Pinning outer synchronization of partially coupled dynamical networks with complex inner coupling matrices

Physica A 515 (2019) 497–509

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Physica A journal homepage: www.elsevier.com/locate/physa

Pinning outer synchronization of partially coupled dynamical networks with complex inner coupling matrices ∗

Xuechen Li a , Nan Wang a , Jianquan Lu b , , Fuad E. Alsaadi c a

School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China School of Mathematics, Southeast University, Nanjing 210096, China c Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia b

highlights • We investigate outer synchronization of partially coupled dynamical networks. • The inner coupling matrices between coupled nodes can be non-full rank and also asymmetric. • Our synchronization criteria can be theoretically proved to be less conservative.

article

info

Article history: Received 29 April 2018 Received in revised form 16 August 2018 Available online xxxx Keywords: Complex dynamical networks Non-full rank and asymmetric inner coupling Outer synchronization Pinning impulsive control Average impulsive interval

a b s t r a c t This paper investigates outer synchronization of partially coupled dynamical networks with complex inner coupling matrices via pinning impulsive controller. At first, we establish more realistic drive–response partially coupled networks, where the inner coupling matrices between coupled nodes can be non-full rank and also asymmetric. Then, a weighted-norm-based method is given to select the nodes that should be pinning controlled, and our synchronization criteria derived by this method can be proved to be less conservative. By using the regrouping method and our new average impulsive interval method, some efficient and less conservative synchronization criteria are derived. Our results show that the outer synchronization can be achieved by impulsively controlling a crucial fraction of nodes in the response network. Finally, numerical examples are exploited to illustrate the effectiveness of our theoretical results. © 2018 Elsevier B.V. All rights reserved.

1. Introduction In recent years, complex networks have attracted increasing attentions due to their wide applications in machine learning, information transmission, optimization, and so on. Complex dynamical networks consist of a large set of coupled nodes, in which each node represents an individual element and the edges represent the relations between the nodes [1–3]. Many research on complex dynamical networks have been addressed, such as synchronization, stability, and Hopf bifurcation analysis. Synchronization, as a typical collective behavior in networks, has received considerable attention in last two decades, since it is not only a common phenomenon occurring in many natural systems, but also has many applications in engineering. Synchronization means that all subsystems in coupled systems with different initial values give rise to a common behavior. Till now, many results about synchronization of coupled systems or complex dynamical networks have been addressed in various angles [4–16]. ∗ Corresponding author. E-mail address: [email protected] (J. Lu). https://doi.org/10.1016/j.physa.2018.09.095 0378-4371/© 2018 Elsevier B.V. All rights reserved.

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Most of the research on synchronization of networks aforementioned focused on the inner synchronization, which is related to the collective behavior among all nodes within single network. Actually, there exist many other kinds of network synchronization, such as projective synchronization [17], cluster synchronization [18]. Different from the inner synchronization, there exists another kind of synchronization, regardless of happening of the inner synchronization, the corresponding nodes of two or more coupled networks could realize synchronization. This synchronization has been regarded as ‘‘outer synchronization’’. Outer synchronization is also ubiquitous in our daily life, such as, the spread of infectious diseases. Outer synchronization problems have attracted significant attention [19–21] since they were firstly studied by Li et al. [22]. Tang et al. [19] investigated two complex networks obtained outer synchronization by designing an effective adaptive controller. Lei et al. [21] studied the generalized matrix projective outer synchronization of non-dissipatively coupled time-varying complex dynamical networks. Wu et al. [20] investigated the generalized outer synchronization between two completely different networks under a nonlinear control scheme. In complex dynamical networks, the initial states of the nodes are different. Thus nodes of the networks need to exchange their information with their neighbors to achieve synchronization. Note that most of the literature on network dynamical behaviors need the assumption that full state information of the nodes can be transmitted via the connections. Namely, there is no communication constraint. In fact, communication constraint exists widely due to environmental and physical limitation. As each node of the real network has multiple levels of information, the connections among any pairs of nodes should have multiple channels to transmit the corresponding state information. However, in many cases, only part of the channels can successfully transmit signals normally [23,24]. For example, in wireless networks, not all channel state information of users are well known to the transmitter. Hence, it is necessary and desirable to model and study partially coupled networks. To overcome the difficulties from communication constraint, regrouping method has been developed [23,24]. In [23], a class of stochastic dynamical networks with partial states, transmissions has been firstly established by introducing the concept of channel matrices, and some efficient synchronization criteria have been derived. In [25], Lu et al. investigated the outer synchronization of drive–response partially coupled networks. In [24,25], the channel matrix Rij = diag{rij1 , rij2 , . . . , rijn } was defined as follows: if the sth channel of the connection from node j to node i is active, then rijs = 1; otherwise, rijs = 0. However, under many circumstances, this simplification is not in accord with the peculiarities of real networks. It means that there may exist communication from the lth component of node j to the kth component of node i. Thus, it is necessary to model this kind of more general networks and study the outer synchronization of the drive– response partially coupled networks. In this paper, we investigate outer synchronization of partially coupled dynamical networks with more complex inner coupling matrices, where the matrices can be non-diagonal and also asymmetric. So our network models will be more general, extensive and consistent with the real world than that of [25]. In order to achieve synchronization while the network itself cannot realize synchronization, many kinds of techniques have been developed in recent years. An effective method is adding controllers to the nodes in the response network, such as impulsive control. The stability of impulsive systems has been extensively investigated [26,27]. Impulsive control is an effective method for systems that cannot endure continuous control inputs. Due to its good properties, such as simplicity and flexibility, impulsive control scheme has been successfully applied in many disciplines, including neural networks, and population-growth models. When the network own a large set of high dimensional nodes, controlling all nodes becomes quite difficult to implement. Motivated by this practical consideration, only a small fraction of nodes is directly controlled has been proposed [28], which is called pinning control scheme [29–31]. This control scheme has been widely applied in complex networks [32,28,33–35]. Combining impulsive control and pinning control together, pinning impulsive control scheme has been introduced [36–40]. 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. presented an analytical study of outer synchronization of partially coupled dynamical networks via pinning impulsive controller [25]. In [25], pinning impulsive controllers are chosen to be acted according to the norm of the error states, while in this paper, a weighted-norm-based method is given to select the nodes that should be pinning controlled. It can be proved that our weighted-norm-based method can make our synchronization criteria less conservative. Using lower bound or upper bound of the impulsive intervals to characterize the frequency of impulses would lead to very conservative results. Hence, Lu et al. [41] has sought out a description about impulses’ occurrence with the novel concept named average impulsive interval. Motivated by [41], furthermore, we introduced a new average impulsive interval in the form of limit [42], which can be used to characterize much wider range of impulsive sequences. In this paper, we will use our novel method named average impulsive interval to study outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. Compared with existing results about synchronization of networks, the main contributions of this paper are listed as follows: (1) the channel matrix can be non-full rank, non-diagonal and asymmetric, and this makes our model more general. (2) a weighted-norm-based method is given to select the nodes that should be pinning controlled, and based on this method our results can be theoretically proved to be less conservative. (3) the impulsive sequence is characterized by our new concept of average impulsive interval, and this concept can be used to characterize much wider range of impulsive sequences. The remainder of this paper is arranged as follows: In Section 2, we propose the problem of outer synchronization for two partially coupled dynamical networks and give some necessary preliminaries. In Section 3, several efficient outer synchronization criteria are established for partially coupled dynamical networks. 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. The

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arithmetic roots of semi-positive matrix X , denoted as X 1/2 , X −1/2 , such that X 1/2 X 1/2 = X , especially when X is a positivedefinite matrix, one has X −1/2 X −1/2 = X −1 . 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 ∑n 1 a vector. σ is the permutation. ∥x∥ indicates the 2-norm of a vector x, i.e., ∥x∥ = ( i=1 x2i ) 2 . |x| = (|x1 |, |x2 |, . . . , |xn |)⊤ . ♯D denotes the number of elements of a finite set D, and ⊗ denotes the Kronecker-product. Matrices, if not explicitly stated, are assumed to have compatible dimensions. Any symmetric matrix can be decomposed into difference of two positive semi-definite matrix. For example, a symmetric matrix W can be composed into W = W+ − W− . 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 Rij (xj (t) − xi (t)),

(1)

j=1,j̸ =i

where xi (t) = (xi1 (t), . . . , xin (t))⊤ ∈ Rn is the state vector of the ith node at time t; 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; the outer coupling matrix G = (gij ) ∈ RN ×N is defined as follows: if there is a connection from node j to node i(j ̸ = i), then gij > 0; otherwise, gij = 0; and Rij = (rijkl ) ∈ Rn×n is channel matrix, in which rijkl , k, l = 1, 2, . . . , n is defined as follows: if the klth channel of the connection from the lth component of node j to the kth component of node i is active, then rijkl > 0; otherwise, rijkl = 0. Let R = (r kl ) represents the klth a completely communication channel matrix. Specifically, R is the connection rule of component of the node, which is defined by completely communication of the lth component of node j to the kth component of ( node i. For ) example, k ∈ σ (12 · · · n), l ∈ σ (12 · · · n),(then R =)In . If k = σ (12...n), l = σ (12...n), for n = 3, then R = 1 0 0 0 0 1 2l 1l nl 0 1 0 . If k = σ (12...n), l = σ (23...n1), then R = 1 0 0 . Further, let Rij = diag(rij 1 , rij 2 , . . . , rij n )R. 0 0 1 0 1 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: y˙ i (t) = Ayi (t) + Bf (yi (t)) + c

N ∑

gij Rij (yj (t) − yi (t)) + ui (t),

(2)

j=1,j̸ =i

where yi (t) = (yi1 (t), . . . , yin (t))⊤ ∈ Rn is the response state vector of the ith node, and ui (t) are controllers to be designed. Remark 1. In [25], the channel matrix Rij = diag{rij1 , rij2 , . . . , rijn } was defined as follows: if the sth channel of the connection from node j to node i is active, then rijs = 1; otherwise, rijs = 0. However, in many circumstances, this simplification does not match the peculiarities of real networks. It means that there may exist communication from the lth component of node j to the kth component of node i. Thus, it is necessary to model this kind of more general networks and study synchronization of the drive–response partially coupled dynamical networks. Our network models are more general, extensive and consistent with the real world than that of [25]. Let Cij = gij Rij ≜ (cijkl ) ∈ Rn×n (j ̸ = i) and Cii = − networks can be presented in the following form:

∑N

j=1,j̸ =i

Cij , where cijkl = gij rijkl . Then, the drive–response partially coupled

⎧ N ⎪ ∑ ⎪ ⎪ ⎪ x˙ i (t) = Axi (t) + Bf (xi (t)) + c Cij (xj (t) − xi (t)), ⎪ ⎨ j=1

N ∑ ⎪ ⎪ ⎪ ⎪ ˙ y (t) = Ay (t) + Bf (y (t)) + c Cij (yj (t) − yi (t)) + ui (t). i i ⎪ ⎩ i

(3)

j=1

Due to the existence of channel matrices Rij , the synchronization analysis of the drive–response partially coupled dynamical networks (3) become more difficult. To deal with the difficulties raising from the partial communication , we use the regrouping method. We collect the klth element Cijkl from Cij (i, j = 1, 2, . . . , N) and regroup them in a new matrix C kl = (Cijkl )N ×N . That is, the regrouping matrices C kl denote all of the communicated information at the klth level. Hence, each C kl represents the communicated information with the same level. C kl can be regarded as the network topology at the klth level. Moreover, one can see that C kl has the zero-row-sum property. To achieve outer synchronization of partially coupled dynamical networks, we need the following definitions, assumptions and lemmas.

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Assumption 1. Let k ∈ σ (12...n), lk ∈ σ (l1 l2 ...ln ). Matrices C klk (k = 1, 2, . . . , n) are assumed to be irreducible, and the others C kl = 0. Assumption 2. For nonlinear function f (x(t)) = [f1 (x(t)), . . . , fn (x(t))]T , we assume that there exist constants liκ > 0 (i = n 1, 2, . . . , n) such that, for any x1 , x2 ∈ Rn , |fi (x1 ) − fi (x2 )| ≤ Σκ= 1 liκ |x1κ − x2κ |. l11 ⎜l21

l12 l22

.. .

··· ··· .. .

l1n l2n ⎟ .⎟ ⎠.

ln1

ln2

···

lnn

⎛

Denote L = ⎜ ⎝ .. .

⎞

..

Definition 1 (Average Impulsive Interval [42]). The average impulsive interval Ta of the impulsive sequence ζ = {t1 , t2 , . . .} is defined as follow: Ta = lim

t →+∞

t − t0 Nζ (t , t0 )

,

(4)

where Nζ (t , t0 ) denotes the number of impulsive times of the impulsive sequence ζ on the interval (t0 , t). Remark 2. This concept was firstly proposed in [42]. This concept has more intuitive feeling and is weaker than the concept of average impulsive interval in [25]. Based on this concept, we gave unified synchronization criteria when average impulsive interval Ta < ∞ and Ta = ∞. Further, this concept can be used to characterize much wider range of impulsive sequences. Lemma 1 ([43,44]). Suppose A = (aij )Ni,j=1 ∈ RN ×N . If (1) aij ≥ 0, i ̸ = j, aii = − (2) A is irreducible,

∑N

j=1,j̸ =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 negative real parts; ∑N (b) Suppose ξ = (ξ1 , ξ2 , . . . , ξN )⊤ ∈ RN is the left eigenvector of A corresponding to eigenvalue 0 satisfying i=1 ξi = 1. Then ξi > 0 holds for all i = 1, . . . , N; (c) Let Ξ = diag{ξ1 , . . . , ξN }. Then Ξ A + A⊤ Ξ is a symmetric matrix with all eigenvalues are real and satisfy 0 = λ1 > λ2 ≥ · · · ≥ λN . kl

kl

kl

Suppose that ξ klk = (ξ1 k , ξ2 k , . . . , ξN k )⊤ (k = 1, 2, . . . , n) is the left eigenvector of the regrouping matrix C klk corre∑N kl kl kl kl sponding to eigenvalue 0 satisfying i=1 ξi k = 1. Denote Ξ klk = diag{ξ1 k , ξ2 k , . . . , ξN k }, Ξ ∗ = diag {Ξ 1l1 , Ξ 2l2 , . . . , Ξ nln } 1l1 2l2 nl and Ξ i = diag{ξi , ξi , . . . , ξi n } . According to Assumption 1 and Lemma 1, we can conclude that Ξ klk > 0 for ⊤

k, l = 1, . . . , n , and Ξ i > 0 for i = 1, . . . , N. From Lemma 1, we can also denote the eigenvalue of Ξ klk C klk + C klk Ξ klk by kl kl kl 0 = λ1 k > λ2 k ≥ · · · ≥ λN k . Let ei (t) = yi (t) − xi (t), the error dynamical system can be obtained from (3) as follows: e˙ i (t) = Aei (t) + Bf˜ (ei (t)) + c

N ∑

Cij ej (t) + ui (t),

(5)

j=1

where f˜ (ei (t)) = f (yi (t)) − f (xi (t)). In order to ensure that the drive–response partially coupled dynamical networks (3) achieve the outer synchronization, the following impulsive controllers are constructed for sm nodes:

⎧ ∞ ∑ ⎪ ⎨− δ (t − tm )qm ei (t), i ∈ Dm , ui (t) = m=1 ⎪ ⎩ 0, i∈ / Dm .

♯Dm = sm ,

(6)

where δ (·) is the Dirac delta function, qm ∈ (0, 1) (1, 2) is the impulsive control gain, tm is the impulsive instant sequence satisfying 0 = t0 < t1 < t2 < · · · tm < · · · , and limm→∞ tm = +∞, and sm is the number of nodes to be controlled at impulsive instant tm . The index set Dm is defined as follows: at the impulsive instant tm , we reorder the ⊤ ⊤ error states e1 (tm ), e2 (tm ), . . . , eN (tm ) such that e⊤ p1 (tm )Ξ p1 ep1 (tm ) ≥ ep2 (tm )Ξ p2 ep2 (tm ) ≥ · · · ≥ eps (tm )Ξ psj epsj (tm ) ≥

⋃

e⊤ ps

j+1

(tm )Ξ psj+1 epsj+1 (tm ) ≥ · · · e⊤ ps (tm )Ξ psN epsN (tm ), then Dk = {p1 , p2 , . . . , psm }, and ♯Dm = sm . N

j

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Remark 3. In [25], Lu et al. chose pinning impulsive controller according to the norm of the error states. while in this paper, a weighted-norm-based method is given to select the nodes that should be pinning controlled. By using the weightednorm-based method, we can prove that the obtained synchronization criteria are less conservative and this point can be numerically illustrated. − Define ei (tm ) = ei (tm ). According to the properties of the Dirac delta function δ (·), by adding the pinning impulsive controllers (6) to the nodes in Dm , the controlled error dynamical networks can be rewritten in the following form:

⎧ N ⎪ ⎨e˙ (t) = Ae (t) + Bf˜ (e (t)) + c ∑ C e (t), i i i ij j ⎪ ⎩

t ̸ = tm ,

(7)

j=1

△ei (tm ) = ei (tm+ ) − ei (tm− ) = −qm ei (tm− ),

i ∈ Dm ,

♯Dm = sm .

Thus, the solution of (7) is piecewise left-hand continuous with discontinuities at t = tm for m ∈ N. Definition 2. The coupled dynamical networks (3) are said to be globally exponentially outer synchronized if there exist ϑ > 0, T > 0 and M0 > 0, such that for any initial values ei (0) (i = 1, 2, . . . , N),

∥ei (t)∥ ≤ M0 e−ϑ t hold for all t > T , and for any i = 1, 2, . . . , N.

3. Main results In this section, we will investigate the outer synchronization of drive–response partially coupled networks. Based on the above-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. ˜i = For any matrix W with proper dimension, we denote the symbol ‘‘∗’’ and ‘‘∼’’ by Wi∗ = (Ξ i )1/2 W (Ξ i )−1/2 , W ¯ ˜ i L∗ )| Wi∗ + (Wi∗ )⊤ . Let λ = λmax {(A˜ i )(i=1,2,...,N) } + 12 (λmax {(B˜ i )(i=1,2,...,N) } + max{|λ(L∗⊤ } ) + c λ ( M), with M = B max i i (i=1,2,...,N) ∑n klk C ⊗ R(k), where R(k) denotes the k-row-elements from the k row of the matrix R, the rests are zero. k=1 Theorem 1. Suppose that Assumptions 1 and 2 hold, and the average impulsive interval of impulsive sequence ζ = {t1 , t2 , . . .} is equal to Ta < ∞, then the drive–response partially coupled dynamical networks (3) can achieve globally exponentially outer synchronization if

ϑ=

lnη Ta

+ λ < 0,

where η = limm→+∞

(8)

|η1 |+|η2 |+···+|ηm | m

> 0, and ηm = 1 −

sm q (2 N m

− qm ).

Proof. Consider the following Lyapunov function: V (t) =

N ∑

e⊤ i (t)Ξ i ei (t).

i=1

For t ∈ (tm , tm+1 ), by calculating the derivative of V (t) along the trajectory of (7), we can obtain V˙ (t)

=2

N ∑

˙ i (t) e⊤ i (t)Ξ i e

i=1

=2

N ∑

˜ e⊤ i (t)Ξ i [Aei (t) + Bf (ei (t)) + c

∑

e⊤ i (t)Ξ i Aei (t) + 2

i=1

+2c

Cij ej (t)]

j=1

i=1 N

=2

N ∑

N N ∑ ∑ i=1 j=1

N ∑ i=1

e⊤ i (t)Ξ i Cij ej (t).

˜ e⊤ i (t)Ξ i Bf (ei (t))

(9)

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By Assumption 1, we can obtain that

2

N ∑

e⊤ i (t)Ξ i Aei (t)

i=1

=

N ∑

⊤ e⊤ i (t)(Ξ i A + A Ξ i )ei (t)

i=1

=

N ∑

1/2 (Ξ i )−1/2 (Ξ i A + A⊤ Ξ i )(Ξ i )−1/2 (Ξ i )1/2 ei (t) e⊤ i (t)(Ξ i )

(10)

i=1

=

N ∑

1/2 ˜ e⊤ Ai (Ξ i )1/2 ei (t) i (t)(Ξ i )

i=1

≤ λmax {(A˜ i )(i=1,2,...,N) }

N ∑

e⊤ i (t)Ξ i ei (t).

i=1

For a symmetric matrix W , by symmetric matrix decomposition theory [43], there exists orthogonal matrix U(W ), such that W = U(W )Λ(W )U ⊤ (W ), where Λ(W ) = diag {λ1 (W ), λ2 (W ), . . . , λn (W )}. Suppose that λ1 (W ) ≥ λ2 (W ) ≥ · · · ≥ λn (W ), let p(W ) denote the positive inertia exponent of matrix W , then W = W+ − W− , where W+ = U(W )(Λ(W ))+ U ⊤ (W ), W− = U(W )(Λ(W ))− U ⊤ (W ), with (Λ(W ))+ = diag {λ1 (W ), . . . , λp(W ) (W ), 0, . . . , 0}, (Λ(W ))− = −diag {0, . . . , 0, λp(W )+1 (W ), . . . , λn (W )}. Then, we have 2

N ∑

˜ e⊤ i (t)Ξ i Bf (ei (t))

i=1

=

N ∑

⊤ ˜ e⊤ i (t)(Ξ i B + B Ξ i )f (ei (t))

i=1

=

N ∑

⊤ ˜ e⊤ i (t)(Ξ i B + B Ξ i )+ f (ei (t)) −

N ∑

⊤ ˜ e⊤ i (t)(Ξ i B + B Ξ i )− f (ei (t))

i=1

i=1

=

N ∑

⊤ 1/2 e⊤ ((Ξ i B + B⊤ Ξ i )+ )1/2 f˜ (ei (t)) i (t)((Ξ i B + B Ξ i )+ )

i=1

−

N ∑

⊤ 1/2 e⊤ ((Ξ i B + B⊤ Ξ i )− )1/2 f˜ (ei (t)) i (t)((Ξ i B + B Ξ i )− )

i=1 N

∑ ⊤ 1/2 ((Ξ i B + B⊤ Ξ i )+ )1/2 f˜ (ei (t))| ≤| e⊤ i (t)((Ξ i B + B Ξ i )+ ) i=1

+|

N ∑

⊤ 1/2 e⊤ ((Ξ i B + B⊤ Ξ i )− )1/2 f˜ (ei (t))|. i (t)((Ξ i B + B Ξ i )− )

i=1 1/2 ˜ ∗ Let yi (t) = U ⊤ (B˜ i )(Ξ i )1/2 ei (t) and zi (t) = U ⊤ (L∗⊤ ei (t). i Bi Li )(Ξ i ) 1/2 1/2 ˜ ˜ ∗ Then one has (Ξ i ) ei (t) = U(Bi )yi (t), (Ξ i ) ei (t) = U(L∗⊤ i Bi Li )zi (t), it follows that N ∑ ⊤ 1/2 | e⊤ ((Ξ i B + B⊤ Ξ i )+ )1/2 f˜ (ei (t))| i (t)((Ξ i B + B Ξ i )+ ) i=1

≤ ≤

N 1∑

2

i=1

N 1∑

2

⊤ e⊤ i (t)((Ξ i B + B Ξ i )+ )ei (t) +

i=1

ei (t)((Ξ i B + B Ξ i )+ )ei (t) + ⊤

⊤

N 1∑

2

i=1

N 1∑

2

f˜ (ei (t))⊤ ((Ξ i B + B⊤ Ξ i )+ )f˜ (ei (t))

i=1

|ei (t)|⊤ L⊤ ((Ξ i B + B⊤ Ξ i )+ )L|ei (t)|

(11)

X. Li et al. / Physica A 515 (2019) 497–509

= =

N 1∑

2

1/2 ˜ (Bi )+ (Ξ i )1/2 ei (t) + e⊤ i (t)(Ξ i )

i=1 N

1∑ 2

2

i=1

˜

=

N 1∑

˜ y⊤ i (t)Λ(Bi )+ yi (t) +

N p(Bi ) 1 ∑∑

2

λm (B˜i )y2im +

i=1 m=1

2

∗ 1/2 ˜ |ei (t)| |ei (t)|⊤ (Ξ i )1/2 (L∗⊤ i (Bi )+ Li )(Ξ i )

i=1

˜ ∗ |zi (t)|⊤ Λ((L∗⊤ i Bi Li )+ )|zi (t)|

i=1 ∗⊤ ˜ ∗

N p(Li Bi Li ) 1∑ ∑

2

N 1∑

503

i=1

˜ ∗ 2 λm (L∗⊤ i Bi L )zim ,

m=1

1/2 where (B˜ i )+ = (Ξ i )1/2 B+ (Ξ i )−1/2 + (Ξ i )−1/2 B⊤ . + (Ξ i ) Similarly, one can get N ∑ ⊤ 1/2 ((Ξ i B + B⊤ Ξ i )− )1/2 f˜ (ei (t))| | e⊤ i (t)((Ξ i B + B Ξ i )− ) i=1

≤−

N 1∑

2

n ∑

λm (B˜i )y2im −

i=1 m=p(B˜ )+1 i

N 1∑

2

n ∑

(12)

˜ ∗ 2 λm (L∗⊤ i Bi L )zim .

i=1 m=p(L∗⊤ B˜ L∗ )+1 i i i

1/2 . where (B˜ i )− = (Ξ i )1/2 B− (Ξ i )−1/2 + (Ξ i )−1/2 B⊤ − (Ξ i ) Then

2

≤

N ∑

1

˜ e⊤ i (t)Ξ i Bf (ei (t))

i=1

(max{|λ(B˜ i )|, i = 1, 2, . . . , N }

2

(13)

˜ ∗ +max{|λ(L∗⊤ i Bi Li )|, i = 1, 2, . . . , N })

N ∑

e⊤ i (t)Ξ i ei (t).

i=1

Let em (t) = (e1m (t), e2m (t), . . . , eNm (t)) ∈ R (m = 1, 2, . . . , n), e(t) = (e1 (t), e2 (t), . . . , en (t))⊤ ∈ RnN . Hence, by ∑N ∑n ⊤ ⊤ klk Assumption 2 and Lemma 1, considering that i=1 e⊤ ek (t) = e (t)Ξ ∗ e(t) and the construction i (t)Ξ i ei (t) = k=1 ek (t)Ξ klk of the matrices Cij and C , one has ⊤

2c

N N ∑ ∑

N

⊤

⊤

⊤

e⊤ i (t)Ξ i Cij ej (t)

i=1 j=1

= 2c

N N n ∑ ∑ ∑

kl

kl

kl

kl

eik (t)ξi k Cij k ejlk (t)

i=1 j=1 k=1

= 2c

n N N ∑ ∑ ∑

eik (t)ξi k Cij k ejlk (t)

(14)

k=1 i=1 j=1 n

= 2c

∑

ek (t)Ξ klk C klk elk (t) ⊤

k=1

= ce⊤ (t)(Ξ ∗ M + M ⊤ Ξ ∗ )e(t) N ∑ ¯ ≤ c λmax (M) e⊤ i (t)Ξ i ei (t). i=1

Recalling (9), it follows from inequalities (10)–(14) that V˙ (t)

≤λ

N ∑

e⊤ i (t)Ξ i ei (t)

i=1

= λV (t). Then, for t ∈ (tm , tm+1 ), + λ(t −tm ) V (t) ≤ V (tm )e .

(15)

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When t = tm+1 , by the continuity of V (t) in (tm , tm+1 ), we get V (tm+1 )

= lim V (t) − t →tm +1

≤ lim V (tm+ )eλ(t −tm )

(16)

− t →tm +1

= V (tm+ )eλ(tm+1 −tm ) . On the other hand, for any m ∈ N, let αm = min{e⊤ (1, 2), we get 0 < ηm < 1 and i (tm )Ξ i ei (tm ) : i ∈ Dm }. Since qm ∈ (0, 1) (1 − ηm )(N − sm ) = [ηm − (1 − qm )2 ]sm . Considering the selection of nodes in set Dm , we obtain

⋃

(1 − ηm )

∑

e⊤ i (tm )Ξ i ei (tm )

i∈ / Dm

≤ (1 − ηm )(N − sm )αm2 2 = [ηm − (1 − qm )2 ]s∑ m αm 2 ≤ [ηm − (1 − qm ) ] e⊤ i (tm )Ξ i ei (tm ), i∈Dm

which further implies that

∑

(1 − qm )2

e⊤ i (tm )Ξ i ei (tm ) +

N ∑

e⊤ i (tm )Ξ i ei (tm )

i∈ / Dm

i∈Dm

≤ ηm

∑

e⊤ i (tm )Ξ i ei (tm ).

i=1

Therefore, for any m ∈ N, it follows from the second equality of (7) that + V (tm )

= =

N ∑ i=1 ∑

+ + e⊤ i (tm )Ξ i ei (tm ) + + e⊤ i (tm )Ξ i ei (tm ) +

∑

+ + e⊤ i (tm )Ξ i ei (tm )

i∈ / Dm

i∈Dm

=

∑

(1 − qm )2 e⊤ i (tm )Ξ i ei (tm ) +

e⊤ i (tm )Ξ i ei (tm )

(17)

i∈ / Dm

i∈Dm

≤ ηm

∑

N ∑

e⊤ i (tm )Ξ i ei (tm )

i=1

= ηm V (tm ). From inequalities (15)–(17), for t ∈ (tm , tm+1 ], we have V (t)

≤ V (tm+ )eλ(t −tm ) ≤ V (tm )ηm eλ(t −tm ) ≤ V (tm+−1 )eλ(tm −tm−1 ) ηm eλ(t −tm ) ≤ V (tm−1 )ηm−1 eλ(tm −tm−1 ) ηm eλ(t −tm ) ··· ≤ V (t0 )η1 eλ(t1 −t0 ) η2 eλ(t2 −t1 ) . . . ηm eλ(t −tm ) m ∏ ≤ V (t0 ) ηi eλ(t −t0 ) .

(18)

i=1

From inequalities (18), we can obtain V (t)

≤ η1 η2 · · · ηm eλ(t −t0 ) V (t0 ) ( η1 + η2 + · · · ηm )m λ(t −t ) 0 V (t ) ≤ e 0 m

= emln

η1 +η2 +···ηm m

eλ(t −t0 ) V (t0 ).

By referring to the construction of η and the definition of Ta , for any ϑ1 , ϑ < ϑ1 < 0, there exists a sufficiently large scalar T > 0, such that when t > T , we have V (t)

lnη

≤ e( Ta +ϑ1 −ϑ )(t −t0 ) eλ(t −t0 ) V (t0 ) lnη

≤ e( Ta +ϑ1 −ϑ+λ)(t −t0 ) V (t0 ) = eϑ1 (t −t0 ) V (t0 ).

X. Li et al. / Physica A 515 (2019) 497–509

505

Since ϑ1 < 0, globally exponential outer synchronization of impulsively coupled dynamical networks (3) can be achieved according to Definition 2. Hence, Theorem 1 is proved. □ If the completely communication channel matrix R = (r kl ) = In , we have the following result. Corollary 1. Suppose that Assumptions 1 and 2 hold. Let lk = k, i.e. the klk −th level is a completely communication channel ˜˜ matrix R = In , and denote λ = λmax {(A˜ i )(i=1,2,...,N) } + 21 (λmax {(B˜ i )(i=1,2,...,N) } + λmax {(L˜ ⊤ i Bi Li )(i=1,2,...,N) }). If

ϑ=

lnη Ta

+ λ < 0,

(19)

then the drive–response partially coupled dynamical networks (3) are globally exponentially outer synchronized with convergence rate −ϑ1 , for any 0 > ϑ1 > ϑ .

∑N

⊤ ⊤ ∈ RN (m = Proof. Consider the Lyapunov function V (t) = i=1 ei (t)Ξ i ei (t). Let em (t) = (e1m (t), e2m (t), . . . , eNm (t)) 1, 2, . . . , n), by the specific structure of the matrices Cij and C kk , one has

2c

N N ∑ ∑

e⊤ i (t)Ξ i Cij ej (t)

i=1 j=1

= 2c

N N n ∑ ∑ ∑

eik (t)ξikk Cijkk ejk (t)

i=1 j=1 k=1

= 2c

n N N ∑ ∑ ∑

eik (t)ξikk Cijkk ejk (t)

(20)

k=1 i=1 j=1 n

= 2c

∑

ek (t)Ξ kk C kk ek (t) ⊤

k=1

=c

n ∑

⊤

ek (t)(Ξ kk C kk + C kk Ξ kk )ek (t) ⊤

k=1

≤ 0. The other detailed proof is similar to the proof of Theorem 1, and hence omitted here. □ m 2 describes average impulse gain [42]. The new concept average Remark 4. The quantity η = limm→+∞ 1 m impulsive gain can be utilized to well study hybrid impulsive sequences, where |ηm | > 1 and |ηm | < 1 can happen simultaneously.

|η |+|η |+···+|η |

Remark 5. If the completely communication channel matrix R = (r kl ) = In , for example, k ∈ σ (12 . . . n), l ∈ σ (12 . . . n), then our model reduced to the model discussed in [25]. In this paper, more generalized and less conservative results are given than that of [25]. Remark 6. One of the important restrictions in [25] is that ξM /ξm is not allowed to be too large, otherwise the conditions in Theorems [25] are not easy to satisfy. Thus, we used the method of weighted-norm to study the outer synchronization problem in partially coupled dynamical networks and the quantity of ξM /ξm is not necessary used in our result. When matrix ln(ξ /ξ )η lnη A, B, η are given: we have Ta < − Mλ m in [25], and Ta < − λ in our paper. In the proof of Theorem 1, the technique of using the square root of positive definite real symmetric matrix avoid the appearance of ξM /ξm . It means that the value of ξM /ξm is replaced by one in our results, and this would make our results less conservative than [25]. Remark 7. Some interesting results concerning impulsive control strategy of dynamical networks have been studied [25,35,45]. Unfortunately, some of the results are too restrictive to some extent. In [45,46], at different impulsive instants, the number of nodes to be controlled is fixed. In [47], supk∈Z {tk − tk−1 } is used to obtain the result and this would bring some conversation comparing with our average impulsive interval method. Therefore, the control strategy in this paper is more general and efficient. 4. Example In this section, two numerical examples are given to illustrate our theoretical results.

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X. Li et al. / Physica A 515 (2019) 497–509

Example 1. Consider a drive–response partially coupled dynamical networks with six nodes and each node is a three dimensional system. The parameters of the network are given as follows:

[ −0.2

0 0 .4 0

0 0

A=

−11.3 ⎢ 4.2 ⎢ ⎢ 0 G=⎢ ⎢ 0 ⎣ 0

⎡

0 0 , −0.1

]

7 −11.2 4.1 0.1 0 0

7

0.08 0 −0.05

B=

0.2 0 7 −11.2 3.9 0

0 7 −11.1 4 .1 0 0

] −0.1

0 0.02 0

[

0

−0.1

, c = 1,

4.1 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ 7 ⎦ −11

⎤

0 0 0 7 −10.9 4

and k = σ (12...n), l = σ (23...n1). Let the channel matrices be: R12 =

R61 =

0.3 0 , R16 = 0

[

0 0 0

0 0 0

0 2 0

0 0 0

0.5 0 , R34 = 0

[

0 0 0

0 0 , R14 = 0

[

0 0 0

0 0 0.7

0 0 , R32 = 0

[

[

0 0.5 0

[

0 0 0

R43 =

0 0 0

0 0 0.2

[ R23 =

0 0 0

0 0 0

[

]

]

0.3 0 , R45 = 0

0 0 0 0 0 2

]

0 0 , R65 = 0

]

[

0 0 0

0 0.2 0

[

]

]

0 0 0.6 0 0 0

0 0 , R54 = 0

]

0.5 0 , R21 = 0

[

0 0 0.8

0 0 , R42 = 0

[

0 0 0

2 0 , R56 = 0

[

0 3 0

[

]

]

]

0 0.3 0

0 0 0.5

0 3 0

0 0 0

0 0 0

0 0 0.9

1 0 , 0

]

0 0 , 0

]

0 0 , 0

]

0.3 0 . 0

]

Obviously, the channel matrices here are non-full rank, non-diagonal and asymmetric, and hence our network models are more general. Nonlinear function is assumed to be f (xi (t)) = (tanh(xi1 ), tanh(xi2 ), tanh(xi3 ))⊤ . L = diag {1, 1, . . . , 1}. The initial values of these systems are chosen uniformly randomly from real number interval [-100,100]. Using the regrouping kl method, we first let Cij = gij Rij , then collect the klk −th elements Cij k of Cij (i, j = 1, 2, . . . , N) and regroup them in a new kl

matrix C klk = (Cij k )N ×N :

C 13

⎡ −2.11 ⎢ 4.2 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎣ 0

0 −4.2 2.05 0 0 0

0 0 −2.05 12.3 0 0

0 −1.26 8 .2 0 .3 0 0

0 0 −8.2 2.05 0 0

0

C 21

⎡ −4.1 ⎢ 1.26 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎣ 0 0

C 32

⎡ −1.4 ⎢ 2.1 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎣ 0 14

1.4 −7 0 0 0 0

0 4.9 −5.6 0 0 0

0.06 0 0 −12.3 7.8 0 0 0 0 −2.35 11.7 0

0 0 5.6 −4.2 0 0

0 0 0 0 −7.8 1.2

2.05 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ 0 ⎦ −1.2

0 0 0 0 −11.7 0.8

4.1 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ 0 ⎦ −0.8

0 0 0 4.2 −6.3 0

i=1 j=1

⎤

⎤

0 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ 6.3 ⎦ −14

The total synchronization error is defined below:

  6 3 ∑ ∑ (xij (t) − yij (t))2 . E(t) = √

⎤

X. Li et al. / Physica A 515 (2019) 497–509

507

Fig. 1. Dynamical behaviors of the error E(t) under : l = 1, 2, 3.

Fig. 2. Dynamical behaviors of the error E(t) under : Ta = 0.05, 0.07, 0.08.

Fig. 3. Dynamical behaviors of the error E(t) under : η = 0.5, 0.7, 0.9.

Figs. 1–3 show how the total synchronization error E(t) evolves with time under different impulsive controllers, in which the number of nodes to be controlled, average impulsive interval, and average impulsive control gains are different. One can observe that the number of nodes to be controlled, average impulsive interval, and average impulsive control gains together determine a controller’s ability to achieve synchronization. In order to display the advantage of our results, Fig. 4 displays the outer synchronization of the drive–response networks with same parameters of the example in [25] except for Ta . Here, we just need Ta = 0.8, while in [25] Ta should be 0.05. It means that according to our results, less number of impulsive controllers is required. This illustrates the advantage of our method and results very well. 5. Conclusion In this paper, we studied outer synchronization of drive–response partially dynamical networks with complex inner coupling matrices. Our approach was based on a regrouping method. We proposed a weighted-norm-based impulsive control

508

X. Li et al. / Physica A 515 (2019) 497–509

Fig. 4. Topology of the partially coupled drive network with six nodes: Ta = 0.8, η = 0.9, R = In .

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