Exponential Synchronization for Impulsive Dynamical Networks

This paper is devoted to exponential synchronization for complex dynamical networks with delay and impulsive effects. The coupling configuration matrix is assumed to be irreducible. By using impulsive differential inequality and the Kronecker product techniques, some criteria are obtained to guarantee the exponential synchronization for dynamical networks. We also extend the delay fractioning approach to the dynamical networks by constructing a Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical examples are given to demonstrate the theoretical results.


Introduction
In the past two decades, complex dynamical networks have attracted lot of attention in different areas, such as physical science, engineering, mathematics, biology, and sociology 1-3 .The synchronization of all dynamical nodes is an important and interesting phenomena mostly because the synchronization can well explain many natural phenomena.Consequently, the synchronization has been actively investigated due to past physics and potential engineering applications.Recently, there has been an increasing interest in the investigation of synchronization of complex dynamical networks, then many synchronization results have been derived for complex dynamical networks 4-9 .
Impulsive effects widely exist in the networks.Such systems are described by impulsive differential systems which have been used efficiently in modelling many practical problems that arise in the fields of engineering, physics, and science as well.So the theory of impulsive differential equations is also attracting much attention in recent years 10-13 .Correspondingly, based on the theory of impulsive differential equations, a lot of synchronization results of dynamical networks with impulsive effects have been obtained 13-20 .As is well known, two kinds of impulses in terms of synchronization in complex dynamical networks are considered.One is desynchronizing impulse, the other is synchronizing impulse.An impulsive sequence is said to be desynchronizing if the impulsive effect can suppress the synchronization of complex dynamical networks.An impulsive sequence is said to be synchronizing if a corresponding impulsive effect can enhance the synchronization of the complex dynamical networks.According to the previous literature, complex dynamical networks with delay and impulses can reach synchronization provided that delayed dynamical networks are synchronized.In this paper, by impulsive differential inequality 21 , the Lyapunov functional method and the Kronecker product techniques, some sufficient conditions are derived for the globally exponential synchronization of dynamical networks.We also extend the delay fractioning method 22, 23 to dynamical networks by constructing Lyapunov-Krasovskii functional and comparing to a linear discrete system.Meanwhile, numerical simulations are given to show that our derived criteria can easily be used to make judgements on synchronization for the delayed dynamical networks with impulsive effects and show that impulsive effects play an important role in the delay dynamical networks.The rest of this paper is organized as follows.In Section 2, the network model is presented, together with some definitions and lemmas.In Section 3, some synchronization criteria are derived for general dynamical networks with delay and impulsive effects.In Section 4, two numerical examples are given to demonstrate that our results are relevant to not only linear coupling but also delay and impulsive effects.Finally, some conclusions are given in Section 5.
Notations.Throughout this paper, the superscript T represents the transpose.I n stands for the identity matrix of order n.For x x 1 , x 2 , . . ., x n T ∈ R n , the norm is defined as For matrix A, λ max A and λ min A denote the maximum and minimum eigenvalues of matrix A, respectively.For real symmetric matrices X and Y , the notation X ≤ Y resp., X < Y means that the matrix X − Y is negative semidefinite resp., negative definite .For a sequence {t k , k 0, 1, . ..} satisfying 0

Model Description and Preliminaries
We consider a delayed complex dynamical network consisting of N-coupled identical nodes.Each node is an n-dimensional dynamical system composed of linear term and nonlinear term.The ith node can be described as follows: where x i t x i1 t , x i2 t , . . ., x in t T is the state vector of the ith node at time t, C, B 1 , The dynamical behavior of the dynamical network with delay can be described by the following linearly coupled systems: where Γ diag{γ 1 , γ 2 , . . ., γ n } is the inner coupling positive definite matrix between two connected nodes i and j, c is the coupling strength, and a ij is defined as follows: if there is a connection from node j to node i j / i , then a ij > 0; otherwise, a ij 0.
In the process of signal transmission, due to the impulsive effects, the states x i t , i 1, 2, . . ., N are suddenly changed in the form of impulses at discrete times t k .That is, Thus, the dynamical network with delay and impulsive effects can be obtained by the following form: x i t of system 2.3 , we mean a real function on t 0 − τ, ∞ such that x i t 0 ϕ i t for t ∈ t 0 − τ, t 0 , and x i t satisfies system 2.3 for t ≥ t 0 , and x i t is continuous everywhere except for some t k and left continuous at t t k , and the right limit x t k k 1, 2, . . .exists.Here, we always assume that system 2.3 has a unique solution.
Remark 2.1.If |d k | < 1, the impulsive sequence is of synchronizing impulse, which may enhance the synchronization of the networks.But if |d k | > 1, the impulsive sequence can suppress the synchronization, which is said to be desynchronizing impulse.
Definition 2.2.The dynamical networks 2.3 are said to be globally exponentially synchronized if there exist η > 0 and M > 0 such that for any initial values ϕ i t i 1, 2, . . ., N : hold all t > t 0 , and for any i, j 1, 2, . . ., N.
Definition 2.3.For A a ij m×n ∈ R m×n , B b ij p×q ∈ R p×q , the Kronecker product between two matrices is defined by Assumption 2.5.The coupling configuration matrix A is irreducible, and the real parts of the eigenvalues of A are all negative except an eigenvalue 0 with multiplicity 1.
To derive our main results, we need the following lemmas.

2.7
Then where u t Remark 2.8.The condition of Lemma 2.7 does not need −p > q due to the effects α, which implies that the above inequality is less conservative than the results in 25 .
Lemma 2.9.For any vectors x, y ∈ R n , scalar > 0, and positive definite matrix Q ∈ R n×n , the following inequality holds: Lemma 2.10.Let A ∈ R n×n be a positive definite matrix, then for x ∈ R n ,

Synchronization Analysis
In this section, the globally exponential synchronization will be analyzed for delayed dynamical networks with impulsive effects.We assume that the network topology is strongly connected, then the corresponding Laplacian coupling matrix A is irreducible.
Then, the delayed dynamical network 2.3 can be rewritten in the following Kronecker product form: x t ϕ t , t ∈ t 0 − τ, t 0 .

3.1
Suppose that ξ ξ 1 , ξ 2 , . . ., ξ N T is the left eigenvector of the configuration coupling matrix A with respect to eigenvalue 0 satisfying N i 1 ξ i 1.Since the coupling configuration matrix A is irreducible, by Lemma 2.6, we can see that Theorem 3.1.Suppose that Assumptions 2.4 and 2.5 hold.Also suppose that there exist a diagonal positive-definite matrix P and scalars η > 0, ε > 0, γ > 0, μ 2 ≥ 0, μ 1 , δ such that Then the complex dynamical networks 3.1 are exponentially synchronized, where d A is defined to be the second largest eigenvalue of A.
Proof.We define a Lyapunov function V t x T t W ⊗ P x t .Since W Ξ − ξξ T , we have w ij −ξ i ξ j for i / j and w ii ξ i − ξ 2 i .In view of Calculating the Dini derivative of V t along the trajectories of the systems 3.1 , we have for t / t k , k 1, 2, . ..: ΞA, we can obtain that

3.3
Since the matrix A ΞA A T Ξ has the following property:

3.4
By Perron-Frobenius theorem see 24 , we can arrange the eigenvalues of matrix A as follows: 0 y T i t P Γy i t .

3.5
In view of matrix W is a zero row sum irreducible symmetric matrix with negative offdiagonal elements, we see that λ max W > 0 and Wu 1 0, 0, . . ., 0 T .Hence by Lemma 2.10, we have y T i t P Γy i t .

3.6
It follows from condition

3.7
By Assumption 2.4 and Lemma 2.10, there exists ε > 0 such that

3.8
Similarly, we have the following estimation: 3.9 where γ > 0. Substituting these into 3.3 , we have for t / t k

3.10
For t t k , we have

3.11
By Lemma 2.7, there exist M > 0 such that V t 0 s e −η t−t 0 , 3.12 which implies that

3.13
Consequently, the complex dynamical network 3.1 can reach globally exponential synchronization.
Remark 3.2.When the impulsive effects are desynchronizing, that is, |d k | > 1, the condition H 4 in Theorem 3.1 yields −μ 1 > μ 2 , which means that the delayed complex networks without impulsive effects of 2.2 is exponentially synchronized.But when the impulsive effects are synchronizing, that is, |d k | < 1, we do not need the condition −μ 1 > μ 2 due to the effect of impulses.
Theorem 3.3.Suppose that Assumptions 2.4 and 2.5 hold and Δ sup < ∞.Also suppose that there exist a diagonal positive definite matrix P and scalars η > 0, ε > 0, γ > 0, μ 1 > 0, μ 2 ≥ 0 such that for all k ≥ m, and the discrete system: is globally exponentially stable with decay λ > 0, where H 5 there exists a constant T 0 such that the average dwell time T a satisfies where N t 0 , t is the number of impulsive times of the impulsive sequence on the interval t 0 , t ; Then the complex dynamical networks 3.1 are exponentially synchronized with decay rate λ/2T a .
Proof.Consider a Lyapunov-Krasovskii functional: x T s W ⊗ P x s ds.

3.18
Similar to the proof of Theorem 3.1, for t ∈ t k , t k 1 , we get

3.19
For t ∈ t k , t k 1 , we have

3.22
By 3.11 , for t t k , we have

3.23
It follows from condition iii that there exists some t k−j 1 ∈ t k−j , t k−j 1 such that

3.24
Then from 3.22 , we have 3.25 Together with 3.22 , 3.23 and the above inequality, we have

3.28
We consider the discrete system:

3.29
Then, by the comparison principle, we see that for k ≥ m − 1

3.30
Note that the system 3.29 is globally exponential stable with decay λ > 0, then there exists constant M > 0 such that From 3.17 and 3.22 , we have Δ s V t s e ζΔ s , j 0, 1, . . ., m − 1.

3.32
Furthermore, it follows that V t 0 s .

3.35
From 3.31 and the above inequality, we see that for all k 0, 1, . ..,

3.39
This completes the proof of the theorem.
Remark 3.4.Theorem 3.3 presents a new delay-dependent exponential synchronization criterion for complex dynamical networks by using the Lyapunov-Krasovskii functional.Note that, for d k < 1, σ 0, the proposed result demonstrates its superiority to Theorem 3.1, which will be well illustrated via an example in the next section.
Proof.Choose a Lyapunov-Krasovskii functional candidate V x t as x T s W ⊗ P x s ds.

3.42
By the proof of Theorem 3.3, for t ∈ t k , t k 1 , we have

3.43
Note that τ < t k − t k−1 , then there exists some t k ∈ t k − τ, t k such that

3.45
Then from condition H 4 , we obtain where there exists a constant 0 < < 1 such that all roots λ j j 1, 2, . . ., m of the characteristic polynomial: satisfy that |λ j | ≤ < 1, then the complex dynamical networks 3.1 are exponentially synchronized.Remark 3.9.From Theorems 3.1 and 3.3, when the delayed network dynamics are desynchronizing and the impulsive effects are synchronizing, in order to ensure synchronization, it should be naturally assumed that the frequency of impulses should not be too low.Usually, we always use condition t k −t k−1 ≤ T 1 T 1 > 0 to ensure that the frequency of impulses should not be too low.Conversely, when the delayed network dynamics are synchronizing but the impulsive effects are desynchronizing, the impulses should not occur too frequently in order to guarantee synchronization.To ensure that the impulses do not occur too frequently, we always assume that t k − t k−1 ≥ T 2 T 2 > 0 .

Examples and Simulations
In this section, some examples and numerical simulations are provided to illustrate our results.
We consider the following linear coupled delayed networks:   Figure 2 shows the synchronization of networks of 4.3 .At last, we consider the following linear coupled delayed networks with impulsive effects:    1.5626 and P diag{1.3782,0.9991, 1.4307}.We can verify that the synchronization criteria proposed by Theorem 3.1 are not satisfied.However, we conclude that the complex dynamical networks 4.4 are exponentially synchronized by Corollary 3.5.Figure 4 depicts the synchronization state variables x i1 t , x i2 t , and x i3 t with impulsive effects.Figure 5 depicts the synchronization state variables x i1 t , x i2 t , and x i3 t without impulsive effects.

Conclusions
In this paper, by establishing some lemmas of new impulsive differential inequality and by using the Lyapunov functional method and the Kronecker product techniques, exponential synchronization for impulsive dynamical networks with irreducible coupling matrix is derived.Some criteria are obtained not only relevant to delay but also to impulsive effects.In particular, the results can be extended to the case of one reducible coupling matrix A, which implies that the network topology may be a weakly connected graph containing a rooted spanning tree.

Figure 2 :
Figure 2:The state variables x i1 t and x i2 t without impulsive effects.
Applying matrix decomposition theory see 24 , there exists unitary matrix U, such that A UΛU T , where Λ diag{0, λ 2 A , . . ., λ N A } and U {u 1 , u 2 , . . ., u N } with u 1 t 0 , 3.46 for all k 1, 2, . ... The remainder proof of the theorem is similar to Theorem 3.3.By Corollary 3.5, under the case that τ < t k − t k−1 for all k 1, 2, . .., we see that the estimations of maximal time-delay τ and maximal dwell time Δ sup as Suppose that Assumptions 2.4 and 2.5 hold and Δ sup < ∞.If there exist positive definite matrix P and scalars η > 0, ε > 0, γ > 0, μ 1 > 0, μ 2 ≥ 0 such that H 1 -H 3 and H 5 -H 6 of Theorem 3.3 hold and condition H 4 of Theorem 3.3 is replaced by one of the following two conditions: By Theorem 3.1, we see that the complex dynamical networks 4.3 are exponentially synchronized.Figure3shows the synchronization of networks with delay and impulsive effects.
The state variables x i1 t and x i2 t with impulsive effects.
ij Γx j t , t ≥ 0, t / k, k 1, 2, . . ., f 2 x i2 t , f 3 x i3 t T , g x i t g 1 x i1 t , g 2 x i2 t , g 3 x i3 t T , The state variables x i1 t , x i2 t , and x i3 t with impulses.The state variables x i1 t , x i2 t , and x i3 t without impulsive effects.Remark 4.3.In Example 4.2, if we take t k − t k−1 0.1 and d k 0.2, τ t 0.9 sin t, it is easy to see that the synchronization criteria proposed by Corollary 3.8 are not satisfied.However, we conclude that the networks 4.