On Asymptotic Properties of Stochastic Neutral-Type Inertial Neural Networks with Mixed Delays

Abstract

This article studies the stability problem of a class of stochastic neutral-type inertial delay neural networks. By introducing appropriate variable transformations, the second-order differential system is transformed into a first-order differential system. Using homeomorphism mapping, standard stochastic analyzing technology, the Lyapunov functional method and the properties of a neutral operator, we establish new sufficient criteria for the unique existence and stochastically globally asymptotic stability of equilibrium points. An example is also provided, to show the validity of the established results. From our results, we find that, under appropriate conditions, random disturbances have no significant impact on the existence, stability, and symmetry of network systems.

1. Introduction

Inertial neural network systems (INNs) can be understood as damped neural networks. When the damping exceeds a certain critical value, the dynamic properties of each neuron state will also change. Therefore, studying inertial neural networks can help us to understand the complex changes in neural network systems. NNs are second-order systems, which were first introduced by Wheeler and Schieve [1]. Subsequently, INNs have received increasing attention from scholars, whose dynamic behavior research has received considerable attention. In [2,3,4], the authors studied bifurcation, chaos and the stability of periodic solutions in a single INN. The dynamics of an inertial two-neuron system with delay were studied in [5]. Draye, Winters and Cheron [6] considered self-selected modular recurrent neural networks with postural and inertial subnetworks. For the global exponential stability in the Lagrange sense for INNs, see [7,8]; for the global convergence problem of impulsive inertial neural networks with variable delay and complex neural networks, see [9,10]; for the antiperiodic problem of INNs, see [11]; for inertial Cohen–Grossberg neural networks, see [12,13,14]; for inertial BAM neural networks, see [15,16].

In practice, neural network systems are not only affected by damping (inertia) factors, but also by random factors. The research on stochastic neural network systems is relatively mature and has achieved many results. Zhang and Kong [17] considered photovoltaic power prediction based on the hybrid modeling of neural networks and stochastic differential equations. Shu et al. [18] studied the stochastic stabilization of Markov jump quaternion-valued neural networks, by using a sampled-data control strategy. Guo [19] investigated globally robust stability analysis for stochastic Cohen–Grossberg neural networks with impulse control and time-varying delays. For the stability of random cellular neural networks, see [20,21]; for mean square exponential stability and periodic solutions of stochastic Hopfield neural networks, see [22,23]; for the stability problem of recurrent neural networks with random delays, see [24].

Neutral-type neural networks are nonlinear systems, which show neutral properties by involving derivatives with delays. Due to extensive applications in ecology, control theory, biology and physics for neutral-type neural networks, many results have been obtained. He et al. [25] addressed the problem of synchronization control of neutral-type neural networks with sampled data. Using an adaptive event-triggered communication scheme, they obtained some weak synchronization conditions for synchronization control. In [26], a new control scheme was studied for exponential synchronization of coupled neutral-type neural networks with mixed delays. Si, Xie and Lie [27] investigated the global exponential stability of recurrent neural networks with piecewise constant arguments and neutral terms subject to uncertain connection weights. For more results of neutral-type neural networks, see, e.g., [28,29,30,31,32,33].

Stochastic neutral-type INNs provide more useful models in practical applications. In this paper, we will study the stability problem of a class of stochastic neutral-type inertial neural networks with mixed delays via stochastic analyzing technology and by constructing a suitable Lyapunov–Krasovskii functional. A simulation example is used, to demonstrate the usefulness of our theoretical results. The contributions of this paper are threefold:

(1)

In this paper, we introduce a new class of stochastic neutral-type INNs with D-operators, which is different from the existing models (see, e.g., [25,26,27]).

(2)

For constructing a suitable Lyapunov–Krasovskii functional, the mixed delays and the neutral terms are taken into consideration.

(3)

Unlike the previous papers, we introduce a new unified framework, to deal with mixed delays, inertia terms and D-operators. It is noted that our main results are also valid in cases of non-neutral systems.

The following sections are organized as follows: Section 2 gives preliminaries and model formulation. In Section 3, sufficient conditions are established for the unique existence of equilibrium points of system (3). The stochastically globally asymptotic stability of equilibrium points is given in Section 4. In Section 5, a numerical example is given, to show the feasibility of our results. Finally, some conclusions are given about this paper.

2. Preliminaries and Problem Formulation

Consider a class of neutral-type INNs with mixed delays as follows:

$$\begin{array}{cc}\hfill \frac{d^2\left[\left(A_i{x}_i\right)\left(t\right)\right]}{d{t}^2}& =-a_i\frac{d\left[\left(A_i{x}_i\right)\left(t\right)\right]}{dt}-b_i{x}_i\left(t\right)+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)\hfill \\ & +\sum _{j=1}^n{e}_{ij}{\int }_{t-{\gamma }_1}^t{f}_j\left(x_j\left(s\right)\right)ds+I_i,\hfill \end{array}$$

(1)

where $t\ge 0,i=1,2,\cdots ,n$, $A_i$ is a difference operator defined by

$$\left(A_i{x}_i\right)\left(t\right)=x_i\left(t\right)-c_i\left(t\right)x_i(t-{\gamma }_0).$$

(2)

If a random disturbance term is added to system (1), we obtain the following stochastic INNs:

$$\begin{array}{cc}\hfill d\left[{\left(A_i{x}_i\right)}^{\prime }\left(t\right)\right]& =-a_i{\left(A_i{x}_i\right)}^{\prime }\left(t\right)dt-b_i{x}_i\left(t\right)dt+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)dt+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)dt\hfill \\ & +\left(\sum _{j=1}^n{e}_{ij}{\int }_{t-{\gamma }_1}^t{f}_j\left(x_j\left(s\right)\right)ds\right)dt+I_{i}dt+\sum _{j=1}^n{h}_{ij}{g}_j\left(x_j\left(t\right)\right)d{B}_i\left(t\right),\hfill \end{array}$$

(3)

where $x_i\left(t\right)$ denotes the neuron state, $c_i\left(t\right)\in C(\mathbb{R},\mathbb{R})$ is neutral parameter, $a_i>0$ is the damping (inertia) coefficient, $b_i>0$ is a constant, $c_{ij},d_{ij}$ and $e_{ij}$ represent the output feedback weight values, $I_i$ represents the threshold or bias of the system, ${\gamma }_0,{\gamma }_1,{\tau }_j\left(t\right)>0$ are delays with ${\tau }_j^{\prime }\left(t\right)<1$, and where ${\sum }_{j=1}^n{h}_{ij}{g}_j\left(x_j\left(t\right)\right)d{B}_i\left(t\right)$ represents random perturbation and $B\left(t\right)={(B_1\left(t\right),B_2\left(t\right),\cdots ,B_n\left(t\right))}^T$ is defined as an $n-$dimensional Brownian motion with natural filtering ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ on a complete probability space $(\mathrm{\Omega },\mathcal{F},P)$. The initial conditions of system (1) are given by

$$\left\{\begin{array}{c}{x}_i\left(s\right)={\varphi }_i\left(s\right),s\in (-\mu ,0],i=1,2,\cdots ,n,\hfill \\ x_i^{\prime }\left(s\right))={\psi }_i\left(s\right),s\in (-\mu ,0],i=1,2,\cdots ,n,\hfill \end{array}\right.$$

where $\mu =max\{{\gamma }_0,{\gamma }_1,\widehat{\tau }\},\widehat{\tau }={max}_{t\in \mathbb{R}}{\tau }_j\left(t\right)$. Let

$$y_i\left(t\right)=\frac{d\left[A_i{x}_i\left(t\right)\right]}{dt}+{\xi }_i\left(A_i{x}_i\right)\left(t\right),i=1,2,\cdots ,n,$$

where ${\xi }_i>0$ is a constant. Then, system (3) is changed into the following form:

$$\left\{\begin{array}{c}d\left[\left(A_i{x}_i\right)\left(t\right)\right]=-{\xi }_i\left(A_i{x}_i\right)\left(t\right)dt+y_i\left(t\right)dt,\hfill \\ y_i^{\prime }\left(t\right)=-(a_i-{\xi }_i)y_i\left(t\right)dt+\left[(a_i-{\xi }_i){\xi }_i\right]\left(A_i{x}_i\right)\left(t\right)dt-b_i{x}_i\left(t\right)dt\hfill \\ +\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)dt+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)dt\hfill \\ +\left(\sum _{j=1}^n{e}_{ij}{\int }_{t-{\gamma }_1}^t{f}_j\left(x_j\left(s\right)\right)ds\right)dt+I_i\left(t\right)dt+\sum _{j=1}^n{h}_{ij}{g}_j\left(x_j\left(t\right)\right)d{B}_i\left(t\right).\hfill \end{array}\right.$$

(4)

Remark 1. 

The operator $A_i$ in (2) is called $D-$operator (see [34]). $D-$operator has many excellent properties. On the one hand, it can help us better understand the characteristics of neutral equations and, on the other hand, the above properties can be used to study the dynamic behavior of the solution of the system. Recently, the authors [33] studied a class of neutral inertial neural networks. The differences between this article and [33] are mainly reflected in two aspects: firstly, the neutral-type system in this article has the form of $D-$operator, while the system in [33] does not; secondly, the system in this article is stochastic, while the system in [33] is deterministic.

Lemma 1 

([35]). For $t\in \mathbb{R}$, if ${\widehat{c}}_i<1$, then, for the operator $A_i$ there exists an inverse operator $A_i^{-1}$, which satisfies

$$\left|\right|A_i^{-1}\left|\right|\le \frac{1}{1-{\widehat{c}}_i},$$

where $i=1,2,\cdots ,n,{\widehat{c}}_i={max}_{t\in \mathbb{R}}\left|c_i\left(t\right)\right|$, $A_i$ is defined by (2).

Definition 1. 

If $X^{*}={(x_1^{*},x_2^{*},\cdots ,x_n^{*})}^T\in {\mathbb{R}}^n$ satisfies

$$-b_i{x}_i^{*}+\sum _{j=1}^n\left(c_{ij}+d_{ij}+{\gamma }_1{e}_{ij}\right)f_j\left(x_j^{*}\right)+I_i=0,$$

then $X^{*}$ is called the equilibrium point of system (1). If $g_j\left(x_j^{*}\right)=0$, then system (3) reaches equilibrium state and no longer sends noise interference to other units, indicating that the equilibrium point $X^{*}$ of system (1) is also the equilibrium point of system (3).

Let $C^{1,2}({\mathbb{R}}^{+}\times S_h,{\mathbb{R}}^{+})$ be the non-negative function $V(t,X)$ on ${\mathbb{R}}^{+}\times S_h$, where $S_h=\{X:|\left|X\right||<h\}\subset {\mathbb{R}}^n$, $V(t,X)$ has continuous first and second derivatives for $(t,X)$, respectively.

Definition 2 

([36]). Give the following system:

$$\left\{\begin{array}{c}dX\left(t\right)=f(t,X\left(t\right))dt+g(t,X\left(t\right))dB\left(t\right),t\ge t_0,\hfill \\ X\left(t_0\right)=X_0.\hfill \end{array}\right.$$

(5)

Define the operator:

$$\mathcal{L}V(t,X)=V_t(t,X)+V_X(t,X)f(t,X)+\frac{1}{2}\mathrm{trace}\left[g^T(t,X)V_{XX}(t,X)g(t,X)\right],$$

where $V(t,X)\in C^{1,2}({\mathbb{R}}^{+}\times S_h,{\mathbb{R}}^{+}),V_t(t,X)=\frac{\partial V(t,X)}{\partial t},$

$$V_X(t,X)=\left(\frac{\partial V(t,X)}{\partial x_1},\frac{\partial V(t,X)}{\partial x_2},\cdots ,\frac{\partial V(t,X)}{\partial x_n}\right),V_{XX}(t,X)={\left(\frac{{\partial }^{2}V(t,X)}{\partial x_i\partial x_j}\right)}_{n\times n}.$$

Definition 3 

([37]). Let $\mathrm{\Omega }\subset {\mathbb{R}}^n,$ where Ω is an $n-$dimensional open field containing the origin. The function $V(t,X)$ has an infinitesimal upper bound, if there exists a positive definite function $W\left(X\right)$, such that $\left|V\right(t,X\left)\right|\le W\left(X\right)$.

Definition 4 

([37]). The function $W\left(X\right)\in C({\mathbb{R}}^n,\mathbb{R})$ is an infinite positive definite function, if $W\left(X\right)$ is positive definite and $W\left(X\right)\to +\infty$ for $\left|\right|X\left|\right|\to \infty$.

Lemma 2 

([37]). If $H\left(u\right)\in C^0$ satisfies

(i) 

$H\left(u\right)$ is injective on ${\mathbb{R}}^n$;

(ii) 

$\left|\right|H\left(u\right)\left|\right|\to \infty$ for $\left|\right|u\left|\right|\to \infty$, then $H\left(u\right)$ is homeomorphic on ${\mathbb{R}}^n$.

Lemma 3 

([37]). If there is a radial unbounded positive definite function $V(t,X)\in C^{1,2}([t_0,+\infty )\times S_h,{\mathbb{R}}^{+})$ with an infinitesimal upper bound, such that $\mathcal{L}V(t,X)$ is negative definite, then the zero solution of system (5) is stochastically globally asymptotically stable.

Throughout the paper, the following assumptions hold:

(H₁)

There exist constants $l_j\ge 0$, such that

$$|f_j\left(x\right)-f_j\left(y\right)|\le l_j|x-y|,j=1,2,\cdots ,n,\forall x,y\in \mathbb{R}.$$

(H₂)

There exist constants ${\tilde{l}}_j\ge 0$, such that

$$|g_j\left(x\right)-g_j\left(y\right)|\le {\tilde{l}}_j|x-y|,j=1,2,\cdots ,n,\forall x,y\in \mathbb{R}.$$

(H₃)

There exist constants $N_j\ge 0$, such that

$$|f_j\left(x\right)|\le \frac{1}{2}{N}_j,j=1,2,\cdots ,n,\forall x\in \mathbb{R}.$$

3. Existence of Equilibrium Points

Theorem 1. 

Suppose that assumption (H${}_1$) holds. Then, system (1) has a unique equilibrium point, provided that

$$-b_i+\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j+\frac{1}{2}\sum _{j=1}^n\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i<0.$$

(6)

Proof. 

For $u={(u_1,u_2,\cdots ,u_n)}^T$, consider the mapping $H\left(u\right)=(H_1\left(u\right),H_2\left(u\right),\cdots ,$$H_n{\left(u\right))}^T$, where

$$H_i\left(u\right)=-b_i{u}_i+\sum _{j=1}^n\left(c_{ij}+d_{ij}+{\gamma }_1{e}_{ij}\right)f_j\left(u_j\right)+I_i.$$

If $H\left(u\right)$ is homeomorphic on ${\mathbb{R}}^n$, there is a unique equilibrium point $u^{*}$, such that $H\left(u^{*}\right)=0$. Now, we show that $H\left(u\right)$ is homeomorphic on ${\mathbb{R}}^n$. We first show that $H\left(u\right)$ is injective on ${\mathbb{R}}^n$. Assume that there exist $u,v\in {\mathbb{R}}^n$ with $u\ne v$, such that $H\left(u\right)=H\left(v\right)$. Then, we have

$$-b_i(u_i-v_i)+\sum _{j=1}^n\left(c_{ij}+d_{ij}+{\gamma }_1{e}_{ij}\right)[f_j\left(u_j\right)-f_j\left(v_j\right)]=0,i=1,2,\cdots ,n.$$

(7)

Multiplying both sides of system (7) by $u_i-v_i$ yields

$$-b_i{(u_i-v_i)}^2+(u_i-v_i)\sum _{j=1}^n\left(c_{ij}+d_{ij}+{\gamma }_1{e}_{ij}\right)[f_j\left(u_j\right)-f_j\left(v_j\right)]=0.$$

(8)

Using $a^2+b^2\ge 2ab,$ (H${}_1$) and (8), we have

$$-b_i{(u_i-v_i)}^2+\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j[{(u_i-v_i)}^2+{(u_j-v_j)}^2]\ge 0$$

and

$$\sum _{i=1}^n\left(-b_i+\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j+\frac{1}{2}\sum _{j=1}^n\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i\right){(u_i-v_i)}^2\ge 0.$$

(9)

From (6) and (9), we have $u_i=v_i$, which contradicts the assumption $u_i\ne v_i$. Therefore, $H\left(u\right)$ is injective on ${\mathbb{R}}^n$.

Furthermore, let $\tilde{H}\left(u\right)=H\left(u\right)-H\left(0\right)$. We have

$$\begin{array}{cc}\hfill u^T\tilde{H}\left(u\right)& =\sum _{i=1}^n{u}_i{\tilde{H}}_i\left(u\right)=\sum _{i=1}^n\left(-b_i{u}_i^2+u_i\sum _{j=1}^n\left(c_{ij}+d_{ij}+{\gamma }_1{e}_{ij}\right)(f_j\left(u_j\right)-f_j\left(0\right))\right)\hfill \\ & \le \sum _{i=1}^n\left(-b_i+\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j+\frac{1}{2}\sum _{j=1}^n\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i\right)u_i^2\hfill \\ & \le -\underset{1\le i\le n}{min}\left(b_i-\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j-\frac{1}{2}\sum _{j=1}^n\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i\right){\left|\right|u\left|\right|}^2.\hfill \end{array}$$

Using inequality $-X^{T}Y\le |X^{T}Y|\le |\left|X\right|\left|\right|\left|Y\right||$, we obtain

$$\left|\right|\tilde{H}\left(u\right)\left|\right|\ge \underset{1\le i\le n}{min}\left(b_i-\frac{1}{2}\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j-\frac{1}{2}\sum _{j=1}^n\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i\right)\left|\right|u\left|\right|$$

and $\left|\right|\tilde{H}\left(u\right)\left|\right|\to \infty$ for $\left|\right|u\left|\right|\to \infty$, i.e., $\left|\right|H\left(u\right)\left|\right|\to \infty$ for $\left|\right|u\left|\right|\to \infty$. Hence, $H\left(u\right)$ is homeomorphic on ${\mathbb{R}}^n$, and there is a unique equilibrium point for system (1). □

Remark 2. 

The proof of Theorem 1 is similar to the one in [38]. For the convenience of readers, we have provided detailed proof of Theorem 1.

Remark 3. 

Since system (1) contains neutral-type terms and inertial terms, some research techniques, such as topological degree methods and fixed point theorems, are not easy for studying the existence of the equilibrium point of system (1). However, under the assumptions of this article, we can easily study the existence of the equilibrium point of system (1), by using homeomorphism mapping.

4. Stochastically Globally Asymptotic Stability

Theorem 2. 

Assume that all conditions of Theorem 1 hold and that assumptions (H${}_2$) and (H${}_3$) hold. Then, the equilibrium point of system (3) is stochastically globally asymptotically stable, provided that

$$\begin{array}{cc}& -2{\xi }_i+1+|(a_i-{\xi }_i){\xi }_i|+b_i(1-|c_i{\left|\right)}^{-2}+\sum _{j=1}^n\left|d_{ij}\right|{\widehat{\omega }}_j{L}_j^2{(1-{\widehat{c}}_i)}^{-2}\hfill \\ & +\sum _{j=1}^n\left|e_{ij}\right|{\gamma }_1{L}_j^2{(1-{\widehat{c}}_i)}^{-2}+\sum _{k=1}^n\sum _{j=1}^n{h}_{kj}^2{\tilde{l}}_i^2{(1-{\widehat{c}}_i)}^{-2}<0\hfill \end{array}$$

(10)

and

$$-2(a_i-{\xi }_i)+1+\left|(a_i-{\xi }_i){\xi }_i\right|+b_i+\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+|e_{ij}|\right)<0,$$

(11)

where ${\widehat{c}}_i={max}_{t\in \mathbb{R}}\left|c_i\left(t\right)\right|,{\widehat{\omega }}_j={max}_{t\in \mathbb{R}}{\omega }_j\left(t\right),{\omega }_j\left(t\right)=\frac{1}{1-{\tau }_j^{\prime }\left({\mu }_j\left(t\right)\right)},{\mu }_j\left(t\right)$ is an inverse function of $t-{\tau }_j\left(t\right)$.

Proof. 

From Theorem 1, system (1) exists a unique equilibrium point. When $g_j\left(x_j^{*}\right)=0$, systems (1) and (3) have the same equilibrium point. Let $X^{*}={(x_1^{*},x_2^{*},\cdots ,x_n^{*})}^T\in {\mathbb{R}}^n$ be the unique equilibrium point for system (3). Let $\left(A_i{\tilde{x}}_i\right)\left(t\right)=\left(A_i{x}_i\right)\left(t\right)-A_i{x}_i^{*}$ and ${\tilde{y}}_i\left(t\right)=y_i\left(t\right)-y_i^{*}$, where $y_i^{*}={\xi }_i{A}_i{x}_i^{*}$. By (4), we have

$$\left\{\begin{array}{c}d\left[\left(A_i{\tilde{x}}_i\right)\left(t\right)\right]=-{\xi }_i\left(A_i{\tilde{x}}_i\right)\left(t\right)dt+{\tilde{y}}_i\left(t\right)dt,\hfill \\ {\tilde{y}}_i^{\prime }\left(t\right)=-(a_i-{\xi }_i){\tilde{y}}_i\left(t\right)dt+\left[(a_i-{\xi }_i){\xi }_i\right]\left(A_i{\tilde{x}}_i\right)\left(t\right)dt-b_i{\tilde{x}}_i\left(t\right)dt\hfill \\ +\sum _{j=1}^n{c}_{ij}{\tilde{f}}_j\left({\tilde{x}}_j\left(t\right)\right)dt+\sum _{j=1}^n{d}_{ij}{\tilde{f}}_j\left({\tilde{x}}_j(t-{\tau }_j\left(t\right))\right)dt\hfill \\ +\left(\sum _{j=1}^n{e}_{ij}{\int }_{t-{\gamma }_1}^t{\tilde{f}}_j\left({\tilde{x}}_j\left(s\right)\right)ds\right)dt+\sum _{j=1}^n{h}_{ij}{\tilde{g}}_j\left({\tilde{x}}_j\left(t\right)\right)d{B}_i\left(t\right),\hfill \end{array}\right.$$

(12)

where

$${\tilde{f}}_j\left({\tilde{x}}_j\left(t\right)\right)=f_j({\tilde{x}}_j\left(t\right)+x_j^{*})-f_j\left(x_j^{*}\right),{\tilde{g}}_j\left({\tilde{x}}_j\left(t\right)\right)=g_j({\tilde{x}}_j\left(t\right)+x_j^{*})-g_j\left(x_j^{*}\right).$$

Let

$$\tilde{Z}=(A\tilde{X},\tilde{Y})=(A_1{\tilde{x}}_1,A_2{\tilde{x}}_2,\cdots ,A_n{\tilde{x}}_n,{\tilde{y}}_1,{\tilde{y}}_2,\cdots ,{\tilde{y}}_n)$$

and

$$\begin{array}{cc}\hfill V(t,\tilde{Z})& =\sum _{i=1}^n\left({\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)\right)+\sum _{i=1}^n\sum _{j=1}^n\left|d_{ij}\right|{\int }_{t-{\tau }_j\left(t\right)}^t{\omega }_j\left(s\right){\tilde{f}}_j^2\left({\tilde{x}}_j\left(s\right)\right)ds\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|{\int }_0^{{\gamma }_1}{\int }_{t-s}^t{\tilde{f}}_j^2\left({\tilde{x}}_j\left(\theta \right)\right)d\theta ds.\hfill \end{array}$$

(13)

By (13), we obtain

$$\begin{array}{cc}\hfill V_t(t,\tilde{Z})& =\sum _{i=1}^n\sum _{j=1}^n\left|d_{ij}\right|\left({\omega }_j\left(t\right){\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\tilde{f}}_j^2\left({\tilde{x}}_j(t-{\tau }_j\left(t\right))\right)\right)\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|\left({\gamma }_1{\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\int }_0^{{\gamma }_1}{\tilde{f}}_j^2\left({\tilde{x}}_j(t-s)\right)ds\right)\hfill \end{array}$$

(14)

$$V_{\tilde{Z}}(t,\tilde{Z})=2(A_1{\tilde{x}}_1,A_2{\tilde{x}}_2,\cdots ,A_n{\tilde{x}}_n,{\tilde{y}}_1,{\tilde{y}}_2,\cdots ,{\tilde{y}}_n),V_{\tilde{Z}\tilde{Z}}(t,\tilde{Z})=2E_{2n\times 2n},$$

(15)

where $E_{2n\times 2n}$ is a $2n\times 2n$ identity matrix. From Definition 2, (12), (14) and (15), we obtain

$$\begin{array}{cc}\hfill \mathcal{L}V(t,\tilde{Z})& =\sum _{i=1}^n\sum _{j=1}^n\left|d_{ij}\right|\left({\omega }_j\left(t\right){\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\tilde{f}}_j^2\left({\tilde{x}}_j(t-{\tau }_j\left(t\right))\right)\right)\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|\left({\gamma }_1{\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\int }_0^{{\gamma }_1}{\tilde{f}}_j^2\left({\tilde{x}}_j(t-s)\right)ds\right)\hfill \\ & +2\sum _{i=1}^n{A}_i{\tilde{x}}_i\left(-{\xi }_i\left(A_i{\tilde{x}}_i\right)\left(t\right)+{\tilde{y}}_i\left(t\right)\right)\hfill \\ & +2\sum _{i=1}^n{\tilde{y}}_i(-(a_i-{\xi }_i){\tilde{y}}_i\left(t\right)+\left[(a_i-{\xi }_i){\xi }_i\right]\left(A_i{\tilde{x}}_i\right)\left(t\right)-b_i{\tilde{x}}_i\left(t\right)\hfill \\ & +\sum _{j=1}^n{c}_{ij}{\tilde{f}}_j\left({\tilde{x}}_j\left(t\right)\right)+\sum _{j=1}^n{d}_{ij}{\tilde{f}}_j\left({\tilde{x}}_j(t-{\tau }_j\left(t\right))\right)+\sum _{j=1}^n{e}_{ij}{\int }_{t-{\gamma }_1}^t{\tilde{f}}_j\left({\tilde{x}}_j\left(s\right)\right)ds)\hfill \\ & +\sum _{i=1}^n{\left(\sum _{j=1}^n{h}_{ij}{\tilde{g}}_j\left({\tilde{x}}_j\left(t\right)\right)\right)}^2.\hfill \end{array}$$

(16)

Using inequalities $a^2+b^2\ge 2ab,{\left({\sum }_{i=1}^n{a}_i{b}_i\right)}^2\le {\sum }_{i=1}^n{a}_i^2{\sum }_{i=1}^n{b}_i^2$, (16) and Lemma 1, we obtain

$$\begin{array}{cc}\hfill \mathcal{L}V(t,\tilde{Z})& \le \sum _{i=1}^n\sum _{j=1}^n\left|d_{ij}\right|\left({\omega }_j\left(t\right){\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\tilde{f}}_j^2\left({\tilde{x}}_j(t-{\tau }_j\left(t\right))\right)\right)\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|\left({\gamma }_1{\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)-{\int }_0^{{\gamma }_1}{\tilde{f}}_j^2\left({\tilde{x}}_j(t-s)\right)ds\right)\hfill \\ & +2\sum _{i=1}^n\left(-{\xi }_i{\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+\frac{{\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)}{2}\right)\hfill \\ & +2\sum _{i=1}^n(-(a_i-{\xi }_i){\tilde{y}}_i^2\left(t\right)+\left|(a_i-{\xi }_i){\xi }_i\right|\frac{{\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)}{2}+b_i\frac{{\tilde{y}}_i^2\left(t\right)+{(1-{\widehat{c}}_i)}^{-2}{\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)}{2}\hfill \\ & +\sum _{j=1}^n|c_{ij}|\frac{{\tilde{y}}_i^2\left(t\right)+{\tilde{f}}_j^2\left({\tilde{x}}_j\left(t\right)\right)}{2}+\sum _{j=1}^n\left|d_{ij}\right|\frac{{\tilde{f}}_j\left({\tilde{x}}_j^2(t-{\tau }_j\left(t\right))\right)+{\tilde{y}}_i^2\left(t\right)}{2}\hfill \\ & +\sum _{j=1}^n\left|e_{ij}\right|\frac{{\tilde{y}}_i^2\left(t\right)+{\gamma }_1{\int }_{t-{\gamma }_1}^t{\tilde{f}}^2\left({\tilde{x}}_j\left(s\right)\right)ds}{2})+\sum _{i=1}^n\sum _{k=1}^n\sum _{j=1}^n{h}_{kj}^2{\tilde{l}}_i^2{\tilde{x}}_i^2\left(t\right)\hfill \\ & \le \sum _{i=1}^n[-2{\xi }_i+1+|(a_i-{\xi }_i){\xi }_i|+b_i{(1-{\widehat{c}}_i)}^{-2}+\sum _{j=1}^n\left|d_{ij}\right|{\omega }_j\left(t\right)L_j^2{(1-{\widehat{c}}_i)}^{-2}\hfill \\ & +\sum _{j=1}^n\left|e_{ij}\right|{\gamma }_1{L}_j^2{(1-{\widehat{c}}_i)}^{-2}+\sum _{k=1}^n\sum _{j=1}^n{h}_{kj}^2{\tilde{l}}_i^2{(1-{\widehat{c}}_i)}^{-2}]{\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)\hfill \\ & +\sum _{i=1}^n\left[-2(a_i-{\xi }_i)+1+\left|(a_i-{\xi }_i){\xi }_i\right|+b_i+\sum _{j=1}^n\left(|c_{ij}|+|d_{ij}|+|e_{ij}|\right)\right]{\tilde{y}}_i^2\left(t\right).\hfill \end{array}$$

(17)

In view of (10), (11) and (17), we have $\mathcal{L}V(t,\tilde{Z})<0$, i.e., $\mathcal{L}V(t,\tilde{Z})$ is a negative definite. Evidently, $V(t,\tilde{Z})$ is a positive definite. Now, we show that $V(t,\tilde{Z})$ has an infinitesimal upper bound. By assumption (H${}_3$), we obtain $|{\tilde{f}}_j\left(x\right)|\le N_j$ for all $x\in \mathbb{R},j=1,2,\cdots ,n$. Thus,

$$\begin{array}{cc}\hfill V(t,\tilde{Z})& =\sum _{i=1}^n\left({\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)\right)+\sum _{i=1}^n\sum _{j=1}^n\left|d_{ij}\right|{\int }_{t-{\tau }_j\left(t\right)}^t{\omega }_j\left(s\right){\tilde{f}}_j^2\left({\tilde{x}}_j\left(s\right)\right)ds\hfill \\ & +\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|{\int }_0^{{\gamma }_1}{\int }_{t-s}^t{\tilde{f}}_j^2\left({\tilde{x}}_j\left(\theta \right)\right)d\theta ds\hfill \\ & \le \sum _{i=1}^n\left({\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)\right)+\sum _{i=1}^n\sum _{j=1}^n|d_{ij}{\widehat{\tau }}_j{\widehat{\omega }}_j{N}_j^2+\sum _{i=1}^n\sum _{j=1}^n\left|e_{ij}\right|\frac{{\gamma }_1^2}{2}{N}_j^2.\hfill \end{array}$$

In view of Definition 3, $V(t,X)$ has an infinitesimal upper bound. Furthermore, by (13), we obtain

$$V(t,\tilde{Z})\ge \sum _{i=1}^n\left({\left(A_i{\tilde{x}}_i\right)}^2\left(t\right)+{\tilde{y}}_i^2\left(t\right)\right).$$

Evidently, we have $V(t,\tilde{Z})\to +\infty$ for $\left|\right|\tilde{Z}\left|\right|\to \infty$. Hence, by Definition 4, $V(t,\tilde{Z})$ is an infinite positive definite function for $\tilde{Z}$. In view of Lemma 3, the equilibrium point of system (4) is stochastically globally asymptotically stable, i.e., the equilibrium point of system (3) is also stochastically globally asymptotically stable. □

Remark 4. 

Constructing an appropriate Lyapunov functions is the key to proving Theorem 2. We innovatively construct Lyapunov functions, based on fully considering neutral operators which are different from the corresponding ones of [9,10,11,12,13].

5. Examples

Consider the following example:

$$\begin{array}{cc}\hfill d\left[{\left(A_1{x}_1\right)}^{\prime }\left(t\right)\right]& =-a_1{\left(A_i{x}_1\right)}^{\prime }\left(t\right)dt-b_1{x}_1\left(t\right)dt+\sum _{j=1}^2{c}_{1j}{f}_j\left(x_j\left(t\right)\right)dt+\sum _{j=1}^2{d}_{1j}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)dt\hfill \\ & +\left(\sum _{j=1}^2{e}_{1j}{\int }_{t-{\gamma }_1}^t{f}_j\left(x_j\left(s\right)\right)ds\right)dt+I_{1}dt+\sum _{j=1}^2{h}_{1j}{g}_j\left(x_j\left(t\right)\right)d{B}_1\left(t\right),\hfill \\ \hfill d\left[{\left(A_2{x}_2\right)}^{\prime }\left(t\right)\right]& =-a_2{\left(A_2{x}_2\right)}^{\prime }\left(t\right)dt-b_2{x}_2\left(t\right)dt+\sum _{j=1}^2{c}_{2j}{f}_j\left(x_j\left(t\right)\right)dt+\sum _{j=1}^2{d}_{2j}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)dt\hfill \\ & +\left(\sum _{j=1}^n{e}_{2j}{\int }_{t-{\gamma }_1}^t{f}_j\left(x_j\left(s\right)\right)ds\right)dt+I_{i}dt+\sum _{j=1}^2{h}_{2j}{g}_j\left(x_j\left(t\right)\right)d{B}_2\left(t\right),\hfill \end{array}$$

(18)

where

$$i,j=1,2,c_1\left(t\right)=0.1sin\frac{t}{10},c_2\left(t\right)=0.1cos\frac{t}{10},a_i=1.425,b_1=b_2=0.3,$$

$$c_{ij}=d_{ij}=0.15,e_{ij}=h_{ij}=-0.25,{\tau }_j\left(t\right)=\frac{1}{10}cos10t,f_j\left(u\right)=\frac{0.01{sin}^{2}u}{u^2+1},$$

$$g_j\left(u\right)=\frac{2.5}{2\pi }\times {10}^{-2}sin\frac{u\pi }{2.5},{\gamma }_0={\gamma }_1=0.02,I_1=0.25,I_2\left(t\right)=0.35.$$

Evidently, we obtain

$$l_j=N_j=0.01,{\tilde{l}}_j=0.005,{\widehat{c}}_i=0.1,{\widehat{\omega }}_i=1.$$

Choosing ${\xi }_i=0.745$, we obtain

$$-b_i+\frac{1}{2}\sum _{j=1}^2\left(|c_{ij}|+|d_{ij}|+{\gamma }_1\left|e_{ij}\right|\right)l_j+\frac{1}{2}\sum _{j=1}^2\left(|c_{ji}|+|d_{ji}|+{\gamma }_1\left|e_{ji}\right|\right)l_i\approx -0.2952<0,$$

$$\begin{array}{cc}& -2{\xi }_i+1+|(a_i-{\xi }_i){\xi }_i|+b_i(1-|c_i{\left|\right)}^{-2}+\sum _{j=1}^2\left|d_{ij}\right|{\widehat{\omega }}_j{L}_j^2{(1-{\widehat{c}}_i)}^{-2}\hfill \\ & +\sum _{j=1}^2\left|e_{ij}\right|{\gamma }_1{L}_j^2{(1-{\widehat{c}}_i)}^{-2}+\sum _{k=1}^2\sum _{j=1}^2{h}_{kj}^2{\tilde{l}}_i^2{(1-{\widehat{c}}_i)}^{-2}\approx -0.1032<0\hfill \end{array}$$

and

$$-2(a_i-{\xi }_i)+1+\left|(a_i-{\xi }_i){\xi }_i\right|+b_i+\sum _{j=1}^2\left(|c_{ij}|+|d_{ij}|+|e_{ij}|\right)\approx -0.0175<0.$$

Then, all conditions of Theorems 1 and 2 hold, and system (18) has a unique equilibrium point, which is stochastically globally asymptotically stable. Through simple calculations, system (18) has a unique equilibrium point $(2.5,2.5)$. Give any three sets of initial values:

$$(x_i\left(0\right),x_i^{\prime }\left(0\right))=(3.45,2.91);(2.46,1.25);(1.67,3.53),i=1,2.$$

Figure 1 and Figure 2 show that for system (18) there exists a stochastically globally asymptotically stable equilibrium point $(2.5,2.5)$.

6. Conclusions

In this article, we studied the stochastically globally asymptotic stability for a class of stochastic inertial delay neural networks, and provided sufficient conditions for determining their existence and stability. This has a certain significance for practical applications and theoretical exploration. The research methods of this article are based on homeomorphism mapping, standard stochastic analyzing technology and the Lyapunov functional method. Using the discussion ideas and methods in this article, the stability problems of other types of stochastic inertial neural networks can be further studied: examples include stochastic inertial neural networks with impulses and stochastic inertial neural networks with a Markov switch.