IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 2, FEBRUARY 2012 285

Synchronization Control for Nonlinear StochasticDynamical Networks: Pinning Impulsive Strategy

Jianquan Lu, Jürgen Kurths, Jinde Cao, Senior Member, IEEE, Nariman Mahdavi, and Chi Huang

Abstract— In this paper, a new control strategy is proposedfor the synchronization of stochastic dynamical networks withnonlinear coupling. Pinning state feedback controllers have beenproved to be effective for synchronization control of state-coupled dynamical networks. We will show that pinning impulsivecontrollers are also effective for synchronization control of theabove mentioned dynamical networks. Some generic mean squarestability criteria are derived in terms of algebraic conditions,which guarantee that the whole state-coupled dynamical networkcan be forced to some desired trajectory by placing impulsivecontrollers on a small fraction of nodes. An effective method isgiven to select the nodes which should be controlled at eachimpulsive constants. The proportion of the controlled nodesguaranteeing the stability is explicitly obtained, and the synchro-nization region is also derived and clearly plotted. Numericalsimulations are exploited to demonstrate the effectiveness of thepinning impulsive strategy proposed in this paper.

Index Terms— Nonlinear coupling, pinning impulsive control,state-coupled dynamical network, synchronization.

I. INTRODUCTION

COMPLEX dynamical networks are composed of a largenumber of interconnected dynamical nodes, in which

each node is a unit with specific contents [1]–[3]. Typical

Manuscript received August 8, 2011; revised October 26, 2011; acceptedDecember 3, 2011. Date of publication January 6, 2012; date of currentversion February 8, 2012. The work of J. Q. Lu was supported by theNational Natural Science Foundation of China under Grant 61175119, theNatural Science Foundation of the Jiangsu Province of China under GrantBK2010408, the Program for New Century Excellent Talents in University,under Project NCET-10-0329, and the Alexander von Humboldt Foundation ofGermany. The work of J. Kurths was supported by SUMO (EU), GSDP (EU),and ECONS (WGL). The work of J. D. Cao was supported by the NationalNatural Science Foundation of China under Grant 11072059, the SpecializedResearch Fund for the Doctoral Program of Higher Education under Grant20070286003, and the Natural Science Foundation of the Jiangsu Province ofChina under Grant BK2009271.

J. Lu is with the Department of Mathematics, Southeast University, Nanjing210096, China. He is also with the Potsdam Institute for Climate ImpactResearch, Potsdam D-14415, Germany (e-mail: jqluma@seu.edu.cn).

J. Kurths is with the Potsdam Institute for Climate Impact Research,Potsdam D-14415, Germany. He is also with the Department of Physics,Humboldt University Berlin, Berlin 10117, Germany, and with the Institutefor Complex Systems and Mathematical Biology, University of Aberdeen,Aberdeen AB24 3UE, U.K. (e-mail: Juergen.Kurths@pik-potsdam.de).

J. Cao is with the Department of Mathematics, Southeast University,Nanjing 210096, China (e-mail: jdcao@seu.edu.cn).

N. Mahdavi is with the Department of Electrical Engineering, AmirkabirUniversity of Technology, Tehran 16846–13114, Iran. He is also with thePotsdam Institute for Climate Impact Research, Potsdam 14412, Germany(e-mail: nariman.mahdavi@gmail.com).

C. Huang is with the Department of Mathematics, City University of HongKong, Kowloon 2788, Hong Kong (e-mail: huangchima@gmail.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNNLS.2011.2179312

examples of complex networks include the Internet, the WorldWide Web, neural networks, food webs, cellular and metabolicnetworks, etc., [4], [5]. Since the seminal papers on “small-world” and “scale-free” properties [4], [6], complex networkshave become a focus of research and have received increasingattention from various fields of science and engineering. Com-plex networks often exhibit complex and interesting dynamicalbehavior including synchronization [3], [7], consensus [8],flocking etc. As one of the most interesting and importantcollective behavior in dynamical networks, synchronizationhas attracted special attention of researchers in different fields[9]–[13].

Synchronization in dynamical networks is realized via asufficient information exchange among the nodes’ interconnec-tions [8], [14], [15], which makes the final synchronous statedifficult to predict. However, for many biological, physical andsocial dynamical networks, there exists a common requirementto regulate the behavior of large ensembles of interacting units.Some regulatory mechanisms have been uncovered in the con-text of biological, physiological, and cellular processes [16],which are fundamental to guarantee the correct functioningof the whole network. Examples include the control of therespiratory rhythm played by synaptically coupled pacemakerneurons in the medulla in physiology [17], and opinion leaderin social networks. Hence, in many cases, controllers are nec-essary to be designed to force the unpredicted final synchro-nous state into a certain required objective trajectory [18]–[22].

It has been revealed that, in the process of controlling vari-ous networks, feedback control serves as a simple and effectivestrategy for stabilization and synchronization. Different kindsof effective methods, including adaptive controllers [23], [24],impulsive controllers [25]–[27] and pinning state feedbackcontrollers [19], [20], have been designed for the stabiliza-tion and synchronization of complex dynamical networks. In[23], [24], the feedback strength is asymptotically enhancedaccording to a certain update law for the stabilization andsynchronization of dynamical networks. In [26], distributedimpulsive controllers are properly designed for the synchro-nization control of dynamical networks. Pinning state feedbackcontrollers were first proposed to control multi-mode lasersystems in [28], and have recently been used for the synchro-nization of complex networks by controlling a small fractionof nodes [19], [20], [22]. These methods have been shown tobe effective for the synchronization control of networks.

For many realistic networks, the state of nodes is oftensubject to instantaneous perturbations and experience abruptchange at certain instants which may be caused by switching

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286 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 2, FEBRUARY 2012

phenomena, frequency change or sudden noise, i.e., it exhibitsimpulsive effects. On the other hand, each individual nodein dynamical networks is often subject to various types ofnoise and uncertainty, which can have a great influence on thebehavior of dynamical networks. Impulsive control strategyhas also been shown to be an effective control strategy in manyfields due to its potential advantages over general continuouscontrol schemes [26], [29]. However, in previous studies, whenimpulsive controllers are designed for the synchronizationcontrol of dynamical networks with state-coupling, all of thenodes should be controlled, which means that the controllingcost is very high. Pinning state feedback control, which meansthat only a small fraction of nodes is directly controlled,has been proved to be effective for the synchronization ofdynamical networks with state-coupling. Then one may ask:1) can the stochastic dynamical network be synchronized byimpulsively controlling a small fraction of nodes; 2) that is,can we also design a certain pinning impulsive control strategyfor the synchronization of stochastic dynamical networks; and3) this paper is devoted to solving this problem. Some geneticcriteria are given to judge whether dynamical networks canbe globally exponentially forced to a desired equilibrium byimpulsively controlling a small fraction of nodes. Numericalexamples are finally given to demonstrate the effectiveness ofthe proposed impulsive strategy.

Notations: The standard notations will be used in this paper.In is the identity matrix of order n. λmax(·) is used to denotethe maximum eigenvalue of a real symmetric matrix. R

n

denotes the n-dimensional Euclidean space. Rn×n are n×n real

matrices. ‖x‖ denotes the Euclidean norm of vector x ∈ Rn .

Let R+ = [0,+∞), N = {1, 2, 3, . . .}. The superscript “T ”

represents the transpose. For any random variable ζ , let E(ζ )be the expectation value of ζ . #G denotes the number ofelements of a finite set G.

II. PRELIMINARIES

In this paper, we consider the following stochastic dynam-ical network with nonlinear coupling:

dxi (t) = [Cxi (t)+ B f (xi(t))

]dt + g(t, xi (t))dw(t)

+ cN∑

j=1

ai j�h(x j (t))dt, i = 1, 2, . . . , N (1)

where xi (t) = (xi1(t), xi2(t), . . . , xin(t))T ∈ Rn is the state

vector of the i -th node at time t , C ∈ Rn×n , B ∈ R

n×n ,w(t) ∈ R

m is an m-dimensional Brownian motion, f (xi (t)) =[ f1(xi1(t)), f2(xi2(t)), . . . , fn(xin(t))]T satisfying f (0) = 0,g :R+ × R

n → Rn×m is the noise intensity function matrix

satisfying g(t, 0) = 0n×m . The nonlinear function h(x j (t)) =(h(x j1(t)), h(x j2(t)), . . . , h(x jn(t)))T satisfies the followingconditions: [(h(u)− h(v))/(u −v)] ≥ ϑ > 0 for any u, v ∈ R.The configuration coupling matrix A = (ai j )N×N is definedas follows: if there is a connection between node i and nodej ( j �= i ), then ai j = a j i > 0, otherwise, ai j = a j i = 0, andthe diagonal elements are defined as aii = − ∑N

j=1, j �=i ai j .� = diag{γ1, γ2, . . . , γn} > 0 is the inner coupling positivedefinite matrix between two connected nodes i and j , and cis the coupling strength of the network.

We have the following assumptions and lemma for thederivation of the main results.

Assumption 1: The nonlinear function f (·) is assumed tosatisfy a Lipschitz condition, that is, there exists a constantκ > 0 such that ‖ f (u) − f (v)‖ ≤ κ‖u − v‖ holds for anyu, v ∈ R

n .Assumption 2: Assume that the noise intensity function

matrix g : R+×R

n → Rn×m is uniformly Lipschitz continuous

in terms of the norm induced by the trace inner product onthe matrices

trace

[(g(t, u)− g(t, v)

)T · (g(t, u)− g(t, v))]

≤ ‖M(u − v)‖2 ∀u, v ∈ Rn (2)

where M is a known constant matrix with compatible dimen-sions.

Lemma 1 ([30]): Consider the following stochastic systemwith impulses:{

dx(t) = φ(t, x(t))dt + η(t, x(t))dw(t), t ≥ t0, t �= tk,x(t+k )− x(t−k ) = Ik(x(t

−k )) k ∈ N.

(3)

Assume that there exist a Lyapunov function V (t, x(t)), andfunctions ϕ, ψk with ϕ(t, 0) = ψk(0) = 0 for any t ≥ 0,k ∈ N, such that:

1) there exist positive constants c1 and c2 such that for allt ≥ t0, c1‖x(t)‖ ≤ V (t, x(t)) ≤ c2‖x(t)‖;

2) there exists continuous function ϕ : R+ × R

+ → R,and ϕ(t, s) is concave on s for each t ∈ R

+, suchthat LV (t, x) ≤ ϕ(t, V (t, x)), where the operator Lis defined as LV (t, x) = Vt (t, x) + Vx(t, x)φ(t, x) +(1/2)trace[ηT (t, x)Vx xη(t, x)];

3) there exist continuous and concave functions ψk :R+ →R

+, k ∈ N, such that V (t+k , x(t+k )) ≤ ψk(V (t−k , x(t−k )))

then the exponential stability of the trivial solution of thefollowing comparison systems:

⎧⎨

⎩

w(t) = ϕ(t, w(t)), t ≥ t0, t �= tk,w(t+k ) = ψk(w(t

−k )), k ∈ N,

w(t0) = E(V (t0, x0))(4)

implies the exponential stability of the trivial solution of thestochastic impulsive system (3).

Let s(t) be a solution of an isolated node described by

ds(t) = [Cs(t) + B f (s(t))

]dt + g(t, s(t))dw(t) (5)

with initial condition s0 ∈ Rn . In this paper, we want to control

the nonlinear dynamical network (1) into the desired trajectorys(t).

Let ei (t) = xi (t)− s(t) be the error state of the node i . Inorder to force the whole network (1) into the desired trajectorys(t), the following impulsive controllers are designed for lnodes:

Ii (t) =⎧⎨

⎩

+∞∑k=1

μei (t)δ(t − tk), i ∈ D(tk), #D(tk) = l,

0, i /∈ D(tk)(6)

where the constant μ ∈ (−2, 0), δ(·) is the Dirac deltafunction, the time series {t1, t2, t3, . . .} is a sequence of strictlyincreasing impulsive instants satisfying limk→∞ tk = +∞,

LU et al.: SYNCHRONIZATION CONTROL FOR NONLINEAR STOCHASTIC DYNAMICAL NETWORKS 287

and the index set of D(tk) is defined as follows: at time instanttk , for the vectors e1(tk), e2(tk), . . . , eN (tk), one can reorderthe states of the nodes such that ‖ep1(tk)‖ ≥ ‖ep2(tk)‖ ≥· · · ≥‖epl(tk)‖ ≥ ‖ep,l+1(tk)‖ ≥ · · · ≥ ‖epN (tk)‖. Then theindex set of l controlled nodes D(tk) is defined as D(tk) ={p1, p2, . . . , pl}, and #D(tk) = l.

Since c∑N

j=1 ai j�h(s(t)) = 0, after adding the pinningimpulsive controllers (6) to the dynamical network (1), onecan obtain the following impulsively controlled dynamicalnetwork:

⎧⎪⎪⎨

⎪⎪⎩

dei (t) = [Cei (t)+ B f (ei (t))]dt + g(t, ei (t))dw(t)

+cN∑

j=1ai j�h(e j (t))dt, t �= tk, k ∈ N,

ei (t+k ) = ei (t

−k )+ μei (t

−k ), i ∈ D(tk), #D(tk) = l

(7)

where f (ei (t)) = f (xi (t))− f (s(t)), g(t, ei (t)) = g(t, xi (t))−g(t, s(t)), h(ei (t)) = h(xi (t)) − h(s(t)). Since [(h(u) −h(v))/(u − v)] ≥ ϑ > 0, we have [(h(u)− h(v))/(u − v)] ≥ϑ > 0 for any u, v ∈ R.

Throughout this paper, we always assume that ei (t) is left-hand continuous at t = tk , i.e., e(tk) = e(t−k ). Therefore, thesolutions of (7) are piecewise left-hand continuous functionswith discontinuities at t = tk for k ∈ N.

Definition 1: The trivial solution of the dynamical system(7) is said to be exponentially mean square stable if forany initial condition ei (t0) (i = 1, 2, . . . , N), there existpositive constants W0 and ω such that E{∑N

i=1 ‖xi (t)‖2} ≤W0e−ω(t−t0).

By referring to the concept of average dwell time [31], [32],a new concept named average impulsive interval has beenproposed by the authors to describe wider class of impulsivesignal, and has been utilized for the derivation of a unifiedsynchronization criterion of dynamical networks in [7]. Sinceμ ∈ (−2, 0), which means that the impulsive effects arestabilizing, the frequency of impulses should not be too low.In order to guarantee that the frequency of impulses is not toolow, the following definition is presented.

Definition 2 ([7] average impulsive interval): The averageimpulsive interval of the impulsive sequence ζ = {t1, t2, . . .} isless than Ta , if there exist a positive integer N0 and a positivenumber Ta , such that

Nζ (T, t) ≥ T − t

Ta− N0 ∀T ≥ t ≥ 0 (8)

where Nζ (T, t) denotes the number of impulsive times of theimpulsive sequence ζ in the time interval (t, T ).

Remark 1: According to Definition 2, there is no strictrequirement for the impulsive sequence on the upper boundof the impulsive intervals, which is normally necessary in thereferences concerning impulsive control. For very large ς > 0and any Ta > 0, many impulsive sequences {t1, t2, . . .} canbe constructed such that the upper bound of the impulsiveintervals is not less than to ς and simultaneously the averageimpulsive intervals are less than Ta . Let k = ς/Ta� and ε > 0very small. One simple example is ζ ∗ = {t0 + Ta + ε, t0 +Ta + 2ε, . . . , t0 + Ta + kε, t0 + Ta + kε + ς, t0 + Ta + (k +1)ε + ς, . . . , t0 + Ta + 2kε + ς, t0 + Ta + 2kε + 2ς, . . .}. Forthe impulsive sequence ζ ∗, the upper bound of the impulsive

interval is ς , which can be very large. Since the upper boundis used to represent the frequency of the impulsive sequence in[26], and [33]–[35], or identical impulsive interval is used in[36], the results obtained in these references are not availablefor the impulsive sequence ζ ∗ with very large upper bound ofimpulsive intervals, for which our results may be applicable.

Remark 2: By using the special example ζ ∗ presented inRemark 1, the idea behind this concept can be explained asfollows: low-density impulses (such as “t0 +Ta +kε, t0 +Ta +kε+ς”) are allowed to happen in a certain interval, and high-density impulses (such as “t0 + Ta + (k + 1)ε + ς, . . . , t0 +Ta + 2kε + ς”) should follow for compensation.

III. MAIN RESULTS

In this section, we will derive the main results about ourpinning impulsive strategy for synchronization control of thestochastic dynamical network (1) with nonlinear coupling.Based on the above-mentioned assumptions and definitions,we can obtain the following theorem to show that the state-coupled dynamical network can be successfully stabilizedto an objective state by only impulsive controlling a smallfraction of nodes.

Theorem 1: Consider the controlled dynamical network (7)with an irreducible coupling matrix A. Let #D(tk) = l, ρ =1+(l/N) ·μ(μ+2) ∈ (0, 1) and δ = λmax(C +CT + MT M)+2√λmax(BT B)κ . Suppose that Assumptions 1 and 2 hold, and

the average impulsive interval of the impulsive sequence ζ ={t1, t2, . . .} is less than Ta . Then, the controlled dynamicalnetwork (7) is globally exponentially stable in mean square, if

lnρ

Ta+ δ < 0. (9)

It means that the nonlinear stochastic dynamical network (1)can be exponentially controlled to the objective trajectorys(t) by using pinning impulsive controllers (6).

Proof: Consider the following Lyapunov functions:

V (t) =N∑

i=1

eTi (t)ei (t). (10)

For t ∈ (tk−1, tk], k ∈ N, we have

LV (t)

= 2N∑

i=1

eTi (t)

[Cei (t)+ B f (ei (t))+ c

N∑

j=1

ai j�h(e j (t))]

+N∑

i=1

trace[gT (t, ei (t))g(t, ei (t))

]

= 2N∑

i=1

[eT

i (t)Cei (t)+ eTi (t)B f (ei (t))

]

+2cN∑

i=1

N∑

j=1

ai j eTi (t)�h(e j (t))

+N∑

i=1

trace[gT (t, ei (t))g(t, ei (t))

]. (11)

288 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 2, FEBRUARY 2012

By Assumptions 1 and 2, the following inequalities can beobtained:

2eTi (t)B f (ei (t))

≤ 2‖ei (t)‖ · ‖B f (ei (t))‖≤ 2‖ei (t)‖ ·

√λmax(BT B) · ‖ f (ei (t))‖

≤ 2√λmax(BT B)κ‖ei (t)‖2

= 2√λmax(BT B)κeT

i (t)ei (t) (12)

and

trace[gT (t, ei (t))g(t, ei (t))

] ≤ ‖Mei (t)‖2

= eTi (t)M

T Mei (t). (13)

Since [(h(u)− h(v))/(u − v)] ≥ ϑ > 0, it follows from thediffusive property of symmetric matrix A that

2cN∑

i=1

N∑

j=1

ai j eTi (t)�h(e j (t))

= 2cN∑

i=1

N∑

j=1

ai j

[n∑

θ=1

eiθ (t)γθh(e jθ (t))

]

= 2cn∑

θ=1

γθ

⎡

⎣N∑

i=1

N∑

j=1

eiθ (t)ai j h(e jθ (t))

⎤

⎦

= −cn∑

θ=1

γθ

N∑

i=1

N∑

j=1j �=i

ai j (eiθ (t)− e jθ (t))

×(h(eiθ (t))− h(e jθ (t)))

≤ −cn∑

θ=1

N∑

i=1

N∑

j=1j �=i

ϑγθai j (eiθ (t)− e jθ (t))2

≤ 0. (14)

Considering (12)–(14), it follows from (11) that

LV (t)

≤N∑

i=1

eTi (t)

[C + CT + MT M

]ei (t)

+N∑

i=1

2√λmax(BT B)κeT

i (t)ei (t)

≤(λmax(C + CT + MT M)+ 2

√λmax(BT B)κ

)

×N∑

i=1

eTi (t)ei (t)

= δ · V (t), for t ∈ (tk−1, tk], k ∈ N. (15)

For any k ∈ N, let α(tk) = min{‖ei (tk)‖ : i ∈ D(tk)} andβ(tk) = max{‖ei (tk)‖ : i �∈ D(tk)}. According to the selectionof nodes in set D(tk), we have α(tk) ≥ β(tk). Since ρ =1 + (l/N) · μ(μ + 2) ∈ (0, 1), we get (1 − ρ)(N − l) =

[ρ − (1 + μ)2]l. Hence, one has

(1 − ρ)∑

i �∈D(tk)

eTi (t

−k )ei (t

−k )

≤ (1 − ρ)(N − l)(β(tk))2

≤ (1 − ρ)(N − l)(α(tk))2

≤ l[ρ − (1 + μ)2

](α(tk))

2

≤ [ρ − (1 + μ)2

] ∑

i∈D(tk)

eTi (t

−k )ei (t

−k ) (16)

which follows that

(1 + μ)2∑

i∈D(tk )

eTi (t

−k )ei (t

−k )+

∑

i �∈D(tk )

eTi (t

−k )ei (t

−k )

≤ ρ

N∑

i=1

eTi (t

−k )ei (t

−k ). (17)

Then, for any k ∈ N, we yield

V (t+k )

=N∑

i=1

eTi (t

+k )ei (t

+k )

=∑

i∈D(tk )

eTi (t

+k )ei (t

+k )+

∑

i �∈D(tk )

eTi (t

+k )ei (t

+k )

=∑

i∈D(tk )

(1 + μ)2eTi (t

−k )ei (t

−k )+

∑

i �∈D(tk)

eTi (t

−k )ei (t

−k )

≤ ρ

N∑

i=1

eTi (t

−k )ei (t

−k )

= ρV (t−k ). (18)

By (15) and (18), we can obtain the following comparisonsystem (19) for the controlled dynamical network (7):

⎧⎨

⎩

w(t) = δw(t), t ≥ t0, t �= tk,w(t+k ) = ρw(t−k ), ρ ∈ (0, 1), k ∈ N,w(t0) = E(V (t0)).

(19)

According to (19), for any t ∈ R+, one has

w(t) = E(V (t0)) · eδ(t−t0)ρNζ (t,t0) (20)

where Nζ (t, t0) means the number of impulses of the impul-sive sequence ζ in the time interval (t0, t).

According to the facts that ρ ∈ (0, 1) and that the averageimpulsive interval of the impulsive sequence ζ = {t1, t2, . . .}is less than Ta , it follows from Definition 2 that

w(t) = E(V (t0)) · eδ(t−t0)ρNζ (t,t0)

≤ E(V (t0)) · eδ(t−t0)ρt−t0Ta

−N0

= E(V (t0))ρ−N0 · eδ(t−t0) · e

lnρTa(t−t0)

= E(V (t0))ρ−N0 · e(

lnρTa

+δ)(t−t0). (21)

Since (lnρ/Ta) + δ < 0, the trivial solution of the compar-ison system (19) is exponentially stable. By Lemma 1, wecan conclude that the controlled dynamical network (7) isexponential stable, which further implies that the dynamicalnetwork (1) can be exponentially stabilized to the objective

LU et al.: SYNCHRONIZATION CONTROL FOR NONLINEAR STOCHASTIC DYNAMICAL NETWORKS 289

trajectory s(t) by only impulsively controlling a small fractionof nodes. Theorem 1 is proved.

Remark 3: By using the Lyapunov method combined withthe comparison principle, the exponential stability criterionof the pinning impulsively controlled dynamical network hasbeen obtained. It means that the state-coupled dynamicalnetwork can be efficiently forced to the objective trajectoryby using pinning impulsive controllers. Our result displaysanother kind of effective and relatively cheap control strategyfor the synchronization of complex dynamical networks.

Remark 4: The criterion presented in Theorem 1 is closelyrelated to the system parameters, average impulsive interval,impulsive strength, and the proportion of the controlled nodes.The criterion can be easily judged without large computa-tion. In the following, a theorem will be given to explicitlyshow how many nodes should be controlled for a successfulsynchronization control of the nonlinear stochastic dynamicalnetwork (1).

Theorem 2: Consider the controlled dynamical network (7)with an irreducible coupling matrix A. Let δ = λmax(C+CT +MT M) + 2

√λmax(BT B)κ . Suppose that Assumptions 1 and

2 hold, and the average impulsive interval of the impulsivesequence ζ = {t1, t2, . . .} is less than Ta . Then, the controlleddynamical network (7) is globally exponentially stable in meansquare, if

l

N>

1

μ(μ+ 2)

(e−δTa − 1

)(22)

where l N is the number of nodes to be controlled.Proof: Since ρ = 1 + (l/N) · μ(μ + 2) ∈ (0, 1), this

theorem can be proved by using Theorem 1. The detailed proofis omitted here.

Remark 5: Since l N is the number of nodes to becontrolled, (l/N) 1 is the proportion of the controllednodes. The numerical example illustrates that the stochasticdynamical network can be successfully synchronized to acertain objective trajectory by impulsively controlling 10%of the nodes. It means that our pinning impulsive strategyis effective for the synchronization of networks with a smallfraction of nodes controlled.

Remark 6: Similar with Theorem 2, we can conclude that:the stochastic dynamical network (1) can be forced to theobjective trajectory s(t) by pinning impulsive controllers (6)if one of the following inequalities is satisfied:

1) Ta < − 1δ ln(μ(μ+ 2) · l

N + 1);

2) −√

e−δTa −1l/N + 1 − 1 < μ <

√e−δTa −1

l/N + 1 − 1.

Remark 7: The synchronization problem for discrete-timestochastic dynamical networks has drawn much research atten-tion [36], [37]. Our pinning impulsive strategy obtained in thispaper is also applicable to the case of discrete-time dynamicalnetworks.

IV. NUMERICAL EXAMPLE

In this section, numerical example will be given to demon-strate the effectiveness of our main results. A chaotic systemwith Brownian noise is selected as the isolated node of the

−2

0

2

−2

0

2−1.5

−1

−0.5

0

0.5

1

1.5

(a)

−40−20

020

−50

0

50−30

−20

−10

0

10

20

(b)

Fig. 1. Phase trajectories of single dynamical system (a) without noise and(b) with noise.

Synchronization region

00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.5

1 0−1−0.5

−1.5−2

µ

l/N

Ta

Fig. 2. Estimation of boundaries of synchronization region with respect toμ, Ta , and (l/N).

dynamical network, and the i -th node is described as follows:

dxi(t) = [Cxi (t)+ B f (xi(t))

]dt + g(t, xi (t))dwi (t) (23)

where xi (t) = (xi1(t), xi2(t), xi3(t))T ∈ R3, dwi (t) is an

3-D Brownian motion, and the parameters are given as C =−1.2 · I3, B =

⎛

⎝1.16 −1.5 −1.5−1.5 1.16 −2.0−1.2 2.0 1.16

⎞

⎠, nonlinear function

f (xi (t)) = (tanh(xi1), tanh(xi2), tanh(xi3))T , and the noise

intensity function matrix g(t, xi(t)) = 0.5 · ‖xi (t)‖ · I3. Thenwe have κ = 1 and M = 0.5 · I3 for Assumptions 1 and 2.System (23) without Brownian motion noise has a chaoticattractor [38] with initial value [0.3,−0.1,−0.4] as shownin Fig. 1.

In this example, a Newman-Watts small-world network with100 nodes will be considered [39]. The small-world networkis generated by taking initial neighboring nodes k = 4 andthe edge adding probability p = 0.1. The coupling matrix Ais defined as follows: if there is a connection between nodesi and j , then ai j = a j i = 1, otherwise ai j = a j i = 0. Thenonlinear coupling function h is taken as h(z) = z + tanh(z)

290 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 2, FEBRUARY 2012

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ta

l/N

µ�−0.3µ�−0.5µ�−0.7µ�−0.9

Synchronization region

Fig. 3. Estimation of the synchronization region about Ta and (l/N) withdifferent μ.

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 00

0.02

0.04

0.06

0.08

0.1

0.12

0.14

µ

Ta

l/N�0.1l/N�0.2l/N�0.3l/N�0.4l/N�0.5

Synchronization region

Fig. 4. Estimation of the synchronization region about μ and Ta with different(l/N).

with ϑ = 1, coupling strength c = 1 and the inner couplingmatrix � = I3. By some simple calculations, we can obtainthat δ = 5.0542. Theorem 2 and Remark 6 have been given toshow the explicit relationship between three quantities (l/N),μ and Ta . Fig. 2 shows the synchronization region of thecontrolled dynamical network for μ, Ta , and (l/N). Fig. 3displays the estimation of boundaries of the synchronizationregions for different μ with respect to Ta and (l/N).

Remark 8: According to the property of the impulsivecontrol, one knows that if μ = −1, the error state ofthe controlled system becomes zero immediately after theimpulsive controller. Hence the impulsive interval can be +∞when μ = −1 [40], [41]. However, since only a small fractionof nodes is controlled at each impulsive instant tk , the errorstates of the controlled nodes would become nonzero due tothe interconnections with some other uncontrolled nodes evenfor μ = −1. Therefore, when μ = −1, the average impulsiveinterval has an upper bound as shown in Fig. 4. In some realapplications, one may choose μ = −1 to have the maximumimpulsive interval.

Now, we take special values of μ, Ta , and (l/N) fornumerical illustration. Let μ = −0.9 and Ta = 0.02, one canget that [1/μ(μ + 2)](e−δTa − 1) = 0.0971. By Theorem 2,it can be concluded that the nonlinear stochastic dynamicalnetwork can be synchronized to the objective trajectory if(l/N) = 10% of the nodes is controlled. In other words,ten nodes should be impulsively controlled in the generatedsmall-world network containing 100 nodes. The trivial points(t) = 0 is taken as the objective trajectory in this example.

0 0.2 0.4 0.6 0.8 1−10

−8

−6

−4

−2

0

2

4

6

8

10

t

x i(t),

i �

1,2,

…,1

00

Fig. 5. Pinning impulsive synchronization of small-world coupled dynamicalnetworks by controlling 10% nodes.

Fig. 5 presents the numerical process for the synchronizationcontrol, in which the impulsive sequence is generated withN0 = 10 and ε = 0.01. All initial values of the dynamicalnetwork are uniformly randomly selected from [−10, 10].

V. CONCLUSION

In this paper, the synchronization control problem of sto-chastic dynamical networks with nonlinear coupling has beenstudied by pinning a small fraction of nodes with impulsivecontrollers. The uncontrolled nodes can be virtually forced tothe desired synchronization trajectory by the pinned nodes viathe inter-connections. Some stability criteria have been estab-lished to guarantee the success of synchronization via pinningcontrollers, and moreover the stable region can be explicitlyrevealed and plotted. Numerical examples are also given todemonstrate the effectiveness of our proposed control strategy.

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Jianquan Lu received the B.S. degree in math-ematics from Zhejiang Normal University, Zhe-jiang, China, the M.S. degree in mathematics fromSoutheast University, Nanjing, China, and the Ph.D.degree from the City University of Hong Kong,Kowloon, Hong Kong, in 2003, 2006, and 2009,respectively.

He is currently an Associate Professor with theDepartment of Mathematics, Southeast University.He has published over 25 papers in refereed interna-tional journals. His current research interests include

collective behavior in complex dynamical networks, multi-agent systems,hybrid systems, synchronization, nonlinear systems and controls, and neuralnetworks.

Dr. Lu is an Associate Editor of Neural Processing Letters (Springer) and aGuest Editor of Mathematics and Computers in Simulation (Elsevier). He isthe recipient of the Jiangsu Provincial Outstanding Master’s Thesis Award in2007, Top 100 Most Cited Chinese Papers Published in International Journalsin 2007, an Alexander von Humboldt Fellowship in 2010, a Program for NewCentury Excellent Talents in University by the Ministry of Education, China,in 2010, and the First Award of Jiangsu Provincial Progress in Science andTechnology as the Second Project Member in 2010.

Jürgen Kurths received the degree in mathematicsfrom the University of Rostock, Rostock, Germany,and the Ph.D. and Dr.Habil. degrees from the GDRAcademy of Sciences, Berlin, Germany in 1983 and1990, respectively.

He was a Full Professor with the University ofPotsdam, Potsdam, Germany, from 1994 to 2008.Currently, he is a Professor of nonlinear dynamicswith Humboldt University, Berlin, Germany, andthe Chair of the research domain transdisciplinaryconcepts of the Potsdam Institute for Climate Impact

Research. He has published more than 450 papers and two monographswhich are cited more than 16 000 times. His current research interestsinclude complex synchronization phenomena, complex networks, time seriesanalysis and their applications in climatology, sustainability, physiology, andengineering.

He is a fellow of the American Physical Society and the Fraunhofer Societyin Germany, and a member of Academia Europaea. He received a Humboldt-CSIR Research Prize in 2005 and a Dr.h.c. in 2008. He is on the editorialboards of more than ten journals, among them CHAOS, Philosophical Trans.Royal Soc. A, European Journal of Physics ST, and Nonlinear Processes inGeophysics.

292 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 2, FEBRUARY 2012

Jinde Cao (M’07–SM’07) received the B.S.degree from Anhui Normal University, Wuhu, China,the M.S. degree from Yunnan University, Kun-ming, China, and the Ph.D. degree from SichuanUniversity, Chengdu, China, in 1986, 1989, and1998, respectively, all in mathematics/applied math-ematics.

He was with Yunnan University from 1989 to2000. In 2000, he joined the Department of Mathe-matics, Southeast University, Nanjing, China. From2001 to 2002, he was a Post-Doctoral Research

Fellow with the Department of Automation and Computer-Aided Engineering,Chinese University of Hong Kong, Shatin, Hong Kong. From 2006 to 2008,he was a Visiting Research Fellow and a Visiting Professor with the Schoolof Information Systems, Computing and Mathematics, Brunel University,Middlesex, U.K. He is the author or co-author of more than 160 researchpapers and five edited books. His current research interests include nonlinearsystems, neural networks, complex systems, complex networks, stabilitytheory, and applied mathematics.

Dr. Cao was an Associate Editor of the IEEE TRANSACTIONS ON NEURAL

NETWORKS from 2006 to 2009. He is an Associate Editor of the Journal ofthe Franklin Institute, Mathematics and Computers in Simulation, Neurocom-puting, Discrete Dynamics in Nature and Society, the International Journalof Differential Equations, and Differential Equations and Dynamical Systems.He is a reviewer of Mathematical Reviews and Zentralblatt-Math.

Nariman Mahdavi received the B.Sc. degree fromthe Isfahan University of Technology, Isfahan, Iran,in 2004, and the M.Sc. (first class) and Ph.D.degrees from the Amirkabir University of Technol-ogy, Tehran, Iran, in 2007 and 2011, respectively, allin electrical engineering.

He was a Guest Researcher with the PotsdamInstitute for Climate Impact Research, Potsdam,Germany, from August 2010 to February 2011. Hiscurrent research interests include complex systemsand their synchronization, adaptive and impulsive

controls, computational intelligence, and their applications in controls.

Chi Huang received the M.S. degree in math-ematics from the City University of Hong Kong,Kowloon, Hong Kong, in 2009. He is currentlypursuing the Ph.D. degree with the Department ofMathematics, City University of Hong Kong.

His current research interests include multiagentsystems, complex dynamical networks, and sensornetworks.

Mr. Huang was the recipient of the Best PaperAward in the Eighth Asian Control Conferencein 2011.