## Abstract

We discuss stability analysis for uncertain stochastic neural networks (SNNs) with time delay in this letter. By constructing a suitable Lyapunov-Krasovskii functional (LKF) and utilizing Wirtinger inequalities for estimating the integral inequalities, the delay-dependent stochastic stability conditions are derived in terms of linear matrix inequalities (LMIs). We discuss the parameter uncertainties in terms of norm-bounded conditions in the given interval with constant delay. The derived conditions ensure that the global, asymptotic stability of the states for the proposed SNNs. We verify the effectiveness and applicability of the proposed criteria with numerical examples.

## 1  Introduction

The significance of neural networks (NNs) cannot be limited to being a class of mathematical models and information processing systems. Their application is far-reaching in many areas, among them automatic control, signal processing, pattern recognition, and quadric recognition (Haykin, 2007). The stability of NNs has been discussed by many researchers (Anbuvithya, Mathiyalagan, Sakthivel, & Prakash, 2016; Cichocki & Unbehauen, 1993; Lakshmanan, Prakash, Rakkiyappan, & Joo, 2020; Liu, Zeng, & Wang, 2017; Liu, Wang, & Liu, 2006; Li, Zheng, & Lin, 2011; Lv et al., 2017; Zhang, Liu, & Zhou, 2012; Wong & Selvi, 1998; Zheng, Zhang, & Wang, 2009). However, in many practical NNs, time delays are unavoidable, and they lead to NN instability, oscillation, and poor performance. Due to this, stability investigation of NNs with time delays has become an important area for research, and many levant reports have been published (Chen & Rong, 2003; Chen & Wu, 2009; Chen, Sun, Liu, & Rees, 2010; Fu & Li, 2011; Lakshmanan et al., 2018; Li, Wang, Yang, Zhang, & Wang, 2008; Li & Chen, 2009; Qiu, Cui, & Wu, 2009; Shao, Huang, & Zhou, 2009; Yu, Zhang, & Quan, 2015; Zhang, Cao, Wu, Chen, & Alsaadi, 2018; Zhang & Quan, 2015). A global exponential stability condition and inequality based on a linear matrix inequality (LMI) that forms an global exponential stability condition for inertial Cohen-Grossberg NNs with time delays is discussed in Yu et al. (2015). Projective synchronization of fractional-order NNs with multiple time delays was studied in Zhang et al. (2018), Zhang and Yu (2016), and Zhang and Quan (2015). Zhang and Quan (2015) sought to obtain sufficient LMI-based conditions for the existence and global exponential stability of inertial bidirectional associative memory NNs with time delays. Therefore, it becomes imperative to include the factor of time delays in the dynamical analysis of NNs.

Stochastic disturbance generally is affected by network models. Thus, when the stability of NNs is analyzed, stochastic disturbance becomes unavoidable. This happens due to the common factor that synaptic transmission is a noisy process, and the neurons' connection weights rely on certain values of resistance and capacitance where there are uncertainties. In this regard, a great deal of work has been conducted on stability analysis for delayed SNNs and robust stability for uncertain stochastic neural networks (SNNs). As a result, scientific results have been published in relation to the stability of NNs with stochastic disturbance (Balasubramaniam & Lakshmanan, 2011; Blythe, Mao, & Liao, 2001; Chen & Wu, 2009; Liao & Mao, 1996; Mao, 1997; Muralisankar, Manivannan, & Balasubramaniam, 2015; Xia, Yu, Li, & Zheng, 2012; Zhao, Gao, & Mou, 2008; Zhu & Cao, 2010a, 2010b, 2014). Stability analysis for NNs by using specific stochastic inputs was discussed in Blythe et al. (2001) and Liao and Mao (1996). For Markovian jump impulsive stochastic Cohen-Grossberg NNs with mixed time delays, Zhu and Cao (2010b) used the Lyapunov-Krasovskii functional (LKF) method for structuring a novel robust exponential stability criterion and known or unknown parameters to be achieved. Zhu and Cao (2014) investigated the stability of stochastic delayed recurrent NNs with the use of an augmented LKF method. This leads to the need for increased attention to the issue of stability investigation for SNNs with time delays.

However, there are also inevitable uncertainties in modeling NNs due to errors in modeling and fluctuating parameters at the time of execution, resulting in instability and poor performance. There have also been many interesting results recently (Chen & Qin, 2010; Deng, Hua, Liu, Peng, & Fei, 2011; Hua, Liu, Deng, & Fei, 2010; Huang & Cao, 2007; Li, Chen, Zhou, & Fang, 2008; Wang, Shu, Fang, & Liu, 2006; Wu, Su, Chu, & Zhou, 2009; Zhang, Shi, & Qiu, 2007; Zhang, Shi, Qiu, & Yang, 2008) on the stability of uncertain SNNs with delay. Chen and Qin (2010), Hua et al. (2010), Huang and Cao (2007), Li et al. (2008), and Zhang et al. (2008) investigated uncertain SNNs with robust stability and time-varying delays in terms of LKF and stochastic analysis approaches. The robust stability in terms of stochastic Hopfield NNs with time delays was examined by using the LKF functional and conducting stochastic analysis by, Wang, Shu, Fang, and Liu (2006) and Zhang et al. (2007). Deng et al. (2011) studied delay-dependent exponential stability of uncertain where SNNs with mixed delays, based on the LKF method. Wu, Su, Chu, and Zhou (2009) discussed some novel delay-dependent conditions, sufficient to ensure the global exponential stability of discrete, recurrent NNs with time-varying delays. Thus, it is evident that many researchers have contributed to the analysis of the stability of time-delayed NNs. A number of methods have been developed to minimize the conservatism of stability criteria: the multiple integral approach (Fang & Park, 2013), model transformation (Kwon & Park, 2004), free-weighting matrix techniques (He, Liu, Rees, & Wu, 2007; Liu, Wu, Martin, & Tang, 2007), park inequality (Park, 1999), the convex combination technique (Park & Ko, 2007), and reciprocally convex optimization (Park, Ko, & Jeong, 2011). Most important, since estimating a lower bound of the quadratic integral term such as $∫t-ϑtxT(s)Dx(s)ds,(D>0)$ is one of the major research topics on time-delay systems, Jensen's inequality has been used widely as a key lemma in obtaining delay-dependent stability criteria. The Wirtinger-based integral inequality, introduced recently in Seuret and Gouaisbaut (2013), also reduced the conservatism of Jensen's inequality, and its advantage was reflected in the comparisons of delay bounds for numerous systems, such as systems with constant, known, and time-varying delay. However, some new LKFs were not considered, and use of the Wirtinger-based integral inequality was concentrated only in Seuret and Gouaisbaut (2013). Therefore, further improvement on the reduction of conservatism in stability analysis for a system with time delays can be achieved, the motivation behind the research we present in this letter.

This letter discusses robust stability analysis for SNNs with time delay. We also consider parameter uncertainties in the system matrices of delayed SNNs. Based on suitable LKF, we derive the delay stability conditions in line with LMIs.

This letter focuses on the following points:

• Parameter uncertainties and stochastic disturbance are taken into account.

• Integral terms are estimated based on Wirtinger's integral inequalities. With appropriate LKF and stochastic stability theory, the delay-dependent stability conditions are attained to ensure the global asymptotic stability of the proposed system. We have employed well-known software to identify the effectiveness of the intended LMIs. Finally, we provide a number of figures to check the effectiveness of our intended method.

We use the following notations:
$Rnn-dimensionalEuclideanspaceRn×nn×nrealmatrices·EuclideannorminRn(Ω,F,P)Completeprobabilityspacewithafiltration{Ft}t≥0ATTransposeofamatrixA*Symmetricblockinasymmetricmatrix$

## 2  Problem Formulation and Preliminaries

We consider the following Hopfield NNs with time delays,
$dηi(t)dt=-di(ηi(t))+∑j=1nbij0σj(ηj(t))+∑j=1ncij1σj(ηj(t-ϑ)+Ji,i=1,2,…,n,$
(2.1)
or, equivalently, the vector form,
$η˙i(t)=-G0η(t)+G1σ(η(t))+G2σ(η(t-ϑ)+J,$
(2.2)
where $η(t)=[η1(t),η2(t),…,ηn(t)]T∈Rn$ is the neuron state vector; $J=[J1,J2,…,Jn]$ denotes the external input; $σ(η)=[σ1(η1(t)),σ2(η2(t)),…,σn(ηn(t))]T$ denotes the neuron activation function; $G0=diag(d1,d2,…,dn)$, $G1=(bij0)n×n$, $G2=(cij1)n×n$ are the connection weight matrix; and $ϑ>0$ denotes the discrete time delay.

We make following assumptions throughout this letter.

Assumption 1.
For any $j=1,2,…,n,σj(·)$ satisfies the following inequality:
$0≤σj(β1)-σj(β2)β1-β2≤pj,∀β1,β2∈R,β1≠β2,$
where $P=diag(p1,p2,…,pn)>0$.
Assuming that $η*=(η1*,η2*,…,ηn*)T$ is an equilibrium point of system 2.2, one can derive from that system $ξ(t)=η(t)-η*$, which transforms system 2.2 as follows:
$ξ˙(t)=-G0ξ(t)+G1f(ξ(t))+G2f(ξ(t-ϑ)),$
(2.3)
where $ξ(t)$ is the state vector of the transformed system, $fj(ξj(t))=σj(ξj(t)+ηj*)-σj(ηj*)$. Consider that the function $fj(·)$, $j=1,2,…,n$, satisfies the following condition:
$0≤fj(ξj)ξj≤pj,fj(0)=0,∀ξj≠0,j=1,2,…,n.$
(2.4)
We consider parameter uncertainties and stochastic perturbations as follows:
$dξ(t)=-G0(t)ξ(t)+G1(t)f(ξ(t))+G2(t)f(ξ(t-ϑ))dt+G3(t)ξ(t)+G4(t)ξ(t-ϑ)dw(t),ξ(t)=Ψ(t),∀t∈[-ϑ,0],$
(2.5)
where $w(t)$ indicates a one-dimensional Brownian motion satisfying $E{dw(t)}=0$ and $E{dw(t)2}=dt$. $G0(t)=G0+ΔG0(t)$, $G1(t)=G1+ΔG1(t)$, $G2(t)=G2+ΔG2(t)$, $G3(t)=G3+ΔG3(t)$, and $G4(t)=G4+ΔG4(t)$, where $G3$ and $G4$ are connection weight matrices with appropriate dimensions. In equation 2.5, the parametric uncertainties are assumed to have the form
$[ΔG0(t)ΔG1(t)ΔG2(t)ΔG3(t)ΔG4(t)]=EF(t)[H1H2H3H4H5],$
(2.6)
where $E$ and $Hi(i=1,…,5)$ are known, real, constant matrices:
$FT(t)F(t)≤I.$
(2.7)
It is assumed that all elements of $F(t)$ are Lebesque measurable. The matrices $ΔG0(t)$, $ΔG1(t)$, $ΔG2(t)$, $ΔG3(t)$, and $ΔG4(t)$ are said to be admissible if equations 2.5 to 2.7 hold. The initial condition of equation 2.5 is given as $ξ(t)=Ψ(t),t∈[-ϑ,0]$.
Remark 1.

The structure of the parameter uncertainty as in equations 2.6 and 2.7 was extensively exploited in the analysis of robust control and filtering of uncertain systems (Wang, Xie, & De Souza, 1992; Wang & Qiao, 2002). Many practical systems have unknown parameters that can either be modeled exactly or overbound by equation 2.7.

The following lemmas are useful in deriving the stability results for SNNs, equation 2.5:

Lemma 1
(Boyd, El Ghaoui, Feron, & Balakrishnan, 1994). Given constant matrices $μ2,μ3$, and $μ4$ with appropriate dimensions, where $μ2T=μ2$ and $μ3T=μ3$, $μ2+μ4Tμ3-1μ4<0$, if and only if
$μ2μ4Tμ4-μ3<0.$
Lemma 2

(Yue, Tian, Zhang, & Peng, 2009). Let $B,F,N0,N1$ and $M$ be real matrices of appropriate dimensions with $M>0$, $FT(t)F(t)≤I$. Then for any scalar $ε>0$ satisfying $M-1-ε-1N1N1T>0$, we have

1. $N1F(t)N0+N0TFT(t)N1T≤ε-1N1N1T+εN0TN0$

2. $(B+N1F(t)N0)TP(B+N1F(t)N0)≤BT(M-1-ε-1N1N1T)-1B+εN0TN0$.

Lemma 3
(Seuret & Gouaisbaut, 2013). For any constant matrix $M1>0$, the following inequality holds for all continuously differentiable function $φ$ in $[b,c]→Rn$:
$(c-b)∫bcφT(s)M1φ(s)ds≥∫bcφ(s)dsTM1∫bcφ(s)ds+3ΘTM1Θ,whereΘ=∫bcφ(s)ds-2c-b∫bc∫bsφ(u)duds.$

## 3  Main Results

In this section, we derive a delay-dependent stochastic stability condition based on suitable LKF and LMI approaches.

We introduce two new state variables for the SNNs, equation 2.5,
$γ(t)=-G0(t)ξ(t)+G1(t)f(ξ(t))+G2(t)f(ξ(t-ϑ))$
(3.1)
and
$ζ(t)=G3(t)ξ(t)+G4(t)ξ(t-ϑ),$
(3.2)
and have
$dξ(t)=γ(t)dt+ζ(t)dw(t).$
(3.3)
Moreover, the following equality holds …
$ξ(t)-ξ(t-ϑ)=∫t-ϑtdξ(s)=∫t-ϑtγ(s)ds+∫t-ϑtζ(s)dw(s).$
(3.4)
The following theorem provides the mean-square asymptotic stability results for SNNs, equation 2.5.
Theorem 1.
SNNs, equation 2.5, are globally asymptotically stable in the mean square if there exist positive-definite matrices $Q=QT>0$, $Z1=Z1T>0$, $Z2=Z2T>0$, and $Rl=RlT>0$, $l=1,2$, and diagonal matrices $U0>0$ and $U1>0$, such that the following LMIs hold:
$Π=Π1,10Π1,3Π1,40000Π1,9Π1,10Π1,11*-R10Π2,40000Π2,90Π2,11**Π3,3000000Π3,100***Π4,400000Π4,100****Π5,5Π5,600000*****Π6,600000******Π7,7Π7,8000*******Π8,8000********-Q00*********-ϑZ10**********-ϑZ2<0,$
(3.5)
where
$Π1,1=-2QG0+R1,Π1,3=QG1+U0P,Π1,4=QG2,Π1,9=G3TQ,Π1,10=-G0TϑZ1,Π1,11=G3TϑZ2,Π2,4=U1P,Π2,9=G4TQ,Π2,11=G4TϑZ2,Π3,3=R2-2U0,Π3,10=G1TϑZ1,Π4,4=-R2-2U1,Π4,10=G2TϑZ1,Π5,5=-4ϑZ1,Π5,6=6Z1ϑ2,Π6,6=-12Z1ϑ3,Π7,7=-4ϑZ2,Π7,8=6Z2ϑ2,Π8,8=-12Z2ϑ3.$
Proof.
In order to prove the asymptotically stable criteria, we consider the following LKF,
$V(t)=∑i=13Vi(t),$
(3.6)
where
$V1(t)=ξT(t)Qξ(t),V2(t)=∫t-ϑtξT(s)R1ξ(s)ds+∫t-ϑtfT(ξ(s))R2f(ξ(s))ds,V3(t)=∫-ϑ0∫t+θtγT(s)Z1γ(s)dsdθ+∫-ϑ0∫t+θtζT(s)Z2ζ(s)dsdθ.$
Then it can be obtained by Ito's differential formula (Mao, 1997) that
$dV(t)=LV(t)dt+2ξT(t)Qζ(t)dw(t),$
(3.7)
where
$LV1(t)=2ξT(t)Qγ(t)+ζT(t)Qζ(t),$
(3.8)
$LV2(t)=ξT(t)R1ξ(t)-ξT(t-ϑ)R1ξ(t-ϑ)+fT(ξ(t))R2f(ξ(t))-fT(ξ(t-ϑ))R2f(ξ(t-ϑ)),$
(3.9)
$LV3(t)≤ϑγT(t)Z1γ(t)-∫t-ϑtγT(s)Z1γ(s)ds+ϑζT(t)Z2ζ(t)-∫t-ϑtζT(s)Z2ζ(s)ds.$
(3.10)
By lemma 5,
$-∫t-ϑtγT(s)Z1γ(s)ds≤-1ϑ∫t-ϑtγ(s)dsTZ1∫t-ϑtγ(s)ds-3ϑ∫t-ϑtγ(s)ds-2ϑ∫t-ϑt∫stγ(u)dudsTZ1×∫t-ϑtγ(s)ds-2ϑ∫t-ϑt∫stγ(u)duds$
(3.11)
$-∫t-ϑtζT(s)Z2ζ(s)ds≤-1ϑ∫t-ϑtζ(s)dsTZ2∫t-ϑtζ(s)ds-3ϑ∫t-ϑtζ(s)ds-2ϑ∫t-ϑt∫stζ(u)dudsTZ2×∫t-ϑtζ(s)ds-2ϑ∫t-ϑt∫stζ(u)duds.$
(3.12)
From condition 2.4, for any
$U0=diag{e11,e21,…,en1}>0andU1=diag{e12,e22,…,en2}>0,$
it may be noted that
$0≤-2∑j=1nej1fjξj(t)fj(ξj(t))-pjξj(t)-2∑j=1nej2fjξj(t-ϑ)×fj(ξj(t-ϑ))-pjξj(t-ϑ)=2ξT(t)U0Pf(ξ(t))-2fT(ξ(t))U0f(ξ(t))+2ξT(t-ϑ)U1Pf(ξ(t-ϑ))-2fT(ξ(t-ϑ))U1f(ξ(t-ϑ)).$
(3.13)
Substituting equations 3.8 to 3.13 into 3.7, we have
$dV(t)≤χT(t)Πχ(t)dt+2ξT(t)Qζ(t)dw(t).$
(3.14)
Taking the mathematical expectation of both sides of equation 3.14, there exists a positive scalar $α1>0$ satisfying
$EdV(t)≤EχT(t)Πχ(t)≤-α1Eξ(t)2.$
(3.15)
$Π$ is defined in theorem 6 with
$χT(t)=[ξT(t),ξT(t-ϑ),fT(ξ(t)),fT(ξ(t-ϑ)),∫t-ϑtγ(s)dsT,∫t-ϑt∫stγ(u)dudsT,∫t-ϑtζ(s)dsT,∫t-ϑt∫stζ(u)dudsT].$
Thus, if $Π<0$, the SNNs, equation 2.5, are globally asymptotically stable in the mean square.

Now we can study the robust stability analysis for SNNs, equation 2.5, with parameter uncertainties. Based on theorem 6, we provide a delay-dependent criterion:

Theorem 2.
SNNs, equation 2.5, are globally robustly asymptotically stable in the mean square if there exist positive-definite matrices $Q=QT>0,Z1=Z1T>0,Z2=Z2T>0,Rl=RlT>0,l=1,2$; diagonal matrices $U0>0$ and $U1>0$; and scalars $εi>0,(i=1,2,3)$ such that the following LMIs hold:
$ΞQ^EΓ1ϑZ10ε1Γ2Γ3Q0ε2Γ4Γ5ϑZ20ε3Γ6*-ε1I000000000**-ϑZ1ϑZ1E0000000***-ε1I0000000****-ε1I000000*****-QQE0000******-ε2I0000*******-ε2I000********-ϑZ2ϑZ2E0*********-ε3I0**********-ε3I<0,$
(3.16)
where
$Ξ=Π^1,10Π1,3Π1,40000*-R10Π2,40000**Π^3,300000***Π^4,40000****Π5,5Π5,600*****Π6,600******Π7,7Π7,8*******Π8,8,Q^=[Q0000000]T,Γ1=[-G0T0G1TG2T0000]T,Γ2=[-H10H2H30000]T,Γ3=[G3TG4T000000]T,Γ4=[H4H5000000]T,Γ5=[G3TG4T000000]T,Γ6=[H4H5000000]T,Π^1,1=Π1,1+ε1H1TH1,Π^3,3=Π3,3+ε1H2TH2,Π^4,4=Π4,4+ε1H3TH3.$
Proof.

Replacing $G0,G1,G2,G3,G4$ in LMI, equation 3.5, with $G0+EF(t)H1,G1+EF(t)H2,G2+EF(t)H3,G3+EF(t)H4,G4+EF(t)H5$ and using lemmas 3 and 4, we obtain the LMI, equation 3.16.

Remark 2.

Theorem 7 presents a sufficient condition to test the global robust stability for uncertain SNNs with time delay. Therefore, it is straightforward to test the feasibility of equation 3.16 without tuning any parameters using the Matlab LMI toolbox.

To show that our major results are sufficiently general to cover certain cases that have been discussed in the literature, we give a few corollaries.

Case 1. In the case that there are no stochastic disturbances in (2.5), we can get the following deterministic system,
$ξ˙(t)=-(G0+ΔG0(t))ξ(t)+(G1+ΔG1(t))f(ξ(t))+(G2+ΔG2(t))f(ξ(t-ϑ)),$
(3.17)
then we have the given corollary.
Corollary 1.
If there exist positive-definite matrices $Q=QT>0,Z1=Z1T>0,Rl=RlT>0,l=1,2$, diagonal matrices $U0>0$ and $U1>0$, scalar $ε1>0,$ such that the below LMIs
$ΞQ^EΓ1ϑZ10ε1Γ2*-ε1I000**-ϑZ1ϑZ1E0***-ε1I0****-ε1I<0$
(3.18)
where
$Ξ=Π^1,10QG1+U0PQG200*-R10U1P00**R2-2U0+ε1H2TH2000***-R2-2U1+ε1H3TH300****-4ϑZ16Z1ϑ2*****-12Z1ϑ3Π^1,1=Π1,1+ε1H1TH1,Q^=[Q00000]T,Γ1=[-G0T0G1TG2T00]T,Γ2=[-H10H2H300]T$
hold, then the system (3.17) is globally robustly asymptotically stable. Corollary 9 provide the stability condition of delayed NNs without stochastic disturbance in terms of LMI.
Case 2. In the absence of uncertainties in equation 3.10, we can get the following systems:
$ξ˙(t)=-G0ξ(t)+G1f(ξ(t))+G2f(ξ(t-ϑ)).$
(3.19)
The corresponding stability condition is derived in the following corollary:
Corollary 2.
If there exist positive-definite matrices $Q=QT>0,Z1=Z1T>0,Rl=RlT>0,l=1,2$, and diagonal matrices $U0>0$ and $U1>0$, such that the following LMIs hold,
$-2QG0+R10QG1+U0PQG200-G0TϑZ1*-R10U1P000**R2-2U0000G1TϑZ1***-R2-2U100G2TϑZ1****-4ϑZ16Z1ϑ20*****-12Z1ϑ30******-ϑZ1<0.$
(3.20)
hold, then the system, equation 3.19, is globally asymptotically stable.

## 4  Numerical Examples

### 4.1  Example 1

System 2.5 without uncertainties, may be considered with the given matrices:
$G0=4.50005.20003.6,G1=-10.4-0.50-0.70.70.20.60.8,G2=0.50.71.1-0.10.400-0.2-0.8,G3=1.20.4-0.8-1.5-1.80.90.51.12.1,G4=0.20.1-0.400.20.50.600,P=0.4I,f(ξ(t))=0.4tanh(ξ(t)).$
By using the Matlab LMI toolbox, setting $ϑ=1.07$, and solving the LMI condition in theorem 6, the following feasible solutions may be obtained:
$Q=63.516928.511012.257428.511032.000810.111312.257410.111341.8480,R1=247.393539.144552.776139.144542.98614.615452.77614.615430.6158,R2=67.84059.3455-6.31869.345552.0187-16.1782-6.3186-16.178212.7751,Z1=3.44871.65770.76201.65771.64850.05790.76200.05790.4856,Z2=9.40275.50211.94705.50215.58670.44251.94700.44250.9293,U0=90.165700090.165700090.1657,U1=70.239700070.239700070.2397.$
Therefore, it follows from theorem 6 that the delayed stochastic neural network, equation 2.5, is globally asymptotically stable in the mean square.

### 4.2  Example 2

Consider the following uncertain stochastic NNs,
$dξ(t)=[-(G0+ΔG0(t))ξ(t)+(G1+ΔG1(t))f(ξ(t))+(G2+ΔG2(t))f(ξ(t-ϑ))]dt+(G3+ΔG3(t))ξ(t)+(G4+ΔG4(t))ξ(t-ϑ)dw(t),$
(4.1)
where
$G0=211.23,G1=-1.50.60.6-1.5,G2=0.511.20.6,G3=0.2000.2,G4=0.2000.2,E=0.5000.5,P=0.6000.2,Hi=0.4000.4,i=1,2,…,5.$
By using the Matlab LMI toolbox, setting $ϑ=1.07$ and solving the LMI condition in theorem 7, the following feasible solutions may be obtained:
$Q=0.8138-0.1295-0.12950.7336,R1=0.83190.32150.32151.0335,R2=1.15060.17630.17630.8954,Z1=0.1333-0.0283-0.02830.1264,Z2=0.21240.00770.00770.2186,U0=2.0680002.0680,U1=0.7096000.7096,ε1=0.8241,ε2=0.6937,ε3=0.5805.$
Feng, Zhang, and Wu (2008) showed that the uncertain SNNs are globally, robustly, and asymptotically stable in mean square for the maximum time delay allowed, 0.6. However, using theorem 7, the maximum allowable bound can be obtained as $ϑ=1.07$. Hence, the results provided in this example are less conservative compared to those of Feng et al. (2008), and it follows from theorem 7 that the delayed SNNs, equation 4.1, are globally, robustly, and asymptotically stable in the mean square.

## 5  Conclusion

This letter has discussed robust, asymptotic stability analysis for uncertain, stochastic-delayed NNs. In theorem 6, by constructing a suitable LKF and utilizing Wirtinger-based inequality, we derived the sufficient condition for asymptotic stability of the system with time delay. An LMI approach has been proposed to check the mean square stability of stochastic uncertain neural networks, which can be tested easily using Matlab's LMI toolbox. We provided examples to illustrate the effectiveness of our main results.

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