Fixed-Time Synchronization of Reaction-Diffusion Fuzzy Neural Networks with Stochastic Perturbations

Abstract

In this paper, we investigated the fixed-time synchronization problem of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations by developing simple control schemes. First, some generalized fixed-time stability results are introduced for stochastic nonlinear systems. Based on these results, some generic fixed-time stability criteria are established and upper bounds of settling time are directly calculated by using several special functions. Then, the fixed-time synchronization of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations is analysed by designing a type of controller which is more simple and thus have a better applicability. Finally, one numerical example with its Matlab simulations is provided to show the feasibility of developed theoretical results.

1. Introduction

Distributed parameter systems are kind of infinite dimensional dynamical systems described by partial differential equations, inverse differential equations or integral equations in abstract spaces. For instance, pattern dynamics in biology, the movement of electrons in non-uniform magnetic fields in physics, etc. These diffusion phenomena undoubtedly effect the stability of the system. Therefore, reaction-diffusion systems have been investigated since the 1960s, and has become a hot research topic due to their amazing applications in environment system, mechanical system, population system, and other fields [1,2,3].

In recent years, neural networks have been successfully applied in different field of application such as signal processing, parallel computing, pattern classification and nonlinear functions approximation. But these applications rely heavily on the stability, passivity, and dissipativity of the considered system. As mentioned above, the phenomenon of reaction-diffusion is common in nature. Therefore, it is of great significance to analyse the dynamical behaviours of reaction-diffusion neural networks and its application. Till now, many interesting results have emerged regarding the dynamic behavior of reaction-diffusion neural networks. For instance, in [4], several stability and passivity criteria for reaction-diffusion neural networks were established by exploiting some inequality techniques. Ref. [5] studied the asymptotic stability of reaction-diffusion neural networks with time-varying delays. In addition to the reaction-diffusion term, the influence of stochastic perturbations on the stability of the neural networks can not be ignored. As a result, many interesting results have emerged regarding the dynamic behavior of neural networks with stochastic perturbations or reaction-diffusion terms [3,6]. In [3], authors studied the exponential stability of the periodic solutions of impulsive stochastic reaction-diffusion neural networks with mixed time delays. Ref. [6] investigated the stochastic stability of delayed neural networks with local impulses. Additionally, uncertainty and fuzziness are inevitable in application of nonlinear dynamics. In order to consider the fuzziness, fuzzy cellular neural networks were proposed by Yang and Yang in 1986 [7,8]. Afterward, quite a few researchers have found that fuzzy neural networks can provide a useful paradigm for pattern recognition and image processing. Therefore, researchers have extensively considered the stability, synchronization and chaotic bifurcation of fuzzy neural networks with or without time delays, and published many excellent papers, such as [9,10,11,12,13,14], etc.

It is well known that fixed-time (FXT) synchronization or stability means that the system is global finite-time stable, and that the settling time (ST) is independent of the initial conditions of the system and can be bounded for any initial values of systems [15,16]. Compared with classical asymptotic stability, the biggest advantage of FXT stability is that it allows the system to respond to constraints in a finite time, thereby improving product quality or obtaining more reliable anti-interference ability in engineering applications, such as sliding mode controller design [17] and robot control [18]. As a result, FXT stability and synchronization of various systems has been extensively studied such as discontinuous systems [19,20], fuzzy systems [21] and impulsive systems [22], etc. However, till now, there are very few works on the study of FXT synchronization of reaction-diffusion fuzzy neural networks with stochastic perturbations.

Motivated by above discussions, in this paper, we considered the FXT synchronization of reaction-diffusion fuzzy neural networks with stochastic perturbations. The main innovations of this paper are as follows: (1) For the first time, the FXT synchronization issue of neural networks with three main factors such as reaction diffusion, fuzzy term and stochastic perturbations is considered. (2) Compared to some early published works, the introduced controller in this paper is more simple since it has only consisted of the power-law terms and not include linear term (3) The upper bound of ST is estimated more accurately by using some special function approach.

The rest of the paper is structured as follows. In Section 2, some basic definitions, corollaries, and lemmas for FXT synchronization with stochastic perturbations are presented. Section 3 gives the main results on FXT synchronization of reaction-diffusion fuzzy neural networks with stochastic perturbations. Section 4 provides one numerical example to verify the validity of obtained results. A conclusion of the paper and an outlook on the next research steps are given in Section 5.

2. Preliminaries

In this paper, we consider the following model:

$$\begin{array}{c}\begin{array}{cc}\hfill d{\upsilon }_i(t,x)=& [\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{\upsilon }_i(t,x)}{\partial x_l^2}-c_i{\upsilon }_i(t,x)+\sum _{j=1}^n{a}_{ij}{f}_j\left({\upsilon }_j(t,x)\right)+\sum _{j=1}^n{b}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{f}_j\left({\upsilon }_j(t,x)\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{f}_j\left({\upsilon }_j(t,x)\right)+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ \hfill & +I_i]dt+{\rho }_i(t,{\upsilon }_i(t,x))dw\left(t\right),i=1,2,\cdots ,n,\hfill \end{array}\hfill \end{array}$$

(1)

where $i\in \mathcal{I}=\{1,2,\cdots ,n\}$, $x={(x_1,x_2,\cdots ,x_m)}^T\in X\in R^m$, and $X=\left\{x\right|\left|x_l\right|\le h_l$ for $l=1,2,\cdots ,m\}$ is a bounded compact set with smooth boundary $\partial X$ and $mesX\ge 0$ in space $R^m$, where $mesX$ is the measure of the set X; $D_{il}\ge 0$ stand for the transmission diffusion coefficients; ${\upsilon }_i(t,x)$ corresponds to state variable of the ith neuron at time t in space x; $c_i$ denotes the passive decay rates to the $ith$ neuron; $a_{ij}$ are the connections weights among neurons; $b_{ij}$ are elements of fuzzy feed-forward template; $f_j$ is the activation function; $T_{ij}$, $S_{ij}$, $P_{ij}$, $Q_{ij}$ respectively represent the elements of fuzzy feed-forward MIN template, fuzzy feed-forward MAX template, fuzzy feedback MIN template and fuzzy feedback MAX template; ${\nu }_i$ and $I_i$ respectively denote the bias and input value of the ith neuron; ${\rho }_i(t,·):R^{+}\times R\to R^{\ell }$ denotes the noise strength function; $w\left(t\right)={(w_1\left(t\right),w_2\left(t\right),\cdots ,w_{\ell }\left(t\right))}^T$ is a ℓ-dimensional Brown motion defined on a complete probability space $(\mathsf{\Omega },F,P)$; ⋁ and ⋀ are the fuzzy OR and fuzzy AND operation, respectively.

We will study model (1) under the following Dirichlet-type boundary conditions and initial conditions:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\displaystyle {\upsilon }_i(t,x)}& =0,(t,x)\in (t_0,+\infty )\times \partial X,\hfill \\ \hfill {\displaystyle {\upsilon }_i(t_0,x)}& ={\varphi }_i\left(x\right),x\in X,\hfill \end{array}\hfill \end{array}\right.$$

where ${\varphi }_i\left(x\right)$ is a bounded continuous function defined on X and satisfies compatibility condition. In this article, to investigate the FXT synchronization of system (1), we make the following assumptions:

Assumption 1 ([23]).

The activation functions $f_j$ satisfy the Lipschitz condition, i.e., there exist constants $L_j>0$ such that $|f_j\left(z_1\right)-f_j\left(z_2\right)|\le L_j|z_1-z_2|$, $z_1,z_2\in R$.

Assumption 2.

For $\rho (t,·)$, there exists positive constant ${\mu }_i\ge 0$ such that the inequality ${[{\rho }_i(t,z_1)-{\rho }_i(t,z_2)]}^T[{\rho }_i(t,z_1)-{\rho }_i(t,z_2)]\le {\mu }_i{(z_1-z_2)}^2$ holds true for any $z_1,z_2\in R$.

We give the corresponding slave system for the master system (1) as:

$$\begin{array}{c}\begin{array}{cc}\hfill d{\vartheta }_i(t,x)=& [\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{\vartheta }_i(t,x)}{\partial x_l^2}-c_i{\vartheta }_i(t,x)+\sum _{j=1}^n{a}_{ij}{f}_j\left({\vartheta }_j(t,x)\right)+\sum _{j=1}^n{b}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{f}_j\left({\vartheta }_j(t,x)\right)+\underset{j=1}{\overset{n}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{f}_j\left({\vartheta }_j(t,x)\right)+\underset{j=1}{\overset{n}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ \hfill & +u_i(t,x)]dt+{\rho }_i(t,{\vartheta }_i(t,x))dw\left(t\right),\hfill \end{array}\hfill \end{array}$$

(2)

where $u_i(t,x)$ indicates the controller input and other parameters are the same as defined in the system (1). The boundary condition and initial value of system (2) are listed as:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\displaystyle }& {\vartheta }_i(t,x)=0,(t,x)\in (t_0,+\infty )\times \partial X,\hfill \\ \hfill {\displaystyle }& {\vartheta }_i(t_0,x)={\phi }_i\left(x\right),x\in X.\hfill \end{array}\hfill \end{array}\right.$$

Now consider the following general stochastic nonlinear system:

$$dz\left(t\right)=A\left(z\left(t\right)\right)dt+B\left(z\left(t\right)\right)dw\left(t\right),z\left(0\right)=z_0,$$

(3)

where $n_0\in N^{+}$, $z\left(t\right)\in R^{n_0}$ is the state variable of stochastic system (3), $w\left(t\right)=(w_1\left(t\right),w_2\left(t\right)$, $\cdots ,w_{\ell }{\left(t\right))}^T$ is an ℓ-dimensional Brown motion on the probability space $(\mathsf{\Omega },F,P)$. Functions $A(·):R^{n_0}\to R^{n_0}$ and $B(·):R^{n_0}\to R^{n_0\times \ell }$ are continuous functions and they satisfy $A(·)=B(·)=0$.

For simplicity, we let $z\left(t\right)=z(t,z_0)$ be the solution of system (3) which satisfies the initial condition $z\left(0\right)=z_0$. The following definitions and lemmas will be used to obtain the main results of the paper:

Definition 1 ([24]).

The zero solution of system (3) is said to be FXT stable in probability, if the solution $z(t,z_0)$ exists for any initial condition of $z_0\in R^{n_0}$, and the following statements are true:

(i) For any initial value $z_0(\ne 0)\in R^{n_0}$, it holds that Pro$\{T(z_0,w)<\infty \}=1$, where $T(z_0,w)=inf\{T\ge 0|z(t,z_0)=0,t\ge T\}$ is called the stochastic ST.

(ii) For every pair of $\varrho \in (0,1)$ and $\tau >0$, there exists a $\delta =\delta (\varrho ,\tau )>0$ such that $Pro\left\{\right|z(t,z_0)|\le \tau ,\forall t\ge t_0\}\ge 1-\varrho ,$ for all $|z_0|\le \delta$.

(iii) Expectation of ST function $T(z_0,w)$ is nothing to the initial states $z_0$ of (3) and it is bounded by a positive constant $T_{max}$. That is, $E\left(T(z_0,w)\right)\le T_{max}$ for all $z_0\in R^{n_0}$.

In this paper, let $C^{2,1}(R^{+}\times R^{n_0};R^{+})$ be the set of all non-negative function $V(t,z)$ on $R^{+}\times R^{n_0}$ its partial derivative for t continuous and second-order partial derivatives for z exist, then for each $V\in C^{2,1}(R^{+}\times R^{n_0};R^{+})$, the operator $\pounds V(t,z)$ is defined as:

$$\begin{array}{cc}\hfill \pounds V(t,z)& =V_t(t,z)+V_z(t,z)f(t,z\left(t\right))\hfill \\ \hfill & +\frac{1}{2}trace\left[{\rho }^T(t,z\left(t\right))V_{zz}\rho (t,z\left(t\right))\right],\hfill \end{array}$$

where $V_t(t,z)=\frac{\partial V(t,z)}{\partial t}$, $V_z(t,z)=\left[\frac{\partial V(t,z)}{\partial z_1},\cdots ,\frac{\partial V(t,z)}{\partial z_{n_0}}\right]$, $V_{zz}(t,z)={\left(\frac{{\partial }^{2}V(t,z)}{\partial z_i\partial z_j}\right)}_{n_0\times n_0}$.

Lemma 1 ([25]).

Assume that $V\left(z\right):R^{n_0}\to R^{+}$ is a positive definite Lyapunov function, function $r\left(\nu \right)\in V_{\mathfrak{M}}^0$ is continuous, where $V_{\mathfrak{M}}^0$ denotes set of the bounded functions which is defined as

$$V_{\mathfrak{M}}^0=\left\{\varphi \left(z\right)|{\int }_0^{\infty }\frac{1}{\varphi \left(z\right)}dz\le \mathfrak{M},{\varphi }^{\prime }\left(z\right)\ge 0\right\},$$

where $\mathfrak{M}>0$ is positive constant. For any $z\left(t\right)=z(t,z_0)$ with $z_0\in R^{n_0}$, if the following differential inequality holds

$$\pounds V\left(z\right)\le -\varphi \left(V\right(z\left)\right),$$

(4)

then the zero solution $z=0$ of the system (3) is FXT stable in probability with a ST $T(z_0,w)$ which satisfies $E\left[T(z_0,w)\right]\le \mathfrak{M}.$

Lemma 2 ([19,25,26]).

Suppose $V\left(z\right):R^{n_0}\to R$ is a C-regular function such that

$$\pounds V\left(z\right)\le kV\left(z\left(t\right)\right)-\alpha V^p\left(z\left(t\right)\right)-\beta V^q\left(z\left(t\right)\right),z\left(t\right)\in R^{n_0}\setminus \left\{0\right\},$$

(5)

where $\alpha >0$, $\beta >0$, $p>1$ and $0<q<1$, then the origin of system (3) is FXT stable in probability with a ST $T(z_0,w)$ which satisfy $E\left[T(z_0,w)\right]<T_{max}$, where

$${\displaystyle T_{max}\triangleq \left\{\begin{array}{cc}{T}_{max}^1\hfill & {\displaystyle k<0,}\hfill \\ T_{max}^2\hfill & {\displaystyle k=0,}\hfill \\ T_{max}^3\hfill & {\displaystyle 0<k<min\{\alpha ,\beta \},}\hfill \end{array}\right.}$$

$$\begin{array}{cc}{T}_{max}^1=\hfill & {\displaystyle \frac{1}{k\tau (1-p)}ln\left(1-\frac{k}{\beta }{\left(\frac{\beta }{\alpha }\right)}^{\tau }\right)}\hfill \\ T_{max}^2=\hfill & {\displaystyle \frac{\pi }{(p-q)}{\left(\frac{\beta }{\alpha }\right)}^{\tau }csc\left(\tau \pi \right),}\hfill \\ T_{max}^3=\hfill & {\displaystyle \frac{\pi csc\left(\tau \pi \right)}{\alpha (p-q)}{\left(\frac{\alpha }{\beta -k}\right)}^{1-\tau }I\left(\frac{\alpha }{\gamma },\tau ,1-\tau \right)}\hfill & \\ \hfill & {\displaystyle +\frac{\pi csc\left(\tau \pi \right)}{\beta (p-q)}{\left(\frac{\beta }{\alpha -k}\right)}^{\tau }I\left(\frac{\beta }{\gamma },1-\tau ,\tau \right),}\hfill \end{array}$$

(6)

where $\tau =\frac{1-q}{p-q}$, $\gamma =\alpha +\beta -k$. Especially when $p+q=2$, the estimation of ST can be given more precisely as $E\left[T(y_0,w)\right]<{\tilde{T}}_{max}$, where

$${\displaystyle {\tilde{T}}_{max}\triangleq \left\{\begin{array}{cc}{T}_{max}^4\hfill & {\displaystyle -2\sqrt{\alpha \beta }<k<2\sqrt{\alpha \beta }}\hfill \\ T_{max}^5\hfill & {\displaystyle k=-2\sqrt{\alpha \beta },}\hfill \\ T_{max}^6\hfill & {\displaystyle k<-2\sqrt{\alpha \beta },}\hfill \end{array}\right.}$$

$$\begin{array}{cc}{T}_{max}^4=\hfill & {\displaystyle \frac{1}{p-1}\frac{2}{\sqrt{\sigma }}\left(\frac{\pi }{2}+arctan\left(\frac{k}{\sqrt{\sigma }}\right)\right),}\hfill \\ T_{max}^5=\hfill & {\displaystyle \frac{2}{k(p-1)},}\hfill & {\displaystyle }\hfill \\ T_{max}^6=\hfill & {\displaystyle \frac{1}{(p-1)\sqrt{-\sigma }}ln\frac{k+\sqrt{-\sigma }}{k-\sqrt{-\sigma }},}\hfill \end{array}$$

(7)

where $\sigma =4\alpha \beta -k^2$.

Lemma 3 ([27]).

Suppose $z_1$ and $z_2$ are two real numbers, then we have

$$\begin{array}{cc}\hfill & |\underset{j=1}{\overset{\mathrm{n}}{\bigwedge }}{P}_{ij}{f}_j\left(z_1\right)-\underset{j=1}{\overset{\mathrm{n}}{\bigwedge }}{P}_{ij}{f}_j\left(z_2\right)|\le \sum _{j=1}^{\mathrm{n}}|P_{ij}|·|f_j\left(z_1\right)-f_j\left(z_2\right)|,\hfill \\ \hfill & |\underset{j=1}{\overset{\mathrm{n}}{\bigvee }}{Q}_{ij}{f}_j\left(z_1\right)-\underset{j=1}{\overset{\mathrm{n}}{\bigvee }}{Q}_{ij}{f}_j\left(z_2\right)|\le \sum _{j=1}^{\mathrm{n}}\left|Q_{ij}\right|·\left|f_j\left(z_1\right)-f_j\left(z_2\right)\right|.\hfill \end{array}$$

Lemma 4 ([28]).

Let X be a cube $|x_l|<h_l(l=1,2,\cdots ,m)$ and assume that $z\left(x\right)$ is a real-valued function belonging to $C^1\left(X\right)$ which vanishes on the boundary $\partial X$ of X, i.e, $z\left(x\right){\mid }_{\partial X}=0$. Then

$${\int }_X{z}^2\left(x\right)dx\le h_l^2{\int }_X|\frac{\partial z}{\partial x_l}{|}^{2}dx,l=1,2,\cdots ,m.$$

Lemma 5 ([29]).

If $z_1,z_2,\cdots ,z_{n_0}\ge 0,0<\eta \le 1<\kappa$, then

$$\begin{array}{ccc}\hfill \sum _{s=1}^{n_0}{z}_s^{\eta }& \ge {\left(\sum _{s=1}^{n_0}{z}_s\right)}^{\eta },\sum _{s=1}^{n_0}{z}_s^{\kappa }\hfill & \hfill \ge m^{1-\kappa }{\left(\sum _{s=1}^{n_0}{z}_s\right)}^{\kappa }.\end{array}$$

3. Main Results

In this section, we will derive some criteria to guarantee the FXT synchronization between master-slave systems (1) and (2). First, let $e_i(t,x)={\vartheta }_i(t,x)-{\upsilon }_i(t,x)$ be the synchronization error between master system (1) and slave system (2). In addition, according to the boundary conditions of (1) and (2), one has $e_i(t,x)=0$ for $(t,x)\in (t_0,+\infty )\times \partial X$, thus the error dynamical system can be listed as follows:

$$\begin{array}{c}\begin{array}{cc}\hfill d{e}_i(t,x)=& [\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{e}_i(t,x)}{\partial x_l^2}-c_i{e}_i(t,x)+\sum _{j=1}^n{a}_{ij}{g}_j\left(e_j(t,x)\right)+\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{g}_j\left(e_j(t,x)\right)\hfill \\ & +\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{g}_j\left(e_j(t,x)\right)+u_i(t,x)]dt+{\tilde{\rho }}_i(t,e_i(t,x))dw\left(t\right),\hfill \end{array}\hfill \end{array}$$

(8)

where $g_j\left(e_j(t,x)\right)=f_j\left({\vartheta }_j(t,x)\right)-f_j\left({\upsilon }_j(t,x)\right)$, ${\tilde{\rho }}_i(t,e_i(t,x))={\rho }_i(t,{\vartheta }_i(t,x))-{\rho }_i(t,{\upsilon }_i(t,x))$.

Now, to achieve FXT synchronization aim, we deign the controller $u_i(t,x)$ in slave system (2) as follows:

$$\begin{array}{c}{u}_i(t,x)=-{\gamma }_i{e}_i^{\left[\delta \right]}(t,x)-{\lambda }_i{e}_i^{\left[\theta \right]}(t,x),\hfill \end{array}$$

(9)

where ${\gamma }_i>0,{\lambda }_i>0,$$\delta$ and $\theta$ are positive constants such that $0<\delta <1$, $\theta >1$, and

$$\begin{array}{c}{e}_i^{\left[r\right]}(t,x)\triangleq sign\left(e_i(t,x)\right){\left|e_i(t,x)\right|}^r,r>0.\hfill \end{array}$$

Denote

$$\begin{array}{cc}\hfill k_i=& -{\eta }_i-2c_i+{\mu }_i+\sum _{j=1}^m\left[\right(|a_{ij}|+|P_{ij}|+|Q_{ij}\left|\right)L_j)+(|a_{ji}|+|P_{ji}|+|Q_{ji}\left|\right)L_i\left)\right],\hfill \end{array}$$

where ${\eta }_i={\sum }_{l=1}^m\frac{2D_{il}}{h_l^2}$, then the following theorem can be derived via the FXT controller (9).

Theorem 1.

Assume that the Assumptions 1 and 2 are satisfied, and the control parameters ${\gamma }_i$ and ${\lambda }_i$ in (9) satisfy the following condition

$$\underset{i=1}{\overset{n}{max}}\left\{k_i\right\}<min\{{\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2\},$$

(10)

then the master-slave systems (1) and (2) can achieve FXT synchronization in probability via controller (9), and it ST can be estimated by $T_{max}$, where $T_{max}$ is defined in (6) with the parameters ${\mathsf{\Lambda }}_1=2{min}_{i=1}^n\left\{{\gamma }_i\right\}$, ${\mathsf{\Lambda }}_2=2n^{\frac{1-\theta }{2}}{min}_{i=1}^n\left\{{\lambda }_i\right\}$, $\alpha =2\gamma$, $\beta =2n^{\frac{1-\theta }{2}}\lambda$, $p=\frac{1+\delta }{2}$, $q=\frac{1+\theta }{2}$, $\gamma ={min}_i\left\{{\gamma }_i\right\}$ and $\lambda ={min}_i\left\{{\lambda }_i\right\}$.

Proof. 

Consider the following Lyapunov function:

$${\displaystyle V\left(t\right)={\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx.}$$

Calculating the derivative of $\pounds V\left(t\right)$ along the solution of system (8), we have

$$\begin{array}{cc}\hfill \pounds V\left(t\right)=& {\int }_X[\sum _{i=1}^n\sum _{l=1}^{m}2e_i(t,x)D_{il}\frac{{\partial }^2{e}_i(t,x)}{\partial x_l^2}-\sum _{i=1}^{n}2e_i(t,x)c_i{e}_i(t,x)\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^{n}2e_i(t,x)a_{ij}{g}_j\left(e_j(t,x)\right)+\sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{g}_j\left(e_j(t,x)\right)\hfill \\ & +\sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{g}_j\left(e_j(t,x)\right)-\sum _{i=1}^{n}2e_i(t,x){\gamma }_i{e}_i^{\left[\delta \right]}(t,x)\hfill \\ & -\sum _{i=1}^{n}2e_i(t,x){\lambda }_i{e}_i^{\left[\theta \right]}(t,x)+trace\left[{\tilde{\rho }}_i^T(t,e_i(t,x)){\tilde{\rho }}_i(t,e_i(t,x))\right]]dx.\hfill \end{array}$$

(11)

By the Green formula and boundary condition, one can has

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{X}2e_i(t,x)\left(\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{e}_i(t,x)}{\partial x_l^2}\right)dx=}& 2{\int }_X{e}_i(t,x)\nabla ·{\left(D_{il}\frac{\partial e_i(t,x)}{\partial x_l}\right)}_{l=1}^{m}dx\hfill \\ \hfill =& 2{\int }_X\nabla ·{\left(e_i(t,x)D_{il}\frac{\partial e_i(t,x)}{\partial x_l}\right)}_{l=1}^{m}dx\hfill \\ & -2{\int }_X{\left(D_{il}\frac{\partial e_i(t,x)}{\partial x_l}\right)}_{l=1}^m·\nabla e_i(t,x)dx\hfill \\ \hfill =& 2{\int }_{\partial X}{\left(e_i(t,x)D_{il}\frac{\partial e_i(t,x)}{\partial x_l}\right)}_{l=1}^{m}dS\hfill \\ & -2{\int }_X\sum _{l=1}^m{D}_{il}{\left(\frac{\partial e_i(t,x)}{\partial x_l}\right)}^{2}dx\hfill \\ \hfill =& -2{\int }_X\sum _{l=1}^m{D}_{il}{\left(\frac{\partial e_i(t,x)}{\partial x_l}\right)}^{2}dx,\hfill \end{array}$$

(12)

where ${\left(D_{il}\frac{\partial e_i(t,x)}{\partial x_l}\right)}_{l=1}^m={\left(D_{i1}\frac{\partial e_i(t,x)}{\partial x_1},\dots ,D_{im}\frac{\partial e_i(t,x)}{\partial x_m}\right)}^T$, and $\nabla ={\left(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_m}\right)}^T$ is the gradient operator.

From Lemma 4

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{X}2e_i(t,x)\left(\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{e}_i(t,x)}{\partial x_l^2}\right)dx}& =-2{\int }_X\sum _{l=1}^m{D}_{il}{\left(\frac{\partial e_i(t,x)}{\partial x_l}\right)}^{2}dx\hfill \\ & \le -\sum _{l=1}^m\frac{2D_{il}}{h_l^2}{\int }_X{e}_i^2(t,x)dx\hfill \\ & =-{\eta }_i{\int }_X{e}_i^2(t,x)dx,\hfill \end{array}$$

(13)

in which ${\eta }_i={\sum }_{l=1}^m\frac{2D_{il}}{h_l^2}$.

Using the Assumption 1 and inequality $2z_1{z}_2\le z_1^2+z_2^2$ for any $z_1>0,z_2>0$, one has

$$\begin{array}{cc}\hfill \sum _{i=1}^n\sum _{j=1}^{n}2e_i(t,x)a_{ij}{g}_j\left(e_j(t,x)\right)& \le \sum _{i=1}^n\sum _{j=1}^{n}2|e_i(t,x)\left|\right|a_{ij}\left|\right|g_j\left(e_j(t,x)\right)|\hfill \\ \hfill & \le \sum _{i=1}^n\sum _{j=1}^{n}2L_j|a_{ij}\left|\right|e_i(t,x)\left|\right|e_j(t,x)|\hfill \\ \hfill & \le \sum _{i=1}^n\sum _{j=1}^n{L}_j\left|a_{ij}\right|\left(e_i^2(t,x)+e_j^2(t,x)\right)\hfill \\ \hfill & \le \sum _{i=1}^n\sum _{j=1}^n\left(L_j|a_{ij}|+L_i\left|a_{ji}\right|\right)e_i^2(t,x).\hfill \end{array}$$

(14)

Similarly, by Lemma 3, we get

$$\begin{array}{cc}\hfill \sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{g}_j\left(e_j(t,x)\right) & \le \sum _{i=1}^n\sum _{j=1}^n\left(L_j|P_{ij}|+L_i\left|P_{ji}\right|\right)e_i^2(t,x),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{g}_j\left(e_j(t,x)\right) & \le \sum _{i=1}^n\sum _{j=1}^n\left(L_j|Q_{ij}|+L_i\left|Q_{ji}\right|\right)e_i^2(t,x).\hfill \end{array}$$

(16)

Also, by simple observation, we can have

$$\begin{array}{cc}\hfill \sum _{i=1}^n& 2e_i(t,x)\left[-{\gamma }_i{e}_i^{\left[\delta \right]}(t,x)\right]=\sum _{i=1}^n-2{\gamma }_i{\left|e_i(t,x)\right|}^{1+\delta },\hfill \end{array}$$

(17)

$$\begin{array}{c}\hfill \sum _{i=1}^{n}2e_i(t,x)\left[-{\lambda }_i{e}_i^{\left[\theta \right]}(t,x)\right]=\sum _{i=1}^n-2{\lambda }_i{\left|e_i(t,x)\right|}^{1+\theta }.\end{array}$$

(18)

Finally, by Assumption 2, one can has

$$trace\left[{\rho }_i^T(t,e_i(t,x)){\rho }_i(t,e_i(t,x))\right]\le \sum _{i=1}^n{\mu }_i{e}_i^2(t,x).$$

(19)

Then in the view of (13)–(19), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \pounds \left(V\right(t\left)\right)\le }& {\int }_X\sum _{i=1}^n(-{\eta }_i-2c_i+{\mu }_i+\sum _{j=1}^n{L}_j|Q_{ij}|+\sum _{j=1}^n{L}_i|Q_{ji}|+\sum _{j=1}^n{L}_j\left|P_{ij}\right|\hfill \\ & +\sum _{j=1}^n{L}_i|P_{ji}|+\sum _{j=1}^n{L}_j|a_{ij}|+\sum _{j=1}^n{L}_i\left|a_{ji}\right|)e_i^2(t,x)dx\hfill \\ & -2{\int }_X\sum _{i=1}^n{\gamma }_i|e_i{(t,x)|}^{1+\delta }dx-2{\int }_X\sum _{i=1}^n{\lambda }_i{\left|e_i(t,x)\right|}^{1+\theta }dx\hfill \\ \hfill \le & {\int }_X{k}_i\sum _{i=1}^n{e}_i^2(t,x)dx-2{\int }_X\sum _{i=1}^n{\gamma }_i{\left|e_i(t,x)\right|}^{1+\delta }dx\hfill \\ \hfill & -2{\int }_X\sum _{i=1}^n{\lambda }_i{\left|e_i(t,x)\right|}^{1+\theta }dx.\hfill \end{array}$$

(20)

Furthermore, by Lemma 5, we have

$$\begin{array}{cc}\hfill {\int }_X\sum _{i=1}^n{\left|e_i(t,x)\right|}^{1+\delta }dx& \ge {\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{1+\delta }{2}}\hfill \\ \hfill & =V^{\frac{1+\delta }{2}}\left(t\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\int }_X\sum _{i=1}^n{\left|e_i(t,x)\right|}^{1+\theta }dx& \ge n^{\frac{1-\theta }{2}}{\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{1+\theta }{2}}\hfill \\ \hfill & =n^{\frac{1-\theta }{2}}{V}^{\frac{1+\theta }{2}}\left(t\right).\hfill \end{array}$$

(22)

Furthermore, since ${\gamma }_i$ and ${\lambda }_i$ are positive, in view of (21) and (22), we can obtain

$$\begin{array}{cc}\hfill {\displaystyle \pounds \left(V\right(t\left)\right)\le }& {\int }_X{k}_i\sum _{i=1}^n{e}_i^2(t,x)dx-2{\gamma }_i{\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{1+\delta }{2}}\hfill \\ \hfill & -2{\lambda }_i{n}^{\frac{1-\theta }{2}}{\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{1+\theta }{2}}\hfill \\ \hfill \le & kV\left(t\right)-2\gamma V^{\frac{1+\delta }{2}}\left(t\right)-2n^{\frac{1-\theta }{2}}\lambda V^{\frac{1+\theta }{2}}\left(t\right),\hfill \end{array}$$

(23)

where $k={max}_i^n\left\{k_i\right\},\gamma ={min}_i^n\left\{{\gamma }_i\right\}>0$ and $\lambda ={min}_i^n\left\{{\lambda }_i\right\}>0$.

Let $\alpha =2\gamma$,$\beta =2n^{\frac{1-\theta }{2}}\lambda$,$p=\frac{1+\delta }{2}$ and $q=\frac{1+\theta }{2}$, then from the inequality (10) we get that $k<min\{\alpha ,\beta \}$. Therefore, based on Lemma 2, the master-slave systems (1) and (2) will realize FXT synchronization with the ST $T_{max}$, which is given in (6). The proof is completed. □

Corollary 1.

Assume that the inequality (10), Assumptions 1 and 2 holds true. If $\delta +\theta =2$ and $k<2\sqrt{\gamma \lambda }$, where the control gains $k,\gamma$ and λ are introduced in Theorem 1, then the master-slave systems (1) and (2) will realize FXT synchronization in probability with the ST ${\tilde{T}}_{max}$, where ${\tilde{T}}_{max}$ is given in (7).

If the elements of fuzzy feedback MIN template and fuzzy feedback MAX template are removed from system (1), i.e., $b_{ij}=P_{ij}=Q_{ij}=T_{ij}=S_{ij}=0$, then it is reduced to the following form

$$\begin{array}{c}\begin{array}{cc}\hfill d{\upsilon }_i(t,x)=& \left[\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{\upsilon }_i(t,x)}{\partial x_l^2}-c_i{\upsilon }_i(t,x)+\sum _{j=1}^n{a}_{ij}{f}_j\left({\upsilon }_i(t,x)\right)+I_i\right]dt\hfill \\ & +{\rho }_i(t,{\upsilon }_i(t,x))dw\left(t\right).\hfill \end{array}\hfill \end{array}$$

(24)

Accordingly, the corresponding slave system (2) degenerates as

$$\begin{array}{c}\begin{array}{cc}\hfill d{\vartheta }_i(t,x)=& \left[\sum _{l=1}^m{D}_{il}\frac{{\partial }^2{\vartheta }_i(t,x)}{\partial x_l^2}-c_i{\vartheta }_i(t,x)+\sum _{j=1}^n{a}_{ij}{f}_j\left({\vartheta }_i(t,x)\right)+I_i+u_i(t,x)\right]dt\hfill \\ & +{\rho }_i(t,{\vartheta }_i(t,x))dw\left(t\right).\hfill \end{array}\hfill \end{array}$$

(25)

For this case, we denote

$${\tilde{k}}_i=-{\eta }_i-2c_i+{\mu }_i+\sum _{j=1}^n\left(|a_{ij}|L_j+\left|a_{ji}\right|L_i\right).$$

Then, from Theorem 1 and Corollary 1, we can have the following Corollary.

Corollary 2.

Suppose that Assumptions 1 and 2 hold true. If the control parameters ${\gamma }_i$ and ${\lambda }_i$ in the controller (9) satisfy the condition (10), where $k_i$ is substituted by above given $\widetilde{k_i}$, then the master-slave systems (24) and (25) can achieve FXT synchronization in probability under the controller (9), and its ST is bounded by $T_{max}$. Especially, if $\delta +\theta =2$ and $k\triangleq {max}_i\left\{{\tilde{k}}_i\right\}<2\sqrt{\gamma \lambda }$, where γ and λ are defined in Theorem 1, then the systems (24) and (25) will achieve FXT synchronization with the ST ${\tilde{T}}_{max}$, where ${\tilde{T}}_{max}$ is introduced in (7).

When the synchronization errors approach to 0, the discontinuous signum function in (9) may bring some poor chattering effects. To reduce or suppress this undesired influence, now we will achieve the FXT synchronization between master-slave systems (1) and (2) by introducing the following novel controller

$$u_i(t,x)=-{\gamma }_i{e}_i^{\frac{{\delta }_1}{{\delta }_2}}(t,x)-{\lambda }_i{e}_i^{\frac{{\theta }_1}{{\theta }_2}}(t,x),$$

(26)

where ${\gamma }_i\ge 0,{\lambda }_i>0$, and ${\delta }_1,{\delta }_2,{\theta }_1,{\theta }_2$ are positive odd integers satisfying ${\delta }_1<{\delta }_2$ and ${\theta }_1>{\theta }_2$.

Theorem 2.

The systems (1) and (2) will achieve FXT synchronization in probability via the continuous controller (26), and its ST can be bounded by $T_{max}$, as if Assumptions 1 and 2 hold true and the control gains ${\gamma }_i$ and ${\lambda }_i$ of (26) satisfy the inequality of (10). Especially, if ${\delta }_1{\theta }_2+{\delta }_2{\theta }_1=2{\delta }_2{\theta }_2$ and $k<2\sqrt{\gamma \lambda }$, where parameters $k,\gamma$ and λ are introduced in Theorem 1, then the master-slave systems (1) and (2) will achieve FXT synchronization in probability with the ST ${\tilde{T}}_{max}$, where $T_{max}$ and ${\tilde{T}}_{max}$ are given in (6) and (7) respectively, and their parameters are defined as $\alpha =2\gamma$, $\beta =2\lambda n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}},p=\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}$ and $q=\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}$.

Proof. 

Similar to the proof of Theorem 1, we construct the Lyapunov function as ${\displaystyle V\left(t\right)={\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx}$, then we have

$$\begin{array}{cc}\hfill \pounds V\left(t\right)=& {\int }_X[\sum _{i=1}^n\sum _{l=1}^{m}2e_i(t,x)D_{il}\frac{{\partial }^2{e}_i(t,x)}{\partial x_l^2}-\sum _{i=1}^{n}2e_i(t,x)c_i{e}_i(t,x)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^{n}2e_i(t,x)a_{ij}{g}_j\left(e_i(t,x)\right)+\sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigwedge }}{P}_{ij}{g}_j\left(e_j(t,x)\right)\hfill \\ \hfill & +\sum _{i=1}^{n}2e_i(t,x)\underset{j=1}{\overset{n}{\bigvee }}{Q}_{ij}{g}_j\left(e_j(t,x)\right)-\sum _{i=1}^{n}2e_i(t,x){\gamma }_i{e}_i^{\frac{{\delta }_1}{{\delta }_2}}(t,x)\hfill \\ \hfill & -\sum _{i=1}^{n}2e_i(t,x){\lambda }_i{e}_i^{\frac{{\theta }_1}{{\theta }_2}}(t,x))+trace\left[{\tilde{\rho }}_i^T(t,e_i(t,x)){\tilde{\rho }}_i(t,e_i(t,x))\right]]dx.\hfill \end{array}$$

(27)

From Lemma 5, we get

$$\begin{array}{cc}\hfill -{\int }_X\sum _{i=1}^{n}2e_i(t,x){\gamma }_i{e}_i^{\frac{{\delta }_1}{{\delta }_2}}(t,x)dx& \le -2\underset{i=1}{\overset{n}{min}}\left\{{\gamma }_i\right\}{\int }_X\sum _{i=1}^n{e}_i^{\frac{{\delta }_1+{\delta }_2}{{\delta }_2}}(t,x)dx\hfill \\ \hfill & =-2\gamma {\int }_X\sum _{i=1}^n{e}_i^{\frac{{\delta }_1+{\delta }_2}{{\delta }_2}}(t,x)dx\hfill \\ \hfill & \le -2\gamma {\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}}\hfill \\ \hfill & =-2\gamma V^{\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}}\left(t\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill -{\int }_X\sum _{i=1}^{n}2e_i(t,x){\lambda }_i{e}_i^{\frac{{\theta }_1}{{\theta }_2}}(t,x)dx& \le -2\underset{i=1}{\overset{n}{min}}\left\{{\lambda }_i\right\}{\int }_X\sum _{i=1}^n{e}_i^{\frac{{\theta }_1+{\theta }_2}{{\theta }_2}}(t,x)dx\hfill \\ \hfill & =-2\lambda {\int }_X\sum _{i=1}^n{e}_i^{\frac{{\theta }_1+{\theta }_2}{{\theta }_2}}(t,x)dx\hfill \\ \hfill & \le -2\lambda n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}{\left({\int }_X\sum _{i=1}^n{e}_i^2(t,x)dx\right)}^{\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}}\hfill \\ \hfill & =-2\lambda n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}{V}^{\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}}\left(t\right).\hfill \end{array}$$

(29)

Then in view of (13)–(19), (28) and (29), we can obtain

$$\begin{array}{cc}\hfill {\displaystyle \pounds \left(V\right(t\left)\right)\le }& {\int }_X\sum _{i=1}^n(-{\eta }_i-2c_i+{\mu }_i+\sum _{j=1}^n{L}_j|Q_{ij}|+\sum _{j=1}^n{L}_i\left|Q_{ji}\right|\hfill \\ & +\sum _{j=1}^n{L}_j|P_{ij}|+\sum _{j=1}^n{L}_i|P_{ji}|+\sum _{j=1}^n{L}_j|a_{ij}|+\sum _{j=1}^n{L}_i\left|a_{ji}\right|)e_i^2(t,x)dx\hfill \\ & -2\gamma V^{\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}}\left(t\right)-2n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}\lambda V^{\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}}\left(t\right)\hfill \\ \hfill \le & kV\left(t\right)-2\gamma V^{\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}}\left(t\right)-2\lambda n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}{V}^{\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}}\left(t\right),\hfill \end{array}$$

(30)

where $k={max}_i^n\left\{k_i\right\}$.

Thus, based on Lemma 2, the master system (1) and slave system (2) will realize FXT synchronization in probability via controller (26) with the ST $T_{max}$. Especially, when ${\delta }_1{\theta }_2+{\delta }_2{\theta }_1=2{\delta }_2{\theta }_2$ ( for $p+q=2)$ and ${max}_i\left\{k_i\right\}<2\sqrt{{min}_i\left\{{\gamma }_i\right\}\times {min}_i\left\{{\lambda }_i\right\}{n}^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}}$ (for $k<2\sqrt{\gamma \lambda })$, then the master system (1) and the slave system (2) will achieve FXT synchronization in probability with ST ${\tilde{T}}_{max}$, where ${\tilde{T}}_{max}$ is given in (7). □

From the results of Corollary 2 and Theorem 2, we can also obtain the following Corollary.

Corollary 3.

Assume that the Assumptions 1 and 2 are satisfied and control parameters ${\gamma }_i$ and ${\lambda }_i$ in (26) satisfy the condition (10), where $k_i$ is substituted by $\widetilde{k_i}$, then the systems (24) and (25) can be FXT synchronized under the continuous controller (26), and its ST is bounded by $T_{max}$. Especially, when ${\delta }_1{\theta }_2+{\delta }_2{\theta }_1=2{\delta }_2{\theta }_2$ and $k\triangleq {max}_i\left\{{\tilde{k}}_i\right\}<2\sqrt{\gamma \lambda }$, then the master-slave systems (24) and (25) can achieve FXT synchronized in probability via the continuous controller (26) with ST ${\tilde{T}}_{max}$, where ${\tilde{T}}_{max}$ is introduced in (7), and its parameters are defined as $\alpha =2\gamma$, $\beta =2\lambda n^{\frac{{\theta }_2-{\theta }_1}{2{\theta }_2}}$, $p=\frac{{\delta }_1+{\delta }_2}{2{\delta }_2}$ and $q=\frac{{\theta }_1+{\theta }_2}{2{\theta }_2}$.

Remark 1.

In the previous studies [21,23,27,30,31,32], authors achieved FXT synchronization of deterministic fuzzy neural networks by employing a controller that uses a discontinuous signum function. However, this may bring poor chattering effects when synchronization errors approach to 0, and this may somewhat influences the synchronization property of the original master-slave systems. In Theorem 2 and Corollary 3, to overcome this sophisticated issue, we introduce a novel controller which does not use a signum function.

Remark 2.

In the previous studies [23,27,30,31,32,33], authors achieved FXT synchronization of various types of fuzzy neural networks with memristors, stochastic perturbations, or discontinuous neuron activations. However, till now there are seldom results on the FXT synchronization of fuzzy neural networks with reaction-diffusion terms. In this paper, for the first time, we have considered this issue by designing two types of controllers that did not include linear feedback term, but the results of previous works [27,30,31,32,33] were based on the controllers that use the linear feedback term as a main component. From this point, we can see that the obtained results of this paper are more general and thus have better applicability.

4. Numerical Examples and Simulations

In this section, we will provide one example to illustrate the effectiveness of the theoretical results derived above.

Example 1.

For $n=3$, consider the following reaction-diffusion fuzzy neural networks with stochastic perturbations

$$\begin{array}{c}\begin{array}{cc}\hfill d{\upsilon }_i(t,x)=& [\sum _{l=1}^3{D}_{il}\frac{{\partial }^2{\upsilon }_i(t,x)}{\partial x_l^2}-c_i{\upsilon }_i(t,x)+\sum _{j=1}^3{a}_{ij}{f}_j\left({\upsilon }_i(t,x)\right)+\sum _{j=1}^3{b}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{3}{\bigwedge }}{P}_{ij}{f}_j\left({\upsilon }_j(t,x)\right)+\underset{j=1}{\overset{3}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{3}{\bigvee }}{Q}_{ij}{f}_j\left({\upsilon }_j(t,x)\right)+\underset{j=1}{\overset{3}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ & +I_i]dt+{\rho }_i(t,{\upsilon }_i(t,x))dw\left(t\right),\hfill \end{array}\hfill \end{array}$$

(31)

where the activation function $f_j\left(r\right)=tanhr$, $D_{il}=1$, $h_l=5$, ${\rho }_i(t,r)=0.15r$, $c_i=1.128$, the other parameters are assumed as $a_{11}=1.815,a_{12}=-4.6464,a_{13}=-4.6464,a_{21}=-4.6464,a_{22}=1.5972,a_{23}=-6.3888,a_{31}=-4.6464,a_{32}=6.3888,a_{33}=1.452$, $P_{11}=-0.12,P_{12}=-0.18,P_{13}=-0.24,P_{21}=-0.24,P_{22}=-0.12,P_{23}=1.92,P_{31}=-0.18,P_{32}=-0.336,P_{33}=-0.24$, $Q_{11}=-0.18,Q_{12}=-2.268,Q_{13}=0.81,Q_{21}=-0.18,Q_{22}=-0.36,Q_{23}=-0.576,Q_{31}=-0.468,Q_{32}=0.936,Q_{33}=-0.18$, $b_{ij}=T_{ij}=S_{ij}=0$ and $I_i=0$ for $i,j,l=1,2,3$. Its corresponding slave system is described as

$$\begin{array}{c}\begin{array}{cc}\hfill d{\vartheta }_i(t,x)=& [\sum _{l=1}^3{D}_{il}\frac{{\partial }^2{\vartheta }_i(t,x)}{\partial x_l^2}-c_i{\vartheta }_i(t,x)+\sum _{j=1}^3{a}_{ij}{f}_j\left({\vartheta }_i(t,x)\right)+\sum _{j=1}^3{b}_{ij}{\nu }_j\hfill \\ & +\underset{j=1}{\overset{3}{\bigwedge }}{P}_{ij}{f}_j\left({\vartheta }_j(t,x)\right)+\underset{j=1}{\overset{3}{\bigwedge }}{T}_{ij}{\nu }_j+\underset{j=1}{\overset{3}{\bigvee }}{Q}_{ij}{f}_j\left({\vartheta }_j(t,x)\right)+\underset{j=1}{\overset{3}{\bigvee }}{S}_{ij}{\nu }_j\hfill \\ & +I_i+u_i(t,x)]dt+{\rho }_i(t,{\vartheta }_i(t,x))dw\left(t\right),\hfill \end{array}\hfill \end{array}$$

(32)

where $u_i(t,x)$ is the controller and other parameters $D_{il},c_i,a_{ij},b_{ij},P_{ij},T_{ij},Q_{ij},S_{ij},I_i$ and ${\varrho }_i(·,·),i,j=1,2,3$ are defined in system (31). Figure 1, Figure 2 and Figure 3 show the MATLAB simulations of the system (31) with initial values ${\upsilon }_1=0.0517$, ${\upsilon }_2=0.2146$, ${\upsilon }_3=0.4863$, $j=1,2,3$ which indicates that system (31) has a chaotic attractor.

On the basis of the parameters given above, choosing ${\gamma }_1={\gamma }_2={\gamma }_3=6,{\lambda }_1={\lambda }_2={\lambda }_3=6.3$, $p=0.4$ and $q=1.9$. It is not difficult to check that $k_1=6.056,k_2=9.8028,k_3=10.0224$ and $L_i=1$ for $i=1,2,3$. Since $k=max\{k_1,k_2,k_3\}=10.0224$, then condition (10) of Theorem 1 holds true. Thus, we can get from Theorem 1 that the slave and master systems (31) and (32) are FXT synchronized in probability with the ST via controller (9). The corresponding simulation results are given in Figure 4, Figure 5 and Figure 6.

5. Conclusions

In this paper, the FXT synchronization issue of reaction-diffusion fuzzy neural networks with stochastic perturbations is analysed via using some inequality techniques and designing two types of novel controllers. It is worth to mention that the considered model in this paper is very general since it combines fuzzy logic, reaction-diffusion and stochastic perturbations. In addition, the introduced controller in this paper is much simpler since it does not include linear term, which was commonly used in most of the previous published works. We believe that the methods used in this paper can provide some insights to analyse the FXT synchronization of other types of reaction diffusion neural networks with complex-valued coefficients and impulsive effects. In future work, we will analyse the FXT synchronization of discontinuous reaction–diffusion BAM neural networks with stochastic perturbation and impulsive effects due to their applicability in symmetry-related problems.