# New Stability Criterion for Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks with Probabilistic Time-Varying Delays

Authors: Xiongrui Wang; Ruofeng Rao; Shouming Zhong
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A new global asymptotic stability criterion of Takagi-Sugeno fuzzy Cohen-Grossberg neural networks with probabilistic time-varying delays was derived, in which the diffusion item can play its role. Owing to deleting the boundedness conditions on amplification functions, the main result is a novelty to some extent. Besides, there is another novelty in methods, for Lyapunov-Krasovskii functional is the positive definite form ofp powers, which is different from those of existing literature. Moreover, a numerical example illustrates the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Cohen-Grossberg neural networks (CGNNs) have many practical applications, like artificial intelligence, parallel computing, image processing and recovery, and so on ([1–6]). But the success of these applications largely depends on whether the system has some stability, and so people began to be interested in the stability analysis of the system. In recent decades, reaction-diffusion neural networks have received much attention ([7–13]), including various Laplacian diffusion ([6, 14–20]). Besides, people are paying more and more attention to fuzzy neural network system ([21–34]), due to encountering always some inconveniences such as the complicity, the uncertainty, and vagueness ([27, 35–37]). For example, in [27], Zhu and Li investigated the following fuzzy CGNNs model: (1)dxit=-aixitbixit-⋀j=1naijfjxjt-⋁j=1nbijgjxjt-⋀j=1ncijfjxjt-τ-⋁j=1ndijgjxjt-τdt+∑j=1nσijxjt,xjt-τdwjt,xit=ϕit,-τ⩽t⩽0.In [36], Muralisankar and Gopalakrishnan studied the following T-S fuzzy neutral type CGNNs with distributed delays: (2)dxit=∑j=1rhjωt-AjxtBjxt-Cjfxt-τt-Mj∫t-ρttfxsds-Djx˙t-rt.Besides, Balasubramaniam and Syed Ali discussed Takagi-Sugeno fuzzy Cohen-Grossberg BAM neural networks with discrete and distributed time-varying delays in [37].Note that there is the following bounded condition on amplification functions in many literatures (see, e.g., [38, Theorem 4]) related to CGNNs: (3)0<a_i⩽air⩽a¯i,r∈R,i=1,2,…,n.So, in this paper, we try to delete this bounded condition on amplification functions. This is the main purpose of this paper.


## 2. Preliminaries

Consider the following fuzzy Takagi-Sugenop-Laplace partial differential equations with distributed delay.Fuzzy Rule
j.IFω1(t) is μj1 and⋯  ωs(t) is μjsTHEN(4)∂u∂t=∇·Dt,x,u∘∇pu-AuBu-Cjfut-τt,x-Mj∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈Rn,t,x∈R×∂Ω,where Ω is an arbitrary open bounded subset in Rm. ωk(t)(k=1,2,…,s) is the premise variable and μjk(j=1,2,…,r;k=1,2,…,s) is the fuzzy set that is characterized by membership function. And r is the number of theIF-THEN rules; s is the number of the premise variables. u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T∈Rn, where ui(t,x) is the state variable of the ith neuron and the jth neuron at time t and in space variable x. Matrix D(t,x,u)=Dijt,x,un×m with each Dij(t,x,u)⩾0, and Dij(t,x,u) is diffusion operator. D(t,x,u)∘∇pu=Djkt,x,u∇uip-2∂ui/∂xkn×m denotes the Hadamard product of matrix D(t,x,u) and ∇pu (see [39] for details). Matrices A(u)=diag(a1(u1),a2(u2),…,an(un)) and B(u)=diag(b1(u1),b2(u2),…,bn(un)), where ai(ui) and bi(ui) represent an amplification function at time t and an appropriate behavior function at time t. Cj=cikjn×n is the connection matrix. Time delays τ(t)∈[0,+∞). fut-τt,x=(f1(u1(t-τ(t)t,x)),f2(u2(t-τ(t),x)),…,fn(un(t-τ(t),x)))T is the activation function of the neurons. And the second and third equations of (4) imply the initial condition and the Dirichlet boundary condition, respectively.By way of a standard fuzzy inference method, (4) can be inferred as follows. (5)∂u∂t=∇·Dt,x,u∘∇pu-Aut,xBut,x-∑j=1rhjωtCjfut-τt,x+Mj∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈Rn,t,x∈R×∂Ω,where ω(t)=[ω1(t),ω2(t),…,ωs(t)]T and hj(ω(t))=wj(ω(t))/∑k=1rwk(ω(t)),wj(ω(t)):Rs→[0,1](j=1,2,…,r) is the membership function of the system with respect to the fuzzy rule j. hj can be regarded as the normalized weight of eachIF-THEN rule, satisfying hj(ω(t))⩾0 and ∑j=1rhj(ω(t))=1.Next, we consider the following information for probability distribution of time delaysτ(t): (6)P0⩽τt⩽τ1=c0,Pτ1<τt⩽τ2=1-c0.Here the nonnegative scalar c0⩽1. Define a random variable as follows: (7)Ct=1,0⩽τt⩽τ1;0,τ1<τt⩽τ2.So, in this paper, we consider the following Takagi-Sugeno (T-S) fuzzy system with probabilistic time-varying delays:(8)∂u∂t=∇·Dt,x,u∘∇pu-Aut,xBut,x-∑j=1rhjωtc0Cjfut-τ1t,x+1-c0Cjfut-τ2t,x+C-c0Cjfut-τ1t,x-Cjfut-τ2t,x+Mj∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈Rn,t,x∈R×∂Ω.System (8) includes the following integrodifferential equations: (9)dxtdt=-AxtBxt-∑j=1rhjωtc0Cjfxt-τ1t+1-c0Cjfxt-τ2t+C-c0Cjfxt-τ1t-Cjfxt-τ2t+Mj∫t-ρttfxsds,t⩾0,xθ=ϕθ,θ∈-∞,0.Particularly whenp=2, system (8) degenerates into the so-called reaction-diffusion CGNNs:(10)∂u∂t=∇·Dt,x,u∘∇u-Aut,xBut,x-∑j=1rhjωtc0Cjfut-τ1t,x+1-c0Cjfut-τ2t,x+C-c0Cjfut-τ1t,x-Cjfut-τ2t,x+Mj∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈Rn,t,x∈R×∂Ω.Throughout this paper, we assumep=p1/p2>1 with p1 being even number and p2 being odd number. Besides, suppose that the following conditions hold:(H1)
There exist positive definite matricesA_=diag(a_1,a_2,…,a_n) and A¯=diag(a¯1,a¯2,…,a¯n) such that (11)0<a_i⩽aissp-2⩽a¯i,0≠s∈R,i=1,2,…,n,whereA(u)=diag(a1(u1),a2(u2),…,an(un))  and u=(u1,u2,…,un)T∈Rn.(H2)
There exists a positive definite matrixB=diag(b1,b2,…,bn) such that bi(0)=0 and (12)biss⩾bi,0≠s∈R,i=1,2,…,n.(H3)
There is a positive definite matrixF=diag(F1,F2,…,Fn) such that (13)fis⩽Fis,s∈R,i=1,2,…,n.From (H1)–(H3), we know thatbi(0)=fi(0)=0 and u=0 is an equilibrium of fuzzy system (8).Remark 1.
There are numerous functions satisfying (H1). For example, ifp=4/3, we may set (14)ais=0.11+e-s2s23,∀0≠s∈R,ai0=0.2.It is obvious that (15)lims→0⁡ais=lims→0⁡0.11+e-s2s23=+∞.So the function ai(s) is unbounded for s∈R. Moreover, (16)0.1⩽aissp-2=0.11+e-s2⩽0.2.One can know from (16) that 0.1⩽ai(s)/sp-2⩽0.2 with a_i=0.1 and a¯i=0.2.Remark 2.
The amplification functionai(s) defined as (7) is actually unbounded for s∈R. However, various bounded conditions always imposed restrictions on the amplification functions of existing literature ([3–6, 9, 10, 24, 27, 28]). Hence, our condition (H1) is weaker, which will make a corollary with regard to ordinary integrodifferential equations (9) become novel.For convenience’s sake, we need to introduce the following standard notations similarly as [38]:(17)Q=qijn×n>0<0,Q=qijn×n⩾0⩽0,Q1⩾Q2Q1⩽Q2,Q1>Q2Q1<Q2,λmaxΦ,λminΦ,C=cijn×n,ut,x,andtheidentitymatrixwithcompatibleI.(i)
The Sobolev space=u∈Lp:Du∈Lp (see [40] for details).(ii)
Denote byλ1 the lowest positive eigenvalue of the boundary value problem (see [40] for details). (18)-Δpφt,x=λφt,x,x∈Ωφt,x=0,x∈∂Ω.Lemma 3.
One has(19)aq-1b⩽q-1qaq+bqq,∀a,b∈0,+∞,q>1.Note that Lemma3 is the particular case of the famous Young inequality.


## 3. Results and Discussion

Lemma 4.
LetP=diag(p1,p2,…,pn) be a positive definite matrix and u be a solution of the fuzzy system (8). Then one has (20)∫ΩuTP∇·Dt,x,u∘∇pudx⩽-λ1p_Dupp, where D=minik(inft,x,u∣Dij(t,x,u)), upp=∑i=1n∫Ωuipdx, and p_ is a positive scalar, satisfying P>p_I.Proof.
Sinceu is a solution of system (8), it follows by Gauss formula and the Dirichlet zero-boundary condition that (21)∫Ω∑j=1npjuj∑k=1m∂∂xkDjk∇ujp-2∂uj∂xkdx=-∑k=1m∑j=1n∫ΩpjDjk∇ujp-2∂uj∂xk2dx⩽-λ1Dp_∑j=1n∫Ωujpdx=-λ1p_Dupp.Remark 5.
Lemma4 extends the conclusion of [2, Lemma 2.1] and [10, Lemma 2.4] from Hilbert space H01(Ω) to Banach space W01,p(Ω). Particularly, in the case of Ω=(0,T)⊂R1 or W01,p(0,T), the first eigenvalue λ1=(2/T∫0p-11/pdt/1-tp/p-11/p)p (see, e.g., [40]).Theorem 6.
If there exists a positive definite matrixP=diag(p1,p2,…,pn) and two positive scalars p_, p¯ such that the following inequalities hold: (22)λ1Dp_+p_λminA_B>np¯p∑j=1rp-1cj+ρp-1mj+c0cj1-τ1+1-c0cj1-τ2+ρmjλmaxA¯λmaxF,(23)P>p_I,P<p¯I,then the null solution of fuzzy system (8) is globally asymptotically stable, where matrices Cj=(cik(j))n×n,Mj=(mik(j))n×n, cj=maxikcikj, mj=maxikmikj, and τ1′(t)⩽τ1<1,τ2′(t)⩽τ2<1,0⩽ρ(t)⩽ρ.Proof.
Firstly, we can conclude from (H1)–(H3) thatu=0 is an equilibrium point for system (8).
Next, consider the Lyapunov-Krasovskii functional:(24)Vt=V1t+11-τ1V2t+11-τ2V3t+V4t,where (25)V1t=∫ΩuTt,xPut,xdx=∑i=1n∫Ωpiui2dx,V2t=2c0np¯p∑j=1rcjλmaxA¯λmaxF∑k=1n∫t-τ1tt∫Ωuks,xpdxds,V3t=21-c0np¯p∑j=1rcjλmaxA¯λmaxF∑k=1n∫t-τ2tt∫Ωuks,xpdxds,V4t=2p¯p∑j=1rmjλmaxA¯λmaxF∑i=1n∑k=1n∫Ω∫-ρ0dζ∫t+ζtuks,xpdsdx.
Here,u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T is a solution for stochastic fuzzy system (8). Below, we may denote u(t,x) by u and ui(t,x) by ui for simplicity.Remark 7.
It is obvious that our Lyapunov-Krasovskii functional is the positive definite form ofp powers, which is different from those of existing literature ([41–43]). For example, in [41], the model is also neural networks with discrete time delay and distributed delays: (26)dxt=-Cixt+Aifyt-τt,i+Bi∫t-ρttfys,rsdt+σtdw1t,dyt=-C~iyt+A~igxt-τ~t,i+B~i∫t-ρ~ttgxs,rsdt+σ~tdw2t.In [42, Theorem 1], the corresponding Lyapunov-Krasovskii functional is as follows: (27)V~2=∫-ρ0dθ∫t+θtfTys,rsLfys,rsds+∫-ρ~0dθ∫t+θtgTxs,rsLgxs,rsds, which is the positive definite form of 2 powers. And the conclusion of [42, Theorem 1] is the asymptotical stability in the mean square, which is also similar to that of our Theorem 6. However, by means of our Lyapunov-Krasovskii functional with the positive definite form of p powers, we shall derive the asymptotical stability in the mean square for nonlinear p-Laplacian diffusion system (8).Evaluating the time derivation ofV1(t) along the trajectory of the fuzzy system (8), we can get by [38, Lemma 6] and Lemma 4(28)V1′t=2∫ΩuTP∇·Dt,x,u∘∇pu-uTPAuBudx+2∑j=1rhjωt∫ΩuTPAuc0Cjfut-τ1t,xdx+∫ΩuTPAu1-c0Cjfut-τ2t,xdx+∫ΩuTPAuMj∫t-ρttfus,xdsdx⩽-2λ1p_Dupp-2∫ΩuTPAuBudx+2c0∑j=1r∫ΩuTPAuCjfut-τ1t,xdx+21-c0∑j=1r∫ΩuTPAuCjfut-τ2t,xdx+2∑j=1r∫ΩuTPAuMj∫t-ρttfus,xdsdx.Besides, gathering (H1) and (H2) gives (29)∫ΩuTPAuBudx⩾p_λminA_Bupp.It follows by (H1), (H3), and Lemma3 that (30)c0∫ΩuTPAuCjfut-τ1t,xdx=c0∑k=1n∑i=1n∫Ωpiuiaiuicikjfkukt-τ1t,xdx⩽c0p¯∑k=1n∑i=1n∫Ωa¯icjuip-1Fkukt-τ1t,xdx⩽c0p¯cjλmaxA¯λmaxF∑k=1n∑i=1n∫Ωp-1puip+ukt-τ1t,xppdx=c0p-1p¯pncjλmaxA¯λmaxFupp+c0p¯pncjλmaxA¯λmaxF∑k=1n∫Ωukt-τ1t,xpdx.Similarly, (31)1-c0∫ΩuTPAuCjfut-τ2t,xdx⩽1-c0p-1p¯pncjλmaxA¯λmaxFupp+1-c0p¯pncjλmaxA¯λmaxF∑k=1n∫Ωukt-τ2t,xpdx,∫ΩuTPAuMj∫t-ρttfus,xdsdx=∑k=1n∑i=1n∫Ωpiuiaiuimikj∫t-ρttfkuks,xdsdx⩽p¯∑k=1n∑i=1n∫Ω∫t-ρtta¯imjuit,xp-1Fkuks,xdsdx⩽p¯mjλmaxA¯λmaxF∑k=1n∑i=1n∫Ω∫t-ρttp-1puit,xp+uks,xppdsdx⩽ρp-1p¯pnmjλmaxA¯λmaxFupp+p¯pmjλmaxA¯λmaxF∑i=1n∑k=1n∫Ω∫t-ρttuks,xpdsdx.On the other hand,(32)V2′t=2c0np¯p∑j=1rcjλmaxA¯λmaxF∑k=1n∫Ωukt,xpdx-∑k=1n1-τ1′t∫Ωukt-τ1t,xpdx⩽2c0np¯p∑j=1rcjλmaxA¯λmaxFupp-2c0np¯p∑j=1rcjλmaxA¯λmaxF1-τ1∑k=1n∫Ωukt-τ1t,xpdx.Similarly, (33)V3′t⩽21-c0np¯p∑j=1rcjλmaxA¯λmaxFupp-21-c0np¯p∑j=1rcjλmaxA¯λmaxF1-τ2∑k=1n∫Ωukt-τ2t,xpdx.Next, we need to recall some facts derived by mathematical analysis. Assume thatη(t,s) is continuous on variables t and s, and ∂η/∂t exists, utilizing the integral middle value theorem reaches (34)ddt∫ξtϖtηt,sds=ϖ′tηt,ϖt-ξ′tηt,ξt+∫ξtϖt∂ηt,s∂tds,where both ξ(·) and ϖ(·) are differentiable.Moreover, we can derive by employing (32) time and again (35)V4′t=2p¯p∑j=1rmjλmaxA¯λmaxF∑i=1n∑k=1n∫Ω∫-ρ0ukt,xpds-∫-ρ0ukt+s,xpdsdx=2p¯p∑j=1rmjλmaxA¯λmaxF∑i=1n∑k=1n∫Ωρukt,xp-∫t-ρttuks,xpdsdx=2p¯p∑j=1rmjλmaxA¯λmaxFnρupp-∑i=1n∑k=1n∫Ω∫t-ρttuks,xpdsdx.Combining (28)–(35) results in (36)V′t⩽-2λ1Dp_+p_λminA_B-np∑j=1rp¯p-1cj+ρp-1p¯mj+c0p¯1-τ1cj+1-c0p¯1-τ2cj+ρp¯mjλmaxA¯λmaxFupp⩽0.Now the standard Lyapunov functional theory derives that the null solution of the fuzzy system (8) is globally asymptotically stable.Remark 8.
In the case of Takagi-Sugeno fuzzy model, our Theorem6 is better than [38, Theorem 4] because the condition (H1) is weaker than the bounded assumption (2).Remark 9.
In Theorem6, (22) illustrates the influence of nonlinear diffusion on the stability of system (8) while its role was always ignored in existing results (see, e.g., [5, 39, 44]).Theorem6 derives the following corollary.Corollary 10.
If there exists a positive definite matrixP=diag(p1,p2,…,pn) and two positive scalars p_, p¯ such that the following inequalities hold: (37)p_λminA_B>np¯p∑j=1rp-1cj+ρp-1mj+c0cj1-τ1+1-c0cj1-τ2+ρmjλmaxA¯λmaxF,P>p_I,P<p¯I,then the null solution of the ordinary integrodifferential equations (9) is globally asymptotically stable.Furthermore, if both diffusion behaviors and distributed delay are ignored, we derive from Corollary10.Corollary 11.
If there exists a positive definite matrixP=diag(p1,p2,…,pn) and two positive scalars p_, p¯ such that the following inequalities hold: (38)p_λminA_B>np¯p∑j=1rp-1cj+c0cj1-τ1+1-c0cj1-τ2λmaxA¯λmaxF,P>p_I,P<p¯I,then the null solution of the following fuzzy system (39)dxtdt=-AxtBxt-∑j=1rhjωtc0Cjfxt-τ1t+1-c0Cjfxt-τ2t+C-c0Cjfxt-τ1t-Cjfxt-τ2t,t⩾0,xθ=ϕθ,θ∈-∞,0is globally asymptotically stable.Remark 12.
Condition (H1) is weaker than the bounded conditions on amplification functions of existing literature ([3–6, 9, 10, 24, 27, 28]).Discussion 1.
In recent related literature ([27, 45–51]), some new conditions and methods were presented, and their results were very good. However, some of the methods and conditions are not applicable for system (8) with nonlinear p-Laplacian diffusion. How to apply the new conditions and methods of [45–49] to our system (8) is an interesting problem.


## 4. Methods and Numerical Example

### 4.1. Methods

In this paper, Lyapunov functional method is employed to derive the stability criterion. In this process, the integral middle value theorem together with the derivation formula on integral upper limit functions plays the important roles.Example 1.
Consider the following Takagi-Sugenop-Laplace fuzzy T-S dynamic equations.
Fuzzy Rule 1.IFω1(t) is μ11, and ω2(t) is μ12,THEN(40)∂u∂t=∇·Dt,x,u∘∇pu-AuBu-c0C1fut-τ1t,x-1-c0C1fut-τ2t,x-C-c0C1fut-τ1t,x-C1fut-τ2t,x-M1∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈R2,t,x∈R×∂Ω.
Fuzzy Rule 2.IFω1(t) is μ21, and ω2(t) is μ22,THEN(41)∂u∂t=∇·Dt,x,u∘∇pu-AuBu-c0C2fut-τ1t,x-1-c0C2fut-τ2t,x-C-c0C2fut-τ1t,x-C2fut-τ2t,x-M2∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈R2,t,x∈R×∂Ω,where u(t,x)=(u1(t,x),u2(t,x))T,Ω=(0,π),p=4/3, and then Remark 1 gives (42)λ1=2π∫0p-11/pdt1-tp/p-11/pp=0.7915. Let τ1(t)=t/3,τ2(t)=t/2, and then τ1=1/3,τ2=1/2. Let ai(ui)=0.1ui-2/3(1+e-iui2),i=1,2,b1(u1)=2u1(1+sin2u1),b2(u2)=1.95u2,  f1(u1(t-τ(t)))=0.16u1t-τtsin⁡u1(t-τ(t)),f2(u2(t-τ(t)))=0.166u2(t-τ(t)),τ=0.5, D=0.003, c0=0.75,c1=0.2,c2=0.3,m1=0.02,m2=0.03, and (43)A_=0.01000.02,A¯=0.1000.2,B=2001.95,Dt,x,u=0.0030.0050.0040.006,M1=0.020.0100.01,F=0.16000.166,C1=0.20.100.15,C2=0.20.100.3,M2=0.010.0100.03. Now we use MATLAB to solve LMIs (22)-(23), obtaining the feasibility data (44)P=0.9381001.013,p_=0.9103,p¯=1.023.
Now Theorem6 derives that the null solution of this Takagi-Sugeno fuzzy equations is globally asymptotically stable (see Figures 1 and 2).Figure 1
Computer simulations of the stateu1(t,x).Figure 2
Computer simulations of the stateu2(t,x).


## 4.1. Methods

In this paper, Lyapunov functional method is employed to derive the stability criterion. In this process, the integral middle value theorem together with the derivation formula on integral upper limit functions plays the important roles.Example 1.
Consider the following Takagi-Sugenop-Laplace fuzzy T-S dynamic equations.
Fuzzy Rule 1.IFω1(t) is μ11, and ω2(t) is μ12,THEN(40)∂u∂t=∇·Dt,x,u∘∇pu-AuBu-c0C1fut-τ1t,x-1-c0C1fut-τ2t,x-C-c0C1fut-τ1t,x-C1fut-τ2t,x-M1∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈R2,t,x∈R×∂Ω.
Fuzzy Rule 2.IFω1(t) is μ21, and ω2(t) is μ22,THEN(41)∂u∂t=∇·Dt,x,u∘∇pu-AuBu-c0C2fut-τ1t,x-1-c0C2fut-τ2t,x-C-c0C2fut-τ1t,x-C2fut-τ2t,x-M2∫t-ρttfus,xds,uθ,x=ϕθ,x,θ,x∈-∞,0×Ω,ut,x=0∈R2,t,x∈R×∂Ω,where u(t,x)=(u1(t,x),u2(t,x))T,Ω=(0,π),p=4/3, and then Remark 1 gives (42)λ1=2π∫0p-11/pdt1-tp/p-11/pp=0.7915. Let τ1(t)=t/3,τ2(t)=t/2, and then τ1=1/3,τ2=1/2. Let ai(ui)=0.1ui-2/3(1+e-iui2),i=1,2,b1(u1)=2u1(1+sin2u1),b2(u2)=1.95u2,  f1(u1(t-τ(t)))=0.16u1t-τtsin⁡u1(t-τ(t)),f2(u2(t-τ(t)))=0.166u2(t-τ(t)),τ=0.5, D=0.003, c0=0.75,c1=0.2,c2=0.3,m1=0.02,m2=0.03, and (43)A_=0.01000.02,A¯=0.1000.2,B=2001.95,Dt,x,u=0.0030.0050.0040.006,M1=0.020.0100.01,F=0.16000.166,C1=0.20.100.15,C2=0.20.100.3,M2=0.010.0100.03. Now we use MATLAB to solve LMIs (22)-(23), obtaining the feasibility data (44)P=0.9381001.013,p_=0.9103,p¯=1.023.
Now Theorem6 derives that the null solution of this Takagi-Sugeno fuzzy equations is globally asymptotically stable (see Figures 1 and 2).Figure 1
Computer simulations of the stateu1(t,x).Figure 2
Computer simulations of the stateu2(t,x).


## 5. Conclusions

By constructing a novel Lyapunov function, we employed Young inequality and LMI technique to derive the asymptotic stability criteria for CGNNs with distributed delays and nonlinear diffusion. Since the stability of nonlinearp-Laplacian diffusion neural networks was originally investigated in [2], various p-Laplacian diffusion neural networks have attracted a lot of interest ([6, 17, 34, 39, 44]). As pointed out in Discussion 1, some new conditions and methods may not be applicable to CGNNs model with nonlinear p-Laplacian diffusion. So our results are a novelty to some extent.



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*Source: 3793157-2017-11-16.xml*