# Global Robust Exponential Synchronization of Multiple Uncertain Neural Networks Subject to Event-Triggered Strategy

Authors: Jin-E Zhang; Huan Liu
Journal: Complexity
 (2019)

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## Abstract

This paper proposes the event-triggered strategy (ETS) for multiple neural networks (NNs) with parameter uncertainty and time delay. By establishing event-triggered mechanism and using matrix inequality techniques, several sufficient criteria are obtained to ensure global robust exponential synchronization of coupling NNs. In particular, the coupling matrix need not be the Laplace matrix in this paper. In addition, the lower bounds of sampling time intervals are also found by the established event-triggered mechanism. Eventually, three numerical examples are offered to illustrate the obtained results.


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## Body

## 1. Introduction

Multisystem network is a complex dynamical network, which has received great attraction due to its many applications in various fields, such as secure communications and biological systems. The dynamical characteristics of multisystem networks have been paid more and more attention in the fields of science and engineering (see [1–4]). As we know, synchronization of a coupling system means that its all subsystems produce common behavior under different initial values. The coupling scheme describing the interaction rules between subsystems plays an important role in ensuring the synchronization. In particular, the synchronization of multisystem networks has become a hotpot of nonlinear scientific research because the synchronization of multisystem networks can better describe many observed natural phenomena and can create ordered multisystem networks. As a kind of control, coupling is the key to ensure the synchronization of complex dynamical networks; hence, several coupling schemes are proposed to realize the synchronization. However, most of the existing works require that the coupling matrix is always the Laplace matrix; that is, the row sums of the coupling matrix all are zero and the nondiagonal elements are nonnegative, which greatly restrict us to design a good performance coupling controller. In fact, we notice that there are some non-Laplacian coupling in complex dynamical networks, so we can try to achieve the synchronization of multisystem networks by constructing non-Laplacian coupling matrix. In addition, in recent decades, NNs have become a hot research topic because of its rich content and wide application; therefore, there are many results in the research of NNs, such as in [1, 5–12]. The authors in [1] proposed a new ETS to achieve the synchronization of multiple NNs (MNNs). The dynamical characteristics of nonautonomous fractional-order delay NNs were studied in [5]. Wu and Zeng [9] derived two anti-synchronization algorithms to realize exponential anti-synchronization of memristive recurrent NNs. In [10], some results were established to ensure the Lagrange stability of NNs with memristive synapses. Wu and Zeng [11] discussed a class of memristive NNs, and several sufficient criteria were established for exponential stabilization by using matrix inequality techniques. Wu et al. [12] proposed a novel and efficacious method to study the periodic NNs, and some new results for the periodic NNs were obtained. Therefore, it is an interesting and far-reaching research topic to study the synchronization of MNNs by constructing a proper non-Laplacian coupling matrix.Parametric uncertainty arises from a partial understanding of mathematical models, for instance, empirical quantities and constitutive laws (see [13–22]). The uncertainty of parameters must be considered in actual system because the parameters of the model in the process of industrial control are often uncertain. It is fortunate that uncertain parameters have been considered in many models in order to describe practical problems more accurately. In [13], the dynamical characteristics of stochastic nonlinear systems with parametric uncertainty were concerned. Maharajan et al. [14] investigated the problem of enhanced results on robust finite-time passivity for uncertain discrete-time Markovian jumping BAM delayed NNs. The authors in [17] provided several novel delay-dependent stability criteria to ensure robust stability of uncertain stochastic systems. Huang et al. [20] investigated robust state estimation of uncertain neural networks by designing robust state estimator and using new bounding technique. Zhu et al. [22] discussed the stability of uncertain neutral systems, and some new criteria were provided to guarantee the stability of the model. Moreover, fractional-order system is a charming research field, which describes the real world more accurately than integer-order system. Recently, Zhang [21] studied uncertain fractional-order system and its application. The uncertain parameters of the above models are required to be norm-bounded; it is a common way to deal with these uncertain parameters by using inequality techniques. With the mature of the technology, some problems in MNNs have been well solved; however, it is still a great challenge to design a feedback controller with good performance when we analyze multisystem networks. There are many papers about the topic of parametric uncertainty, but the issue about trusted control in the MNNs with uncertain parameters still has a long way to go and much further work is worth studying. In addition, time delay is often inevitable in practical systems, which may lead to instability of the system; in other words, time delay is the important reason for instability of the model. Although parameter uncertainty and time delay bring difficulties to the theoretical analysis of dynamical characteristics of system, in practical application, the analysis of model is often unreasonable without considering uncertain parameter and time delay. Hence, theoretical knowledge and practical experience urge us to study the dynamics problems of MNNs with parameter uncertainty and time delay.Event-triggered mechanism is a very important sampling mechanism, that is, the emergence of certain events rely on the state of the system and this is also the difference between event-triggered feedback control and the traditional sampling mechanism. Because continuous-time control requires the continuous information, it is expensive and unrealistic in the real world. Unlike continuous-time control, event-triggered mechanism is a discontinuous-time control, which only requires the local communication data. Considering the sampling period, the sampling data control scheme usually adopts zero-order hold to keep the last sampling system state and control signal which is sent to the next event. In event-triggered mechanism, the sampling will not start and the controller will not be updated, unless its size reaches the specified threshold; therefore, the greatest advantage of event-triggered mechanism is to reduce communication data. Furthermore, event-triggered mechanism has been widely studied due to its effectiveness in practical systems, for example, [23–26] and references therein. Dolk and Heemels [23] investigated a networked control system subject to event-triggered control and its application to packet losses. In [24], a new decentralized event-triggered item for distributed networked systems was proposed to reduce the waste of network resources. The event-triggered rule was proposed to solve the problem of excessive use of communication resources in [25]. The authors in [26] studied neural networks by the proposed event-triggered rule, and the rule effectively solved a large number of computational problems. It is not difficult to find that event-triggered mechanism is very useful and meaningful sampling mechanism because many practical problems can be solved via ETS, but there are few research results on the combination of event-triggered mechanism and MNNs.Based on the above discussion, the synchronization of MNNs with parameter uncertainty and time delay is studied and emphasized. By establishing the event-triggered mechanism, this paper derives several sufficient criteria to guarantee the synchronization of the systems; it shows that these results are different from the previous ones. Furthermore, the lower bounds of sampling time intervals are also obtained; that is, if the triggering timetk is known, we can predict the next triggering time tk+1. Roughly speaking, this paper has three highlights: (1) Most existing works on multisystem networks require that the coupling matrix is always the Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative. Because of the particularity of Laplacian matrix, the design of coupling controller is greatly limited. In fact, we have found some non-Laplacian coupling in coupled dynamic networks. In this paper, the coupling matrix need not be the Laplace matrix; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary. (2) Some criteria are proposed to ensure the synchronization of MNNs with parameter uncertainty and time delay by establishing event-triggered mechanism and using matrix inequality techniques. Compared with the most of the existing papers, the results obtained are more simple and convenient in this paper. (3) We can find the lower bounds of the sampling time intervals by the established event-triggered mechanism, and then, we can know that the Zeno behavior will not happen.The remaining part of this paper is designed as follows. Section2 introduces the model of MNNs with parameter uncertainty and time delay and some preliminary knowledge. Section 3 gives several sufficient criteria to achieve the global robust exponential synchronization, and the lower bounds of the sampling time intervals are also found. Section 4 provides three numerical examples to demonstrate the obtained results. In the end, Section 5 summarizes the paper’s relevant conclusion.


## 2. Preliminaries and Problem Formulation

Throughout the paper,Rn represents the n-dimensional Euclidean space. I stands for an identity matrix with proper dimensionality. Let Cτ≔C−τ,0,Rn be a Banach space that composed of all the continuous functions φ:−τ,0 to Rn. ⋅2 and ⋅inf stand for 2-norm and inf-norm, respectively, and let us write ⋅2 in terms of ⋅ for the sake of simplicity. For a matrix A=aijn×n, amax=maxaij, and amin=minaij; furthermore, AT stands for its transpositive, and A>0, A<0, A≥0, and A≤0 mean that A are positive definite, negative definite, positive semidefinite, and negative semidefinite, respectively. diag⋅ is representative of diagonal matrix. ϕ refers to empty set. If no otherwise specified, matrices are supposed to have compatible dimensions.Consider a group ofr NNs with parameter uncertainty and time delay, and the model of kth NNs with parameter uncertainty and time delay is given as follows:(1)dzktdt=−D+ΔDzkt+B+ΔBfzkt+C+ΔCgzkt−τ+Jt+ukt,where k∈ℐ=1,2,…,r; t≥0; zkt=zk1t,zk2t,…,zkntT∈Rn refers to the state vector; D=diagd1,d2,…,dn is the real diagonal positive-definite matrix standing for the neuron self-inhibitions; Jt∈Rn is the input or bias; τ>0 represents the transmission delay; fzkt=f1zk1t,f2zk2t,…,fnzkntT and gzkt−τ=g1zk1t−τ,g2zk2t−τ,…,gnzknt−τT are the neuron activation functions; B=bkmn×n is the weight matrix and C=ckmn×n is the delay weight matrix; ΔB, ΔC, and ΔD are the norm-bounded uncertainty terms; and ukt∈Rn denotes the control input.The distributed event-triggered controller is given below:(2)ukt=λ∑m=1rykmzmtms,t∈tks,tks+1,for k∈ℐ and s≥1, where tks refers to the sampling time sequence, λ>0 represents coupling gain, and K=ykmr×r stands for the coupling matrix such that(3)∑m=1rykm=ϱ∈R.Next, we define the following distributed event-triggered function:(4)Hkst=zkt−zktks−αk∑m∈ℐkzktms−zktks,and the sampling time sequence tks satisfies ETS:(5)tks+1=inft>tks,Hks>0,for k∈ℐ and s≥1, where αk>0 refers to the control parameter and ℐk⊆ℐ with ℐk=m,m≠k,ykm≠0.Remark 1.
LetΦkm=αk∑m∈ℐkzktms−zktks be the sth threshold on the kth neuron. It is not difficult to find that Φkm determines sampling time of the kth neuron, and Φkm is closely related to the control parameter αk and subset ℐk. In particular, if αk=0 or ℐk=ϕ, the kth neuron will have sampling time at any t≥t0; that is to say, the Zeno behavior does happen when αk=0 or ℐk=ϕ. It should be noted that the transmission event is triggered and the controller is updated when the measurement error zkt−zktks exceeds the threshold Φkm in the event-triggered controller; unlike the event-triggered controller, the impulse controller samples on the determined impulse time sequence tks. Thus, we can know that the sampling time sequence tks satisfies tks+1=inft>tks,Hks>0 in the event-triggered controller, and the sampling time sequence tks is known in the impulse controller.
Then, we can rewritte model (1) as follows:(6)dzktdt=−D+ΔDzkt+B+ΔBfzkt+C+ΔCgzkt−τ+Jt+λ∑m=1rykmzmtms,t∈tks,tks+1.
The initial conditions of model (6) are assumed to be(7)zkt0+ϑ=φkϑ,where φk∈Cτ,ϑ∈−τ,0,andk=1,2,…,n.
Letzt,t0,φ stands for the solution of model (6); in order to avoid this paper being too long, we assume that there exists a unique solution of system (6).
System (6) is called to be globally robustly exponentially synchronized, namely, each subsystem in (6) can achieve global robust exponential synchronization if there are constants M≥1 and h>0 satisfying(8)zkt−zmt≤Mmaxi,j∈ℐφi−φje−ht−t0,for any t≥0 and k,m∈ℐ, where h stands for convergence rate.
For the sake of discussion, we give two basic assumptions.Assumption 1.
(9)
Δ
D
=
M
1
R
1
t
N
1
,
Δ
B
=
M
2
R
2
t
N
2
,
Δ
C
=
M
3
R
3
t
N
3
,where Mi and Ni (i=1,2,3) are constant matrices with proper dimensionality and Rit are the unknown matrix with RiTtRit≤I (i=1,2,3).Assumption 2.
For anyu,v∈R and k∈ℐ, the activation functions fku and gku are all continuous in R and satisfy(10)0<fku−fkvu−v≤pk,0<gku−gkvu−v≤qk,where pk>0 and qk>0 represent Lipschitz constants. Let P=diagp12,p22,…,pn2 and Q=diagq12,q22,…,qn2.Remark 2.
Assumption 1 guarantees the boundedness ofΔB, ΔC, and ΔD. Assumption 2 ensures the speciality of the activation functions fku and gku. The Lipschitz constants pk>0 and qk>0 are dependent on the activation functions fku and gku rather than fixed. Lipschitz continuity is a smoother condition than uniform continuity; intuitively, Lipschitz continuity restricts the speed of function change. Furthermore, the slope of the function satisfying Lipschitz condition must be less than a real number called Lipschitz constant. In differential equation theory, Lipschitz condition is a core condition in the existence and uniqueness theorem of solutions under initial conditions.
Next, two important lemmas are introduced as follows.Lemma 1 (see [19]).
LetF, G, and St be real matrices and η>0 is a constant; if STtSt≤I, then(11)FTSTtGT+GStF≤1ηFTF+ηGGT.Lemma 2 (see [18]).
LetF, G, P, and Qt be real matrices; if P>0 and QTtQt≤I, then(1)
For scalarξ>0 and vectors z1 and z2 of proper dimensionality(12)2z1TFTQtGz2≤1ξz1TFTFz1+ξz2TGTGz2.(2)
For matrixP>0 of suitable dimensions:(13)2z1Tz2≤z1TP−1z1+z2TPz2.


## 3. Main Results

Letεkmt=zkt−zmt be the synchronization error and μkt=zkt−zktks be the measurement error for t∈tks,tks+1. Then, by establishing event-triggered mechanism and using matrix inequality techniques, we obtain several novel conditions to achieve global robust exponential synchronization of system (6), and the lower bounds of sampling time interval are also found.Theorem 1.
Under ETS (5), for model (6), if Assumptions 1 and 2 hold, η>0 and ξi>0 (i=1,2,…,5) are constants, αk>0, and ℐk≠ϕ for k∈ℐ, then infs≥1tks+1−tks>0 for any s≥1.Proof.
Fromμkt=zkt−zktks, we have(14)dμktdt=−D+ΔDzkt+B+ΔBfzkt+C+ΔCgzkt−τ+ukt,t∈tks,tks+1,then(15)dμkt2dt=2μkTt−D+ΔDzkt+B+ΔBfzkt+C+ΔCgzkt−τ+ukt=2μkTtD+ΔDzktks−2μkTtD+ΔDμkt+2μkTtB+ΔBfzkt−fzktks+2μkTtC+ΔCgzkt−τ−gzktks¯+2μkTtukt+2μkTtB+ΔBfzktks+2μkTtC+ΔCgzktks¯.
For eacht∈tks,tks+1, there exists a s˜∈0,s satisfying t−τ∈tks˜,tks˜+1; according to the event-triggered function, for any t∈tks,tks+1, there exists a s¯∈0,s satisfying(16)μkt−τ≤αk∑m∈ℐkzmtms¯−zktks¯.
According to Lemmas 1 and 2, we get(17)2μkTtΔDzktks=2μkTtM1R1tN1zktks≤1ξ1μkTtM1M1Tμkt+ξ1zkTtksN1TN1zktks,(18)−2μkTtΔDμkt≤μkTtηM1M1T+1ηN1TN1μkt,(19)2μkTtΔBfzkt−fzktks=2μkTtM2R2tN2fzkt−fzktks≤1ξ2μkTtM2M2Tμkt+ξ2fzkt−fzktksTN2TN2fzkt−fzktks,(20)2μkTtΔCgzkt−τ−gzktks¯=2μkTtM3R3tN3gzkt−τ−gzktks¯≤1ξ3μkTtM3M3Tμkt+ξ3gzkt−τ−gzktks¯TN3TN3gzkt−τ−gzktks¯,(21)2μkTtΔBfzktks=2μkTtM2R2tN2fzktks≤1ξ4μkTtM2M2Tμkt+ξ4fTzktksN2TN2fzktks,(22)2μkTtΔCgzktks¯=2μkTtM3R3tN3gzktks¯≤1ξ5μkTtM3M3Tμkt+ξ5gTzktks¯N3TN3gzktks¯.
Combining (15)–(22), we obtain(23)dμkt2dt≤2μkTtDzktks+1ξ1μkTtM1M1Tμkt+ξ1zkTtksN1TN1zktks−2μkTtDμkt+μkTtηM1M1T+1ηN1TN1μkTt+2μkTtBfzkt−fzktks+1ξ2μkTtM2M2Tμkt+ξ2fzkt−fzktksTN2TN2fzkt−fzktks+2μkTtCgzkt−τ−gzktks¯+1ξ3μkTtM3M3Tμkt+ξ3gzkt−τ−gzktks¯TN3TN3gzkt−τ−gzktks¯+2μkTtukt+2μkTtBfzktks+2μkTtCgzktks¯+1ξ4μkTtM2M2Tμkt+ξ4fTzktksN2TN2fzktks+1ξ5μkTtM3⋅M3Tμkt+ξ5gTzktks¯N3TN3gzktks¯≤D+1ξ1M1M1T−2D+ηM1M1T+1ηN1TN1+B+1ξ2M2M2T+C+1ξ3M3M3T+B+1ξ4M2M2T+C+1ξ5M3M3T+1μkt2+D+ξ1N1TN1zktks2+B+ξ2N2TN2fzkt−fzktks2+C+ξ3N3TN3gzkt−τ−gzktks¯2+B+ξ4N2TN2fzktks2+C+ξ5N3TN3gzktks¯2+μkt2+C+ξ5N3TN3gzktks¯2+ukt2≤−D+2B+2C+1ξ1+ηM1M1T+1ηN1TN1+1ξ2+1ξ4M2M2T+1ξ3+1ξ5M3M3T+BP+ξ2N2TN2P+1μkt2+CQ+ξ3N3TN3Q⋅μkt−τ2+D+ξ1N1TN1zktks2+B+ξ4N2TN2fzktks2+C+ξ5N3TN3gzktks¯2+ukt2.
Thus,(24)dμkt2dt≤Λμkt2+Ωks,where(25)Ωks=CQ+ξ3N3TN3Qαk∑m∈ℐkzmtms¯−zktks¯2+D+ξ1N1TN1zktks2M2M2T+B+ξ4N2TN2⋅fzktks2+C+ξ5N3TN3gzktks¯2+λ∑m=1rykmzmtms2,and Λ>0 such that (26)Λ≥−D+2B+2C+1ξ1+ηM1M1T+1ηN1TN1+1ξ2+1ξ4M2M2T+1ξ3+1ξ5M3M3T+BP+ξ2N2TN2P+1.
Sinceμktks=0, according to (24), we have (27)μktks+12+ΩksΛ≤ΩksΛeΛtks+1−tks.From ETS (5), we get(28)eΛtks+1−tks≥1+Λαk∑m∈ℐkzmtms¯−zktks¯2Ωks,then(29)tks+1−tks≥1Λln1+Λαk∑m∈ℐkzmtms¯−zktks¯2Ωks;hence, we can know infs≥1tks+1−tks>0 when ℐk≠ϕ. The proof is finished.Remark 3.
By Theorem 1, we can knowαk>0 and ℐk≠ϕ implying that the Zeno behavior does not happen for sampling time sequence tks. In addition, if αk>0 and ℐk≠ϕ, there is no Zeno behavior for model (6) under ETS (5).Theorem 2.
Let condition (5), Assumption 1, and Assumption 2 hold; there is l∈ℐ satisfying αk>0 and ℐk≠ϕ k∈ℐ\l. Then, model (6) is globally robustly exponentially synchronized under ETS (5) if there exist r−1 constants ζk>0 k∈ℐ\l and a constant ν>0 satisfying(30)ΨKl+KlTΨ2+νI≤0,and there is 0<αmax<1/r+rmax−1 and three constants η>0, ξ6>0, and ξ7>0 satisfying(31)−2dmin−2λνζmax−αmaxr−1r+rmax−1ζmaxylmax1−αmaxr+rmax−1ζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+CQ+ξ7N3TN3Q<0,where dmin=min1≤k≤ndk, Kl=ykm−ylmk,m∈ℐ\l, Ψ=diagζl1,ζl2,…,ζlr−1, l1,l2,…,lr−1 denotes for rearrangement of ℐ\l with l1<l2<⋯<lr−1, ylmax=maxk,m∈ℐ\lykm−ylm, rk stands for the number of elements in ℐk, and rmax=max1≤k≤rrk.Proof.
By Theorem 1, for anyk∈ℐ\l, tks⟶∞ as s⟶∞, then, there exists a s≥1 satisfying t∈tks,tks+1:(32)dεkltdt=dzktdt−dzltdt=−D+ΔDzkt−zlt+B+ΔBfzkt−fzlt+C+ΔCgzkt−τ−gzlt−τ+ukt−ult,where(33)ukt−ult=λ∑m=1,m≠lrykm−ylmzmtms−zlt,since(34)ukt=λϱzlt+λ∑m=1,m≠lrykmzmtms−zlt,ult=λϱzlt+λ∑m=1,m≠lrylmzmtms−zlt.
LetWkt=εklTtεkltandWt=∑k∈ℐ\lζkWkt, then (35)dWktdt=2εklTtdεkltdt≤−2εklTtD+ΔDεklTt+2εklTtB+ΔB⋅fzkt−fzlt+2εklTtC+ΔC⋅gzkt−τ−gzlt−τ+2εklTtukt−ult.
By Lemma 2, we have(36)2εklTtΔBfzkt−fzlt≤1ξ6εklTtM2M2Tεklt+ξ6fzkt−fzltTN2TN2⋅fzkt−fzlt,(37)2εklTtΔCgzkt−τ−gzlt−τ≤1ξ7εklTtM3M3Tεklt+ξ7gzkt−τ−gzlt−τT⋅N3TN3gzkt−τ−gzlt−τ.
Substituting (36) and (37) into (35), we get(38)dWtdt≤−2dmin+ηM1M1T+1ηN1TN1+B+BP+1ξ6M2M2T+ξ6N2TN2P+C+1ξ7M3M3TWt+CQ+ξ7N3TN3QWt−τ+2ζkεklTtukt−ult.
According to (2) and (5), we obtain(39)εklTtukt−ult=εklTtλ∑m=1,m≠lrykm−ylmzmtms−zlt=εklTtλ∑m=1,m≠lrykm−ylmεmlt−μmt=λ∑m=1,m≠lrykm−ylmεklTtεmlt−λεklTt∑m=1,m≠lrykm−ylmμmt.
From ETS (5), one has(40)μkt≤αk∑m∈ℐkzmtms−zktks≤αk∑m∈ℐkμkt−μmt+εmlt−εklt≤αmaxrkμkt+∑m∈ℐkμmt+∑m∈ℐkεmlt+rkεklt≤αmaxrmaxμkt+r−1μmt+r−1εmlt+rmaxεklt,then(41)∑m∈ℐkrμmt≤Gl∑m∈ℐkrεmlt,k∈ℐ,where(42)Gl=αmaxr+rmax−11−αmaxr+rmax−1.
By (41), we have(43)∑k,m∈ℐ\lζkykm−ylmεkltμmt≤ζmaxylmax∑k,m∈ℐ\lεkltμmt≤Glζmaxylmax∑k,m∈ℐ\lεkltεmlt≤Glζmaxylmaxr−1∑k,m∈ℐ\lεklt2≤Glr−1ζmaxylmaxζminWt.
According to (18), one gets(44)∑k,m∈ℐ\lζkykm−ylmεklTtεmlt=∑k,m∈ℐ\l12ykmζk+ymkζm−ylmζk−ylkζmεklTtεmlt≤−ν∑k∈ℐ\lεklTtεklt≤−νζmaxWt.
Combining (39), (43), and (44), we obtain(45)∑k,m∈ℐ\lζkεklTtukt−ult≤λ−νζmax+Glr−1ζmaxylmaxζminWt.
From the definition ofWt, one has (46)dWtdt≤−2dmin−2λνζmax−Glr−1ζmaxylmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3TWt+CQ+ξ7N3TN3QWt−τ.
By (31), there is a constant h>0 such that(47)2h−2dmin−2λνζmax−Glr−1ζmaxylmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+1+e2hτCQ+ξ7N3TN3Q≤0.
LetVt=e2ht−t0Wt, then(48)dVtdt≤2h−2dmin−2λνζmax−Glr−1ζmaxylmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3TVt+e2hτCQ+ξ7N3TN3QVt−τ≤2h−2dmin−2λνζmax−Glr−1ζmaxylmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+1+e2hτCQ+ξ7N3TN3QVt≤0,when Vt+ϑ≤Vt for any ϑ∈−τ,0.
Let(49)V˜t=supϑ∈−τ,0Vt+ϑ,W˜t=supϑ∈−τ,0Wt+ϑ,then, we have (50)dV˜tdt≤0,∀t≥t0;hence,(51)Wt≤∑m∈ℐ\lζmε˜lmt02e−2ht−t0,which implies(52)εklt≤ζmaxζminε˜lt0e−ht−t0,for all k∈ℐ\l, where εlt=εl1lTt,εl2lTt,…,εlr−1lTt and(53)ε˜klt=supϑ∈−τ,0εklt+ϑ,ε˜lt=supϑ∈−τ,0εlt+ϑ.
Then, for anyk,m∈ℐ and t≥t0, we obtain (54)εkmt≤εklt+εmlt≤2ζmaxζminε˜lt0e−ht−t0.
Thus, model (6) is globally robustly exponentially synchronized with convergence rate h. The proof is finished.Theorem 3.
Let condition (5), Assumption 1, and Assumption 2 hold; there is l∈ℐ satisfying αk>0 and ℐk≠ϕ k∈ℐ\l. Moreover, if there exist r−1 constants ζk>0 satisfying(55)βk=−ykk−ylk−1ζk∑m=1,m≠k,lrζmykm−ylm>0,and there is 0<αmax<1/2rmax and three constants η>0, ξ6>0, and ξ7>0 satisfying(56)−2dmin−2λβmin−2αmaxrmaxζmaxKlinf1−2αmaxrmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+CQ+ξ7N3TN3Q<0,then model (6) is globally robustly exponentially synchronized under ETS (5).Proof.
LetW¯t=maxk∈ℐ\l1/ζkεklt2, and fixing k∈ℐ\l satisfying(57)W¯t=1ζkεklt2,then(58)dW¯tdt=2ζkεkltdεkltdt=2ζkεkltsignεkltTdεkltdt≤−2dmin+ηM1M1T+1ηN1TN1+B+BP+1ξ6M2M2T+ξ6N2TN2P+C+1ξ7M3M3TW¯t+CQ+ξ7N3TN3QW¯t−τ+2ζkεkltsignεkltTukt−ult,where(59)ukt−ult=λ∑m∈ℐ\lykm−ylmzmtms−zlt.
Then, one gets(60)signεkltTukt−ult=λsignεkltT∑m∈ℐ\lykm−ylmεmlt−μmt≤λykk−ylkεklt+λ∑m∈ℐ\l,kykm−ylmεmlt+λ∑m∈ℐ\lykm−ylmμmt.
By Theorem 1, we obtain(61)μtmax≤Gl˜εltmax,where Gl˜=2αmaxrmax/1−2αmaxrmax, μtmax=maxk∈ℐ\lμkt, and εltmax=maxm∈ℐ\lεklt.
By (60) and (61), we have(62)1ζkεkltsignεkltTukt−ult≤λykk−ylk+1ζk∑m∈ℐ\l,krζmykm−ylmW¯t+1ζkεkltλ∑m∈ℐ\lrykm−ylmμtmax≤−λβkW¯t+λGl˜ζmaxKlinfζminW¯t;thus, one has (63)dW¯tdt≤−2dmin−2λβmin−Gl˜ζmaxKlinfζmin+ηM1M1T+1ηN1TN1+B+BP+1ξ6M2M1T+ξ6N2TN2P+C+1ξ7M3M3TW¯t+CQ+ξ7N3TN3QW¯t−τ.
From (56), we can know that there is a h>0 such that(64)2h−2dmin−2λβmin−2αmaxrmaxζmaxKinf1−2αmaxrmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+1+e2hτCQ+ξ7N3TN3Q≤0.
Similar to Theorem 2, by (63), we get(65)W¯t≤1ζminε˜lt02e−2ht−t0,then(66)εklt≤ζmaxζminε˜lt0e−ht−t0,k∈ℐ\l.
Thus, model (6) is globally robustly exponentially synchronized with convergence rate h under ETS (5). The proof is finished.Remark 4.
Comparing Theorem 2 to Theorem 3, it is not hard to see that whenr is large enough and rmax is relatively small, condition (30) is better than condition (55), but condition (31) is not as good as condition (56).Theorem 4.
Let condition (4), Assumption 1, and Assumption 2 hold; αk>0 and ℐk≠ϕ. Then model (6) is globally robustly exponentially synchronized under ETS (5) if there exist r constants ζk>0 k∈ℐ and a constant ν>0 satisfying(67)Ψ¯K+KTΨ¯2+νI≤0,and there is 0<αmax<1/r+rmax and three constants η>0, ξ8>0, and ξ9>0 satisfying(68)−2dmin−2λνζmax−αmaxr+rmaxrζmaxymax1−αmaxr+rmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ8M2M2T+ξ8N2TN2P+1ξ9M3M3T+CQ+ξ9N3TN3Q<0,where dmin=min1≤k≤ndk, K=ykmk,m∈ℐ, Ψ¯=diagζ1,ζ2,…,ζr, and ymax=maxk,m∈ℐykm.Proof.
A reference system is given by(69)dz0tdt=−D+ΔDz0t+B+ΔBfz0t+C+ΔC⋅gz0t−τ+Jt+γρz0t,t≥t0,where u0=λϱz0t, and there exists a s≥1 satisfying t∈tks,tks+1.
According to (6) and (69), we have(70)dεk0tdt=dzktdt−dz0tdt=−D+ΔDzkt−z0t+B+ΔBfzkt−fz0t+C+ΔCgzkt−τ−gz0t−τ+ukt−u0t,where(71)ukt−u0t=λ∑m=1,m≠lrykmzmtms−z0t.
LetW^kt=εk0Ttεk0tandW^t=∑k∈ℐζkW^kt, then(72)dW^ktdt=2εk0Ttdεk0tdt≤−2εk0TtD+ΔDεk0Tt+2εk0TtB+ΔB⋅fzkt−fz0t+2εk0TtC+ΔC⋅gzkt−τ−gz0t−τ+2εk0Ttukt−u0t.
By Lemma 2, we have(73)2εk0TtΔBfzkt−fz0t≤1ξ8εk0TtM2M2Tεk0t+ξ8fzkt−fz0tTN2TN2⋅fzkt−fz0t,(74)2εk0TtΔCgzkt−τ−gz0t−τ≤1ξ9εk0TtM3M3Tεk0t+ξ9gzkt−τ−gz0t−τT⋅N3TN3gzkt−τ−gz0t−τ.
Substituting (73) and (74) into (72), we get(75)dW^tdt≤−2dmin+ηM1M1T+1ηN1TN1+B+BP+1ξ8M2M2T+ξ8N2TN2P+C+1ξ9M3M3TW^t+CQ+1ξ9N3TN3Q⋅W^t−τ+2ζkεk0Ttukt−u0t.
Similar to Theorem 2, we obtain(76)εk0t≤ζmaxζminε˜0t0e−ht−t0,for all k∈ℐ, where ε0t=ε10T,…,εr0TT, ε˜0t0=supϑ∈−τ,0ε0t+ϑ, and h is a position constant such that(77)2h−2dmin−2λνζmax−αmaxr+rmaxrζmaxymax1−αmaxr+rmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ8M2M2T+ξ8N2TN2P+1ξ9M3M3T+1+e2hτCQ+ξ9N3TN3Q≤0.
Then, for anyk,m∈ℐ and t≥t0, we get (78)εkmt≤εk0t+εm0t≤2ζmaxζminε˜0t0e−ht−t0.
Thus, the proof is finished.Theorem 5.
Let condition (4), Assumption 1, and Assumption 2 hold; αk>0 and ℐk≠ϕ k∈ℐ, and if there exist r constants ζk>0 k∈ℐ satisfying(79)βk=−ykk−1ζk∑m=1rζmykm>0,and there is 0<αmax<1/2rmax and three constants η>0, ξ8>0, and ξ9>0 satisfying(80)−2dmin−2λβmin−2αmaxrmaxζmaxKinf1−2αmaxrmaxζmin+B+C+BP+ηM1M1T+1ηN1TN1+1ξ8M2M2T+ξ8N2TN2P+1ξ9M3M3T+CQ+ξ9N3TN3Q<0,then model (6) is globally robustly exponentially synchronized under ETS (5).Remark 5.
From Theorems 2–5, it is easy to find that they are closely related. In Theorems 2 and 3, the coupling matrix can be unstable. However, in Theorems 4 and 5, the coupling matrix must be stable. Theorems 4 and 5 can be derived from Theorems 2 and 3, respectively, when the coupling matrix is stable, but Theorems 3 and 4 are more simpler and convenient than Theorems 2 and 3.Remark 6.
Most existing works on multisystem networks (see [2–4]) require that the coupling matrix is always the Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative. In this paper, the coupling matrix need not be the Laplace matrix; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary. Compared with the previous works, the advantage of this paper is that the coupling matrix need not be the Laplace matrix, and the criteria obtained are more simple and convenient.Remark 7.
Actually, if we consider MNNs with non-Laplace coupling matrix and norm-unbounded uncertainty terms in this paper, we can get Theorem 1 by changing (17)–(22) and get Theorems 2 and 3 by changing (36) and (37). Similarly, we also get Theorems 4 and 5 by changing (72) and (74). In addition, if we consider time-varying delay instead of constant delay into system (6), when the time-varying delay τkst is bounded, namely, there exists a constant σ satisfying 0<τkst<σ, and several similar conclusions can be obtained.


## 4. Three Numerical Examples

We consider model (1) with n=2, where activation function fz=gz and fz=f1z1,f2z2T such that(81)fmzm=zm+1−zm−12,m=1,2,the weight matrices(82)D=4004,B=0.20.10.10.2,C=−0.20.10.1−0.2,ΔD=sin1001,ΔB=cos1111,ΔC=sin1111.Example 1.
Consider system (6) with r = 4. Let τ=0.001, η=0.5, ξ6=0.5, ξ7=0.5, and coupling matrix(83)K=−1.1900.40.80−1.190.2100.2−1.19110.20−1.19,we know that ϱ=0.01 and K has a positive eigenvalue 0.01. Choosing l=1, then(84)K1=−1.19−0.20.20.2−1.590.20.2−0.4−1.99.
Letν=1.0546, then(85)K1+K1T2+νI≤0,which implies that condition (30) holds, where ζi=12≤i≤4; let λ=10 and αk=0.012, then(86)−2dmin−2λν−αmaxr−1r+rmax−1glmax1−αmaxr+rmax−1+B+C+BP+ηM1M1T+1ηN1TN1+1ξ6M2M2T+ξ6N2TN2P+1ξ7M3M3T+CQ+ξ7N3TN3Q=−0.8420<0;hence, condition (31) holds; furthermore, we choose the initial states z1ϑ=0.6,0.5T, z2ϑ=−1,0.8T, z3ϑ=0.7,−1T, and z4ϑ=−1.2,0.2T for ϑ∈−0.001,0. Figure 1 describes the evolutions of the event-triggered controller and synchronization errors. Figure 2 depicts the threshold and the evolutions of measurement errors. By Theorem 2, we can know that system (6) is globally robustly exponentially synchronized under ETS (5).Figure 1
Time evolutions of the event-triggered controller and the synchronization errors in Example 1.
(a)
(b)Figure 2
Time evolutions of measurement errorsμkt and the thresholds Φks (k=1,2,3,4) in Example 1.
(a)
(b)
(c)
(d)Example 2.
Letτ=0.001, η=0.5, ξ8=0.5, ξ9=0.5, and coupling matrix(87)K=−1.0600.020.040−1.03−0.020.06−0.020−1.080.010.0200.04−1.06,we know that ϱ=−1 and K is a stable matrix. Let ν=1.25, then(88)K+KT2+νI≤0;hence, condition (67) holds, where ζi=11≤i≤4; choosing λ=10 and αk=0.021, then(89)−2dmin−2λν−αmaxr+rmaxrymax1−αmaxr+rmax+B+C+BP+ηM1M1T+1ηN1TN1+1ξ8M2M2T+ξ8N2TN2P+1ξ9M3M3T+CQ+ξ9N3TN3Q=−0.64<0,thus, condition (68) holds; furthermore, we choose the initial states z1ϑ=0.6,0.5T, z2ϑ=−1,0.8T, z3ϑ=0.7,−1T, and z4ϑ=−1.2,0.2T for ϑ∈−0.001,0. Figure 3 describes the evolutions of the event-triggered controller and synchronization errors. Figure 4 depicts the threshold and the evolutions of measurement errors. By Theorem 4, we can know that system (6) is globally robustly exponentially synchronized under ETS (5).Figure 3
Time evolutions of the event-triggered controller and the synchronization errors in Example 2.
(a)
(b)Figure 4
Time evolutions of measurement errorsμkt and the thresholds Φks (k=1,2,3,4) in Example 2.
(a)
(b)
(c)
(d)Example 3.
Letτ=0.001, η=0.5, ξ8=0.5, ξ9=0.5, and coupling matrix(90)K=−2.600.2000.40−2.3−0.3000.60−0.2−2.3000.50−0.30−2.701000.10−2.60.50.40000.2−2.6,we know that ϱ=−2 and K is a stable matrix. Let ν=0.99, then(91)βk=−ykk−1ζk∑m=16ζmykm≥0.4;hence, condition (79) holds, where ζi=11≤i≤6, choosing λ=18 and αk=0.015, then(92)−2dmin−2λβmin−2αmaxrmaxKinf1−2αmaxrmax+B+C+BP+ηM1M1T+1ηN1TN1+1ξ8M2M2T+ξ8N2TN2P+1ξ9M3M3T+CQ+ξ9N3TN3Q=−0.2636<0,thus, condition (80) holds; in addition, we choose the initial states z1ϑ=0.6,0.5T, z2ϑ=−1,0.8T, z3ϑ=0.7,−1T, z4ϑ=−1.2,0.2T, z5ϑ=−1.5,−0.8T, and z6θ=−0.1,0.6T for ϑ∈−0.01,0. Figure 5 describes the evolutions of the event-triggered controller and synchronization errors. Figure 6 depicts the threshold and the evolutions of measurement errors. By Theorem 5, we can know that system (6) is globally robustly exponentially synchronized under ETS (5).Figure 5
Time evolutions of the event-triggered controller and the synchronization errors in Example 3.
(a)
(b)Figure 6
Time evolutions of measurement errorsμkt and the thresholds Φks (k=1,2,3,4,5,6) in Example 3.
(a)
(b)
(c)
(d)
(e)
(f)Remark 8.
It can be seen that the row sums of coupling matrices in Examples 1, 2, and 3 all are 0.01,−1, and −2, respectively, and there are nondiagonal elements in Examples 2 and 3. Thus, our coupling matrices all are non-Laplacian.


## 5. Concluding Remarks

This paper has introduced a novel ETS for MNNs with parameter uncertainty and time delay. By using matrix inequality techniques and establishing the event-triggered mechanism, some sufficient criteria are derived for judging global robust exponential synchronization of the system; in addition, the lower bounds of the sampling time intervals are also obtained. In particular, the most existing papers on multisystem networks require that the coupling matrix is Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative. However, the coupling matrix need not be the Laplace matrix in this paper; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary. In the future, we can do the following works: (1) MNNs will be extended to fractional-order MNNs. (2) It is meaningful to study MNNs with parameter uncertainty and time delay via centralized and decentralized data-sampling approaches. (3) MNNs with parameter uncertainty and time-varying delay may be valuable research topic. (4) It is a challenge to study MNNs with non-Laplace coupling matrix and norm-unbounded uncertainty terms.



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