On pth moment exponential stability of stochastic Cohen–Grossberg neural networks with time-varying delays

Abstract

With the help of Lyapunov function, stochastic analysis technique, generalized Halanay inequality and Hardy inequality, a set of novel sufficient conditions on pth moment exponential stability for non-autonomous stochastic Cohen–Grossberg neural networks is given, which modifies and generalizes some corresponding published results.

Introduction

In real nervous systems, there are various stochastic perturbations to the networks and it is important to understand how these perturbations affect the networks. Especially, it is very critical to know whether the networks are stable or not under the perturbations [1]. Recently, a great deal of results on stochastic neural networks with discrete delays have been reported in the literature, see e.g. [2], [3], [4], [5], [6], [7], [8], [9], [10]. pth moment exponential stability analysis is one of the most challenging, yet difficult problems in the field of stochastic neural networks [11]. In the aforementioned paper [11], the authors took the Razumikhin-type theorem to study pth moment exponential stability of the following system and obtained some interesting results:

$$\mathrm{d}x(t)=-\alpha (x(t))[\beta (x(t))-\mathit{Ag}(x(t))-\mathit{Bg}(x(t-\tau (t)))]\mathrm{d}t+\sigma (t,x(t),x(t-\tau (t)))\mathrm{d}w(t)\text{,}$$

where $x(t)=(x_1(t),x_2(t),\dots ,x_n(t){)}^T\in R^n$ is the neuron states vector; $\alpha (x(t))=\mathrm{diag}({\alpha }_1(x_1(t)),\dots ,{\alpha }_n(x_n(t)))$; $\beta (x(t))=({\beta }_1(x_1(t)),\dots ,{\beta }_n(x_n(t)){)}^T$; $A=(a_{\mathit{ij}}{)}_{n\times n}$ and $B=(b_{\mathit{ij}}{)}_{n\times n}$ are the connection weight matrix and delayed connection weight matrix, respectively; $g(x(t))=(g_1(x_1(t)),\dots ,g_n(x_n(t)){)}^T$ is the activation functions vector; $g(x(t-\tau (t)))=(g_1(x_1(t-\tau (t))),\dots ,g_n(x_n(t-\tau (t))){)}^T,0\le \tau (t)\le \tau$; $\sigma (t)=({\sigma }_{\mathit{ij}}(t){)}_{n\times n}$ is the diffusion coefficient matrix and $\omega (t)=({\omega }_1(t),\cdots ,{\omega }_n(t){)}^T$ is an n-dimensional Brownian motion defined on a complete probability space $(\mathbf{\Omega },\mathcal{F},\mathbf{P})$ with a natural filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ (i.e. ${\mathcal{F}}_t=\sigma \{w(s):0\le s\le t\}$). In order to obtain pth moment exponential stability of system (1), the following conditions are established in [11].

• For each $i\in \{1,\dots ,n\}$, there exist positive constants ${\underline{\alpha }}_i,{\overline{\alpha }}_i$, such that ${\underline{\alpha }}_i\le a_i(x_i(t))\le {\overline{\alpha }}_i$.

• For each $i\in \{1,\dots ,n\}$, $x_i^{p-1}(t){\beta }_i(x_i(t))\ge {\gamma }_i{x}_i^p(t)$.

• For each $i\in \{1,\dots ,n\}$, there exists positive constant $G_i$, such that

  $$|g_i(x)-g_i(y)|\le G_i|x-y|,\forall x,y,\in R\text{.}$$

• For each $i,j\in \{1,\dots ,n\}$, there are non-negative constants $c_{\mathit{ij}}^0$, $c_{\mathit{ij}}^1$, such that

  $${\sigma }_{\mathit{ij}}^2\triangleq {\sigma }_{\mathit{ij}}^2(t,x_i(t),x_j(t-\tau (t)))\le c_{\mathit{ij}}^0{x}_i^2(t)+c_{\mathit{ij}}^1{x}_j^2(t-\tau (t))\text{.}$$

  We can read the following main theorem which was obtained in [11].

Theorem A Zhu [11]

Under assumptions $(H_1)–(H_4)$, if there exists a positive diagonal matrix $Q=\mathrm{diag}(q_1,\dots ,q_n)$, such that

$${\lambda }_1>{\lambda }_2\text{,}$$

where

$${\lambda }_1=\underset{1\le i\le n}{\min }\left\{p\underline{{\alpha }_i}{\gamma }_i-(p-1)\sum _{j=1}^n\overline{{\alpha }_i}\left|a_{\mathit{ij}}\right|G_j-\frac{1}{q_i}\sum _{j=1}^n\overline{{\alpha }_j}{q}_j\left|a_{\mathit{ji}}\right|G_i-(p-1)\sum _{j=1}^n\overline{{\alpha }_i}\left|b_{\mathit{ij}}\right|G_j-\frac{p(p-1)}{2}\underset{1\le i\le n}{\max }\{q_i\}\frac{1}{q_i}\sum _{j=1}^n{c}_{\mathit{ij}}^0-(p-1)(p-2)\underset{1\le i\le n}{\max }\{q_i\}\frac{1}{q_i}\sum _{j=1}^n{c}_{\mathit{ij}}^1\right\}\text{,}$$

and

$${\lambda }_2=\underset{1\le i\le n}{\max }\left[\frac{1}{q_i}\sum _{j=1}^n\left|b_{\mathit{ji}}\right|\overline{{\alpha }_j}{q}_j{G}_i+(p-1)\underset{1\le i\le n}{\max }\{q_i\}\frac{1}{q_i}\sum _{j=1}^n{c}_{\mathit{ji}}^1\right]\text{,}$$

then for all $\xi \in L_{{\mathcal{F}}_0}^p([-\tau ,0],R^n)$, the trivial solution of system (1) is pth moment exponentially stable.

However, a defect appearing in the main result in [11] when $p=2k+1,k\in Z^{+},x(t)<0$, just from the constructed Lyapunov function, one can find the term $\text{“}{x}^{p/2}(t)\text{”}$ is a blemish. In order to modify this imperfection, the constructed Lyapunov function should be replaced with $V(t,x(t))={\sum }_{i=1}^n{q}_i|x_i(t){|}^p$. Noticing that $\partial |x_i(t){|}^p/\partial x_i=p|x_i{|}^{p-1}\mathrm{sgn}\{x_i\}=p|x_i{|}^{p-2}{x}_i$, we have $\partial |x_i(t){|}^p/\partial x_i{\beta }_i(x_i(t))=p|x_i{|}^{p-2}{x}_i{\beta }_i(x_i(t))$, it is easy to see that the assumed condition $(H_2)$ is not correct, which should be revised as the below $(H_2^{\prime })$. On the other hand, there is an error appear in (3), the coefficient of ${\lambda }_1$, the term $\text{“}(p-1)(p-2)\text{”}$ should be replaced with $(p-1)(p-2)/2$.

$(H_2^{\prime })$ For each $i\in \{1,2,\dots ,n\}$, there exist positive constant ${\gamma }_i>0$, such that

$$x_i(t){\beta }_i(x_i(t))\ge {\gamma }_i{x}_i^2(t)\text{.}$$

In this paper, we will generalize system (1) and further study the following non-autonomous stochastic functional differential equations:

$$\mathrm{d}{x}_i(t)=-a_i(x_i(t))\left[{\beta }_i(x_i(t))-\sum _{j=1}^n{a}_{\mathit{ij}}(t)g_j(x_j(t))-\sum _{j=1}^n{b}_{\mathit{ij}}(t)g_j(x_j(t-{\tau }_j(t)))\right]\mathrm{d}t+\sum _{j=1}^n{\sigma }_{\mathit{ij}}(t,x_i(t),x_j(t-{\tau }_j(t)))\mathrm{d}{\omega }_j(t)\text{.}$$

We will improve and generalize the corresponding results published in [11]. We assume the following generalized conditions are satisfied:

• There exist positive functions ${\gamma }_j(t)$, such that

  $$x_j(t){\beta }_j(x_j(t))\ge {\gamma }_j(t)x_j^2(t)\text{.}$$

• There are non-negative functions $c_{\mathit{ij}}^0(t),c_{\mathit{ij}}^1(t)$, for $\forall t,u,v\in R$, such that

  $${\sigma }_{\mathit{ij}}^2(t,u,v)\le c_{\mathit{ij}}^0(t)u^2+c_{\mathit{ij}}^1(t)v^2\text{.}$$

Remark 1.1

Obviously, choosing ${\gamma }_j(t)={\gamma }_j>0$, $c_{\mathit{ij}}^0(t)=c_{\mathit{ij}}^0>0,c_{\mathit{ij}}^1(t)=c_{\mathit{ij}}^1>0$, we can see that conditions $(H_2^{″})$ and $(H_4^{\prime })$ generalize the conditions $(H_2^{\prime })$ and $(H_4)$, the significant of these generalization may provide engineer with a wider selection on the parameters in practice. For simplicity and avoiding lengthy statements, some symbols and notations with the same meanings as those in [11] are employed in the following derivations.