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Exponential synchronization for a class of complex networks of networks with directed topology and time delay
Communicated by Prof. Zidong Wang
Accepted Manuscript
Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay Mohmmed Alsiddig Alamin Ahmed, Yurong Liu, Wenbing Zhang, Ahmed Alsaedi, Tasawar Hayat PII: DOI: Reference:
S0925-2312(17)30889-5 10.1016/j.neucom.2017.05.039 NEUCOM 18457
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
23 March 2017 18 May 2017 18 May 2017
Please cite this article as: Mohmmed Alsiddig Alamin Ahmed, Yurong Liu, Wenbing Zhang, Ahmed Alsaedi, Tasawar Hayat, Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.05.039
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Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay
In practice, time delay can not be overlooked in the networked environment due to the finite switching speed of amplifiers [4], and time delay may result in undesirable dynamic behaviors such as instability, oscillations or poor performances. In recent years, many excellent papers have been reported on synchronization of complex networks with time delays caused by diverse factors in different forms, i.e., node delays [5], [6], coupling delay [7]–[9], delays in both nodes and couplings [10]–[12]. As we know, in real-world networks, there exists a common requirement to regulate the complex networks. Thus, it is important to study the synchronization control of complex networks. However, due to the fact that the complex networks consist of a large number of nodes and it is very difficult to apply controllers on all the nodes. Thus, the pinning control strategy that applies controllers to only a fraction of nodes of complex networks is very helpful for controlling complex networks. During the past decades, numerous results have been reported on pinning control and synchronization of complex dynamical networks [13]–[27]. For instance, pinning synchronization of complex networks has been investigated in [14], in which, the authors have considered both specifically and randomly pinning schemes for scale-free networks and then they concluded that the nodes with highly connected nodes should be pinned first. Pinning synchronization was investigated for a class of general complex networks with hybrid coupling in [21], where constant coupling, discrete time delayed coupling and distributed time delayed coupling are considered. Moreover, in [22], by means of the M matrix theory, pinning synchronization of complex networks has been discussed deeply, where both undirected and directed network topology have been investigated, respectively. Some criteria were obtained for synchronization of delayed discrete nonlinear dynamical networks with the identical or nonidentical topology structure by pinning control in [24]. In [25], by proposing a new kind of Lyapunov function, synchronization of complex networks with switched directed network topologies has been extensively studied. Very recently, cluster synchronization was investigated for a class of general complex dynamical networks under pinning control scheme in [26]. In [27], the cluster synchronization in network of linear
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Abstract—The pinning synchronization problem for a class of complex networks of networks is investigated in this paper, in which the network topology is assumed to be directed. The concerned complex networks of networks are composed of both leaders’ network and followers’ networks (also called “subnetworks”), where the leaders’ network and followers’ networks are regarded as the nodes of the networks of networks, followers’ networks can receive the information from leaders’ network, but not the reverse. Then, by means of the Lyapunov stability theory, graph theory, Barbalat’s Lemma and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization for the general complex networks of networks. In addition, theoretical results are verified through an illustrative example.
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Mohmmed Alsiddig Alamin Ahmed, Yurong Liu∗ , Wenbing Zhang, Ahmed Alsaedi, and Tasawar Hayat
I. I NTRODUCTION
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Index Terms—Complex network; Network of networks; Exponential synchronization; Pinning control; Directed topology; Time delay.
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Complex networks are composed of a large number of nodes and the interactions between them. Many real-world networks can be can be abstracted as a network composed by interactive individuals such as the protein interaction networks, WWW, Power grids, biological networks, citation networks, communication networks, social networks and so on. Complex networks can exhibit many interesting collective phenomena, including self-organization, traveling waves, defect propagation, synchronization, and spatiotemporal chaos [1], [2]. Among the above various collective phenomena, synchronization have received a lot of research attention, in which the main objective of complex networks is to seek a common behavior [3]. This work is supported by National Natural Science Foundation of China under Grants 61374010, 61503328, 11671008, and Top Talent Plan of Yangzhou University. M. A. A. Ahmed, Y. Liu, and W. Zhang are with the Department of Mathematics, Yangzhou University, Yangzhou 225002, P. R. China; and Y. Liu is also with the Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. A. Alsaedi is with the NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. T. Hayat is with the NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia, and also with the Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. ∗ Corresponding author. (Email: [email protected] (Y.Liu).)
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Although, there are fruitful results have been published on pinning synchronization of complex networks, in most existing result, it is implicitly assumed that each node can be synchronized to an isolated node, i.e., the control objective is to control complex networks to an isolated node. However, with the development of information technology, the network size becomes larger and larger, and therefore complex networks may contain many clusters. Actually, in reality, different kinds of critical infrastructure may be coupled together, such as systems of water and food supply, communications, fuel, financial transactions and power generation and transmission [28]. If we think of each cluster as a subnetwork, then complex networks become the so called complex networks of networks [29]–[31]. In complex networks of networks, subnetworks can be seen as nodes of complex networks and the interactions can be seen as the interactions among the subnetworks. Recently, a complex networks of networks model is proposed in [30] and some synchronization criteria are established on pinning synchronization of complex networks of networks. More recently, exponential synchronization was investigated for a class of undirected complex networks of networks with time delays under pinning adaptive control scheme in [31]. The hybrid combinatorial synchronization on multiple dynamical subnetworks with different dimensions of nodes, where the nodes in each subnetwork have different dynamical behaviors has been investigated in [32]. It should be kept in mind that in [30], [32], time delay has been overlooked due to the difficulties in handling the coexistence of time delay and the pinning controller. Motivated by the aforementioned discussions, we aim to solve the exponential synchronization problem for complex networks of networks with directed topology and time delay by applying pinning control. The concerned complex networks of networks are made up of a leader’s network and many followers’ networks. By using the Lyapunov stability theory, graph theory, Barbalat’s Lemma and the Kronecker product techniques, sufficient conditions are derived to ensure the exponential synchronization of the general complex networks of networks. The main contribution of this paper can be summarized as follows: (1) the pinning synchronization control problem for complex networks of networks with both directed topology and time delay will be first addressed. Pinning control scheme is implemented to achieve the exponential synchronization of complex networks of networks. (2) one quantitative measure is distilled from the network’s coupling matrix to characterize the synchronization of the leaders’ network. The calculation of such a quantity can easily and conveniently be realized by using a normal computer. Furthermore, Barbalats Lemma is used to ensure that the error between the subnetworks states and the average
of leaders can converge to zero exponentially. The paper is organized as follows. In Section II, some preliminaries and problem formulation are presented. The main results for the pinning synchronization on complex networks of networks with directed topology and delayed nodes are given in Section III. Section IV provides a numerical simulation example to validate theoretical results. Conclusions are finally drawn in Section V. Notations: The notations used throughout this paper are fairly standard. Rn denotes the n−dimensional Euclidean space and Rm×n represents m × n real matrices. X > 0 (X ≥ 0) is used to denote the positive definite (semi positive definite) matrix, while X > Y (X ≥ Y ) means X − Y > 0 (X −Y ≥ 0). The ith row and the ith column of G is called the ith row-column pair of G [14]. We denote Gl ∈ R(N −l)×(N −l) as a minor matrix of G ∈ RN ×N by removing arbitrary l (1 ≤ l < N ) row-column pairs of G [14]. Let In be an n−dimensional identity matrix. Denote 1N ∈ RN be a vector with each entry being 1. Let 1M ×M be a matrix with each entry being 1. The superscript ”T ” represents the transpose, diag{· · · } stands for a diagonal matrix. k · k denotes the Euclidean norm of a vector. Denote λmax (X) (λmin (X)) as the maximum (minimum) eigenvalue of the matrix X, respectively. Let X ⊗ Y be the Kronecker product of two matrices X and Y [34]. Denote λ2 (X) as the second largest eigenvalue of the matrix X.
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systems via pinning control technique in a general framework that the couplings among nodes are of partial-state and directed has been studied.
II. PRELIMINARIES AND PROBLEM FORMULATION Suppose that there is a network containing one leaders’ network and m subnetworks (followers’ networks) Gk = {υk , εk }, with N (k) nodes in each subnetworks k, where k = 1, 2, ..., m. Let N = N (1) +N (2) +...+N (m) represent the total nodes in the subnetworks. The leaders’ network is made up of M identical nodes with linearly diffusive coupling and is described as follows: S˙ i (t) = f (Si (t), Si (t − τ )) +c
M X
j=1,j6=i
(s)
gij Γ (Sj (t) − Si (t)) , i = 1, 2, · · · , M.
(1)
Here Si (t) = (Si1 (t), Si2 (t), · · · , Sin (t))T ∈ Rn is the state vector of the ith leader, τ > 0 is the constant time delay, f : Rn × Rn → Rn is a continuous vector-valued function, c > 0 is the coupling strength, Γ = diag(γ1 , γ2 , · · · , γn ) ∈ Rn×n is a non-negative definite inner coupling matrix where γi > 0 if two leaders can communicate through the ith state, and γi = 0 (s) otherwise. G(s) = gij is the coupling configuration M ×M matrix representing the topological structure for the network, where if there exists a connection from leader j to leader i, (s) (s) then gij > 0; otherwise, gij = 0 (j 6= i). It means that the leaders’ network topology could be directed. The diagonal entries of G(s) are determined by the following diffusive
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= −
N P
(s)
j=1,j6=i
gij . Equivalently, we
can rewrite the network (1) in a simpler form as follows: S˙ i (t) = f (Si (t), Si (t − τ )) + c
M X
(s)
(2)
gij ΓSj (t),
j=1
i = 1, 2, ..., M . We denote the synchronization error for the leaders by ej = M e e = P ξj Sj (t) is the average state of Sj (t) − S(t), where S(t) j=1
all the leaders’ states, where ξ = (ξ1 , · · · , ξM )T is the positive left eigenvector of G(s) associated with the eigenvalue zero e can be described satisfying ξ T 1M = 1. The dynamics of S(t) by M
M
i=1
i=1
j=1
M P
M P
M
X X X (s) e˙ S(t) = ξi f (Si (t), Si (t − τ )) + c ξi gij ΓSj (t). =
0, one has
i=1
[(ξ T G(s) ) ⊗ Γ] = 0, it then follows that
ξi
j=1
(s)
gij ΓSj (t)
M
X e˙ S(t) = ξi f (Si (t), Si (t − τ )).
=
(3)
i=1
j=1
i=1
(s)
ξi f (Si (t), Si (t − τ ))
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+c
M X
M X
M
Therefore, the error dynamics of the leaders’ network (2) can be obtained as follows: e˙ i (t) = f (Si (t), Si (t − τ )) −
gij Γej (t), i = 1, 2, · · · , M.
(4)
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Let e(t) = (eT1 (t), · · · , eTM (t))T , S(t) = T T T (S1 (t), · · · , SM (t)) and f (S(t), S(t − τ )) = T T T (f (S1 (t), S1 (t − τ )), · · · , f (SM (t), SM (t − τ ))) . Then, the general leaders’ network (5) can be recast in a compact matrix form as follows:
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e(t) ˙ = [(IM − 1M ξ T ) ⊗ In ]f (S(t), S(t − τ ))
(k)
(k)
(k)
j=1,j6=i
the ith node in the kth subnetwork, and (k)
(k)
(k)
e ui (t) = −di Γ(xi (t) − S(t)) ∈ Rn ,
i = 1, 2, · · · , l(k) , (7)
are n-dimensional linear feedback controllers with all the (k) control gains di > 0, j = 1, 2, · · · , M , k = 1, 2, · · · , m. Denote the synchronization error for each subnetwork k by (k) (k) e ei (t) = xi (t) − S(t), where i = 1, 2, · · · , N (k) and k = 1, 2, · · · , m. Definition 1: The leaders’ network (1) and the followers’ networks (6) are said to be exponentially synchronized if there exists constants α1 > 0, α2 > 0, β2 > 0 and β2 > 0 such that for any initial conditions:
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Since ξ T G(s)
(k)
where xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn is the state vector of the ith node inthe kth subnetwork, τ > 0 is the (k) constant time delay, G(k) = gij is the coupling N (k) ×N (k) matrix representing the topological structure of the kth subnetwork, where if there exists a connection between nodes i and (k) (k) j, then gij > 0; otherwise gij = 0 (j 6= i). It means that the subnetwork topology could be directed. The diagonal entries of G(k) are determined by the following diffusive coupling N P (k) (k) gij = deg(k) (i) is the degree of condition: gii = −
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(s)
coupling condition: gii
+ c(G(s) ⊗ Γ)e(t). (5)
e
kSi (t) − S(t)k ≤ α1 exp(−β1 t), i = 1, 2, · · · , M, (k) e kxi (t) − S(t)k ≤ α2 exp(−β2 t), i = 1, 2, · · · , N (k) ,
k = 1, 2, · · · , m. Definition 2: For an N × N irreducible Laplacian matrix G with non-negative off-diagonal elements, which satisfies the diffusive coupling, the quantity a(G) is defined as follows. Let ξ = (ξ1 , ξ2 , · · · , ξN )T ∈ RN be the unique normalized left eigenvector of G corresponding to eigenvalue zero satisfying N P ξi = 1 and Ξ = diag{ξ1 , ξ2 , · · · , ξN }. Then a(G) is i=1
defined to be the second largest eigenvalue of the symmetric b = ΞG + GT Ξ, that is matrix G a(G) = λ2 (ΞG + GT Ξ).
(8)
In the following, a useful assumption and lemmas are introduced. Assumption 1: For the vector function f (x(t), x(t − τ )), suppose that the uniform Lipschitz condition holds, i.e., for any xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T and s(t) = (s1 (t), s2 (t), · · · , sn (t))T . There exists a positive constants γ > 0 and β > 0, such that
For followers’ networks, we assume that the first l(k) nodes are controlled in the kth subnetwork for k = 1, 2, · · · , m. Considering the leaders’ network (2), the pinning controlled kth subnetwork with delayed nodes can be described by: (k) N X (k) (k) kf (xi (t), xi (t − τ )) − f (s(t), s(t − τ ))k (k) (k) (k) x˙ i (t) = f (xi (t), xi (t − τ )) + c gij Γxj (t) ≤ γkxi (t) − s(t)k + βkxi (t − τ ) − s(t − τ )k, (9) j=1 (k) (k) where i = 1, 2, · · · , N . + ui (t), i = 1, 2, · · · , l , (6) Lemma 1 (Schur Complement [35]): The following linear (k) N X (k) (k) (k) (k) (k) matrix inequality: x˙ i (t) = f (xi (t), xi (t − τ )) + c gij Γxj (t), j=1 Q(x) S(x) >0 S(x)T R(x) i = l(k) + 1, l(k) + 2, · · · , N (k) ,
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1) Q(x) > 0, R(x) − S(x)T Q(x)−1 S(x) > 0; 2) R(x) > 0, Q(x) − S(x)R(x)−1 S(x)T > 0.
Lemma 2 ( [37]): For matrices A, B, C and D with appropriate dimensions, the Kronecker product ⊗ satisfies: 1) 2) 3) 4)
(ρA) ⊗ B = A ⊗ (ρB), where ρ is a constant; (A + B) ⊗ C = A ⊗ C + B ⊗ C; (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD); (A ⊗ B)T = AT ⊗ B T .
Lemma 3 ( [36]): For an irreducible matrix L with nonnegative off-diagonal elements, which satisfies the diffusive coupling, we have the following propositions:
H (k) =
cα G(k) + G(k)T IN (k) + c , λmin (Γ) 2
(10)
where α is positive scalar. (k) (k) (k) (k) Let D(k) = diag(d1 , d2 , · · · , dl(k) , 0, · · · , 0) with di > | {z } N(k) −l(k)
0, i = 1, 2, · · · , l(k) . Using matrix decomposition, we have (k) ˜ (k) B (k) U −D H (k) − D(k) = , (11) B (k)T C (k) (k)
(k)
(k)
(k)
where U (k) = (uij )l(k) ×l(k) , uij = uji = hij , i, j = ˜ (k) = diag(d(k) , d(k) , · · · , d(k) 1, 2, · · · , l(k) , D ), B (k) = 1 2 l(k) (k) (k) (k) (bij )l(k) ×(N (k) −l(k) ) , bij = hij , i = 1, 2, · · · , l(k) , j = l(k) + 1, · · · , N (k) , and cα G(k) + G(k)T IN (k) + c C (k) = λmin (Γ) 2 l(k)
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1) If λ is an eigenvalue of L and λ 6= 0, then Re(λ) < 0. 2) L has an eigenvalue 0 with multiplicity 1 and the right eigenvector [1, 1, · · · , 1]T . 3) Suppose that ξ = (ξ1 , ξ2 , · · · , ξN )T ∈ RN satisfy N P ξi = 1 is the normalized left eigenvector of L
the Kronecker product techniques, some criteria ensuring the exponential synchronization of the general complex networks of networks will be derived. Construct the following symmetric matrix H (k) = (k) (hij )N (k) ×N (k) :
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where Q(x) = Q(x)T , R(x) = R(x)T , is equivalent to one of the following condition: 1)
i=1
corresponding to eigenvalue 0. Then, ξi > 0 hold for all i = 1, 2, · · · , N . Furthermore, if L is symmetric, then we have ξi = 1/N for i = 1, 2, · · · , N .
Lemma 4 ( [22]): Assume that Φ = (φij )N ×N is symmetric. Let D = diag(d1 , · · · , dl , 0, · · · , 0) in which di > 0, | {z } N −l
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i = 1, 2, · · · , l. When di > 0, i = 1, 2, · · · , l are sufficiently large, Φ − D < 0 is equivalent to Φl , where Φl is the minor matrix of Φ by removing its first l row-column pairs. Lemma 5 ( [22]): Assume that U , B are N by N Hermitian matrices. Let α1 ≥ α2 ≥ · · · ≥ αN , β1 ≥ β2 ≥ · · · ≥ βN , and ζ1 ≥ ζ2 ≥ · · · ≥ ζN be eigenvalues of U , B and U + B, respectively. Then, one has αi + βN ≤ ζi ≤ αi + β1 , i = 1, 2, · · · , N . Lemma 6 (Barbalat’s Lemma [39]): Let f be non-negative function defined on [0, +∞). if f Lebesgue integrable on [0, +∞) and is uniformly continuous on [0, +∞), then lim f (t) = 0. t→+∞ In the rest of this paper, our main aim is to deal with the exponential synchronization problem of the complex networks of networks (1) and (6) with directed topology and delayed nodes via pinning feedback control. By utilizing Lyapunov-Krasoviskii functional, Barbalat’s Lemma and Kronecker product techniques, we aim to derive sufficient conditions such that the complex networks of networks (1) and (6) with directed topology and delayed nodes are globally exponentially synchronized. III. M AIN R ESULTS In this section, we investigate the exponential synchronization of the general complex networks of networks with delayed nodes in the case that the coupling matrices G(s) and G(k) are irreducible. based on the Lyapunov stability method and
is the minor matrix of H (k) by removing its first l(k) rowcolumn pairs. Assume that ξ = (ξ1 , ξ2 , · · · , ξM )T and ξ (k) = (k) (k) (k) (ξ1 , ξ2 , · · · , ξN (k) )T are the normalized left eigenvectors of the configuration coupling matrices G(s) and G(k) with respect (k) M NP P (k) to zero eigenvalue satisfying ξi = 1 and ξi = 1, i=1
i=1
respectively. Since the coupling configuration matrices G(s) and G(k) , k = 1, 2, · · · , m are irreducible, according to lemma 3, we can conclude that ξj > 0, j = 1, 2, · · · , M and (k) ξi > 0, i = 1, 2, · · · , N (k) . Let Ξ = diag{ξ1 , ξ2 , · · · , ξM } is a positive diagonal matrix of the leaders’ network, Ξ(k) = (k) (k) (k) diag(ξ1 , ξ2 , · · · , ξN(k) ) is a positive diagonal matrix of the kth subnetwork. We are now in a position to give the main results of this paper as follows. Theorem 1: Suppose that Assumption 1 holds, and that there exists a positive scalar α such that the following conditions are satisfied: [2γ + 1 − cρ1 ]Ξ + ξξ T βΞ < 0, (12) βΞ −Ξ γ+1+
(k) di
β2 δ + − cα < 0, 2 4
(13)
(k) (k) (k) −1 (k)T > λmax U − B C B , λmax
G(k) + G(k)T 2
l(k)
i = 1, · · · , l(k) , (14)
<−
α , λmin (Γ)
(15)
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It follows from LMIs (12) and (16) that there exists a positive constant , which may be very small, such that [ + 2γ + exp(τ ) − cρ1 ]Ξ + ξξ T βΞ < 0, (17) Φ1 = βΞ −Ξ (k)
Φ2
[ + γ + exp(τ ) − cα + 2δ ]IN (k) = β 2 IN (k)
β 2 IN (k)
−IN (k)
< 0, (18)
where k = 1, 2, · · · , m. Consider the following Lyapunov functional:
e e − τ ))] eT (t)(Ξ ⊗ In )[1M ⊗ f (S(t), S(t e e − τ ))](Ξ ⊗ In ) = [1TM ⊗ f T (S(t), S(t
× [(IM − 1M ξ T ) ⊗ In ]S(t) e e − τ )) S(t) = [1TM Ξ(IM − 1M ξ T )] ⊗ f T (S(t), S(t e e − τ )) S(t) = 0, = [ξ T (IM − 1M ξ T )] ⊗ f T (S(t), S(t (23)
eT (t)(Ξ ⊗ In )[(1M ξ T ) ⊗ In ]f (S(t), S(t − τ ))
= f T (S(t), S(t − τ ))[(ξ1TM ) ⊗ In ](Ξ ⊗ In )
× [(IM − 1M ξ T ) ⊗ In ]S(t) = f T (S(t), S(t − τ )) [ξ1TM Ξ(IM − 1M ξ T )] ⊗ In S(t) = f T (S(t), S(t − τ )) [ξξ T (IM − 1M ξ T )] ⊗ In S(t) = 0.
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In addition,
(24)
2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t)
(19)
VL (t) = VL1 (t) + VL2 (t),
= 2ceT (t)(ΞG(s) ⊗ Γ)e(t)
where VL1 (t) = eT (t)(Ξ ⊗ In )e(t) exp(t), Zt
t−τ
eT (s)(Ξ ⊗ In )e(s) exp((s + τ ))ds.
M
VL2 (t) =
Then, by calculating the derivative of VL (t) along the trajectory of (5) one can obtain that
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+ eT (t)(Ξ ⊗ In )e(t) exp(t) = 2eT (t)(Ξ ⊗ In ) [(IM − 1M ξ T ) ⊗ In ] × f (S(t), S(t − τ )) exp(t)
+ 2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t) exp(t)
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+ eT (t)(Ξ ⊗ In )e(t) exp(t),
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V˙ L2 (t) = eT (t)(Ξ ⊗ In )e(t) exp(τ ) − eT (t − τ )(Ξ ⊗ In )e(t − τ ) exp(t).
(20)
(25)
where G = ΞG + G Ξ. T Since ξ = (ξ1 , ξ2 , · · · , ξM ) is the normalized left eigenvector of the configuration coupling matrix G(s) with respect to zero M P eigenvalue satisfying ξi = 1. It can be easily see that (s)
(s)T
i=1
By Assumption 1, one obtains e e 2eT (t)(Ξ⊗In ) f (S(t), S(t−τ ))−1M ⊗f (S(t), S(t−τ )) =2
M X i=1
e e − τ )) ξi eTi (t) f (Si (t), Si (t − τ )) − f (S(t), S(t
≤2 (21)
Combination of (20) and (21) yields ˙ VL (t) = exp(t) 2eT (t)(Ξ ⊗ In ) f (S(t), S(t − τ )) e e − τ )) + eT (t)(Ξ ⊗ In )e(t) − 1M ⊗ f (S(t), S(t
+ 2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t) + eT (t)(Ξ ⊗ In ) T
× e(t) exp(τ ) − e (t − τ )(Ξ ⊗ In )e(t − τ )
b (s)
= ceT (t)[(ΞG(s) + G(s)T Ξ) ⊗ Γ]e(t) b (s) ⊗ Γ]e(t), = ceT (t)[G
b (s) is a zero-sum symmetric matrix with nonnegative offG diagonal elements. Moreover, from the irreducibility of G(s) , b (s) = ΞG(s) + G(s)T Ξ irreducible. we conclude that G
V˙ L1 (t) = 2e (t)(Ξ ⊗ In )e(t) ˙ exp(t) T
and
Following [38], since e(t) = [(IM − 1M ξ T ) ⊗ In ]S(t) and ξ T 1M = 1, one has
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where ρ1 = −a G(s) λmin (Γ)/λmax (Ψ) and k = 1, 2, · · · , m. Then, the directed controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization. Proof: By Lemma 1, the condition (13) is equivalent to that [γ + 1 − cα + 2δ ]IN (k) β2 IN (k) (k) e Φ2 = < 0. (16) β −IN (k) 2 IN (k)
− 2eT (t)(Ξ ⊗ In )[(1M ξ T ) ⊗ In ]f (S(t), S(t − τ )) e e − τ ))] . (22) + 2eT (t)(Ξ ⊗ In )[1M ⊗ f (S(t), S(t
M X i=1
ξi kei (t)k γkei (t)k + βkei (t − τ )k . (26)
Combining (23)-(26), and adding a vanishing terms −eT (t)(Ξ ⊗ cρ1 In )e(t) + eT (t)(Ξ ⊗ cρ1 In )e(t), −eT (t)(ξξ T ⊗ cρ1 In )e(t) + eT (t)(ξξ T ⊗ cρ1 In )e(t), we can obtain that V˙ L (t) ≤ exp(t)η T (t)Φ1 η(t) b (s) ⊗ Γ) + (Ψ ⊗ cρ1 In ) e(t) + exp(t)eT (t) c(G ≤ exp(t)η T (t)Φ1 η(t) + exp(t)eT (t) cρ1 (s) b Γ e(t), × c(G ⊗ Γ) + Ψ ⊗ λmin (Γ)
(27)
where η(t) = (eT (t), eT (t − τ ))T , Φ1 is defined in (17) and Ψ = Ξ − ξξ T .
ACCEPTED MANUSCRIPT 6
b (s) = G b (s)T is an irreducible symmetric matrix, there Since G exist a unitary matrix U = (u1 , u2 , · · · , uM ) such that
−a G(s) λmin (Γ)/λmax (Ψ), it follows from (27) that V˙ L (t) ≤ exp(t)η T (t)Φ1 η(t) + exp(t) ca G(s) +
b (s) = U Λ1 U T , G
U T U = I,
b (s) ), λ2 (G b (s) ), · · · , λM (G b (s) )}, Λ1 = diag{λ1 (G
b (s) ) > λ2 (G b (s) ) ≥ · · · ≥ λM (G b (s) ). 0 = λ1 (G Now, introducing transformation y(t) = (U T ⊗ In )e(t),
e(t) = (U ⊗ In )y(t),
b (s) ⊗ Γ]e(t) = cy T (t)(U T ⊗ In )[G b (s) ⊗ Γ] ceT (t)[G × (U ⊗ In )y(t) b (s) U ⊗ Γ)y(t) = cy T (t)(U T G = cy T (t)(Λ1 ⊗ Γ)y(t) M X i=1
b (s) )y T (t)Γyi (t) λi (G i
b (s) ≤ cλ2 G
yiT (t)Γyi (t)
i=2 M X
yiT (t)Γyi (t).
(28)
i=2
ED
= ca G(s)
M X
M
=c
CE
PT
According to the structure of matrix Ψ, we can conclude that Ψ is an irreducible symmetric matrix with negative off-diagonal elements, and also, Ψ satisfies the diffusive coupling condition. Therefore, we can conclude that λmax (Ψ) > 0. From the construction of Ψ, it follows that Ψ.u1 = (0, 0, · ·"· , 0)T := 0n #∈ 0 0Tn RN . Further, we can obtain that U T ΨU = T e ΨU e , 0n U e = [u2 , u3 , · · · , uM ] satisfying U eT U e = IM −1 . Using where U the above analysis, one has
AC
cρ1 Γ]e(t) λmin (Γ) cρ1 y T (t)[U T ΨU ⊗ Γ]y(t) λmin (Γ) cρ1 e T ΨU e ⊗ Γ]e yeT (t)[U y (t) λmin (Γ) cρ1 eT U e ⊗ Γ]e λmax (Ψ)e y T (t)[U y (t) λmin (Γ) M X cρ1 λmax (Ψ) yiT (t)Γyi (t), λmin (Γ) i=2
eT (t)[Ψ ⊗ = = ≤ =
T where ye = [y2T (t), · · · , yM (t)]T . By using (28), (29) and
equality
ρ1
yiT (t)Γyi (t)
i=2
= exp(t)η T (t)Φ1 η(t).
(30)
Using (17), the inequality (27) gives that V˙ L (t) ≤ 0, which implies that VL (t) is bounded. Therefore, we can conclude that ei (t) exponentially approach to zero. (k) (k) e On the other hand, since ei (t) = xi (t) − S(t), one can get the following error system between the nodes in the followers’ networks and the leaders: M X (k) (k) (k) e ˙ (t) = f (x (t), x (t − τ )) − ξi f (Si (t), Si (t − τ )) i i i i=1 (k) N M X (k) (k) X (k) (k) + c g Γe (t) − dij Γei (t), ij j j=1 j=1 i = 1, 2, · · · , l(k) M X (k) (k) (k) e ˙ (t) = f (x (t), x (t − τ )) − ξi f (Si (t), Si (t − τ )) i i i i=1 (k) N X (k) (k) +c gij Γej (t), j=1 i = l(k) + 1, l(k) + 2, · · · , N (k) . (31) (k)T (k)T Let e(k) (t) = (e1 (t), · · · , eN (k) (t))T , f (x(k) (t), x(k) (t − (k) (k) (k) (k) τ )) = (f T (x1 (t), x1 (t − τ )), · · · , f T (xN (k) (t), xN (k) (t − T τ ))) and k = 1, 2, · · · , m. Then, we can rewrite (31) in a compact form as follows: e˙ (k) (t) = f (x(k) (t), x(k) (t − τ )) − 1N (k) e e − τ )) + c G(k) ⊗ Γ e(k) (t) ⊗ f (S(t), S(t − D(k) ⊗ Γ e(k) (t) + F (e, S), (32)
e e − τ )) − [(1M ξ T ) ⊗ where F (e, S) = 1N (k) ⊗ f (S(t), S(t (k) In ]f (S(t), S(t − τ )) and D is defined above. Construct the following Lyapunov functional for system (32): Vs (t) = Vs1 (t) + Vs2 (t),
(33)
where m
Vs1 (t) = (29)
=
1 X (k)T e (t)(Ξ(k) ⊗ In )e(k) (t) exp(t), 2 k=1
Vs2 (t) = the
M X
AN US
T T where y(t) = y1T (t), y2T (t), · · · , yM (t) , yr (t) ∈ Rn , r = 1, 2, · · · , M . Then, we have
×
CR IP T
where
cρ1 λmax (Ψ) λmin (Γ)
m Zt X
k=1t−τ
e(k)T (s)(Ξ(k) ⊗ In )e(k) (s) exp((s + τ ))ds.
ACCEPTED MANUSCRIPT 7
Then, the derivative of Vs (t) along the trajectory of (32) yields
In )e(k) (t) + e(k)T (t)(cαΞ(k) ⊗ In )e(k) (t), one can obtain (k)
V˙ s (t) ≤ exp(t) e(k)T (t)(Ξ(k) ⊗ In )e˙ (k) (t) exp(t)
k=1 m X
+
k=1
= x
k=1 (k)
e(k)T (t)(Ξ(k) ⊗ In ) f (x(k) (t),
(t − τ )) − 1N (k) (k)T
+ ce
(t)(Ξ
(k)
(k)
⊗ In )(G
k=1
⊗ Γ)e
+
m X
(k)
≤ exp(t)
(t)
(k)
k=1
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t) exp(t),
e(k)T (t)(Ξ(k) ⊗ In ) (cαIN (k) ⊗ In )
k=1
⊗ Γ)e (t) + e(k)T (t)(Ξ(k) ⊗ In )F (e, S) exp(t) (t)(Ξ
⊗ In )(D
(k)
(k)
k=1 i=1 m X
+ exp(t)
m N X X
(k) (k)T
ξi ηi
k=1 i=1
m X
e
(k)T
(k)
(t)(Ξ
k=1 (k)T
(k) (k)
(t)Φ2 ηi (t)
⊗ In )
cα I (k) λmin (Γ) N
+G − D(k) ⊗ Γ e(k) (t) 2 m X
2
1 + (Ξ(k) ⊗ In ) F (e, S) exp(t) 2δ (k)
+c (34)
G
AN US
−e
(k)
e e − τ )) ⊗ f (S(t), S(t
(k) (k)
(t)Φ2 ηi (t)
(k) G + G(k)T (k) −D ⊗ Γ e(k) (t) + c 2 m
2
1 X (k) (Ξ ⊗ In ) F (e, S) exp(t) + 2δ
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t)
m X
(k)T
+ exp(t)
(k) (k)T
ξi ηi
CR IP T
V˙ s1 (t) =
m X
m N X X
k=1
(k)
= exp(t)
and
m N X X
(k) (k)T
ξi ηi
(k) (k)
(t)Φ2 ηi (t)
k=1 i=1
+ exp(t)
M
k=1 (k)T
(t − τ )(Ξ(k) ⊗ In )e(k) (t − τ ) exp(t). (35)
m X
=
≤
(k)
m N 1XX
2 δ 2
k=1 i=1 m X (k)T
e
k=1
(k)
ξi
m
2
1 X (k) (Ξ ⊗ In ) F (e, S) exp(t), 2δ
(37)
k=1
(k)
(k)T
(k)T
(k)
where ηi (t) = (ei (t), ei (t − τ ))T , Φ2 is defined in (18) and k = 1, 2, · · · , m. By Lemma 4, from (11) and the given condition (14), we know that H (k) − D(k) < 0 is equivalent to G(k) + G(k)T cα (k) C = I (k) + c <0 λmin (Γ) N 2 l(k) (k)
di
(k)T 2ei (t)F (e, S)
> λmax U (k) − B (k) C (k)
−1
B (k)T ,
i = 1, · · · , l(k) , k = 1, · · · , m.
(t)(Ξ(k) ⊗ In )e(k) (t)
m
2 1 X (k) (Ξ ⊗ In ) F (e, S) . + 2δ
+
when the pinning feedback gains satisfy
e(k)T (t)(Ξ(k) ⊗ In )F (e, S)
AC
k=1
CE
For δ > 0, we have
e(k)T (t)(Ξ(k) ⊗ In )
k=1 (k) × H − D(k) ⊗ Γ e(k) (t)
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t) exp(τ )
ED
−e
m X
PT
V˙ s2 (t) =
m X
(36)
k=1
Combining (34) with (35), and then by using Assumption 1, (36) and adding a vanishing term −e(k)T (t)(cαΞ(k) ⊗
Therefore, we only need to show that cα G(k) + G(k)T C (k) = IN (k) + c < 0. λmin (Γ) 2 l(k) By lemma 5, we have cα G(k) + G(k)T λmax IN (k) + c ≤ λmin (Γ) 2 l(k) (k) cα G + G(k)T + cλmax . (38) λmin (Γ) 2 l(k)
ACCEPTED MANUSCRIPT 8
indicates that C (k) =
l(k)
cα G(k) + G(k)T IN (k) + c λmin (Γ) 2
Therefore, we can conclude that H
(k)
−D
(k)
(39)
Considering that Γ > 0 in (37), then (13), (14) and (15) can guarantees that m X V˙ s (t) ≤ − (Ξ ⊗ In )ke(k) (t)k2 exp(t) k=1
+
m
2 1 X (k) (Ξ ⊗ In ) F (e, S) exp(t) 2δ k=1
+
(k) 2 ˜ ξi kxi (t) − S(t)k exp(t)
k=1 i=1 m 1 X (k)
(Ξ
2δ
k=1
Therefore, we have
Vs (t) − Vs (0) ≤ −
2 ⊗ In ) F (e, S) exp(t).
(k) Zt X m N X
0 k=1 i=1
(k)
2 ˜ ξi kxi (θ) − S(θ)k
M
=−
which implies that
0 k=1 i=1
(k)
2 ˜ ξi kxi (θ) − S(θ)k exp(θ)dθ ≤ Vs (0)
Zt X m
2 (Ξ(k) ⊗ In ) F (e, S) exp(θ)dθ.
CE
1 + 2δ
PT
(k) Zt X m N X
ED
× exp(θ)dθ Zt X m
2 1 + (Ξ(k) ⊗ In ) F (e, S) exp(θ)dθ 2δ 0 k=1
0 k=1
AC
From (12), it is easy to see that F (e, S) exponentially approaches to zero. Then (k) Zt X m N X (k) 2 ˜ ξi kxi (θ) − S(θ)k dθ < +∞, 0 k=1 i=1
which implies that
(k)
m N X X
k=1 i=1
(k) 2 ˜ ξi kxi (θ) − S(θ)k < M0 eαθ ,
where M0 > 0 and α > 0. Therefore, by choosing < α, we can find that (k) Zt X m N X (k) 2 ˜ ξi kxi (θ) − S(θ)k exp(θ)dθ < +∞. 0 k=1 i=1
(k)
(k)
2 ˜ ξi kxi (t) − S(t)k exp(t) → 0
when t → ∞.
Thus, from the conditions (12), (13), (14) and (15), it is easy to observe that the directed controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization under the given controllers. Thus, the proof is completed. Remark 1: In Theorem 1, the condition (12) ensures the synchronization of leaders’ network. When the leaders reach synchronization exponentially, then the conditions (13), (14) and (15) provide the control design procedure for synchronization in each subnetworks. It should be mention that, the quantity a G(s) and the scalar α play a key role in the conditions (12), (13), respectively. And the maximum eigenvalue also plays a key role in the conditions (14) and (15). Remark 2: In the literature [32], the considered complex network divided into multiple sub-networks, such that the nodes belonging to the same sub-network are identical, while the ones belonging to different sub-networks are non-identical, which are different from the complex networks of networks. Remark 3: In this paper, the networks under consideration are different from the networks in the literatures [30], [31]. Specifically, in [30], pinning synchronization is investigated for the undirected complex networks of networks without considering the time delays. In [31], the time delays are taken into account, and adaptive controller is designed to ensure the synchronization of the controlled network, but the concerned network of networks is still undirected. In this paper, the considered networks have directed topological structure, which are more general compared with [30], [31]. In Theorem 1, we focus on the exponential synchronization for the directed controlled networks. In the following, we consider the special case when the complex networks of networks is undirected. In this case, the left eigenvectors ξ = 1/M [1, 1, · · · , 1]T ∈ RM and ξ (k) = (k) 1/N (k) [1, 1, · · · , 1]T ∈ RN . Then, we have Ξ = (1/M )IM and Ξ(k) = (1/N (k) )IN (k) . Moreover, since G(s) is symmetric, one can find a(G(s) ) = (2/M )λ2 (G(s) ). Therefore, for the undirected complex networks of networks, the quantity ρ1 = −2λ2 (G(s) )λmin (Γ)/M λmax (Ξ). Then, we can get the following corollary. Corollary 1: Suppose that Assumption 1 holds, and that there exist a positive scalar α such that the following conditions are satisfied: [2γ + 1 − cρ1 ]IM + (1/M )1M ×M βIM (1/M ) < 0, βIM −IM (40)
AN US
(k)
m N X X
(k)
ξi kxi (t) −
2 ˜ S(t)k exp(t) is uniformly continuous on [0, +∞). From Lemma 6, we can conclude that
k=1 i=1
l(k)
m NP P
k=1 i=1
m N X X
< 0.
< 0.
(k)
Moreover, it is not difficult to verify that
CR IP T
cα From the condition (15), it is easy to have λmin (Γ) + (k) (k)T G +G cλmax < 0. Then, in view l(k) of(38), we know 2 G(k) +G(k)T cα < 0, which that λmax λmin (Γ) IN (k) + c 2
ACCEPTED MANUSCRIPT 9
(k)
di
β2 δ + − cα < 0, 2 4
(41)
−1 (k)T B > λmax U (k) − B (k) C (k) , λmax
G(k)
l(k)
i = 1, · · · , l(k) , (42)
<−
α , λmin (Γ)
(43)
where ρ1 = −2λ2 (G(s) )λmin (Γ)/M λmax (Ψ) and k = 1, 2, · · · , m. Then, the undirected controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization. IV. N UMERICAL E XAMPLE
We consider the time delay τ = 1. Obviously, the nonlinear function (44) satisfies Assumption 1 with γ = 1 and β = 1, 2 then α = 4.6029 is chosen such that α > 1c γ + 1 + 2δ + β4 . According to [22], we rearrange the subnetwork nodes such that the out-degree are bigger than their in-degree. Therefore, we assume that two nodes in each subnetwork are chosen as the candidate for applying pinning feedback controllers. The coupling strength is c = 0.49 and we select δ = 0.001. Therefore, according to Theorem 1, we can conclude that the leaders’ network and the followers’ network are exponentially stable. Furthermore, an illustrative example is also carried out to demonstrate the effectiveness of the theoretical results. For the leaders’ network (1) and the followers’ network (6) with the above parameters, we choose the following initial values, for any t ∈ [0, 1], T T S1 (0) = −2 1 0.2 , S2 (0) = −4 1.2 1 , T T S3 (0) = −2 0.5 1 , S4 (0) = −4 1 0.5 , T T (1) S5 (0) = 1 −2 0 , x1 (0) = 0.5 −1.1 0.2 , T T (1) (1) x2 (0) = −1 1 0 , x3 (0) = −0.1 0.1 0 , T (1) (1) x4 (0) = x5 (0) = 0.1 0.2 −0.3 , T T (2) = −0.5 0.2 0.1 , x1 (0) 0.2 −0.6 0.1 , T (2) T (2) x2 (0) = −0.5 0.2 0.2 , x3 (0) = 0.1 −0.2 0 , T (2) (2) x4 (0) = = −4.5 0.2 −0.1 , x5 (0) T (3) T (3) 0.1 0.2 −0.5 , x1 (0) = 0.5 −2 1 , x2 (0) = T (3) T (3) 1 −1 0.4 , x3 (0) = 0.5 −1.1 0.7 , x4 (0) = T T (3) −0.6 0.5 0.3 , x5 (0) = −0.2 0.1 0.1 , and simulation result in Fig. 1 shows that the evolution of the synchronization error dynamics for both leaders’ network and subnetworks approaches zero, which matches well with the theoretical results.
AN US
In this section, a numerical simulation example is given to demonstrate our theoretical analysis. Consider the leaders’ network (1) with five leaders. The coupling matrix is chosen as follows: −3 1 1 1 0 1 −3 0 1 1 (s) G = 1 1 −3 0 1 . 0 1 1 −3 1 1 0 1 1 −3
where i = 1, 2, 3, 4, 5, −0.6 0.3 0.3 −0.6 0.3 0.3 D1 = 0.3 −1.5 0.2 , D2 = 0.3 −1.7 0.2 0.3 1.2 −0.5 0.3 1.5 −0.6 9.9 0 0 and Λ = 0 0.3 0 . 0 0 0.3
CR IP T
γ+1+
G(3)
1 −2 0 0 0
0 0 −1 1 1
0 0 1 −2 0
1 −3 1 0 1
1 0 −2 1 1
0 1 1 −3 0
CE
AC
G(2)
−1 1 = 0 0 1
PT
ED
M
Suppose that there are also three subnetworks with five nodes in each subnetworks. The coupling matrices of three subnetworks are given, respectively, by −1 0 0 0 1 1 −4 1 1 1 (1) G = 0 1 −2 1 0 , 1 0 1 −3 1 1 1 1 0 −3
−2 1 = 0 1 1
0 1 0 , 1 −2 0 1 0 . 1 −3
V. C ONCLUSIONS
Γ is assumed to be diag{2, 2, 2}. The nonlinear vector-valued function in the delayed dynamical equation of each leader and node is described by f (xi (t), xi (t − τ )) = D1 tanh(Λxi (t))
+ D2 tanh(Λxi (t − τ )),
(44)
The exponential synchronization on complex networks of networks with directed topology and time delay has been studied in this paper, where the leader’s network and the followers’ network form the networks of networks. The coupling configuration matrix is not assumed to be symmetric. Based on this model, some sufficient conditions of the exponential synchronization are then deduced by introducing the left eigenvector to the construction of Lyapunov functionals. An illustrative example has been provided to demonstrate the effectiveness of the proposed results. The future topic for the
ACCEPTED MANUSCRIPT 10
5
[13]
4
3
2 (k) e (t)
[14]
1
[15]
0
−1
[16]
−2
−3
0
2
4
6
8
10
t
12
14
16
[17] (k)
States of errors between the nodes xi i = 1, 2, 3, 4, 5, k = 1, 2, 3 and the average leaders Se in the directed network with time delay.
Fig. 1.
[19]
[20]
AN US
synchronization would consider more general cases involving network-induced phenomena [40]–[46].
[18]
[21]
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CE
PT
ED
M
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stochastic delays and its application to multi-agent systems”, Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 1175–1192, 2013. Q. Song, F. Liu, J. Cao and W. Yu, “Pinning-controllablility analysis of complex networks: An M-matrix approach”, IEEE Transactions on Circuits and Systems I, Regular papers, vol. 59, no. 11, pp. 2692–2701, 2012. X.F. Wang and G. Chen, “Pinning control of scale-free dynamical networks”, Physica A, vol. 310, pp. 521–531, 2002. X. Li, X. Wang and G. Chen, “Pinning a complex dynamical network to its equilibrium”, IEEE Transactions Circuits Systems I, Regular Papers, vol. 51, no. 10, pp. 2074-2087, 2004. M. Porfiri and M. di Bernardo, “Criteria for global pinningcontrollability of complex networks”, Automatica, vol. 44, no. 12, pp. 3100–3106, 2008. W. Yu, G. Chen and J. L¨ u, “On pinning synchronization of complex dynamical networks”, Automatica, vol. 45, no. 2, pp. 429–435, 2009. W. Yu, J. Cao, J. L¨ u and J. Kurths, “Synchronization via pinning control on general complex dynamical networks”, SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 1395–1416, 2013. J. Zhou, X. Wu, W. Yu, M. small and J. Lu, “Pinning synchronization of delayed neural networks”, Chaos, vol. 18, no. 4, article no. 043111, 2008. L. Xiang, Z. Liu, Z. Chen, F. Chen and Z. Yuan, “Pinning control of complex dynamical networks with general topology”, Physica A: Statistical Mechanics and its Applications, vol. 379, no. 1, pp. 298– 306, 2007. Q. Song, J. Cao and F. Liu, “Pinning-controlled synchronization of hybrid-coupled complex dynamical networks with mixed time-delays”, International Journal of Robust and Nonlinear Control, vol. 22, no. 6, pp. 690–706, 2011. Q. Song and J. Cao, “On pinning synchronization of directed and undirected complex dynamical networks”, IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 57, no. 3, pp. 672–680, 2010. Z. H. Liu, Z. Q. Chen and Z. Z. Yuan, “Pinning control of weighted general complex dynamical networks with time delay”, Physica A, vol. 375, pp. 345–354, 2006. R. Cheng, M. S. Peng and J. Zuo, “Pinning synchronization of discrete dynamical networks with delay coupling”, Physica A, vol. 450, pp. 444C-453, 2016. G. Wen, W. Yu, G. Hu, J. Cao and X. Yu, “Pinning synchronization of directed networks with switching topologies: A multiple Lyapunov functions approach”, IEEE Transactions on Neural Networks and Learning Systems, vol. 26, No. 12, pp. 3239–3250, 2015. Q. Ma and J. Lu, “Cluster synchronization for directed complex dynamical networks via pinning control”, Neurocomputing, vol. 101, pp. 354–360, 2013. C. Yu, J. Qin and H. Gao, “Cluster synchronization in directed networks of partial-state coupled linear systems under pinning control”, Automatica, vol. 40, pp. 2341–2349, 2014. J. Gao, S. V. Buldyrev, H. E. Stanley and S. Havlin, “Networks formed from interdependent networks”, Nature Physics, vol. 8, pp. 40–48, 2011. J. Gao, S. V. Buldyrev, S. Havlin and H. E. Stanley, “Robustness of a network of networks”, Physical Review Letters., vol. 107, no. 19, pp. 195701-1-195701-5, 2011. R. Lu, W. Yu, J. L¨ u and A. Xue, “Synchronization on complex networks of networks”, IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 11, pp. 2110–2118, 2014. M. A. A Ahmed, Y. R. Liu, W. B. Zhang and F. E. Alsaadi, “Exponential synchronization via pinning adaptive control for complex networks of networks with time delays”, Neurocomputing, vol. 225, pp. 198–204, 2017. L. Zhou and C. Wang, “Hybrid combinatorial synchronization on multiple sub-networks of complex network with unknown boundaries of uncertainties”, Optik, vol. 127, no. 22, pp. 11037–11048, 2016. R. Yu, H. G. Zhang, Z. L. Wang and J. Y. Wang, “Synchronization of complex dynamical networks via pinning scheme design under hybrid topologies”, Neurocomputing, vol. 214, pp. 210–217, 2016. A. N. Langville and W. J. Stewart, “The Kronecker product and
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Yurong Liu’s Biography:
Yurong Liu was born in China in 1964. He received his B.Sc. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.Sc. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2001. Dr. Liu is currently a professor with the Department of Mathematics at Yangzhou University, China. He also serves as an Associate Editor of Neurocomputing. So far, he has published more than 50 papers in refereed international journals. His current interests include stochastic control, neural networks, complex networks, nonlinear dynamics, time-delay systems, multi-agent systems, and chaotic dynamics.
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Ph.D. degree in Applied Mathematics at Yangzhou University, Yangzhou, China. His research interests include complex networks, nonlinear dynamics and complex networks of networks.
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stochastic automata networks”, Journal of Computational and Applied Mathematics, vol. 167, no. 2, pp. 429–447, 2004. S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, “Linear Matrix Inequalities in System and Control Theory”, Philadelphia, PA, USA: SIAM, 1994. J. Lu and D. W. C. Ho, “Globally exponential synchronization and synchronizability for general dynamical networks”, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, no. 2, pp. 350–361, 2010. R. A. Horn and C. R. Johnson, “Topics in Matrix Anlysis”, Cambridge, U. K.: Cambridge University Press, 1991. W. Yu, G. Chen, M. Cao and J. Kurths, “Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics”, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, no. 3, pp. 881–891, 2010. J. E. Slotine and W. Li, “Applied Nonlinear Control”, Prentice-Hall, New Jersey, 1991. D. Liu, Y. Liu, and F. E. Alsaadi, A new framework for output feedback controller design for a class of discrete-time stochastic nonlinear system with quantization and missing measurement, International Journal of General Systems, vol. 45, no. 5, pp. 517–531, 2016. Q.-L. Han, Y. Liu, and F. Yang, Optimal communication network-based Hi nf ty quantized control with packet dropouts for a class of discretetime neural networks with distributed time delay, IEEE Transactions on Neural Networks and Learning Systems, 27 (2) (2016) 426–434. Y. Liu, W. Liu, M. A. Obaid, and I. A. Abbas, Exponential stability of Markovian jumping Cohen-Grossberg neural networks with mixed mode-dependent time-delays, Neurocomputing, vol. 177, pp. 409–415, 2016. Y. Luo, Z. Wang, G. Wei, F. E. Alsaadi, T. Hayat, “State estimation for a class of artificial neural networks with stochastically corrupted measurements under Round-Robin protocol”, Neural Networks, vol. 77, pp. 70–79, 2016. Y. Luo, Z. Wang, G. Wei, F. E. Alsaadi, H-infinity Fuzzy Fault Detection for Uncertain 2-D Systems Under Round-Robin Scheduling Protocol,IEEE Transactions on Systems, Man, and Cybernetics: Systems, (in press), DOI: http://dx.doi.org/10.1109/TSMC.2016.2632043. H. Shu, S. Zhang, B. Shen, and Y. Liu, Unknown input and state estimation for linear discrete-time systems with missing measurements and correlated noises, International Journal of General Systems, vol. 45, no. 5, pp. 648–661, 2016. Q. Li, B. Shen, Y. Liu, and F. E. Alsaadi, Event-triggered H∞ state estimation for discrete-time stochastic genetic regulatory networks with Markovian jumping parameters and time-varying delays, vol. 174, pp. 912–920, 2016.
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Biographies
Mohmmed Alsiddig Alamin Ahmed’s Biography: Mohmmed Alsiddig Alamin Ahmed received the B.SC degree in Mathematics from Sudan University of Sciences and Technology, Khartoum, Sudan and the M.SC degree in Mathematics from Khartoum University, Khartoum, Sudan, in 2007 and 2009, respectively. In 2011, he received Postgraduate Diploma from African Institute for Mathematical Sciences AIMS, Cape Town, South Africa. Currently, he is pursuing the
Wenbing Zhang’s Biography: Wenbing Zhang received the M.S. degree in applied mathematics from Yangzhou University, Jiangsu, China, and the Ph.D. degree in pattern recognition and intelligence systems from Donghua University, Shanghai, China, in 2009 and 2012, respectively. He was a Research Associate with The Hong Kong Polytechnic University, Kowloon, Hong Kong, from 2012 to 2013. From July 2014 to Aug 2014, he was a DAAD fellow with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. He is currently an associated professor with the Department of Mathematics, Yangzhou University. His current research interests include synchronization/consensus, networked control systems, and genetic regulatory networks. Dr. Zhang is a very active reviewer for many international journals.
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Ahmed Alsaedi’s Biography: Ahmed Alsaedi obtained his Ph.D. degree from Swansea University (UK) in 2002. He has a broad experience of research in applied mathematics. His fields of interest include dynamical systems, nonlinear analysis involving ordinary differential equations, fractional differential equations, boundary value problems, mathematical modeling, biomathematics, Newtonian and Non-Newtonian fluid mechanics. He has published several articles in peer-reviewed journals. He has supervised several M.S. students and executed many research projects successfully. He is reviewer of several international journals. He served as the chairman of the mathematics department at KAU and presently he is serving as director of the research program at KAU. Under his great leadership, this program is running quite successfully and it has attracted a large number of highly rated researchers and distinguished professors from all over the world. He is also the head of NAAM international research group at KAU.
by the membership of international and national Committees, leadership and motivation, numerous scholarships and fellowships, conducted research projects, convened many national and international conferences, seminars delivered and attended conferences the world over, established research collaboration with leading international scientists, associate editor/editorial membership of the international journals including ISI, reviewer of the international journals and MS and PhD students produced. His publications in diverse areas are in high impact factor journals. His research work has total ISI WEB citations (11730) and h-index (52) at present. He has received many national and international awards including Tamgha-i-Imtiaz, Sitara-i-Imtiaz, Khwarizmi Int. award, ISESCO Int. award, TWAS prize for young scientists, Alexander-Von-Humboldt fellowship etc.
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Tasawar Hayat’s Biography: Dr. Hayat (born in Khanewal, Punjab), Distinguished National Professor and Chairperson of Mathematics Department at Quaid-I-Azam University is renowned worldwide for his seminal, diversified and fundamental contributions in models relevant to physiological systems, control engineering, climate change, renewable energy, low-carbon technologies, environmental issues, non-Newtonian fluids, wave mechanics, homotopic solutions, stability, nanofluids and in several other areas. He has a honor of being fellow of Pakistan Academy of Sciences, Third World Academy of Sciences (TWAS) and Islamic World Academy of Sciences in the mathematical Sciences. His national and international recognition is evident
Accepted Manuscript
Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay Mohmmed Alsiddig Alamin Ahmed, Yurong Liu, Wenbing Zhang, Ahmed Alsaedi, Tasawar Hayat PII: DOI: Reference:
S0925-2312(17)30889-5 10.1016/j.neucom.2017.05.039 NEUCOM 18457
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
23 March 2017 18 May 2017 18 May 2017
Please cite this article as: Mohmmed Alsiddig Alamin Ahmed, Yurong Liu, Wenbing Zhang, Ahmed Alsaedi, Tasawar Hayat, Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.05.039
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Exponential Synchronization for a Class of Complex Networks of Networks with Directed Topology and Time Delay
In practice, time delay can not be overlooked in the networked environment due to the finite switching speed of amplifiers [4], and time delay may result in undesirable dynamic behaviors such as instability, oscillations or poor performances. In recent years, many excellent papers have been reported on synchronization of complex networks with time delays caused by diverse factors in different forms, i.e., node delays [5], [6], coupling delay [7]–[9], delays in both nodes and couplings [10]–[12]. As we know, in real-world networks, there exists a common requirement to regulate the complex networks. Thus, it is important to study the synchronization control of complex networks. However, due to the fact that the complex networks consist of a large number of nodes and it is very difficult to apply controllers on all the nodes. Thus, the pinning control strategy that applies controllers to only a fraction of nodes of complex networks is very helpful for controlling complex networks. During the past decades, numerous results have been reported on pinning control and synchronization of complex dynamical networks [13]–[27]. For instance, pinning synchronization of complex networks has been investigated in [14], in which, the authors have considered both specifically and randomly pinning schemes for scale-free networks and then they concluded that the nodes with highly connected nodes should be pinned first. Pinning synchronization was investigated for a class of general complex networks with hybrid coupling in [21], where constant coupling, discrete time delayed coupling and distributed time delayed coupling are considered. Moreover, in [22], by means of the M matrix theory, pinning synchronization of complex networks has been discussed deeply, where both undirected and directed network topology have been investigated, respectively. Some criteria were obtained for synchronization of delayed discrete nonlinear dynamical networks with the identical or nonidentical topology structure by pinning control in [24]. In [25], by proposing a new kind of Lyapunov function, synchronization of complex networks with switched directed network topologies has been extensively studied. Very recently, cluster synchronization was investigated for a class of general complex dynamical networks under pinning control scheme in [26]. In [27], the cluster synchronization in network of linear
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Abstract—The pinning synchronization problem for a class of complex networks of networks is investigated in this paper, in which the network topology is assumed to be directed. The concerned complex networks of networks are composed of both leaders’ network and followers’ networks (also called “subnetworks”), where the leaders’ network and followers’ networks are regarded as the nodes of the networks of networks, followers’ networks can receive the information from leaders’ network, but not the reverse. Then, by means of the Lyapunov stability theory, graph theory, Barbalat’s Lemma and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization for the general complex networks of networks. In addition, theoretical results are verified through an illustrative example.
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Mohmmed Alsiddig Alamin Ahmed, Yurong Liu∗ , Wenbing Zhang, Ahmed Alsaedi, and Tasawar Hayat
I. I NTRODUCTION
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Index Terms—Complex network; Network of networks; Exponential synchronization; Pinning control; Directed topology; Time delay.
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Complex networks are composed of a large number of nodes and the interactions between them. Many real-world networks can be can be abstracted as a network composed by interactive individuals such as the protein interaction networks, WWW, Power grids, biological networks, citation networks, communication networks, social networks and so on. Complex networks can exhibit many interesting collective phenomena, including self-organization, traveling waves, defect propagation, synchronization, and spatiotemporal chaos [1], [2]. Among the above various collective phenomena, synchronization have received a lot of research attention, in which the main objective of complex networks is to seek a common behavior [3]. This work is supported by National Natural Science Foundation of China under Grants 61374010, 61503328, 11671008, and Top Talent Plan of Yangzhou University. M. A. A. Ahmed, Y. Liu, and W. Zhang are with the Department of Mathematics, Yangzhou University, Yangzhou 225002, P. R. China; and Y. Liu is also with the Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. A. Alsaedi is with the NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. T. Hayat is with the NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia, and also with the Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. ∗ Corresponding author. (Email: [email protected] (Y.Liu).)
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Although, there are fruitful results have been published on pinning synchronization of complex networks, in most existing result, it is implicitly assumed that each node can be synchronized to an isolated node, i.e., the control objective is to control complex networks to an isolated node. However, with the development of information technology, the network size becomes larger and larger, and therefore complex networks may contain many clusters. Actually, in reality, different kinds of critical infrastructure may be coupled together, such as systems of water and food supply, communications, fuel, financial transactions and power generation and transmission [28]. If we think of each cluster as a subnetwork, then complex networks become the so called complex networks of networks [29]–[31]. In complex networks of networks, subnetworks can be seen as nodes of complex networks and the interactions can be seen as the interactions among the subnetworks. Recently, a complex networks of networks model is proposed in [30] and some synchronization criteria are established on pinning synchronization of complex networks of networks. More recently, exponential synchronization was investigated for a class of undirected complex networks of networks with time delays under pinning adaptive control scheme in [31]. The hybrid combinatorial synchronization on multiple dynamical subnetworks with different dimensions of nodes, where the nodes in each subnetwork have different dynamical behaviors has been investigated in [32]. It should be kept in mind that in [30], [32], time delay has been overlooked due to the difficulties in handling the coexistence of time delay and the pinning controller. Motivated by the aforementioned discussions, we aim to solve the exponential synchronization problem for complex networks of networks with directed topology and time delay by applying pinning control. The concerned complex networks of networks are made up of a leader’s network and many followers’ networks. By using the Lyapunov stability theory, graph theory, Barbalat’s Lemma and the Kronecker product techniques, sufficient conditions are derived to ensure the exponential synchronization of the general complex networks of networks. The main contribution of this paper can be summarized as follows: (1) the pinning synchronization control problem for complex networks of networks with both directed topology and time delay will be first addressed. Pinning control scheme is implemented to achieve the exponential synchronization of complex networks of networks. (2) one quantitative measure is distilled from the network’s coupling matrix to characterize the synchronization of the leaders’ network. The calculation of such a quantity can easily and conveniently be realized by using a normal computer. Furthermore, Barbalats Lemma is used to ensure that the error between the subnetworks states and the average
of leaders can converge to zero exponentially. The paper is organized as follows. In Section II, some preliminaries and problem formulation are presented. The main results for the pinning synchronization on complex networks of networks with directed topology and delayed nodes are given in Section III. Section IV provides a numerical simulation example to validate theoretical results. Conclusions are finally drawn in Section V. Notations: The notations used throughout this paper are fairly standard. Rn denotes the n−dimensional Euclidean space and Rm×n represents m × n real matrices. X > 0 (X ≥ 0) is used to denote the positive definite (semi positive definite) matrix, while X > Y (X ≥ Y ) means X − Y > 0 (X −Y ≥ 0). The ith row and the ith column of G is called the ith row-column pair of G [14]. We denote Gl ∈ R(N −l)×(N −l) as a minor matrix of G ∈ RN ×N by removing arbitrary l (1 ≤ l < N ) row-column pairs of G [14]. Let In be an n−dimensional identity matrix. Denote 1N ∈ RN be a vector with each entry being 1. Let 1M ×M be a matrix with each entry being 1. The superscript ”T ” represents the transpose, diag{· · · } stands for a diagonal matrix. k · k denotes the Euclidean norm of a vector. Denote λmax (X) (λmin (X)) as the maximum (minimum) eigenvalue of the matrix X, respectively. Let X ⊗ Y be the Kronecker product of two matrices X and Y [34]. Denote λ2 (X) as the second largest eigenvalue of the matrix X.
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systems via pinning control technique in a general framework that the couplings among nodes are of partial-state and directed has been studied.
II. PRELIMINARIES AND PROBLEM FORMULATION Suppose that there is a network containing one leaders’ network and m subnetworks (followers’ networks) Gk = {υk , εk }, with N (k) nodes in each subnetworks k, where k = 1, 2, ..., m. Let N = N (1) +N (2) +...+N (m) represent the total nodes in the subnetworks. The leaders’ network is made up of M identical nodes with linearly diffusive coupling and is described as follows: S˙ i (t) = f (Si (t), Si (t − τ )) +c
M X
j=1,j6=i
(s)
gij Γ (Sj (t) − Si (t)) , i = 1, 2, · · · , M.
(1)
Here Si (t) = (Si1 (t), Si2 (t), · · · , Sin (t))T ∈ Rn is the state vector of the ith leader, τ > 0 is the constant time delay, f : Rn × Rn → Rn is a continuous vector-valued function, c > 0 is the coupling strength, Γ = diag(γ1 , γ2 , · · · , γn ) ∈ Rn×n is a non-negative definite inner coupling matrix where γi > 0 if two leaders can communicate through the ith state, and γi = 0 (s) otherwise. G(s) = gij is the coupling configuration M ×M matrix representing the topological structure for the network, where if there exists a connection from leader j to leader i, (s) (s) then gij > 0; otherwise, gij = 0 (j 6= i). It means that the leaders’ network topology could be directed. The diagonal entries of G(s) are determined by the following diffusive
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= −
N P
(s)
j=1,j6=i
gij . Equivalently, we
can rewrite the network (1) in a simpler form as follows: S˙ i (t) = f (Si (t), Si (t − τ )) + c
M X
(s)
(2)
gij ΓSj (t),
j=1
i = 1, 2, ..., M . We denote the synchronization error for the leaders by ej = M e e = P ξj Sj (t) is the average state of Sj (t) − S(t), where S(t) j=1
all the leaders’ states, where ξ = (ξ1 , · · · , ξM )T is the positive left eigenvector of G(s) associated with the eigenvalue zero e can be described satisfying ξ T 1M = 1. The dynamics of S(t) by M
M
i=1
i=1
j=1
M P
M P
M
X X X (s) e˙ S(t) = ξi f (Si (t), Si (t − τ )) + c ξi gij ΓSj (t). =
0, one has
i=1
[(ξ T G(s) ) ⊗ Γ] = 0, it then follows that
ξi
j=1
(s)
gij ΓSj (t)
M
X e˙ S(t) = ξi f (Si (t), Si (t − τ )).
=
(3)
i=1
j=1
i=1
(s)
ξi f (Si (t), Si (t − τ ))
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+c
M X
M X
M
Therefore, the error dynamics of the leaders’ network (2) can be obtained as follows: e˙ i (t) = f (Si (t), Si (t − τ )) −
gij Γej (t), i = 1, 2, · · · , M.
(4)
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Let e(t) = (eT1 (t), · · · , eTM (t))T , S(t) = T T T (S1 (t), · · · , SM (t)) and f (S(t), S(t − τ )) = T T T (f (S1 (t), S1 (t − τ )), · · · , f (SM (t), SM (t − τ ))) . Then, the general leaders’ network (5) can be recast in a compact matrix form as follows:
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e(t) ˙ = [(IM − 1M ξ T ) ⊗ In ]f (S(t), S(t − τ ))
(k)
(k)
(k)
j=1,j6=i
the ith node in the kth subnetwork, and (k)
(k)
(k)
e ui (t) = −di Γ(xi (t) − S(t)) ∈ Rn ,
i = 1, 2, · · · , l(k) , (7)
are n-dimensional linear feedback controllers with all the (k) control gains di > 0, j = 1, 2, · · · , M , k = 1, 2, · · · , m. Denote the synchronization error for each subnetwork k by (k) (k) e ei (t) = xi (t) − S(t), where i = 1, 2, · · · , N (k) and k = 1, 2, · · · , m. Definition 1: The leaders’ network (1) and the followers’ networks (6) are said to be exponentially synchronized if there exists constants α1 > 0, α2 > 0, β2 > 0 and β2 > 0 such that for any initial conditions:
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Since ξ T G(s)
(k)
where xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn is the state vector of the ith node inthe kth subnetwork, τ > 0 is the (k) constant time delay, G(k) = gij is the coupling N (k) ×N (k) matrix representing the topological structure of the kth subnetwork, where if there exists a connection between nodes i and (k) (k) j, then gij > 0; otherwise gij = 0 (j 6= i). It means that the subnetwork topology could be directed. The diagonal entries of G(k) are determined by the following diffusive coupling N P (k) (k) gij = deg(k) (i) is the degree of condition: gii = −
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(s)
coupling condition: gii
+ c(G(s) ⊗ Γ)e(t). (5)
e
kSi (t) − S(t)k ≤ α1 exp(−β1 t), i = 1, 2, · · · , M, (k) e kxi (t) − S(t)k ≤ α2 exp(−β2 t), i = 1, 2, · · · , N (k) ,
k = 1, 2, · · · , m. Definition 2: For an N × N irreducible Laplacian matrix G with non-negative off-diagonal elements, which satisfies the diffusive coupling, the quantity a(G) is defined as follows. Let ξ = (ξ1 , ξ2 , · · · , ξN )T ∈ RN be the unique normalized left eigenvector of G corresponding to eigenvalue zero satisfying N P ξi = 1 and Ξ = diag{ξ1 , ξ2 , · · · , ξN }. Then a(G) is i=1
defined to be the second largest eigenvalue of the symmetric b = ΞG + GT Ξ, that is matrix G a(G) = λ2 (ΞG + GT Ξ).
(8)
In the following, a useful assumption and lemmas are introduced. Assumption 1: For the vector function f (x(t), x(t − τ )), suppose that the uniform Lipschitz condition holds, i.e., for any xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T and s(t) = (s1 (t), s2 (t), · · · , sn (t))T . There exists a positive constants γ > 0 and β > 0, such that
For followers’ networks, we assume that the first l(k) nodes are controlled in the kth subnetwork for k = 1, 2, · · · , m. Considering the leaders’ network (2), the pinning controlled kth subnetwork with delayed nodes can be described by: (k) N X (k) (k) kf (xi (t), xi (t − τ )) − f (s(t), s(t − τ ))k (k) (k) (k) x˙ i (t) = f (xi (t), xi (t − τ )) + c gij Γxj (t) ≤ γkxi (t) − s(t)k + βkxi (t − τ ) − s(t − τ )k, (9) j=1 (k) (k) where i = 1, 2, · · · , N . + ui (t), i = 1, 2, · · · , l , (6) Lemma 1 (Schur Complement [35]): The following linear (k) N X (k) (k) (k) (k) (k) matrix inequality: x˙ i (t) = f (xi (t), xi (t − τ )) + c gij Γxj (t), j=1 Q(x) S(x) >0 S(x)T R(x) i = l(k) + 1, l(k) + 2, · · · , N (k) ,
ACCEPTED MANUSCRIPT 4
1) Q(x) > 0, R(x) − S(x)T Q(x)−1 S(x) > 0; 2) R(x) > 0, Q(x) − S(x)R(x)−1 S(x)T > 0.
Lemma 2 ( [37]): For matrices A, B, C and D with appropriate dimensions, the Kronecker product ⊗ satisfies: 1) 2) 3) 4)
(ρA) ⊗ B = A ⊗ (ρB), where ρ is a constant; (A + B) ⊗ C = A ⊗ C + B ⊗ C; (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD); (A ⊗ B)T = AT ⊗ B T .
Lemma 3 ( [36]): For an irreducible matrix L with nonnegative off-diagonal elements, which satisfies the diffusive coupling, we have the following propositions:
H (k) =
cα G(k) + G(k)T IN (k) + c , λmin (Γ) 2
(10)
where α is positive scalar. (k) (k) (k) (k) Let D(k) = diag(d1 , d2 , · · · , dl(k) , 0, · · · , 0) with di > | {z } N(k) −l(k)
0, i = 1, 2, · · · , l(k) . Using matrix decomposition, we have (k) ˜ (k) B (k) U −D H (k) − D(k) = , (11) B (k)T C (k) (k)
(k)
(k)
(k)
where U (k) = (uij )l(k) ×l(k) , uij = uji = hij , i, j = ˜ (k) = diag(d(k) , d(k) , · · · , d(k) 1, 2, · · · , l(k) , D ), B (k) = 1 2 l(k) (k) (k) (k) (bij )l(k) ×(N (k) −l(k) ) , bij = hij , i = 1, 2, · · · , l(k) , j = l(k) + 1, · · · , N (k) , and cα G(k) + G(k)T IN (k) + c C (k) = λmin (Γ) 2 l(k)
AN US
1) If λ is an eigenvalue of L and λ 6= 0, then Re(λ) < 0. 2) L has an eigenvalue 0 with multiplicity 1 and the right eigenvector [1, 1, · · · , 1]T . 3) Suppose that ξ = (ξ1 , ξ2 , · · · , ξN )T ∈ RN satisfy N P ξi = 1 is the normalized left eigenvector of L
the Kronecker product techniques, some criteria ensuring the exponential synchronization of the general complex networks of networks will be derived. Construct the following symmetric matrix H (k) = (k) (hij )N (k) ×N (k) :
CR IP T
where Q(x) = Q(x)T , R(x) = R(x)T , is equivalent to one of the following condition: 1)
i=1
corresponding to eigenvalue 0. Then, ξi > 0 hold for all i = 1, 2, · · · , N . Furthermore, if L is symmetric, then we have ξi = 1/N for i = 1, 2, · · · , N .
Lemma 4 ( [22]): Assume that Φ = (φij )N ×N is symmetric. Let D = diag(d1 , · · · , dl , 0, · · · , 0) in which di > 0, | {z } N −l
AC
CE
PT
ED
M
i = 1, 2, · · · , l. When di > 0, i = 1, 2, · · · , l are sufficiently large, Φ − D < 0 is equivalent to Φl , where Φl is the minor matrix of Φ by removing its first l row-column pairs. Lemma 5 ( [22]): Assume that U , B are N by N Hermitian matrices. Let α1 ≥ α2 ≥ · · · ≥ αN , β1 ≥ β2 ≥ · · · ≥ βN , and ζ1 ≥ ζ2 ≥ · · · ≥ ζN be eigenvalues of U , B and U + B, respectively. Then, one has αi + βN ≤ ζi ≤ αi + β1 , i = 1, 2, · · · , N . Lemma 6 (Barbalat’s Lemma [39]): Let f be non-negative function defined on [0, +∞). if f Lebesgue integrable on [0, +∞) and is uniformly continuous on [0, +∞), then lim f (t) = 0. t→+∞ In the rest of this paper, our main aim is to deal with the exponential synchronization problem of the complex networks of networks (1) and (6) with directed topology and delayed nodes via pinning feedback control. By utilizing Lyapunov-Krasoviskii functional, Barbalat’s Lemma and Kronecker product techniques, we aim to derive sufficient conditions such that the complex networks of networks (1) and (6) with directed topology and delayed nodes are globally exponentially synchronized. III. M AIN R ESULTS In this section, we investigate the exponential synchronization of the general complex networks of networks with delayed nodes in the case that the coupling matrices G(s) and G(k) are irreducible. based on the Lyapunov stability method and
is the minor matrix of H (k) by removing its first l(k) rowcolumn pairs. Assume that ξ = (ξ1 , ξ2 , · · · , ξM )T and ξ (k) = (k) (k) (k) (ξ1 , ξ2 , · · · , ξN (k) )T are the normalized left eigenvectors of the configuration coupling matrices G(s) and G(k) with respect (k) M NP P (k) to zero eigenvalue satisfying ξi = 1 and ξi = 1, i=1
i=1
respectively. Since the coupling configuration matrices G(s) and G(k) , k = 1, 2, · · · , m are irreducible, according to lemma 3, we can conclude that ξj > 0, j = 1, 2, · · · , M and (k) ξi > 0, i = 1, 2, · · · , N (k) . Let Ξ = diag{ξ1 , ξ2 , · · · , ξM } is a positive diagonal matrix of the leaders’ network, Ξ(k) = (k) (k) (k) diag(ξ1 , ξ2 , · · · , ξN(k) ) is a positive diagonal matrix of the kth subnetwork. We are now in a position to give the main results of this paper as follows. Theorem 1: Suppose that Assumption 1 holds, and that there exists a positive scalar α such that the following conditions are satisfied: [2γ + 1 − cρ1 ]Ξ + ξξ T βΞ < 0, (12) βΞ −Ξ γ+1+
(k) di
β2 δ + − cα < 0, 2 4
(13)
(k) (k) (k) −1 (k)T > λmax U − B C B , λmax
G(k) + G(k)T 2
l(k)
i = 1, · · · , l(k) , (14)
<−
α , λmin (Γ)
(15)
ACCEPTED MANUSCRIPT 5
It follows from LMIs (12) and (16) that there exists a positive constant , which may be very small, such that [ + 2γ + exp(τ ) − cρ1 ]Ξ + ξξ T βΞ < 0, (17) Φ1 = βΞ −Ξ (k)
Φ2
[ + γ + exp(τ ) − cα + 2δ ]IN (k) = β 2 IN (k)
β 2 IN (k)
−IN (k)
< 0, (18)
where k = 1, 2, · · · , m. Consider the following Lyapunov functional:
e e − τ ))] eT (t)(Ξ ⊗ In )[1M ⊗ f (S(t), S(t e e − τ ))](Ξ ⊗ In ) = [1TM ⊗ f T (S(t), S(t
× [(IM − 1M ξ T ) ⊗ In ]S(t) e e − τ )) S(t) = [1TM Ξ(IM − 1M ξ T )] ⊗ f T (S(t), S(t e e − τ )) S(t) = 0, = [ξ T (IM − 1M ξ T )] ⊗ f T (S(t), S(t (23)
eT (t)(Ξ ⊗ In )[(1M ξ T ) ⊗ In ]f (S(t), S(t − τ ))
= f T (S(t), S(t − τ ))[(ξ1TM ) ⊗ In ](Ξ ⊗ In )
× [(IM − 1M ξ T ) ⊗ In ]S(t) = f T (S(t), S(t − τ )) [ξ1TM Ξ(IM − 1M ξ T )] ⊗ In S(t) = f T (S(t), S(t − τ )) [ξξ T (IM − 1M ξ T )] ⊗ In S(t) = 0.
AN US
In addition,
(24)
2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t)
(19)
VL (t) = VL1 (t) + VL2 (t),
= 2ceT (t)(ΞG(s) ⊗ Γ)e(t)
where VL1 (t) = eT (t)(Ξ ⊗ In )e(t) exp(t), Zt
t−τ
eT (s)(Ξ ⊗ In )e(s) exp((s + τ ))ds.
M
VL2 (t) =
Then, by calculating the derivative of VL (t) along the trajectory of (5) one can obtain that
ED
PT
+ eT (t)(Ξ ⊗ In )e(t) exp(t) = 2eT (t)(Ξ ⊗ In ) [(IM − 1M ξ T ) ⊗ In ] × f (S(t), S(t − τ )) exp(t)
+ 2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t) exp(t)
CE
+ eT (t)(Ξ ⊗ In )e(t) exp(t),
AC
V˙ L2 (t) = eT (t)(Ξ ⊗ In )e(t) exp(τ ) − eT (t − τ )(Ξ ⊗ In )e(t − τ ) exp(t).
(20)
(25)
where G = ΞG + G Ξ. T Since ξ = (ξ1 , ξ2 , · · · , ξM ) is the normalized left eigenvector of the configuration coupling matrix G(s) with respect to zero M P eigenvalue satisfying ξi = 1. It can be easily see that (s)
(s)T
i=1
By Assumption 1, one obtains e e 2eT (t)(Ξ⊗In ) f (S(t), S(t−τ ))−1M ⊗f (S(t), S(t−τ )) =2
M X i=1
e e − τ )) ξi eTi (t) f (Si (t), Si (t − τ )) − f (S(t), S(t
≤2 (21)
Combination of (20) and (21) yields ˙ VL (t) = exp(t) 2eT (t)(Ξ ⊗ In ) f (S(t), S(t − τ )) e e − τ )) + eT (t)(Ξ ⊗ In )e(t) − 1M ⊗ f (S(t), S(t
+ 2ceT (t)(Ξ ⊗ In )(G(s) ⊗ Γ)e(t) + eT (t)(Ξ ⊗ In ) T
× e(t) exp(τ ) − e (t − τ )(Ξ ⊗ In )e(t − τ )
b (s)
= ceT (t)[(ΞG(s) + G(s)T Ξ) ⊗ Γ]e(t) b (s) ⊗ Γ]e(t), = ceT (t)[G
b (s) is a zero-sum symmetric matrix with nonnegative offG diagonal elements. Moreover, from the irreducibility of G(s) , b (s) = ΞG(s) + G(s)T Ξ irreducible. we conclude that G
V˙ L1 (t) = 2e (t)(Ξ ⊗ In )e(t) ˙ exp(t) T
and
Following [38], since e(t) = [(IM − 1M ξ T ) ⊗ In ]S(t) and ξ T 1M = 1, one has
CR IP T
where ρ1 = −a G(s) λmin (Γ)/λmax (Ψ) and k = 1, 2, · · · , m. Then, the directed controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization. Proof: By Lemma 1, the condition (13) is equivalent to that [γ + 1 − cα + 2δ ]IN (k) β2 IN (k) (k) e Φ2 = < 0. (16) β −IN (k) 2 IN (k)
− 2eT (t)(Ξ ⊗ In )[(1M ξ T ) ⊗ In ]f (S(t), S(t − τ )) e e − τ ))] . (22) + 2eT (t)(Ξ ⊗ In )[1M ⊗ f (S(t), S(t
M X i=1
ξi kei (t)k γkei (t)k + βkei (t − τ )k . (26)
Combining (23)-(26), and adding a vanishing terms −eT (t)(Ξ ⊗ cρ1 In )e(t) + eT (t)(Ξ ⊗ cρ1 In )e(t), −eT (t)(ξξ T ⊗ cρ1 In )e(t) + eT (t)(ξξ T ⊗ cρ1 In )e(t), we can obtain that V˙ L (t) ≤ exp(t)η T (t)Φ1 η(t) b (s) ⊗ Γ) + (Ψ ⊗ cρ1 In ) e(t) + exp(t)eT (t) c(G ≤ exp(t)η T (t)Φ1 η(t) + exp(t)eT (t) cρ1 (s) b Γ e(t), × c(G ⊗ Γ) + Ψ ⊗ λmin (Γ)
(27)
where η(t) = (eT (t), eT (t − τ ))T , Φ1 is defined in (17) and Ψ = Ξ − ξξ T .
ACCEPTED MANUSCRIPT 6
b (s) = G b (s)T is an irreducible symmetric matrix, there Since G exist a unitary matrix U = (u1 , u2 , · · · , uM ) such that
−a G(s) λmin (Γ)/λmax (Ψ), it follows from (27) that V˙ L (t) ≤ exp(t)η T (t)Φ1 η(t) + exp(t) ca G(s) +
b (s) = U Λ1 U T , G
U T U = I,
b (s) ), λ2 (G b (s) ), · · · , λM (G b (s) )}, Λ1 = diag{λ1 (G
b (s) ) > λ2 (G b (s) ) ≥ · · · ≥ λM (G b (s) ). 0 = λ1 (G Now, introducing transformation y(t) = (U T ⊗ In )e(t),
e(t) = (U ⊗ In )y(t),
b (s) ⊗ Γ]e(t) = cy T (t)(U T ⊗ In )[G b (s) ⊗ Γ] ceT (t)[G × (U ⊗ In )y(t) b (s) U ⊗ Γ)y(t) = cy T (t)(U T G = cy T (t)(Λ1 ⊗ Γ)y(t) M X i=1
b (s) )y T (t)Γyi (t) λi (G i
b (s) ≤ cλ2 G
yiT (t)Γyi (t)
i=2 M X
yiT (t)Γyi (t).
(28)
i=2
ED
= ca G(s)
M X
M
=c
CE
PT
According to the structure of matrix Ψ, we can conclude that Ψ is an irreducible symmetric matrix with negative off-diagonal elements, and also, Ψ satisfies the diffusive coupling condition. Therefore, we can conclude that λmax (Ψ) > 0. From the construction of Ψ, it follows that Ψ.u1 = (0, 0, · ·"· , 0)T := 0n #∈ 0 0Tn RN . Further, we can obtain that U T ΨU = T e ΨU e , 0n U e = [u2 , u3 , · · · , uM ] satisfying U eT U e = IM −1 . Using where U the above analysis, one has
AC
cρ1 Γ]e(t) λmin (Γ) cρ1 y T (t)[U T ΨU ⊗ Γ]y(t) λmin (Γ) cρ1 e T ΨU e ⊗ Γ]e yeT (t)[U y (t) λmin (Γ) cρ1 eT U e ⊗ Γ]e λmax (Ψ)e y T (t)[U y (t) λmin (Γ) M X cρ1 λmax (Ψ) yiT (t)Γyi (t), λmin (Γ) i=2
eT (t)[Ψ ⊗ = = ≤ =
T where ye = [y2T (t), · · · , yM (t)]T . By using (28), (29) and
equality
ρ1
yiT (t)Γyi (t)
i=2
= exp(t)η T (t)Φ1 η(t).
(30)
Using (17), the inequality (27) gives that V˙ L (t) ≤ 0, which implies that VL (t) is bounded. Therefore, we can conclude that ei (t) exponentially approach to zero. (k) (k) e On the other hand, since ei (t) = xi (t) − S(t), one can get the following error system between the nodes in the followers’ networks and the leaders: M X (k) (k) (k) e ˙ (t) = f (x (t), x (t − τ )) − ξi f (Si (t), Si (t − τ )) i i i i=1 (k) N M X (k) (k) X (k) (k) + c g Γe (t) − dij Γei (t), ij j j=1 j=1 i = 1, 2, · · · , l(k) M X (k) (k) (k) e ˙ (t) = f (x (t), x (t − τ )) − ξi f (Si (t), Si (t − τ )) i i i i=1 (k) N X (k) (k) +c gij Γej (t), j=1 i = l(k) + 1, l(k) + 2, · · · , N (k) . (31) (k)T (k)T Let e(k) (t) = (e1 (t), · · · , eN (k) (t))T , f (x(k) (t), x(k) (t − (k) (k) (k) (k) τ )) = (f T (x1 (t), x1 (t − τ )), · · · , f T (xN (k) (t), xN (k) (t − T τ ))) and k = 1, 2, · · · , m. Then, we can rewrite (31) in a compact form as follows: e˙ (k) (t) = f (x(k) (t), x(k) (t − τ )) − 1N (k) e e − τ )) + c G(k) ⊗ Γ e(k) (t) ⊗ f (S(t), S(t − D(k) ⊗ Γ e(k) (t) + F (e, S), (32)
e e − τ )) − [(1M ξ T ) ⊗ where F (e, S) = 1N (k) ⊗ f (S(t), S(t (k) In ]f (S(t), S(t − τ )) and D is defined above. Construct the following Lyapunov functional for system (32): Vs (t) = Vs1 (t) + Vs2 (t),
(33)
where m
Vs1 (t) = (29)
=
1 X (k)T e (t)(Ξ(k) ⊗ In )e(k) (t) exp(t), 2 k=1
Vs2 (t) = the
M X
AN US
T T where y(t) = y1T (t), y2T (t), · · · , yM (t) , yr (t) ∈ Rn , r = 1, 2, · · · , M . Then, we have
×
CR IP T
where
cρ1 λmax (Ψ) λmin (Γ)
m Zt X
k=1t−τ
e(k)T (s)(Ξ(k) ⊗ In )e(k) (s) exp((s + τ ))ds.
ACCEPTED MANUSCRIPT 7
Then, the derivative of Vs (t) along the trajectory of (32) yields
In )e(k) (t) + e(k)T (t)(cαΞ(k) ⊗ In )e(k) (t), one can obtain (k)
V˙ s (t) ≤ exp(t) e(k)T (t)(Ξ(k) ⊗ In )e˙ (k) (t) exp(t)
k=1 m X
+
k=1
= x
k=1 (k)
e(k)T (t)(Ξ(k) ⊗ In ) f (x(k) (t),
(t − τ )) − 1N (k) (k)T
+ ce
(t)(Ξ
(k)
(k)
⊗ In )(G
k=1
⊗ Γ)e
+
m X
(k)
≤ exp(t)
(t)
(k)
k=1
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t) exp(t),
e(k)T (t)(Ξ(k) ⊗ In ) (cαIN (k) ⊗ In )
k=1
⊗ Γ)e (t) + e(k)T (t)(Ξ(k) ⊗ In )F (e, S) exp(t) (t)(Ξ
⊗ In )(D
(k)
(k)
k=1 i=1 m X
+ exp(t)
m N X X
(k) (k)T
ξi ηi
k=1 i=1
m X
e
(k)T
(k)
(t)(Ξ
k=1 (k)T
(k) (k)
(t)Φ2 ηi (t)
⊗ In )
cα I (k) λmin (Γ) N
+G − D(k) ⊗ Γ e(k) (t) 2 m X
2
1 + (Ξ(k) ⊗ In ) F (e, S) exp(t) 2δ (k)
+c (34)
G
AN US
−e
(k)
e e − τ )) ⊗ f (S(t), S(t
(k) (k)
(t)Φ2 ηi (t)
(k) G + G(k)T (k) −D ⊗ Γ e(k) (t) + c 2 m
2
1 X (k) (Ξ ⊗ In ) F (e, S) exp(t) + 2δ
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t)
m X
(k)T
+ exp(t)
(k) (k)T
ξi ηi
CR IP T
V˙ s1 (t) =
m X
m N X X
k=1
(k)
= exp(t)
and
m N X X
(k) (k)T
ξi ηi
(k) (k)
(t)Φ2 ηi (t)
k=1 i=1
+ exp(t)
M
k=1 (k)T
(t − τ )(Ξ(k) ⊗ In )e(k) (t − τ ) exp(t). (35)
m X
=
≤
(k)
m N 1XX
2 δ 2
k=1 i=1 m X (k)T
e
k=1
(k)
ξi
m
2
1 X (k) (Ξ ⊗ In ) F (e, S) exp(t), 2δ
(37)
k=1
(k)
(k)T
(k)T
(k)
where ηi (t) = (ei (t), ei (t − τ ))T , Φ2 is defined in (18) and k = 1, 2, · · · , m. By Lemma 4, from (11) and the given condition (14), we know that H (k) − D(k) < 0 is equivalent to G(k) + G(k)T cα (k) C = I (k) + c <0 λmin (Γ) N 2 l(k) (k)
di
(k)T 2ei (t)F (e, S)
> λmax U (k) − B (k) C (k)
−1
B (k)T ,
i = 1, · · · , l(k) , k = 1, · · · , m.
(t)(Ξ(k) ⊗ In )e(k) (t)
m
2 1 X (k) (Ξ ⊗ In ) F (e, S) . + 2δ
+
when the pinning feedback gains satisfy
e(k)T (t)(Ξ(k) ⊗ In )F (e, S)
AC
k=1
CE
For δ > 0, we have
e(k)T (t)(Ξ(k) ⊗ In )
k=1 (k) × H − D(k) ⊗ Γ e(k) (t)
e(k)T (t)(Ξ(k) ⊗ In )e(k) (t) exp(τ )
ED
−e
m X
PT
V˙ s2 (t) =
m X
(36)
k=1
Combining (34) with (35), and then by using Assumption 1, (36) and adding a vanishing term −e(k)T (t)(cαΞ(k) ⊗
Therefore, we only need to show that cα G(k) + G(k)T C (k) = IN (k) + c < 0. λmin (Γ) 2 l(k) By lemma 5, we have cα G(k) + G(k)T λmax IN (k) + c ≤ λmin (Γ) 2 l(k) (k) cα G + G(k)T + cλmax . (38) λmin (Γ) 2 l(k)
ACCEPTED MANUSCRIPT 8
indicates that C (k) =
l(k)
cα G(k) + G(k)T IN (k) + c λmin (Γ) 2
Therefore, we can conclude that H
(k)
−D
(k)
(39)
Considering that Γ > 0 in (37), then (13), (14) and (15) can guarantees that m X V˙ s (t) ≤ − (Ξ ⊗ In )ke(k) (t)k2 exp(t) k=1
+
m
2 1 X (k) (Ξ ⊗ In ) F (e, S) exp(t) 2δ k=1
+
(k) 2 ˜ ξi kxi (t) − S(t)k exp(t)
k=1 i=1 m 1 X (k)
(Ξ
2δ
k=1
Therefore, we have
Vs (t) − Vs (0) ≤ −
2 ⊗ In ) F (e, S) exp(t).
(k) Zt X m N X
0 k=1 i=1
(k)
2 ˜ ξi kxi (θ) − S(θ)k
M
=−
which implies that
0 k=1 i=1
(k)
2 ˜ ξi kxi (θ) − S(θ)k exp(θ)dθ ≤ Vs (0)
Zt X m
2 (Ξ(k) ⊗ In ) F (e, S) exp(θ)dθ.
CE
1 + 2δ
PT
(k) Zt X m N X
ED
× exp(θ)dθ Zt X m
2 1 + (Ξ(k) ⊗ In ) F (e, S) exp(θ)dθ 2δ 0 k=1
0 k=1
AC
From (12), it is easy to see that F (e, S) exponentially approaches to zero. Then (k) Zt X m N X (k) 2 ˜ ξi kxi (θ) − S(θ)k dθ < +∞, 0 k=1 i=1
which implies that
(k)
m N X X
k=1 i=1
(k) 2 ˜ ξi kxi (θ) − S(θ)k < M0 eαθ ,
where M0 > 0 and α > 0. Therefore, by choosing < α, we can find that (k) Zt X m N X (k) 2 ˜ ξi kxi (θ) − S(θ)k exp(θ)dθ < +∞. 0 k=1 i=1
(k)
(k)
2 ˜ ξi kxi (t) − S(t)k exp(t) → 0
when t → ∞.
Thus, from the conditions (12), (13), (14) and (15), it is easy to observe that the directed controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization under the given controllers. Thus, the proof is completed. Remark 1: In Theorem 1, the condition (12) ensures the synchronization of leaders’ network. When the leaders reach synchronization exponentially, then the conditions (13), (14) and (15) provide the control design procedure for synchronization in each subnetworks. It should be mention that, the quantity a G(s) and the scalar α play a key role in the conditions (12), (13), respectively. And the maximum eigenvalue also plays a key role in the conditions (14) and (15). Remark 2: In the literature [32], the considered complex network divided into multiple sub-networks, such that the nodes belonging to the same sub-network are identical, while the ones belonging to different sub-networks are non-identical, which are different from the complex networks of networks. Remark 3: In this paper, the networks under consideration are different from the networks in the literatures [30], [31]. Specifically, in [30], pinning synchronization is investigated for the undirected complex networks of networks without considering the time delays. In [31], the time delays are taken into account, and adaptive controller is designed to ensure the synchronization of the controlled network, but the concerned network of networks is still undirected. In this paper, the considered networks have directed topological structure, which are more general compared with [30], [31]. In Theorem 1, we focus on the exponential synchronization for the directed controlled networks. In the following, we consider the special case when the complex networks of networks is undirected. In this case, the left eigenvectors ξ = 1/M [1, 1, · · · , 1]T ∈ RM and ξ (k) = (k) 1/N (k) [1, 1, · · · , 1]T ∈ RN . Then, we have Ξ = (1/M )IM and Ξ(k) = (1/N (k) )IN (k) . Moreover, since G(s) is symmetric, one can find a(G(s) ) = (2/M )λ2 (G(s) ). Therefore, for the undirected complex networks of networks, the quantity ρ1 = −2λ2 (G(s) )λmin (Γ)/M λmax (Ξ). Then, we can get the following corollary. Corollary 1: Suppose that Assumption 1 holds, and that there exist a positive scalar α such that the following conditions are satisfied: [2γ + 1 − cρ1 ]IM + (1/M )1M ×M βIM (1/M ) < 0, βIM −IM (40)
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(k)
m N X X
(k)
ξi kxi (t) −
2 ˜ S(t)k exp(t) is uniformly continuous on [0, +∞). From Lemma 6, we can conclude that
k=1 i=1
l(k)
m NP P
k=1 i=1
m N X X
< 0.
< 0.
(k)
Moreover, it is not difficult to verify that
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cα From the condition (15), it is easy to have λmin (Γ) + (k) (k)T G +G cλmax < 0. Then, in view l(k) of(38), we know 2 G(k) +G(k)T cα < 0, which that λmax λmin (Γ) IN (k) + c 2
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(k)
di
β2 δ + − cα < 0, 2 4
(41)
−1 (k)T B > λmax U (k) − B (k) C (k) , λmax
G(k)
l(k)
i = 1, · · · , l(k) , (42)
<−
α , λmin (Γ)
(43)
where ρ1 = −2λ2 (G(s) )λmin (Γ)/M λmax (Ψ) and k = 1, 2, · · · , m. Then, the undirected controlled networks (1) and (6) with delayed nodes can reach global exponential synchronization. IV. N UMERICAL E XAMPLE
We consider the time delay τ = 1. Obviously, the nonlinear function (44) satisfies Assumption 1 with γ = 1 and β = 1, 2 then α = 4.6029 is chosen such that α > 1c γ + 1 + 2δ + β4 . According to [22], we rearrange the subnetwork nodes such that the out-degree are bigger than their in-degree. Therefore, we assume that two nodes in each subnetwork are chosen as the candidate for applying pinning feedback controllers. The coupling strength is c = 0.49 and we select δ = 0.001. Therefore, according to Theorem 1, we can conclude that the leaders’ network and the followers’ network are exponentially stable. Furthermore, an illustrative example is also carried out to demonstrate the effectiveness of the theoretical results. For the leaders’ network (1) and the followers’ network (6) with the above parameters, we choose the following initial values, for any t ∈ [0, 1], T T S1 (0) = −2 1 0.2 , S2 (0) = −4 1.2 1 , T T S3 (0) = −2 0.5 1 , S4 (0) = −4 1 0.5 , T T (1) S5 (0) = 1 −2 0 , x1 (0) = 0.5 −1.1 0.2 , T T (1) (1) x2 (0) = −1 1 0 , x3 (0) = −0.1 0.1 0 , T (1) (1) x4 (0) = x5 (0) = 0.1 0.2 −0.3 , T T (2) = −0.5 0.2 0.1 , x1 (0) 0.2 −0.6 0.1 , T (2) T (2) x2 (0) = −0.5 0.2 0.2 , x3 (0) = 0.1 −0.2 0 , T (2) (2) x4 (0) = = −4.5 0.2 −0.1 , x5 (0) T (3) T (3) 0.1 0.2 −0.5 , x1 (0) = 0.5 −2 1 , x2 (0) = T (3) T (3) 1 −1 0.4 , x3 (0) = 0.5 −1.1 0.7 , x4 (0) = T T (3) −0.6 0.5 0.3 , x5 (0) = −0.2 0.1 0.1 , and simulation result in Fig. 1 shows that the evolution of the synchronization error dynamics for both leaders’ network and subnetworks approaches zero, which matches well with the theoretical results.
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In this section, a numerical simulation example is given to demonstrate our theoretical analysis. Consider the leaders’ network (1) with five leaders. The coupling matrix is chosen as follows: −3 1 1 1 0 1 −3 0 1 1 (s) G = 1 1 −3 0 1 . 0 1 1 −3 1 1 0 1 1 −3
where i = 1, 2, 3, 4, 5, −0.6 0.3 0.3 −0.6 0.3 0.3 D1 = 0.3 −1.5 0.2 , D2 = 0.3 −1.7 0.2 0.3 1.2 −0.5 0.3 1.5 −0.6 9.9 0 0 and Λ = 0 0.3 0 . 0 0 0.3
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γ+1+
G(3)
1 −2 0 0 0
0 0 −1 1 1
0 0 1 −2 0
1 −3 1 0 1
1 0 −2 1 1
0 1 1 −3 0
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G(2)
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Suppose that there are also three subnetworks with five nodes in each subnetworks. The coupling matrices of three subnetworks are given, respectively, by −1 0 0 0 1 1 −4 1 1 1 (1) G = 0 1 −2 1 0 , 1 0 1 −3 1 1 1 1 0 −3
−2 1 = 0 1 1
0 1 0 , 1 −2 0 1 0 . 1 −3
V. C ONCLUSIONS
Γ is assumed to be diag{2, 2, 2}. The nonlinear vector-valued function in the delayed dynamical equation of each leader and node is described by f (xi (t), xi (t − τ )) = D1 tanh(Λxi (t))
+ D2 tanh(Λxi (t − τ )),
(44)
The exponential synchronization on complex networks of networks with directed topology and time delay has been studied in this paper, where the leader’s network and the followers’ network form the networks of networks. The coupling configuration matrix is not assumed to be symmetric. Based on this model, some sufficient conditions of the exponential synchronization are then deduced by introducing the left eigenvector to the construction of Lyapunov functionals. An illustrative example has been provided to demonstrate the effectiveness of the proposed results. The future topic for the
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Fig. 1.
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synchronization would consider more general cases involving network-induced phenomena [40]–[46].
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Yurong Liu’s Biography:
Yurong Liu was born in China in 1964. He received his B.Sc. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.Sc. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2001. Dr. Liu is currently a professor with the Department of Mathematics at Yangzhou University, China. He also serves as an Associate Editor of Neurocomputing. So far, he has published more than 50 papers in refereed international journals. His current interests include stochastic control, neural networks, complex networks, nonlinear dynamics, time-delay systems, multi-agent systems, and chaotic dynamics.
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Ph.D. degree in Applied Mathematics at Yangzhou University, Yangzhou, China. His research interests include complex networks, nonlinear dynamics and complex networks of networks.
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Biographies
Mohmmed Alsiddig Alamin Ahmed’s Biography: Mohmmed Alsiddig Alamin Ahmed received the B.SC degree in Mathematics from Sudan University of Sciences and Technology, Khartoum, Sudan and the M.SC degree in Mathematics from Khartoum University, Khartoum, Sudan, in 2007 and 2009, respectively. In 2011, he received Postgraduate Diploma from African Institute for Mathematical Sciences AIMS, Cape Town, South Africa. Currently, he is pursuing the
Wenbing Zhang’s Biography: Wenbing Zhang received the M.S. degree in applied mathematics from Yangzhou University, Jiangsu, China, and the Ph.D. degree in pattern recognition and intelligence systems from Donghua University, Shanghai, China, in 2009 and 2012, respectively. He was a Research Associate with The Hong Kong Polytechnic University, Kowloon, Hong Kong, from 2012 to 2013. From July 2014 to Aug 2014, he was a DAAD fellow with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. He is currently an associated professor with the Department of Mathematics, Yangzhou University. His current research interests include synchronization/consensus, networked control systems, and genetic regulatory networks. Dr. Zhang is a very active reviewer for many international journals.
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Ahmed Alsaedi’s Biography: Ahmed Alsaedi obtained his Ph.D. degree from Swansea University (UK) in 2002. He has a broad experience of research in applied mathematics. His fields of interest include dynamical systems, nonlinear analysis involving ordinary differential equations, fractional differential equations, boundary value problems, mathematical modeling, biomathematics, Newtonian and Non-Newtonian fluid mechanics. He has published several articles in peer-reviewed journals. He has supervised several M.S. students and executed many research projects successfully. He is reviewer of several international journals. He served as the chairman of the mathematics department at KAU and presently he is serving as director of the research program at KAU. Under his great leadership, this program is running quite successfully and it has attracted a large number of highly rated researchers and distinguished professors from all over the world. He is also the head of NAAM international research group at KAU.
by the membership of international and national Committees, leadership and motivation, numerous scholarships and fellowships, conducted research projects, convened many national and international conferences, seminars delivered and attended conferences the world over, established research collaboration with leading international scientists, associate editor/editorial membership of the international journals including ISI, reviewer of the international journals and MS and PhD students produced. His publications in diverse areas are in high impact factor journals. His research work has total ISI WEB citations (11730) and h-index (52) at present. He has received many national and international awards including Tamgha-i-Imtiaz, Sitara-i-Imtiaz, Khwarizmi Int. award, ISESCO Int. award, TWAS prize for young scientists, Alexander-Von-Humboldt fellowship etc.
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Tasawar Hayat’s Biography: Dr. Hayat (born in Khanewal, Punjab), Distinguished National Professor and Chairperson of Mathematics Department at Quaid-I-Azam University is renowned worldwide for his seminal, diversified and fundamental contributions in models relevant to physiological systems, control engineering, climate change, renewable energy, low-carbon technologies, environmental issues, non-Newtonian fluids, wave mechanics, homotopic solutions, stability, nanofluids and in several other areas. He has a honor of being fellow of Pakistan Academy of Sciences, Third World Academy of Sciences (TWAS) and Islamic World Academy of Sciences in the mathematical Sciences. His national and international recognition is evident