Synchronization for Delayed Fractional-Order Memristive Neural Networks Based on Intermittent-Hold Control with Application in Secure Communication

Abstract

This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of the systems exponentially stable. Furthermore, two new intermittent-hold controllers are designed to improve the robustness of the system and reduce the cost of the controller. One intermittent-hold controller is based on the feedback control strategy, while the other one integrates an adaptive control strategy. Moreover, two crucial theorems are derived from the proposed lemma and controllers, guaranteeing the exponential synchronization between drive and response systems. Finally, the superior performance of the controllers in achieving exponential synchronization is demonstrated through simulations.

1. Introduction

A memristor is a kind of basic circuit element introduced by Chua [1]. In 2008, a physical model was proposed, making memristors appropriate for practical applications [2]. Owing to their unique nonlinear characteristics and considerable advantages, memristors have been extensively studied. Inspired by biological concepts, memristors have been applied in neural networks due to their abilities to learn and forget [3]. Therefore, memristors have become a hot topic in various fields, including electrical engineering [4], chaotic circuits [5] and neural networks [6]. Memristor neural networks (MNNs) are complex network systems that integrate the characteristics of memristors and neural networks. The authors of [7] investigated the multistability of MNNs using nonlinear theory and demonstrated the chaotic behavior of memristor-based Hopfield neural networks. Finite time synchronization was proposed through an adaptive control strategy for memristive neural networks in [8]. MNNs can perform chaotic behavior when appropriate parameters are selected, which has attracted significant attention [9,10].

Cellular neural networks (CNNs) were proposed in 1988 in [11], with the connection weights determined according to the characteristics and dynamical behaviors of neurons, reflecting the intrinsic nature of CNNs. CNNs have attracted a lot of attention, as seen in [12,13,14]. Fuzzy cellular neural networks (FCNNs) are a specialized form of CNNs incorporating fuzzy operations that were first proposed in [15]. Fuzzy logic can provide effective features of dynamic behavior, such as increased robustness and stability. The uncertainties in the FCNNs make them well-suited to address nonlinear filtering problems. Furthermore, ref. [16] demonstrated that FCNNs behave well in image processing and recognition. The authors of [17], investigated the mean square stability of FCNNs with mixed delays. The authors of [18] introduced memristive cellular neural networks (MCNNs) by integrating memristors with cellular neural networks. Compared with CNNs, their improved storage capacity makes MCNNs more popular Furthermore, the authors of [19] investigated MCNNs, with fixed-time synchronization conditions established by using feedback controllers and Lyapunov methods. Additionally, ref. [20] reported the synchronization results of memristive neural networks with the fuzzy cellular neural networks (MFCNNs) through feedback controllers. The combination of memristors with FCNNs can expand the advantages of memory, in addition to exhibiting chaotic characteristics.

Fractional-order calculus originated three hundred years ago, although it was not widely promoted due to the lack of physical meanings. Scientists and engineers began to realize that fractional derivatives and integrals can more accurately describe practical phenomena [21]. For instance, the model of a DC–DC converter is a typical fractional-order (FO) system, and its performance is affected if a traditional integer-order model is adopted [22]. In [23], Stamova and Stamov studied fractional-order neural networks (FNNs) with delays and reaction–diffusion. The concept of FO cellular neural networks was proposed by the authors of [24]. The authors of [25] integrated fuzzy logic into FO cellular neural networks. With the efforts of scholars, FNNs have become increasingly significant, with their dynamic properties, such as stability and synchronization, having received extensive attention. The introduction of FO memristive neural networks (FMNNs) is discussed in [26]. Since then, many efforts have been made in the study of FMNNs. For instance, FO memristive fuzzy cellular neural networks (FMFCNNs), which integrate fuzzy logic with fractional-order MNNs, have been explored, with finite-time stability results reported in [27]. The authors of [28,29] studied FMFCNNs for finite-time synchronization and fixed-time synchronization, respectively. However, the authors of [28] did not employ the Lyapunov method, and the authors of [29] used an integer-order derivative to handle the Lyapunov function. Research on FMFCNNs, particularly concerning exponential synchronization, remains limited. This is one of the motivations of this paper.

Synchronization is a critical dynamic behavior that has been extensively studied. Controller design is a core problem, and there are many effective traditional controllers [30,31,32,33]. Intermittent controllers stand out as an economical option that can reduce control resource loss and is easy to achieve. From the perspective of engineering, the transmitted information may be interrupted by disturbances, which means discontinuous controllers are more suitable for practical applications. The authors of [34,35] investigated the synchronization of neural networks using an intermittent controller. In [36], a novel intermittent-hold controller was proposed that extends traditional intermittent control by incorporating holding time, allowing the system to retain state information when communication is interrupted. This enhancement increases the robustness of the system. Traditional intermittent control strategies set the control input to zero during communication interruptions, which may cause a decreased convergence rate. However, intermittent-hold control strategies can improve this phenomenon, as demonstrated in [37]. Moreover, intermittent-hold control strategies simplify the design of controllers by not updating the control input during communication interruptions, thereby reducing the cost of controllers. But this kind of controller has not been applied in FNNs. For delayed FNNs, achieving exponential synchronization is challenging due to the distinct principles of fractional derivatives compared to integer-order derivatives. To address this, we present an inequality to handle delay terms and study exponential synchronization problems of FMFCNNs.

Moreover, in practical applications, chaotic synchronization can be utilized in secure communications. For example, the authors of [38] considered the synchronization problems of the MNNs and achieved encryption and decryption. Due to the enriched dynamical behaviors and the expanded parameter spaces of FO systems, the neural networks represented by FO calculus have more unpredictable dynamics. These characteristics make secure communication problems based on FO systems more complicated [39]. The study reported in [40] exhibits the robustness of FO systems, implying that such systems can ensure the stability and reliability of secure communication, even in unstable communication environments and under external interference. Inspired by the above discussion, this paper designs an intermittent-hold controller of FO systems and applies the synchronization results to secure communication problems.

The exponential synchronization of the FMFCNNs is achieved in this paper, and a novel intermittent-hold controller is designed. The main contributions are summarized as follows:

(1)

In this paper, a novel intermittent-hold controller is designed. The controller can chose control time flexibly with less control loss. Furthermore, adaptive control is integrated with intermittent-hold control to handle more complicated situations such as uncertainties and chaos.

(2)

A new inequality is introduced to provide the exponential stability condition for fractional-order (FO) systems, overcoming the challenges in constructing Lyapunov functions for FO systems. Based on this novel inequality, the exponential synchronization of FMFCNNs is achieved.

(3)

Some examples are provided to demonstrate the effectiveness of the proposed conditions. The control time and resting time can be chosen as required, and larger delays can be tackled by using the proposed conditions and controller, as shown in simulations.

(4)

The conditions can be applied to secure communication problems, as shown in Example 3. The chaotic signal generated by the FO drive system is mixed with the original signal for encryption. The decrypted signal is recovered through synchronized response systems.

The remainder of this paper is structured as follows. Some lemmas, definitions and models used in this paper are described in Section 2. Two types of intermittent-hold controllers are designed and some new lemmas and theorems are provided in Section 3. Three examples of three-dimensional and two-dimensional cases, as well as secure communications, are presented in Section 4. The last part discusses future work and our proposed conclusions.

2. Preliminaries and Models

This section contains a description of FMFCNNs and introduces some lemmas used in this paper.

2.1. Model Description

Consider the following fuzzy cellular FMNNs:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{x}_i\left(t\right)& =-c_i{x}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}\left(x_i\left(t\right)\right)f_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}\left(x_i\left(t\right)\right)g_j\left(x_j(t-\tau )\right)\hfill \\ \hfill & +\sum _{j=1}^m{d}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(x_j(t-\tau )\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ \hfill & +\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j\left(x_j(t-\tau )\right)+I_i,\hfill \end{array}$$

(1)

where $\varphi \left(s\right)\in C([-\tau ,0],{\mathcal{R}}^n)$ denotes the initial condition, $c_i$ is the self-feedback coefficient, and $a_{ij}\left(x_i\left(t\right)\right)$ and $b_{ij}\left(x_i\left(t\right)\right)$ are the memristor’s connective weights. $\tau$ is the time delay, and $f_j$ and $g_j$ are feedback functions. In the fuzzy cellular neural network model, ${\alpha }_{ij}$ and ${\beta }_{ij}$ are elements of fuzzy feedback, ${\alpha }_{ij}$ is MIN and other is MAX. The MIN of a fuzzy feed-forward neural network is $T_{ij}$, and MAX is $S_{ij}$. ${\nu }_j$, ⋀ and ⋁ are bias of the ith neuron and fuzzy AND and OR, respectively. The memristive weights of $a_{ij}\left(x_i\left(t\right)\right)$ and $b_{ij}\left(x_i\left(t\right)\right)$ are satisfied as follows:

$$a_{ij}\left(x_i\left(t\right)\right)=\left\{\begin{array}{c}{\widehat{\mathcal{A}}}_{ij},\left|x_i\left(t\right)\right|\le T_i\hfill \\ {\check{\mathcal{A}}}_{ij},\left|x_i\left(t\right)\right|>T_i,\hfill \end{array}\right.b_{ij}\left(x_i\left(t\right)\right)=\left\{\begin{array}{c}{\widehat{\mathcal{B}}}_{ij},\left|x_i\left(t\right)\right|\le T_i\hfill \\ {\check{\mathcal{B}}}_{ij},\left|x_i\left(t\right)\right|>T_i,\hfill \end{array}\right.$$

(2)

where $T_i>0$ is a switching jump and ${\hat{\mathcal{A}}}_{ij}$, ${\check{\mathcal{A}}}_{ij}$, ${\widehat{\mathcal{B}}}_{ij}$ and ${\check{\mathcal{B}}}_{ij}$ represent memristance-related constants.

Regarding (1) as drive systems, the response systems of fuzzy cellular FMNNs can be described as follows:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{y}_i\left(t\right)& =-c_i{y}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}\left(y_i\left(t\right)\right)f_j\left(y_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}\left(y_i\left(t\right)\right)g_j\left(y_j(t-\tau )\right)\hfill \\ \hfill & +\sum _{j=1}^m{d}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(y_j(t-\tau )\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j\left(y_j(t-\tau )\right)+I_i+u_i\left(t\right),\hfill \end{array}$$

(3)

where the initial condition is $\phi \left(s\right)\in C([-\tau ,0],{\mathcal{R}}^n)$, and $u_i\left(t\right)$ is the intermittent hold controller. The memristor’s connection weights ($a_{ij}\left(y_i\left(t\right)\right)$ and $b_{ij}\left(y_i\left(t\right)\right)$) are denoted as follows:

$$a_{ij}\left(y_i\left(t\right)\right)=\left\{\begin{array}{c}{\widehat{\mathcal{A}}}_{ij}\prime ,\left|y_i\left(t\right)\right|\le T_i\hfill \\ {\check{\mathcal{A}}}_{ij}\prime ,\left|y_i\left(t\right)\right|>T_i,\hfill \end{array}\right.b_{ij}\left(y_i\left(t\right)\right)=\left\{\begin{array}{c}{\widehat{\mathcal{B}}}_{ij}\prime ,\left|y_i\left(t\right)\right|\le T_i\hfill \\ {\check{\mathcal{B}}}_{ij}\prime ,\left|y_i\left(t\right)\right|>T_i.\hfill \end{array}\right.$$

(4)

According to Filippov and differential inclusion theories, there exist measurable functions, i.e., $a_{ij}^{*}\in \overline{co}[{\underline{a}}_{ij},{\overline{a}}_{ij}]$, $b_{ij}^{*}\in \overline{co}[{\underline{b}}_{ij},{\overline{b}}_{ij}]$, where ${\underline{a}}_{ij}$ represents the minimum of $\{{\widehat{\mathcal{A}}}_{ij},{\check{\mathcal{A}}}_{ij}\}$, ${\overline{a}}_{ij}$ denotes the maximum of $\{{\widehat{\mathcal{A}}}_{ij},{\check{\mathcal{A}}}_{ij}\}$ and ${\underline{b}}_{ij}$ and ${\overline{b}}_{ij}$ stand for $min\{{\widehat{\mathcal{B}}}_{ij},{\check{\mathcal{B}}}_{ij}\}$ and $max\{{\widehat{\mathcal{B}}}_{ij},{\check{\mathcal{B}}}_{ij}\}$, respectively. Then the FO neural networks (1) can be rewritten as follows:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{x}_i\left(t\right)& =-c_i{x}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}^{*}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}^{*}{g}_j\left(x_j(t-\tau )\right)\hfill \\ \hfill & +\sum _{j=1}^m{d}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(x_j(t-\tau )\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ \hfill & +\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j\left(x_j(t-\tau )\right)+I_i.\hfill \end{array}$$

(5)

There exist measurable functions, i.e., $a_{ij}^{**}\in \overline{co}[{\underline{a}}_{ij}\prime ,{\overline{a}}_{ij}\prime ]$, $b_{ij}^{**}\in \overline{co}[{\underline{b}}_{ij}\prime ,{\overline{b}}_{ij}\prime ]$, where ${\underline{a}}_{ij}$ represents the minimum of $\{{\widehat{\mathcal{A}}}_{ij}\prime ,{\check{\mathcal{A}}}_{ij}\prime \}$ and ${\overline{a}}_{ij}$ denotes the maximum of $\{{\widehat{\mathcal{A}}}_{ij}\prime ,{\check{\mathcal{A}}}_{ij}\prime \}$. ${\underline{b}}_{ij}$ and ${\overline{b}}_{ij}$ stand for $min\{{\widehat{\mathcal{B}}}_{ij}\prime ,{\check{\mathcal{B}}}_{ij}\prime \}$ and $max\{{\widehat{\mathcal{B}}}_{ij}\prime ,{\check{\mathcal{B}}}_{ij}\prime \}$, respectively. The response systems can be described as follows:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{y}_i\left(t\right)& =-c_i{y}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}^{**}{f}_j\left(y_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}^{**}{g}_j\left(y_j(t-\tau )\right)\hfill \\ \hfill & +\sum _{j=1}^m{d}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(y_j(t-\tau )\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j\left(y_j(t-\tau )\right)+I_i+u_i\left(t\right).\hfill \end{array}$$

(6)

The error is denoted as $e_i\left(t\right)=y_i\left(t\right)-x_i\left(t\right)$, and the error system of (5) and (6) is expressed as follows:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{e}_i(t)& =-c_i{e}_i(t)+\sum _{j=1}^n[\left(a_{ij}^{**}{f}_j(y_j(t))-a_{ij}^{*}{f}_j(x_j(t))\right)+(b_{ij}^{**}{g}_j(y_j(t-\tau ))\hfill \\ \hfill & -b_{ij}^{*}{g}_j(x_j(t-\tau ))]+\underset{j=1}{\overset{n}{\bigwedge }}\left[{\alpha }_{ij}\left(g_j(y_j(t-\tau ))-g_j(x_j(t-\tau ))\right)\right]\hfill \\ \hfill & +\underset{j=1}{\overset{n}{\bigvee }}\left[{\beta }_{ij}\left(g_j(y_j(t-\tau ))-g_j(x_j(t-\tau ))\right)\right]+u_i(t).\hfill \end{array}$$

(7)

When the error system is stable under controller $u_i\left(t\right)$, systems (5) and (6) can achieve synchronization. Therefore, the subsequent sections focus on designing controllers and investigating the synchronization problems.

2.2. Preliminaries

In this subsection, some useful assumption and lemmas are presented, along with the definition of a Caputo derivative. The models proposed in this paper are described using the Caputo derivative.

Definition 1 

([41]). For $n+1$-order continuous differentiable functions ($f\in C^{n+1}\left([t_0,+\infty ]\right)$), the Caputo fractional-order derivative is defined as follows:

$${}_{0}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\int }_{t_0}^t{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds,$$

(8)

where $n-1\le \alpha <n$ and $n>0$ is an integer.

Assumption A1. 

Activation functions $f_j$ and $g_j$ satisfy the following for all $j=1,2,\dots ,n$, $x,y\in \mathbf{R}$, $x\ne y$:

$$\begin{array}{cc}\hfill & \left(1\right)f_j\left(0\right)=g_j\left(0\right)=0,\hfill \\ \hfill & \left(2\right)|f_j\left(y\right)-f_j\left(x\right)|\le F_j|y-x|,|g_j\left(y\right)-g_j\left(x\right)|\le G_j|y-x|,\hfill \end{array}$$

(9)

where $F_j$ and $G_j>0$ are Lipschitz constants.

Lemma 1 

([42]). Based on Assumption 1, for measurable functions ($a_{ij}^{*},a_{ij}^{**}\in \overline{co}[{\underline{a}}_{ij},{\overline{a}}_{ij}]$, $b_{ij}^{*},b_{ij}^{**}\in \overline{co}[{\underline{b}}_{ij},{\overline{b}}_{ij}]$), then

$$\begin{array}{cc}\hfill & |a_{ij}^{**}{f}_j\left(y_j\left(t\right)\right)-a_{ij}^{*}{f}_j\left(x_j\left(t\right)\right)|\le {\gamma }_{ij}^{*}{F}_j|y_j\left(t\right)-x_j\left(t\right)|,\hfill \\ \hfill & |b_{ij}^{**}{g}_j\left(y_j\left(t\right)\right)-b_{ij}^{*}{g}_j\left(x_j\left(t\right)\right)|\le {\delta }_{ij}^{*}{G}_j|y_j\left(t\right)-x_j\left(t\right)|,\hfill \end{array}$$

(10)

where ${\gamma }_{ij}^{*}=max\left\{\right|{\widehat{\mathcal{A}}}_{ij}|,|{\check{\mathcal{A}}}_{ij}\left|\right\}$, ${\delta }_{ij}^{*}=max\left\{\right|{\widehat{\mathcal{B}}}_{ij}|h,|{\check{\mathcal{B}}}_{ij}\left|\right\}$.

Lemma 2 

([16]). For system (5), we have the following:

$$\begin{array}{cc}\hfill & |\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}\left(g_j\left(y_j\left(t\right)\right)-g_j\left(x_j\left(t\right)\right)\right)|\le \sum _{j=1}^n{G}_j|{\alpha }_{ij}\left|\right|y_j\left(t\right)-x_j\left(t\right)|,\hfill \\ \hfill & |\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}\left(g_j\left(y_j\left(t\right)\right)-g_j\left(x_j\left(t\right)\right)\right)|\le \sum _{j=1}^n{G}_j|{\beta }_{ij}\left|\right|y_j\left(t\right)-x_j\left(t\right)|,\hfill \end{array}$$

(11)

where $x_j\left(t\right)$ and $y_j\left(t\right)$ are states of system (5).

Lemma 3 

([43]). For a continuous function ($V\left(t\right)$), for $0<\alpha <1$, there exist constants (ρ and δ) such that

$${}_{0}D_t^{\alpha }V\left(t\right)\le -\rho V\left(t\right)+\delta ,t\ge 0.$$

(12)

Then, we can obtain

$${}_{0}D_t^{\alpha }V\left(t\right)-{\displaystyle \frac{\delta }{\rho }}\le (V\left(0\right)-{\displaystyle \frac{\delta }{\rho }})e^{-{\displaystyle \frac{\rho }{\mathrm{\Gamma }(\alpha +1)}}}{t}^{\alpha }.$$

(13)

3. Main Results

A novel intermittent-hold controller is designed and the control strategies are given in this section. Feedback control and adaptive control are integrated with intermittent-hold control. Some sufficient conditions for achieving exponential synchronization of FO systems are provided in this part.

3.1. Controller Design

Consider following intermittent controller:

$$u_i\left(t\right)=\left\{\begin{array}{cc}\hfill & {\tilde{u}}_i\left(t\right),t\in [t_k,t_{k1}),\hfill \\ \hfill & {\tilde{u}}_i\left(t_k\right),t\in [t_{k1},t_{k2}),\hfill \\ \hfill & 0,t\in [t_{k2},t_{k+1}),\hfill \end{array}\right.$$

(14)

In contrast with traditional intermittent control, the intermittent-hold control strategy remains the last received state information of the system instead of stopping control input. By selecting the appropriate holding interval, drive and response systems can achieve synchronization faster. In addition, the intermittent-hold control strategy is less conservative than traditional intermittent control and can achieve synchronization even with shorter communication intervals [44]. The time intervals are $t_{k+1}-t_k=T$, $t_{k1}-t_k={\theta }_1$, $t_{k2}-t_{k1}={\theta }_2$ and $t_{k+1}-t_{k2}={\theta }_3$. The controller can be designed as follows:

$$u_i\left(t\right)=\left\{\begin{array}{cc}\hfill & -k_i{e}_i\left(t\right)-{\eta }_i\left|e_i(t-\tau )\right|,t\in [t_k,t_{k1}),\hfill \\ \hfill & -k_i{e}_i\left(t_{k1}^{-}\right)-{\eta }_i\left|e_i(t_{k1}^{-}-\tau )\right|,t\in [t_{k1},t_{k2}),\hfill \\ \hfill & 0,t\in [t_{k2},t_{k+1}),\hfill \end{array}\right.$$

(15)

where the control gains ($k_i$ and ${\eta }_i$) need to be designed later. Controller (14) can be designed as an adaptive intermittent-hold controller as follows:

$$u_i\left(t\right)=\left\{\begin{array}{cc}\hfill & -k_i{e}_i\left(t\right)-{\displaystyle \frac{1}{2}}{\xi }_i{\lambda }_i\left(t\right),t\in [t_k,t_{k1}),\hfill \\ \hfill & -k_i^{\prime }{e}_i\left(t_{k1}^{-}\right)-{\xi }_i^{\prime }{\lambda }_i\left(t_{k1}^{-}\right),t\in [t_{k1},t_{k2}),\hfill \\ \hfill & 0,t\in [t_{k2},t_{k+1}),\hfill \end{array}\right.$$

(16)

where ${\lambda }_i\left(t\right)={\displaystyle \frac{1}{e_i\left(t\right)}}{e}_i^2(t-\tau )$, ${}_{0}D_t^{\alpha }{k}_i={\widehat{k}}_i{e}_i^2\left(t\right)-{\displaystyle \frac{1}{2}}{\widehat{k}}_i{q}_1{k}_i$, ${}_{0}D_t^{\alpha }{\xi }_i={\widehat{\xi }}_i{e}_i^2(t-\tau )-{\displaystyle \frac{1}{2}}{\widehat{\xi }}_i{q}_2{\xi }_i$ and ${\widehat{k}}_i,{\widehat{\xi }}_i,q_1,q_2>0$ are constants.

Remark 1. 

Controller (14) adopts an intermittent-hold control strategy, allowing for flexible selection of control time and holding time. By using this controller, larger delays can be tackled as shown in Figure 8. Controller (15) is feedback-based, and controller (16) is adaptive-based so that it can handle more complex situations. The design of control gains is addressed subsequently.

Remark 2. 

The use of an adaptive controller is an intelligent control strategy. The control law includes an update rule to adjust the control parameters. These parameters are dynamically updated based on errors and state changes between drive and response systems. The design of control parameters ${\widehat{k}}_i$ and ${\widehat{\xi }}_i$ needs to be determined by the conditions outlined in Theorem 2.

3.2. Exponential Synchronization of FMNNs with Fuzzy Cellular Neural Networks

In this subsection, a lemma for a delayed FO system is presented that addresses the exponential stability and synchronization problems. The results for FMFCNNs with two kinds of intermittent-hold control are reported.

Lemma 4. 

For following delayed FO system with $V\left(t\right)>0$, $V\left(t\right)$ is exponentially stable if ${\rho }_1>0$, ${\rho }_2>0$ and $V\left(0\right)$ is bounded:

$${}_{0}D_t^{\alpha }V\left(t\right)\le -{\rho }_{1}V\left(t\right)+{\rho }_{2}V(t-\tau )+\delta .$$

Then, there exist positive constants (b and ${\sigma }_1$) such that

$$V\left(t\right)\le b{e}^{-{\rho }_2{\sigma }_1{t}^{\alpha }}.$$

Proof of Lemma 4. 

For a positive function ($m\left(t\right)$), we have

$${}_{0}D_t^{\alpha }V\left(t\right)=-{\rho }_{1}V\left(t\right)+{\rho }_{2}V(t-\tau )+\delta -m\left(t\right),$$

(17)

and, taking the FO integral of (17), we can obtain

$$V\left(t\right)=V\left(0\right)+{}_{0}D_t^{-\alpha }(-{\rho }_{1}V\left(t\right)+{\rho }_{2}V(t-\tau )+\delta -m\left(t\right)).$$

(18)

Then, taking the Laplace transform of (18), we derive the following equation:

$$\overline{V}\left(s\right)=s^{-1}V\left(0\right)+s^{-\alpha }(-{\rho }_1\overline{V}\left(s\right))+s^{-\alpha }\left({\rho }_2\overline{V}(s-\tau )\right)+s^{-\alpha }\delta s^{-1}-s^{-\alpha }\overline{m}\left(s\right),$$

(19)

where $\mathcal{L}\{V\left(t\right);s\}=\overline{V}\left(s\right)$, $\mathcal{L}\{m\left(t\right);s\}=\overline{m}\left(s\right)$. From (19), we have

$$\overline{V}\left(s\right)={\displaystyle \frac{s^{-1}}{1+s^{-\alpha }{\rho }_1}}V\left(0\right)+{\displaystyle \frac{s^{-\alpha }}{1+s^{-\alpha }{\rho }_1}}\left({\rho }_2\overline{V}(s-\tau )\right)+{\displaystyle \frac{s^{-\alpha -1}}{1+s^{-\alpha }{\rho }_1}}\delta -{\displaystyle \frac{s^{-\alpha }}{1+s^{-\alpha }{\rho }_1}}\overline{m}\left(s\right),$$

(20)

Then, taking the inverse Laplace transform of (20), we can obtain

$$\begin{array}{cc}\hfill V\left(t\right)& \le V\left(0\right)E_{\alpha }(-{\rho }_1{t}^{\alpha })+\delta t^{\alpha }{E}_{\alpha ,\alpha +1}(-{\rho }_1{t}^{\alpha })\hfill \\ \hfill & +{\rho }_2{\int }_0^t{(t-s)}^{\alpha -1}{E}_{\alpha ,\alpha }(-{\rho }_1{(t-s)}^{\alpha })V(s-\tau )ds.\hfill \end{array}$$

(21)

According to [45], we have $0<E_{\alpha }(-{\rho }_1{t}^{\alpha })<1$, and $0<t^{\alpha }{E}_{\alpha ,\alpha +1}(-{\rho }_1{t}^{\alpha })<{\displaystyle \frac{1}{{\rho }_1}}$. Then, we can obtain

$$\begin{array}{c}\hfill V\left(t\right)\le V\left(0\right)+{\displaystyle \frac{\delta }{{\rho }_1}}+{\rho }_2{\int }_0^t{(t-s)}^{\alpha -1}{E}_{\alpha ,\alpha }(-{\rho }_1{(t-s)}^{\alpha })V(s-\tau )ds.\end{array}$$

(22)

Letting ${\sigma }^{*}={sup}_{-\tau \le s\le t}\{V(s-\tau )+V\left(s\right)\}$, we have

$$\begin{array}{c}\hfill V\left(t\right)\le V\left(0\right)+{\displaystyle \frac{\delta +{\sigma }^{*}{\rho }_2}{{\rho }_1}}-{\rho }_2{\int }_0^t{(t-s)}^{\alpha -1}{E}_{\alpha ,\alpha }(-{\rho }_1{(t-s)}^{\alpha })V\left(s\right)ds.\end{array}$$

(23)

By utilizing grown-wall inequality we obtain

$$V\left(t\right)\le b{e}^{-{\rho }_2{t}^{\alpha }{E}_{\alpha ,\alpha +1}(-{\rho }_1{t}^{\alpha })},$$

(24)

where $b=V\left(0\right)+{\displaystyle \frac{\delta +{\sigma }^{*}{\rho }_2}{{\rho }_1}}$ and, because of $t^{\alpha }{E}_{\alpha ,\alpha +1}(-{\rho }_1{t}^{\alpha })>0$, there exists ${\sigma }_1>0$ such that $E_{\alpha ,\alpha +1}(-{\rho }_1{t}^{\alpha })>{\sigma }_1$. Therefore, we can obtain

$$V\left(t\right)\le b{e}^{-{\rho }_2{\sigma }_1{t}^{\alpha }}.$$

(25)

□

Remark 3. 

When FO systems satisfy the requirements of Lemma 4, $V\left(t\right)$ decays exponentially. If $V\left(t\right)$ is a Lyapunov functional, the results of exponential stability or exponential synchronization can be obtained under Lemma 4.

Theorem 1. 

Under the assumption 1, if there exists a positive control gain ($k_i$) such that

$$\rho =\underset{i}{min}\{c_i+k_i-\sum _{j=1}^n{\gamma }_{ji}^{*}{F}_i\}>0,$$

(26)

$${\rho }_1^{*}=\underset{i}{min}\{2c_i-\sum _{j=1}^n({\gamma }_{ij}^{*}{F}_j+{\gamma }_{ji}^{*}{F}_i)-{\rho }_2\}>0,$$

(27)

$${\rho }_2=\underset{i}{max}\{\sum _{j=1}^n({\delta }_{ji}^{*}+|{\alpha }_{ji}|+|{\beta }_{ji}\left|\right)G_i\}>0$$

(28)

and

$${\eta }_i=\sum _{j=1}^n({\delta }_{ji}^{*}+|{\alpha }_{ij}|+|{\beta }_{ij}\left|\right)G_i$$

(29)

hold, where $i=1,2,\dots ,n$ and other constants are previously defined system parameters, fuzzy cellular FMNNs (5) and (6) can achieve exponential synchronization with controller (15).

The proof of Theorem 1 is shown in Appendix A.

If controller (15) is changed into an adaptive intermittent-hold controller as (16), a different synchronization result can be obtained.

Theorem 2. 

Fuzzy cellular FMNNs (5) and (6) can achieve exponential synchronization with adaptive intermittent-hold controller (16) if there exist control gains (${\widehat{k}}_i$, ${\widehat{\xi }}_i$ and $k_i^{\prime }$) ${\xi }_i^{\prime }$ such that

$$\rho =\underset{i}{min}\{2c_i+2{\widehat{k}}_i-\sum _{j=1}^n\left[{\gamma }_{ij}^{*}{F}_j+{\gamma }_{ji}^{*}{F}_i\right]-\sum _{j=1}^n\left[{\delta }_{ij}^{*}+|{\alpha }_{ij}|+|{\beta }_{ij}|\right]G_j\}>0,$$

$${\rho }_1=\underset{i}{min}\{2c_i+k_i^{\prime }+{\displaystyle \frac{1}{2}}{\xi }_i^{\prime }-\sum _{j=1}^n\left[{\gamma }_{ij}^{*}{F}_j+{\gamma }_{ji}^{*}{F}_i\right]-\sum _{j=1}^n\left[{\delta }_{ij}^{*}+|{\alpha }_{ij}|+|{\beta }_{ij}|\right]G_j\}>0,$$

$$\sum _{j=1}^n\left({\delta }_{ji}^{*}+|{\alpha }_{ji}|+|{\beta }_{ji}|\right)G_i-{\widehat{\xi }}_i\le 0,$$

and assumption 1 holds, where $i=1,2,\dots ,n$ and other constants are previously defined system parameters.

The proof of Theorem 2 is given in Appendix B.

Remark 4. 

The gains of controllers are determined by the conditions of theorems. For controller (15), the gains ($k_i(i=1,2,\dots ,n)$) need to satisfy the conditions of Theorem 1, and ${\eta }_i(i=1,2,\dots ,n)$ are given in (29). For controller (16), the ${\widehat{k}}_i,{\widehat{\xi }}_i(i=1,2,\dots ,n)$ of adaptive laws need to satisfy the conditions of Theorem 2. When inequalities in theorems are satisfied with system parameters and control gains, the drive and response systems can achieve synchronization.

4. Simulation

Example 1. 

Consider a three-dimensional FMNN with a fuzzy cellular neural network with the following parameters:

$$\begin{array}{cc}\hfill & a_{11}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}2,|x_1\left(t\right)|\le 1\hfill \\ 1.9,|x_1\left(t\right)|>1\hfill \end{array}\right.,a_{12}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}-1.3,|x_1\left(t\right)|\le 1\hfill \\ -1.1,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & a_{13}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}0.2,|x_1\left(t\right)|\le 1\hfill \\ -0.2,|x_1\left(t\right)|>1\hfill \end{array}\right.,a_{21}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}1.8,|x_2\left(t\right)|\le 1\hfill \\ 1.8,|x_2\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & a_{22}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}1.7,|x_2\left(t\right)|\le 1\hfill \\ 1.72,|x_2\left(t\right)|>1\hfill \end{array}\right.,a_{23}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}1.1,|x_2\left(t\right)|\le 1\hfill \\ 1.2,|x_2\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & a_{31}\left(x_3\left(t\right)\right)=\left\{\begin{array}{c}-4.8,|x_3\left(t\right)|\le 1\hfill \\ -4.7,|x_3\left(t\right)|>1\hfill \end{array}\right.,a_{32}\left(x_3\left(t\right)\right)=\left\{\begin{array}{c}0.5,|x_3\left(t\right)|\le 1\hfill \\ -0.5,|x_3\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & a_{33}\left(x_3\left(t\right)\right)=\left\{\begin{array}{c}1,|x_3\left(t\right)|\le 1\hfill \\ 1.2,|x_3\left(t\right)|>1\hfill \end{array}\right.,b_{11}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}-0.1,|x_1\left(t\right)|\le 1\hfill \\ -0.3,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{12}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}0.25,|x_1\left(t\right)|\le 1\hfill \\ 0.35,|x_1\left(t\right)|>1\hfill \end{array}\right.,b_{13}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}0.5,|x_1\left(t\right)|\le 1\hfill \\ -0.5,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{21}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}-0.25,|x_2\left(t\right)|\le 1\hfill \\ -0.15,|x_2\left(t\right)|>1\hfill \end{array}\right.,b_{22}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}-0.18,|x_2\left(t\right)|\le 1\hfill \\ -0.2,|x_2\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{23}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}0.15,|x_2\left(t\right)|\le 1\hfill \\ 0.15,|x_2\left(t\right)|>1\hfill \end{array}\right.,b_{31}\left(x_3\left(t\right)\right)=\left\{\begin{array}{c}0.15,|x_3\left(t\right)|\le 1\hfill \\ 0.7,|x_3\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{32}\left(x_3\left(t\right)\right)=\left\{\begin{array}{c}0.2,|x_3\left(t\right)|\le 1\hfill \\ -0.2,|x_3\left(t\right)|>1\hfill \end{array}\right.,b_{33}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}-0.28,|x_3\left(t\right)|\le 1\hfill \\ -0.12,|x_3\left(t\right)|>1\hfill \end{array}\right.,\hfill \end{array}$$

where $c_1=c_2=c_3=2$, $\tau =0.1$, $\alpha =0.9$ and $k_1=k_2=k_3=5$ and the activation functions are $f_i\left(x\right)=g_i\left(x\right)={\displaystyle \frac{1}{2}}\left(\right|1+x|+|1-x\left|\right)$.

Other parameters are denoted as follows:

$$\begin{array}{cc}\hfill & {\left({\alpha }_{ij}\right)}_{3\times 3}=\left(\begin{array}{ccc}-0.1& -0.01& 0.1\\ -0.2& -0.1& 0.1\\ -0.04& -0.2& 0.4\end{array}\right),{\left({\beta }_{ij}\right)}_{3\times 3}=\left(\begin{array}{ccc}-0.1& -0.01& 0.3\\ -0.1& -0.2& 0.2\\ -0.1& -0.2& 0.3\end{array}\right),\hfill \\ \hfill & D={\left(d_{ij}\right)}_{3\times 3}=\left(\begin{array}{ccc}0.1& 0.1& -0.1\\ 0.1& 0.1& -0.2\\ 0.2& 0.1& 0.2\end{array}\right),T={\left(T_{ij}\right)}_{3\times 3}=\left(\begin{array}{ccc}0.2& 0.1& 0.2\\ 0.2& 0.2& 0.1\\ 0.1& 0.1& 0.2\end{array}\right),\hfill \\ \hfill & S={\left(S_{ij}\right)}_{3\times 3}=\left(\begin{array}{ccc}0.2& 0.1& 0.2\\ 0.3& 0.1& 0.2\\ 0.1& 0.1& 0.2\end{array}\right),V={\left({\nu }_j\right)}_{3\times 1}=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right).\hfill \end{array}$$

Let $F_i=0.1$ and $G_i=1$; then, according to Lemma 2 and the above parameters, we can obtain

$${\left({\delta }_{ij}^{*}\right)}_{3\times 3}=\left(\begin{array}{ccc}0.3& 0.35& 0.5\\ 0.25& 0.2& 0.15\\ 0.7& 0.2& 0.28\end{array}\right),{\left({\gamma }_{ij}^{*}\right)}_{3\times 3}=\left(\begin{array}{ccc}2& 1.3& 0.2\\ 1.8& 1.72& 1.2\\ 4.8& 0.5& 1.2\end{array}\right).$$

We can calculate that ${\rho }_2={max}_i\{{\sum }_{j=1}^n({\delta }_{ji}^{*}+|{\alpha }_{ji}|+|{\beta }_{ji}\left|\right)G_i\}=2.37>0$, ${\rho }_1^{*}={min}_i\{2c_i-{\sum }_{j=1}^n({\gamma }_{ij}^{*}{F}_j+{\gamma }_{ji}^{*}{F}_i)-{\rho }_2\}=0.42>0$ and $\rho ={min}_i\{c_i+k_i-{\sum }_{j=1}^n{\gamma }_{ji}^{*}{F}_i\}=6.4>0$ for $i=1,2,3$ and that ${\eta }_i={\sum }_{j=1}^n({\delta }_{ji}^{*}+|{\alpha }_{ij}|+|{\beta }_{ij}\left|\right)G_i$, ${\eta }_1=1.87$, ${\eta }_2=1.65$ and ${\eta }_3=2.17$. The control gains of controller (15) satisfy the conditions of Theorem 1 with the above system parameters. Therefore, the drive and response systems can achieve synchronization.

Figure 1 shows that the trajectories of $x_1\left(t\right)$ and $y_1\left(t\right)$ can achieve synchronization under intermittent-hold controller (15) with initial values of $x_1\left(0\right)=-0.8626$ and $y_1\left(0\right)=-0.9978$. Figure 2 shows that the error systems can converge to zero with initial values of $x\left(0\right)={(x_1\left(0\right),x_2\left(0\right),x_3\left(0\right))}^T={(-0.8626,0.0387,0.1755)}^T$ and $y\left(0\right)={(y_1\left(0\right),y_2\left(0\right),y_3\left(0\right))}^T={(-0.9978,-1.1277,-0.3095)}^T$.

Letting $t_{k+1}-t_k=0.1$, the control time is chosen as ${\theta }_1=0.05$, the holding time is ${\theta }_2=0.025$ and the rest time is ${\theta }_3=0.025$, as represented in Figure 3 and Figure 4. Furthermore, by selecting a delay of $\tau =1$, FMNs (5) and (6) with controller (15) can achieve exponential synchronization, which is illustrated in Figure 5. The trajectories of error between FMNs (5) and (6) are below $3.4*exp(-1.085x)$, which is consistent with the results of Theorem 1.

To compare the performance of intermittent-hold control with that of traditional intermittent control, we compare convergence time using three different sets of initial values. The first set of initial values is $x\left(0\right)={(-0.3175,-2.5191,-0.1228)}^T$ and $y\left(0\right)={(1.0146,1.7545,0.3030)}^T$. The second set is $x\left(0\right)={(0.5871,-0.2314,-0.1870)}^T$ and $y\left(0\right)={(-0.5449,-0.2592,-0.9586)}^T$. The third set is $x\left(0\right)={(-0.9639,0.4034,-0.4632)}^T$ and $y\left(0\right)={(0.5931,0.6637,-0.8803)}^T$. The intermittent-hold controller becomes a traditional intermittent controller when the holding time is ${\theta }_2=0$, letting $t_{k+1}-t_k=0.5$, $\tau =0.2$ and $\tau =1.5$.

Table 1. The error convergence time with different intermittent holding times and an error bound of $0.02$.

Time Delay	Initial Value Set	Control Strategy	Convergence Time (s)
$\tau =0.2$	first set	${\theta }_1=0.25$, ${\theta }_2=0.175$, ${\theta }_3=0.075$	2.18
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	4.05
	second set	${\theta }_1=0.25$, ${\theta }_2=0.175$, ${\theta }_3=0.075$	2.36
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	4.06
	third set	${\theta }_1=0.25$, ${\theta }_2=0.175$, ${\theta }_3=0.075$	2.37
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	4.03
$\tau =1.5$	first set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	3.76
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	5.07
	second set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	5.01
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	5.08
	third set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	4.95
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	5.06

show that the intermittent-hold control strategy achieves a faster convergence rate than traditional intermittent control. Moreover, the intermittent-hold controller performs better than the traditional one when dealing with large time delays.

Remark 5. 

Based on the FO system parameters in Example 2 in [46], we can calculate that ${\rho }_2=5.72>0$, ${\rho }_1^{*}=1.33>0$, $k_i=10$, ${\eta }_1=4.44$, ${\eta }_2=5.9$ and ${\eta }_3=3.74$, indicating that the presented results still work under the parameters reported in [46]. Figure 6 shows the synchronized states ($x_1\left(t\right)$ and $y_1\left(t\right)$), along with the trajectories of error between two states. Under the parameters of Example 1 in [28], the theorems proposed in this paper are also valid. Moreover, the proposed results can be applied to handle larger delays, such as $\tau =1.5$. Therefore, the results obtained in this paper are less conservative.

Example 2. 

Consider a two-dimensional FMNN with a fuzzy cellular neural network with the following parameters:

$$\begin{array}{cc}\hfill & a_{11}\left(x_1\left(t\right)\right)=1,a_{22}\left(x_2\left(t\right)\right)=1.8,\hfill \\ \hfill & a_{12}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}7,|x_1\left(t\right)|\le 1\hfill \\ 5,|x_1\left(t\right)|>1\hfill \end{array}\right.,a_{21}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}0.8,|x_1\left(t\right)|\le 1\hfill \\ 1,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{11}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}-1.5,|x_1\left(t\right)|\le 1\hfill \\ -1.2,|x_1\left(t\right)|>1\hfill \end{array}\right.,b_{12}\left(x_1\left(t\right)\right)=\left\{\begin{array}{c}1,|x_1\left(t\right)|\le 1\hfill \\ 0.8,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \\ \hfill & b_{21}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}0.8,|x_1\left(t\right)|\le 1\hfill \\ 1,|x_1\left(t\right)|>1\hfill \end{array}\right.,b_{22}\left(x_2\left(t\right)\right)=\left\{\begin{array}{c}-1.4,|x_1\left(t\right)|\le 1\hfill \\ -1.6,|x_1\left(t\right)|>1\hfill \end{array}\right.,\hfill \end{array}$$

where $c_1=c_2=2$, $\tau =0.8$, $\alpha =0.98$ and ${\widehat{k}}_i=4$ and the activation functions are $f_i\left(x\right)=g_i\left(x\right)={\displaystyle \frac{1}{2}}\left(\right|1+x|+|1-x\left|\right)$. Other parameters are denoted as follows:

$$\begin{array}{cc}\hfill & {\left({\alpha }_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.6& -0.2\\ -0.4& -2.8\end{array}\right),{\left({\beta }_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.2& -0.4\\ -0.1& -2.4\end{array}\right),\hfill \\ \hfill & D={\left(d_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}0.1& 0.1\\ 0.1& 0.1\end{array}\right),T={\left(T_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}0.2& 0.1\\ 0.2& 0.2\end{array}\right),\hfill \\ \hfill & S={\left(S_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}0.2& 0.1\\ 0.3& 0.1\end{array}\right),V={\left({\nu }_j\right)}_{2\times 1}=\left(\begin{array}{c}0.5\\ 0.5\end{array}\right).\hfill \end{array}$$

Let $F_i=0.1$ and $G_i=1$; then, according to Lemma 2 and the above parameters, we can obtain

$${\left({\delta }_{ij}^{*}\right)}_{2\times 2}=\left(\begin{array}{cc}1.5& 1\\ 1& 1.6\end{array}\right),{\left({\gamma }_{ij}^{*}\right)}_{2\times 2}=\left(\begin{array}{cc}1& 7\\ 1& 1.8\end{array}\right).$$

We can calculate that $\rho ={min}_i\{2c_i+2{\widehat{k}}_i-{\sum }_{j=1}^n\left[{\gamma }_{ij}^{*}{F}_j+{\gamma }_{ji}^{*}{F}_i\right]-{\sum }_{j=1}^n\left[{\delta }_{ij}^{*}+|{\alpha }_{ij}|+|{\beta }_{ij}|\right]G_j\}=2.54>0$ and ${\sum }_{j=1}^n\left({\delta }_{ji}^{*}+|{\alpha }_{ji}|+|{\beta }_{ji}|\right)G_i-{\widehat{\xi }}_i\le 0$, for $i=1,2$, ${\widehat{\xi }}_1\ge 0.58$, ${\widehat{\xi }}_2\ge 0.84$. The control gains of controller (16) satisfy the conditions of Theorem 2 with the above system parameters. Therefore, the drive and response systems can achieve synchronization.

According to Theorem 2, we find that ${\rho }_1^{*}=1.38>0$, $\rho >0$, ${\eta }_1=1.18$ and ${\eta }_2=1.46$. Figure 7 shows a control time of ${\theta }_1=0.07$, a holding time of ${\theta }_2=0.015$ and a rest time of ${\theta }_3=0.015$. To demonstrate the trajectory and effectiveness of the controller, we select three sets of states ($x\left(t\right)$ and $y\left(t\right)$) with different initial values of $x_1\left(0\right)={(-1.0292,2.6821)}^T$, $y_1\left(0\right)={(-1.6697,0.5007)}^T$, $x_2\left(0\right)={(-1.8612,0.9289)}^T$, $y_2\left(0\right)={(-0.2962,-0.3957)}^T$, $x_3\left(0\right)={(0.4037,0.5386)}^T$ and $y_3\left(0\right)={(0.6360,-0.8386)}^T$. Figure 8 shows the results obtain using the intermittent-hold controller integrated with an adaptive strategy and a delay of $\tau =0.8$.

Figure 9 presents the error between drive and response systems based on three sets of states ($x\left(t\right)$ and $y\left(t\right)$) with different initial values. $e_i$ is the first dimension of $y_i\left(t\right)-x_i\left(t\right)$. Figure 9 shows that the conservation of Theorem 2 is reduced due to the application of intermittent-hold controller (16). In this simulation, we use three different initial values, with delays set to $\tau =0.1$, $\tau =0.5$, $\tau =1$ and $\tau =1.5$. The errors between drive and response systems can converge to zero, which means that the results are applicable even in situations with larger delays.

Figure 10 shows the trajectories of controller (16) with four different delays. The lager the time delays, the larger the controller adjustment range. Comparisons of convergence times for three sets of initial values are provided. The first set of initial values is $x\left(0\right)={(0.1873,0.4373)}^T$ and $y\left(0\right)={(1.5822,-0.9426)}^T$. The second set is $x\left(0\right)={(-0.0323,0.0224)}^T$ and $y\left(0\right)={(-0.6654,-1.2561)}^T$. The third set is $x\left(0\right)={(0.3669,-1.2666)}^T$ and $y\left(0\right)={(-0.3175,0.3563)}^T$. Figure 11 shows a comparison between an intermittent-hold controller and a traditional intermittent controller. The delays are set as $\tau =1.5$, $t_{k+1}-t_k=0.5$, and Figure 11a shows that FMNs (5) and (6) can achieve synchronization, while Figure 11b displays the errors between two states under intermittent-hold control and intermittent control.

Table 2. The error converge time with different intermittent holding times and an error bound of $0.02$.

Time Delay	Initial Value Set	Control Strategy	Convergence Time (s)
$\tau =0.2$	first set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	1.01
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	1.11
	second set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	0.87
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	1.09
	third set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	0.62
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	0.95
$\tau =1.5$	first set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	5.11
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	7.12
	second set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	5.01
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	7.03
	third set	${\theta }_1=0.25$, ${\theta }_2=0.125$, ${\theta }_3=0.125$	4.01
		${\theta }_1=0.25$, ${\theta }_3=0.25$ (without hold)	6.52

demonstrates that the intermittent-hold control strategy can converge faster than the traditional intermittent control.

Example 3. 

Memristive neural networks can exhibit chaotic behaviors. Therefore, the results reported in this paper can be employed in secure communication problems.

System (1) can be degenerated as follows:

$$\begin{array}{cc}\hfill {}_{0}D_t^{\alpha }{x}_i\left(t\right)& =-c_i{x}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}\left(x_i\left(t\right)\right)f_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}\left(x_i\left(t\right)\right)g_j\left(x_j(t-\tau )\right)+I_i,\hfill \end{array}$$

(30)

The memristive weights can be described as uncertainties, for instance, $A+\mathcal{A}\left(t\right)={\left(a_{ij}\left(x_i\left(t\right)\right)\right)}_{n\times n}$, $\mathcal{A}\left(t\right)=EFM$, $F^{T}F\le I$. The parameters are set as

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccc}2& -1.2& 0\\ 1.8& 1.71& 1.15\\ -4.75& 0& 1.1\end{array}\right),B=\left(\begin{array}{ccc}-10& 10& 0\\ 28& -1& 0\\ 0& 0& -{\displaystyle \frac{8}{3}}\end{array}\right),\end{array}$$

and $f_j\left(x\right)={\displaystyle \frac{1}{2}}\left(\right|1+x|+|1-x\left|\right)$, $g_j\left(x\right)=cos\left(x\right)$, $c_i=2(i=1,2,3)$, $\tau =0.1$, ${\widehat{k}}_i=3(i=1,2,3)$, ${\theta }_1=0.5$, ${\theta }_2={\theta }_3=0.25$.

The response system has the same parameters. In the following simulation, the synchronization results are applied to the secure communication problem. First, the transmitter and receiver construct the same memristive neural networks to achieve signal synchronization. Based on Figure 12, we conclude that synchronization can be achieved by using controller (16). The transmitter combines the information signal with fractional-order MNN signals for encryption. The transmitted signal described as $S\left(t\right)=sin\left(t\right)+sin\left(5t\right)+sin\left(7t\right)$ is shown in Figure 13a. The encryption function is ${\Phi }_1(x,s)=({\phi }_1\left(x\right)+{\phi }_2\left(x\right))S$, where ${\phi }_1\left(x\right)=x_2{x}_3$ and ${\phi }_2\left(x\right)=x_2+x_3$. The mixed signal is shown in Figure 13b.

In this way, we can obtain the encryption signal ($SE$). The receiver uses the synchronized signal to recover the original signal as $Sd={\Phi }_2(y,SE)$. Figure 14 shows that the signal can be decoded successfully.

Remark 6. 

In the process of encryption and decryption through the use of the synchronization method, the initial errors are usually caused by the time required for the transmitter and receiver to establish synchronization. However, the two systems gradually adjust, reducing errors over time. These initial errors usually do not affect the long-term decryption effect because once the systems are synchronized, encryption and decryption can be proceed accurately, ensuring the security and reliability of information transmission.

5. Conclusions

In this paper, the exponential synchronization problems of FMFCNNs are investigated. A novel lemma is introduced to tackle the delay terms of FO systems via inequality techniques and the FO Laplace transform method. A new intermittent-hold controller is designed to address the synchronization problems. Furthermore, to handle more complicated situations, an intermittent-hold controller integrated with an adaptive control strategy is proposed. Additionally, two significant theorems are obtained on the basis of the proposed lemma and the two controllers. The simulation results confirm that the two proposed controllers can achieve exponential synchronization and effectively handle larger time delays. Compared with the traditional intermittent controllers, the proposed controllers have a faster convergence rate. Moreover, an application in secure communication is exhibited, which demonstrates the effectiveness of the results. For in future works, we will concentrate on the chaotic synchronization problems of FO systems.