On p th moment exponential stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses

Abstract

This paper is concerned with the p th moment exponential stability of fuzzy cellular neural networks with time-varying delays under impulsive perturbations and stochastic noises. Based on the Lyapunov function, stochastic analysis and differential inequality technique, a set of novel sufficient conditions on p th moment exponential stability of the system are derived. These results generalize and improve some of the existing ones. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.

1 Introduction

In the last decades, cellular neural networks [1, 2] have been extensively studied and applied in many different fields such as associative memory, signal processing and some optimization problems. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network and thus should be incorporated into the model equation. In recent years, the dynamical behaviors of cellular neural networks with constant delays or time-varying delays or distributed delays have been studied; see, for example, [3–11] and the references therein.

In addition to the delay effects, recently, studies have been intensively focused on stochastic models. It has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks or dynamical system described by stochastic functional differential equations (see [12–23]). On the other hand, most neural networks can be classified as either continuous or discrete. Therefore most of the investigations focused on the continuous or discrete systems, respectively. However, there are many real-world systems and neural processes that behave in piecewise continuous style interlaced with instantaneous and abrupt change (impulses). Motivated by this fact, several new neural networks with impulses have been recently proposed and studied (see [24–33]).

In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [34] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [34–40]. However, to the best of our knowledge, few author investigated the stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses.

Motivated by the above discussions, in this paper, we consider the following stochastic fuzzy cellular neural networks with time-varying delays and impulses

$$\{\begin{array}{l}d{x}_i(t)=[-c_i{x}_i(t)+{\sum }_{j=1}^n{a}_{ij}{f}_j(x_j(t))+{\bigwedge }_{j=1}^n{\alpha }_{ij}{g}_j(x_j(t-{\tau }_j(t)))\\ \phantom{d{x}_i(t)=}+{\bigvee }_{j=1}^n{\beta }_{ij}{g}_j(x_j(t-{\tau }_j(t)))+I_i]dt\\ \phantom{d{x}_i(t)=}+{\sum }_{j=1}^n{\sigma }_{ij}(t,x_i(t),x_i(t-{\tau }_i(t)))d{\omega }_j(t),t\ne k_k,\\ \mathrm{\Delta }{x}_i(t_k)=I_{ik}(x(t_k))=x_i(t_k)-x_i(t_k^{-}),k\in Z^{+},i=1,2,\dots ,n,\end{array}$$

(1)

where n corresponds to the number of units in the neural networks. For $i=1,2,\dots ,n$, $x_i(t)$ corresponds to the state of the i th neuron. $f_j(\cdot )$, $g_j(\cdot )$ are signal transmission functions. $c_i>0$ denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; ${\tau }_j(t)$ corresponds to the transmission delay. $a_{ij}$ represents the elements of the feedback template. $I_i=\widetilde{I_i}+\bigwedge T_{ij}{u}_j+\bigvee H_{ij}{u}_j$, ${\alpha }_{ij}$, ${\beta }_{ij}$, $T_{ij}$ and $H_{ij}$ are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operation, respectively; $u_j$ denotes the external input of the i th neurons. $\widetilde{I_i}$ is the external bias of the i th unit. ${\tau }_i(t)$ is a transmission delay satisfying $0\le {\tau }_i(t)\le \tau$; $\sigma (t,x,y)={({\sigma }_{ij}(t,x_i,y_i))}_{n\times n}\in R^{n\times n}$ is the diffusion coefficient matrix and ${\sigma }_i(t,x_i,y_i)=({\sigma }_{i1}(t,x_i,y_i),{\sigma }_{i2}(t,x_i,y_i),\dots ,{\sigma }_{in}(t,x_i,y_i))$ is the i th row vector of $\sigma (t,x,y)$: $\omega (t)={({\omega }_1(t),{\omega }_2(t),\dots ,{\omega }_n(t))}^T$ is an n-dimensional Brownian motion defined on a complete probability space $(\mathrm{\Omega },F,{\{F_t\}}_{t\ge 0},P)$ with a filtration ${\{F_t\}}_{t\ge 0}$ satisfying the usual conditions (i.e., it is right continuous and $F_0$ contains all P-null sets). $\mathrm{\Delta }{x}_i(t_k)=x_i(t_k^{+})-x_i(t_k^{-})$ is the impulses at moment $t_k$, the fixed moments of time $t_k$ satisfy $0=t_0<t_1<t_2<\dots$ , ${lim}_{k\to +\mathrm{\infty }}{t}_k=+\mathrm{\infty }$; $x_t(s)=x(t+s)$, $s\in [-\tau ,0]$.

System (1) is supplemented with the initial condition given by

$$x_{t_0}(s)=\phi (s),s\in [-\tau ,0],$$

(2)

where $\phi (s)$ is $F_0$-measurable and continuous everywhere except at a finite number of points $t_k$, at which $\phi (t_k^{+})$ and $\phi (t_k^{-})$ exist and $\phi (t_k^{+})=\phi (t_k)$.

Let $P{C}^{1,2}([t_k,t_{k+1})\times R^n;R^{+})$ denote the family of all nonnegative functions $V(t,x)$ on $[t_k,t_{k+1})\times R^n$ which are continuous once differentiable in t and twice differentiable in x. If $V(t,x)\in P{C}^{1,2}([t_k,t_{k+1})\times R^n;R^{+})$, define an operator LV associated with (1) as

$$\begin{array}{rl}LV(t,x)=& V_t(t,x)+\sum _{i=1}^n{V}_{x_i}(t,x)[-c_i{x}_i(t)+\sum _{j=1}^n{a}_{ij}{f}_j(x_j(t))+\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(x_j(t-{\tau }_j(t))\right)\\ +\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j\left(x_j(t-{\tau }_j(t))\right)+I_i]dt+\frac{1}{2}trace[{\sigma }^T{V}_{xx}(t,x)\sigma ],\end{array}$$

(3)

where

$$V_t(t,x)=\frac{\partial V(t,x)}{\partial t},V_{x_i}(t,x)=\frac{\partial V(t,x)}{\partial x_i},V_{xx}(t,x)={\left(\frac{\partial V(t,x)}{\partial x_i\partial x_j}\right)}_{n\times n}.$$

For convenience, we introduce several notations. $x={(x_1,x_2,\dots ,x_n)}^T\in R^n$ denotes a column vector. $\parallel x\parallel$ denotes a vector norm defined by $\parallel x\parallel ={({\sum }_{i=1}^n{|x_i|}^p)}^{1/p}$; $C[X,Y]$ denotes the space of continuous mappings from topological space X to topological space Y. Denoted by $C_{F_0}^b[[-\tau ,0),R^n]$ is the family of all bounded $F_0$-measurable, $C[[-\tau ,0),R^n]$-valued random variables ϕ, satisfying ${\parallel \varphi \parallel }_{L^P}={sup}_{-\tau \le \theta \le 0}E\parallel \varphi (\theta )\parallel <+\mathrm{\infty }$, where $E(\cdot )$ denotes the expectation of a stochastic process.

Throughout the paper, we give the following assumptions.

(A1) The signal transmission functions $f_j(\cdot )$, $g_j(\cdot )$ ($j=1,2,\dots ,n$) are Lipschitz continuous on R with Lipschitz constants ${\mu }_j$ and ${\nu }_j$, namely, for any $u,v\in R$,

$$|f_j(u)-f_j(v)|\le {\mu }_j|u-v|,|g_j(u)-g_j(v)|\le {\nu }_j|u-v|,f_j(0)=g_j(0)=0.$$

(A2) There exist non-negative numbers $s_i$, $w_i$ such that for all $x,y,x^{\prime },y^{\prime }\in R$, $i=1,2,\dots ,n$,

$$[{\sigma }_i(t,x^{\prime },y^{\prime })-{\sigma }_i(t,x,y)]{[{\sigma }_i(t,x^{\prime },y^{\prime })-{\sigma }_i(t,x,y)]}^T\le s_i{|x^{\prime }-x|}^2+w_i{|y^{\prime }-y|}^2.$$

(A3) $I_{ik}(x_i(t_k))=-{\gamma }_{ik}(x_i(t_k)-x_i^{\ast })$, where $x_i^{\ast }$ is the equilibrium point of (1) with the initial condition (2), ${\gamma }_{ik}$ satisfies $0<{\gamma }_{ik}\le 2$.

Definition 1.1 The equilibrium point $x^{\ast }={(x_1^{\ast },x_2^{\ast },\dots ,x_n^{\ast })}^T$ of system (1) is said to be p th moment exponentially stable if there exist positive constants $M>0$, $\lambda >0$ such that

$$E\left({\parallel x(t)-x^{\ast }\parallel }^p\right)\le M{\parallel \phi -x^{\ast }\parallel }^p{e}^{-\lambda (t-t_0)},t\ge t_0,$$

where $x(t)={(x_1(t),x_2(t),\dots ,x_n(t))}^T$ is any solution of system (1) with initial value $x_i(t+s)={\phi }_i(s)\in PC([-\tau ,0],R)$, $i=1,2,\dots ,n$.

When $p=2$, it is usually said to be exponentially stable in mean square.

Lemma 1.1 [34]

Suppose x and y are two states of system (1), then we have

$$|\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j(x)-\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j(y)|\le \sum _{j=1}^n|{\alpha }_{ij}||g_j(x)-g_j(y)|$$

and

$$|\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j(x)-\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ij}{g}_j(y)|\le \sum _{j=1}^n|{\beta }_{ij}||g_j(x)-g_j(y)|.$$

Lemma 1.2 If $a_i>0$ ($i=1,2,\dots ,m$) denote p nonnegative real numbers, then

$$a_1{a}_2\cdots a_m\le \frac{a_1^p+a_2^p+\cdots +a_m^p}{p},$$

(4)

where $p\ge 1$ denotes an integer. A particular form of (4), namely

$$a_1^{p-1}{a}_2\le \frac{(p-1)a_1^p}{p}+\frac{a_2^p}{p}.$$

2 Main results

In this section, we consider the existence and global p th moment exponential stability of system (1).

Lemma 2.1 For two positive real numbers a and b, assume that there exists a constant number $0\le \eta <1$ such that $0<b\le \eta a$. Then the equation

$$\lambda =a-b{e}^{\lambda \tau }$$

(5)

has a unique solution $\lambda >0$.

Proof Let $F(\lambda )=\lambda -a+b{e}^{\lambda \tau }$. It is easy to see $F(0)=-a+b<0$, $F(a)=b{e}^{a\tau }>0$, $F^{\prime }(\lambda )=1+b\tau e^{\lambda \tau }>0$. Thus, $F(\lambda )$ is strictly increasing on $[0,\mathrm{\infty })$. Therefore, Eq. (5) has a unique positive solution $\lambda >0$. □

Lemma 2.2 [18]

For two positive real numbers a and b, assume that there exists a constant number $0\le \eta <1$ such that $0<b\le \eta a$. Assume that $z(t)$ is a nonnegative continuous function on $[t_0-\tau ,t_0]$ and satisfies the following inequality:

$$D^{+}z(t)\le -az(t)+b\parallel z_t\parallel ,\mathit{\text{for}}t>0,$$

(6)

then $z(t)\le \parallel z_{t_0}\parallel e^{-\lambda (t-t_0)}$, where λ is a solution of (5) and the upper right Dini derivative of $z(t)$ is defined as

$$D^{+}z(t)=\underset{\delta \to 0^{+}}{lim}sup\frac{z(t+\delta )-z(t)}{\delta }.$$

Theorem 2.1 Under conditions (A1)-(A3), if there exist constants $K_i>0$ ($i=1,2$), $0<\eta <1$ such that

$$0<K_2<\eta K_1,$$

(7)

where

$$\begin{array}{r}{K}_1=\underset{1\le i\le n}{min}\{p{c}_i-(p-1)\sum _{j=1}^n({\mu }_j|a_{ij}|+{\nu }_j(|{\alpha }_{ij}|+|{\beta }_{ij}|))\\ \phantom{K_1=}-\sum _{j=1}^n|a_{ji}|{\mu }_i-\frac{p(p-1)}{2}{s}_i-\frac{(p-1)(p-2)}{2}{w}_i\},\\ K_2=\underset{1\le i\le n}{max}\{{\nu }_i\sum _{j=1}^n(|{\alpha }_{ji}|+|{\beta }_{ji}|)+(p-1)s_i\}.\end{array}$$

Then $x^{\ast }={(x_1^{\ast },x_2^{\ast },\dots ,x_n^{\ast })}^T$ is a unique equilibrium which is globally pth moment exponential stable.

Proof The proof of existence and uniqueness of equilibrium for the system is similar to that of [39]. So we omit it. Suppose that $x^{\ast }={(x_1^{\ast },x_2^{\ast },\dots ,x_n^{\ast })}^T$ is the unique equilibrium of system (1).

Let

$$\begin{array}{c}{y}_i(t)=x_i(t)-x_i^{\ast },\hfill \\ \widetilde{{\sigma }_{ij}}(t,y_i(t),y_i(t-{\tau }_i(t)))={\sigma }_{ij}(t,y_i(t)+x_i^{\ast },y_i(t-{\tau }_i(t)+x_i^{\ast }))-{\sigma }_{ij}(t,x_i^{\ast },x_i^{\ast }),\hfill \end{array}$$

then system (1) can be transformed into the following equation, for $i=1,2,\dots ,n$:

$$\{\begin{array}{l}d{y}_i(t)=[-c_i{y}_i(t)+{\sum }_{j=1}^n{a}_{ij}(f_j(y_j(t-{\tau }_{ij}(t))+x_j^{\ast })-f_j(x_j^{\ast }))\\ \phantom{d{y}_i(t)=}+({\bigwedge }_{j=1}^n{\alpha }_{ij}{g}_j(y_j(s)+x_j^{\ast })-{\bigwedge }_{j=1}^n{\alpha }_{ij}{g}_j(x_j^{\ast }))\\ \phantom{d{y}_i(t)=}+({\bigvee }_{j=1}^n{\beta }_{ji}{g}_j(y_j(s)+x_j^{\ast })-{\bigvee }_{j=1}^n{\beta }_{ji}{g}_j(x_j^{\ast }))]dt\\ \phantom{d{y}_i(t)=}+{\sum }_{j=1}^n\widetilde{{\sigma }_{ij}}(t,y_i(t),y_i(t-{\tau }_i(t)))d{\omega }_j(t),t\ne t_k,k=1,2,\dots ,\\ \mathrm{\Delta }{y}_i(t_k)=\widetilde{I_{ik}}(y(t_k))=I_{ik}(y(t_k)+x^{\ast })-I_{ik}(x^{\ast }).\end{array}$$

(8)

We define a Lyapunov function $V(t,y(t))={\sum }_{i=1}^n{|y_i(t)|}^p$. Let $t\ge t_0$, $t\in [t_{k-1},t_k)$, then we can get the operator $LV(t,y(t))$ associated with system (8) of the following form:

$$\begin{array}{rcl}LV(t,y(t))& =& p\sum _{i=1}^n{|y_i(t)|}^{p-1}sgn(y_i(t))[-c_i{y}_i(t)+\sum _{j=1}^n{a}_{ij}(f_j(y_j(t-{\tau }_{ij}(t))+x_j^{\ast })-f_j\left(x_j^{\ast }\right))\\ +(\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j(y_j(s)+x_j^{\ast })-\underset{j=1}{\overset{n}{\bigwedge }}{\alpha }_{ij}{g}_j\left(x_j^{\ast }\right))\\ +(\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ji}{g}_j(y_j(s)+x_j^{\ast })-\underset{j=1}{\overset{n}{\bigvee }}{\beta }_{ji}{g}_j\left(x_j^{\ast }\right))]\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{|y_i(t)|}^{p-2}\sum _{j=1}^n\widetilde{{\sigma }_{ij}}(t,y_i(t),y_i(t-{\tau }_i(t)))\\ \le & -p\sum _{i=1}^n{c}_i{|y_i(t)|}^p+p\sum _{i=1}^n\sum _{j=1}^n|a_{ij}|{\mu }_j{|y_i(t)|}^{p-1}|y_j(t)|\\ +p\sum _{i=1}^n\sum _{j=1}^n(|{\alpha }_{ij}|+|{\beta }_{ij}|){\nu }_j{|y_i(t)|}^{p-1}\left|y_j(t-{\tau }_j(t))\right|\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{|y_i(t)|}^{p-2}(s_i{|y_i(t)|}^2+w_i{\left|y_i(t-{\tau }_i(t))\right|}^2)\\ \le & -p\sum _{i=1}^n{c}_i{|y_i(t)|}^p+\sum _{i=1}^n\sum _{j=1}^n|a_{ij}|{\mu }_j((p-1){|y_i(t)|}^p+{|y_j(t)|}^p)\\ +\sum _{i=1}^n\sum _{j=1}^n(|{\alpha }_{ij}|+|{\beta }_{ij}|){\nu }_j((p-1){|y_i(t)|}^p+{\left|y_j(t-{\tau }_j(t))\right|}^p)\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{s}_i{|y_i(t)|}^p+\frac{p-1}{2}\sum _{i=1}^n{w}_i((p-1){|y_i(t)|}^p+2{\left|y_i(t-{\tau }_i(t))\right|}^p)\\ =& -\sum _{i=1}^n[p{c}_i-(p-1)\sum _{j=1}^n({\mu }_j|a_{ij}|+{\nu }_j(|{\alpha }_{ij}|+|{\beta }_{ij}|))\\ -\sum _{j=1}^n|a_{ji}|{\mu }_i-\frac{p(p-1)}{2}{s}_i-\frac{(p-1)(p-2)}{2}{w}_i]{|y_i(t)|}^p\\ +\sum _{i=1}^n[{\nu }_i\sum _{j=1}^n(|{\alpha }_{ji}|+|{\beta }_{ji}|)+(p-1)s_i]{\left|y_i(t-{\tau }_i(t))\right|}^p\\ \le & -K_{1}V(t,y(t))+K_2\underset{t-\tau \le s\le t}{sup}V(s,y(s)),\end{array}$$

(9)

where

$$\begin{array}{r}{K}_1=\underset{1\le i\le n}{min}\{p{c}_i-(p-1)\sum _{j=1}^n({\mu }_j|a_{ij}|+{\nu }_j(|{\alpha }_{ij}|+|{\beta }_{ij}|))\\ \phantom{K_1=}-\sum _{j=1}^n|a_{ji}|{\mu }_i-\frac{p(p-1)}{2}{s}_i-\frac{(p-1)(p-2)}{2}{w}_i\},\\ K_2=\underset{1\le i\le n}{max}\{{\nu }_i\sum _{j=1}^n(|{\alpha }_{ji}|+|{\beta }_{ji}|)+(p-1)s_i\}.\end{array}$$

Firstly, for $t\in [t_0,t_1)$, applying the Ito formula, we obtain that

$$\begin{array}{r}V(t+\delta ,y(t+\delta ))-V(t,y(t))\\ ={\int }_t^{t+\delta }LV(s,y(s))ds+{\int }_t^{t+\delta }{V}_y(s,y(s))\sigma (s,y(s),y(s-\tau (s)))d\omega (s).\end{array}$$

(10)

Since $E[V_y(s,y(s))\sigma (s,y(s),y(s-\tau (s)))d\omega (s)]=0$, taking expectations on both sides of equality (10) and applying the inequality (9) yields

$$\begin{array}{r}E\left(V(t+\delta ,y(t+\delta ))\right)-E\left(V(t,y(t))\right)\\ \le {\int }_t^{t+\delta }[-K_{1}E\left(V(s,y(s))\right)+K_{2}E\left(\underset{s-\tau \le \theta \le s}{sup}V(\theta ,y(\theta ))\right)]ds.\end{array}$$

(11)

Since the Dini derivative $D^{+}$ is

$$D^{+}E\left(V(t,y(t))\right)=\underset{\delta \to 0^{+}}{lim}sup\frac{E(V(t+\delta ,y(t+\delta )))-E(V(t,y(t)))}{\delta },$$

(12)

denote $z(t)=E(V(t,y(t)))$, the preceding result (11) leads directly to

$$D^{+}z(t)\le -K_{1}z(t)+K_2{\parallel z_t\parallel }^p.$$

(13)

Hence, from Lemma 2.2, we have

$$z(t)\le {\parallel z(t_0)\parallel }^p{e}^{-\lambda (t-t_0)}.$$

Namely,

$$E\left[{\parallel x(t)-x^{\ast }\parallel }^p\right]\le M{\parallel \phi -x^{\ast }\parallel }^p{e}^{-\lambda (t-t_0)},$$

where $M=1$, λ is the unique positive solution of the following equation:

$$\lambda =K_1-K_2{e}^{\lambda \tau }.$$

Next, suppose that for $k=1,2,\dots ,m$, the inequality

$$E\left(V(t,y(t))\right)\le M{\parallel \phi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},t\in [t_{k-1},t_k),k=1,2,\dots ,$$

(14)

holds. From (A3), we get

$$\begin{array}{rcl}E\left(V(t_m,y(t_m))\right)& =& \sum _{i=1}^{n}E{|y_i\left(t_m^{-}\right)+\widetilde{I_i}(y_i\left(t_m^{-}\right)+x_i^{\ast })|}^p=E\sum _{i=1}^n{|1-{\gamma }_{im}|}^p{|y_i(t_m^{-})|}^p\\ \le & E\sum _{i=1}^n{\left|y_i\left(t_m^{-}\right)\right|}^p=E(V\left(t_m^{-}\right),y\left(t_m^{-}\right))\le M{\parallel \phi \parallel }_{\tau }^p{e}^{-\lambda (t_m-t_0)}.\end{array}$$

This, together with (14), leads to

$$E\left(V(t,y(t))\right)\le M{\parallel \phi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},t\in [-\tau ,t_m].$$

(15)

On the other hand, for $t\in [t_m,t_{m+1})$, applying the Ito formula, we get

$$\begin{array}{r}V(t,y(t))-V(t_m,y(t_m))\\ ={\int }_{t_m}^{t}LV(s,y(s))ds+{\int }_{t_m}^t{V}_y(s,y(s))\sigma (s,y(s),y(s-\tau (s)))d\omega (s).\end{array}$$

Then, we have

$$EV(t,y(t))-EV(t_m,y(t_m))={\int }_{t_m}^{t}ELV(s,y(s))ds.$$

(16)

So, for small enough $\delta \to 0^{+}$, we have

$$EV(t+\delta ,y(t+\delta ))-EV(t_m,y(t_m))={\int }_{t_m}^{t+\delta }ELV(s,y(s))ds.$$

(17)

From (16) and (17), we have

$$\begin{array}{r}EV(t+\delta ,y(t+\delta ))-EV(t,y(t))\\ ={\int }_t^{t+\delta }ELV(s,y(s))ds\\ \le {\int }_t^{t+\delta }[-K_{1}E\left(V(s,y(s))\right)+K_{2}E\left(\underset{s-\tau \le \theta \le s}{sup}V(\theta ,y(\theta ))\right)]ds.\end{array}$$

(18)

Similarly, we obtain

$$E\left(V(t,y(t))\right)\le M{\parallel \phi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},t\in [t_m,t_{m+1}).$$

(19)

Hence, by the mathematical induction, for any $k=1,2,\dots$ , we conclude that

$$E\left(V(t,y(t))\right)\le M{\parallel \phi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},t\ge t_0,$$

which implies that the equilibrium point of the impulsive system (1) is p th moment exponentially stable. This completes the proof of the theorem. □

3 Comparisons and remarks

It can be easily seen that many neural networks are special cases of system (1). Thus, in this section, we give some comparisons and remarks.

Suppose that $I_{ik}(x)=x\in R^n$, system (1) becomes the stochastic fuzzy cellular neural networks with time-varying delays.

$$\{\begin{array}{l}d{x}_i(t)=[-c_i{x}_i(t)+{\sum }_{j=1}^n{a}_{ij}{f}_j(x_j(t))+{\bigwedge }_{j=1}^n{\alpha }_{ij}{g}_j(x_j(t-{\tau }_j(t)))\\ \phantom{d{x}_i(t)=}+{\bigvee }_{j=1}^n{\beta }_{ij}{g}_j(x_j(t-{\tau }_j(t)))+I_i]dt\\ \phantom{d{x}_i(t)=}+{\sum }_{j=1}^n{\sigma }_{ij}(t,x_i(t),x_i(t-{\tau }_i(t)))d{\omega }_j(t),t\ge t_0,\\ x_i(t_0+s)={\phi }_i(s),-\mathrm{\infty }<s<0,i=1,2,\dots ,n.\end{array}$$

(20)

For (20), we have the following corollary by Theorem 2.1.

Corollary 3.1 If (A1)-(A2) hold, if there exist constants $K_i>0$ ($i=1,2$), $0<\eta <1$ such that

$$0<K_2<\eta K_1,$$

where

$$\begin{array}{r}{K}_1=\underset{1\le i\le n}{min}\{p{c}_i-(p-1)\sum _{j=1}^n({\mu }_j|a_{ij}|+{\nu }_j(|{\alpha }_{ij}|+|{\beta }_{ij}|))\\ \phantom{K_1=}-\sum _{j=1}^n|a_{ji}|{\mu }_i-\frac{p(p-1)}{2}{s}_i-\frac{(p-1)(p-2)}{2}{w}_i\}>0,\\ K_2=\underset{1\le i\le n}{max}\{{\nu }_i\sum _{j=1}^n(|{\alpha }_{ji}|+|{\beta }_{ji}|)+(p-1)s_i\},\end{array}$$

then the unique equilibrium $x^{\ast }={(x_1^{\ast },x_2^{\ast },\dots ,x_n^{\ast })}^T$ of system (20) is globally pth moment exponential stable.

If we do not consider fuzzy AND and fuzzy OR operations and when $I_{ik}(x)=x\in R^n$ in system (1), then system (1) becomes impulsive stochastic cellular neural networks with time-varying delays

$$\{\begin{array}{l}d{x}_i(t)=[-c_i{x}_i(t)+{\sum }_{j=1}^n{a}_{ij}{f}_j(x_j(t))+{\sum }_{j=1}^n{\alpha }_{ij}{g}_j(x_j(t-{\tau }_j(t)))\\ \phantom{d{x}_i(t)=}+{\sum }_{j=1}^n{\beta }_{ij}{g}_j(x_j(t-{\tau }_j(t)))+I_i]dt\\ \phantom{d{x}_i(t)=}+{\sum }_{j=1}^n{\sigma }_{ij}(t,x_i(t),x_i(t-{\tau }_i(t)))d{\omega }_j(t),t\ge t_0,\\ x_i(t_0+s)={\phi }_i(s),-\mathrm{\infty }<s<0,i=1,2,\dots ,n.\end{array}$$

(21)

Remark 3.1 The stability of system (21) has been investigated in [31]. In [31], authors required the differentiability and monotonicity of a time delays function which satisfied ${\tau }_j^{\prime }(t)\le \xi <1$. Hence, this assumption may impose a very strict constraint on the model because time delays sometimes vary dramatically with time in real circuits. Obviously, Theorem 2.1 does not require these conditions.

Remark 3.2 In Theorem 2.1, if we do not consider fuzzy AND and OR operation, it becomes traditional cellular neural networks. The results in [32] are the corollary of Theorem 2.1. Therefore the results of this paper extend the previous known publication.

4 An example

Example 4.1 Consider the following stochastic fuzzy neural networks with time-varying delays and time-varying delays and impulses

$$\{\begin{array}{l}d{x}_1(t)=[-11x_1(t)+0.1f_1(x_1(t))+0.7f_1(x_2(t))+{\bigwedge }_{j=1}^2{\alpha }_{1j}{f}_j(x_j(t-{\tau }_j(t)))\\ \phantom{d{x}_1(t)=}+{\bigvee }_{j=1}^2{\beta }_{1j}{f}_j(x_j(t-{\tau }_j(t)))+{\bigwedge }_{j=1}^2{T}_{1j}{u}_j+{\bigvee }_{j=1}^2{H}_{1j}{u}_j]dt\\ \phantom{d{x}_1(t)=}+{\sigma }_{11}(t,x_1(t),x_1(t-{\tau }_1(t)))d{\omega }_1\\ \phantom{d{x}_1(t)=}+{\sigma }_{12}(t,x_1(t),x_1(t-{\tau }_1(t)))d{\omega }_2,t\ne t_k,\\ d{x}_2(t)=[-17x_2(t)-0.6f_2(x_1(t))+0.3f_2(x_2(t))+{\bigwedge }_{j=1}^2{\alpha }_{2j}{f}_j(x_j(t-{\tau }_j(t)))\\ \phantom{d{x}_2(t)=}+{\bigvee }_{j=1}^2{\beta }_{2j}{f}_j(x_j(t-{\tau }_j(t)))+{\bigwedge }_{j=1}^2{T}_{2j}{u}_j+{\bigvee }_{j=1}^2{H}_{2j}{u}_j]dt\\ \phantom{d{x}_2(t)=}+{\sigma }_{21}(t,x_2(t),x_2(t-{\tau }_2(t)))d{\omega }_1\\ \phantom{d{x}_2(t)=}+{\sigma }_{22}(t,x_2(t),x_2(t-{\tau }_2(t)))d{\omega }_2,t\ne t_k,\\ \mathrm{\Delta }{x}_1(t_k)=-(1+0.3sin(1+k^2)x_1(t_k^{-})),\\ \mathrm{\Delta }{x}_2(t_k)=-(1+0.6sin(1+k)x_2(t_k^{-})),\end{array}$$

(22)

where $t_0=0$, $t_k=t_{k-1}+0.2k$, $k=1,2,\dots$ , and

$$\begin{array}{c}{f}_i(r)=g_i(r)=\frac{1}{2}(|r+1|-|r-1|),{\tau }_j(t)=0.3|sint|+0.1\le 0.4,i,j=1,2,\hfill \\ {\alpha }_{11}=\frac{5}{3},{\alpha }_{21}=\frac{1}{3},{\alpha }_{12}=-\frac{1}{4},{\alpha }_{22}=\frac{3}{4};\hfill \\ {\beta }_{11}=\frac{1}{3},{\beta }_{21}=\frac{2}{3},{\beta }_{12}=-\frac{1}{4},{\beta }_{22}=\frac{3}{4};\hfill \\ {\sigma }_{11}(x,y)=0.2x-0.1y,{\sigma }_{12}(x,y)=0.3x+0.1y,\hfill \\ {\sigma }_{21}(x,y)=0.1x+0.2y,{\sigma }_{22}(x,y)=0.2x+0.1y,\hfill \end{array}$$

and $T_{ij}=H_{ij}=S_{ij}=L_{ij}=u_i=u_j=1$ ($i,j=1,2$).

Let $p=2$, obviously, ${\mu }_i={\mu }_i=1$, $i=1,2$. By simple computation, we can easily get that $K_1=min\{19.4,31.6\}=19.4$, $K_2=max\{6,2\}=6$. Letting $\eta =0.4$, we have

$$\eta K_1>K_2=6>0.$$

(23)

Thus, system (22) satisfies assumptions (A1)-(A3). It follows from Theorem 2.1 that system (22) is exponentially stable in mean square.