# Global Stability Analysis of Neural Networks with Constant Time Delay via Frobenius Norm

Authors: N. Mohamed Thoiyab; P. Muruganantham; Grienggrai Rajchakit; Nallappan Gunasekaran; Bundit Unyong; Usa Humphries; Pramet Kaewmesri; Chee Peng Lim
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper deals with the global asymptotic robust stability (GARS) of neural networks (NNs) with constant time delay via Frobenius norm. The Frobenius norm result has been utilized to find a new sufficient condition for the existence, uniqueness, and GARS of equilibrium point of the NNs. Some suitable Lyapunov functional and the slope bounded functions have been employed to find the new sufficient condition for GARS of NNs. Finally, we give some comparative study of numerical examples for explaining the advantageous of the proposed result along with the existing GARS results in terms of network parameters.


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## Body

## 1. Introduction

Neural networks (NNs) operate on principles similar to the human nervous system. It has a huge number of processors. These types of processors operate in parallel and are organized in layers. The initiating layer takes raw input, similar to the raw information received by humans. All subsequent layers receive input from the layer before it. Again, pass the output to the next layer. Finally, the end layer sends the final output. Most nodes are interconnected in layers.NNs have been studied by many of the researchers because of their applications in different fields. The modern technology is mostly based on computational models which are known as artificial neural networks (ANNs). Nowadays artificial intelligence plays the most important role in electrical and electronics world. ANNs are the backbone of this artificial intelligence. In recent years, the role of ANNs has been developed due to their applications in various disciplines. The machine learning uses the distinct variety of NNs such as Feedforward NNs—artificial neuron, radial basis function NNs, multilayer perceptron, convolutional NNs, recurrent neural networks (RNNs), modular NNs, and sequence-to-sequence models. Moreover, NNs have wide applications in engineering areas [1–6] such as radar systems, signal classification, 3D reconstruction, face identification, object recognition, medical diagnosis, visualization, machine translation, combinatorial optimization, and signal processing. Also they may have great applications in nonengineering areas such as sales forecasting, risk management, and target marketing.In NNs, time delay plays an important role in different areas such as video lip reading and speech recognition. Due to this delay parameter, the convergence of solution of the given system can be affected. The convergence of solution of neural system is to make the NNs to be stable. So, the concept of global stability analysis plays an important role in the convergence of solution of NNs. Also that the different kinds of stability analysis such as global asymptotic robust stability (GARS), exponential stability, and complete stability of NNs, have been studied by many researchers in [7–14]. These different types of stability results of the delayed NNs have been discussed based on the methods of Lyapunov stability theory, linear matrix inequalities, nonsmooth analysis, and M-matrix theory in the previous literature. Therefore, the GARS analysis of NNs under parameter uncertainties is the most important problem. Recently, it has been exclusively studied by many authors in [15–34].From the motivation of the above concepts, GARS of NNs has been investigated in this paper. The objective of this paper is to obtain a new sufficient condition for the GARS of the equilibrium point of the delayed neural system using the Lyapunov stability theory and Frobenius norm under parameter uncertainties. The Frobenius norm is always an upper bound for the spectral norm (⋅2). Moreover, the Frobenius norm is much easier to compute than the spectral norm. For calculating the Frobenius norm, it is not necessary to find the eigenvalues rather by using the traces (sum of the diagonals of a matrix). So, the calculation of finding eigenvalues is pretty difficult for higher dimensional matrices. Moreover, some matrices having real entries may give complex eigenvalues. By utilizing the Frobenius norm, we avoid such situations. Therefore, the Frobenius norm is important for calculating the upper bound of connection weight matrices. By utilizing the concept of homeomorphism, we find a new sufficient condition for the existence and uniqueness of the equilibrium point of NNs. Finally, we will give some comparative studies of numerical example to illustrate the effectiveness of our results for the NNs.We apply the following notations for the norm of vectors and matrices: letw=w1,w2,…,wnT∈ℝn. The most common vector norms w1,w2,andw∞ which are defined as follows: w1=∑i=1nwi, w2=∑i=1nwi21/2, and w∞=max1≤i≤nwi. Let S=sijn×n. The following are the definitions for S1,S2, S∞, and SF. S1=max1≤j≤n∑i=1nsij, S2=λmaxSTS1/2, S∞=max1≤i≤n∑j=1nsij, and SF=trSTS=trSST. For any vector w=w1,w2,…,wnT, w is defined as w=w1,w2,…,wnT. For any matrix S=sijn×nwith real entries S will be defined as S=sijn×n. The minimum and maximum eigenvalues of S are denoted by λminS and λmaxS, respectively. trS be the trace of the matrix S. That is, the sum of the diagonal values of the matrix S. If S=sijn×n is a symmetric matrix and wTSw>0≥0, for any real vector w=w1,w2,…,wnT, then S is said to be positive definite matrix (positive semidefinite matrix). Consider the two positive definite matrices H=hijn×n and S=sijn×n. Then, H<S implies that wTHw<wTSw for any real vector w=w1,w2,…,wnT.


## 2. Problem Statement and Fundamentals

In this paper, we consider the following delayed neural networks:(1)dwitdt=−ciwit+∑j=indijfjwjt+∑j=inrijfjwjt−τ+Ji,where i=1,2,…,n and n denotes the total neurons. wit denotes the ith neuron state of the vector at time t. ci represents the rate of charge for the ith neuron. rij and dij are the connection weight matrices with and without time delay, respectively. fj⋅ denotes the activation functions at time t and t−τ. Here τ denotes the constant time delay. Ji represents the vector with constant input between the neurons.The matrix vector form of equation (1) is as follows:(2)w.t=−Cwt+Dfwt+Rfwt−τ+J,where C=diagci>0, wt=w1t,w2t,…,wntT∈ℝn, R=rij∈ℝn×n, D=dij∈ℝn×n, fwt=f1w1t,f2w2t,…,fnwntT∈ℝn, and fwt−τ=f1w1t−τ,f2w2t−τ,…,fnwnt−τ​T∈ℝn, J=J1,J2,…,JnT∈ℝn. The initial condition is wt=ϕt∈C−τ,0,ℝn, and C−τ,0,ℝn denotes all continuous functions from −τ,0 to ℝn.The most common approach for handling the delayed neural system is to make the connection weight matricesD=dijn×n and R=rijn×n, and the matrix C=diagci>0 in an interval as follows:(3)CI=C=diagci:0<C≤C≤C¯,ie.,0<c¯i≤ci≤c¯i,i=1,2,…,n,DI=D=dij:D¯≤D≤D¯,ie.,d¯ij≤dij≤d¯ij,i,j=1,2,…,n,RI=R=rij:R¯≤R≤R¯,ie.,r¯ij≤rij≤r¯ij,i,j=1,2,…,n.Assumption 1.
(see [24]). The activation functions fi are assumed to be slope bounded; that is, there exist some positive constants ki such that the following conditions hold:(4)0≤fiw−fivw−v≤ki,∀w,v∈ℝ,w≠v,i=1,2,…,n.
This class of functions will be denoted byf∈k. The functions of this class do not require to be bounded, differentiable, and monotonically increasing.Lemma 1.
(see [29]). Let wt=w1t,w2t,…,wntT∈ℝn. If D∈DI be the matrix defined as in equation (3), then for any positive diagonal matrix and a nonnegative diagonal matrix , the following inequality holds:(5)wTℳD+DTℳw≤wTℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2Iw,where D∗=1/2D¯+D¯ and D∗=1/2D¯−D¯.Lemma 2 (see [24]).
IfGx∈C0 (C0 means that the set of continuous functions on ℝn) satisfies the following conditions, then Gx is a homeomorphism on ℝn:(i)
Gx is an injective mapping on ℝn(ii)
Gx⟶∞ as x⟶∞


## 3. Existence and Uniqueness of Equilibrium Point

This section will focus on the new sufficient condition for the existence of equilibrium point of our model (2) which is unique. By using the Frobenius norm, we prove a new sufficient condition for existence of equilibrium point of our NNs model (2) which is unique.Theorem 1.
Suppose thatf∈k and there exist matrices K=diagki>0,ℳ=diagmi>0, and ϒ=diagγi≥0 such that(6)Ω1=2C¯ℳK−1−ℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2I−2ℳmaxR^FI>0,where ℳmax=maximummi,D∗=1/2D¯+D¯,D∗=1/2D¯−D¯, and R^=r^ij with r^ij=maxr¯ij,r¯ij. Then, for each constant vector J, the neural network model (2) satisfying (3) has a unique equilibrium point.Proof.
Define the following map:(7)Gw=−Cw+Dfw+Rfw+J.
Gw=0 is equivalent to w˙=0. Here every solution of Gw=0 is an equilibrium point of system (2). For proving this theorem, it is enough to prove that Gw is a homeomorphism on ℝn.
Letw,v∈ℝn be the two vectors such that w≠v.
Then,(8)Gw−Gv=−Cw−v+Dfw−fv+Rfw−fv.
Iff∈k, w≠v then, fw−fv=0 or fw−fv≠0. First, let us consider w≠v and fw−fv=0. From equation (8), we have(9)Gw−Gv=−Cw−v.
Sincew≠v and C=diagci>0. Therefore, from equation (9), Gw≠Gv. Next, let us consider w≠v and fw−fv≠0 and multiply (9) by 2fw−fvTℳ; we get(10)2fw−fvTℳGw−Gv=−2fw−fvTℳCw−v+2fw−fvTℳDfw−fv+2fw−fvTℳℛfw−fv=−2fw−fvTℳCw−v+fw−fvTℳD+DTℳfw−fv+2fw−fvTℳℛfw−fv.
Then, we get(11)2fw−fvTℳCw−v=2∑i=1nmicifiwi−fiviwi−vi≥2∑i=1nmic¯ikifiwi−fivi=2fw−fvTC¯ℳK−1fw−fv,−2fw−fvTℳCw−v≤−2fw−fvTC¯ℳK−1fw−fv.
From Lemma1, we get(12)fw−fvTℳD+DTℳfw−fv≤fw−fvTℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2Ifw−fv.(13)2fw−fvTℳℛfw−fv=∑i=1n∑j=1n2mirijfiwi−fivifjwj−fjvj≤2ℳmax∑i=1n∑j=1nr^ijfiwi−fivifjwj−fjvj≤2ℳmaxR^Ffw−fvTfw−fv.
Applying the results (11)–(13) in (10), we get(14)2fw−fvTℳGw−Gv≤−2fw−fvTC¯ℳK−1fw−fv+fw−fvTℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2Ifw−fv+2ℳmax‖R^‖Ffw−fvTfw−fv=−fw−fvTΩ1fw−fv.
Given thatΩ1>0, we have(15)2fw−fvTℳGw−Gv≤−λminΩ1fw−fv22,where λminΩ1 is the smallest eigen value of the positive definite matrix Ω1. If fw−fv≠0 and λminΩ1>0, then it follows from (14) that 2fw−fvTℳGw−Gv<0.
Therefore,Gw≠Gv for all w≠v. Hence G is injective on ℝn. Now, we prove that Gw⟶∞ as w⟶∞. For this, let v=0 in (15) which implies that(16)2fw−f0TℳGw−G0≤−λminΩ1fw−f022,2fw−f0TℳGw−G0≥λminΩ1fw−f022.
From the above inequality, we write(17)2ℳ∞fw−f0∞Gw−G01>λminΩ1fw−f022.
By applying the properties of norm for the above inequalities, we have(18)fw−f0∞≤fw−f02,Gw−G01≤Gw1+G01,fw−f02≥fw2−f02.
Using the above inequalities in (20), we have the following inequality:(19)Gw1>λminΩ1fw2−λminΩ1f02−2ℳ∞G012ℳ∞,where f02, G01, and ℳ∞ are finite. Moreover, Gw⟶∞ as fw⟶∞ or equivalently Gw⟶∞asw⟶∞. Therefore, from the result of Lemma 2, for each constant vector J, the neural networks model (2) has a unique equilibrium point. Hence, the proof.


## 4. Global Stability Analysis

In this section, we prove that the obtained sufficient conditions for the existence and uniqueness of the equilibrium point in the previous Theorem1 will also give the sufficient conditions for the GARS of the neural system (2). Furthermore, we denote the equilibrium point of 1 by w∗ and use some proper transformation say yi⋅=wi⋅−wi∗,i=1,2,…,n. After giving such transformation, the network model (1) can be put in the following form:(20)y˙it=−ciyit+∑j=1ndijgjyjt+∑j=1nrijgjyjt−τ,where giyi⋅=fiyi⋅+wi∗−fiwi∗,i=1,2,…,n Moreover, Assumption 1 holds for the function g, i.e., f∈k gives that g∈k with gi0=0,i=1,2,…,n. By using this transformation, the equilibrium point w∗ of 2 is shifted to the origin of (20).Now, our focus is to show that GARS for the origin of the transformed model (20) instead of focusing the GARS for w∗.The matrix form of (20) is as follows:(21)y˙t=−Cyt+Dgyt+Rgyt−τ,where yt=y1t,y2t,…,yntT∈ℝn is the new state vector, gyt=g1y1t,g2y2t,…,gnyntT∈ℝn, and gyt−τ=g1y1t−τ,g2y2t−τ,…,gnynt−τT∈ℝn.Theorem 2.
Suppose thatg∈k and there exist matrices K=diagki>0,ℳ=diagmi>0, and ϒ=diagγi≥0, such that(22)Ω1=2C¯ℳK−1−ℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2I−2ℳmaxR^FI>0,where ℳmax=maximummi,D∗=1/2D¯+D¯,D∗=1/2D¯−D¯,andR^=r^ij with r^ij=maxr¯ij,r¯ij. Then, origin of the neural network model (21) satisfying the network parameters 3 is globally asymptotically robust stable.Proof.
Consider the following positive definite Lyapunov functional:(23)Vyt=yTtyt+2δ∑i=1n∫0yitmigiξdξ+δμ+η∑i=1n∫t−τitgi2yiζdζ,where mi, δ, η, and μ are the positive constants which will be determined later. The following equation is the time derivative of equation (23) along the trajectories of the model (21):(24)V˙yt=−2δgTytℳCyt+2δgTytℳDgyt+2δgTytℳℛgyt−τ+2yTtRgyt−τ−2yTtCyt+2yTtDgyt+δμgyt22−δμgyt−τ22+ηgyt22−ηgyt−τ22.
Also,(25)−yTtCyt+2yTtDgyt≤gTytDTC−1Dgyt≤D22C−12gyt22,(26)−yTtCyt+2yTtRgyt−τ≤gTyt−τRTC−1Rgyt−τ≤R22C−12gyt−τ22,(27)2δgTytℳℛgyt−τ≤2δℳmaxR2gyt2gyt−τ2≤δℳmaxR2gyt22+δℳmaxR2gyt−τ22≤δℳmaxR^Fgyt22+δℳmaxR^Fgyt−τ22,(28)−2δgTytℳCyt≤−2δgTytℳC¯K−1gyt.
By applying equations (25)–(28) in (24), we have(29)V˙yt≤D22C−12gyt22+R22C−12gyt−τ22−2δgTytℳC¯K−1gyt+δgTytℳD+DTℳgyt+δℳmaxR^Fgyt22+δℳmaxR^Fgyt−τ22+δμgyt22−δμgyt−τ22+ηgyt22−ηgyt−τ22.
SinceD2≤D^F,R2≤R^F,andC−12≤C¯−12.
Therefore,(30)V˙yt≤D^F2C¯−12gyt22+R^F2C¯−12gyt−τ22−2δgTytℳC¯K−1gyt+δgTytℳD+DTℳgyt+δℳmaxR^Fgyt22+δℳmaxR^Fgyt−τ22+δμgyt22−δμgyt−τ22+ηgyt22−ηgyt−τ22.
By takingη=R^F2C¯−12 and μ=ℳmaxR^F, V˙yt can be written in the following form:(31)V˙yt≤D^F2+R^F2C¯−12gyt22−2δgTytℳC¯K−1gyt+δgTytℳD+DTℳgyt+2δℳmaxR^Fgyt22.
Using the result of Lemma2, we write(32)gTytℳD+DTℳgyt≤gTytℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2Igyt.
Then,V˙yt becomes(33)V˙yt≤D^F2+R^F2C¯−12gyt22−2δgTytℳC¯K−1gyt+δgTytℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2Igyt+2δℳmaxR^Fgyt22=D^F2+R^F2C¯−12gyt22−δgTytΩ1gyt.
SinceΩ1>0, it follows from (33) that(34)V˙yt≤D^F2+R^F2C¯−12gyt22−δλminΩ1gyt22.
If we takeδ>D^F2+R^F2C¯−12/λminΩ1, then it follows that V˙yt<0, ∀gyt≠0 since gyt≠0 gives that yt≠0. Also if gyt=0 and yt≠0, then V˙yt can be written in the following form:(35)V˙yt=−ηgTyt−τgyt−τ−δμgTyt−τgyt−τ−2yTtCyt+2yTtRgt−τ≤−2yTtCyt+2yTtRgt−τ−ηgTyt−τgyt−τ.
Since−yTtCyt+2yTtRgt−τ−ηgTyt−τgyt−τ≤0, then we have V˙yt=−yTtCyt. Therefore V˙yt<0, ∀yt≠0. Finally, let us assume that gyt=0 and yt=0. Then, V˙yt=−ηgTyt−τgyt−τ−δμgTyt−τgyt−τ.
It is obvious thatV˙yt<0, ∀gyt−τ≠0. Hence, V˙yt=0 if and only if yt=gyt=gyt−τ=0; otherwise, V˙yt<0. Moreover, Vyt is radially unbounded provided Vyt⟶∞ as y⟶∞. Hence, origin of system (21), or the equilibrium point of the neural system (2), is GARS.


## 5. Comparisons with Numerical Examples

In this section, we will compare our sufficient conditions of GARS with the previous existing results of GARS. For the comparison, the previous results of GARS are restated as follows.Theorem 3.
(see [24]). Suppose that g∈k and there exist matrices K=diagki>0 and ℳ=diagmi>0 such that(36)Ω2=2C¯ℳK−1−ℳD∗+D∗Tℳ+ℳD∗+D∗Tℳ2I−2ℳ2R^FI>0.Then, origin of system (21) satisfying the network parameters 3 is globally asymptotically robust stable.Theorem 4 (see [15]).
Suppose thatg∈k and there exist matrices K=diagki>0 and ℳ=diagmi>0 such that(37)Ω3=2C¯ℳK−1+Z−2ℳ2R^FI>0.Then origin of system (21) satisfying the network parameters 3 is globally asymptotically robust stable., where Z=zijn×n with zii=−2mid¯ii and zij=−maximummid¯ij+mjd¯ji,mid¯ij+mjd¯ji, for i≠j.Now we demonstrate the advantages of our result with some examples as follows.Example 1.
Consider the following network parameters of the neural network model 2:(38)D¯=1−1−1−1010−1−2−11−1−2−201,D¯=1111212101110021,R¯=−10−100−1−11−10−111−11−1,R¯=110101−111011111−1.
Letk1=k2=k3=k4=1 and c¯1=c¯2=c¯3=c¯4=c. From the above matrices, we get(39)D∗=10001110−1010−1−111,D∗=a0111101111011110,R^=1111111111111111,ϒ=1000010000100001.
Using the above parameters,R^F=4.
In this example, we compare our sufficient conditionΩ1 with the result Ω3 by taking ℳ as an identity matrix. Now, Ω1andΩ3 are calculated as follows:(40)Ω1=2C¯ℳK−1−ℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2I−2ℳmaxR^FI=2c−16−111−12c−16−111−12c−16−111−12c−16.
Ω1>0, provided c>9.12. For the sufficient condition Ω1>0, system 2 will become GARS whenever c>9.12. For calculating Ω3, we need the matrix Z, and it is calculated using the matrices ℳ, D¯, and D¯. The matrix Z is given as follows:(41)Z=−2−3−3−3−3−2−3−3−3−3−2−3−3−3−3−2.,Ω3=2C¯ℳK−1−+Z−2ℳ2R^FI=22c−10−3−3−3−32c−10−3−3−3−32c−10−3−3−3−32c−10.
Ω3>0, provided c>9.5. For the sufficient condition Ω3>0, system 2 will become GARS whenever c>9.5. Consider the fixed network parameters C=20I,D=D¯,andR=R¯, τ is a constant, and the activation function gyt=tanhyt. The state trajectories are depicted in Figures 1 and 2 under the initial state y0=−0.5,1,0.3,−0.2 and for the different initial states, respectively. The activation function gyt=e−y2t−1/e−y2t+1, and the state trajectories are depicted in Figures 3 and 4 under the initial state y0=−0.5,1,0.3,−0.2 and for the different initial states, respectively.Figure 1
System solution for the initial statey0=−0.5,1,0.3,−0.2.Figure 2
System solution for the different initial states.Figure 3
System solution for the initial statey0=−0.5,1,0.3,−0.2.Figure 4
System solution for the different initial states.Remark 1.
From the above Example1, Ω3 is valid for c>9.5. Moreover, our result Ω1 is valid in the range 9.12<c<9.5, but Ω3 does not hold. Therefore, we conclude that Ω1 is less conservative than Ω3 for the network parameters of this example.Example 2.
Consider the neural network model 1 with the following network parameters:(42)D¯=−2−1−1−1−10−1−1−1−10−1−1−1−10,D¯=0111101111011110,R¯=−20−1−20−1−1−2−2−2−111−21−1,R¯=1221222110222122.
Letk1=k2=k3=k4=1 and c¯1=c¯2=c¯3=c¯4=c. From the above matrices, we get(43)D∗=−1000000000000000,D∗=a1111101111011110,R^=2222222222222222,ϒ=0000010000100001.
Using the above parameters,R^F=8. In this example, we compare our sufficient condition Ω1 with the result Ω2 by taking ℳ as an identity matrix. Now, Ω1andΩ2 are calculated as follows:(44)Ω1=2C¯ℳK−1−ℳD∗−ϒ+D∗−ϒTℳ+ℳD∗+ϒ+D∗+ϒTℳ2I−2ℳmaxR^FI=2c−2200002c−2200002c−2200002c−22.
Ω1>0, provided c>11. For the sufficient condition Ω1>0, system 2 will become GARS whenever c>11. Now, Ω2 is calculated as follows:(45)Ω2=2C¯ℳK−1−ℳD∗+D∗Tℳ+ℳD∗+D∗Tℳ2I−2ℳ2R^FI=2c−20.600002c−22.600002c−22.600002c−22.6.
Ω2>0, provided c>11.3. For the sufficient condition Ω2>0, system 2 will become GARS whenever c>11.3. Consider the fixed network parameters C=10I,D=D¯,andR=R¯, τ is a constant, and the activation function gyt=tanhyt. The state trajectories are depicted in Figures 5 and 6 under the initial state y0=−0.4,0.3,0.5,0.1 and for the different initial states, respectively. The activation function gyt=e−y2t−1/e−y2t+1, and the state trajectories are depicted in Figures 7 and 8 under the initial state y0=−0.4,0.3,0.5,0.1 and for the different initial states, respectively.Figure 5
System solution for the initial statey0=−0.4,0.3,0.5,0.1.Figure 6
System solution for the different initial states.Figure 7
System solution for the initial statey0=−0.4,0.3,0.5,0.1.Figure 8
System solution for the different initial states.Remark 2.
From Example2, Ω2 is valid for c>11.3. Moreover, our result Ω1 is valid in the range 11<c<11.3, but Ω2 does not hold. Therefore, we conclude that Ω1 is less conservative than Ω2 for the network parameters of this example.
From the above examples, our sufficient conditionsΩ1 is less conservative than those imposed by the results Ω2 and Ω3 . Hence, our sufficient condition is more advantageous than the previous results for the above network parameters. Our sufficient condition may have less advantage than the existing stability conditions for the different sets of network parameters. However, all such results will give sufficient conditions.


## 6. Conclusion

In this paper, global stability of NNs has been studied by using the Frobenius norm result under parameter uncertainties. Since the Frobenius norm is the easiest norm by its calculation when compared with spectral norm. By using this Frobenius norm, we have obtained a new sufficient condition for the GARS of NNs model 2. Finally, we discussed some numerical examples to illustrate the effectiveness of our result with the previous results.



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