Bipartite Synchronization of Cooperation–Competition Neural Networks Using Asynchronous Sampling Scheme

Abstract

This paper investigates the bipartite synchronization problem for cooperation–competition neural networks (CCNNs) under asynchronous sampling control. First, using signed graph theory to characterize the interrelationships between cooperation and competition, a mathematical model for cooperative–competitive neural networks is established. To formulate the error systems of such networks, a Laplacian matrix with zero row sum is derived through coordinate transformation techniques. Considering network complexity and deception attack impacts, an asynchronous sampling-based secure control scheme is designed while preserving performance guarantees. By relaxing positive definiteness constraints, a class of looped functionals is introduced. State norm estimations are utilized to derive criteria for achieving bipartite synchronization. The feedback gain matrices of the asynchronous sampling controller are obtained by solving linear matrix inequalities. Finally, numerical simulations validate the effectiveness of the proposed method.

1. Introduction

Traditional artificial neural network models are predominantly governed by single mechanisms—pure cooperation (e.g., federated learning) or pure competition (e.g., generative adversarial networks) [1,2,3]. While these models excel in specific tasks, their limitations become apparent when addressing complex real-world scenarios where cooperation and competition coexist. To address this challenge, biological neural networks offer insights. The nervous system achieves diverse functions through synergistic interactions between excitation and inhibition. For instance, neurons in the visual cortex suppress redundant signals via lateral inhibition mechanisms to dominate perceptual regions while collaborating through excitatory connections for image recognition. This biological mechanism demonstrates that cooperation and competition are not mutually exclusive but form the core of complex system operations. Recent neuroscience research has shown that the competitive interactions among inhibitory neurons can enhance memory formation and synaptic changes, which are crucial for memory consolidation. Therefore, it is important to represent social networks through cooperation–competition dynamics, specifically using cooperation–competition neural networks. Inspired by this, cooperation–competition neural networks simulate the interplay of cooperation among neurons, providing a biomimetic foundation for modeling complex systems. In recent years, these networks have been widely applied in pattern recognition [4], smart grids [5], secure communication [6], and other domains.

Based on the above discussion, this paper studies the bipartite synchronization of CCNNs under asynchronous sampling control. The key contributions are listed as follows:

(1)

Different from previous works [7,8], an asynchronous sampling scheme is proposed. By integrating the sampling instants of an asynchronous sampling mechanism and coordinate transformation techniques, an error system model for cooperative–competitive neural networks is established.

(2)

Unlike existing works [9,10,11,12], we consider the asynchronous sampling scheme in the presence of deception attacks. Using the characteristics of an asynchronous sampling scheme, we constructed a class of associated loop functionals that can fully utilize the state information at each node’s sampling instants.

2. Discussion

In recent years, research on cooperation–competition neural networks (CCNNs) has made significant progress. For example, ref. [13] investigated multi-delay CCNNs in memristive neural networks using state feedback control, while [14] studied switching cooperation–competition neural networks using the same method. Ref. [15] employed adaptive control to analyze CCNNs. However, these studies relied on traditional point-to-point control, which demands substantial control resources and consumes excessive communication bandwidth for networked systems, leading to congestion. To address this issue, networked control approaches have emerged. For instance, ref. [7] explored bipartite synchronization in coupled Lur’e systems using sampling control, and ref. [8] addressed second-order multi-agent system synchronization through sampling-based methods. For CCNNs, key phenomena are synchronization [16,17] and anti-synchronization [18,19]. CCNNs require signed graphs with both positive and negative edges. Their synchronization splits nodes into two anti-phase groups (i.e., bipartite synchronization) [20,21,22,23], further classified into leader-based and leaderless synchronization. Leader–follower bipartition is practical for real-world efficiency. In contrast to traditional point-to-point control [24,25,26,27], sampling control is undoubtedly a superior approach for achieving bipartite synchronization in CCNNs. However, real-world sensors often exhibit varying sampling periods, making asynchronous sampling mechanisms more practical for studying cooperation–competition coupled neural networks. In recent years, extensive research has been conducted on asynchronous sampling control [9,10,11,12]. Asynchronous sampling control, as a flexible strategy, allows nodes to adopt varying sampling intervals, adapting to network heterogeneity and resource constraints. The advantages of asynchronous sampling control lie in its flexibility and efficiency compared to synchronous sampling. Unlike synchronous systems governed by a fixed global clock, asynchronous control allows individual sensors, actuators, or subsystems to operate and communicate based on their own local timing requirements or triggered by specific events. This eliminates the delays and resource wastage associated with waiting for the next global clock tick when data is ready earlier, enabling more responsive handling of critical events without being constrained by a rigid clock cycle.

3. Preliminaries and Problem Formulation

Consider a CCNN with N nodes as follows:

$${\displaystyle {\dot{\varphi }}_i\left(t\right)=-C{\varphi }_i\left(t\right)+Bf\left({\varphi }_i\left(t\right)\right)-\tau \sum _{j=1}^N\left|a_{ij}^p\right|({\varphi }_i\left(t\right)-\mathrm{sgn}\left(a_{ij}\right){\varphi }_j\left(t\right))+u_i\left(t\right),}$$

(1)

where ${\varphi }_i\left(t\right)={({\varphi }_{i1}\left(t\right),{\varphi }_{i2}\left(t\right),{\varphi }_{i3}\left(t\right),\dots ,{\varphi }_{in}\left(t\right))}^T$ is the state vector of the ith node. C and B denote the connection weight matrices. $\tau$ denotes the coupling strength. $p\in 1,2$ implies that N nodes can be divided into two parts. $a_{i,j}^p$ denotes the adjacency information. $f\left({\varphi }_i\left(t\right)\right)$ is the activation function, and $u_i\left(t\right)\in \mathbb{R}$ represents the control input.

The leader node of (1) is derived as follows:

$$\dot{s}\left(t\right)=-Cs\left(t\right)+Bf\left(s\left(t\right)\right),$$

(2)

where $s\left(t\right)={(s_1\left(t\right),s_2\left(t\right),s_3\left(t\right),\dots ,s_n\left(t\right))}^T$.

In the following, some definitions and lemmas will be used.

Assumption 1

([7]). The vertex array of a directed graph can be divided into two non-adjacent subsets ${\nu }_1$ and ${\nu }_2$, and it satisfies the following:

(1)

If ${\nu }_i,{\nu }_j\in {\nu }_p(p\in 1,2)$, $a_{ij}^p\ge 0$.

(2)

If ${\nu }_i\in {\nu }_q,{\nu }_j\in {\nu }_p(p\ne q,p,q\in 1,2)$, $a_{ij}^p\le 0$.

Definition 1

([7]). The activation $f_j(·)$ function that is monotonically non-decreasing is an odd function and Lipschiz continuous, such that

$$f_{ij}(-\epsilon )=-f_{ij}\left(\epsilon \right)$$

$$0\le {\displaystyle \frac{f_{ij}\left(p\right)-f_{ij}\left(p\right)}{p-q}}\ge F_{ij},$$

for all $p,q\in \mathbb{R}$, $p\ne q$, $i=1,2,\dots ,N$, and $j=1,2,\dots ,n.$

Lemma 1

([28]). If the signed graph is balanced, $\exists \mathsf{\Xi }=({\lambda }_1,{\lambda }_2,{\lambda }_3\dots {\lambda }_N)$ and ${\lambda }_i\in \left\{-1,1\right\}$, then $\mathsf{\Xi }{A}^p\mathsf{\Xi }=A^q$, where $A^p={\left(a_{ij}^p\right)}_{N\times N}$, $a_{ij}^q={\lambda }_i{a}_{ij}^p{\lambda }_i=\left|a_{ij}^p\right|$.

Lemma 2

([29]). Consider a scalar $\varpi \in (0,1)$, $n\times n$ matrices $R_1>0$ and $R_2>0$, and the following function

$$\phi \left(\varpi \right)={\displaystyle \frac{1}{\varpi }}{\upsilon }_1^T{R}_1{\upsilon }_1+{\displaystyle \frac{1}{1-\varpi }}{\upsilon }_2^T{R}_2{\upsilon }_2,$$

where ${\upsilon }_1$ and ${\upsilon }_2$ are arbitrary vectors. If there exists a matrix H with $\left[\begin{array}{cc}{R}_1& H\\ \ast & R_2\end{array}\right]$, one can obtain

$$\underset{\varpi \in (0,1)}{min}\phi \left(\varpi \right)\ge {\left[\begin{array}{c}{\upsilon }_1\\ {\upsilon }_2\end{array}\right]}^T\left[\begin{array}{cc}{R}_1& H\\ \ast & R_2\end{array}\right]\left[\begin{array}{c}{\upsilon }_1\\ {\upsilon }_2\end{array}\right].$$

Let ${\dot{\overline{\varphi }}}_i\left(t\right)={\lambda }_i{\varphi }_i\left(t\right)$, ${\lambda }_i\in \{-1,1\}$, $i=1,\dots ,N$, and then we can derive that ${\varphi }_i\left(t\right)=n_i{\dot{\overline{\varphi }}}_i\left(t\right)$. From (1) and Definition 1, we obtain

$$\begin{array}{cc}\hfill \dot{\overline{\varphi }}\left(t\right)& {\displaystyle =-C{\overline{\varphi }}_i\left(t\right)+B{\lambda }_{i}f\left({\lambda }_i{\overline{\varphi }}_i\left(t\right)\right)-\tau {\lambda }_i\sum _{j=1}^N\left|a_{ij}^p\right|({\lambda }_i{\overline{\varphi }}_i\left(t\right)-\mathrm{sgn}\left(a_{ij}^p\right){\lambda }_j{\overline{\varphi }}_j\left(t\right))+u_i\left(t\right)}\hfill \\ & {\displaystyle =-C{\overline{\varphi }}_i\left(t\right)+Bf\left({\overline{\varphi }}_i\left(t\right)\right)-\tau \sum _{j=1}^N\left|a_{ij}^p\right|({\overline{\varphi }}_i\left(t\right)-{\lambda }_i\mathrm{sgn}\left(a_{ij}^p\right){\lambda }_j{\overline{\varphi }}_j\left(t\right))+u_i\left(t\right).}\hfill \end{array}$$

Define $L^p$ as the Laplacian matrix of the $A^p$, and then we can derive that

$$\begin{array}{c}\hfill {\displaystyle \dot{\overline{\varphi }}\left(t\right)=-C{\overline{\varphi }}_i\left(t\right)+Bf\left({\overline{\varphi }}_i\left(t\right)\right)-\tau \sum _{j=1}^N{L}_{ij}^p{\overline{\varphi }}_j\left(t\right)+u_i\left(t\right).}\end{array}$$

In setting $\sigma \left(t\right)={\overline{\varphi }}_i\left(t\right)-s\left(t\right)$, one has

$$\begin{array}{c}\hfill {\displaystyle {\dot{\sigma }}_i\left(t\right)=-C{\sigma }_i\left(t\right)+Bf\left({\overline{\varphi }}_i\left(t\right)\right)-Bf\left(s\left(t\right)\right)-\tau \sum _{j=1}^N{L}_{ij}^p{\sigma }_j\left(t\right)+u_i\left(t\right).}\end{array}$$

(3)

where ${\sigma }_i\left(t\right)={({\sigma }_{i1}\left(t\right),{\sigma }_{i2}\left(t\right),\dots {\sigma }_{in}\left(t\right))}^T$.

Definition 2.

Systems (1) and (3) are said to achieve bipartite synchronization if ${\lim }_{t\to +\infty }\left|\right|{\overline{\varphi }}_i\left(t\right)-s\left(t\right)\left|\right|=0$ holds, for $i=1,\dots ,N$. In other words, $\forall i\in {\nu }_1,{\lambda }_i=1$, ${\overline{\varphi }}_i\left(t\right)\to s\left(t\right)$, and $\forall i\in {\nu }_2,{\lambda }_i=-1$, ${\overline{\varphi }}_i\left(t\right)\to -s\left(t\right)$.

Remark 1.

In traditional coupled neural networks for bipartite synchronization, the adjacency matrix is strictly positive because connection weights are non-negative. Consequently, the Laplacian matrix becomes a zero-row-sum matrix, which simplifies error system derivation. However, in CCNNs, connection weights can be positive or negative, which leads to non-zero-row-sum Laplacian matrices. This complicates error system construction using conventional methods. A key challenge in this paper is establishing an error system for CCNNs. To resolve this, we employ coordinate transformation to convert the Laplacian matrix into a zero-row-sum instruction, enabling the derivation of a stable error dynamics model.

In the following, we will design the controller under multi-rate sampled-data control under deception attack. On the one hand, we consider m sensors $S_i$, $i=1,2,\dots ,m$. These sensors will be sampled in different sampling intervals. The sampling interval of the ith sensor is $d_{k_i}^i=t_{k_i+1}^i-t_{k_i}^i$, satisfying $d_{k_i}^i\in (0,{\overline{d}}^i]$, where ${\overline{d}}^i$ is the maximum sampling interval. Then, we define a function between the present time and the last sampling time $p^i\left(t\right)=t-t_{k_i}^i$, and one can obtain

$$\begin{array}{c}\hfill 0\le p^i\left(t\right)\le d_{k_i}^i\le {\overline{d}}^i\le \overline{d},\end{array}$$

where $\overline{d}$ = max$\overline{d^i}$. Then, we can define the state ${\overline{\sigma }}_i\left(t\right)$ as follows:

$$\begin{array}{c}\hfill {\overline{\sigma }}_i\left(t\right)=({\sigma }_{i1}(t-p^1\left(t\right)),{\sigma }_{i2}(t-p^2\left(t\right)),\dots ,{\sigma }_{im}(t-p^m\left(t\right))).\end{array}$$

Remark 2.

In practical engineering, sensors often have different sampling periods due to their distinct functional requirements. Additionally, in networked control systems, factors like communication delays, packet loss, or deception attacks further exacerbate mismatches in sampling intervals across sensors. Asynchronous sampling thus better reflects real-world scenarios. To address these challenges, we recombine the sampling period as shown in Figure 1.

On the other hand, the sampled package ${\overline{\sigma }}_i\left(t\right)$ may suffer deception attack $\beta \left(t\right)\in \{-1,1\}$; then, the false data $g\left({\overline{\sigma }}_i\left(t\right)\right)$ will substitute the ${\overline{\sigma }}_i\left(t\right)$. Then, consider the multi-rate sampled-data scheme with a deception attack; the controller can be designed as follows:

$$u_i\left(t\right)=(1-\beta \left(t\right))K_i{\overline{\sigma }}_i\left(t\right)+\beta \left(t\right)K_{i}g\left({\overline{\sigma }}_i\left(t\right)\right),$$

(4)

where $K_i$ is the control gain that needs to be designed, and $\beta \left(t\right)$ is the attack signal defined on $(\omega ,{\mathcal{F}}_t,{\mathcal{F}}_{t\ge 0},\mathbb{P})$, with $\mathcal{F}$ satisfying the usual conditions (i.e., it is increasing and right continuous, while $F_0$ contains all P-null sets), and satisfying $P\left\{\beta \right(t)=1\}=\gamma$ and $P\left\{\beta \right(t)=0\}=1-\gamma$, $\alpha \in [0,1]$. It should be mentioned that the deception attack $\beta \left(t\right)$ has limited energy. Accordingly, we give the following assumption:

Assumption 2.

The deception attack satisfies the following inequality:

$${\left|\right|g\left(\epsilon \right)\left|\right|}_2\le {\left|\right|L\epsilon \left|\right|}_2,$$

where L is the constant matrix.

Based on the above discussion, one can obtain the following error system:

$$\begin{array}{cc}\hfill \dot{\sigma }\left(t\right)=& -(I_N\otimes C)\sigma \left(t\right)+(I_N\otimes B)f\left(\sigma \left(t\right)\right)-\tau (L^p\otimes I_n)\sigma \left(t\right)\hfill \\ \hfill & +(1-\beta \left(t\right))K\overline{\sigma }\left(t\right)+\beta \left(t\right)Kg\left(\overline{\sigma }\left(t\right)\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill =& -\overline{C}\sigma \left(t\right)+\overline{B}f\left(\sigma \left(t\right)\right)+(1-\beta \left(t\right))K\overline{\sigma }\left(t\right)+\beta \left(t\right)Kg\left(\overline{\sigma }\left(t\right)\right),t\in [t_k,t_{k+1}),\hfill \end{array}$$

(6)

where $\sigma \left(t\right)=({\sigma }_1\left(t\right),{\sigma }_2\left(t\right),\dots ,{\sigma }_N\left(t\right))$, $f\left(\sigma \left(t\right)\right)=(f\left({\sigma }_1\left(t\right)\right),f\left({\sigma }_2\left(t\right)\right),\dots ,f\left({\sigma }_N\left(t\right)\right))$, $K=(K_1,K_2,\dots ,K_N)$, $g\left(\overline{\sigma }\left(t\right)\right)=(g\left({\overline{\sigma }}_1\left(t\right)\right),g\left({\overline{\sigma }}_2\left(t\right)\right),\dots ,g\left({\overline{\sigma }}_N\left(t\right)\right)$, $\overline{C}=-(I_N\otimes C+\tau (L^p\otimes I_n)$, and $\overline{B}=I_N\otimes B$.

4. Main Results

Theorem 1.

For given scalars $d_2>d_1>0$, $\tau >0$,$\alpha >0$, $\gamma >0$, and $\overline{h}>0$ and $Nn\times Nn$ diagonal matrix K, if there exist $Nn\times Nn$ matrices $P>0$, $Y>0$, $D>0$, $Q>0$, $S_i>0$, and $Z_i>0$; arbitrary matrices R, H, and W; $2Nn\times 2Nn$ matrices X and R; $3Nn\times 3Nn$ matrix M; and $Nn\times Nn$ diagonal matrices $U_1$, $U_2$, and Λ, such that for $d_k\in \left\{d_1,d_2\right\}$,

$$\begin{array}{c}\hfill \Theta \left(d_k\right)<0,\end{array}$$

(7)

$$\begin{array}{c}\hfill \Pi \left(d_k\right)<0,\end{array}$$

(8)

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{Z}_i& W_i\\ \ast & Z_i\end{array}\right]>0,i\in \left\{1,2,\dots ,m\right\},\end{array}$$

$$\begin{array}{c}\hfill \left[\begin{array}{cc}D& H\\ \ast & Q\end{array}\right]>0,\end{array}$$

where ${\Theta }_{11}=-D+H+H^T-Q-Z+S-U_1^T\overline{C}-2R_1-2R_2-2R_3-2R_4$, ${\Theta }_{12}=P+Y+Y^T-{\displaystyle \frac{1}{2}}{U}_1^T-{\displaystyle \frac{1}{2}}{\overline{C}}^T{U}_2+d_k(R_1+R_1^T)+d_k(R_2+R_2^T)$, ${\Theta }_{13}=D-{\displaystyle \frac{1}{2}}(H+H^T)+R_1+R_1^T+R_2^T-R_3-X_1+X_3$, ${\Theta }_{14}=-H+{\displaystyle \frac{1}{2}}(Q+Q^T)+R_2-R_3^T-R_4-R_4^T-X_2+X_4$, ${\Theta }_{15}={\displaystyle \frac{1}{2}}{U}_1^T(1-\gamma )K$, ${\Theta }_{16}=Z-W$, ${\Theta }_{17}=W$, ${\Theta }_{18}={\displaystyle \frac{1}{2}}{U}_1^T\overline{B}+F^T\Lambda$, ${\Theta }_{19}={\displaystyle \frac{1}{2}}\gamma U_1^{T}K$, ${\Theta }_{22}=d_k^{2}D+{\mu }^{2}Z-U_2^T$, ${\Theta }_{23}=-Y^T+d_k(R_1-R_1^T)-d_k{R}_2^T+d_k{X}_1$, ${\Theta }_{24}=-Y-d_k{R}_2+d_k{X}_2$, ${\Theta }_{25}={\displaystyle \frac{1}{2}}{U}_2^T(1-\gamma )K$, ${\Theta }_{28}={\displaystyle \frac{1}{2}}{U}_2^T\overline{B}$, ${\Theta }_{29}={\displaystyle \frac{1}{2}}\gamma U_2^{T}K$, ${\Theta }_{33}=-D-2R_1+2X_1+d_k{M}_1$, ${\Theta }_{34}=H+d_k{M}_2-R_2+R_3^T+X_2-X_3^T$, ${\Theta }_{35}=d_k{M}_3$, ${\Theta }_{44}=-Q+d_k{M}_4+2R_4-2X_4$, ${\Theta }_{45}=d_k{M}_5$, ${\Theta }_{55}=d_k{M}_6$, ${\Theta }_{56}={\displaystyle \frac{1}{2}}\alpha L^{T}L$, ${\Theta }_{66}=-2Z+W^T+W$, ${\Theta }_{67}=Z-W$, ${\Theta }_{77}=-Z-S$${\Theta }_{88}=2\Lambda$, ${\Theta }_{99}=-\alpha$, ${\Pi }_{11}=-D+H+H^T-Q-Z+S-U_1^T\overline{C}-2R_1-2R_2+2R_3+2R_4$, ${\Pi }_{12}=P+2Y-{\displaystyle \frac{1}{2}}{U}_1^T-{\displaystyle \frac{1}{2}}{\overline{C}}^T{U}_2+d_k(R_3+R_3^T)+d_k(R_4+R_4^T)$, ${\Pi }_{13}=D-{\displaystyle \frac{1}{2}}(H+H^T)+R_1+R_1^T+R_2^T-R_3+X_1+X_3$, ${\Pi }_{14}=-{\displaystyle \frac{1}{2}}(H+H^T)+Q+R_2-R_3^T-{\displaystyle \frac{1}{2}}(R_4+R_4^T)$, ${\Pi }_{15}={\displaystyle \frac{1}{2}}{U}_1^T(1-\beta \left(t\right))K$, ${\Pi }_{16}=Z-W$, ${\Pi }_{17}=W$, ${\Pi }_{18}={\displaystyle \frac{1}{2}}{U}_1^T\overline{B}+F^T\Lambda$, ${\Pi }_{19}={\displaystyle \frac{1}{2}}{U}_1^T\beta \left(t\right)K$, ${\Pi }_{22}=d_k^{2}Q+{\mu }^{2}Z-U_2^T$, ${\Pi }_{23}=-Y^T+d_k(X_3-R_3)$, ${\Pi }_{24}=-Y-d_k(R_4+R_4^T)-d_k{R}_3^T+d_k{X}_4$, ${\Pi }_{25}={\displaystyle \frac{1}{2}}{U}_2^T(1-\beta \left(t\right))K$, ${\Pi }_{28}={\displaystyle \frac{1}{2}}{U}_2^T\overline{B}$, ${\Pi }_{29}={\displaystyle \frac{1}{2}}{U}_2^T\beta \left(t\right)K$, ${\Pi }_{33}=-D-2R_1+2X_1+d_k{M}_1$, ${\Pi }_{34}=H+d_k{M}_2-R_2+R_3^T+X_2-X_3^T$, ${\Pi }_{35}=d_k{M}_3$, ${\Pi }_{44}=-Q+d_k{M}_4+2R_4-2X_4$, ${\Pi }_{45}=d_k{M}_5$, ${\Pi }_{55}=d_k{M}_6$, ${\Pi }_{56}={\displaystyle \frac{1}{2}}\alpha L^{T}L$, ${\Pi }_{66}=-2Z+W^T+W$, ${\Pi }_{67}=Z-W$, ${\Pi }_{67}=-Z-S$, ${\Pi }_{88}=2\Lambda$, and ${\Pi }_{99}=-\alpha$, then error system (5) is exponentially stable. That is, the bipartite synchronization of CCNNs (1) and (3) can be achieved using controller (4).

Proof. 

Construct the Lyapunov–Krasovskii functional as

$$V\left(t\right)=\sum _{i=1}^6{V}_i\left(t\right),t\in [t_k,t_{k+1})$$

with

$$\begin{array}{cc}\hfill V_1\left(t\right)=& {\sigma }^T\left(t\right)P\sigma \left(t\right),\hfill \\ \hfill V_2\left(t\right)=& 2{\zeta }_1^T\left(t\right)Y{\zeta }_2\left(t\right),\hfill \\ \hfill V_3\left(t\right)=& (t_{k+1}-t)(t-t_k){\zeta }_3^{T}M{\zeta }_3,\hfill \\ \hfill V_4\left(t\right)=& (t_{k+1}-t)(t_{k+1}-t_k){\int }_{t_k}^t{\dot{\sigma }}^T\left(s\right)D\dot{\sigma }\left(s\right)ds\hfill \\ & -(t-t_k)(t_{k+1}-t_k){\int }_t^{t_{k+1}}{\dot{\sigma }}^T\left(s\right)Q\dot{\sigma }\left(s\right)ds,\hfill \\ \hfill V_5\left(t\right)=& {\displaystyle \sum _{i=1}^m{\int }_{t-{\overline{d}}^i}^t{\sigma }_i^T\left(s\right)S_i{\sigma }_i\left(s\right)ds+\sum _{i=1}^m{\overline{d}}^i{\int }_{-{\overline{d}}^i}^0{\int }_{t+\theta }^t{\sigma }_i^T\left(s\right)Z_i{\sigma }_i\left(s\right)dsd\theta ,}\hfill \\ \hfill V_6\left(t\right)=& 2{\left[\begin{array}{c}(t_{k+1}-t){\zeta }_1\left(t\right)\\ (t-t_k){\zeta }_2\left(t\right)\end{array}\right]}^T\left(R\left[\begin{array}{c}{\zeta }_1\left(t\right)\\ {\zeta }_2\left(t\right)\end{array}\right]+X\left[\begin{array}{c}\sigma \left(t_k\right)\\ \sigma \left(t_{k+1}\right)\end{array}\right]\right),\hfill \end{array}$$

and ${\zeta }_1\left(t\right)=\sigma \left(t\right)-\sigma \left(t_k\right),{\zeta }_2\left(t\right)=\sigma \left(t\right)-\sigma \left(t_{k+1}\right),{\zeta }_3=\mathrm{col}\left\{\sigma \left(t_k\right),\sigma \left(t_{k+1}\right),\overline{\sigma }\left(t_k\right)\right\},$

$$M=\left[\begin{array}{ccc}{M}_1& M_2& M_3\\ \ast & M_4& M_5\\ \ast & \ast & M_6\end{array}\right],R=\left[\begin{array}{cc}{R}_1& R_2\\ R_3& R_4\end{array}\right],X=\left[\begin{array}{cc}{X}_1& X_2\\ X_3& X_4\end{array}\right].$$

□

Next, the proof is divided into two parts.

When $t\in [t_k,t_{k+1})$, we will prove the stability at the sampling points.

$$\begin{array}{cc}\hfill \mathbb{E}\left\{{\dot{V}}_1\left(t\right)\right\}=& \mathbb{E}\left\{2\sigma \left(t\right)P\dot{\sigma }\left(t\right)\right\}\hfill \\ \hfill \mathbb{E}\left\{{\dot{V}}_2\left(t\right)\right\}=& \mathbb{E}\left\{2{\dot{\sigma }}^T\left(t\right)Y{\zeta }_2\left(t\right)+2{\zeta }_1\left(t\right)Y\dot{\sigma }\left(t\right)\right\},\hfill \\ \hfill \mathbb{E}\left\{{\dot{V}}_3\left(t\right)\right\}=& \mathbb{E}\left\{(t_{k+1}-t){\zeta }_3^{T}M{\zeta }_3-(t-t_k){\zeta }_3^{T}M{\zeta }_3\right\},\hfill \\ \hfill \mathbb{E}\left\{{\dot{V}}_4\left(t\right)\right\}=& \mathbb{E}\left\{-h_k{\int }_{t_k}^t{\dot{\sigma }}^T\left(s\right)D\dot{\sigma }\left(s\right)ds+(t_{k+1}-t)(t_{k+1}-t_k){\dot{\sigma }}^T\left(t\right)D\dot{\sigma }\left(t\right)\right.\hfill \\ & \left.-h_k{\int }_t^{t_{k+1}}{\dot{\sigma }}^T\left(s\right)Q\dot{\sigma }\left(s\right)ds+(t-t_k)(t_{k+1}-t_k){\dot{\sigma }}^T\left(t\right)Q\dot{\sigma }\left(t\right)\right\},\hfill \\ \hfill \mathbb{E}\left\{\dot{V_5}\left(t\right)\right\}=& \mathbb{E}\left\{{\displaystyle \sum _{i=1}^m\left[{\sigma }_i^T\left(t\right)S_i{\sigma }_i\left(t\right)-{\sigma }_i^T(t-{\overline{d}}^i)S_i{\sigma }_i(t-{\overline{d}}^i)\right]}\right.\hfill \\ & \left.{\displaystyle +\sum _{i=1}^m{\left({\overline{d}}^i\right)}^2{\sigma }_i^T\left(t\right)Z_i{\sigma }_i\left(t\right)-\sum _{i=1}^m\overline{d^i}{\int }_{t-{\overline{d}}^i}^t{\dot{\sigma }}_i^T\left(s\right)Z_i{\dot{\sigma }}_i\left(s\right)ds}\right\}\hfill \\ \hfill =& \mathbb{E}\{{\sigma }^T\left(t\right)S\sigma \left(t\right)-{\tilde{\sigma }}^T\left(t\right)S\tilde{\sigma }\left(t\right)+{\mu }^2{\dot{\sigma }}^T\left(t\right)Z\dot{\sigma }\left(t\right)\hfill \\ \hfill & {\displaystyle -\sum _{i=1}^m{\overline{d}}^i{\int }_{t-p_i\left(t\right)}^t{\dot{\sigma }}_i^T\left(s\right)Z_i{\dot{\sigma }}_i\left(s\right)ds-\sum _{i=1}^m{\overline{d}}^i{\int }_{t-{\overline{d}}^i}^{t-p_i\left(t\right)}{\dot{\sigma }}_i^T\left(s\right)Z_i{\dot{\sigma }}_i\left(s\right)ds\},}\hfill \\ \hfill \mathbb{E}\left\{{\dot{V}}_6\left(t\right)\right\}=& \mathbb{E}\left\{2\left[\begin{array}{c}-{\zeta }_1\left(t\right)+\left(t_{k+1}\right)\dot{\sigma }\left(t\right)\\ {\zeta }_2\left(t\right)+(t-t_k)\dot{\sigma }\left(t\right)\end{array}\right]\left(R\left[\begin{array}{c}{\zeta }_1\left(t\right)\\ {\zeta }_2\left(t\right)\end{array}\right]+X\left[\begin{array}{c}\sigma \left(t_k\right)\\ \sigma \left(t_{k+1}\right)\end{array}\right]\right)\right.\hfill \\ \hfill & \left.+\left[\begin{array}{c}(t_{k+1}-t){\zeta }_1\left(t\right)\\ (t-t_k){\zeta }_2\left(t\right)\end{array}\right]R\left[\begin{array}{c}\dot{\sigma }\left(t\right)\\ \dot{\sigma }\left(t\right)\end{array}\right]\right\},\hfill \end{array}$$

where $\mu =\mathrm{diag}\left\{{\overline{d}}^1,{\overline{d}}^1,\dots ,{\overline{d}}^m\right\}$ and $\tilde{\sigma }\left(t\right)=\left(\sigma (t-{\overline{d}}^1),\sigma (t-{\overline{d}}^2),\dots ,\sigma (t-{\overline{d}}^m)\right)$.

Based on the Jensen inequality and Lemma 2, one can derive that

$$\begin{array}{cc}\hfill & -d_k{\int }_{t_k}^t{\dot{\sigma }}^T\left(s\right)D\dot{\sigma }\left(s\right)ds-d_k{\int }_t^{t_{k+1}}{\dot{\sigma }}^T\left(s\right)Q\dot{\sigma }\left(s\right)ds\hfill \\ \hfill \le & -{\displaystyle \frac{d_k}{t-t_k}}{\left({\int }_{t_k}^t\dot{\sigma }\left(s\right)ds\right)}^{T}D\left({\int }_{t_k}^t\dot{\sigma }\left(s\right)ds\right)-{\displaystyle \frac{d_k}{t_{k+1}-t}}{\left({\int }_t^{t_{k+1}}\dot{\sigma }\left(s\right)ds\right)}^{T}Q\left({\int }_t^{t_{k+1}}\dot{\sigma }\left(s\right)ds\right)\hfill \\ \hfill \le & -{\displaystyle \frac{d_k}{t-t_k}}{\left(\sigma \left(t\right)-\sigma \left(t_k\right)\right)}^{T}D\left(\sigma \left(t\right)-\sigma \left(t_k\right)\right)-{\displaystyle \frac{d_k}{t_{k+1}-t}}{\left(\sigma (t_{k+1}-\sigma \left(t\right))\right)}^{T}Q\left(\sigma (t_{k+1}-\sigma \left(t\right))\right)\hfill \\ \hfill \le & -{\left[\begin{array}{c}\sigma \left(t\right)-\sigma \left(t_k\right)\\ \sigma \left(t_{k+1}\right)-\sigma \left(t\right)\end{array}\right]}^T\left[\begin{array}{cc}D& H\\ \ast & Q\end{array}\right]\left[\begin{array}{c}\sigma \left(t\right)-\sigma \left(t_k\right)\\ \sigma \left(t_{k+1}\right)-\sigma \left(t\right)\end{array}\right].\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\displaystyle -\sum _{i=1}^m{\overline{d}}^i{\int }_{t-p_i\left(t\right)}^t{\dot{\sigma }}_i^T\left(s\right)Z_i{\dot{\sigma }}_i\left(s\right)ds-\sum _{i=1}^m{\overline{d}}^i{\int }_{t-{\overline{d}}^i}^{t-p_i\left(t\right)}{\dot{\sigma }}_i^T\left(s\right)Z_i{\dot{\sigma }}_i\left(s\right)ds}\hfill \\ \hfill \le & {\displaystyle -\sum _{i=1}^m\frac{{\overline{d}}^i}{p_i\left(t\right)}{\left({\sigma }_i\left(t\right)-{\sigma }_i(t-p_i\left(t\right))\right)}^T{Z}_i\left({\sigma }_i\left(t\right)-{\sigma }_i(t-p_i\left(t\right))\right)}\hfill \\ \hfill & {\displaystyle -\sum _{i=1}^m\frac{{\overline{d}}^i}{{\overline{d}}^i-p_i\left(t\right)}{\left({\sigma }_i(t-p_i\left(t\right))-{\sigma }_i(t-{\overline{d}}^i)\right)}^T{Z}_i\left({\sigma }_i(t-p_i\left(t\right))-{\sigma }_i(t-{\overline{d}}^i)\right)}\hfill \\ \hfill \le & {\displaystyle -\sum _{i=1}^m{\left[\begin{array}{c}{\sigma }_i\left(t\right)-{\sigma }_i(t-p_i\left(t\right))\\ {\sigma }_i(t-p_i\left(t\right))-{\sigma }_i(t-{\overline{d}}^i)\end{array}\right]}^T\left[\begin{array}{cc}{Z}_i& W_i\\ \ast & Z_i\end{array}\right]\left[\begin{array}{c}{\sigma }_i\left(t\right)-{\sigma }_i(t-p_i\left(t\right))\\ {\sigma }_i(t-p_i\left(t\right))-{\sigma }_i(t-{\overline{d}}^i)\end{array}\right]}\hfill \\ \hfill \le & -{\left[\begin{array}{c}\sigma \left(t\right)-\overline{\sigma }\left(t\right)\\ \sigma \left(t\right)-\tilde{\sigma }\left(t\right)\end{array}\right]}^T\left[\begin{array}{cc}Z& W\\ \ast & Z\end{array}\right]\left[\begin{array}{c}\sigma \left(t\right)-\overline{\sigma }\left(t\right)\\ \sigma \left(t\right)-\tilde{\sigma }\left(t\right)\end{array}\right].\hfill \end{array}$$

(10)

Based on the feature of the attack signal and the activation function, this yields

$$\begin{array}{c}\hfill \mathbb{E}\left\{2f^T\left(\sigma \left(t\right)\right)\Lambda F\sigma \left(t\right)-2f^T\left(\sigma \left(t\right)\right)\Lambda f\left(\sigma \left(t\right)\right)\right\}\ge 0,\end{array}$$

(11)

$$\begin{array}{c}\hfill \mathbb{E}\left\{\alpha {\overline{\sigma }}^T\left(t\right)L^{T}L{\overline{\sigma }}^T\left(t\right)-\alpha g^T\left(\overline{\sigma }\left(t\right)\right)g\left(\overline{\sigma }\left(t\right)\right)\right\}\ge 0.\end{array}$$

(12)

For arbitrary matrices $U_1$ and $U_2$, one can obtain

$$\begin{array}{cc}\hfill 0=& \mathbb{E}\left\{2\left[{\sigma }^T\left(t\right)U_1^T+{\dot{\sigma }}^T\left(t\right)U_2^T\right]\left[-\dot{\sigma }\left(t\right)-(I_N\otimes C)\sigma \left(t\right)+(I_N\otimes B)f\left(\sigma \left(t\right)\right)\right.\right.\hfill \\ \hfill & \left.\left.-(L^p\otimes I_n)\sigma \left(t\right)+(1-\beta \left(t\right))K\overline{\sigma }\left(t\right)+\beta \left(t\right)Kg\left(\overline{\sigma }\left(t\right)\right)\right]\right\}.\hfill \end{array}$$

(13)

Based on the ${\displaystyle \dot{V}\left(t\right)=\sum _{i=1}^6{\dot{V}}_i\left(t\right)}$ and (9)–(13), one has

$$\begin{array}{c}\hfill \mathbb{E}\left\{\dot{V}\left(t\right)\right\}\le \mathbb{E}\left\{{\displaystyle \frac{t_{k+1}-t}{d_k}}{\varsigma }^T\left(t\right)\Theta \left(d_k\right)\varsigma \left(t\right)+{\displaystyle \frac{t-t_k}{d_k}}{\varsigma }^T\left(t\right)\Pi \left(d_k\right)\varsigma \left(t\right)\right\}.\end{array}$$

where $\varsigma \left(t\right)=\mathrm{col}\left\{\sigma \left(t\right),\dot{\sigma }\left(t\right),\sigma \left(t_k\right),\sigma \left(t_{k+1}\right),\overline{\sigma }\left(t_k\right),\overline{\sigma }\left(t\right),\tilde{\sigma }\left(t\right),f\left(\sigma \left(t\right)\right),g\left(\overline{\sigma }\left(t\right)\right)\right\}$. From (7) and (8), $\Theta \left(d_1\right)<0$ and $\Theta \left(d_2\right)<0$, and $\Theta \left(d_k\right)<0$ is guaranteed based on the convex combination technique. Similarly, $\Pi \left(d_k\right)<0$ can be guaranteed if $\Pi \left(d_1\right)<0$ and $\Pi \left(d_2\right)<0$. Then, we can know that

$$\begin{array}{c}\hfill \mathbb{E}\left\{\dot{V}\left(t\right)\right\}<0,t\in [t_k,t_{k+1}).\end{array}$$

So, based on (20), we can know that

$$\begin{array}{c}\hfill \mathbb{E}\left\{V\left(t\right)\right\}\le \mathbb{E}\left\{V\left(t_k\right)\right\}\le \mathbb{E}\left\{V\left(t_{k-1}\right)\right\}\le \dots \le \mathbb{E}\left\{V\left(0\right)\right\}.\end{array}$$

When $t\in [t_k,t_{k+1})$, we can derive the following norm estimate:

$$\begin{array}{cc}\hfill \mathbb{E}\left\{∥\sigma \left(t\right)∥\right\}& \le \mathbb{E}\left\{∥\sigma \left(t_{k_i}^i\right)∥+∥{\int }_{t_{k_i}^i}^t\overline{C}\sigma \left(s\right)ds∥+∥{\int }_{t_{k_i}^i}^t\overline{B}f\left(\sigma \left(s\right)\right)ds∥+∥{\int }_{t_{k_i}^i}^{t}K\overline{\sigma }\left(s\right)ds∥\right.\hfill \\ \hfill & \left.+∥{\int }_{t_{k_i}^i}^{t}Kg\left(\overline{\sigma }\left(s\right)\right)ds∥\right\}.\hfill \end{array}$$

Using the Cauchy–Schwarz inequality, we can know that

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{{∥\sigma \left(t\right)∥}^2\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{5{∥\sigma \left(t_{k_i}^i\right)∥}^2+5{∥{\int }_{t_{k_i}^i}^t\overline{C}\sigma \left(s\right)ds∥}^2+5{∥{\int }_{t_{k_i}^i}^t\overline{B}f\left(\sigma \left(s\right)\right)ds∥}^2+5{∥{\int }_{t_{k_i}^i}^{t}K\overline{\sigma }\left(s\right)ds∥}^2+5{∥{\int }_{t_{k_i}^i}^{t}Kg\left(\overline{\sigma }\left(s\right)\right)ds∥}^2\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{5{∥\sigma \left(t_{k_i}^i\right)∥}^2+5{\left({\int }_{t_{k_i}^i}^t∥\overline{C}∥∥\sigma \left(s\right)∥ds\right)}^2+5{\left({\int }_{t_{k_i}^i}^t∥\overline{B}∥∥f\left(\sigma \right(s\left)\right)∥ds\right)}^2+5{\left({\int }_{t_{k_i}^i}^t∥K∥∥\overline{\sigma }\left(s\right)∥ds\right)}^2\right.\hfill \\ \hfill & \left.+5{∥{\int }_{t_{k_i}^i}^{t}Kg\left(\overline{\sigma }\left(s\right)\right)ds∥}^2\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{5{∥\sigma \left(t_{k_i}^i\right)∥}^2+5\overline{d}{∥\overline{C}∥}^2{\int }_{t_{k_i}^i}^t{∥\sigma \left(s\right)∥}^{2}ds+5\overline{d}∥\overline{B}∥{∥F∥}^2{\int }_{t_{k_i}^i}^t{∥\sigma \left(s\right)∥}^{2}ds+5{\overline{d}}^2{∥K∥}^2{∥\overline{\sigma }\left(t\right)∥}^2+5{\overline{d}}^2{∥K∥}^2{∥L∥}^2{∥\overline{\sigma }\left(t\right)∥}^2\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{5{∥\sigma \left(t_{k_i}^i\right)∥}^2+5\overline{d}{∥K∥}^2{∥\overline{\sigma }\left(t\right)∥}^2+5\overline{d}{∥K∥}^2{∥L∥}^2{∥\overline{\sigma }\left(t\right)∥}^2+5\overline{d}\left({∥\overline{C}∥}^2+{∥\overline{B}∥}^2{∥\overline{F}∥}^2\right){\int }_{t_{k_i}^i}^t{∥\sigma \left(s\right)∥}^{2}ds\right\}.\hfill \end{array}$$

Add up the m inequalities for i = 1, 2, …, m:

$$\begin{array}{cc}\hfill \mathbb{E}\left\{{∥\sigma \left(t\right)∥}^2\right\}\le & \mathbb{E}\left\{{\displaystyle \frac{5}{m}}\left({∥\sigma \left(t_{k_1}^1\right)∥}^2+{∥\sigma \left(t_{k_2}^2\right)∥}^2+\dots +{∥\sigma \left(t_{k_i}^i\right)∥}^2\right)+5\overline{d}{∥K∥}^2\left(1+{∥L∥}^2\right){∥\overline{\sigma }\left(t\right)∥}^2\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{{\displaystyle \frac{5\overline{d}}{m}}\left({∥\overline{C}∥}^2+{∥\overline{B}∥}^2{∥\overline{F}∥}^2\right)\left({\int }_{t_{k_1}^1}^t∥\sigma \left(s\right)∥ds+{\int }_{t_{k_2}^2}^t∥\sigma \left(s\right)∥ds+\dots +{\int }_{t_{k_i}^i}^t∥\sigma \left(s\right)∥ds\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{\left({\displaystyle \frac{5}{m}}+5\overline{d}{∥K∥}^2\left(1+{∥L∥}^2\right)\right)\left({∥\sigma \left(t_{k_1}^1\right)∥}^2+{∥\sigma \left(t_{k_2}^2\right)∥}^2+\dots +{∥\sigma \left(t_{k_i}^i\right)∥}^2\right)\right.\hfill \\ \hfill & \left.+5\overline{d}\left({∥\overline{C}∥}^2+{∥\overline{B}∥}^2{∥\overline{F}∥}^2\right){\int }_{\underset{1\le t\le m}{min}{t}_{k_i}^i}^t{∥\sigma \left(s\right)∥}^{2}ds\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{\left({\displaystyle \frac{5}{m}}+5\overline{d}{∥K∥}^2\left(1+{∥L∥}^2\right)\right)e^{5{\overline{d}}^2\left({∥\overline{C}∥}^2+{∥\overline{B}∥}^2{∥\overline{F}∥}^2\right)}\left({∥\sigma \left(t_{k_1}^1\right)∥}^2+{∥\sigma \left(t_{k_2}^2\right)∥}^2+\dots +{∥\sigma \left(t_{k_i}^i\right)∥}^2\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{\omega \left({∥\sigma \left(t_{k_1}^1\right)∥}^2+{∥\sigma \left(t_{k_2}^2\right)∥}^2+\dots +{∥\sigma \left(t_{k_i}^i\right)∥}^2\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{{\displaystyle \frac{\omega {\lambda }_{\min }\left(P\right)}{{\lambda }_{\min }\left(P\right)}}\left({∥\sigma \left(t_{k_1}^1\right)∥}^2+{∥\sigma \left(t_{k_2}^2\right)∥}^2+\dots +{∥\sigma \left(t_{k_i}^i\right)∥}^2\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{{\displaystyle \frac{\omega }{{\lambda }_{\min }\left(P\right)}}\left({\sigma }^T\left(t_{k_1}^1\right)P\sigma \left(t_{k_1}^1\right)+{\sigma }^T\left(t_{k_2}^2\right)P\sigma \left(t_{k_2}^2\right)+\dots +{\sigma }^T\left(t_{k_m}^m\right)P\sigma \left(t_{k_m}^m\right)\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{{\displaystyle \frac{\omega e^{2\alpha t_k}}{{\lambda }_{\min }\left(P\right)}}\left(V_1\left(t_{k_1}^1\right),V_1\left(t_{k_2}^2\right),\dots ,V_m\left(t_{k_m}^m\right)\right)e^{-2\alpha t}{e}^{-2\alpha (t-t_k)}\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{{\displaystyle \frac{m\omega e^{2\alpha (d_1+t_k)}}{{\lambda }_{\min \left(P\right)}}}V\left(0\right)e^{-2\alpha t}\right\}.\hfill \end{array}$$

where $\omega =\left({\displaystyle \frac{5}{m}}+5\overline{d}{∥K∥}^2\left(1+{∥L∥}^2\right)\right)e^{5{\overline{d}}^2\left({∥\overline{C}∥}^2+{∥\overline{B}∥}^2{∥\overline{F}∥}^2\right)}.$

$$\begin{array}{c}\hfill \mathbb{E}\left\{V\left(0\right)\right\}=\mathbb{E}\left\{{\sigma }^T\left(0\right)P\sigma \left(0\right)\right\}\le \mathbb{E}\left\{{\lambda }_{max}\left(P\right){∥\sigma \left(0\right)∥}^2\right\}.\end{array}$$

Then, one can obtain

$$\begin{array}{c}\hfill \mathbb{E}\left\{∥\sigma \left(t\right)∥\right\}\le \mathbb{E}\left\{\varrho ∥\sigma \left(0\right)∥e^{-\alpha t}\right\}.\end{array}$$

where $\varrho =e^{\alpha (d_1+t_k)}\sqrt{{\displaystyle \frac{m\omega {\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}}}$. So, system (5) is exponentially stable; that is, CCNNs (1) and (3) achieve bipartite synchronization under controller (4).

Theorem 2.

For given scalars $d_2>d_1>0$, $\tau >0$, $\alpha >0$, $\gamma >0$, and $\overline{h}>0$, if there exist $Nn\times Nn$ matrices $P>0$, $Y>0$, $D>0$, $Q>0$, $S_i>0$, and $Z_i>0$; arbitrary matrices R, H, and W; $2Nn\times 2Nn$ matrices X and R; $3Nn\times 3Nn$ matrix M; and $Nn\times Nn$ diagonal matrices $U_1$, $U_2$, and Λ, $U_2=\varsigma U_1$ and $G_1=U_1^{T}K$ for $d_k\in \left\{d_1,d_2\right\}$. The control gain can be solved by $K=U_1^{-1}{G}_1$.

Proof. 

Let $U_2=\varsigma U_1$, $G_1=U_1^{T}K$ in LMIs (7) and (8); then, we can solve the feedback gain K using the MATLAB LMI toolbox. The proof is complete. □

5. Numerical Examples

Consider the CCNNs ${\varphi }_i\left(t\right),i=1,\dots ,4$. The directed graph of the CCNNs is shown in Figure 2. The parameters and the activation function of each node are as follows:

$$C=\left[\begin{array}{ccc}-0.9712& 0& 0\\ 0& -0.9712& 0\\ 0& 0& -0.9712\end{array}\right],B=\left[\begin{array}{ccc}0.05& -0.03& 0.0125\\ 0.0625& 0.0425& 0.0287\\ -0.2225& 0& -0.0113\end{array}\right],$$

$$f\left({\varphi }_i\left(t\right)\right)=tanh\left({\varphi }_i\left(t\right)\right),$$

Then, let $F_{ij}=1$.

Based on Assumption 1 and Figure 2, we can know that the directed graph of the CCNNs is structurally balanced and the node set can be divided into ${\nu }_1=\{1,3\}$ and ${\nu }_2=\{2,4\}$. Here, we set a four-node CCNNs to show the effectiveness of the control scheme. In the future, we will design a 100-node CCNN, and then, it can be used in the simulation section.

Set the initial conditions as follows: $S\left(0\right)=(0,-0.76,0)$, ${\varphi }_1\left(0\right)=(0.5,-0.5,-1.6)$, ${\varphi }_2\left(0\right)=(1.2,-1.5,0)$, ${\varphi }_3\left(0\right)=(-0.5,0.6,-0.5)$, and ${\varphi }_4\left(0\right)=(-0.5,-0.6,1.5)$. With some calculation, we can derive some parameters:

$$A^P=\left[\begin{array}{cccc}0& 0& 0& 0.2\\ -0.3& 0& 0& -0.3\\ 0& 0.4& 0& 0\\ 0& 0& -0.2& 0\end{array}\right],L^P=\left[\begin{array}{cccc}0.2& 0& 0& -0.2\\ 0.3& 0.6& 0& 0.3\\ 0& -0.4& 0.4& 0\\ 0& 0& 0.2& 0.2\end{array}\right].$$

Based on Lemma 1 and Figure 2, we can know that there exists a diagonal matrix

$$\mathsf{\Xi }=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right],$$

and then,

$$L^q=\mathsf{\Xi }{L}^p\mathsf{\Xi }=\left[\begin{array}{cccc}0.2& 0& 0& -0.2\\ -0.3& 0.6& 0& -0.3\\ 0& -0.4& 0.4& 0\\ 0& 0& -0.2& 0.2\end{array}\right].$$

As long as Theorem 2 can be solved in MATLAB, the bipartite synchronization behavior can be achieved by choosing specific parameters. Set $\alpha =0.2,$ $d_1=0.01,$ $d_2=0.2,$ and $\gamma =0.5$. Two sensors with different sampling rates, ${\overline{d}}^1=0.2$ and ${\overline{d}}^2=0.15$, are selected for each node. It can be seen that the sampling schemes are different between the two sensors. Assign the first two state variables of each node to ${\overline{d}}^1$ and the third state variable to ${\overline{d}}^2=0.15$; then, we derive $\tau =I_4\otimes diag(0.2,0.2,0.15)$. Here, the deception signals are considered as follows:

$$g\left({\overline{\sigma }}_i\left(t\right)\right)=col(tanh(-0.2{\overline{\sigma }}_{i1}\left(t\right)),tanh(-0.1{\overline{\sigma }}_{i2}\left(t\right)),tanh(-0.2{\overline{\sigma }}_{i3}\left(t\right))).$$

From Assumption 2, one can obtain

$$L=I_4\otimes diag(-0.2,-0.2,-0.2).$$

Based on Theorem 2, we can derive the feedback gain by the MATLAB LMI toolbox (2016 version) as follows:

$$K=diag\{K_1,K_2,K_3,K_4\}$$

where

$$K_1=diag\{0,0,0\},K_2=diag\{-5.5,-5.5,-5.5\},K_3=diag\{0,0,0\},K_4=diag\{-5.5,-5.5,-5.5\}.$$

From Figure 3, one can see that node 2 and node 3 have the same trajectory with the leader $s\left(t\right)$ as times goes by. On the other hand, node 1 and node 4 have the same trajectory with $-s\left(t\right)$. Also, Figure 4 displays the responses of ${\sigma }_i\left(t\right)$, ${\sigma }_i\left(t_k\right)$, $i=1,2,\dots ,4$. It can be seen that ${\sigma }_i\left(t\right)$ and ${\sigma }_i\left(t_k\right)$ converge to 0. It is clear why the states of systems (1) and (3) are said to achieve bipartite synchronization under theorem 2. Hence, we can draw a conclusion that CCNNs (1) and (3) achieve bipartite synchronization under controller (4).

Implementation Considerations: The purpose of the proposed LMIs is to identify matrices that ensure the LMIs in Theorem 1 are satisfied. It is worth noting that LMI optimization is an active field in applied mathematics and optimization. The MATLAB toolbox facilitates solving the unknown matrices in Theorem 1. If these unknown matrices can be determined using the MATLAB toolbox, it indicates that the LMIs are solvable.

6. Conclusions

This paper uses signed graph theory to model CCNNs for describing cooperation–competition interactions. To tackle the challenge of constructing an error system caused by non-positive, definite adjacency matrices, a zero-row-sum Laplacian matrix is obtained through coordinate transformation, facilitating error system modeling. In order to counter deception attacks while maintaining performance, a security control scheme is designed, which integrates asynchronous sampling control. A bilateral loop function is suggested to reduce conservatism by encapsulating state information. By employing Lyapunov-based lemmas and inequality techniques, sufficient conditions for bipartite synchronization under deception attacks are derived. Simulations confirm the effectiveness of the synchronization criteria.