# Passivity and Synchronization of Coupled Different Dimensional Delayed Reaction-Diffusion Neural Networks with Dirichlet Boundary Conditions

Authors: Shanrong Lin; Yanli Huang; Erfu Yang
Journal: Complexity
 (2020)

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## Abstract

Two types of coupled different dimensional delayed reaction-diffusion neural network (CDDDRDNN) models without and with parametric uncertainties are analyzed in this paper. On the one hand, passivity and synchronization of the raised network model with certain parameters are studied through exploiting some inequality techniques and Lyapunov stability theory, and some adequate conditions are established. On the other hand, the problems of robust passivity and robust synchronization of CDDDRDNNs with parameter uncertainties are solved. Finally, two numerical examples are given to testify the effectiveness of the derived passivity and synchronization conditions.


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## Body

## 1. Introduction

In recent years, complex networks (CNs) have attracted much attention since they are ubiquitous under the circumstance of our daily life, for instance, communication networks, food webs, and social networks. Coupled neural networks (CNNs), as a particular type of CNs, have been put into use successfully in various fields, e.g., pattern recognition, chaos generators design, and brain science [1–3]. Strictly speaking, these applications in a large extent are depending on some properties of dynamics in CNNs (e.g., synchronization). Thus, the problem of synchronization for CNNs has attracted comprehensive attention and been developed into a hot research topic. So far, many important and interesting results have been derived on this topic recently [4–10]. Through exploiting the properties of random variables, Yang et al. [4] acquired some synchronization conditions for randomly delayed CNNs. In [7], several synchronization criteria were gained for delayed CNNs in accordance with the Lyapunov functional strategy.Nevertheless, most of the synchronization results in aforesaid literatures [4–10] neglected reaction-diffusion phenomena. In fact, the reaction-diffusion phenomena are unavoidable for CNs such as neural networks and cellular networks when they are implemented by means of electric circuits in practical situations [11]. Hence, it is necessary to research coupled reaction-diffusion neural networks (CRDNNs). Recently, some significant synchronization results for CRDNNs have been acquired [12–17]. In [12], the authors researched synchronization of CRDNNs and presented some synchronization criteria. By designing appropriate pinning controllers, several synchronization criteria were established for CRDNNs in [17]. Furthermore, owing to the existence of external interferences, the noises of environment, and equipment restrictions, it is very difficult to ensure network models containing the certain parameter values in some practice situations. Consequently, some authors studied parametric uncertainties of neural networks [18–22], and a few interesting results have been derived regarding robust synchronization of CNNs with uncertain parameters [23–25]. In [24], the scholars presented the memristive CNNs model with parametric uncertainties and obtained some adequate conditions for guaranteeing the robust synchronization for the considered network. By utilizing impulsive functional strategy combined with the stability theory, Li et al. [25] presented some robust synchronization criteria for the CNNs with uncertainties.Actually, passivity is also one of the most important behaviors of dynamics for CNs which can guarantee a system’s internal stabilization in system theory. Due to the potential applications in plenty of fields, e.g., fuzzy control and sliding mode control, the passivity problem of CNNs has been investigated extensively in the past years [26–33]. In [27], the authors proposed several passivity criteria for delayed CNNs. Ren et al. [28] analyzed the model of CRDNNs and established some passivity and pinning passivity conditions for the considered network. Unfortunately, the passivity results in aforementioned studies [26–33] are based on the case that the dimension of input is identical with output. As far as we know, only few scholars have addressed the problem of passivity for the network with nonidentical dimensional output and input [34–36]. The authors in [34] established some conditions for ensuring that the CRDNNs with the input and output in different dimensions achieve passivity. Ren et al. [36] discussed the (pinning) passivity problems of CNNs with nonidentical dimensional output and input and acquired some corresponding passivity criteria.Note that the networks are composed of identical nodes in the aforementioned works [4–36]. Unfortunately, this case is very rare in the real-world networks. Consequently, the CNs consisting of nonidentical nodes in the same dimension [37–41] have been discussed firstly by researchers. Zhao et al. [39] dealt with the synchronization problem of CNs with nonidentical nodes. As a matter of fact, the networks constructed by nonidentical nodes of different dimensions can reflect more real networks in many circumstances. Note that the networks constructed by nonidentical nodes in different dimensions can exhibit different and even more sophisticated dynamical behaviors, which makes the passivity and synchronization methods for the networks with the same dimensional nonidentical nodes or identical nodes in the abovementioned works [4–41] invalid. Hence, it is necessary and meaningful to develop some new stabilization and synchronization strategies for the network with nonidentical nodes of different dimensions [42–45]. In [43], by designing appropriate decentralized controllers, the authors devoted to establishing stabilization and synchronization criteria for CNs consisting of nonidentical nodes. Up to now, only a few researchers considered CNNs constructed by the nonidentical nodes of different dimensions [46–48]. In [47], the authors investigated generalized synchronization of delayed CNNs with different dimensional nodes by making use of the Lyapunov functional method. To the best of knowledge, the problems of synchronization and passivity for CDDDRDNNs have not yet been investigated. Consequently, it is essential to put forth some efforts to study passivity and synchronization of CDDDRDNNs.In terms of the above introduction, the main aim in this paper is to address passivity and synchronization of CDDDRDNNs without and with parametric uncertainties. On the one hand, we discuss the CDDDRDNNs without parametric uncertainties, and several conditions are derived to guarantee the considered network to achieve passivity and synchronization. On the other hand, the problems of robust synchronization and robust passivity for CDDDRDNNs with parametric uncertainties are also studied.


## 2. Preliminaries

Let the matrixG∈ℝn×n, the notation G<0G>0,G≤0,G≥0 signifies G is symmetric and negative (positive, seminegative, and semipositive) definite. λm⋅λM⋅ represents the minimum (maximum) eigenvalue of the corresponding matrix. An open-bounded domain in ℝϱ with smooth boundary ∂Ω is defined by Ω=m=m1,m2,⋯,mϱTmσ<ζσ,σ=1,2,⋯,ϱ. For any zm,t=z1m,t,z2m,t,⋯,znm,tT∈ℝn, we have(1)z⋅,t2=∫Ω∑i=1nzi2m,tdm1/2.Definition 1 (see [49]).
Letum,t∈ℝp and ym,t∈ℝq denote the input and output of a system. Assume that there exists a storage function S:0,+∞⟶0,+∞ which satisfies(2)∫t0tlΠu,ydt≥Stl−St0,for any t0,tl∈0,+∞ and t0≤tl, then the system with supply rate Πu,y is dissipative. Moreover, a system is passive if the system is dissipative with(3)Πu,y=∫ΩyTm,tQum,tdm,where the matrix Q∈ℝq×p. Furthermore, assume that a system is dissipative with(4)Πu,y=∫ΩyTm,tQum,tdm−∫ΩyTm,tM1ym,tdm−∫ΩuTm,tM2um,tdm,in which M1∈ℝq×q⩾0,M2∈ℝp×p⩾0, λmM1+λmM2>0, and Q∈ℝq×p, then the system is strictly passive. Especially, if M1>0, then the system is called to be output-strictly passive; if M2>0, then the system is called to be input-strictly passive.Lemma 2.1 (see [50]).
LetΩ be a cube mσ<ζσσ=1,2,⋯,ϱ and real-valued function wm∈C1Ω satisfy wm∂Ω=0. Then,(5)∫Ωw2mdm⩽ζσ2∫Ω∂wm∂mσ2dm,where m=m1,m2,⋯,mϱT.


## 3. Passivity and Synchronization of CDDDRDNNs

### 3.1. Network Model

The CDDDRDNNs considered in this section is stated as follows:(6)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=1NcijHijwjm,t,in which i=1,2,⋯,N, wim,t=w1im,t,w2im,t,⋯,wξiim,tT∈ℝξi denotes the state vector of ith neuron; Bi∈ℝξi×ξi=diagb1i,b2i,⋯,bξii>0;Ai=aghiξi×ξi,Zi=zghiξi×ξi; fiwim,t=f1iw1im,t,f2iw2im,t,⋯,fξiiwξiim,tT; Di=diagd1i,d2i,⋯,dξii>0; φiwim,t−τit=φ1iw1im,t−τit,φ2iw2im,t−τit,⋯,φξiiwξiim,t−τitT, fli⋅, and φli⋅l=1,2,⋯,ξi denote the activation functions for the lth neuron in neural network i; Ji=J1i,J2i,⋯,JξiiT; uim,t∈ℝpi denotes the control input; the inner coupling matrix is defined by Hij∈ℝξi×ξj; Gi∈ℝξi×pi is a known matrix; C=cijN×N is the coupling configuration matrix denoting coupling weight, which satisfies cij≠0i≠j if node i and node j are connected, or else cij=0. In addition, the time-varying delay τit satisfies 0⩽τit⩽τi and τ˙it⩽δi<1.Remark 1.
As a special type of CNs, CNNs have attracted much attention due to their extensive applications on plenty of fields. So far, a large number of scholars have acquired some interesting research results about synchronization and passivity of CNNs [4–36]. Nevertheless, the considered CNs in the aforementioned literature are made up of identical nodes. As a matter of fact, it is utterly impractical that the networks have totally identical nodes in many practical situations. For instance, due to the differences of the parameters, it is impossible that the neurons in the nervous system of neural networks are entirely the same as each other. Consequently, it is significant to study CNNs composed of nonidentical nodes. To our knowledge, a few investigators have discussed CNs consisting of the same dimensional nonidentical nodes in recent years [37–41]. However, the networks with nonidentical nodes of different dimensions can describe more practical networks. In addition, the considered networks in these existing works did not take the reaction-diffusion terms into account. Therefore, we pay our attention on the CRDNNs with different dimensional nodes. As far as we know, this is our first step toward addressing the passivity and synchronization problems of CDDDRDNNs.
For network (6), the initial value condition and boundary value condition are described by(7)wim,t=ϑim,t,m,t∈Ω×−ρ,0,(8)wim,t=0,m,t∈∂Ω×−ρ,+∞,where i=1,2,⋯,N,ϑim,t is bounded and continuous on Ω×−ρ,0 and ρ=maxi=1,2,⋯,Nτi.
Suppose that the functionsfli⋅ and φli⋅ satisfy the following global Lipschitz condition; there exist positive constants ψil and ψ^il such that(9)fliυ1−fliυ2⩽ψilυ1−υ2,φliυ1−φliυ2⩽ψ^ilυ1−υ2,hold for all υ1,υ2∈ℝ,l=1,2,⋯,ξi,i=1,2,⋯,N.
Let the constant vectorO^=o^1T,o^2T,⋯,o^NTT be an equilibrium point of an isolated node of network (6). Then,(10)Di∑σ=1ϱ∂2o^i∂mσ2−Bio^i+Aifio^i+Ji+Ziφio^i+∑j=1NcijHijo^j=0,where o^i=o^1i,o^2i,⋯,o^ξiiT∈ℝξi,i=1,2,⋯,N.
For the error vectorzim,t=z1im,t,z2im,t,⋯,zξiim,tT=wim,t−o^i, we have(11)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NcijHijzjm,t,where i=1,2,⋯,N,f^izim,t=fiwim,t−fio^i=f1iw1im,t−f1io^1i,f2iw2im,t−f2io^2i,⋯,fξiiwξiim,t−fξiio^ξiiT and φ^izim,t−τit=φiwim,t−τit−φio^i=φ1iw1im,t−φ1io^1i,φ2iw2im,t−φ2io^2i,⋯,φξiiwξiim,t−φξiio^ξiiT.
The output vectoryim,t∈ℝqi of the network (11) is given as follows:(12)yim,t=K1izim,t+K2iuim,t,where K1i∈ℝqi×ξi and K2i∈ℝqi×pi are the known matrices.
In the whole paper, we denote(13)zm,t=z1Tm,t,z2Tm,t,⋯,zNTm,tT,um,t=u1Tm,t,u2Tm,t,⋯,uNTm,tT,ym,t=y1Tm,t,y2Tm,t,⋯,yNTm,tT,L=11−δ1Iξ1,11−δ2Iξ2,⋯,11−δNIξN,p=p1+p2+⋯+pN,q=q1+q2+⋯+qN,ξ=ξ1+ξ2+⋯+ξN,D=diagD1,D2,⋯,DN,Z=diagZ1,Z2,⋯,ZN,G=diagG1,G2,⋯,GN,B=diagB1,B2,⋯,BN,A=diagA1,A2,⋯,AN,Ψi=diagψi12,ψi22,⋯,ψiξi2,Ψ=diagΨ1,Ψ2,⋯,ΨN,Ψ^i=diagψ^i12,ψ^i22,⋯,ψ^iξi2,Ψ^=diagΨ^1,Ψ^2,⋯,Ψ^N,f^zm,t=f^1z1m,t,f^2z2m,t,⋯,f^NzNm,tT,φ^zm,t−τt=φ^1z1m,t−τ1t,φ^2z2m,t−τ2t,⋯,φ^NzNm,t−τNtT,H^=H^ijN×N,whereH^ij=cijHij,K^1=diagK11,K12,⋯,K1N,K^2=diagK21,K22,⋯,K2N.


### 3.2. Passivity Analysis

It follows from (11) and (12) that the following system with output yim,t∈ℝqi and input uim,t∈ℝpi can be obtained as follows:(14)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t,yim,t=K1izim,t+K2iuim,t,where H^ij=cijHij and i=1,2,⋯,N.Theorem 1.
System (14) reaches output-strict passivity if there are matrices Q∈ℝq×p,P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi, and 0<M1∈ℝq×q satisfying(15)PD+DP⩾0,(16)Φ1Λ1Λ1TΛ2⩽0,where Φ1=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1,Λ1=PG+K^1TM1K^2−K^1TQ,Λ2=K^2TM1K^2−K^2TQ−QTK^2.Proof.
Construct the Lyapunov functional for system (14) as follows:(17)Vt=∑i=1N∫ΩziTm,tPizim,tdm+∑i=1N∑j=1ξiψ^ij21−δi∫t−τitt∫Ωzjim,h2dmdh.
Then,(18)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,tdm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t+Giuim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm=2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2−Bzm,t+Af^zm,t+Gum,t+Zφ^zm,t−τt+H^zm,tdm+∫ΩzTm,tΨ^Lzm,tdm−∫ΩzTm,t−τtΨ^zm,t−τtdm.
By means of the boundary condition and Green’s formula, one can obtain the following:(19)∫Ωzgim,t∑σ=1ϱ∂2zhim,t∂mσ2dm=−∑σ=1ϱ∫Ω∂zgim,t∂mσ∂zhim,t∂mσdm,where h=1,2,⋯,ξi,i=1,2,⋯,N,andg=1,2,⋯,ξi. Let Pi=pghiξi×ξi, one has(20)∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm=∑i=1N∑g=1ξi∑h=1ξipghidhi∫Ωzgim,t∑σ=1ϱ∂2zhim,t∂mσ2dm=−∑i=1N∑σ=1ϱ∑g=1ξi∑h=1ξipghidhi∫Ω∂zgim,t∂mσ∂zhim,t∂mσdm=−∑σ=1ϱ∫Ω∂zm,t∂mσTPD∂zm,t∂mσdm.
Then, we obtain(21)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm=−∑σ=1ϱ∫Ω∂zm,t∂mσTPD+DP∂zm,t∂mσdm.
Obviously, there is a real matrixΓ∈ℝξ×ξ satisfying(22)PD+DP=ΓTΓ.
Thus,(23)∂zm,t∂mσTPD+DP∂zm,t∂mσ=∂zm,t∂mσTΓTΓ∂zm,t∂mσ=∂Γzm,t∂mσT∂Γzm,t∂mσ.
Letϕm,t=Γzm,t, for m,t∈∂Ω×−ρ,+∞, it follows from boundary condition (8) that ϕm,t=Γzm,t=0. In terms of Lemma 2.1, we can get(24)∑σ=1ϱ∫Ω∂ϕm,t∂mσT∂ϕm,t∂mσdm≥∑σ=1ϱ1ζσ2∫ΩϕTm,tϕm,tdm=∑σ=1ϱ1ζσ2∫ΩzTm,tPD+DPzm,tdm.
Therefore,(25)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm≤−∑σ=1ϱ1ζσ2∫ΩzTm,tPD+DPzm,tdm.
In addition, we can derive that(26)2∫ΩzTm,tPAf^zm,tdm=2∑i=1N∫ΩziTm,tPiAif^izim,tdm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫Ωf^iTwim,tf^iwim,tdm=∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∑l=1ξi∫Ωfliwlim,t−flio^li2dm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∑l=1ξi∫Ωψil2zlim,t2dm=∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm=∫ΩzTm,tPAATP+Ψzm,tdm.
Similarly, we have(27)2∫ΩzTm,t−τtPZφ^zm,t−τtdm=2∑i=1N∫ΩziTm,t−τitPiZiφ^izim,t−τitdm≤∑i=1N∫ΩziTm,tPiZiZiTPizim,tdm+∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm=∫ΩzTm,tPZZTPzm,tdm+∫ΩzTm,t−τtΨ^zm,t−τtdm.
By (25)–(27), one has(28)V˙t≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm+2∫ΩzTm,tPGum,tdm.
Thus,(29)V˙t−2∫ΩyTm,tQum,tdm+∫ΩyTm,tM1ym,tdm≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm+∫ΩzTm,tK^1TM1K^1zm,tdm+∫ΩzTm,tK^1TM1K^2um,tdm+∫ΩuTm,tK^2TM1K^1zm,tdm+∫ΩuTm,tK^2TM1K^2um,tdm−2∫ΩzTm,tK^1TQum,tdm−2∫ΩuTm,tK^2TQum,tdm+2∫ΩzTm,tPGum,tdm=∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1zm,tdm+2∫ΩzTm,tPG+K^1TM1K^2−K^1TQum,tdm+∫ΩuTm,tK^2TM1K^2−K^2TQ−QTK^2um,tdm=∫ΩϖTm,tΦ1Λ1Λ1TΛ2ϖm,tdm,where ϖm,t=zTm,t,uTm,tT. On the basis of (16), one gets(30)2∫ΩyTm,tQum,tdm−∫ΩyTm,tM1ym,tdm≥V˙t.
Then,(31)2∫t0tl∫ΩyTm,tQum,tdmdt−∫t0tl∫ΩyTm,tM1ym,tdmdt≥Vtl−Vt0,for any t0,tl∈0,+∞ and tl⩾t0. In other words,(32)∫t0tl∫ΩyTm,tQum,t−yTm,tM12ym,tdmdt≥Stl−St0,where St=Vt/2.
The following results can be deduced by using the similar method.Corollary 1.
System (14) realizes passivity if there are matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi and Q∈ℝq×p satisfying(33)PD+DP≥0,(34)Φ2PG−K^1TQGTP−QTK^1−K^2TQ−QTK^2≤0,where Φ2=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP.Corollary 2.
System (14) realizes input-strict passivity if there exist matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi,Q∈ℝq×p, and 0<M2∈ℝp×p satisfying(35)PD+DP≥0,(36)Φ2PG−K^1TQGTP−QTK^1M2−K^2TQ−QTK^2≤0,where Φ2=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP.Remark 2.
To the best of our knowledge, the concept of passivity is proposed for the first time in circuit analysis and has discovered comprehensive potential applications in lots of areas after that. Over the past few decades, some scholars have investigated the passivity of CNNs and CRDNNs and derived many meaningful results [26–33]. Note that the passivity problem is solved in the abovementioned works based on the case that the input has identical dimension as the output. As far as we know, only a few passivity results have been obtained for the CRDNNs with different dimensional input and output until now [34–36]. However, the networks considered in these works are coupled by identical nodes. Therefore, we devote ourselves to studying the passivity of CRDNNs consisting of different dimensional nodes (i.e., CDDDRDNNs), in which the input has different dimensions as output. Hence, the acquired passivity results are more general and less conservative in this paper.


### 3.3. Synchronization Analysis

Definition 2.
Network (6) achieves synchronization if for all i=1,2,⋯,N,(37)limt⟶+∞wi⋅,t−o^i2=0,under the condition uim,t=0.Theorem 2.
Network (6) is synchronized if there are matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi satisfying(38)PD+DP≥0,(39)W1+PH^+H^TP<0,where W1=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L.Proof.
Choose the same Lyapunov functional as (17) for system (11), then,(40)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2+Aif^izim,t+Ziφ^izim,t−τit−Bizim,t+∑j=1NH^ijzjm,tdm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2+Aif^izim,t+Ziφ^izim,t−τit−Bizim,t+∑j=1NH^ijzjm,tdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm≤ϒ1z⋅,t22,where ϒ1=λM−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP<0.
From (40), we have(41)z⋅,t22⩽V˙tϒ1.
From (17) and (40), one knows that all terms of Vt are bounded and Vt is nonincreasing. Hence, Vt converges to a finite nonnegative real number. According to (41), we can infer that limt⟶+∞∫0tz⋅,t22dt exists and is finite. Furthermore, since τit is bounded, it is easily derived that(42)limt⟶+∞∑i=1N∑j=1ξiψ^ij21−δi∫t−τitt∫Ωzjim,h2dmdh=0.
Hence,limt⟶+∞∫ΩzTm,tPzm,tdm is not only existing but also is a nonnegative real number. Next, we will prove that limt⟶+∞∫ΩzTm,tPzm,tdm=0. If not, one can get(43)limt⟶+∞∫ΩzTm,tPzm,tdm=ς>0.
Obviously, there is a real constantθ>0 such that(44)∫ΩzTm,tPzm,tdm>ς2,t≥θ.
Hence,(45)z⋅,t22>ς2κ,t≥θ,where κ=λMP. According to (40) and (45), we obtain(46)V˙t<ϒ1ς2κ,t≥θ.
It follows from (46) that(47)−Vθ≤V+∞−Vθ=∫θ+∞V˙tdt<∫θ+∞ϒ1ς2κdt=−∞.
This is unreasonable. Consequently,(48)limt⟶+∞∫ΩzTm,tPzm,tdm=0.
Then, we havelimt⟶+∞z⋅,t2=0. Hence, network (6) is synchronized.


## 3.1. Network Model

The CDDDRDNNs considered in this section is stated as follows:(6)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=1NcijHijwjm,t,in which i=1,2,⋯,N, wim,t=w1im,t,w2im,t,⋯,wξiim,tT∈ℝξi denotes the state vector of ith neuron; Bi∈ℝξi×ξi=diagb1i,b2i,⋯,bξii>0;Ai=aghiξi×ξi,Zi=zghiξi×ξi; fiwim,t=f1iw1im,t,f2iw2im,t,⋯,fξiiwξiim,tT; Di=diagd1i,d2i,⋯,dξii>0; φiwim,t−τit=φ1iw1im,t−τit,φ2iw2im,t−τit,⋯,φξiiwξiim,t−τitT, fli⋅, and φli⋅l=1,2,⋯,ξi denote the activation functions for the lth neuron in neural network i; Ji=J1i,J2i,⋯,JξiiT; uim,t∈ℝpi denotes the control input; the inner coupling matrix is defined by Hij∈ℝξi×ξj; Gi∈ℝξi×pi is a known matrix; C=cijN×N is the coupling configuration matrix denoting coupling weight, which satisfies cij≠0i≠j if node i and node j are connected, or else cij=0. In addition, the time-varying delay τit satisfies 0⩽τit⩽τi and τ˙it⩽δi<1.Remark 1.
As a special type of CNs, CNNs have attracted much attention due to their extensive applications on plenty of fields. So far, a large number of scholars have acquired some interesting research results about synchronization and passivity of CNNs [4–36]. Nevertheless, the considered CNs in the aforementioned literature are made up of identical nodes. As a matter of fact, it is utterly impractical that the networks have totally identical nodes in many practical situations. For instance, due to the differences of the parameters, it is impossible that the neurons in the nervous system of neural networks are entirely the same as each other. Consequently, it is significant to study CNNs composed of nonidentical nodes. To our knowledge, a few investigators have discussed CNs consisting of the same dimensional nonidentical nodes in recent years [37–41]. However, the networks with nonidentical nodes of different dimensions can describe more practical networks. In addition, the considered networks in these existing works did not take the reaction-diffusion terms into account. Therefore, we pay our attention on the CRDNNs with different dimensional nodes. As far as we know, this is our first step toward addressing the passivity and synchronization problems of CDDDRDNNs.
For network (6), the initial value condition and boundary value condition are described by(7)wim,t=ϑim,t,m,t∈Ω×−ρ,0,(8)wim,t=0,m,t∈∂Ω×−ρ,+∞,where i=1,2,⋯,N,ϑim,t is bounded and continuous on Ω×−ρ,0 and ρ=maxi=1,2,⋯,Nτi.
Suppose that the functionsfli⋅ and φli⋅ satisfy the following global Lipschitz condition; there exist positive constants ψil and ψ^il such that(9)fliυ1−fliυ2⩽ψilυ1−υ2,φliυ1−φliυ2⩽ψ^ilυ1−υ2,hold for all υ1,υ2∈ℝ,l=1,2,⋯,ξi,i=1,2,⋯,N.
Let the constant vectorO^=o^1T,o^2T,⋯,o^NTT be an equilibrium point of an isolated node of network (6). Then,(10)Di∑σ=1ϱ∂2o^i∂mσ2−Bio^i+Aifio^i+Ji+Ziφio^i+∑j=1NcijHijo^j=0,where o^i=o^1i,o^2i,⋯,o^ξiiT∈ℝξi,i=1,2,⋯,N.
For the error vectorzim,t=z1im,t,z2im,t,⋯,zξiim,tT=wim,t−o^i, we have(11)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NcijHijzjm,t,where i=1,2,⋯,N,f^izim,t=fiwim,t−fio^i=f1iw1im,t−f1io^1i,f2iw2im,t−f2io^2i,⋯,fξiiwξiim,t−fξiio^ξiiT and φ^izim,t−τit=φiwim,t−τit−φio^i=φ1iw1im,t−φ1io^1i,φ2iw2im,t−φ2io^2i,⋯,φξiiwξiim,t−φξiio^ξiiT.
The output vectoryim,t∈ℝqi of the network (11) is given as follows:(12)yim,t=K1izim,t+K2iuim,t,where K1i∈ℝqi×ξi and K2i∈ℝqi×pi are the known matrices.
In the whole paper, we denote(13)zm,t=z1Tm,t,z2Tm,t,⋯,zNTm,tT,um,t=u1Tm,t,u2Tm,t,⋯,uNTm,tT,ym,t=y1Tm,t,y2Tm,t,⋯,yNTm,tT,L=11−δ1Iξ1,11−δ2Iξ2,⋯,11−δNIξN,p=p1+p2+⋯+pN,q=q1+q2+⋯+qN,ξ=ξ1+ξ2+⋯+ξN,D=diagD1,D2,⋯,DN,Z=diagZ1,Z2,⋯,ZN,G=diagG1,G2,⋯,GN,B=diagB1,B2,⋯,BN,A=diagA1,A2,⋯,AN,Ψi=diagψi12,ψi22,⋯,ψiξi2,Ψ=diagΨ1,Ψ2,⋯,ΨN,Ψ^i=diagψ^i12,ψ^i22,⋯,ψ^iξi2,Ψ^=diagΨ^1,Ψ^2,⋯,Ψ^N,f^zm,t=f^1z1m,t,f^2z2m,t,⋯,f^NzNm,tT,φ^zm,t−τt=φ^1z1m,t−τ1t,φ^2z2m,t−τ2t,⋯,φ^NzNm,t−τNtT,H^=H^ijN×N,whereH^ij=cijHij,K^1=diagK11,K12,⋯,K1N,K^2=diagK21,K22,⋯,K2N.


## 3.2. Passivity Analysis

It follows from (11) and (12) that the following system with output yim,t∈ℝqi and input uim,t∈ℝpi can be obtained as follows:(14)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t,yim,t=K1izim,t+K2iuim,t,where H^ij=cijHij and i=1,2,⋯,N.Theorem 1.
System (14) reaches output-strict passivity if there are matrices Q∈ℝq×p,P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi, and 0<M1∈ℝq×q satisfying(15)PD+DP⩾0,(16)Φ1Λ1Λ1TΛ2⩽0,where Φ1=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1,Λ1=PG+K^1TM1K^2−K^1TQ,Λ2=K^2TM1K^2−K^2TQ−QTK^2.Proof.
Construct the Lyapunov functional for system (14) as follows:(17)Vt=∑i=1N∫ΩziTm,tPizim,tdm+∑i=1N∑j=1ξiψ^ij21−δi∫t−τitt∫Ωzjim,h2dmdh.
Then,(18)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,tdm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t+Giuim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm=2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2−Bzm,t+Af^zm,t+Gum,t+Zφ^zm,t−τt+H^zm,tdm+∫ΩzTm,tΨ^Lzm,tdm−∫ΩzTm,t−τtΨ^zm,t−τtdm.
By means of the boundary condition and Green’s formula, one can obtain the following:(19)∫Ωzgim,t∑σ=1ϱ∂2zhim,t∂mσ2dm=−∑σ=1ϱ∫Ω∂zgim,t∂mσ∂zhim,t∂mσdm,where h=1,2,⋯,ξi,i=1,2,⋯,N,andg=1,2,⋯,ξi. Let Pi=pghiξi×ξi, one has(20)∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm=∑i=1N∑g=1ξi∑h=1ξipghidhi∫Ωzgim,t∑σ=1ϱ∂2zhim,t∂mσ2dm=−∑i=1N∑σ=1ϱ∑g=1ξi∑h=1ξipghidhi∫Ω∂zgim,t∂mσ∂zhim,t∂mσdm=−∑σ=1ϱ∫Ω∂zm,t∂mσTPD∂zm,t∂mσdm.
Then, we obtain(21)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm=−∑σ=1ϱ∫Ω∂zm,t∂mσTPD+DP∂zm,t∂mσdm.
Obviously, there is a real matrixΓ∈ℝξ×ξ satisfying(22)PD+DP=ΓTΓ.
Thus,(23)∂zm,t∂mσTPD+DP∂zm,t∂mσ=∂zm,t∂mσTΓTΓ∂zm,t∂mσ=∂Γzm,t∂mσT∂Γzm,t∂mσ.
Letϕm,t=Γzm,t, for m,t∈∂Ω×−ρ,+∞, it follows from boundary condition (8) that ϕm,t=Γzm,t=0. In terms of Lemma 2.1, we can get(24)∑σ=1ϱ∫Ω∂ϕm,t∂mσT∂ϕm,t∂mσdm≥∑σ=1ϱ1ζσ2∫ΩϕTm,tϕm,tdm=∑σ=1ϱ1ζσ2∫ΩzTm,tPD+DPzm,tdm.
Therefore,(25)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm≤−∑σ=1ϱ1ζσ2∫ΩzTm,tPD+DPzm,tdm.
In addition, we can derive that(26)2∫ΩzTm,tPAf^zm,tdm=2∑i=1N∫ΩziTm,tPiAif^izim,tdm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫Ωf^iTwim,tf^iwim,tdm=∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∑l=1ξi∫Ωfliwlim,t−flio^li2dm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∑l=1ξi∫Ωψil2zlim,t2dm=∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm=∫ΩzTm,tPAATP+Ψzm,tdm.
Similarly, we have(27)2∫ΩzTm,t−τtPZφ^zm,t−τtdm=2∑i=1N∫ΩziTm,t−τitPiZiφ^izim,t−τitdm≤∑i=1N∫ΩziTm,tPiZiZiTPizim,tdm+∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm=∫ΩzTm,tPZZTPzm,tdm+∫ΩzTm,t−τtΨ^zm,t−τtdm.
By (25)–(27), one has(28)V˙t≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm+2∫ΩzTm,tPGum,tdm.
Thus,(29)V˙t−2∫ΩyTm,tQum,tdm+∫ΩyTm,tM1ym,tdm≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm+∫ΩzTm,tK^1TM1K^1zm,tdm+∫ΩzTm,tK^1TM1K^2um,tdm+∫ΩuTm,tK^2TM1K^1zm,tdm+∫ΩuTm,tK^2TM1K^2um,tdm−2∫ΩzTm,tK^1TQum,tdm−2∫ΩuTm,tK^2TQum,tdm+2∫ΩzTm,tPGum,tdm=∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1zm,tdm+2∫ΩzTm,tPG+K^1TM1K^2−K^1TQum,tdm+∫ΩuTm,tK^2TM1K^2−K^2TQ−QTK^2um,tdm=∫ΩϖTm,tΦ1Λ1Λ1TΛ2ϖm,tdm,where ϖm,t=zTm,t,uTm,tT. On the basis of (16), one gets(30)2∫ΩyTm,tQum,tdm−∫ΩyTm,tM1ym,tdm≥V˙t.
Then,(31)2∫t0tl∫ΩyTm,tQum,tdmdt−∫t0tl∫ΩyTm,tM1ym,tdmdt≥Vtl−Vt0,for any t0,tl∈0,+∞ and tl⩾t0. In other words,(32)∫t0tl∫ΩyTm,tQum,t−yTm,tM12ym,tdmdt≥Stl−St0,where St=Vt/2.
The following results can be deduced by using the similar method.Corollary 1.
System (14) realizes passivity if there are matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi and Q∈ℝq×p satisfying(33)PD+DP≥0,(34)Φ2PG−K^1TQGTP−QTK^1−K^2TQ−QTK^2≤0,where Φ2=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP.Corollary 2.
System (14) realizes input-strict passivity if there exist matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi,Q∈ℝq×p, and 0<M2∈ℝp×p satisfying(35)PD+DP≥0,(36)Φ2PG−K^1TQGTP−QTK^1M2−K^2TQ−QTK^2≤0,where Φ2=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP.Remark 2.
To the best of our knowledge, the concept of passivity is proposed for the first time in circuit analysis and has discovered comprehensive potential applications in lots of areas after that. Over the past few decades, some scholars have investigated the passivity of CNNs and CRDNNs and derived many meaningful results [26–33]. Note that the passivity problem is solved in the abovementioned works based on the case that the input has identical dimension as the output. As far as we know, only a few passivity results have been obtained for the CRDNNs with different dimensional input and output until now [34–36]. However, the networks considered in these works are coupled by identical nodes. Therefore, we devote ourselves to studying the passivity of CRDNNs consisting of different dimensional nodes (i.e., CDDDRDNNs), in which the input has different dimensions as output. Hence, the acquired passivity results are more general and less conservative in this paper.


## 3.3. Synchronization Analysis

Definition 2.
Network (6) achieves synchronization if for all i=1,2,⋯,N,(37)limt⟶+∞wi⋅,t−o^i2=0,under the condition uim,t=0.Theorem 2.
Network (6) is synchronized if there are matrices P=diagP1,P2,⋯,PN>0Pi∈ℝξi×ξi satisfying(38)PD+DP≥0,(39)W1+PH^+H^TP<0,where W1=−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L.Proof.
Choose the same Lyapunov functional as (17) for system (11), then,(40)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2+Aif^izim,t+Ziφ^izim,t−τit−Bizim,t+∑j=1NH^ijzjm,tdm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2+Aif^izim,t+Ziφ^izim,t−τit−Bizim,t+∑j=1NH^ijzjm,tdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm≤∫ΩzTm,t−∑σ=1ϱ1ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TPzm,tdm≤ϒ1z⋅,t22,where ϒ1=λM−∑σ=1ϱ1/ζσ2PD+DP−PB−BP+PAATP+PZZTP+Ψ+Ψ^L+PH^+H^TP<0.
From (40), we have(41)z⋅,t22⩽V˙tϒ1.
From (17) and (40), one knows that all terms of Vt are bounded and Vt is nonincreasing. Hence, Vt converges to a finite nonnegative real number. According to (41), we can infer that limt⟶+∞∫0tz⋅,t22dt exists and is finite. Furthermore, since τit is bounded, it is easily derived that(42)limt⟶+∞∑i=1N∑j=1ξiψ^ij21−δi∫t−τitt∫Ωzjim,h2dmdh=0.
Hence,limt⟶+∞∫ΩzTm,tPzm,tdm is not only existing but also is a nonnegative real number. Next, we will prove that limt⟶+∞∫ΩzTm,tPzm,tdm=0. If not, one can get(43)limt⟶+∞∫ΩzTm,tPzm,tdm=ς>0.
Obviously, there is a real constantθ>0 such that(44)∫ΩzTm,tPzm,tdm>ς2,t≥θ.
Hence,(45)z⋅,t22>ς2κ,t≥θ,where κ=λMP. According to (40) and (45), we obtain(46)V˙t<ϒ1ς2κ,t≥θ.
It follows from (46) that(47)−Vθ≤V+∞−Vθ=∫θ+∞V˙tdt<∫θ+∞ϒ1ς2κdt=−∞.
This is unreasonable. Consequently,(48)limt⟶+∞∫ΩzTm,tPzm,tdm=0.
Then, we havelimt⟶+∞z⋅,t2=0. Hence, network (6) is synchronized.


## 4. Robust Passivity and Robust Synchronization of CDDDRDNNs

### 4.1. Network Model

Taking the parameter uncertainties into account, a CDDDRDNN with uncertain parameters is stated as follows:(49)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=1NcijHijwjm,t,in which i=1,2,⋯,N,wim,t,Ji,Gi,uim,t,fiwim,t,φiwim,t−τit,Hij,andcij represent the same senses as in model (6). The parameters Di,Bi,Ai,andZi can be changed within a certain parameter range of precisions as follows:(50)DP≔Di=diagdsi:D¯i≤Di≤D¯i,0<d¯si≤dsi≤d¯si,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Di∈DP;BP≔Bi=diagbsi:B¯i≤Bi≤B¯i,0<b¯si≤bsi≤b¯si,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Bi∈BP;AP≔Ai=aghiξi×ξi:a¯ghi≤aghi≤a¯ghi,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Ai∈AP;ZP≔Zi=zghiξi×ξi:z¯ghi≤zghi≤z¯ghi,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Zi∈ZP.In addition, for convenience, we denote(51)a˜ghi=maxa¯ghi,a¯ghi,z˜ghi=maxz¯ghi,z¯ghi,ϑiA=∑g=1ξi∑h=1ξia˜ghi2,ϑiZ=∑g=1ξi∑h=1ξiz˜ghi2,ϑA=diagϑ1AIξ1,ϑ2AIξ2,⋯,ϑNAIξN∈ℝξ×ξ,ϑZ=diagϑ1ZIξ1,ϑ2ZIξ2,⋯,ϑNZIξN∈ℝξ×ξ,D¯=diagD¯1,D¯2,⋯,D¯N,B¯=diagB¯1,B¯2,⋯,B¯N,where g=1,2,⋯,ξiandh=1,2,⋯,ξi,i=1,2,⋯,N.According to (10), we can get the error system zim,t of network (49) as follows:(52)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NcijHijzjm,t,where the ranges of Di,Bi,Ai, and Zi are described by using (50).


### 4.2. Robust Passivity Analysis

The output vectoryim,t of system (52) is defined similarly as (12). In terms of (12) and (52), the error system with output yim,t∈ℝqi and input uim,t∈ℝpi can be described by(53)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t,yim,t=K1izim,t+K2iuim,t,where H^ij=cijHij and i=1,2,⋯,N.Theorem 3.
If there are matricesP=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi,Q∈ℝq×p, and 0<M1∈ℝq×q satisfying(54)Φ3Λ1Λ1TΛ2≤0,where Φ3=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1,Λ1=PG+K^1TM1K^2−K^1TQ,Λ2=K^2TM1K^2−K^2TQ−QTK^2, then, system (53) under the given ranges of parameters (50) is output-robustly passive.Proof.
Choose the same Lyapunov functional as (17) for system (52). Then,(55)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,tdm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm⩽2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2−Bzm,t+Af^zm,t+Gum,t+Zφ^zm,t−τt+H^zm,tdm+∫ΩzTm,tΨ^Lzm,tdm−∫ΩzTm,t−τtΨ^zm,t−τtdm.
Obviously,(56)−2∫ΩzTm,tPBzm,tdm=−2∑i=1N∑h=1ξiphibhi∫Ωzhim,t2dm≤−2∑i=1N∑h=1ξiphib¯hi∫Ωzhim,t2dm=−2∫ΩzTm,tPB¯zm,tdm.
Similar to the deduction of (25), one can get(57)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm≤−∑σ=1ϱ2ζσ2∫ΩzTm,tPDzm,tdm=−∑i=1N∑h=1ξi∑σ=1ϱ2ζσ2phidhi∫Ωzhim,t2dm≤−∑i=1N∑h=1ξi∑σ=1ϱ2ζσ2phid¯hi∫Ωzhim,t2dm=−∑σ=1ϱ2ζσ2∫ΩzTm,tPD¯zm,tdm.
In addition,(58)2∫ΩzTm,tPAf^zm,tdm=2∑i=1N∫ΩziTm,tPiAif^izim,tdm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm≤∑i=1N∫ΩziTm,tϑiAPi2zim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm=∫ΩzTm,tϑAP2+Ψzm,tdm,(59)2∫ΩzTm,tPZφ^zm,t−τtdm=2∑i=1N∫ΩziTm,tPiZiφ^izim,t−τitdm⩽∑i=1N∫ΩziTm,tPiZiZiTPizim,tdm+∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm=∫ΩzTm,tϑZP2zm,tdm+∫ΩzTm,t−τtΨ^zm,t−τtdm.
According to (56)–(59), one gets(60)V˙t≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TPzm,tdm+2∫ΩzTm,tPGum,tdm.
Hence,(61)V˙t−2∫ΩyTm,tQum,tdm+∫ΩyTm,tM1ym,tdm≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1zm,tdm+2∫ΩzTm,tPG+K^1TM1K^2−K^1TQum,tdm+∫ΩuTm,tK^2TM1K^2−K^2TQ−QTK^2um,tdm=∫ΩϖTm,tΦ3Λ1Λ1TΛ2ϖm,tdm.
From (54), one can get(62)2∫ΩyTm,tQum,tdm−∫ΩyTm,tM1ym,tdm≥V˙t.
Then,(63)2∫t0tl∫ΩyTm,tQum,tdmdt−∫t0tl∫ΩyTm,tM1ym,tdmdt≥Vtl−Vt0,for any t0,tl∈0,+∞ and tl≥t0. In other words,(64)∫t0tl∫ΩyTm,tQum,t−yTm,tM12ym,tdmdt≥Stl−St0,in which St=Vt/2.
The following results can be deduced by using the similar method.Corollary 3.
System (53) under the given ranges of parameters (50) is robustly passive if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξiandQ∈ℝq×p satisfying(65)Φ4PG−K^1TQGTP−QTK^1−K^2TQ−QTK^2≤0,where Φ4=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP.Corollary 4.
System (53) under the given ranges of parameters (50) is input-robustly passive if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi,Q∈ℝq×p, and 0<M2∈ℝP×P satisfying(66)Φ4PG−K^1TQGTP−QTK^1M2−K^2TQ−QTK^2≤0,where Φ4=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP.


### 4.3. Robust Synchronization Analysis

Definition 3.
Network (49) under the given ranges of parameter (50) realizes robust synchronization if for all Di∈DP,Bi∈BP,Ai∈AP,andZi∈ZP, i=1,2,⋯,N,(67)limt⟶+∞wi⋅,t−o^i2=0,under the condition uim,t=0.Theorem 4.
Network (49) under the given ranges of parameters (50) achieves robust synchronization if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi satisfying(68)W2+PH^+H^TP<0,where W2=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L.Proof.
Select the same Lyapunov functional as (17) for system (52); then,(69)V˙t≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Ziφ^izim,t−τit+Aif^izim,t+∑j=1NH^ijzjm,tdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TPzm,tdm≤ϒ2z⋅,t22,where ϒ2=λM−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ^L+Ψ+PH^+H^TP<0.
Then,limt⟶+∞z⋅,t2=0 can be proved similarly as in Theorem 2. Consequently, network (49) under the given ranges of parameters (50) realizes robust synchronization.Remark 3.
In Section3, we deal with the passivity and synchronization problems of CDDDRDNNs and establish some adequate conditions to achieve the passivity and synchronization of network (6). Note that the parameters in matrices Di,Bi,Ai,andZi of network model (6) are fixed. However, it is utterly unrealistic for the networks with some certain parameters due to the noises of environment and equipment restrictions in some practical situations [18–25]. Thus, it is necessary to consider the case that the parameters in matrices Di,Bi,Ai,andZi of network model (6) belong to some given ranges and investigate robust dynamical properties of the considered network. As far as we know, the robust synchronization and robust passivity of CDDDRDNNs with parametric uncertainties have never been studied. In this section, we present several robust synchronization and robust passivity criteria of CDDDRDNNs with uncertain parameters in Theorems 3 and 4 and Corollaries 3 and 4, respectively.


## 4.1. Network Model

Taking the parameter uncertainties into account, a CDDDRDNN with uncertain parameters is stated as follows:(49)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=1NcijHijwjm,t,in which i=1,2,⋯,N,wim,t,Ji,Gi,uim,t,fiwim,t,φiwim,t−τit,Hij,andcij represent the same senses as in model (6). The parameters Di,Bi,Ai,andZi can be changed within a certain parameter range of precisions as follows:(50)DP≔Di=diagdsi:D¯i≤Di≤D¯i,0<d¯si≤dsi≤d¯si,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Di∈DP;BP≔Bi=diagbsi:B¯i≤Bi≤B¯i,0<b¯si≤bsi≤b¯si,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Bi∈BP;AP≔Ai=aghiξi×ξi:a¯ghi≤aghi≤a¯ghi,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Ai∈AP;ZP≔Zi=zghiξi×ξi:z¯ghi≤zghi≤z¯ghi,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Zi∈ZP.In addition, for convenience, we denote(51)a˜ghi=maxa¯ghi,a¯ghi,z˜ghi=maxz¯ghi,z¯ghi,ϑiA=∑g=1ξi∑h=1ξia˜ghi2,ϑiZ=∑g=1ξi∑h=1ξiz˜ghi2,ϑA=diagϑ1AIξ1,ϑ2AIξ2,⋯,ϑNAIξN∈ℝξ×ξ,ϑZ=diagϑ1ZIξ1,ϑ2ZIξ2,⋯,ϑNZIξN∈ℝξ×ξ,D¯=diagD¯1,D¯2,⋯,D¯N,B¯=diagB¯1,B¯2,⋯,B¯N,where g=1,2,⋯,ξiandh=1,2,⋯,ξi,i=1,2,⋯,N.According to (10), we can get the error system zim,t of network (49) as follows:(52)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NcijHijzjm,t,where the ranges of Di,Bi,Ai, and Zi are described by using (50).


## 4.2. Robust Passivity Analysis

The output vectoryim,t of system (52) is defined similarly as (12). In terms of (12) and (52), the error system with output yim,t∈ℝqi and input uim,t∈ℝpi can be described by(53)∂zim,t∂t=Di∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,t,yim,t=K1izim,t+K2iuim,t,where H^ij=cijHij and i=1,2,⋯,N.Theorem 3.
If there are matricesP=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi,Q∈ℝq×p, and 0<M1∈ℝq×q satisfying(54)Φ3Λ1Λ1TΛ2≤0,where Φ3=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1,Λ1=PG+K^1TM1K^2−K^1TQ,Λ2=K^2TM1K^2−K^2TQ−QTK^2, then, system (53) under the given ranges of parameters (50) is output-robustly passive.Proof.
Choose the same Lyapunov functional as (17) for system (52). Then,(55)V˙t=2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Aif^izim,t+Giuim,t+Ziφ^izim,t−τit+∑j=1NH^ijzjm,tdm−∑i=1N∑j=1ξiψ^ij21−τ˙it1−δi∫Ωzjim,t−τit2dm+∑i=1N∑j=1ξiψ^ij21−δi∫Ωzjim,t2dm⩽2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2−Bzm,t+Af^zm,t+Gum,t+Zφ^zm,t−τt+H^zm,tdm+∫ΩzTm,tΨ^Lzm,tdm−∫ΩzTm,t−τtΨ^zm,t−τtdm.
Obviously,(56)−2∫ΩzTm,tPBzm,tdm=−2∑i=1N∑h=1ξiphibhi∫Ωzhim,t2dm≤−2∑i=1N∑h=1ξiphib¯hi∫Ωzhim,t2dm=−2∫ΩzTm,tPB¯zm,tdm.
Similar to the deduction of (25), one can get(57)2∫ΩzTm,tPD∑σ=1ϱ∂2zm,t∂mσ2dm≤−∑σ=1ϱ2ζσ2∫ΩzTm,tPDzm,tdm=−∑i=1N∑h=1ξi∑σ=1ϱ2ζσ2phidhi∫Ωzhim,t2dm≤−∑i=1N∑h=1ξi∑σ=1ϱ2ζσ2phid¯hi∫Ωzhim,t2dm=−∑σ=1ϱ2ζσ2∫ΩzTm,tPD¯zm,tdm.
In addition,(58)2∫ΩzTm,tPAf^zm,tdm=2∑i=1N∫ΩziTm,tPiAif^izim,tdm≤∑i=1N∫ΩziTm,tPiAiAiTPizim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm≤∑i=1N∫ΩziTm,tϑiAPi2zim,tdm+∑i=1N∫ΩziTm,tΨizim,tdm=∫ΩzTm,tϑAP2+Ψzm,tdm,(59)2∫ΩzTm,tPZφ^zm,t−τtdm=2∑i=1N∫ΩziTm,tPiZiφ^izim,t−τitdm⩽∑i=1N∫ΩziTm,tPiZiZiTPizim,tdm+∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm=∫ΩzTm,tϑZP2zm,tdm+∫ΩzTm,t−τtΨ^zm,t−τtdm.
According to (56)–(59), one gets(60)V˙t≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TPzm,tdm+2∫ΩzTm,tPGum,tdm.
Hence,(61)V˙t−2∫ΩyTm,tQum,tdm+∫ΩyTm,tM1ym,tdm≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP+K^1TM1K^1zm,tdm+2∫ΩzTm,tPG+K^1TM1K^2−K^1TQum,tdm+∫ΩuTm,tK^2TM1K^2−K^2TQ−QTK^2um,tdm=∫ΩϖTm,tΦ3Λ1Λ1TΛ2ϖm,tdm.
From (54), one can get(62)2∫ΩyTm,tQum,tdm−∫ΩyTm,tM1ym,tdm≥V˙t.
Then,(63)2∫t0tl∫ΩyTm,tQum,tdmdt−∫t0tl∫ΩyTm,tM1ym,tdmdt≥Vtl−Vt0,for any t0,tl∈0,+∞ and tl≥t0. In other words,(64)∫t0tl∫ΩyTm,tQum,t−yTm,tM12ym,tdmdt≥Stl−St0,in which St=Vt/2.
The following results can be deduced by using the similar method.Corollary 3.
System (53) under the given ranges of parameters (50) is robustly passive if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξiandQ∈ℝq×p satisfying(65)Φ4PG−K^1TQGTP−QTK^1−K^2TQ−QTK^2≤0,where Φ4=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP.Corollary 4.
System (53) under the given ranges of parameters (50) is input-robustly passive if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi,Q∈ℝq×p, and 0<M2∈ℝP×P satisfying(66)Φ4PG−K^1TQGTP−QTK^1M2−K^2TQ−QTK^2≤0,where Φ4=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TP.


## 4.3. Robust Synchronization Analysis

Definition 3.
Network (49) under the given ranges of parameter (50) realizes robust synchronization if for all Di∈DP,Bi∈BP,Ai∈AP,andZi∈ZP, i=1,2,⋯,N,(67)limt⟶+∞wi⋅,t−o^i2=0,under the condition uim,t=0.Theorem 4.
Network (49) under the given ranges of parameters (50) achieves robust synchronization if there are matrices P=diagP1,P2,⋯,PN>0Pi=diagp1i,p2i,⋯,pξii∈ℝξi×ξi satisfying(68)W2+PH^+H^TP<0,where W2=−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L.Proof.
Select the same Lyapunov functional as (17) for system (52); then,(69)V˙t≤2∑i=1N∫ΩziTm,tPiDi∑σ=1ϱ∂2zim,t∂mσ2−Bizim,t+Ziφ^izim,t−τit+Aif^izim,t+∑j=1NH^ijzjm,tdm+∑i=1N∫ΩziTm,tΨ^i1−δizim,tdm−∑i=1N∫ΩziTm,t−τitΨ^izim,t−τitdm≤∫ΩzTm,t−∑σ=1ϱ2ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ+Ψ^L+PH^+H^TPzm,tdm≤ϒ2z⋅,t22,where ϒ2=λM−∑σ=1ϱ2/ζσ2PD¯−2PB¯+ϑA+ϑZP2+Ψ^L+Ψ+PH^+H^TP<0.
Then,limt⟶+∞z⋅,t2=0 can be proved similarly as in Theorem 2. Consequently, network (49) under the given ranges of parameters (50) realizes robust synchronization.Remark 3.
In Section3, we deal with the passivity and synchronization problems of CDDDRDNNs and establish some adequate conditions to achieve the passivity and synchronization of network (6). Note that the parameters in matrices Di,Bi,Ai,andZi of network model (6) are fixed. However, it is utterly unrealistic for the networks with some certain parameters due to the noises of environment and equipment restrictions in some practical situations [18–25]. Thus, it is necessary to consider the case that the parameters in matrices Di,Bi,Ai,andZi of network model (6) belong to some given ranges and investigate robust dynamical properties of the considered network. As far as we know, the robust synchronization and robust passivity of CDDDRDNNs with parametric uncertainties have never been studied. In this section, we present several robust synchronization and robust passivity criteria of CDDDRDNNs with uncertain parameters in Theorems 3 and 4 and Corollaries 3 and 4, respectively.


## 5. Numerical Examples

Example 1.
Consider the following CDDDRDNNs:(70)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=13cijHijwjm,t,where i=1,2,3,ξ1=3,ξ2=ξ3=2,fliυ=φliυ=υ+1−υ−1/8,l=1,2,3,Ω=m−0.5<m<0.5, D1=diag0.6,0.8,0.9,D2=diag0.5,0.7,D3=diag0.8,0.9,B1=diag0.8,0.9,1.2,B2=diag0.7,0.9, B3=diag0.6,0.9,τit=0.04i−0.02ie−t,τ1=0.04,δ1=0.02,τ2=0.08,δ2=0.04,τ3=0.12,δ3=0.06 and J1=0,0,0T,J2=J3=0,0T,u1m,t=8tcosπm,5tcosπm,10tcosπmT,u2m,t=16tcosπm,10tcosπmT,u3m,t=24tcosπm,15tcosπmT. The matrices Ai=aghiξi×ξi,Zi=zghiξi×ξi,C=cij3×3,Gi,Hij,K1i,andK2i are chosen as follows:(71)A1=0.50.60.70.60.40.50.30.20.8,A2=0.40.60.30.4,A3=0.30.60.50.4,Z1=0.40.60.70.60.30.50.50.20.6,Z2=0.30.60.50.4,Z3=0.50.60.60.4,G1=0.20.10.20.30.10.20.10.20.2,G2=0.20.30.30.1,G3=0.30.20.20.3,H11=0.20.10.30.10.30.20.30.20.4,H12=0.30.20.30.50.20.4,H13=0.30.20.20.30.10.2,H21=0.20.30.10.10.40.2,H22=0.30.20.10.3,H23=0.20.30.30.4,H31=0.10.20.40.30.20.5,H32=0.30.10.20.4,H33=0.40.30.20.4,K11=0.10.20.30.20.40.10.30.20.1,K12=0.10.20.20.30.20.1,K13=0.20.30.30.40.20.1,K21=0.20.20.30.10.30.20.20.10.3,K22=0.30.20.10.20.20.4,K23=0.10.20.10.30.30.2,C=−0.10.30.20.2−0.40.20.30.2−0.8.Case 1.
Obviously, the equilibrium point of an isolated node of network (70) is O^=0,0,0,0,0,0,0T and ψil=ψ^il=0.25. It is easy to calculate the following matrices P,Q, and M1 satisfying (15) and (16):(72)M1=4.9725−0.7644−0.7186−0.0353−0.0591−0.0363−0.0508−0.0707−0.0289−0.76444.3790−0.9676−0.0454−0.0765−0.0477−0.0609−0.0850−0.0352−0.7186−0.96764.5415−0.0363−0.0613−0.0386−0.0533−0.0742−0.0306−0.0353−0.0454−0.03635.36160.0154−0.0287−0.0215−0.0299−0.0122−0.0591−0.0765−0.06130.01545.3503−0.0393−0.0360−0.0501−0.0206−0.0363−0.0477−0.0386−0.0287−0.03935.3398−0.0222−0.0309−0.0128−0.0508−0.0609−0.0533−0.0215−0.0360−0.02225.51260.23420.0883−0.0707−0.0850−0.0742−0.0299−0.0501−0.03090.23425.66880.1327−0.0289−0.0352−0.0306−0.0122−0.0206−0.01280.08830.13275.4320,Q=4.33455.84415.2880−0.06400.0404−0.02080.0671−3.75237.7721−3.5249−0.0102−0.0813−0.0252−0.11252.9109−10.79223.1831−0.0047−0.0718−0.0219−0.08900.02180.00850.036711.6387−10.21330.05460.0150−0.1185−0.3761−0.2903−2.93893.7925−0.0623−0.18720.03790.17980.1093−0.900211.25760.00670.0789−0.5730−0.9531−1.1102−0.1287−0.088122.6726−64.92020.27520.48710.54490.08140.0072−16.926150.43790.09930.15530.1884−0.0423−0.00478.02043.4236,and P=diagP1,P2,P3, where(73)P1=1.4434−0.4099−0.3575−0.40991.1014−0.1331−0.3575−0.13311.3827,P2=1.8096−0.1693−0.16931.5437,P3=1.4530−0.0737−0.07371.2896.
On the basis of Theorem 1, the network (70) with the input um,t∈ℝ7 and output ym,t∈ℝ9 as described in (12) realizes output-strictly passivity. Figure 1 displays the simulation results.Figure 1
Change processes ofzim,t, uim,t and yim,t in network (70).Case 2.
ψ
i
l
,
ψ
^
i
l, and O^ are similar as those in Case 1. By exploiting the MATLAB Toolbox, the matrix P satisfying (38) and (39) can be computed as follows:(74)P=diagP1,P2,P3,where(75)P1=0.2165−0.0111−0.0091−0.01110.1750−0.0054−0.0091−0.00540.1493,P2=0.2567−0.0117−0.01170.1929,P3=0.1780−0.0118−0.01180.1520.
According to Theorem 2, the network (70) realizes synchronization. Figures 2–4 show the simulation results.Figure 2
Change processes ofzj1m,t in network (70).Figure 3
Change processes ofzj2m,t in network (70).Figure 4
Change processes ofzj3m,t in network (70).Example 2.
Consider the following CDDDRNNs:(76)∂wim,t∂t=Di∑σ=1ϱ∂2wim,t∂mσ2−Biwim,t+Aifiwim,t+Ji+Giuim,t+Ziφiwim,t−τit+∑j=13cijHijwjm,t,in which i=1,2,3,ξ1=3,ξ2=ξ3=2,Ω=m−0.5<m<0.5,fliυ=φliυ=υ+1−υ−1/4, l=1,2,3,τit=0.02i−0.04ie−t, τ1=0.02,δ1=0.04,τ2=0.04,δ2=0.08,τ3=0.06,δ3=0.12, and J1=0,0,0T,J2=J3=0,0T,u1m,t=3tcosπm,4tcosπm,6tcosπmT,u2m,t=6tcosπm,8tcosπmT,u3m,t=9tcosπm,12tcosπmT. The matrices C=cij3×3,Gi,Hij,K1i,andK2i are chosen as follows: (77)G1=0.20.10.40.30.40.20.20.30.1,G2=0.20.30.40.5,G3=0.30.40.20.3,H11=0.20.30.40.30.20.10.20.30.3,H12=0.20.40.40.20.30.3,H13=0.30.60.30.40.20.5,H21=0.20.30.40.40.50.3,H22=0.30.40.50.6,H23=0.30.40.50.4,H31=0.30.20.30.40.30.2,H32=0.30.40.30.4,H33=0.30.40.40.2,K11=0.20.10.20.20.30.10.20.50.3,K12=0.30.40.40.50.10.3,K13=0.40.20.20.30.30.2,K21=0.40.50.30.10.30.20.30.40.1,K22=0.20.30.30.20.30.1,K23=0.30.20.40.50.30.4,C=0.2−0.80.5−0.70.10.20.40.3−0.6.
The parametersDi=diagd1i,d2i,⋯,dξii,Bi=diagb1i,b2i,⋯,bξii,Ai=aghiξi×ξi,andZi=zghiξi×ξii=1,2,3 in network (76) can be changed in the precisions given as follows:(78)DP≔Di=diagdsi:D¯i≤Di≤D¯i,0<0.4is≤dsi≤0.6is,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Di∈DP;BP≔Bi=diagbsi:B¯i≤Bi≤B¯i,0<1.2is≤bsi≤1.6is,i=1,2,⋯,N,s=1,2,⋯,ξi,∀Bi∈BP;AP≔Ai=aghiξi×ξi:1g+h+0.2i≤aghi≤1g+h+0.3i,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Ai∈AP;ZP≔Zi=zghiξi×ξi:1g+h+0.3i≤zghi≤1g+h+0.4i,i=1,2,⋯,N,g=1,2,⋯,ξi,h=1,2,⋯,ξi,∀Zi∈ZP.Case 1.
Obviously,ψil=ψ^il=0.5 and the equilibrium point of an isolated node of network (76) is O^=0,0,0,0,0,0,0T. It is not difficult to calculate the matrices P,Q, and M1 satisfying (54) as follows:(79)M1=1.2297−0.0937−0.11830.03180.04130.0159−0.0142−0.0103−0.0115−0.09371.2081−0.11950.04100.05300.0217−0.0207−0.0166−0.0171−0.1183−0.11951.17760.06220.08030.0340−0.0330−0.0281−0.02770.03180.04100.06221.2876−0.01920.0102−0.0097−0.0074−0.00790.04130.05300.0803−0.01921.27100.0095−0.0124−0.0093−0.01010.01590.02170.03400.01020.00951.3233−0.0062−0.0049−0.0051−0.0142−0.0207−0.0330−0.0097−0.0124−0.00621.35340.04550.0449−0.0103−0.0166−0.0281−0.0074−0.0093−0.00490.04551.35850.0431−0.0115−0.0171−0.0277−0.0079−0.0101−0.00510.04490.04311.3386,Q=0.99040.01970.4052−0.0358−0.00440.0347−0.0164−1.75631.42522.80330.09460.0507−0.2257−0.00201.36500.3042−1.71680.04070.03570.0309−0.0368−0.0357−0.1531−0.0033−1.48613.93520.0425−0.07180.09290.36810.01931.0186−2.3689−0.10920.0390−0.0214−0.21210.04122.51310.27590.0837−0.0048−0.0274−0.04980.0196−0.0190−0.06304.6554−0.9467−0.01330.12130.0760−0.0019−0.02403.5739−0.05270.0312−0.1258−0.10580.00450.0832−6.84132.9637,and P=diagP1,P2,P3, where(80)P1=0.28420000.28860000.1766,P2=0.2473000.1490,P3=0.1724000.0778.
In terms of Theorem 3, network (76) with the input um,t∈ℝ7 and output ym,t∈ℝ9 as described in (12) under the given parameters defined in (78) realizes output-strictly passivity. Figure 5 displays the simulation results.Figure 5
Change processes ofzim,t, uim,t and yim,t in network (76).Case 2.
ψ
i
l
,
ψ
^
i
l, and O^ are the same as those in Case 1. By exploiting the MATLAB Toolbox, the matrices P satisfying (68) can be computed as follows:(81)P=diagP1,P2,P3,where(82)P1=0.18760000.10070000.0677,P2=0.1015000.0512,P3=0.0672000.0340.
According to Theorem 4, network (76) under the given ranges of parameters defined in (78) achieves robust synchronization. Figures 6–8 show the simulation results.Figure 6
Change processes ofzj1m,t in network (76).Figure 7
Change processes ofzj2m,t in network (76).Figure 8
Change processes ofzj3m,t in network (76).Remark 4.
Section3 is devoting to investigating the synchronization and passivity of CDDDRDNNs. First, the network model of CDDDRDNNs is presented in Section 3.1. In Section 3.2, the passivity of CDDDRDNNs with certain parameters is studied. Moreover, we establish some adequate conditions to ensure the network being output-strictly passive in Theorem 1, passive in Corollary 1, and input-strictly passive in Corollary 2, respectively. Then, in Theorem 2 of Section 3.3, a synchronization criterion is obtained for the considered network. Because the precise values of parameters are difficult to acquire because of noises of environment and equipment limitations, we address the problems of robust synchronization and passivity for CDDDRDNNs with parametric uncertainties in Section 4. The main distinction on two kinds of CDDDRDNNs in Section 3 and Section 4 is whether Di,Bi,Ai,andZi in the network model of CDDDRDNNs is uncertain or not. More precisely, in Section 4.1, we give the CDDDRDNN model with parametric uncertainties firstly. After that, a robust output-strict passivity criterion is proposed for CDDDRDNNs with parameter uncertainties in Theorem 3 of Section 4.2. Meanwhile, we establish the related robust passivity condition in Corollary 3 and robust input-strict passivity condition in Corollary 4, respectively. Then, a robust synchronization criterion is obtained for the considered network in Theorem 4 of Section 4.3. In Section 5, we select the most intricate output-strict passivity condition for verifying the validity of the obtained passivity results. Actually, the related passivity or input-strict passivity condition can be illustrated similarly. In order to avoid repetition, we omit the simulation results for showing the effectiveness of the obtained passivity results in corollaries. Therefore, in Case 1 and Case 2 of Example 1, the correctness of the output-strict passivity and synchronization conditions in Theorems 1 and 2 for CDDDRDNNs (70) with parametric certainties are demonstrated, respectively. Similarly, in Case 1 and Case 2 of Example 2, the robust output-strict passivity criterion and robust synchronization criterion in Theorem 3 and 4 for CDDDRDNNs (76) with uncertain parameters described by (78) are demonstrated, respectively.


## 6. Conclusion

The synchronization and passivity of CDDDRDNNs with and without parameter uncertainties have been investigated in this paper. First, several new criteria for CDDDRDNNs with parametric certainties have been derived to guarantee the passivity and synchronization by taking advantage of the Lyapunov functional method. Second, we have also studied the problems of robust synchronization and robust passivity for CDDDRDNNs with uncertain parameters. Third, two numerical examples have been provided to display the effectiveness of the obtained passivity and synchronization results. In our future work, it would be very interesting to study the pinning adaptive passivity and synchronization of CDDDRDNNs.



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