The p th moment exponential stability of stochastic cellular neural networks with impulses

Abstract

This paper studies the p th moment exponential stability of stochastic cellular neural networks with time-varying delays under impulsive perturbations. Based on the Lyapunov function, Razumikhin theory, stochastic analysis and differential inequality technique, criteria on the p th moment exponential stability of this model are derived. These results generalize and improve some of the existing ones. A numerical example illustrates the effectiveness and improvements of our results.

1 Introduction

Since Chua and Yang [1, 2] introduced a cellular neural network in 1988, it has received great attention because of its various applications such as classification of patterns, associative memories and optimization, etc. It should be pointed out that time delays are commonly encountered in real systems, which are the source of oscillation and instability both in biological and artificial neural networks, hence it is necessary and important to discuss the delayed cellular neural networks models. Up to now, many results on the stability of delayed neural networks have been developed [3–5]. In fact, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, noise is unavoidable and should be taken into consideration in modeling. Some recent results of stochastic cellular neural networks with delays can be found in [6–15].

On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks. In recent years, the dynamic analysis of neural networks with impulsive and stochastic effects has been an attractive topic for many researchers, and a large number of stability criteria of these systems have been reported; see [3, 4, 10–12, 16, 17].

In [8], Sun et al. investigated the following stochastic cellular neural networks model with time-varying delays:

$$\begin{array}{rcl}d{x}_i(t)& =& [-c_i{x}_i(t)+\sum _{j=1}^n{a}_{ij}{f}_j(x_j(t))+\sum _{j=1}^n{b}_{ij}{g}_j\left(x_j(t-{\tau }_j(t))\right)+I_i]dt\\ +\sum _{l=1}^m{\sigma }_{il}(t,x_i(t),x_i(t-{\tau }_i(t)))d{w}_l(t),\end{array}$$

(1.1)

where $x_i(t)$ denotes the potential (or voltage) of a cell i at time t; $\mathrm{\Lambda }=\{1,2,\dots ,n\}$, n corresponds to the number of units in a neural network; $f_j(\cdot )$, $g_j(\cdot )$ are activation 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; $a_{ij}$ and $b_{ij}$ denote the strengths of connectivity between cells i and j, respectively; $I_i$ denotes the external bias on the i th unit, ${\tau }_i(t)$ satisfies $0\le {\tau }_i(t)\le \tau$ and it is a transmission delay. $\sigma (t,x,y)={({\sigma }_{il}(t,x_i,y_i))}_{n\times m}\in {\mathbb{R}}^{n\times m}$ is the diffusion coefficient matrix and ${\sigma }_i(t,x_i,y_i)=({\sigma }_{i1}(t,x_i,y_i),\dots ,{\sigma }_{im}(t,x_i,y_i))$ is the i th row vector of $\sigma (t,x,y)$. $w(t)={(w_1(t),w_2(t),\dots ,w_m(t))}^T$ is an m-dimensional Brownian motion defined on a complete probability space $(\mathrm{\Omega },\mathcal{F},P)$ with a natural filtration ${\{F_t\}}_{t\ge 0}$.

They investigated the p th moment exponential stability with the help of the method of variation parameter and inequality technique, where $p\ge 2$ denotes a positive constant. More precisely, they established the following fundamental assumptions:

• (H) For each $j=1,2,\dots ,n$, ${\tau }_j(t)$ is a differentiable function, namely, there exists ζ such that

  $${\dot{\tau }}_j(t)\le \zeta <1.$$

(H1) Functions $f_j(\cdot )$ and $g_j(\cdot )$ are Lipschitz-continuous on ℝ with Lipschitz constants $L_i>0$, $N_i>0$. That is, for all $x,y\in \mathbb{R}$, $i\in \mathrm{\Lambda }$,

$$|f_i(x)-f_i(y)|\le L_i|x-y|,|g_i(x)-g_i(y)|\le N_i|x-y|.$$

(H2) There exist nonnegative constants $l_i$, $e_i$, such that for all $x,y,x^{\mathrm{\prime }},y^{\mathrm{\prime }}\in \mathbb{R}$, $i\in \mathrm{\Lambda }$,

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

Huang et al. [6] studied (1.1) and obtained the p th moment exponential stability by using Dini-derivative and Halanay-type inequality without assumption (H). When $k_1^{\mathrm{\prime }}>k_2^{\mathrm{\prime }}$, the equilibrium point of the system (1.1) is p th moment exponentially stable, where

(1.2)

where ${\mu }_i$ ($i\in \mathrm{\Lambda }$) are positive constants.

Very recently, Li [11] generalized (1.1); he considered a stochastic cellular neural network under impulsive perturbations. The condition $k_1^{\mathrm{\prime }}>k_2^{\mathrm{\prime }}$ is also needed to ensure exponential stability in mean square.

We have a question whether the condition $k_1^{\mathrm{\prime }}>k_2^{\mathrm{\prime }}$ in the theorems [6, 11, 12] is an essential condition or not for the equilibrium point of (1.1) to be p th moment exponentially stable.

In this paper, we solve this question and obtain the improved version of the p th moment exponential stability by applying Lyapunov functions, Razumikhin theory and inequality technique. An example is also provided to illustrate the effectiveness of the new results.

2 Preliminaries

In this paper, we study stochastic cellular neural networks with impulses described by the delayed differential equations

$$\{\begin{array}{c}d{x}_i(t)=[-c_i{x}_i(t)+{\sum }_{j=1}^n{a}_{ij}{f}_j(x_j(t))+{\sum }_{j=1}^n{b}_{ij}{g}_j(x_j(t-{\tau }_j(t)))+I_i]dt\hfill \\ \phantom{d{x}_i(t)=}+{\sum }_{l=1}^m{\sigma }_{il}(t,x_i(t),x_i(t-{\tau }_i(t)))d{w}_l(t),t\ne t_k,\hfill \\ \mathrm{\Delta }{x}_i(t_k)=p_{ik}(x(t_k))=x_i(t_k)-x_i(t_k^{-}),k\in {\mathbb{Z}}^{+},i\in \mathrm{\Lambda },\hfill \end{array}$$

(2.1)

where $\{t_k\}$ is the time sequence and satisfies $0=t_0<t_1<t_2<\cdots <t_k<t_{k+1}\cdots$ , ${lim}_{k\to \mathrm{\infty }}{t}_k=\mathrm{\infty }$; $x_t(s)=x(t+s)$, $s\in [-\tau ,0]$. For $k=1,2,\dots$ , $p_{ik}(x(t_k))$ represents the abrupt change of the state $x_i(t)$ at the impulsive moments $t_k$.

System (2.1) is supplemented with the initial condition given by

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

(2.2)

where $\psi (s)$ is ${\mathcal{F}}_0$-measurable and continuous everywhere except at a finite number of points $t_k$, at which $\psi (t_k^{+})$ and $\psi (t_k^{-})$ exist and $\psi (t_k^{+})=\psi (t_k)$.

Let $P{C}^{1,2}([t_k,t_{k+1})\times {\mathbb{R}}^n;{\mathbb{R}}^{+})$ denote the family of all nonnegative functions $V(t,x)$ on $[t_k,t_{k+1})\times {\mathbb{R}}^n$ which are continuous once differentiable in t and twice differentiable in x. If $V(t,x)\in P{C}^{1,2}([t_k,t_{k+1})\times {\mathbb{R}}^n;{\mathbb{R}}^{+})$, define an operator $\mathcal{L}V$ associated with (2.1) as

$$\begin{array}{rcl}\mathcal{L}V(t,x)& =& V_t(t,x)+\sum _{i=1}^n{V}_{x_i}(t,x)[-c_i{x}_i(t)+\sum _{j=1}^n{a}_{ij}{f}_j(x_j(t))+\sum _{j=1}^n{b}_{ij}{g}_j\left(x_j(t-{\tau }_j(t))\right)+I_i]\\ +\frac{1}{2}trace[{\sigma }^T{V}_{xx}(t,x)\sigma ],\end{array}$$

(2.3)

where

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

Throughout this paper, the following standard hypothesis is needed:

(H3) $p_i(x_i(t_k))=-{\beta }_{ik}(x_i(t_k)-x_i^{\ast })$, where $x_i^{\ast }$ is the equilibrium point of (2.1) with the initial condition (2.2), ${\beta }_{ik}$ satisfies $|1-{\beta }_{ik}|\le d_k$, $d_k$ is a positive constant.

Let $y_i(t)=x_i(t)-x^{\ast }$, then (2.1) can be written by

$$\{\begin{array}{c}d{y}_i(t)=[-c_i{y}_i(t)+{\sum }_{j=1}^n{a}_{ij}{\tilde{f}}_j(y_j(t))+{\sum }_{j=1}^n{b}_{ij}{\tilde{g}}_j(y_j(t-{\tau }_j(t)))]dt\hfill \\ \phantom{d{y}_i(t)=}+{\sum }_{l=1}^m{\tilde{\sigma }}_{il}(t,y_i(t),y_i(t-{\tau }_i(t)))d{w}_l(t),t\ne t_k,\hfill \\ \mathrm{\Delta }{y}_i(t_k)={\tilde{p}}_{ik}(y(t_k)),k\in Z^{+},i\in \mathrm{\Lambda },\hfill \end{array}$$

(2.4)

where

$$\begin{array}{c}{\tilde{f}}_j(y_j(t))=f_j(y_j(t)+x_j^{\ast })-f_j\left(x_j^{\ast }\right),{\tilde{g}}_j\left(y_j(t-{\tau }_j(t))\right)=g_j(y_j(t-{\tau }_j(t))+x_j^{\ast })-g_j\left(x_j^{\ast }\right),\hfill \\ {\tilde{\sigma }}_{ij}(t,y_i(t),y_i(t-{\tau }_i(t)))={\sigma }_{ij}(t,y_i(t)+x_i^{\ast },y_i(t-{\tau }_i(t))+x_i^{\ast })-{\sigma }_{ij}(t,x_i^{\ast },x_i^{\ast }),\hfill \\ {\tilde{p}}_{ik}(y(t_k))=p_{ik}(y(t_k)+x^{\ast })-p_{ik}\left(x^{\ast }\right).\hfill \end{array}$$

In the following, for further study, we first give the following definitions and lemmas.

$\parallel x\parallel$ denotes a vector norm defined by

$${\parallel x\parallel }^p=\sum _{i=1}^n{|x_i(t)|}^p,{\parallel \psi \parallel }_{\tau }=\underset{s\in [-\tau ,0]}{sup}\parallel \psi (s)\parallel .$$

Definition 2.1 (Mao [18])

The equilibrium point $x^{\ast }={(x_1^{\ast },x_2^{\ast },\dots ,x_n^{\ast })}^T$ of the system (2.1) is said to be p th moment exponentially stable if there exist $\lambda >0$ and $M>0$ such that

$$E{\parallel x(t)-x^{\ast }\parallel }^p\le M{\parallel x_0-x^{\ast }\parallel }^p{e}^{-\lambda (t-t_0)},\mathrm{\forall }t\ge t_0,\mathrm{\forall }{x}_0\in {\mathbb{R}}^n,x(t_0)=x_0.$$

In such a case,

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{t}lnE\left({\parallel x(t)-x^{\ast }\parallel }^p\right)\le -\lambda .$$

(2.5)

The right-hand side of (2.5) is commonly known as the p th moment Lyapunov exponent of this solution.

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

Lemma 2.2 If $a_i$ ($i=1,2,\dots ,p$) denote p nonnegative real numbers, then

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

(2.6)

where $p\ge 1$ denotes an integer.

A particular form of (2.6), namely

$$a_1^{p-1}{a}_2\le \frac{(p-1)a_1^p}{p}+\frac{a_2^p}{p},\mathit{\text{for }}p=1,2,3,\dots .$$

3 Main result

Theorem 3.1 Assume that (H1)-(H3) hold; furthermore, let

• (i) there exist $\sigma >0$, $\lambda >0$ such that $-k_1+\frac{k_2{e}^{\lambda \tau }}{d_{k-1}^p}\le \sigma -\lambda$;

• (ii) $pln{d}_{k-1}<-(\sigma +\lambda )(t_k-t_{k-1})$, $k\in \mathbb{N}$, then the equilibrium point of (2.1) is pth moment exponentially stable.

Proof We define a Lyapunov function $V(t,y(t))={\sum }_{i=1}^n{|y_i(t)|}^p={\parallel y(t)\parallel }^p$. Let $t\ge t_0$ and $t\in [t_{k-1},t_k)$, then we can get the operator $\mathcal{L}V(t,y)$ associated with the system (2.4) of the form as follows:

$$\begin{array}{rcl}\mathcal{L}V(t,y)& =& p\sum _{i=1}^n{|y_i(t)|}^{p-1}sgn(y_i(t))[-c_i{y}_i(t)+\sum _{j=1}^n{a}_{ij}{\tilde{f}}_j(y_j(t))+\sum _{j=1}^n{b}_{ij}{\tilde{g}}_j\left(y_j(t-{\tau }_j(t))\right)]\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{|y_i(t)|}^{p-2}\sum _{l=1}^m{\tilde{\sigma }}_{il}^2(t,y_i(t),y_i(t-{\tau }_i(t)))\\ \le & -p\sum _{i=1}^n{c}_i{|y_i(t)|}^p+p\sum _{i=1}^n\sum _{j=1}^n|a_{ij}|L_j{|y_i(t)|}^{p-1}|y_j(t)|\\ +p\sum _{i=1}^n\sum _{j=1}^n|b_{ij}|N_j{|y_i(t)|}^{p-1}\left|y_j(t-{\tau }_j(t))\right|\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{|y_i(t)|}^{p-2}(e_i{|y_i(t)|}^2+l_i{\left|y_i(t-{\tau }_i(t))\right|}^2)\\ \le & -p\sum _{i=1}^n{c}_i{|y_i(t)|}^p+\sum _{i=1}^n\sum _{j=1}^n|a_{ij}|L_j((p-1){|y_i(t)|}^p+{|y_j(t)|}^p)\\ +\sum _{i=1}^n\sum _{j=1}^n|b_{ij}|N_j((p-1){|y_i(t)|}^p+{\left|y_j(t-{\tau }_j(t))\right|}^p)\\ +\frac{p(p-1)}{2}\sum _{i=1}^n{e}_i{|y_i(t)|}^p+\frac{(p-1)}{2}\sum _{i=1}^n{l}_i((p-2){|y_i(t)|}^p+2{\left|y_i(t-{\tau }_i(t))\right|}^p)\\ =& -\sum _{i=1}^n[p{c}_i-(p-1)\sum _{j=1}^n(L_j|a_{ij}|+N_j|b_{ij}|)-\sum _{j=1}^n|a_{ji}|L_i-\frac{p(p-1)}{2}{e}_i]{|y_i(t)|}^p\\ -\sum _{i=1}^n\frac{(p-1)(p-2)}{2}{l}_i{|y_i(t)|}^p+\sum _{i=1}^n[N_i\sum _{j=1}^n|b_{ji}|+(p-1)l_i]{\left|y_i(t-{\tau }_i(t))\right|}^p\\ \le & -k_{1}V(t,y(t))+k_2\underset{t-\tau \le s\le t}{sup}V(s,y(s)),\end{array}$$

(3.1)

where

Let $\gamma ={inf}_{k\in \mathrm{\Lambda }}\frac{1}{d_{k-1}^p}$, there exist $\sigma >0$, $\lambda >0$ such that

$$-k_1+k_2\gamma e^{\lambda \tau }\le -k_1+\frac{k_2{e}^{\lambda \tau }}{d_{k-1}^p}\le \sigma -\lambda$$

(3.2)

and

$$ln\gamma +\lambda \tau -(\sigma +\lambda )(t_k-t_{k-1})>0.$$

(3.3)

Hence, we can choose $M\ge 1$ such that

$$e^{(\sigma +\lambda )(t_1-t_0)}\le M\le \gamma e^{\lambda \tau }.$$

(3.4)

For convenience, we denote that $\varphi (s)=\psi (s)-x^{\ast }$ for $s\in [-\tau ,0)$.

It is obvious that

$${\parallel \varphi \parallel }_{\tau }^p\le {\parallel \varphi \parallel }_{\tau }^p{e}^{\sigma (t_1-t_0)}\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}.$$

(3.5)

Now, we should prove

$$E{\parallel y(t)\parallel }^p\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},\mathrm{\forall }t\ge t_0.$$

(3.6)

Firstly, we prove when $t\in [t_0,t_1)$,

$$EV(t,y(t))=E{\parallel y(t)\parallel }^p\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)}.$$

(3.7)

If (3.7) is not true, there exists $\overline{t}\in [t_0,t_1)$ such that

$$\begin{array}{rcl}EV(\overline{t},y(\overline{t}))& >& M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}>{\parallel \varphi \parallel }_{\tau }^p{e}^{\sigma (t_1-t_0)}\\ >& {\parallel \varphi \parallel }_{\tau }^p\ge EV(t_0+s,y(t_0+s)),s\in [-\tau ,0].\end{array}$$

(3.8)

Since $V(t,y(t))$ is continuous on $[t_0,t_1)$, which implies that there exists $\hat{t}\in [t_0,\overline{t})$ such that

$$EV(\hat{t},y(\hat{t}))=M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}$$

and

$$EV(t,y(t))\le EV(\hat{t},y(\hat{t})),\mathrm{\forall }t\in [t_0-\tau ,\hat{t}),$$

then there exists some $\tilde{t}\in [t_0,\hat{t})$ satisfying

$$EV(\tilde{t},y(\tilde{t}))={\parallel \varphi \parallel }_{\tau }^p$$

and

$$EV(t,y(t))\ge EV(\tilde{t},y(\tilde{t})),\mathrm{\forall }t\in (\tilde{t},\hat{t}).$$

Hence, for any $s\in [-\tau ,0]$, $t\in (\tilde{t},\hat{t})$,

$$EV(t+s,y(t+s))\le EV(\hat{t},y(\hat{t}))<\gamma e^{\lambda \tau }{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}<\gamma e^{\lambda \tau }{\parallel \varphi \parallel }_{\tau }^p\le \gamma e^{\lambda \tau }\mathbb{E}V(t,y(t)).$$

By (3.1) and (3.2), we get

$$E\mathcal{L}V(t,y)\le (-k_1+k_2\gamma e^{\lambda \tau })EV(t,y(t))\le (\sigma -\lambda )EV(t,y(t)),\mathrm{\forall }t\in (\tilde{t},\hat{t}).$$

Then

$$EV(\hat{t},y(\hat{t}))\le EV(\tilde{t},y(\tilde{t}))e^{(\sigma -\lambda )(\hat{t}-\tilde{t})}<{\parallel \varphi \parallel }_{\tau }^p{e}^{\sigma (t_1-t_0)}\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_1-t_0)}=EV(\hat{t},y(\hat{t})),$$

which is a contradiction. Hence, (3.7) holds.

Next, we will show

$$EV(t,y(t))\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},t\in [t_{k-1},t_k),k\in \mathrm{\Lambda }.$$

(3.9)

Assuming (3.9) holds for $k=1,2,\dots ,m$, we shall show that it holds for $k=m+1$, i.e.,

$$EV(t,y(t))=E{\parallel y(t)\parallel }^p\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},\mathrm{\forall }t\in [t_m,t_{m+1}).$$

(3.10)

Suppose (3.10) is not true. Then we define $\overline{t}\in [t_m,t_{m+1})$ such that

$$EV(\overline{t},y(\overline{t}))>M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)}.$$

From (H3) we get

$$\begin{array}{rcl}EV(t_m,y(t_m))& =& \sum _{i=1}^{n}E{|y_i\left(t_m^{-}\right)+{\tilde{p}}_i(y_i\left(t_m^{-}\right)+x_i^{\ast })|}^p=E\sum _{i=1}^n{|1-{\beta }_{im}|}^p{\left|y_i\left(t_m^{-}\right)\right|}^p\\ \le & E\sum _{i=1}^n{d}_m^p{\left|y_i\left(t_m^{-}\right)\right|}^p=d_m^{p}EV(t_m^{-},y\left(t_m^{-}\right))\le d_m^{p}M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_m-t_0)}\\ =& d_m^{p}M{\parallel \varphi \parallel }_{\tau }^p{e}^{\lambda (\overline{t}-t_m)}{e}^{-\lambda (\overline{t}-t_0)}<d_m^{p}M{\parallel \varphi \parallel }_{\tau }^p{e}^{\lambda (t_{m+1}-t_m)}{e}^{-\lambda (\overline{t}-t_0)}\\ <& M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (\overline{t}-t_0)}\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t_m-t_0)},\end{array}$$

which implies $\overline{t}\in (t_m,t_{m+1})$. Let

$$\overline{t}=inf\{t\in (t_m,t_{m+1}):EV(\overline{t},y(\overline{t}))=M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (\overline{t}-t_0)}\},$$

then for $t\in (t_m-\tau ,\overline{t})$, we can get $EV(t,y(t))\le EV(\overline{t},y(\overline{t}))$.

Hence, there exists $t^{\ast }\in (t_m,\overline{t})$ such that

$$EV(t^{\ast },y\left(t^{\ast }\right))=d_m^{p}M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (\overline{t}-t_0)}{e}^{\lambda (t_{m+1}-t_m)}$$

and

$$EV(t,y(t))\ge EV(t^{\ast },y\left(t^{\ast }\right)),t\in [t^{\ast },\overline{t}).$$

On the other hand, for any $t\in [t^{\ast },\overline{t})$, $s\in [-\tau ,0]$, either $t+s\in [t_m-\tau ,t_m)$ or $t+s\in [t_m,\overline{t}]$.

If $t+s\in [t_m-\tau ,t_m)$, we can obtain

$$EV(t+s,y(t+s))\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t+s-t_0)}<M{\parallel \varphi \parallel }_{\tau }^p{e}^{\lambda \tau }{e}^{-\lambda (\overline{t}-t_0)}{e}^{\lambda (t_{m+1}-t_m)}=\frac{e^{\lambda \tau }}{d_m^p}EV(t^{\ast },y\left(t^{\ast }\right)).$$

If $t+s\in [t_m,\overline{t}]$, we can get

$$EV(t+s,y(t+s))\le EV(\overline{t},y(\overline{t}))<\frac{e^{\lambda \tau }}{d_m^p}EV(t^{\ast },y\left(t^{\ast }\right)).$$

Then for any $s\in [-\tau ,0]$, we get

$$EV(t+s,y(t+s))\le \frac{e^{\lambda \tau }}{d_m^p}EV(t^{\ast },y\left(t^{\ast }\right))\le \frac{e^{\lambda \tau }}{d_m^p}EV(t,y(t)),t\in [t^{\ast },\overline{t}).$$

Hence,

$$\mathcal{L}V(t,y)\le (-k_1+\frac{k_2{e}^{\lambda \tau }}{d_m^p})EV(t^{\ast },y\left(t^{\ast }\right))\le (\sigma -\lambda )EV(t,y(t)).$$

Then

$$EV(\overline{t},y(\overline{t}))\le EV(t^{\ast },y\left(t^{\ast }\right))e^{(\sigma -\lambda )(\overline{t}-t^{\ast })}=d_m^{p}M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (\overline{t}-t_0)}{e}^{\lambda (t_{m+1}-t_m)}{e}^{(\sigma -\lambda )(\overline{t}-t^{\ast })}.$$

From the condition (ii), it is obvious that

$$d_m^p{e}^{\lambda (t_{m+1}-t_m)}{e}^{(\sigma -\lambda )(\overline{t}-t^{\ast })}<1.$$

Hence,

$$EV(\overline{t},y(\overline{t}))<M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (\overline{t}-t_0)}=EV(\overline{t},y(\overline{t})),$$

which is a contradiction. Hence, (3.10) holds.

By induction, we can obtain that (3.9) holds for any $k\in \mathrm{\Lambda }$, i.e.,

$$EV(t,y(t))\le M{\parallel \varphi \parallel }_{\tau }^p{e}^{-\lambda (t-t_0)},\mathrm{\forall }t\ge t_0,$$

which implies that the equilibrium point of the impulsive system (2.1) is p th moment exponentially stable. This completes the proof of the theorem. □

Theorem 3.2 Assume that (H1)-(H3) hold, ${\mu }_i(i\in \mathrm{\Lambda })>0$,

• (i) if there exist $\sigma >0$, $\lambda >0$ such that $-k_1+{\frac{k_2}{e}}^{\lambda \tau }{d}_{k-1}^p\le \sigma -\lambda$;

• (ii) $pln{d}_{k-1}<-(\sigma +\lambda )(t_k-t_{k-1})$, $k\in \mathbb{N}$,

where

then the equilibrium point of the system (2.1) is pth moment exponentially stable.

Proof Let $V(t,y(t))={\sum }_{i=1}^n{\mu }_i{|y_i(t)|}^p$, the proof of the theorem is similar to that of Theorem 3.1 hence it is omitted. □

Corollary 3.3 Assume that (H1)-(H3) hold, ${\mu }_i(i\in \mathrm{\Lambda })>0$,

$$\begin{array}{c}{k}_1=\underset{1\le i\le n}{min}\{2c_i-\sum _{j=1}^n(L_j|a_{ij}|+N_j|b_{ij}|)-\sum _{j=1}^n\frac{{\mu }_j}{{\mu }_i}|a_{ji}|L_i-e_i\}>0,\hfill \\ k_2=\underset{1\le i\le n}{max}\{N_i\sum _{j=1}^n\frac{{\mu }_j}{{\mu }_i}|b_{ji}|+l_i\},\hfill \end{array}$$

• (i) if there exist $\sigma >0$, $\lambda >0$ such that $-k_1+\frac{k_2{e}^{\lambda \tau }}{d_{k-1}^2}\le \sigma -\lambda$;

• (ii) $2ln{d}_{k-1}<-(\sigma +\lambda )(t_k-t_{k-1})$, $k\in \mathbb{N}$,

then the equilibrium point of the system (2.1) is exponentially stable in mean square.

Remark 3.4 In many stability results for stochastic cellular neural networks, $E\mathcal{L}V\le 0$ is an important condition for their conclusions [13–15], which means that the origin systems without impulses need to be stable. However, by constructing the impulses, we do not need this condition to ensure the equilibrium point of the impulsive system (2.1) is p th moment exponentially stable. Our results show that impulses play an important role in the p th moment exponential stability for the stochastic cellular neural network with time delay, even if the corresponding systems may be unstable themselves. It should be mentioned that our results develop an effective impulse control strategy to stabilize underlying retarded cellular neural networks. And it is particularly meaningful for some practical applications.

Remark 3.5 It is important to emphasize that, in contrast to some existing exponential stability results, see [6, 11, 12, 19], the condition $k_1>k_2$ is needed to ensure the equilibrium point of the system (2.1) is p th moment exponentially stable, while in our paper we omit it and obtain the results.

4 Illustrative example

In the following, we will give an example to illustrate the advantages of our results.

Example 1 Consider the following model:

$$\{\begin{array}{c}d{x}_i(t)=[-c_i{x}_i(t)+{\sum }_{j=1}^2{a}_{ij}{f}_j(x_j(t))+{\sum }_{j=1}^2{b}_{ij}{g}_j(x_j(t-{\tau }_j(t)))]dt\hfill \\ \phantom{d{x}_i(t)=}+{\sigma }_i(t,x_i(t),x_i(t-{\tau }_i(t)))dw(t),t\ne t_k,\hfill \\ x_i(t_k)=(1-{\beta }_{ik})x_i(t_k^{-})k\in {\mathbb{Z}}^{+},i\in \mathrm{\Lambda }=(1,2),\hfill \end{array}$$

(4.1)

where $f_i(x)=g_i(x)=tanh(x)$, $0\le {\tau }_i(t)\le \tau =0.5$, $t_k-t_{k-1}=0.1$, $x_1(t_k)=\frac{x_1(t_k^{-})}{3}$, $x_2(t_k)=\frac{x_2(t_k^{-})}{4}$. $C_{2\times 1}=\left(\begin{array}{c}2\\ 2\end{array}\right)$, ${(a_{ij})}_{2\times 2}=\left(\begin{array}{cc}0.5& -0.8\\ 0.3& 0.6\end{array}\right)$, ${(b_{ij})}_{2\times 2}=\left(\begin{array}{cc}0.2& -0.3\\ -0.1& 0.4\end{array}\right)$.

Obviously, $L_i=N_i=1$, $e_i=0$, $l_i=1$ ($i=1,2$), ${\beta }_{1k}=2/3$, ${\beta }_{2k}=3/4$.

Let $d_k=2/3$, $\sigma =3.4$, $\lambda =0.2$, ${\mu }_1={\mu }_2$. Then for $p=2$, we can get $k_1=min(1.4,1.2)=1.2>0$, $k_2=max(1.3,1.7)=1.7$, $-k_1+k_2\frac{e^{\lambda \tau }}{d_k^p}=-1.2+1.7\times \frac{e^{0.2\times 0.5}}{{(2/3)}^2}=3.027<\sigma -\lambda =3.4-0.2=3.2$, $2ln{d}_{k-1}=2ln(2/3)=-0.811<-(\sigma +\lambda )(t_k-t_{k_1})=-3.6\times 0.1=-0.36$.

All conditions of Corollary 3.3 are satisfied, then the equilibrium point is exponentially stable in mean square.

Remark 4.1 If ${\mu }_1={\mu }_2$, we have computed $k_1=1.2$, $k_2=1.7$. If ${\mu }_1\ne {\mu }_2$, set $\frac{{\mu }_2}{{\mu }_1}=\alpha$, then $k_1=min\{1.7-0.3\alpha ,2-0.8/\alpha \}$, $k_2=max\{1.2+0.1\alpha ,1.4+0.3/\alpha \}$. If ${\mu }_1<{\mu }_2$, then $\alpha >1$, we can compute $k_1<1.4$ and $k_2>1.4$; if ${\mu }_1>{\mu }_2$, then $0<\alpha <1$, $k_1<1.2$ and $k_2>1.4$. Hence, in either case, we always have $k_1<k_2$, so the exponential stability in mean square of the system (4.1) cannot be derived by applying the corresponding exponential stability result for cellular neural networks given in the literature [6, 11, 12, 19], since $k_1>k_2$ is not satisfied.

Remark 4.2 Since $\rho [C^{-1}(M{M}_{1}K+M{M}_{2}K+N{N}_1+N{N}_2)]=33.635$, where $\rho [C^{-1}(M{M}_{1}K+M{M}_{2}K+N{N}_1+N{N}_2)]$ was defined in [8], the condition $\rho [C^{-1}(M{M}_{1}K+M{M}_{2}K+N{N}_1+N{N}_2)]\le 1$ is not satisfied. Hence, the results in [8] are useless to judge the exponential stability of the system (4.1).