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Synchronization control for the competitive complex networks with time delay and stochastic effects
Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
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Synchronization control for the competitive complex networks with time delay and stochastic effects Wuneng Zhou a, Tianbo Wang a,b,⇑, Jinping Mou a a b
College of Information Sciences and Technology, Donghua University, Shanghai 200051, PR China College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China
a r t i c l e
i n f o
Article history: Received 27 January 2011 Received in revised form 14 August 2011 Accepted 25 December 2011 Available online 8 January 2012 Keywords: Competitive complex networks Synchronization control Time delay Stochastic effects
a b s t r a c t The synchronization control problem for the competitive complex network with time delay and stochastic effects is investigated by using the stochastic technique and Lyapunov stability theory. The competitive complex network means that the dynamical varying rate of a part of nodes is faster than other nodes. Some synchronization criteria are derived by the full controller and pinning controller, respectively, and these criteria are convenient to be used for concision. A numerical example is provided to illustrate the effectiveness of the method proposed in this paper. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, complex networks have received increasing research attention from the fields of the technological practice to theoretical study [1–5]. One of the important reasons is that complex networks can be found in almost everywhere in the real world, such as the Internet, the genetic networks, and the social networks [6–11]. Among all of the dynamical behaviors of complex networks, synchronization is one of the most interesting topics and has been extensively investigated (see [12–18]). The contributions of these efforts are mainly composed of deriving the synchronization criteria and designing different controllers to attain synchronization of complex networks. The common synchronization control methods include the adaptive control, pinning control, hybrid control, impulsive control and so on. On the other hand, it is known that time delays and the stochastic disturbances are important factors that effect the synchronization ability of complex networks in the real world. So there exist many literatures such as [19–24] to study the synchronization problem of complex networks with time delays and the stochastic disturbances. Besides the study for some common complex networks models, there also exist a lot of literatures to study special complex networks models. For instance, in [25], the authors considered the synchronization of complex dynamical networks under successful but recoverable attacks utilizing the framework of switching systems, and obtained the upper bounds of the attack frequency and the average recovering time to ensure that the whole network achieves global synchronization under attacks. In [26], the authors studied the synchronization problem for complex dynamical networks with switching topology from a switched system point of view, and established synchronization criteria for general connection topology by constructing multiple Lyapunov functions. In [27], the authors constructed a model of complex dynamical networks with multi-links by splitting time delay and achieved synchronization between two complex networks with different structures by designing effective controllers. ⇑ Corresponding author at: College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China. E-mail address: [email protected] (T. Wang). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.12.021
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
As a special case of dynamical complex networks, when part of nodes in the complex networks are disturbed by some singular parameters, the varying rate of different nodes exists diversity. Many practical examples on competitive dynamical systems and the reason to bring about these kinds of competitive phenomenon can be found in [28]. However, for the complex networks with this character, there exists less research yet. In this paper, we intend to investigate the synchronization control problem for competitive complex networks by using full controller and pinning controller, respectively. The rest of this letter is organized as follows. In Section 2, the proposed competitive complex network model and some preliminaries are given. In Section 3, the synchronization criteria are derived by the full controller and pinning controller, respectively. In Section 4, a numerical example is provided to illustrate the effectiveness of the method proposed. In the last section, conclusions are presented. Notation. Rn and Rnm, respectively, denote the n-dimensional Euclidean space and the set of all n m real matrices, kk represents the norm of a vector. C([a, b],Rn) denotes the set of continuous vector functions defined on the interval [a, b]. In is an n n identical matrix, kmax(H) stands for the largest eigenvalue of matrix H. is the Kronecker product, L2(R+; Rn) is the set of square integrable functions defined on R+ = (0, +1). 2. Problem formulation In this paper, we consider the following competitive complex networks
"
ei dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
# g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
j¼1
" dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
i ¼ 1; 2; . . . ; l;
#
ð1Þ
g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
i ¼ l þ 1; . . . ; N;
j¼1
where N denotes the number of nodes in the networks, ei is a small positive constant satisfying 0 < ei 6 1. xi(t) 2 Rn denotes the state of node i, f(xi(t), t) : Rn R+ ? Rn is a nonlinear vector valued function. ci denotes the coupling strength of node i with other nodes, C is the inner-coupling matrix, ui(t) is the control input. dwi(t) and ui(t) 2 Rn denote the white noise and noise intensity, respectively. gij is defined as follows: if there exists a connection between node i and node j then gij = 1, else gij = 0. P The matrix G = [gij]NN is called outer-coupling matrix, in particular, g ii ¼ Nj¼1;j–i g ij . The time varying delay s(t) is differentiable with 0 6 s(t) 6 s and s_ ðtÞ 6 l < 1. Let el+1 = el+2 = = eN = 1, then the complex network (1) can be written as
"
ei dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
#
g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
i ¼ 1; 2; . . . ; N:
ð2Þ
j¼1
Let the target node model and the designed control law be
ds ¼ f ðsðtÞ; tÞ dt þ /ðtÞ drðtÞ
ð3Þ
ui ðtÞ ¼ ki ðxi ðtÞ yi ðtÞÞ;
ð4Þ
and
respectively, where
yi ðtÞ ¼ sðtÞ þ wi ðtÞ dmi ðtÞ;
i ¼ 1; 2; . . . ; N
ð5Þ
is the measurement of node i on the target node. dr(t) and dmi(t) denote the noise disturbance of target node and measure noise from target node, respectively. /(t) 2 Rn and wi(t) 2 Rn denote the corresponding noise intensities, respectively. Denoting ei(t) = xi(t) s(t), then
"
ei dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
#
g ij Cej ðt sðtÞÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ
j¼1
ei /ðtÞ drðtÞ
ð6Þ
for i = 1, 2, . . . , N. In what follows, we will provide some assumptions and a lemma.
Assumption 1. The function f(x(t), t) is Lipschitz continuous on x(t) and in L2(Rn R+; Rn), i.e. there exists a positive constant L > 0 such that
kf ðxðtÞ; tÞ f ðzðtÞ; tÞk 6 LkxðtÞ zðtÞk;
8xðtÞ;
zðtÞ 2 Rn
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and
Z
1
kf ðxðhÞ; hÞk2 dh < 1:
0
From Assumption 1, it is easy to see that
kf ðxðtÞ; tÞ f ðsðtÞ; tÞk 6 LkeðtÞk: Assumption 2. The noises intensities /(t), wi(t) and ui(t) are in L2(R+; Rn) for i = 1, 2, . . . , N. Assumption 3. dxi(t), dmi(t) and dr(t) are independent 1-dimensional Brownian motion with
Efdxi ðtÞg ¼ Efdmi ðtÞg ¼ EfdrðtÞg ¼ 0; Ef½dxi ðtÞ2 g ¼ Ef½dmi ðtÞ2 g ¼ Ef½drðtÞ2 g ¼ dt and
Efdxi ðtÞ dmj ðtÞg ¼ Efdxi ðtÞ drðtÞg ¼ EfdrðtÞ dmj ðtÞg ¼ 0 for i, j = 1, 2, . . . , N. The objective of this paper is to design controller (4) such that all of the nodes in the complex network (1) synchronize with target node (3) for any initial state, i.e. all of the states satisfy
limt!1 kxi ðtÞ sðtÞk ¼ limt!1 kei ðtÞk ¼ 0;
8i ¼ 1; 2; . . . ; N:
Remark 1. Different from some existing complex network models [12–17], complex network (1) is a competitive network in which a part of nodes vary faster than other nodes. The first l nodes are called fast-varying nodes and the later N l nodes are called slow-varying nodes. Especially, while ei ? 0, i = 1, 2, . . . , l, the first l subsystems will become algebra systems, which will cause abrupt change in the dynamical properties of these nodes. Definition 1 [29]. For an n-dimensional stochastic differential system
dxðtÞ ¼ f ðxðtÞ; xðt sÞ; tÞ dt þ gðxðtÞ; xðt sÞ; tÞ dtðtÞ
ð7Þ
n
n
n
n
with initial state n(t) 2 R on t > 0, where dt(t) is an m-dimensional Brownian motion, f(x(t), x(t s), t) : R R R+ ? R and g(x(t), x(t s), t) : Rn Rn R+ ? Rnm are continuous differentiable functions. Let V(x(t), t) 2 C2,1(Rn R+; R+). The operator LVðxðtÞ; tÞ is defined as
1 LVðxðtÞ; tÞ ¼ V t ðxðtÞ; tÞ þ V x ðxðtÞ; tÞf ðxðtÞ; xðt sÞ; tÞ þ trace fg T ðxðtÞ; xðt sÞ; tÞV xx gðxðtÞ; xðt sÞ; tÞg; 2 2
where V t ðxðtÞ; tÞ ¼ @V ðxðtÞ;tÞ ; V x ðxðtÞ; tÞ ¼ @t
@V ðxðtÞ;tÞ @VðxðtÞ;tÞ ; @x2 ; . . . ; @V ðxðtÞ;tÞ @x1 @xn
and V xx ¼
@ V ðxðtÞ;tÞ @xi @xj
nn
.
Lemma 1 [30]. Assume that the stochastic differential system (7) exists a unique solution x(t, n(t)) on t P 0 for any given initial state nðtÞ 2 C bF 0 ð½s; 0; Rn Þ. Moreover, both f(x(t), x(t s), t) and g(x(t), x(t s), t) are locally bounded and uniformly bounded on t. If there exists a function V(x(t), t) 2 C2,1(Rn R+; R+), b(t) 2 L1(R+; R+) and x1, x2 2 C(Rn; R+) such that
LVðxðtÞ; xðt sÞ; tÞ 6 bðtÞ x1 ðxðtÞÞ þ x2 ðxðt sÞÞ;
8xðtÞ 2 Rn ;
x1 ðxðtÞÞ > x2 ðxðtÞÞ; 8xðtÞ – 0; limkxk!1 inf 06t61 VðxðtÞ; tÞ ¼ 1; then
limt!1 xðt; nðtÞÞ ¼ 0; for every nðtÞ 2
C bF 0 ð½
a:s: n
s; 0; R Þ.
3. Main results In this section, we derive some synchronization schemes by using the full controller and pinning controller, respectively. 3.1. Synchronization criteria with full controller Theorem 1. Under Assumptions 1–3, for given constants L > 0 and s > 0, the complex network (1) synchronizes with the target node (3) under the control law (4) if there exists positive scalars j > 0 and ki > 0 such that
L þ 2 þ s ki þ kmax fðCGGT CÞ ðCCT Þg þ j < 0
ð8Þ
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
for i = 1, 2, . . . , N. Furthermore, the control gain satisfies
ki > L þ 2 þ s þ kmax fðCGGT CÞ ðCCT Þg; where C = diag (c1, c2, . . . , cN), G = [gij]NN. Proof. Choosing a Lyapunov functional candidate N N Z t N Z 0 Z t X X 1X ei eTi ðtÞei ðtÞ þ eTi ðsÞei ðsÞ ds þ eTi ðhÞei ðhÞ dh ds: 2 i¼1 tsðtÞ s tþs i¼1 i¼1
VðtÞ ¼
By the Itö differential formula, the derivative of V(t) along the state trajectories of system (6) is N X
dVðtÞ ¼ LVðtÞ dt þ
eTi ðtÞ½ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ ei /ðtÞ drðtÞ:
i¼1
By Definition 1, we have LVðtÞ ¼
Z N X ½ei eTi ðtÞe_ i ðtÞ þ eTi ðtÞei ðtÞ ð1 s_ ðtÞÞeTi ðt sðtÞÞei ðt sðtÞÞ þ seTi ðtÞei ðtÞ
ts
i¼1
þ
¼
t
eTi ðsÞei ðsÞds
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei i¼1
N X
(
" eTi ðtÞ
f ðxi ðtÞ;tÞ f ðsðtÞ;tÞ þ ð1 ei Þf ðsðtÞ;tÞ þ ci
i¼1
N X
# g ij Cej ðt sðtÞÞ ki ei ðtÞ
j¼1
N X 1 T þ 1 þ sÞeTi ðtÞei ðtÞ ð1 lÞeTi ðt sðtÞÞei ðt sðtÞÞ þ ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2 e i i¼1
6
N X
eTi ðtÞðL þ 1 þ s ki Þei ðtÞ þ ð1 ei Þkf ðsðtÞ;tÞk kei ðtÞk
i¼1
þ
N X N X
ci g ij eTi ðtÞCej ðt sðtÞÞ ð1 lÞ
i¼1 j¼1
N X
eTi ðt sðtÞÞei ðt sðtÞÞ þ
i¼1
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ : 2 e i i¼1
ð9Þ
Noticing that N X
N pffiffiffiffi X kei ðtÞk 6 N kei ðtÞk2
i¼1
!12 ð10Þ
i¼1
and N X N X i¼1
ci g ij eTi ðtÞCej ðt sðtÞÞ 6 kmax fðCGGT CÞ ðCCT Þg
j¼1
N X
eTi ðtÞei ðtÞ þ
i¼1
N X
eTi ðt sðtÞÞei ðt sðtÞÞ;
ð11Þ
i¼1
we obtain LVðtÞ 6 max16i6N fL þ 1 þ s ki g
N X
N X pffiffiffiffi kei ðtÞk2 þ max16i6N ð1 ei Þ N kf ðsðtÞ;tÞk kei ðtÞk2
i¼1
!12
i¼1
N X 1 T þ kmax fðCGGT CÞ ðCCT Þg eTi ðtÞei ðtÞ þ l eTi ðt sðtÞÞei ðt sðtÞÞ þ ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ : 2ei i¼1 i¼1 i¼1 N X
N X
ð12Þ
From the inequality (8), one gets
2 LVðtÞ 6 j4
N X i¼1
þ
!12 2
kei ðtÞk
32
N N X aðtÞ5 a2 ðtÞ X þ eTi ðtÞei ðtÞ þ l eTi ðt sðtÞÞei ðt sðtÞÞ 2j 4j i¼1 i¼1
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ ; 2 e i i¼1
ð13Þ
pffiffiffiffi where aðtÞ ¼ max16i6N ð1 ei Þ N kf ðsðtÞ; tÞk. By Lemma 1, we know that complex network (1) is synchronization with the target node (3). The proof of Theorem 1 is complete. h
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
3421
Remark 2. When ei = 1, i = 1, 2, . . . , l, the complex network (1) still synchronizes with the target node (3) under the action of controller (4) if the inequality (8) holds. For this case, all the nodes can synchronize each other if only the nonlinear function f(s(t), t) satisfies Lipschitz condition. Different from the results in [20,22,26], we only need to verify the inequality (8) according to the parameters of complex network (1) and not to solve some complex linear or nonlinear matrix inequalities depended on some unknown variables. Which also shows that synchronization is an essential phenomenon of the complex network and only depends on its parameters. Remark 3. It can be seen from the synchronization condition (8) that the control gain only depends on the time delay, the Lipschitz constant, the coupling matrices and the coupling strength, which implies that the synchronization of a complex network tightly depends on its parameters. Furthermore, we can obtain the smallest synchronization control gain from (8). Next, we will consider the synchronization problem for the complex network (1) and the target node (3) by pinning control method. We will study this control problem in two cases, one is only controlling fast-varying nodes and the other is only controlling slow-varying nodes. 3.2. Synchronization criteria with pinning controller While using control law
( ui ðtÞ ¼
ki ðxi ðtÞ yi ðtÞÞ; 0;
i ¼ 1; 2; . . . ; l;
ð14Þ
i ¼ l þ 1; l þ 2; . . . ; N;
to control complex network (1) without time delay, we obtain the following dynamical system
ei dei ðtÞ ¼ ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
g ij Cej ðtÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ
j¼1
þ ki wi ðtÞdmi ðtÞ ei /ðtÞdrðtÞ; dei ðtÞ ¼ ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci
N X
i ¼ 1; 2; . . . ; l;
g ij Cej ðtÞ dt þ ui ðtÞ dxi ðtÞ /ðtÞ drðtÞ; i ¼ l þ 1; l þ 2; . . . ; N:
ð15Þ
j¼1
Theorem 2. Under Assumptions 1–3, for a given scalar L > 0, the complex network (1) without time delay synchronizes with the target node (3) under the pinning controller (14) if there exist positive constants g > 0, j > 0 and ki > 0 such that
e In þ ðCGÞ C < 0 ðL þ jÞIN In þ K 0
ð16Þ
1
e ¼ diag @g k1 ; g k2 ; . . . ; g kl ; 0; . . . ; 0A. for i = 1, 2, . . . , l, where K |fflfflfflffl{zfflfflfflffl} Nl
Proof. Choose a Lyapunov functional candidate
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; P P where V 1 ðtÞ ¼ 12 li¼1 ei eTi ðtÞei ðtÞ and V 2 ðtÞ ¼ 12 Ni¼lþ1 eTi ðtÞei ðtÞ. By the Itö differential formula, the derivative of V(t) along the state trajectories of system (15) is dVðxðtÞÞ ¼ LV 1 ðtÞdt þ LV 2 ðtÞdt þ
l X
eTi ðtÞ½ui ðtÞdxi ðtÞ þ ki wi ðtÞdmi ðtÞ ei /ðtÞdrðtÞ þ
i¼1
N X
eTi ðtÞ½ui ðtÞdxi ðtÞ /ðtÞdrðtÞ:
i¼lþ1
Using Definition 1, we get
LV 1 ðtÞ ¼
l X
ei eTi ðtÞe_ i ðtÞ þ
i¼1
¼
l X
l X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei i¼1
eTi ðtÞ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
i¼1
N X j¼1
g ij Cej ðtÞ ki ei ðtÞ þ
l X 1 T u ðtÞui ðtÞ 2 ei i i¼1
l X T 2 ei ðtÞðL ki Þei ðtÞ þ ð1 ei Þkf ðsðtÞ; tÞk kei ðtÞk þ ki wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 6 i¼1
þ and
l X
N X
i¼1
j¼1
ci g ij eTi ðtÞCej ðtÞ þ
l X i¼1
1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei
ð17Þ
3422
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426 N X 1 T ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ 2 i¼lþ1 i¼lþ1 " # N N N X X X 1 T T ¼ ei ðtÞ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci g ij Cej ðtÞ þ ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ 2 j¼1 i¼lþ1 i¼lþ1
LV 2 ðtÞ ¼
6
N X
N X
eTi ðtÞe_ i ðtÞ þ
LeTi ðtÞei ðtÞ þ
N N X X
ci g ij eTi ðtÞCej ðtÞ þ
i¼lþ1 j¼1
i¼lþ1
N X 1 T ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ : 2 i¼lþ1
ð18Þ
From (17) and (18), we obtain
2 LVðtÞ 6 g4
l X
!12 kei ðtÞk2
i¼1
32 bðtÞ5 e In þ ðCGÞ CeðtÞ þ UðtÞ; þ eT ðtÞ½LðIN In Þ þ K 2g
pffiffiffiffi T where bðtÞ ¼ max16i6l ð1 ei Þ N kf ðsðtÞ; tÞk; eðtÞ ¼ eT1 ðtÞ; eT2 ðtÞ; . . . ; eTN ðtÞ ; UðtÞ ¼ T PN T 2 T 1 þei / ðtÞ/ðtÞÞ þ i¼lþ1 2 ui ðtÞui ðtÞ þ / ðtÞ/ðtÞ . From (16), one gets
b2 ðtÞ 4g
þ
ð19Þ
Pl
1 i¼1 2ei
uTi ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ
LVðtÞ 6 jeT ðtÞeðtÞ þ UðtÞ:
ð20Þ
Noticing that
eT ðtÞeðtÞ P 2VðtÞ; yields
20
20
0
0 −20 −40
−40
ei2(t)
ei1(t)
−20
−60
−80
−80
−100
−100 −120
−60
−120 0
20
40
60
80
−140
100
0
20
40
Time(sec)
60
80
100
Time(sec)
(a)
(b) 80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 1. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the full controller.
3423
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
20
20
0
0 −20 −40
−40
ei2(t)
ei1(t)
−20
−60
−80
−80
−100
−100 −120
−60
−120 0
20
40
60
80
−140
100
0
20
40
Time(sec)
60
80
100
Time(sec)
(a)
(b) 80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 2. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the pinning fast-varying nodes controller.
LVðtÞ 6 2jVðtÞ þ UðtÞ:
ð21Þ
Integrating t on the both sides of (21) from 0 to T > 0, we get
VðTÞ 6 Vð0Þ 2j
Z
T
VðtÞ dt þ M;
ð22Þ
0
where M ¼
R1 0
UðtÞ dt. By the Gronwall–Bellman inequality [28], we obtain
VðTÞ 6 ðM þ Vð0ÞÞ exp ð2jTÞ:
ð23Þ
Since T is arbitrary, limt?1kei(t)k = 0 for i = 1, 2, . . . , N. The proof of Theorem 2 is complete. h While using control law
ui ðtÞ ¼
0;
i ¼ 1; 2; . . . ; l;
ki ðxi ðtÞ yi ðtÞÞ;
ð24Þ
i ¼ l þ 1; l þ 2; . . . ; N;
to control complex network (1) without time delay, we obtain the following closed-loop system
"
ei dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
# g ij Cej ðtÞ dt þ ui ðtÞ dxi ðtÞ ei /ðtÞ drðtÞ;
j¼1
i ¼ 1; 2; . . . ; l; " dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci
N X j¼1
i ¼ l þ 1; l þ 2; . . . ; N:
# g ij Cej ðtÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ /ðtÞ drðtÞ;
ð25Þ
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
20
20
0
0
−20
−20 −40
−60
ei2(t)
ei1(t)
−40
−80
−80
−100
−100
−120
−120
−140 −160
−60
0
20
40
60
80
−140
100
0
20
40
60
Time(sec)
Time(sec)
(a)
(b)
80
100
80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 3. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the pinning slow-varying nodes controller.
Using the same idea in Theorem 2, we obtain the following theorem. Theorem 3. Under Assumptions 1–3, for a given scalar L > 0, the complex network (1) without time delay synchronizes with the target node (3) under the pinning controller (24) if there exist positive constants g > 0, j > 0 and ki > 0 such that
b In þ ðCGÞ 0 ðL þ jÞIN In þ K C<0
1
ð26Þ
b ¼ diag @g; . . . ; g; klþ1 ; klþ2 ; . . . ; kN A. for i = l + 1, l + 2, . . . , N, where K |fflfflfflffl{zfflfflfflffl} l
Remark 4. Theorems 2 and 3 are derived by using the method of only controlling fast-varying nodes and slow-varying nodes, respectively. It can be seen from our results that the control effectiveness is better if the number of controlled nodes becomes more, so the control effectiveness by using the full controller is better than by using the pinning controller. Remark 5. We have only considered the synchronization problem for competitive complex network without time delay by pinning controller. However, it is still a problem on the synchronization of competitive complex networks with time delay using the pinning controller, which is the next research topic for us.
4. A numerical example Consider the complex network (1) with five nodes (N = 5) and each node being a 3-dimensional dynamical system (n = 3).
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
2
0:3 0:5 We choose f ðxðtÞ; tÞ ¼ Aðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ , where A ¼ 4 0:6 1 0:8 0:3 It can be seen that there exists a positive constant L = 2 such that T
1 1þt 2
3425
3 0:9 0:7 5. 1
kf ðxðtÞ; tÞ f ðyðtÞ; tÞk 6 LkxðtÞ yðtÞk for x(t), y(t) 2 R3 and f(x(t), t) 2 L2([0, 1), R3) for kx(t)k < 1. The other parameters of the complex network (1) are given as follows: e1 = 0.001, e2 = 0.01, e3 = 0.1, C = diag (0.2, 0.25, 0.3, 0.35, 0.4), C = diag (3, 4, 5),
2
4 1 1 1 6 1 6 1 3 0 6 G¼6 0 2 0 6 1 6 1 0 3 4 1 1
1
1
1
3 1 7 1 7 7 1 7 7; 7 1 5 4
sðtÞ ¼ 0:5 sin t þ 1; s ¼ 1:5; l ¼ 0:5:
pffi The noise strength coefficients are ui ðtÞ ¼ lnðtþ1Þ ; /ðtÞ ¼ wi ðtÞ ¼ 1þtt2 for i = 1, 2, 3, 4, 5, respectively. The inequality (8) holds tþ1 when j = 1 and ki = 96.5675 for i = 1, 2, 3, 4, 5. Therefore, from Theorem 1, the five nodes can synchronize with the target node. The simulation results on the synchronization error with initial state
eð0Þ ¼ ð10; 80; 65; 120; 140; 40; 40; 60; 55; 90; 10; 70; 95; 20; 10ÞT are shown in Fig. 1. This figure shows that all the nodes synchronize well. If we only control fast-varying nodes, the inequality (16) holds when j = 0.1, g = 0.5, ki = 96.5675 for i = 1, 2, 3. From Theorem 2, the five nodes synchronize with the target node. The simulation results on the synchronization error with the same initial state e(0) are shown in Fig. 2. In this example, there does not exist ki for i = 4, 5 satisfying the inequality (26). So the five nodes cannot synchronize with the target node by pinning controller (24). Taking j = 0.1, g = 0.5, ki = 96.5675 for i = 4, 5, the simulation results on the synchronization error with the same initial state e(0) are shown in Fig. 3. This figure implies that the five nodes do not synchronize with the target node if only controlling slow-varying nodes. 5. Conclusions The synchronization of the competitive complex networks with time delay and stochastic effect has been studied by the Lyapunov stability theory and stochastic analysis technique. The full controller and pinning controller have been designed respectively to obtain the synchronization schemes. A numerical example has shown that the proposed method are effective. Acknowledgements This work was supported by the National Natural Science Foundation of China (61075060), the National 863 Key Program of China (2008AA042902), the Innovation Program of Shanghai Municipal Education Commission (12zz064,11xk11). References [1] Wang Y, Wang ZD, Liang JL. A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances. Phys Lett A 2008;372:6066–73. [2] Yang JM, Lu LP, Xie WD, et al. On competitive relationship networks: a new method for industrial competition analysis. Physica A 2007;382:704–14. [3] Lu XB, Qin BZ. Adaptive cluster synchronization in complex dynamical networks. Phys Lett A 2009;373:3650–8. [4] Hu AH, Xu ZY. Pinning a complex dynamical network via impulsive control. Phys Lett A 2009;374:186–90. [5] Yang JM, Yao CZ, Ma WC, Chen GR. A study of the spreading scheme for viral marketing based on a complex network model. Physica A 2010;389:859–70. [6] Watts D, Strogatz S. Collective dynamic of small world network. Nature 1998;393:440–2. [7] Barabási AL, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [8] Newman MEJ. The structure and function of complex networks. SIAM Rev 2003;45:167–256. [9] Wang XF, Chen G. Complex networks: small-world, scale-free and beyond. IEEE Circ Syst Mag 2003;3:6–20. [10] Wang X, Li X, Chen G. Theory and application of complex networks. Beijing: Tsinghua University Press; 2006. [11] Zhu HL, Luo H, Peng HP, Li LX, Luo Q. Complex networks-based energy-efficient evolution model for wireless sensor networks. Chaos Solitons Fract 2009;41:1828–35. [12] Gao HJ, Lam J, Chen GR. New criteria for synchronization stability of general complex dynamical networks with coupling delays. Phys Lett A 2006;360:263–73. [13] Chen TP, Liu XW, Lu WL. Pinning complex networks by a single controller. IEEE Trans Circ Syst I-Regular Paper 2007;54(6):1317–26. [14] Chen MY. Chaos synchronization in complex networks. IEEE Trans Circ Syst I-Regular Paper 2008;55(5):1335–46. [15] Sun W, Chen Z, Lü YB, Chen SH. An intriguing hybrid synchronization phenomenon of two coupled complex networks. Appl Math Comput 2010;216:2301–9. [16] Huang C, Ho Daniel WC, Lu JQ. Synchronization analysis of a complex network family. Nonlinear Anal Real World Appl 2010;11:1933–45. [17] Wu W, Zhou WJ, Chen TP. Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans Circ Syst I-Regular Paper 2009;56(4):829–39. [18] Yu WW, Chen GR, Lü JH. On pinning synchronization of complex dynamical networks. Automatica 2009;45:429–35. [19] Wang QG, Duan ZS, Chen GR, Feng ZS. Synchronization in a class of weighted complex networks with coupling delays. Physica A 2008;387:5616–22. [20] Tu LL, Lu JA. Delay-dependent synchronization in general complex delayed dynamical networks. Comput Math Appl 2009;57:28–36.
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[21] Zheng S, Bi QS, Cai GL. Adaptive projective synchronization in complex networks with time-varying coupling delay. Phys Lett A 2009;373:1553–9. [22] Wang BX, Guan ZH. Chaos synchronization in general complex dynamical networks with coupling delays. Nonlinear Anal Real World Appl 2010;11:1925–32. [23] Wang ZD, Wang Y, Liu YR. Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans Neural Netw 2010;21(1):11–25. [24] Yang M, Wang YW, Xiao JW, Wang HO. Robust synchronization of impulsively-coupled complex switched networks with parametric uncertainties and time-varying delays. Nonlinear Anal Real World Appl 2010;11:3008–20. [25] Wang YW, Wang HO, Xiao JW, Guan ZH. Synchronization of complex dynamical networks under recoverable attacks. Automatica 2010;46:197–203. [26] Zhao J, Hill David J, Liu T. Synchronization of complex dynamical networks with switching topology: A switched system point of view. Automatica 2009;45:2502–11. [27] Peng HP, Wei N, Li LX, Xie WS, Yang YX. Models and synchronization of time-delayed complex dynamical networks with multi-links based on adaptive control. Phys Lett A 2010;374:2335–9. [28] Khalil HK. Nonlinear Systems. third ed. Prentice-Hall; 2002. [29] Gu HB. Adaptive synchronization for competitive neural networks with different time scales and stochastic perturbation. Neurocomputing 2009;73:350–6. [30] Mao XR. A note on the LaSalle-type theorems for stochastic differential delay equations. J Math Anal Appl 2002;268:125–42.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization control for the competitive complex networks with time delay and stochastic effects Wuneng Zhou a, Tianbo Wang a,b,⇑, Jinping Mou a a b
College of Information Sciences and Technology, Donghua University, Shanghai 200051, PR China College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China
a r t i c l e
i n f o
Article history: Received 27 January 2011 Received in revised form 14 August 2011 Accepted 25 December 2011 Available online 8 January 2012 Keywords: Competitive complex networks Synchronization control Time delay Stochastic effects
a b s t r a c t The synchronization control problem for the competitive complex network with time delay and stochastic effects is investigated by using the stochastic technique and Lyapunov stability theory. The competitive complex network means that the dynamical varying rate of a part of nodes is faster than other nodes. Some synchronization criteria are derived by the full controller and pinning controller, respectively, and these criteria are convenient to be used for concision. A numerical example is provided to illustrate the effectiveness of the method proposed in this paper. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, complex networks have received increasing research attention from the fields of the technological practice to theoretical study [1–5]. One of the important reasons is that complex networks can be found in almost everywhere in the real world, such as the Internet, the genetic networks, and the social networks [6–11]. Among all of the dynamical behaviors of complex networks, synchronization is one of the most interesting topics and has been extensively investigated (see [12–18]). The contributions of these efforts are mainly composed of deriving the synchronization criteria and designing different controllers to attain synchronization of complex networks. The common synchronization control methods include the adaptive control, pinning control, hybrid control, impulsive control and so on. On the other hand, it is known that time delays and the stochastic disturbances are important factors that effect the synchronization ability of complex networks in the real world. So there exist many literatures such as [19–24] to study the synchronization problem of complex networks with time delays and the stochastic disturbances. Besides the study for some common complex networks models, there also exist a lot of literatures to study special complex networks models. For instance, in [25], the authors considered the synchronization of complex dynamical networks under successful but recoverable attacks utilizing the framework of switching systems, and obtained the upper bounds of the attack frequency and the average recovering time to ensure that the whole network achieves global synchronization under attacks. In [26], the authors studied the synchronization problem for complex dynamical networks with switching topology from a switched system point of view, and established synchronization criteria for general connection topology by constructing multiple Lyapunov functions. In [27], the authors constructed a model of complex dynamical networks with multi-links by splitting time delay and achieved synchronization between two complex networks with different structures by designing effective controllers. ⇑ Corresponding author at: College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China. E-mail address: [email protected] (T. Wang). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.12.021
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
As a special case of dynamical complex networks, when part of nodes in the complex networks are disturbed by some singular parameters, the varying rate of different nodes exists diversity. Many practical examples on competitive dynamical systems and the reason to bring about these kinds of competitive phenomenon can be found in [28]. However, for the complex networks with this character, there exists less research yet. In this paper, we intend to investigate the synchronization control problem for competitive complex networks by using full controller and pinning controller, respectively. The rest of this letter is organized as follows. In Section 2, the proposed competitive complex network model and some preliminaries are given. In Section 3, the synchronization criteria are derived by the full controller and pinning controller, respectively. In Section 4, a numerical example is provided to illustrate the effectiveness of the method proposed. In the last section, conclusions are presented. Notation. Rn and Rnm, respectively, denote the n-dimensional Euclidean space and the set of all n m real matrices, kk represents the norm of a vector. C([a, b],Rn) denotes the set of continuous vector functions defined on the interval [a, b]. In is an n n identical matrix, kmax(H) stands for the largest eigenvalue of matrix H. is the Kronecker product, L2(R+; Rn) is the set of square integrable functions defined on R+ = (0, +1). 2. Problem formulation In this paper, we consider the following competitive complex networks
"
ei dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
# g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
j¼1
" dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
i ¼ 1; 2; . . . ; l;
#
ð1Þ
g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
i ¼ l þ 1; . . . ; N;
j¼1
where N denotes the number of nodes in the networks, ei is a small positive constant satisfying 0 < ei 6 1. xi(t) 2 Rn denotes the state of node i, f(xi(t), t) : Rn R+ ? Rn is a nonlinear vector valued function. ci denotes the coupling strength of node i with other nodes, C is the inner-coupling matrix, ui(t) is the control input. dwi(t) and ui(t) 2 Rn denote the white noise and noise intensity, respectively. gij is defined as follows: if there exists a connection between node i and node j then gij = 1, else gij = 0. P The matrix G = [gij]NN is called outer-coupling matrix, in particular, g ii ¼ Nj¼1;j–i g ij . The time varying delay s(t) is differentiable with 0 6 s(t) 6 s and s_ ðtÞ 6 l < 1. Let el+1 = el+2 = = eN = 1, then the complex network (1) can be written as
"
ei dxi ðtÞ ¼ f ðxi ðtÞ; tÞ þ ci
N X
#
g ij Cxj ðt sðtÞÞ þ ui ðtÞ dt þ ui ðtÞ dxi ðtÞ;
i ¼ 1; 2; . . . ; N:
ð2Þ
j¼1
Let the target node model and the designed control law be
ds ¼ f ðsðtÞ; tÞ dt þ /ðtÞ drðtÞ
ð3Þ
ui ðtÞ ¼ ki ðxi ðtÞ yi ðtÞÞ;
ð4Þ
and
respectively, where
yi ðtÞ ¼ sðtÞ þ wi ðtÞ dmi ðtÞ;
i ¼ 1; 2; . . . ; N
ð5Þ
is the measurement of node i on the target node. dr(t) and dmi(t) denote the noise disturbance of target node and measure noise from target node, respectively. /(t) 2 Rn and wi(t) 2 Rn denote the corresponding noise intensities, respectively. Denoting ei(t) = xi(t) s(t), then
"
ei dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
#
g ij Cej ðt sðtÞÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ
j¼1
ei /ðtÞ drðtÞ
ð6Þ
for i = 1, 2, . . . , N. In what follows, we will provide some assumptions and a lemma.
Assumption 1. The function f(x(t), t) is Lipschitz continuous on x(t) and in L2(Rn R+; Rn), i.e. there exists a positive constant L > 0 such that
kf ðxðtÞ; tÞ f ðzðtÞ; tÞk 6 LkxðtÞ zðtÞk;
8xðtÞ;
zðtÞ 2 Rn
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
and
Z
1
kf ðxðhÞ; hÞk2 dh < 1:
0
From Assumption 1, it is easy to see that
kf ðxðtÞ; tÞ f ðsðtÞ; tÞk 6 LkeðtÞk: Assumption 2. The noises intensities /(t), wi(t) and ui(t) are in L2(R+; Rn) for i = 1, 2, . . . , N. Assumption 3. dxi(t), dmi(t) and dr(t) are independent 1-dimensional Brownian motion with
Efdxi ðtÞg ¼ Efdmi ðtÞg ¼ EfdrðtÞg ¼ 0; Ef½dxi ðtÞ2 g ¼ Ef½dmi ðtÞ2 g ¼ Ef½drðtÞ2 g ¼ dt and
Efdxi ðtÞ dmj ðtÞg ¼ Efdxi ðtÞ drðtÞg ¼ EfdrðtÞ dmj ðtÞg ¼ 0 for i, j = 1, 2, . . . , N. The objective of this paper is to design controller (4) such that all of the nodes in the complex network (1) synchronize with target node (3) for any initial state, i.e. all of the states satisfy
limt!1 kxi ðtÞ sðtÞk ¼ limt!1 kei ðtÞk ¼ 0;
8i ¼ 1; 2; . . . ; N:
Remark 1. Different from some existing complex network models [12–17], complex network (1) is a competitive network in which a part of nodes vary faster than other nodes. The first l nodes are called fast-varying nodes and the later N l nodes are called slow-varying nodes. Especially, while ei ? 0, i = 1, 2, . . . , l, the first l subsystems will become algebra systems, which will cause abrupt change in the dynamical properties of these nodes. Definition 1 [29]. For an n-dimensional stochastic differential system
dxðtÞ ¼ f ðxðtÞ; xðt sÞ; tÞ dt þ gðxðtÞ; xðt sÞ; tÞ dtðtÞ
ð7Þ
n
n
n
n
with initial state n(t) 2 R on t > 0, where dt(t) is an m-dimensional Brownian motion, f(x(t), x(t s), t) : R R R+ ? R and g(x(t), x(t s), t) : Rn Rn R+ ? Rnm are continuous differentiable functions. Let V(x(t), t) 2 C2,1(Rn R+; R+). The operator LVðxðtÞ; tÞ is defined as
1 LVðxðtÞ; tÞ ¼ V t ðxðtÞ; tÞ þ V x ðxðtÞ; tÞf ðxðtÞ; xðt sÞ; tÞ þ trace fg T ðxðtÞ; xðt sÞ; tÞV xx gðxðtÞ; xðt sÞ; tÞg; 2 2
where V t ðxðtÞ; tÞ ¼ @V ðxðtÞ;tÞ ; V x ðxðtÞ; tÞ ¼ @t
@V ðxðtÞ;tÞ @VðxðtÞ;tÞ ; @x2 ; . . . ; @V ðxðtÞ;tÞ @x1 @xn
and V xx ¼
@ V ðxðtÞ;tÞ @xi @xj
nn
.
Lemma 1 [30]. Assume that the stochastic differential system (7) exists a unique solution x(t, n(t)) on t P 0 for any given initial state nðtÞ 2 C bF 0 ð½s; 0; Rn Þ. Moreover, both f(x(t), x(t s), t) and g(x(t), x(t s), t) are locally bounded and uniformly bounded on t. If there exists a function V(x(t), t) 2 C2,1(Rn R+; R+), b(t) 2 L1(R+; R+) and x1, x2 2 C(Rn; R+) such that
LVðxðtÞ; xðt sÞ; tÞ 6 bðtÞ x1 ðxðtÞÞ þ x2 ðxðt sÞÞ;
8xðtÞ 2 Rn ;
x1 ðxðtÞÞ > x2 ðxðtÞÞ; 8xðtÞ – 0; limkxk!1 inf 06t61 VðxðtÞ; tÞ ¼ 1; then
limt!1 xðt; nðtÞÞ ¼ 0; for every nðtÞ 2
C bF 0 ð½
a:s: n
s; 0; R Þ.
3. Main results In this section, we derive some synchronization schemes by using the full controller and pinning controller, respectively. 3.1. Synchronization criteria with full controller Theorem 1. Under Assumptions 1–3, for given constants L > 0 and s > 0, the complex network (1) synchronizes with the target node (3) under the control law (4) if there exists positive scalars j > 0 and ki > 0 such that
L þ 2 þ s ki þ kmax fðCGGT CÞ ðCCT Þg þ j < 0
ð8Þ
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
for i = 1, 2, . . . , N. Furthermore, the control gain satisfies
ki > L þ 2 þ s þ kmax fðCGGT CÞ ðCCT Þg; where C = diag (c1, c2, . . . , cN), G = [gij]NN. Proof. Choosing a Lyapunov functional candidate N N Z t N Z 0 Z t X X 1X ei eTi ðtÞei ðtÞ þ eTi ðsÞei ðsÞ ds þ eTi ðhÞei ðhÞ dh ds: 2 i¼1 tsðtÞ s tþs i¼1 i¼1
VðtÞ ¼
By the Itö differential formula, the derivative of V(t) along the state trajectories of system (6) is N X
dVðtÞ ¼ LVðtÞ dt þ
eTi ðtÞ½ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ ei /ðtÞ drðtÞ:
i¼1
By Definition 1, we have LVðtÞ ¼
Z N X ½ei eTi ðtÞe_ i ðtÞ þ eTi ðtÞei ðtÞ ð1 s_ ðtÞÞeTi ðt sðtÞÞei ðt sðtÞÞ þ seTi ðtÞei ðtÞ
ts
i¼1
þ
¼
t
eTi ðsÞei ðsÞds
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei i¼1
N X
(
" eTi ðtÞ
f ðxi ðtÞ;tÞ f ðsðtÞ;tÞ þ ð1 ei Þf ðsðtÞ;tÞ þ ci
i¼1
N X
# g ij Cej ðt sðtÞÞ ki ei ðtÞ
j¼1
N X 1 T þ 1 þ sÞeTi ðtÞei ðtÞ ð1 lÞeTi ðt sðtÞÞei ðt sðtÞÞ þ ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2 e i i¼1
6
N X
eTi ðtÞðL þ 1 þ s ki Þei ðtÞ þ ð1 ei Þkf ðsðtÞ;tÞk kei ðtÞk
i¼1
þ
N X N X
ci g ij eTi ðtÞCej ðt sðtÞÞ ð1 lÞ
i¼1 j¼1
N X
eTi ðt sðtÞÞei ðt sðtÞÞ þ
i¼1
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ : 2 e i i¼1
ð9Þ
Noticing that N X
N pffiffiffiffi X kei ðtÞk 6 N kei ðtÞk2
i¼1
!12 ð10Þ
i¼1
and N X N X i¼1
ci g ij eTi ðtÞCej ðt sðtÞÞ 6 kmax fðCGGT CÞ ðCCT Þg
j¼1
N X
eTi ðtÞei ðtÞ þ
i¼1
N X
eTi ðt sðtÞÞei ðt sðtÞÞ;
ð11Þ
i¼1
we obtain LVðtÞ 6 max16i6N fL þ 1 þ s ki g
N X
N X pffiffiffiffi kei ðtÞk2 þ max16i6N ð1 ei Þ N kf ðsðtÞ;tÞk kei ðtÞk2
i¼1
!12
i¼1
N X 1 T þ kmax fðCGGT CÞ ðCCT Þg eTi ðtÞei ðtÞ þ l eTi ðt sðtÞÞei ðt sðtÞÞ þ ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ : 2ei i¼1 i¼1 i¼1 N X
N X
ð12Þ
From the inequality (8), one gets
2 LVðtÞ 6 j4
N X i¼1
þ
!12 2
kei ðtÞk
32
N N X aðtÞ5 a2 ðtÞ X þ eTi ðtÞei ðtÞ þ l eTi ðt sðtÞÞei ðt sðtÞÞ 2j 4j i¼1 i¼1
N X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ ; 2 e i i¼1
ð13Þ
pffiffiffiffi where aðtÞ ¼ max16i6N ð1 ei Þ N kf ðsðtÞ; tÞk. By Lemma 1, we know that complex network (1) is synchronization with the target node (3). The proof of Theorem 1 is complete. h
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
3421
Remark 2. When ei = 1, i = 1, 2, . . . , l, the complex network (1) still synchronizes with the target node (3) under the action of controller (4) if the inequality (8) holds. For this case, all the nodes can synchronize each other if only the nonlinear function f(s(t), t) satisfies Lipschitz condition. Different from the results in [20,22,26], we only need to verify the inequality (8) according to the parameters of complex network (1) and not to solve some complex linear or nonlinear matrix inequalities depended on some unknown variables. Which also shows that synchronization is an essential phenomenon of the complex network and only depends on its parameters. Remark 3. It can be seen from the synchronization condition (8) that the control gain only depends on the time delay, the Lipschitz constant, the coupling matrices and the coupling strength, which implies that the synchronization of a complex network tightly depends on its parameters. Furthermore, we can obtain the smallest synchronization control gain from (8). Next, we will consider the synchronization problem for the complex network (1) and the target node (3) by pinning control method. We will study this control problem in two cases, one is only controlling fast-varying nodes and the other is only controlling slow-varying nodes. 3.2. Synchronization criteria with pinning controller While using control law
( ui ðtÞ ¼
ki ðxi ðtÞ yi ðtÞÞ; 0;
i ¼ 1; 2; . . . ; l;
ð14Þ
i ¼ l þ 1; l þ 2; . . . ; N;
to control complex network (1) without time delay, we obtain the following dynamical system
ei dei ðtÞ ¼ ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
g ij Cej ðtÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ
j¼1
þ ki wi ðtÞdmi ðtÞ ei /ðtÞdrðtÞ; dei ðtÞ ¼ ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci
N X
i ¼ 1; 2; . . . ; l;
g ij Cej ðtÞ dt þ ui ðtÞ dxi ðtÞ /ðtÞ drðtÞ; i ¼ l þ 1; l þ 2; . . . ; N:
ð15Þ
j¼1
Theorem 2. Under Assumptions 1–3, for a given scalar L > 0, the complex network (1) without time delay synchronizes with the target node (3) under the pinning controller (14) if there exist positive constants g > 0, j > 0 and ki > 0 such that
e In þ ðCGÞ C < 0 ðL þ jÞIN In þ K 0
ð16Þ
1
e ¼ diag @g k1 ; g k2 ; . . . ; g kl ; 0; . . . ; 0A. for i = 1, 2, . . . , l, where K |fflfflfflffl{zfflfflfflffl} Nl
Proof. Choose a Lyapunov functional candidate
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; P P where V 1 ðtÞ ¼ 12 li¼1 ei eTi ðtÞei ðtÞ and V 2 ðtÞ ¼ 12 Ni¼lþ1 eTi ðtÞei ðtÞ. By the Itö differential formula, the derivative of V(t) along the state trajectories of system (15) is dVðxðtÞÞ ¼ LV 1 ðtÞdt þ LV 2 ðtÞdt þ
l X
eTi ðtÞ½ui ðtÞdxi ðtÞ þ ki wi ðtÞdmi ðtÞ ei /ðtÞdrðtÞ þ
i¼1
N X
eTi ðtÞ½ui ðtÞdxi ðtÞ /ðtÞdrðtÞ:
i¼lþ1
Using Definition 1, we get
LV 1 ðtÞ ¼
l X
ei eTi ðtÞe_ i ðtÞ þ
i¼1
¼
l X
l X 1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei i¼1
eTi ðtÞ½f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
i¼1
N X j¼1
g ij Cej ðtÞ ki ei ðtÞ þ
l X 1 T u ðtÞui ðtÞ 2 ei i i¼1
l X T 2 ei ðtÞðL ki Þei ðtÞ þ ð1 ei Þkf ðsðtÞ; tÞk kei ðtÞk þ ki wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 6 i¼1
þ and
l X
N X
i¼1
j¼1
ci g ij eTi ðtÞCej ðtÞ þ
l X i¼1
1 T ui ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ þ e2i /T ðtÞ/ðtÞ 2ei
ð17Þ
3422
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426 N X 1 T ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ 2 i¼lþ1 i¼lþ1 " # N N N X X X 1 T T ¼ ei ðtÞ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci g ij Cej ðtÞ þ ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ 2 j¼1 i¼lþ1 i¼lþ1
LV 2 ðtÞ ¼
6
N X
N X
eTi ðtÞe_ i ðtÞ þ
LeTi ðtÞei ðtÞ þ
N N X X
ci g ij eTi ðtÞCej ðtÞ þ
i¼lþ1 j¼1
i¼lþ1
N X 1 T ui ðtÞui ðtÞ þ /T ðtÞ/ðtÞ : 2 i¼lþ1
ð18Þ
From (17) and (18), we obtain
2 LVðtÞ 6 g4
l X
!12 kei ðtÞk2
i¼1
32 bðtÞ5 e In þ ðCGÞ CeðtÞ þ UðtÞ; þ eT ðtÞ½LðIN In Þ þ K 2g
pffiffiffiffi T where bðtÞ ¼ max16i6l ð1 ei Þ N kf ðsðtÞ; tÞk; eðtÞ ¼ eT1 ðtÞ; eT2 ðtÞ; . . . ; eTN ðtÞ ; UðtÞ ¼ T PN T 2 T 1 þei / ðtÞ/ðtÞÞ þ i¼lþ1 2 ui ðtÞui ðtÞ þ / ðtÞ/ðtÞ . From (16), one gets
b2 ðtÞ 4g
þ
ð19Þ
Pl
1 i¼1 2ei
uTi ðtÞui ðtÞ þ k2i wTi ðtÞwi ðtÞ
LVðtÞ 6 jeT ðtÞeðtÞ þ UðtÞ:
ð20Þ
Noticing that
eT ðtÞeðtÞ P 2VðtÞ; yields
20
20
0
0 −20 −40
−40
ei2(t)
ei1(t)
−20
−60
−80
−80
−100
−100 −120
−60
−120 0
20
40
60
80
−140
100
0
20
40
Time(sec)
60
80
100
Time(sec)
(a)
(b) 80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 1. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the full controller.
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
20
20
0
0 −20 −40
−40
ei2(t)
ei1(t)
−20
−60
−80
−80
−100
−100 −120
−60
−120 0
20
40
60
80
−140
100
0
20
40
Time(sec)
60
80
100
Time(sec)
(a)
(b) 80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 2. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the pinning fast-varying nodes controller.
LVðtÞ 6 2jVðtÞ þ UðtÞ:
ð21Þ
Integrating t on the both sides of (21) from 0 to T > 0, we get
VðTÞ 6 Vð0Þ 2j
Z
T
VðtÞ dt þ M;
ð22Þ
0
where M ¼
R1 0
UðtÞ dt. By the Gronwall–Bellman inequality [28], we obtain
VðTÞ 6 ðM þ Vð0ÞÞ exp ð2jTÞ:
ð23Þ
Since T is arbitrary, limt?1kei(t)k = 0 for i = 1, 2, . . . , N. The proof of Theorem 2 is complete. h While using control law
ui ðtÞ ¼
0;
i ¼ 1; 2; . . . ; l;
ki ðxi ðtÞ yi ðtÞÞ;
ð24Þ
i ¼ l þ 1; l þ 2; . . . ; N;
to control complex network (1) without time delay, we obtain the following closed-loop system
"
ei dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ð1 ei Þf ðsðtÞ; tÞ þ ci
N X
# g ij Cej ðtÞ dt þ ui ðtÞ dxi ðtÞ ei /ðtÞ drðtÞ;
j¼1
i ¼ 1; 2; . . . ; l; " dei ðtÞ ¼ f ðxi ðtÞ; tÞ f ðsðtÞ; tÞ þ ci
N X j¼1
i ¼ l þ 1; l þ 2; . . . ; N:
# g ij Cej ðtÞ ki ei ðtÞ dt þ ui ðtÞ dxi ðtÞ þ ki wi ðtÞ dmi ðtÞ /ðtÞ drðtÞ;
ð25Þ
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W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
20
20
0
0
−20
−20 −40
−60
ei2(t)
ei1(t)
−40
−80
−80
−100
−100
−120
−120
−140 −160
−60
0
20
40
60
80
−140
100
0
20
40
60
Time(sec)
Time(sec)
(a)
(b)
80
100
80 60 40
i3
e (t)
20 0
−20 −40 −60 −80
0
20
40
60
80
100
Time(sec)
(c) Fig. 3. Synchronization error ei(t) = (ei1(t), ei2(t), ei3(t))T between the target state s(t) = (s1(t), s2(t), s3(t))T and the state xi(t) = (xi1(t), xi2(t), xi3(t))T (i = 1, 2, 3, 4, 5) under the pinning slow-varying nodes controller.
Using the same idea in Theorem 2, we obtain the following theorem. Theorem 3. Under Assumptions 1–3, for a given scalar L > 0, the complex network (1) without time delay synchronizes with the target node (3) under the pinning controller (24) if there exist positive constants g > 0, j > 0 and ki > 0 such that
b In þ ðCGÞ 0 ðL þ jÞIN In þ K C<0
1
ð26Þ
b ¼ diag @g; . . . ; g; klþ1 ; klþ2 ; . . . ; kN A. for i = l + 1, l + 2, . . . , N, where K |fflfflfflffl{zfflfflfflffl} l
Remark 4. Theorems 2 and 3 are derived by using the method of only controlling fast-varying nodes and slow-varying nodes, respectively. It can be seen from our results that the control effectiveness is better if the number of controlled nodes becomes more, so the control effectiveness by using the full controller is better than by using the pinning controller. Remark 5. We have only considered the synchronization problem for competitive complex network without time delay by pinning controller. However, it is still a problem on the synchronization of competitive complex networks with time delay using the pinning controller, which is the next research topic for us.
4. A numerical example Consider the complex network (1) with five nodes (N = 5) and each node being a 3-dimensional dynamical system (n = 3).
W. Zhou et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3417–3426
2
0:3 0:5 We choose f ðxðtÞ; tÞ ¼ Aðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ , where A ¼ 4 0:6 1 0:8 0:3 It can be seen that there exists a positive constant L = 2 such that T
1 1þt 2
3425
3 0:9 0:7 5. 1
kf ðxðtÞ; tÞ f ðyðtÞ; tÞk 6 LkxðtÞ yðtÞk for x(t), y(t) 2 R3 and f(x(t), t) 2 L2([0, 1), R3) for kx(t)k < 1. The other parameters of the complex network (1) are given as follows: e1 = 0.001, e2 = 0.01, e3 = 0.1, C = diag (0.2, 0.25, 0.3, 0.35, 0.4), C = diag (3, 4, 5),
2
4 1 1 1 6 1 6 1 3 0 6 G¼6 0 2 0 6 1 6 1 0 3 4 1 1
1
1
1
3 1 7 1 7 7 1 7 7; 7 1 5 4
sðtÞ ¼ 0:5 sin t þ 1; s ¼ 1:5; l ¼ 0:5:
pffi The noise strength coefficients are ui ðtÞ ¼ lnðtþ1Þ ; /ðtÞ ¼ wi ðtÞ ¼ 1þtt2 for i = 1, 2, 3, 4, 5, respectively. The inequality (8) holds tþ1 when j = 1 and ki = 96.5675 for i = 1, 2, 3, 4, 5. Therefore, from Theorem 1, the five nodes can synchronize with the target node. The simulation results on the synchronization error with initial state
eð0Þ ¼ ð10; 80; 65; 120; 140; 40; 40; 60; 55; 90; 10; 70; 95; 20; 10ÞT are shown in Fig. 1. This figure shows that all the nodes synchronize well. If we only control fast-varying nodes, the inequality (16) holds when j = 0.1, g = 0.5, ki = 96.5675 for i = 1, 2, 3. From Theorem 2, the five nodes synchronize with the target node. The simulation results on the synchronization error with the same initial state e(0) are shown in Fig. 2. In this example, there does not exist ki for i = 4, 5 satisfying the inequality (26). So the five nodes cannot synchronize with the target node by pinning controller (24). Taking j = 0.1, g = 0.5, ki = 96.5675 for i = 4, 5, the simulation results on the synchronization error with the same initial state e(0) are shown in Fig. 3. This figure implies that the five nodes do not synchronize with the target node if only controlling slow-varying nodes. 5. Conclusions The synchronization of the competitive complex networks with time delay and stochastic effect has been studied by the Lyapunov stability theory and stochastic analysis technique. The full controller and pinning controller have been designed respectively to obtain the synchronization schemes. A numerical example has shown that the proposed method are effective. Acknowledgements This work was supported by the National Natural Science Foundation of China (61075060), the National 863 Key Program of China (2008AA042902), the Innovation Program of Shanghai Municipal Education Commission (12zz064,11xk11). References [1] Wang Y, Wang ZD, Liang JL. A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances. Phys Lett A 2008;372:6066–73. [2] Yang JM, Lu LP, Xie WD, et al. On competitive relationship networks: a new method for industrial competition analysis. Physica A 2007;382:704–14. [3] Lu XB, Qin BZ. Adaptive cluster synchronization in complex dynamical networks. Phys Lett A 2009;373:3650–8. [4] Hu AH, Xu ZY. Pinning a complex dynamical network via impulsive control. Phys Lett A 2009;374:186–90. [5] Yang JM, Yao CZ, Ma WC, Chen GR. A study of the spreading scheme for viral marketing based on a complex network model. Physica A 2010;389:859–70. [6] Watts D, Strogatz S. Collective dynamic of small world network. Nature 1998;393:440–2. [7] Barabási AL, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [8] Newman MEJ. The structure and function of complex networks. SIAM Rev 2003;45:167–256. [9] Wang XF, Chen G. Complex networks: small-world, scale-free and beyond. IEEE Circ Syst Mag 2003;3:6–20. [10] Wang X, Li X, Chen G. Theory and application of complex networks. Beijing: Tsinghua University Press; 2006. [11] Zhu HL, Luo H, Peng HP, Li LX, Luo Q. Complex networks-based energy-efficient evolution model for wireless sensor networks. Chaos Solitons Fract 2009;41:1828–35. [12] Gao HJ, Lam J, Chen GR. New criteria for synchronization stability of general complex dynamical networks with coupling delays. Phys Lett A 2006;360:263–73. [13] Chen TP, Liu XW, Lu WL. Pinning complex networks by a single controller. IEEE Trans Circ Syst I-Regular Paper 2007;54(6):1317–26. [14] Chen MY. Chaos synchronization in complex networks. IEEE Trans Circ Syst I-Regular Paper 2008;55(5):1335–46. [15] Sun W, Chen Z, Lü YB, Chen SH. An intriguing hybrid synchronization phenomenon of two coupled complex networks. Appl Math Comput 2010;216:2301–9. [16] Huang C, Ho Daniel WC, Lu JQ. Synchronization analysis of a complex network family. Nonlinear Anal Real World Appl 2010;11:1933–45. [17] Wu W, Zhou WJ, Chen TP. Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans Circ Syst I-Regular Paper 2009;56(4):829–39. [18] Yu WW, Chen GR, Lü JH. On pinning synchronization of complex dynamical networks. Automatica 2009;45:429–35. [19] Wang QG, Duan ZS, Chen GR, Feng ZS. Synchronization in a class of weighted complex networks with coupling delays. Physica A 2008;387:5616–22. [20] Tu LL, Lu JA. Delay-dependent synchronization in general complex delayed dynamical networks. Comput Math Appl 2009;57:28–36.
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