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Pinning a complex dynamical network via impulsive control
Physics Letters A 374 (2009) 186–190
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Pinning a complex dynamical network via impulsive control ✩ Aihua Hu a,b,∗ , Zhenyuan Xu a a b
School of Science, Jiangnan University, Wuxi 214122, PR China School of Information Technology, Jiangnan University, Wuxi 214122, PR China
a r t i c l e
i n f o
Article history: Received 8 July 2009 Received in revised form 12 October 2009 Accepted 19 October 2009 Available online 24 October 2009 Communicated by A.R. Bishop PACS: 05.45.+b
a b s t r a c t Complex dynamical networks are being studied across many fields of science and engineering today. The issue of controlling a network to the desired state has attracted increasing attention. In this Letter, we investigate the problem of pinning a complex dynamical network to the solution of an uncoupled system. Our strategy is to apply impulsive control to a small fraction of network nodes. Based on the Lyapunov stability theory, we prove that the theoretical results derived here are effective. In addition, a B-A scalefree network with 20 nodes is taken for illustration and verification. © 2009 Elsevier B.V. All rights reserved.
Keywords: Impulsive control Scale-free network Chaotic system
1. Introduction Complex dynamical networks exist everywhere in the real world, such as the Internet, which is a huge-scale network of routers and computers connected by various physical or wireless links; the World Wide Web, which is an enormous virtual network of web sites connected by hyperlinks; and food webs, biological neural networks, telephone cell graphs, etc. Complex networks consist of a large number of dynamical nodes, and have attracted increasing attention from various fields, including physical, economical, social and biological sciences [1–4]. For over a century, the question of how to model complex networks is a point of great interest. In 1960, the theory of random graph [5] was introduced by Paul Erdös and Alfréd Rényi (E-R), which made a breakthrough in the completely regular graph theory. However, many real-world complex networks cannot be described by E-R random graph, therefore, researchers have been proposing the new network models. For example, recently, Watts and Strogatz (W-S) introduced the so-called small world network [6], which exhibits a high degree of
✩ Project supported by the National Natural Science Foundation of China (Grant No. 10901073) and the Youth Foundation of Jiangnan University (Grant No. 31400052210756). Corresponding author at: School of Science, Jiangnan University, Wuxi 214122, PR China. Tel.: +86 51085910656. E-mail address: [email protected] (A. Hu).
*
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.10.049
clustering as in the regular networks and a small average distance among nodes as in the random networks. Another significant discovery is that a number of real-world complex networks have the scale-free feature. It means that the degree distributions of these networks follow a power law form, which was first pointed out by Barabási and Albert (B-A) [7]. Scale-free networks are inhomogeneous in nature, that is, most nodes have very few connections but a small number of particular nodes have many connections. The characters of the small-world and scale-free networks have been verified to fit many real-world complex networks [8–10]. More recently, synchronization and control of complex networks are being seriously studied. In particular, control of scalefree networks or random networks were investigated by applying local linear feedback, which is injected to a small fraction of networks nodes [11,12]. However, the cost of controlling is expensive for the feedback is continuous and the gain is large. This Letter is a further investigation of this subject, in which we want to control a complex dynamical network onto the solution of an uncoupled system. Our strategy is applying impulsive control to pin a small fraction of network nodes. Compared with the method of local linear feedback, impulsive control is attractive because it is discrete and only needs small control gain. The Letter is organized as follows: Problem formulation is introduced in Section 2.1. The theoretical results are presented in Section 2.2. In Section 3, we take a B-A scale-free network with 20 nodes for illustration and verification. Finally, Section 4 concludes the investigation.
A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
For t = tk (k = 1, 2, . . .),
2. Pinning a complex network via impulsive control 2.1. Problem formulation Consider a complex dynamical network consisting of N identical coupled nodes, with each node being an m-dimensional dynamical system. Recently, Wang and Chen [2] introduced the following simple uniform network model:
x˙ i = f (xi ) +
N
(1)
j =1 j =i
j =1 j =i
ai j =
N
a ji = ki
(i = 1, 2, . . . , N ).
For t = tk (k = 1, 2, . . .),
+
= xi tk− + b i xi tk− − x¯ ,
(5a)
i = 1, 2, . . . , l ,
+ xi tk = xi tk− ,
(5b)
where xi (tk+ ) = limt →t + xi (t ), xi (tk− ) = limt →t − xi (t ). k
k
In general, it is assumed that xi (tk− ) = xi (tk ) (i = 1, 2, . . . , N ), b i ∈ R m×m (i = 1, 2, . . . , l) denote the matrix of control gain, and b i (xi (tk− ) − x¯ ) (i = 1, 2, . . . , l; k = 1, 2, . . .) stand for the controller. Let the error vector e i = xi − x¯ (i = 1, 2, . . . , N ), from Eqs. (4), (5a) and (5b), we have: For t = tk (k = 1, 2, . . .),
e˙ i = x˙ i − x˙¯
= f ( xi ) + c
j =1 j =i
N
ai j x j − f (¯x)
j =1
Let the diagonal elements be aii = −ki . For simplicity, we assume that c i j = c , Γ = I m (I m denotes an m × m-dimensional identity matrix). Then, Eq. (1) can be rewritten as follow:
x˙ i = f (xi ) + c
(4)
i = l + 1, l + 2, . . . , N ,
where xi = (xi1 , xi2 , . . . , xim ) T ∈ R m are the state variables of node i , i = 1, 2, . . . , N, the constant c i j 0 represents the coupling strength between nodes i and j, Γ = (si j )m×m is a matrix linking coupled variables, and if some pairs (i , j) with si j = 0, then it means two coupled nodes are linked through their ith and jth state variables, respectively. If there is a connection between nodes i and j (i = j ), then ai j = a ji = 1; otherwise ai j = a ji = 0. If the degree ki of node i is defined to be the number of its outreaching connections, then N
⎧ N ⎪ ⎨ x˙ = f (x ) + c a x , i i ij j j =1 ⎪ ⎩ i = 1, 2, . . . , N . xi t k
c i j ai j Γ (x j − xi ),
187
N
ai j x j .
(2)
j =1
We want to apply impulsive control to a small fraction of the nodes in network (2), so that the network can be controlled onto the solution of an uncoupled system defined by
x1 = x2 = · · · = x N = x¯ ,
x˙¯ = f (¯x).
(3)
ai j e j .
j =1
For simplicity, we just take the first-order term of Taylor series of f (xi ) − f (¯x), i.e., we assume that f (xi ) − f (¯x) is equal to J (xi − x¯ ), where J = D f (¯x) ∈ R m×m is the Jacobian of f (x) at x¯ , then
⎧ N ⎪ ⎨ e˙ = J e + c a e , i i ij j j =1 ⎪ ⎩ i = 1, 2, . . . , N .
Definition 1 (Kronecker product (⊗)). If matrix A = (ai j )n×m , B = (bi j ) p ×q , then
⎛a B ··· a B ⎞ 11 1m ⎜ . ⎟ np ×mq .. A ⊗ B = ⎝ .. . ⎠∈R .
(6)
For t = tk (k = 1, 2, . . .),
e i tk+ = e i tk− + b i e i tk− ,
First of all, we present a useful definition:
+
e i tk
(7a)
−
= e i tk ,
i = l + 1, l + 2, . . . , N .
(7b)
Define E = (e 11 , e 12 , . . . , e 1m , e 21 , e 22 , . . . , e 2m , . . . , e N1 , e N2 , . . . , e Nm ) T ∈ R Nm , then For t = tk (k = 1, 2, . . .),
E˙ = ( I N ⊗ J ) E + c ( A ⊗ I m ) E .
(8)
For t = tk (k = 1, 2, . . .),
· · · anm B
Suppose that we select l (l > 1) nodes in network (2) for controlling, where l = [δ N ] (0 ≺ δ ≺≺ 1, [ ] stands for the Integer function, i.e., l is integral part of the real number δ N). Without loss of generality, we assume that the first l nodes are chosen. Impulsive control is discrete, so consider that a discrete instant set {t i } satisfies
t 1 ≺ t 2 ≺ · · · ≺ t k ≺ t k +1 ≺ · · · ,
N
i = 1, 2, . . . , l ,
2.2. Theoretical analysis
an1 B
= f (xi ) − f (¯x) + c
lim tk = ∞.
k→∞
Define impulsive intervals are τk = tk − tk−1 (k = 1, 2, . . .). After introducing the control to the network (2), we can get the controlled model as follow:
E tk+ = E tk− + B E tk− ,
(9)
where A = (ai j ) N × N , B = diag{b1 , b2 , . . . , bl , 0, . . . , 0} ∈ R Nm× Nm , I N and I m represent an N × N-dimensional identity matrix and an m × m-dimensional identity matrix, respectively. Suppose the network is connected in the sense of having no isolated clusters. Then, the matrix A is symmetric and irreducible. Since all rows of A sum to zero, A always admits the max eigenvalue λmax = 0 [2]. Let G = diag{−a11 , −a22 , . . . , −all , 0, . . . , 0} ∈ R N × N , then the matrix G includes the degrees of the first l nodes. Define R = A − G.
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A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
Theorem 1. If there exists an ξ 1 to satisfy the following conditions, then the network (2) will be controlled onto the desired state x¯ by applying impulsive control to l nodes.
Since
l
i =1
E 2i (t 1− )
N
E 2i (t 1− ) = V ( E (t 1 )), then
i =1
V E t 1+ d exp r (t 1 − t 0 )
l
(1) The coupling strength c − 2λλ1 ; (2) r
1 max{τk }
ln
i =1
2
1 ξd ,
N
E 2i t 1+ d exp r (t 1 − t 0 )
i =1
Since Proof. Choose the form of the Lyapunov function as V ( E (t )) = E T (t ) E (t ). When t ∈ (tk−1 , tk ] (k = 1, 2, . . .), the derivative of V ( E (t )) along the solution of error system (8) is
= E T (t ) ( I N ⊗ J )T + ( I N ⊗ J ) E (t )
l
T
T
E 2i (t ),
V E t 2+ d
i =1
N
E 2i tk− +
E 2i tk−
l
E 2i tk− .
E 2i (t 0 ),
i =1 l
V E (t 1 ) exp r (t 1 − t 0 )
E 2i (t 0 ),
i =1
E 2i t 1− +
E 2i t 2+ d2 exp r (t 2 − t 0 )
l
E 2i (t 0 ).
i =1
N i =l+1
E 2i t 1− .
V E (t ) (13)
i =l+1
V E (t ) exp r (t − t 0 )
i =1
i =l+1
l
E 2i (t 0 ).
In virtue of t − t 0 = t 1 − t 0 + t 2 − t 1 + · · · + tk − tk−1 + t − tk = τ1 + τ2 + · · · + τk + t − tk and the condition (2) in Theorem 1, we know that
i =l+1
i =1
V E t 1+ d
i =l+1
In general, for t ∈ (tk , tk+1 ],
Thus, let k = 1, i.e., t ∈ (t 0 , t 1 ], from Eqs. (12) and (13), we have:
l
E 2i t 2−
i =1
N
di E 2i tk− +
N
V E (t ) dk exp r (t − t 0 )
= E T tk− ( I Nm + B )T ( I Nm + B ) E tk−
E 2i t 2− +
i =1
(12)
V E tk+ = E T tk+ E tk+
l
i =1
When t = tk (k = 1, 2, . . .),
E 2i (t 0 ),
i =1
l
i =1
E 2i t 0 .
Then,
l V E (t ) exp r (t − tk−1 ) E 2i tk+−1 .
l
(11)
l
t 1 , then
l N d2 exp r (t 2 − t 0 ) E 2i (t 0 ) + E 2i t 2− .
where = It is easy to see that r 0, therefore
d
i =l+1
E 2i t 1+
V E (t 2 ) d exp r (t 2 − t 0 )
i =1
2 j =1 e i j .
l
−
i =1
(10)
m
l
E 2i t 1− .
l d exp r (t − t 0 ) E 2i (t 0 ),
i =1
E 2i
2 i =l+1 E i
N
i =1
2c (−aii + λ2 ) + λ1 E 2i (t )
l
t1 =
N
l
i =1
r
E 2i (t 0 ) +
i =1
+
V E (t ) exp r (t − t 1 )
where U = diag{2c (−a11 + λ2 ) + λ1 , . . . , 2c (−all + λ2 ) + λ1 , 2c λ2 + λ1 , . . . , 2c λ2 + λ1 } ∈ R N × N . According to the condition (1) in Theorem 1, i.e. 2c λ2 + λ1 ≺ 0. Then
l
i =1
T
= E T (t )(U ⊗ I m ) E (t ),
2 i =l+1 E i
i =1
λ1 E (t ) E (t ) + 2c E (t )(G ⊗ I m ) E (t ) + 2c λ2 E (t ) E (t )
N
E 2i t 1+ d exp r (t 1 − t 0 )
+ 2c E T (t )( A ⊗ I m ) E (t )
V˙ E (t )
i =l+1
Let k = 2, i.e., t ∈ (t 1 , t 2 ],
V˙ E (t ) = E˙ T (t ) E (t ) + E T (t ) E˙ (t )
l
E 2i t 1− ,
i.e.,
where λ1 = λmax ( J T + J ), λ2 = λmax ( R ), r = max{2c (−aii +λ2 )+λ1 }, d = max{di }, di = ρ 2 ( I m + b i ), ρ ( A ) denotes the spectral radius of A, i = 1, 2, . . . , l, k = 1, 2, . . . .
N
E 2i (t 0 ) +
1
ξk
l
exp r (t − tk )
E 2i (t 0 ).
(14)
i =1
Thus, for k → ∞, V ( E (t )) = 0, i.e., x1 (t ) = x2 (t ) = · · · = x N (t ) = x¯ . The proof of Theorem 1 is completed. 2 3. Numerical simulation In order to illustrate the aforementioned theoretical analysis clearly, we take a B-A scale-free network with 20 nodes for example, and each node is the chaotic Lorenz system [13]. Firstly, we will give the algorithm of the B-A scale-free model [14] as follows. (1) Growth: Starting with a small number (m0 ) of nodes, at every time step, add a new node with m ( m0 ) edges that link the new node to m different nodes already presented in the network.
A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
189
Fig. 1. The plot of Lorenz’s attractor.
(2) Preferential attachment: When choosing the nodes to which the new node connects, assume that the probability Πi that a new node will be connected to node i depends on the degree ki of node i, in such a way that Πi = ki / j k j . After n time steps, we get a network having N = n + m0 nodes and mn edges. This network evolves into a scale-invariant state with the probability that a node has k edges following a powerlaw distribution P (k) ∼ 2m2 k−γ with an exponent γ = 3. The Lorenz system is described in the form by
⎧ ⎨ x˙ 1 = σ (x2 − x1 ), x˙ = ρ x1 − x1 x3 − x2 , ⎩ 2 x˙ 3 = x1 x2 − bx3 .
(15)
Fig. 2. Graphical representation of nodes’ variables when two nodes with the largest degrees being controlled.
node, a state variable attaches the value of 27, and another two variables are 8.4853, which means that 20 nodes are controlled to the point x¯ = (8.4853, 8.4853, 27) T . Now we control 2 nodes with the smallest degrees, such as nodes 19 and 20, both of them have 3 degrees. Then,
G = diag{0, 0, . . . , 0, 3, 3} ∈ R 20×20 ,
λ2 = λmax ( R ) = λmax ( A − G ) = −0.1502. Choose two sets of parameters:
Simulation 1. Pinning the network to its equilibrium x¯ ( f (¯x) = 0).
(1) c = 5, which cannot satisfy the condition (1) in Theorem 1, so the control scheme is useless. Even if we introduce the impulsive control, the network will not be controlled onto the state x¯ . For example, select that b19 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b20 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01. The result is shown in Fig. 3a, which means that whenever it is, 20 nodes in the network cannot reach x¯ = (8.4853, 8.4853, 27) T . (2) c = 18, b19 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b20 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1, then two conditions in Theorem 1 are satisfied. The result is shown in Fig. 3b. In addition, Fig. 3b is similar to Fig. 2.
In the above set of system parameters, one unstable equilibrium point of the system (15) is x¯ = (8.4853, 8.4853, 27) T . Thus we have:
Comparing the results of Fig. 2 with Fig. 3, we will easily find out the nodes with larger degrees are selected, the smaller coupling strength c needed to be.
When the parameters are taken as σ = 10, ρ = 28, b = 8/3, the Lorenz system has a chaotic attractor, which is shown in Fig. 1. A B-A scale-free network is generated by m = m0 = 3, n = 17. Then we attain the matrix A of the network, the diagonal elements of it are (−1, −6, −9, −12, −13, −7, −11, −4, −4, −4, −3, −3, −3, −4, −3, −3, −3, −3, −3, −3). In the following simulations, suppose that only 2 nodes are selected (i.e., l = 2), and let the impulsive intervals τk = τ 0 (k = 1, 2, . . .).
⎡
J =⎣
−10
10
1
−1
8.4853 8.4853
0
⎤
−8.4853 ⎦ , −8/3
Simulation 2. Pinning the network to x¯ ( f (¯x) = 0).
20
Firstly, we control 2 nodes with the largest degrees, i.e., nodes 4 and 5, which have 12 degrees and 13 degrees, respectively. Consequently, we can get
1 We select x¯ = 20 i =1 xi , and control the nodes 4 and 5. The parameters are chosen as c = 7, b4 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b5 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1. Fig. 4 shows that the state variables of 20 nodes will synchronize at 20 1 x¯ = 20 i =1 xi .
G = diag{0, 0, 0, 12, 13, 0, 0, . . . , 0} ∈ R 20×20 ,
4. Conclusions
T
λ1 = λmax J + J = 4.8231.
λ2 = λmax ( R ) = λmax ( A − G ) = −0.6073. Select the parameters as follows: c = 5, b4 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b5 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1, thus two conditions in Theorem 1 can be satisfied, Fig. 2 shows the result. We can see that after t = 1.2, for any arbitrary
In this Letter, we have proposed some conditions for controlling a complex dynamical network to desired state by pinning a small fraction of the network nodes. The method of impulsive control has been used. The numerical results verify the theoretical analysis, and we can find that the cost of controlling is economical.
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A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
Fig. 4. Graphical representation of nodes’ variables when the network being pinned 20 1 to x¯ = 20 i =1 xi .
Acknowledgements The authors would like to thank the reviewers for their insightful suggestions.
References
Fig. 3. Graphical representations of nodes’ variables when two nodes with the smallest degrees being controlled.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Pinning a complex dynamical network via impulsive control ✩ Aihua Hu a,b,∗ , Zhenyuan Xu a a b
School of Science, Jiangnan University, Wuxi 214122, PR China School of Information Technology, Jiangnan University, Wuxi 214122, PR China
a r t i c l e
i n f o
Article history: Received 8 July 2009 Received in revised form 12 October 2009 Accepted 19 October 2009 Available online 24 October 2009 Communicated by A.R. Bishop PACS: 05.45.+b
a b s t r a c t Complex dynamical networks are being studied across many fields of science and engineering today. The issue of controlling a network to the desired state has attracted increasing attention. In this Letter, we investigate the problem of pinning a complex dynamical network to the solution of an uncoupled system. Our strategy is to apply impulsive control to a small fraction of network nodes. Based on the Lyapunov stability theory, we prove that the theoretical results derived here are effective. In addition, a B-A scalefree network with 20 nodes is taken for illustration and verification. © 2009 Elsevier B.V. All rights reserved.
Keywords: Impulsive control Scale-free network Chaotic system
1. Introduction Complex dynamical networks exist everywhere in the real world, such as the Internet, which is a huge-scale network of routers and computers connected by various physical or wireless links; the World Wide Web, which is an enormous virtual network of web sites connected by hyperlinks; and food webs, biological neural networks, telephone cell graphs, etc. Complex networks consist of a large number of dynamical nodes, and have attracted increasing attention from various fields, including physical, economical, social and biological sciences [1–4]. For over a century, the question of how to model complex networks is a point of great interest. In 1960, the theory of random graph [5] was introduced by Paul Erdös and Alfréd Rényi (E-R), which made a breakthrough in the completely regular graph theory. However, many real-world complex networks cannot be described by E-R random graph, therefore, researchers have been proposing the new network models. For example, recently, Watts and Strogatz (W-S) introduced the so-called small world network [6], which exhibits a high degree of
✩ Project supported by the National Natural Science Foundation of China (Grant No. 10901073) and the Youth Foundation of Jiangnan University (Grant No. 31400052210756). Corresponding author at: School of Science, Jiangnan University, Wuxi 214122, PR China. Tel.: +86 51085910656. E-mail address: [email protected] (A. Hu).
*
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.10.049
clustering as in the regular networks and a small average distance among nodes as in the random networks. Another significant discovery is that a number of real-world complex networks have the scale-free feature. It means that the degree distributions of these networks follow a power law form, which was first pointed out by Barabási and Albert (B-A) [7]. Scale-free networks are inhomogeneous in nature, that is, most nodes have very few connections but a small number of particular nodes have many connections. The characters of the small-world and scale-free networks have been verified to fit many real-world complex networks [8–10]. More recently, synchronization and control of complex networks are being seriously studied. In particular, control of scalefree networks or random networks were investigated by applying local linear feedback, which is injected to a small fraction of networks nodes [11,12]. However, the cost of controlling is expensive for the feedback is continuous and the gain is large. This Letter is a further investigation of this subject, in which we want to control a complex dynamical network onto the solution of an uncoupled system. Our strategy is applying impulsive control to pin a small fraction of network nodes. Compared with the method of local linear feedback, impulsive control is attractive because it is discrete and only needs small control gain. The Letter is organized as follows: Problem formulation is introduced in Section 2.1. The theoretical results are presented in Section 2.2. In Section 3, we take a B-A scale-free network with 20 nodes for illustration and verification. Finally, Section 4 concludes the investigation.
A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
For t = tk (k = 1, 2, . . .),
2. Pinning a complex network via impulsive control 2.1. Problem formulation Consider a complex dynamical network consisting of N identical coupled nodes, with each node being an m-dimensional dynamical system. Recently, Wang and Chen [2] introduced the following simple uniform network model:
x˙ i = f (xi ) +
N
(1)
j =1 j =i
j =1 j =i
ai j =
N
a ji = ki
(i = 1, 2, . . . , N ).
For t = tk (k = 1, 2, . . .),
+
= xi tk− + b i xi tk− − x¯ ,
(5a)
i = 1, 2, . . . , l ,
+ xi tk = xi tk− ,
(5b)
where xi (tk+ ) = limt →t + xi (t ), xi (tk− ) = limt →t − xi (t ). k
k
In general, it is assumed that xi (tk− ) = xi (tk ) (i = 1, 2, . . . , N ), b i ∈ R m×m (i = 1, 2, . . . , l) denote the matrix of control gain, and b i (xi (tk− ) − x¯ ) (i = 1, 2, . . . , l; k = 1, 2, . . .) stand for the controller. Let the error vector e i = xi − x¯ (i = 1, 2, . . . , N ), from Eqs. (4), (5a) and (5b), we have: For t = tk (k = 1, 2, . . .),
e˙ i = x˙ i − x˙¯
= f ( xi ) + c
j =1 j =i
N
ai j x j − f (¯x)
j =1
Let the diagonal elements be aii = −ki . For simplicity, we assume that c i j = c , Γ = I m (I m denotes an m × m-dimensional identity matrix). Then, Eq. (1) can be rewritten as follow:
x˙ i = f (xi ) + c
(4)
i = l + 1, l + 2, . . . , N ,
where xi = (xi1 , xi2 , . . . , xim ) T ∈ R m are the state variables of node i , i = 1, 2, . . . , N, the constant c i j 0 represents the coupling strength between nodes i and j, Γ = (si j )m×m is a matrix linking coupled variables, and if some pairs (i , j) with si j = 0, then it means two coupled nodes are linked through their ith and jth state variables, respectively. If there is a connection between nodes i and j (i = j ), then ai j = a ji = 1; otherwise ai j = a ji = 0. If the degree ki of node i is defined to be the number of its outreaching connections, then N
⎧ N ⎪ ⎨ x˙ = f (x ) + c a x , i i ij j j =1 ⎪ ⎩ i = 1, 2, . . . , N . xi t k
c i j ai j Γ (x j − xi ),
187
N
ai j x j .
(2)
j =1
We want to apply impulsive control to a small fraction of the nodes in network (2), so that the network can be controlled onto the solution of an uncoupled system defined by
x1 = x2 = · · · = x N = x¯ ,
x˙¯ = f (¯x).
(3)
ai j e j .
j =1
For simplicity, we just take the first-order term of Taylor series of f (xi ) − f (¯x), i.e., we assume that f (xi ) − f (¯x) is equal to J (xi − x¯ ), where J = D f (¯x) ∈ R m×m is the Jacobian of f (x) at x¯ , then
⎧ N ⎪ ⎨ e˙ = J e + c a e , i i ij j j =1 ⎪ ⎩ i = 1, 2, . . . , N .
Definition 1 (Kronecker product (⊗)). If matrix A = (ai j )n×m , B = (bi j ) p ×q , then
⎛a B ··· a B ⎞ 11 1m ⎜ . ⎟ np ×mq .. A ⊗ B = ⎝ .. . ⎠∈R .
(6)
For t = tk (k = 1, 2, . . .),
e i tk+ = e i tk− + b i e i tk− ,
First of all, we present a useful definition:
+
e i tk
(7a)
−
= e i tk ,
i = l + 1, l + 2, . . . , N .
(7b)
Define E = (e 11 , e 12 , . . . , e 1m , e 21 , e 22 , . . . , e 2m , . . . , e N1 , e N2 , . . . , e Nm ) T ∈ R Nm , then For t = tk (k = 1, 2, . . .),
E˙ = ( I N ⊗ J ) E + c ( A ⊗ I m ) E .
(8)
For t = tk (k = 1, 2, . . .),
· · · anm B
Suppose that we select l (l > 1) nodes in network (2) for controlling, where l = [δ N ] (0 ≺ δ ≺≺ 1, [ ] stands for the Integer function, i.e., l is integral part of the real number δ N). Without loss of generality, we assume that the first l nodes are chosen. Impulsive control is discrete, so consider that a discrete instant set {t i } satisfies
t 1 ≺ t 2 ≺ · · · ≺ t k ≺ t k +1 ≺ · · · ,
N
i = 1, 2, . . . , l ,
2.2. Theoretical analysis
an1 B
= f (xi ) − f (¯x) + c
lim tk = ∞.
k→∞
Define impulsive intervals are τk = tk − tk−1 (k = 1, 2, . . .). After introducing the control to the network (2), we can get the controlled model as follow:
E tk+ = E tk− + B E tk− ,
(9)
where A = (ai j ) N × N , B = diag{b1 , b2 , . . . , bl , 0, . . . , 0} ∈ R Nm× Nm , I N and I m represent an N × N-dimensional identity matrix and an m × m-dimensional identity matrix, respectively. Suppose the network is connected in the sense of having no isolated clusters. Then, the matrix A is symmetric and irreducible. Since all rows of A sum to zero, A always admits the max eigenvalue λmax = 0 [2]. Let G = diag{−a11 , −a22 , . . . , −all , 0, . . . , 0} ∈ R N × N , then the matrix G includes the degrees of the first l nodes. Define R = A − G.
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Theorem 1. If there exists an ξ 1 to satisfy the following conditions, then the network (2) will be controlled onto the desired state x¯ by applying impulsive control to l nodes.
Since
l
i =1
E 2i (t 1− )
N
E 2i (t 1− ) = V ( E (t 1 )), then
i =1
V E t 1+ d exp r (t 1 − t 0 )
l
(1) The coupling strength c − 2λλ1 ; (2) r
1 max{τk }
ln
i =1
2
1 ξd ,
N
E 2i t 1+ d exp r (t 1 − t 0 )
i =1
Since Proof. Choose the form of the Lyapunov function as V ( E (t )) = E T (t ) E (t ). When t ∈ (tk−1 , tk ] (k = 1, 2, . . .), the derivative of V ( E (t )) along the solution of error system (8) is
= E T (t ) ( I N ⊗ J )T + ( I N ⊗ J ) E (t )
l
T
T
E 2i (t ),
V E t 2+ d
i =1
N
E 2i tk− +
E 2i tk−
l
E 2i tk− .
E 2i (t 0 ),
i =1 l
V E (t 1 ) exp r (t 1 − t 0 )
E 2i (t 0 ),
i =1
E 2i t 1− +
E 2i t 2+ d2 exp r (t 2 − t 0 )
l
E 2i (t 0 ).
i =1
N i =l+1
E 2i t 1− .
V E (t ) (13)
i =l+1
V E (t ) exp r (t − t 0 )
i =1
i =l+1
l
E 2i (t 0 ).
In virtue of t − t 0 = t 1 − t 0 + t 2 − t 1 + · · · + tk − tk−1 + t − tk = τ1 + τ2 + · · · + τk + t − tk and the condition (2) in Theorem 1, we know that
i =l+1
i =1
V E t 1+ d
i =l+1
In general, for t ∈ (tk , tk+1 ],
Thus, let k = 1, i.e., t ∈ (t 0 , t 1 ], from Eqs. (12) and (13), we have:
l
E 2i t 2−
i =1
N
di E 2i tk− +
N
V E (t ) dk exp r (t − t 0 )
= E T tk− ( I Nm + B )T ( I Nm + B ) E tk−
E 2i t 2− +
i =1
(12)
V E tk+ = E T tk+ E tk+
l
i =1
When t = tk (k = 1, 2, . . .),
E 2i (t 0 ),
i =1
l
i =1
E 2i t 0 .
Then,
l V E (t ) exp r (t − tk−1 ) E 2i tk+−1 .
l
(11)
l
t 1 , then
l N d2 exp r (t 2 − t 0 ) E 2i (t 0 ) + E 2i t 2− .
where = It is easy to see that r 0, therefore
d
i =l+1
E 2i t 1+
V E (t 2 ) d exp r (t 2 − t 0 )
i =1
2 j =1 e i j .
l
−
i =1
(10)
m
l
E 2i t 1− .
l d exp r (t − t 0 ) E 2i (t 0 ),
i =1
E 2i
2 i =l+1 E i
N
i =1
2c (−aii + λ2 ) + λ1 E 2i (t )
l
t1 =
N
l
i =1
r
E 2i (t 0 ) +
i =1
+
V E (t ) exp r (t − t 1 )
where U = diag{2c (−a11 + λ2 ) + λ1 , . . . , 2c (−all + λ2 ) + λ1 , 2c λ2 + λ1 , . . . , 2c λ2 + λ1 } ∈ R N × N . According to the condition (1) in Theorem 1, i.e. 2c λ2 + λ1 ≺ 0. Then
l
i =1
T
= E T (t )(U ⊗ I m ) E (t ),
2 i =l+1 E i
i =1
λ1 E (t ) E (t ) + 2c E (t )(G ⊗ I m ) E (t ) + 2c λ2 E (t ) E (t )
N
E 2i t 1+ d exp r (t 1 − t 0 )
+ 2c E T (t )( A ⊗ I m ) E (t )
V˙ E (t )
i =l+1
Let k = 2, i.e., t ∈ (t 1 , t 2 ],
V˙ E (t ) = E˙ T (t ) E (t ) + E T (t ) E˙ (t )
l
E 2i t 1− ,
i.e.,
where λ1 = λmax ( J T + J ), λ2 = λmax ( R ), r = max{2c (−aii +λ2 )+λ1 }, d = max{di }, di = ρ 2 ( I m + b i ), ρ ( A ) denotes the spectral radius of A, i = 1, 2, . . . , l, k = 1, 2, . . . .
N
E 2i (t 0 ) +
1
ξk
l
exp r (t − tk )
E 2i (t 0 ).
(14)
i =1
Thus, for k → ∞, V ( E (t )) = 0, i.e., x1 (t ) = x2 (t ) = · · · = x N (t ) = x¯ . The proof of Theorem 1 is completed. 2 3. Numerical simulation In order to illustrate the aforementioned theoretical analysis clearly, we take a B-A scale-free network with 20 nodes for example, and each node is the chaotic Lorenz system [13]. Firstly, we will give the algorithm of the B-A scale-free model [14] as follows. (1) Growth: Starting with a small number (m0 ) of nodes, at every time step, add a new node with m ( m0 ) edges that link the new node to m different nodes already presented in the network.
A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
189
Fig. 1. The plot of Lorenz’s attractor.
(2) Preferential attachment: When choosing the nodes to which the new node connects, assume that the probability Πi that a new node will be connected to node i depends on the degree ki of node i, in such a way that Πi = ki / j k j . After n time steps, we get a network having N = n + m0 nodes and mn edges. This network evolves into a scale-invariant state with the probability that a node has k edges following a powerlaw distribution P (k) ∼ 2m2 k−γ with an exponent γ = 3. The Lorenz system is described in the form by
⎧ ⎨ x˙ 1 = σ (x2 − x1 ), x˙ = ρ x1 − x1 x3 − x2 , ⎩ 2 x˙ 3 = x1 x2 − bx3 .
(15)
Fig. 2. Graphical representation of nodes’ variables when two nodes with the largest degrees being controlled.
node, a state variable attaches the value of 27, and another two variables are 8.4853, which means that 20 nodes are controlled to the point x¯ = (8.4853, 8.4853, 27) T . Now we control 2 nodes with the smallest degrees, such as nodes 19 and 20, both of them have 3 degrees. Then,
G = diag{0, 0, . . . , 0, 3, 3} ∈ R 20×20 ,
λ2 = λmax ( R ) = λmax ( A − G ) = −0.1502. Choose two sets of parameters:
Simulation 1. Pinning the network to its equilibrium x¯ ( f (¯x) = 0).
(1) c = 5, which cannot satisfy the condition (1) in Theorem 1, so the control scheme is useless. Even if we introduce the impulsive control, the network will not be controlled onto the state x¯ . For example, select that b19 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b20 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01. The result is shown in Fig. 3a, which means that whenever it is, 20 nodes in the network cannot reach x¯ = (8.4853, 8.4853, 27) T . (2) c = 18, b19 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b20 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1, then two conditions in Theorem 1 are satisfied. The result is shown in Fig. 3b. In addition, Fig. 3b is similar to Fig. 2.
In the above set of system parameters, one unstable equilibrium point of the system (15) is x¯ = (8.4853, 8.4853, 27) T . Thus we have:
Comparing the results of Fig. 2 with Fig. 3, we will easily find out the nodes with larger degrees are selected, the smaller coupling strength c needed to be.
When the parameters are taken as σ = 10, ρ = 28, b = 8/3, the Lorenz system has a chaotic attractor, which is shown in Fig. 1. A B-A scale-free network is generated by m = m0 = 3, n = 17. Then we attain the matrix A of the network, the diagonal elements of it are (−1, −6, −9, −12, −13, −7, −11, −4, −4, −4, −3, −3, −3, −4, −3, −3, −3, −3, −3, −3). In the following simulations, suppose that only 2 nodes are selected (i.e., l = 2), and let the impulsive intervals τk = τ 0 (k = 1, 2, . . .).
⎡
J =⎣
−10
10
1
−1
8.4853 8.4853
0
⎤
−8.4853 ⎦ , −8/3
Simulation 2. Pinning the network to x¯ ( f (¯x) = 0).
20
Firstly, we control 2 nodes with the largest degrees, i.e., nodes 4 and 5, which have 12 degrees and 13 degrees, respectively. Consequently, we can get
1 We select x¯ = 20 i =1 xi , and control the nodes 4 and 5. The parameters are chosen as c = 7, b4 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b5 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1. Fig. 4 shows that the state variables of 20 nodes will synchronize at 20 1 x¯ = 20 i =1 xi .
G = diag{0, 0, 0, 12, 13, 0, 0, . . . , 0} ∈ R 20×20 ,
4. Conclusions
T
λ1 = λmax J + J = 4.8231.
λ2 = λmax ( R ) = λmax ( A − G ) = −0.6073. Select the parameters as follows: c = 5, b4 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , b5 = diag{−0.5, −0.5, −0.5} ∈ R 3×3 , τ = 0.01, ξ = 1.1, thus two conditions in Theorem 1 can be satisfied, Fig. 2 shows the result. We can see that after t = 1.2, for any arbitrary
In this Letter, we have proposed some conditions for controlling a complex dynamical network to desired state by pinning a small fraction of the network nodes. The method of impulsive control has been used. The numerical results verify the theoretical analysis, and we can find that the cost of controlling is economical.
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A. Hu, Z. Xu / Physics Letters A 374 (2009) 186–190
Fig. 4. Graphical representation of nodes’ variables when the network being pinned 20 1 to x¯ = 20 i =1 xi .
Acknowledgements The authors would like to thank the reviewers for their insightful suggestions.
References
Fig. 3. Graphical representations of nodes’ variables when two nodes with the smallest degrees being controlled.
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