Robust Fixed-Time Synchronization for Coupled Delayed Neural Networks with Discontinuous Activations Subject to a Quadratic Polynomial Growth

In this paper, we focus on the robust fixed-time synchronization for discontinuous neural networks (NNs) with delays and hybrid couplings under uncertain disturbances, where the growth of discontinuous activation functions is governed by a quadratic polynomial. New state-feedback controllers, which include integral terms and discontinuous factors, are designed. By Lyapunov–Krasovskii functional method and inequality analysis technique, some sufficient criteria, which ensue that networks can realize the robust fixed-time synchronization, are addressed in terms of linear matrix inequalities (LMIs). Moreover, the upper bound of the settling time, which is independent on the initial values, can be determined to any desired values in advance by the configuration of parameters in the proposed control law. Finally, two examples are provided to illustrate the validity of the theoretical results.


Introduction
In recently years, coupled neural networks (CNNs), as a special sort of complex dynamic networks, have attracted the widespread attention from a lot of scholars due to its potential applications in parallel computation, multiagent cooperative control, cryptography, nuclear magnetic resonance instrument, and other aspects [1][2][3][4][5]. Particularly, the synchronization with respect to CNNs has been extensively studied in many science fields [6][7][8][9][10][11] and the references therein.
Time delay often arises in the transmission of signals in CNNs [12,13]. In [13], Shao and Zhang considered the delay-dependent stabilization for CNNs with two additive input delays. In [14], He et al. investigated the pinning synchronization for CNNs with hybrid couplings and delays by an adoptive approach. In [15], Wang and Huang discussed the pining synchronization of delayed CNNs with reaction-diffusion effects.
Recently, many works are devoted to the synchronization behaviors of NNs with discontinuous activations. For example, in [16], by applying the state-feedback control strategies, the global finite-time synchronization conditions are addressed for delayed NNs with discontinuous activations. In [17], the authors discussed the global synchronization for NNs with time-varying delays and discontinuous right-hand side. e exponential synchronization for discontinuous NNs with delays has been considered in [18]. In [19], the discrete nonfragile control strategy was designed to achieve the synchronization for fractional-order NNs by adjusting the coupling gain.
In engineering applications, the synchronization is required to be achieved in a finite time [20,21]. In [22], the authors pointed out that the finite-time control can demonstrate better disturbance and robustness rejection properties. erefore, it is meaningful to investigate the global synchronization in finite time for CNNs [23][24][25][26][27].
However, it should be noted that the upper bound of the settling time greatly depends on the initial states of CNNs under the finite-time synchronization. In generality, it is difficult to obtain the initial values in the real world.
is shows that there exists a constraint to practical applications of CNNs [28]. To solve this problem, the fixed-time synchronization control strategy is considered in [28], where the settling time is independent on the initial conditions of CNNs. At present, as a kind of efficient control strategies, the fixed-time synchronization control has received tremendous attention from many researchers. e global fixed-time synchronization was discussed for semi-Markovian jumping neural networks with time-varying delays and discontinuous activation functions in [29]. e authors discussed the global fixed-time stability of dynamical nonlinear systems and realized the global fixed-time synchronization for CNNs with discontinuous activations in [30]. In [31], the nonsingular fixed-time consensus tracking was considered for second-order multiagent networks. en, Zheng et al. discussed the fixed-time synchronization of memristive fuzzy BAM cellular neural networks with time-varying delays based on feedback controllers in [32]. Some criterions were obtained for the fixed-time synchronization of Cohen-Grossberg CNNs with and without parameter uncertainties in [33]. By fixed-time controllers, the global synchronization of delayed hybrid CNNs is studied in [34]. In [35,36], Zhu et al. discussed the synchronization in fixed time with hybrid couplings and delays for semi-Markovian switching complex NNs, and the upper bound of settling time could be determined under the designed controller. e authors considered the global synchronization in fixed time for CNNs with delays and discontinuous/continuous activations and proposed two discontinuous control protocols in [37]. Event-triggered synchronization in fixed time was investigated for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity in [38]. It is worth pointing out, to the best of authors' knowledge, there are few results on the global synchronization for hybrid CNNs with discontinuous activations subject to a nonlinear function growth. Inspired by the above discussion, in this paper, we focus on the global robust fixed-time synchronization for hybrid CNNs with delays and discontinuous activations in the presence of disturbances. Under the designed controllers with integral terms and discontinuous factors, the global synchronization conditions in fixed time are addressed in the terms of LMIs. Moreover, the upper bound of the settling time is estimated explicitly. e primary contributions of this paper are listed as follows: (1) e neuron activation functions are modeled to be discontinuous and subject to a quadratic polynomial growth. e rest of this paper is organized as follows. In Section 2, some preliminaries and CNNs model are provided. In Section 3, the state-feedback discontinuous controllers are designed, and the global fixed-time synchronization conditions are addressed in the form of LMIs. In Section 4, two numerical simulations verifying the theoretic findings are presented. Conclusion is received in Section 5.
Notation. R refers to the set of real numbers. R n represents the n-dimensional Euclidean space, and R n×n denotes the set of all n × n real matrices. Given A ∈ R n×n , λ max(A) (λ min(A) ) stands for the maximal (minimal) eigenvalues of A.

Preliminaries and System Description
Consider an array of hybrid CNNs described by where i � 1, 2, . . . , N, x i (t) � (x i1 (t), x i2 (t), . . . , x in (t)) T is the state of the ith network at time t; . . , q n ), q i > 0; A � (a lr ) n×n represents the connection weight matrix; B � (b lr ) n×n denotes the delayed connection matrix; c 1 and c 2 are . . , f n (x in (t))) T intends neuron activation function; τ > 0 is delay. u i (t) stands for the control input; J(t) � (J 1 (t), J 2 (t), . . . , J n (t)) T is an external input; and D � (d ij ) N×N and H � (h ij ) N×N indicate coupling configuration matrix and delayed coupling configuration matrix, respectively. If there exists an edge from node i to j, then d ij � d ji > 0; otherwise, d ij � d ji � 0(i ≠ j). Laplacian matrix L � (l ij ) N×N of a graph corresponding to D is given by In system (1), f i (·) is made to satisfy the following assumptions: (A1) f i (·) is continuous expect on a countable set of isolate points ρ i k , and f i (·) has at most a limited number of discontinuous points on any compact 2 Mathematical Problems in Engineering interval of R; in addition, at the discontinuous points ρ i k , the finite right limit f i (ρ i+ k ) and left limit f i (ρ i− k ) exist. (A2) Let D ∈ R n be a domain containing the origin. ere exist positive real constants ] l , ω l , and g l for each l � 1, 2, . . . , n, i � 1, 2, . . . , N, ∀ t ≥ 0, such that . Under A1, system (1) is a functional differential equation with discontinuous right-hand side [38]. In this paper, analogous to [39,40], we use the definition of Filippov solutions for system (1).
Definition 1 (see [41]). x i (t) is solution of system (1) in Filippov sense, if the following holds: Noting that set-valued map F(x i ) has a nonempty, compact, and convex value and is upper semicontinuous, so it is measurable. By measurable selection theorem [41], there exists measurable function Definition 2 (IVP [42]). For any Consider the following isolated neural network: Analogous to Definition 2, the IVP associated with system (6) is obtained as follows: In this paper, our objective is to design new feedback controllers to realize the robust fixed-time synchronization between CNNs (1) and isolated network (6). where According to Definitions 2 and 3, IVP of error system can be written as Mathematical Problems in Engineering In order to derive the robust synchronization results of CNNs (1), for the terms Δ i (t, x), we make the following assumption: where Δ max is a known nonnegative constant. (1) is said to be globally robust finite-time synchronized with system (6). Moreover, if there exists scalar T max > 0 such that T(e i , η i ) ≤ T max , then system (1) is said to be globally robust fixed-time synchronized with system (6). T(e i , η i ) is called as the settling time function, and T max is the upper bound of the settling time function.

Main Results
4 Mathematical Problems in Engineering e controller u i (t) is designed as follows: where ere exists a path between network i and (6), if and only if, β i > 0.
Note that controller (20) is discontinuous, we have where e i (t) > 0, co[sign(e i (t))] � 1; e i (t) � 0, co[sign(e i (t))] � [− 1, 1]; e i (t) < 0, co[sign(e i (t))] � − 1. Let , then there exists Sign(e i (t)) ∈ co[sign(e i (t))], such that Theorem 1. Suppose that (A 1 ), (A 2 ), and (A 3 ) are satisfied, and the coupling interaction topology is undirected and connected. If the following conditions  (20). And the upper bound T max of settling time is estimated by Proof. Construct the Lyapunov-Krasovskii functional Calculating the derivative of V(t) at time t along the trajectories of error system (9), it follows that Substituting (22) into (31), we can obtain Mathematical Problems in Engineering Due to d ij � d ji , it follows that � − e T (t) L ⊗ I n e(t).
It is easy to derive that Similar to (31), one obtains Sign e j (t) − e i (t) Sign e j (t) − e i (t) By Assumption (A 2 ), one has 6 Mathematical Problems in Engineering n l�1 e il (t) .
n l�1 e il (t) e inequality above can be rewritten as where η � [e i (t), e i (t − τ)] T . Together (24)- (26) and (37), one has where m is the real number nodes of β i , n � n(N(N − 1) + m), and Noting that ∧ + L is positive definite, we can get where By Lemma 3, we can conclude that error system (9) is globally robust fixed-time stable. is shows that system (1) can achieve the global robust fixed-time synchronization with system (6) under the controller (20). e upper bound T max of settling time is estimated by T max � (3/2K)(K/2K) (2/3) . e proof is completed. □ Remark 1. In [30,[33][34][35][36], the global fixed-time synchronization issues were considered for delayed CNNs with discontinuous activations, where activation function f(·) is subject to linear growth. However, in eorem 1, discontinuous activations f(·) are nonlinear growth and subject to a quadratic polynomial function. In addition, in [37], the Lyapunov function V(t) � N i�1 e T i (t)e i (t) is used to achieve the global fixed-time synchronization conditions. In this paper, the integral item N i�1 n l�1 t t− τ e 2 il (s)ds is introduced in Lyapunov functional (28). Obviously, compared with the above works in [30,[33][34][35][36], the result in this paper is more general.

Remark 2.
It should be pointed out that the upper bound of the settling time, T max � (3/2K)(K/2K) (2/3) , is independent on initial conditions. In addition, it is easy to see that, on the basis of the configuration for parameters K in the proposed control law, the upper bound T max can be determined in advance to any desired values.
In the designed controller (20), the integral item n l�1 ( t t− τ e 2 il (s)ds) (3/2) is used, which may bring difficulties in implement. We remove this integral item and design the following controller: Applying controller (43), we can obtain the following result.

Conclusion
In this paper, the global robust synchronization in fixed time and global synchronization in finite time have been investigated for a class of hybrid coupled delayed NNs with discontinuous activation functions. Under the designed discontinuous feedback controller, the global synchronization conditions have been presented in the forms of LMIs, and the settling time, which is independent on initial conditions, has been also evaluated. Compared with the existing works, where the neuron activations are supposed to be linear growth, the results proposed in this paper are more general.
It is worth noting that the designed feedback controllers contain the sign function and the integral term, which may bring the chatter in actual implement, indicating that the designed control schemes have certain limitation.
Future work will be focused on how to remove the chatter of the designed controller and to extend the results here obtained for stochastic sampled-date synchronization control for CNNs with delays and discontinuous activations.
Data Availability e underlying data supporting the results of our study can be found in the original paper.

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