The Fast Fixed-Time Bipartite Synchronization of Coupled Delayed Neural Networks with Signed Graphs

In this paper, the ﬁxed-time bipartite synchronization problem for coupled delayed neural networks with signed graphs is discussed. Diﬀerent from traditional neural networks, the interactions between nodes of delayed neural networks can be either collaborative or antagonistic. Furthermore, compared with the initial-condition based ﬁnite-time synchronization, the settling time is bounded by a constant within ﬁxed-time regardless of the initial condition. It is worth noting that the ﬁxed-time stable network for bipartite synchronization in this paper achieves more faster convergence than most existing publications. By applying constructing comparison system method, Lyapunov stability theory and inequality techniques, some suﬃcient criteria for ﬁxed-time bipartite synchronization are obtained. Finally, two numerical examples are granted to display the performance of the obtained results.


Introduction
As we know, the control problem of neural networks plays an important role due to its broad applications, which has been capturing considerable attention from various fields, such as parallel computation, signal and image processing, nonlinear optimization, pattern recognition, among many others [1][2][3][4][5][6][7][8]. Generally speaking, according to different interactions between nodes of neural networks, the networks are classified into two types: unsigned networks and signed networks. If the interaction digraphs are assumed to be unsigned, then we get an unsigned networks, where all the communication links are positive. If the network with signed interaction digraphs, where the communication links between neighboring nodes of network can be either positive or negative, then it is a signed network. The pioneering works in [9][10][11][12] have been investigated many dynamical behaviors of unsigned networks. However, the analysis method is difficult to be applied to signed networks because these proofs depend on the symmetry of the communication topology. Yet different from the traditional unsigned neural networks, the signed neural networks can exhibit an interesting synchronization phenomenon named as bipartite synchronization. Bipartite means the nodes of these networks can be divided into two disjoint nonempty sets such that every edge only connects a pair of nodes which belong to different sets. Many real-world networks can be described by bipartite networks with signed graphs, such as the papersscientists networks [13], producer-consumer networks [14], the actors-films networks [15] and so on.
Over the past years, the synchronization problem have been investigated by applying different control methods, such as adaptive synchronization [16], impulsive synchronization [17], pinning synchronization [18] and so on. Recently, the researchers pay more attentions to investigate the bipartite synchronization for some networks. For example, Bian investigated the adaptive synchronization of bipartite dynamical networks with distributed delays and nonlinear derivative coupling by constructing effective adaptive feedback controllers and update laws in [19]. In [20], the authors study the bipartite synchronization in a network of nonlinear systems with collaborative and antagonistic interactions by using contraction theory to obtain some sufficient conditions. Liu investigated the bipartite synchronization in coupled delayed neural networks by utilizing the pinning control strategy and M-matrix theory in [21]. Guo derived the bipartite consensus for multi-agent Notations: R is the set of real numbers. R n and R n×m denote the n-dimensional Euclidean space and the set of n × m matrices. A = (a ij ) n×n represents a matrix of n × n-dimension. A s = 1 2 (A + A T ). A ≥ 0 (A ≤ 0) implies A is symmetric and semi-positive (semi-negative) definite matrix. · is the 2-norm of a vector or a matrix. I N stands for N -dimension identity matrix. sign(·) is sign function and sign(s) = (sign(s 1 ), sign(s 2 ), · · · , sign(sn)) T for s = (s 1 , s 2 , · · · , sn) T ∈ R n .

Preliminaries
In this section,we will formulate the considered problem in this paper and give some useful mathematical preliminaries.
This paper investigates the following coupled delayed neural networks: where i ∈ N = {1, 2, . . . , N }, z i (t) = (z i1 (t), z i2 (t), · · · , z in (t)) T ∈ R n is the state vector of the ith note; C =diag(c 1 , c 2 · · · , cn) > 0, and A = (a ij ) n×n ∈ R n×n , B = (b ij ) n×n ∈ R n×n represent the weight and delayed weight matrices for the ith neural-network node, respectively; f (z i (t)) = (f 1 (z i1 (t)), f 2 (z i2 (t)), . . . , fn(z in (t))) T ∈ R n is continuous vector function, f j (·), j = 1, 2, · · · , n is odd function, that is f j (−y) = −f j (y), τ ≥ 0 is the node-delay; the constant σ > 0 is the coupling strength, L = (ℓ ij ) N ×N is the adjacency matrix associated with the signed graph G of the network (1), satisfying ℓ ii = 0 for i ∈ N ; ℓ ij = 0 (i = j) if there is a directed communication link from nodes j to i, otherwise ℓ ij = 0. If ℓ ij > 0, the coupling term is presented by ℓ ij (z i (t) − z j (t)), in this case, the interaction between nodes i and j is cooperative; if ℓ ij < 0, the coupling term becomes −ℓ ij (z i (t) + z j (t)) implying that the nodes i and j are competitive. u i (t) is a controller that we will design later. z i (0) ∈ R n , i ∈ N is the initial value of system.
To derive the main results of this paper, we will make the following assumptions: The signed graph G of network(1) is structurally balanced if it admits a bipartition of the nodes . . , vn} denotes a set which concludes all the nodes of the signed graph G.
Assumption 2 In network (1), Nonlinearity f (·) is an odd function satisfied a Lipshitz condition, that is, there exists an unknown constant η such that Next, the definition of the fixed-time bipartite synchronization is proposed.

Definition 1
The network (1) is said to achieve fixed-time bipartite synchronization with (2) if there exists a constant t 1 > 0 (t 1 is independent of z(0) = (z 1 (0), z 2 (0)), . . . , z N (0)) T and y 0 ) such that There is no doubt that when W = I N , the fixed-time bipartite synchronization reduces to the traditional fixed-time synchronization. Therefore, this paper can improve some existing fixed-time synchronization.
Lemma 1 (Young's inequality) Suppose a, b, u, v are all positive scalars, and 1 where " = " holds if and only if a u + b v = 1.

Fixed-time bipartite synchronization of the delayed neural networks
In this section, we focus on studying the fixed-time bipartite synchronization of complex networks by constructing comparison system. Under Assumption 1, we design the following controller u i (t) to achieve bipartite synchronization.
where λ i , θ i > 0 is constant to be determined, γ 1 > 0 and γ 2 > 0 are tunable constants. c 1 > c 2 > 0 and Remark 2 In the designed controller, ω i is brought to overcome the difficulties induced by signed graphs.
Let L denote the Laplacian matrix of signed graph G, then we have Clearly, if there is a ℓ ij < 0, L can not guarantee the sum of row is zero, which means the Laplacian matrix is different from this generated by unsigned graph. Furthermore, one can derive thaṫ Now we defineL = (l ij ) N ×N = (|ℓ ij |) N ×N . LetĜ andL = (l ij ) N ×N represent the graph and Laplacian matrix associated withL. Obviously,Ĝ is an unsigned graph andL is a zero-row-sum matrix. Takeẑ ) and by considering the fact that f j (·) is odd function. Furthermore, we can generate thaṫ That isė Therefore, if there exists a T > 0 such that e i (T ) = 0 and e i (t) ≡ 0 for t ≥ T , then according to ω 2 i = 1, it follows that lim t→T z i (t) − ω i y(t) = 0 and z i (t) − ω i y(t) ≡ 0, for t > T . This implies the fast fixed-time bipartite synchronization is realized.
With the help of controller (5), the following theorem presents some conditions for achieving fast fixed-time bipartite synchronization in the signed network (1).
According to the similar analysis of Lemma 1 in [30] and Theorem 1 in [31], let χ(t) = ν By use the similar calculation with Lemma 1 in [30] and Theorem 1 in [31], we will obtain the estimation of settling time is 2c 2 . Therefore, the control algorithm (5) solves the bipartite synchronization problem of signed network (1). that is, the coupled delayed neural network (1) is fixed-time bipartite synchronization with system (2) in T , which can be described by (1). The proof is completed.

Remark 3
The results reported in [25][26][27][28][29] have been discussed the fixed-time control problems, the convergence rate in this paper is faster than most existing fixed-time results since 1 α n m + q q−p 1 which is presented in [30,31]. In addition, the estimation of settling time (12) is also more accurate than the previous works in [25][26][27][28][29], where the settling time of fixed-time synchronization is independent of system's initial value and it is related to parameters of system and controllers.

Remark 4
In the previous investigation, the fast fixed-time synchronization of delayed neural networks have been discussed in [31], but this paper have not considered the fast fixed-time bipartite synchronization of delayed neural network with signed graphs. In addition, although the authors have considered the fast fixedtime synchronization in [25,26], but the fast fixed-time bipartite synchronization problem is much more general, which is still challenging.
In the following, we will discuss some special cases by applying Theorem 1. When there is no delays in neural networks, system (1) reduces tȯ The goal node of network (20) is presented bẏ where y(t) = (y 1 (t), y 2 (t), . . . , yn(t)) T ∈ R n . The definitions of other parameters are similar with system (1).
Then the control law u i (t) is given We can obtain the following corollary by applying the similar analysis from Theorem 1.
Corollary 1 Let Assumptions 1 and 2 hold, Under the control protocol defined by (22), if the control gaiñ λ i satisfy the following conditions where the definitions of corresponding parameters are similar with Theorem 1. Then the neural network (20) can achieve fast fixed-time bipartite synchronization with (21). Moreover, the settling time is estimated as Proof Consider the following Lyapunov function: One can derive thaṫ According to the similar analysis with Theorem 1, we can construct the following comparison system 3 +c 4 2c 4 (t), 0 <ν(t) < 1, 0,ν(t) = 0, Using the similar discussion with Theorem 1, if the constantT satisfiesν(t) ≡ 0, for any t ≥T , then it also hold V (t) ≡ 0, for any t ≥T . Therefore, there is no difficult to derive that when Therefore, the control algorithm (22) solves the bipartite synchronization problem of signed network (20). that is, the coupled neural network (20) is fixed-time bipartite synchronization with system (21) inT . (5) and (22), one can see that −θ i

Remark 5 From the controllers
) is used to deal with the delays introduced in (1).

Numerical Simulations
In this section, two numerical examples will be proposed to demonstrate the correctness of the bipartite synchronization criteria. Considering signed network (1) including nine nodes, which is shown in Fig. 1. It can be seen from Fig. 1 that the topology of the signed network (1) Example 1. Consider a single delayed neural network is described aṡ where y(t) = (y 1 (t), y 2 (t)) T , f (y(t)) = (tanh(y 1 (t)), tanh(y 2 (t))) T , the parameters are given as follows: Obviously, f (·) is odd function, Assumption 2 is satisfied with η = 1. To illustrate the effectiveness of the proposed method, let node-delay τ = 1 and the initial value is y(0) = (0.5, 0.6) T , Fig. 2 shows the the chaotic trajectory of goal node (2).
The values of the parameters of the fixed-time control laws are designed by λ i = 8, θ i = 1, γ 1 = 10, γ 2 = 5, c 1 = 5, c 2 = 3, c 3 = 1, c 4 = 3, and the coupling strength σ = 0.2 which satisfy the conditions in Theorem 1. By simple computation, we can get the settling time is about 1.2176. Obviously, the settling time is less than the theoretical estimate in this control protocol as illustrated in Fig. 3. The trajectories of y(t) and z i (t), i = 1, 2, . . . , 9 are presented in Fig. 3, where the red line denotes the trajectories of y 1 (t), y 2 (t), and the blue line represents the trajectories of z i1 (t), z i2 (t), from which it can be observed that the fixed-time bipartite synchronization can be achieved for the delayed neural network, which verifies the theoretical result in Theorem 1 well. Example 2. Consider the following single delayed neural networks: where y(t) = (y 1 (t), y 2 (t)) T , The activation functions are assumed to be f (y(t)) = 0.5(|y(t) + 1| − |y(t) − 1|). (29) Obviously, the function f (·) satisfies Assumption 2 with η = 0.5. Similarly, the chaotic trajectory of system (28) with τ = 1 and the initial value y(0) = (0.2, 0.1) T is shown in Fig. 4. Considering the following network with signed graphs which are coupled by chaotic system (28) described byż |ℓ ij |(z i (t) − sign(ℓ ij )z j (t)), i = 1, 2, . . . , 9.
In this example, if λ i ≥ C + η A + 1 2 η 2 B 2 + σλmax(Φ s ), θ min ≥ 1 2 , where λmax is the maximum eigenvalue of Φ s , θ min = min{θ 1 , θ 2 , · · · , θ N }, i = 1, 2, . . . , 9, then the Theorem 1 can be satisfied. For numerical simulation, the parameters of the controller are selected by λ i = 4, θ i = 0.5, γ 1 = 20, γ 2 = 10, c 1 = 5, c 2 = 3, c 3 = 1, c 4 = 3, and the coupling strength σ = 0.2, which satisfy the conditions in Theorem 1. By simple computation, we can get the settling time is about 0.5853. Obviously, the settling time is less than the theoretical estimate in this control protocol as illustrated in Fig. 5. The trajectories of y(t) and z i (t), i = 1, 2, . . . , 9 are presented in Fig. 5, where the red line denotes the trajectories of y 1 (t), y 2 (t), and the blue line represents the trajectories of z i1 (t), z i2 (t), from which it can be observed that the fixed-time bipartite synchronization can be achieved for the delayed neural network, which verifies the theoretical result in Theorem 1 well.

Conclusions
In this paper, the fast fixed-time bipartite synchronization problem for delayed neural networks with signed graphs has been investigated. By using algebraic graph theory and Lyapunov theory, an efficacious controller has been designed to ensure the fixed-time bipartite synchronization. The settling time of fixed-time bipartite synchronization is proposed accurately, which is more faster convergence sufficient condition than the previous results [25][26][27][28][29][30]. Some numerical examples are given to validate the correctness of the our obtained theoretical results. Note that node-delays in neural networks are always affected by time-varying, it is expected to extend the presented method to time-varying delayed systems. In addition, impulsive effects always exist in complex networks, thus the bipartite synchronization of complex networks under impulsive control will be considered in our future works.