Nonlinear DynDOI 10.1007/s11071-013-0873-0

O R I G I NA L PA P E R

Adaptive exponential synchronization in pth momentfor stochastic time varying multi-delayed complex networks

Yuhua Xu · Hongzheng Yang · Dongbing Tong ·Yuling Wang

Received: 2 December 2012 / Accepted: 19 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract In this paper, the analysis problem of adap-tive exponential synchronization in pth moment isconsidered for stochastic complex networks withtime varying multi-delayed coupling. By using theLyapunov–Krasovskii functional, stochastic analysistheory, several sufficient conditions to ensure the modeadaptive exponential synchronization in pth momentfor stochastic delayed complex networks are derived.To illustrate the effectiveness of the synchronization

The first author and the second author contributed equally tothis work.

Y. Xu (�)Computer School of Wuhan University, Wuhan 430079,P.R. Chinae-mail: yuhuaxu2004@163.com

Y. XuDepartment of Maths, Yunyang Teachers’ College, Shiyan,Hubei 442000, P.R. China

H. YangRenhe Hospital, China Three Gorges University, Yichang,Hubei 443001, P.R. Chinae-mail: yhwndb@163.com

D. TongCollege of Information Science and Technology, DonghuaUniversity, Shanghai 201620, P.R. China

Y. WangSchool of Economics, South-Central Universityfor Nationalities, Wuhan 430074, P.R. China

conditions derived in this paper, a numerical exampleis finally provided.

Keywords Adaptive synchronization · Time-varyingdelays · Stochastic · Complex network

1 Introduction

As is known to all, complex dynamical networkswidely exist in the real world, including food webs,communication networks, social networks, powergrids, cellular networks, World Wide Web, metabolicsystems, disease transmission networks, and so on[1–6]. As obstructions to the transmission of signalsare inevitable in a biological neural network, in anepidemiological model, in a communications network,or in an electrical power grid, one important consid-eration in practical networks is the existence of timedelays; under many circumstances, there exists infor-mation communication of nodes not only at time t ,but also at time t − τ [7–14]. It is noticed that most ofthe studies on synchronization of a dynamical networkhave been performed under some implicit assumptionsthat there exists the information communication ofnodes via the edges only at time t or at time t − τ .Especially, in some cases, the multi-delayed couplingconsists of providing more information about the dy-namics in nodes to the other nodes in the network [15].

Apart from networks with time delays, complexsystems are susceptible to stochastic perturbation. In

Y. Xu et al.

[16], the adaptive lag synchronization issue of un-known chaotic delayed neural networks with noiseperturbation is considered. An adaptive feedback con-troller is designed to achieve complete synchroniza-tion of unidirectionally coupled delayed neural net-works with stochastic perturbation in [17]. In [18],adaptive synchronization condition under almost ev-ery initial data for stochastic neural networks withtime-varying delays and distributed delays is derived.In [19], the issues of lag synchronization of coupledchaotic delayed neural networks are investigated. In[20], Lu et al. investigated globally exponential syn-chronization for linearly coupled neural networks withtime varying delay and impulsive disturbances. How-ever, all of the mentioned works consider the synchro-nization for neural networks is to achieve the accor-dance of the states of the drive system and the re-sponse system in a moment. That is to say, the stateof the error system of the drive systems and the re-sponse system can achieve to zero eventually whenthe time approaches infinity. But on the other hand,the pth moment exponential stability on neural net-works has been widely studied by many authors; forinstance, see [21–27] and the references therein. Thus,it is also interesting to the pth moment exponentialsynchronization on neural networks.

It should be pointed out that, up to now, the prob-lem of adaptive exponential synchronization in thepth moment for stochastic multi-delayed complex net-works has received little research attention. In this pa-per, we are concerned with the analysis issue for adap-tive exponential synchronization of networks withstochastic delayed. The main purpose of this paperis to establish stability criteria for testing whether thestochastic delayed complex networks is stochasticallyexponentially synchronization in the pth moment. Wewill use a simple example to illustrate the usefulnessof synchronization conditions.

The rest of this work is organized as follows. Sec-tion 2 gives the problem formulation. Section 3 givessome theoretical analyses. Section 4 gives illustrativeexample. Section 5 gives the conclusion of the paper.

2 Problem formulation

The multi-delayed coupled complex network can bedescribed by

xi (t) = f(xi(t)

) +N∑

j=1

aij xj (t)

+N∑

j=1

(b1)ij xj(t − τ1(t)

) + · · ·

+N∑

j=1

(bn)ij xj(t − τn(t)

), (1)

where xi = (x1, x2, . . . , xn)T ∈ Rn, f : Rn → Rn

standing for the activity of an individual subsystemis a vector value function. A = (aij )N×N ∈ RN×N

and Bi = ((bi)ij )N×N ∈ RN×N are the coupling ma-trix, and aij and (bi)ij are the weight or couplingstrength. If there exists a link from node i to j (i �= j),then aij �= 0 and (bi)ij �= 0. Otherwise, aij = 0 and(bi)ij = 0. τi(t) is the time-vary delay satisfying that

0 < τi(t) ≤ τi and τi (t) ≤�τ i< 1, where τi ,

�τ i are con-

stants, i = 1,2, . . . , n.Let x(t) = ((x1(t))T , (x2(t))T , . . . , (xN(t))T )T ∈

RnN , F(x(t)) = (f (x1(t))T , f (x2(t))T , . . . ,

f (xN(t))T )T , rewrite Eq. (1) in the following com-pact form:

dx(t) = [F

(x(t)

) + A ⊗ Inx(t) + B1 ⊗ Inx(t − τ1(t)

)

+ · · · + Bn ⊗ Inx(t − τn(t)

)]dt. (2)

For the drive systems (2), a response system is con-structed as follows:

dy(t) = [F

(y(t)

) + A ⊗ Iny(t) + B1 ⊗ Iny(t − τ1(t)

)

+ · · · + Bn ⊗ Iny(t − τn(t)

) + U(t)]dt

+ σ(t, y(t) − x(t), y

(t − τ1(t)

)

− x(t − τ1(t)

), . . . , y

(t − τn(t)

)

− x(t − τn(t)

))dω(t), (3)

where y(t) is the state vector of the response sys-tem (3), w(t) = (w1(t),w2(t), . . . ,wn(t))

T is an n-dimensional Brown moment defined on a completeprobability space (Ω,F,P ) with a natural filtration{Ft }t≥0, and σ : R+ × Rn × Rn → Rn×n is the noiseintensity matrix and can be regarded as a result fromthe occurrence of eternal random fluctuation and otherprobabilistic causes.

Let ei(t) = yi(t)−xi(t), e(t) = ((e1(t))T , (e2(t))T ,

. . . , (eN(t))T )T ∈ RnN . For the purpose of simplicity,we mark e(t − τi(t)) = eτi

(t). From the drive system(2) and the response system (3), the error system oftheirs can be represented as follows:

de(t) = [F

(y(t)

) − F(x(t)

) + A ⊗ Ine(t)

+ B1 ⊗ Ineτ1(t) + · · · + Bn ⊗ Ineτn(t)

Adaptive exponential synchronization in pth moment for stochastic time varying multi-delayed

+ U(t)]dt

+ σ(t, e(t), eτ1(t), . . . , eτn(t)

)dω(t), (4)

where U(t) = (u1(t), u2(t), . . . , un(t))T .The initial condition associated with (4) is given in

the following form:

e(s) = ξ(s), s ∈ [−τ ,0]for any ξ(s) ∈ L2

F0([−τ ,0],Rn), where L2

F0([−τ ,0],

Rn) is the family of all F0-measurable C([−τ ,0],Rn)-value random variables satisfying thatsup−τ≤s≤0 E|ξ(s)|2 < ∞, and C([−τ ,0],Rn) de-notes the family of all continuous Rn-valued functionsξ(s) on [−τ ,0] with the norm ‖ξ(s)‖ =sup−τ≤s≤0 |ξ(s)|.

To obtain the main result, we need the followingpreliminaries.

Definition 1 The trivial solution e(t, ξ(s)) of the errorsystem (4) is said to be exponential stability in pthmoment if

lim supt→∞

1

tlog

(ε∣∣e

(t, ξ(s)

)∣∣p)< 0

for any ξ(s) ∈ Lp

L0([−τ ,0];Rn), where p ≥ 2,p ∈ Z.

When p = 2, it is said to be exponential stability inmean square.

The drive system (2) and the response system (3)are said to be exponential synchronized in pth mo-ment, if the error system (4) is exponential stabilityin pth moment.

Definition 2 For an n-dimensional stochastic differ-ential system

dx(t) = f(t, x(t), x(t − τ)

)dt

+ g(t, x(t), x(t − τ)

)dω(t).

Let V ∈ C2,1(Rn × R+;R+), define an operator Lfrom R+ × Rn × Rn to R by

LV(t, x(t), x(t − τ)

)

= Vt

(t, x(t)

) + Vx

(t, x(t)

)f

(t, x(t), x(t − τ)

)

+ 1

2trace

(gT

(t, x(t), x(t − τ)

)

× Vxx

(t, x(t)

)g(t, x(t), x(t − τ)

)),

where

Vt

(t, x(t)

) = ∂V (t, x(t))

∂t,

Vx

(t, x(t)

) =(

∂V (t, x(t))

∂x1,∂V (t, x(t))

∂x2,

. . . ,∂V (t, x(t))

∂xn

),

Vxx

(t, x(t)

) =(

∂2V (t, x(t))

∂xj xk

)

n×n

.

Assumption 1 We also assume that f is Lipschitzwith respect to its argument, i.e.,∣∣f

(yi(t)

) − f(xi(t)

)∣∣ ≤ ηei(t), η ∈ R.

Assumption 2 The noise intensity matrix σ(·, ·, . . . , ·)satisfies the linear growth condition. That is, there ex-ist positives Ψ,γ1, . . . , γn, such that

trace(σT (t, e, eτ1 , . . . , eτn)σ (t, e, eτ1 , . . . , eτn)

)

≤ (Ψ |e|2 + γ1|eτ1 |2 + · · · + γn|eτn |2

),

where trace(·) denotes the trace of matrix.

Lemma 1 [28] Let x ∈ Rn and y ∈ Rn. Then

xT y + yT x ≤ xT x + −1yT y

for any > 0.

Lemma 2 (Yong inequality) Let a, b ∈ R and β ∈[0,1]. Then |a|β |b|(1−β) ≤ β|a| + (1 − β)|b|.

Lemma 3 (Dynkin formula) [29] Let V ∈ C2,1(Rn ×R+;R+) and τ1, τ2 be bounded stopping times suchthat 0 ≤ τi ≤ τj a.s. If V (t, x(t)) and LV (t, x(t)) arebounded on t ∈ [τi, τj ] with probability 1, then

εV(τj , x(τj )

) − εV(τi, x(τi)

)

= ε

∫ τj

τi

LV(u,x(s)

)ds.

Lemma 4 (Gronwally’s inequality) [30] Let T > 0and u(·) be a Borel measurable bounded non-negativefunction on [0, T ]. If

u(t) ≤ c + v

∫ t

0u(s) ds, ∀0 ≤ t ≤ T

for some constants c, v, then

u(t) ≤ c exp(vt), ∀0 ≤ t ≤ T .

Y. Xu et al.

3 Main results

In this section, we give some criterion of adaptive ex-ponential synchronization in pth moment for the drivesystem (2) and the response system (3).

Theorem 1 Let p ≥ 2. Under Assumptions 1–2, sup-pose that the following condition holds:

δ > max1≤i≤n

{1

1− �τ i

} n∑

i=1

βi, (5)

where

δ = p

(

H − η − Ξ − 1

2(p − 1)

[

Ψ + p − 2

p

n∑

i=1

γi

]

−n∑

i=1

Θi

)

, Ξ = λmax(A ⊗ In),

Θi = λmax

(1

2(Bi ⊗ In)(Bi ⊗ In)

T

),

βi =(

(p − 1)γi + 1

2

).

Ψ , γi , and H are positive constants. Then the errorsystem (4) is exponential stability in the pth momentby the following adaptive controller:

U(t) = (u1(t), u2(t), . . . , un(t)

)T,

ui(t) = ki

(yi(t) − xi(t)

),

(6)

kj = −1

2djp|e|p−2(ej

)2, dj > 0. (7)

Proof Choose a non-negative function as follows:

V (t, e) = |e|p +n∑

j=1

1

dj

(kj + H)2,

where H is a sufficiently large positive constant whichis to be determined.

Computing LV (t, e, eτ ) along the trajectory of er-ror system (4), and using (6) and (7), one can obtainthat

LV (t, e, eτ ) = Vt (t, e) + Ve(t, e)[F

(y(t)

) − F(x(t)

)

+ A ⊗ Ine(t) + B1 ⊗ Ineτ1(t) + · · ·+ Bn ⊗ Ineτn(t) + U(t)

]

+ 1

2trace

(σT (t, e, eτ1, . . . , eτn)

× Vee(t, e)σ (t, e, eτ1 , . . . , eτn))

= 2n∑

j=1

1

dj

(kj + H)kj + p|e|p−2eT

× [F

(y(t)

) − F(x(t)

) + A ⊗ Ine(t)

+ B1 ⊗ Ineτ1(t) + · · · + Bn ⊗ Ineτn(t)

+ U(t)]

+ 1

2trace

(σT (t, e, eτ1 , . . . , eτn)

× p(p − 1)|e|p−2σ(t, e, eτ1, . . . , eτn))

= p|e|p−2eT[F

(y(t)

) − F(x(t)

)

+ A ⊗ Ine(t) + B1 ⊗ Ineτ1(t) + · · ·+ Bn ⊗ Ineτn(t) − He(t)

]

+ 1

2trace

(σT (t, e, eτ1 , . . . , eτn)

× p(p − 1)|e|p−2σ(t, e, eτ1, . . . , eτn)).

(8)

Now, using Assumptions 1 and 2 together withLemma 1 yield

eT[F

(y(t)

) − F(x(t)

)] ≤ η|e|2, (9)

eT Bi ⊗ Ineτi≤ 1

2eT (Bi ⊗ In)(Bi ⊗ In)

T e + 1

2eTτieτi

,

(10)1

2trace

(σT (t, e, eτ1 , . . . , eτn)

× p(p − 1)|e|p−2σ(t, e, eτ1 , . . . , eτn))

≤ 1

2p(p − 1)|e|p−2

× (Ψ |e|2 + γ1|eτ1 |2 + · · · + γn|eτn |2

). (11)

Also, making use of Yong inequality (Lemma 2),we have

|e|p−2|eτi|2 ≤ p − 2

p|e|p + 2

p|eτi

|p. (12)

Substituting (9)–(12) into (8), yields

LV ≤ p|e|p−2eT[ηe(t) + A ⊗ Ine(t) + B1 ⊗ Ineτ1(t)

+ · · · + Bn ⊗ Ineτn(t) − He(t)]

+ 1

2p(p − 1)|e|p−2

× (Ψ |e|2 + γ1|eτ1 |2 + · · · + γn|eτn |2

)

≤ p

(η + Ξ − H

+ 1

2(p − 1)

[Ψ + p − 2

p(γ1 + · · · + γn)

]

Adaptive exponential synchronization in pth moment for stochastic time varying multi-delayed

+ Θ1 + · · · + Θn

)|e|p

+(

(p − 1)γ1 + 1

2

)|eτ1 |p + · · ·

+(

(p − 1)γn + 1

2

)|eτn |p

= −δ|e|p + β1|eτ1 |p + · · · + βn|eτn |p,

where

δ = p

(

H − η − Ξ − 1

2(p − 1)

[

Ψ + p − 2

p

n∑

i=1

γi

]

−n∑

i=1

Θi

)

, Ξ = λmax(A ⊗ In),

Θi = λmax

(1

2(Bi ⊗ In)(Bi ⊗ In)

T

),

βi =(

(p − 1)γi + 1

2

).

On the other hand, for the function V (t, e), apply-ing Lemma 3 and using the above conditions (5), weobtain that

ε|e|p≤ εV

(t, e(t)

) ≤ εV(0, ξ(0)

)

+ ε

∫ t

0LV

(s, e(s), eτ1(s), . . . , eτn(s)

)ds

≤ εV(0, ξ(0)

)

+ ε

∫ t

0

(−δ|e|p + β1|eτ1 |p + · · · + βn|eτn |p)ds.

For∫ t

0 |eτi|p ds, let u = s − τi(s), then du = (1 −

τi (s)) ds and∫ t

0|eτi

|p ds =∫ t−τi (t)

−τi (0)

1

1 − τi (s)

∣∣e(s)∣∣p ds

≤ 1

1− �τ i

∫ t

−τi

∣∣e(s)∣∣p ds

= 1

1− �τ i

∫ 0

−τi

∣∣e(s)∣∣p ds

+ 1

1− �τ i

∫ t

0

∣∣e(s)∣∣p ds

≤ τi

1− �τ i

maxτi≤s≤0

∣∣ξ(s)∣∣p

+ 1

1− �τ i

∫ t

0

∣∣e(s)∣∣p ds

≤ max1≤i≤n

{τi

1− �τ i

}max

τi≤s≤0

∣∣ξ(s)

∣∣p

+ max1≤i≤n

{1

1− �τ i

}∫ t

0

∣∣e(s)∣∣p ds.

So,

ε|e|p≤ εV

(0, ξ(0)

)

+ max1≤i≤n

{τi

1− �τ i

} n∑

i=1

βi maxτi≤s≤0

∣∣εξ(s)∣∣p

+∫ t

0

[

−δ + max1≤i≤n

{1

1− �τ i

} n∑

i=1

βi

]

ε∣∣e(s)

∣∣p ds

that is,

ε|e|p ≤ c +∫ t

0υε|x|p ds,

where

c = εV(0, ξ(0)

) + max1≤i≤n

{τi

1− �τ i

}

×n∑

i=1

βi maxτ≤s≤0

∣∣ξ(s)∣∣p,

ν = −δ + max1≤i≤n

{1

1− �τ i

} n∑

i=1

βi.

It can be seen that c, v are constants and c > 0and v < 0. By using the Gronwally’s inequality(Lemma 4), we have

ε|e|p ≤ c exp(vt).

Therefore,

lim supt→∞

1

tlog

(ε∣∣e(t, ξ)

∣∣p) ≤ v < 0.

Thus, the error system (4) is exponential stability inpth moment. This completes the proof. �

Remark 1 Systems (1)–(3) are the inner time-invariantcomplex networks. As discussed in [31, 32], if systems(1)–(3) are the inner time-varying complex networks,which is a more complicated research issue.

Remark 2 For complex networks (1), some approachesof multi-agent systems can be used in this paper, whichare the next research topic for us; see [33, 34].

Y. Xu et al.

When we use the delayed and non-delayed con-troller. Then we have the following Theorem 2.

Theorem 2 Let p ≥ 2. Under Assumptions 1–2, sup-pose that the following condition holds:

δ > max1≤i≤n

{1

1− �τ i

} n∑

i=1

βi,

where

δ = p

(

H − η + 1

2Jρ − Ξ

− 1

2(p − 1)

[

Ψ + p − 2

p

n∑

i=1

γi

]

−n∑

i=1

Θi

)

,

Ξ = λmax(A ⊗ In),

Θi = λmax

(1

2(Bi ⊗ In)(Bi ⊗ In)

T

),

βi =(

(p − 1)γi + 1

2(1 + Jρ)

), ρ = 1,2, . . . , n.

Ψ , γi and H are positive constants. Then the errorsystem (4) is exponential stability in pth moment bythe following adaptive controller:

U(t) = (u1(t), u2(t), . . . , un(t)

)T,

ui(t) = ki

(yi(t) − xi(t)

)

+n∑

ρ=1

(mρ)i(yi(t − τρ) − xi(t − τρ)

),

kj = −1

2djp|e|p−2(ej

)2,

(mρ)j = −1

2(χρ)jp|e|p−2(ej

τρ

)2, dj > 0, χρ > 0.

Proof Choose the following non-negative function:

V (t, e) = |e|p +n∑

j=1

1

dj

(kj + H)2

+n∑

j=1

n∑

ρ=1

1

(χρ)j

((mρ)j + Jρ

)2,

where H is a sufficiently large positive constant whichis to be determined.

The proof is similar to that of Theorem 1, and henceomitted. �

When we only using the adaptive controller withdelayed. Then we have the following corollary.

Corollary 1 Let p ≥ 2. Under Assumptions 1–2, sup-pose that the following condition hold.

δ > max1≤i≤n

{1

1− �τ i

} n∑

i=1

βi,

where

δ = p

(1

2Jρ − η − Ξ

− 1

2(p − 1)

[

Ψ + p − 2

p

n∑

i=1

γi

]

−n∑

i=1

Θi

)

,

Ξ = λmax(A ⊗ In),

Θi = λmax

(1

2(Bi ⊗ In)(Bi ⊗ In)

T

),

βi =(

(p − 1)γi + 1

2(1 + Jρ)

), ρ = 1,2, . . . , n.

Ψ , γi and H are positive constants. Then the errorsystem (4) is exponential stability in the pth momentby the following adaptive controller:

U(t) = (u1(t), u2(t), . . . , un(t)

)T,

ui(t) =n∑

ρ=1

(mρ)i(yi(t − τρ) − xi(t − τρ)

),

(mρ)j = −1

2(χρ)jp|e|p−2(ej

τρ

)2.

The proof is similar to that of Theorem 1, and henceomitted.

4 Illustrative example

In the section, we present an example to illustrate theusefulness of the main results obtained in this paper.The adaptive exponential stability in pth moment isexamined for given stochastic delayed complex net-works.

We select the Lü chaotic system as the dynamicalnode of complex network [35], that is,

s =⎛

⎝−a a 00 c 00 0 −b

⎞

⎠

⎛

⎝s1

s2

s3

⎞

⎠ +⎛

⎝0

−s1s3

s1s2

⎞

⎠ ,

where a = 36, c = 20, b = 3.According to Sect. 3, we show that the network

with four nodes described by

Adaptive exponential synchronization in pth moment for stochastic time varying multi-delayed

Fig. 1 Synchronizationerrors with τ = 0.2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xi1(t) = a(xi

2(t) − xi1(t)) + ∑4

j=1 aij xj

1 (t)

+ ∑4j=1 bij x

j

1 (t − τ),

xi2(t) = bxi

2(t) − xi1(t)x

i3(t) + ∑4

j=1 aij xj

2 (t)

+ ∑4j=1 bij x

j

2 (t − τ),

xi3(t) = −cxi

3(t) + xi1(t)x

i2(t) + ∑4

j=1 aij xj

3 (t)

+ ∑4j=1 bij x

j

3 (t − τ),

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

yi1(t) = a(yi

2(t) − yi1(t)) + ∑4

j=1 aij yj

1 (t)

+ ∑4j=1 bij (y

j

1 (t − τ) + 0.1(yi1(t) − xi

1(t))

+ 0.1(yi1(t − τ) − xi

1(t − τ))

+ ki(yi1(t) − xi

1(t)),

yi2(t) = byi

2(t) − yi1(t)y

i3(t) + ∑4

j=1 aij yj

2 (t)

+ ∑4j=1 bij (y

j

2 (t − τ) + 0.1(yi2(t) − xi

2(t))

+ 0.1(yi2(t − τ) − xi

2(t − τ))

+ ki(yi2(t) − xi

2(t)),

yi3(t) = −cyi

3(t) + yi1(t)y

i2(t) + ∑4

j=1 aij yj

3 (t)

+ ∑4j=1 bij (y

j

3 (t − τ) + 0.1(yi3(t) − xi

3(t))

+ 0.1(yi3(t − τ) − xi

3(t − τ))

+ ki(yi3(t) − xi

3(t)),

ki = −1

2dip|e|p−2(ei

)2.

In numerical simulation, let

A =

⎛

⎜⎜⎝

4 −2 −2 0−2 2 1 −1−2 1 2 −10 −1 0 1

⎞

⎟⎟⎠ ,

B =

⎛

⎜⎜⎝

−1 2 −1 02 −2 1 −1

−1 1 0 00 −1 0 1

⎞

⎟⎟⎠ ,

we choose τ = 0.2, p = 2, and di = 1. The initialvalues are xi(0) = (2 − i,−3 + 2i,2 − 2i), yi(0) =(−2+ i,1+ i,2+2i), ki(0) = 2−2i. We introduce the

quantity E(t) =√∑N

i=1 ‖yi(t) − xi(t)‖2/N which isused to measure the quality of the control process. Itis obvious that when E(t) no longer increases, twonetworks achieve synchronization. Figure 1 shows thevariance of the synchronization errors. Figure 2 showsthe feedback gain ki (i = 1,2,3,4). All numericalsimulations illustrate the effectiveness of Theorem 1.

Numerical simulations of Theorem 2 and Corol-lary 1 can be illustrated in a similar way as shown inTheorem 1. Thus, we leave out numerical simulationshere.

Y. Xu et al.

Fig. 2 Dynamic curve ofthe feedback gain ki

(i = 1,2,3,4)

5 Conclusion

In this paper, we have dealt with the problem of adap-tive exponential synchronization in pth moment forcomplex networks with stochastic delayed. The con-ditions for the adaptive exponential synchronization inthe pth moment have been derived in terms of somealgebraical inequalities. These synchronization condi-tions are much different to those of linear matrix in-equality. Finally, a simple example has been used todemonstrate the effectiveness of the main results ob-tained in this paper.

Acknowledgements The authors would like to thank the ref-erees and the editor for their valuable comments and sugges-tions. This research is supported by the National Natural Sci-ence Foundation of China (61075060), the Science and Tech-nology Research Key Program for the Education Departmentof Hubei Province of China (D20105001, D20126002), theResearch Fund for the Technology Development Program ofHigher School of Tianjin (20111004), the Excellent Young KeyTalents Cultivation Plan of Hubei Province and China Postdoc-toral Science Foundation (2012M511663).

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