Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

Abstract

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

1. Introduction

Since the 1990s, theoretical research on fractional-order neural networks (FONNs) [1] has gradually emerged and has made remarkable progress in recent years. The development of FONNs stems from the improvement and expansion of the traditional integer-order neural network model. Integer-order neural networks have limitations in dealing with some complex systems with memory and genetic characteristics, while FONNs do not have them [2,3,4,5], so FONNs can describe the dynamic behavior of the system more accurately by introducing fractional-order differential equations. There are many types of FONNs, such as fractional-order bidirectional associative memory neural networks [6], fractional-order Hopfield neural networks [7], fractional-order cellular neural networks (FOCNNs) [8], etc. Among them, the fractional-order cellular neural network model has attracted much attention in recent years due to its wide application in the field of pattern recognition, prediction analysis and secret communication [9]. The existing results show that fractional-order increases the degree of freedom of the system and makes FOCNNs more flexible in dealing with complex systems [10,11,12,13], so fractional-order fuzzy cellular neural networks (FOFCNNs) will have broader development prospects and wider potential applications in the future.

Fuzzy systems have experienced remarkable development since they were put forward by Zadeh in the 1960s. Nowadays, fuzzy systems have developed into mature intelligent systems, which have wide applications [14,15,16,17]. These applications show the unique advantages of fuzzy systems in dealing with complex system control and uncertainty. In recent years, fuzzy factors have been considered in fractional-order cellular neural network models [15]. These systems not only combine the powerful learning ability of neural networks, but also integrate the fuzzy processing ability of fuzzy systems, so that the systems can make intelligent decisions in uncertain environment [18]. For example, in [19], the finite-time stability of FOFCNNs with time delay is studied, the global asymptotic stability of complex-valued FOFCNNs with impulsive effects and time-varying delays is studied in [20], and the asymptotic stability of FOFCNNs with fixed-time impulse and time delay is investigated in [21]. At the same time, it can be foreseen that the application prospects of fuzzy systems will be broader with the continuous application of fuzzy systems to other fields, such as in image recognition [22], natural language processing [23] and other fields. Furthermore, the research into FOFCNNs will be more and more in-depth. In this paper, the properties of FOFCNNs are studied and corresponding conclusions are given.

The synchronization of FOCNNs is a widely studied problem and many results have been obtained. However, sometimes synchronization cannot be achieved by only the dynamics of the system itself; therefore, some synchronization control methods are needed. The main control methods include continuous-time feedback control and discontinuous-time feedback control. Discontinuous-time feedback control is more resource-saving in practice, so it is widely used. As a typical kind of discontinuous-time feedback control, impulsive control was originally applied in the field of mechanical control [24], later it was gradually used in chaotic synchronization control [25], and now it is also frequently used in FOCNNs [26,27,28,29]. In this paper, the synchronization problem of FOFCNNs is studied based on impulsive control.

In cellular neural networks, due to the different functions and the differences of environment, the model parameters are usually different. At this time, complete synchronization cannot be achieved by static linear feedback control alone—only quasi-synchronization can be achieved. With the in-depth study of systems with parameters mismatch, faster quasi-synchronization speed and smaller system error have particularly been considered. Although the quasi-synchronization problem of FONNs has been studied by many people, and fuzzy systems have also been considered in neural networks, few researchers have paid attention to controlling FOFCNNs by impulsive control to achieve exponential quasi-synchronization.

Based on the issues highlighted in the above discussion, and considering the general cellular neural network model, this paper studies the exponential quasi-synchronization of FOFCNNs with parameters mismatch under impulsive control and gives the relevant controller. The main contributions of this paper are as follows:

• Compared with the general cellular neural network model [30,31,32], fuzzy terms and parameters mismatch are considered in the model of cellular neural networks, which has a deep connection with other models. The results of this paper are all based on fuzzy cellular neural networks with parameters mismatch, so these conclusions can be generalized and they have strong application potential, especially in the case of high uncertainty.
• In this paper, a new fractional-order impulsive differential inequality is proposed. The differential inequality is analyzed by using the Laplace transform and related properties. It can be found that the inequality can be used as a tool for the stability analysis of fractional-order impulsive systems. Compared with reference [33], the impulsive differential inequality proposed in this paper is more general, which is of great significance for future research on the exponential quasi-synchronization of fractional-order systems based on impulsive control.
• Based on impulsive control, this paper obtains the quasi-synchronization criteria of FOFCNNs with parameters mismatch. As a corollary, the complete synchronization criterion of FOFCNNs with parameters match is given. The quasi-synchronization error bound, which is very close to the real error of the system, is estimated.

The rest of the article is organized as follows: In Section 2, the definition of a fractional-order derivative and the system model are introduced. Some assumptions about the system and a description of the impulsive controller are also given in this section. In Section 3, by introducing the definition of a Dini derivative, the Caputo fractional-order derivative is generalized, and its corresponding properties are also generalized. In Section 4, a new fractional-order impulsive differential inequality is proposed; by analyzing the inequality, the estimation of the solution of the inequality is obtained. In Section 5, the criteria for FOFCNNs to achieve exponential quasi-synchronization are derived. In order to prove the effectiveness of the results of this paper, some simulation examples are given in Section 6. Section 7 is a summary of the content of this paper and the prospects for future work.

2. Preliminaries and Problem Formulation

This section reviews the definition of fractional-order derivatives, and the definition and related properties of the Mittag–Leffler function are introduced. Related models, assumptions and lemmas are also introduced in this section.

2.1. Preliminaries of Fractional-Order Calculus

Definition 1

([34]). For a Lebesgue-integrable function ψ: $[a,b]\to \mathbb{R}$, the fractional-order integral of order $\alpha \in (0,+\infty )$ is defined by

$${}_a\mathcal{I}_t^{\alpha }\psi \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-\tau )}^{\alpha -1}\psi \left(\tau \right)\mathrm{d}\tau .$$

Definition 2

([34]). Let function ψ: $[a,b]\to \mathbb{R}$ be differentiable; the Caputo fractional-order derivative of order $\alpha \in (0,1)$ for ψ is defined as

$${}_{t_0}{}^C\mathcal{D}_t^{\alpha }\psi \left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_{t_0}^t{\displaystyle \frac{\dot{\psi }\left(\tau \right)}{{(t-\tau )}^{\alpha }}}\mathrm{d}\tau .$$

Remark 1.

The Caputo fractional-order derivative defined by Definition 2 requires that the function to be derived must be differentiable, which limits the application of the Caputo fractional-order derivative. For example, for the absolute value function, the Caputo fractional-order derivative can not be used, which greatly limits the research on the consistency of FOCNNs. In Section 3, the Dini derivative, which can relax this constraint, will be introduced.

In the study of fractional-order differential equations, the Mittag–Leffler function not only provides the basis of the theoretical framework, but also plays an irreplaceable role in solving and approximating these equations. In the following, the definition and properties of the Mittag–Leffler function are introduced.

Definition 3

([29]). The two-parameter Mittag–Leffler function is defined by

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}},$$

where $\alpha >0$, $\beta >0$, $z\in \mathbb{C}$.

For $\beta =1$, its one-parameter form is

$$E_{\alpha }=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+1)}}=E_{\alpha ,1}\left(z\right).$$

The Laplace transform is an important tool for studying fractional-order differential equations. Next, the Laplace transform formulas of the Mittag–Leffler function are introduced and the monotonicity of the Mittag–Leffler function will be given in special cases.

Lemma 1

([35]). For $\forall p_i,r_i\in \mathbb{R}$, $n\in \mathbb{N}$, the following statements are correct:

1. 

if $\beta =\alpha$, then the following equality holds:

$${\mathcal{L}}^{-1}\left[\sum _{i=1}^n{\displaystyle \frac{r_i}{s^{\alpha }+p_i}}\right]=\sum _{i=1}^n{r}_i{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-p_i{t}^{\alpha }),$$

2. 

if $\beta =\alpha +1$, the following equality holds:

$${\mathcal{L}}^{-1}\left[\sum _{i=1}^n{\displaystyle \frac{r_i}{s(s^{\alpha }+p_i)}}\right]=\sum _{i=1}^n{r}_i{t}^{\alpha }{E}_{\alpha ,\alpha +1}(-p_i{t}^{\alpha })=\sum _{i=1}^n{\displaystyle \frac{r_i}{p_i}}[1-E_{\alpha }(-p_i{t}^{\alpha })].$$

Lemma 2

([36]). Let $0<\alpha <1$ and $t\ge t_0$, then function $E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)$ is non-negative and

1. 

$E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)$ is monotonically non-increasing and $0\le E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)\le 1$ for $t\ge t_0$ when $\mu \le 0$;

2. 

$E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)$ is monotonically non-decreasing and $E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)\ge 1$ for $t\ge t_0$ when $\mu \ge 0$.

2.2. System Model and Assumptions

Next, the specific model of FOFCNNs is introduced and the related system properties are explained.

Considering the following m-dimensional FOFCNNs [31] as:

$$\begin{array}{cc}\hfill {}_{t_n}{}^C\mathcal{D}_t^{\alpha }{x}_i\left(t\right)=& -c_i{x}_i\left(t\right)+\sum _{k=1}^m{a}_{ik}{f}_k\left(x_k\left(t\right)\right)+\sum _{k=1}^m{b}_{ik}{v}_k\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(x_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigwedge }}{w}_{ik}{v}_k+\underset{k=1}{\overset{m}{\bigvee }}{q}_{ik}{v}_k+{\tilde{I}}_i,\hfill \end{array}$$

(1)

under the initial conditions $x_i\left(t_0\right)={\psi }_i$, where $i\in I\triangleq \{1,2,\dots ,m\}$, $x_i\left(t\right)$ represents the state variable of the $i\mathrm{th}$ neuron at time t. $a_{ik}$, $b_{ik}$ denote the elements of the fuzzy feedback templates, $c_i$ represents the passive decay rate of the $i\mathrm{th}$ neuron; ⋀ and ⋁, denote the fuzzy AND and fuzzy OR operations, ${\alpha }_{ik}$ and ${\beta }_{ik}$ denote the fuzzy minimum and maximum feedback templates, respectively; $w_{ik}$ and $q_{ik}$ denote the fuzzy minimum and maximum feed-forward templates, respectively. ${\tilde{I}}_i$ and $v_i$ denote the bias and input of $i\mathrm{th}$ neuron, $f_k$ and $h_k$ represent the activation functions and interaction functions of the $k\mathrm{th}$ neurons at time t, and $\{t_n,n\in {\mathbb{N}}^{+}\}$ is the impulsive sequence. Here, $\mathit{\psi }\left(t\right)={({\psi }_1\left(t\right),{\psi }_2\left(t\right),\dots ,{\psi }_m\left(t\right))}^T\in {\mathbb{R}}^m$ represents the Banach space of all continuous functions with $p-\mathrm{norm}$ ($p>0$ is an integer) given as

$${\left|\right|\mathit{\psi }\left|\right|}_p={\left[\underset{t\in \mathbb{N}}{sup}\sum _{i=1}^m{\left|{\psi }_i\left(t\right)\right|}^p\right]}^{{\displaystyle \frac{1}{p}}}.$$

(2)

Definition 4

([29]). Let $r>0$ and

$$\chi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{r}},\hfill & 0\le t<r\hfill \\ 0,\hfill & t\ge r,\hfill \end{array}\right.$$

the limiting form of $\chi \left(t\right)$, denoted by $\delta \left(t\right)$, is called the $Dirac$ delta function, delta impulse, Dirac impulse, or unit impulse, that is, $\delta \left(t\right)=\underset{r\to 0}{lim}\chi \left(t\right)$.

This paper considers system (1) as the drive system and the response system is

$$\begin{array}{cc}\hfill {}_{t_n}{}^C\mathcal{D}_t^{\alpha }{y}_i\left(t\right)=& -{\tilde{c}}_i{y}_i\left(t\right)+\sum _{k=1}^m{\tilde{a}}_{ik}{f}_k\left(y_k\left(t\right)\right)+\sum _{k=1}^m{\tilde{b}}_{ik}{v}_k\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigwedge }}{w}_{ik}{v}_k+\underset{k=1}{\overset{m}{\bigvee }}{q}_{ik}{v}_k+{\tilde{I}}_i+u_i\left(t\right),\hfill \end{array}$$

(3)

under the initial conditions $y_i\left(t_0\right)={\phi }_i$, where ${\tilde{c}}_i$ represents the passive decay rate for the ith neuron, ${\tilde{a}}_{ik}$, ${\tilde{b}}_{ik}$ are the elements of the feed-back templates, and $u_i\left(t\right)$ is the control input.

Here, this paper assumes ${\theta }_i\left(t\right)=y_i\left(t\right)-x_i\left(t\right)$, $i\in I$, that is,

$$\mathit{\theta }\left(t\right)={[{\theta }_1\left(t\right),{\theta }_2\left(t\right),\dots ,{\theta }_m\left(t\right)]}^T,$$

denotes the error vector, and the error system is

$$\begin{array}{cc}\hfill {}_{t_n}{}^C\mathcal{D}_t^{\alpha }{\theta }_i\left(t\right)=& -{\tilde{c}}_i{\theta }_i\left(t\right)+\sum _{k=1}^m{\tilde{a}}_{ik}\left(f_k\left(y_k\left(t\right)\right)-f_k\left(x_k\left(t\right)\right)\right)+\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\left(t\right)\right)\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(x_k\left(t\right)\right)\hfill \\ \hfill & +g_i\left(x_i\left(t\right)\right)+u_i\left(t\right),\hfill \end{array}$$

(4)

where $g_i\left(x_i\left(t\right)\right)=\Delta c_i{x}_i\left(t\right)+{\displaystyle \sum _{k=1}^m}\Delta a_i{f}_k\left(x_k\left(t\right)\right)$, $\Delta c_i={\tilde{c}}_i-c_i$ and $\Delta a_i=a_{ik}-{\tilde{a}}_{ik}$.

Using Definition 4, the impulsive controller $u_i\left(t\right)$ is defined by

$$u_i\left(t\right)=-d_i\delta (t-t_n)\times {\theta }_i\left(t_n\right),$$

(5)

in which $i=1,2,\dots ,m$, $d_i\ge 0$ is the control gain.

Definition 5

([30]). The average impulsive interval of the impulsive sequence $\{t_n,n\in {\mathbb{N}}^{+}\}$ is equal to $T_a$ if there is a non-negative number $N_0$ and a positive number $T_a$ such that

$${\displaystyle \frac{t-t_0}{T_a}}-N_0\le N(t,t_0)\le {\displaystyle \frac{t-t_0}{T_a}}+N_0,$$

where $N(t,t_0)$ denotes the number of impulsive times of the impulsive sequence on the interval $(t_0,t)$.

Assumption 1.

The average impulsive interval of the impulsive sequence is $(t_{n-1},t_n)$ and there are two constants $0<T_{min}<T_{max}<+\infty$ such that $T_{min}\le t_n-t_{n-1}\le T_{max}$ for all $n\in {\mathbb{N}}^{+}$.

Note that:

$${}_{t_n}{}^C\mathcal{D}_t^{\alpha }{\theta }_i\left(t\right)={\sigma }_i\left(t\right)+u_i\left(t\right),$$

(6)

where

$$\begin{array}{cc}\hfill {\sigma }_i\left(t\right)=& -{\tilde{c}}_i{\theta }_i\left(t\right)+\sum _{k=1}^m{\tilde{a}}_{ik}\left(f_k\left(y_k\left(t\right)\right)-f_k\left(x_k\left(t\right)\right)\right)+\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\left(t\right)\right)\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(x_k\left(t\right)\right)\hfill \\ \hfill & +g_i\left(x_i\left(t\right)\right).\hfill \end{array}$$

In order to better analyze the impulsive system, similar to the method in  [29], we can use the following lemma to transform the impulsive differential equality into the impulsive differential equation.

Lemma 3

([29]). This paper considers the following fractional-order controlled system

$$\left\{\begin{array}{cc}{}_{t_n}{}^C\mathcal{D}_t^{\alpha }z\left(t\right)=f(t,z\left(t\right))+u\left(t\right),\hfill & t_n<t\le t_{n+1}\hfill \\ u\left(t\right)=I_{n+1}\left(z\left(t_{n+1}\right)\right)\delta (t-t_{n+1}),\hfill & n\in \mathbb{N},\hfill \end{array}\right.$$

(7)

where δ is the function defined in Definition 4, $t_n<t_{n+1}$ for each $n\in {\mathbb{N}}^{+}$ and $\underset{n\to +\infty }{lim}{t}_n=+\infty$, $I_n:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is called the impulsive function satisfying $I_n\left(0\right)=0$ for $n\in {\mathbb{N}}^{+}$.

Then, the above system (7) can be rewritten as the following impulsive differential equations:

$$\left\{\begin{array}{cc}{}_{t_n}{}^C\mathcal{D}_t^{\alpha }z\left(t\right)=f(t,z\left(t\right)),\hfill & t_n<t\le t_{n+1}\hfill \\ \Delta z\left(t_{n+1}\right)=z\left(t_{n+1}^{+}\right)-z\left(t_{n+1}\right)={\displaystyle \frac{I\left(z\left(t_{n+1}\right)\right)}{\Gamma (\alpha +1)}},\hfill & n\in \mathbb{N},\hfill \end{array}\right.$$

where $z\left(t_0\right)=z\left(t_0^{+}\right)$ and $z\left(t_n^{-}\right)=z\left(t_n\right)$ for $n\in {\mathbb{N}}^{+}$.

According to Lemma 3, the error system (6) can be rewritten as follows:

$$\left\{\begin{array}{cc}{}_{t_n}{}^C\mathcal{D}_t^{\alpha }{\theta }_i\left(t\right)={\sigma }_i\left(t\right),\hfill & t_n<t\le t_{n+1}\hfill \\ \Delta {\theta }_i\left(t_{n+1}\right)={\theta }_i\left(t_{n+1}^{+}\right)-{\theta }_i\left(t_{n+1}\right)=-{\displaystyle \frac{d_i}{\Gamma (\alpha +1)}}{\theta }_i\left(t_{n+1}\right),\hfill & n\in \mathbb{N}.\hfill \end{array}\right.$$

(8)

Assumption 2.

The neuron activation functions $f_k$, $h_k$ have Lipschitz continuity, then $\forall k\in I$, $\exists L_k>0$ and $H_k>0$, such that

$$|f_k\left(x\right)-f_k\left(y\right)|\le L_k|x-y|,\forall x,y\in \mathbb{R},$$

$$|h_k\left(x\right)-h_k\left(y\right)|\le H_k|x-y|,\forall x,y\in \mathbb{R}.$$

Assumption 3.

Suppose that $x\left(t\right)$ is bounded, then $\forall t>0$ and $p\in {\mathbb{N}}^{+}$, $\exists M>0$, such that

$${\left|\right|x\left(t\right)\left|\right|}_p\le M,\forall t>0.$$

Remark 2.

Therefore, because of the continuity of $f_i$, there is ${\overline{M}}_i>0$ in the interval $(t_n,t_{n+1}]$, $i\in I$, and there is

$$\left|g_i\left(x_i\left(t\right)\right)\right|\le {\overline{M}}_i,\forall t\in (t_n,t_{n+1}].$$

Assumption 4.

Suppose the input of the ith neuron is bounded. Then, $\forall i\in I$, $\exists {\tilde{M}}_i>0$, such that

$$\left|v_i\right|\le {\tilde{M}}_i,\forall i\in I.$$

The following result is significant in fuzzy systems. Although it has been proved in [33], there is another way to prove it.

x and y are the i-th neurons of system 1 and 2, respectively.

Lemma 4.

If $x_i$ and $y_i$ are the ith neurons of the FOFCNN (1) and (3), respectively, $i=1,2,\cdots ,m$, then one has

$$\left|\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\right)-\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\right)\right|\le \sum _{k=1}^m\left|{\alpha }_{ik}\right|\left|h_k\left(x_k\right)-h_k\left(y_k\right)\right|,$$

$$\left|\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(x_k\right)-\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\right)\right|\le \sum _{k=1}^m\left|{\beta }_{ik}\right|\left|h_k\left(x_k\right)-h_k\left(y_k\right)\right|,$$

where the meaning of each symbol is the same as that in (1).

Proof. 

First, proving the first inequality. According to the definition of the fuzzy intersection operator, suppose $\exists j,l\in I$, such that

$$\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\right)=\underset{1\le k\le m}{min}\left\{{\alpha }_{ik}{h}_k\left(x_k\right)\right\}={\alpha }_{ij}{h}_j\left(x_j\right),$$

$$\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\right)=\underset{1\le k\le m}{min}\left\{{\alpha }_{ik}{h}_k\left(y_k\right)\right\}={\alpha }_{il}{h}_l\left(y_l\right).$$

It may be assumed here that ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(y_l\right)$, then we have:

Case

1: ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(x_l\right)\le {\alpha }_{il}{h}_l\left(y_l\right)$,

By the definition of ${\alpha }_{il}{h}_l\left(y_l\right)$: ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(x_l\right)\le {\alpha }_{il}{h}_l\left(y_l\right)\le {\alpha }_{ij}{h}_j\left(y_j\right)$,

and we have

$$\begin{array}{cc}\hfill \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|& \le \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|\hfill \\ \hfill & \le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\};\hfill \end{array}$$

Case

2: ${\alpha }_{il}{h}_l\left(y_l\right)\le {\alpha }_{il}{h}_l\left(x_l\right)\le {\alpha }_{ij}{h}_j\left(y_j\right)$,

By the definition of ${\alpha }_{il}{h}_l\left(y_l\right)$: ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(y_l\right)\le {\alpha }_{il}{h}_l\left(x_l\right)\le {\alpha }_{ij}{h}_j\left(y_j\right)$,

and we have:

$$\begin{array}{cc}\hfill \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|& \le \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|\hfill \\ \hfill & \le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\};\hfill \end{array}$$

Case

3: ${\alpha }_{il}{h}_l\left(x_l\right)\ge {\alpha }_{ij}{h}_j\left(y_j\right)$,

By the definition of ${\alpha }_{il}{h}_l\left(y_l\right)$: ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(y_l\right)\le {\alpha }_{ij}{h}_j\left(y_j\right)\le {\alpha }_{il}{h}_l\left(x_l\right)$,

and we can obtain:

$$\begin{array}{cc}\hfill \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|& \le \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|\hfill \\ \hfill & \le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\}.\hfill \end{array}$$

To sum up, when ${\alpha }_{ij}{h}_j\left(x_j\right)\le {\alpha }_{il}{h}_l\left(y_l\right)$, there is:

$$\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\}.$$

Similarly, when ${\alpha }_{il}{h}_l\left(y_l\right)\le {\alpha }_{ij}{h}_j\left(x_j\right)$, there is:

$$\begin{array}{cc}\hfill \left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|& \le \left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\hfill \\ \hfill & \le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\},\hfill \end{array}$$

then,

$$\begin{array}{cc}\hfill \left|\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(x_k\right)-\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\right)\right|& =\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\hfill \\ \hfill & \le max\left\{\left|{\alpha }_{ij}{h}_j\left(x_j\right)-{\alpha }_{ij}{h}_j\left(y_j\right)\right|,\left|{\alpha }_{il}{h}_l\left(x_l\right)-{\alpha }_{il}{h}_l\left(y_l\right)\right|\right\}\hfill \\ \hfill & \le \sum _{k=1}^m\left|{\alpha }_{ik}\right|\left|h_k\left(x_k\right)-h_k\left(y_k\right)\right|.\hfill \end{array}.$$

This conclusion is proved.    □

3. Generalized Caputo Fractional-Order Derivative

In Remark 1, the limitation of the Caputo derivative has been analyzed. In order to relax the requirement of the derivative function, the generalized Caputo fractional-order derivative is introduced and the related properties are generalized in this section. At the same time, with regard to the generalized Caputo derivative, a very important lemma is introduced, which is about the estimation of the solution of the generalized fractional-order differential inequality.

Definition 6

([37]). For $0<\alpha <1$, the generalized Caputo fractional-order derivative is defined as

$${}_{t_0}{}^C\mathcal{D}_t^{{\alpha }^{+}}\psi \left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_{t_0}^t{\displaystyle \frac{D^{+}\psi \left(\tau \right)}{{(t-\tau )}^{\alpha }}}\mathrm{d}\tau ,$$

in which $D^{+}\psi \left(t\right)=lim\underset{h\to 0^{+}}{sup}{\displaystyle \frac{\psi (t+h)-\psi \left(t\right)}{h}}$ is the upper right-hand Dini derivative of $\psi \left(t\right)$.

Remark 3.

In this definition, the so-called Dini derivative of the function is the left/right limit of the upper/lower supremum of this function, which usually exists, even if the function is not differentiable. Therefore, the Caputo fractional-order derivative defined by the Dini derivative reduces the dependence on the differentiability of the function and has been used in the analysis of various fractional-order systems [31,38,39,40].

In order to better apply the generalized Caputo fractional-order derivative in system analysis, some of its basic properties are introduced.

Lemma 5

([29]). Let $f\left(t\right)$ be a continuous and differential function on an interval $I\subseteq \mathbb{R}$, then

$${}_{t_0}{}^C\mathcal{D}_t^{{\alpha }^{+}}\left|f\left(t\right)\right|\le \mathrm{sign}\left(f\left(t\right)\right){}_{t_0}{}^C\mathcal{D}_t^{\alpha }f\left(t\right),$$

where $t_0$, $t\in I$, and $t\ge t_0$.

Lemma 6

([29]). Let function $f\left(t\right)$ be continuous on an interval $I\subseteq \mathbb{R}$. Then, for $t_0\in I$ and $0<\alpha <1$

$${}_{t_0}\mathcal{I}_t^{\alpha }{}_{t_0}{}^C\mathcal{D}_t^{{\alpha }^{+}}f\left(t\right)=f\left(t\right)-f\left(t_0\right),$$

where $t\in I$ and $t\ge t_0$.

Lemma 7.

Let $V\left(t\right)$ be a continuous and non-negative function defined on $[t_0,T]$ and which satisfies

$${}_{t_0}{}^C\mathcal{D}_t^{{\alpha }^{+}}V\left(t\right)\le \rho V\left(t\right)+\lambda ,$$

where $0<\alpha <1$, $T\ge t\ge t_0$, and ρ are constants. Then,

$$V\left(t\right)\le V\left(t_0\right)E_{\alpha }\left(\rho {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\rho }}[1-E_{\alpha }\left(\rho {(t-t_0)}^{\alpha }\right)].$$

Remark 4.

The above results are also valid for Caputo fractional-order derivatives, because the results of the integration of these two kinds of fractional-order derivative are the same, and the proof process is roughly the same, which is not proved further in detail here. And if $\lambda =0$, then the result in [29] can also be given. Namely, let $V\left(t\right)$ be a continuous and non-negative function defined on $[t_0,T]$ and which satisfies

$${}_{t_0}{}^C\mathcal{D}_t^{{\alpha }^{+}}V\left(t\right)\le \rho V\left(t\right),$$

where $0<\alpha <1$, $T\ge t\ge t_0$, and ρ are constants. Then,

$$V\left(t\right)\le V\left(t_0\right)E_{\alpha }\left(\rho {(t-t_0)}^{\alpha }\right).$$

It shows that this is the special case of Lemma 7 when $\lambda =0$. In this paper, the case of parameters mismatch is considered, so $\lambda \ne 0$, which makes the proof process more complicated, but the obtained result is more general and practical than the conclusion in [29].

4. A Novel Impulsive Fractional-Order Differential Inequality

In this section, a new fractional-order impulsive differential inequality is proposed. By analyzing the solution of this inequality, the solution of this differential inequality can be estimated by the initial value condition, so it can be applied to the exponential quasi-synchronization analysis of FOFCNNs.

Theorem 1.

Under the premise of Assumption 1, suppose that $V\left(t\right)$ is a piecewise continuous and non-negative function which satisfies the following inequalities:

$$\left\{\begin{array}{cc}{}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}V\left(t\right)\le \mu V\left(t\right)+\lambda ,\hfill & t_n<t\le t_{n+1}\hfill \\ V\left(t_{n+1}^{+}\right)\le \ell V\left(t_{n+1}\right),\hfill & n\in \mathbb{N},\hfill \end{array}\right.$$

(9)

where $V\left(t_0^{+}\right)=V\left(t_0\right)$, $0<\alpha <1$, $\mu \in \mathbb{R}$, $\ell \in {\mathbb{R}}^{+}$, $\lambda \ge 0$. Then, the following statement is true:

1. 

If $\mu \le 0$ and $\ell a<1$, then $V\left(t\right)\le {(\ell a)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln(\ell a)}{T_a}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell a)}^{N_0}{\mathrm{e}}^{\frac{ln(\ell a)}{T_a}(t-t_0)}}{1-\ell a}}+b$,

2. 

If $\mu >0$ and $\ell \overline{a}<1$, then $V\left(t\right)\le \left[\overline{a}{(\ell \overline{a})}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln(\ell \overline{a})}{T_a}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell \overline{a})}^{N_0}{\mathrm{e}}^{\frac{ln(\ell \overline{a})}{T_a}(t-t_0)}}{1-\ell \overline{a}}}+b$,

where $a=E_{\alpha }\left(\mu T_{min}^{\alpha }\right)<1$, $\overline{a}=E_{\alpha }\left(\mu T_{max}^{\alpha }\right)>1$, $b={\displaystyle \frac{\lambda }{\mu }}[E_{\alpha }\left(\mu T_{max}^{\alpha }\right)-1]>0$, and $N_0$ are positive integers defined by Definition 5.

Proof. 

On the one hand, based on Lemma 7, for $\forall t\ge t_0$

$$V\left(t\right)\le V\left(t_0\right)E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)].$$

For $\forall t\ge t_1$, $\exists \overline{n}\in {\mathbb{N}}^{+}$ such that $t_{\overline{n}}<t\le t_{\overline{n}+1}$, with Assumption 1 and Lemma 2.

When $\mu \le 0$, for $t\ge t_0$, $E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)\le 1$; so, we have:

$$\begin{array}{cc}\hfill V\left(t\right)& \le V\left(t_{\overline{n}}^{+}\right)E_{\alpha }\left(\mu {(t-t_{\overline{n}})}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu {(t-t_{\overline{n}})}^{\alpha }\right)]\hfill \\ \hfill & \le V\left(t_{\overline{n}}^{+}\right)-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu T_{max}^{\alpha }\right)]\hfill \\ \hfill & =V\left(t_{\overline{n}}^{+}\right)+b,\hfill \end{array}$$

where $b=-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu T_{max}^{\alpha }\right)]>0$.

On the other hand, for $\forall n\in {\mathbb{N}}^{+}$,

$$\begin{array}{cc}\hfill V\left(t_n^{+}\right)& \le \ell V\left(t_n\right)\hfill \\ \hfill & \le \ell [V\left(t_{n-1}^{+}\right)E_{\alpha }\left(\mu {(t_n-t_{n-1})}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\mu }}\left(1-E_{\alpha }\left(\mu {(t_n-t_{n-1})}^{\alpha }\right)\right)]\hfill \\ \hfill & \le \ell [aV\left(t_{n-1}^{+}\right)+b],\hfill \end{array}$$

where $a=E_{\alpha }\left(\mu T_{min}^{\alpha }\right)<1$.

Hence, by the use of recursion

$$\begin{array}{cc}\hfill V\left(t_{\overline{n}}^{+}\right)& \le \ell [aV\left(t_{\overline{n}-1}^{+}\right)+b]\hfill \\ \hfill & \le a\ell [a\ell V\left(t_{\overline{n}-2}^{+}\right)+b\ell ]+\ell b\hfill \\ \hfill & \le a^{\overline{n}}{\ell }^{\overline{n}}V\left(t_0\right)+{\ell }^{\overline{n}}{a}^{\overline{n}-1}b+\cdots +{\ell }^{2}ab+\ell b\hfill \\ \hfill & ={\ell }^{\overline{n}}{a}^{\overline{n}}V\left(t_0\right)+b{\displaystyle \frac{\ell -{\ell }^{\overline{n}+1}{a}^{\overline{n}}}{1-\ell a}}\hfill \\ \hfill & ={\mathrm{e}}^{\overline{n}ln(\ell a)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{\mathrm{e}}^{\overline{n}ln(\ell a)}}{1-\ell a}}+b.\hfill \end{array}$$

Based on the Definition 5, there exists a positive integer $N_0$ such that for any $t_{\overline{n}}<t\le t_{\overline{n}+1}$,

$${\displaystyle \frac{t-t_0}{T_a}}-N_0\le \overline{n}\le {\displaystyle \frac{t-t_0}{T_a}}+N_0,$$

then by $\ell a=\ell E_{\alpha }\left(\mu T_{min}^{\alpha }\right)<1$, there is:

$$V\left(t\right)\le {(\ell a)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln(\ell a)}{T_a}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell a)}^{N_0}{\mathrm{e}}^{\frac{ln(\ell a)}{T_a}(t-t_0)}}{1-\ell a}}+b.$$

When $\mu >0$, for $t\ge t_0$, $E_{\alpha }\left(\mu {(t-t_0)}^{\alpha }\right)\ge 1$, there is:

$$\begin{array}{cc}\hfill V\left(t\right)& \le V\left(t_{\overline{n}}^{+}\right)E_{\alpha }\left(\mu {(t-t_{\overline{n}})}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu {(t-t_{\overline{n}})}^{\alpha }\right)]\hfill \\ \hfill & \le V\left(t_{\overline{n}}^{+}\right)E_{\alpha }\left(\mu T_{max}^{\alpha }\right)-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu T_{max}^{\alpha }\right)]\hfill \\ \hfill & =\overline{a}V\left(t_{\overline{n}}^{+}\right)+b,\hfill \end{array}$$

where $\overline{a}=E_{\alpha }\left(\mu T_{max}^{\alpha }\right)>1$, $b=-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu T_{max}^{\alpha }\right)]>0$.

Based on $\ell \overline{a}=\ell E_{\alpha }\left(\mu T_{max}^{\alpha }\right)<1$, in a similar way, $\exists N_0\in {\mathbb{N}}^{+}$, such that:

$$V\left(t\right)\le \left[\overline{a}{(\ell \overline{a})}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln(\ell \overline{a})}{T_a}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell \overline{a})}^{N_0}{\mathrm{e}}^{\frac{ln(\ell \overline{a})}{T_a}(t-t_0)}}{1-\ell \overline{a}}}+b.$$

Finally, the conclusion is proved.    □

Remark 5.

By using the recursive method to prove Theorem 1, we can see that there are still similar results in the general Caputo fractional-order derivative and the integer-order derivative, which can be guaranteed by the fact that the Remark 4 and the integer order is a special fractional order. At the same time, Theorem 2 in [29] is a special case of the above theorem when $\lambda =0$.

The result of Theorem 1 is obtained when the impulsive sequence is not periodic. As a special case, the result that the impulsive sequence is periodic can still be obtained, which is expressed in the following corollary:

Corollary 1.

The impulsive sequence is $\{t_n,n\in {\mathbb{N}}^{+}\}$ and there is a constant $T<+\infty$ such that $t_n-t_{n-1}=T$ for all $n\in \mathbb{N}$. Suppose that $V\left(t\right)$ is a piecewise continuous and non-negative function which satisfies the inequalities (9) and the corresponding conditions. Then, the following statements are true:

1. 

If $\mu \le 0$ and $\ell a<1$, then $V\left(t\right)\le {(\ell a)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln(\ell a)}{T}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell a)}^{N_0}{\mathrm{e}}^{\frac{ln(\ell a)}{T}(t-t_0)}}{1-\ell a}}+b$,

2. 

If $\mu >0$ and $\ell a<1$, then $V\left(t\right)\le \left[a{(\ell a)}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln(\ell a)}{T}}(t-t_0)}V\left(t_0\right)+b\ell {\displaystyle \frac{1-{(\ell a)}^{N_0}{\mathrm{e}}^{\frac{ln(\ell a)}{T}(t-t_0)}}{1-\ell a}}+b$,

where $a=E_{\alpha }\left(\mu T^{\alpha }\right)$ and $b=-{\displaystyle \frac{\lambda }{\mu }}[1-E_{\alpha }\left(\mu T^{\alpha }\right)]>0$, $N_0$ is a positive integer.

For the Theorem 1, using the norm to limit the function $V\left(t\right)$ when the $V\left(t\right)$ is a vector of error $\mathit{\theta }\left(t\right)$, the following theorem is right:

Theorem 2.

Under Assumption 1, if there exists a piece-wise continuous function $V(t,\mathit{\theta }(t\left)\right)$: $[t_0,+\infty )\times {\mathbb{R}}^m\to \mathbb{N}$ with $V(t,0)=0$ and constants ${\mathsf{ȷ}}_1>0$, ${\mathsf{ȷ}}_2>0$, $\mathsf{ı}>0$ $\hslash >0$, and $\lambda ,\eta \in {\mathbb{R}}^{+}$, such that

$${\mathsf{ȷ}}_1{\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}}\le V(t,\mathit{\theta }\left(t\right))\le {\mathsf{ȷ}}_2{\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}},$$

(10)

$${}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}V(t,\mathit{\theta }\left(t\right))\le {\hslash \left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}}+\lambda ,t_n<t\le t_{n+1},$$

(11)

$$V(t_{n+1}^{+},\mathit{\theta }\left(t_{n+1}^{+}\right))\le \eta V(t_{n+1},\mathit{\theta }\left(t_{n+1}\right)),n\in \mathbb{N},$$

(12)

then, the following statements are true:

1. 

If $\hslash \le 0$ and $\eta a<1$, then

$${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}}\le {\displaystyle \frac{1}{{\mathsf{ȷ}}_1}}{\left(\eta a\right)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln\left(\eta a\right)}{T_a}}(t-t_0)}V(t_0,\mathit{\theta }\left(t_0\right))+{\displaystyle \frac{1}{{\mathsf{ȷ}}_1}}b\eta {\displaystyle \frac{1-{\left(\eta a\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\eta a\right)}{T_a}(t-t_0)}}{1-\eta a}}+{\displaystyle \frac{b}{{\mathsf{ȷ}}_1}},$$

2. 

If $\hslash >0$ and $\eta \overline{a}<1$, then

$${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}}\le {\displaystyle \frac{1}{{\mathsf{ȷ}}_1}}\left[\overline{a}{\left(\eta \overline{a}\right)}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln\left(\eta \overline{a}\right)}{T_a}}(t-t_0)}V(t_0,\mathit{\theta }\left(t_0\right))+{\displaystyle \frac{1}{{\mathsf{ȷ}}_1}}b\ell {\displaystyle \frac{1-{\left(\eta \overline{a}\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\eta \overline{a}\right)}{T_a}(t-t_0)}}{1-\eta \overline{a}}}+{\displaystyle \frac{b}{{\mathsf{ȷ}}_1}},$$

and the corresponding systems can realize exponential quasi-synchronization.

Proof. 

Based on (10) and (11), the following inequalities can be easily derived:

$$\left\{\begin{array}{c}{}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}V(t,\mathit{\theta }\left(t\right))\le {\displaystyle \frac{\hslash }{{\mathsf{ȷ}}_2}}V(t,\mathit{\theta }\left(t\right))+{\displaystyle \frac{\lambda }{{\mathsf{ȷ}}_2}},\hslash \le 0,n\in \mathbb{N}\hfill \\ {}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}V(t,\mathit{\theta }\left(t\right))\le {\displaystyle \frac{\hslash }{{\mathsf{ȷ}}_1}}V(t,\mathit{\theta }\left(t\right))+{\displaystyle \frac{\lambda }{{\mathsf{ȷ}}_1}},\hslash >0,n\in \mathbb{N}.\hfill \end{array}\right.$$

For $\forall t\ge t_1$, $\exists \overline{n}\in {\mathbb{N}}^{+}$, $t_{\overline{n}}<t\le t_{\overline{n}+1}$, based on Theorem 1 and Definition 5, we have:

If $\hslash \le 0$, then, when $\eta a=\eta E_{\alpha }\left({\displaystyle \frac{\hslash }{{\mathsf{ȷ}}_2}}{T}_{min}^{\alpha }\right)<1$, we have

$$V(t,\mathit{\theta }\left(t\right))\le {\left(\eta a\right)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln\left(\eta a\right)}{T_a}}(t-t_0)}V(t_0,\mathit{\theta }\left(t_0\right))+b\eta {\displaystyle \frac{1-{\left(\eta a\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\eta a\right)}{T_a}(t-t_0)}}{1-\eta a}}+b,$$

where $b=-{\displaystyle \frac{\lambda }{\hslash }}[1-E_{\alpha }\left({\displaystyle \frac{\hslash }{{\mathsf{ȷ}}_2}}{T}_{max}^{\alpha }\right)]$ and $N_0$ is a positive integer defined by Definition 5.

Similarly, if $\hslash >0$, then, when $\eta \overline{a}=\eta E_{\alpha }\left({\displaystyle \frac{\hslash }{{\mathsf{ȷ}}_1}}{T}_{max}^{\alpha }\right)<1$, we have

$$V(t,\mathit{\theta }\left(t\right))\le \left[\overline{a}{\left(\eta \overline{a}\right)}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln\left(\eta \overline{a}\right)}{T_a}}(t-t_0)}V(t_0,\mathit{\theta }\left(t_0\right))+b\ell {\displaystyle \frac{1-{\left(\eta \overline{a}\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\eta \overline{a}\right)}{T_a}(t-t_0)}}{1-\eta \overline{a}}}+b.$$

Based on (10),

$${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p^{\mathsf{ı}}\le {\displaystyle \frac{1}{{\mathsf{ȷ}}_1}}V(t,\mathit{\theta }\left(t\right)),$$

then the conclusions are proved.    □

Remark 6.

Theorem 2 generalizes the existed Lyapunov method for the stability analysis of impulsive fractional-order systems and plays an important role in synchronization analysis. Compared with the previous conclusions, the conclusion of Theorem 2 is more operable in practice.

5. Main Results

In this section, the exponential quasi-synchronization problem of FOFCNNs is analyzed. By providing a suitable impulsive controller to control the system, the conditions under which the system can achieve exponential quasi-synchronization are obtained.

Definition 7

([32]). It is said that a drive system (1) and response system (3) with impulsive control (5) can achieve exponential quasi-synchronization with an error bound $\delta \ge 0$ if there exist constants $M>0$ and $r>0$ such that

$${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_p\le M{\mathrm{e}}^{-r(t-t_0)}+\delta ,$$

for any $t\ge 0$; here, r is said to be the exponential convergence rate and $\mathit{\theta }\left(t\right)$ is the system error. In particular, a drive system (1) and response system (3) with impulsive control (5) are said to be exponentially synchronized when $\delta =0$.

In order to more clearly show the overall framework of this article, this article presents the flowchart below. From Figure 1, the key steps required for the processing system which are conducive to the application of this article and subsequent simulation can be clearly seen.

Now, based on the rewritten error system (8), the exponential quasi-synchronization problem of FOFCNNs under the controller (5) is studied. The results are as follows:

Theorem 3.

Let $\lambda ={\displaystyle \sum _{i=1}^m}\left\{|{\displaystyle \sum _{k=1}^m}\left({\tilde{b}}_{ik}-b_{ik}\right)v_k|+{\overline{M}}_i\right\}$ and $\varphi =\underset{i\in I}{max}\left\{1-{\displaystyle \frac{d_i}{\Gamma (\alpha +1)}}\right\}$, then under the Assumptions 1–4, drive-response systems (1) and (3) can achieve exponential quasi-synchronization with the impulsive controller (5) and the corresponding quasi-synchronization error bound δ can also be obtained if one of the following conditions holds:

1. 

If $\hslash \le 0$, and $\varphi E_{\alpha }(\hslash T_{min}^{\alpha })<1$;

2. 

If $\hslash >0$, and $\varphi E_{\alpha }(\hslash T_{max}^{\alpha })<1$,

where $\hslash =\underset{i=1,2,\dots ,n}{max}\left\{-{\tilde{c}}_i+{\displaystyle \sum _{k=1}^m}|{\tilde{a}}_{ik}|L_k+{\displaystyle \sum _{k=1}^m}\left(\right|{\alpha }_{ik}|+|{\beta }_{ik}\left|\right)H_k\right\}$,

and the corresponding upper bound of the quasi-synchronization errors is also estimated:

1. 

If $\hslash \le 0$, and $\varphi E_{\alpha }(\hslash T_{min}^{\alpha })<1$, then the quasi-synchronization error bound δ is ${\displaystyle \frac{b\varphi }{1-\varphi a}}+b$;

2. 

If $\hslash >0$, and $\varphi E_{\alpha }(\hslash T_{max}^{\alpha })<1$, then the quasi-synchronization error bound δ is ${\displaystyle \frac{b\varphi }{1-\varphi \overline{a}}}+b$.

Proof. 

Choose a candidate Lyapunov function as follows:

$$V\left(t\right)=\sum _{i=1}^m\left|{\theta }_i\left(t\right)\right|.$$

(13)

For any $t\in (t_n,t_{n+1}]$, $n\in \mathbb{N}$, taking the derivative of $V\left(t\right)$ along the trajectories of the error system (4) and using Lemma 5 with regard to the inequalities of the generalized Caputo derivatives:

$$\begin{array}{cc}\hfill {}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}V\left(t\right)& ={}_{t_n}{}^C\mathcal{D}_t^{{\alpha }^{+}}\sum _{i=1}^m\left|{\theta }_i\left(t\right)\right|\hfill \\ \hfill & \le \sum _{i=1}^m\mathrm{sign}\left({\theta }_i\left(t\right)\right)·{}_{t_n}{}^C\mathcal{D}_t^{\alpha }{\theta }_i\left(t\right)\hfill \\ \hfill & =\sum _{i=1}^m\mathrm{sign}\left({\theta }_i\left(t\right)\right)\left(-{\tilde{c}}_i{\theta }_i\left(t\right)+\sum _{k=1}^m{\tilde{a}}_{ik}\left(f_k\left(y_k\left(t\right)\right)-f_k\left(x_k\left(t\right)\right)\right)+\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k\right.\hfill \\ \hfill & \left.+\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k(x_k\left(t\right)+\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\left(t\right)\right)-\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(x_k\left(t\right)\right)+g_i\left(x_i\left(t\right)\right)\right)\hfill \\ \hfill & \le \sum _{i=1}^m\mathrm{sign}\left({\theta }_i\left(t\right)\right)\left(-{\tilde{c}}_i{\theta }_i\left(t\right)+\sum _{k=1}^m|{\tilde{a}}_{ik}|L_k|y_k\left(t\right)-x_k\left(t\right)|+\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k\right.\hfill \\ \hfill & \left.+\sum _{k=1}^m|{\alpha }_{ik}\left|\right|h_k\left(y_k\right)-h_k\left(x_k\right)|+\sum _{k=1}^m|{\beta }_{ik}\left|\right|h_k\left(y_k\left(t\right)\right)-h_k\left(x_k\left(t\right)\right)|+g_i\left(x_i\left(t\right)\right)\right)\hfill \\ \hfill & \le \sum _{i=1}^m\mathrm{sign}\left({\theta }_i\left(t\right)\right)\left(-{\tilde{c}}_i{\theta }_i\left(t\right)+\sum _{k=1}^m|{\tilde{a}}_{ik}|L_k\left|{\theta }_k\left(t\right)\right|+\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k\right.\hfill \\ \hfill & \left.+\sum _{k=1}^m|{\alpha }_{ik}|H_k|y_k-x_k|+\sum _{k=1}^m|{\beta }_{ik}|H_k|y_k-x_k|+g_i\left(x_i\left(t\right)\right)\right)\hfill \\ \hfill & \le \sum _{i=1}^m\left(-{\tilde{c}}_i+\sum _{k=1}^m|{\tilde{a}}_{ik}|L_k+\sum _{k=1}^m|{\alpha }_{ik}|H_k+\sum _{k=1}^m\left|{\beta }_{ik}\right|H_k\right)\left|{\theta }_i\left(t\right)\right|\hfill \\ \hfill & +\sum _{i=1}^m\left\{|\sum _{k=1}^m({\tilde{b}}_{ik}-b_{ik})v_k|+{\overline{M}}_i\right\}.\hfill \end{array}$$

Let

$$\hslash =\underset{i=1,2,\cdots ,m}{max}\left\{-{\tilde{c}}_i+\sum _{k=1}^m|{\tilde{a}}_{ik}|L_k+\sum _{k=1}^m|{\alpha }_{ik}|H_k+\sum _{k=1}^m\left|{\beta }_{ik}\right|H_k\right\},$$

$$\lambda =\sum _{i=1}^m\left\{|\sum _{k=1}^m\left({\tilde{b}}_{ik}-b_{ik}\right)v_k|+{\overline{M}}_i\right\},$$

so $\lambda$ is bounded based on Remark 2 and Assumption 4.

Then, the following form is obtained:

$${}_{t_n}{}^C\mathcal{D}_t^{\alpha }V\left(t\right)\le \hslash \sum _{i=1}^m\left|{\theta }_i\left(t\right)\right|+\lambda .$$

By the definition of p-norm in Formula (2), we have:

$${}_{t_n}{}^C\mathcal{D}_t^{\alpha }V\left(t\right)\le {\hslash \left|\right|\mathit{\theta }\left(t\right)\left|\right|}_1+\lambda .$$

(14)

When $\forall n\in \mathbb{N}$, it can be obtained from Equation (8):

$$V\left(t_{n+1}^{+}\right)=\sum _{i=1}^m|{\theta }_i\left(t_{n+1}^{+}\right)|=\sum _{i=1}^m\left(1-{\displaystyle \frac{d_i}{\Gamma (\alpha +1)}}\right)\left|{\theta }_i\left(t\right)\right|=\left(1-{\displaystyle \frac{d_i}{\Gamma (\alpha +1)}}\right)V\left(t_{n+1}\right).$$

Based on the definition of $\varphi$, we have:

$$V\left(t_{n+1}^{+}\right)\le \varphi V\left(t_{n+1}\right).$$

(15)

Then, from Theorem 2, let ${\mathsf{ȷ}}_1={\mathsf{ȷ}}_2=1$ and $\mathsf{ı}=p=1$, according to Formulas (14) and (15), we can obtain:

• If $\hslash \le 0$ and $\varphi a<1$, then ${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_1\le {\left(\varphi a\right)}^{-N_0}{\mathrm{e}}^{{\displaystyle \frac{ln\left(\varphi a\right)}{T_a}}(t-t_0)}V\left(t_0\right)+b\varphi {\displaystyle \frac{1-{\left(\varphi a\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\varphi a\right)}{T_a}(t-t_0)}}{1-\varphi a}}+b$ and the quasi-synchronization error bound $\delta$ is ${\displaystyle \frac{b\varphi }{1-\varphi a}}+b$;
• If $\hslash >0$ and $\varphi \overline{a}<1$, then, ${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_1\le \left[\overline{a}{\left(\varphi \overline{a}\right)}^{-N_0}\right]{\mathrm{e}}^{{\displaystyle \frac{ln\left(\varphi \overline{a}\right)}{T_a}}(t-t_0)}V\left(t_0\right)+b\varphi {\displaystyle \frac{1-{\left(\varphi \overline{a}\right)}^{N_0}{\mathrm{e}}^{\frac{ln\left(\varphi \overline{a}\right)}{T_a}(t-t_0)}}{1-\varphi \overline{a}}}+b$ and the quasi-synchronization error bound $\delta$ is ${\displaystyle \frac{b\varphi }{1-\varphi \overline{a}}}+b$

where $a=E_{\alpha }\left(\hslash T_{min}^{\alpha }\right)<1$, $\overline{a}=E_{\alpha }(\hslash T_{max}^{\alpha })>1$ and $b={\displaystyle \frac{\lambda }{\hslash }}[E_{\alpha }(\hslash T_{max}^{\alpha })-1]>0$ are constants.

Furthermore, when $\hslash \le 0$, and $\varphi a<1$, let $M={\left(\varphi a\right)}^{-N_0}V\left(t_0\right)-{\displaystyle \frac{b\varphi {\left(\varphi a\right)}^{N_0}}{1-\varphi a}}$, $r=-{\displaystyle \frac{ln\left(\varphi a\right)}{T_a}}$ and $\delta ={\displaystyle \frac{b\varphi }{1-\varphi a}}+b$, then

$${\left|\right|\mathit{\theta }\left(t\right)\left|\right|}_1\le M{\mathrm{e}}^{-r(t-t_0)}+\delta ,$$

so, by Definition 7, the system (1) and (3) can achieve exponential quasi-synchronization.

Or else, when $\hslash >0$, and $\varphi \overline{a}<1$, let $M=\overline{a}{\left(\varphi \overline{a}\right)}^{-N_0}V\left(t_0\right)-{\displaystyle \frac{b\varphi {\left(\varphi \overline{a}\right)}^{N_0}}{1-\varphi \overline{a}}}$, $r=-{\displaystyle \frac{ln\left(\varphi \overline{a}\right)}{T_a}}$ and $\delta ={\displaystyle \frac{b\varphi }{1-\varphi \overline{a}}}+b$, the conclusion is still valid.    □

Remark 7.

For the above results, we can see that when $\hslash \le 0$, the system itself can have exponential quasi-synchronization. At this time, the impulsive controller (5) acts as an impulsive interference with the exponential quasi-synchronization of the system, but finally, the system can still reach exponential quasi-synchronization. When $\hslash >0$, the system itself does not have exponential quasi-synchronization, but it can achieve exponential quasi-synchronization under the action of impulsive control.

Remark 8.

When the minimum impulsive interval of the system increases, $T_{min}$ becomes larger. When $\hslash \le 0$, $a=E_{\alpha }(\hslash T_{min}^{\alpha })$ does not increase at this time; that is, as a decreases with the increase in $T_{min}$, then ${\displaystyle \frac{b\varphi }{1-\varphi a}}+b$ decreases. That is, for the impulsive interference, the number of impulsive times decreases, and the exponential quasi-synchronization effect of the system is better. When $\hslash >0$, $\overline{a}=E_{\alpha }(\hslash T_{max}^{\alpha })$ and $\overline{a}$ is not reduced, so $\overline{a}$ increases with increase in $T_{max}$, then ${\displaystyle \frac{b\varphi }{1-\varphi \overline{a}}}+b$ increases; that is, for impulsive control, the number of impulsive times decreases, and the exponential quasi-synchronization effect of the system becomes weak. The above two cases are in line with common sense.

Obviously, if the impulsive time is periodic, which as a special case, using the Corollary 1, it can be proved that the system has exponential quasi-synchronization under the $\varphi E_{\alpha }\left(\mu T^{\alpha }\right)<1$. Otherwise, if the drive system (1) and response system (3) are a parameters match, that is, ${\tilde{c}}_i=c_i$, ${\tilde{a}}_{ik}=a_{ik}$ and ${\tilde{b}}_{ik}=b_{ik}$, at this point, it is easy to obtain $\lambda =0$, and then $b=0$, so $\delta =0$ in Theorem 3. That is, complete exponential synchronization is obtained and the result is expressed in the following corollary:

Corollary 2.

Considering system (1) as the drive system and let the response system be

$$\begin{array}{cc}\hfill {}_{t_n}{}^C\mathcal{D}_t^{\alpha }{y}_i\left(t\right)=& -c_i{y}_i\left(t\right)+\sum _{k=1}^m{a}_{ik}{f}_k\left(y_k\left(t\right)\right)+\sum _{k=1}^m{b}_{ik}{v}_k\hfill \\ \hfill & +\underset{k=1}{\overset{m}{\bigwedge }}{\alpha }_{ik}{h}_k\left(y_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigvee }}{\beta }_{ik}{h}_k\left(y_k\left(t\right)\right)+\underset{k=1}{\overset{m}{\bigwedge }}{w}_{ik}{v}_k+\underset{k=1}{\overset{m}{\bigvee }}{q}_{ik}{v}_k+{\tilde{I}}_i+u_i\left(t\right).\hfill \end{array}$$

(16)

Under Assumption 3 and Lemma 4 with the impulsive controller (5), the exponential synchronization between the drive system (1) and the response system (16) can be realized, if one of the following conditions holds:

1. 

$\hslash \le 0$, and $\varphi E_{\alpha }(\hslash T_{min}^{\alpha })<1$;

2. 

$\hslash >0$, and $\varphi E_{\alpha }(\hslash T_{max}^{\alpha })<1$,

where $=\underset{i=1,2,\dots ,m}{max}\left\{-c_i+{\displaystyle \sum _{k=1}^m}|a_{ik}|L_k+{\displaystyle \sum _{k=1}^m}\left(\right|{\alpha }_{ik}|+|{\beta }_{ik}\left|\right)H_k\right\}$, $\varphi =\underset{i\in I}{max}\{1-{\displaystyle \frac{d_i}{\Gamma (\alpha +1)}}\}$.

In order to better simulate the system and increase the readability of the article, the algorithm of Theorem 3 is shown in Algorithm 1.

Algorithm 1 The algorithm to implement the Theorem 3

Calculating the $L_k,H_k,M,{\tilde{M}}_i$ such that $f_k(·)$, $h_k(·)$ can satisfy the Assumption 1, and $x\left(t\right)$ and $v_i$ satisfy the Assumptions 3 and 4, respectively;

Input: Parameters of the drive-response systems (1) and (3) as follows:

constants $c_i,{\tilde{c}}_i,d_i,a_{ik},{\tilde{a}}_{ik},b_{ik},{\tilde{b}}_{ik},{\alpha }_{ik},{\beta }_{ik},w_{ik}$ and $q_{ik}$;

Step1: The Conversion of Impulsive Differential Equality

Using Lemma 3, the error system (4) can be transformed into the impulsive differential equation;

Step2: Impulsive Control Strategy

The Mittag–Leffler function and Gamma function are calculated by MATLAB (R2018b) to obtain $a\left(or\overline{a}\right)$, b and $\varphi$, then we can estimate the error $\delta$.

6. Numerical Simulation

In this section, two specific numerical simulations are given to verify the validity of the results.

Example 1.

Contemplating the FOFCNNs (1) and (3) using the given parameters: $m=2$, $\alpha =0.9$, $f_k\left(x_k\right)=sin\left(x_k\right)$, $h_k\left(x_k\right)=cos\left(x_k\right)$, $I_k=0$, $k=1,2$,

$$\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=\left(\begin{array}{c}0.1\\ 0.1\end{array}\right),\left(\begin{array}{c}{\tilde{c}}_1\\ {\tilde{c}}_2\end{array}\right)=\left(\begin{array}{c}0.11\\ 0.09\end{array}\right),\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right),$$

$$\left(\begin{array}{c}{d}_1\\ d_2\end{array}\right)=\left(\begin{array}{c}0.6\\ 1.2\end{array}\right),{\left(a_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),{\left({\tilde{a}}_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.9& 0\\ 0& 1.1\end{array}\right),$$

$${\left(b_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}1.1& 0.8\\ -1.3& 0.7\end{array}\right),{\left({\tilde{b}}_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}1& 0.7\\ -1.6& 0.75\end{array}\right),{\left({\alpha }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.1& -2\\ 0.3& -1.5\end{array}\right),$$

$${\left({\beta }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.2& 0.5\\ 0.4& 0.7\end{array}\right),{\left(w_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}-0.9& -0.2\\ -0.3& -1.5\end{array}\right),{\left(q_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.8& 0.4\\ 1.2& -3\end{array}\right).$$

Then, the constants in Assumption 2–4 and Remark 2 can be calculated as: $\forall i,k=1,2$, $L_k=H_k=1$, ${\tilde{M}}_i=0$, let $M=2$, then ${\overline{M}}_1=0.1200$, ${\overline{M}}_2=0.1200$, $\lambda =0.2400$, and $\varphi =0.3761$ and $\hslash =4.5900$ by the definition of ϕ and ℏ. Select $T_{min}=0.02$ and $T_{max}=0.1$.

When $\hslash =4.5900>0$, there are $\varphi E_{\alpha }(\hslash T_{max}^{\alpha })=0.6980<1$, $b=-{\displaystyle \frac{\lambda }{\hslash }}[1-E_{\alpha }(\hslash T_{max}^{\alpha })]=0.0447$ by Theorem 3, the above systems can achieve exponential quasi-synchronization and eventually, the system error will converge to δ, where $\delta ={\displaystyle \frac{b\varphi }{1-\varphi E_{\alpha }(\hslash T_{max}^{\alpha })}}+b=0.1003$. Select the impulsive period to be $T=0.5$ and the initial value to ${\psi }_1\left(0\right)=1$, ${\psi }_2\left(0\right)=-1$ and ${\phi }_1\left(0\right)=-1$, ${\phi }_2\left(0\right)=1$. The quasi-synchronous evolution of the system is shown in Figure 2; both the control and control outcomes are included in Figure 2a,b. It can be seen that after impulsive control, the divergent error system can converge to a region of no more than 0.03733, which is close to the error region $\delta =0.1003$ estimated by Theorem 3, which also shows that the selected impulsive controller has good performance. The state error of each system is shown in Figure 3; both the control and non-control outcomes are included in Figure 3a,b. It can be seen that the effect of the controller (5) on the state error of each system is obvious. Further, the trajectory of each subsystem is shown in Figure 4a, and the trajectory under the action of the controller is shown in Figure 4b.

Remark 9.

The results are still right when $\hslash <0$ by changing the parameters. In order to obtain the result of $\hslash <0$, some parameters are changed. The changed parameters are as follows:

$$\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=\left(\begin{array}{c}0.2\\ 0.1\end{array}\right),\left(\begin{array}{c}{\tilde{c}}_1\\ {\tilde{c}}_2\end{array}\right)=\left(\begin{array}{c}0.19\\ 0.11\end{array}\right),{\left(a_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.02& 0\\ 0& 0.01\end{array}\right),$$

$${\left({\tilde{a}}_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.021& 0\\ 0& 0.009\end{array}\right),{\left({\alpha }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}-0.01& -0.02\\ 0.025& -0.015\end{array}\right),{\left({\beta }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.03& 0.01\\ 0.015& 0.015\end{array}\right).$$

Then, $\hslash =-0.031<0$, there are $\lambda =0.042$, $E_{\alpha }(\hslash T_{min}^{\alpha })=0.9970$, $\varphi =0.3761$ and $\varphi E_{\alpha }(\hslash T_{min}^{\alpha })=0.3757<1$, so, $b=0.0014$; the above two FOFCNNs can achieve exponential quasi-synchronization by Theorem 3 and eventually, the system error will converge to δ, where $\delta ={\displaystyle \frac{b\varphi }{1-\varphi E_{\alpha }(\hslash T_{min}^{\alpha })}}+b=0.0022$.

When $\hslash =-0.031<0$, the system itself can achieve quasi-synchronization, as shown in Figure 5a, and for this system, the quasi-synchronization error does not exceed 0.01239, but it can be found that it takes a long time to achieve this process. However, after the controller (5) works, as shown in Figure 5b, the system can achieve quasi-synchronization faster, and the quasi-synchronization error can be controlled within the range of 0.002114, which shows that the controller (5) is very useful. The state error of each system is shown in Figure 6; both the control and non-control outcomes are included in Figure 6a,b. Comparing the two Figures, it can be found that the controller makes the state error of each system decrease rapidly. Further, the trajectory of each subsystem is shown in Figure 7a, and the trajectory under the action of the controller is shown in Figure 7b.

Figure 8 shows the impulsive inference on the system, which makes the system diverge. Comparing the two, Figure 8a and Figure 8b of Figure 8, it can be found that the system error can converge to a certain bounded value, but after the action of controller (5), the system error increases or even diverges. Of course, by comparing with Figure 5, it can be found that not all impulsive control will have a negative impact on the system, because this is related to the impulsive period T and the impulsive gain d.

Remark 10.

The existing synchronous control of FOFCNNs includes nonlinear control [41], nonlinear feedback control [42], adaptive control [43], sliding mode control [44] and so on. These control methods are based on continuous-time feedback, and the control cost is relatively high. The impulsive control designed in this paper is only controlled at some moments, and it is discontinuous control, which can greatly reduce the consumption of control resources. And impulsive control has low cost and simple operation, so has been widely considered and has developed rapidly in the control field.

By changing the parameters of the above systems, the system when the parameters are matched is obtained, and the results of the response can also be derived. The specific process is shown in Example 2.

Example 2.

Contemplating the FOFCNNs (1) and (3) with parameters match, that is, ${\tilde{c}}_i=c_i$, ${\tilde{a}}_{ik}=a_{ik}$ and ${\tilde{b}}_{ik}=b_{ik}$ and using the given parameters: $m=2$, $\alpha =0.9$, $f_k\left(x_k\right)=sin\left(x_k\right)$, $h_k\left(x_k\right)=cos\left(x_k\right)$, $I_k=0$, $k=1,2$,

$$\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=\left(\begin{array}{c}0.1\\ 0.1\end{array}\right),\left(\begin{array}{c}{d}_1\\ d_2\end{array}\right)=\left(\begin{array}{c}0.6\\ 1.2\end{array}\right),\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right),$$

$${\left(a_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),{\left(b_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}1.1& 0.8\\ -1.3& 0.7\end{array}\right),{\left({\alpha }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.1& -2\\ 0.3& -1.5\end{array}\right),$$

$${\left({\beta }_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.2& 0.5\\ 0.4& 0.7\end{array}\right),{\left(w_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}-0.9& -0.2\\ -0.3& -1.5\end{array}\right),{\left(q_{ik}\right)}_{2\times 2}=\left(\begin{array}{cc}0.8& 0.4\\ 1.2& -3\end{array}\right).$$

It is easy to obtain $\lambda =0$, and then $b=0$, so $\delta =0$ in Theorem 3. Let $\forall i,k=1,2$, $L_k=H_k=1$, ${\tilde{M}}_i=0$, let $M=2$, then ${\overline{M}}_1={\overline{M}}_2=0$, and $\varphi =0.3761$ and $\hslash =4.7000$ by the definition of ϕ and ℏ. Select the impulsive period to be $T=1$ and set the initial values to ${\psi }_1\left(0\right)=1$, ${\psi }_2\left(0\right)=-1$ and ${\phi }_1\left(0\right)=-1$, ${\phi }_2\left(0\right)=1$.

When $\hslash =4.7000>0$, there are $\varphi E_{\alpha }(\hslash T_{max}^{\alpha })=0.7088<1$, and the above two FOFCNNs can eventually achieve exponential synchronization. The corresponding errors and model trajectories are shown in Figure 9, Figure 10 and Figure 11. Among them, Figure 9 shows the evolution of the global error when the system is controlled and not controlled, and Figure 10 and Figure 11 show the evolution of the error and state of each subsystem, respectively. By comparison, the results reported in this paper are still valid when the parameters are matched. As a special result of parameters mismatch, this is obviously also in line with our cognition.

7. Conclusions

In this paper, the exponential quasi-synchronization of FOFCNNs with parameters mismatch under impulsive control was studied, and the corresponding criteria were given. Firstly, the generalized fractional-order derivative was introduced, and some properties were introduced, and different from other articles, based on the general fractional-order Caputo derivative, this paper mainly deduced a differential inequality, and the obtained result was more valuable than that in [29]. Secondly, the fractional-order impulsive control system was transformed into an impulsive differential equation, which was the classical method to deal with an impulsive control system. Then, based on some theoretic tools, the criterion of exponential quasi-synchronization of parameters mismatch systems was given, and the adaptability of the impulsive controller to various systems was proved.

The occurrence of Dos attacks is very widespread, such as in drone team performance, driverless cars and so on. How to resist such attacks encountered in the process of information transmission is a problem that we must pay attention to.This is especially so for fractional-order fuzzy cellular neural networks, because human cells can be invaded by viruses at any time, and in order to maintain the body’s function, the brain needs to make corresponding decisions. In order to better describe this process, it is necessary for the FOFCNN model proposed in this paper to consider the attack. In recent years, although research on fuzzy cellular neural networks has attracted more and more attention by researchers, due to the limitations of inequality derivation, there are few attack results in the case of parameters mismatch. In the research process, we can still use the impulsive control method that is proposed in [45], still use impulsive control, or use event-triggered control [46,47], intermittent control [48] and other methods. Interestingly, we can combine event-triggered control with impulsive control or intermittent control to further determine the trigger time and control time, which will be the focus of our future work.