# Sampled-Data-Based Adaptive Group Synchronization of Second-Order Nonlinear Complex Dynamical Networks with Time-Varying Delays

Authors: Bo Liu; Jiahui Bai; Yue Zhao; Chao Liu; Xuemin Yan; Hongtao Zhou
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper studies the adaptive group synchronization of second-order nonlinear complex dynamical networks with sampled-data and time-varying delays by designing a new adaptive strategy to feedback gains and coupling strengths. According to Lyapunov stability properties, it is shown that the agents of subgroups can converge the given synchronous states, respectively, under some conditions on the sampled period. Moreover, some simulation results are given.


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## Body

## 1. Introduction

Complex dynamical networks are used to describe the large size and complexity of the research object to solve the practical application problem by constructing the mathematical models in essence. In nature, synchronization is a ubiquitous phenomenon, such as the synchronization of beating rhythm of cardiac myocytes and consistency of fireflies twinkling. Recently, the synchronization problem of complex systems with nonlinear dynamical has attracted increasing attention and wide application including physics, mathematics, chemistry, biology, information science, electronics, and medicine [1–16]. Because of the extensive application value of synchronization in engineering technology, complex network synchronization has become a hot issue in the field of nonlinearity science, for example, the evolutionary origin of asymptotically stable consensus in [7] and the application of synchronization in engineering was introduced in [8].In order to achieve network synchronization, some advisable methods are introduced in outstanding works (e.g., [17–28]), such as pinning control [17–19] and adaptive strategies [20–28]. In [25], the authors introduced the adaptive coupling strengths and studied the adaptive synchronization of two heterogeneous second-order nonlinear coupled dynamical systems. The synchronization of fractional-order complex networks were well considered in [26–28] and applying decentralized adaptive strategies, pinning control and adaptive control strategy, respectively. The authors [29–31] investigated the synchronization of complex dynamical systems with time-varying delays. Works [32, 33] discussed adaptive consensus of networks with single-integrator nonlinear dynamics and adaptive synchronization of networks with double-integrator nonlinear dynamics, respectively. In [34], the author investigated the adaptive synchronization for first-order complex systems with local Lipschitz nonlinearity. Su et al. [35] also researched the adaptive flocking of multiagent networks with local Lipschitz nonlinearity. In engineering practice, the whole network (group) can be partitioned into several subnetworks (subgroups) to study the synchronization problems, called as group synchronization. Li et al. [36] investigated the group synchronization for complex systems with nonlinear dynamics. Some conditions were established in [37] for solving consensus problem of multiagent complex systems with double-integrator and sampled control. The consensus of complex networks with sampled data and time-delay topology was studied in [38].Inspired by these works, the adaptive group synchronization of second-order nonlinear complex dynamical undirected networks with sampled-data and time-varying delays will be discussed in this paper. And its main contributions are threefold: (1) the new second-order model with sampled-data and time-varying delays is established; (2) the communication delays of all the neighboring agents’ positions and velocities are time varying; (3) adaptive laws for solving the group synchronization of second-order nonlinear complex dynamical systems are introduced.The rest of this paper is arranged as follows. The mathematical model with time delays and sampled data and some necessary preliminaries are given in Section2. Section 3 presents the main results. Some numerical simulations are given in Section 4. Finally, Section 5 shows the conclusion.


## 2. Problem Formulation and Preliminaries

A second-order complex network with nonlinear dynamics consists ofN nodes and each node obeys(1)x˙it=vit,v˙it=fvits,viTs+∑j∈ℳ1icijtsaijxjTs−xiTs+∑j∈ℳ2idijtsbijxjTs+∑j∈ℳ1iαijtspijvjTs−viTs+∑j∈ℳ2iβijtsqijvjTs+ui,∀i∈ℓ1,∀t∈ts,ts+1fvits,viTs+∑j∈ℳ2icijtsaijxjTs−xiTs+∑j∈ℳ1idijtsbijxjTs+∑j∈ℳ2iαijtspijvjTs−viTs+∑j∈ℳ1iβijtsqijvjTs+ui,∀i∈ℓ2,∀t∈ts,ts+1,where xit∈Rn is the position vector of agent i; vit∈Rn is its velocity vector, for i=1,…,N as t∈0,+∞; f:Rn⟶Rn is a continuous differentiable function; Ts=ts−τts and τ>0; ℳi is the neighbor set of node i, ℳi∈ℳ1i∪ℳ2i with ℳ1i=xj∈X1:aij≥0,i,j∈ℓ1 and ℳ2i=xj∈X2:aij≥0,i,j∈ℓ2, where X=X1∪X2, ℓ=ℓ1∪ℓ2 with X1=x1,…,xL, X2=xL+1,…,xN, ℓ1=1,…,L, ℓ2=L+1,…,N, and L<N; cijts,dijts,αijts,andβijts are the position’s and velocity’s coupling strengths between agent i and agent j; and nonnegative numbers aij,bij,pij,andqij are the edge-weights connecting agent i and agent j.Design the control input as(2)ui=−citshixiTs−x¯1Ts−ditsliviTs−v¯1Ts,i∈ℓ1,−citshixiTs−x¯2Ts−ditsliviTs−v¯2Ts,i∈ℓ2,where hiandli are on-off controls, if node i is steered, then hi=1 and li=1, otherwise hi=0 and li=0, citsanddits represent the position’s and velocity’s feedback gains, respectively, x¯1t∈Rn and x¯2t∈Rn are the given synchronous positions, and v¯1t∈Rnandv¯2t∈Rn are their velocities, respectively.According to (1) and (2), we design the adaptive laws for coupling strengths respectively as(3)c˙ijts=aijkijxiTs−xjTsTxiTs−xjTs+x˙it−x˙jtTx˙it−x˙jt,i,j∈ℓ1,aijkijxiTs−xjTsTxiTs−xjTs+x˙it−x˙jtTx˙it−x˙jt,i,j∈ℓ2,α˙ijts=pijεijviTs−vjTsTviTs−vjTs+v˙it−v˙jtTv˙it−v˙jt,i,j∈ℓ1,pijεijviTs−vjTsTviTs−vjTs+v˙it−v˙jtTv˙it−v˙jt,i,j∈ℓ2,(4)d˙ijts=bijkijxjTs−x¯2TsTxjTs−x¯2Ts,i∈ℓ1,j∈ℓ2,bijkijxjTs−x¯1TsTxjTs−x¯1Ts,i∈ℓ2,j∈ℓ1,β˙ijts=qijεijvjTs−v¯2TsTvjTs−v¯2Ts,i∈ℓ1,j∈ℓ2,qijεijvjTs−v¯1TsTvjTs−v¯1Ts,i∈ℓ2,j∈ℓ1.where kij>0andεij>0 are the weights of cijtsandαijts, respectively.Similarly, we design the adaptive laws for the feedback gains, respectively, as(5)c˙its=hikixiTs−x¯1TsTxiTs−x¯1Ts+x˙it−x¯˙1tTx˙it−x¯˙1t,i∈ℓ1,hikixiTs−x¯2TsTxiTs−x¯2Ts+x˙it−x¯˙2tTx˙it−x¯˙2t,i∈ℓ2,d˙its=liεiviTs−v¯1TsTviTs−v¯1Ts+v˙it−v¯˙1tTv˙it−v¯˙1t,i∈ℓ1,liεiviTs−v¯2TsTviTs−v¯2Ts+v˙it−v¯˙2tTv˙it−v¯˙2t,i∈ℓ2,in which ki>0 and εi>0 are the weights of cits and dits, respectively.The position’s and velocity’s weighted coupling configuration matrices of system (1) can represented as(6)A=A11L×LB12L×N−LB21N−L×LA22N−L×N−L,P=P11L×LQ12L×N−LQ21N−L×LP22N−L×N−L,where(7)A11=a11−∑j=1La1j⋯a1L⋮⋱⋮aL1⋯aLL−∑j=1LaLj,A22=aL+1L+1−∑j=L+1NaL+1j⋯aL+1N⋮⋱⋮aNL+1⋯aNN−∑j=L+1NaNj,P11=p11−∑j=1Lp1j⋯p1L⋮⋱⋮pL1⋯pLL−∑j=1LpLj,P22=pL+1L+1−∑j=L+1NpL+1j⋯pL+1N⋮⋱⋮pNL+1⋯pNN−∑j=L+1NpNj.In order to solve the synchronization problem, we briefly give some assumptions, lemmas, and definitions used in this paper.Assumption 1.
(see [39]). The coupling strengths and feedback gains are all bounded, that is,(8)cijts≤cij,dijts≤dij,cits≤ci,αijts≤αij,βijts≤βij,dits≤di,where ⋅ is the Euclidean norm and cij,dij,ci,αij,βij,anddi are positive constants. In fact, the coupling strengths and feedback gains are usually bounded.Assumption 2.
(see [39]). 0≤τt≤τ, when t≥0 and τ>0 are constants.
Unlike some existing works, such as [40], 0≤τ˙t≤1 is required; however, this paper does not need to know any information about the derivative of τt.Assumption 3.
∃ρ1>0,ρ2>0 such that(9)fα,β−fγ,δ≤ρ1α−γ+ρ2β−δ,∀α,β,γ,δ∈Rn,which can guarantee the boundedness of the nonlinear term for system (1).Lemma 1.
(see [39]). Suppose that x,y∈RN are arbitrary vectors and matrix Q∈RN×N is positive definite; then, the inequality satisfies(10)2xTy≤xTQx+yTQ−1y.Lemma 2.
(see [39]). If A=aij∈RN×N is symmetric irreducible, each eigenvalue of A−B is negative, where aii=−∑j=1,j≠iNaij and B=diagb,0,…,0 with b>0.Lemma 3.
(see [39]). For an undirected graph G, its corresponding coupling matrix A is irreducible iff G is connected.Lemma 4.
(see [39]). If ∀xt∈Rn is real differentiable and W=WT>0 is a constant matrix, we can have(11)∫t−τttxkdkTW∫t−τttxkdk≤τ∫t−τttxTkWxkdk,t≥0,where 0≤τt≤τ.


## 3. Main Results

Forα>0 and the sample periodic T, we assume that(12)ti+1−ti=αTi,∀i=0,1,2,…,where t0<t1<⋯ are the discrete time periods and integer Ti>0 is a sampled time with Ti≤T. Inspired by [36], we design a linear synchronization protocol under the sampling period as(13)v˙its+α=v˙its−1Tv˙its=1−1Tv˙its,v˙its+2α=v˙its+α+v˙its+α−v˙its=1−2Tv˙its,⋮v˙its+1−α=v˙its+Tsα−α=1−Ts−1Tv˙its.Leth=0,1,…,Ts−1; thus, we have(14)v˙it=1−hT×fvits,viTs+∑j∈ℳ1icijtsaijxjTs−xiTs+∑j∈ℳ2idijtsbijxjTs+∑j∈ℳ1iαijtspijvjTs−viTs+∑j∈ℳ2iβijtsqijvjTs+ui,∀i∈ℓ1,∀t∈ts,ts+1,1−hT×fvits,viTs+∑j∈ℳ2icijtsaijxjTs−xiTs+∑j∈ℳ1idijtsbijxjTs+∑j∈ℳ2iαijtspijvjTs−viTs+∑j∈ℳ1iβijtsqijvjTs+ui,∀i∈ℓ2,∀t∈ts,ts+1.Theorem 1.
Consider connected network (1) with control input (2) steered by (3)–(5) under Assumptions 1–3 and Lemmas 1–4; then, each node’s position and velocity can asymptotically synchronize.Proof.
Let(15)x˜its≜xits−x¯1ts,v˜its≜vits−v¯1ts,for i∈ℓ1, and(16)x˜its≜xits−x¯2ts,v˜its≜vits−v¯2ts,for i∈ℓ2; then, we obtain(17)v˜˙it=1−hT×fvits,viTs−fv¯1ts,v¯1Ts+∑j∈ℳ1icijtsaijx˜jTs−x˜iTs+∑j∈ℳ2idijtsbijx˜jTs+∑j∈ℳ1iαijtspijv˜jTs−v˜iTs+∑j∈ℳ2iβijtsqijv˜jTs+ui,∀i∈ℓ1,∀t∈ts,ts+1,1−hT×fvits,viTs−fv¯2ts,v¯2Ts+∑j∈ℳ2icijtsaijx˜jTs−x˜iTs+∑j∈ℳ1idijtsbijx˜jTs+∑j∈ℳ2iαijtspijv˜jTs−v˜iTs+∑j∈ℳ1iβijtsqijv˜jTs+ui,∀i∈ℓ2,∀t∈ts,ts+1.
Construct a Lyapunov function as(18)Vts=V1ts+V2ts+V3ts,where(19)V1ts=12∑i∈ℳ1ix˜iTtsx˜its+∑i∈ℳ1i∑j∈ℳ1icijts−2cij−p24kij+∑i∈ℳ1i∑j∈ℳ2idijts−2dij−124kij+∑i∈ℳ1icits−3/2ci−p22ki+12∑i∈ℳ1iv˜iTtsv˜its+∑i∈ℳ1i∑j∈ℳ1ipijts−2pij−p24εij+∑i∈ℳ1i∑j∈ℳ2iqijts−2qij−124εij∑i∈ℳ1idits−3/2di−p22εi,V2ts=12∑i∈ℳ2ix˜iTtsx˜its+∑i∈ℳ2i∑j∈ℳ2icijts−2cij−p24kij+∑i∈ℳ2i∑j∈ℳ1idijts−2dij−124kij+∑i∈ℳ2icits−3/2ci−p22ki+12∑i∈ℳ2iv˜iTtsv˜its+∑i∈ℳ2i∑j∈ℳ2iαijts−2αij−p24εij+∑i∈ℳ2i∑j∈ℳ1iβijts−2βij−124εij+∑i∈ℳ2idits−3/2di−p22εi,V3ts=τ∑i∈ℳ1i2ρ1+ρ2+1+∑i∈ℳ1icijtsaij+∑i∈ℳ1iαijtspij+∑i∈ℳ2idijtsbij+∑i∈ℳ2iβijtsqij+citshi+ditsli∫Tstsk−ts+τx˜˙iTkx˜˙ikdk+τ∑i∈ℳ1i∫Tstsk−ts+τx˜˙iTkx˜˙ikdk+τ∑i∈ℳ2i2ρ3+ρ4+1+∑i∈ℳ2icijtsaij+∑i∈ℳ2iαijtspij+∑i∈ℳ1idijtsbij+∑i∈ℳ1iβijtsqij+citshi+ditsli∫Tstsk−ts+τx˜˙iTkx˜˙ikdk+τ∑i∈ℳ2i∫Tstsk−ts+τx˜˙iTkx˜˙ikdk,and p>0 is sufficiently large. Next, there are two cases to discuss.Case 1.
A11,A22,P11,andP22 are symmetric.
DifferentiatingV1ts, under Assumptions 1–3 and Lemmas 1–4, we can have(20)V˙1ts≤12∑i∈ℳ1ix˜iTtx˜it+12ρ2∑i∈ℳ1iv˜iTTsv˜iTs+12∑i∈ℳ1iξ1iv˜iTtv˜it−p2∑i∈ℳ1i∑j∈ℳ1iaijx˜iTs−x˜jTsTx˜iTs−x˜jTs+x˜˙it−x˜˙jtTx˜˙it−x˜˙jt−p2∑i∈ℳ1i∑j∈ℳ1ipijv˜iTs−v˜jTsTv˜iTs−v˜jTs+v˜˙it−v˜˙jtTv˜˙it−v˜˙jt−12∑i∈ℳ1i∑j∈ℳ2ibijx˜jTTsx˜jTs−12∑i∈ℳ1i∑j∈ℳ2iqijv˜jTTsv˜jTs−p2∑i∈ℳ1ihix˜iTTsx˜iTs+x˜˙iTtx˜˙it−p2∑i∈ℳ1iliv˜iTTsv˜iTs+v˜˙iTtv˜˙it,where ξ1i=1+2ρ1+ρ2+∑j∈ℳ1icijtsaij+∑j∈ℳ1iαijtspij+∑j∈ℳ2idijtsbij+∑j∈ℳ2iβijtsqij+citshi+ditsli.
Using Leibniz–Newton formula,(21)xts−xTs=∫Tstsx˙kdk,then(22)xtsT=xTsT+∫Tstsx˙kdkT,and then(23)x˜tsTx˜ts=x˜TsT+∫Tstsx˜˙kdkTx˜Ts+∫Tstsx˜˙kdk≤2x˜TsTx˜Ts+2∫Tstsx˜˙kdkT∫Tstsx˜˙kdk≤2x˜TsTx˜Ts+2τ∫Vstsx˜˙iTkx˜˙ikdk.
Similarly,(24)v˜tsTv˜ts≤2v˜TsTv˜Ts+2τ∫Tstsv˜˙iTkv˜˙ikdk.
Substituting (23) and (24) into (20), we obtain(25)V˙1ts≤∑i∈ℳ1ix˜iTTsx˜iTs+τ∑i∈ℳ1i∫Tstsx˜˙iTkx˜˙ikdk−12∑i∈ℳ1i∑j∈ℳ2ibijx˜jTTsx˜jTTs+∑i∈ℳ1iξ1i+12ρ2v˜iTTsv˜iTs+τ∑i∈ℳ1iξ1i∫Tstsv˜˙iTkv˜˙ikdk−12∑i∈ℳ1i∑j∈ℳ2iqijv˜jTTsv˜jTs+∑i∈ℳ1ipλ1xi˜Ts2+∑i∈ℳ1ipλ1xi˜.ts2+∑i∈ℳ1ipλ3vi˜Ts2+∑i∈ℳ1ipλ3vi˜.ts2,where ξ1i=1+2ρ1+ρ2+∑j∈ℳ1icijtsaij+∑j∈ℳ1iαijtspij+∑j∈ℳ2idijtsbij+∑j∈ℳ2iβijtsqij+citshi+ditsli and λ1andλ3 are the minimum eigenvalues of A11−H1andP11−L1, respectively.
Similarly, differentiatingV2ts and V3ts, we can obtain(26)V˙2ts≤∑i∈ℳ2ix˜iTTsx˜iTs+τ∑i∈ℳ2i∫Tstsx˜˙iTkx˜˙ikdk−12∑i∈ℳ2i∑j∈ℳ1ibijx˜jTTsx˜jTs+∑i∈ℳ2iξ2i+12ρ4v˜iTTsv˜iTs+τ∑i∈ℳ2iξ2i∫Tstsv˜˙iTkv˜˙ikdk−12∑i∈ℳ2i∑j∈ℳ1iqijv˜jTTsv˜jTs+∑i∈ℳ2ipλ2xi˜Ts2+∑i∈ℳ2ipλ2xi˜.ts2+∑i∈ℳ2ipλ4vi˜Ts2+∑i∈ℳ2ipλ4vi˜.ts2,where ξ2i=1+2ρ3+ρ4+∑j∈ℳ2icijtsaij+∑j∈ℳ2iαijtspij+∑j∈ℳ1idijtsbij+∑j∈ℳ1iβijtsqij+citshi+ditsli and λ2andλ4 are the minimum eigenvalues of A22−H2andP22−L2, respectively:(27)V˙3ts=τ2∑i∈ℳ1ix˜˙iTtsx˜˙its−τ∑i∈ℳ1i∫Tstsx˜˙iTkx˜˙ikdk+τ2∑i∈ℳ1iξ1iv˜˙iTtsv˜˙its−τ∑i∈ℳ1iξ1i∫Tstsv˜˙iTkv˜˙ikdk+τ2∑i∈ℳ2ix˜˙iTtsx˜˙its−τ∑i∈ℳ2i∫Tstsx˜˙iTkx˜˙ikdk+τ2∑i∈ℳ2iξ2iv˜˙iTtsv˜˙its−τ∑i∈ℳ2iξ2i∫Tstsv˜˙iTkv˜˙ikdk.
CombiningV˙1ts,V˙2ts, and V˙3ts, we obtain(28)V˙ts≤∑i∈ℳ1i1+pλ1x˜iTs2+∑i∈ℳ1iτ2+pλ1x˜˙its2+∑i∈ℳ2i1+pλ2x˜iTs2+∑i∈ℳ2iτ2+pλ2x˜˙its2+∑i∈ℳ1iξ1i+12ρ2−12∑j∈ℳ2ibji−12∑j∈ℳ2iqji+pλ3v˜iTs2+∑i∈ℳ1iτ2ξ1i+pλ3v˜˙its2+∑i∈ℳ2iξ2i+12ρ4−12∑j∈ℳ1ibji−12∑j∈ℳ1iqji+pλ4v˜iTs2+∑i∈ℳ2iτ2ξ2i+pλ4v˜˙its2,where λ1,λ2,λ3,andλ4 are the minimum eigenvalues of A11−H1,A22−H2,P11−L1,andP22−L2, respectively; with H1=diaghiandL1=diagli for ∀i∈ℓl, and H2=diaghiandL2=diagli for ∀i∈ℓ2. Furthermore, since A11,A22,P11,andP22 are both symmetric and diagonal matrices H1,H2,L1,andL2 have at least one element being 1, λi<0,i=1,2,3,4 based on Lemma 2. Therefore, V˙ts<0, if p>0 is sufficiently large.Case 2.
A11,A22,P11,andP22 are asymmetric.
It is known thatA11+A11T/2,A22+A22T/2,P11+P11T/2,andP22+P22T/2 are symmetric, even if A11,A22,P11,andP22 are asymmetric. Therefore, all eigenvalues of A11+A11T/2−H1,A22+A22T/2−H2,P11+P11T/2−L1,andP22+P22T/2−L2 are negative from Lemma 2. Similarly, we can have(29)V˙1ts≤∑i∈ℳ1ix˜iTTsx˜iTs+τ∑i∈ℳ1i∫Tstsx˜˙iTkx˜˙ikdk−12∑i∈ℳ1i∑j∈ℳ2ibijx˜jTTsx˜jTs+∑i∈ℳ1iξ1i+12ρ2v˜iTTsv˜iTs+τ∑i∈ℳ1iξ1i∫Tstsv˜˙iTkv˜˙ikdk−12∑i∈ℳ1i∑j∈ℳ2iqijv˜jTTsv˜jTs+∑i∈ℳ1ipλ1xi˜Ts2+∑i∈ℳ1ipλ1xi˜.ts2+∑i∈ℳ1ipλ3vi˜Ts2+∑i∈ℳ1ipλ3vi˜.ts2,where ξ2i=1+2ρ3+ρ4+∑j∈ℳ2icijtsaij+∑j∈ℳ2iαijtspij+∑j∈ℳ1idijtsbij+∑j∈ℳ1iβijtsqij+citshi+ditsli, and λ2andλ4 are the minimum eigenvalue of A22+A22T/2−H2andP22+P22T/2−L2, respectively.(30)V˙2ts≤∑i∈ℳ2ix˜iTTsx˜iTs+τ∑i∈ℳ2i∫Tstsx˜˙iTkx˜˙ikdk−12∑i∈ℳ2i∑j∈ℳ1ibijx˜jTTsx˜jTs+∑i∈ℳ2iξ2i+12ρ4v˜iTTsv˜iTs+τ∑i∈ℳ2iξ2i∫Tstsv˜˙iTkv˜˙ikdk−12∑i∈ℳ2i∑j∈ℳ1iqijv˜jTTsv˜jTs+∑i∈ℳ2ipλ2xi˜Ts2+∑i∈ℳ2ipλ2xi˜.ts2+∑i∈ℳ2ipλ4vi˜Ts2+∑i∈ℳ2ipλ4vi˜.ts2,where ξ1i=1+2ρ1+ρ2+∑j∈ℳ1icijtsaij+∑j∈ℳ1iαijtspij+∑j∈ℳ2idijtsbij+∑j∈ℳ2iβijtsqij+citshi+ditsli. and λ1andλ3 are the minimum eigenvalue of A11+A11T/2−H1,P11+P11T/2−L1, respectively:
CombiningV˙1ts,V˙2ts, and V˙3ts, we obtain(31)V˙ts≤∑i∈ℳ1i1+pλ1x˜iTs2+∑i∈ℳ1iτ2+pλ1x˜˙its2+∑i∈ℳ2i1+pλ2x˜iTs2+∑i∈ℳ2iτ2+pλ2x˜˙its2+∑i∈ℳ1iξ1i+12ρ2−12∑j∈ℳ2ibji−12∑j∈ℳ2iqji+pλ3v˜iTs2+∑i∈ℳ1iτ2ξ1i+pλ3v˜˙its2+∑i∈ℳ2iξ2i+12ρ4−12∑j∈ℳ1ibji−12∑j∈ℳ1iqji+pλ4v˜iTs2+∑i∈ℳ2iτ2ξ2i+pλ4v˜˙its2,where λ1,λ2,λ3,andλ4 are the minimum eigenvalue of A11+A11T/2−H1,A22+A22T/2−H2,P11+P11T/2−L1,andP22+P22T/2−L2, respectively. Even though matrix A11,A22,P11,andP22 are asymmetric, matrix A11+A11T/2,A22+A22T/2,P11+P11T/2,andP22+P22T/2 are symmetric; thus, from Lemma 2, we can obtain all eigenvalues of A11+A11T/2−H1,A22+A22T/2−H2,P11+P11T/2−L1,andP22+P22T/2−L2 are negative. So, λi<0,∀i∈1,2,3,4 and V˙ts<0, if p>0 is sufficiently large.
Therefore, all the agents of sampled-data based network (1) with time-varying delays can achieve the given synchronous states asymptotically.Remark 1.
When the topology structure is connected regardless of the coupled weighted matrices, the sampled-data-based network (1) with time-varying delays can be asymptotically group synchronized by controller (2).


## 4. Simulations

A complex dynamical system withN=6andL=3. Let the initial values be X0=2912201725−7229 and cij0=dij0=αij0=βij0=ci0=di0=0.01.LetA11,A22,P11,andP22 be symmetric as(32)A11=−2101−3101−1∗0.1,A22=−3212−4111−3∗0.05,P11=020201010∗0.01,P22=021202120∗0.005,and A11,A22,P11,andP22 be asymmetric as(33)A11=−3032−3101−1∗0.1,A22=−2203−6312−3∗0.05,P11=030201010∗0.01,P22=020303020∗0.005,respectively.Take(34)B12=010200000.5∗0.1,B21=120000000∗0.1,H1=diag0.100,H2=diag00.10.Figure1 presents that the effects of adaptive strategies for the synchronization of complex networks with nonlinear dynamical. Figure 1(a) shows the position and velocity of all nodes without adaptive strategies, and Figure 1(b) shows the position and velocity of all nodes with adaptive laws, respectively, in which the subgroups’ coupling matrices are symmetric. It is obvious to see the fact that all the nodes with adaptive laws can achieve their given synchronous states asymptotically, while all the nodes without adaptive laws cannot converge. Figures 2 and 3 show the simulations of network (1) with τ=0.1, in which the subnetworks’ coupling matrices are symmetric or asymmetric descried as Figures 2 and 3, respectively. Figures 2(a) and 2(b) present that the position and velocity of all nodes of network (1) with τ=0.1, where subgroups’ coupling matrices are symmetric, and the coupling strengths cij,dijαij,andβij and the feedback gains cianddi are presented in Figures 2(c)–2(h), respectively. Similarly, Figures 3(a) and 3(b) present that the position and velocity of all nodes of network (1) with τ=0.1, where the subgroups’ coupling matrices are asymmetric, the coupling strengths cij,dijαij,andβij and the feedback gains cianddi presented as Figures 3(c)–3(h), respectively. From Figures 2 and 3, we can find that all nodes of network (1) can achieve synchronization and the coupling strengths and the feedback gains also converge to be consistent. However, compared with Figure 3, the system in Figure 2 can achieve synchronization faster than that in Figure 3. Figure 4 is the simulation of network (1) with adaptive laws, in which the subgroups’ coupling matrices are symmetric, where Figures 4(a) and 4(b) are the positions and velocities of all nodes of network (1) with τ=0.1 and τ=1, respectively. We can know that, with the time delay τ increasing, the system cannot achieve synchronization.Figure 1
(a) Without adaptive strategies. (b) With adaptive strategies.
(a)(b)Figure 2
Intragroup coupling matrix is symmetric and time delayτ=0.1. (a) Positions. (b) Velocities. (c) cij. (d) dij. (e) ci. (f) αij. (g) βij. (h) di.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 3
Intragroup coupling matrix is asymmetric and time delayτ=0.1. (a) Positions. (b) Velocities. (c) cij. (d) dij. (e) ci. (f) αij. (g) βij. (h) di.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 4
Intragroup coupling matrix is symmetric and with adaptive strategies. (a) Time delayτ=0.1. (b) Time delay τ=1.
(a)(b)


## 5. Conclusion

The adaptive group synchronization of second-order nonlinear complex dynamical networks with time-varying delays and sampled data has been researched in this paper. A new adaptive law has been designed, and we have proved that the second-order system with sampled data can achieve group synchronization no matter whether the coupling matrix is symmetric or not. Moreover, we have discussed the influences of time-varying delays and adaptive laws for group synchronization of complex networks with nonlinear dynamics in the different simulations. Finally, some simulations have been represented.



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*Source: 9819156-2020-06-27.xml*