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

Yu, Lina; Ma, Yunfei; Yang, Yuntong; Zhang, Jingchao; Wang, Chunwei

doi:https://doi.org/10.1155/2021/6614150

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

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Synchronization, Estimation, Observation and Applications of Dynamical Network Systems

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Research Article | Open Access

Volume 2021 | Article ID 6614150 | https://doi.org/10.1155/2021/6614150

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

Lina Yu,¹Yunfei Ma,¹Yuntong Yang,^1,2Jingchao Zhang,¹and Chunwei Wang^1,3

Academic Editor: Sanbo Ding

Received26 Nov 2020

Revised22 Jan 2021

Accepted02 Feb 2021

Published17 Feb 2021

Abstract

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.

1. 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 [1–5]. Particularly, the synchronization with respect to CNNs has been extensively studied in many science fields [6–11] 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. The 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. Therefore, it is meaningful to investigate the global synchronization in finite time for CNNs [23–27].

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. This 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. The global fixed-time synchronization was discussed for semi-Markovian jumping neural networks with time-varying delays and discontinuous activation functions in [29]. The 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. Then, 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. The 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. The primary contributions of this paper are listed as follows:(1)The neuron activation functions are modeled to be discontinuous and subject to a quadratic polynomial growth.(2)On the basis of the configuration of parameters in the designed control law, the upper bound of the settling time can be determined to any desired values in advance.(3)The robust synchronization conditions in fixed time are achieved in the form of LMIs.

The 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. refers to the set of real numbers. represents the -dimensional Euclidean space, and denotes the set of all real matrices. Given , stands for the maximal (minimal) eigenvalues of . represents is a positive (negative) definite matrix. Set , where the superscript is the transpose operator. Define , , and , , where and are positive constants and denotes the absolute real value. For a set , represents the closure of the convex hull of . Let , and be matrices with appropriate dimensions. denotes the Kronecker product.

2. Preliminaries and System Description

Consider an array of hybrid CNNs described by where , is the state of the th network at time ; , , ; represents the connection weight matrix; denotes the delayed connection matrix; and are couple strength; intends neuron activation function; is delay. stands for the control input; is an external input; and and indicate coupling configuration matrix and delayed coupling configuration matrix, respectively. If there exists an edge from node to , then ; otherwise, . Laplacian matrix of a graph corresponding to is given by , . expresses the uncertain disturbance.

In system (1), is made to satisfy the following assumptions: (A1) is continuous expect on a countable set of isolate points , and has at most a limited number of discontinuous points on any compact interval of ; in addition, at the discontinuous points , the finite right limit and left limit exist. (A2) Let be a domain containing the origin. There exist positive real constants , , and for each , , , such thatholds for , where , .

Based on A1, it follows that , , where .

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]). is solution of system (1) in Filippov sense, if the following holds:(i) is continuous on , and is absolutely continuous on .(ii)for . .

Noting that set-valued map has a nonempty, compact, and convex value and is upper semicontinuous, so it is measurable. By measurable selection theorem [41], there exists measurable function , and , such that, for a.e., ,

Definition 2. (IVP [42]). For any and any measurable selection , where for a.e . Absolute continuous function associated with measurable function is said to be solution of IVP of system (1) on with initial value if

Consider the following isolated neural network:

Analogous to Definition 2, the IVP associated with system (6) is obtained as follows:

Definition 3. (IVP [42]). For any and any measurable selection , where for a.e. . Absolute continuous function associated with measurable function is said to be solution of IVP of system (6) on with initial value if

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).

Set synchronization error . Then,where and .

According to Definitions 2 and 3, IVP of error system can be written aswhere , , and .

In order to derive the robust synchronization results of CNNs (1), for the terms , we make the following assumption: (A3) The uncertain disturbances are bounded by where is a known nonnegative constant.

Definition 4. Under the designed controller , if there exists time function such that and , , then system (1) is said to be globally robust finite-time synchronized with system (6). Moreover, if there exists scalar such that , then system (1) is said to be globally robust fixed-time synchronized with system (6). is called as the settling time function, and is the upper bound of the settling time function.

Definition 5. (see [43]). For function , if(i) is regular in ,(ii)for , , and for , ,(iii) as ,then is called as -regular.

Lemma 1. (chain rule [44]). If function is -regular and is absolutely continuous on , then is differentiable for andwhere is the Clarke generalized gradient of at .

Lemma 2. (see [44]). Suppose that is -regular and is absolutely continuous. Let . If there exist a continuous function with , , such thatthen and , . Especially, if for all and , where and , then can be calculated by

Lemma 3. (see [30]). Suppose that is a -regular function and is solution with initial value of error system (9). If there exist constants , and , such thatthen the upper bound settling time is estimated by

Lemma 4. (see [45, 46]). Let , , and . And then,

Lemma 5. (see [47, 48]). Let , . Then,

3. Main Results

Set , , , and . Define and , , where

The controller is designed as follows:where , , , , , , . There exists a path between network and (6), if and only if, .

Note that controller (20) is discontinuous, we havewhere , ; , ; , .

Let , then there exists , such that

Theorem 1. Suppose that (A₁), (A₂), and (A₃) are satisfied, and the coupling interaction topology is undirected and connected. If the following conditionshold, where , then CNNs (1) are globally robust fixed-time synchronized with system (6) under the designed controller (20). And the upper bound of settling time is estimated by

Proof. Construct the Lyapunov–Krasovskii functionalCalculating the derivative of at time along the trajectories of error system (9), it follows thatSubstituting (22) into (31), we can obtainDue to , it follows thatIt is easy to derive thatSimilar to (31), one obtainsBy means of Assumptions (A₁), (A₂), and (A₃), we getBy Assumption (A₂), one hasUnder Assumption (A₃), combining (30) with (35), it follows thatThe inequality above can be rewritten aswhere .
Together (24)–(26) and (37), one hasUsing Lemma 4 yieldsBy Lemma 3 and (18), one haswhere is the real number nodes of , , and . Then, . Combining (39) with (40), we haveNoting that is positive definite, we can getwhere .
By Lemma 3, we can conclude that error system (9) is globally robust fixed-time stable. This shows that system (1) can achieve the global robust fixed-time synchronization with system (6) under the controller (20). The upper bound of settling time is estimated by . The proof is completed.

Remark 1. In [30, 33–36], the global fixed-time synchronization issues were considered for delayed CNNs with discontinuous activations, where activation function is subject to linear growth. However, in Theorem 1, discontinuous activations are nonlinear growth and subject to a quadratic polynomial function. In addition, in [37], the Lyapunov function is used to achieve the global fixed-time synchronization conditions. In this paper, the integral item is introduced in Lyapunov functional (28). Obviously, compared with the above works in [30, 33–36], the result in this paper is more general.

Remark 2. It should be pointed out that the upper bound of the settling time, , is independent on initial conditions. In addition, it is easy to see that, on the basis of the configuration for parameters in the proposed control law, the upper bound can be determined in advance to any desired values.

In the designed controller (20), the integral item 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.

Corollary 1. Suppose that Assumptions (A₁), (A₂), and (A₃) are satisfied and couple interaction topology is connected and undirected. If the following conditionshold, where , then CNNs (1) are globally robust fixed-time synchronized with system (6) under controller (43). Also, the settling time is estimated by

Proof. Consider and follow the similar proof process of Theorem 1, and Corollary 1 can easily be proved, hence omitted here.
Next, we developed the global robust finite-time synchronization conditions for CNNs (1). Design the following controller:where , , , , and .

Theorem 2. Suppose that Assumptions (A₁), (A₂), and (A₃) are satisfied and couple interaction topology is connected and undirected. If the following conditionshold, where , then CNNs (1) are globally robust finite-time synchronized under the controller (46). And the setting time is estimated by

Proof. Consider Lyapunov–Krasovskii functional (28). Analogous as the proof of Theorem 1, we can getwhere .
By (47)–(49) and (51), it follows thatFurthermore, we haveApplying Lemma 2, we can conclude that error system (9) is globally robust finite-time stable. This shows that system (1) can achieve the global robust finite-time synchronization with system (6) under the controller (46). The settling time is estimated by . The proof is completed.

4. Numerical Examples

Example 1. Consider an array CNNs (1) with five NNs, in which the dynamical equation of each network is described byThe corresponding isolated network (6) is given bywhere Take The initial values are selected as , , , , and . The initial condition of the corresponding isolated network is . Take activation function as , where , then .Set , , , , , , , and . By simple calculation, , , , and .
Then, we calculate . Moreover, it is easy to verify that condition (23) in Theorem 1 is satisfied. As shown in Figures 1–3, the simulation results agree well with the theoretical analysis. Specifically, Figures 1 and 2 depict the state variables of system (54) and (55). Figure 3 shows the evolutions of , . From Figure 3, it is easy to view that systems (54) and (55) can achieve the global robust synchronization in fixed time under controller (20).

Example 2. Consider an array delayed CNNs (1) with three NNs, in which the dynamical equation of each network described byThe corresponding isolated network (6) is given bywhere Take The initial values are selected as , , and . The trivial point is taken as the objective trajectory.
We take the activation functions as , , where , then .
Let , . Set , , , , , , and . By simple computation, , and .
From (50), . Then, we calculate . Moreover, it is easy to verify that condition (50) in Theorem 2 also satisfied. As shown in Figure 4, the simulation results agree well with the theoretical analysis. Figure 4 shows the evolutions of , , and it is obvious to view that systems (57) and (58) can achieve THE global robust finite-time synchronization before under controller (46).

5. 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

The underlying data supporting the results of our study can be found in the original paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Lina Yu 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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