#### Abstract

This paper investigates the passivity of multiple weighted coupled memristive neural networks (MWCMNNs) based on the feedback control. Firstly, a kind of memristor-based coupled neural network model with multiple weights is presented for the first time. Furthermore, a novel passivity criterion for MWCMNNs is established by constructing an appropriate Lyapunov functional and developing a suitable feedback controller. In addition, with the assistance of some inequality techniques, sufficient conditions for ensuring the input strict passivity and output strict passivity of MWCMNNs are derived. Finally, the validity of the theoretical results is verified by a numerical example.

#### 1. Introduction

Neural networks (NNs) have aroused widespread attention since they have been applied in numerous fields including machine learning, deep learning, and engineering data prediction [1–3]. As the fourth two-terminal circuit element, the memristor was predicted to exist by Chua in 1971, and the prototype of memristor was obtained by the research team of HP for the first time [4–6]. Memristor is considered to be an excellent candidate for imitating biological synapses in circuit implementation of NNs owing to its characteristics of nanometer size, high storage capacity, and low energy consumption [7]. Through replacing the resistors with memristors in NNs circuit implementation, a new type of NN called memristive NN (MNN) has been successfully introduced [8]. Recently, it is reported that MNNs have many potential applications in face detection, bioengineering, pattern recognition, feature extraction, and associative memory [9–11]. To our knowledge, these applications for MNNs were to a great extent from the dynamical behaviors of MNNs. Particularly, the stability as a significant dynamical behavior for MNNs is one of the hot research topics [12]. Zhang et al. [12] obtained several sufficient conditions to insure the stability for MNNs.

The theory of passivity is valid and robust when studying the stability for nonlinear systems since passivity properties of a system can keep the internal stability of the system [13–15]. Up to now, a lot of interesting results about passivity of MNNs have been reported [16–18]. In [16], Meng and Xiang conducted the passivity analysis for a kind of complex-valued MNNs. Xiao et al. [17] obtained a new passivity criterion by utilizing set-valued mapping as well as transforming MNNs into traditional NNs. Based on the Lyapunov–Krasovskii method, Wu and Zeng [18] acquired an exponential passivity criterion for MNNs with mixed time-varying delays.

Complex networks (CNs) have attracted more and more interests of researchers in recent years, and CNs are ubiquitous in our life, such as communication networks, metabolic system networks, and food networks. Coupled NNs (CNNs) are a special class of CNs, which are composed of many NNs through mutually coupling [19, 20]. Considering the fact that the passivity of CNNs has been broadly applied to many fields including chaos generators and brain science. Consequently, it is interesting to research the passivity for coupled MNNs (CMNNs) [21]. In [21], Yue et al. investigated the passivity of delayed CMNNs with reaction-diffusion terms with the aid of two pinning control schemes and some inequality techniques.

It is well known that most of networks in the practical world are supposed to be described as multiple weighted CNs, such as human social networks and public transport networks. However, only a few researchers have discussed multiple weighted CNNs (MWCNNs) in recent years [22]. Chen et al. [22] dealt with the dissipativity problem of MWCNNs via dynamic event-triggered pinning control. It should be noted that the passivity of multiple weighted CMNNs has only been discussed by few researchers so far.

It is worth noticing that the passivity of MNNs usually cannot be achieved on their own [23]. In consequence, it is essential to make use of some control strategies to make MNNs passive [24, 25]. To ensure exponential synchronization for MNNs, Lin et al. [24] developed a nonlinear feedback controller. Zhang et al. [25] derived some sufficient conditions for achieving finite time synchronization based on the feedback control. To our knowledge, the problem of passivity for multiple weighted CMNNs (MWCMNNs) under the feedback control has never been considered.

Motivated by the above analyses, this paper considers the passivity of MWCMNNs via the feedback control. The primary contributions of this paper are displayed as follows:(1)A kind of memristor-based coupled neural network model with multiple weights is firstly proposed.(2)It is first time that the feedback control strategy is adopted to ensure the passivity, output strict passivity, and input strict passivity of MWCMNNs.(3)Several new passivity criteria are established according to linear matrix inequalities that can be checked through utilizing standard numerical packages.

#### 2. Preliminaries

Let be the set of real matrices of order . stands for that the matrix is symmetric and positive (negative) definite. stands for that the matrix is symmetric and semipositive (seminegative) definite. represents the transpose of matrix . For any . and mean the minimal as well as the maximal eigenvalue of matrix , respectively.

*Definition 1. *(see [26]). A system with supply rate is dissipative if there is a nonnegative storage function , such thatfor any and , where , are the output and input of the system, respectively.

*Definition 2. *(see [27]). If a system is dissipative and satisfyingin which is a constant matrix, the system can achieve the passivity.

*Definition 3. *(see [27]). If a system is dissipative and satisfyingin which , , , and , then the system can achieve the strict passivity.

If , the system is output-strictly passive, and if , the system is input-strictly passive.

#### 3. Passivity of MWCMNNs

##### 3.1. Network Model

The model of coupled neural network with multiple weights is given bywhere , indicates the state vector of the th node; diag , and means the activation function of th neuron; denotes a constant matrix; is the constant external input vector; is a constant matrix; denotes the external input vector; stands for the coupling strength of the th coupling form; diag means the inner coupling matrix in the th coupling form; is the external coupling matrix for the th coupling form, where satisfies the following conditions:if there is a link between node and node , then or else .

Consider the following multiple weighted coupled memristive neural network (MWCMNN) consisting of identical MNNs with multiple weights:where have the same meanings as in network (1); , ; stands for the voltage for capacitor ; , and means the activation function of th neuron; indicates the propagation delay; is the control input; and are described bywhere and represent the memductances of memristors and , respectively. indicates the memristor between and the function , indicates the memristor between and the function . In the light of the traits of voltage and current of memristor, we can obtain thatwhere the switching jumps are constants, .

Define diag diag .

*Remark 1. *Considering the fact that plenty of networks in the actual world ought to be described by CNs with multiple weights, for instance, human social networks and public transport networks. Nevertheless, only some researchers have investigated the passivity of MWCNNs [22]. Therefore, it is interesting to discuss the passivity of MWCMNNs. In this paper, a kind of memristor-based coupled neural network model with multiple weights is proposed.

Throughout this paper, we put forward the following assumption.

*Assumption 1. *If there are positive constants , and , such thatfor any .

Consider that is an equilibrium point for network (6), thenLetting error vector , we getwhereAccording to the network (11), a feedback controller is developed as follows:where diag.

From (11) and (13), one hasThe output vector of the network (14) is given byin which and .

For convenience, we denote

##### 3.2. Passivity Criteria

Theorem 1. *If there is a matrix satisfyingwhere , then the network (14) is passive.*

*Proof. *Select the following Lyapunov functional for the network (14):Then, one hasBased on Assumption 1, one can obtainSimilarly, one hasIt can be obtained from Lemma 2.1in [28] and Assumption 1 thatSimilarly, we getAccording to (19)–(23), one getsFurther, one obtainswhere .

From (17), one can deriveBy (26), one hasfor any and .

Theorem 2. *If there exist matrices and satisfyingwhere , then the network (14) is input-strictly passive.*

*Proof. *For the network (14), choosing the identical Lyapunov functional as (18), then one can obtainwhere .

From (28), we can getBy (30), one hasfor any and .

Theorem 3. *If there exist matrices and satisfyingwherethen the network (14) is output-strictly passive.*

*Proof. *For the network (14), choosing the identical Lyapunov functional as (18), then we can derivewhere .

From (32), one can getBy (35), one hasfor any and .

*Remark 2. *It is a key issue that NNs are unable to achieve the passivity by themselves in some circumstances [23]. As a consequence, it is necessary to utilize an appropriate control method to make NNs passive. For all we know, the passivity problem of MWCMNNs via feedback control has not been researched. In the above discussion, with the help of a developed feedback controller, several criteria are established to ensure that the proposed network is passive, output-strictly passive, and input-strictly passive, respectively.

#### 4. Numerical Example

*Example 1. *The MWCMNN is considered as follows:in which diag diag diag diag , and the matrices are chosen as follows:Obviously, as well as satisfy Assumption 1 with and . Besides, is an equilibrium point of the isolated node for MWCMNN (37).

The output vector is chosen as follows:in which

*Case 1. *The following matrix which satisfies (17) can be derived with the assistance of the MATLAB YALMIP Toolbox:In accordance with Theorem 1, the MWCMNN (37) is passive via the controller (13).

*Case 2. *The following matrices and which satisfy (28) can be derived with the assistance of the MATLAB YALMIP Toolbox:By Theorem 2, the MWCMNN (37) is input-strictly passive via the controller (13).

*Case 3. *The following matrices and which satisfy (32) can be derived with the assistance of the MATLAB YALMIP Toolbox:On account of Theorem 3, the MWCMNN (37) is output-strictly passive via the controller (13). Figure 1 exhibits the simulation results.

#### 5. Conclusion

It is the first time that the passivity of MWCMNNs has been discussed in this paper. Based on the Lyapunov stability theory, feedback control theory, and functional differential equations, several novel criteria have been set up to guarantee that the considered network is passive, output-strictly passive, and input-strictly passive. Finally, one numerical simulation example has been presented to demonstrate the effectiveness of the theoretical results. In the future work, the investigation of the synchronization and passivity for multiple weighted CMNNs with time-varying delays via adaptive control will be considered.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Scientific Research Foundation of Chongqing University of Technology under Grant 2020ZDZ028, Grant 2021ZDZ010, and Grant 2021ZDZ001.