Improved results on synchronization of stochastic delayed networks under aperiodically intermittent control

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Abstract

This paper addresses the synchronization of stochastic complex networks with time-varying delay via aperiodically intermittent control (AIC). By proposing the concepts of average control ratio and average control frequency for AIC, some new synchronization conditions are obtained, which relax the constraints of the lower bound of control widths and the upper bound of control periods. And the proportion of rest widths can be any value in (0,1). So the constraints on AIC are loosened and thus the conservativeness is reduced compared with the existing related results. Two types of time delay are investigated: (i) the upper bound of time-varying delay should be smaller than the average control width but can be larger than the lower bound of control widths; (ii) the upper bound of time-varying delay has no relationship with control and rest widths. An example of coupled stochastic oscillators systems is presented to show the effectiveness and superiority of our results.

Introduction

During the past few decades, synchronization of complex networks has attracted widespread interest since it plays a significant role in many areas such as sensor networks, pattern recognition and so on [1], [2], [3], [4], [5]. It ought to be noted that the impacts of time delay are indispensable, which may give rise to undesirable performance. Therefore, it is of significance to investigate the effect of time delay on the synchronization of complex networks [6], [7], [8], [9]. On the other hand, in real life, complex networks often suffer from environment noises that might arise from random fluctuations during the processes of transmission. Until now, a great deal of results on synchronization of stochastic delayed complex networks have been developed, see [10], [11], [12], [13] for example.

Usually, it is difficult for a complex network to achieve synchronization by itself. Hence, various control strategies have been designed to actuate networks to the objective states, such as impulsive control [14], feedback control [15], sampled-data control [10], [16], [17], event-triggered control [18], [19], [20], [21], intermittent control [22], [23], [24]. It is noteworthy that in recent years, intermittent control, especially the aperiodically intermittent control (AIC), has drawn considerable attention in the study of synchronization issues since this kind of control strategy is more economical and can well deal with the case that the feedback information is interrupted intermittently [24], [25], [26], [27], [28], [29]. In [27], [28], [29], Liu and Chen firstly carried out the synchronization analysis of complex networks with and without time delay by AIC. Then based on [27], [28], [29], many authors continued to study the synchronization of stochastic complex networks via AIC [11], [26], [30], [31], [32], [33], [34].

Nevertheless, it should be noted that in current study of AIC, the following two kinds of constraints were imposed to restrict the lower bound of control widths and upper bound of control periods or the maximum proportion of rest widths:

(i) (see e.g., [26], [27], [28], [29], [30], [31], [32], [35]) infl(sltl)=ϑ˜ and supl(tl+1tl)=ϖ˜ with 0<ϑ˜<ϖ˜<+;

(ii) (see e.g., [11], [26], [29], [33], [34]) lim supl+tl+1sltl+1tl=ψ(0,1), where [tl,sl) and [sl,tl+1) represent the control interval and rest interval, sltl and tl+1sl are the control width and rest width, respectively. From above two constraints, one can see that although the intermittent control in previous work is aperiodic, the distribution of control intervals and rest intervals is essentially uniform, i.e., the AIC satisfies quasi periodicity, which is not reasonable in the implementations. For example, in the generation of wind power, it is usually unreasonable to require the length of each wind time since the wind is usually completely irregular. Therefore, it is highly desirable to relax above two constraints and to explore more general AIC that satisfies complete aperiodicity.

To our knowledge, up to now, there have been rare work addressing the synchronization of networks via AIC with complete aperiodicity. In this paper, we will investigate this kind of AIC. And in this regard, certain control widths and rest widths of AIC are allowed to be very large or very small. The concepts of average control ratio and average control frequency (see Definitions 2 and 3) are proposed for AIC with complete aperiodicity for the first time. It should be noted that it is reasonable. Taking the generation of wind power for instance again, although each wind time cannot be obtained, the average wind time and average wind period of one month or one quarter can be estimated. Additionally, in the current study of synchronization of networks via AIC, time delay is usually considered to be time-varying and divided into two types: the first is that the upper bound of time-varying delay is less than the infimum of control widths [27], [29] (this case is noted as small time-varying delay in [29]); the second is that the upper bound of time-varying delay has no relationship with the control and rest widths [11], [29], [33], [34] (this case is noted as large time-varying delay in [29]). The current synchronization analyses were carried out based on condition (i) or condition (ii) in above paragraph. In this paper, we will continue to consider the two types of time-varying delay. And for the case that the upper bound of time-varying delay is small, it only needs that the upper bound of time-varying delay is less than the average control width but can be larger than the lower bound of control widths. Compared with previous related work, the main contributions of this paper are outlined as follows:

(I) The distribution for control widths and rest widths of AIC can be more flexible, which makes the controller closer to the realistic situation.

(II) The constraints of quasi periodicity for AIC are relaxed and thus the conservativeness is relaxed compared with the existing works on AIC.

(III) Average control width of AIC instead of the lower bound of all control intervals of AIC, is used to measure the small time-varying delay and large time-varying delay. And for the case of small time-varying delay, the upper bound of time-varying delay can be larger than the lower bound of control widths.

Section snippets

Preliminaries and model descriptions

Let R=(,+), R+=[0,+), Rt0+=[t0,+) for certain t0R+, N={1,2,,N} and N={0,1,}. Let |·| and · stand for the Euclidean norm for vectors and the trace norm for matrices, respectively. For a matrix A=(aij)N×N, the Laplacian matrix of A is noted as L(A)=(lij)N×N, where lii=j=1,jiNaij and lij=aij for ij. Denote by (G,A) a digraph G with weight matrix A. Let (Ω,F,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions and B(t) be an m-dimensional

Main results

In this section, some novel sufficient conditions under two kinds of time delay: small time-varying delay and large time-varying delay, are separately presented to verify the synchronization of stochastic complex networks via more general AIC with complete aperiodicity. Different from the measurement index (the lower bound of control widths) for time delay used in [29], we here use the average control width (i.e., averagecontrolratio×1averagecontrolfrequency) to measure the small time-varying

Numerical examples

Recently, there has been a heated discussion among coupled oscillators because of its widespread applications in various areas [13], [43], [44], [45], [46]. In this section, to apply the theoretical results above, we focus on the synchronization of stochastic coupled oscillators via AIC. Firstly, we consider a coupled oscillators on network as follows:x¨i(t)+bix˙i(t)+xi(t)+f˜i(t,xi(tτ(t)))+j=1Naijxj(t)=Ii(t),where iN, bi>0 denotes the damping coefficient, τ(t) is time-varying delay

Conclusion

In this paper, the exponential synchronization of stochastic complex networks with time-varying delay was studied by AIC with complete aperiodicity. Some new synchronization criteria are given, which relax the constraints of the lower bound of control widths and the upper bound of control periods. And the proportion of rest widths can be any value in (0,1). It should be noted that the AIC of each vertex system is activated or not activated synchronously. In the future, we will proceed to

Declaration of Competing Interest

The authors do not have any conflict of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province (Nos. ZR2017MA008, ZR2017BA007), Project of Shandong Province Higher Educational Science and Technology Program of China (No. J16LI09), and the Innovation Technology Funding Project in Harbin Institute of Technology (No.HIT.NSRIF.201703).

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