Fixed-Time Synchronization for Dynamical Complex Networks with Nonidentical Discontinuous Nodes

This article investigates the ﬁxed-time synchronization issue for linearly coupled complex networks with discontinuous nonidentical nodes by employing state-feedback discontinuous controllers. Based on the ﬁxed-time stability theorem and linear matrix inequality techniques, novel conditions are proposed for concerned complex networks, under which the ﬁxed-time synchronization can be realized onto any target node by using a set of newly designed state-feedback discontinuous controllers. To some extent, this article extends and improves some existing results on the synchronization of complex networks. In the ﬁnal numerical example section, the Chua circuit network is introduced to indicate the eﬀectiveness of our method by showing its ﬁxed-timely synchronization results with the proposed control scheme.


Introduction
As we know, in the last few decades, complex networks have been widely presented in our real world, for example, electrical power grids, metabolic pathways, neural networks, food webs, and World Wide Web [1][2][3][4][5]. Synchronization is a well-known crucial collective behavior for complex networks, so the synchronization of complex networks has received more attention due to many crucial applications [6][7][8] in information processing, secure communication, and biological systems [9][10][11][12]. Up to now, there are many research studies on complex network synchronization, most of which focus on asymptotic synchronization, mainly on asymptotic synchronization behavior [13][14][15] and exponential synchronization results [16], but the two kinds of synchronization belong to the infinite-time category [17][18][19].
Since it has been found that finite-time control ways will further enhance the rate of convergence greatly and synchronization will be performed in a settling time by designing appropriate finite-time synchronization controllers, the finite-time synchronization research [20][21][22][23][24][25] in complex networks has been carried out one after another [26][27][28][29][30]. In [20], the issue of the finite-time synchronization is studied between complex networks with nondelay and delay coupling by using pulse control and periodic intermittent control. By use of aperiodically intermittent control, Liu et al. [21] considered the finite-time synchronization problem in dynamic networks with time delay. e global random finite-time synchronization issue is investigated in [22] for discontinuous semi-Markov switched neural networks with time delay and noise interference. e finite-time synchronization analysis of linear coupled complex networks is discussed [23] with discontinuous nonidentical nodes.
e convergence rate of classical finite-time synchronization is relatively fast in contrast to asymptotic synchronization and exponential synchronization. However, it has an obvious disadvantage that the synchronization convergence rate of complex networks depends on the initial states of all nodes. Unfortunately, it is very difficult or even impossible for some chaotic systems to know their state previously. In these results, the finite-time control methods may be ineffective. Taking advantage of the benefits of finite-time control, a special finite-time synchronization is proposed in [20]. As for the novel fixed-time synchronization, the settling time has no relation with the initial conditions of the network system and only depends on the control parameters of the system controller, see [21,22]. us, synchronization can be accomplished by using the fixed-time controller within a specified time.
Moreover, if the dynamics of the nodes are different, then the synchronization issue will be more complex and challenging than the same node condition. By using the free matrix, the equilibrium solution synchronization is concerned on all alone nodes together with the average state trajectory synchronization of different nodes in [35]. e intermittent controller is employed to fix the complex network with different nodes in [36]. In [37], the cluster synchronization problem is investigated for complex dynamic networks with time-delay coupling and nonidentical nodes by the pinning control method. Furthermore, the finite-time synchronization issue is considered for coupled complex networks with discontinuous nonidentical nodes in [23].
Recently, the complex networks with perturbations have attracted more attention for their wide applications [38][39][40][41][42][43]. In [40], the global exponential synchronization issue is studied for linear coupled neural networks with impulsive disturbance and time-varying delay. e clustering synchronization scheme is deeply concerned with regard to uncertain delayed complex networks in [41]. e adaptive pinning control design is proposed in [42] for the clustering synchronization problem of coupled complex networks with uncertain disturbances.
Until now, there are several research results on the finitetime synchronization of complex networks with different nodes or uncertain disturbances, mostly about the asymptotic or exponential synchronization. However, it has not been fully investigated for the fixed-time synchronization analysis of heterogeneous networks with uncertain disturbances, and the relevant research results are rarely covered. In a word, it is indispensable and significant to consider the fixed-time synchronization problem of complex networks with different nodes and uncertain disturbances, which has profound theoretical and practical significance. From the above analysis, we face two difficulties: (i) what conditions are applicable and easy to verify for general complex networks with different nodes and uncertain disturbances? (ii) How to design the controller to overcome heterogeneity and uncertain disturbance of network nodes? is paper tries to conquer these two difficulties and realize the fixed-time synchronization of a certain kind of complex linear coupled networks with different nodes and uncertain disturbances, and then the theoretical results of network synchronization can be further enriched.
Applying the discontinuous control scheme, the fixedtime synchronization problem is analysed for complex networks with uncertain disturbances and nonidentical nodes. Our main contributions here can be concluded as follows: (1) for a class of heterogeneous networks with uncertain disturbances, a novel state-feedback discontinuous controller is designed to get over the influence on the fixed-time synchronization from heterogeneous nodes and uncertain disturbances simultaneously; (2) several criteria are proposed to deduce the fixed-time synchronization for the considered networks. Unlike most existing results, the obtained fixed-time synchronization conditions are expressed by linear matrix inequality, which is easy to be verified; (3) as special cases, the fixed-time synchronization of complex networks without uncertain disturbances is also considered by employing some existing controllers, respectively, and the corresponding results are given in some corollaries. e rest of the paper is arranged as follows. A network model is established with uncertain disturbances and nonidentical nodes, and then the problem of the fixedtime synchronization is described; meanwhile, some necessary definitions and assumptions are given in Section 2. e fixed-time synchronization conditions are achieved in Section 3. Several numerical examples are introduced in Section 4 to indicate the effectiveness of the proposed results. Section 5 summarizes the research conclusions of this paper and puts forward the future research directions.

Problem Formulation and Preliminaries
A kind of nonlinear system including N nonidentical nodes with diffusion linear coupling is considered, in which each node can be regarded as an n-dimensional dynamic system, as shown in the following: where x i (t) � [x i1 (t), . . . , x in (t)] T ∈ R n denotes the state vector of the ith dynamical node; the dynamics of the ith uncoupled node is _ x i (t))] T : R + × R n ⟶ R n is a uncertain vector and represents the disturbance. e constant c > 0 can be considered as the coupling strength of the concerned networks, and Γ � (c ij ) ij ∈ R n×n is a matrix and denotes the inner coupling relation between the network nodes and indicates how the components of each pair of nodes are connected with each other, and c ij ≥ 0; G � (G ij ) N×N is a coupling configuration constant matrix, which describes the topological structure and can be exhibited as the diffusion structure, i.e., G ij ≥ 0 and G ii � − N j�1,j≠i G ij . In this paper, the driven dynamical node of (1) satisfies where A 0 ∈ R n×n , f 0 (t, x 0 (t)) ∈ R n , and h 0 (t, x 0 (t)) ∈ R n . In fact, most of the well-known chaotic systems can be described by the above dynamical equation, such as Sprott circuit, Chua circuit, Rössler's systems, and Chen system [42].
Definition 1 (see [44]). Complex network (1) is said to be synchronized onto (2) in finite time if there exist a designed feedback controller to system (1) and a constant t * > 0 such that where t * > 0 is called the settling time and often depends on the initial state vector value Definition 2 (see [44]). Complex network (1) is said to be synchronized onto (2) in fixed time if there exists a fixed settling time T * > 0 such that where T * > 0 is called the settling time and is independent of the initial synchronization error In this paper, the goal is to fixed-timely synchronize the state of network (1) onto the driven one (2) by designing feedback controllers.
Obviously, controlled complex network (1) can be rewritten as follows: Introduce the synchronization errors e i (t), which are defined as (5), the error dynamical network model can be given by In order to obtain our main results, some necessary assumptions are listed as follows.

Assumption 1.
ere exist constant M i > 0 and uniformly symmetric positive definite matrix ere exist constant M 0 > 0 and uniformly symmetric positive definite matrix L 0 such that f 0 (t, x) satisfies Assumption 3 (see [43]). ere exists a time-varying function μ(t) ≥ 0 such that Assumption 4. For any i, i �� 1, 2, . . . , n, the uncertain function vector h i (t, x i (t)) is assumed to be continuous at t, x i (t) ≥ 0 and bounded. Moreover, there is a known nonnegative number h max such that Remark 1. Assumptions 1 and 2 are general and satisfied with most of the well-known chaotic systems, for instance, Chua circuit [43], Rössler's systems, and discontinuous Chen system. In fact, the above systems meet the following conditions: there exist some positive constants . , x n ) T ∈ R n , and y � (y 1 , y 2 , . . . , y n ) T ∈ R n . Using condition (11), we have Mathematical Problems in Engineering where Clearly, if α � max 1≤i≤n l i jj , j � 1, 2, . . . , n and M � max 1≤i≤n β i , then Assumptions 1 and 2 involve conditions (H2) and (H3) in [25]. Moreover, the continuous chaotic system is also a special case by setting M i � 0 in Assumption 1 or M 0 � 0 in Assumption 2, for instance, the continuous Rössler system, Chua's circuit, Chen system, Lorenz system, and logistic differential system. Hence, Assumptions 1 and 2 are more general, and most popular chaotic systems are applicable. Assumptions 3 and 4 take advantage of conditions on the activation function, and it is seen that they are diffusely imposed in the literature [23,[35][36][37].
According to Definition 2, it is clear that the fixed-time synchronization of dynamical network (5) onto (2) can be degenerated into the fixed-time stabilization of error dynamic system (6).

Fixed-Time Synchronization Analysis
In this part, the controllers are designed for the fixed-time synchronization problem of complex network (1), and concerned complex network (5) can realize the fixed-timely synchronization under the appropriate designed controllers. Firstly, we give the synchronization controller design of complex network (1), and then fixed-time synchronization criteria can be obtained based on error system (6). Several corollaries are also obtained for (5) and (2) with identical nodes. For concerned complex network (1), the control input u i (t) ∈ R n , i � 1, . . . , N, is designed as follows: where t)), . . . , sign(e in (t))), and the real numbers p, q follow 0 < q < 1, p > 1. e following lemmas are necessary and given to derive the subsequent main results.

Denote υ(t) � V(x(t)) if there exists a continuous function
for any t > 0 that υ(t) > 0, and υ(t) is differentiable at t and satisfies en, we have υ(t) � 0 for t ≥ t 1 . In particular, if c(υ) � Qυ μ , where μ ∈ (0, 1) and Q > 0, then the setting time is estimated by Lemma 2 (see [47]). For matrices A, B, C, and D with appropriate dimensions and a scalar α, the following assertions hold: where ⊗ is the Kronecker product.

Theorem 1. For concerned complex network (1) with the control input, if Assumptions 1-4 hold and
, then drivenresponse complex networks (1) and (2) can achieve fixed-time synchronization under controller (13), with the settling time where

For error dynamical system (6), a Lyapunov function is listed by
and the derivative of the above Lyapunov function along the trajectory of system (6) can be computed as with According to Lemma 2, we know the following equation is true: For the term I 2 (t), using condition (10) in Assumption 4, we get the following inequality:

Mathematical Problems in Engineering
Also, it can be seen that with . From (7) in Assumption 1, it is known that where L � diag(L s 1 , L s 2 , . . . , L s N ). By (9) in Assumption 3, one can get Submitting (28)-(32) into (27) yields that where By use of Lemma 2 and (24), it gives that where D � diag (d 1 , d 2 , . . . , d N ).

Mathematical Problems in Engineering
Submitting (36), (37), and (43) to (33), we can get According to Lemma 4,V(t) converges to zero within a settling time T * , which is defined in Definition 2, and one can obtain that, by use of controller (13), the considered complex network (1) is fixed-timely synchronized onto driven node (2) within the fixed time T * , which is given by . (45) erefore, it can be concluded that the error vector e i (t) converges to zero within T * , and driven-response complex networks (1) and (2) are fixed-timely synchronized under controller (13) within the fixed time T * . e proof is completed.

Remark 2.
In recent years, a lot of extensive research has been conducted on the finite-time synchronization and fixed-time synchronization of complex networks, and many breakthroughs have been made. However, as far as we know, there are few published papers that deal with the fixed-time synchronization of heterogeneous complex networks. eorem 1 suggests a way to choose the controller to realize the fixed-time synchronization for the heterogeneous complex network. e controller consists of three sections: the first two terms are used to overcome the influence from the linear condition of the nonlinear function, the second one − η i (t)sign(e i (t)) is introduced to compensate the influence of disturbance h i (t, x i (t)), and finally, the last section sign(e i (t))(a|e i (t)| p + b|e i (t)| q ) is employed to force the considered networks achieve the fixed-time synchronization. Now, if M � max 1≤i≤n M i and η(t) � max 1≤i≤n η i (t), then the controllers u i (t) ∈ R n can be designed as follows (i � 1, . . . , N): where the parameters d i , a, b, p, and q are defined as the same as in (13). erefore, by using the same analysis method in eorem 1, we can obtain Corollary 1 that is a similar conclusion with [25]. (1) and (2) under controller (46), if Assumptions 1-4 hold and the control parameters η i (t) and d i in (46) satisfy the following inequalities,

Corollary 1. For concerned complex networks
where , then driven-response complex networks (1) and (2) can achieve fixed-time synchronization under controller (46), with the settling time If the uncertain disturbance is not considered in the complex network model, i.e., h 1 � h 2 � · · · � h N � 0, then the following network model is degenerated as Let x 0 | t�0 � x 0 (0), and then the driven network node is governed by en, the corresponding error dynamical system can be rewritten as follows: where e controllers are the same as before, and then a criterion can be obtained on the fixed-time synchronization of the concerned complex networks with nonidentical nodes. By taking h max � 0 in eorem 1, one can easily get the following corollary, and its proof is omitted here. (46). If Assumptions 1-3 hold and the controller parameters satisfy the following matrix inequalities,

Corollary 2. Consider complex network (49) with drive node (50) under the set of controllers
, then (49) can be synchronized to the state of drive node (50) within a fixed time T * and the settling time Furthermore, if h i � 0, A i � 0, and f i � f for i � 0, 1, . . . , N in (49), then complex network (1) is further reduced to 8 Mathematical Problems in Engineering and driven network node (2) is changed correspondingly into the following form: In the issue, the proposed fixed-time synchronization scheme can be applied to the corresponding complex networks with identical nodes here, and the criteria are given in the following corollary.

Conclusions
e fixed-time synchronization problem is studied for a type of dynamic complex networks with nonidentical nodes and uncertain disturbances. By employing the Lyapunov function theory, some novel sufficient conditions are provided and further applied to some special cases, such as the identical node issue. Future work may be centered on synchronous applications of complex networks with nonidentical nodes and uncertain disturbances.

Data Availability
e data used to support the findings of this study are included within the article. No other data are used beyond this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.