# Exponential Synchronization of Memristive Neural Networks with Discrete and Distributed Time-Varying Delays via Event-Triggered Control

Authors: Biwen Li; Wenbo Zhou
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we investigate the exponential synchronization problem of memristive neural networks (MNNs) with discrete and distributed time-varying delays under event-triggered control. An event-triggered controller with the static and dynamic event-triggering conditions is designed to improve the efficiency of resource utilization. By constructing a new Lyapunov function, some sufficient criteria are obtained to realize the exponential synchronization of considered drive-response MNNs under the designed event-triggered controller. In addition, the Zeno behavior will not occur by proving that the event-triggering interval has a positive lower bound under different event-triggering conditions. Finally, a numerical example is provided to prove the validity of our theoretical results.


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## Body

## 1. Introduction

Memristor, as the fourth fundamental circuit element, was predicted to exist by Chua [1] in 1971. But the physical implementation of memristor was very difficult by the immature nanotechnology at that time. It was not until 2008 that the prototype of memristor was manufactured firstly by the Hewlett-Packard research team [2, 3]. The memristor has a memory property because it can remember the amount of charge passing through. As an important part of neural networks, synapses have plasticity in terms of information storage and computing performance, and their previous activities need to be memorized according to presynaptic input and postsynaptic activities. Therefore, the memristor can simulate synaptic function perfectly in artificial neural networks [4]. In recent ten years, memristor-based neural networks, also known as memristive neural networks (MNNs), have attracted growing attention by the advantage of human brain simulation. The stability of neural networks as a crucial prerequisite to the application is very important, and the problem for stability analysis of MNNs has been widely studied by many researchers [5–12].However, in the process of neural network implementation, because of the immanent communication time and the finite speed of information processing of neurons, signal transmission among neurons will inevitably encounter time delay, especially time-varying delay, which will cause divergence, oscillation, or even instability of neural systems [13]. At the same time, neural networks generally have a spatial extent due to the existence of a large number of parallel pathways with different lengths and sizes of axons. Thus, there could exist either a distribution of conduction velocities along those pathways or a distribution of propagation delays over a spell under some circumstances, which may result in another kind of time delays in neural networks, that is, distributed time delays [14]. Recently, some outstanding works on the research of stability for MNNs with both discrete and distributed delays have been reported [15–18].As another critical dynamical behavior of neural networks, synchronization has equivalent importance as stability, which is also an active topic of research due to its extensively potential applications in information science, combinatorial optimization, biological systems, and secure communication [19–21]. It is noteworthy that MNNs are sensitive to the initial states than the traditional neural networks, which will lead to the more complicated chaotic path. So, the study for synchronization of MNNs has significantly practical value in science and engineering. Some excellent works dealing with the synchronization problem of MNNs have also appeared [22–26]. For example, lag synchronization of coupled MNNs with or without parameter mismatch was studied by Li and Cao [22] via ω-measure method. Wu et al. [25] proposed some new criteria to achieve the weak, modified, and function projective synchronization of chaotic delayed MNNs by using Lyapunov–Krasovskii function approach and generalized Halanay inequality. Based on fractional-order Leibniz rule and Razumikhin-type approach, Chen et al. [26] proved that a class of fractional-order complex-valued MNNs with delays is globally asymptotically stable and can achieve adaptive Mittag-Leffler synchronization under the differential inclusion theory and the framework of Fillippov solution, respectively.In addition, MNNs can be seen as the type of switching system depending on the state, and they often cannot achieve synchronization by themselves. Therefore, it is indispensable to find a more appropriate approach to synchronize the MNNs. Up to now, so many effective approaches for synchronization control have been found, such as sampled-data control [27], pining control [28], adaptive control [29], and impulsive control [30]. Under the assumption that the communication bandwidth is large enough, these synchronization control approaches based on continuous or discontinuous information exchange have good performance. However, the communication bandwidth is generally limited in practice. On the other hand, sampling-data control can efficiently save communication bandwidth. But on the premise of guaranteeing the synchronization performance, periodic sampled-data control usually chooses the small enough sampling period, which may lead to unnecessary utilization of the communication bandwidth. Accordingly, event-triggered control has been proposed to overcome the limitations of the aforementioned control approaches [31]. Event-triggered control only transmits information when the measured error exceeds the threshold, and it can decrease the frequency of information exchange and improve the efficiency of resource utilization. Very recently, the dynamic behaviors of various neural networks have been widely studied via event-triggered control [32–38]. For instance, by designing a dynamic event-triggered output feedback controller, Fei et al. [32] investigated the exponential stability for linear switched systems with frequent asynchronism. Li et al. [33] studied the tracking control problem of parameter strict-feedback systems with actuator faults and uncertain disturbances via event-triggered co-design method. Zhou and Zheng [36] were concerned with quasi-synchronization problem for MNNs with time-varying delays by event-triggered impulsive control. Nevertheless, we find that event-triggered exponential synchronization problem of MNNs with discrete and distributed time-varying delays has not been explored.Based on the above elaborations, we aim to solve the exponential synchronization problem for MNNs with discrete and distributed time-varying delays via event-triggered control. The main results of this paper can be listed as follows.(1)
We studied the event-triggered exponential synchronization of MNNs with two types of delays. Based on the idea of extending neural network model to get more practical results, the discrete and distributed delays in this paper are considered to be time-varying, which makes our network model more complex and comprehensive than previous works.(2)
Without using linear matrix inequalities (LMIs), a novel Lyapunov function is constructed to deal with the two different delayed terms, and thus the computational time and the conservation of the results are greatly reduced. Some sufficient criteria achieving exponential synchronization of drive-response MNNs with discrete and distributed time-varying delays are obtained.(3)
In order to ensure the synchronization performance of the system while reducing unnecessary waste of resources, we designed an event-triggered controller with the static and dynamic event-triggering conditions, which is composed of discontinuous sign term and linear diffusion term. Moreover, the positive lower bound of the interevent times under the two event-triggering conditions can be computed, which guarantees that the Zeno behavior does not happen.The rest of this paper is arranged as follows. Some preliminaries including model description, event-triggered control strategy, synchronization error system, and related definitions and assumptions are introduced in Section2. The event-triggered exponential synchronization on MNNs with discrete and distributed time-varying delays is analyzed in Section 3, respectively. In Section 4, an example is presented to illustrate our theoretical results. Finally, Section 5 concludes this paper.


## 2. Preliminaries

Consider a mathematical model of MNNs with discrete and distributed time-varying delays described by the following equation:(1)x˙it=−dixit+∑j=1naijxjtfjxjt+∑j=1nbijxjtfjxjt−τjt∑j=1ncijxjt∫t−σjttfjxjsds+Ii,where i=1,2,…,n, t≥0, xit is a state variable of the ith neuron; aijxjt, bijxjt and cijxjt are the memristive feedback connection weights without delay, with discrete delay and with distributed delay for i,j=1,2,…,n, respectively; di>0 denotes the self-inhibition of the ith neuron; fj· represents the activation function of the jth neuron; τjt and σjt are discrete and distributed time-varying delays satisfying 0≤τjt≤τ and 0≤σjt≤σ, where τ>0 and σ>0 are constants; and Ii is the external input variable to the ith neuron. Assume that the initial condition of system (1) is xiϵ=ϕiϵ,ϵ∈−ρ,0, where ρ=maxτ,σ, ϕiϵ∈C−ρ,0,ℝ.Moreover, the memristive connection weightsaijxjt, bijxjt, and cijxjt satisfy the following conditions:(2)aijxjt=a¯ij,D−fij>0;a¯ij,D−fij<0;aijt−,D−fij=0,bijxjt=b¯ij,D−fijτ>0;b¯ij,D−fijτ<0;bijt−,D−fijτ=0,cijxjt=c¯ij,D−fijσ>0;c¯ij,D−fijσ<0;cijt−,D−fijσ=0,where a¯ij,a¯ij,b¯ij,b¯ij,c¯ij, and c¯ij are all constants; D−fij, D−fijτ, and D−fijσ are the left Dini derivatives of fij, fijτ, and fijσ; and fij=fjxjt−xit, fijτ=fjxjt−τjt−xit and fijσ=∫t−σjttfjxjsds−xit. Throughout this paper, denote a^ij=maxa¯ij,a¯ij, b^ij=maxb¯ij,b¯ij, and c^ij=maxc¯ij,c¯ij for i,j=1,2,…,n.According to the drive-response concept, let MNNs (1) be the drive system; then, we can get the following response system:(3)y˙it=−diyit+∑j=1naijyjtfjyjt+∑j=1nbijyjtfjyjt−τjt+∑j=1ncijyjt∫t−σjttfjyjsds+Ii+uit,where yit denotes a state variable of the ith neuron, uit is the controller, and other parameters have the same definition as those in system (1). Assume that the initial condition of system (3) is yiϵ=ψiϵ,ϵ∈−ρ,0, where ρ=maxτ,σ, ψiϵ∈C−ρ,0,ℝ.Remark 1.
MNNs (1) and (3) can be seen as a type of switching system because the connection weights aij·, bij·, and cij· are state-dependent parameters. When the connection weights of MNNs (1) and (3) are fixed, i.e., a¯ij=a¯ij, b¯ij=b¯ij, and c¯ij=c¯ij, MNNs (1) and (3) will degrade into the traditional recurrent neural networks.
Let synchronization errorzit=yit−xit. Then, the following synchronization error system can be obtained:(4)z˙it=−dizit+∑j=1naijyjtfjyjt−aijxjtfjxjt+bijyjtfjyjt−τjt−bijxjtfjxjt−τjt+cijyjt∫t−σjttfjyjsds−cijxjt∫t−σjttfjxjsds+uit=−dizit+∑j=1naijyjtgjzjt+bijyjtgjzjt−τjt+cijyjt∫t−σjttgjzjsds+aijyjt−aijxjtfjxjt+bijyjt−bijxjtfjxjt−τjt+cijyjt−cijxjt∫t−σjttfjxjsds+uit,where gizi·=fiyi·−fixi· and gizit−τjt=fiyit−τjt−fixit−τjt.
In addition, the synchronization error system (4) can be expressed in a more compact form:(5)z˙t=−Dzt+Aygzt+Ay−Axfxt+Bygzt−τt+By−Bxfxt−τt+Cy∫t−σttgzsds+Cy−Cx∫t−σttfxsds+ut,where zt=z1t,z2t,…,zntT, Ax=aijxjtn×n, Ay=aijyjtn×n, Bx=bijxjtn×n, By=bijyjtn×n, Cx=cijxjtn×n, Cy=cijyjtn×n, D=diagd1,d2,…,dn, τt=τ1t,τ2t,…,τntT, σt=σ1t,σ2t,…,σntT, ut=u1t,u2t,…,untT, gz·=g1z1·,g2z2·,…,gnzn·T, and fx·=f1x1·,f2x2·,…,fnxn·T.
The following event-triggered controller is designed to synchronize MNNs (1) and (3), which can decrease the frequency of information exchange:(6)ut=−Pztk−Qsignztk,t∈tk,tk+1,where P=diagp1,p2,…,pn is the positive definite control gain matrix, Q=diagq1,q2,…,qn is the control gain matrix with switched elements, and tkt0=0 represents a release instant and satisfies(7)tk+1=inft≥tk:h>0,k∈N,where ‘inf’ is the infimum of a set and h denotes the event-triggering function.
Figure1 shows a schematic of the proposed event-triggered control strategy with a communication network. The measured error can be defined as Δzt=ztk−zt for t∈tk,tk+1. An event is triggered when the prescribed threshold is exceeded by the measured error, and then the controller will be updated. And the event-triggering function h is related to the measured error, which will be designed in Section 3. That is to say, the synchronization error signal can be sent to update the controller by judging whether h satisfies the condition. Thus, the measured error plays a key role in designing event-triggering controller.
Next, we give a definition and some assumptions for the theoretical derivation of this paper.Figure 1
Schematic of event-triggered control strategy.Definition 1.
For any initial conditionsϕiϵ, ψiϵ∈C−ρ,0,ℝ, if there are scalars S>0 and r>0 such that(8)zt1≤Se−rtzϵ1,t>0,where z·1=∑i=1nzi·, ziϵ=ψiϵ−ϕiϵ, ϵ∈−ρ,0, then the drive-response MNNs (1) and (3) are said to realize exponential synchronization.Assumption 1.
The discrete time-varying delayτjt and distributed time-varying delay σjt are bounded and differentiable. Moreover, their derivatives satisfy the conditions of 0≤τ˙jt≤τ^<1 and 0≤σ˙jt≤σ^<1, j=1,2,…,n, where τ^ and σ^ are positive constants.Assumption 2.
The activation functionsfi·:R⟶R satisfy boundedness and the Lipschitz condition, which means that there are two constants Mi>0 and li>0, i=1,2,…,n, such that(9)fiv1≤Mi,fiv1−fiv2≤liv1−v2,where v1,v2∈R.Remark 2.
Assumption2 is very important in the analysis of event-triggered exponential synchronization problem for the considered system because the drive MNNs (1) and the response MNNs (3) show parameter mismatch. Meanwhile, gizi·≤liyi·−xi· can be derived from the second inequality of Assumption 2. By adopting this assumption, the problems for synchronization of different MNNs have been solved in [31, 35]. In fact, there already exist some activation functions that meet Assumption 2, such as fix=x+1−x−1/2, fix=1/1+e−x, and fix=tanhx, respectively.


## 3. Theoretical Results

In this section, static and dynamic event-triggering functions will be presented to ensure that the event-triggered exponential synchronization between drive MNNs (1) and response MNNs (3) with discrete and distributed time-varying delays can be realized. In addition, the Zeno behavior has been guaranteed not to occur because the positive lower bound of the interevent times under the designed event-triggered controller with different triggering conditions can be computed.Theorem 1.
When Assumptions1 and 2 hold, if the parameters pi and qi of the control gain matrices satisfy(10)pi≥−di+∑j=1nlja^ij+b^ij1−τ^+c^ijσ1−σ^,(11)qi>πi,signzitsignzitk>0,qi≤−πi,signzitsignzitk≤0,then the drive-response MNNs (1) and (3) can be said to be exponentially synchronized under the controller (5) with the following static event-triggering function:(12)h=pmaxΔzt1−ωη−Hmaxzt1.On the other hand, the event-triggering intervalTk has a lower positive bound and satisfies the following inequality:(13)Tk≥ln1+ωD1η−ℋmaxzt1/pmaxKD1,where ω∈0,1, πi>∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj, pmax=maxipi, η=∑i=1nπi−∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj, Hmax=maxir−di−pi+∑j=1nlja^ij+erτb^ij/1−τ^+erσc^ijσ/1−σ^, r is a positive constant, K=2A^1+B^1+σC^1M1+∑i=1nqi+P1+D1ztk1, M=M1,M2,…,MnT, A^=a^ijn×n, B^=b^ijn×n, and C^=c^ijn×n.Proof.
Consider the following Lyapunov function:(14)Vt=V1t+V2t+V3t,with(15)V1t=ert∑i=1nzit,V2t=∑i=1n∑j=1nb^ij1−τ^∫t−τjtters+τgjzjsds,V3t=∑i=1n∑j=1nc^ij1−σ^∫−σjt0∫t+θters+σgjzjsdsdθ,in which r is a positive constant.
Then, fort∈tk,tk+1, the right upper Dini derivative of Vλt,λ=1,2,3, along with the solutions to synchronization error system (4), is calculated as(16)D+V1t≤ert∑i=1nr−dizit+∑j=1naijyjtgjzjt+∑j=1naijyjt−aijxjtfjxjt+∑j=1nbijyjtgjzjt−τjt+∑j=1nbijyjt−bijxjtfjxjt−τjt+∑j=1ncijyjt∫t−σjttgjzjsds+∑j=1ncijyt−cijxjt∫t−σjttfjxjsds−signzitpizitk+qisignzitk,(17)D+V2t=∑i=1n∑j=1nb^ij1−τ^ert+τgjzjt−b^ij1−τ˙t1−τ^ert−τjt+τgjzjt−τjt,(18)D+V3t=∑i=1n∑j=1nc^ijσ˙jt1−σ^∫t−σjtters+σgjzjsds+c^ijσjt1−σ^ert+σgjzjt−c^ij1−σ^∫−σjt0ert+θ+σgjzjt+θdθ=∑i=1n∑j=1nc^ijσjt1−σ^ert+σgjzjt−c^ij1−σ˙jt1−σ^∫t−σjtters+σgjzjsds.
Based on Assumption1 and 0≤τjt≤τ, 0≤σjt≤σ, under equalities (17) and (18), we have(19)D+V2t≤ert∑i=1n∑j=1nerτb^ij1−τ^gjzjt−b^ijgjzjt−τjt,(20)D+V3t≤∑i=1n∑j=1nert+σc^ijσ1−σ^gjzjt−c^ij∫t−σjtters+σgjzjsds≤ert∑i=1n∑j=1nerσc^ijσ1−σ^gjzjt−c^ij∫t−σjttgjzjsds.
According to the definition of measured errorΔzt=ztk−zt for t∈tk,tk+1, the following condition can be easily obtained:(21)ztk=zt+Δzt.
Combining (16)–(21) and Assumption 2, it follows that(22)D+Vt≤ert∑i=1nr−dizit+∑j=1na^ij+erτb^ij1−τ^+erσc^ijσ1−σ^gjzjt+aijyjt−aijxjtfjxjt+bijyjt−bijxjtfjxjt−τjt+cijyjt−cijxjt∫t−σjttfjxjsds+signzit−pizitk−qisignzitk≤ert∑i=1nr−di−pi+∑j=1nlja^ij+erτb^ij1−τ^+erσc^ijσ1−σ^zit+∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj+pmaxΔzit−qisignzitsignzitk.
In order to facilitate the subsequent derivation, we consider the following function:(23)Hir˜≐r˜−di−pi+∑j=1nlja^ij+er˜τb^ij1−τ^+er˜σc^ijσ1−σ^,where i=1,2,…,n, r˜>0.
Thus, we obtain(24)H˙ir˜=1+∑j=1nljτer˜τb^ij1−τ^+σer˜σc^ijσ1−σ^>0.
From (10), we have(25)Hi0=−di−pi+∑j=1nlja^ij+b^ij1−τ^+c^ijσ1−σ^≤0.
Forr˜∈0,+∞, Hir˜ is continuous and when r˜⟶+∞, Hir˜⟶+∞. Therefore, we can infer that there is a small enough constant rˇi>0 such that Hirˇi≤0, and for r˜i∈0,rˇi, Hir˜i≤0. Thus, for rˇ=minrˇ1,rˇ2,…,rˇn and i=1,2,…,n, we have(26)Hirˇ=rˇ−di−pi+∑j=1nlja^ij+erˇτb^ij1−τ^+erˇσc^ijσ1−σ^≤0.
Next, we can chooser∈0,rˇ, so(27)Hir≤0.
From condition (7) and the static event-triggering function (12) and for t∈tk,tk+1, we can know that h≤0, namely,(28)Δzt1≤ωη−Hmaxzt1pmax.
According to (11) and (27), we can combine (28) to simplify (22) to the following form:(29)D+Vt≤ert∑i=1nHirzit+pmaxΔzit−πi+∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj≤ertHmaxzt1+pmaxΔzt1−η≤ertω−1η−Hmaxzt1≤0,and then Vt can be proved to be monotonically decreasing, and we can easily deduce that Vt≤V0.
On the one hand, the above condition is derived by (14):(30)Vt≥ert∑i=1nzit=ertzt1.
On the other hand, letlmax=maxili; then,(31)V0=∑i=1nziϵϵ∈−ρ,0+∑i=1n∑j=1nb^ij1−τ^∫−τj00ers+τgjzjsds+c^ij1−σ^∫−σj00∫θ0ers+τgjzjsdsdθ≤∑i=1nsupϵ∈−ρ,0ziϵ+∑j=1nljerτb^ijτ1−τ^supϵ∈−ρ,0ziϵ+∑j=1nljerσc^ijσ1−σ^supϵ∈−ρ,0ziϵ≤1+lmaxmax1≤i≤n∑j=1nerτb^ijτ1−τ^+erσc^ijσ1−σ^×supϵ∈−ρ,0zϵ1.
From (30) and (31), it is concluded that(32)zt1≤1+lmaxmax1≤i≤n∑j=1nerτb^ijτ1−τ^+erσc^ijσ1−σ^×e−rtsupϵ∈−ρ,0zϵ1≤Se−rtsupϵ∈−ρ,0zϵ1,where S=1+lmaxmax1≤i≤nerτ∑j=1nb^ijτ/1−τ^+erσ∑j=1nc^ijσ/1−σ^.
Consequently, under controller (6) with the static event-triggering function (12), we can get that the response MNNs (3) can achieve exponential synchronization with the drive MNNs (1) from Definition 1.
Furthermore, we will prove that the event-triggering intervalTk has a positive lower bound, which can exclude the Zeno behavior.
As we all know, if the event-triggered condition is violated, the synchronization error will be sent to the controller. Thus, fort∈tk,tk+1, from condition (7) and static event-triggering function (12), we have(33)Δztk+11≥ωη−Hmaxzt1pmax.
In addition, from (4) and (5), we can obtain(34)dΔzt1dt≤dΔztdt1=dztdt1≤−Dzt+Ayfyt−Axfxt+Byfyt−τt−Bxfxt−τt+Cy∫t−σttfysds−Cx∫t−σttfxsds−Pztk−Qsignztk1≤D1zt1+2A^1+B^1+σC^1M1+P1+D1ztk1+∑i=1nqi≤D1Δzt1+2A^1+B^1+σC^1M1+P1+D1ztk1+∑i=1nqi≤D1Δzt1+K,where A^=a^ijn×n, B^=b^ijn×n, C^=c^ijn×n, M=M1,M2,…,MnT, and K=2A^1+B^1+σC^1M1+P1+D1ztk1+∑i=1nqi.
Fort∈tk,tk+1, according to (34) with the state Δztk=0, we get(35)Δzt1≤KD1eD1t−tk−1.
Combining with inequalities (33) and (35), it follows that(36)ωη−ℋmaxzt1pmax≤KD1eD1t−tk−1,which implies(37)Tk=tk+1−tk≥ln1+ωD1η−ℋmaxzt1/pmaxKD1.
Therefore, we have completed the proof.Remark 3.
According to the definition ofπi>∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj and condition (11), it is easy to know that chattering behaviors derived from the symbolic function in event-triggered controller (6) can be effectively reduced by choosing small parameters πi.Theorem 2.
When Assumptions1 and 2 hold, if the parameters pi and qi of the control gain matrices satisfy conditions (10) and (11), then the drive-response MNNs (1) and (3) can be said to be exponentially synchronized under controller (6) with the following dynamic event-triggering function:(38)h=pmaxΔzt1−ζt−ωη−Hmaxzt1,ζ˙t=−ζt−pmaxΔzt1+ωη−Hmaxzt1,where ζt is an internal dynamic function with the initial value ζ0=ζ0≥0.On the side, the event-triggering intervalTk has a lower positive bound and satisfies the following inequality:(39)Tk≥ln1+D1pmaxζt+ωη−Hmaxzt1/pmaxKD1,where ω,Hmax,η, and pmax are defined in Theorem 1.Proof.
From conditions (7) and (38), we can know that the event-triggering function h≤0 for t∈tk,tk+1, which means that(40)Δzt1≤ζt+ωη−Hmaxzt1pmax.
Then, from (38) and (40), we have(41)ζ˙t≥−1+pmaxζt.
Consider the following equality:(42)μ˙t=−1+pmaxμt,with initial value μ0=ζ0≥0. And the solution μt of the equality (42) satisfies(43)μt=ζ0e−1+pmaxt>0,ζ0>0,0,ζ=0.
According to the comparison principle, we can obtainζt≥0 from (41)–(43).
The following Lyapunov function is constructed:(44)Wt=Vt+ertζt,where r∈0,1 and the definition of Vt is the same as in (11).
Next, fort∈tk,tk+1, we can calculate the right upper Dini derivative of Wt along with the solutions to synchronization error system (4):(45)D+Wt=D+Vt+ertζ˙t+rertζt≤ertHmaxzt1−η+pmaxΔzt1−ζt−pmaxΔzt1+ωη−Hmaxzt1+rζt≤ertω−1η−Hmaxzt1+r−1ζt≤0.
So, similar to Theorem1, we get(46)zt1≤e−rtSsupε∈−ρ,0zɛ1+ζ0.
Consequently, under controller (6) with dynamic event-triggering function (38), we can get that the drive MNNs (1) and the corresponding response system (3) can achieve exponential synchronization from Definition 1.
In addition, from the second equation of (37), it is easy to see that(47)ζt=e−tζ0+∫0te−t−s−pmaxΔzs1+ωη−Hmaxzs1ds.
Therefore, we can draw the conclusion thatζt converges to a constant because Δzt1 and zt1 are bounded.
Moreover, similar to Theorem1, we obtain(48)Tk=tk+1−tk≥ln1+D1pmaxζt+ωη−Hmaxzt1/pmaxKD1.
So, we have completed the proof.Remark 4.
Guo et al. [31] studied the event-based synchronization problem of MNNs with time-varying delay, and Li et al. [35] investigated the exponential synchronization of complex-valued MNNs with time-varying delays via event-triggered control. Compared with the Lyapunov functions in [31, 35], Vt in Theorem 1 and Wt in Theorem 2 added the term V3t which can deal with distributed time-varying delay and the term ert which can realize exponential synchronization. Therefore, the activation functions in this paper are more specific when analyzing the dynamic behaviors of the considered model.Remark 5.
Zeno behavior refers to an infinite number of triggers in a limited time. In the proposed event-triggered control strategy of this paper, Zeno behavior will occur when the event-triggering conditions (12) and (38) are constantly satisfied, which would lead to controller (6) not effectively adjusting the trigger and the desired control effect not being achieved. Therefore, the Zeno behavior is not expected in the actual stimulation. By demonstrating that the event-triggering interval Tk is always larger than a positive constant, it has been proved that the Zeno behavior does not occur in Theorems 1 and 2.Remark 6.
In the event-triggering condition of Theorem2, a dynamic function ζt is added. Then, the threshold value under the dynamic event-triggering condition (38) can be more higher because ζt is always non-negative, so the frequency for information exchange between the controller and the sensor will be further reduced. Thus, the control strategy in Theorem 2 is better than that in Theorem 1 from the perspective of saving resources.


## 4. Illustrative Examples

In this section, an example and its simulations are provided to support our theoretical results.Example 1.
We consider the 2-D drive MNNs with discrete and distributed time-varying delays as follows:(49)x˙it=−dixit+∑j=12aijxjtfjxjt+∑j=12bijxjtfjxjt−τjt+∑j=12cijxjt∫t−σjttfjxjsds+Ii,where i=1,2, t≥0. Let d1=1.5, d2=1.3, and(50)a11x1t=1.1,D−f11>0;1.3,D−f11<0;a11t−,D−f11=0,a12x1t=2,D−f12>0;1.9,D−f12<0;a12t−,D−f12=0,a21x2t=0.2,D−f21>0;0.1,D−f21<0;a21t−,D−f21=0,a22x2t=1.5,D−f22>0;1.7,D−f22<0;a22t−,D−f22=0,b11x1t=−1.8,D−f11τ>0;−1.6,D−f11τ<0;b11t−,D−f11τ=0,b12x1t=0.25,D−f12τ>0;0.4,D−f12τ<0;b12t−,D−f12τ=0,b21x2t=0.1,D−f21τ>0;0.3,D−f21τ<0;b21t−,D−f21τ=0,b22x2t=−1.5,D−f22τ>0;−1.4,D−f22τ<0;b22t−,D−f22τ=0,c11x1t=−0.5,D−f11σ>0;−0.3,D−f11σ<0;c11t−,D−f11σ=0,c12x1t=0.57,D−f12σ>0;0.35,D−f12σ<0;c12t−,D−f12σ=0,c21x2t=0.7,D−f21σ>0;0.3,D−f21σ<0;c21t−,D−f21σ=0,c22x2t=−0.1,D−f22σ>0;−0.2,D−f22σ<0;c22t−,D−f22σ=0.
The discrete and distributed time-varying delays areτjt=σjt=et/1+et, the activation functions are selected as fixi·=tanhxi·, and the input signals are I1=I2=0. Therefore, we can get τ=σ=1, τ^=σ^=0.25, l1=l2=1, M1=M2=1, and a^11=1.3, a^12=2, a^21=0.2, a^22=1.7, b^11=1.8, b^12=0.4, b^21=0.3, b^22=1.5, c^11=0.5, c^12=0.57, c^21=0.7, and c^22=0.2. From the aforementioned concept, the corresponding 2-D response MNNs can be given as(51)y˙it=−diyit+∑j=12aijyjtfjyjt+∑j=12bijyjtfjyjt−τjt+∑j=12cijyjt∫t−σjttfjyjsds+Ii+uit,where i=1,2, t≥0, di,aij,bij,cij,τjt,σjt,Ii, and the activation function fiyi· are the same as (49).
When the controllerut=0, the trajectories of the synchronization errors z1t=y1t−x1t and z2t=y2t−x2t for drive-response systems (49) and (50) with the initial conditions x1ϵ=−4.3, x2ϵ=1.5, y1ϵ=−1.8, and y2ϵ=−2.5ϵ∈−1,0 are drawn in Figure 2, and it is easy to see that z1≠0 and z2≠0, that is, the considered drive MNNs (49) and the corresponding response MNNs (50) are not synchronized. Therefore, we will show that the two systems can achieve event-triggered exponential synchronization under certain controller.
Now, from condition (10) that the parameters pi need to satisfy, we can work it out that p1≥6.11 and p2≥4.2. For convenience of calculation, we choose p1=8.8 and p2=6.6. Then, pmax=8.8 can be obtained, and we can get that rˇ=0.4126 from (26). Besides, let r=0.25, so we can derive Hmax=−1.2575. Since πi>∑j=1na¯ij−a¯ij+b¯ij−b¯ij+σc¯ij−c¯ijMj, then π1>1.07 and π2>1.2. Next, we choose π1=1.27 and π2=1.6, from which we can see that η=0.6. Based on condition (11) that the parameters qi need to satisfy, we can obtain(52)q1=1.27,signz1tsignz1tk>0;q1=−1.27,signz1tsignz1tk≤0,(53)q2=1.6,signz2tsignz2tk>0,q2=−1.6,signz2tsignz2tk≤0.
Then, fort∈tk,tk+1, and according to the event-triggering functions (12) and (38), we can obtain(54)h=8.8Δzt1−ω0.6+1.1275zt1,(55)h=8.8Δzt1−ζt−ω0.6+1.1275zt1,ζ˙t=−ζt−8.8Δzt1+ω0.6+1.1275zt1,where ζ0=ζ0=0. Therefore, all the parameters satisfy the conditions in Theorems 1 and 2.
Whenω=0.25 and ω=0.75, the state trajectories of synchronization error system, trigger parameters W1t=Δz1 and W2t=ωη−Hmaxzt1/pmax, event trigger points, and event intervals are depicted in Figures 3–8.
Figures3(a), 4(a), 5(a), and 6(a) are the state trajectories of synchronization error system. In event interval tk,tk+1, we can conclude that ztk remains constant. And drive MNNs (49) and response MNNs (50) can achieve event-triggered exponential synchronization under controller (6) with the static/dynamic event-triggered functions (53) and (54).
The trigger parametersW1t=Δz1 and W2t=ωη−Hmaxzt1/pmax are shown in Figures 3(b), 4(b), 5(b), and 6(b). We can know that the event is triggered when the prescribed threshold is exceeded by the measured error. Therefore, the considered event-triggered mechanism can decrease the frequency of information exchange between the controller and the sensor while guaranteeing that MNNs (49) and (50) achieve exponential synchronization.
In addition, Figures3(c), 4(c), 5(c), and 6(c) are the event trigger points. It is not difficult to find that the higher the value of ω is, the lower the frequency of event-triggering is under the same event-triggering condition; in other words, the frequency of event-triggering is inversely proportional to the parameter ω. In addition, when the value of ω is fixed, the frequency of event-triggering under trigger function (54) is more higher than the case under function (53).
Figures7 and 8 depict the event intervals. Under the controller (6) with the static event-triggered function (54) and ω=0.25, we can get the maximum event-triggering interval of 0.202 and 0.42. Under the controller (6) with the dynamic event-triggering function (55) and ω=0.75, we can get the maximum event-triggering interval of 0.17 and 0.291. Hence, the Zeno behavior has been excluded due to the finite event-triggering interval.
Table1 gives some information about times of event-triggering TET, average release time ART, and data transmission rate DTR under static (53) and dynamic (54) triggering conditions with different values of ω. On the basis of Table 1, we can get that the times of event triggering with dynamic threshold are lower than those with static threshold for different values of ω∈0,1, and a larger value of ω can take more less data. When ω=1, it only needs to transmit 19 sampled signals under static triggering condition (54) and 14 sampled signals under dynamic triggering condition (55) to the controller, which economize 98.42% and 99.08% of resources.Figure 2
State trajectories of synchronization error system without control.Figure 3
The static event-triggering function (53) with ω=0.25. (a) State trajectories of synchronization error system. (b) Trigger parameters W1t=Δz1 and W2t=ωη−Hmaxzt1/pmax. (c) Event trigger points.
(a)(b)(c)Figure 4
The static event-triggering function (53) with ω=0.75. (a) State trajectories of synchronization error system. (b) Trigger parameters W1t=Δz1 and W2t=ωη−Hmaxzt1/pmax. (c) Event trigger points.
(a)(b)(c)Figure 5
The dynamic event-triggering function (54) with ω=0.25. (a) State trajectories of synchronization error system. (b) Trigger parameters W1t=Δz1 and W2t=ζt+ωη−Hmaxzt1/pmax. (c) Event trigger points.
(a)(b)(c)Figure 6
The dynamic event-triggering function (54) with ω=0.75. (a) State trajectories of synchronization error system. (b) Trigger parameters W1t=Δz1 and W2t=ζt+ωη−Hmaxzt1/pmax. (c) Event trigger points.
(a)(b)(c)Figure 7
Release instants and event intervals under triggering condition (53) with ω=0.25 and ω=0.75.
(a)(b)Figure 8
Release instants and event intervals under triggering condition (54) with ω=0.25 and ω=0.75.
(a)(b)Table 1
For different values ofω and fixed t = 1.2, h = 0.001, times of event-triggering TET, average release time ART, and data transmission rate DTR under static and dynamic triggering conditions (53) and (54).
ω0.250.50.751TETARTDTR (%)TETARTDTR (%)TETARTDTR (%)TETARTDTR (%)Static750.0166.25450.0273.75300.042.5190.0631.58Dynamic490.0244.08340.0352.83200.061.67110.1090.92Remark 7.
According to the example and simulations, we can further validate the conclusion in Remark6. But under dynamic event-triggering function (54), higher computational speed is required to deal with the increasing computational burden of the controller. From Table 1, we can see that the average release time ART under static and dynamic event-triggering conditions is basically the same, so static event-triggering condition is undemanding and efficient in view of the complexity of implementation. Then, we can choose the appropriate condition to realize the desired control effect from different perspectives.


## 5. Conclusion

The realization of synchronization for MNNs is always a widely concerned issue. To our knowledge, the exponential synchronization problem of MNNs with discrete and distributed time-varying delays has scarcely been studied. Therefore, we investigated the aforementioned problem by using the event-triggered control in this paper. Based on the drive-response concept, the synchronization error system has been derived. Then, an event-triggered controller with static and dynamic event-triggering conditions has been designed to improve the efficiency of resource utilization. Besides, some sufficient conditions achieving event-triggered exponential synchronization for drive-response MNNs with discrete and distributed time-varying delays are obtained by using Lyapunov functional approach. Furthermore, the positive lower bound of the event-triggering interval can be derived, which ensures that the Zeno behavior does not happen. Finally, an example is shown to illustrate our theoretical results. In order to further study the dynamic behaviors including finite-time/fixed-time synchronization and multimode function synchronization of MNNs with discrete and distributed time-varying delays, we will aim at designing more effective event-triggered control strategy with different event-triggering conditions in the future.



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*Source: 5575849-2021-04-05.xml*