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Exponential synchronization of the coupling delayed switching complex dynamical networks via impulsive control
Advances in Difference Equations volume 2013, Article number: 195 (2013)
In this paper, we investigate the exponential synchronization issue of coupling delayed switching complex dynamical networks via impulsive control. Basing on the Lyapunov functional method and establishing a new impulsive delay differential inequality, we derive some sufficient conditions which depend on delay and impulses to guarantee the exponential synchronization of the coupling delay switching complex dynamical network. Finally, numerical simulations are given to illustrate the effectiveness of the obtained results.
During the last two decades, synchronization and control problems of complex dynamical networks have been focused on in many different fields such as mathematics, engineering, social and economic science, etc. [1–8]. Many effective methods, like feedback control, adaptive control, sampled-data control and impulsive control, are used to stabilize and synchronize a coupled complex dynamical network. At the same time, a wide variety of synchronization criteria have also been presented for different network coupling such as switch topology, time delays, impulsive characters, etc.
Up to now, plenty of researchers have devoted much effort to guarantee synchronization of complex dynamical networks with fixed topology [9–16]. However, in real situations, many complex systems may be subject to abrupt changes in their connection structure or network mode switching caused by some phenomena such as link failures, component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbance, etc. Although some synchronization criteria of networks with uncertain topological structure and continuous time-varying topology have been studied, those methods may not work for the network topology when it becomes discontinued or changes very quickly [17, 18]. Hence, to study the synchronization of the switched networks is still very useful and meaningful. Because of this reason, the synchronization of a complex network with switching topology has attracted researchers’ interest [17–21]. Wang et al.  provided several synchronization criteria for switched networks, in which synchronization could be evaluated by the time average of the second smallest eigenvalue that corresponded to the Laplacians matrix of switching topology. Authors in  studied the local and global exponential synchronization of switched networks with time-varying coupling delays, whose inner and outer coupling matrices take values in two finite sets of matrices via a switching signal. An adaptive controller was designed to synchronize a switched network under arbitrary switching in . Yu et al.  explored the synchronization of switched neural networks, and some sufficient conditions were given to guarantee the global synchronization. Jia et al.  investigated the leader-following problem of network, in which the network topology is assumed to be arbitrarily switched among a finite set of topologies, and time-varying delay exists in the coupling of agents.
In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments. In general, there are two kinds of impulse in terms of synchronization in complex dynamical networks: desynchronizing impulse and synchronizing impulse . In previous literature, most of the results were devoted to investigating the desynchronizing impulse (the impulsive effect can suppress the synchronization of the complex dynamical networks) [23–26]. The global exponential synchronization was studied for linear coupled neural networks with impulsive disturbances in . Zhu et al. gave some global impulsive exponential synchronization criteria for time-delayed coupled chaotic systems . In , some impulsive control schemes were given to guarantee the consensus of nonlinear multi-agent systems with switching topology. Yang and Cao  studied the exponential synchronization of a coupling delay complex dynamical network with impulsive effects and proved that the network can achieve synchronization for a desynchronizing impulse. All of them have a common feature that the network must be synchronous. As we all know that the network is not always synchronous, there are some factors that will lead to an unstable network such as the change of topology structure, time delays and low strength of the coupling. Impulsive control (synchronizing impulse) may give an efficient method to deal with a dynamical system which is unstable. It is worth mentioning that synchronization and the control problems in complex networks with fix topology and synchronizing impulse have been widely studied [23, 27–29], but research into switched topology and synchronizing impulse is rare.
In this paper, we investigate the problem of exponential synchronization of a switching complex dynamical network via impulsive control. The contribution of this paper is to propose a new impulsive delay differential inequality. By utilizing the Lyapunov stability and impulsive control theory on delayed dynamical networks, some sufficient conditions of exponential synchronization for a switching complex dynamical network are presented. It shows that impulsive controller (synchronizing impulsive) can control the coupling delay switching complex dynamical network to a homogenous solution. Numerical simulations are given to show the validity of the developed results.
2 Model and preliminaries
The switching complex dynamical networks investigated in this paper consist of N nodes, whose state is described as
where is the state vector of node i; is a continuous vector value function, is the coupling strength, τ is a coupling delay; is an inner coupling matrix between the two connected nodes; is a switching signal, which is a piecewise constant function; is a Laplacian matrix associated with the switching function , in which the entries of matrix are defined as follows: if nodes i and j () are connected, then ; otherwise, , and the diagonal entries of matrix are defined by . Note that the coupling matrix is not assumed to be irreducible; is the i th node impulsive gain at . The discrete set satisfies , as , note , and .
We assume that the network (1) satisfies the following initial conditions: .
To discuss exponential synchronization, we define the set
which is the synchronization state for the network (1), where and .
Remark 1 In general, the synchronization state may be an equilibrium point, a periodic orbit, or a chaotic attractor. In this paper, we did not need to be the left eigenvector of coupling matrix G corresponding to eigenvalue 0.
Definition 1 The network (1) is said to achieve exponentially synchronization if there exist some constants and such that
for all initial conditions and .
Let be a positive definite diagonal matrix, and let be a diagonal matrix. denotes a class of continuous functions satisfying
for some , all and .
Remark 2 It is easy to verify that the function class QUAD exists in almost all the well-known chaotic systems with or without time delays such as Lorenz systems, Rössler system, Chen system, Chua’s circuit, delayed Hopfield neural networks and delayed cellular neural networks (cNN), etc.
Define error state (). It is easy to verify that and the dynamical equation of and satisfies
where and .
In order to derive the main results, it is necessary to propose the following lemmas.
Lemma 1 Let satisfy the scalar impulsive differential inequality
where , , is continuous at , , and exists, . Then
The proof is given in the Appendix.
Lemma 2 
For real constant matrices with , , and diagonal matrix , where , the matrices and
share the same eigenvalues, where indicates the N dimensional identity matrix and .
3 Main result
In this section, we investigate the exponential synchronization of error system (4), in which coupling matrix is divided into two cases: symmetric or asymmetric. Some new criteria are presented for the exponential synchronization of the network (1) based on the Lyapunov functional method, linear matrix inequality approach and establishing an impulsive delay differential inequality.
Case 1. Asymmetric connected of switching topology
Theorem 1 The network (1) is exponential synchronization if there exist positive definite diagonal matrices and , and positive constants β and η such that
where , , , , and .
Proof Condition (3) of Theorem 1 implies that the impulsive gains .
Choose the Lyapunov function as follows:
Then the derivative of with respect to time t along the solution of Eq. (4) can be calculated as follows:
Since , we have
Considering the time intervals in which the σ th topology is being activated and using the QUAD condition, we have
where and .
According to Condition (2) of Theorem 1 and linear matrix inequality, it is not difficult to verify that
Substituting (7) into (6) yields
On the other hand, from the construction of , we have
Let , , and , then
Using Condition (3) of Theorem 1, we get
From the construction of , we have
Hence, , where .
The proof of Theorem 1 is completed. □
If the switching signal , then the network (1) has only one coupling matrix G. Suppose G is irreducible and is the left eigenvector of coupling matrix G corresponding to eigenvalue 0. By the proof of Theorem 1, we can derive the exponential synchronization criteria of the network (1) with only one topology, which is given as follows.
Corollary 1 The network (1) with only one topology is exponential synchronization if there exist positive definite diagonal matrices and and positive constants β and η such that
where , , , , and .
Remark 3 The result of Theorem 2 in  must satisfy , where . However, for almost chaotic systems there exists j such that . It means that the condition of Theorem 2 (in ) is not true. In Corollary 1 of this paper, there exists such that . So, Corollary 1 is more common than Theorem 2 in .
Case 2. Symmetric connected of switching topology
Theorem 2 Suppose that is a symmetric matrix. If there exist positive constants β and η and positive definite diagonal matrices and such that
then the network (1) is exponential synchronization, where , , , , , and are the eigenvalues of matrices .
Proof Construct the following Lyapunov function:
Then, taking the derivative of with respect to time t along the solution of Eq. (4), we have
Consider the properties of a symmetric matrix. There exists an orthogonal matrix such that and . Let . According to Eq. (11) and the properties of the Kronecker product, we can get
Basing on Condition (2) of Theorem 2 and , where are the eigenvalues of matrices , for all and , we have
By the linear matrix inequality, for all , , one gets
Then, applying Lemma 2, we obtain
Using Condition (3) of Theorem 2, we get
It is clear that .
So, , where .
The proof of Theorem 2 is completed. □
Let impulsive gains , and choose the synchronization state . By the proof of Theorem 2, we can derive the exponential synchronization criteria of the network (1) with the fixed impulsive gain, which is given as follows.
Corollary 2 The network (1) with the fixed impulsive gain is exponential synchronization if there exist positive definite diagonal matrices and , and positive constants β and η such that
where , , , and are the eigenvalues of matrices .
4 Numerical simulation
In this section, we give two numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections.
where and the function was chosen as follows:
where , and .
Example 1 Consider a network model consisting of five nodes and three connective topology. Each node in the network is three-order Chua’s circuit described by
where , and .
If the coupling matrices are selected as follows and (without impulsive controller), then the network (13) is not synchronized (see Figure 2).
If we choose and , then the function satisfies the condition of the function class , where . The switch time is . Let , , , and let the synchronization state be , then all the conditions in Theorem 1 are satisfied, and , , so the asymmetric coupled network (13) can achieve exponential synchronization. The simulation results are given in Figures 3-5. It can be seen clearly from Figures 3-5 that all states of the asymmetric coupled network (13) tend to the synchronization state .
Example 2 The network model is the same as Example 1. If the coupling matrices are chosen as follows and (without impulsive controller), then the network (13) is not synchronized (see Figure 6).
Hence, . Choose the synchronization state and switch time . If , , , and , then all the conditions in Theorem 2 are satisfied, and , , so the symmetric coupled network (13) can achieve exponential synchronization. The simulation results are given in Figures 7-9. It can be seen clearly from Figures 7-9 that all states of the symmetric coupled network (13) tend to the synchronization state .
In this paper, by establishing an impulsive delay differential inequality, the exponential synchronization of the coupling delay switching complex networks has been investigated. Based on Lyapunov stability theory, some simple yet generic criteria for exponential synchronization have been derived. It shows that criteria can provide an effective impulsive control scheme to synchronize for an arbitrary given switch topology. Furthermore, the effectiveness of the presented method has been verified by numerical simulations.
Proof of Lemma 1 For , integrating both sides of equation (5) from to t, we can get
It is easy to obtain that
Now, we begin to prove that
We shall show this by induction.
For , by Lemma 3 in , we have
In view of (16), we see that (15) holds when . Under the inductive assumption that (15) holds for some , we shall show that (15) still holds for .
For , without any loss of generality, we assume that there are l first-class intermittent points, then (14) can be rewritten as
Noting , then the derivative of can be calculated as follows:
It is not difficult to show that
Hence, we have
Since , one has
Substituting (18) into (17) yields
That is, (15) holds for . Hence, by induction, (15) holds for all .
The proof is complete. □
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This work is supported by the National Natural Science Foundation of China (61075060 and 61273220), the Key Foundation Project of Shanghai (12JC1400400) and the Innovation Program of Shanghai Municipal Education Commission (12zz064, 13zz050).
The authors declare that they have no competing interests.
AD carried out the main part of this manuscript. WZ participated in the discussion and corrected the main theorem. All authors read and approved the final manuscript.
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