Research ArticleSampled-Data-Based Adaptive Group Synchronization ofSecond-Order Nonlinear Complex Dynamical Networks withTime-Varying Delays

Bo Liu 1 Jiahui Bai2 Yue Zhao1 Chao Liu 3 Xuemin Yan4 and Hongtao Zhou 5

1School of Information Engineering Minzu University of China Beijing 100081 China2College of Science North China University of Technology Beijing 100144 China3School of Mathematics Shanghai University of Finance and Economics Shanghai 200433 China4National Institute of Education Sciences Beijing 100088 China5Key Laboratory of Imaging Processing and Intelligence Control School of Artificial Intelligence and AutomationHuazhong University of Science and Technology Wuhan 430074 China

Correspondence should be addressed to Bo Liu boliuncuteducn

Received 23 April 2020 Accepted 5 June 2020 Published 27 June 2020

Academic Editor Zhiwei Gao

Copyright copy 2020 Bo Liu et al is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Complex dynamical networks are used to describe the large sizeand complexity of the research object to solve the practicalapplication problem by constructing the mathematical modelsin essence In nature synchronization is a ubiquitous phe-nomenon such as the synchronization of beating rhythm ofcardiac myocytes and consistency of fireflies twinkling Re-cently the synchronization problem of complex systems withnonlinear dynamical has attracted increasing attention andwide application including physics mathematics chemistrybiology information science electronics and medicine [1ndash16]Because of the extensive application value of synchronization inengineering technology complex network synchronization hasbecome a hot issue in the field of nonlinearity science forexample the evolutionary origin of asymptotically stableconsensus in [7] and the application of synchronization inengineering was introduced in [8]

In order to achieve network synchronization someadvisable methods are introduced in outstanding works

(eg [17ndash28]) such as pinning control [17ndash19] and adaptivestrategies [20ndash28] In [25] the authors introduced theadaptive coupling strengths and studied the adaptive syn-chronization of two heterogeneous second-order nonlinearcoupled dynamical systems e synchronization of frac-tional-order complex networks were well considered in[26ndash28] and applying decentralized adaptive strategiespinning control and adaptive control strategy respectivelye authors [29ndash31] investigated the synchronization ofcomplex dynamical systems with time-varying delaysWorks [32 33] discussed adaptive consensus of networkswith single-integrator nonlinear dynamics and adaptivesynchronization of networks with double-integrator non-linear dynamics respectively In [34] the author investi-gated the adaptive synchronization for first-order complexsystems with local Lipschitz nonlinearity Su et al [35] alsoresearched the adaptive flocking of multiagent networkswith local Lipschitz nonlinearity In engineering practice thewhole network (group) can be partitioned into severalsubnetworks (subgroups) to study the synchronization

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9819156 14 pageshttpsdoiorg10115520209819156

problems called as group synchronization Li et al [36]investigated the group synchronization for complex systemswith nonlinear dynamics Some conditions were establishedin [37] for solving consensus problem of multiagent complexsystems with double-integrator and sampled control econsensus of complex networks with sampled data and time-delay topology was studied in [38]

Inspired by these works the adaptive group synchro-nization of second-order nonlinear complex dynamicalundirected networks with sampled-data and time-varyingdelays will be discussed in this paper And its main con-tributions are threefold (1) the new second-order modelwith sampled-data and time-varying delays is established(2) the communication delays of all the neighboring agentsrsquo

positions and velocities are time varying (3) adaptive lawsfor solving the group synchronization of second-ordernonlinear complex dynamical systems are introduced

e rest of this paper is arranged as follows emathematical model with time delays and sampled dataand some necessary preliminaries are given in Section 2Section 3 presents the main results Some numericalsimulations are given in Section 4 Finally Section 5 showsthe conclusion

2 Problem Formulation and Preliminaries

A second-order complex network with nonlinear dynamicsconsists of N nodes and each node obeys

_xi(t) vi(t)

_vi(t)

f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944jisinM1i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM1i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873

+ 1113944jisinM2i

βij ts( 1113857qijvj Ts( 1113857 + ui foralli isin ℓ1 forallt isin ts ts+11113858 1113859

f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944jisinM2i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM2i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873

+ 1113944jisinM1i

βij ts( 1113857qijvj Ts( 1113857 + ui foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where xi(t) isin Rn is the position vector of agent i vi(t) isin Rn

is its velocity vector for i 1 N as t isin [0 +infin)f Rn⟶ Rn is a continuous differentiable function Ts

ts minus τ(ts) and τ gt 0 Mi is the neighbor set of node iMi isinM1i cupM2i with M1i xj isin X1 aij ge 0 i j isin ℓ11113966 1113967 andM2i xj isin X2 aij ge 0 i j isin ℓ21113966 1113967 where X X1 cupX2 ℓ

ℓ1 cup ℓ2 with X1 x1 xL1113864 1113865 X2 xL+1 xN1113864 1113865 ℓ1

1 L ℓ2 L + 1 N and LltN cij(ts) dij(ts)

αij(ts) and βij(ts) are the positionrsquos and velocityrsquos couplingstrengths between agent i and agent j and nonnegativenumbers aij bij pij and qij are the edge-weights connectingagent i and agent j

Design the control input as

ui

minus ci ts( 1113857hi xi Ts( 1113857 minus x1 Ts( 1113857( 1113857 minus di ts( 1113857li vi Ts( 1113857 minus v1 Ts( 1113857( 1113857 i isin ℓ1

minus ci ts( 1113857( 1113857hi xi Ts( 1113857 minus x2 Ts( 1113857( 1113857 minus di ts( 1113857li vi Ts( 1113857 minus v2 Ts( 1113857( 1113857 i isin ℓ2

⎧⎪⎨

⎪⎩(2)

where hi and li are on-off controls if node i is steered thenhi 1 and li 1 otherwise hi 0 and li 0 ci(ts) and di(ts)

represent the positionrsquos and velocityrsquos feedback gains

respectively x1(t) isin Rn and x2(t) isin Rn are the given syn-chronous positions and v1(t) isin Rn and v2(t) isin Rn are theirvelocities respectively

2 Mathematical Problems in Engineering

According to (1) and (2) we design the adaptive laws forcoupling strengths respectively as

_cij ts( 1113857

aijkij xi Ts( 1113857 minus xj Ts( 11138571113872 1113873T

xi Ts( 1113857 minus xj Ts( 11138571113872 1113873 + _xi(t) minus _xj(t)1113872 1113873T

_xi(t) minus _xj(t)1113872 11138731113876 1113877 i j isin ℓ1

aijkij xi Ts( 1113857 minus xj Ts( 11138571113872 1113873T

xi Ts( 1113857 minus xj Ts( 11138571113872 1113873 + _xi(t) minus _xj(t)1113872 1113873T

_xi(t) minus _xj(t)1113872 11138731113876 1113877 i j isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

_αij ts( 1113857

pijεij vi Ts( 1113857 minus vj Ts( 11138571113872 1113873T

vi Ts( 1113857 minus vj Ts( 11138571113872 1113873 + _vi(t) minus _vj(t)1113872 1113873T

_vi(t) minus _vj(t)1113872 11138731113876 1113877 i j isin ℓ1

pijεij vi Ts( 1113857 minus vj Ts( 11138571113872 1113873T

vi Ts( 1113857 minus vj Ts( 11138571113872 1113873 + _vi(t) minus _vj(t)1113872 1113873T

_vi(t) minus _vj(t)1113872 11138731113876 1113877 i j isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(3)

_dij ts( 1113857 bijkij xj Ts( 1113857 minus x2 Ts( 11138571113872 1113873

Txj Ts( 1113857 minus x2 Ts( 11138571113872 1113873 i isin ℓ1 j isin ℓ2

bijkij xj Ts( 1113857 minus x1 Ts( 11138571113872 1113873T

xj Ts( 1113857 minus x1 Ts( 11138571113872 1113873 i isin ℓ2 j isin ℓ1

⎧⎪⎨

⎪⎩

_βij ts( 1113857 qijεij vj Ts( 1113857 minus v2 Ts( 11138571113872 1113873

Tvj Ts( 1113857 minus v2 Ts( 11138571113872 1113873 i isin ℓ1 j isin ℓ2

qijεij vj Ts( 1113857 minus v1 Ts( 11138571113872 1113873T

vj Ts( 1113857 minus v1 Ts( 11138571113872 1113873 i isin ℓ2 j isin ℓ1

⎧⎪⎨

⎪⎩

(4)

where kij gt 0 and εij gt 0 are the weights of cij(ts) and αij(ts)respectively

Similarly we design the adaptive laws for the feedbackgains respectively as

_ci ts( 1113857

hiki xi Ts( 1113857 minus x1 Ts( 1113857( 1113857T

xi Ts( 1113857 minus x1 Ts( 1113857( 1113857 + _xi(t) minus _x1(t)1113872 1113873T

_xi(t) minus _x1(t)1113872 11138731113876 1113877 i isin ℓ1

hiki xi Ts( 1113857 minus x2 Ts( 1113857( 1113857T

xi Ts( 1113857 minus x2 Ts( 1113857( 1113857 + _xi(t) minus _x2(t)1113872 1113873T

_xi(t) minus _x2(t)1113872 11138731113876 1113877 i isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

_di ts( 1113857

liεi vi Ts( 1113857 minus v1 Ts( 1113857( 1113857T

vi Ts( 1113857 minus v1 Ts( 1113857( 1113857 + _vi(t) minus _v1(t)1113872 1113873T

_vi(t) minus _v1(t)1113872 11138731113876 1113877 i isin ℓ1

liεi vi Ts( 1113857 minus v2 Ts( 1113857( 1113857T

vi Ts( 1113857 minus v2 Ts( 1113857( 1113857 + _vi(t) minus _v2(t)1113872 1113873T

_vi(t) minus _v2(t)1113872 11138731113876 1113877 i isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

in which ki gt 0 and εi gt 0 are the weights of ci(ts) and di(ts)respectively

e positionrsquos and velocityrsquos weighted coupling config-uration matrices of system (1) can represented as

A ALtimesL11 B

Ltimes(Nminus L)12

B(Nminus L)timesL21 A

(Nminus L)times(Nminus L)22

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

P PLtimesL11 Q

Ltimes(Nminus L)12

Q(Nminus L)timesL21 P

(Nminus L)times(Nminus L)22

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

(6)

where

A11

a11 minus 1113944L

j1a1j middot middot middot a1L

⋮ ⋱ ⋮

aL1 middot middot middot aLL minus 1113944L

j1aLj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A22

a(L+1)(L+1) minus 1113944N

jL+1a(L+1)j middot middot middot a(L+1)N

⋮ ⋱ ⋮

aN(L+1) middot middot middot aNN minus 1113944N

jL+1aNj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P11

p11 minus 1113944L

j1p1j middot middot middot p1L

⋮ ⋱ ⋮

pL1 middot middot middot pLL minus 1113944L

j1pLj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P22

p(L+1)(L+1) minus 1113944N

jL+1p(L+1)j middot middot middot p(L+1)N

⋮ ⋱ ⋮

pN(L+1) middot middot middot pNN minus 1113944

N

jL+1pNj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

Mathematical Problems in Engineering 3

In order to solve the synchronization problemwe briefly givesome assumptions lemmas and definitions used in this paper

Assumption 1 (see [39]) e coupling strengths and feed-back gains are all bounded that is

cij ts( 1113857

le cij dij ts( 1113857

ledij ci ts( 1113857

le ci αij ts( 1113857

le αij βij ts( 1113857

le βij di ts( 1113857

ledi(8)

where middot is the Euclidean norm and cij dij ci αij βij and di

are positive constants In fact the coupling strengths andfeedback gains are usually bounded

Assumption 2 (see [39]) 0le τ(t)le τ when tge 0 and τ gt 0are constants

Unlike some existing works such as [40] 0le _τ(t)le 1 isrequired however this paper does not need to know anyinformation about the derivative of τ(t)

Assumption 3 existρ1 gt 0 ρ2 gt 0 such thatf(α β) minus f(c δ)le ρ1α minus c + ρ2β minus δ forallα β c δ isin R

n

(9)

which can guarantee the boundedness of the nonlinear termfor system (1)

Lemma 1 (see [39]) Suppose that x y isin RN are arbitraryvectors and matrix Q isin RNtimesN is positive definite then theinequality satisfies

2xTyle x

TQx + y

TQ

minus 1y (10)

Lemma 2 (see [39]) If A (aij) isin RNtimesN is symmetric ir-reducible each eigenvalue of A minus B is negative where aii

minus 1113936Nj1jnei aij and B diag(b 0 0) with bgt 0

Lemma 3 (see [39]) For an undirected graph G its corre-sponding coupling matrix A is irreducible iff G is connected

Lemma 4 (see [39]) If forallx(t) isin Rn is real differentiable andW WT gt 0 is a constant matrix we can have

1113946t

tminus τ(t)x(k)dk1113890 1113891

T

W 1113946t

tminus τ(t)x(k)dk1113890 1113891

le τ 1113946t

tminus τ(t)x

T(k)Wx(k)dk tge 0

(11)

where 0le τ(t)le τ

3 Main Results

For αgt 0 and the sample periodic T we assume that

ti+1 minus ti αTi foralli 0 1 2 (12)

where t0 lt t1 lt middot middot middot are the discrete time periods and integerTi gt 0 is a sampled timewithTi leT Inspired by [36] we designa linear synchronization protocol under the sampling period as

_vi ts + α( 1113857 _vi ts( 1113857 minus1T

_vi ts( 1113857 1 minus1T

1113874 1113875 _vi ts( 1113857

_vi ts + 2α( 1113857 _vi ts + α( 1113857 + _vi ts + α( 1113857 minus _vi ts( 1113857( 1113857 1 minus2T

1113874 1113875 _vi ts( 1113857

⋮

_vi ts+1 minus α( 1113857 _vi ts + Tsα minus α( 1113857 1 minusTs minus 1T

1113888 1113889 _vi ts( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

Let h 0 1 Ts minus 1 thus we have

_vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM1i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM2i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM2i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM1i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

4 Mathematical Problems in Engineering

Theorem 1 Consider connected network (1) with controlinput (2) steered by (3)ndash(5) under Assumptions 1ndash3 andLemmas 1ndash4 then each nodersquos position and velocity canasymptotically synchronize

Proof Let1113957xi ts( 1113857≜ xi ts( 1113857 minus x1 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857 minus v1 ts( 1113857(15)

for i isin ℓ1 and1113957xi ts( 1113857≜ xi ts( 1113857 minus x2 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857( minus v2 ts( 1113857(16)

for i isin ℓ2 then we obtain

_1113957vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v1 ts( 1113857 v1 Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM1i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM2i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v2 ts( 1113857 v2 Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM2i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM1i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Construct a Lyapunov function asV ts( 1113857 V1 ts( 1113857 + V2 ts( 1113857 + V3 ts( 1113857 (18)

where

V1 ts( 1113857 12

1113944iisinM1i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM1i

1113944jisinM2i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM1i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM1i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

pij ts( 1113857 minus 2pij minus p1113872 11138732

4εij

+ 1113944iisinM1i

1113944jisinM2i

qij ts( 1113857 minus 2qij minus 11113872 11138732

4εij

1113944iisinM1i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V2 ts( 1113857 12

1113944iisinM2i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM2i

1113944jisinM1i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM2i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM2i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

αij ts( 1113857 minus 2αij minus p1113872 11138732

4εij

+ 1113944iisinM2i

1113944jisinM1i

βij ts( 1113857 minus 2βij minus 11113872 11138732

4εij

+ 1113944iisinM2i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V3 ts( 1113857 τ 1113944iisinM1i

2ρ1 + ρ2 + 1 + 1113944iisinM1i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM1i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM2i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM2i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM1i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

+ τ 1113944iisinM2i

2ρ3 + ρ4 + 1 + 1113944iisinM2i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM2i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM1i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM1i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM2i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

(19)

Mathematical Problems in Engineering 5

and pgt 0 is sufficiently large Next there are two cases todiscuss

Case 1 A11 A22 P11 andP22 are symmetric

Differentiating V1(ts) under Assumptions 1ndash3 andLemmas 1ndash4 we can have

_V1 ts( 1113857le12

1113944iisinM1i

1113957xTi (t)1113957xi(t) +

12ρ2 1113944

iisinM1i

1113957vTi Ts( 11138571113957vi Ts( 1113857 +

12

1113944iisinM1i

ξ1i1113957vTi (t)1113957vi(t)

minusp

21113944

iisinM1i

1113944jisinM1i

aij 1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873T

1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873 + _1113957xi(t) minus _1113957xj(t)1113872 1113873T _1113957xi(t) minus _1113957xj(t)1113872 11138731113876 1113877

minusp

21113944

iisinM1i

1113944jisinM1i

pij 1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873T

1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873 + _1113957vi(t) minus _1113957vj(t)1113872 1113873T _1113957vi(t) minus _1113957vj(t)1113872 11138731113876 1113877

minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857 minus

12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

minusp

21113944

iisinM1i

hi 1113957xTi Ts( 11138571113957xi Ts( 1113857 + _1113957x

T

i (t) _1113957xi(t)1113876 1113877 minusp

21113944

iisinM1i

li 1113957vTi Ts( 11138571113957vi Ts( 1113857 + _1113957v

T

i (t) _1113957vi(t)1113876 1113877

(20)

where ξ1i 1 + 2ρ1 + ρ2 +1113936jisinM1icij (ts)aij + 1113936jisin M1iαij(ts)

pij+ 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liUsing LeibnizndashNewton formula

x ts( 1113857 minus x Ts( 1113857 1113946ts

Ts

_x(k)dk (21)

then

x ts( 1113857( 1113857T

x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_x(k)dk1113888 1113889

T

(22)

and then

1113957x ts( 1113857( 1113857T

1113957x ts( 1113857( 1113857 1113957x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

⎡⎣ ⎤⎦ 1113957x Ts( 1113857( 1113857 + 1113946ts

Ts

_1113957x(k)dk1113888 11138891113890 1113891

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

1113946ts

Ts

_1113957x(k)dk1113888 1113889

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2τ 1113946ts

Vs

_1113957xT

i (k) _1113957xi(k)dk

(23)

Similarly

1113957v ts( 1113857( 1113857T

1113957v ts( 1113857( 1113857le 2 1113957v Ts( 1113857( 1113857T

1113957v Ts( 1113857( 1113857 + 2τ 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(24)

Substituting (23) and (24) into (20) we obtain

6 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

According to (1) and (2) we design the adaptive laws forcoupling strengths respectively as

_cij ts( 1113857

aijkij xi Ts( 1113857 minus xj Ts( 11138571113872 1113873T

xi Ts( 1113857 minus xj Ts( 11138571113872 1113873 + _xi(t) minus _xj(t)1113872 1113873T

_xi(t) minus _xj(t)1113872 11138731113876 1113877 i j isin ℓ1

aijkij xi Ts( 1113857 minus xj Ts( 11138571113872 1113873T

xi Ts( 1113857 minus xj Ts( 11138571113872 1113873 + _xi(t) minus _xj(t)1113872 1113873T

_xi(t) minus _xj(t)1113872 11138731113876 1113877 i j isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

_αij ts( 1113857

pijεij vi Ts( 1113857 minus vj Ts( 11138571113872 1113873T

vi Ts( 1113857 minus vj Ts( 11138571113872 1113873 + _vi(t) minus _vj(t)1113872 1113873T

_vi(t) minus _vj(t)1113872 11138731113876 1113877 i j isin ℓ1

pijεij vi Ts( 1113857 minus vj Ts( 11138571113872 1113873T

vi Ts( 1113857 minus vj Ts( 11138571113872 1113873 + _vi(t) minus _vj(t)1113872 1113873T

_vi(t) minus _vj(t)1113872 11138731113876 1113877 i j isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(3)

_dij ts( 1113857 bijkij xj Ts( 1113857 minus x2 Ts( 11138571113872 1113873

Txj Ts( 1113857 minus x2 Ts( 11138571113872 1113873 i isin ℓ1 j isin ℓ2

bijkij xj Ts( 1113857 minus x1 Ts( 11138571113872 1113873T

xj Ts( 1113857 minus x1 Ts( 11138571113872 1113873 i isin ℓ2 j isin ℓ1

⎧⎪⎨

⎪⎩

_βij ts( 1113857 qijεij vj Ts( 1113857 minus v2 Ts( 11138571113872 1113873

Tvj Ts( 1113857 minus v2 Ts( 11138571113872 1113873 i isin ℓ1 j isin ℓ2

qijεij vj Ts( 1113857 minus v1 Ts( 11138571113872 1113873T

vj Ts( 1113857 minus v1 Ts( 11138571113872 1113873 i isin ℓ2 j isin ℓ1

⎧⎪⎨

⎪⎩

(4)

where kij gt 0 and εij gt 0 are the weights of cij(ts) and αij(ts)respectively

Similarly we design the adaptive laws for the feedbackgains respectively as

_ci ts( 1113857

hiki xi Ts( 1113857 minus x1 Ts( 1113857( 1113857T

xi Ts( 1113857 minus x1 Ts( 1113857( 1113857 + _xi(t) minus _x1(t)1113872 1113873T

_xi(t) minus _x1(t)1113872 11138731113876 1113877 i isin ℓ1

hiki xi Ts( 1113857 minus x2 Ts( 1113857( 1113857T

xi Ts( 1113857 minus x2 Ts( 1113857( 1113857 + _xi(t) minus _x2(t)1113872 1113873T

_xi(t) minus _x2(t)1113872 11138731113876 1113877 i isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

_di ts( 1113857

liεi vi Ts( 1113857 minus v1 Ts( 1113857( 1113857T

vi Ts( 1113857 minus v1 Ts( 1113857( 1113857 + _vi(t) minus _v1(t)1113872 1113873T

_vi(t) minus _v1(t)1113872 11138731113876 1113877 i isin ℓ1

liεi vi Ts( 1113857 minus v2 Ts( 1113857( 1113857T

vi Ts( 1113857 minus v2 Ts( 1113857( 1113857 + _vi(t) minus _v2(t)1113872 1113873T

_vi(t) minus _v2(t)1113872 11138731113876 1113877 i isin ℓ2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

in which ki gt 0 and εi gt 0 are the weights of ci(ts) and di(ts)respectively

e positionrsquos and velocityrsquos weighted coupling config-uration matrices of system (1) can represented as

A ALtimesL11 B

Ltimes(Nminus L)12

B(Nminus L)timesL21 A

(Nminus L)times(Nminus L)22

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

P PLtimesL11 Q

Ltimes(Nminus L)12

Q(Nminus L)timesL21 P

(Nminus L)times(Nminus L)22

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

(6)

where

A11

a11 minus 1113944L

j1a1j middot middot middot a1L

⋮ ⋱ ⋮

aL1 middot middot middot aLL minus 1113944L

j1aLj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A22

a(L+1)(L+1) minus 1113944N

jL+1a(L+1)j middot middot middot a(L+1)N

⋮ ⋱ ⋮

aN(L+1) middot middot middot aNN minus 1113944N

jL+1aNj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P11

p11 minus 1113944L

j1p1j middot middot middot p1L

⋮ ⋱ ⋮

pL1 middot middot middot pLL minus 1113944L

j1pLj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P22

p(L+1)(L+1) minus 1113944N

jL+1p(L+1)j middot middot middot p(L+1)N

⋮ ⋱ ⋮

pN(L+1) middot middot middot pNN minus 1113944

N

jL+1pNj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

Mathematical Problems in Engineering 3

In order to solve the synchronization problemwe briefly givesome assumptions lemmas and definitions used in this paper

Assumption 1 (see [39]) e coupling strengths and feed-back gains are all bounded that is

cij ts( 1113857

le cij dij ts( 1113857

ledij ci ts( 1113857

le ci αij ts( 1113857

le αij βij ts( 1113857

le βij di ts( 1113857

ledi(8)

where middot is the Euclidean norm and cij dij ci αij βij and di

are positive constants In fact the coupling strengths andfeedback gains are usually bounded

Assumption 2 (see [39]) 0le τ(t)le τ when tge 0 and τ gt 0are constants

Unlike some existing works such as [40] 0le _τ(t)le 1 isrequired however this paper does not need to know anyinformation about the derivative of τ(t)

Assumption 3 existρ1 gt 0 ρ2 gt 0 such thatf(α β) minus f(c δ)le ρ1α minus c + ρ2β minus δ forallα β c δ isin R

n

(9)

which can guarantee the boundedness of the nonlinear termfor system (1)

Lemma 1 (see [39]) Suppose that x y isin RN are arbitraryvectors and matrix Q isin RNtimesN is positive definite then theinequality satisfies

2xTyle x

TQx + y

TQ

minus 1y (10)

Lemma 2 (see [39]) If A (aij) isin RNtimesN is symmetric ir-reducible each eigenvalue of A minus B is negative where aii

minus 1113936Nj1jnei aij and B diag(b 0 0) with bgt 0

Lemma 3 (see [39]) For an undirected graph G its corre-sponding coupling matrix A is irreducible iff G is connected

Lemma 4 (see [39]) If forallx(t) isin Rn is real differentiable andW WT gt 0 is a constant matrix we can have

1113946t

tminus τ(t)x(k)dk1113890 1113891

T

W 1113946t

tminus τ(t)x(k)dk1113890 1113891

le τ 1113946t

tminus τ(t)x

T(k)Wx(k)dk tge 0

(11)

where 0le τ(t)le τ

3 Main Results

For αgt 0 and the sample periodic T we assume that

ti+1 minus ti αTi foralli 0 1 2 (12)

where t0 lt t1 lt middot middot middot are the discrete time periods and integerTi gt 0 is a sampled timewithTi leT Inspired by [36] we designa linear synchronization protocol under the sampling period as

_vi ts + α( 1113857 _vi ts( 1113857 minus1T

_vi ts( 1113857 1 minus1T

1113874 1113875 _vi ts( 1113857

_vi ts + 2α( 1113857 _vi ts + α( 1113857 + _vi ts + α( 1113857 minus _vi ts( 1113857( 1113857 1 minus2T

1113874 1113875 _vi ts( 1113857

⋮

_vi ts+1 minus α( 1113857 _vi ts + Tsα minus α( 1113857 1 minusTs minus 1T

1113888 1113889 _vi ts( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

Let h 0 1 Ts minus 1 thus we have

_vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM1i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM2i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM2i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM1i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

4 Mathematical Problems in Engineering

Theorem 1 Consider connected network (1) with controlinput (2) steered by (3)ndash(5) under Assumptions 1ndash3 andLemmas 1ndash4 then each nodersquos position and velocity canasymptotically synchronize

Proof Let1113957xi ts( 1113857≜ xi ts( 1113857 minus x1 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857 minus v1 ts( 1113857(15)

for i isin ℓ1 and1113957xi ts( 1113857≜ xi ts( 1113857 minus x2 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857( minus v2 ts( 1113857(16)

for i isin ℓ2 then we obtain

_1113957vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v1 ts( 1113857 v1 Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM1i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM2i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v2 ts( 1113857 v2 Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM2i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM1i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Construct a Lyapunov function asV ts( 1113857 V1 ts( 1113857 + V2 ts( 1113857 + V3 ts( 1113857 (18)

where

V1 ts( 1113857 12

1113944iisinM1i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM1i

1113944jisinM2i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM1i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM1i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

pij ts( 1113857 minus 2pij minus p1113872 11138732

4εij

+ 1113944iisinM1i

1113944jisinM2i

qij ts( 1113857 minus 2qij minus 11113872 11138732

4εij

1113944iisinM1i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V2 ts( 1113857 12

1113944iisinM2i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM2i

1113944jisinM1i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM2i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM2i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

αij ts( 1113857 minus 2αij minus p1113872 11138732

4εij

+ 1113944iisinM2i

1113944jisinM1i

βij ts( 1113857 minus 2βij minus 11113872 11138732

4εij

+ 1113944iisinM2i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V3 ts( 1113857 τ 1113944iisinM1i

2ρ1 + ρ2 + 1 + 1113944iisinM1i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM1i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM2i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM2i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM1i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

+ τ 1113944iisinM2i

2ρ3 + ρ4 + 1 + 1113944iisinM2i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM2i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM1i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM1i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM2i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

(19)

Mathematical Problems in Engineering 5

and pgt 0 is sufficiently large Next there are two cases todiscuss

Case 1 A11 A22 P11 andP22 are symmetric

Differentiating V1(ts) under Assumptions 1ndash3 andLemmas 1ndash4 we can have

_V1 ts( 1113857le12

1113944iisinM1i

1113957xTi (t)1113957xi(t) +

12ρ2 1113944

iisinM1i

1113957vTi Ts( 11138571113957vi Ts( 1113857 +

12

1113944iisinM1i

ξ1i1113957vTi (t)1113957vi(t)

minusp

21113944

iisinM1i

1113944jisinM1i

aij 1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873T

1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873 + _1113957xi(t) minus _1113957xj(t)1113872 1113873T _1113957xi(t) minus _1113957xj(t)1113872 11138731113876 1113877

minusp

21113944

iisinM1i

1113944jisinM1i

pij 1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873T

1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873 + _1113957vi(t) minus _1113957vj(t)1113872 1113873T _1113957vi(t) minus _1113957vj(t)1113872 11138731113876 1113877

minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857 minus

12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

minusp

21113944

iisinM1i

hi 1113957xTi Ts( 11138571113957xi Ts( 1113857 + _1113957x

T

i (t) _1113957xi(t)1113876 1113877 minusp

21113944

iisinM1i

li 1113957vTi Ts( 11138571113957vi Ts( 1113857 + _1113957v

T

i (t) _1113957vi(t)1113876 1113877

(20)

where ξ1i 1 + 2ρ1 + ρ2 +1113936jisinM1icij (ts)aij + 1113936jisin M1iαij(ts)

pij+ 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liUsing LeibnizndashNewton formula

x ts( 1113857 minus x Ts( 1113857 1113946ts

Ts

_x(k)dk (21)

then

x ts( 1113857( 1113857T

x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_x(k)dk1113888 1113889

T

(22)

and then

1113957x ts( 1113857( 1113857T

1113957x ts( 1113857( 1113857 1113957x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

⎡⎣ ⎤⎦ 1113957x Ts( 1113857( 1113857 + 1113946ts

Ts

_1113957x(k)dk1113888 11138891113890 1113891

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

1113946ts

Ts

_1113957x(k)dk1113888 1113889

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2τ 1113946ts

Vs

_1113957xT

i (k) _1113957xi(k)dk

(23)

Similarly

1113957v ts( 1113857( 1113857T

1113957v ts( 1113857( 1113857le 2 1113957v Ts( 1113857( 1113857T

1113957v Ts( 1113857( 1113857 + 2τ 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(24)

Substituting (23) and (24) into (20) we obtain

6 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

In order to solve the synchronization problemwe briefly givesome assumptions lemmas and definitions used in this paper

Assumption 1 (see [39]) e coupling strengths and feed-back gains are all bounded that is

cij ts( 1113857

le cij dij ts( 1113857

ledij ci ts( 1113857

le ci αij ts( 1113857

le αij βij ts( 1113857

le βij di ts( 1113857

ledi(8)

where middot is the Euclidean norm and cij dij ci αij βij and di

are positive constants In fact the coupling strengths andfeedback gains are usually bounded

Assumption 2 (see [39]) 0le τ(t)le τ when tge 0 and τ gt 0are constants

Unlike some existing works such as [40] 0le _τ(t)le 1 isrequired however this paper does not need to know anyinformation about the derivative of τ(t)

Assumption 3 existρ1 gt 0 ρ2 gt 0 such thatf(α β) minus f(c δ)le ρ1α minus c + ρ2β minus δ forallα β c δ isin R

n

(9)

which can guarantee the boundedness of the nonlinear termfor system (1)

Lemma 1 (see [39]) Suppose that x y isin RN are arbitraryvectors and matrix Q isin RNtimesN is positive definite then theinequality satisfies

2xTyle x

TQx + y

TQ

minus 1y (10)

Lemma 2 (see [39]) If A (aij) isin RNtimesN is symmetric ir-reducible each eigenvalue of A minus B is negative where aii

minus 1113936Nj1jnei aij and B diag(b 0 0) with bgt 0

Lemma 3 (see [39]) For an undirected graph G its corre-sponding coupling matrix A is irreducible iff G is connected

Lemma 4 (see [39]) If forallx(t) isin Rn is real differentiable andW WT gt 0 is a constant matrix we can have

1113946t

tminus τ(t)x(k)dk1113890 1113891

T

W 1113946t

tminus τ(t)x(k)dk1113890 1113891

le τ 1113946t

tminus τ(t)x

T(k)Wx(k)dk tge 0

(11)

where 0le τ(t)le τ

3 Main Results

For αgt 0 and the sample periodic T we assume that

ti+1 minus ti αTi foralli 0 1 2 (12)

where t0 lt t1 lt middot middot middot are the discrete time periods and integerTi gt 0 is a sampled timewithTi leT Inspired by [36] we designa linear synchronization protocol under the sampling period as

_vi ts + α( 1113857 _vi ts( 1113857 minus1T

_vi ts( 1113857 1 minus1T

1113874 1113875 _vi ts( 1113857

_vi ts + 2α( 1113857 _vi ts + α( 1113857 + _vi ts + α( 1113857 minus _vi ts( 1113857( 1113857 1 minus2T

1113874 1113875 _vi ts( 1113857

⋮

_vi ts+1 minus α( 1113857 _vi ts + Tsα minus α( 1113857 1 minusTs minus 1T

1113888 1113889 _vi ts( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

Let h 0 1 Ts minus 1 thus we have

_vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM1i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM2i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij xj Ts( 1113857 minus xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bijxj Ts( 1113857 + 1113944jisinM2i

αij ts( 1113857pij vj Ts( 1113857 minus vi Ts( 11138571113872 1113873⎡⎢⎢⎢⎣

+ 1113944jisinM1i

βij ts( 1113857qijvj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

4 Mathematical Problems in Engineering

Theorem 1 Consider connected network (1) with controlinput (2) steered by (3)ndash(5) under Assumptions 1ndash3 andLemmas 1ndash4 then each nodersquos position and velocity canasymptotically synchronize

Proof Let1113957xi ts( 1113857≜ xi ts( 1113857 minus x1 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857 minus v1 ts( 1113857(15)

for i isin ℓ1 and1113957xi ts( 1113857≜ xi ts( 1113857 minus x2 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857( minus v2 ts( 1113857(16)

for i isin ℓ2 then we obtain

_1113957vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v1 ts( 1113857 v1 Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM1i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM2i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v2 ts( 1113857 v2 Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM2i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM1i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Construct a Lyapunov function asV ts( 1113857 V1 ts( 1113857 + V2 ts( 1113857 + V3 ts( 1113857 (18)

where

V1 ts( 1113857 12

1113944iisinM1i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM1i

1113944jisinM2i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM1i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM1i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

pij ts( 1113857 minus 2pij minus p1113872 11138732

4εij

+ 1113944iisinM1i

1113944jisinM2i

qij ts( 1113857 minus 2qij minus 11113872 11138732

4εij

1113944iisinM1i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V2 ts( 1113857 12

1113944iisinM2i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM2i

1113944jisinM1i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM2i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM2i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

αij ts( 1113857 minus 2αij minus p1113872 11138732

4εij

+ 1113944iisinM2i

1113944jisinM1i

βij ts( 1113857 minus 2βij minus 11113872 11138732

4εij

+ 1113944iisinM2i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V3 ts( 1113857 τ 1113944iisinM1i

2ρ1 + ρ2 + 1 + 1113944iisinM1i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM1i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM2i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM2i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM1i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

+ τ 1113944iisinM2i

2ρ3 + ρ4 + 1 + 1113944iisinM2i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM2i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM1i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM1i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM2i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

(19)

Mathematical Problems in Engineering 5

and pgt 0 is sufficiently large Next there are two cases todiscuss

Case 1 A11 A22 P11 andP22 are symmetric

Differentiating V1(ts) under Assumptions 1ndash3 andLemmas 1ndash4 we can have

_V1 ts( 1113857le12

1113944iisinM1i

1113957xTi (t)1113957xi(t) +

12ρ2 1113944

iisinM1i

1113957vTi Ts( 11138571113957vi Ts( 1113857 +

12

1113944iisinM1i

ξ1i1113957vTi (t)1113957vi(t)

minusp

21113944

iisinM1i

1113944jisinM1i

aij 1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873T

1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873 + _1113957xi(t) minus _1113957xj(t)1113872 1113873T _1113957xi(t) minus _1113957xj(t)1113872 11138731113876 1113877

minusp

21113944

iisinM1i

1113944jisinM1i

pij 1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873T

1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873 + _1113957vi(t) minus _1113957vj(t)1113872 1113873T _1113957vi(t) minus _1113957vj(t)1113872 11138731113876 1113877

minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857 minus

12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

minusp

21113944

iisinM1i

hi 1113957xTi Ts( 11138571113957xi Ts( 1113857 + _1113957x

T

i (t) _1113957xi(t)1113876 1113877 minusp

21113944

iisinM1i

li 1113957vTi Ts( 11138571113957vi Ts( 1113857 + _1113957v

T

i (t) _1113957vi(t)1113876 1113877

(20)

where ξ1i 1 + 2ρ1 + ρ2 +1113936jisinM1icij (ts)aij + 1113936jisin M1iαij(ts)

pij+ 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liUsing LeibnizndashNewton formula

x ts( 1113857 minus x Ts( 1113857 1113946ts

Ts

_x(k)dk (21)

then

x ts( 1113857( 1113857T

x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_x(k)dk1113888 1113889

T

(22)

and then

1113957x ts( 1113857( 1113857T

1113957x ts( 1113857( 1113857 1113957x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

⎡⎣ ⎤⎦ 1113957x Ts( 1113857( 1113857 + 1113946ts

Ts

_1113957x(k)dk1113888 11138891113890 1113891

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

1113946ts

Ts

_1113957x(k)dk1113888 1113889

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2τ 1113946ts

Vs

_1113957xT

i (k) _1113957xi(k)dk

(23)

Similarly

1113957v ts( 1113857( 1113857T

1113957v ts( 1113857( 1113857le 2 1113957v Ts( 1113857( 1113857T

1113957v Ts( 1113857( 1113857 + 2τ 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(24)

Substituting (23) and (24) into (20) we obtain

6 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

Theorem 1 Consider connected network (1) with controlinput (2) steered by (3)ndash(5) under Assumptions 1ndash3 andLemmas 1ndash4 then each nodersquos position and velocity canasymptotically synchronize

Proof Let1113957xi ts( 1113857≜ xi ts( 1113857 minus x1 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857 minus v1 ts( 1113857(15)

for i isin ℓ1 and1113957xi ts( 1113857≜ xi ts( 1113857 minus x2 ts( 1113857

1113957vi ts( 1113857≜ vi ts( 1113857( minus v2 ts( 1113857(16)

for i isin ℓ2 then we obtain

_1113957vi(t)

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v1 ts( 1113857 v1 Ts( 1113857( 1113857 + 1113944

jisinM1i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM2i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM1i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM2i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ1 forallt isin ts ts+11113858 1113859

1 minush

T1113888 1113889 times f vi ts( 1113857 vi Ts( 1113857( 1113857 minus f v2 ts( 1113857 v2 Ts( 1113857( 1113857 + 1113944

jisinM2i

cij ts( 1113857aij 1113957xj Ts( 1113857 minus 1113957xi Ts( 11138571113872 1113873 + 1113944jisinM1i

dij ts( 1113857bij1113957xj Ts( 1113857⎡⎢⎢⎢⎣

+ 1113944jisinM2i

αij ts( 1113857pij 1113957vj Ts( 1113857 minus 1113957vi Ts( 11138571113872 1113873 + 1113944jisinM1i

βij ts( 1113857qij1113957vj Ts( 1113857 + ui⎤⎥⎥⎥⎦ foralli isin ℓ2 forallt isin ts ts+11113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Construct a Lyapunov function asV ts( 1113857 V1 ts( 1113857 + V2 ts( 1113857 + V3 ts( 1113857 (18)

where

V1 ts( 1113857 12

1113944iisinM1i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM1i

1113944jisinM2i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM1i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM1i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM1i

1113944jisinM1i

pij ts( 1113857 minus 2pij minus p1113872 11138732

4εij

+ 1113944iisinM1i

1113944jisinM2i

qij ts( 1113857 minus 2qij minus 11113872 11138732

4εij

1113944iisinM1i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V2 ts( 1113857 12

1113944iisinM2i

1113957xTi ts( 11138571113957xi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

cij ts( 1113857 minus 2cij minus p1113872 11138732

4kij

+ 1113944iisinM2i

1113944jisinM1i

dij ts( 1113857 minus 2dij minus 11113872 11138732

4kij

+ 1113944iisinM2i

ci ts( 1113857 minus (32)ci minus p( 11138572

2ki

+12

1113944iisinM2i

1113957vTi ts( 11138571113957vi ts( 1113857 + 1113944

iisinM2i

1113944jisinM2i

αij ts( 1113857 minus 2αij minus p1113872 11138732

4εij

+ 1113944iisinM2i

1113944jisinM1i

βij ts( 1113857 minus 2βij minus 11113872 11138732

4εij

+ 1113944iisinM2i

di ts( 1113857 minus (32)di minus p( 11138572

2εi

V3 ts( 1113857 τ 1113944iisinM1i

2ρ1 + ρ2 + 1 + 1113944iisinM1i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM1i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM2i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM2i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM1i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

+ τ 1113944iisinM2i

2ρ3 + ρ4 + 1 + 1113944iisinM2i

cij ts( 1113857aij⎛⎝ ⎞⎠ + 1113944

iisinM2i

αij ts( 1113857pij⎛⎝ ⎞⎠ + 1113944

iisinM1i

dij ts( 1113857bij⎛⎝ ⎞⎠ + 1113944

iisinM1i

βij ts( 1113857qij⎛⎝ ⎞⎠⎡⎢⎢⎢⎣

+ ci ts( 1113857hi( 1113857 + di ts( 1113857li( 11138571113859 1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk + τ 1113944iisinM2i

1113946ts

Ts

k minus ts + τ( 1113857 _1113957xT

i (k) _1113957xi(k)dk

(19)

Mathematical Problems in Engineering 5

and pgt 0 is sufficiently large Next there are two cases todiscuss

Case 1 A11 A22 P11 andP22 are symmetric

Differentiating V1(ts) under Assumptions 1ndash3 andLemmas 1ndash4 we can have

_V1 ts( 1113857le12

1113944iisinM1i

1113957xTi (t)1113957xi(t) +

12ρ2 1113944

iisinM1i

1113957vTi Ts( 11138571113957vi Ts( 1113857 +

12

1113944iisinM1i

ξ1i1113957vTi (t)1113957vi(t)

minusp

21113944

iisinM1i

1113944jisinM1i

aij 1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873T

1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873 + _1113957xi(t) minus _1113957xj(t)1113872 1113873T _1113957xi(t) minus _1113957xj(t)1113872 11138731113876 1113877

minusp

21113944

iisinM1i

1113944jisinM1i

pij 1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873T

1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873 + _1113957vi(t) minus _1113957vj(t)1113872 1113873T _1113957vi(t) minus _1113957vj(t)1113872 11138731113876 1113877

minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857 minus

12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

minusp

21113944

iisinM1i

hi 1113957xTi Ts( 11138571113957xi Ts( 1113857 + _1113957x

T

i (t) _1113957xi(t)1113876 1113877 minusp

21113944

iisinM1i

li 1113957vTi Ts( 11138571113957vi Ts( 1113857 + _1113957v

T

i (t) _1113957vi(t)1113876 1113877

(20)

where ξ1i 1 + 2ρ1 + ρ2 +1113936jisinM1icij (ts)aij + 1113936jisin M1iαij(ts)

pij+ 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liUsing LeibnizndashNewton formula

x ts( 1113857 minus x Ts( 1113857 1113946ts

Ts

_x(k)dk (21)

then

x ts( 1113857( 1113857T

x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_x(k)dk1113888 1113889

T

(22)

and then

1113957x ts( 1113857( 1113857T

1113957x ts( 1113857( 1113857 1113957x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

⎡⎣ ⎤⎦ 1113957x Ts( 1113857( 1113857 + 1113946ts

Ts

_1113957x(k)dk1113888 11138891113890 1113891

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

1113946ts

Ts

_1113957x(k)dk1113888 1113889

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2τ 1113946ts

Vs

_1113957xT

i (k) _1113957xi(k)dk

(23)

Similarly

1113957v ts( 1113857( 1113857T

1113957v ts( 1113857( 1113857le 2 1113957v Ts( 1113857( 1113857T

1113957v Ts( 1113857( 1113857 + 2τ 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(24)

Substituting (23) and (24) into (20) we obtain

6 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

and pgt 0 is sufficiently large Next there are two cases todiscuss

Case 1 A11 A22 P11 andP22 are symmetric

Differentiating V1(ts) under Assumptions 1ndash3 andLemmas 1ndash4 we can have

_V1 ts( 1113857le12

1113944iisinM1i

1113957xTi (t)1113957xi(t) +

12ρ2 1113944

iisinM1i

1113957vTi Ts( 11138571113957vi Ts( 1113857 +

12

1113944iisinM1i

ξ1i1113957vTi (t)1113957vi(t)

minusp

21113944

iisinM1i

1113944jisinM1i

aij 1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873T

1113957xi Ts( 1113857 minus 1113957xj Ts( 11138571113872 1113873 + _1113957xi(t) minus _1113957xj(t)1113872 1113873T _1113957xi(t) minus _1113957xj(t)1113872 11138731113876 1113877

minusp

21113944

iisinM1i

1113944jisinM1i

pij 1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873T

1113957vi Ts( 1113857 minus 1113957vj Ts( 11138571113872 1113873 + _1113957vi(t) minus _1113957vj(t)1113872 1113873T _1113957vi(t) minus _1113957vj(t)1113872 11138731113876 1113877

minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857 minus

12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

minusp

21113944

iisinM1i

hi 1113957xTi Ts( 11138571113957xi Ts( 1113857 + _1113957x

T

i (t) _1113957xi(t)1113876 1113877 minusp

21113944

iisinM1i

li 1113957vTi Ts( 11138571113957vi Ts( 1113857 + _1113957v

T

i (t) _1113957vi(t)1113876 1113877

(20)

where ξ1i 1 + 2ρ1 + ρ2 +1113936jisinM1icij (ts)aij + 1113936jisin M1iαij(ts)

pij+ 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liUsing LeibnizndashNewton formula

x ts( 1113857 minus x Ts( 1113857 1113946ts

Ts

_x(k)dk (21)

then

x ts( 1113857( 1113857T

x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_x(k)dk1113888 1113889

T

(22)

and then

1113957x ts( 1113857( 1113857T

1113957x ts( 1113857( 1113857 1113957x Ts( 1113857( 1113857T

+ 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

⎡⎣ ⎤⎦ 1113957x Ts( 1113857( 1113857 + 1113946ts

Ts

_1113957x(k)dk1113888 11138891113890 1113891

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2 1113946ts

Ts

_1113957x(k)dk1113888 1113889

T

1113946ts

Ts

_1113957x(k)dk1113888 1113889

le 2 1113957x Ts( 1113857( 1113857T

1113957x Ts( 1113857( 1113857 + 2τ 1113946ts

Vs

_1113957xT

i (k) _1113957xi(k)dk

(23)

Similarly

1113957v ts( 1113857( 1113857T

1113957v ts( 1113857( 1113857le 2 1113957v Ts( 1113857( 1113857T

1113957v Ts( 1113857( 1113857 + 2τ 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(24)

Substituting (23) and (24) into (20) we obtain

6 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957x

Tj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2

(25)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij(ts)

pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts)qij + ci(ts)hi + di(ts)liand λ1 and λ3 are the minimum eigenvalues of A11minus

H1 andP11 minus L1 respectively

Similarly differentiating V2(ts) and V3(ts) we canobtain

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2

(26)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)li

and λ2 and λ4 are the minimum eigenvalues of A22 minus H2andP22 minus L2 respectively

_V3 ts( 1113857 τ2 1113944iisinM1i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM1i

ξ1i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

+ τ2 1113944iisinM2i

_1113957xT

i ts( 1113857 _1113957xi ts( 1113857 minus τ 1113944iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk + τ2 1113944iisinM2i

ξ2i_1113957v

T

i ts( 1113857 _1113957vi ts( 1113857 minus τ 1113944iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk

(27)

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(28)

Mathematical Problems in Engineering 7

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalues ofA11 minus H1 A22 minus H2 P11 minus L1 andP22 minus L2 respectivelywith H1 diag hi1113864 1113865 an d L1 diag li1113864 1113865 for foralli isin ℓl and H2

diag hi1113864 1113865 and L2 diag li1113864 1113865 for foralli isin ℓ2 Furthermore sinceA11 A22 P11 andP22 are both symmetric and diagonalmatrices H1 H2 L1 andL2 have at least one element being1 λi lt 0 i 1 2 3 4 based on Lemma 2erefore _V(ts)lt 0if pgt 0 is sufficiently large

Case 2 A11 A22 P11 andP22 are asymmetricIt is known that (A11 + AT

11)2 (A22 + AT22)2

(P11 + PT11)2 and (P22 + PT

22)2 are symmetric even ifA11 A22 P11 andP22 are asymmetric erefore all eigen-values of (A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 +

PT112) minus L1 and (P22 + PT

222) minus L2 are negative fromLemma 2 Similarly we can have

_V1 ts( 1113857le 1113944iisinM1i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM1i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM1i

ξ1i +12ρ21113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM1i

ξ1i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM1i

1113944jisinM2i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM1i

pλ1 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

pλ1 1113957xi

ts( 1113857

2+ 1113944

iisinM1i

pλ3 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

pλ3 1113957vi

ts( 1113857

2 (29)

where ξ2i 1 + 2ρ3 + ρ4 + 1113936jisinM2icij(ts)aij + 1113936jisinM2i

αij(ts)

pij + 1113936jisinM1idij(ts)bij + 1113936jisinM1i

βij(ts)qij + ci(ts)hi + di(ts)liand λ2 and λ4 are the minimum eigenvalue of (A22+

AT222) minus H2 and (P22 + PT

222) minus L2 respectively

_V2 ts( 1113857le 1113944iisinM2i

1113957xTi Ts( 11138571113957xi Ts( 1113857 + τ 1113944

iisinM2i

1113946ts

Ts

_1113957xT

i (k) _1113957xi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

bij1113957xTj Ts( 11138571113957xj Ts( 1113857

+ 1113944iisinM2i

ξ2i +12ρ41113874 11138751113957v

Ti Ts( 11138571113957vi Ts( 1113857 + τ 1113944

iisinM2i

ξ2i 1113946ts

Ts

_1113957vT

i (k) _1113957vi(k)dk minus12

1113944iisinM2i

1113944jisinM1i

qij1113957vTj Ts( 11138571113957vj Ts( 1113857

+ 1113944iisinM2i

pλ2 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

pλ2 1113957xi

ts( 1113857

2+ 1113944

iisinM2i

pλ4 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

pλ4 1113957vi

ts( 1113857

2 (30)

where ξ1i 1 + 2ρ1 + ρ2 + 1113936jisinM1icij(ts)aij + 1113936jisinM1i

αij

(ts)pij + 1113936jisinM2idij(ts)bij + 1113936jisinM2i

βij(ts) qij + ci(ts)hi + di

(ts)li and λ1 and λ3 are the minimum eigenvalue of(A11 + AT

112) minus H1 (P11 + PT112) minus L1 respectively

Combining _V1(ts)_V2(ts) and _V3(ts) we obtain

_V ts( 1113857le 1113944iisinM1i

1 + pλ1( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM1i

τ2 + pλ11113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM2i

1 + pλ2( 1113857 1113957xi Ts( 1113857

2

+ 1113944iisinM2i

τ2 + pλ21113872 1113873 _1113957xi ts( 1113857

2

+ 1113944iisinM1i

ξ1i +12ρ2 minus

12

1113944jisinM2i

bji minus12

1113944jisinM2i

qji + pλ3⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM1i

τ2ξ1i + pλ31113872 1113873 _1113957vi ts( 1113857

2

+ 1113944iisinM2i

ξ2i +12ρ4 minus

12

1113944jisinM1i

bji minus12

1113944jisinM1i

qji + pλ4⎛⎝ ⎞⎠ 1113957vi Ts( 1113857

2

+ 1113944iisinM2i

τ2ξ2i + pλ41113872 1113873 _1113957vi ts( 1113857

2

(31)

8 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

where λ1 λ2 λ3 and λ4 are the minimum eigenvalue of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 respectively Even though matrix

A11 A22 P11 andP22 are asymmetric matrix (A11 + AT11) 2

(A22 + AT22)2 (P11 + PT

11)2 and (P22 + PT22)2 are sym-

metric thus from Lemma 2 we can obtain all eigenvalues of(A11 + AT

112) minus H1 (A22 + AT222) minus H2 (P11 + PT

112) minus L1

and (P22 + PT222) minus L2 are negative So λi lt 0foralli isin 1 2 3 4

and _V(ts)lt 0 if pgt 0 is sufficiently largeerefore all the agents of sampled-data based network

(1) with time-varying delays can achieve the given syn-chronous states asymptotically

Remark 1 When the topology structure is connected re-gardless of the coupled weightedmatrices the sampled-data-based network (1) with time-varying delays can be as-ymptotically group synchronized by controller (2)

4 Simulations

A complex dynamical system with N 6 andL 3 Let theinitial values be X(0) 29 12 20 17 25 minus 7 22 91113858 1113859 andcij(0) dij(0) αij(0) βij(0) ci(0) di(0) 001

Let A11 A22 P11 andP22 be symmetric as

A11

minus 2 1 01 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 3 2 12 minus 4 11 1 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 2 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 12 0 21 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(32)

and A11 A22 P11 andP22 be asymmetric as

A11

minus 3 0 32 minus 3 10 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 01

A22

minus 2 2 03 minus 6 31 2 minus 3

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 005

P11

0 3 02 0 10 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 001

P22

0 2 03 0 30 2 0

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦lowast 0005

(33)

respectivelyTake

B12

0 1 0

2 0 0

0 0 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

B21

1 2 0

0 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lowast 01

H1 diag 01 0 01113864 1113865

H2 diag 0 01 01113864 1113865

(34)

Figure 1 presents that the effects of adaptive strategiesfor the synchronization of complex networks with non-linear dynamical Figure 1(a) shows the position andvelocity of all nodes without adaptive strategies andFigure 1(b) shows the position and velocity of all nodeswith adaptive laws respectively in which the subgroupsrsquocoupling matrices are symmetric It is obvious to see thefact that all the nodes with adaptive laws can achieve theirgiven synchronous states asymptotically while all thenodes without adaptive laws cannot converge Figures 2and 3 show the simulations of network (1) with τ 01 inwhich the subnetworksrsquo coupling matrices are symmetricor asymmetric descried as Figures 2 and 3 respectivelyFigures 2(a) and 2(b) present that the position and ve-locity of all nodes of network (1) with τ 01 wheresubgroupsrsquo coupling matrices are symmetric and thecoupling strengths cij dijαij and βij and the feedbackgains ci and di are presented in Figures 2(c)ndash2(h) re-spectively Similarly Figures 3(a) and 3(b) present that theposition and velocity of all nodes of network (1) withτ 01 where the subgroupsrsquo coupling matrices areasymmetric the coupling strengths cij dijαij and βij andthe feedback gains ci and di presented as Figures 3(c)ndash3(h)respectively From Figures 2 and 3 we can find that allnodes of network (1) can achieve synchronization and thecoupling strengths and the feedback gains also converge tobe consistent However compared with Figure 3 thesystem in Figure 2 can achieve synchronization faster thanthat in Figure 3 Figure 4 is the simulation of network (1)with adaptive laws in which the subgroupsrsquo couplingmatrices are symmetric where Figures 4(a) and 4(b) arethe positions and velocities of all nodes of network (1)with τ 01 and τ 1 respectively We can know thatwith the time delay τ increasing the system cannotachieve synchronization

Mathematical Problems in Engineering 9

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

Position convergence Velocity convergence

ndash20

0

20

40

60

80

xi

5 100t

ndash5

0

5

vi

5 100t

(a)

Position convergence Velocity convergence

5 10 15 200t

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

ndash1000

ndash500

0

500

1000

vi

(b)

Figure 1 (a) Without adaptive strategies (b) With adaptive strategies

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

(a)

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

500

1000

1500

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 2 Continued

10 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

50

100

150

200

250

ci

(e)

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

100

200

300

400

500

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

20

40

60

80

100

di

(h)

Figure 2 Intragroup coupling matrix is symmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash100

0

100

200

xi

(a)

Velocity convergence

5 10 15 200t

ndash4000

ndash2000

0

2000

4000

vi

(b)

Adaptive coupling strengths of position

5 10 15 200t

0

2000

4000

6000

cij

(c)

Adaptive coupling strengths of position

5 10 15 200t

0

50

100

150

dij

(d)

Figure 3 Continued

Mathematical Problems in Engineering 11

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

Adaptive feedback gain of position

5 10 15 200t

0

5

10

15

ci

(e)

times104Adaptive coupling strengths of velocity

5 10 15 200t

0

2

4

6

αij

(f )

Adaptive coupling strengths of velocity

5 10 15 200t

0

2000

4000

6000

8000

βij

(g)

Adaptive feedback gain of velocity

5 10 15 200t

0

200

400

600

800

di

(h)

Figure 3 Intragroup couplingmatrix is asymmetric and time delay τ 01 (a) Positions (b) Velocities (c) cij (d) dij (e) ci (f) αij (g) βij (h) di

Position convergence

5 10 15 200t

ndash200

ndash100

0

100

200

300

xi

Velocity convergence

5 10 15 200t

ndash1000

ndash500

0

500

1000

vi

(a)

Position convergence

5 10 15 200t

ndash400

ndash200

0

200

400

xi

Velocity convergence

5 10 15 200t

ndash2000

ndash1000

0

1000

2000

vi

(b)

Figure 4 Intragroup coupling matrix is symmetric and with adaptive strategies (a) Time delay τ 01 (b) Time delay τ 1

12 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

5 Conclusion

e adaptive group synchronization of second-order non-linear complex dynamical networks with time-varying de-lays and sampled data has been researched in this paper Anew adaptive law has been designed and we have provedthat the second-order system with sampled data can achievegroup synchronization no matter whether the couplingmatrix is symmetric or not Moreover we have discussed theinfluences of time-varying delays and adaptive laws forgroup synchronization of complex networks with nonlineardynamics in the different simulations Finally some simu-lations have been represented

Data Availability

No data were used in this study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the achievements of the im-portant project ldquoClass Teaching Experiment Research inBasic Education Schoolrdquo of Beijing Education Science ldquo12thFive Year Planrdquo in 2015 (no ABA15009) and NationalNatural Science Foundation of China (Grant no 61773023)

References

[1] G Chen and Z Duan ldquoNetwork synchronizability analysis agraph theoretic approachrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 18 no 3 Article ID 371022008

[2] C Wu and L Chua ldquoApplication of graph theory to thesynchronization in an array of coupled nonlinear oscillatorsrdquoIEEE Transactions on Circuits and Systems I FundamentalIeory and Applications vol 42 no 8 pp 494ndash497 1995

[3] A Papachristodoulou A Jadbabaie and U Munz ldquoEffects ofdelay in multi-agent consensus and oscillator synchroniza-tionrdquo IEEE Transactions on Automatic Control vol 55 no 6pp 1471ndash1477 2010

[4] X L Wang H Su M Z Q Chen X F Wang and G ChenldquoReaching non-negative edge consensus of networked dy-namical systemsrdquo IEEE Transactions on Cybernetics vol 48no 9 pp 2712ndash2722 2018

[5] X Liu H Su and M Z Q Chen ldquoA switching approach todesigning finite-time synchronization controllers of coupledneural networksrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 2 pp 471ndash482 2016

[6] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[7] C Tang BWu JWang and L Xiang ldquoEvolutionary origin ofasymptotically stable consensusrdquo Scientific Reports vol 4p 4590 2014

[8] I Z Kiss ldquoSynchronization engineeringrdquo Current Opinion inChemical Engineering vol 21 pp 1ndash9 2018

[9] H Su Y Sun and Z Zeng ldquoSemiglobal observer-based non-negative edge consensus of networked systems with actuatorsaturationrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2827ndash2836 2020

[10] H Su Y Liu and Z Zeng ldquoSecond-order consensus formultiagent systems via intermittent sampled position datacontrolrdquo IEEE Transactions on Cybernetics vol 50 no 5pp 2063ndash2072 2020

[11] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[12] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[13] S Liu Z Ji H Ma and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[14] B Liu H Su L Wu X Li and X Lu ldquoFractional-ordercontrollability of multi-agent systems with time-delayrdquoNeurocomputing 2020

[15] X Wang H Su X Wang and G Chen ldquoFully distributedevent-triggered semiglobal consensus of multi-agent systemswith input saturationrdquo IEEE Transactions on IndustrialElectronics vol 64 no 6 pp 5055ndash5064 2017

[16] W Yu L Zhou X Yu J Lu and R Lu ldquoConsensus in multi-agent systems with second-order dynamics and sampleddatardquo IEEE Transactions on Industrial Informatics vol 9no 4 pp 2137ndash2146 2013

[17] B Liu P Wei and X Wang ldquoGroup synchronization ofcomplex network with nonlinear dynamics via pinningcontrolrdquo in Proceeding of the 32nd Chinese Control Confer-ence IEEE Xirsquoan China pp 235ndash240 July 2013

[18] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2pp 429ndash435 2009

[19] Y Lu J Lu J Liang and J Cao ldquoPinning synchronization ofnonlinear coupled Lure networks under hybrid impulsesrdquoIEEE Transactions on Circuits and Syatems II-Express Briefsvol 66 no 3 pp 432ndash436 2018

[20] H Su Z Rong M Chen X Wang G Chen and H WangldquoDecentralized adaptive pinning control for cluster syn-chronization of complex dynamical networksrdquo IEEE Trans-actions on Systems Man and Cybernetics Cybernetics vol 43no 1 pp 394ndash399 2013

[21] P Delellis M Bemnardo and F Garofalo ldquoSynchronizationof complex networks through local adaptive couplingrdquo Chaosvol 18 no 3 p 175 2008

[22] J Hu and W X Zheng ldquoAdaptive tracking control of leader-follower systems with unknown dynamics and partial mea-surementsrdquo Automatica vol 50 no 5 pp 1416ndash1423 2014

[23] B Liu S Li and L Wang ldquoAdaptive synchronization of twotime-varying delay nonlinear coupled networksrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 3800ndash3804 IEEE Nanjing China July 2014

[24] Y Zhao G Wen Z Duan and G Chen ldquoAdaptive consensusfor multiple nonidentical matching nonlinear systems anedge-based frameworkrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 62 no 1 pp 85ndash89 2015

[25] B Liu S Li and X Tang ldquoAdaptive second-order syn-chronization of two heterogeneous nonlinear coupled net-worksrdquo Mathematical Problems in Engineering vol 2015Article ID 578539 7 pages 2015

Mathematical Problems in Engineering 13

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering

[26] Q Xu S Zhuang Y Zeng and J Xiao ldquoDecentralizedadaptive strategies for synchronization of fractional-ordercomplex networksrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 543ndash550 2017

[27] D Ding J Yan N Wang and D Liang ldquoAdaptive syn-chronization of fractional order complex-variable dynamicalnetworks via pinning controlrdquo Communications in Ieoret-ical Physics vol 68 no 9 pp 366ndash374 2017

[28] L Du Y Yang and Y Lei ldquoSynchronization in a fractional-order dynamic network with uncertain parameters using anadaptive control strategyrdquo Applied Mathematics and Me-chanics vol 39 no 3 pp 353ndash364 2018

[29] S Wang H Yao S Zheng and Y Xie ldquoA novel criterion forcluster synchronization of complex dynamical networks withcoupling time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 7 pp 2997ndash3004 2012

[30] Z-J Tang T-Z Huang J-L Shao and J-P Hu ldquoConsensusof second-order multi-agent systems with nonuniform time-varying delaysrdquo Neurocomputing vol 97 no 1 pp 410ndash4142012

[31] B Liu X Wang H Su H Zhou Y Shi and R Li ldquoAdaptivesynchronization of complex dynamical networks with time-varying delaysrdquo Circuits Systems and Signal Processingvol 33 no 4 pp 1173ndash1188 2014

[32] Y Liang and X Wang ldquoAdaptive synchronization in complexnetworks through nonlinearly couplingrdquo Computer Engi-neering and Applications vol 48 no 10 pp 25ndash28 2012

[33] H Su G Chen X Wang and Z Lin ldquoAdaptive second-orderconsensus of networked mobile agents with nonlinear dy-namicsrdquo Automatica vol 47 no 2 pp 368ndash375 2011

[34] B Liu X Wang Y Gao G Xie and H Su ldquoAdaptivesynchronization of complex dynamical networks governed bylocal lipschitz nonlinearlity on switching topologyrdquo Journal ofApplied Mathematics vol 2013 Article ID 818242 7 pages2013

[35] H Su N Zhang M Z Chen H Wang and X WangldquoAdaptive flocking with a virtual leader of multiple agentsgoverned by locally Lipschitz nonlinearityrdquo NonlinearAnalysis Real World Applications vol 23 no 9 pp 978ndash9902013

[36] M Li B Liu Y Zhu L Wang and M Zhou ldquoGroup syn-chronization of nonlinear complex dynamics networks withsampled datardquo Mathematical Problems in Engineeringvol 2014 Article ID 142061 8 pages 2014

[37] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010

[38] Y Gao and L Wang ldquoSampled-data based consensus ofcontinuous-time multi-agent systems with time-varying to-pologyrdquo IEEE Transactions on Automatic Control vol 56no 5 pp 1226ndash1231 2011

[39] B Liu X Wang H Su Y Gao and L Wang ldquoAdaptivesecond-order consensus of multi-agent systems with het-erogeneous nonlinear dynamics and time-varying delaysrdquoNeurocomputing vol 118 pp 289ndash300 2013

[40] J Zhang M Li and B Liu ldquoSynchronization of a complexnetwork for neighboring agents with time-varying delaysrdquoCommunications in Computer and Information Sciencevol 225 pp 73ndash81 2011

14 Mathematical Problems in Engineering