Pinning impulsive control algorithms for complex network.

Pinning impulsive control algorithms for complex network Wen Sun, Jinhu Lü, Shihua Chen, and Xinghuo Yu Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 013141 (2014); doi: 10.1063/1.4869818 View online: http://dx.doi.org/10.1063/1.4869818 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Adaptive synchronization and pinning control of colored networks Chaos 22, 043137 (2012); 10.1063/1.4769991 Pinning control of complex networks via edge snapping Chaos 21, 033119 (2011); 10.1063/1.3626024 Generalized synchronization of complex dynamical networks via impulsive control Chaos 19, 043119 (2009); 10.1063/1.3268587 On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems Chaos 18, 037103 (2008); 10.1063/1.2944235 Adaptive dynamical networks via neighborhood information: Synchronization and pinning control Chaos 17, 023122 (2007); 10.1063/1.2737829

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 81.140.71.206 On: Sun, 06 Apr 2014 12:24:30

CHAOS 24, 013141 (2014)

Pinning impulsive control algorithms for complex network €,2 Shihua Chen,3 and Xinghuo Yu4 Wen Sun,1 Jinhu Lu 1

School of Information and Mathematics, Yangtze University, Jingzhou 434023, People’s Republic of China Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 3 College of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China 4 School of Electrical and Computer Engineering, RMIT University, Melbourne VIC 3001, Australia 2

(Received 25 March 2013; accepted 17 March 2014; published online 28 March 2014) In this paper, we further investigate the synchronization of complex dynamical network via pinning control in which a selection of nodes are controlled at discrete times. Different from most existing work, the pinning control algorithms utilize only the impulsive signals at discrete time instants, which may greatly improve the communication channel efficiency and reduce control cost. Two classes of algorithms are designed, one for strongly connected complex network and another for non-strongly connected complex network. It is suggested that in the strongly connected network with suitable coupling strength, a single controller at any one of the network’s nodes can always pin the network to its homogeneous solution. In the non-strongly connected case, the location and minimum number of nodes needed to pin the network are determined by the Frobenius normal form of the coupling matrix. In addition, the coupling matrix is not necessarily symmetric or irreducible. Illustrative examples are then given to validate the proposed pinning impulsive control C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869818] algorithms. V

Since it is impractical to design a controller for each node in a large scale network, pinning control is a good choice for many natural and man-made systems. Impulsive control, which is active only in discrete instants, is very effective, robust, and low-cost. In this paper, we propose two pinning impulsive control algorithms for complex network. We prove that a single controller at any one of the network’s nodes can pin a strongly connected network. While in the non-strongly connected case, the location and number of nodes needed to pin the network are determined by the Frobenius normal form of the network’s coupling matrix. Numerical simulations are presented to demonstrate the effectiveness of the proposed scheme.

I. INTRODUCTION

Many natural, social, and man-made systems are complex networks.1 Hence complex networks are relevant to diverse disciplines including sociology, biology, mathematics, physics, and so on.2 Several mathematical models, such as the random graph model, small-world network model,3,4 and scale-free network model5 have been developed to describe real-world complex network. Since many networks can spontaneously synchronize even in the absence of any control, the topic of network synchronization becomes an especially active research area. The “master stability function” was constructed to investigate the local stability of the synchronization manifold.6,7 A distance was introduced from the collective spatial states of the coupled system to the synchronization manifold based on which some results were obtained for global synchronization of coupled systems.8,9 L€u et al. introduced a time-varying 1054-1500/2014/24(1)/013141/10/$30.00

complex dynamical network model and further investigated its synchronization.10–13 Wu et al. pointed out that chaos synchronization can be obtained if and only if the network topology has a directed spanning tree.14 In the case when the whole network cannot achieve synchronization by itself, various approaches have been developed to guide the network to reach a desired goal, such as linear feedback control,8,9 adaptive control,11 and pinning control,16–24 to name just a few. Since most real-world complex networks have numerous nodes, it may be costly or even impossible to control every node in a network. Hence, pinning control, in which a selection of the network’s nodes is controlled, is especially welcome. There have been many recent works on pinning control. Grigoriev et al.15 discussed pinning control of spatiotemporal chaos in coupled map lattices. Wang et al. proved that the degree strategy is much more effective than the random strategy for some typical complex network.16,17 Chen et al. proved that, if the coupling strength is large enough, even one single pinning controller is able to control a large network.18 Some other recent advances can be found in Refs. 19–24. Note, however, the aforementioned references focus on continuous communication, i.e., information exchange between nodes is incessant and the communication channels are occupied all the time, which may cause information jam. In order to overcome the limitations brought by the continuous communication, communications between nodes only at some discrete time instants (impulsive signals) are introduced in many realistic networks, which results in occasional occupation of the communication channels. It is obvious that the impulsive communication mode may save communication channel capacity. In addition, the impulsive communication has the advantage of reducing cost, improving system efficiency, and has been widely applied into

24, 013141-1

C 2014 AIP Publishing LLC V


013141-2

Sun et al.

Chaos 24, 013141 (2014)

various disciplines,25–35 including space science, information science, and life science over the last few decades. Zhou et al., for instance, considered the synchronization of delayed dynamical networks with impulsive effects.25 Liu et al. investigated robust impulsive synchronization of uncertain dynamical network.28 Guan et al. introduced the concept of control topology and investigated the problem of distributed impulsive synchronization of complex dynamical networks with system delay and multiple coupling delays.32 To utilize the advantages of impulsive control and pinning control, Zhou et al. pinned complex delayed dynamical networks by a single impulsive controller.34 It is assumed that the network under consideration is symmetric and strongly connected. Unfortunately, the assumption is not realistic. Therefore, in this paper, without assuming that the communication topology is symmetric and strongly connected, we aim at exploring the collective behavior of complex dynamical network by combining the ideas of pinning control and impulsive control. In particular, we develop a novel method for analyzing the synchronization of complex network with impulsive signals in parts of the network nodes. Furthermore, we prove that for a strongly connected complex network with the suitable coupling strength, a single controller at any one of the network’s nodes can always pin complex network to its homogenous solution. While for non-strongly connected complex network, the location and the minimum number of nodes to pin the network are exactly determined by the Frobenius normal form of its coupling matrix, which is not clearly stated in the existing literature. The rest of this paper is organized as follows. In Sec. II, we formulate the basic framework of this paper. Some sufficient conditions are then deduced for the pinning impulsive synchronization of complex network in Sec. III. In Sec. IV, numerical simulations are then given to verify the above theoretical results. And the concluding remarks are drawn in Sec. V. Throughout this paper, we use the following notations. R(C) and Z þ denote the set of real (complex) numbers and positive integer numbers, respectively. For the vector u 2 Rn ðCn Þ, uT ðu Þ denotes its transpose (Hermite conjugate). In is the identity matrix of order n, and 0n is the zero matrix of order n. RNN denotes N N real matrices, for M 2 RNN ; Ms ¼ 12 ðM þ M Þ is the symmetric part of M, and the spectral pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi norm of M is defined as jjMjj ¼ kmax ðMT MÞ. is the Kronecker product, for matrices A; B; C, and D with appropriate dimensions, ðA BÞðC DÞ ¼ ðACÞ ðBDÞ.

In the paper, we only consider the situation that C ¼ diag fc1 ; :::; cn g with ci 0; i ¼ 1; :::; n, i.e., C ¼ I. It must be pointed out that networks with the off-diagonal inner connection matrix are more challenging. It is well known that the coupled systems (1) can be characterized by a directed graph G ¼ ðV; EÞ with the set of nodes V ¼ fv1 ; v2 ; :::; vN g and the set of directed edges E V V. A ¼ ðaij ÞNN is the coupling matrix. If there exists an edge P from vj to vi, then aij > 0. Otherwise, aij ¼ 0. Let aii ¼ Nj¼1 aij ; i 2 f1; :::; Ng. A directed path from node vj to node vi is defined by a sequence of edges ðvj ; vi1 Þ; ðvi1 ; vi2 Þ; :::; ðvil ; vi Þ in the set of directed edges E. Definition 1. A directed graph G is strongly connected if there exists a directed path from any vertex vi to any other vertex vj. Definition 2. A directed graph G has a directed spanning tree if there is a vertex called root from which there exists a directed path to all the other vertices. From Definitions 1 and 2, a strongly connected directed graph must have a directed spanning tree, and any vertex can be regarded as root of directed graph G, which implies that any vertex can indirectly influences all the other vertices. Definition 3. Matrix A is reducible if it can be rewritten as

II. PROBLEM FORMULATIONS

where Nf ðT; tÞ denotes the number of elements of the impulsive sequence f that are contained in the interval ðt; TÞ. To begin with, the necessary assumption and lemma are given in the following. Assumption 1. Assume that34

Consider the linearly coupled complex network xi ðtÞ ¼ f ðt; xi ðtÞÞ þ c

N X aij Cðxj ðtÞ xi ðtÞÞ;

i ¼ 1; ; N;

j¼1

(1) where xi ¼ ðxi1 ; :::; xin ÞT 2 Rn is the state vector of ith node, f : ½0; 1 Rn ! Rn is continuous vector function, c is the coupling strength, and C is the inner connection matrix.

D A¼P 1 0

D12 T P ; D2

where P is a permutation matrix, D1 ; D12 ; D2 are the matrices with suitable dimensions, and 0 is a zero matrix. A matrix is irreducible if it is not reducible. Definition 4. A1 is a set of matrices A ¼ ðaij Þ 2 RNN , where A ¼ ðaij Þ satisfies the following two conditions: P (i) aij 0ði 6¼ jÞ, and aii ¼ Nj¼1;j6¼i aij , for i ¼ 1; 2; :::; N; (ii) A is irreducible. A2 is a set of matrices A ¼ ðaij Þ 2 RNN , where A is symmetric in addition to the above conditions (i) and (ii). Definition 5. (Average Impulsive Interval)33 The average impulsive interval of the impulsive sequence f ¼ ft1 ; t2 ; …g is a positive number Ta, such that there exist positive integer N0 Tt Tt N0 Nf ðT; tÞ

þ N0 ; Ta Ta

8T t 0 ;

(2)

jjDf ðt; sðtÞÞjj < 1; where Df ðt; sðtÞÞ is the Jacobian of f ðt; sðtÞÞ with respect to s(t). Lemma 1. If A 2 A1 , then one gets9


013141-3

(i) (ii)

Sun et al.

Chaos 24, 013141 (2014)

8 N X > > > > x_ 1 ðtÞ ¼ f ðt; x1 ðtÞÞ þ c a1j xj ðtÞ; > > > j¼1 > > > < 1 1 Dx ðtÞ ¼ bk ðx ðtÞ sðtÞÞ; t ¼ tk ; N > X > > i i > _ ðtÞ ¼ f ðt; x ðtÞÞ þ c aij xj ðtÞ; x > > > > j¼1 > > : i ¼ 2; :::; N;

If k is an eigenvalue of A and k 6¼ 0, then ReðkÞ < 0; A has an eigenvalue 0 with multiplicity 1 and the right eigenvector ½1; 1; :::; 1T . Lemma 2. If A 2 A1 , then all eigenvalues of the matrix A~ ¼ A diagf1 ; 0; ; 0g

~ < 0, where 1 is a have negative real parts, i.e., ReðkðAÞÞ positive constant. Proof. See Appendix. Lemma 3. Let k2 ; k3 ; :::; kl be the non-zero eigenvalues of the coupling matrix A 2 A1 , if variational equations z_ 1 ðtÞ ¼ ðDf ðt; sðtÞÞ þ ckj In Þz1 ðtÞ; j ¼ 2; :::; l are locally exponentially stable, then variational equations z_ pþ1 ðtÞ ¼ ðDf ðt; sðtÞÞ þ ckj In Þzpþ1 ðtÞ þ czp ðtÞ; p ¼ 1; ; mj 1; j ¼ 2; :::; l

(5)

where bk is a negative constant impulse gain. Let us introduce the synchronization error ei ðtÞ i ¼ x ðtÞ sðtÞ. Then, the variation equations are given by 8 N X > > > > e_ 1 ðtÞ ¼ Df ðt; sðtÞÞe1 ðtÞ þ c a1j ej ðtÞ; > > > j¼1 > > > < 1 1 De ðtÞ ¼ bk e ðtÞ; t ¼ tk ; (6) N > X > > i i j > e_ ðtÞ ¼ Df ðt; sðtÞÞe ðtÞ þ c aij e ðtÞ; > > > > j¼1 > > : i ¼ 2; :::; N:

are also locally exponentially stable. The proof of Lemma 3 is similar to that of Theorem 1.9 We omit it here.

Let eðtÞ ¼ ½e1 ðtÞT ; …; eN ðtÞT T 2 RnN . Rewrite Eq. (6) into the following matrix form: ( _ ¼ ðIN Df ðt; sðtÞÞÞeðtÞ þ cðA In ÞeðtÞ; t 6¼ tk ; eðtÞ (7) DeðtÞ ¼ ðBk In Þeðt k Þ; t ¼ tk ;

III. SUFFICIENT CONDITIONS

where Bk ¼ diagfbk ; 0; …; 0g 2 RNN . Since A 2 A1 , the eigenvalues of matrix A can be arranged as follows:

In this section, we present our main results in three parts: first, we consider the case that the graph G keeps strongly connected; and next, we take into account the case that the graph G does not keep strongly connected but contains a directed spanning tree, and then we deal with the case that it does not contain a spanning tree.

A. Directed graph G is strongly connected

Now we shall address pinning synchronization in linearly coupled dynamical network (1) with the strongly connected graph G via a single impulsive controller. By designing a suitable impulsive controller, we prove that all nodes of the controlled network can reach synchronization at a homogenous solution s(t), i.e., _ ¼ f ðt; sðtÞÞ; sðtÞ

(3)

in the sense of jjxi ðtÞ sðtÞjj M0 et ; i ¼ 1; :::; N

(4)

for any t > T > 0, where > 0 and M0 > 0 are given constants. Hereafter, s(t) may be an equilibrium point, a periodic orbit, or a chaotic attractor. Without loss of generality, let the first node be the pinned node by rearranging all the network’s nodes. Thus, the pinning-controlled network is described by

0 ¼ k1 > Reðk2 Þ Reðk3 Þ ReðkN Þ:

(8)

If A 2 A2 , then all eigenvalues of A are real and denoted by 0 ¼ k1 > k2 k3 kN . Suppose A ¼ SJS1 be the Jordan decomposition of A, where J ¼ diagfJ1 ; …; Jl g is a block diagonal matrix. By Lemma 1, k1 ¼ 0, and J1 is a 1 1 matrix. Denote dðtÞ ¼ ðS1 In ÞeðtÞ. Then, one gets the following variational equation in terms of dðtÞ: ( _ ¼ ðIN Df ðt; sðtÞÞÞdðtÞ þ cðJ In ÞdðtÞ; t 6¼ tk ; dðtÞ DdðtÞ ¼ ½ðS1 Bk SÞ In dðt k Þ;

t ¼ tk : (9)

Equivalently, one has 8 1 > d_ ðtÞ ¼ Df ðt; sðtÞÞd1 ðtÞ; t 6¼ tk ; > > > > > > i1 _ i1 > > < d ðtÞ ¼ ðDf ðt; sðtÞÞ þ cki In Þd ðtÞ; t 6¼ tk ; ipþ1 d_ ðtÞ ¼ ðDf ðt; sðtÞÞ þ cki In Þdipþ1 ðtÞ þ cdip ðtÞ; > > > > > > 1 p mi 1; 2 i l; > > > : DdðtÞ ¼ ½ðS1 Bk SÞ In dðt k Þ; t ¼ tk ;

(10)

where di ðtÞ ¼ ½di1 ðtÞT ; di2 ðtÞT ; …; dimi ðtÞT T ; dðtÞ ¼ ½d1 ðtÞT ; …; dl ðtÞT T :


013141-4

Sun et al.

Chaos 24, 013141 (2014)

t t0 t t0 N0 Nf ðt; t0 Þ

þ N0 ; Ta Ta

Then, the main results are summarized as follows. Theorem 1. Suppose A 2 A1 , Assumption 1 holds, and there exist constant ci > 0ði ¼ 2; :::; NÞ such that for all k 2 Z þ , the following conditions are satisfied: (i) (ii)

If 0 < d < 1, from (16) and (17), one gets

lnðdÞ Ta

s

8t t0 0: (17)

jjDf ðt; sðtÞÞ jj þ cReðki Þ ci ; gi ¼ 2ci < 0; 2 1 d ¼ jjS ðIN þ Bk ÞSjj ; i ¼ 2; :::; l. 0Þ jjDf ðt; sðtÞÞs jj < c0 ; g0 ¼ lnðd Ta þ c0 < 0, 2 0 < d0 ¼ ð1 þ bk Þ < 1.

tt0

Vi1 ðtÞ d Ta N0 e2ci ðtt0 Þ Vi1 ðt0 Þ

d

N0

lnðdÞ ðt t0 Þ e Ta e2ci ðtt0 Þ Vi1 ðt0 Þ lnðdÞ

dN0 eð Ta 2ci Þðtt0 Þ Vi1 ðt0 Þ

Then, the pinning-controlled network (5) can realize locally asymptotically exponentially synchronization at a homogenous solution via a arbitrary selection impulsive controller. Proof. To begin with, one needs to prove the exponential stability of zero solution of error systems (10). Consider the Lyapunov candidate

dN0 egi ðtt0 Þ Vi1 ðt0 Þ:

(18)

If d 1, one gets tt0

VðtÞ ¼ V1 ðtÞ þ

mi l X X

Vi1 ðtÞ d Ta þN0 e2ci ðtt0 Þ Vi1 ðt0 Þ Vij ðtÞ;

lnðdÞ

dN0 e Ta ðtt0 Þ e2ci ðtt0 Þ Vi1 ðt0 Þ

(11)

i¼2 j¼1

lnðdÞ

dN0 eð Ta 2ci Þðtt0 Þ Vi1 ðt0 Þ

where V1 ðtÞ ¼ d1 ðtÞ d1 ðtÞ; Vij ðtÞ ¼ dij ðtÞ dij ðtÞ; j ¼ 1; …; mi . Calculating the Dini derivative of Vi1 ðtÞði ¼ 2; …; lÞ along the second equation of Eq. (10), according to condition (i) of Theorem 1, one deduces

dN0 egi ðtt0 Þ Vi1 ðt0 Þ:

(19)

By the condition ðiÞ of Theorem 1, one obtains Vi1 ðtÞ ¼ Oðegi ðtt0 Þ Þ;

DVi1 ðtÞ ¼ di1 ðtÞ ½ðDf ðt; sðtÞÞ þ cki In Þ

(20)

þðDf ðt; sðtÞÞ þ cki In Þ di1 ðtÞ

or

¼ 2di1 ðtÞ ½Df ðt; sðtÞÞs þ cReðki ÞIn di1 ðtÞ

2ðjjDf ðt; sðtÞÞs jj þ cReðki ÞÞdi1 ðtÞ di1 ðtÞ

gi

di1 ðtÞ ¼ Oðe 2 ðtt0 Þ Þ;

2ci di1 ðtÞ di1 ðtÞ ¼ 2ci Vi1 ðtÞ:

t 2 ðtk1 ; tk :

(12)

On the other hand, for t ¼ tk , from the last equation of Eq. (10), one has þ 1 dðtþ k Þ dðtk Þ ¼ dðtk Þ ½ðS ðIN þ Bk ÞSÞ In ;

½ðS1 ðIN þ Bk ÞSÞ In dðt kÞ

ddðtk Þ dðtk Þ;

gi

dipþ1 ðtÞ ¼ Oðe 2 ðtt0 Þ Þ;

(21)

i ¼ 2; …; l:

Vipþ1 ðtÞ ¼ Oðegi ðtt0 Þ Þ;

1 p mi 1;

i ¼ 2; …; l: (23)

Denote g ¼ maxfg2 ; :::; gl g, one has (14)

g

jjdip ðtÞjj ¼ Oðe2ðtt0 Þ Þ;

i ¼ 2; …; l;

1 p mi

i ¼ 2; …; l;

1 p mi : (25)

(24)

and (15)

Denote Nf ðt; t0 Þ be the impulsive times of impulsive sequence f on interval ðt0 ; tÞ. Thus, for any t 2 R, one obtains Vi1 ðtÞ dNf ðt;t0 Þ e2ci ðtt0 Þ Vi1 ðt0 Þ:

i ¼ 2; …; l: (22)

(13)

According to (12) and (14), for t 2 ðtk ; tkþ1 , one has Vi1 ðtÞ d k e2ci ðtt0 Þ Vi1 ðt0 Þ :

1 p mi 1;

It follows that

where d ¼ jjS1 ðIN þ Bk ÞSjj2 > 0. One gets i1 þ i1 þ Vi1 ðtþ k Þ ¼ d ðtk Þ d ðtk Þ dVi1 ðtk Þ;

i ¼ 2; …; l:

Combining this with the third equation of (10), by Lemma 3, one gets

Therefore, one obtains Vi1 ðtÞ e2ci ðttk1 Þ Vi1 ðtk1 Þ;

gi < 0;

(16)

Since the average impulsive interval of the impulsive sequence f ¼ ft1 ; t2 ; …g is equal to Ta, we have

ip

g

jjd_ ðtÞjj ¼ Oðe2ðtt0 Þ Þ;

From Lemma 1, S1 ¼ ð1; :::; 1ÞT 2 RN is an eigenvector with respect to the eigenvalue k1 ¼ 0, by eðtÞ ¼ ðS In ÞdðtÞ, one gets e1 ðtÞ ¼ d1 ðtÞ þ

N X

s1j dj ðtÞ

(26)

i¼2


013141-5

Sun et al.

Chaos 24, 013141 (2014) 1

jje1 ðtÞjj ¼ Oðe2g0 ðtt0 Þ Þ:

or d1 ðtÞ ¼ e1 ðtÞ

N X

s1j dj ðtÞ:

(27)

i¼2

N N X X j s1j d_ ðtÞ e_ 1 ðtÞ ¼ Df ðt; sðtÞÞe1 ðtÞ Df ðt; sðtÞÞ s1j dj ðtÞ þ i¼2 T

1

¼ Df ðt; sðtÞÞe ðtÞ þ ð1; :::; 1Þ Oðe

i¼2 g2ðtt0 Þ

Þ:

(28)

Consider the function W1 ðtÞ ¼ e1 ðtÞT e1 ðtÞ

(29)

for t 2 ðtk ; tkþ1 , applying Assumption 1, then there exists a positive constant Q such that W_ 1 ðtÞ ¼ e1 ðtÞT ½Df ðt; sðtÞÞT þ Df ðt; sðtÞÞe1 ðtÞ g

þ 2ð1; :::; 1Þe1 ðtÞOðe2ðtt0 Þ Þ pffiffiffiffiffiffiffiffiffiffiffi g

c0 W1 ðtÞ þ Q W1 ðtÞe2ðtt0 Þ ; which leads to pffiffiffiffiffiffiffiffiffiffiffi g d W1 ðtÞ 1 1

c0 W1 ðtÞ þ Qe2ðtt0 Þ : dt 2 2

(30)

(31)

On the other hand, for t ¼ tk , one can get 2 1 T 1 W1 ðtþ k Þ ¼ ð1 þ bk Þ e ðtk Þ e ðtk Þ d0 W1 ðtk Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi d0 W1 ðt W1 ðtþ kÞ

k Þ:

(32)

(33)

Then, from the condition (ii), there exists a small enough real number g0 > 0 such that h ¼ infþ k2Z

lnd0 : tkþ1 tk

According to the representation of the solutions for the impulsive differential inequalities,37 by some elementary computation, one gets for t t0 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi W1 ðtÞ W1 ðt0 Þ

Y pffiffiffiffiffi Ð t 1c ds d0 e t0 2 0 t0 x h ðtÞ þ cðAh In Þ~ x h ðtÞ > x~_ ðtÞ ¼ ðImh Df ðt; sðtÞÞ~ > > > > l X > > > > þ cðAh;j In Þ~ x j ðtÞ; h ¼ 1; :::; l 1; > > > j¼hþ1 > > > > ml > X > < x_ l1 ðtÞ ¼ Df ðt; sðtÞÞxl1 ðtÞ þ c A xlj ðtÞ; t 6¼ t ; l1j k j¼1 > > > l1 l1 > > > Dx ðtÞ ¼ bk ðx ðtÞ sðtÞÞ; t ¼ tk ; > > > ml X > > li li > > _ ðtÞ ¼ Df ðt; sðtÞÞx ðtÞ þ c Alij xlj ðtÞ; x > > > j¼1 > > > > : i ¼ 2; :::; ml :

From Lemma 4, one knows Al1;l 6¼ 0; and A a zero row sums matrix, we decompose Al1 as Fl1 þ Hl1 , where Fl1 is an irreducible and a zero row sum matrix, while Hl1 is a diagonal matrix with nonpositive diagonal elements. By Lemma 2, all eigenvalues of Al1 have negative real parts. Let Al1 ¼ Sl1 JS1 l1 be the Jordan decomposition of Al1 , where Jl1 ¼ diagfJl1;1 ; …; Jl1;q g is a block diagonal l1 matrix. Denote ~d ðtÞ ¼ ðS1 In Þ~ e l1 ðtÞ. Then, one gets l1

the following variational equation: l1 ~d_ ðtÞ ¼ ðIm Df ðt; sðtÞÞÞ~d l1 ðtÞ l1 l1 þ cðJml1 In Þ~d ðtÞ;

(37)

(40)

l1

where ~d ðtÞ ¼ ½dl1;1 ðtÞT ; …; dl1;ml1 ðtÞT T . Therefore, one has 8 l1;1 > > ðtÞ ¼ ðDf ðt; sðtÞÞ þ ck1 ðAl1 ÞIn Þdl1;1 ðtÞ; > d_
> > : þcdl1;p ðtÞ; 1 p m 1: l1

Theorem 2. Let matrix A has the Frobenius normal form (36), Assumption 1 holds, and there exist constants ch > 0ðh ¼ 1; :::; l 1Þ; cli > 0ðli ¼ 2; :::; ml Þ, such that for all k 2 Zþ , the following conditions are satisfied: (i) (ii) (iii)

lÞ jjDf ðt; sðtÞÞs jj þ cReðkli ðAl ÞÞ cli ; gli ¼ lnðd Ta 2cli 2 < 0; dl ¼ jjS1 l ðIN þ Bk ÞSl jj ; li ¼ 2; :::; ml . 0Þ jjDf ðt; sðtÞÞs jj c0 ; g0 ¼ lnðd Ta þ c0 < 0, 2 0 < d0 ¼ ð1 þ bk Þ < 1. jjDf ðt; sðtÞÞs jj þ cReðkmax ðAh ÞÞ ch ; 1 h l 1.

Then, the pinning-controlled network (37) can realize locally asymptotically exponentially synchronization at a homogenous solution via one impulsive controller located in Vl. Proof. Note that the matrix A has the Frobenius normal form (36), its root must be located in Vl. We first consider the nodes in the part Vl. If the conditions (i) and (ii) of Theorem 2 are satisfied and there exists at least one node in Vl to be pinned, from the previous arguments in Theorem 1, all the nodes in the part Vl can be locally exponentially synchronized to a homogenous solution. Second, we consider the nodes xsl2 þ1 ; :::; xsl1 in the part Al1 , and the dynamics of xsl2 þ1 ; :::; xsl1 can be written as l1

l1 x~_ ðtÞ ¼ f~

ðtÞ þ cðAl1 In Þ~ x l1 ðtÞþcðAl1;l In Þ~ x l ðtÞ: (38)

Since any system in the part Vl is locally exponentially synchronized to a homogenous solution s(t), one gets gp

elp ðtÞ ¼ Oðe 2 ðtt0 Þ Þ;

gp < 0; 1 p ml :

In this case, we have e~_

l1

l1

ðtÞ ¼ ðIml1 Df ðt; sðtÞÞÞ~ e l1

þ cðAl1 In Þ~ e

ðtÞ:

ðtÞ (39)

From Lemma 3, one can choose suitable c such that the condition (iii) of Theorem 2 is satisfied, it is not difficult to derive dl1;p ðtÞ ¼ Oðecl1 ðtt0 Þ Þ; 1 p ml1 . By using the same method, one can prove that the same results also hold for the nodes in the parts V1 ; :::; Vl2 . Therefore, we conclude that any node in the graph G can be locally exponentially synchronized to a homogenous solution. Remark 3. If the directed graph G is not strongly connected but contains a directed spanning tree with root node located in Vl, only the system in Vl can directly or indirectly influence all other system. Therefore, to achieve the aim that the states of all nodes can be locally exponentially synchronized to a homogenous solution, the pinned node is selective and not arbitrary. Remark 4. Zhou et al. pinned a given complex delayed dynamical network to a homogenous solution.34 Unfortunately, they considered the case that the coupling matrix is irreducible and symmetric, but other case that the coupling matrix is reducible or asymmetric, is missing. If the directed graph G does not contain a directed spanning tree, we assume there exists at least l r þ 1 separate groups, and A has the Frobenius normal form 2 6 6 6 6 6 A¼6 6 6 6 4

A1

A12 A2

.. . Ar

3 A1l A2l 7 7 7 ⯗ 7 7 7; 0 0 7 7 .. 7 . 0 5 Al

(42)

where Ah ðh ¼ 1; :::; lÞ are irreducible square matrices. In general, these different groups of systems will not synchronize with each other. We will show that all the states of the network will be synchronized to a homogenous


013141-7

Sun et al.

Chaos 24, 013141 (2014)

solution via at least one impulsive controller in every separate part Vh ; r h l. The pinning controlled network can be written as 8 h h > > x h ðtÞ x~_ ðtÞ ¼ f~ ðtÞ þ cðAh In Þ~ > > > l X > > > þ > cðAh;j In Þ~ x j ðtÞ; h ¼ 1; :::; r 1; > > > j¼hþ1 > > > mh > X < h1 x_ ðtÞ ¼ f ðt; xh1 ðtÞÞ þ c Ah1j xhj ðtÞ; t 6¼ tk ; (43) > j¼1 > > > > Dxh1 ðtÞ ¼ bk ðxh1 ðtÞ sðtÞÞ; t ¼ tk ; > > mh > X > > hi hi > _ ðtÞ ¼ f ðt; x ðtÞÞ þ c Ahij xhj ðtÞ; x > > > > j¼1 > : h ¼ r; :::; l; i ¼ 2; :::; mh : Theorem 3. Let matrix A has the Frobenius normal form (42), Assumption 1 holds, and there exist constants chi > 0 such that for all k 2 Zþ , the following conditions are satisfied: (i) (ii) (iii)

hÞ jjDf ðt;sðtÞÞs jj þ cReðkhi ðAh ÞÞ chi ;ghi ¼ lnðd Ta 2chi 2 1 < 0;dh ¼ jjSh ðIN þ Bk ÞSh jj ; r h l;hi ¼ 2;:::;mh . 0Þ jjDf ðt; sðtÞÞs jj c0 ; g0 ¼ lnðd 0 < d0 ¼ Ta þ c0 < 0, 2 ð1 þ bk Þ < 1. jjDf ðt; sðtÞÞs jj þ cReðkhi ðAh ÞÞ chi ; 1 h r 1; i ¼ 1; ; mh .

Then, the pinning-controlled network (43) can realize locally asymptotically exponentially synchronization at a homogenous solution via at least one impulsive controller in every separate part Vh ; r h l. Proof. If A has the Frobenius normal form (42), we first consider the nodes in the parts Vh ; h ¼ r; :::; l. From the conditions (i) and (ii) of Theorem 3, for each h ¼ r; :::; l, there exists at least one impulsive controller in every separate part Vh ; r h l, by using the previous arguments in Theorem 1, all the nodes in the parts Vh ; h ¼ r; :::; l can be locally asymptotically exponentially synchronized to a homogenous solution. Second, we consider the nodes xsr2 þ1 ; :::; xsr1 in the part Vr1 . The dynamics of xsr2 þ1 ; :::; xsr1 can be written as r1 r1 x r1 ðtÞ x~_ ðtÞ ¼ f~ ðtÞ þ cðAr1 In Þ~

þ

l X cðAr1;j In Þ~ x j ðtÞ:

Since any system in the parts Vh ; h ¼ r; :::; l is synchronized to a homogenous solution s(t), namely gh i

1 i mh ;

r1

variational equation r1 ~d_ ðtÞ ¼ ðIm Df ðt; sðtÞÞÞ~d r1 ðtÞ r1 r1 þ cðJmr1 In Þ~d ðtÞ

or 8 r1;1 > ðtÞ ¼ ðDf ðt; sðtÞÞ þ ckr1;i ðAr1 ÞIn Þdr1;1 ðtÞ; < d_ r1;pþ1 ðtÞ ¼ ðDf ðt; sðtÞÞ þ ckr1;i ðAr1 ÞIn Þdr1;pþ1 ðtÞ d_ > : þcdr1;p ðtÞ; 1 p mr1 1: (47) One can choose suitable c such that the condition (iii) of Theorem 3 are satisfied, and it is easy to derive dr1;p ðtÞ ¼ Oðechr1 ðtt0 Þ Þ; 1 p mr1 . By using the same method, one can prove that the results also hold for the nodes in the part V1 ; :::; Vr2 : Therefore, any node in the graph G can be locally exponentially synchronized to a homogenous solution. If the directed graph G does not contain a directed spanning tree and A has the Frobenius normal form 2 3 A1 6 7 .. (48) A¼4 5; . Al where Ah ðh ¼ 1; :::; lÞ are irreducible square matrices, the pinning controlled network can be written as 8 mh X > > h1 h1 > _ ðtÞ ¼ f ðx ðtÞÞ þ c Ah1j xhj ðtÞ; t 6¼ tk ; x > > > > j¼1 > < h1 Dx ðtÞ ¼ bk ðxh1 ðtÞ sðtÞÞ; t ¼ tk ; (49) mh X > > > x_ hi ðtÞ ¼ f ðxhi ðtÞÞ þ c Ahij xhj ðtÞ; > > > > j¼1 > : h ¼ 1; :::; l; i ¼ 2; :::; mh :

r h l:

According to Theorem 3, one can obtain: Corollary 1. Suppose matrix A has the Frobenius normal form (48), Assumption 1 holds, and there exist constants chi > 0; 1 h l; i ¼ 2; ; mh , such that for all k 2 Z þ , the following conditions are satisfied: (i)

Then, one obtains (ii)

r1 e~_ ðtÞ ¼ ðImr1 Df ðt; sðtÞÞÞ~ e r1 ðtÞ

þcðAr1 In Þ~ e r1 ðtÞ:

(46)

(44)

j¼r

eh;i ðtÞ ¼ Oðe 2 ðtt0 Þ Þ;

where Fr1 is an irreducible and a zero row sum matrix, while Hr1 is a diagonal matrix with nonpositive diagonal elements. From Lemma 2, all eigenvalues of Ar1 have negative real parts. Let Ar1 ¼ Sr1 JS1 r1 be the Jordan decomposition of Al1 , where Jr1 ¼ diagfJr1;1 ; …; Jr1;q g is a block diagr1 e r1 ðtÞ, one has onal matrix. Denote d~ ðtÞ ¼ ðS1 In Þ~

(45)

Noting that there exists at least a Ar1;j ¼ 6 0; r j l, and A a zero row sums matrix. We decompose Ar1 as Fr1 þ Hr1 ,

hÞ jjDf ðt;sðtÞÞs jj þcReðkhi ðAh ÞÞ chi ;ghi ¼ lnðd Ta 2chi 2 < 0;dh ¼ jjS1 h ðIN þBk ÞSh jj ;1 h l;i ¼ 2;:::;mh . 0Þ 0 < d0 jjDf ðt; sðtÞÞs jj c0 ; g0 ¼ lnðd Ta þ c0 < 0; 2 ¼ ð1 þ bk Þ < 1.

Then, the pinning-controlled network (49) can realize locally asymptotically exponentially synchronization at a homogenous solution via at least one impulsive controller in every separate part Vh ð1 h lÞ.


013141-8

Sun et al.

Chaos 24, 013141 (2014)

IV. NUMERICAL RESULTS

In this section, numerical simulations are given to illustrate the above theoretical results. Consider the classic Lorenz system as network’s node, which is described by 8 > < x_1 ¼ aðx2 x1 Þ; x_2 ¼ cx1 x1 x3 x2 ; > : x_3 ¼ x1 x2 bx3 ;

(50)

when a ¼ 10; b ¼ 8=3; c ¼ 28, system (50) has a typical chaotic attractor. Let the initial value be ðx1 ð0Þ; x2 ð0Þ; x3 ð0ÞÞT ¼ ð0:001; 0:001; 0:001ÞT . From Ref. 33, one deduces jjx1 ðtÞjj 20; jjx2 ðtÞjj 30; jjx3 ðtÞjj 50. Therefore, one has 2jjDf ðt; sðtÞÞs jj 2jjDf ðt; sðtÞÞjj 37:5284. Without loss of generality, we pin a strongly connected network with 20 Lorenz oscillators by a single impulsive controller. The network is obtained by adding directed edges with probability five percent to a regular network, which consists of 20 nodes, with each node connecting to the nearest 2 nodes. The second largest real part of the coupling

matrix A2020 is Reðk2 ðAÞÞ ¼ 0:3785. For simplicity, we consider the equidistant impulsive interval. Let c0 ¼ 19, 0 bk ¼ 0:9; d0 ¼ 0:01, d ¼ 1, c ¼ 100, it is clear that lnd Ta þ c0 < 0 and c0 þ cReðk2 ðAÞÞ < 0 for Ta ¼ 0:005; 0:01; 0:02. Therefore, the conditions of Theorem 1 are satisfied. The network synchronization error is calculated by EðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P20 P3 2 i ¼ i¼1 j¼1 ðxj ðtÞ sj ðtÞÞ . Figure 1 shows how the synchronization error changes over time for several impulsive controllers that satisfy the sufficient conditions of Theorem 1. In particular, the coupling strength, average impulsive interval, and impulsive control gains are varied to produce multiple controllers. Figure 1 also shows that E(t) converges to zero. Hence, a single controller located at any one of the network’s nodes can pin this strongly connected complex network to a homogenous solution. We can see that the coupling strength, the impulsive control gains, and the impulsive intervals determine a controller’s ability to achieve synchronization, but these characteristics of the control are less important than the number of nodes that are subject to control.

FIG. 1. Pin a strongly connected network with 20 Lorenz oscillators. (a) tk tk1 ¼ 0:01; bk ¼ 0:9, (b) bk ¼ 0:9; c ¼ 100, (c) tk tk1 ¼ 0:01; c ¼ 100, (d) tk tk1 ¼ 0:01; c ¼ 100; bk ¼ 0:9.


013141-9

Sun et al.

Chaos 24, 013141 (2014)

Next, we consider an example in which the network graph is not strongly connected, but contains a directed spanning tree. For convenience, we take 2 3 A1 A12 A13 A14 6 7 6 A2 A23 A24 7 7; A¼6 6 A3 A34 7 4 5 A4 where 2

3

6 6 1 6 A1 ¼ 6 6 0 6 4 0 1 2 2 6 6 1 6 A4 ¼ 6 6 0 6 4 0 1

A12 ¼ A23

1

0

1

0

2 1

1 2

0 1

0 0

3

7 7 7 7; 7 7 0 1 2 1 5 0 0 1 2 3 1 0 1 0 7 2 1 0 0 7 7 1 2 1 0 7 7; 7 0 1 2 1 5 0 0 1 2 2 3 1 0 0 0 0 6 7 60 0 0 0 07 6 7 7 ¼ A34 ¼ 6 6 0 0 0 0 0 7; 6 7 40 0 0 0 05 0 0 0 0 0

homogenous solution, and the location of the pinned node influences the synchronization performance heavily. This is because the root node, which is located in V4, is the only node which is able to influence every other node in the network. Remark 5. The location of the pinned nodes influences the performance heavily as shown in Figure 2. If a directed graph G does not contain a spanning directed tree, we assume 2 3 A01 6 7 6 7 A02 7; A0 ¼ 6 6 7 0 A3 4 5 A04 where A01 ¼ A02 ¼ A03 ¼ A04 ¼ A4 . In order to satisfy the conditions of Theorem 3, at least one node in each strongly connected component Vh ðh ¼ 1; 2; 3; 4Þ must be controlled. Figure 3 shows that the location of the controlled nodes is important for synchrony, and some controls (the random

A2 ¼ A3 ¼ A1 ; A13 ¼ A14 ¼ A24 ¼ 055 , and the rest parameters are the same as those in simulation 1. Figure 2 shows the synchronization error E(t) of network pinned by one impulsive controller in the part Vj ðj ¼ 1; 2; 3; 4Þ. It can be seen from simulations that one impulsive controller cannot always pin a non-strongly connected network containing a directed spanning tree to the

FIG. 2. Pin a non-strongly connected Lorenz network with a directed spanning tree by one impulsive controller located in Vi ði ¼ 1; :::; 4Þ, where tk tk1 ¼ 0:01; c ¼ 100.

FIG. 3. Pin 20 Lorenz oscillator network without a directed spanning tree by 4 impulsive controllers. (a) tk tk1 ¼ 0:01; c ¼ 100, (b) tk tk1 ¼ 0:005; c ¼ 300.


013141-10

Sun et al.

selection) may fail to achieve synchronization. However, when the location of the controlled nodes is chosen according to Theorem 3 (the particular selection), synchrony is quickly achieved. We also see that the coupling strength, the impulsive control gains and the impulsive intervals can influence a control ability to synchronize a network of oscillators. V. CONCLUSION

In this paper, we propose a novel approach for analyzing synchronization of complex network via pinning impulsive control. We establish sufficient conditions of pinning impulsive synchronization for strongly connected and nonstrongly connected complex network. Our results show that the nodes to add impulsive controllers needed at least are arbitrary for a strongly connected complex network, and selective for non-strongly connected complex network. It should be especially pointed out that the Frobenius normal form of the coupling matrix plays an important role in deciding the location and the number of the pinned nodes. What is more, the coupling matrix A may be asymmetric or reducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed pinning impulsive control algorithm. It sheds some light on the control of the communication network with limited communication channel capacity. ACKNOWLEDGMENTS

This research was supported by The National Science Fund for Distinguished Young Scholars (61025017), the National Natural Science Foundation of China (61273215, 61273179, and 11201039), Young Project of Hubei Provincial Department of Education (Q20121216), the China Postdoctoral Science Foundation (2012M520417), and Basic Subjects Development Fund for Scientific Research of Yangtze University (2013cjp07). APPENDIX: PROOF OF LEMMA 2

~ by the Proof. Assume ~k is the eigenvalue of A, 36 ~ Gerschgorin theorem, then the eigenvalue k of A~ lies in a Gershgorin disc described by fDða11 1 ; ja11 jÞ : j~k ða11 1 Þj ja11 jg; fDðaii ; jaii jÞ : j~k aii j jaii j; i ¼ 2; :::; Ng; ~ 0. Noting that the property which suggests that ReðkðAÞÞ ~ i.e., ~k 1 ~k 2 ; :::; ~k N ¼ jAj, ~ of the characteristic polynomial of A, ~ it suffices to show 0 is not the eigenvalue of A. Since A 2 A1 , from (i) in Lemma 1 and the Gershgorin Theorem, or Proposition 1,18 it is clear that A~ is nonsingular, which ~ The proof of Lemma 2 implies 0 is not the eigenvalue of A. is completed. 1

S. H. Strogatz, “Exploring complex networks,” Nature 410, 268 (2001). R. Albert and A. L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47 (2002). 3 Y. Wu, Y. Shang, M. Chen, C. Zhou, and J. Kurths, “Synchronization in small-world networks,” Chaos 18, 037111 (2008). 2

Chaos 24, 013141 (2014) 4

D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world networks,” Nature 393, 440 (1998). A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science 286, 509 (1999). 6 L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821 (1990). 7 L. M. Pecora and T. L. Carroll, “Master stability function for synchronized coupled systems,” Phys. Rev. Lett. 80, 2109 (1998). 8 C. W. Wu and L. O. Chua, “Synchronization in an array of linearly coupled dynamical systems,” IEEE Trans. Circuits Syst. I 42, 430 (1995). 9 W. Lu and T. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Physica D 213, 214 (2006). 10 J. L€ u and G. Chen, “A time-varying dynamical network model and its controlled synchronization criteria,” IEEE Trans. Automat. Control 50, 841 (2005). 11 J. Zhou, J. Lu, and J. L€ u, “Adaptive synchronization of an uncertain complex dynamical network,” IEEE Trans. Automat. Control 51, 652 (2006). 12 J. L€ u, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I 51, 787 (2004). 13 P. Wang, J. L€ u, and M. J. Ogorzalek, “Global relative parameter sensitivities of the feed-forward loops in genetic networks,” Neurocomputing 78, 155 (2012). 14 C. W. Wu, “Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,” Nonlinearity 18, 1057 (2005). 15 R. Grigoriev, M. Cross, and H. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795 (1997). 16 X. Wang and G. Chen, “Pinning control of scale-free dynamical networks,” Physica A 310, 521 (2002). 17 X. Wang, X. Li, and J. L€ u, “Control and flocking of networked systems via pinning,” IEEE Circuits Syst. Mag. 10, 83 (2010). 18 T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Trans. Circuits Syst. I 54, 1317 (2007). 19 Z. Liu, Z. Chen, and Z. Yuan, “Pinning control of weighted general complex dynamical networks with time delay,” Physica A 375, 345 (2007). 20 F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, “Controllability of complex networks via pinning,” Phys. Rev. E 75, 046103 (2007). 21 W. Yu, G. Chen, and J. L€ u, “On pinning synchronization of complex dynamical networks,” Automatica 45, 429 (2009). 22 J. Xiang and G. Chen, “Analysis of pinning-controlled networks: A renormalization approach,” IEEE Trans. Automat. Control 54, 1869 (2009). 23 W. Lu, X. Li, and Z. Rong, “Global stabilization of complex networks with diagraph topologies via a local pinning algorithm,” Automatica 46, 116 (2010). 24 Y. Chen, J. L€ u, F. Han, and X. Yu, “On the cluster consensus of discretetime multi-agent systems,” Syst. Control Lett. 60, 517 (2011). 25 J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,” Physica A 384, 684 (2007). 26 Z. Yang and D. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Trans. Automat. Control 52, 1448 (2007). 27 J. Sun, Y. Zhang, and Q. Wu, “Less conservative conditions for asymptotic stability of impulsive control systems,” IEEE Trans. Automat. Control 48, 829 (2003). 28 B. Liu, X. Liu, and G. Chen, “Robust impulsive synchronization of uncertain dynamical networks,” IEEE Trans. Circuits Syst. I 52, 1431 (2005). 29 T. Yang and L. O. Chua, “Impulsive stability for control and synchronization of chaotic systems: Theory and application to secure communication,” IEEE Trans. Circuits Syst. I 44, 976 (1997). 30 T. Yang, “Impulsive control,” IEEE Trans. Automat. Control 44, 1081 (1999). 31 W. Sun, Z. Yan, S. Chen, and J. L€ u, “Outer synchronization of complex networks by impulse,” Commun. Theor. Phys. 56, 885 (2011). 32 Z. Guan, Z. Liu, G. Feng, and Y. Wang, “Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control,” IEEE Trans. Circuits Syst. I 57, 2182 (2010). 33 J. Lu, D. W. C. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatica 46, 1215 (2010). 34 J. Zhou, Q. Wu, and L. Xiang, “Pinning complex delayed dynamical networks by a single impulsive controller,” IEEE Trans. Circuits Syst. I 58, 2882 (2011). 35 W. Sun, F. Austin, J. L€ u, and S. Chen, “Synchronization of impulsively coupled complex systems with delay,” Chaos 21, 033123 (2011). 36 P. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, New York, 1985). 37 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989). 5


Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy.

Optimizing Dynamical Network Structure for Pinning Control.

Active noise cancellation algorithms for impulsive noise.

Impulsive control for synchronizing delayed discrete complex networks with switching topology.

Metaheuristic Algorithms for Convolution Neural Network.

Control of multidimensional systems on complex network.

Pinning synchronization of hybrid-coupled directed delayed dynamical network via intermittent control.

Criteria for stochastic pinning control of networks of chaotic maps.

Pinning Impulsive Synchronization of Reaction-Diffusion Neural Networks With Time-Varying Delays.

Pinning consensus in networks of multiagents via a single impulsive controller.

Impulsive action: emotional impulses and their control.

Private algorithms for the protected in social network search.

Hybrid algorithms for fuzzy reverse supply chain network design.

Pathway-Dependent Effectiveness of Network Algorithms for Gene Prioritization.

Netmes: assessing gene network inference algorithms by network-based measures.

Network models. Comment on "Control profiles of complex networks".

Current algorithms for the diagnosis of wide QRS complex tachycardias.

Advanced Algorithms for Local Routing Strategy on Complex Networks.

Interventional Algorithms for the Control of Coagulopathic Bleeding in Surgical, Trauma, and Postpartum Settings: Recommendations From the Share Network Group.

Algorithms for the derivation of Flux and Concentration Control Coefficients.

An OFDM Receiver with Frequency Domain Diversity Combined Impulsive Noise Canceller for Underwater Network.

Network-based biomarkers for complex diseases.

Pinning Control Strategies for Synchronization of Linearly Coupled Neural Networks With Reaction-Diffusion Terms.

BCI Control of Heuristic Search Algorithms.