# Bipartite Synchronization of Heterogeneous Signed Networks with Distributed Impulsive Control

Authors: Guizhen Feng; Jian Ding; Jinde Cao; Qingqing Cao
Journal: Complexity
 (2021)

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## Abstract

This study investigates the bipartite synchronization of heterogeneous signed networks with distributed impulsive control. Leader-follower bipartite synchronization within a nonzero error bound is analyzed when the average impulsive interval isTa<∞ or Ta=∞. Some sufficient conditions to achieve the bipartite quasi-synchronization are presented, and the synchronization error level is estimated by the specific mathematical expression. The correctness of the theoretical results is verified by numerical examples.


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## Body

## 1. Introduction

There are some cooperative behaviors of interconnected dynamic systems, such as image processing [1], pattern recognition [2], and secure communication [3], which have been eliciting increasing attention recently. In particular, synchronization, as a typical collective behavior, has been widely studied by many researchers through different control methods, for example, feedback control [4, 5], adaptive control [6–8], event-triggered control [9–11], and impulsive control [12–18]. Among these methods, impulsive control, as a form of discontinuous control [19], has advantages of low cost and high efficiency and so has been widely used in many fields, such as the population growth model [19] and neural networks [20]. Particularly, a distributed impulsive controller was proposed in [21] and successfully applied to synchronization of a leaderless network. Later, the method of distributed impulsive control was also used to investigate the consensus of multiagent systems [22, 23].In most early studies, in order to make the system achieve the desired trajectory, the controller is usually applied to all nodes. However, this is difficult to implement or even is inapplicable, especially for large networks. Pinning control is a kind of control strategy which only needs control finite nodes of the system [24, 25]. Considering the advantages of pinning control and impulsive control, the pinning impulsive controller was proposed in [26]. Lu et al. [27] studied the synchronization of complex dynamic networks by using a pinning impulsive controller. In [28], a distributed pinning impulsive controller was developed to investigate the synchronization of heterogeneous networks.In most of the aforementioned literature, the upper bound and lower bound of the impulsive interval determine the frequency of the impulse occurrence. Lu et al. [29] proposed the concept of average impulsive interval, and then it was used to describe a sparse impulsive sequence with an infinite average impulsive interval in [30]. On the contrary, most studies on interactions among individual systems are assumed to be unsigned, where the weights of the adjacency matrix are nonnegative. In fact, it is more reasonable to consider signed networks since the agents in a network may be cooperative or competitive so that interaction weights between agents can either be positive or negative. Such weights are called signed interaction digraphs. Coordination control on signed networks has become a hot topic in recent years [31, 32]. Proskurnikov et al. [31] investigated opinion dynamics of social networks. Xia et al. [32] studied the separation of opinion in trust-mistrust social networks. Meng et al. [33] achieved bipartite consensus on a signed network. The concept of bipartite synchronization was presented in [34] under a signed network. In [35], bipartite synchronization under a pinning controller was studied on signed and switching networks.Based on the aforementioned studies, we intend to study the bipartite synchronization of a signed network with distributed impulsive control. The following factors are integrated into our study: (i) interaction graphs are signed; in other words, the weights of the adjacency matrix can be positive or negative; (ii) two error levels are given for bipartite quasi-synchronization of asymmetric networks; (iii) more general impulsive sequences whose average impulsive interval is finite or infinite are considered. By designing a distributed impulsive controller, this study aims to develop the heterogeneous signed network that can be bipartitely quasi-synchronized.The rest of the paper is organized as follows: in Section2, we formulate the problem of bipartite quasi-synchronization between the leader and the heterogeneous systems under a distributed impulsive controller and give some necessary preparations. In Section 3, we establish some bipartite quasi-synchronization criteria for the signed network. In Section 4, we design numerical examples to verify the correctness of our theoretical results. In Section 5, we summarize all results in a conclusion.Notations: throughout the paper,In denotes an n-dimensional identity matrix; we write A>0 if matrix A is positive definite; σmaxB is the maximum singular value for matrix B; ⋅ is the Euclidean vector norm; ⊗ is the Kronecker product.


## 2. Problem Descriptions and Preliminaries

### 2.1. Problem Formulation

For a heterogeneous system, we use a signed graphAsU,V,Gs to denote its underlying topology, where U=1,2,…,N is the set of nodes, V⊆U×U is the set of edges, Gs=aijs∈ℝℕ×ℕ:aijs≠0⇔j,i∈V is the weighted adjacency matrix, and aijs can be positive or negative.There are some necessary definitions and lemmas in the process of theoretical derivation.Definition 1.
(see [36]). The sighed graph AsU,V,Gs is called structurally balanced if its node set U can be partitioned into two disjoint subsets U1 and U2 such that their induced subgraphs GU1 and GU2 are both unsigned, and the weights of all links between GU1 and GU2 are negative.Lemma 1.
(see [36]). If a signed graph AsU,V,Gs is structurally balanced, then there exists a diagonal matrix of form Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, such that all entries of ΣGsΣ are nonnegative.Assume that the nodes of the heterogeneous system have the following dynamics:(1)x˙it=Aixit+Bifxit+uit,where xit=xi1t,…,xintT∈ℝn is the state of the ith node; fx=f1x,f2x,…,fnxT is a nonlinear function on ℝn; Ai,Bi∈ℝn×n are constant matrices; and uit is a control protocol.The leader node of nonlinear system (1) is described by(2)s˙t=Ast+Bfst,where st∈ℝn is bounded.Assumption 1.
The nonlinear functionfx=f1x,f2x,…,fnxT is odd and satisfies the Lipschitz condition. Specifically,(3)fi−x=−fix,i,j=1,2,…,n,and there are nonnegative constants δij such that, for any x1,x2∈ℝn,(4)fix1−fix2≤∑j=1nδijx1j−x2j.Since (1) is heterogeneous, it is difficult to realize full synchronization by a static controller. In recent years, quasi-synchronization about heterogeneous networks received wide attention [28, 35], but this is not applicable for signed heterogeneous networks. According to dynamic properties of network (1) and leader (2), a more general synchronization concept, called bipartite quasi-synchronization, is introduced below.Definition 2.
Suppose that the signed graphAsU,V,Gs of (1) is structurally balanced, Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, is a diagonal matrix such that all entries of ΣGsΣ are nonnegative. Signed network (1) is said to achieve bipartite quasi-synchronization if the error eit=xit−σisti=1,2,…,N converges to some compact set M1=eit∈ℝn|eit≤ϵ,ϵ>0 when t⟶∞.Obviously, whenσi=1 for i=1,2,…,n, bipartite quasi-synchronization becomes quasi-synchronization.Suppose that the signed graphAsU,V,Gs of (1) is structurally balanced, Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, is a fixed diagonal matrix such that all entries of ΣGsΣ are nonnegative. The distributed impulsive protocol with pinning control of (1) is designed as(5)uit=∑k=1∞−c∑j=1Naijsxit−signaijsxjt−cdikxit−σistδt−tk,where c is a constant and dik>0 if node i is pinned; otherwise, dik=0. δ is the Dirac function, and tkk=1∞ is the impulse sequence satisfying 0=t0<t1<t2⋯<tk<⋯,limk⟶∞tk=∞.Remark 1.
Controller (5) indicates that if two nodes i and j are cooperative, aijs>0, and the coupling term is expressed as aijsxit−xjt; if two nodes i and j are competitive, aijs<0, and the coupling term is expressed as −aijsxit+xjt [36]. On the contrary, if the graph As is unsigned, controller (5) degenerates into a traditional distributed impulsive controller defined by previous studies [21, 23, 28].Based on the aforementioned control protocol (5), the impulsive signed network reads as(6)x˙it=Aixit+Bifxit,t≠tk,Δxit=−c∑j=1Naijsxitk−−signaijsxjtk−−cdikxitk−−σistk−,where Δxitk=xitk+−xitk−, xitk−=limh⟶0−xitk+h, and xitk=xitk+=limh⟶0+xitk+h.Remark 2.
According to Lemma2, it is necessary that As is structurally balanced during investigating the bipartite synchronization. In fact, Meng et al. [33] demonstrated that a network cannot obtain bipartite synchronization when the signed network is structurally unbalanced.Definition 3.
Letζ≜t1,t2,… be an impulsive sequence on t0,+∞. For any t∈t0,+∞, Nζt,t0 denotes the impulsive times of ζ on the interval t0,t. Then, the average impulsive interval, denoted by Ta, is defined by(7)Ta=limt⟶+∞t−t0Nζt,t0.Remark 3.
Obviously,0≤Ta≤∞. For example, let Nζt,t0=t3/2; then, Ta=0; let Nζt,t0=t1/2; then, Ta=∞. When 0<Ta<∞, it was proposed in [29] and then was applied in [19–23, 26–28]. The case of Ta=∞ was proposed in [30] when impulses occur infinitely but sparsely.


### 2.2. Synchronization Error Dynamics

LetL¯=l¯ijN×N be the Laplacian matrix of network (6); then, L¯=l¯ijN×N=ℬ−Gs, where ℬ=diagb1,b2,…,bN with bi=∑k=1Naiks. Let AU,V,G be the unsigned graph corresponding to As. Then, G=aijsN×N, and so, the corresponding Laplacian matrix L=lijN×N=ℬ−G with lij=−aijs for i≠j and lii=∑k=1,k≠iaiks.SinceAsU,V,Gs is structurally balanced and the diagonal matrix Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, make sure that ΣGsΣ=G. By L¯=ℬ−Gs and L=ℬ−G, it is easy to see that L=ΣL¯Σ holds with lij=−aijs for i≠j and lii=∑k=1,k≠iaiks.Letx¯it=σixit, i=1,…,N, that is, xit=σix¯it. Then, it follows from (6) that(8)x¯⋅it=Aix¯it+Biσifσix¯it,t≠tk,Δx¯it=−c∑j=1Naijsx¯itk−−signaijsx¯jtk−−cdikx¯itk−−stk−.By Assumption 1,fi⋅, i∈1,…,N is odd. Then, σifσix¯it=fx¯it holds for any σi∈−1,1 and i∈1,…,N. Then, from (8), we get(9)x¯⋅it=Aix¯it+Bifx¯it,t≠tk,Δx¯it=−c∑j=1Nlijx¯jtk−−cdikx¯itk−−stk−.From (2) and (9), based on the fact that L is a matrix with zero row sum, the following error system can be obtained:(10)e˙it=Aieit+Bigeit+Wist,t≠tk,Δeit=−c∑j=1Nlijejtk−−cdikeitk−,where eit=x¯it−st, geit=feit+st−fst, and Wist=Ai−Ast+Bi−Bfst. Since st is bounded and f⋅ is a Lipschitz function, it is easy to know that Wist is bounded. Suppose supt≥t0Wist=wi, i=1,…,n.Now, we rewrite the error system with a matrix form at the impulse instantt=tk:(11)etk=IN−cL+Dk⊗Inetk−,where et=e1T,e2T,…,eNTT and Dk=diagd1k,d2k,…,dNk.Lemma 2.
(see [37]; Schur complement). Suppose that QxandRx are both symmetric time-varying matrices; then, for any time-varying matrix Sx,(12)QxSxSTxRx>0is equivalent to(1)
Qx>0, Rx−STxQ−1xSx>0,or(2)
Rx>0, Qx−STxR−1xSx>0.Definition 4.
Symmetric time-varying matrixPt∈ℝn×n is called to be uniformly bounded and positive definite if there exist constants mˇ>0 and m^>0 such that(13)0<mˇ≤λminPt≤λmaxPt≤m^,∀t≥0.Lemma 3.
(see [38]). Suppose that 0≤τit≤τ, i=1,2,…,n, Ft,w,w1,…,wn:ℝ+×ℝn+1⟶ℝ is nondecreasing in wi for each fixed t,w,w1,…,wi−1,wi+1,…,wn,i=1,2,…,n, and Ikw:ℝ⟶ℝ is nondecreasing in w,k∈ℕ+. If utandvt satisfy(14)D+ut≤Ft,ut,ut−τ1t,…,ut−τnt,t≠tk,utk+≤Ikutk−,k∈ℕ+,D+vt≥Ft,vt,vt−τ1t,…,vt−τnt,t≠tk,vtk+≥Ikvtk−,k∈ℕ+,then ut≤vt for −τ≤t≤0 implies that ut≤vt for t≥0.


## 2.1. Problem Formulation

For a heterogeneous system, we use a signed graphAsU,V,Gs to denote its underlying topology, where U=1,2,…,N is the set of nodes, V⊆U×U is the set of edges, Gs=aijs∈ℝℕ×ℕ:aijs≠0⇔j,i∈V is the weighted adjacency matrix, and aijs can be positive or negative.There are some necessary definitions and lemmas in the process of theoretical derivation.Definition 1.
(see [36]). The sighed graph AsU,V,Gs is called structurally balanced if its node set U can be partitioned into two disjoint subsets U1 and U2 such that their induced subgraphs GU1 and GU2 are both unsigned, and the weights of all links between GU1 and GU2 are negative.Lemma 1.
(see [36]). If a signed graph AsU,V,Gs is structurally balanced, then there exists a diagonal matrix of form Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, such that all entries of ΣGsΣ are nonnegative.Assume that the nodes of the heterogeneous system have the following dynamics:(1)x˙it=Aixit+Bifxit+uit,where xit=xi1t,…,xintT∈ℝn is the state of the ith node; fx=f1x,f2x,…,fnxT is a nonlinear function on ℝn; Ai,Bi∈ℝn×n are constant matrices; and uit is a control protocol.The leader node of nonlinear system (1) is described by(2)s˙t=Ast+Bfst,where st∈ℝn is bounded.Assumption 1.
The nonlinear functionfx=f1x,f2x,…,fnxT is odd and satisfies the Lipschitz condition. Specifically,(3)fi−x=−fix,i,j=1,2,…,n,and there are nonnegative constants δij such that, for any x1,x2∈ℝn,(4)fix1−fix2≤∑j=1nδijx1j−x2j.Since (1) is heterogeneous, it is difficult to realize full synchronization by a static controller. In recent years, quasi-synchronization about heterogeneous networks received wide attention [28, 35], but this is not applicable for signed heterogeneous networks. According to dynamic properties of network (1) and leader (2), a more general synchronization concept, called bipartite quasi-synchronization, is introduced below.Definition 2.
Suppose that the signed graphAsU,V,Gs of (1) is structurally balanced, Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, is a diagonal matrix such that all entries of ΣGsΣ are nonnegative. Signed network (1) is said to achieve bipartite quasi-synchronization if the error eit=xit−σisti=1,2,…,N converges to some compact set M1=eit∈ℝn|eit≤ϵ,ϵ>0 when t⟶∞.Obviously, whenσi=1 for i=1,2,…,n, bipartite quasi-synchronization becomes quasi-synchronization.Suppose that the signed graphAsU,V,Gs of (1) is structurally balanced, Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, is a fixed diagonal matrix such that all entries of ΣGsΣ are nonnegative. The distributed impulsive protocol with pinning control of (1) is designed as(5)uit=∑k=1∞−c∑j=1Naijsxit−signaijsxjt−cdikxit−σistδt−tk,where c is a constant and dik>0 if node i is pinned; otherwise, dik=0. δ is the Dirac function, and tkk=1∞ is the impulse sequence satisfying 0=t0<t1<t2⋯<tk<⋯,limk⟶∞tk=∞.Remark 1.
Controller (5) indicates that if two nodes i and j are cooperative, aijs>0, and the coupling term is expressed as aijsxit−xjt; if two nodes i and j are competitive, aijs<0, and the coupling term is expressed as −aijsxit+xjt [36]. On the contrary, if the graph As is unsigned, controller (5) degenerates into a traditional distributed impulsive controller defined by previous studies [21, 23, 28].Based on the aforementioned control protocol (5), the impulsive signed network reads as(6)x˙it=Aixit+Bifxit,t≠tk,Δxit=−c∑j=1Naijsxitk−−signaijsxjtk−−cdikxitk−−σistk−,where Δxitk=xitk+−xitk−, xitk−=limh⟶0−xitk+h, and xitk=xitk+=limh⟶0+xitk+h.Remark 2.
According to Lemma2, it is necessary that As is structurally balanced during investigating the bipartite synchronization. In fact, Meng et al. [33] demonstrated that a network cannot obtain bipartite synchronization when the signed network is structurally unbalanced.Definition 3.
Letζ≜t1,t2,… be an impulsive sequence on t0,+∞. For any t∈t0,+∞, Nζt,t0 denotes the impulsive times of ζ on the interval t0,t. Then, the average impulsive interval, denoted by Ta, is defined by(7)Ta=limt⟶+∞t−t0Nζt,t0.Remark 3.
Obviously,0≤Ta≤∞. For example, let Nζt,t0=t3/2; then, Ta=0; let Nζt,t0=t1/2; then, Ta=∞. When 0<Ta<∞, it was proposed in [29] and then was applied in [19–23, 26–28]. The case of Ta=∞ was proposed in [30] when impulses occur infinitely but sparsely.


## 2.2. Synchronization Error Dynamics

LetL¯=l¯ijN×N be the Laplacian matrix of network (6); then, L¯=l¯ijN×N=ℬ−Gs, where ℬ=diagb1,b2,…,bN with bi=∑k=1Naiks. Let AU,V,G be the unsigned graph corresponding to As. Then, G=aijsN×N, and so, the corresponding Laplacian matrix L=lijN×N=ℬ−G with lij=−aijs for i≠j and lii=∑k=1,k≠iaiks.SinceAsU,V,Gs is structurally balanced and the diagonal matrix Σ=diagσ1,σ2,…,σN,σi∈±1,i=1,2,…,N, make sure that ΣGsΣ=G. By L¯=ℬ−Gs and L=ℬ−G, it is easy to see that L=ΣL¯Σ holds with lij=−aijs for i≠j and lii=∑k=1,k≠iaiks.Letx¯it=σixit, i=1,…,N, that is, xit=σix¯it. Then, it follows from (6) that(8)x¯⋅it=Aix¯it+Biσifσix¯it,t≠tk,Δx¯it=−c∑j=1Naijsx¯itk−−signaijsx¯jtk−−cdikx¯itk−−stk−.By Assumption 1,fi⋅, i∈1,…,N is odd. Then, σifσix¯it=fx¯it holds for any σi∈−1,1 and i∈1,…,N. Then, from (8), we get(9)x¯⋅it=Aix¯it+Bifx¯it,t≠tk,Δx¯it=−c∑j=1Nlijx¯jtk−−cdikx¯itk−−stk−.From (2) and (9), based on the fact that L is a matrix with zero row sum, the following error system can be obtained:(10)e˙it=Aieit+Bigeit+Wist,t≠tk,Δeit=−c∑j=1Nlijejtk−−cdikeitk−,where eit=x¯it−st, geit=feit+st−fst, and Wist=Ai−Ast+Bi−Bfst. Since st is bounded and f⋅ is a Lipschitz function, it is easy to know that Wist is bounded. Suppose supt≥t0Wist=wi, i=1,…,n.Now, we rewrite the error system with a matrix form at the impulse instantt=tk:(11)etk=IN−cL+Dk⊗Inetk−,where et=e1T,e2T,…,eNTT and Dk=diagd1k,d2k,…,dNk.Lemma 2.
(see [37]; Schur complement). Suppose that QxandRx are both symmetric time-varying matrices; then, for any time-varying matrix Sx,(12)QxSxSTxRx>0is equivalent to(1)
Qx>0, Rx−STxQ−1xSx>0,or(2)
Rx>0, Qx−STxR−1xSx>0.Definition 4.
Symmetric time-varying matrixPt∈ℝn×n is called to be uniformly bounded and positive definite if there exist constants mˇ>0 and m^>0 such that(13)0<mˇ≤λminPt≤λmaxPt≤m^,∀t≥0.Lemma 3.
(see [38]). Suppose that 0≤τit≤τ, i=1,2,…,n, Ft,w,w1,…,wn:ℝ+×ℝn+1⟶ℝ is nondecreasing in wi for each fixed t,w,w1,…,wi−1,wi+1,…,wn,i=1,2,…,n, and Ikw:ℝ⟶ℝ is nondecreasing in w,k∈ℕ+. If utandvt satisfy(14)D+ut≤Ft,ut,ut−τ1t,…,ut−τnt,t≠tk,utk+≤Ikutk−,k∈ℕ+,D+vt≥Ft,vt,vt−τ1t,…,vt−τnt,t≠tk,vtk+≥Ikvtk−,k∈ℕ+,then ut≤vt for −τ≤t≤0 implies that ut≤vt for t≥0.


## 3. Main Results

In this section, we derive some bipartite quasi-synchronization criteria for error system (10). As it is known, it is very difficult to achieve bipartite quasi-synchronization when we considered different average impulsive intervals. Here, we focus on two cases of average impulsive intervals. One is the case of Ta<∞, and the other is the case of Ta=∞.


### 3.1. The Case ofTa<∞

Theorem 1.
Suppose that Assumption1 holds and the average impulsive interval Ta<∞. Let Pit∈ℝn×n,i=1,2,…,N, be a sequence of symmetric time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers mˇi and m^i satisfying mˇi≤λminPit≤λmaxPit≤m^i, i=1,2,…,N. If there exist diagonal matrices Σj>0j=1,2, Dkk∈ℕ+, and scalars α, μkk=1,2,…,Nζt,s, c>0, such that(15)P˙i+PiAi+AiTPi−αPi+ΔTΣ1ΔPiBiPiBiTPi−Σ10Pi0−Σ2<0,(16)μkPtk−ΠkTPtk+Ptk+ΠkPtk+>0,(17)η≜lnμTa+α<0,where Pt=diagP1t,P2t,…,PNt, Πk=IN−cL+Dk⊗In, μ=limt⟶∞μ1+μ2+⋯μNζt,s/Nζt,s>0, and Δ=δijn×n, then the bipartite quasi-synchronization can be achieved. Specifically, for 0>η1>η, error system (10) between nonlinear system (1) and leader (2) exponentially converges into(18)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2η1minimˇi,and the convergence rate is −η1/2.Construct the Lyapunov function(19)Vet=∑i=1NeiTtPiteit.Then, the derivative ofVet is(20)V˙et=∑i=1NeiTtP˙iteit+2PiAieit+2PiBigeit+2PiWist.By Assumption1,(21)2∑i=1NeiTtPiBigeit≤∑i=1NeiTtPiBiΣ1−1BiTPieit+∑i=1NgeitTΣ1geit≤∑i=1NeiTtPiBiΣ1−1BiTPieit+∑i=1NeiTtΔTΣ1Δeit=∑i=1NeiTtPiBiΣ1−1BiTPi+ΔTΣ1Δeit,(22)2∑i=1NeiTtPiWist≤∑i=1NeiTtPiΣ2−1Pieit+∑i=1NWistTΣ2Wist≤∑i=1NeiTtPiΣ2−1Pieit+λmaxΣ2∑i=1Nωi2,where Δ=δijn×n.Substituting (21) and (22) into (20) yields(23)V˙et=∑i=1NeiTtP˙it+2PiAi+PiBiΣ1−1BiTPi+ΔTΣ1Δ+PiΣ2−1Pieit+λmaxΣ2∑i=1Nωi2.By Lemma2 and (15), we obtain(24)V˙et≤αVt+λmaxΣ2∑i=1Nωi2.Whent=tk, (11) gives(25)etk=IN−cL+Dk⊗Inetk−=Πkeitk−.By Lemma2 and (16), we have μkPtk−−ΠkTPtk+Πk>0. Thus,(26)Vtk+=eTtkPtk+etk=eTtk−ΠkTPtk−Πketk−≤μkeTtk−Ptk−etk−≤μkVtk−.For anyε>0, consider the following comparison system:(27)v˙et=αvt+λmaxΣ2∑i=1Nωi2+ε,t≠tk,vtk+=μkvtk−.v0=maxim^ie02.Letvt be the unique solution of (27); then, we obtain from Lemma 3 that Vt≤vt for all t>0. Using the idea of the variation of parameters, we can write the form of vt as(28)vt=Wt,0v0+∫0tWt,sλmaxΣ2∑i=1Nωi2+εds,where Wt,s, 0≤s≤t, satisfies(29)Wt,s=eαt−sμ1μ2⋯μNζt,s≤αt−sμ1+μ2+⋯μNζt,sNζt,sNζt,s=eαt−seNζt,slnμ1+μ2+⋯μNζt,s/Nζt,s=eαt−selnμ1+μ2+⋯μNζt,s/Nζt,st−s/Nζt−st−s.Due to the construction ofμ and the definition of Ta, for any η1, η<η1<0, there is T>0 such that, for t>T and 0≤s≤t,(30)Wt,s≤αt−selnμ/Ta+η1−ηt−s=elnμ/Ta+η1−η+αt−s=eη1t−s.Substituting (30) into (28) yields(31)vt≤η1tv0+∫0teη1t−sλmaxΣ2∑i=1Nωi2+εds≤η1tv0+λmaxΣ2∑i=1Nωi2+εη1eη1t−1≤η1tv0−λmaxΣ2∑i=1Nωi2+εη1.Letε⟶0; one has(32)minimˇiet2≤∑i=1NeitPiteit=Vt≤vt≤η1tv0−λmaxΣ2∑i=1Nωi2η1.Consequently,(33)et≤v0minimˇieη1/2t+−λmaxΣ2∑i=1Nωi2η1minimˇi,which implies that (10) converges exponentially into S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2/η1minimˇi at convergence rate −η1/2 when t⟶∞.Remark 4.
It is noticed that the impulses in Theorem1 may be desynchronizing μk>1, synchronizing μk<1, or inactive μk=1. However, most of the previous studies are devoted to investigate these impulses separately. The problem of bipartite quasi-synchronization is to find an impulsive interval Ta, a coupling strength c, and a pinning matrix Dk so that all stable or unstable isolated nodes in a network synchronize into a bounded region.Corollary 1.
Suppose that Assumption1 holds and the average impulsive interval Ta<∞. If there exist matrix P>0, diagonal matrices Σj>0j=1,2, D, and scalars c>0, 0<μ<∞, α, and mˇ>0 such that(34)PAi+AiTP−αP+ΔTΣ1ΔPBiPBiTP−Σ10P0−Σ2<0,(35)P≥mˇI,(36)η≜lnμTa+α<0,where Δ=δijn×n and μ=σmax2IN−cL+D, then (10) exponentially converges into(37)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2η1mˇat convergence rate −η1/2 for 0>η1>η.Proof.
Let(38)Vet=∑i=1NeiTtPeit.
Ift=tk, one has(39)Vtk+=eTtkIN⊗Petk=eTtk−IN−cL+DT⊗InIN⊗PIN−cL+D⊗Inetk−=eTtk−IN−cL+DTIN−cL+D⊗Petk−≤σmax2IN−cL+DTeTtk−Petk−≜μVtk−,where μ=σmax2IN−cL+D.
Using a similar proof as Theorem1, we have(40)Wt,s=eαt−s∏s≤tk≤tμ≤αt,sμNζt,s=eαt,seNζt,slnμ=eαt,selnμ/t,s/Nζt,st,s.
By the definition ofTa and (36), for any η1, η<η1<0, there exists sufficiently large T such that(41)Wt,s≤eαt,selnμ/Ta+η1−ηt−s=elnμ/Ta+η1−η+αt−s=eη1t−s,when t>T and 0≤s≤t.
Using Theorem1, we obtain the conclusion.


### 3.2. The Case ofTa=∞

This section concerns the bipartite quasi-synchronization of system (1) with average impulsive interval Ta=∞.Theorem 2.
Suppose that Assumption1 holds and the average impulsive interval Ta=∞. Let Piti=1,2,…,N be a sequence of time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers mˇi and m^i satisfying mˇi≤λminPt≤λmaxPt≤m^ii=1,2,…,N. If there exist diagonal matrices Σj>0j=1,2, Dkk∈ℕ+, and scalars α<0, c>0, and μii=1,2,…,Nζt,s such that(42)PiAi+AiTPi−αPi+ΔTΣ1ΔPiBiPiBiTPi−Σ10Pi0−Σ2<0,(43)μkPtk−ΠkTPtk+Ptk+ΠkPtk+>0,where Pt=diagP1t,P22,…,PNt, Πk=IN−cL+Dk⊗In, Δ=δijn×n, and μ=limt⟶∞μ1+μ2+⋯+μNζt,s/Nζt,s>0, then error system (10) exponentially converges into(44)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2α1mˇat convergence rate −α1/2 for 0>α1>α.Proof.
Using the similar proof of Theorem1, we obtain(45)Wt,s=eαt−sμ1μ2⋯+μNζt,s≤eαt−selnμ1+μ2+⋯μNζt,s/Nζt,s/t−s/Nζt,st−s.
Recalling the construction ofμ and Ta=∞, we can get(46)limt⟶∞lnμ1+μ2+⋯μNζt,s/Nζt,st−s/Nζt,s=0.
Thus, for anyα1, α<α1<0, there exists sufficiently large T such that(47)lnμ1+μ2+⋯+μNζt,s/Nζt,st−s/Nζt,s<α1−α,when t>T.
Then, by (45) and (47), we have(48)Wt,s≤eα1−αt−seαt−s=eα1t−s.
Using Theorem1, we obtain the conclusion.Remark 5.
From the conclusion of Theorem2, we can observe that there is no relationship between the convergence rate and the average impulsive interval when Ta=∞. In other words, the impulsive control, when it occurs infinitely but very sparsely, does not influence the bipartite quasi-synchronization of the networks, but it is necessary to put forward high requirements for their dynamics of the leader and five nodes, which can be seen from Example 2.Corollary 2.
Suppose that Assumption1 holds and the average impulsive interval Ta=∞. If there exist matrix P>0, diagonal matrices Σj>0j=1,2, D, and scalars α<0, c>0, mˇ>0, and 0<μ<∞ such that(49)PAi+AiTP−αP+ΔTΣ1ΔPBiPBiTP−Σ10P0−Σ2<0,(50)P≥mˇI,where Δ=δijn×n and μ=σmax2IN−cL+D, then error system (10) exponentially converges into(51)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2α1mˇ.at convergence rate −α1/2 for 0>α1>α.


## 3.1. The Case ofTa<∞

Theorem 1.
Suppose that Assumption1 holds and the average impulsive interval Ta<∞. Let Pit∈ℝn×n,i=1,2,…,N, be a sequence of symmetric time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers mˇi and m^i satisfying mˇi≤λminPit≤λmaxPit≤m^i, i=1,2,…,N. If there exist diagonal matrices Σj>0j=1,2, Dkk∈ℕ+, and scalars α, μkk=1,2,…,Nζt,s, c>0, such that(15)P˙i+PiAi+AiTPi−αPi+ΔTΣ1ΔPiBiPiBiTPi−Σ10Pi0−Σ2<0,(16)μkPtk−ΠkTPtk+Ptk+ΠkPtk+>0,(17)η≜lnμTa+α<0,where Pt=diagP1t,P2t,…,PNt, Πk=IN−cL+Dk⊗In, μ=limt⟶∞μ1+μ2+⋯μNζt,s/Nζt,s>0, and Δ=δijn×n, then the bipartite quasi-synchronization can be achieved. Specifically, for 0>η1>η, error system (10) between nonlinear system (1) and leader (2) exponentially converges into(18)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2η1minimˇi,and the convergence rate is −η1/2.Construct the Lyapunov function(19)Vet=∑i=1NeiTtPiteit.Then, the derivative ofVet is(20)V˙et=∑i=1NeiTtP˙iteit+2PiAieit+2PiBigeit+2PiWist.By Assumption1,(21)2∑i=1NeiTtPiBigeit≤∑i=1NeiTtPiBiΣ1−1BiTPieit+∑i=1NgeitTΣ1geit≤∑i=1NeiTtPiBiΣ1−1BiTPieit+∑i=1NeiTtΔTΣ1Δeit=∑i=1NeiTtPiBiΣ1−1BiTPi+ΔTΣ1Δeit,(22)2∑i=1NeiTtPiWist≤∑i=1NeiTtPiΣ2−1Pieit+∑i=1NWistTΣ2Wist≤∑i=1NeiTtPiΣ2−1Pieit+λmaxΣ2∑i=1Nωi2,where Δ=δijn×n.Substituting (21) and (22) into (20) yields(23)V˙et=∑i=1NeiTtP˙it+2PiAi+PiBiΣ1−1BiTPi+ΔTΣ1Δ+PiΣ2−1Pieit+λmaxΣ2∑i=1Nωi2.By Lemma2 and (15), we obtain(24)V˙et≤αVt+λmaxΣ2∑i=1Nωi2.Whent=tk, (11) gives(25)etk=IN−cL+Dk⊗Inetk−=Πkeitk−.By Lemma2 and (16), we have μkPtk−−ΠkTPtk+Πk>0. Thus,(26)Vtk+=eTtkPtk+etk=eTtk−ΠkTPtk−Πketk−≤μkeTtk−Ptk−etk−≤μkVtk−.For anyε>0, consider the following comparison system:(27)v˙et=αvt+λmaxΣ2∑i=1Nωi2+ε,t≠tk,vtk+=μkvtk−.v0=maxim^ie02.Letvt be the unique solution of (27); then, we obtain from Lemma 3 that Vt≤vt for all t>0. Using the idea of the variation of parameters, we can write the form of vt as(28)vt=Wt,0v0+∫0tWt,sλmaxΣ2∑i=1Nωi2+εds,where Wt,s, 0≤s≤t, satisfies(29)Wt,s=eαt−sμ1μ2⋯μNζt,s≤αt−sμ1+μ2+⋯μNζt,sNζt,sNζt,s=eαt−seNζt,slnμ1+μ2+⋯μNζt,s/Nζt,s=eαt−selnμ1+μ2+⋯μNζt,s/Nζt,st−s/Nζt−st−s.Due to the construction ofμ and the definition of Ta, for any η1, η<η1<0, there is T>0 such that, for t>T and 0≤s≤t,(30)Wt,s≤αt−selnμ/Ta+η1−ηt−s=elnμ/Ta+η1−η+αt−s=eη1t−s.Substituting (30) into (28) yields(31)vt≤η1tv0+∫0teη1t−sλmaxΣ2∑i=1Nωi2+εds≤η1tv0+λmaxΣ2∑i=1Nωi2+εη1eη1t−1≤η1tv0−λmaxΣ2∑i=1Nωi2+εη1.Letε⟶0; one has(32)minimˇiet2≤∑i=1NeitPiteit=Vt≤vt≤η1tv0−λmaxΣ2∑i=1Nωi2η1.Consequently,(33)et≤v0minimˇieη1/2t+−λmaxΣ2∑i=1Nωi2η1minimˇi,which implies that (10) converges exponentially into S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2/η1minimˇi at convergence rate −η1/2 when t⟶∞.Remark 4.
It is noticed that the impulses in Theorem1 may be desynchronizing μk>1, synchronizing μk<1, or inactive μk=1. However, most of the previous studies are devoted to investigate these impulses separately. The problem of bipartite quasi-synchronization is to find an impulsive interval Ta, a coupling strength c, and a pinning matrix Dk so that all stable or unstable isolated nodes in a network synchronize into a bounded region.Corollary 1.
Suppose that Assumption1 holds and the average impulsive interval Ta<∞. If there exist matrix P>0, diagonal matrices Σj>0j=1,2, D, and scalars c>0, 0<μ<∞, α, and mˇ>0 such that(34)PAi+AiTP−αP+ΔTΣ1ΔPBiPBiTP−Σ10P0−Σ2<0,(35)P≥mˇI,(36)η≜lnμTa+α<0,where Δ=δijn×n and μ=σmax2IN−cL+D, then (10) exponentially converges into(37)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2η1mˇat convergence rate −η1/2 for 0>η1>η.Proof.
Let(38)Vet=∑i=1NeiTtPeit.
Ift=tk, one has(39)Vtk+=eTtkIN⊗Petk=eTtk−IN−cL+DT⊗InIN⊗PIN−cL+D⊗Inetk−=eTtk−IN−cL+DTIN−cL+D⊗Petk−≤σmax2IN−cL+DTeTtk−Petk−≜μVtk−,where μ=σmax2IN−cL+D.
Using a similar proof as Theorem1, we have(40)Wt,s=eαt−s∏s≤tk≤tμ≤αt,sμNζt,s=eαt,seNζt,slnμ=eαt,selnμ/t,s/Nζt,st,s.
By the definition ofTa and (36), for any η1, η<η1<0, there exists sufficiently large T such that(41)Wt,s≤eαt,selnμ/Ta+η1−ηt−s=elnμ/Ta+η1−η+αt−s=eη1t−s,when t>T and 0≤s≤t.
Using Theorem1, we obtain the conclusion.


## 3.2. The Case ofTa=∞

This section concerns the bipartite quasi-synchronization of system (1) with average impulsive interval Ta=∞.Theorem 2.
Suppose that Assumption1 holds and the average impulsive interval Ta=∞. Let Piti=1,2,…,N be a sequence of time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers mˇi and m^i satisfying mˇi≤λminPt≤λmaxPt≤m^ii=1,2,…,N. If there exist diagonal matrices Σj>0j=1,2, Dkk∈ℕ+, and scalars α<0, c>0, and μii=1,2,…,Nζt,s such that(42)PiAi+AiTPi−αPi+ΔTΣ1ΔPiBiPiBiTPi−Σ10Pi0−Σ2<0,(43)μkPtk−ΠkTPtk+Ptk+ΠkPtk+>0,where Pt=diagP1t,P22,…,PNt, Πk=IN−cL+Dk⊗In, Δ=δijn×n, and μ=limt⟶∞μ1+μ2+⋯+μNζt,s/Nζt,s>0, then error system (10) exponentially converges into(44)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2α1mˇat convergence rate −α1/2 for 0>α1>α.Proof.
Using the similar proof of Theorem1, we obtain(45)Wt,s=eαt−sμ1μ2⋯+μNζt,s≤eαt−selnμ1+μ2+⋯μNζt,s/Nζt,s/t−s/Nζt,st−s.
Recalling the construction ofμ and Ta=∞, we can get(46)limt⟶∞lnμ1+μ2+⋯μNζt,s/Nζt,st−s/Nζt,s=0.
Thus, for anyα1, α<α1<0, there exists sufficiently large T such that(47)lnμ1+μ2+⋯+μNζt,s/Nζt,st−s/Nζt,s<α1−α,when t>T.
Then, by (45) and (47), we have(48)Wt,s≤eα1−αt−seαt−s=eα1t−s.
Using Theorem1, we obtain the conclusion.Remark 5.
From the conclusion of Theorem2, we can observe that there is no relationship between the convergence rate and the average impulsive interval when Ta=∞. In other words, the impulsive control, when it occurs infinitely but very sparsely, does not influence the bipartite quasi-synchronization of the networks, but it is necessary to put forward high requirements for their dynamics of the leader and five nodes, which can be seen from Example 2.Corollary 2.
Suppose that Assumption1 holds and the average impulsive interval Ta=∞. If there exist matrix P>0, diagonal matrices Σj>0j=1,2, D, and scalars α<0, c>0, mˇ>0, and 0<μ<∞ such that(49)PAi+AiTP−αP+ΔTΣ1ΔPBiPBiTP−Σ10P0−Σ2<0,(50)P≥mˇI,where Δ=δijn×n and μ=σmax2IN−cL+D, then error system (10) exponentially converges into(51)S=e∈ℝn×N|e≤−λmaxΣ2∑i=1Nωi2α1mˇ.at convergence rate −α1/2 for 0>α1>α.


## 4. Simulation Examples

In this section, two numerical examples are used to simulate the theoretical results in the above section.Example 1.
Bipartite quasi-synchronization withTa<∞.
Suppose that the leaderst satisfies the following Chua’s circuit:(52)s˙t=Ast+Bfst,where st=s1t,s2t,s3tT, A=−am1a01−110−b0, B=am1−m000000000, parameters m0=−1/7, m1=2/7, a=9, and b=14.286, and fst=0.5s1+1−s1−1,0,0T.
Consider a signed network shown in Figure1, consisting of five followers described by(53)x˙it=Aixit+Bifxit,i=1,2,3,4,5,where xit=xi1t,xi2t,xi3tT, fxit=0.5xi1+1−xi1−1,0,0T, and(54)A1=−am1+0.3a+0.101−110−b+0.40,B1=am1−m0−10000.20001,A2=−am1−0.5a−0.101−110−b−0.20,B2=am1−m0+10000.50001,A3=−am1+0.1a−0.101−1−0.110−b+0.10,B3=am1−m0+0.80000.50001,A4=−am1−0.1a01−1+0.110−b−0.10,B4=am1−m0−0.600010001,A5=−am1−0.1a−0.101−110−b+0.3−0.1,B5=am1−m0−1000−0.5000−1.
The dynamics of the five followers may be chaotic, stable, or periodic. From Figure1, one sees that the signed network is balanced with U1=1,2,3 and U2=4,5. Lemma 1 holds for Γ=diag1,1,1,−1,−1, and the Laplacian matrix is(55)L=1−100002−1−10−102−100001−100−101.
In addition,s0=0.2,0.1,0.1T is the initial condition of the leader node, and 0.1,0.2,0.2;0.1,0.1,0.2;−0.1,0.1,0.2;0.1,0.1,0.2;0.1,0.1,0.2 is the initial condition of the five nodes in the signed network. By simulation, one gets ω1=0.7554, ω2=0.5447, ω3=1.0092, ω4=0.8248, and ω5=1.2946.
Note that the node functionsfk⋅k=1,2,3 are odd functions satisfying the Lipschitz condition with Δ=diag1,0,0. Hence, Assumption 1 is satisfied.
LetD=diag3,0.9,3,0,0, i.e., agents 1, 2, and 3 are pinned. Let c=0.33; then, it is found that μ=σmax2IN−cL+D=0.8734. According to Corollary 1, by solving generalized eigenvalue problems (34)–(36), we can find a feasible solution with Ta=0.01, mˇ=3.13, α=23.812, P=3.2067−0.2836−0.0563−0.08464.69261.4635−0.03761.32164.4915, and Σ2=diag0.446,0.4376,0.4419. Then, λmaxΣ2=0.446. From (43), one obtains that η1=−3.25. Consequently, the error level is 0.4305. From Figures 2 and 3, one easily sees that the heterogeneous systems achieve bipartite leader-following quasi-synchronization with the distributed impulsive control, where e1,2,3t=e1T,e2T,e3TtT and e4,5t=e4T,e5TT.Figure 1
A signed network with five nodes.Figure 2
e1,2,3t on U1=1,2,3 and the error level 0.4305 in the signed network.Figure 3
e4,5t on U2=4,5 and the error level 0.4305 in the signed network.Example 2.
Bipartite quasi-synchronization withTa=∞.
According to the conditionα<0 in Corollary 2, the dynamics of the leader and the five followers need to be stable. The leader st has the following parameters: A=10−am1a01−110−b0, B=0.01am1−m000000000, parameters m0=−1/7, m1=2/7, a=9, and b=14.286, and fst=0.01s1+1−s1−1,0,0T.
The signed network is the same as that of Example1, in which five followers have the following parameters:(56)A1=8−am1−1a−101−110−b−1−0.3,B1=0.1am1−m0−1000−0.5000−1,A2=8−am1−0.9a−0.101−110−b−1−0.2,B2=am1−m0−0.9000−0.5000−0.9,A3=8−am1−0.8a−0.901−110−b−1−0.2,B3=0.1am1−m0−0.9000−0.5000−0.9,A4=10−am1−0.7a−101−110−b−0.8−0.2,B4=am1−m0−0.8000−0.5000−0.9,A5=−am1−0.6a−101−110−b−1−0.2,B5=am1−m0−0.7000−0.5000−1,and fxit=0.5xi1+1−xi1−1,0,0T, i=1,2,3,4,5.
By simulation, one getsω1=0.345, ω2=0.351, ω3=0.348, ω4=0.074, and ω5=0.099. In addition, the node function satisfies the Lipschitz condition with Δ=diag0.01,0,0. By solving generalized eigenvalue problems (49) and (50) of Corollary 2, one can find a feasible solution with Ta=∞ for Nζt,t0=t1/2, mˇ=3.3975, α=−0.1, P=15.3805−3.98226.4789−3.982283.6290−2.71296.4789−2.71296.9045, Σ2=I3, and α1=−0.099. Theoretically, the error level is 0.0113. From Figures 4 and 5, we can observe that the heterogeneous systems achieve bipartite leader-following quasi-synchronization with lower level, where e1,2,3t=e1T,e2T,e3TtT and e4,5t=e4T,e5TT.Figure 4
e1,2,3t on U1=1,2,3.Figure 5
e4,5t on U2=4,5.


## 5. Conclusions

This work investigates bipartite synchronization of heterogeneous signed networks. A pinning impulsive control strategy is proposed to realize bipartite synchronization. A leader-follower system within a nonzero error bound is analyzed simultaneously with these two kinds of impulsesTa<∞ and Ta=∞ by using the concept on average impulsive gain. Sufficient conditions on bipartite quasi-synchronization are obtained, and synchronization error is derived to show the synchronization level. However, when Ta=∞, synchronization is necessary to put forward high requirements for their dynamics of the leader and the followers with α<0. Simulation examples show that the theoretical results are effective.



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