Synchronization for stochastic Lévy noise systems on a time-varying multi-weights network via delay intermittent control

Abstract

In previous papers, either the time-varying coupling structure or multi-weights have been considered into networks. However, few scholars have paid attention to networks with both time-varying coupling structure and multi-weights. In this paper, we formulate and probe stochastic Lévy noise delayed systems on a time-varying multi-weights network (SLDSTN) for the first time. In order to solve the synchronization problem of SLDSTN, we design a novel class of delay intermittent control. Different from previous intermittent control based on current state, delay intermittent control is based on past state. Then, by means of Lyapunov method, graph theory and some techniques of inequalities, sufficient conditions for exponential synchronization in mean square of SLDSTN are proposed. Therein, we relax the condition of processing the time-varying coupling term successfully. Furthermore, for presenting the superiorities of delay intermittent control, delay feedback control and aperiodically intermittent control also are applied to solve the synchronization problem of SLDSTN. To demonstrate the effectiveness of the theoretical results, a class of single-link robot arms is considered as a practical application. Finally, some numerical simulations are provided.

Introduction

With the progress of science and technology, artificial intelligence develops faster and faster. Therein, neural networks, deep learning, complex networks, etc. have made great progress, which are parts of applications of artificial intelligence methods in engineering (Lopez-Garcia et al., 2020, Huang et al., 2020, Zhang and Shi, 2021). For example, in order to predict the deteriorated characteristics of gas turbine, the authors in Talaat et al. (2018) designed an artificial neural network to analyze the gas path measurement. In fact, these can all be formulated as a class of coupled systems on networks (CSNs). Moreover, CSNs have extensive applications developed in many fields, such as mathematics, physics and biology (Li et al., 2011, Luo and Yao, 2019, Wu et al., 2021). These applications are mainly dependent on dynamical behaviors of CSNs including synchronization, consensus and stability (Xu et al., 2021a, Zhou et al., 2020d, Liu et al., 2021, Li and Yang, 2017). Among them, as a typical collective behavior of dynamical networks, synchronization is the focus of scholars and lots of results on synchronization of CSNs have been reported (see Selvaraj et al., 2018, Alsaedi et al., 2020, Ding et al., 2021 and Zhou et al. (2020b)). For example, in Alsaedi et al. (2020), finite-time synchronization of sampled-data Markovian jump complex dynamical networks with additive time-varying delays has been investigated based on dissipative theory.

Moreover, in previous studies, time delays are considered as an important and unavoidable factor which may cause desynchronization of networks. In fact, a signal or influence traveling through networks is often associated with time delay because of traffic congestion and the finite speeds of transmission and spreading (Hu et al., 2019, Zhu, 2019). There exist two types of time delays in general networks in reality. One is internal delay occurring inside the networks (Zhou et al., 2020e) and the other is coupling delay caused by exchanging of information among nodes of networks (Aadhithiyan et al., 2021). For instance, in queuing networks, there exists random time delay of customers waiting in a line before departing from the service (Dshalalow, 1989).

Besides, another inevitable factor which may affect synchronization of networks is stochastic disturbances, such as white noise, Markovian switching and Lévy noise (Zhou et al., 2020e, Wu et al., 2017, Guo et al., 2021, Xu et al., 2021c, Tong et al., 2020b). Therein, as a discontinuous noise generated by a fluctuating driving process with heavy tails, Lévy noise has advantages over white noise and Markovian switching in describing some sudden shocks, especially the very rare and extreme sudden events, such as earthquake, tsunami and epidemics (Zhou et al., 2020d, Zhou et al., 2020e). As a matter of fact, CSNs with Lévy noise have been applied in many fields such as neural networks, disease spreading, climate dynamics and so on Zhou et al., 2020b, Sokolov et al., 2011. For instance, the neutral-type neuron models are driven by white noise in general (Gerstner and Kistler, 2002). But in real neural systems, fewer inputs coming from synapses near the trigger zone of postsynaptic neurons might cause a large impulse in noisy amplitudes due to the higher concentration of voltage sensitive sodium-channel of the trigger zone (Pacut, 2001). This situation is more suitable to be modeled by Lévy noise systems rather than white noise systems. Thus, it is meaningful to investigate the effects of time delays and Lévy noise on synchronization of CSNs.

Most of previous literature on CSNs consider the topology structure under the time invariance assumption. However, the topology structure of networks in practical systems may change with time because networks are inevitably influenced by the external environment (see, for example, the work of Zhou et al. (2020d) and Olfati-Saber and Murray (2004)). For instance, bidirectional associative memory neural networks always vary with time and weights of state transmission in oscillators networks are ubiquitously changed with time in mechanical systems and epidemic models (Wu et al., 2019). Therefore, CSNs with time-varying coupling structure have attracted much research attention and some useful results have been reported (Zhou et al., 2020d, Wu et al., 2019, Liu et al., 2018). For instance, synchronization of stochastic coupled systems with time-varying coupling structure on networks was investigated in Liu et al. (2018); Zhou et al. researched the stabilization of stochastic time-varying coupled systems with delays and Lévy noise on networks in Zhou et al. (2020d).

It is worth pointing out that above work on time-varying coupling structure is focused on the single weight. However, numerous networks in real world are multi-weights, such as complex biology networks, transportation network and social network (Li et al., 2021, An et al., 2014, Zheng et al., 2016, Zhao et al., 2021), in which more than one weight is linked to vertices. That is, the interconnecting forms between the vertices in the highly complex network are many (Aadhithiyan et al., 2021). For example, in social network, taking persons as nodes, people can communicate with others through many channels such as letters, email and telephone, where each contact method represents different coupling having different weights. Hence, using CSNs with multi-weights to model the real world networks is more reasonable. To the best of our knowledge, the synchronization problem of stochastic Lévy noise delayed systems on a time-varying multi-weights network (SLDSTN) has not been investigated yet. Therefore, we attempt to research the exponential synchronization problem of SLDSTN. Since considering both time-varying coupling structure and multi-weights into CSNs leads to the increase in complexity of topology structure of network, it is difficult for SLDSTN to achieve synchronization spontaneously. In view of this, we devote to seeking for an effective technique.

Up till now, many control strategies have been proposed to force CSNs to reach synchronization or stability, including continuous feedback control, intermittent control and so on (Zheng et al., 2016, Yang et al., 2020, Xu et al., 2021b, Rakkiyappan et al., 2017, Tong et al., 2020a). It is worth pointing out that the synchronization problem of networks with time-varying coupling structure or multi-weights under various controls has been investigated (Zhou et al., 2020d, Liu et al., 2018, Zhang and Yang, 2019, Zhou et al., 2019). In fact, continuous feedback control is a classical and valid control strategy to make networks achieve synchronization. Note that continuous feedback control is activated at all times. In contrast, intermittent control is an effective discontinuous control strategy, which is activated in some certain time intervals and off at other time intervals (Zhou et al., 2020b). Therefore, it is more economical and reduce the amount of transmitted information greatly. In fact, intermittent control can be divided into periodically intermittent control (PIC) and aperiodically intermittent control (AIC) (Zhou et al., 2020d, Wu et al., 2020). Compared with PIC, AIC does not require fixed control periods or fixed control widths, which implies that AIC is more general, more flexible and with less restriction (Li et al., 2021). On the other hand, there may exist the time delay between the time of state observation and the time when control arrives at the system (Mao et al., 2008). For instance, there is a time delay (i.e. 1.28 s) of a radio signal from the moon to reach the earth (Hu et al., 2020). As a result, it is more reasonable if the control depends on a past state instead of a current state in practice. Consequently, some researchers have paid attention to the stabilization problem of CSNs based on delay control, such as delay feedback control (DFC) (Li and Mao, 2020), delay impulse control (Wang et al., 2020a), etc. It is shown that in Li and Mao (2020), authors designed a DFC to make the controlled hybrid stochastic differential delay equation achieve asymptotically stable. Motivated by these, by combining with the advantages of AIC and DFC, a class of delay intermittent control (DIC) is designed in this paper.

Inspired by the aforementioned discussions, the main purpose of this paper is employing DIC, DFC and AIC to solve the synchronization problem of SLDSTN. Furthermore, our theoretical results are applied to study synchronization of single-link time-varying multi-weights robot arms network with Lévy noise and time delays (STRNLT). In addition, the primary contributions of the present paper are listed as follows

(1) This paper considers coupled systems on a time-varying multi-weights network for the first time and relax the conditions of processing the time-varying coupling term successfully compared with Zhou et al. (2020d) and Wu et al. (2019).

(2) In this paper, a novel class of DIC is designed. Therein, delay in DIC is time-varying which is different from that in previous delay controls (Mao et al., 2008, Hu et al., 2020, Li and Mao, 2020, Wang et al., 2020a). Meanwhile, DIC has the advantages of both AIC and DFC.

(3) Compared with the existing results for the synchronization problem of networks with time-varying coupling structure or multi-weights under various controls (Zhou et al., 2020d, Liu et al., 2018, Zhang and Yang, 2019, Zhou et al., 2019), DIC we designed is based on past state rather than current state.

(4) The synchronization problems of SLDSTN and STRNLT are researched via AIC, DFC and DIC, respectively. And our theoretical results show that the control law depends on perturbed intensity of noise and time-varying multi-weights coupling strength, time delays and the maximum rest time ratio of DIC.

The rest of this paper is organized as follows. Section 2 introduces preliminaries and model formulation. In Section 3, some synchronization criteria of SLDSTN based on AIC, DFC and DIC, respectively, are presented. Then, the synchronization problem of STRNLT is studied and corresponding numerical simulations are given to show the effectiveness of our theoretical results in Section 4. Finally, Section 5 completes the paper by giving our conclusions (see Table 1).