New criteria on event-triggered cluster synchronization of neutral-type neural networks with Lévy noise and non-Lipschitz condition☆
Introduction
Synchronization is not only a basic phenomenon but also common in artificial systems. Synchronous behavior is one of the important research directions in the field of complex network science. Among them, cluster synchronization has received more and more attention due to its versatility. The so-called cluster synchronization means that all nodes in a complex network can be divided into many clusters, so that nodes belonging to the same cluster are synchronized separately, but nodes in different clusters are not synchronized. Obviously, cluster synchronization is more practical and secure in some application areas, such as biological sciences and communication engineering.
Recently, some interesting research advances on cluster synchronization of networks have been implemented (see e.g., [1], [2], [3], [4], [5], [6], [7]). In [1], Zhou et al. solved the exponential cluster synchronization problem of directed community networks via adaptive nonperiodically pinning control. In [2], Liu et al. focus on the cluster synchronization problem for complex dynamical networks with switching signal which is characterized by average dwell-time constraint. Both literatures use a way to update the controller, the former uses the time-triggered mechanism, and the latter uses the event-triggered mechanism. Obviously the controller trigger timing in [2] is more reasonable than [1], and the number of times is less. However, it is a pity that neither considers the effect of time delay on the state of the system. Tang et al. studied cluster synchronization of dynamic networks consisted of idential or nonidential Lur’e systems in [3]. The advantage of this article is that it not only synchronizes all Lur’e systems in the same cluster but also decreases the negative influence among different clusters. But the controller in [3] is designed to be randomly triggered so that its update time is independent of state changes, and there is no advantage compared to event triggering. Literature [4] extends cluster synchronization into a more general class of Markovian switching complex networks consisting of nonidentical nodes with hybrid couplings. These networks are also influenced by coupling delays and disturbed by stochastic noise perturbations. In light of Lyapunov’s theorem and Lipschitz condition, the identification rules of the network controller and uncertain parameters are designed, and the cluster synchronization and uncertain parameters are effectively realized. However, many neural network activation functions do not satisfy the Lipschitz condition, so it is meaningful to discuss cluster synchronization under the non-Lipschitz condition.
In many practical situations, the network cannot synchronize spontaneously. External control is an effective method that has been used to design appropriate controllers for synchronization. Often, a complex network can include a large number of nodes. Therefore, synchronizing complex networks by adding controllers to all nodes is neither practical nor economical. Hence, an effective method is put forward to add a controller to only a small number of network nodes to achieve synchronization at a lower cost. This strategy is called pinning control [8]. Inspired by this idea, many works on pinning control strategies have recently been developed to achieve synchronization (see e.g., [1], [2], [9], [10], [11], [12], [13], [14]). To reduce energy usage, only the first node in each cluster is booted while the other nodes are working.
In order to reduce the number of controller updates and thus reduce resource consumption, the event trigger mechanism has received extensive attention (see e.g., [15], [16], [17], [18], [19], [20], [21], [22], [23]). There are two basic elements in the trigger mechanism, namely the feedback controller that calculates the control amount and the trigger condition when the control law is updated. A basic introduction to the event triggering mechanism can be found in [15], [16]. At the same time, there are many interesting research results worth noting. For example, An event-triggered synchronization algorithm is proposed in [17], which can asymptotically synchronize all agents. For the proposed event triggering mechanism, the unexpected Zeno phenomenon can be avoided because there is a lower limit on the time of two consecutive transmissions. However, the shortcoming of this article also comes from the lower limit because it is difficult to find. The event-triggered strategy in [20] is that the event detection is only applicable to the network topology switching moment, which can significantly reduce the node’s communication frequency and save network resources. But we also need to consider that the state error caused by hardware or time delay is too large at the time of non-switching, and the controller needs to be adjusted.
Among a large number of the existing stochastic cluster synchronization literatures like we mentioned above, according to the authors’ knowledge, the clustering synchronization of neural networks of neutral-type or governed by neutral functional differential equations has not been taken into account. Neutral neural networks have had a large use in practice because their system models involve current state derivatives and past state derivatives and have proven to be more suitable for describing and simulating the dynamics of this characteristic in neural response processes and nerve cells. The ubiquity of neutral models has attracted widespread attention in exploring the synchronicity and stability of random neutral neural networks (see e.g., [24], [25], [26], [27], [28], [29], [30], [31]).
At the same time, the activation function of the neural network usually has a strict restriction condition, that is, the Lipschitz condition is satisfied. However, the too strict condition makes the result unapplied and practical. Many neural networks without Lipschitz continuous neuron activation function appear in the dynamics of neural networks and other equations. Therefore, it is necessary to obtain a wider cluster synchronization criterion that is not based on Lipshitz conditions, especially for neutral neural networks with time-varying and Lévy noise.
Motivated by the above discussions, this paper considers the cluster synchronization exponentially in mean square of stochastic neutral-type neural networks with time-varying delay and Lévy noise under non-Lipschitz condition. By contrast of the existing literatures, the main contributions of this paper can be highlighted into three points. (1) Consider the cluster synchronization problem of neutral neural networks, and achieve mean square exponential cluster synchronization and almost surely exponential cluster synchronization. (2) In order to achieve the expected cluster synchronization, the update of the event triggering method depends on the dynamic evolution of the system and considers the impact of the system target time delay to system state change and event triggering conditions. This helps to increase the feasibility of practical applications and avoid unnecessary energy consumption. (3) Non-Lipschitz conditions adapted to cluster synchronization analysis is established. Our new condition can change into Lipschitz condition when we choose some special function to limit the boundary. Thus we can say our condition has much weaker requirement than usual Lipschitz condition.
The remaining part of this paper is organized as follows. In Section 2, some mathematical preliminaries are given. In Section 3, an event-triggered controller is designed and the detailed analysis for the cluster synchronization with non-Lipschitz is presented. Numerical example is provided to illustrate our theoretical results in Section 4. Conclusions are drawn in Section 5.
To facilitate the presentation, some notations are necessary. Throughout the paper, represents as the set of natural numbers and refers to positive integers. While describes n × m dimensional real matrix spaces and is an identity matrix. If is a vector or matrix, stands for the transposition of . and are the minimum and maximum eigenvalue of . Set . The symbol ⊗ means the Kronecker product. Let τ > 0 and denotes the family of all the bounded continuous -valued functions on then is the family of all -measurable -valued random variables . Unless otherwise specified, all matrices are supposed to have compatible dimensions.
Section snippets
Graph theory
The interactions topology for the system nodes can be described as a directed graph which is made up of a node set . Whether there exist a connection between nodes can be shown by an edge set . indicates the node vj can obtain the node vi’s information. The coupling weight matrix of graph is represented as belonging to the space with the elements ωij. ωij ≠ 0 if and j ≠ i while otherwise .
According to the idea of cluster
Main results
The goal of this section is to design a pinning controller which meets the appropriate event trigger condition, making all nodes in the same cluster synchronized, while different clusters are synchronized to different values. In order to make the subsequent content easier to understand, we first denote
Numerical example
In this section, an example will be given to illustrate the effectiveness of developed results. Example 1 Let the state space of Markov chain {r(t)}t ≥ 0 be with generator . Consider a neural network containing three clusters with ten nodes like and . Each cluster has an isolated head which we want all ten nodes can achieve synchronization with correspondingly. The set of pining node is . Set . Let the activation
Conclusions
Based on the general Its formula and the nonnegative semi-martingale convergence theorem, we have investigated two kinds of exponential cluster synchronization for stochastic neutral-type neural networks with time-varying delay and Lévy noise under non-Lipschitz condition, namely mean square exponential synchronization and almost surely exponential synchronization. The non-Lipschitz condition has much weaker re- quirement than the usual Lipschitz condition, so the neuron activation functions
Declaration of Competing Interest
We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further conform that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due
Yuqing Sun received the B.S. degrees and she is currently pursuing the Ph.D. degree in control science and engineering from Donghua University, Shanghai, China, in 2014. Her current research interests include synchronization and stability research of stochastic neural network systems and optimal control.
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Yuqing Sun received the B.S. degrees and she is currently pursuing the Ph.D. degree in control science and engineering from Donghua University, Shanghai, China, in 2014. Her current research interests include synchronization and stability research of stochastic neural network systems and optimal control.
Yihong Zhang, Assistant Dean of College of Information Science & Technology, DongHua university. Supervisor of postgraduates, Associate professor, IEEE member. He received his Ph.d. degree on HongKong polytechnic university. His mainly research area focus on intelligent system and control, Pattern recognition and robot control, intelligent sensor technology, intelligent digital textile and garment technology. Intelligent wearable technology, and etc.
Wuneng Zhou received the B.S. degree in mathematics from Huazhong Normal University, China, in 1982 and the Ph.D.degree in control science and engineering from Zhejiang University, China, in 2005. Now he is a professor in Donghua University, Shanghai, China. His current research interests include the stability, the synchronization and control for neural networks, wireless sensor networks and complex networks.
Xin Zhang received the B.S. and M.S. degrees in mathematics from Liaocheng University, Liaocheng, China, in 2012 and 2015, respectively. She is currently pursuing the Ph.D. degree in control science and engineering from Donghua University, Shanghai, China. Her current research interests include stability and control for neural networks, stochastic control with semi-Markov parameters.
Xiaofeng Wang is currently a Lecturer with the Department of Automation, College of Information Science & Technology, Donghua University, Shanghai. He received the B.S. degree from Hangzhou Dianzi University, Hangzhou, China, in 2009, and the Ph.D degree from Xidian University, Xian, China, in 2015. His research interests cover modeling, analysis and control of dynamical systems, especially of the complex systems driven by evolutionary games.
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This work was partially supported by the Natural Science Foundation of China (Grant nos. 61573095 and 61903077), Shanghai Sailing Program (No. 19YF1402500) and Fundamental Research Funds for the Central Universities (NO. 223201903-56).