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

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 finitetime 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.


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][2][3] and many practical systems, including sensor network, communication network, neural networks, social network, and so on [4][5][6], can be modeled by complex networks.Therefore, many valuable and meaningful results for synchronization dynamics problems of complex networks have been obtained [7][8][9][10][11][12][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][17][18][19][20][21][22][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  ∞ 2 Complexity 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][27][28].Hence, a lot of results about finitetime synchronization dynamics problems for coupling systems have been obtained [29][30][31][32][33][34].For example, based on finite-time control theory, Wang et al. [29] designed finitetime 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][38][39][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][36][37][38][39][40][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 finitetime synchronization dynamics for a class of nonlinear coupling multi-weighted complex networks (NCMWCNs) with switching topology based on stability and a novel finitetime 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 finitetime 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 2 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. ≥  > 0 (respectively,  >  > 0), where ,  ∈  × are symmetric matrices, means that the matrix  −  is positive semidefinite (respectively, positive definite).If  is a matrix,  max () and  min () denote its maximal eigenvalue and its minimum eigenvalue, respectively.The Kronecker product of matrices  ∈  × and  ∈  × is a matrix in  × denoted as  ⊗ .The matrix diag() represents diagonal matrix.If the dimensions of matrices are not explicitly indicated that means they are suitable for any algebraic operations.

Model and Preliminaries
Firstly, consider a class of NCMWCNs with switching topology presented by where () : [0, ∞) →  = {1, 2, . . ., } is the topology switching signal, which is defined as the switching sequence where  0 is the initial time and   denotes the serial number of the activated subsystem at   .  ( = 1, 2, . . ., ) represents the coupling strength and   > 0, Γ  = diag( 1 ,  2 , . . .,   ) > 0 is inner-coupling matrix and Γ  ∈  × and  , = ( ,  ) × is outer-coupling matrix with the coupling weights in the th coupling form and the th topology.For each  ∈ , there is  ,  =  ,  > 0 if node i and node j are connected.Otherwise,  ,  =  ,  = 0.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 .
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][15][16][17][18][19][20][21][22][23].However, it should be emphasized that, in these existing works [14][15][16][17][18][19][20][21][22][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 .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 (  ()) =   (), () : [0, ∞) →  = {1} and  ,  = − ∑  =1, ̸ =  ,  , 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).
Remark .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].

Main Results
In this section, we, respectively, design two classes of finitetime 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).

. Based on Lemma , the Design of Finite-Time Synchronization Control Rule for the Network ( )
Theorem 12.Under Assumptions and , if there exists the network ( ) can realize finite-time synchronization using the following controller where  > 0 and 0 <  < 1.Moreover, the settling time of synchronization  * 1 satisfies where  = (1 + )/2.
Proof.Consider the Lyapunov-Krasovskii functional for the network (1) as The next proof is similar to that of Theorem 12.
Remark .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 sectorbound 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 .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  , ⊗ Γ  = ( , ⊗ Γ  )  , according to LMIs ( 17) and (28), it is derived that , where A , =  , ⊗ Γ  .Letting Ψ  ≤ 0 hold, then it is clear that Ψ  ≤ 0 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 Ψ  ≤ 0 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.
Similar to the proof of Corollary 18, we can obtain Corollaries 19-21.
Remark .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.Proof.From the proof of Corollaries 18 and 20, there must be  + 3 ((), ) <  + 1 ((), ) ≤ 0. 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

Corollary 24. Under Corollaries and , global synchronization dynamics of the network ( ) within finite time
, where  > 0 and (]) > 0, we can make  * 1 >  * 2 > 0 hold.The proof is completed.Remark .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 (()) > 0 but also closely connected with  , > 0, () < 0, and (()) > 0, 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 (0), 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.

Corollary 25. Under Corollaries and , global synchronization dynamics of the network ( ) within finite-time
Similar to the proof of Corollary 18 and letting  1 ∈ , ( 1 + 1) ∈ , then we can also obtain the switching of  , being how to affect finite-time synchronization dynamics of the network (1).
Remark .Note that there is no function relationship between  * 2 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  * 2 .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  , > 0 or  , < 0. Therefore, one has ( , ) > 0 or ( , ) < 0. Letting  , = [−2, 1, 1; 1, −2, 1; 1, 1, −2], then one gets that ( , ) is -3, -3, and 0, respectively.This shows that if  ,  = − ∑  =1, ̸ =  ,  ,  , > 0 and  , < 0 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
. 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 multiagent 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 discretetime 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 .It is worth noting that because finite time control is mainly dependent on the initial conditions and fixedtime 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 multiagent 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][51][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 .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.

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 Complexity 13 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 outercoupling matrix [54], how to get the related results is still an open problem.
* 1 is poorer than that of the network ( ) within finite-time  * 2 and synchronization convergence time  * 1 of the network ( ) is larger than synchronization convergence time  * 2 of the network ( ).Proof.Similar to the proof of Corollary 24, we can derive Corollary 25.

Figure 1 :
Figure 1: Synchronization state trajectories of the network (36) for the case I of Corollary 20.

Figure 2 :
Figure 2: Synchronization total error trajectories of the network (36) for the case I of Corollary 20.

Figure 3 :
Figure 3: Synchronization state trajectories of the network (36) for the case II of Corollary 20.

Figure 4 :
Figure 4: Synchronization total error trajectories of the network (36) for the case II of Corollary 20.

( 11 −
k− Î) (  ()), where k = 3,4 and Î = ,.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 Complexity

Figure 5 :
Figure 5: Synchronization state trajectories of the network (36) for the case I of Corollary 21.

Figure 6 :
Figure 6: Synchronization total error trajectories of the network (36) for the case I of Corollary 21.

Figure 7 :
Figure 7: Synchronization state trajectories of the network (36) for the case II of Corollary 21.

Figure 8 :
Figure 8: Synchronization total error trajectories of the network (36) for the case II of Corollary 21.

Figure 9 :
Figure 9: Synchronization state trajectories of the network (36) for the case I-1 and I-2 of Corollary 27.

Figure 10 :
Figure 10: Synchronization total error trajectories of the network (36) for the case I-1 and I-2 of Corollary 27.

Figure 11 :
Figure 11: Synchronization state trajectories of the network (36) for the case I-1 and I-2 of Corollary 29.

Figure 12 :
Figure 12: Synchronization total error trajectories of the network (36) for the case I-1 and I-2 of Corollary 29.