# Passivity Analysis of Coupled Stochastic Neural Networks with Multiweights

Authors: Min Cao; Xun-Wu Yin; Wen-He Song; Xue-Mei Sun; Cheng-Dong Yang; Shun-Yan Ren
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we devote to the investigation of passivity in two types of coupled stochastic neural networks (CSNNs) with multiweights and incompatible input and output dimensions. First, some new definitions of passivity are proposed for stochastic systems that may have incompatible input and output dimensions. By utilizing stochastic analysis techniques and Lyapunov functional method, several sufficient conditions are respectively developed for ensuring that CSNNs without and with multiple delay couplings can realize passivity. Besides, the synchronization criteria for CSNNs with multiweights are established by employing the results of output-strictly passivity. Finally, two simulation examples are given to illustrate the validity of the theoretical results.


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## Body

## 1. Introduction

In recent decades, neural networks (NNs) have potential applications in the image encryption, pseudorandom number generators, optimization, and other areas [1–3], which depend on the dynamical behaviors of NNs including stability and passivity. Therefore, the stability [4–8] and passivity [9–14] for various NNs have received special attention in recent years. Mou et al. [4] considered the asymptotic stability problem for Hopfield NNs with time delay via combining the Lyapunov functional and delay fractioning approach. Yang et al. [6] discussed the stability for a kind of NNs with time-varying delays and gave several delay-dependent stability conditions by taking advantage of integral inequality. In [9], a class of NNs with time-varying delays and parameter uncertainties was took into account, and some exponential passivity criteria were established by exploiting weighted integral inequalities. Xiao et al. [11] studied the passivity for a type of memristive NNs with inertial term, obtained some criteria of asymptotic stability by utilizing the passivity, and discussed the case that parameters are uncertain but bounded.As it is known to all, stochastic perturbations are unavoidable in the implementation of NNs and may cause undesirable dynamical behaviors in NNs [15, 16]. Therefore, the dynamical behaviors including the stability [17–21] and passivity [22–27] have been widely investigated by numerous researchers for NNs with stochastic perturbations in recent years. In [18], several sufficient conditions on mean square stability for stochastic neural networks (SNNs) with local impulses were derived by using the mathematical induction method. Yang and Li [20] coped with the stability problem for switched SNNs with parameter uncertainties, and derived several conditions to guarantee the robust stability by utilizing the state-dependent switching method. In [24], the authors took into account one type of uncertain SNNs with distributed and discrete time-varying delays and gave some passivity criteria with the help of integral inequality technique. Nagamani et al. [25], respectively, discussed the passivity and dissipativity for Markovian jump stochastic NNs with two types of time-varying delays and obtained several delay-dependent passivity and dissipativity criteria by taking a suitable Lyapunov functional.Coupled neural networks (CNNs) comprised of a number of NNs have tremendous potential applications in many areas of engineering [28–30]. Hence, the dynamical behaviors of CNNs have attracted much attention; especially, the passivity [31–35] and synchronization [36–40] for many types of CNNs have been deeply discussed. In [34], the authors not only obtained several passivity criteria for the directed CNNs based on the developed adaptive control strategies but also discussed the case that topologies are undirected. Qi et al. [37] derived several global exponential synchronization criteria for quaternion-valued CNNs with impulses by exploiting Lyapunov functional and matrix inequalities. In [38], the authors considered CNNs with mixed impulses and developed some exponential synchronization conditions for the proposed network model through using delayed impulsive differential inequality. Particularly, a number of authors have discussed the dynamical behavior of coupled SNNs (CSNNs) in recent years [41–45]. Chen et al. [44] utilized the adaptive feedback controller to deal with the exponential synchronization for CSNNs. In [45], the authors respectively employed time-triggered and event-triggered impulsive control methods to investigate the synchronization of discrete time CSNNs with multidelays. Unfortunately, the passivity of CSNNs has not yet been investigated.It should be pointed out that the results in [31–45] only focused on CNNs with single weight. Considering the diversity of influencing factors, some researchers discussed the synchronization and passivity for multiweighted CNNs (MWCNNs) [46–48]. In [46], the authors proposed MWCNNs without and with coupling delays and dealt with the passivity and synchronization for these network models by employing the impulsive control method. Wang and Zhao [48] not only discussed the passivity for MWCNNs by utilizing the designed proportional-integral and proportional-derivative controllers but also derived several synchronization criteria by virtue of output-strict passivity. However, the passivity and synchronization for multiweighted CSNNs (MWCSNNs) have not been investigated.In this paper, the passivity for two types of MWCSNNs with incompatible input and output dimensions is investigated. The main contributions have three aspects. First, we present several new definitions of passivity for stochastic systems with incompatible dimensions of output and input. Second, some sufficient conditions to ensure the passivity of MWCSNNs are obtained by taking advantage of the Lyapunov functional method and stochastic analysis techniques, and a synchronization criterion is also developed by utilizing the result of output-strictly passivity. Third, we further address the passivity and synchronization for CSNNs with multiple delay couplings (CSNNMDCs).


## 2. Preliminaries

### 2.1. Notations

Ω,ℱ,ℱtt≥0,P is a complete probability space with the natural filtration ℱtt≥0 satisfying the usual conditions. C1,2ℝ+×ℝn;ℝ+ represents the family of all nonnegative functions Vt,κt on ℝ+×ℝn, which are once differentiable in t and twice continuously differentiable in κt. Tr⋅ stands for the trace of a matrix. P≥0 is used to denote a symmetric semipositive definite matrix. λM⋅ and λm⋅, respectively, denote the maximum and minimum eigenvalue of a real symmetric matrix.


### 2.2. Lemmas

Lemma 1.
(Ito^ formula, see [49]). A stochastic system can be described by(1)dκt=ft,κtdt+gt,κtdwt,in which κt∈ℝn represents the state of system, f⋅:ℝ+×ℝn⟶ℝn is continuous nonlinear function, g⋅:ℝ+×ℝn⟶ℝn×n is noise intensity function, and wt is an n-dimensional Brownian motion (Wiener process) defined on Ω,ℱ,ℱtt≥0,P.For anyVt,κt∈C1,2ℝ+×ℝn;ℝ+, define the operator ℒV for system (1) as follows:(2)ℒVt,κt=Vtt,κt+Vκt,κtft,κt+12TrgTt,κtVκκt,κtgt,κt,where Vtt,κt=∂Vt,κt/∂t,Vκt,κt=∂Vt,κt/∂κ1,∂Vt,κt/∂κ2,…,∂Vt,κt/∂κn,Vκκt,κt=∂2Vt,κt/∂κiκjn×n.IfVt,κt∈C1,2ℝ+×ℝn;ℝ+, one has(3)EVt,κt=EVt0,κt+E∫t0tℒVt,κtdt,for all t>t0≥0.Lemma 2 (see [50]).
For any matricesM∈ℝm×n and 0≤P∈ℝm×m, one obtains(4)TrMTPM≤λMPTrMTM.


### 2.3. Definitions

Definition 1.
A stochastic system with inputβt∈ℝp and output ηt∈ℝq is passive if(5)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p and S is a nonnegative function.Definition 2.
A stochastic system with inputβt∈ℝp and output ηt∈ℝq is input-strictly passive if(6)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0+E∫ϑ0ϑpβTsA1βsds,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p,0<A1∈ℝp×p, and S is a nonnegative function.Definition 3.
A stochastic system inputβt∈ℝp and output ηt∈ℝq is output-strictly passive if(7)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0+E∫ϑ0ϑpηTsA2ηsds,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p,0<A2∈ℝq×q, and S is a nonnegative function.


## 2.1. Notations

Ω,ℱ,ℱtt≥0,P is a complete probability space with the natural filtration ℱtt≥0 satisfying the usual conditions. C1,2ℝ+×ℝn;ℝ+ represents the family of all nonnegative functions Vt,κt on ℝ+×ℝn, which are once differentiable in t and twice continuously differentiable in κt. Tr⋅ stands for the trace of a matrix. P≥0 is used to denote a symmetric semipositive definite matrix. λM⋅ and λm⋅, respectively, denote the maximum and minimum eigenvalue of a real symmetric matrix.


## 2.2. Lemmas

Lemma 1.
(Ito^ formula, see [49]). A stochastic system can be described by(1)dκt=ft,κtdt+gt,κtdwt,in which κt∈ℝn represents the state of system, f⋅:ℝ+×ℝn⟶ℝn is continuous nonlinear function, g⋅:ℝ+×ℝn⟶ℝn×n is noise intensity function, and wt is an n-dimensional Brownian motion (Wiener process) defined on Ω,ℱ,ℱtt≥0,P.For anyVt,κt∈C1,2ℝ+×ℝn;ℝ+, define the operator ℒV for system (1) as follows:(2)ℒVt,κt=Vtt,κt+Vκt,κtft,κt+12TrgTt,κtVκκt,κtgt,κt,where Vtt,κt=∂Vt,κt/∂t,Vκt,κt=∂Vt,κt/∂κ1,∂Vt,κt/∂κ2,…,∂Vt,κt/∂κn,Vκκt,κt=∂2Vt,κt/∂κiκjn×n.IfVt,κt∈C1,2ℝ+×ℝn;ℝ+, one has(3)EVt,κt=EVt0,κt+E∫t0tℒVt,κtdt,for all t>t0≥0.Lemma 2 (see [50]).
For any matricesM∈ℝm×n and 0≤P∈ℝm×m, one obtains(4)TrMTPM≤λMPTrMTM.


## 2.3. Definitions

Definition 1.
A stochastic system with inputβt∈ℝp and output ηt∈ℝq is passive if(5)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p and S is a nonnegative function.Definition 2.
A stochastic system with inputβt∈ℝp and output ηt∈ℝq is input-strictly passive if(6)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0+E∫ϑ0ϑpβTsA1βsds,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p,0<A1∈ℝp×p, and S is a nonnegative function.Definition 3.
A stochastic system inputβt∈ℝp and output ηt∈ℝq is output-strictly passive if(7)E∫ϑ0ϑpηTsFβsds≥ESϑp−ESϑ0+E∫ϑ0ϑpηTsA2ηsds,for any ϑp,ϑ0∈ℝ+ and ϑp≥ϑ0, in which F∈ℝq×p,0<A2∈ℝq×q, and S is a nonnegative function.


## 3. Passivity for MWCSNNs

### 3.1. Network Model

The MWCSNNs in this paper is considered as follows:(8)dκzt=−Dκzt+Gfκzt+B+∑n=1s∑h=1MbnCzhnΓnκht+Hβztdt+σκztdωt,z=1,2,…,M,where n=1,2,…,s, G=Gzjm×m∈ℝm×m,Gzj∈ℝ, represents the strength of the j th neuron on the z th neuron, H∈ℝm×p is the known matrix, κzt=κz1t,κz2t,…,κzmtT∈ℝm is the state vector of the z th node, βzt∈ℝp represents the external input, 0<D=diagd1,d2,…,dm∈ℝm×m,0<dz∈ℝ, represents the rate with which the z th neuron will reset its potential to the resting state when disconnected from the network and external input, B=B1,B2,…,BmT∈ℝm,fκzt=f1κz1t,f2κz2t,…,fmκzmtT∈ℝm,0<bn∈ℝ, denotes coupling strength, and Cn=CzhnM×M∈ℝM×M represents the outer coupling matrix, where Czhn satisfies the following conditions: if there exists a connection between nodes z and h, then ℝ∋Czhn=Chzn>0z≠h, otherwise, ℝ∋Czhn=Chzn=0, and Czzn=−∑h=1h≠zMCzhn,z=1,2,…,M; 0<Γn=diagΓ1n,Γ2n,…,Γmn∈ℝm×m denotes the inner coupling matrix; ωt=ω1t,ω2t,…,ωmtT∈ℝm is an m-dimensional Brownian motion defined on Ω,ℱ,ℱtt≥0,P; σκzt=diagσκz1t,σκz2t,…,σκzmt∈ℝm×m is the noise intensity matrix.In this paper, the following assumptions are made.Assumption 1.
fl⋅l=1,2,…,m satisfies(9)flj1−flj2≤ξlj1−j2,for any j1,j2∈ℝ, where ξl>0. Let ξ=diagξ12,ξ22,…,ξm2∈ℝm×m.Assumption 2.
There exists a positive constantμ such that σ⋅ satisfies the following inequality:(10)Trσk1−σk2Tσk1−σk2≤μ2k1−k22,for any k1,k2∈ℝn.Remark 1.
On the one hand, the passivity for various CNNs has been investigated and some meaningful results have been obtained [31–35]. However, the passivity of CSNNs has not yet been discussed. On the other hand, some researchers have dealt with the synchronization problem for CSNNs [41–45]. Given that passivity has been developed as a powerful tool to solve the synchronization problem of CNNs, the investigation on synchronization for CSNNs based on the passivity is apparently very valuable. Regrettably, the result about this topic has not yet been reported.


### 3.2. Passivity Analysis

Suppose thatst=s1t,s2t,…,smtT∈ℝm is an arbitrary desired solution of the isolated node of system (8), then it satisfies(11)dst=−Dst+Gfst+Bdt+σstdωt.Lettingαzt=αz1t,αz2t,…,αzmtT=κzt−st, we can obtain from (8) and (11) that(12)dαzt=−Dαzt+Gfκzt−Gfst+∑n=1s∑h=1MbnCzhnΓnκht−st+Hβztdt+σκzt−σstdωt.The output vectorηzt∈ℝq of network (12) is defined as follows:(13)ηzt=Z1αzt+Z2βzt,where Z1∈ℝq×mandZ2∈ℝq×p.DefineD^=diagD,D,…,D∈ℝMm×Mm,G^=diagG,G,…,G∈ℝMm×Mm,H^=diagH,H,…,H∈ℝMm×Mp,Z^1=diagZ1,Z1,…,Z1∈ℝMq×Mm,Z^2=diagZ2,Z2,…,Z2∈ℝMq×Mp,ξ^=diagξ,ξ,…,ξ∈ℝMm×Mm, and ω^t=ωTt,ωTt,…,ωTtT∈ℝMm.Take(14)αt=α1Tt,α2Tt,…,αMTtT,βt=β1Tt,β2Tt,…,βMTtT,ηt=η1Tt,η2Tt,…,ηMTtT.According to (12), we have(15)dαt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαtdt+σ^tdω^t,ηt=Z^1αt+Z^2βt,where σ^t=IM⊗σκt−σst.Theorem 1.
Network (15) is passive if there exist F∈ℝMq×Mp, and 0<P^∈ℝMm×Mm satisfying(16)W1E1E1TM1≤0,where W1=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^,E1=P^H^−Z^1TF/2,M1=−Z^2TF+FTZ^2/2, and q=λMP^.Proof.
For convenience, we denote(17)yt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt,
then network (15) can be rewritten as(18)dαt=ytdt+σ^tdw^t.
Choose the following Lyapunov functional for network (15):(19)V1t=αTtP^αt.
In light of Lemma1, we can obtain(20)ℒV1t=Vt1t+Vα1tyt+12Trσ^TtVαα1tσ^t=2αTtP^−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt+Trσ^TtP^σ^t.
Obviously,(21)2αTtP^G^fκt−fst≤αTtP^G^G^TP^+ξ^αt.
From Lemma2 and Assumption 2, we have(22)Trσ^TtP^σ^t≤q∑z=1MTrσκzt−σstTσκzt−σst≤q∑z=1Mμ2κzt−st2=qμ2αt2.
Furthermore,(23)2αTtP^∑n=1sbnCn⊗Γnαt=αTt∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt.
By (20)–(23), one obtains(24)ℒV1t≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt.
From (24), we have(25)ℒV1t−ηTtFβt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt−αTtZ^1T+βTtZ^2TFβt=ζTtW1E1E1TM1ζt,where ζt=αTt,βtTT.
By (16), one obtains(26)ηTtFβt≥ℒV1t.
From (26), we obtain(27)∫t0tpηTtFβtdt≥∫t0tpℒV1tdt.
According to (27) and Lemma 1, we can acquire(28)E∫t0tpηTtFβtdt≥EV1tp−EV1t0.Theorem 2.
Network (15) is input-strictly passive if there exist matrices F∈ℝMq×Mp,0<A1∈ℝMp×Mp, and 0<P^∈ℝMm×Mm satisfying(29)W1E1E1TM2≤0,where W1 and E1 have the same meanings as these in Theorem 1 and M2=M1+A1.Proof.
Construct the sameV1t as (19) for network (15), and we can easily obtain(30)ℒV1t−ηTtFβt+βTtA1βt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt+βTtA1βt−αTtZ^1T+βTtZ^2TFβt=ζTtW1E1E1TM2ζt.
From (29), we have(31)ηTtFβt≥ℒV1t+βTtA1βt.
Similarly, we can derive(32)E∫t0tpηTtFβtdt≥EV1tp−EV1t0+E∫t0tpβTtA1βtdt.Theorem 3.
Network (15) is output-strictly passive if there exist matrices F∈ℝMq×Mp,0<A2∈ℝMq×Mq, and 0<P^∈ℝMm×Mm satisfying(33)W2E2E2TM3≤0,where W2=W1+Z^1TA2Z^1,E2=E1+Z^1TA2Z^2, and M3=M1+Z^2TA2Z^2.Proof.
We select the sameV1t as (19) for network (15), and we can easily obtain(34)ℒV1t−ηTtFβt+ηTtA2ηt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt+αTtZ^1T+βTtZ^2TA2Z^1αt+Z^2βtβt−αTtZ^1T+βTtZ^2TFβt=ζTtW2E2E2TM3ζt.
From (47), one has(35)ηTtFβt≥ℒV1t+ηTtA2ηt.
Similarly, we can derive(36)E∫t0tpηTtFβtdt≥EV1tp−EV1t0+E∫t0tpηTtA2ηtdt.


### 3.3. Synchronization in Passive MWCSNNs

Definition 4.
The network (8) with input βzt=0z=1,2,…,M achieves synchronization if(37)limt⟶+∞Eκzt−st=0,z=1,2,…,M.Theorem 4.
If network (15) is output-strictly passive with regard to storage function Kt=V1t/2 and Z1∈ℝm×m is nonsingular, then MWCSNNs (8) can achieve synchronization.Proof.
If network (15) is output-strictly passive with respect to storage function Kt, then there exist matrices ℝMm×Mm∋A2>0 and F∈ℝMm×Mm such that(38)E∫tt+δηTsFβsds≥EKt+δ−EKt+E∫tt+δηTsA2ηsds,for any t∈ℝ+ and δ>0. Then, we can easily obtain(39)EKt+δ−Ktε≤E∫tt+δηTsFβsdsε−∫tt+δηTsA2ηsdsε.
By taking limitδ⟶0 in (39), one has(40)EK˙t≤EηTsFβs−ηTsA2ηs.
Lettingβzt=0z=1,2,…,M, we can get from (40) that(41)EK˙t≤−EαTtZ^1TA2Z1^αt≤−λmZ1TZ1λmA2Eαt2.
From (41) and definition of V1t, we can get that limt⟶+∞EKt exists. Therefore, we can conclude that(42)limt⟶+∞Eαt2=θ≥0.
Suppose that(43)limt⟶+∞Eαt2=θ>0.
Then, there obviously exists0<t∗∈ℝ such that(44)Eαt2>θ2,t≥t∗.
By (41) and (44), we have(45)EK˙t<−λMZ1TZ1λMA2θ2,t≥t∗.
From (45), we derive(46)E−Kt∗≤EK+∞−Kt∗=∫t∗+∞EK˙tdt<−∫t∗+∞λMZ1TZ1λMA2θ2=−∞,which results in a contradiction.
Hence,limt⟶+∞Eαt=0. That is, network (8) realizes synchronization.
The following conclusion can be obtained from Theorems3 and 4.Corollary 1.
Network (8) achieves synchronization if there exist matrices F∈ℝMm×Mm,0<A2∈ℝMm×Mm, and 0<P^∈ℝMm×Mm satisfying(47)W2E2E2TM3≤0,where W2=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^+Z^1TA2Z1^,E2=P^H^−Z^1TF/2+Z^1TA2Z^2,M3=−Z^2TF+FTZ^2/2+Z^2TA2Z^2, and q=λMP^.


## 3.1. Network Model

The MWCSNNs in this paper is considered as follows:(8)dκzt=−Dκzt+Gfκzt+B+∑n=1s∑h=1MbnCzhnΓnκht+Hβztdt+σκztdωt,z=1,2,…,M,where n=1,2,…,s, G=Gzjm×m∈ℝm×m,Gzj∈ℝ, represents the strength of the j th neuron on the z th neuron, H∈ℝm×p is the known matrix, κzt=κz1t,κz2t,…,κzmtT∈ℝm is the state vector of the z th node, βzt∈ℝp represents the external input, 0<D=diagd1,d2,…,dm∈ℝm×m,0<dz∈ℝ, represents the rate with which the z th neuron will reset its potential to the resting state when disconnected from the network and external input, B=B1,B2,…,BmT∈ℝm,fκzt=f1κz1t,f2κz2t,…,fmκzmtT∈ℝm,0<bn∈ℝ, denotes coupling strength, and Cn=CzhnM×M∈ℝM×M represents the outer coupling matrix, where Czhn satisfies the following conditions: if there exists a connection between nodes z and h, then ℝ∋Czhn=Chzn>0z≠h, otherwise, ℝ∋Czhn=Chzn=0, and Czzn=−∑h=1h≠zMCzhn,z=1,2,…,M; 0<Γn=diagΓ1n,Γ2n,…,Γmn∈ℝm×m denotes the inner coupling matrix; ωt=ω1t,ω2t,…,ωmtT∈ℝm is an m-dimensional Brownian motion defined on Ω,ℱ,ℱtt≥0,P; σκzt=diagσκz1t,σκz2t,…,σκzmt∈ℝm×m is the noise intensity matrix.In this paper, the following assumptions are made.Assumption 1.
fl⋅l=1,2,…,m satisfies(9)flj1−flj2≤ξlj1−j2,for any j1,j2∈ℝ, where ξl>0. Let ξ=diagξ12,ξ22,…,ξm2∈ℝm×m.Assumption 2.
There exists a positive constantμ such that σ⋅ satisfies the following inequality:(10)Trσk1−σk2Tσk1−σk2≤μ2k1−k22,for any k1,k2∈ℝn.Remark 1.
On the one hand, the passivity for various CNNs has been investigated and some meaningful results have been obtained [31–35]. However, the passivity of CSNNs has not yet been discussed. On the other hand, some researchers have dealt with the synchronization problem for CSNNs [41–45]. Given that passivity has been developed as a powerful tool to solve the synchronization problem of CNNs, the investigation on synchronization for CSNNs based on the passivity is apparently very valuable. Regrettably, the result about this topic has not yet been reported.


## 3.2. Passivity Analysis

Suppose thatst=s1t,s2t,…,smtT∈ℝm is an arbitrary desired solution of the isolated node of system (8), then it satisfies(11)dst=−Dst+Gfst+Bdt+σstdωt.Lettingαzt=αz1t,αz2t,…,αzmtT=κzt−st, we can obtain from (8) and (11) that(12)dαzt=−Dαzt+Gfκzt−Gfst+∑n=1s∑h=1MbnCzhnΓnκht−st+Hβztdt+σκzt−σstdωt.The output vectorηzt∈ℝq of network (12) is defined as follows:(13)ηzt=Z1αzt+Z2βzt,where Z1∈ℝq×mandZ2∈ℝq×p.DefineD^=diagD,D,…,D∈ℝMm×Mm,G^=diagG,G,…,G∈ℝMm×Mm,H^=diagH,H,…,H∈ℝMm×Mp,Z^1=diagZ1,Z1,…,Z1∈ℝMq×Mm,Z^2=diagZ2,Z2,…,Z2∈ℝMq×Mp,ξ^=diagξ,ξ,…,ξ∈ℝMm×Mm, and ω^t=ωTt,ωTt,…,ωTtT∈ℝMm.Take(14)αt=α1Tt,α2Tt,…,αMTtT,βt=β1Tt,β2Tt,…,βMTtT,ηt=η1Tt,η2Tt,…,ηMTtT.According to (12), we have(15)dαt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαtdt+σ^tdω^t,ηt=Z^1αt+Z^2βt,where σ^t=IM⊗σκt−σst.Theorem 1.
Network (15) is passive if there exist F∈ℝMq×Mp, and 0<P^∈ℝMm×Mm satisfying(16)W1E1E1TM1≤0,where W1=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^,E1=P^H^−Z^1TF/2,M1=−Z^2TF+FTZ^2/2, and q=λMP^.Proof.
For convenience, we denote(17)yt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt,
then network (15) can be rewritten as(18)dαt=ytdt+σ^tdw^t.
Choose the following Lyapunov functional for network (15):(19)V1t=αTtP^αt.
In light of Lemma1, we can obtain(20)ℒV1t=Vt1t+Vα1tyt+12Trσ^TtVαα1tσ^t=2αTtP^−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt+Trσ^TtP^σ^t.
Obviously,(21)2αTtP^G^fκt−fst≤αTtP^G^G^TP^+ξ^αt.
From Lemma2 and Assumption 2, we have(22)Trσ^TtP^σ^t≤q∑z=1MTrσκzt−σstTσκzt−σst≤q∑z=1Mμ2κzt−st2=qμ2αt2.
Furthermore,(23)2αTtP^∑n=1sbnCn⊗Γnαt=αTt∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt.
By (20)–(23), one obtains(24)ℒV1t≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt.
From (24), we have(25)ℒV1t−ηTtFβt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt−αTtZ^1T+βTtZ^2TFβt=ζTtW1E1E1TM1ζt,where ζt=αTt,βtTT.
By (16), one obtains(26)ηTtFβt≥ℒV1t.
From (26), we obtain(27)∫t0tpηTtFβtdt≥∫t0tpℒV1tdt.
According to (27) and Lemma 1, we can acquire(28)E∫t0tpηTtFβtdt≥EV1tp−EV1t0.Theorem 2.
Network (15) is input-strictly passive if there exist matrices F∈ℝMq×Mp,0<A1∈ℝMp×Mp, and 0<P^∈ℝMm×Mm satisfying(29)W1E1E1TM2≤0,where W1 and E1 have the same meanings as these in Theorem 1 and M2=M1+A1.Proof.
Construct the sameV1t as (19) for network (15), and we can easily obtain(30)ℒV1t−ηTtFβt+βTtA1βt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt+βTtA1βt−αTtZ^1T+βTtZ^2TFβt=ζTtW1E1E1TM2ζt.
From (29), we have(31)ηTtFβt≥ℒV1t+βTtA1βt.
Similarly, we can derive(32)E∫t0tpηTtFβtdt≥EV1tp−EV1t0+E∫t0tpβTtA1βtdt.Theorem 3.
Network (15) is output-strictly passive if there exist matrices F∈ℝMq×Mp,0<A2∈ℝMq×Mq, and 0<P^∈ℝMm×Mm satisfying(33)W2E2E2TM3≤0,where W2=W1+Z^1TA2Z^1,E2=E1+Z^1TA2Z^2, and M3=M1+Z^2TA2Z^2.Proof.
We select the sameV1t as (19) for network (15), and we can easily obtain(34)ℒV1t−ηTtFβt+ηTtA2ηt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^αt+2αTtP^H^βt+αTtZ^1T+βTtZ^2TA2Z^1αt+Z^2βtβt−αTtZ^1T+βTtZ^2TFβt=ζTtW2E2E2TM3ζt.
From (47), one has(35)ηTtFβt≥ℒV1t+ηTtA2ηt.
Similarly, we can derive(36)E∫t0tpηTtFβtdt≥EV1tp−EV1t0+E∫t0tpηTtA2ηtdt.


## 3.3. Synchronization in Passive MWCSNNs

Definition 4.
The network (8) with input βzt=0z=1,2,…,M achieves synchronization if(37)limt⟶+∞Eκzt−st=0,z=1,2,…,M.Theorem 4.
If network (15) is output-strictly passive with regard to storage function Kt=V1t/2 and Z1∈ℝm×m is nonsingular, then MWCSNNs (8) can achieve synchronization.Proof.
If network (15) is output-strictly passive with respect to storage function Kt, then there exist matrices ℝMm×Mm∋A2>0 and F∈ℝMm×Mm such that(38)E∫tt+δηTsFβsds≥EKt+δ−EKt+E∫tt+δηTsA2ηsds,for any t∈ℝ+ and δ>0. Then, we can easily obtain(39)EKt+δ−Ktε≤E∫tt+δηTsFβsdsε−∫tt+δηTsA2ηsdsε.
By taking limitδ⟶0 in (39), one has(40)EK˙t≤EηTsFβs−ηTsA2ηs.
Lettingβzt=0z=1,2,…,M, we can get from (40) that(41)EK˙t≤−EαTtZ^1TA2Z1^αt≤−λmZ1TZ1λmA2Eαt2.
From (41) and definition of V1t, we can get that limt⟶+∞EKt exists. Therefore, we can conclude that(42)limt⟶+∞Eαt2=θ≥0.
Suppose that(43)limt⟶+∞Eαt2=θ>0.
Then, there obviously exists0<t∗∈ℝ such that(44)Eαt2>θ2,t≥t∗.
By (41) and (44), we have(45)EK˙t<−λMZ1TZ1λMA2θ2,t≥t∗.
From (45), we derive(46)E−Kt∗≤EK+∞−Kt∗=∫t∗+∞EK˙tdt<−∫t∗+∞λMZ1TZ1λMA2θ2=−∞,which results in a contradiction.
Hence,limt⟶+∞Eαt=0. That is, network (8) realizes synchronization.
The following conclusion can be obtained from Theorems3 and 4.Corollary 1.
Network (8) achieves synchronization if there exist matrices F∈ℝMm×Mm,0<A2∈ℝMm×Mm, and 0<P^∈ℝMm×Mm satisfying(47)W2E2E2TM3≤0,where W2=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗Γn+∑n=1sbnCn⊗ΓnP^+Z^1TA2Z1^,E2=P^H^−Z^1TF/2+Z^1TA2Z^2,M3=−Z^2TF+FTZ^2/2+Z^2TA2Z^2, and q=λMP^.


## 4. Passivity for CSNNMDCs

### 4.1. Network Model

The CSNNMDCs in this paper is considered as follows:(48)dκzt=−Dκzt+Gfκzt+B+∑n=1s∑h=1MbnCzhnΓnκht−τn+Hβztdt+σκztdωt,z=1,2,…,M,where τnn=1,2,…,s are coupling delays and κzt,βzt,fκzt,D,G,B,H,bn,Czhn, and Γn have the same meanings as those in Section 3.Suppose thatst=s1t,s2t,…,smtT∈ℝm is an arbitrary desired solution of the isolated node of system (48), then it satisfies(49)dst=−Dst+Gfst+Bdt+σstdωt.Lettingαzt=αz1t,αz2t,…,αzmtT=κzt−st, we can obtain from (48) and (49) that(50)dαzt=−Dαzt+Gfκzt−Gfst+∑n=1s∑j=1MbnCzjnΓnαjt−τn+Hβztdt+σκzt−σstdωt.The output vectorηzt∈ℝq of network (48) is defined as follows:(51)ηzt=Z1αzt+Z2βzt,where Z1∈ℝq×mandZ2∈ℝq×p.According to (50), we have(52)dαt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt−τndt+σ^tdω^t,ηt=Z1^αt+Z2^βt,where σ^t=IM⊗σκt−σst.Theorem 5.
Network (51) is passive if there exist F∈ℝMq×Mp,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(53)W3E3E3TM4≤0,where W3=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^n,E3=P^H^−Z^1TF/2,M4=−Z^2TF+FTZ^2/2, and q=λMP^.Proof.
For convenience, we denote(54)yt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt−τn,
then network (51) can be rewritten as(55)dαt=ytdt+σ^tdw^t.
Choose the following Lyapunov functional for network (51):(56)V2t=αTtP^αt+∑n=1sbn∫t−τntαThN^nαhdh.
In light of Lemma1, we can obtain(57)ℒV2t=Vt2t+Vα2tyt+12Trσ^TtVαα2tσ^t=∑n=1sbnαTtN^nαt+2αTtP^−D^αt+H^βt+G^fκt−fst+∑n=1sbnCn⊗Γnαt−τn−∑n=1sbnαTt−τnN^nαt−τn+Trσ^TtP^σ^t.
Moreover,(58)2αTtP^∑n=1sbnCn⊗Γnαt−τn⩽∑n=1sbnαTtP^Cn⊗ΓnN^n−1Cn⊗ΓnP^αt+∑n=1sbnαTt−τnN^nαt−τn.
From (57) and (58), we have(59)ℒV2t≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+μ2qIMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^nαt+2αTtP^H^βt.
From (59), we have(60)ℒV2t−ηTtFβt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+μ2qIMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^nαt+2αTtP^H^βt−αTtZ^1T+βTtZ^2TFβt=ζTtW3E3E3TM4ζt.
By (53), we can acquire(61)E∫t0tpηTtFβtdt≥EV2tp−EV2t0.
By employing similar proof methods in Theorem5, we can get the following conclusions.Theorem 6.
Network (51) is input-strictly passive if there exist matrices F∈ℝMq×Mp,0<A1∈ℝMp×Mp,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(62)W3E3E3TM5≤0,where W3 and E3 have the same meanings as those in Theorem 5 and M5=A1+M4.Theorem 7.
Network (51) is output-strictly passive if there exist matrices F∈ℝMq×Mp,0<A2∈ℝMq×Mq,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(63)W4E4E4TM6≤0,where W4=W3+Z^1TA2Z^1,E4=E3+Z^1TA2Z^2, and M6=M4+Z^2TA2Z^2.


### 4.2. Synchronization in Passive CSNNMDCs

Theorem 8.
The CSNNMDCs (48) can achieve synchronization if network (51) is output-strictly passive with regard to storage function Kt=V2t/2, and Z1∈ℝm×m is nonsingular.Proof.
The results can be easily obtained by employing similar proof method in the Theorem4.
The following conclusion can be obtained from Theorems7 and 8.Corollary 2.
Network (48) achieves synchronization if there exist matrices F∈ℝMm×Mm,0<A2∈ℝMm×Mm,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(64)W4E4E4TM6≤0,where W4=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^n+Z^1TA2Z^1,E4=P^H^−Z^1TF/2+Z^1TA2Z^2,M6=−Z^2TF+FTZ^2/2+Z^2TA2Z^2, and q=λMP^.Remark 2.
In this paper, two types of network models are proposed (see (8) and (48)), some sufficient conditions for ensuring the passivity of networks (8) and (48) are acquired by employing the stochastic analysis techniques and Lyapunov functional method (see Theorems 1–3 and Theorems 5–7), and several synchronization criteria for networks (8) and (48) are established in view of the output-strictly passivity (see Corollaries 1 and 2).


## 4.1. Network Model

The CSNNMDCs in this paper is considered as follows:(48)dκzt=−Dκzt+Gfκzt+B+∑n=1s∑h=1MbnCzhnΓnκht−τn+Hβztdt+σκztdωt,z=1,2,…,M,where τnn=1,2,…,s are coupling delays and κzt,βzt,fκzt,D,G,B,H,bn,Czhn, and Γn have the same meanings as those in Section 3.Suppose thatst=s1t,s2t,…,smtT∈ℝm is an arbitrary desired solution of the isolated node of system (48), then it satisfies(49)dst=−Dst+Gfst+Bdt+σstdωt.Lettingαzt=αz1t,αz2t,…,αzmtT=κzt−st, we can obtain from (48) and (49) that(50)dαzt=−Dαzt+Gfκzt−Gfst+∑n=1s∑j=1MbnCzjnΓnαjt−τn+Hβztdt+σκzt−σstdωt.The output vectorηzt∈ℝq of network (48) is defined as follows:(51)ηzt=Z1αzt+Z2βzt,where Z1∈ℝq×mandZ2∈ℝq×p.According to (50), we have(52)dαt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt−τndt+σ^tdω^t,ηt=Z1^αt+Z2^βt,where σ^t=IM⊗σκt−σst.Theorem 5.
Network (51) is passive if there exist F∈ℝMq×Mp,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(53)W3E3E3TM4≤0,where W3=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^n,E3=P^H^−Z^1TF/2,M4=−Z^2TF+FTZ^2/2, and q=λMP^.Proof.
For convenience, we denote(54)yt=−D^αt+G^fκt−fst+H^βt+∑n=1sbnCn⊗Γnαt−τn,
then network (51) can be rewritten as(55)dαt=ytdt+σ^tdw^t.
Choose the following Lyapunov functional for network (51):(56)V2t=αTtP^αt+∑n=1sbn∫t−τntαThN^nαhdh.
In light of Lemma1, we can obtain(57)ℒV2t=Vt2t+Vα2tyt+12Trσ^TtVαα2tσ^t=∑n=1sbnαTtN^nαt+2αTtP^−D^αt+H^βt+G^fκt−fst+∑n=1sbnCn⊗Γnαt−τn−∑n=1sbnαTt−τnN^nαt−τn+Trσ^TtP^σ^t.
Moreover,(58)2αTtP^∑n=1sbnCn⊗Γnαt−τn⩽∑n=1sbnαTtP^Cn⊗ΓnN^n−1Cn⊗ΓnP^αt+∑n=1sbnαTt−τnN^nαt−τn.
From (57) and (58), we have(59)ℒV2t≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+μ2qIMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^nαt+2αTtP^H^βt.
From (59), we have(60)ℒV2t−ηTtFβt≤αTt−P^D^−D^P^+P^G^G^TP^+ξ^+μ2qIMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^nαt+2αTtP^H^βt−αTtZ^1T+βTtZ^2TFβt=ζTtW3E3E3TM4ζt.
By (53), we can acquire(61)E∫t0tpηTtFβtdt≥EV2tp−EV2t0.
By employing similar proof methods in Theorem5, we can get the following conclusions.Theorem 6.
Network (51) is input-strictly passive if there exist matrices F∈ℝMq×Mp,0<A1∈ℝMp×Mp,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(62)W3E3E3TM5≤0,where W3 and E3 have the same meanings as those in Theorem 5 and M5=A1+M4.Theorem 7.
Network (51) is output-strictly passive if there exist matrices F∈ℝMq×Mp,0<A2∈ℝMq×Mq,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(63)W4E4E4TM6≤0,where W4=W3+Z^1TA2Z^1,E4=E3+Z^1TA2Z^2, and M6=M4+Z^2TA2Z^2.


## 4.2. Synchronization in Passive CSNNMDCs

Theorem 8.
The CSNNMDCs (48) can achieve synchronization if network (51) is output-strictly passive with regard to storage function Kt=V2t/2, and Z1∈ℝm×m is nonsingular.Proof.
The results can be easily obtained by employing similar proof method in the Theorem4.
The following conclusion can be obtained from Theorems7 and 8.Corollary 2.
Network (48) achieves synchronization if there exist matrices F∈ℝMm×Mm,0<A2∈ℝMm×Mm,0<P^∈ℝMm×Mm, and N^n=diagNn1,Nn2,…,NnM∈ℝMm×Mm,n=1,2,…,s, satisfying(64)W4E4E4TM6≤0,where W4=−P^D^−D^P^+P^G^G^TP^+ξ^+qμ2IMm+∑n=1sbnP^Cn⊗ΓnN^n−1Cn⊗ΓnP^+∑n=1sbnN^n+Z^1TA2Z^1,E4=P^H^−Z^1TF/2+Z^1TA2Z^2,M6=−Z^2TF+FTZ^2/2+Z^2TA2Z^2, and q=λMP^.Remark 2.
In this paper, two types of network models are proposed (see (8) and (48)), some sufficient conditions for ensuring the passivity of networks (8) and (48) are acquired by employing the stochastic analysis techniques and Lyapunov functional method (see Theorems 1–3 and Theorems 5–7), and several synchronization criteria for networks (8) and (48) are established in view of the output-strictly passivity (see Corollaries 1 and 2).


## 5. Numerical Examples

Example 1.
Take the following MWCSNNs into consideration:(65)dκzt=−Dκzt+Gfκzt+B+0.4∑h=16Czh1Γ1κht+0.2∑h=16Czh2Γ2κht+0.1∑h=16Czh3Γ3κht+Hβztdt+σκztdωt,where z=1,2,…,6,fij=0.25j+1−j−1,i=1,2,3,D=diag0.5,0.2,0.4,B=0,0,0T,Γ1=diag0.9,0.4,0.3,Γ2=diag0.8,0.3,0.4,Γ3=diag0.6,0.3,0.8,σκzt=diag0.2κz1t,0.4κz2t,0.3κz3t,(66)G=0.30.50.10.20.10.20.120.10.4,H=0.40.60.80.30.50.7,C1=0.80.200.10.20.30.2−0.60.10.20.1000.1−0.500.30.10.10.20−0.80.20.30.20.10.30.2−0.90.10.300.10.30.1−0.8,C2=−0.50.200.10.10.10.20.60.10.10.2000.1−0.800.30.40.10.10−0.60.10.30.10.20.30.1−0.80.10.100.40.30.1−0.9,C3=−0.90.400.20.20.10.4−0.80.20.10.1000.2−0.500.10.20.20.10−0.80.40.10.20.10.10.4−0.90.10.100.20.10.1−0.5.
Apparently,st=0,0,0T∈ℝ3 is a desired solution of an isolated node of network (65), and fi⋅ meets Assumption 1 with ξl=0.5.
Theηzt∈ℝ3 is selected as(67)ηzt=Z1αzt+Z2βzt,z=1,2,…,6,where βzt=0.6z∗sint,0.4z∗sintT,(68)Z1=0.30.10.50.80.50.40.40.20.7,Z2=0.70.60.80.50.20.3.Case 1: the following matricesF and P^ can be obtained:(69)F=I6⊗−0.48360.35681.5425−0.93710.08972.7229,P^=I6⊗0.6443−0.0057−0.0834−0.00570.7983−0.2136−0.0834−0.21361.1224.From Theorem1, network (65) is passive and Figures 1 and 2 display the simulation results.Case 2: the following matricesF,P^, and A2 that satisfy the condition of Theorem 3 can be obtained:(70)F=I6⊗41.664636.62494.1455−8.3129−30.5560−11.0012,P^=I6⊗1.95250.3124−1.60510.31243.8069−3.2750−1.6051−3.27506.8604,A2=I6⊗20.0009−0.5782−12.8691−0.57821.0321−0.9658−12.8691−0.965810.1673.
From Theorem4, network (65) is synchronized, Figure 3 demonstrates the effectiveness and correctness of the obtained results.Figure 1
αz1t,αz2t, and αz3t,z=1,2,…,6.
(a)(b)(c)Figure 2
ηzt,z=1,2,…,6.Figure 3
αz1t,αz2t, and αz3t,z=1,2,…,6.
(a)(b)(c)Example 2.
Take the following CSNNMDCs into consideration:(71)dκzt=−Dκzt+Gfκzt+B+0.2∑h=16Czh1Γ1κht−0.5+0.1∑h=16Czh2Γ2κht−0.2+0.5∑h=16Czh3Γ3κht−0.3+Hβztdt+σκztdωt,where z=1,2,…,6,fij=0.25j+1−j−1,i=1,2,3,D=diag0.8,0.9,0.7,B=0,0,0T,Γ1=diag0.5,0.3,0.2,Γ2=diag0.2,0.5,0.3,Γ3=diag0.3,0.2,0.5,σκzt=diag0.4κz1t,0.3κz2t,0.5κz3t,(72)G=0.10.30.10.20.10.30.10.20.1,H=0.20.10.50.30.10.4,C1=−0.60.20.100.20.10.2−0.80.10.20.10.20.10.1−0.700.30.200.20−0.50.20.10.20.10.30.2−0.800.10.20.20.10−0.6,C2=−0.40.200.100.10.2−0.50.100.2000.1−0.600.30.20.100−0.50.10.300.20.30.1−0.70.10.100.20.30.1−0.7,C3=−0.50.10.20.100.10.1−0.70.20.100.30.20.2−0.500.100.10.10−0.80.30.3000.10.3−0.50.10.10.300.30.1−0.8.
Apparently,st=0,0,0T∈ℝ3 is a desired solution of an isolated node of network (67), and fi⋅ satisfies Assumption 1 with ξl=0.5.
Theηzt∈ℝ3 is selected as(73)ηzt=Z1αzt+Z2βzt,z=1,2,…,6,where βzt=0.4z∗sint,0.5z∗sintT,(74)Z1=0.50.70.90.60.40.60.20.50.1,Z2=0.80.60.50.90.70.4.
The matricesN^1,N^2, and N^3 are chosen as, respectively,(75)N^1=I6⊗diag0.5,0.6,0.3,N^2=I6⊗diag0.3,0.4,0.2,N^3=I6⊗diag0.2,0.1,0.4.Case 1: the following matrices F and P^ can be obtained:(76)F=I6⊗1.7842−0.9653−2.67263.27073.1171−0.5937,P^=I6⊗1.4460−0.0495−0.0150−0.04951.3677−0.0831−0.0150−0.08311.6143.From Theorem5, network (67) is passive, and Figures 4 and 5 display the simulation results.Case 2: the following matrices F,P^, and A2 that satisfy the condition of Theorem 7 can be obtained.
(77)F=I6⊗1.6654−7.2114−5.473510.485312.45712.6646,P^=I6⊗1.8950−0.24640.1908−0.24642.3726−0.2827−0.1908−0.28271.5491,A2=I6⊗1.4514−1.7789−0.2735−1.77892.5172−0.2573−0.2735−0.25731.7342.From Theorem 8, network (67) is synchronized, and Figure 6 demonstrates the effectiveness and correctness of the obtained results.Figure 4
αz1t,αz2t, and αz3t,z=1,2,…,6.
(a)(b)(c)Figure 5
ηzt,z=1,2,…,6.Figure 6
αz1t,αz2t, and αz3t,z=1,2,…,6.
(a)(b)(c)Remark 3.
More recently, some research results on the dynamical behaviors of CSNNs have been obtained, but they all discussed the single weighted network models [41–45]. In this paper, we respectively consider the passivity and synchronization of MWCSNNs (65) and CSNNMDCs (67), which are apparently different from these network models considered in [41–45]. Figures 1 and 2 (Figures 4 and 5) respectively show the change tendencies of αz1t,αz2t,αz3t,z=1,2,…,6 and ηzt,z=1,2,…,6 for networks (65) and (67). From Figures 3 and 6, we can explicitly see that αz1t,αz2t, and αz3t,z=1,2,…,6, in networks (65) and (67), respectively, converge to 0 after 12s and 5s, which verify the correctness of the obtained synchronization criteria.


## 6. Conclusion

Two kinds of MWCSNNs models have been proposed, in which the dimension of output is incompatible with input. On the one hand, we have analyzed the passivity, input-strict passivity, and output-strict passivity for MWCSNNs by employing stochastic analysis techniques. Moreover, two synchronization criteria for MWCSNNs and CSNNMDCs have been derived on the basis of output-strict passivity. Finally, the correctness of the passivity and synchronization criteria has been verified through two numerical examples.



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