# Finite-Time Passivity of Stochastic Coupled Complex Networks

Authors: Xunwu Yin; Min Cao
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

The finite-time passivity problem is, respectively, investigated for stochastic coupled complex networks (SCCNs) with and without time-varying delay. Firstly, we present several new concepts about finite-time passivity in the sense of expectation on the basis of existing passivity definition. By designing appropriate controllers, the finite-time passivity of SCCNs with and without time-varying delay is obtained. In addition, the definition of finite-time synchronization in the sense of expectation is proposed. Under some sufficient conditions and designed controllers, finite-time passivity derives finite-time synchronization. Finally, two examples are given to demonstrate the effectiveness of finite-time passive and synchronization criteria.


---

## Body

## 1. Introduction

In the real world, complex networks can be seen everywhere such as food webs, communication networks, World Wide Web, and many others [1–3]. Due to various uncertainties in the actual system, complex network systems may be affected by noise. In recent years, the stability of stochastic systems has been extensively studied. At the same time, the synchronization and stability of stochastic complex networks have gradually become a topic of widespread concern for scholars in various fields [4–8].Passivity is one part of dissipativeness. The main property of passivity is keeping the systems internally stable. The passivity theory has been extensively applied in many fields such as stability, complexity, signal processing, chaos control, synchronization fuzzy control, and so on [9–12]. These are the main reasons why the passivity theory has been one of the most active research areas. In [13], the problem of passivity analysis was studied for discrete-time stochastic Markovian jump neural networks with both discrete and distributed delays. In [14], the problem of passivity analysis is investigated for a class of discrete-time stochastic neural networks with time-varying delays.It is well known that passivity theory can provide a powerful tool to analyze synchronization of complex networks. However, in many existing works, synchronization is defined over the infinite time interval. Most of the theoretical methods on the synchronization of complex networks can only realize the network [15] or exponential asymptotical synchronization [16] which guarantees that error tends to 0 when t tends to infinity. That is to say, achieving asymptotically stable convergence will be in infinite time. No further consideration has been given to the time and speed of synchronization. However, in practical engineering, people usually expect faster convergence rate and predict the required convergence time. Consequently, in order to achieve better control, the idea of finite-time synchronization has been proposed, and more and more attention has been paid by researchers. This kind of method can predict the synchronization time in advance and has better robustness, anti-interference, and better control effect. It has important research significance in theory and practice. Therefore, it is more meaningful to study finite-time synchronization [17–21]. In [22], the authors study finite-time passivity of multi-weighted coupled neural networks with and without coupling delays. As far as we know, very few scholars have discussed finite-time passivity of stochastic complex networks in recent years.Motivated by the above discussions, we will investigate finite-time passivity of stochastic coupled complex networks (SCCNs). The main novelty and contributions of this paper can be summarized as follows. Firstly, we give three concepts of finite-time passivity in the sense of expectation. Secondly, we develop several finite-time passivity criteria. Lastly, we establish the relationship between finite-time passivity and finite-time synchronization in the sense of expectation.


## 2. Lemmas and Definitions

In this section, we will give some lemmas and definitions.


### 2.1. Lemmas

Lemma 1.
(see [23]). Assume that a continuous, positive-definite function Wt satisfies the following differential inequality:(1)W˙t≤−ϱWμt,t≥t0,Wt0≥0,where ϱ>0 and 0<μ<1 are constants. Then, for any given t0, Wt satisfies the following inequality:(2)W1−μt≤W1−μt0−ϱ1−μt−t0,t0≤t≤t1,Wt≡0,t≥t1,with t1 given by(3)t1=t0+W1−μt0ϱ1−μ.Lemma 2.
(see [24]). For any bi∈ℝ,i=1,…,n,0<p≤1, the following inequality holds:(4)∑i=1nbip≤∑i=1nbip.Lemma 3.
(see [25]). For any vectors x,y∈ℝn and matrix 0<P∈ℝn×n, the following inequality holds:(5)xTy+yTx≤xTPx+yTP−1y.


### 2.2. Definitions

Next, we will give three definitions about finite-time passivity in the sense of expectation.E⋅ in these definitions stands for the mathematical expectation operator with respect to the given probability.Definition 1.
A stochastic system with inputut∈ℝn and output yt∈ℝn is said to be finite-time passive in the sense of expectation if there exists a nonnegative function V such that(6)EuTtyt≥EdVtdt+βEVαt,for some α∈0,1 and β>0.Definition 2.
A stochastic system with inputut∈ℝn and output yt∈ℝn is finite-time input strictly passive in the sense of expectation if there exists a nonnegative function V such that(7)EuTtyt−γ1EuTtut≥EdVtdt+βEVαt,for some α∈0,1,β>0, and γ1>0.Definition 3.
A stochastic system with inputut∈ℝn and output yt∈ℝn is finite-time output strictly passive in the sense of expectation if there exists a nonnegative function V such that(8)EuTtyt−γ2EyTtyt≥EdVtdt+βEVαt,for some α∈0,1,β>0, and γ2>0.Definition 4.
(see [25]). Let A=aijm×m∈ℝm×n and B=bijp×q∈ℝp×q. Then, the Kronecker product of A and B is defined as the matrix(9)A⊗B=a11Ba12B⋯a2nBa21Ba22B⋯a2nB⋮⋮⋯⋮am1Bam2B⋯amnB∈ℝmp×nq.
Throughout this paper, we make the following assumptions.(H1) (see [26]) The function f⋅ is in the QUAD class, that is, there exist diagonal matrices 0<P=diagp1,p2,…,pn∈ℝn×n and Δ=diagδ1,δ2,…,δn∈ℝn×n, such that(10)x−yTPfx−fy−Δx−y≤−λx−yTx−y,for allx,y∈ℝn and some λ>0.(H2) For arbitrary u,v∈ℝn, there exists a positive constant L such that the following inequality holds:(11)tracehu−hvThu−hv≤Lu−vTu−v.Remark 1.
(see [25]). It can be verified that many of the benchmark chaotic systems belong to “function class QUAD,” such as the Lorenz system, the Chen system, and the Lu¨ system.


## 2.1. Lemmas

Lemma 1.
(see [23]). Assume that a continuous, positive-definite function Wt satisfies the following differential inequality:(1)W˙t≤−ϱWμt,t≥t0,Wt0≥0,where ϱ>0 and 0<μ<1 are constants. Then, for any given t0, Wt satisfies the following inequality:(2)W1−μt≤W1−μt0−ϱ1−μt−t0,t0≤t≤t1,Wt≡0,t≥t1,with t1 given by(3)t1=t0+W1−μt0ϱ1−μ.Lemma 2.
(see [24]). For any bi∈ℝ,i=1,…,n,0<p≤1, the following inequality holds:(4)∑i=1nbip≤∑i=1nbip.Lemma 3.
(see [25]). For any vectors x,y∈ℝn and matrix 0<P∈ℝn×n, the following inequality holds:(5)xTy+yTx≤xTPx+yTP−1y.


## 2.2. Definitions

Next, we will give three definitions about finite-time passivity in the sense of expectation.E⋅ in these definitions stands for the mathematical expectation operator with respect to the given probability.Definition 1.
A stochastic system with inputut∈ℝn and output yt∈ℝn is said to be finite-time passive in the sense of expectation if there exists a nonnegative function V such that(6)EuTtyt≥EdVtdt+βEVαt,for some α∈0,1 and β>0.Definition 2.
A stochastic system with inputut∈ℝn and output yt∈ℝn is finite-time input strictly passive in the sense of expectation if there exists a nonnegative function V such that(7)EuTtyt−γ1EuTtut≥EdVtdt+βEVαt,for some α∈0,1,β>0, and γ1>0.Definition 3.
A stochastic system with inputut∈ℝn and output yt∈ℝn is finite-time output strictly passive in the sense of expectation if there exists a nonnegative function V such that(8)EuTtyt−γ2EyTtyt≥EdVtdt+βEVαt,for some α∈0,1,β>0, and γ2>0.Definition 4.
(see [25]). Let A=aijm×m∈ℝm×n and B=bijp×q∈ℝp×q. Then, the Kronecker product of A and B is defined as the matrix(9)A⊗B=a11Ba12B⋯a2nBa21Ba22B⋯a2nB⋮⋮⋯⋮am1Bam2B⋯amnB∈ℝmp×nq.
Throughout this paper, we make the following assumptions.(H1) (see [26]) The function f⋅ is in the QUAD class, that is, there exist diagonal matrices 0<P=diagp1,p2,…,pn∈ℝn×n and Δ=diagδ1,δ2,…,δn∈ℝn×n, such that(10)x−yTPfx−fy−Δx−y≤−λx−yTx−y,for allx,y∈ℝn and some λ>0.(H2) For arbitrary u,v∈ℝn, there exists a positive constant L such that the following inequality holds:(11)tracehu−hvThu−hv≤Lu−vTu−v.Remark 1.
(see [25]). It can be verified that many of the benchmark chaotic systems belong to “function class QUAD,” such as the Lorenz system, the Chen system, and the Lu¨ system.


## 3. Finite-Time Passivity of SCCNs

### 3.1. Network Model

In this paper, we will consider the following stochastic coupled complex networks model:(12)dzit=fzit+a∑j=1NGijΓzjt+uit+vitdt+hzitdωt,where zit=zi1t,zi2t,…,zintT∈ℝn is the state vector of the ith node; N corresponds to the number of neurons; fzit=f1zi1t,f2zi2t,…,fnzintT∈ℝn denotes the neuron activation function and satisfies assumption (H1); uit=ui1t,ui2t,…,uintT∈ℝn is a varying external input vector; vit=vi1t,vi2t,…,vintT∈ℝn denotes the control input; h⋅∈ℝn×n satisfies assumption (H2); ωt=ω1t,ω2t,…,ωntT∈ℝn is a n−dimensional Brownian motion defined on a complete probability space Ω,P; a is a positive real number which represents the overall coupling strength; Γ denotes the inner coupling matrix; and G=GijN×N represents the topological structure of the network, where Gij is defined as follows: if there exists a connection between node i and node j, then Gij=Gji>0; otherwise, Gij=Gji=0,i≠j, and the diagonal elements of matrix G are defined by(13)Gii=−∑j=1j≠iNGij,i=1,2,…,N.


### 3.2. Finite-Time Passivity

Set synchronization functionz¯t satisfies(14)dz¯t=fz¯tdt+hz¯tdωt,where z¯t=z¯1t,z¯2t,…,z¯nt∈ℝn.Defineeit=zit−z¯t,i=1,2,…,N. Then, we have(15)deit=fzit−fz¯t+a∑j=1NGijΓejt+uit+vitdt+hzit−hz¯tdωt,where i=1,2,…,N.yit∈ℝn refers to the output vector of (15) and is defined as follows:(16)yit=A1eit+A2uit,where A1,A2∈ℝn×n,The controller for network (12) is defined as follows:(17)vit=−Qizit−z¯t−βPα−1/2signzit−z¯tzit−z¯tα,where Qi∈ℝn×n, 0<α<1,β>0,P is defined in (H1), and(18)signzit−z¯t=diagsignzi1t−z¯1t,signzi2t−z¯2t,…,signzint−z¯nt,zit−z¯tα=zi1t−z¯1tα,zi2t−z¯2tα,…,zint−z¯ntαT,Pα−1/2=diagp1α−1/2,p2α−1/2,…,pnα−1/2.Theorem 1.
Under assumptions (H1) and (H2), network model (15) is finite-time passive in the sense of expectation under controller (17) if there exists matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN such that(19)K1E1E1T−IN⊗A2+A2T2≤0,where(20)K1=IN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓP,E1=IN⊗P−IN⊗A1T2.Proof.
For network (15), the Lyapunov functional is chosen as follows:(21)V1t=∑i=1NeiTtPeit.
According to Ito’s lemma, we acquire from (15) and (17)(22)dV1t=ℒV1tdt+Φtdωt.
Here(23)ℒV1t=2∑i=1NeiTtPfzit−fz¯t+a∑j=1NGijΓejt+uit−Qieit−βPα−1/2signeiteitα+∑i=1Ntracehzit−hz¯tTPhzit−hz¯t,Φt=2∑i=1NeiTtPhzit−hz¯t.
According to (H1), we can obtain(24)∑i=1NeiTtPfzit−fz¯t≤∑i=1NeiTtPΔ−λIneit.
We can get the following from (H2):(25)tracehzit−hz¯tTPhzit−hz¯t≤λMPtracehzit−hz¯tThzit−hz¯t≤LλMPzit−z¯tTzit−z¯t=λ0eiTteit.
HereλMP represents maximum eigenvalue of matrix P, λ0=LλMP.
Thus,(26)ℒV1t≤∑i=1NeiTtPΔ+ΔP−2λIn+λ0Ineit+2a∑i=1N∑j=1NGijeiTtPΓejt+2∑i=1NeiTtPuit−2∑i=1NeiTtPQieit−2β∑i=1NeiTtPα+1/2signeiteitα=eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put−2β∑i=1NeiTtPα+1/2signeiteitα,where et=e1Tt,e2Tt,…,eNTtT, ut=u1Tt,u2Tt,…,uNTtT.
Considering signx⋅x=x,∀x∈ℝ and Lemma 2, we can easily conclude that(27)∑i=1NeiTtPα+1/2signeiteitα=∑i=1N∑j=1npjα+1/2eijtα+1≥∑i=1N∑j=1npjeij2tα+1/2=∑i=1NeiTtPeitα+1/2.
Setyt=y1Tt,y2Tt,…,yNTtT; consequently,(28)uTtyt=∑i=1NuiTtyit=∑i=1NuiTtA1eit+A2uit=uTtIN⊗A1et+uTtIN⊗A2ut.
From (19) and (26)–(28),(29)ℒV1t−uTtyt≤−2β∑i=1NeiTtPeitα+1/2+ζTtK1E1E1T−IN⊗A2+A2T2ζt≤−2β∑i=1NeiTtPeitα+1/2≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t,where ζt=eTt,uTtT.
ConsideringEdV1t=EℒV1tdt, consequently(30)EdV1t−uTtytdt=EℒV1t−uTtytdt≤−2βEV1α+1/2tdt.
Then, we can obtain(31)EuTtyt≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time passive in the sense of expectation under controller (17).Theorem 2.
Under assumptions (H1) and (H2), network model (15) is finite-time input strictly passive in the sense of expectation under controller (17) if there exist matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN and a positive real number γ1 such that(32)K1E1E1Tγ1InN−IN⊗A2+A2T2≤0,where K1,E1 have the same meanings as in Theorem 1.Proof.
We will choose the sameV1t as (21) for network (15).
By (26)–(28), one can get(33)ℒV1t−uTtyt+γ1uTtut≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put+γ1uTtut−2β∑i=1NeiTtPα+1/2signeiteitα−uTtIN⊗A1et−uTtIN⊗A2ut=−2β∑i=1NeiTtPα+1/2signeiteitα+ζTtK1E1E1Tγ1InN−IN⊗A2+A2T2ζt≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t.
Taking the mathematical expectation on both sides above, one can derive that(34)EuTtyt−γ1EuTtut≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time input strictly passive in the sense of expectation under controller (17).Theorem 3.
Under assumptions (H1) and (H2), network model (15) is finite-time output strictly passive in the sense of expectation under controller (17) if there exist matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN and a positive real number γ2 such that(35)K2E2E2TK3≤0,where(36)K2=K1+γ2IN⊗A1TIN⊗A1,E2=E1+γ2IN⊗A1TIN⊗A2,K3=γ2IN⊗A2TIN⊗A2−IN⊗A2+A2T2,K1, E1 have the same meanings as in Theorem 1.Proof.
Firstly we calculate the following equality:(37)yTtyt=∑i=1NyiTtyiTt=∑i=1NA1eit+A2uitTA1eit+A2uit=∑i=1NeiTtA1T+uiTtA2TA1eit+A2uit=∑i=1NeiTtA1TA1eit+eiTtA1TA2uit+uiTtA2TA1eit+uiTtA2TA2uit=eTtIN⊗A1TA1et+eTtIN⊗A1TA2ut+uTtIN⊗A2TA1et+uTtIN⊗A2TA2ut=eTtIN⊗A1TIN⊗A1et+eTtIN⊗A1TIN⊗A2ut+uTtIN⊗A2TIN⊗A1et+uTtIN⊗A2TIN⊗A2ut.
For the last step, we utilize the important properties of the Kronecker product:(38)M1⊗M2M3⊗M4=M1M3⊗M2M4.
Select the sameV1t as (21) for network (15). We can obtain(39)ℒV1t−uTtyt+γ2yTtyt≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put−uTtIN⊗A1et−uTtIN⊗A2ut+γ2eTtIN⊗A1TIN⊗A1et+γ2eTtIN⊗A1TIN⊗A2ut+γ2uTtIN⊗A2TIN⊗A1et+γ2uTtIN⊗A2TIN⊗A2ut−2β∑i=1NeiTtPα+1/2signeiteitα=−2β∑i=1NeiTtPα+1/2signeiteitα+ζTtK2E2E2TK3ζt≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t.
Taking the mathematical expectation on both sides above, one can derive that(40)EuTtyt−γ2EyTtyt≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time output strictly passive in the sense of expectation under controller (17).


### 3.3. Finite-Time Synchronization

In this section, we will verify finite-time synchronization in the sense of expectation for SCCNs (12). Firstly, the definition of finite-time synchronization is given as follows.Definition 5.
The SCCN (12) is finite-time synchronized in the sense of expectation if there exists a constant T>0 such that(41)limt⟶T−Ezit−z¯t2=0,Ezit−z¯t2≡0,t≥T,for i=1,2,…,N, where uit=0,i=1,2,…,N.Theorem 4.
Assume that a continuous, positive-definite functionVt satisfies the following inequality:(42)φ1Eet2≤Vt,where φ1:0,+∞⟶0,+∞ is continuous and strictly monotonically increasing function and φ1s is positive for s>0 with φ10=0. If network (15) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to Vt, then SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).Proof.
The network model (15) is finite-time passive in the sense of expectation with respect to Vt under controller (17), that is to say, there exist α∈0,1 and β>0 such that(43)EuTtyt≥EdVtdt+βEVαt.
Consideringut=0, one obtains(44)EdVtdt+βEVαt≤0.
According to the property of mathematical expectation,(45)V˙t≤−βVαt.
Choosingt0=0 in Lemma 1, we can obtain Vt≡0 for t⩾t1, where t1=V1−α0/β1−α. On the one hand, since(46)φ1Eet2≤Vt,one has(47)φ1Eet2≤Vt≡0,for t≥t1. Since φ1s=0 if and only if s=0. Then, we can conclude that(48)Eet2≡0,t≥t1.
On the other hand,Vt is continuous, so(49)limt⟶t1−Vt=limt⟶t1+Vt=0.
Taking the limitt⟶t1− on both sides of (46), we will get(50)limt⟶t1−φ1Eet2=0.
Namely, SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).
Similarly, it is easy to prove that SCCN (12) is also finite-time synchronized in the sense of expectation under controller (17) if network model (15) is finite-time input strictly passive or finite-time output strictly passive in the sense of expectation.


## 3.1. Network Model

In this paper, we will consider the following stochastic coupled complex networks model:(12)dzit=fzit+a∑j=1NGijΓzjt+uit+vitdt+hzitdωt,where zit=zi1t,zi2t,…,zintT∈ℝn is the state vector of the ith node; N corresponds to the number of neurons; fzit=f1zi1t,f2zi2t,…,fnzintT∈ℝn denotes the neuron activation function and satisfies assumption (H1); uit=ui1t,ui2t,…,uintT∈ℝn is a varying external input vector; vit=vi1t,vi2t,…,vintT∈ℝn denotes the control input; h⋅∈ℝn×n satisfies assumption (H2); ωt=ω1t,ω2t,…,ωntT∈ℝn is a n−dimensional Brownian motion defined on a complete probability space Ω,P; a is a positive real number which represents the overall coupling strength; Γ denotes the inner coupling matrix; and G=GijN×N represents the topological structure of the network, where Gij is defined as follows: if there exists a connection between node i and node j, then Gij=Gji>0; otherwise, Gij=Gji=0,i≠j, and the diagonal elements of matrix G are defined by(13)Gii=−∑j=1j≠iNGij,i=1,2,…,N.


## 3.2. Finite-Time Passivity

Set synchronization functionz¯t satisfies(14)dz¯t=fz¯tdt+hz¯tdωt,where z¯t=z¯1t,z¯2t,…,z¯nt∈ℝn.Defineeit=zit−z¯t,i=1,2,…,N. Then, we have(15)deit=fzit−fz¯t+a∑j=1NGijΓejt+uit+vitdt+hzit−hz¯tdωt,where i=1,2,…,N.yit∈ℝn refers to the output vector of (15) and is defined as follows:(16)yit=A1eit+A2uit,where A1,A2∈ℝn×n,The controller for network (12) is defined as follows:(17)vit=−Qizit−z¯t−βPα−1/2signzit−z¯tzit−z¯tα,where Qi∈ℝn×n, 0<α<1,β>0,P is defined in (H1), and(18)signzit−z¯t=diagsignzi1t−z¯1t,signzi2t−z¯2t,…,signzint−z¯nt,zit−z¯tα=zi1t−z¯1tα,zi2t−z¯2tα,…,zint−z¯ntαT,Pα−1/2=diagp1α−1/2,p2α−1/2,…,pnα−1/2.Theorem 1.
Under assumptions (H1) and (H2), network model (15) is finite-time passive in the sense of expectation under controller (17) if there exists matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN such that(19)K1E1E1T−IN⊗A2+A2T2≤0,where(20)K1=IN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓP,E1=IN⊗P−IN⊗A1T2.Proof.
For network (15), the Lyapunov functional is chosen as follows:(21)V1t=∑i=1NeiTtPeit.
According to Ito’s lemma, we acquire from (15) and (17)(22)dV1t=ℒV1tdt+Φtdωt.
Here(23)ℒV1t=2∑i=1NeiTtPfzit−fz¯t+a∑j=1NGijΓejt+uit−Qieit−βPα−1/2signeiteitα+∑i=1Ntracehzit−hz¯tTPhzit−hz¯t,Φt=2∑i=1NeiTtPhzit−hz¯t.
According to (H1), we can obtain(24)∑i=1NeiTtPfzit−fz¯t≤∑i=1NeiTtPΔ−λIneit.
We can get the following from (H2):(25)tracehzit−hz¯tTPhzit−hz¯t≤λMPtracehzit−hz¯tThzit−hz¯t≤LλMPzit−z¯tTzit−z¯t=λ0eiTteit.
HereλMP represents maximum eigenvalue of matrix P, λ0=LλMP.
Thus,(26)ℒV1t≤∑i=1NeiTtPΔ+ΔP−2λIn+λ0Ineit+2a∑i=1N∑j=1NGijeiTtPΓejt+2∑i=1NeiTtPuit−2∑i=1NeiTtPQieit−2β∑i=1NeiTtPα+1/2signeiteitα=eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put−2β∑i=1NeiTtPα+1/2signeiteitα,where et=e1Tt,e2Tt,…,eNTtT, ut=u1Tt,u2Tt,…,uNTtT.
Considering signx⋅x=x,∀x∈ℝ and Lemma 2, we can easily conclude that(27)∑i=1NeiTtPα+1/2signeiteitα=∑i=1N∑j=1npjα+1/2eijtα+1≥∑i=1N∑j=1npjeij2tα+1/2=∑i=1NeiTtPeitα+1/2.
Setyt=y1Tt,y2Tt,…,yNTtT; consequently,(28)uTtyt=∑i=1NuiTtyit=∑i=1NuiTtA1eit+A2uit=uTtIN⊗A1et+uTtIN⊗A2ut.
From (19) and (26)–(28),(29)ℒV1t−uTtyt≤−2β∑i=1NeiTtPeitα+1/2+ζTtK1E1E1T−IN⊗A2+A2T2ζt≤−2β∑i=1NeiTtPeitα+1/2≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t,where ζt=eTt,uTtT.
ConsideringEdV1t=EℒV1tdt, consequently(30)EdV1t−uTtytdt=EℒV1t−uTtytdt≤−2βEV1α+1/2tdt.
Then, we can obtain(31)EuTtyt≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time passive in the sense of expectation under controller (17).Theorem 2.
Under assumptions (H1) and (H2), network model (15) is finite-time input strictly passive in the sense of expectation under controller (17) if there exist matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN and a positive real number γ1 such that(32)K1E1E1Tγ1InN−IN⊗A2+A2T2≤0,where K1,E1 have the same meanings as in Theorem 1.Proof.
We will choose the sameV1t as (21) for network (15).
By (26)–(28), one can get(33)ℒV1t−uTtyt+γ1uTtut≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put+γ1uTtut−2β∑i=1NeiTtPα+1/2signeiteitα−uTtIN⊗A1et−uTtIN⊗A2ut=−2β∑i=1NeiTtPα+1/2signeiteitα+ζTtK1E1E1Tγ1InN−IN⊗A2+A2T2ζt≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t.
Taking the mathematical expectation on both sides above, one can derive that(34)EuTtyt−γ1EuTtut≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time input strictly passive in the sense of expectation under controller (17).Theorem 3.
Under assumptions (H1) and (H2), network model (15) is finite-time output strictly passive in the sense of expectation under controller (17) if there exist matrix Q=diagQ1,Q2,…,QN∈ℝnN×nN and a positive real number γ2 such that(35)K2E2E2TK3≤0,where(36)K2=K1+γ2IN⊗A1TIN⊗A1,E2=E1+γ2IN⊗A1TIN⊗A2,K3=γ2IN⊗A2TIN⊗A2−IN⊗A2+A2T2,K1, E1 have the same meanings as in Theorem 1.Proof.
Firstly we calculate the following equality:(37)yTtyt=∑i=1NyiTtyiTt=∑i=1NA1eit+A2uitTA1eit+A2uit=∑i=1NeiTtA1T+uiTtA2TA1eit+A2uit=∑i=1NeiTtA1TA1eit+eiTtA1TA2uit+uiTtA2TA1eit+uiTtA2TA2uit=eTtIN⊗A1TA1et+eTtIN⊗A1TA2ut+uTtIN⊗A2TA1et+uTtIN⊗A2TA2ut=eTtIN⊗A1TIN⊗A1et+eTtIN⊗A1TIN⊗A2ut+uTtIN⊗A2TIN⊗A1et+uTtIN⊗A2TIN⊗A2ut.
For the last step, we utilize the important properties of the Kronecker product:(38)M1⊗M2M3⊗M4=M1M3⊗M2M4.
Select the sameV1t as (21) for network (15). We can obtain(39)ℒV1t−uTtyt+γ2yTtyt≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+aG⊗PΓ+ΓPet+2eTtIN⊗Put−uTtIN⊗A1et−uTtIN⊗A2ut+γ2eTtIN⊗A1TIN⊗A1et+γ2eTtIN⊗A1TIN⊗A2ut+γ2uTtIN⊗A2TIN⊗A1et+γ2uTtIN⊗A2TIN⊗A2ut−2β∑i=1NeiTtPα+1/2signeiteitα=−2β∑i=1NeiTtPα+1/2signeiteitα+ζTtK2E2E2TK3ζt≤−2β∑i=1NeiTtPeitα+1/2=−2βV1α+1/2t.
Taking the mathematical expectation on both sides above, one can derive that(40)EuTtyt−γ2EyTtyt≥EdV1tdt+2βEV1α+1/2t.
Therefore, network (15) is finite-time output strictly passive in the sense of expectation under controller (17).


## 3.3. Finite-Time Synchronization

In this section, we will verify finite-time synchronization in the sense of expectation for SCCNs (12). Firstly, the definition of finite-time synchronization is given as follows.Definition 5.
The SCCN (12) is finite-time synchronized in the sense of expectation if there exists a constant T>0 such that(41)limt⟶T−Ezit−z¯t2=0,Ezit−z¯t2≡0,t≥T,for i=1,2,…,N, where uit=0,i=1,2,…,N.Theorem 4.
Assume that a continuous, positive-definite functionVt satisfies the following inequality:(42)φ1Eet2≤Vt,where φ1:0,+∞⟶0,+∞ is continuous and strictly monotonically increasing function and φ1s is positive for s>0 with φ10=0. If network (15) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to Vt, then SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).Proof.
The network model (15) is finite-time passive in the sense of expectation with respect to Vt under controller (17), that is to say, there exist α∈0,1 and β>0 such that(43)EuTtyt≥EdVtdt+βEVαt.
Consideringut=0, one obtains(44)EdVtdt+βEVαt≤0.
According to the property of mathematical expectation,(45)V˙t≤−βVαt.
Choosingt0=0 in Lemma 1, we can obtain Vt≡0 for t⩾t1, where t1=V1−α0/β1−α. On the one hand, since(46)φ1Eet2≤Vt,one has(47)φ1Eet2≤Vt≡0,for t≥t1. Since φ1s=0 if and only if s=0. Then, we can conclude that(48)Eet2≡0,t≥t1.
On the other hand,Vt is continuous, so(49)limt⟶t1−Vt=limt⟶t1+Vt=0.
Taking the limitt⟶t1− on both sides of (46), we will get(50)limt⟶t1−φ1Eet2=0.
Namely, SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).
Similarly, it is easy to prove that SCCN (12) is also finite-time synchronized in the sense of expectation under controller (17) if network model (15) is finite-time input strictly passive or finite-time output strictly passive in the sense of expectation.


## 4. Finite-Time Passivity of SCCNs with Time-Varying Delay

### 4.1. Network Model

In this section, the network model is described by(51)dzit=fzit+a∑j=1NGijΓzjt−τt+uit+vitdt+hzitdωt,where τt is the time delay and satisfies 0≤τt≤τ,0≤τ˙t≤d<1.


### 4.2. Finite-Time Passivity

Letz¯t also satisfy(52)dz¯t=fz¯tdt+hz¯tdωt.Defineeit=zit−z¯t,i=1,2,…,N. Then, we have(53)deit=fzit−fz¯t+a∑j=1NGijΓejt−τt+uit+vitdt+hzit−hz¯tdωt.The output vectoryit of network (53) is defined as follows:(54)yit=B1eit+B2uit.Design the following controller for network (51):(55)vit=−βP−1a1−d∫t−τttzih−z¯hTMizih−z¯hdhα+1/2zit−z¯tzit−z¯t22−βPα−1/2signzit−z¯tzit−z¯tα−Qizit−z¯t,where 0<Mi∈ℝn×n,M=diagM1,M2,…,MN,Qi,α,P,β,Pα−1/2,signzit−z¯t,zit−z¯tα have the same meanings as in (17).Theorem 5.
Under assumptions (H1) and (H2), network model (53) is finite-time passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN such that(56)W1Ω1Ω1T−IN⊗B2+B2T2≤0,where(57)W1=IN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+a1−dM+aG⊗PΓM−1G⊗ΓP,Ω1=IN⊗P−IN⊗B1T2.Proof.
Choose the following Lyapunov functional for network (51):(58)V2t=∑i=1NeiTtPeit+a1−d∫t−τtteThMehdh,where et=e1Tt,e2Tt,…,eNTtT.
According to Ito’s lemma, we acquire from (53) and (55)(59)dV2t=ℒV2tdt+Φtdωt.
Here(60)ℒV2t=2∑i=1NeiTtPfzit−fz¯t+a∑j=1NGijΓejt−τt+uit−Qieit−βPα−1/2signeiteitα+∑i=1Ntracehzit−hz¯tTPhzit−hz¯t−2β∑i=1Na1−d∫t−τtteiThMieihdhα+1/2−2β∑i=1NeiTtPα+1/2signeiteitα+a1−deTtMet−eTt−τtMet−τt1−τ˙t,Φt=2∑i=1NeiTtPhzit−hz¯t.
According to Lemma3, we can take(61)x=et−τt,y=G⊗ΓPet.
It is not difficult to obtain(62)2aeTtG⊗PΓet−τt≤aeTtG⊗PΓM−1G⊗ΓPet+aeTt−τtMet−τt.
From the above, one has(63)ℒV2t≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗Pet+2aeTtG⊗PΓet−τt−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2−aeTt−τtMet−τt+a1−deTtMet+2eTtIN⊗Put≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+a1−dM+aG⊗PΓM−1G⊗ΓPet−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2+2eTtIN⊗Put.
Thus,(64)ℒV2t−uTtyt≤ζTtW1Ω1Ω1T−IN⊗B2+B2T2ζt−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2β∑i=1NeiTtPeit+a1−d∫t−τtteThMehdhα+1/2=−2βV2α+1/2t,where ut=u1Tt,u2Tt,…,uNTtT,yt=y1Tt,y2Tt,…,yNTtT,ζt=eTt,uTtT.
Taking the mathematical expectation on (59), we can obtain(65)EuTtyt≥EdV2tdt+2βEV2α+1/2t.
Consequently, network model (53) is finite-time passive in the sense of expectation under controller (55).Theorem 6.
Under assumptions (H1) and (H2), network model (53) is finite-time input strictly passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN and a positive real number γ3 such that(66)W1Ω1Ω1Tγ3InN−IN⊗B2+B2T2≤0,where W1,Ω1 have the same meanings as in Theorem 5.Proof.
We also select the sameV2t as (58) for network (53). By (64), we get(67)ℒV2t−uTtyt+γ3uTtut≤ζTtW1Ω1Ω1Tγ3InN−IN⊗A2+A2T2ζt−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdh+∑i=1NeiTtPeitα+1/2=−2βV2α+1/2t.
Taking the mathematical expectation on (59), we can obtain(68)EuTtyt−γ3EuTtut≥EdV2tdt+2βEV2α+1/2t.
Therefore, network (53) is finite-time input strictly passive in the sense of expectation under controller (55).Theorem 7.
Under assumptions (H1) and (H2), network model (53) is finite-time output strictly passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN and a positive real number γ4 such that(69)W2Ω2Ω2TW3≤0,where(70)W2=W1+γ4IN⊗B1TIN⊗B1,Ω2=Ω1+γ4IN⊗B1TIN⊗B2,W3=γ4IN⊗B2TIN⊗B2−IN⊗B2+B2T2,W1, Ω1 have the same meanings as in Theorem 5.Proof.
Select the sameV2t as (58) for network (53). By (37) and (64), we get(71)ℒV2t−uTtyt+γ4yTtyt≤ζTtW2Ω2Ω2TW3ζt−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdh+∑i=1NeiTtPeitα+1/2=−2βV2α+1/2t.
Taking the mathematical expectation on (59), we can obtain(72)EuTtyt−γ4EyTtyt≥EdV2tdt+2βEV2α+1/2t.
Therefore, network (53) is finite-time output strictly passive in the sense of expectation under controller (55).


### 4.3. Finite-Time Synchronization

Theorem 8.
Assume that a continuous, positive-definite functionV^t satisfies the following inequality:(73)φ2Eet2≤V^t,where φ2:0,+∞⟶0,+∞ is continuous and strictly monotonically increasing function and φ2s is positive for s>0 with φ20=0. If network (51) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to V^t, then SCCN (53) is finite-time synchronized in the sense of expectation under controller (55).Here we omit the proof of the theorem. The readers can refer to the proof of Theorem4.


## 4.1. Network Model

In this section, the network model is described by(51)dzit=fzit+a∑j=1NGijΓzjt−τt+uit+vitdt+hzitdωt,where τt is the time delay and satisfies 0≤τt≤τ,0≤τ˙t≤d<1.


## 4.2. Finite-Time Passivity

Letz¯t also satisfy(52)dz¯t=fz¯tdt+hz¯tdωt.Defineeit=zit−z¯t,i=1,2,…,N. Then, we have(53)deit=fzit−fz¯t+a∑j=1NGijΓejt−τt+uit+vitdt+hzit−hz¯tdωt.The output vectoryit of network (53) is defined as follows:(54)yit=B1eit+B2uit.Design the following controller for network (51):(55)vit=−βP−1a1−d∫t−τttzih−z¯hTMizih−z¯hdhα+1/2zit−z¯tzit−z¯t22−βPα−1/2signzit−z¯tzit−z¯tα−Qizit−z¯t,where 0<Mi∈ℝn×n,M=diagM1,M2,…,MN,Qi,α,P,β,Pα−1/2,signzit−z¯t,zit−z¯tα have the same meanings as in (17).Theorem 5.
Under assumptions (H1) and (H2), network model (53) is finite-time passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN such that(56)W1Ω1Ω1T−IN⊗B2+B2T2≤0,where(57)W1=IN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+a1−dM+aG⊗PΓM−1G⊗ΓP,Ω1=IN⊗P−IN⊗B1T2.Proof.
Choose the following Lyapunov functional for network (51):(58)V2t=∑i=1NeiTtPeit+a1−d∫t−τtteThMehdh,where et=e1Tt,e2Tt,…,eNTtT.
According to Ito’s lemma, we acquire from (53) and (55)(59)dV2t=ℒV2tdt+Φtdωt.
Here(60)ℒV2t=2∑i=1NeiTtPfzit−fz¯t+a∑j=1NGijΓejt−τt+uit−Qieit−βPα−1/2signeiteitα+∑i=1Ntracehzit−hz¯tTPhzit−hz¯t−2β∑i=1Na1−d∫t−τtteiThMieihdhα+1/2−2β∑i=1NeiTtPα+1/2signeiteitα+a1−deTtMet−eTt−τtMet−τt1−τ˙t,Φt=2∑i=1NeiTtPhzit−hz¯t.
According to Lemma3, we can take(61)x=et−τt,y=G⊗ΓPet.
It is not difficult to obtain(62)2aeTtG⊗PΓet−τt≤aeTtG⊗PΓM−1G⊗ΓPet+aeTt−τtMet−τt.
From the above, one has(63)ℒV2t≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗Pet+2aeTtG⊗PΓet−τt−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2−aeTt−τtMet−τt+a1−deTtMet+2eTtIN⊗Put≤eTtIN⊗PΔ+ΔP−2λIn+λ0In−IN⊗PQ−QTIN⊗P+a1−dM+aG⊗PΓM−1G⊗ΓPet−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2+2eTtIN⊗Put.
Thus,(64)ℒV2t−uTtyt≤ζTtW1Ω1Ω1T−IN⊗B2+B2T2ζt−2βa1−d∫t−τtteThMehdhα+1/2−2β∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2β∑i=1NeiTtPeit+a1−d∫t−τtteThMehdhα+1/2=−2βV2α+1/2t,where ut=u1Tt,u2Tt,…,uNTtT,yt=y1Tt,y2Tt,…,yNTtT,ζt=eTt,uTtT.
Taking the mathematical expectation on (59), we can obtain(65)EuTtyt≥EdV2tdt+2βEV2α+1/2t.
Consequently, network model (53) is finite-time passive in the sense of expectation under controller (55).Theorem 6.
Under assumptions (H1) and (H2), network model (53) is finite-time input strictly passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN and a positive real number γ3 such that(66)W1Ω1Ω1Tγ3InN−IN⊗B2+B2T2≤0,where W1,Ω1 have the same meanings as in Theorem 5.Proof.
We also select the sameV2t as (58) for network (53). By (64), we get(67)ℒV2t−uTtyt+γ3uTtut≤ζTtW1Ω1Ω1Tγ3InN−IN⊗A2+A2T2ζt−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdh+∑i=1NeiTtPeitα+1/2=−2βV2α+1/2t.
Taking the mathematical expectation on (59), we can obtain(68)EuTtyt−γ3EuTtut≥EdV2tdt+2βEV2α+1/2t.
Therefore, network (53) is finite-time input strictly passive in the sense of expectation under controller (55).Theorem 7.
Under assumptions (H1) and (H2), network model (53) is finite-time output strictly passive in the sense of expectation under controller (55) if there exist matrices Q=diagQ1,Q2,…,QN∈ℝnN×nN,0<M=diagM1,M2,…,MN∈ℝnN×nN and a positive real number γ4 such that(69)W2Ω2Ω2TW3≤0,where(70)W2=W1+γ4IN⊗B1TIN⊗B1,Ω2=Ω1+γ4IN⊗B1TIN⊗B2,W3=γ4IN⊗B2TIN⊗B2−IN⊗B2+B2T2,W1, Ω1 have the same meanings as in Theorem 5.Proof.
Select the sameV2t as (58) for network (53). By (37) and (64), we get(71)ℒV2t−uTtyt+γ4yTtyt≤ζTtW2Ω2Ω2TW3ζt−2βa1−d∫t−τtteThMehdhα+1/2+∑i=1NeiTtPeitα+1/2≤−2βa1−d∫t−τtteThMehdh+∑i=1NeiTtPeitα+1/2=−2βV2α+1/2t.
Taking the mathematical expectation on (59), we can obtain(72)EuTtyt−γ4EyTtyt≥EdV2tdt+2βEV2α+1/2t.
Therefore, network (53) is finite-time output strictly passive in the sense of expectation under controller (55).


## 4.3. Finite-Time Synchronization

Theorem 8.
Assume that a continuous, positive-definite functionV^t satisfies the following inequality:(73)φ2Eet2≤V^t,where φ2:0,+∞⟶0,+∞ is continuous and strictly monotonically increasing function and φ2s is positive for s>0 with φ20=0. If network (51) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to V^t, then SCCN (53) is finite-time synchronized in the sense of expectation under controller (55).Here we omit the proof of the theorem. The readers can refer to the proof of Theorem4.


## 5. Numerical Examples

Example 1.
The following SCCNs are discussed:(74)dzit=fzit+0.7∑j=16GijΓzjt+uit+vitdt+hzitdωt,i=1,2,…,6,where fsς=1/4ς+1−ς−1,s=1,2,3,Γ=diag0.15,0.05,0.25,hzit=diag0.2,0.4,0.2,(75)G=−0.40.100.10.200.1−0.400.20.1000−0.50.20.10.20.10.20.2−0.600.10.20.10.10−0.50.1000.20.10.1−0.4.
Obviously,H1 holds under the condition that P=I3,λ=0.8, and Δ=diag0.15,0.14,0.12. Choose A2=diag0.6,0.8,0.9 and(76)A1=0.30.20.10.40.30.30.60.10.4.
TakeQ=diag0.2I3,0.6I3,0.7I3,0.5I3,0.3I3,0.4I3. According to Theorem 1, the SCCNs (74) can realize finite-time passivity in the sense of expectation under controller (17). Then, we can easily find the parameters γ1=0.0163 and γ2=0.0300 satisfying the condition of Theorems 2 and 3. The simulation results are shown in Figures 1 and 2.
By Theorem4, network (74) under finite-time output strictly passive can achieve finite-time synchronization. Figure 3 shows the simulation results.Figure 1
The norms of the input and output vectorsuit2,yit2,i=1,2,…,6.Figure 2
The error vectorseit,i=1,2,…,6.Figure 3
The state vectorszit,i=1,2,…,6.Example 2.
The following SCCNs with time-varying delay are discussed:(77)dzit=fzit+0.8∑j=16GijΓzjt−τt+uit+vitdt+hzitdωt,i=1,2,…,6,where fsς=1/4ς+1−ς−1,s=1,2,3,Γ=diag0.15,0.15,0.15,hzit=diag0.1,0.2,0.2. Taking τt=0.5−0.5e−t, we can get d=0.5. The matrix G is chosen as(78)G=−0.80.10.20.10.20.20.1−0.500.20.10.10.20−0.70.20.10.20.10.20.2−0.600.10.20.10.10−0.50.10.20.10.20.10.1−0.7.
Obviously,H1 holds under the condition that P=I3,λ=0.7, and Δ=diag0.18,0.16,0.15. Choose B2=diag0.6,0.8,0.9 and(79)B1=0.60.40.30.20.10.10.30.30.4.
TakeQ=diag0.5I3,0.6I3,0.7I3,0.5I3,0.7I3,0.8I3,M=I6⊗diag0.1,0.2,0.4. According to Theorem 5, the SCCNs (77) can realize finite-time passivity under controller (55). Then, we can easily find the parameters γ3=0.0132 and γ4=0.0369 satisfying the condition of Theorems 6 and 7. The simulation results are shown in Figures 4 and 5.
By Theorem8, network (77) under finite-time output strictly passive can achieve finite-time synchronization. Figure 6 shows the simulation results.Figure 4
The norms of the input and output vectorsuit2,yit2,i=1,2,…,6.Figure 5
The error vectorseit,i=1,2,…,6.Figure 6
The state vectorszit,i=1,2,…,6.



---

*Source: 9905125-2021-07-09.xml*