kurths, Cao, Mahdavi, and Huang (2012) and Mahdavi, Menhaj, Kurths, Lu, and Afshar (2012), effective pinning impulsive strategies were proposed to synchronize dynamic networks. The pinning control strategy in Hu and Xu (2009) is only effective for linear systems. In Lu et al. (2012) and Mahdavi et al. (2012), a novel node selection strategy was proposed according to the norm of the error signal between the follower and the leader. However, the mathematical model of such dynamic networks was delay-free. Pinning impulsive synchronization of delayed dynamic network was studied in Yang, Cao, and Yang (2013) and Lu, Wang, Cao, Ho, and Kurths (2012), while the communication delays were neglected. Realistic modeling of many large networks with communication inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units (He, Qian, Han, & Cao, 2012). Therefore, developing a novel impulse pinning control strategy for a class of coupled delayed networks stands out as one motivation of the current study.

In order to model transmittal delays, coupling schemes involving time delays are proposed such as discrete-delay coupling and distributed-delay coupling. There have been studies on synchronization of coupled systems with discrete-delay coupling (He et al., 2012). Dynamic properties of neural networks with distributed delays such as stability (Liu, Wang, & Liu, 2006), passivity (Wen et al., 2015, Li et al., 2010), almost periodic solutions (Jiang, Zeng, & Chen, 2015) have also been explored, while there is relatively less work on synchronization of coupled neural networks with distributed-delay coupling. In Cao, Chen, and Li (2008) and Yuan (2009), a hybrid coupling including current-state coupling and distributed-delay coupling was proposed and global synchronization was achieved by coupling forces. On one hand, pinning control of coupled neural networks with distributed-delay coupling is seldom considered, especially impulse pinning control (He et al., 2016, He et al., 2015). On the other hand, when the hybrid coupling is proposed, the same network topology is used. It is more reasonable and general to have different network topologies, as in Guan, Liu, Feng, and Wang (2010). However, decoupling coupled neural networks into several subsystems becomes more challenging. Therefore, studying the effect of distributed delays on impulse pinning synchronization and further deriving low-dimensional criteria provide another motivation of the current study.

The paper aims to investigate pinning synchronization of coupled neural networks with distributed-delay coupling by impulsive control. The contribution of this paper can be summarized as follows:

• a novel impulse pinning strategy is proposed, in which the pinning ratio is defined as an index to determine the node selection.

• a general criterion is derived to ensure an array of neural networks with two different topologies synchronizes with the desired trajectory. The effect of distributed delays is further discussed.

• the technique of dimensionality reduction based on simultaneous diagonalization of two matrices is introduced and a low-dimensional criterion is obtained.