Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers

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Outer Synchronization of Partially Coupled Dynamical Networks via Pinning Impulsive Controllers Jianquan Lu, Chengdan Ding, Jungang Lou, Jinde Cao

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S0016-0032(15)00337-3 http://dx.doi.org/10.1016/j.jfranklin.2015.08.016 FI2424

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6 May 2015 22 July 2015 21 August 2015

Cite this article as: Jianquan Lu, Chengdan Ding, Jungang Lou, Jinde Cao, Outer Synchronization of Partially Coupled Dynamical Networks via Pinning Impulsive Controllers, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.08.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Outer Synchronization of Partially Coupled Dynamical Networks via Pinning Impulsive Controllers Jianquan Lu a Chengdan Ding a Jungang Lou b Jinde Cao a,c a Department b School

of Mathematics, Southeast University, Nanjing 210096, China

of Information Engineering, Huzhou University, Huzhou, 313000, China.

c Department

of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract This paper presents an analytical study of outer synchronization of partially coupled dynamical networks via pinning impulsive controller. At first, more realistic driveresponse partially coupled networks are established. Then, based on the regrouping method, some efficient and less conservative synchronization criteria are derived and developed in terms of average impulsive interval. Our results show that, by impulsively controlling a crucial fraction of nodes in the response network, the outer synchronization can be achieved. Finally, illustrated examples are given to verify the effectiveness of the proposed strategy.

Keywords: Complex dynamical networks; Partial coupling; Outer synchronization; Pinning impulsive control; Average impulsive interval.

1

INTRODUCTION

Complex dynamical networks typically consist of a large set of interconnected dynamical nodes, in which each node is a fundamental unit with detailed contents [1,2]. A large number of systems including naturally occurring networks and man-made networks can be modeled by complex networks, which are shown to widely exist in our daily life, such as food webs, communication networks, the Internet, the World Wide Web, and social organizations, etc. [3–6]. Email address: [email protected] (Jianquan Lu). Preprint submitted to Elsevier

27 August 2015

In the past decades, many interesting collective behaviors in complex dynamical networks have attracted increasing attention, e.g., synchronization [7–13], consensus [14], and spatiotemporal chaos spiral waves [15]. Special attention has been focused on the synchronization phenomena of various complex networks. Synchronization, which means the agreement of the states of all nodes, is closely related to many practical applications, such as parallel image processing, flocking of birds, and agreement of opinions [16]. Many important results on synchronization have been obtained in last few years. Cao et al.[20] analyzed the globally exponential synchronization in an array of coupled identical delayed neural networks with delayed coupling. Lu et al.[21] considered globally exponential synchronization problem for general dynamical networks by the Lyapunov functional method and the Kronecker product techniques. Investigations on synchronization of networks mentioned above mainly focused on the inner synchronization, which is concerned with the collective behavior among all nodes in a network. In the real world, there are many other kinds of network synchronization. For example, outer synchronization between two or more complex networks, means that the corresponding nodes of coupled networks could realize synchronization regardless of the inner synchronization. Outer synchronization always exist in our daily life, such as, the spread of infectious diseases and the balance of intestinal microflora for human beings. A great deal of examples about relationships between different networks show that it is necessary and important to study the dynamics between coupled networks. Some interesting results on outer synchronization have been reported. Li et al. [22] studied the outer synchronization between two unidirectionally coupled networks and a synchronization criterion between two networks with same topological connectivities was presented. Shortly after, Tang et al. [23] designed an effective adaptive controller to achieve outer synchronization between two complex networks with identical and nonidentical topologies. Wu et al. [24] investigated the generalized outer synchronization between two completely different networks under a nonlinear control scheme. To guide the network reach synchronization when the network itself cannot achieve synchronization, various approaches have been developed in recent years, such as linear feedback control [25,26], adaptive control [17–19], and impulsive control [27–32]. Compared with continuous time controllers which must be imposed at every time instant t, discontinuous control technique with less costs and more efficiency has come into the limelight. Particularly, impulsive control scheme has been successfully applied in many disciplines, including neural networks [27], and population-growth models [29]. Yet, in most of works, controllers are imposed on all the nodes to tame the node dynamics to arrive desired sometimes trajectory. Unfortunately, controlling all nodes is quite difficult and even inapplicable, especially when network is composed of 2

a large set of high dimensional nodes. Hinted by this practical consideration, pinning control, which means that only a small fraction of nodes is directly controlled, has been proposed [34,36] and then widely applied in complex networks [35–41]. Shortly after, to take advantage of both impulsive control and pinning control, pinning impulsive control scheme has been introduced. That is, the impulsive controllers are imposed on a small fraction of nodes, even only several nodes or a single node in whole network. Obviously, pinning impulsive control is a powerful technique because it reduces the control cost to a certain extent. Therefore, it is necessary to study the synchronization problem under pinning impulsive control strategy. Lu et al. [44] used pinning impulsive control technique to synchronize stochastic dynamical networks with nonlinear coupling. Further Lu et al. [45] studied the synchronization control of impulsive dynamical networks under a single impulsive controller and/or a single negative state-feedback control. In [46], hybrid pinning controller was used in synchronization of stochastic delayed complex network. On the other hand, it is noticed that most of the studies on network dynamical behaviors have been performed under certain implicit assumptions that full states of the nodes can be transmitted via the connections. However, in many circumstances, this simplification does not match satisfactorily the peculiarities of real networks. It means that there exist communication constraints between connected nodes. In fact, the connections among any pairs of nodes have multiple channels to deliver the corresponding states. In many real cases, only part of the channels of the connections can work normally and moreover the valid channels for distinct connections can be different [49,50]. Hence, it is necessary and desirable to investigate partially coupled network. In [49], a class of stochastic dynamical networks with partial states transmitted via the connections has been firstly established by introducing the concept of channel matrices, and some efficient partial synchronization criteria have been derived. However, to the best of our knowledge, the outer synchronization for driveresponse partially coupled networks has not been explicitly considered and studied in the literatures. Motivated by the above discussions, in this paper, we intend to consider outer synchronization of two partially coupled networks under pinning impulsive control. We incorporate the following three factors into our study simultaneously: (i) the information transmission in networks have communication constraints; (ii) the coupling matrices can be asymmetric and weighted; and, (iii) a fraction of nodes should be controlled at discrete time instants. By designing pinning impulsive control methods, our goal of this paper is that the response network can be fully synchronized to the drive network. The remainder of this paper is arranged as follows: In Section 2, we formulate 3

the problem of outer synchronization for two partially coupled dynamical networks and present some necessary preliminaries. In Section 3, several efficient outer synchronization criteria are established for partially coupled dynamical networks under pinning impulsive control. In Section 4, numerical examples are given to illustrate our theoretical results. Finally, Section 5 presents the conclusion. Notations : The standard notations will be used throughout this paper. The notation X > (≥, <, ≤) 0 is used to denote a real symmetric positive-definite (respectively, positive-semidefinite, negative, and negative-semidefinite) matrix. In represents the identity matrix with dimension n. λmin (·) and λmax (·) represent the minimum and maximum eigenvalue of the corresponding matrix, respectively. N = {1, 2, 3, . . .}. Rn denotes the n dimensional Euclidean space. Rn×n denotes the n × n real matrices. The notation “T ” denotes the transpose of a matrix or a vector. x indicates the 2-norm of a vector x,  1 i.e., x = ( ni=1 x2i ) 2 . D denotes the number of elements of a finite set D. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2

Preliminaries

In this section, we consider the following partially coupled dynamical network of N identical nodes:

x˙ i (t) = Axi (t) + Bf (xi (t)) + c

N 

gij DRij (xj (t) − xi (t)),

(1)

j=1,j=i

where xi (t) = (xi1 (t), . . . , xin (t)) ∈ Rn represents the state of the ith node; A = diag{a1 , a2 , . . . , an } ∈ Rn×n , B ∈ Rn×n ; f : Rn → Rn is a nonlinear vector function, which satisfies f (0) = 0; c > 0 is the coupling strength; D = diag{d1 , d2 , . . . , dn } denotes the inner coupling matrix between two connected nodes; the network topology is represented by the outer coupling matrix G = (gij ) ∈ RN ×N , e.g., if there is a connection from node j to node i(j = i), then gij > 0; otherwise, gij = 0; and Rij = diag{rij1 , rij2 , . . . , rijn } is channel matrix, in which rijs , s = 1, 2, . . . , n is defined as follows: if the sth channel of the connection from node j to node i is active, then rijs = 1; otherwise, rijs = 0. Based on the configuration of drive-response system, the network (1) is regarded as the drive network. Then, the corresponding response network is established as follows: 4

y˙ i (t) = Ayi (t) + Bf (yi (t)) N 

+c

j=1,j=i

gij DRij (yj (t) − yi (t)) + ui (t),

(2)

where yi (t) = (yi1 (t), . . . , yin (t)) ∈ Rn is the response state vector of the ith node; ui (t) are controllers to be designed. Definition 1 Network (1) is said to achieve outer synchronization with network (2) if lim yi (t) − xi (t) = 0, f or i = 1, 2, . . . , N. t→∞



Let Cij = gij Rij  diag{c1ij , . . . , cnij }(i = j) and Cii = − N j=1,j=i Cij . Thus, the drive-response partially coupled networks can be rewritten as the following form: ⎧ ⎪ ⎪ ⎪ ⎪ x˙ i (t) ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y˙ i (t)

= Axi (t) + Bf (xi (t)) + c = Ayi (t) + Bf (yi (t)) + c

N  j=1 N 

j=1

DCij (xj (t) − xi (t)) (3)

DCij (yj (t) − yi (t)) + ui (t)

Due to the introduction of channel matrices Rij , the synchronization analysis of the drive-response partially coupled dynamical networks (3) become much harder. Hence, we employ the regrouping method to deal with the difficulties raised from the communication constraint. We collect the sth diagonal element Cijs from Cij (i, j = 1, 2, . . . , N) and regroup them in a new matrix Cs = (Cijs )N ×N . That is, the regrouping matrices Cs reflect all of the communicated information at the sth level. Moreover, it is easy to see that Cs has the zero-row-sum property. Remark 1 Each Cs collects the communicated information with the same level. Hence, Cs can be regarded as the network topology at the sth level. Example 1 will give a specific illustration. Assumption 1 The nonlinear function f (·) is assumed to satisfy the global Lipschitz condition, i.e., there exists a positive constant L such that f (u) − f (v) ≤ Lu − v holds for any u, v ∈ Rn . Assumption 2 Matrices Cs (s = 1, 2, . . . , n) are assumed to be irreducible. Remark 2 In this paper, we study the directed network without assuming the communication topology to be symmetric. Here Assumption 2 means that we are demanding each level sub-network to be strongly connected, and it implies that the whole drive-response partially coupled networks are strongly connected. 5

In the sequel, some necessary lemmas and definitions are introduced. N ×N Lemma 1 [48] Suppose A = (aij )N . If i,j=1 ∈ R

(1) aij ≥ 0, i = j, aii = − (2) A is irreducible,

N  j=1,j=i

aij , i = 1, . . . , N;

then the following items are valid: (a) rank(A) = N − 1, i.e., zero is an eigenvalue of A with multiplicity one, and all nonzero eigenvalues of A have negaive real parts; (b) Suppose ξ = (ξ1 , ξ2 , . . . , ξN ) ∈ RN (without loss of generality, assume N 

i=1

ξi = 1) is the left eigenvector of A corresponding to eigenvalue 0.

Then, ξi > 0 holds for all i = 1, . . . , N; (c) Let Ξ = diag{ξ1 , . . . , ξN }. Then ΞA + A Ξ is a symmetric matrix with all eigenvalues are real and satisfy 0 = λ1 > λ2 ≥ . . . ≥ λN . Definition 2 ([42] Average Impulsive Interval). The average impulsive interval of the impulsive sequence ζ = {t1 , t2 , . . .} is equal to Ta if there exist positive integer N0 and positive number Ta , such that T −t T −t − N0 ≤ Nζ (T, t) ≤ + N0 , Ta Ta where Nζ (T, t) denotes the number of impulsive times of the impulsive sequence ζ on the interval (t, T ). We define the error state as ei (t) = yi (t) − xi (t), then the error dynamical system can be derived from (3) that:

e˙ i (t) = Aei (t) + B f˜(ei (t)) + c

N  j=1

DCij ej (t) + ui (t),

(4)

where f˜(ei (t)) = f (yi (t)) − f (xi (t)). Therefore, the drive-response partially coupled dynamical networks (3) can realize outer synchronization if and only if ei (t) → 0 as t → ∞ for i = 1, 2, . . . , N. In order to force the drive-response partial coupled networks (3) into the outer synchronization, the following impulsive controllers are constructed for lk nodes: 6

⎧∞  ⎪ ⎨

ui (t) = ⎪k=1 ⎩0,

−δ(t − tk )qk ei (t), i ∈ Dk ,

Dk = lk ,

(5)

i∈ / Dk .

where δ(·) is the Dirac delta function, and qk ∈ (0, 1) is the impulsive control gain to be determined. tk is the impulsive instant sequence satisfying 0 = t0 < t1 < t2 < . . . tk < . . ., and limk→∞ tk = +∞. lk is the number of nodes to be controlled at impulsive instant tk . The index set Dk is defined as follows: at the impulsive instant tk , one can reorder the error states e1 (tk ), e2 (tk ), . . . , eN (tk ) such that ep1 (tk ) ≥ ep2 (tk ) ≥ . . . ≥ eplk (tk ) ≥ ep1k+1 (tk ) ≥ . . . epN (tk ), then Dk = {p1 , p2 , . . . , plk }, and Dk = lk . Define ei (t− k ) = ei (tk ). After adding the pinning impulsive controllers (5) to the nodes in Dk and using the properties of the Dirac delta function δ(·), the controlled error dynamical system can be rewritten as follows: ⎧ ⎪ ⎪ ⎨ e˙ i (t) ⎪ ⎪ ⎩ e

N  = Aei (t) + B f˜(ei (t)) + c DCij ej (t),

i (tk )

j=1

=

ei (t+ k)

−

ei (t− k)

=

−qk ei (t− k ),

t = tk ,

i ∈ Dk ,

(6)

Dk = lk .

Thus, the solution of (6) is piecewise left-hand continuous with discontinuities at t = tk for k ∈ N. In the following, we will focus on the stability of system (6). Remark 3 Many results about impulsive control strategy of diverse dynamical networks have been reported [43–47]. However, most of the results are restrictive to some extent. In [43–46], the number of nodes to be controlled l is fixed at different impulsive instants. In [43], supk∈Z{tk − tk−1 } has a upper bound. In [47], the location and the number of the pinned nodes need to be decided by the Frobenius normal form of the coupling matrix. Therefore, the control strategy in this paper is more general.

3

Main results

In this section, we will investigate the outer synchronization of drive-response partially coupled networks. Based on the aboved-mentioned assumptions and lemma, we can get the following theorem to guarantee that the networks (3) can realize outer synchronization by only impulsively controlling a small fraction of network nodes. Suppose that ξs = (ξs1 , ξs2, . . . , ξsN ) (s = 1, 2, . . . , n) is the unique normalized left eigenvector of the regrouping matrix Cs corresponding to eigen7

value 0 satisfying

N  i=1

ξi = 1. Denote Ξs = diag{ξs1 , ξs2, . . . , ξsN } and Ξi =

diag{ξ1i , ξ2i , . . . , ξni } (i = 1, 2, . . . , N). According to Assumption 2 and Lemma 1, we can conclude that Ξs > 0 for s = 1, . . . , n , and Ξi > 0 for i = 1, . . . , N. max {ξsi } and ξm = min {ξsi }. From Lemma 1, we can Let ξM = 1≤s≤n,1≤i≤N

1≤s≤n,1≤i≤N

also denote the eigenvalue of Ξs Cs + Cs Ξs by 0 = λs1 > λs2 ≥ . . . ≥ λsN and λN = min {λsN }. 1≤s≤n

lk qk (2−qk ) Theorem 1 Suppose that Assumptions 1 and 2 hold. Let ηk = 1− N   and λ = [2ξM (a + L λmax (B B)) − cdλN ]/ξm , where a = max {|as |} and d = 1≤s≤n

max {|ds |}. If there exists a constant γ > 1, such that γ(ξM /ξm )ηk eλ(tk+1 −tk ) ≤

1≤s≤n

1 for any k ∈ N, then the drive-response partially coupled dynamical networks (3) can achieve outer synchronization.

Proof. Consider the following Lyapunov function: N 

V (t) =

i=1

e i (t)Ξi ei (t).

For t ∈ (tk , tk+1 ), by calculating the derivative of V (t) along the trajectory of system (6), we can obtain N  V˙ (t) = 2 e i (t)Ξi e˙ i (t)

=2 =2

i=1 N  i=1 N  i=1

+2c

N ˜ e i (t)Ξi [Aei (t) + B f (ei (t)) + c j=1 DCij ej (t)]

e i (t)Ξi Aei (t) + 2 N  N  e (t)Ξ

i=1 j=1

i

N

 ˜ i=1 ei (t)Ξi B f (ei (t))

(7)

i DCij ej (t).

By Assumption 1, the following inequalities can be obtained: 2 ≤2 ≤2

N  e (t)Ξ

i=1 N 

i=1 N  i=1

i

i Aei (t)

e i (t)Ξi Aei (t) ξM ei (t) · aei (t)

= 2aξM

N  e (t)e

i=1

i

8

i (t),

(8)

and 2 ≤2

N  e (t)Ξ

i=1 N 

≤2 ≤2

i=1 N  i=1 N  i=1

i

˜

i B f (ei (t))

˜ e i (t)Ξi B f (ei (t)) 

ξM ei (t) λmax (B  B)f˜(ei (t))

(9)



ξM ei (t) λmax (B  B)Lei (t) 

= 2ξM L λmax (B  B)

N  e (t)e

i=1

i

i (t).

Let es (t) = (e1s (t), e2s (t), . . . , eN s (t)) ∈ RN . Hence, by Assumption 2 and Lemma 1, one has 2c = 2c = = = ≤ ≤ =

N  N  e (t)Ξ

i i=1 j=1 N  N  n 

i DCij ej (t)

eis (t)ξsi ds Cijs ejs (t)

i=1 j=1 s=1 n N  N   ds eis (t)ξsi Cijs ejs (t) 2c s=1 i=1 j=1 n  ds e 2c s (t)Ξs Cs es (t) s=1 n   ds e c s (t)(Ξs Cs + Cs Ξs )es (t) s=1 n   e −cd s (t)(Ξs Cs + Cs Ξs )es (t) s=1 n  e −cdλN s (t)es (t) s=1 N  e −cdλN i (t)ei (t). i=1

(10)

Recalling Eq.(7), it follows from inequalities (8)-(10) that  N  V˙ (t) ≤ [2ξM (a + L λmax (B  B)) − cdλN ] e i (t)ei (t)

≤

1 [2ξM (a ξm

i=1



+ L λmax (B  B)) − cdλN ]

N  e (t)Ξ

i=1

i

i ei (t)

= λV (t). Then, it follows that, for t ∈ (tk , tk+1 ), λ(t−tk ) V (t) ≤ V (t+ . k )e

9

(11)

Consider that t = tk+1 , by the continuity of V (t) in (tk , tk+1), we have λ(t−tk ) V (tk+1 ) = limt→t− V (t) ≤ limt→t− V (t+ k )e k+1

=V

k+1

λ(tk+1 −tk ) (t+ . k )e

(12)

On the other hand, for any k ∈ N, let αk = min{ei (tk ) : i ∈ Dk }. Since qk ∈ (0, 1), we get 0 < ηk < 1 and (1 − ηk )(N − lk ) = [ηk − (1 − qk )2 ]lk . According to the selection of nodes in set Dk , we have 

(1 − ηk )

i∈D / k

e i (tk )ei (tk )

≤ (1 − ηk )(N − lk )αk2 = [ηk − (1 − qk )2 ]lk αk2 ≤ [ηk − (1 − qk )2 ]

 i∈Dk

e i (tk )ei (tk ),

which further implies that (1 − qk )2 ≤ ηk

 i∈Dk

e i (tk )ei (tk ) +

N  e (t

i∈D / k

e i (tk )ei (tk )

k )ei (tk ).

i

i=1



Therefore, for any k ∈ N, it follows from the second equality of (6) that V (t+ k) = = =

N  e (t+ )Ξ

i=1



i∈Dk



i∈Dk

≤ ξM (

i

k

+ i ei (tk )

+ + e i (tk )Ξi ei (tk ) +

 i∈D / k

+ + e i (tk )Ξi ei (tk )

(1 − qk )2 e i (tk )Ξi ei (tk ) +  i∈Dk

≤ ξM ηk

 i∈D / k

(1 − qk )2 e i (tk )ei (tk ) +

N  e (t

i=1

i

≤ (ξM /ξm )ηk

k )ei (tk )

N  e (t

i=1

i

k )Ξi ei (tk )

= (ξM /ξm )ηk V (tk ).

10

e i (tk )Ξi ei (tk )



i∈D / k

e i (tk )ei (tk ))

(13)

Summarizing inequalities (11)-(13), for t ∈ (tk , tk+1 ], we have λ(t−tk ) V (t) ≤ V (t+ k )e

≤ V (tk )(ξM /ξm )ηk eλ(t−tk ) λ(tk −tk−1 ) ≤ V (t+ (ξM /ξm )ηk eλ(t−tk ) k−1 )e

≤ V (tk−1 )(ξM /ξm )ηk−1 eλ(tk −tk−1 ) (ξM /ξm )ηk eλ(t−tk ) .. .

(14)

≤ V (t1 )(ξM /ξm )η1 eλ(t2 −t1 ) . . . (ξM /ξm )ηk eλ(t−tk ) ≤ V (t1 ) ≤

1 V γk

k i=1

(ξM /ξm )ηi eλ(ti+1 −ti )

(t1 ).

Since γ > 1, we have V (t) → 0 as k → ∞. Thus, for i = 1, . . . , N, limt→∞ ei (t) = 0, i.e., the drive-response partially coupled networks (3) reach outer synchronization as t → ∞. Hence, Theorem 1 is proved. Remark 4 It is worth noting that Theorem 1 explicitly present how many nodes should be controlled for a successful synchronization control of the networks (3): 1 ξm −λ(tk+1 −tk ) lk ≥ (1 − e ), (15) N qk (2 − qk ) γξM where lk /N represents the proportion of the controlled nodes at each impulsive instant tk . From inequality (15), we conclude that the proportion lk /N should be greater than certain lower bound at tk in order to achieve outer synchronization of (3). In practice, for convenience, the impulsive control gain qk and the number of nodes to be controlled lk can be selected as constants and the impulsive distances tk+1 − tk (k ∈ N) are set to be a positive constant. Then we have the following corollary. Corollary 1 Assume tk+1 − tk = T > 0, qk = q ∈ (0, 1) and lk = l(k = 1, 2, . . .). Then the drive-response partially coupled network (3) can achieve outer synchronization if one of the following inequalities is satisfied: 1) 0 < T ≤

In(ξm /ξM )−In(1− Nl q(2−q)) ; λ



2) 1 − 3)

l N

≥

1−

1−(ξm /ξM )e−λT l/N

1 (1 q(2−q)

−

≤ q < 1;

ξm −λT e ). ξM

11

Remark 5 The above three conditions, which similarly correspond to the results in the literature [44], show the proportional relationships among the proportion of the controlled nodes, the impulsive control gain and the distances of impulsive instants. The longer the impulsive distances T is, the larger the proportion of the controlled nodes l/N should be. Indeed, the longer the impulsive distances T is, the larger the impulsive control gain q is needed to guarantee the synchronization of drive-response partially coupled network (3). Our numerical example show a clearer relationship of all three. Similarly, we can obtain a corollary from Theorem 1 in the case that the regrouping matrices Cs are symmetric. By Lemma 1, the eigenvalue of matrix Cs can be arranged as follows: 0 = λs1 > λs2 ≥ . . . ≥ λsN . Let λN = min {λsN }. 1≤s≤n

Corollary 2 Suppose that Assumptions 1 and 2 hold, and the regrouping lk qk (2 − qk ) and matrices Cs (s = 1, 2, . . . , n) are symmetric. Let ηk = 1 − N  2  λmax = λmax (2A + BB + L I − cdλN I). If there exists a constant γ > 1, such that γηk eλmax (tk+1 −tk ) ≤ 1 for any k ∈ N, then the drive-response partially coupled networks (3) can achieve outer synchronization. 

 Proof. Consider the Lyapunov function V (t) = N i=1 ei (t)ei (t). The detailed proof is similar to the proof of Theorem 1, and hence omitted here.

Remark 6 The criterion in Theorem 1 is equivalent to γ(ξM /ξm )ηk eλsupk∈N {tk+1 −tk } ≤ 1. Hence, the results in Theorem 1 and Corollary 2 may be invalid when supk∈N {tk+1− tk } is very large. In the following, we apply average impulsive interval technique to analyze the drive-response partially coupled network (3) for a more flexible and less conservative pinning impulsive control law. Theorem 2 Suppose that Assumptions 1 and 2 hold, and the average impulsive interval of impulsive sequence ζ = {t1 , t2 , . . .} is equal to Ta . Let η = 1 − Nl q(2 − q) and λ = [2ξM (a + L λmax (B  B)) − cdλN ]/ξm . If λ+

ln(ηξM /ξm ) < 0, Ta

then the drive-response partially coupled networks (3) can achieve outer synchronization. Proof. Choose the same Lyapunov function as given in Theorem 1. By a 12

similar analysis as Theorem 1, we have V (t) ≤ V (t1 )(ηξM /ξm )k eλ(t−t1 ) ≤ V + (t0 )(ηξM /ξm )k eλ(t−t0 ) = V (t0 )(ηξM /ξm )Nζ (t,t0 )+1 eλ(t−t0 ) . If ηξM /ξm = 1, we have V (t) ≤ V (t0 )eλ(t−t0 ) and ln(ηξM /ξm ) = 0. And, λ + ln(ηξTMa/ξm ) < 0 implies that λ < 0, hence conclusion hold. According to Definition 2, we can obtain that if ηξM /ξm < 1, then V (t) ≤ V (t0 )eλ(t−t0 ) (ηξM /ξm )

t−t0 −N0 +1 Ta

= (ηξM /ξm )1−N0 V (t0 )eλ(t−t0 ) (ηξM /ξm ) = (ηξM /ξm )1−N0 V (t0 )e(λ+

t−t0 Ta

ln(ηξM /ξm ) )(t−t0 ) Ta

.

If ηξM /ξm > 1, we have V (t) ≤ V (t0 )eλ(t−t0 ) (ηξM /ξm )

t−t0 +N0 +1 Ta

= (ηξM /ξm )1+N0 V (t0 )eλ(t−t0 ) (ηξM /ξm ) = (ηξM /ξm )1+N0 V (t0 )e(λ+

t−t0 Ta

ln(ηξM /ξm ) )(t−t0 ) Ta

.

Since λ + ln(ηξTMa/ξm ) < 0, outer synchronization of drive-response partially coupled networks (3) is achieved. The proof is completed. Remark 7 Compared with [44], the model in this paper is more general. When all the channel matrices Rij are identity matrices, all of the channels of the connections are active. Therefore, our results are also applicable to two linearly complete coupled dynamical networks. In other words, the synchronization criterion proposed in this paper improves and extends the previous results when reducing to the outer synchronization of two linearly complete coupled networks.

4

EXAMPLES

In this section, we present two numerical examples to illustrate our main theoretical results. 13

Example 1 In the first example, we consider a drive-response partially coupled network with six nodes and each node is a three dimensional system. The topology can be described in Fig.1. Obviously, each level sub-network is strongly connected. The parameters of the network are listed as follows: ⎡

A = −0.1 · I3 , B =

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

0.08 0

0 −0.1⎥ 0.02

⎥

⎥ , c = 1, D = 0.02 · I3 , 0 ⎥ ⎥

−0.05 0 −0.1

⎡

G=

⎢−11.3 ⎢ ⎢ 4.2 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎣

7

⎦

⎤

7

0

0.2

0

−11.2

7

0

0

7

0

⎥ 0 ⎥ ⎥

7

0 ⎥ ⎥

4.1 −11.1 0.1

4.1 −11.2

0

0

0

0

4.1 ⎥ ⎥

0 ⎥ ⎥ ⎥ ⎥, ⎥ ⎥

3.9 −10.9 7 ⎥ ⎥ 0

4

−11

⎦

and choose the channel matrices:

R12 = diag{0, 0, 1}, R14 = diag{1, 0, 0}, R16 = diag{1, 1, 0}, R21 = diag{1, 1, 0}, R23 = diag{0, 0, 1}, R32 = diag{1, 1, 0}, R34 = diag{0, 0, 1}, R42 = diag{0, 1, 0}, R43 = diag{1, 1, 0}, R45 = diag{0, 0, 1}, R54 = diag{1, 1, 0}, R56 = diag{0, 0, 1}, R61 = diag{0, 0, 1}, R65 = diag{1, 1, 0}, and nonlinear function f (xi (t)) = (tanh(xi1 ), tanh(xi2 ), tanh(xi3 )) . Thus, the Lipschitz constant can be obtained as L = 1. The initial values of these systems are chosen uniformly randomly in the real number interval [-100,100]. The drive and response systems cannot achieve synchronization by itself as shown in Fig.5.

Applying the regrouping method, we first let Cij = gij Rij , then collect the sth diagonal elements Cijs of Cij (i, j = 1, 2, . . . , N) and regroup them in a new matrix Cs = (Cijs )N ×N : 14

⎡

−4.3 ⎢

C1 =

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

4.2 −4.2

0

0

0

0

0

0

0

0

0

4.1 −4.1

0

0

0

0

0

0

0

−4.1 ⎢

0

3.9 −3.9 0 0

−4

4

0

4.2 −4.2 0

0

0

0

0

0

0

0

0

4.1 4.1

0

0.1 4.1 −4.2

4.1 ⎥ ⎥ 0 ⎥ ⎥

3.9 −3.9 0

0

0

0

0

0

0

0

4

0

0

0

0 −7 7

0

0

0 ⎥ ⎥ 0 ⎥ ⎥

0

0 −7 7

0

0

0

0

0 −7 7

0

0

0

0

0 −7 7

7

0

0

0

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

0

−7 7 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

4.1 ⎥ ⎥ 0 ⎥ ⎥

0

⎡

C3 =

0.2

0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

0

4.1 −4.1

0

⎡

C2 =

0

−4

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

0 −7

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

By some simple calculations, one obtains that ξm = 0.1614 and ξM = 0.1714. It follows that λ = 1.0971.

Now, for simplicity, we consider the equidistant impulsive interval. Let the impulsive strength q = 0.9 and the impulsive interval T = 0.1, it is clear that m −λT (1 − ξξM e )/q(2 − q) = 0.1582. Therefore, by Corollary 1, one can conclude that a single controller can pin this drive-response partially coupled network to outer synchronization. 15

Define the total synchronization error as follows: E(t) =

  6 3    (x

ij (t)

i=1 j=1

− yij (t))2 .

Fig.3 shows how the total synchronization error E(t) changes over time under different impulsive controllers, in which the number of nodes to be controlled, impulsive interval, and impulsive control gains are varied. We can find that the number of nodes to be controlled, impulsive interval, and impulsive control gains together determine a controller’s ability to achieve synchronization.

Fig. 1. Topology of the partially coupled drive network with six nodes: the solid (dash) arrows represent the active (respectively, inactive) channels.

100 90

xi1(t),xi2(t),xi3(t)(i=1,2,...,6)

80 70 60 50 40 30 20 10 0

0

5

10 t

15

20

Fig. 2. Dynamical behaviors of the drive system without controllers.

16

300

300 l=1 l=2 l=3

100

0

E(t)

100

0

0.5

1

1.5

q=0.5 q=0.7 q=0.9

200

E(t)

200

E(t)

200

300 T=0.05 T=0.1 T=0.15

0

100

0

0.5

t (a)

1 t (b)

1.5

0

0

0.5

1

1.5

t (c)

Fig. 3. Dynamical behaviors of the synchronization error E(t) under different impulsive controllers. (a) q=0.9, T =0.05, (b) q=0.9, l=1, (c) T =0.05, l=1.

Example 2 Now let us consider a NW directed small-world network[51]. The small-world network is generated by setting N = 100, k = 4, p = 0.1. Other parameters are the same with example 1. To guarantee each level sub-network to be strongly connected, we choose Rij = I except for the edges which are added randomly. In this simulation, one obtains that ξm = 0.0096 and ξM = 0.0103. It follows that λ = 0.7927. Corollary 1 have presented explicitly the proportional relationships among the proportion of the controlled nodes, the impulsive control gain and the impulsive distances. The synchronization region of the controlled partially coupled network for l, T and q are shown in Fig.4.

Fig.5 shows the estimation of the synchronization region about T and l/N with different q. And Fig.6 shows the estimation of the synchronization region about q and T with different l/N. We can see, as mentioned in Remark 5, the longer the impulsive distances T is, the larger the proportion of the controlled nodes l/N should be; the longer the impulsive distances T is, the larger the impulsive control gain q is needed to guarantee the synchronization of drive-response partially coupled network.

m −λT Next, we choose q = 0.9 and T = 0.02, then (1 − ξξM e )/q(2 − q) = 0.0862. Therefore, by Corollary 1, it can be concluded that drive-response partially coupled dynamical networks can be outer synchronized if l/N = 9% of the nodes are controlled. Hence, we choose l = 10. Fig.7 shows the error trajectories of the small-world drive-response partially coupled dynamical network. The simulation results verified our main theoretical results very well.

17

Fig. 4. The synchronization region of the controlled partially coupled network for l, T and q.

1 q=0.2 q=0.4 q=0.6 q=0.8

0.9 0.8 0.7

l/N

0.6 0.5 0.4

Synchronization region

0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 T

1.2

1.4

1.6

1.8

2

Fig. 5. The synchronization region of the controlled partially coupled networks for different q with respect to T and l/N .

18

0.2 l/N=0.1 l/N=0.2 l/N=0.3 l/N=0.4 l/N=0.5

0.18 0.16 0.14 0.12 T

0.1 0.08 0.06 0.04

Synchronization region

0.02 0

0

0.1

0.2

0.3

0.4

0.5 q

0.6

0.7

0.8

0.9

1

Fig. 6. The synchronization region of the controlled partially coupled network for different l/N with respect to q and T . 10 8 6

ei(t), i=1,2,...,100

4 2 0 −2 −4 −6 −8 −10

0

0.2

0.4

0.6

0.8

1

t

Fig. 7. Dynamical behaviors of the synchronization error under pinning on just ten nodes.

5

CONCLUSION

In this paper, outer synchronization of drive-response partially coupled networks with same connection topologies is theoretically and numerically studied. Our approach is based on a regrouping process. The main result in Theorem 1 proposes a pinning impulsive control scheme which is used to guarantee outer synchronization of drive-response partially coupled networks, while Theorem 2 proposes a more flexible impulsive control law by using the concept of average impulsive interval. Finally, two partial coupled small-world networks are given to illustrate the efficiency of the proposed approaches, and moreover the synchronization region is clearly plotted. One of the restrictions in this paper is that ξM /ξm is not allowed to be too large, otherwise the conditions in 19

Theorems are not easy to satisfy. Thus, in our near future work, we will study the outer synchronization problem in partially coupled dynamical networks by avoiding using the quantity of ξM /ξm .

Acknowledgments This work was partially supported by the NNSF of China under Grants 61175119 and 61272530, China Postdoctoral Science Foundation under Grants 2014M560377 and 2015T80483, and the NSF of Zhejiang Province of China under Grant LY15F020018.

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