Periodically intermittent discrete observation control for synchronization of the general stochastic complex network☆
Introduction
In view of the fruitful achievements of the World Wide Web (Broder et al., 2000) and scale-free topology (Ebel, Mielsch, & Bornholdt, 2002), complex networks have become an active and rich research area and their applications can be found in many fields. As a consequence, many researchers have contributed to the researches (Chen, Yu et al., 2016, Meng et al., 2018, Wang et al., 2015, Wu et al., 2016). With the viewpoint of complex networks, a multitude of interesting and essential dynamical properties, such as synchronization, consensus and stability, have been researched (Mao et al., 2014, Saravanakumar et al., 2019, Sheng et al., 2017, Yang et al., 2017, Yang et al., 2019, Zhang et al., 2019). Synchronization, as an interesting collective property of stochastic complex networks, has become a hot topic due to its broad applications in various disciplines, and many relevant results have been reported these years (Liu et al., 2016, Yang et al., 2011, Zhang et al., 2015). However, it is worth noting that most of the aforementioned papers considered a constant coupling structure, while in real world, the coupling structure may change over time. For example, in reality, bidirectional associative memory neural networks ubiquitously change with time (Anbuvithya, Mathiyalagan, Sakthivel, & Prakash, 2015). Apart from that, in Chua’s circuits, channels and parameters can also be time-varying (Chua, Yang, Zhong, & Wu, 1996). Therefore, it would be more significant for us to take time-varying coupling structure into consideration. Besides, due to its generality, recently synchronization analysis for stochastic complex networks with time-varying coupling structure has received great attention from researchers, and many results have been published. For example, in Zhang, Yang, Li, Zhang, and Yang (2018), time-varying coupling structure was taken into consideration by Zhang et al. when they studied stochastic exponential synchronization of delayed memristive neural networks. Besides, considering time-varying coupling structure, some techniques for global asymptotical synchronization problem of delayed fractional-order memristor-based complex-valued neural networks were presented by Yang et al. in Yang, Li, Huang, Song and Huang (2018).
Recently, many scholars have paid attention to the synchronization of complex networks (Li et al., 2018, Liu et al., 2019, Wu et al., 2017, Yang, Cao et al., 2018, Yang, Qiu and He, 2015). It is worth pointing out that most of the aforementioned papers held the assumptions that networks are either strongly connected or with a directed spanning tree, which is a special case in practice. In the real world, a lot of networks are developed on the basis of stochastic complex networks without strong connectivity (SCNWSCs). For example, in social networks formed of the public by social interaction, an individual stands for a node, and the degrees of acquaintance among each other are characterized by the weights of social digraphs. Besides, neural networks which are composed of interconnected neurons, and electric networks which are made up of the substation, transmission and distribution lines, are all established on SCNWSCs. Obviously, SCNWSCs exist everywhere in our life. Hence, SCNWSCs are more applicable and can suit the practical problems in a better way, and it has aroused considerable interest from researchers. Meanwhile, a load of results have been reported (Liu and Li, 2018, Liu et al., 2018). Clearly, to research the SCNWSC is of great significance. Thus in this paper, we tend to research the synchronization problem of the SCNWSC with time-varying coupling structure.
As we all know, in general, a network cannot achieve synchronization spontaneously, which indicates that we are supposed to add some external controllers on the network to realize its synchronization. In fact, these years, many control methods have been put forward. Among diverse control strategies, feedback control and intermittent control are two frequently-used types. The former one requires the control inputs to be activated continuously, while the latter one, discontinuously. In fact, control inputs in the latter one are activated during the work time and off during the rest time. Obviously, intermittent control is more economical than feedback control. Hence, it has received increasing interest from researchers, and a lot of interesting results have been reported (Chen, Zhong and Zheng, 2016, Cheng et al., 2018, Cheng et al., 2019, Ding et al., 2019, Liu and Chen, 2015, Wu, Zhu and Li, 2019, Xu et al., 2018, Xu et al., 2019). We note that the controller in the aforementioned papers were observed continuously in the control time. Considering that as a rule, controller is observed at some discrete points of work time, such as , ( is a positive constant), therefore, it would be more general and economical on condition that a novel periodically intermittent control, which is based on discrete-time state observations in the control time, is established for the need of low cost in practical applications. As a matter of fact, Mao (2013) first came up with a discrete observation control with the aim of synchronizing the controlled systems, which merely requires state observations at discrete times , in the control time. Due to the effectiveness and generality, it has become of high interest these days, and many scholars have contributed to the researches (Mao et al., 2014, Wu et al., 2017, You et al., 2015). Simultaneously, taking into account the high efficiency and lower cost of periodically intermittent control, in this paper, a novel control strategy called periodically intermittent discrete observation control (PIDOC), in which the states in the control time are observed on discrete time points, is firstly introduced. On the one hand, compared with regular feedback control, PIDOC is capable of reducing the cost from the perspective of economy, since it does not work during the rest time. On the other hand, PIDOC is more general and practical on account of the fact that observations for the states of the SCNWSC are at discrete time for the most part. Thus, our paper makes an attempt to research the synchronization problem of SCNWSC with time-varying coupling structure via PIDOC.
Motivated by the aforementioned discussions, in this paper, our main purpose is to research the synchronization of the SCNWSC with time-varying coupling structure. A novel PIDOC is designed in order to achieve synchronization for the first time. Secondly, by employing graph theory in combination with the Tarjan’s algorithm, several sufficient conditions are derived with the aim of ensuring the synchronization. What is more, we apply the main results to stochastic coupled oscillators with time-varying coupling structure. Meanwhile, two corollaries are derived. Finally, a numerical example is shown to testify the feasibility and effectiveness of the acquired theoretical results.
The dominant contributions of this paper are as follows:
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The synchronization of the SCNWSC with time-varying coupling structure is discussed via a novel PIDOC strategy, which is a generalization of intermittent control and feedback control based on discrete-time state observations (FCBDSO).
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The coupling structure that we consider is time-varying. Furthermore, the topology structure is not strongly connected.
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Some novel synchronization criteria are derived through the combination of graph theory and Lyapunov method.
The rest of this paper is structured as follows. In Section 2, some preliminaries and model formulation are introduced. Next, in Section 3, we mainly discuss the synchronization of the strongly connected stochastic complex network (SCSCN) with time-varying coupling structure via PIDOC. Then in Section 4, we discuss the synchronization of the SCNWSC with time-varying coupling structure via PIDOC. Our analytical results are applied to stochastic coupled oscillators in Section 5. A numerical example is presented in Section 6. Finally, some conclusions are drawn in Section 7.
Notations In this paper, unless otherwise specified, the superscript “T” stands for the transpose of a vector. Also, we let be the set of all real numbers, be the -dimensional Euclidean space, and , , . In addition, define as the Euclidean norm for vectors or the trace norm of matrices. Besides, make be a complete probability space with a filtration satisfying usual conditions, and denotes the mathematical expectation with respect to the given probability measure .
Let be a weighted digraph with nodes, where is a weighted matrix for . Furthermore, define the Laplacian matrix of digraph as , where for , and . A directed path in is a subgraph with distinct nodes such that its set of arcs is . If , we call a directed cycle. A digraph is called strongly connected if for any pair of distinct nodes there exists a path from node to node and a path from node to node . Apart from that, the relation among nodes of a digraph as a partial order is defined. For nodes , where represents the node set of a digraph , if there is a directed path from node to node . Then, we define a strongly connected component of a digraph as follows: if a subgraph is strongly connected and for any node , a subgraph consisting of the node set is not strongly connected, then is a strongly connected component. What is more, we compress each strongly connected component of the digraph into a single node. And all of them form a new digraph , which is considered as a condensed graph of . Moreover, for a condensed digraph , a strict partial order is defined as well. Concretely, for , one has , if there are node and node such that in the digraph .
Section snippets
Preliminaries and model formulation
In this section, consider a stochastic complex network with identical coupled nodes, which can be described as where represents the state vector of node . are continuous vector-valued functions representing the inherent nonlinear dynamics of node , are the weights of the edges between node and node , represents one-dimensional normal Brownian motion.
Let system (1)
Synchronization analysis for the SCSCN with time-varying coupling structure
In this section, synchronization of the SCSCN with time-varying coupling structure is discussed. To obtain our main results, the following assumptions are needed.
Assumption 1 For any , digraph is strongly connected, where . is the cofactor of the th diagonal element of Laplacian matrix of digraph , and , where , , are some positive constants.
Assumption 2 For any , there exist positive constants and , such that
Synchronization analysis for the SCNWSC with time-varying coupling structure
In the previous section, synchronization of the SCSCN was studied. Now, we will change our focus to synchronization of the SCNWSC. In fact, for the SCNWSC, the well-known Tarjan’s algorithm (Tarjan, 1972) can be employed to find all strongly connected components of a digraph. If digraph is not strongly connected, we introduce a hierarchical method for it in detail (Liu and Li, 2018, Liu et al., 2018).
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At first, for a digraph , we are supposed to compress each strongly connected
An application to stochastic coupled oscillators with time-varying coupling structure
In this section, by employing the main results, we discuss the synchronization of time-varying stochastic coupled oscillators (TSCOs), which have been researched by many scholars (Guo et al., 2018, Wang et al., 2019, Wu, Li and Li, 2019).
In this paper, we consider the following second-order oscillators with stochastic perturbation where , and are positive constants.
Set , then the above system can be expressed as follows:
Numerical test
In this section, we mainly present a numerical example to illustrate the effectiveness and feasibility of our results.
Here, consider digraph with 20 nodes (see Fig. 2), where is the weight matrix of it. Besides, from Fig. 2, we can see that the network with 20 nodes is not strongly connected. By employing Tarjan’s algorithm, we divide it into three layers (see Fig. 3). Then we set some elements of as follows, while other elements are defined as 0.
Conclusions
In this paper, the exponential synchronization of SCNWSCs was studied. A novel PIDOC was introduced to analyze the exponential synchronization of the general stochastic complex network with time-varying coupling structure. Furthermore, the PIDOC will degenerate into a regular FCBDSO when the control width tends to the control period. Besides, two corollaries about synchronization in mean square and asymptotical synchronization in mean square were obtained. In our future work, we will consider
Acknowledgments
The authors really appreciate the valuable comments of the editors and reviewers. This work was supported by the Shandong Province Natural Science Foundation (Nos. ZR2018MA005, ZR2018MA020, ZR2017MA008); the Key Project of Science and Technology of Weihai (No. 2014DXGJMS08) and the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703).
Yongbao Wu was born in 1992. He received the M.S. degree in applied mathematics from the Harbin Institute of Technology, Weihai, China, in 2017, where he is currently pursuing the Ph.D. degree in applied mathematics. His current research interest includes stability theory for stochastic differential equations, networked control systems, and multiagent systems.
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Yongbao Wu was born in 1992. He received the M.S. degree in applied mathematics from the Harbin Institute of Technology, Weihai, China, in 2017, where he is currently pursuing the Ph.D. degree in applied mathematics. His current research interest includes stability theory for stochastic differential equations, networked control systems, and multiagent systems.
Sixian Zhuang was born in 1999. He is currently pursuing the undergraduate degree in applied mathematics with the Harbin Institute of Technology, Weihai, China. His current research interest includes stability theory for stochastic differential equations.
Wenxue Li was born in 1981. He received the Ph.D. degree in applied mathematics from the Harbin Institute of Technology, Weihai, China, in 2009. He is currently an Associate Professor with the Harbin Institute of Technology. His current research interest includes stability theory for stochastic differential and integral equations.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Florian Dorfler under the direction of Editor Christos G. Cassandras