The Impact of Coupling Function on Finite-Time Synchronization Dynamics of Multi-Weighted Complex Networks with Switching Topology

Yang, Bin; Wang, Xin; Fang, Jian-an; Xu, Yuhua

doi:https://doi.org/10.1155/2019/7276152

Complexity

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Abstract Introduction Model and Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 7276152 | https://doi.org/10.1155/2019/7276152

The Impact of Coupling Function on Finite-Time Synchronization Dynamics of Multi-Weighted Complex Networks with Switching Topology

Bin Yang,¹Xin Wang,²Jian-an Fang,³and Yuhua Xu⁴

Academic Editor: Xianggui Guo

Received12 Jan 2019

Revised08 Feb 2019

Accepted17 Feb 2019

Published21 Mar 2019

Abstract

This paper is not only concerned with the problem of finite-time synchronization control for a class of nonlinear coupling multi-weighted complex networks (NCMWCNs) with switching topology but also an attempt at using the derived results and Lyapunov stability theory to study the impact of nonlinear coupling function on finite-time synchronization dynamics of the raised network model. Firstly, different from the existing related results, based on the existing and new finite-time theories, two finite-time synchronization controllers are, respectively, designed to make the considered network achieve finite-time synchronization. Secondly, according to the obtained results, several finite-time synchronization dynamics criteria are established to show that nonlinear coupled function and the switching of outer-coupling matrix are how to impact finite-time synchronization dynamics. Finally, two illustrated examples are provided to verify the effectiveness of theoretical results proposed in this paper.

1. Introduction

During recent years, many researchers have paid close attention to synchronization dynamics problems of complex networks because synchronization dynamics is one of the most important collective behavior of complex networks [1–3] and many practical systems, including sensor network, communication network, neural networks, social network, and so on [4–6], can be modeled by complex networks. Therefore, many valuable and meaningful results for synchronization dynamics problems of complex networks have been obtained [7–13]. For example, based on passivity theory and Lyapunov stability theory, Kaviarasan et al. [7] investigated robust asymptotic synchronization of complex dynamical networks with uncertain inner coupling and successive delays via state feedback delayed control scheme. In [8], pinning synchronization problem is proposed for nonlinear coupling complex networks by Liu and Chen. Under the pinning control technique, the authors not only derived several global synchronization criteria for considered networks but also discussed the effect of nonlinear coupling function for synchronization dynamics from simulation aspect. By making use of pinning control strategies, Kaviarasan et al. [9] studied the problem of global synchronization on singular complex dynamical networks with Markovian switching and two additive time-varying delays. Moreover, in the real world, a lot of networks such as QQ networks, E-mail networks, and transportation networks, can be more properly modeled by multi-weighted complex networks [14, 15], in which the coupling forms among nodes are multiple. That is to say, the nodes in complex networks with multi-weights are connected by more than one weight. Thus, recently, synchronization and passivity dynamics problems on multi-weighted complex networks have aroused an increasing interest of some researchers [16–23]. For instance, in [16], Qiu et al. made a discussion on finite-time synchronization problem of linear coupling multi-weighted complex networks. Qin et al. [17], respectively, investigated global synchronization and synchronization of linear coupling multi-weighted complex networks with fixed and switching topologies by making use of Lyapunov stability theory and inequality techniques. Besides these, in fact, due to some factors including external disturbance, limited communications, and so on [24, 25], there is inevitable switching in many dynamical systems, in which the switching may affect synchronization dynamics of systems. Therefore, it is interesting to study synchronization dynamics of multi-weighted complex networks with switching topology.

As a matter of fact, in many practical engineering areas, it is necessary and meaningful for a coupling dynamical system to achieve the desired dynamical behaviors in finite time interval [26–28]. Hence, a lot of results about finite-time synchronization dynamics problems for coupling systems have been obtained [29–34]. For example, based on finite-time control theory, Wang et al. [29] designed finite-time control rule to achieve global synchronization within convergence time for a class of linear coupling Markovian jump complex networks. In [30], the authors investigated finite-time synchronization problem of linear coupling complex networks and finite-time synchronization criteria were derived by exploring finite-time stability theory.

In reality, in some cases, nodes of practical coupling networks are entangled by nonlinear function method [35], such as the interactions between different neuron elements in brain dynamical networks and the interactions between different electrical elements in an electrical gird dynamical networks. Therefore, recently, the researches on dynamic behaviors for nonlinear coupling networks including consensus problems of nonlinear coupling multi-agent systems [35, 36], synchronization problems of nonlinear coupling neural networks and complex networks [8, 37–40], and so on [41] have witnessed an increasing interest. Although some valuable results about dynamic behaviors of nonlinear coupling networks have been developed (e.g., [8, 35–41]), in these existing works, the researchers mainly concentrated on how to derive sufficient condition criteria for considered nonlinear coupling systems. It is worth pointing out that nonlinear coupling function is one of important factors affecting synchronization dynamics. Regrettably, few researchers devoted themselves to exploring the impact of nonlinear coupling function on synchronization dynamics from theory aspect.

Motivated by the above analysis, the main purpose of this paper is to investigate the effect of nonlinear coupling function and outer-coupling matrix switching on finite-time synchronization dynamics for a class of nonlinear coupling multi-weighted complex networks (NCMWCNs) with switching topology based on stability and a novel finite-time theory. The contributions of our works include the following three aspects. First, a new class of NCMWCNs with switching topology is considered. Second, in order to address the novel finite-time control method proposed, two finite-time synchronization controllers built on the existing and a new finite-time synchronization theories are, respectively, designed to make the considered network model achieve global synchronization within finite time interval. Third, we not only derive sufficient condition for ensuring finite-time synchronization of the NCMWCNs with switching topology but also give synchronization dynamics criteria on the impact of nonlinear coupling function and outer-coupling matrix switching.

Notations. Some necessary notations that will be used throughout the article are introduced. refers to the standard norm in Euclidean space. The number represents a positive integer. is the set of real matrices and denotes the -dimensional Euclidean space. The superscript denotes the matrix transposition. means an -dimensional identity matrix. (respectively, ), where are symmetric matrices, means that the matrix is positive semidefinite (respectively, positive definite). If is a matrix, and denote its maximal eigenvalue and its minimum eigenvalue, respectively. The Kronecker product of matrices and is a matrix in denoted as . The matrix represents diagonal matrix. If the dimensions of matrices are not explicitly indicated that means they are suitable for any algebraic operations.

2. Model and Preliminaries

Firstly, consider a class of NCMWCNs with switching topology presented by where is the topology switching signal, which is defined as the switching sequence where is the initial time and denotes the serial number of the activated subsystem at . represents the coupling strength and , is inner-coupling matrix and and is outer-coupling matrix with the coupling weights in the th coupling form and the th topology. For each , there is if node i and node j are connected. Otherwise, . Besides these, stands for the activity of th node and is a vector-value function, is nonlinear coupling function, and is the control input of th node.

Remark 1. Recently, because multi-weighted complex networks can more accurately describe some practical engineering networks, e.g., public traffic networks and Mobile phone networks, some researchers began to pay more and more attention to dynamical behaviors of multi-weighted complex networks and have obtained some valuable and meaningful results [14–23]. However, it should be emphasized that, in these existing works [14–23], the authors concentrated on linear coupling multi-weighted complex networks. To the best of our knowledge, until now, there is still no discussion on synchronization dynamics problems for the NCMWCNs with switching topology. Hence, it is very significant to study finite-time synchronization for the NCMWCNs with switching topology. Besides this, in the above network model (1), there is no the restriction which is that outer-coupling matrix with coupling weights must satisfy . Thus, the considered network model (1) is more general. Actually, the similar scheme has been adopted in the literature [21].

Remark 2. From Remark 1, it is obtained that some practical engineering networks can be more accurately described by multi-weighted complex networks. For example, in the network (1), let , and , and the network (1) becomes the addressed network model (1) in [14, 15]. It is clear that the proposed network model (1) in [14, 15] is one special case of the network (1) in this paper. According to [14, 15], it is seen that the effectiveness of the derived results is testified by the public traffic transfer networks. This reflects that the public traffic transfer networks in [14, 15] can be expressed by the network (1) of this paper. Furthermore, compared with the network model of [14, 15], it is not difficult to find that the network (1) of this paper can model more general public traffic transfer networks. Besides this, if the network (1) is composed of some Kuramoto oscillators [42], the network (1) becomes nonlinear coupling Kuramoto oscillator network with multi-weights and switching topology. These above show the significance of considering the multi-weighted coupling term and real physical meaning of the network (1).
The synchronization state of the network (1) satisfies where .
Secondly, in order to derive the main results, we need to give the following Definition 3, some assumptions, and lemmas.

Definition 3. The network (1) is said to achieve global synchronization within finite time interval if there exists a constant such that and for , where and is the synchronization state of the network (1).

Assumption 4 (see [16]). There exist matrices and , such that satisfies the following inequality: where and and .

Assumption 5. The coupling function of the network (1) satisfies the Lipschitz condition and . That means there exists constant such that and , where .

Remark 6. Note that nonlinear functions and can be linearized by Assumptions 4 and 5, respectively. Assumption 4 is so-called the QUAD condition (or one-sided Lipschitz) [43] and Assumption 5 is the Lipschitz condition. In fact, Assumption 4 is more general than Assumption 5. Until now, in the research about synchronization problems of complex networks, the Lipschitz condition and the QUAD condition have been widely used to process nonlinear functions [1, 14–16, 18, 43, 44].

Lemma 7 (see [45]). Suppose that a positive-definite and continuous satisfies the following condition: where and are constants. Then, one has and , , with given by

Lemma 8 (see [30]). Assume that a continuous and positive-definite function satisfies the following condition: where and are two functions, . Then, for any given , one has where , , , and , , with given by

Lemma 9. Assume that a continuous and positive-definite function satisfies the following condition: where , and . Then, one has and , with given by

Proof. Let , , and ; then combining Lemma 8, we can obtain Lemma 9.

Lemma 10 (see [46]). If and , then

Lemma 11 (see [47]). Assuming one positive definite matrix , then where .

3. Main Results

In this section, we, respectively, design two classes of finite-time synchronization control rules to realize global synchronization in finite time for the network (1). Furthermore, based on the obtained finite-time synchronization control rules, several finite-time synchronization dynamics criteria are established to show that nonlinear coupling function and the switching of outer-coupling matrix is how to impact finite-time synchronization dynamics of the network (1).

Letting (1) and (3), we have the following error system of the network (1): where , , and .

3.1. Based on Lemma 7, the Design of Finite-Time Synchronization Control Rule for the Network (1)

Theorem 12. Under Assumptions 4 and 5, if there existsthe network (1) can realize finite-time synchronization using the following controller where and . Moreover, the settling time of synchronization satisfies where .

Proof. Consider the following Lyapunov-Krasovskii functional for the network (1) as Computing along the trajectory of error system (16), we can obtain From Assumption 4, we can get Applying Lemma 11 and Assumption 5, we have where , , , and .
Substituting (22)-(23) into (21) and using the inequality (17) in Theorem 12, we have where .
By Lemma 10, we can obtain Combining (24) and (25), we have By Lemma 7, converges to zero in finite-time and is gotten by where , . That means the error vector will converge to zero within . Thus, we can obtain if . According to Definition 3, we have if . Hence, global synchronization of the network (1) will be achieved in finite-time . This completes the proof.

3.2. Based on Lemma 9, the Design of Finite-Time Synchronization Control Rule for the Network (1)

Theorem 13. Under Assumptions 4 and 5, if there existsthe network (1) can realize finite-time synchronization using the following controller: where , , and . Moreover, the settling time of synchronization satisfies where .

Proof. Consider the Lyapunov-Krasovskii functional for the network (1) as

The next proof is similar to that of Theorem 12.

Remark 14. According to Theorems 12 and 13, although the controllers (18) and (29) can be, respectively, designed to realize finite-time synchronization of the network (1), it is not difficult to find that the synchronization finite-time estimation approach in Theorem 13 is more practical than that of Theorem 12. This can be testified by and . According to inequalities (19) and (30), it is seen that is closely related to ; otherwise, is not. By using Assumption 5, we can obtain that there is . This shows that nonlinearity of can be reflected by . For example, assuming that there are and and nonlinearity of is more serious than that of , then combining Assumption 5, must exist, where . Thus, more feasible can be chosen by using . In order to further investigate the effectiveness of and , the following several interesting and useful Corollaries are derived.

Remark 15. It can be seen from Theorems 12 and 13 that nonlinear coupling function of the network (1) is linearized by the Lipschitz condition in Assumption 5. Actually, from the process of proving Theorems 12 and 13, it is observed that some nonlinearity bound conditions such as the sector-bound nonlinearity condition and the QUAD condition can replace Assumption 5 to process nonlinear coupling function of the network (1). It should be pointed out that the sector-bound nonlinearity condition and the QUAD condition [7, 10, 43, 44] are more general that the Lipschitz condition. Therefore, based on the above two techniques, the derived results have lower conservatism than Theorems 12 and 13 built on the Lipschitz condition. How do we get the related results? This is one of interesting topics in the future.

Remark 16. In fact, it can be further obtained that the conservatism of the proposed method based on Theorems 12 and 13 is closely related to Assumption 4 and Lemmas 7 and 9, the considered Lyapunov-Krasovskii functional, and the controllers (18) and (29). For example, assume the Lyapunov-Krasovskii functional for the network (1) . Thus, in Assumption 4 and Theorems 12 and 13, let ; then sufficient conditions for finite-time synchronization of the network (1) can be obtained. It is clear that the new derived results have higher conservatism than Theorems 12 and 13. From the above, it is seen that, in order to get the results, besides choosing a new , in Assumption 4, is replaced by . Although Assumption 4 with still holds, its conservatism becomes high. Furthermore, comparing Theorems 12 and 13, it can also be found that and , where , , , and . Because of , Theorem 12 must hold if Theorem 13 holds. Otherwise, the conclusion does not hold. That means the conservatism of Theorem 12 is higher than that of Theorem 13. This is caused by the finite-time control scheme of controllers (18) and (29) built on Lemmas 7 and 9, respectively. This validates that more feasible control approach can also reduce the conservatism of the derived results. How to further explore more effective finite-time control method is another valuable problem.

Remark 17. Notice that if the network (1) is large-scaled, the dimension of LMIs (17) and (28) in Theorems 12 and 13 becomes high. This causes that it might not be available to easily realize LMIs (17) and (28) in practice. How do we establish low-dimensional LMIs conditions? Because of , according to LMIs (17) and (28), it is derived that , where . Letting hold, then it is clear that is larger than LMIs (17) and (28). Thus, low-dimensional condition in Theorems 12 and 13 is obtained. It needs to be emphasized that although the condition is more practical than the conditions (17) and (28), its conservatism is higher than LMIs (17) and (28). How do we balance its conservatism and feasibility? This is a more attractive and open question.

3.3. Based on Theorems 12 and 13, the Impact Analysis of Nonlinear Coupling Function and the Switching of Outer-Coupling Matrix on Finite-Time Synchronization Dynamics of the Network (1)

Corollary 18. Under Theorem 12 and the controller (18), if , , and , or if , , and , or if , , and , or if , , and , where and , with increasing nonlinearity of nonlinear coupling function , global synchronization dynamics of the network (1) within finite-time will develop poorer.

Proof. From Theorem 12, there are Similar to the proof of Theorem 12, we have Combining inequalities (23), (24), and (34), there is under Theorem 12.
According to equality (33), it is derived that if , , and , or if , , and , or if , , and , or if , , and , we have Next, we prove the relationship between nonlinearity of nonlinear coupling function and the equality (35). Letting be fixed, then if nonlinearity of nonlinear coupling function is more serious, must become larger. This leads to increasing. Therefore, with increasing nonlinearity of , and will increase and decrease, respectively. This causes that, under , , and , or , , and , or , , and , or , , and , the above equality (35) must be larger if nonlinearity of develops more serious. This makes increase. This completes the proof.

Similar to the proof of Corollary 18, we can obtain Corollaries 19–21.

Corollary 19. Under Theorem 12 and the controller (18), if , , and , or if , , and , or if , , and , or if , , and , where and , with increasing nonlinearity of nonlinear coupling function , global synchronization dynamics of the network (1) within finite-time will become better.

Corollary 20. Suppose that function is decreasing function and . Then, under Theorem 13 and the controller (29), if , , and , or if , , and , or if , , and , or if , , and , where and , with increasing nonlinearity of nonlinear coupling function , global synchronization dynamics of the network (1) within finite-time will be poorer and synchronization convergence time of the network (1) becomes larger.

Corollary 21. Suppose that function is increasing function and . Then, under Theorem 13 and the controller (29), if , , and , or if , , and , or if , , and , or if , , and , where and , with increasing nonlinearity of nonlinear coupling function , global synchronization dynamics of the network (1) within finite-time will become better and synchronization convergence time of the network (1) will be smaller.

Remark 22. In Corollaries 20 and 21, it is seen that function is decreasing function and increasing function, respectively. Why? Similar to the proof of Corollary 18, it is obtained that, in Corollary 21, if , there is , where . Moreover, with increasing nonlinearity of nonlinear coupling function , becomes smaller. This causes to decrease. Thus, synchronization time will be smaller. Therefore, in inequality (30) of Theorem 13, let be increasing function and then it can be realized. In Corollary 20, by using the similar analysis, it is needed to make be decreasing function.

Remark 23. By Theorem 13 and Corollaries 20 and 21, it is obtained that the proposed finite-time computing approach can not only estimate synchronization time of the network (1) but also reflect that nonlinear coupling function , coupling matrix , the initial conditions, and synchronization states are how to impact synchronization dynamics of the network (1). This reveals the relationship between the multi-weighted coupling term and the finite-time synchronization dynamics in the network (1). Recently, although some wonderful works about finite-time synchronization problems of nonlinear coupling systems such as stochastic chaotic neural networks [2], Lur’e networks [41], and so on [27] have been developed, synchronization time of the addressed nonlinear coupling systems can be estimated by the derived finite-time approaches. Unfortunately, it is pity that the obtained finite-time results cannot reflect the effect of coupling term on finite-time synchronization dynamics of the considered coupling systems. This testifies that compared with the existing results [2, 27, 41], the main advantage of the proposed finite-time approach is more feasible.

Corollary 24. Under Corollaries 18 and 20, global synchronization dynamics of the network (1) within finite time is poorer than that of the network (1) within finite time and synchronization convergence time of the network (1) is larger than synchronization convergence time of the network (1).

Proof. From the proof of Corollaries 18 and 20, there must be . This shows that, under Corollaries 18 and 20, finite-time synchronization dynamics of the network (1) with the controller (29) is better than that of the network (1) with the controller (18). Thus, there must be . Because of and , where and , we can make hold. The proof is completed.

Corollary 25. Under Corollaries 19 and 21, global synchronization dynamics of the network (1) within finite-time is poorer than that of the network (1) within finite-time and synchronization convergence time of the network (1) is larger than synchronization convergence time of the network (1).

Proof. Similar to the proof of Corollary 24, we can derive Corollary 25.

Remark 26. By Corollaries 18–21, it is difficult to obtain nonlinearity of nonlinear coupling function being how to impact global synchronization in finite time for the network (1). The reason is that the impact of nonlinear coupling function on finite time synchronization dynamics of the network (1) is not only related to but also closely connected with , , and , where and . All these show that the impact of nonlinear coupling function on finite time synchronization dynamics of the network (1) is decided by the initial state , nonlinear coupling function , synchronization state , and coupling matrix . Furthermore, Corollaries 24 and 25 further testify that synchronization time estimation scheme of Theorem 13 is more feasible and reasonable than that of Theorem 12. Therefore, for nonlinear coupling systems, how to design more scientific and practical controller is very meaningful and valuable.

Similar to the proof of Corollary 18 and letting , , then we can also obtain the switching of being how to affect finite-time synchronization dynamics of the network (1).

Corollary 27. Under Theorem 13 and the controller (29), if and , or if and , where and , finite-time synchronization dynamics of the network (1) based on and is poorer than that of the network (1) built on and or and .

Corollary 28. Under Theorem 13 and the controller (29), if and , or if and , where and , finite-time synchronization dynamics of the network (1) based on and is poorer than that of the network (1) built on and or and .

Corollary 29. Under Theorem 13 and the controller (29), if and , or if and , where and , finite-time synchronization dynamics of the network (1) based on and is better than that of the network (1) built on and or and .

Corollary 30. Under Theorem 13 and the controller (29), if and , or if and , where and , finite-time synchronization dynamics of the network (1) based on and is better than that of the network (1) built on and or and .

Remark 31. Note that there is no function relationship between and switching of . Therefore, the weakness of the application of the proposed method is that the impact of the switching on finite-time synchronization dynamics of the network (1) cannot be reflected by . This shows that synchronization finite time estimation approach located in Theorem 13 still exists in some conservatism. In the future, it would be very interesting to further investigate the issue. Besides this, from Corollaries 18–28, there are or . Therefore, one has or . Letting , then one gets that is -3, -3, and 0, respectively. This shows that if , and may not hold. Therefore, if coupling matrix satisfies diffusive coupled condition, under Theorems 12 and 13, it is difficult to obtain the impact of nonlinear coupling function on finite-time synchronization dynamics of the network (1) from theory aspect.

Remark 32. Compared with the nonfinite-time control, finite-time control can improve robust performance and antidisturbance performance of systems [2]. Therefore, recently, besides finite-time synchronization control of complex networks, finite-time scaled consensus control of multi-agent systems has been paid close attention [48, 49]. In [48, 49], based on linear iterations and graph theory, Shang investigated finite-time scaled consensus control about discrete-time multi-agent system. It is seen that [48, 49] and this paper consider finite-time control problems about coupling systems. Moreover, it can also be found that [48, 49] addressed scaled consensus of linear coupling discrete-time multi-agent system within finite steps and the derived finite time in [48, 49] is a positive integer. In this paper, synchronization of NCMWCNs within finite time is proposed and the obtained finite time is a real number and greater than zero.

Remark 33. It is worth noting that because finite time control is mainly dependent on the initial conditions and fixed-time control does not rely on the initial conditions [50, 51]; in a few years recently, fixed-time control problems such as fixed-time synchronization [52], fixed-time group consensus [50], and fixed-time group tracking [51] began to be widely studied. For example, in [50, 51], according to graph theory and Lyapunov stability theory, fixed-time group consensus and fixed-time group tracking for multi-agent systems were investigated, respectively. Compared with coupling function of the network (1), nonlinear function of the considered multi-agent systems in [50, 51] is more general. Besides this, in [50, 51], multi-agent systems were coupled by linear coupling ways. If there is nonlinear coupling relationship in each agent of multi-agent systems, how to further explore fix-time consensus is a challenging and attractive question. Furthermore, from this paper and [50–52], it is seen that the similarity and difference between finite-time synchronization and fixed-time synchronization are as follows: (I) the similarity is how to get synchronization time for the addressed systems and (II) the difference is that finite time synchronization is closely related to the initial conditions; otherwise, fixed-time synchronization is not.

Remark 34. In the last few years, some valuable and meaningful results about synchronization dynamics problems of linear coupling complex or nonlinear coupling complex networks have been obtained [1–25, 29–34, 37–40]. However, these existing works mainly concentrated on how to derive sufficient conditions for synchronization problems of the considered complex networks. In this paper, the research about sufficient conditions of finite-time synchronization and synchronization dynamics criteria on the impact of nonlinear coupling function and outer-coupling matrix switching for the addressed complex networks is explored. All these show that our derived results enrich and complement the earlier works.

4. Numerical Examples

In order to test the effectiveness of the derived results, two examples are given. Define the synchronization total error being , and assume , , , and ; then the network (1) with the controller (29) is as follows: where

Example 1. According to , , and of Corollaries 20 and 21, it is observed that there exist four cases in Corollaries 20 and 21. Because the simulation process of the other cases including cases III and IV of Corollaries 20 and 21, cases I-IV of Corollaries 18 and 19 are similar to that of cases I and II in Corollaries 20 and 21; this example gives the simulation results of cases I and II in Corollaries 20 and 21.

Case I of Corollary 20. , , and , where and . Let , , , and From Assumptions 4–5, we have , , , , and . Choosing , it is easily testified that inequality (28) in Theorem 13 holds. Thus, we have , where , , , and . Next, making and , we derive that inequality (28) in Theorem 13 still holds, where , , and . Taking , , and , we get and .

Case II of Corollary 20. , , and . Let , , , , and , From the above case I and , there are , , , , , , and . Choosing , , , and , we get , , and .

Case I of Corollary 21. , , and . Taking , , , and , , , , , , , , , , , and , we obtain , , and .

Case II of Corollary 21. , , and . We choose , , , , and . Similar to case I, it is obtained that , , , , , , , , , , , , , and .

Example 2. From Corollaries 27–30, it is seen that the other derived results in Corollaries 27–30 are similar to that of cases I-1 and I-2 in Corollaries 27 and 29, respectively. Therefore, in this example, we only give the simulation results of cases I-1 and I-2 Corollaries 27 and 29. According to Example 1 and Theorem 13, there are , , , , , and .

Case I-1 of Corollary 27. , , , and . Letting , , , and then we have , , , , , and .

Case I-2 of Corollary 27. , , , and . Making , , , , and , we get and , where , , , and .

Case I-1 of Corollary 29. , , , and . Choosing , , , and , there are and , where , , , and .

Case I-2 of Corollary 29. , , , and . Taking , , , , and , we obtain and , where , , , and .

Remark 35. In Figures 1–8, trajectories marked with green, red, and blue, respectively, represent synchronization state and total error trajectories of the network (36) with nonlinear coupling functions , , and , where and . From the simulation results, it is observed that the impact of nonlinear coupling function on finite-time synchronization dynamics of the network (36) is that, in cases I-II of Corollaries 20 and 21, with increasing nonlinearity of nonlinear coupling function , finite-time synchronization dynamics of the network (36) becomes poorer or better, respectively. Furthermore, it can also seen that, in cases I-II of Corollaries 20 and 21, there are , , and , , and , where and . Thus, more feasible is chosen to make and hold, respectively. That means the proposed finite-time estimation approach of Theorem 13 can accurately describe that nonlinearity of is how to impact global synchronization dynamic of the network (1) within finite time interval.

Remark 36. According to Figures 9–12, one can see that, in cases I-1 and I-2 of Corollaries 27 and 29, the effect of the switching of coupling matrix on finite-time synchronization dynamic of the network (1) is not only related to the switching of but also connected with the initial condition and nonlinear coupling function . Moreover, from Example 2, one can obtain , , , and , respectively. This shows that the proposed finite-time estimation method in Theorem 13 cannot reflect that the switching of coupling matrix is how to impact finite-time synchronization dynamic of the network (1). This further testifies there still exists weakness of the application of the proposed method in Theorem 13. Besides this, by using the above similar simulation method, the simulation results of the other cases in Corollaries 27–30 can also be obtained. All these further show that the derived results in this paper are reasonable and valuable.

5. Conclusions

Different from the existing earlier works about synchronization problems for nonlinear coupling complex networks and neural networks, this paper mainly emphasizes the impact of nonlinear coupling function and outer-coupling matrix switching on global synchronization dynamics for a class of NCMWCNs with switching topology in finite time. According to the existing and new finite-time synchronization theories, two finite-time synchronization controllers are, respectively, designed to achieve finite-time synchronization of NCMWCNs with switching topology. Furthermore, based on the obtained controllers, sufficient conditions of the impact of nonlinear coupling function and outer-coupling matrix switching on finite-time synchronization dynamics for NCMWCNs with switching topology are derived. By comparing the results of synchronization convergence time for NCMWCNs, it is testified that synchronization finite time estimation approach built on the new finite-time synchronization theory can more effectively reflect that nonlinear coupling function is how to impact finite-time synchronization dynamics. Numerical simulations further demonstrate the correctness and usefulness of the proposed results.

It should be noted that, in the addressed network of this paper, time delay is not considered. Actually, due to information transmission and finite processing speed, in many real practical systems, time delay is inevitable. Inspired by the delayed consensus analysis of multi-agent networked systems [53], in the future, we will propose nonlinear coupling delayed multi-weighted complex networks with switching topology and investigate its finite-time/fixed-time synchronization dynamics. Besides this, in the network (1), if is a complex-valued connection outer-coupling matrix [54], how to get the related results is still an open problem.

Data Availability

No data were used to support this study. The reason is that this study is about how to prove stability of equation from mathematics aspect.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Xin Wang carried out the main part of this manuscript. Bin Yang participated in the discussion, corrected the derived results, and finished the simulation part. All authors read and approved the final manuscript.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (61673221 and 61603325), the Innovation Program of Shanghai Municipal Education Commission under Grants 13ZZ050, the Natural Science Foundation of Jiangsu Province (BK20181418), the fifteenth batch of Six-Talent Peaks Project in Jiangsu Province (DZXX-019), and “Qing-Lan Engineering” Foundation of Jiangsu Higher Education Institutions.

References

X. Wang, J.-A. Fang, A. Dai, W. Cui, and G. He, “Mean square exponential synchronization for a class of Markovian switching complex networks under feedback control and M-matrix approach,” Neurocomputing, vol. 144, pp. 357–366, 2014.
View at: Publisher Site | Google Scholar
Y. Xu, J. Wang, W. Zhou, and X. Wang, “Synchronization in pth moment for stochastic chaotic neural networks with finite-time control,” Complexity, vol. 2019, 8 pages, 2019.
View at: Publisher Site | Google Scholar
A. Wang, T. Dong, and X. Liao, “Event-triggered synchronization strategy for complex dynamical networks with the Markovian switching topologies,” Neural Networks, vol. 74, pp. 52–57, 2016.
View at: Publisher Site | Google Scholar
X.-J. Li and G.-H. Yang, “Adaptive fault-tolerant synchronization control of a class of complex dynamical networks with general input distribution matrices and actuator faults,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 3, pp. 559–569, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
M. Syed Ali and J. Yogambigai, “Extended dissipative synchronization of complex dynamical networks with additive time-varying delay and discrete-time information,” Journal of Computational and Applied Mathematics, vol. 348, pp. 328–341, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Lei, L. Zhang, Y. Wang, and Y. Fan, “Generalized matrix projective outer synchronization of non-dissipatively coupled time-varying complex dynamical networks with nonlinear coupling functions,” Neurocomputing, vol. 230, pp. 390–396, 2017.
View at: Publisher Site | Google Scholar
B. Kaviarasan, R. Sakthivel, and Y. Lim, “Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory,” Neurocomputing, vol. 186, pp. 127–138, 2016.
View at: Publisher Site | Google Scholar
X. W. Liu and T. P. Chen, “Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 1, pp. 113–126, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
R. Rakkiyappan, B. Kaviarasan, F. A. Rihan, and S. Lakshmanan, “Synchronization of singular Markovian jumping complex networks with additive time-varying delays via pinning control,” Journal of The Franklin Institute, vol. 352, pp. 3178–3195, 2015.
View at: Google Scholar
R. Sakthivel, M. Sathishkumar, B. Kaviarasan, and S. Marshal Anthoni, “Synchronization and state estimation for stochastic complex networks with uncertain inner coupling,” Neurocomputing, vol. 238, pp. 44–55, 2017.
View at: Publisher Site | Google Scholar
X. Wang, J.-a. Fang, A. Dai, Z. Li, and W. Zhou, “Mean square exponential synchronization for two classes of Markovian switching complex networks under feedback control from synchronization control cost viewpoint,” Journal of The Franklin Institute, vol. 352, no. 8, pp. 3221–3242, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Dong, D. Ye, J. Feng, and J. Wang, “Almost sure cluster synchronization of Markovian switching complex networks with stochastic noise via decentralized adaptive pinning control,” Nonlinear Dynamics, vol. 87, no. 2, pp. 727–739, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
R. Sakthivel, R. Sakthivel, B. Kaviarasan, C. Wang, and Y.-K. Ma, “Finite-time nonfragile synchronization of stochastic complex dynamical networks with semi-markov switching outer coupling,” Complexity, vol. 2018, Article ID 8546304, 13 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
X. L. An, L. Zhang, and Y. Z. Li, “Synchronization analysis of complex networks with multi-weights and its application in public traffic network,” Physica A Statistical Mechanics & Its Applications, vol. 412, no. 10, pp. 149–156, 2014.
View at: Publisher Site | Google Scholar
X.-L. An, L. Zhang, and J.-G. Zhang, “Research on urban public traffic network with multi-weights based on single bus transfer junction,” Physica A: Statistical Mechanics and its Applications, vol. 436, no. 15, pp. 748–755, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. Chen, L. Li, and S. Ren, “Finite-time synchronization of multi-weighted complex dynamical networks with and without coupling delay,” Neurocomputing, vol. 275, pp. 1250–1260, 2018.
View at: Publisher Site | Google Scholar
Z. Qin, J.-L. Wang, Y.-L. Huang, and S.-Y. Ren, “Synchronization and H∞ synchronization of multi-weighted complex delayed dynamical networks with fixed and switching topologies,” Journal of The Franklin Institute, vol. 354, no. 15, pp. 7119–7138, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-P. Zhao, P. He, H. Saberi Nik, and J. Ren, “Robust adaptive synchronization of uncertain complex networks with multiple time-varying coupled delays,” Complexity, vol. 20, no. 6, pp. 62–73, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. Yi, J. Feng, J. Wang, C. Xu, Y. Zhao, and Y. Gu, “Pinning synchronization of nonlinear and delayed coupled neural networks with multi-weights via aperiodically intermittent control,” Neural Processing Letters, 2018, https://10.1007/s11063-018-9784-x.
View at: Google Scholar
X.-X. Zhang, J.-L. Wang, Y.-L. Huang, and S.-Y. Ren, “Analysis and pinning control for passivity of multi-weighted complex dynamical networks with fixed and switching topologies,” Neurocomputing, vol. 275, pp. 958–968, 2018.
View at: Publisher Site | Google Scholar
Y.-L. Huang, W.-Z. Chen, and J.-M. Wang, “Finite-time passivity of delayed multi-weighted complex dynamical networks with different dimensional nodes,” Neurocomputing, vol. 312, pp. 74–89, 2018.
View at: Publisher Site | Google Scholar
J. Wang, M. Xu, H. Wu, and T. Huang, “Passivity analysis and pinning control of multi-weighted complex dynamical networks,” IEEE Transactions on Network Science and Engineering, 2017.
View at: Publisher Site | Google Scholar
J.-L. Wang, M. Xu, H.-N. Wu, and T. Huang, “Finite-time passivity of coupled neural networks with multiple weights,” IEEE Transactions on Network Science and Engineering, vol. 5, no. 3, pp. 184–197, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
T. Ma, “Synchronization of multi-agent stochastic impulsive perturbed chaotic delayed neural networks with switching topology,” Neurocomputing, vol. 151, no. 3, pp. 1392–1402, 2015.
View at: Publisher Site | Google Scholar
B.-B. Xu, Y.-L. Huang, J.-L. Wang, P.-C. Wei, and S.-Y. Ren, “Passivity of linearly coupled neural networks with reaction-diffusion terms and switching topology,” Journal of The Franklin Institute, vol. 353, no. 8, pp. 1882–1898, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
X. Liu, J. Cao, W. Yu, and Q. Song, “Nonsmooth finite-time synchronization of switched coupled neural networks,” IEEE Transactions on Cybernetics, vol. 46, no. 10, pp. 2360–2371, 2016.
View at: Publisher Site | Google Scholar
D. Peng, X. Li, C. Aouiti, and F. Miaadi, “Finite-time synchronization for Cohen–Grossberg neural networks with mixed time-delays,” Neurocomputing, vol. 294, pp. 39–47, 2018.
View at: Publisher Site | Google Scholar
G. Mei, X. Wu, D. Ning, and J.-A. Lu, “Finite-time stabilization of complex dynamical networks via optimal control,” Complexity, vol. 21, pp. 417–425, 2016.
View at: Publisher Site | Google Scholar
X. Wang, J.-A. Fang, H. Mao, and A. Dai, “Finite-time global synchronization for a class of Markovian jump complex networks with partially unknown transition rates under feedback control,” Nonlinear Dynamics, vol. 79, no. 1, pp. 47–61, 2015.
View at: Publisher Site | Google Scholar
Y. Xu, W. Zhou, J. Fang, C. Xie, and D. Tong, “Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling,” Neurocomputing, vol. 173, pp. 1356–1361, 2016.
View at: Publisher Site | Google Scholar
M. S. Ali and J. Yogambigai, “Finite-time robust stochastic synchronization of uncertain Markovian complex dynamical networks with mixed time-varying delays and reaction-diffusion terms via impulsive control,” Journal of The Franklin Institute, vol. 354, no. 5, pp. 2415–2436, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
W. Cui, S. Sun, J.-a. Fang, Y. Xu, and L. Zhao, “Finite-time synchronization of Markovian jump complex networks with partially unknown transition rates,” Journal of The Franklin Institute, vol. 351, no. 5, pp. 2543–2561, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Xie, G. Si, Y. Zhang, Y. Yuan, and R. Yao, “Finite-time synchronization and identification of complex delayed networks with Markovian jumping parameters and stochastic perturbations,” Chaos, Solitons & Fractals, vol. 86, pp. 35–49, 2016.
View at: Publisher Site | Google Scholar
X. Liu, D. W. Ho, Q. Song, and W. Xu, “Finite/fixed-time pinning synchronization of complex networks with stochastic disturbances,” IEEE Transactions on Cybernetics, vol. 99, pp. 1–6, 2018.
View at: Publisher Site | Google Scholar
Q. Jia and W. K. S. Tang, “Event-triggered protocol for the consensus of multi-agent systems with state-dependent nonlinear coupling,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 2, pp. 723–732, 2018.
View at: Publisher Site | Google Scholar
Q. Jia and W. K. S. Tang, “Consensus of multi-agents with event-based nonlinear coupling over time-varying digraphs,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 65, no. 12, pp. 1969–1973, 2018.
View at: Publisher Site | Google Scholar
J. Feng, S. Chen, J. Wang, and Y. Zhao, “Quasi-synchronization of coupled nonlinear memristive neural networks with time delays by pinning control,” IEEE Access, vol. 6, pp. 26271–26282, 2018.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, and H. Li, “Exponential synchronization of memristive chaotic recurrent neural networks via alternate output feedback control,” Asian Journal of Control, vol. 20, no. 1, pp. 469–482, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
C. Zhang, X. Wang, C. Luo, J. Li, and C. Wang, “Robust outer synchronization between two nonlinear complex networks with parametric disturbances and mixed time-varying delays,” Physica A: Statistical Mechanics and its Applications, vol. 494, pp. 251–264, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
X.-J. Li and G.-H. Yang, “FLS-based adaptive synchronization control of complex dynamical networks with nonlinear couplings and state-dependent uncertainties,” IEEE Transactions on Cybernetics, vol. 46, no. 1, pp. 171–180, 2016.
View at: Publisher Site | Google Scholar
Z. Tang, J. H. Park, and H. Shen, “Finite-time cluster synchronization of lur'e networks: a nonsmooth approach,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 8, pp. 1213–1224, 2018.
View at: Publisher Site | Google Scholar
X. Li and P. Rao, “Synchronizing a weighted and weakly-connected Kuramoto-oscillator digraph with a pacemaker,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 62, no. 3, pp. 899–905, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
W. Yu, P. DeLellis, G. Chen, M. di Bernardo, and J. Kurths, “Distributed adaptive control of synchronization in complex networks,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2153–2158, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
A. Hu, J. Cao, M. Hu, and L. Guo, “Cluster synchronization in directed networks of non-identical systems with noises via random pinning control,” Physica A: Statistical Mechanics and its Applications, vol. 395, pp. 537–548, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Tang, “Terminal sliding mode control for rigid robots,” Automatica, vol. 34, no. 1, pp. 51–56, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 950–955, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
S. Boyd, L. El Ghaoui, E . Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, Pa, USA, 1994.
View at: Publisher Site | MathSciNet
Y. Shang, “Finite-time scaled consensus in discrete-time networks of agents,” Asian Journal of Control, vol. 20, no. 6, pp. 2351–2356, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Shang, “Finite-time scaled consensus through parametric linear iterations,” International Journal of Systems Science, vol. 48, no. 10, pp. 2033–2040, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Shang, “Fixed-time group consensus for multi-agent systems with non-linear dynamics and uncertainties,” IET Control Theory & Applications, vol. 12, no. 3, pp. 395–404, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Shang and Y. Ye, “Fixed-time group tracking control with unknown inherent nonlinear dynamics,” IEEE Access, vol. 5, pp. 12833–12842, 2017.
View at: Publisher Site | Google Scholar
Y. Xu, X. Wu, and C. Xu, “Synchronization of Time-Varying Delayed Neural Networks by Fixed-Time Control,” IEEE Access, vol. 6, pp. 74240–74246, 2018.
View at: Publisher Site | Google Scholar
Y. Shang, “On the delayed scaled consensus problems,” Applied Sciences, vol. 7, no. 713, pp. 1–10, 2017.
View at: Google Scholar
X. Li, J.-A. Fang, and H. Li, “Master–slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control,” Neural Networks, vol. 93, pp. 165–175, 2017.
View at: Publisher Site | Google Scholar

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Copyright © 2019 Bin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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