Synchronization Problems of Fuzzy Competitive Neural Networks

Zhang, Lingping; Duan, Feng; Du, Bo

doi:https://doi.org/10.1155/2022/5926415

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5926415 | https://doi.org/10.1155/2022/5926415

Synchronization Problems of Fuzzy Competitive Neural Networks

Lingping Zhang,¹Feng Duan,²and Bo Du¹

Academic Editor: Jorge E. Macias-Diaz

Received03 Mar 2022

Revised13 Apr 2022

Accepted27 Apr 2022

Published19 May 2022

Abstract

This paper is devoted to investigating the fixed-time and finite-time synchronization for fuzzy competitive neural networks with discontinuous activation functions. We introduce Filippov solution for overcoming the nonexistence of classical solutions of discontinuous system. Using the fixed-time synchronization theory, inequality technique, we obtain simple robust fixed-time synchronization conditions. Designing proper feedback controllers is a key step for the implementation of synchronization. Furthermore, based on the fixed-time robust synchronization, we design a switching adaptive controller and obtain the finite-time synchronization. It is noted that the settling time is independent on the initial value in the fixed-time robust synchronization. Hence, under the conditions of this paper, the considered system has better stability and feasibility. Finally, the theoretical results of this paper are attested to be indeed feasible in terms of a numerical example.

1. Introduction

Fuzzy cellular neural networks were first proposed in [1]. Fuzzy cellular neural networks can fulfil vagueness or uncertainty for human cognitive processes. Therefore, the use of fuzzy network system can more accurately simulate the situation of the real world. In recent decades, there have been a lot of studies on fuzzy neural network systems. In [2], the authors introduced fuzzy cellular neural network theory and applications. Ali et al. [3] studied global stability analysis of fractional-order fuzzy BAM neural networks with time delay and impulsive effects. Chen, Li, and Yang [4] considered asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects. In [5], the authors studied a fuzzy Cohen-Grossberg neural networks and obtained global exponential stability by using matrix and Liapunov functions. The use of the Lyapunov method and the linear matrix inequality (LMI) approach, existence, uniqueness, and the global asymptotic stability of a class of fuzzy cellular neural networks with mixed delays were obtained in [6]. For discrete-time fuzzy BAM, see [7]; for memristor-based fuzzy cellular neural networks, see [8]; for fuzzy Cohen-Grossberg-type neural networks, see [9]; and for chaotic fuzzy cellular neural networks, see [10].

Synchronization is a widespread phenomenon in nature. Its research has important theoretical significance and practical application value (see [11–19] and related references). Synchronization means that the state of coupled system tends to be consistent with time moving. In finite-time synchronization, the settling time is dependent on the initial conditions which restrict its applications (see [20–24]). In 2012, Polyakov [25] proposed the concept of fixed-time synchronization. In the case of fixed-time synchronization, the settling time is independent on the initial conditions. Hence, fixed-time synchronization has stronger applicability than finite-time synchronization. Compared with many finite-time synchronization problems on the neural networks, the research of fixed-time synchronization is still in a primitive stage, and lots of results have been obtained for the fixed-time synchronization of neural networks (for more results about fixed-time synchronization, see [25–28]).

Competitive neural networks (CNNs) can describe the dynamic behavior of cortical cognitive maps with unsupervised synaptic modifications. In the early studies for CNNs, Meyer-Bse [29] studied CNNs with different time scales and obtained dynamic behavior of CNNs. In the CNNs, there exist two classes of state variables: the short-term memory (STM) variable describing the fast neural activity and the long-term memory (LTM) variable describing the slow neural activity. Therefore, there exist two classes of time scales in the CNNs: one of which describes to the fast change of the state and the other to the slow change of state. CNNs have extensive applications in different industries and have been studied by many researchers. Gu, Jiang, and Teng [30] studied existence and global exponential stability of equilibrium of CNNs with different time scales and multiple delays. Meyer-Bse, Roberts, and Thmmler [31] considered the local uniform stability of CNNs with different time scales under vanishing perturbations. For stochastic stability analysis of CNNs, see [32]; for multistability of CNNs, see [33]; for multistability and instability of CNNs, see [34]; and for robust stability analysis of CNNs, see [34].

To the best of our knowledge, there are few papers studying the finite-time and fixed-time synchronization problems of fuzzy CNNs with discontinuous activations by designing the adaptive controllers. Inspired by the above work, we study the problems of finite-time and fixed-time synchronization of CNNs with discontinuous activation functions. The motivation of this paper is to enrich and develop the research of competitive neural networks. Particularly, we will study a fuzzy competitive neural networks with discontinuous activation functions which is a new model. The main advantages are summarized in the following three aspects: (1)By designing some proper feedback controllers, we obtain simple finite-time and fixed-time synchronization conditions which can be easily tested. Furthermore, the above synchronization conditions are different from the corresponding ones of [20–22](2)We first study a fuzzy CNNs with discontinuous activations which can extend some previous results to the discontinuous case, such as [26, 27, 35, 36]. In addition, the study of this paper enriches the research content of CNNS (see [30, 31])(3)In the synchronization control, adaptive control is often more valuable than state-feedback control. Through designing a proper and simple switching adaptive control, we consider the finite-time synchronization of the addressed drive-response systems. Furthermore, the upper bounds of the settling time are also easily estimated. Hence, our results are more valuable than the corresponding ones of [13, 15, 17]

We organize the following sections as follows: Section 2 gives system description and some preliminaries. In Section 3, we give some sufficient conditions for the finite-time and fixed-time robust synchronization. In Section 4, a numerical example is given to test the feasibility of the obtained results. Finally, some conclusions and discussions are drawn in Section 5.

2. Model Description and Preliminaries

In this paper, we consider the following delayed fuzzy CNNs with discontinuous activations: with initial conditions where , denotes state of neuron current; is synaptic transfer efficiency, is the output of neurons; is constant; denotes the connection weight, is the strength of the external stimulus; is feed-forward template; and are elements of fuzzy feedback template and fuzzy feedback template, respectively; and are fuzzy feed-forward template and fuzzy feed-forward template, respectively; and are fuzzy OR and fuzzy AND operations, respectively; denotes input of the th neuron; and corresponds to the transmission delay along the axon of the th unit with .

In view of drive-response synchronization, take system (1) as the drive system and design the following response system: where and are controllers. System (3) has the following initial values

Throughout this paper, we need the following assumptions:

(H₁) For each is piecewise continuous; i.e., is continuous except on a countable set of isolate points . There exist finite right and left limits and . Moreover, has at most a finite number of discontinuities on any compact interval of .

(H₂) For each , there exist nonnegative constants and such that where

Since system (1) has discontinuous connection strength coefficients, the classic solution is not suitable for system (1); we introduce Fiippov solution for system (1). Consider the following dynamic system where is the state variable. If is locally measurable function but is discontinuous with respect to , Filippov [37] discussed the solution of Cauchy problem (7) and gave the following definition.

Definition 1. Assume that is locally bounded and Lebesgue measurable for . A vector-value function is called to be a Filippov solution of system (7) if is absolutely continuous and satisfying the following differential inclusionwhere or , is the Filippov set-valued map, is the convex closure of set , is the Lebesgue measure, and is the open ball with the center at and the radius .

Let be the solution of system (1) with corresponding initial conditions and be the solution of system (3) with corresponding initial conditions. For and , if and are absolutely continuous on any compact subinterval of and satisfy the following inclusions:

Obviously, the following set-valued maps have nonempty compact convex values. In view of the measurable selection theorem, they are upper semicontinuous and measurable. Thus, if and are the solutions of system (1) and and are the solutions of system (3), there exist measurable functions where for a.e. such that

From (11) and (12), the errors are defined as

Then, the error systems can be obtained by with initial conditions

Let and

Definition 2. The drive system (1) and response system (3) are said to be finite-time robustly synchronized, if there exists a time such that and for .

Definition 3. The origin of error system (14) is said to be globally fixed-time stable if it is globally uniformly finite-time stable and the settling time is globally bounded; i.e., there exists such that for .

Definition 4. The drive-response systems (1)-(3) are said to achieve robust fixed-time synchronization if there exist a fixed time and a settling time function such that where represents the Euclidean norm and is a dimessional continuous function space on .

Definition 5 (see [35]). A function isregular, if it is (1)Regular in (2)Positive definite, i.e., for and (3)Radially unbounded, i.e., as

Lemma 6. [36] If is regular and is absolutely continuous on any compact subinterval of , then and are differential at t for a.e. . Furthermore we have where is the generalized gradient of at and denotes the convex hull. is a set of measure zero and is a set of nondifferentiable points of function .

Lemma 7. [38] If there exists a continuous radially unbounded function such that (1)For some , any solution of system (12) satisfiesand then, the error system (14) is global fixed-time stable at the origin; moreover, the following estimate admits with the settling time bound by where is the upper right-hand Dini derivative and and are defined by Definition 3.

Lemma 8 (see [21]). For assume that . Then

Lemma 9 (see [39]). Assume that and . Then, the following inequalities hold:

3. Main Results

We design the following discontinuous control inputs: where and are positive, and and need to satisfy some conditions.

Theorem 10. Suppose that the conditions (H₁) and (H₂) hold, systems (1) and (3) can be robustly synchronized by the control law (23) in a fixed time, provided that

Furthermore, and for , where the settling time is given as where .

Proof. Construct the following Lyapunov function: where . It is easy to see that is regular. Compute the derivative of along the trajectories of error system (14); then

By assumption (H₂), we have

From Lemma 8 and assumption (H₂), we have

From the first equation of (23), we have

From (30) to (33), we obtain

Similar to the above certificate, we have

According to assumption (H₂), we have

From the second equation of (23), we have

In view of (35)-(37), we obtain

From (25), (34), (38), and Lemma 9, we have

Based on Lemma 7, the error system (14) gets fixed-time stability which yields that systems (11) and (12) achieve the robust fixed-time synchronization, i.e., systems (1) and (3) can be robustly synchronized by the control law (23). In addition, the settling time is given as

Now, we consider robust finite-time synchronization for systems (1) and (3) under discontinuous adaptive controller. Design the following discontinuous control inputs: where and are positive constants. For and , the feedback gains and are adapted according to the updated laws as follows:

For and let and , where and are sufficiently large constants.

Theorem 11. Suppose that the conditions (H₁) and (H₂) hold, systems (1) and (3) can be robustly finite-time synchronized by the control law (41), provided that

Furthermore, the settling time for finite-time robust synchronization can be estimated by where is defined by (46).

Proof. Construct the following Lyapunov function: where and are defined by (42). Recalling the proof of Theorem 10, we need estimate and . From the first equation of (41), we have

From the second equation of (41), we have

Furthermore, we have

From (46) to (50) and the proof of Theorem 10, we have

Condition (43) and the above inequality lead to where is defined by (43). Integrate both sides of the inequality (50) on , then

Thus, which leads to a contradiction. When , we claim that

If , then, there exists some nondegenerate interval such that for all contradicts with (52). Hence, (55) holds. By Definition 2, we obtain the desired result.

Remark 12. In a recent paper, Zhou and Bao [40] considered the fixed-time synchronization for competitive neural networks with Gaussian-wavelet-type activation functions. Gaussian-wavelet-type activation functions are nonmonotonic and continuous. However, the activation functions of this paper are nonmonotonic and discontinuous. Hence, this paper deals with more complicated activation functions and generalizes the corresponding results of [40].

Remark 13. In Theorem 10, a kind of discontinuous control input has been designed for achieving the fixed-time synchronization for the systems (1) and (3). It is noted that the fixed time in Theorem 10 is independent on the system elements.

Remark 14. We note that the control inputs (23) contain the discontinuous sign functions; as a hard switcher, it may be caused to undesirable chattering [20]. For avoiding the chattering, we can replace the sign function by a continuous function to remove undesirable chattering. Hence, the control law (23) can be replaced by

4. Numerical Examples

Example 1. Consider the following discontinuous fuzzy competitive neural networks as the drive system: and the response system:

Let

Then, the error systems can be obtained by where

It is easy to see that the activation function is discontinuous and has a discontinuous point and . Obviously, assumptions (H₁) and (H₂) hold. This fact can be seen in Figure 1.

The initial values of the system (57) satisfy the following conditions:

The initial values of corresponding slave system (58) are

State trajectories of error system (60) without control are shown in Figure 2. From Figure 2, we find that systems (57) and (58) are not robustly synchronized.

Choose as parameters of controller (23). Moreover, we choose

By simple computation, we can have

Hence, all the conditions in Theorem 10 are satisfied. Thus, the response system (58) can robustly synchronize with the drive system (57) in a fixed time under the control law (23). Obviously, Choosing , by Theorem 10, we have

From Figure 3, the error system (60) converges to 0 under the control law (23) which means systems (57) and (58) are robustly synchronized in a fixed time. The simulations show that the main results of robust fixed-time synchronization established in the present paper are correct.

Remark 15. It is well known that Lyapunov method has been widely used for studying dynamic behaviors of neural networks. In this paper, designing some novel discontinuous control inputs and constructing proper Lyapunov-Krasovskii functional, we obtain some sufficient criteria for achieving fixed-time synchronization, and the corresponding setting times are estimated. Our results and the proposed methods are different from for continuous neural network systems (see [6–8]). And the proposed analysis method is also easy to extend to the case of other type neural networks. In the future, we will further study the synchronization problem and/or the Markovian jumping problem of competitive neural networks.

5. Conclusions and Discussions

This paper is devoted to studying the finite-time and fixed-time robust synchronization of fuzzy competitive neural networks with discontinuous activations. For achieving fixed-time synchronization of the competitive neural networks, we consider the fixed-time stability problem of the error system between the drive-response systems which is an effective method to study synchronization problems. We construct a novel discontinuous state-feedback control inputs to the response competitive neural system. Then, based on Filippov solutions for discontinuous differential system, we obtain some new criteria for guaranteeing fixed-time robust synchronization of fuzzy competitive neural networks with discontinuous activations. Fixed-time synchronization is the basis of finite-time synchronization. Hence, we further construct a simple switching adaptive control to the response competitive neural systems which can effectively deal with the finite-time robust synchronization between the response competitive neural systems and the drive competitive neural systems. It should be pointed out that we first study the synchronization control of competitive neural networks with discontinuous activations. Finally, a simulation numerical has been shown to verify the correctness of our theoretical results.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This paper is supported by the Anhui Provincial Department of Education Provincial Quality Process Project No. 2020jyxm2044.

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