Stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay

Abstract

This paper studies stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay. Firstly, the second-order differential equation is converted into the first-order differential equation by appropriate variable substitution. Secondly, the existence of the equilibrium point is derived by using homeomorphic mapping, finite increment formula of Lagrange mean value theorem and linear matrix inequality. The sufficient conditions for the stochastic asymptotic stability of the equilibrium point of the system are derived by defining the appropriate operator, and constructing the appropriate positive Lyapunov function and positive-definite matrix. Thirdly, a numerical example illustrates the correctness of these theorems.

Keywords

Stochastic inertial Cohen-Grossberg neural networks Time-varying delay Homeomorphic mapping Linear matrix inequality Stochastic asymptotic stability

1. Introduction

The conduction process of synapses is a noise process caused by neurotransmitters and stochastic fluctuation, so the actual neural network system is a stochastic dynamic system in practical problems. Therefore, the stability of stochastic neural networks’ delay has attracted much attention in the past decade. The reference [1] studies the mean-square exponential input status stability of multi proportional stochastic delay neural networks. The reference [2, 3] study separately the mean-square exponential input status stability of stochastic delay recurrent neural network stability and fuzzy stochastic Cohen-Grossberg neural network with time-varying delay. The reference [4] gives the conclusion that the drive response stochastic memristor neural network is almost exponentially synchronized. In [5, 6], a new standard for mean-square exponential stability is given under stochastic memristor with leakage delays and stochastic Cohen-Grossberg BAM, respectively. The reference [7] analyzes the moment exponential stability for stochastic Cohen-Grossberg neural networks with time-varying delay. In [8, 9], the exponential stability of stochastic recurrent neural networks with time-delay and stochastic higher-order neural networks with time-varying delay are studied, respectively.

From the research [1, 2, 3, 4, 5, 6, 7, 8, 9], the stochastic neural network model didn’t consider the damping coefficient (inertia term), or the stochastic neural network model studied can be understood as a over damping model (damping tends to infinity). However, the dynamic properties of each neuron state will also change when the damping exceeds the critical value from the perspective of mathematics and physics. Therefore, the dynamic behavior of the stochastic neural network model with inertia term should be considered in practical application. There are also many results on the dynamic behavior of stochastic inertial neural networks.The reference [10] studies the finite and fixed-time synchronization of hybrid delay inertial neural networks. The reference [11] studies inertial Cohen-Grossberg neural networks stability on Markov jump parameters. The stability for stochastic inertial neural network on mean-square exponential input status is studied in [12]. The exponential stability for BAM stochastic inertial delay neural networks is shown in [13]. The stochastic inertial delay neural networks’ stability is shown in [14]. The reference [15] studies the exponential synchronization stability of stochastic inertial neural networks with delay.

According to data, there is little literature to describ stochastic asymptotic stability of the equilibrium point of stochastic inertial Cohen-Grossberg neural networks with time-varying delay, which will be a new research topic. The research process of this paper mainly adopts: 1) transforming second-order differential equation system into first-order differential equation system; 2) Constructing homeomorphic mapping, positive-definite matrix and positive Lyapunov function; 3) Using homeomorphic mapping, linear matrix inequality, properties of differential operators and fixed point theory and knowledge, obtain the sufficient conditions of stochastic asymptotic stability of the equilibrium point of the system. The results have certain research significance for the theoretical exploration and practical application of the dynamic properties of the system.

We consider a class of inertial Cohen-Grossberg neural networks with time-varying delayï¼š

$\displaystyle\ddot{x}_{i}(t)=-d_{i}\dot{x}_{i}(t)-\alpha_{i}(x_{i}(t))\left[h_% {i}(x_{i}(t))-\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j}(t))}-\sum\limits_{j=1}^{% n}{b_{ij}f_{j}(x_{j}(t-\tau(t)))}+I_{i}\right].$ (1)

If add a stochastic disturbance term in system Eq. (1)ï¼Œwe obtain a class of stochastic inertial Cohen-Grossberg neural networks with time-varying delay:

$\displaystyle\text{d}(\dot{x}_{i}(t))=-d_{i}\dot{x}_{i}(t)\text{d}t-\alpha_{i}% (x_{i}(t))\!\!\left[h_{i}(x_{i}(t))\!-\!\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j% }(t))}\!-\!\sum\limits_{j=1}^{n}{b_{ij}f_{j}(x_{j}(t-\tau(t)))}+I_{i}\!\right]% \!\text{d}t+\sum\limits_{j=1}^{n}{g_{j}(x_{j}(t))\text{dB}_{i}(t)}\text{, }i=1% ,2,\ldots,n.$ (2)

Where $d_{i}>0$ is isolated resting state rate of the $i$ ith neuron when the neural network is disconnected and there is no external additional voltage difference; $x_{i}(t)$ is the state variable of the $i$ th neuron at moment $t$ ; $\alpha_{i}(\cdot)$ is an amplification function; $h_{i}(\cdot)$ is a behavior function; $f_{j}$ represents nonlinear activation function of the $j$ th neuron at moment $t$ ; $a_{ij},b_{ij}$ represent weights; $\tau(t)$ represents time lag, and $\tau(t)>0$ ; $I_{i}$ represents external input of the $i$ th neuron at moment $t$ ; $g_{j}(x_{j}(t))\text{d}B_{i}(t)$ is stochastic disturbance, $B(t)=(B_{1}(t),B_{2}(t),\ldots,B_{n}(t))^{T}$ is a Brownian motion on n-dimensional space, $\{\Re_{t}\}_{t\geqslant 0}$ is natural filtering on a complete probability space $(\Omega,\Re,{\rm P})$ . Suppose the initialization of system Eq. (2)

$\displaystyle x_{i}(s)=\varphi_{i}(s),\dot{x}_{i}(s)=\psi_{i}(s),-\tau% \leqslant s\leqslant 0.$ (3)

where $i=1,2,\ldots,n,\tau=\sup\{\tau(t),0<t<+\infty\},\varphi_{i}(s),\psi_{i}(s)$ are bounded continuous functions.

This paper mainly studies the stochastic asymptotic stability of the equilibrium point of system Eq. (2), and gives the sufficient conditions for its determination.

2. Preliminaries

Notations and definitions are used in this article. $E$ is the identity matrix, $A^{T}$ is the transpose matrix of $A$ , and $A^{-1}$ is the inverse matrix of $A$ . $A$ is called a positive-definite matrix if $A>0$ , and $A$ is a negative-definite matrix if $A<0$ . $\lambda_{M}(A),\lambda_{m}(A)$ are respectively the largest and smallest eigenvalue matrix of $A$ .

Let $||X||=(\sum_{i=1}^{n}{x_{i}^{2}})^{\frac{1}{2}}$ for any column vector $X=(x_{1},x_{2},x_{3},\ldots,x_{n})^{T}$ in $R^{n}$ space.

Definition 1. [16] A mapping $H$ is to $R^{n}\to R^{n}$ , if $H$ is continuous and one-to-one mapping, $H^{-1}$ is also continuous, then $H$ is a homeomorphic mapping on $R^{n}\to R^{n}$ .

Lemma 1. [16] Let $H(u)\in C[a,b]$ , then $H(u)$ is a homeomorphic mapping on $R^{n}$ if: (1) $H(u)$ is a monomorphism on $R^{n}$ ; (2) $\left\|{H(u)}\right\|\to+\infty$ when $\left\|u\right\|\to+\infty$ .

Lemma 2. Let $X,Y\in R^{n}$ , the inequality $2X^{T}Y\leqslant\varepsilon X^{T}X+\varepsilon^{-1}Y^{T}Y$ holds for any $\varepsilon>0$ .

The article makes three assumptions as below:

(H ${}_{1}$ ) $\alpha_{i}(x)$ is a continuous and bounded function of $x$ , there exist positive number $\underline{\alpha}_{i},\bar{\alpha}_{i}$ such that $\overline{\alpha}_{i}\geqslant\alpha_{i}(x)\geqslant{\alpha}_{i}>0$ for $x\in R$ .

(H ${}_{2}$ ) $h_{i}(x)$ is a differentiable and bounded of function $x$ , there exist positive number $\underline{h}_{i},\bar{h}_{i}$ such that $\bar{h}_{i}\geqslant{h}^{\prime}_{i}(x)\geqslant\underline{h}_{i}>0$ for $x\in R$ .

(H ${}_{3}$ ) $f_{i},g_{i}$ satisfy Lipschitz condition, there exist $l_{i},\mu_{i}>0$ such that

$\displaystyle\left|{f_{i}(t_{1})-f_{i}(t_{2})}\right|\leqslant l_{i}\left|{t_{% 1}-t_{2}}\right|,\left|{g_{i}(t_{1})-g_{i}(t_{2})}\right|\leqslant\mu_{i}\left% |{t_{1}-t_{2}}\right|$

for $t_{1},t_{2}\in R$ .

Variable substitution: $y_{i}(t)=\dot{x}_{i}(t)+x_{i}(t)$ , then system Eq. (1) becomes:

$\displaystyle\left\{\begin{array}[]{lll}\dot{x}_{i}(t)&=&y_{i}(t)-x_{i}(t),\\ \dot{y}_{i}(t)&=&(d_{i}-1)x_{i}(t)-(d_{i}-1)y_{i}(t)-\alpha_{i}(x_{i}(t))\left% [h_{i}(x_{i}(t))-\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j}(t))}\right.\\ &&-\left.\sum\limits_{j=1}^{n}{b_{ij}f_{j}(x_{j}(t-\tau(t)))+I_{i}}\right].\\ \end{array}\right.$ (4)

With the same substitution, system Eq. (2) and the initial conditions Eq. (3) become:

$\displaystyle\left\{\begin{array}[]{lll}\text{d}(x_{i}(t))&=&(y_{i}(t)-x_{i}(t% ))\text{d}t,\\ \text{d}(y_{i}(t))&=&\left\{(d_{i}-1)x_{i}(t)-(d_{i}-1)y_{i}(t)-\alpha_{i}(x_{% i}(t))\left[h_{i}(x_{i}(t))-\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j}(t))}\right% .\right.\\ &&\left.\left.-\sum\limits_{j=1}^{n}{b_{ij}f_{j}(x_{j}(t-\tau(t)))+I_{i}}% \right]\right\}\text{d}t+\sum\limits_{j=1}^{n}{g_{j}(x_{j}(t))\text{d}B_{i}(t)% .\text{ }}\\ \end{array}\right.$ (5)

$\displaystyle\left\{{\begin{array}[]{l}x_{i}(s)=\varphi_{i}(s),\dot{x}_{i}(s)=% \psi_{i}(s),\\ y_{i}(s)=\varphi_{i}(s)+\psi_{i}(s),\\ \end{array}}\right.-\tau<s\leqslant 0.$ (6)

The vector-matrix form of Eq. (5):

$\displaystyle\left\{{\begin{array}[]{l}\text{d}(X(t))=(Y(t)-X(t))\text{d}t,\\ \text{d}(Y(t))=\{\textit{RX}(t)-\textit{RY}(t)-\alpha(X(t))[H(X(t))-\textit{Af% }(X(t))-\textit{Bf}(X(t-\tau(t)))+I]\}\text{d}t\\ \qquad\qquad\ \ \ +g(X(t))\text{d}B(t),\\ \end{array}}\right.$ (7)

where

$\displaystyle X(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T},Y(t)=(y_{1}(t),y_{2% }(t),\ldots,y_{n}(t))^{T},$ $\displaystyle R=\text{diag}(d_{1}-1,d_{2}-1,\ldots,d_{n}-1),$ $\displaystyle\alpha(X(t))=\text{diag}(\alpha_{1}(x_{1}(t)),\alpha_{2}(x_{2}(t)% ),\ldots,\alpha_{n}(x_{n}(t))),H(X(t))=(h_{1}(x_{1}(t)),$ $\displaystyle\qquad\ \ \ \qquad h_{2}(x_{2}(t)),\ldots,h_{n}(x_{n}(t)))^{T},$ $\displaystyle A=(a_{ij})_{n\times n},B=(b_{ij})_{n\times n},f(X(t))=(f_{1}(x_{% 1}(t),f_{2}(x_{2}(t),\ldots,f_{n}(x_{n}(t)))))^{T},$ $\displaystyle I=(I_{1},I_{2},\ldots,I_{n})^{T}.$ $\displaystyle g(X(t))=(g_{1}(x_{1}(t)),g_{2}(x_{2}(t))\ldots,g_{n}(x_{n}(t)))^% {T},\text{d}B(t)=(\text{d}B_{1}(t),\text{d}B_{2}(t),\ldots,$ $\displaystyle\qquad\ \ \ \qquad\text{d}B_{n}(t))^{T}.$

Definition 2. For $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}\in R^{n}$ , if there is

$\displaystyle-h_{i}(x_{i}^{\ast})+\sum\limits_{j=1}^{n}a_{ij}f_{j}(x_{j}^{\ast% })+\sum\limits_{ij}^{n}b_{ij}f_{j}(x^{\ast}_{j})-I_{i}=0,i=1,2,\ldots,n,$ (8) $\displaystyle\text{or}\ -H(X^{\ast})+\textit{Af}(X^{\ast})+\textit{Bf}(X^{\ast% })-I=0,$ (9)

then $X^{\ast}$ is the equilibrium point of system Eq. (1).

Definition 3. Suppose $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}$ is the equilibrium point of system Eq. (1), if $g_{i}(x_{i}^{\ast})=$ 0 and $-h_{i}(x_{i}^{\ast})+\sum_{j=1}^{n}a_{ij}f_{j}(x_{j}^{\ast})+\sum_{j=1}^{n}b_{% ij}f_{j}(x^{\ast}_{j})-I_{i}=$ 0, $i=$ 1, 2, $\ldots$ , $n$ , then $X^{\ast}$ is the equilibrium point of system Eq. (2).

Remark 1. Condition $g_{i}(x_{i}^{\ast})=$ 0, $j=$ 1, 2, $\ldots$ , $n$ indicate, if the $i$ th unite has reach the equilibrium state, then it no longer sends noise interference to other units, it means $X^{\ast}$ is the equilibrium point of system Eq. (2).

Definition 4. For the system

$\displaystyle\left\{{\begin{array}[]{l}\text{d}(X(t))=f(t,X(t))\text{d}t+g(t,X% (t))\text{d}B(t)\\ X(t_{0})=X_{0},t\geqslant t_{0}\\ \end{array}}\right.$ (10)

Let operator $\textit{LV}(t,X)$ [17]:

$\displaystyle\textit{LV}(t,X)=V_{t}(t,X)+V_{x}(t,X)f(t,X)+\frac{1}{2}\textit{% trac}[g^{T}(t,X)V_{xx}(t,X)g(t,X)]$

here

$\displaystyle V_{t}(t,X)=\frac{\partial V(t,X)}{\partial t},V_{x}(t,X)=\left({% \frac{\partial V(t,X)}{\partial x_{1}},\frac{\partial V(t,X)}{\partial x_{2}}% \cdots,\frac{\partial V(t,X)}{\partial x_{n}}}\right),$ $\displaystyle V_{xx}(t,X)=\left({\frac{\partial^{2}V(t,X)}{\partial x_{i}x_{j}% }}\right)_{n\times n}.$

Definition 5. [17] Let $\Omega\subset R^{n}$ be a n-dimensional open neighborhood containing the origin, then the function $V(t,X)\in C[[0,+\infty)\times R^{n},R]$ has an infinitesimal upper bound if there is a positive-definite function $W(X)\in C[\Omega,R]$ so that $\left|{V(t,X)}\right|\in W(X)$ .

Definition 6. [17] $W(X)\in C[R^{n},R]$ is a unbounded positive-definite function if $W(X)$ is positive-definite and $W(X)\to+\infty$ when $||X||\to+\infty$ .

Lemma 3. [17] The zero solution of system Eq. (10) is stochastic asymptotically stable, if there is a radial unbounded positive-definite function $V(t,X)\in C^{1.2}[(t_{0},\infty)\times S_{h},R_{+}]$ with infinitesimal upper bound so that $\textit{LV}(t,X)<0$ .

Where: note $C^{1.2}[R_{+}\times S_{h},R_{+}]$ is the nonnegative functions $V(t,X)$ defined on $R_{+}\times S_{h}$ , $S_{h}=\{\left.x\right|||x||<h\}\subset R^{n}$ is the neighborhood of the origin in n-dimensional space, $V(t,X)$ has continuous first and second derivatives of $(t,X)$ .

3. Main results

Theorem 1. Under hypotheses $(\text{H}_{1})-(\text{H}_{3})$ , there exist $\beta>0,\delta>0$ , $0<q_{i}<\sqrt{d_{i}-\bar{h}_{i}\bar{\alpha}_{i}}<1$ and $Q=\text{diag}(q_{1},q_{2},\ldots,q_{n})$ , if $1+\underline{h}_{i}\underline{\alpha}_{i}-d_{i}>0$ , so that

$\displaystyle P_{1}=2\underline{H}-(\beta+\delta+1)E>0,$ $\displaystyle P_{2}=(E-Q^{2}-G)L^{-2}-\frac{1}{\beta}A^{T}A>0,$ $\displaystyle P_{3}=(1-\eta)Q^{2}L^{-2}-\frac{1}{\delta}B^{T}B>0,$

then system Eq. (1) has an equilibrium point, where $L=\text{diag}(l_{1},l_{2},\ldots,l_{n}),{H}=\text{diag}({h}_{1},{h}_{2},\ldots% ,{h}_{n})$ , $0<\tau^{\prime}(t)\leqslant\eta<1,G=\text{diag}(1+\bar{h}_{1}\bar{\alpha}_{1}-% d_{1},1+\bar{h}_{2}\bar{\alpha}_{2}-d_{2},\ldots,1+\bar{h}_{n}\bar{\alpha}_{n}% -d_{n}),l_{i},{h}_{i},\bar{h}_{i},{\alpha}_{i},\bar{\alpha}_{i}$ are all given by hypotheses $(\text{H}_{1})-(\text{H}_{3})$ .

Proof: Suppose mapping $W(X)=(W_{1}(X),W_{2}(X),\ldots,W_{n}(X))^{T}$ , whereï¼š

$\displaystyle W_{i}(X)=-h_{i}(x_{i})+\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j})}% +\sum\limits_{j=1}^{n}{b_{ij}f_{j}(x_{j})}-I_{i},i=1,2,\ldots,n.$

Therefore $W(X)=$ 0 is the equilibrium equation of system Eq. (1), if $W(X)$ is a homeomorphic mapping on $R^{n}$ , there exist a unique point $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}$ satisfied $W(X^{\ast})=0$ , namely $X^{\ast}$ is the equilibrium point of system Eq. (1). Next, we prove $W(X)$ is a homeomorphic mapping on $R^{n}$ .

First we prove $W(X)$ is a monomorphism on $R^{n}$ . Suppose vector $X,Z\in R^{n}$ , $X\neq Z$ , we obtain

$\displaystyle W(X)-W(Z)=A[f(X)-f(Z)]+B[f(X)-f(Z)]-[H(X)-H(Z)].$ (11)

Left multiply both sides of Eq. (11) with equation $2(X-Z)^{T}$ , then the formula is:

$\displaystyle 2(X-Z)^{T}W(X)-W(Z)=2(X-Z)^{T}\{A[f(X)-f(Z)]+B[f(X)-f(Z)]-[H(X)-% H(Z)]\}.$ (12)

From Lemma 2, there exist $\beta>0,\delta>0$ ,

$\displaystyle 2(X-Z)^{T}A[f(X)-f(Z)]\leqslant\beta(X-Z)^{T}(X-Z)+\frac{1}{% \beta}[f(X)-f(Z)]^{T}A^{T}A[f(X)-f(Z)],$ (13) $\displaystyle 2(X-Z)^{T}B[f(X)-f(Z)]\leqslant\delta(X-Z)^{T}(X-Z)+\frac{1}{% \delta}[f(X)-f(Z)]^{T}B^{T}B[f(X)-f(Z)].$ (14)

Moreover, under the hypothesis $(\text{H}_{3})$ , if 0 $<q_{i}<$ 1, $i=$ 1, 2, $\ldots,n$ ,

$\displaystyle(X-Z)^{T}(E-Q^{2})(X-Z)\geqslant(f(X)-f(Z))^{T}(E-Q^{2})L^{-2}(f(% X)-f(Z)),$ (15) $\displaystyle(X-Z)^{T}Q^{2}(X-Z)\geqslant(f(X)-f(Z))^{T}Q^{2}L^{-2}(f(X)-f(Z)).$ (16)

There has $0<\underline{h}_{i}\leqslant{h}^{\prime}_{i}(x)\leqslant\bar{h}_{i}$ under the hypothesis $(\text{H}_{2})$ and $h_{i}(x_{i})-h_{i}(z_{i})={h}^{\prime}_{i}(z_{i}+\theta_{i}(x_{i}-z_{i}))(x_{i% }-z_{i}),0<\theta_{i}<1$ , $i=1,2,\ldots,n$ , and according to Lagrange mean value theorem, we obtain:

$\displaystyle(X-Z)^{T}(H(X)-H(Z))=\sum\limits_{i=1}^{n}{(x_{i}-z_{i})(h_{i}(x_% {i})-h_{i}(z_{i}))}=\sum\limits_{i=1}^{n}{{h}^{\prime}_{i}(z_{i}+\theta_{i}(x_% {i}-z_{i}))}(x_{i}-z_{i})^{2}\geqslant\sum\limits_{i=1}^{n}{{h}_{i}}(x_{i}-z_{% i})^{2}=(X-Z)^{T}{H}(X-Z).$ (17)

From Eqs (12)—(17) we have:

$\displaystyle 2(X-Z)^{T}(W(X)-W(Z))\leqslant-(X-Z)^{T}P_{1}(X-Z)-[f(X)-f(Z)]^{% T}-P_{2}^{\ast}[f(X)-f(Z)][f(X)-f(Z)]^{T}P_{3}^{\ast}[f(X)-f(Z)].$ (18)

Here $P_{1}=2\underline{H}-(\beta+\delta+1)E>0$ , $P_{2}^{\ast}=(E-Q^{2})L^{-2}-\frac{1}{\beta}A^{T}A$ , $P_{3}^{\ast}=(Q^{2}L^{-2}-\frac{1}{\delta}B^{T}B)$ are known by conditions:

$\displaystyle P_{2}=(E-Q^{2}-G)L^{-2}-\frac{1}{\beta}A^{T}A>0,P_{3}=(1-\eta)Q^% {2}L^{-2}-\frac{1}{\delta}B^{T}B>0.$

We get $P_{1}>0,P_{2}^{\ast}>GL^{-2}>0,P_{3}^{\ast}>\eta Q^{2}L^{-2}>0$ .

So we get $2(X-Z)^{T}(W(X)-W(Z))<0$ , so $W(X)\neq W(Z)$ , then $W(X)$ is a monomorphism of $R^{n}$ .

Again from Eq. (18):

$\displaystyle 2X^{T}(W(X)-W(0))\leqslant-X^{T}P_{1}X-[f(X)-f(0)]^{T}P_{2}^{% \ast}[f(X)-f(0)]-[f(X)-f(0)]^{T}P_{3}^{\ast}[f(X)-f(0)].$ (19)

According to $-X^{T}Y\leqslant||X^{T}Y||\leqslant\left\|{X^{T}}\right\|\left\|Y\right\|$ , we get from Eq. (19):

$\displaystyle 2\left\|{X^{T}}\right\|\left\|{W(X)-W(0)}\right\|\geqslant 2% \left\|{X^{T}(W(X)-W(0))}\right\|\geqslant\lambda_{m}(P_{1})\left\|X\right\|^{% 2}.$ (20)

So $\left\|{W(X)}\right\|+\left\|{W(0)}\right\|\geqslant\frac{\lambda_{m}(P_{1})}{% 2}\left\|X\right\|$ , then

$\displaystyle\left\|{W(X)}\right\|\geqslant\frac{\lambda_{m}(P_{1})}{2}\left\|% X\right\|-\left\|{W(0)}\right\|.$

So $||W(X)||\to+\infty$ when $||X||\to+\infty$ . We get $W(X)$ is a homeomorphic mapping on $R^{n}$ from Lemma 1. So the conclusion is system Eq. (1) has an equilibrium point.

Inference 1. Suppose Theorem 1 holds, $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}$ is the equilibrium point of system Eq. (1). If $g_{i}(x_{i}^{\ast})=$ s 0 and $-h_{i}(x_{i}^{\ast})+\sum_{j=1}^{n}a_{ij}f_{j}(x_{j}^{\ast})+\sum_{ij}^{n}b_{% ij}f_{j}(x^{\ast}_{j})-I_{i}=$ 0, $i=$ 1, 2, $\ldots$ $m$ , then $X^{\ast}$ is also the equilibrium point of system Eq. (2).

Remark 2. The result of Inference 1 can be obtained directly from Theorem 1 and Definition 3.

Theorem 2. Suppose Theorem 1 holds, $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}$ is the equilibrium point of system Eq. (1). If

$\displaystyle g(x_{i}^{\ast})=0,i=1,2,\ldots,n,$ $\displaystyle P_{4}=-\frac{1}{\beta}A^{T}A+(E-\frac{1}{2}\mu^{2}-Q^{2}-G)L^{-2% }>0,$ $\displaystyle P_{5}=2R-(\beta+\delta)\bar{\alpha}^{2}-E-G>0,$

then system Eq. (2) has an equilibrium point, and the point is stochastic asymptotically stable, where $\bar{\alpha}^{2}=\text{diag}(\bar{\alpha}_{1}^{2},\bar{\alpha}_{2}^{2},\ldots,% \bar{\alpha}_{n}^{2}),\mu^{2}=\text{diag}(\mu_{1}^{2},\mu_{2}^{2},\ldots,\mu_{% n}^{2}),\bar{\alpha}_{i},\mu_{i}$ are all given by hypotheses $(\text{H}_{1})$ , $(\text{H}_{3})$ .

Proof: According to the conclusion of Inference 1, we know system Eq. (2) has equilibrium point $X^{\ast}=(x_{1}^{\ast},x_{2}^{\ast},\ldots,x_{n}^{\ast})^{T}$ .

Suppose $y_{i}(t)=\dot{x}_{i}(t)+x_{i}(t)$ , $u_{i}(t)=x_{i}(t)-x_{i}^{*}$ , $z_{i}(t)=y_{i}(t)-y_{i}^{*}$ , we get from Eq. (7) and (9):

$\displaystyle\left\{\begin{array}[]{l}\text{d}(U(t))=(Z(t)-U(t))\text{d}t,\\ \text{d}(Z(t))=\{\textit{RU}(t)-\textit{RZ}(t)-\alpha(U(t))[H(U(t))-\textit{Af% }(U(t))-\textit{Bf}(U(t-\tau(t)))+I]\}\text{d}t\\ \hskip 51.214961pt+g(U(t))\text{d}B(t),\\ \end{array}\right.$ (21)

where $U(t)=(x_{1}(t)-x_{1}^{*},\ldots,x_{n}(t)-x_{n}^{*})^{T},Z(t)=(y_{1}(t)-y_{1}^{% \ast},\ldots,y_{n}(t)-y_{n}^{\ast})^{T}$ ,

$\displaystyle\alpha(U(t))=\text{diag}(\alpha_{1}(u_{1}(t)+x_{1}^{*}),\ldots,% \alpha_{n}(u_{n}(t)+x_{n}^{*})),H(U(t))=(h_{1}(u_{1}(t))+x_{1}^{*},\ldots,h_{n% }(u_{n}(t))+x_{n}^{*})^{T},$ $\displaystyle f(U(t))=(f_{1}(x_{1}(t))-f\ldots,f_{n}(x_{n}(t))-f_{g}(U(t))=(g_% {1}(x_{1}(t))-g\ldots,g_{n}(x_{n}(t))-g$

The positive-definite Lyapunov function is given as follow:

$\displaystyle V(t,U(t),Z(t))=U^{T}(t)U(t)+Z^{T}(t)Z(t)+\int_{t-\tau(t)}^{t}{U^% {T}(s)Q^{2}U(s)\text{d}s}.$ (22)

then $V_{t}=-U^{T}(t-\tau(t))Q^{2}U(t-\tau(t))(1-\tau^{\prime}(t))+U^{T}(t)Q^{2}U(t)$ , $V_{uz}^{\prime}=2(U^{T}(t),Z^{T}(t))$ , $V_{uz}{{}^{\prime\prime}}=2E_{n\times n}$ , where $V_{uz}^{\prime}$ is the first derivative of $U, Z$ , $V_{uz}{{}^{\prime\prime}}$ is the second derivative of $U, Z$ . Apply $L$ to function $V(t,U(t),Z(t))$ , from Definition 4 we can get:

$\displaystyle\textit{LV}(t,U,Z)=U^{T}(t)Q^{2}U(t)-U^{T}(t-\tau(t))Q^{2}U(t-% \tau(t))(1-\tau^{\prime}(t))+2[-U^{T}(t)U(t)+U^{T}(t)Z(t)+Z^{T}(t)\textit{RU}(% t)-Z^{T}(t)\textit{RZ}(t)-Z^{T}(t)\alpha(U(t))H(U(t))+Z^{T}(t)\alpha(U(t))% \textit{Af}(U(t))+Z^{T}(t)\alpha(U(t))\textit{Bf}(U(t-\tau(t)))]+\frac{1}{2}g^% {T}(U(t))g(U(t)).$ (23)

From Lemma 2:

$\displaystyle 2Z^{T}(t)\textit{RU}(t)-2Z^{T}(t)\alpha(U(t))H(U(t))\leqslant Z^% {T}(t)\textit{GZ}(t)+U^{T}(t)\textit{GU}(t).$ (24) $\displaystyle 2Z^{T}(t)\alpha(U(t))\textit{Af}(U(t))\leqslant\frac{1}{\beta}f^% {T}(U(t))A^{T}\textit{Af}(U(t))+\beta Z^{T}(t)\bar{\alpha}^{2}Z(t).$ (25) $\displaystyle 2Z^{T}(t)\alpha(U(t))\textit{Bf}(U(t-\tau(t)))\leqslant\delta Z^% {T}(t)\bar{\alpha}^{2}Z(t)+\frac{1}{\delta}f^{T}(U(t-\tau(t)))B^{T}\textit{Bf}% (U(t-\tau(t))),$ (26)

where $\bar{\alpha}^{2}=\text{diag}(\bar{\alpha}_{1}^{2},\bar{\alpha}_{2}^{2},\ldots,% \bar{\alpha}_{n}^{2})$ .

$\displaystyle U^{T}(t)Z(t)\leqslant\frac{1}{2}U^{T}(t)U(t)+\frac{1}{2}Z^{T}(t)% Z(t).$ (27)

It can be seen from $(\text{H}_{3})$ that if $f_{i},g_{i}$ satisfied Lipschitz conditions, there is

$\displaystyle U^{T}(t-\tau(t))Q^{2}U(t-\tau(t))\geqslant f^{T}(U(t-\tau(t)))Q^% {2}L^{-2}f(U(t-\tau(t))).$ (28) $\displaystyle g^{T}(U(t))g(U(t))\leqslant U^{T}(t)\mu^{2}U(t),$ (29)

where $\mu^{2}=\text{diag}(\mu_{1}^{2},\mu_{2}^{2},\ldots,\mu_{n}^{2})$ .

And $0<{\tau}^{\prime}(t)<\eta<1$ , bring Eqs (24)–(29) into (23), we get

$\displaystyle\textit{LV}(t,U,Z)\leqslant-f^{T}(U(t))\left[\left(-\frac{1}{2}% \mu^{2}-Q^{2}-G+E\right)L^{-2}-\frac{1}{\beta}A^{T}A\right]f(U(t))-Z^{T}(t)[-(% \beta+\delta)\overline{\alpha}^{2}-E-G+2R]Z(t)-f^{T}(U(t-\tau(t)))\left[(1-% \eta)Q^{2}L^{-2}-\frac{1}{\delta}B^{T}B\right]f(U(t-\tau(t))).$ (30)

Due to $P_{3}=(1-\eta)Q^{2}L^{-2}-\frac{1}{\delta}B^{T}B>0,P_{4}=(-Q^{2}+E-G-\frac{1}{% 2}\mu^{2})L^{-2}-\frac{1}{\beta}A^{\text{T}}A>0,P_{5}=-G+2R-E-(\beta+\delta)% \overline{\alpha}^{2}>0$ , we get $LV(t,U(t),Z(t)))<$ 0 from Eq. (30). And $V(t,U(t),Z(t))\geqslant 0,V(0,0,0)=0$ , then $V(t,U(t),Z(t))$ is a positive-definite function with infinitesimal upper bound. And from $V(t,U(t),Z(t))\geqslant\left\|{U(t)}\right\|+\left\|{Z(t)}\right\|$ , we have $V(t,U(t),Z(t))\to+\infty$ , when $||U||\to+\infty,_{|}|Z||\to+\infty$ . From Definition 6, $V(t,U(t),Z(t))$ is the radial unbounded positive-definite function of $(U^{T}(t),Z^{T}(t))^{T}$ . Then the zero solution of system Eq. (21) is stochastic asymptotically stable from Lemma 3. Due to $u_{i}(t)=x_{i}(t)-x_{i}^{*},z_{i}(t)=y_{i}(t)-y_{i}^{*},i=1,2,\ldots n$ , when $U(t)=0,Z(t)=0$ , there is $X(t)=X^{\ast},Y(t)=Y^{\ast}$ , it means the equilibrium point $X^{*}$ is stochastic asymptotically stable of system Eq. (2).

Remarks 3. If $g_{j}(\cdot)=$ 0, system Eq. (2) is inertial Cohen-Grossberg neural networks with time-varying delay, i.e. system Eq. (1). Suppose Theorem 1 holds, and

$\displaystyle P_{4}=-\frac{1}{\beta}A^{T}A+(E-Q^{2}-G)L^{-2}>0,$ $\displaystyle P_{5}=2R-E-(\beta+\delta)\bar{\alpha}^{2}-G>0,$

so system Eq. (1) has an equilibrium point, and the point is stochastic asymptotically stable, where $\bar{\alpha}^{2}=\text{diag}(\bar{\alpha}_{1}^{2},\bar{\alpha}_{2}^{2},\ldots,% \bar{\alpha}_{n}^{2})$ , $\bar{\alpha}_{i}$ are all given by hypothesis $(\text{H}_{1})$ .

If $\alpha_{i}(x_{i}(t))=\alpha_{i},h_{i}(x_{i}(t))=h_{i}x_{i}(t)$ , system Eq. (2) becomes stochastic inertial neural networks with time-varying delay.

$\displaystyle\text{d}(\dot{x}_{i}(t))=-d_{i}\dot{x}_{i}(t)\text{d}t-\alpha_{i}% \left[h_{i}x_{i}(t)-\sum\limits_{j=1}^{n}{a_{ij}f_{j}(x_{j}(t))}-\sum\limits_{% j=1}^{n}{b_{ij}f_{j}(x_{j}(t-\tau(t)))}+I_{i}\right]\text{d}t+\sum\limits_{j=1% }^{n}{g_{j}(x_{j}(t))\text{d}B_{i}(t)}.$ (31)

4. Numerical simulation

An example is given to illustrate the correctness of the theorems in this paper.

Giving a stochastic inertial Cohen-Grossberg neural networks with time-varying delay ( $n=$ 3):

$\displaystyle\text{d}(\dot{x}_{i}(t))=-d_{i}\dot{x}_{i}(t)\text{d}t-\alpha_{i}% (x_{i}(t))\!\!\left[h_{i}(x_{i}(t))\!-\!\sum_{j=1}^{3}{a_{ij}f_{j}(x_{j}(t))}% \!-\!\sum_{j=1}^{3}{b_{ij}f_{j}(x_{j}(t-\tau(t)))}+I_{i}\right]\!\!\text{d}t+% \sum\limits_{j=1}^{3}{g_{j}(x_{j}(t))\text{d}B_{i}(t)},i=1,2,3.$ (32)

where: $d_{1}=$ 2.3, $d_{2}=$ 2.1, $d_{3}=$ 2.3, $h_{1}(x_{1})=\frac{2}{3}x_{1}$ , $h_{2}(x_{2})=\frac{3}{4}x_{2}$ , $h_{3}(x_{3})=\frac{5}{6}x_{3}$ , $\alpha_{1}(x_{1})=2+\frac{1}{1+x_{1}^{2}}$ , $\alpha_{2}(x_{2})=2.5-\frac{1}{x_{2}^{2}+1}$ , $\alpha_{3}(x_{3})=1.6+\frac{1}{2(x_{3}^{2}+1)}$ , $\tau(t)=\frac{-e^{-t}+2}{3}$ , $f_{j}(x_{j})=\sin(x_{j})$ , $g_{1}(x_{1})=\tan(x_{1})-1$ , $g_{2}(x_{2})=\tan(x_{2})-\sqrt{3}$ , $g_{3}(x_{3})=\tan(x_{3})-\frac{\sqrt{3}}{3}$ , $I_{1}=-\frac{\pi}{6}+\frac{1}{144}(\sqrt{2}+1-\sqrt{3})$ , $I_{2}=-\frac{\pi}{4}+\frac{1}{144}(1-\sqrt{2}-\sqrt{3})$ , $I_{3}=-\frac{5\pi}{36}+\frac{1}{144}(\sqrt{2}+\sqrt{3}-1)$ ,

$\displaystyle A=\frac{1}{144}\left({{\begin{array}[]{*{20}c}1\hfill&{-1}\hfill% &1\hfill\\ {-1}\hfill&{-1}\hfill&1\hfill\\ 1\hfill&1\hfill&{-1}\hfill\\ \end{array}}}\right),\text{ }B=\frac{1}{144}\left({{\begin{array}[]{*{20}c}1% \hfill&{-1}\hfill&1\hfill\\ {-1}\hfill&{-1}\hfill&1\hfill\\ 1\hfill&1\hfill&{-1}\hfill\\ \end{array}}}\right).$

Let $\underline{\alpha}_{1}=$ 2, $\bar{\alpha}_{1}=$ 3, $\underline{\alpha}_{2}=$ 1.5, $\bar{\alpha}_{2}=$ 2.5, $\underline{\alpha}_{3}=$ 1.6, $\bar{\alpha}_{3}=$ 2.1, $\underline{h}_{1}=\bar{h}_{1}=\frac{2}{3}$ , $\underline{h}_{2}=\bar{h}_{2}=\frac{3}{4}$ , $\underline{h}_{3}=\bar{h}_{3}=\frac{5}{6}$ , $\eta=\frac{1}{3}$ , $l_{1}=l_{2}=l_{3}=1$ , $\mu_{1}=\mu_{2}=\mu_{3}=\frac{1}{10}$ , then hypotheses $(\text{H}_{1})-(\text{H}_{3})$ hold. We select $Q=\text{diag}\left(\frac{2}{5},\frac{3}{10},\frac{1}{5}\right)$ , $\delta=\frac{1}{27}$ , $\beta=\frac{1}{81}$ , $L=\text{diag}(1,1,1)$ , $\mu^{2}=\text{diag}(0.01,0.01,0.01)$ .

Through calculating, $0<q_{1}<\sqrt{0.3}<$ 1, $0<q_{2}<\sqrt{0.225}<$ 1, $0<q_{3}<\sqrt{0.55}<$ 1, $\underline{H}=\text{diag}\left(\frac{2}{3},\frac{3}{4},\frac{5}{6}\right)$ , $1+\underline{h}_{i}\underline{\alpha}_{i}-d_{i}>$ 0, $i=$ 1, 2, 3.

$\displaystyle R=\left({{\begin{array}[]{*{20}c}{1.3}&0&0\\ 0&{1.1}&0\\ 0&0&{1.3}\\ \end{array}}}\right),\bar{\alpha}=\left({{\begin{array}[]{*{20}c}3&0&0\\ 0&{2.5}&0\\ 0&0&{2.1}\\ \end{array}}}\right),G=\left({{\begin{array}[]{*{20}c}{0.7}&0&0\\ 0&{0.775}&0\\ 0&0&{0.45}\\ \end{array}}}\right),$ $\displaystyle P_{1}=\left({{\begin{array}[]{*{20}c}{0.284}&0&0\\ 0&{0.4506}&0\\ 0&0&{0.6173}\\ \end{array}}}\right)>0,P_{2}=\left({{\begin{array}[]{*{20}c}{0.14}&0&0\\ 0&{0.135}&0\\ 0&0&{0.51}\\ \end{array}}}\right)>0,$ $\displaystyle P_{3}=\left({{\begin{array}[]{*{20}c}{0.1067}&0&0\\ 0&{0.06}&0\\ 0&0&{0.0267}\\ \end{array}}}\right)>0,P_{4}=\left({{\begin{array}[]{*{20}c}{0.135}&0&0\\ 0&{0.13}&0\\ 0&0&{0.505}\\ \end{array}}}\right)>0,$ $\displaystyle P_{5}=\left({{\begin{array}[]{*{20}c}{0.4556}&0&0\\ 0&{0.1164}&0\\ 0&0&{0.9322}\\ \end{array}}}\right)>0.$

We get the equilibrium equation of system Eq. (32) as follows:

$\displaystyle\left\{{\begin{array}[]{l}\frac{2}{3}x_{1}-\frac{1}{72}\sin(x_{1}% )+\frac{1}{72}\sin(x_{2})-\frac{1}{72}\sin(x_{3})-\frac{\pi}{6}+\frac{1}{144}(% \sqrt{2}+1-\sqrt{3})=0,\\ \frac{3}{4}x_{2}+\frac{1}{72}\sin(x_{1})+\frac{1}{72}\sin(x_{2})-\frac{1}{72}% \sin(x_{3})-\frac{\pi}{4}+\frac{1}{144}(1-\sqrt{2}-\sqrt{3})=0,\\ \frac{5}{6}x_{3}-\frac{1}{72}\sin(x_{1})-\frac{1}{72}\sin(x_{2})+\frac{1}{72}% \sin(x_{3})-\frac{5\pi}{36}+\frac{1}{144}(\sqrt{2}+\sqrt{3}-1)=0.\\ \end{array}}\right.$

And the equilibrium point is

$\displaystyle(x_{1}^{\ast},x_{2}^{\ast},x_{3}^{\ast})^{T}=\left(\frac{\pi}{4},% \frac{\pi}{3},\frac{\pi}{6}\right)^{T}.$

The stochastic term $g_{1}(x_{1}^{\ast})=g_{2}(x_{2}^{\ast})=g_{3}(x_{3}^{\ast})=$ 0. Therefore, when Theorem 2 holds, system Eq. (32) has equilibrium point $\left(\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{6}\right)^{T}$ , and it is stochastic asymptotically stable.

On the other hand, four groups of initial values are given: $[x_{1}(0),x_{2}(0),x_{3}(0),\dot{x}_{1}(0),\dot{x}_{2}(0),\dot{x}_{3}(0)]=$ : [2.95, 3, 2, 2.9, 2.95, 1.95]; [1.5, 1.2, 1, 1.45, 1.2, 1]; [0, 0, 0, 0.1, 0.05, 0.01]; [ $-$ 1, $-$ 1, $-$ 1, $-$ 0.95, $-$ 0.99, $-$ 0.98].

Figure 1.

Transient change of $x_{1}(t)$ .

Figure 2.

Transient change of $x_{2}(t)$ .

Figure 3.

Transient change of $x_{3}(t)$ .

Figures 1–3 describe a respectively instantaneous variation of variables under system Eq. (32). This conclusion is coincident with the results in numerical simulation.

5. Conclusion

This paper studies the existence and stochastic asymptotic stability of stochastic inertial Cohen-Grossberg neural networks with time-varying delay. Given the sufficient conditions that the equilibrium point is stochastic asymptotically stable, namely Theorems 1 and 2, are obtained. And the correctness of the theorems is illustrated by a numerical simulation.

Footnotes

Acknowledgments

The authors acknowledge the Project Foundation of Zhejiang Provincial Department of Education (Grant: Y202145903), Project Foundation of Shaoxing University (Grant: 2020LG1009).

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