Synchronization for stochastic coupled networks with Lévy noise via event-triggered control
Introduction
In the past decade, the analysis and control of the dynamical behaviors for various stochastic networks have been attracting many attentions. This is mainly due to the fact that for many real systems the mutual interaction of states in nature can be modeled as complex networks, such as disease transmission, computer viruses spreading, social behaviors, smart grid systems, internet communication etc (see, e.g., Cherifi, 2014, Qin et al., 2016 and references therein). It is known that the study of synchronization is one important problem of stochastic networks, because the realization of network synchronization is an essential task in many areas, such as sensor network, information processing, and secure communication (see, e.g., Lakshmanan et al., 2018, Shi and Zeng, 2020, Wan et al., 2020, Zhang et al., 2020).
In general, a network itself cannot achieve synchronization without extra control actions. In order for a network to be synchronized, various control methods become essential tools, such as feedback control (Dong et al., 2019, Leng and Aihara, 2020, Zhu and Zhang, 2017), impulsive control (Qi et al., 2020, Wang et al., 2020, Yang et al., 2017), pinning control (Wen, Yu, Hu, Cao, & Yu, 2015), adaptive control (Zhu and Cao, 2010, Zhu and Cao, 2011), periodic intermittent control (Liu, Li, & Chen, 2015), stochastic sampling control (Rakkiyappan, Dharani, & Cao, 2015), and among others. However, for large-scale networked systems, the classical control method may not be suitable since it often requires the frequent or continuous communications between nodes and control actions, in which some actions may not be always necessary under the traditional framework.
Recently, event-triggered control appears to be a better alternative for time-scheduled action in terms of reducing the controller actuation and communication between nodes significantly. During recent years, many event-triggered sampling approaches for various systems have been established due to the success of theoretical developments and the effectiveness to many real systems, including biological systems and wireless networks, among others, in order to gain better performances, as shown in latest works (see, e.g., Almeida et al., 2017, Cao et al., 2019, Ding and Wang, 2020, Lin et al., 2020). It is worth mentioning that the results in Almeida et al., 2017, Cao et al., 2019, Ding and Wang, 2020 and Lin et al. (2020) all focused on deterministic networks without considering stochastic uncertainties, which is in fact unavoidable in modeling network systems. As we know, stochastic coupled network (Long et al., 2020, Zhu and Cao, 2012, Zhu et al., 2010) is an appropriate mathematical framework to capture various stochastic properties of network switching resulted from various uncertainties such as random component failures/repair, altering in subsystem interconnections of the network, and sudden environmental changes, etc. Recently, there have been only limited results on the event-triggered control problem of stochastic system (Åström and Bernhardsson, 2002, Dong et al., 2018, Xie and Zhu, 2020, Zhou, Dong et al., 2017, Zhu, 2019). The work in Zhu (2019) was firstly to solve the event-triggered feedback control problem of stochastic delay systems with exogenous disturbances by developing a novel event-triggered rule. Xie and Zhu in Xie and Zhu (2020) first studied the self-triggered state-feedback control problem for stochastic nonlinear systems with Markovian switching.
As we know, stochastic noise usually are unavoidable in applications due to various uncertainties or unpredictable factors. Most of existing works, such as Dong, Ye, Feng, and Wang (2017) and Dong, Zhou, and Xiao (2020), focuses on the situation of stochastic networks with Brownian noise, in which its sample path is assumed to be continuous. Unfortunately, in many practical applications, stochastic disturbances appear to be not continuous and often involves some possible jumps, for example, when certain subsystem fails to work unexpectedly and the repair time appears to be random. For this situation, the noise cannot be modeled as Brownian motion. Thus, it is necessary and important to introduce a more general framework to model this type of noise/disturbance.
In order to deal with the discontinuity of noise/disturbance, Lévy process is a natural generalization of Brownian motion since it is a linear combination of time, a Brownian motion, and a pure jump process. Lévy process has drawn many attentions for stochastic modeling in recent years due to its wide application in many fields, for instance, mathematical finance, stochastic control, stochastic filtering, and so on (see Ma et al., 2020, Sun et al., 2018, Yang et al., 2015, Zhou et al., 2016, Zhou, Zhu et al., 2017, Zhu, 2014, Zhu, 2017, Zhu, 2018). Particularly, Zhu (2018) discussed the th exponential stability problem for a class of stochastic delay differential equations driven by Lévy processes. Zhu in Zhu (2017) studied the Razumikhin-type theorem for a class of stochastic functional differential equations with Lévy noise and Markov switching. Ma et al. (2020) investigated practical exponential stability problem for a class of stochastic age-dependent capital system with Lévy noise. Zhu (2014) studies the asymptotic stability in the th moment for a class of stochastic differential equations with Lévy noise.
The aforementioned Refs. Ma et al. (2020) and Zhu, 2014, Zhu, 2017, Zhu, 2018 all focus on the various stability of stochastic differential equations with Lévy noise. Recent work in Sun et al., 2018, Yang et al., 2015, Zhou et al., 2016, Zhou, Zhu et al., 2017 and Zhu, 2017, Zhu, 2018 mainly contributed to the study of the exponential synchronization in mean square or in the th moment for neural network systems. Although exponential synchronization in mean square provides the characterization of the convergence under the statistical framework, it does not guarantee the synchronization in each attempt. In contrast, almost sure synchronization appears to be practically preferable since it guarantees the synchronization with a full probability. Technically, when the Lévy noise is considered in network systems, almost sure convergence provides a strong form of dependence between the random variables involved and is thus more desirable for many real applications. However, in terms of almost sure synchronization, to the best of our knowledge, there is little study for stochastic coupled networks with delay and Lévy noise by event-triggered control, leading to our motivation of the present paper.
Inspired by the above discussion, in this paper we will study the almost sure synchronization for the stochastic delay networks associated with Markovian switching and Lévy noise via event-triggering sampling, which to the best of our knowledge is not available in current literature. In summary, the contributions of this paper include the following new developments:
- 1.
To include Lévy noise to network systems can accommodate a more general situation in complexity, and the obtained result contains the often used Brownian noise as a special case. Thus the formulation is more applicable.
- 2.
The developed event-triggered sampling control has the adaptive capacity, which can be set prior through a desirable event-triggered function and provides the network an adequate environment to reduce the network workload.
- 3.
Almost sure convergence in network synchronization is always difficult to develop due to the requirement for estimating the time tail probability, in particular, for the appearance of Lévy noise that allows discontinuous sample path. To the best of our knowledge, the study of almost sure convergence for complex networks associated with Lévy noise is not available yet in current literature.
- 4.
We proved that the lower bound of inter-event intervals is greater than 0 which shows that the Zeno behavior can be excluded, indicating that the proposed approach under our framework is not only feasible but also effective. Moreover, the results have been extended to the directed topology, which removes the limitations of the undirected networks commonly used in literature.
The paper is organized as follows. In Section 2, necessary preliminaries and standard assumptions are introduced. The main results of the paper as well as their detailed proofs are given in Section 3. Numerical simulations, including a comparison with the network under some latest developed feedback controls, are provided in Section 4. The paper ends with concluding remarks in Section 5.
Section snippets
Problem formulation and preliminaries
Notations: stands for the identity matrix with dimension . is a -dimension vector whose entries are . The notions represent minimum and maximum eigenvalue of a given matrix, respectively. For a vector , let denote the transpose vector and denote -vector norm. Let be a complete probability space with a filtration . Let delay and denote the family of continuous function from to with the norm
Undirected coupled network
In this section, the almost sure synchronization criteria of system (1) are established based on the proposed event-triggered strategy, the stochastic Lyapunov–Krasovskii function, and the convergence theorem of non-negative semi-martingales. Moreover, the positive lower bound of the event intervals are given explicitly, which implies that the updating rule will exclude the undesirable Zeno behavior.
Directed coupled network
In practice, undirected graphs consist of a special class of the coupled networks. Most of the graphs in applications appear to be directed, and thus the resultant coupling matrix is asymmetric. This is the issue we will address in this section.
Numerical simulations
In this section, detailed numerical simulations are provided to illustrate the effectiveness of our proposed approach.
Example 1 Consider a stochastically undirected coupled Markovian switching networks with Lévy noise by implementing event-triggered sampling strategy, where represents the state vector of the th node;
Concluding remarks
This paper studies the almost sure synchronous problem of general Markovian switching complex networks associated with Lévy noise and delay by using the event-triggered sampling control. Under our proposed framework, the Markovian coupled networks can achieve the almost sure synchronization, including both direct and undirect coupled networks.
There are some issues remaining such as how to achieve the almost surely exponential synchronization and how to further minimize the triggered numbers
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The first two authors would like to thank Natural Science Foundation of Guangdong, China (2020A1515010372) for its financial support, and the last author is supported in part by National Science Foundation-Division of Mathematical Science (NSF-DMS) (1419028, 1854638).
The first author, Hailing Dong, would like to thank the hospitality from the Department of Mathematics, Southern Illinois University Carbondale during her visit from July 2017–July 2018, and this work was conducted during the
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