# On the Role of Diffusion Behaviors in Stability Criterion forp-Laplace Dynamical Equations with Infinite Delay and Partial Fuzzy Parameters under Dirichlet Boundary Value

Authors: Ruofeng Rao; Zhilin Pu; Shouming Zhong; Jialin Huang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev spaceW01,p(Ω), a global asymptotical stability criterion for p-Laplace partial differential equations with partial fuzzy parameters is derived under Dirichlet boundary condition, which gives a positive answer to an open problem proposed in some related literatures. Different from many previous related literatures, the nonlinear p-Laplace diffusion item plays its role in the new criterion though the nonlinear p-Laplace presents great difficulties. Moreover, numerical examples illustrate that our new stability criterion can judge what the previous criteria cannot do.


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## Body

## 1. Introduction

Very recently, time-delayp-Laplace (p>1) dynamical equations have attracted rapidly growing interest, because the nonlinear Laplace diffusion dynamical equations admit many physics and engineering background [1–6], such as Cohen-Grossberg neural networks and recurrent neural networks. In real world, diffusion phenomena cannot be unavoidable. Particularly, 2-Laplace (p=2) is called the linear Laplace, and the diffusion phenomenon is always simulated by linear Laplace diffusion for simplicity ([7–10] and their references therein). However, diffusion behavior is so complicated that the nonlinear reaction-diffusion models were considered in many other recent literatures [1–6, 11–13]. Even the nonlinear p-Laplace diffusion (p>1) is considered in simulating some diffusion behaviors [1–6]. But the previous related literature lost sight of the role of the nonlinear diffusion in their stability criteria. As pointed out in [2], the problem how the p-Laplace diffusion item plays a role in stability criteria remains open and challenging. And such a situation motivates our present study. Besides, fuzzy logic theory has been shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some local linear subsystems [14–16]. Motivated by some ideas and methods of [17–21], we obtain a global asymptotical stability criterion for fuzzy T-S p-Laplace partial differential equations with Dirichlet boundary value by the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space W01,p(Ω). And in the obtained criterion, the nonlinear p-Laplace diffusion item plays a positive role.


## 2. Model Description and Preliminaries

Let us consider a class of fuzzy Takagi-Sugeno (T-S)p-Laplace partial differential equations described as follows.Fuzzy Rule  j.IF  ω1(t) is μj1 and ⋯ωs(t) is μjs,  THEN(1)∂v∂t=∇·(Γ(t,x,v)∘∇pv)-A(v)×{B(v)-[Cjf(v)+Djg(v(t-τ(t),x))]},v(θ,x)=ϕ(θ,x),(θ,x)∈(-∞,0]×Ω,v(t,x)=0∈Rn,(t,x)∈R×∂Ω,
where Ω is an arbitrary open bounded subset in Rn. ωk(t)(k=1,2,…,s) is the premise variable, 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 the IF-THEN rules; s is the number of the premise variables. Fuzzy partial differential system (1) admits its many physics and engineering background, including the famous Cohen-Grossberg neural networks [1–6]. Consider v(t,x)=(v1(t,x),v2(t,x),…,vn(t,x))T∈Rn, where vi(t,x) is the state variable of the ith neuron and the jth neuron at time t and in space variable x. Matrix Γ(t,x,v)=(Γij(t,x,v))n×m with each Γij(t,x,v)⩾0, and Γij(t,x,v) is diffusion operator. Γ(t,x,v)∘∇pv=(Γik(t,x,v)|∇vi|p-2(∂vi/∂xk))n×m denotes the Hadamard product of matrix Γ(t,x,v) and ∇pv (see [1–6] for details). Matrix A(v)=diag(a1(v1),a2(v2),…,an(vn)) and vector B(v)=(b1(v1),b2(v2),…,bn(vn))T, where ai(vi) and bi(vi) represent an amplification function at time t and an appropriately behaved function at time t. Cj=(cik(j))n×n and Dj=(dik(j))n×n are connection matrices. The time-varying delays are τ(t)∈[0,+∞). f(v)=(f1(v1),f2(v2),…,fn(vn))T and g(v(t-τ(t)))=(g1(v1(t-τ(t))),g2(v2(t-τ(t))),…,gn(vn(t-τ(t))))T are the activation functions of the neurons. And the second and third equations of (1) represent the initial condition and the Dirichlet boundary condition, respectively.With the help of a standard fuzzy inference method, (1) can be inferred as follows:
(2)∂v∂t=∇·(Γ(t,x,v)∘∇pv)-A(v){∑j=1r‍B(v)-∑j=1r‍hj(ω(t))(v)--A(v)∑j=1r‍hj×[Cjf(v)+Djg(v)-)-A(v)∑j=1r‍hj×(v(t-τ(t),x))Cj]∑j=1r‍},v(θ,x)=ϕ(θ,x),(θ,x)∈(-∞,0]×Ω,v(t,x)=0∈Rn,(t,x)∈R×∂Ω,
where ω(t)=[ω1(t),ω2(t),…,ωs(t)], 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 each IF-THEN rule, satisfying hj(ω(t))⩾0 and ∑j=1rhj(ω(t))=1.For convenience’s sake, we need to introduce some standard notations.(i)
Denote|C|=(|cij|)n×n for any matrix C=(cij)n×n.(ii)
Denote|v(t,x)|=(|v1(t,x)|,|v2(t,x)|,…,|vn(t,x)|)T for any v(t,x)=(v1(t,x),v2(t,x),…,vn(t,x))T.(iii)
u⩾v if ui⩾vi for all i=1,2,…,n, where u=(u1,u2,…,un)T, v=(v1,v2,…,vn)T∈Rn.In addition, we also introduce the following standard notations similarly as [2, (iii)–(X)]:
(3)Q=(qij)n×n>0(<0),Q=(qij)n×n⩾0(⩽0),Q1⩾Q2(Q1⩽Q2),Q1>Q2(Q1<Q2),andtheidentitymatrixI.Throughout this paper, we assume the following.(H1)
There exist positive definite matricesA_=diag(a_1,a_2,…,a_n) and A¯=diag(a¯1,a¯2,…,a¯n) such that
(4)0<a_i⩽ai(r)⩽a¯i,r∈R,i=1,2,…,n.(H2)
There exists a positive definite matrixB=diag(b1,b2,…,bn) such that bi(0)=0 and
(5)bi(r)rp-1⩾bi,0≠r∈R,i=1,2,…,n.(H3)
There exist positive definite matricesF=diag(F1,F2,…,Fn) and G=diag(G1,G2,…,Gn) such that
(6)|fi(r)|⩽Fi|r|p-1,|gi(r)|⩽Gi|r|p-1,r∈R,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,…,n.From (H1)–(H3), we know thatf(0)=g(0)=0∈Rn, and v=0 is an equilibrium of (2).Lemma 1.
For anya,b∈(0,+∞), One has
(7)ap-1b⩽p-1pap+bpp.


## 3. Main Result

Before giving the main result of this paper, we have to present the following Lemma via some variational methods in the Sobolev spaceW01,p(Ω), which is the completion of C0∞(Ω) with respect to the norm ∥η∥=(∫Ω‍|∇η(x)|pdx)1/p. Denote by λ1 the first eigenvalue of -Δp in Sobolev space W01,p(Ω), where λ1=min{∫Ω‍|∇η(x)|pdx:η(x)∈W01,p(Ω),∫Ω‍|η(x)|pdx=1} (see [22–24] for details).Lemma 2.
LetQ=diag(q1,q2,…,qn) be a positive definite matrix, and let v be a solution of the fuzzy system (2). Then One has
(8)∫Ω‍vTQ(∇·(Γ(t,x,v)∘∇pv))dx⩽-λ1q_Γ∥v∥pp,
where Γ=mini,k(inft,x,vΓik(t,x,v)), ∥v∥pp=∑i=1n∫Ω‍|vi|pdx, and q_ is a positive scalar, satisfying Q>q_I.Proof.
Sincev is a solution of system (2), it follows by Gauss formula and the Dirichlet boundary condition that
(9)∫Ω‍vTQ(∇·(Γ(t,x,v)∘∇pv))dx=∫Ω‍vTQ(∑k=1m‍∑k=1m‍∂∂xk(Γ1k|∇v1|p-2∂v1∂xk),…,=∫Ω‍vTQ(∑k=1m‍∂∂xk(Γnk|∇vn|p-2∂vn∂xk)∑k=1m‍)Tdx=-∑k=1m‍∑j=1n‍∫Ω‍qjΓjk|∇vj|p-2(∂vj∂xk)2dx⩽-λ1q_Γ∥v∥pp.Remark 3.
Lemma2 actually generalizes the conclusion of [7, Lemma 2.1] and [25, 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)∫0(p-1)1/p‍(dt/(1-(tp/(p-1)))1/p))p (see, e.g., [22]).Theorem 4.
Suppose thatp=m1/m2>1, where m1 is an even number while m2 is an odd number. If, in addition, there exist a positive definite matrix Q=diag(q1,q2,…,qn) and two positive scalars q_, q¯ such that the following inequalities hold:
(10)[λ1Γ+λmin(A_B)]q_>[∑j=1r‍n(λmaxA¯)(λmaxF)>[×∑j=1r‍|cj|+(np+n(p-1)(1-τ)p)>[×(λmaxA¯)(λmaxG)∑j=1r‍|dj|]q¯,(11)qi>q_,i∈𝒩,(12)qi<q¯,i∈𝒩,
then the null solution of (2) is globally asymptotically stable, where 𝒩={1,2,…,n}, τ′(t)⩽τ<1, |cj|=maxi,k|cik(j)| for the matrix Cj=(cik(j))n×n, and |dj|=maxi,k|dik(j)| for the matrix Dj=(dik(j))n×n.Proof.
Define the Lyapunov-Krasovskii functional as follows:(13)V(t)=V1(t)+11-τV2(t),
where
(14)V1(t)=∫Ω‍vTQvdx,V2(t)=2nq¯(λmaxA¯)(λmaxG)p-1p×∑j=1r‍∑k=1n‍|dj|∫t-τ(t)t‍∫Ω‍|vk(s,x)|pdsdx.
Evaluating the time derivation ofV1(t) along the trajectory of the fuzzy system (2), we can get by Lemma 2(15)V1′(t)⩽-2λ1q_Γ∥v∥pp-2∫Ω‍vTQA(v)B(v)dx+2∑j=1r‍hj(ω(t))∫Ω‍vTQ[A(v)Cjf(v)+∑j=1r‍hj(ω(t))∫Ω‍vTQ+A(v)Djg(vτ)]dx⩽-2λ1q_Γ∥v∥pp-2∫Ω‍vTQA(v)B(v)dx+2∑j=1r‍∫Ω‍|vT|Q[A¯|Cj||f(v)|+A¯|Dj||g(vτ)|]dx,
where we denote vτ=v(t-τ(t),x) for convenience.
Then we can get by (H1), (H2) and the restrictive conditions on the parameterp(16)∫Ω‍vTQA(v)B(v)dx⩾q_∫Ω‍|vT|A_B(|v1|p-1,|v2|p-1,…,|vn|p-1)Tdx⩾q_λmin(A_B)∥v∥pp.
It follows by (H3) and Lemma 1 that
(17)∫Ω‍|vT|QA¯|Cj||f(v)|dx⩽∫Ω‍|vTQ|A¯|Cj|F(|v1|p-1,|v2|p-1,…,|vn|p-1)Tdx=q¯∑k=1n‍∑i=1n‍∫Ω‍|vi|a¯i|cik(j)|Fk|vk|p-1dx⩽q¯∑k=1n‍∑i=1n‍∫Ω‍a¯i|cik(j)|Fk(p-1p|vk|p+|vi|pp)dx=q¯p-1p∑k=1n‍∑i=1n‍∫Ω‍a¯i|cik(j)|Fk|vk|pdx+q¯1p∑k=1n‍∑i=1n‍∫Ω‍a¯i|cik(j)|Fk|vi|pdx⩽q¯|cj|(λmaxA¯)(λmaxF)×(p-1p∑k=1n‍∑i=1n‍∫Ω‍|vk|pdx+1p∑k=1n‍∑i=1n‍∫Ω‍|vi|pdx)=nq¯|cj|(λmaxA¯)(λmaxF)×(p-1p∑k=1n‍∫Ω|vk|pdx+1p∑i=1n‍∫Ω‍‍|vi|pdx)=nq¯|cj|(λmaxA¯)(λmaxF)∥v∥pp.
Similarly,(18)∫Ω‍|vT|QA¯|Dj||g(vτ)|dx⩽q¯∑k=1n‍∑i=1n‍∫Ω‍a¯i|dik(j)|Gk×(p-1p|vk(t-τ(t),x)|p+|vi(t,x)|pp)dx⩽nq¯|dj|(λmaxA¯)(λmaxG)×(p-1p∑k=1n‍∫Ω‍|vk(t-τ(t),x)|pdx+1p∥v∥pp).
Combining (15)–(18) results in
(19)V1′(t)⩽-2(∑j=1r‍λ1Γq_+q_λmin(A_B)-nq¯(λmaxA¯)(λmaxF)⩽(-2×∑j=1r‍|cj|-1pnq¯(λmaxA¯)(λmaxG)∑j=1r‍|dj|)×∥v∥pp+2nq¯(λmaxA¯)(λmaxG)p-1p×∑j=1r‍∑k=1n‍|dj|∫Ω‍|vk(t-τ(t),x)|pdx.
On the other hand,(20)V2′(t)=2nq¯(λmaxA¯)(λmaxG)p-1p∥v∥pp∑j=1r‍|dj|-(1-τ′(t))2nq¯(λmaxA¯)(λmaxG)p-1p×∑j=1r‍∑k=1n‍|dj|∫Ω‍|vk(t-τ(t),x)|pdx⩽2nq¯|dj|(λmaxA¯)(λmaxG)p-1p∥v∥pp∑j=1r‍|dj|-(1-τ)2nq¯(λmaxA¯)(λmaxG)p-1p×∑j=1r‍∑k=1n‍|dj|∫Ω‍|vk(t-τ(t),x)|pdx.
So we conclude from (10) that
(21)V′(t)⩽-2(∑j=1r‍λ1q_Γ+q_λmin(A_B)-nq¯(λmaxA¯)(λmaxF)⩽-2(×∑j=1r‍|cj|-1pnq¯(λmaxA¯)(λmaxG)∑j=1r‍|dj|⩽-2(-11-τnq¯(λmaxA¯)(λmaxG)p-1p∑j=1r‍|dj|)×∥v∥pp⩽0.
It follows by the standard Lyapunov functional theory that the null solution of the fuzzy system (2) is globally asymptotically stable.Remark 5.
In many previous related literatures (see, e.g., [1–6]), the nonlinear p-Laplace (p>2) diffusion terms were omitted in the deductions, which results in that their stability criteria do not contain the diffusion terms. In other words, the diffusion terms do not affect their results. In addition, when p=2,2-Laplace is the linear Laplace, and there are many papers (see, e.g., [25–29]) in which the Laplace diffusion item plays its role in their stability criteria, for the linear Laplace PDEs can be considered in the special Hilbert space H1(Ω) that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces. However, the nonlinear p-Laplace (p>1, p≠2) brings great difficulties, for the nonlinear p-Laplace PDEs should be considered in the frame of Sobolev space W1,p(Ω) that is only a reflexive Banach space. Indeed, owing to the great difficulties, the authors only provide in [4] the stability criterion in which the nonlinear p-Laplace items play roles in the case of 1<p<2. However, in this paper, the nonlinear p-Laplace diffusion terms play a positive role in our Theorem 4 for the case of p>2 or p>1, which also gives a positive answer to the open problem proposed in [2] to some extent. Besides, we will provide a numerical example where our Theorem 4 works whereas [2, Corollary 15] do not (see, Example 2).Remark 6.
Particularly whenp=2, Theorem 4 provides a global asymptotical stability criterion for the familiar reaction-diffusion fuzzy CGNNs with infinite delay. Even in this particular case, the result is also good thanks to the infinite allowable upper bounds of time delays.


## 4. Numerical Example

Example 1.
Consider thep-Laplace fuzzy T-S dynamic equations as follows.
Fuzzy Rule  1.
IF  ω1(t) is 1/e-2ω1(t),  THEN(22)∂v∂t=∇·(Γ(t,x,v)∘∇pv)-A(v){B(v)-[C1f(v)+D1g(v(t-τ(t),x))]},v(θ,x)=ϕ(θ,x),(θ,x)∈(-∞,0]×Ω,v(t,x)=0∈R2,(t,x)∈R×∂Ω.
Fuzzy Rule  2.
IF  ω2(t) is 1-(1/e-2ω1(t)),  THEN(23)∂v∂t=∇·(Γ(t,x,v)∘∇pv)-A(v)×{B(v)-[C2f(v)+D2g(v(t-τ(t),x))]},v(θ,x)=ϕ(θ,x),(θ,x)∈(-∞,0]×Ω,v(t,x)=0∈R2,(t,x)∈R×∂Ω,
where v(t,x)=(v1(t,x),v2(t,x))T, Ω=(0,π), p=4/3, and then the first eigenvalue λ1=((2/π)∫0(p-1)1/p‍(dt/(1-(tp/(p-1)))1/p))p=0.7915 (see Remark 3). The initial value function is presented as follows:
(24)ϕ(s,x)=(0.25(1-cos(5πx2))sin398(x2-0.25)e-100s0.2cos2(4πx2)sin508(x2-0.55)e-100s).
Letτ(t)=t/3, and then τ′(t)⩽1/3=τ<1. Besides, A(v)=diag(0.7+0.2sin2(v1),0.6+0.28sin2(v2)), B(v)=(2v11/3,1.9v21/3)T, and f(v)=g(v)=(0.1v11/3,0.2v21/3)T, and correspondingly we assume
(25)A_=(0.7000.6),A¯=(0.9000.88),B=(2001.9),F=(0.1000.2)=G.
In addition, r=2, and
(26)C1=(-0.20.30.280.3)=D1,C2=(0.290.180.2-0.1)=D2,Γ(t,x,v)=(0.0080.0070.0060.006).
Hence, Γ=0.006, |c1|=|d1|=0.3, |c2|=|d2|=0.29, λmin(A_B)=1.1400, λmaxA¯=0.9, and λmaxF=λmaxG=0.2. Then we can use MATLAB LMI toolbox to solve the inequalities (10)–(12), and obtain tmin=-0.5569<0 which implies feasible. Further, extracting the datum shows q_=19.7994, q¯=31.1620, qi=24.3210, i=1,2. Thereby, we can conclude from Theorem 4 that the null solution for this fuzzy dynamic equation is globally asymptotically stable (see Figures 1 and 2).Figure 1
The state variablev1(t,x).Figure 2
The state variablev2(t,x).Example 2.
To compare our Theorem4 with [2, Corollary 15], we may consider the following p-Laplace dynamic equations. 							(27)∂v∂t=∇·(Γ(t,x,v)∘∇3v)-A(v){B(v)-[Cf(v)+Dg(v(t-30,x))]},v(θ,x)=ϕ(θ,x),(θ,x)∈(-∞,0]×Ω,v(t,x)=0∈R2,(t,x)∈R×∂Ω,
where v(t,x)=(v1(t,x),v2(t,x))T, Ω=(0,π/2), p=3, and then the first eigenvalue λ1=7.3104 (see Remark 5). Let τ(t)≡30, and then τ′(t)≡0<1. In addition, r=1 is the number of the IF-THEN. Assume
(28)A_=(0.7000.6),A¯=(0.9000.88),B=(0.2000.1),F=(0.1000.2)=G,C=(-0.20.30.30.3)=D,Γ(t,x,v)=(0.080.070.060.06).Below we shall employ [2, Corollary 15] and our Theorem 4 to judge the stability of System (27), respectively.In [2, Corollary 15], letting λ=0.001, we use MATLAB LMI toolbox to solve the LMI conditions [2, (37)] and obtain tmin=7.1357×10-13>0, which implies that these LMI constraints are not strictly feasible. Again, let λ=1 and λ=10 in [2, Corollary 15], respectively. And then tmin=2.1479×10-20>0 and tmin=2.1026×10-16>0, respectively. Hence, we cannot judge the stability of System (27) from [2, Corollary 15]. However, the LMIs of [2, Corollary 15] are only the sufficient condition for the stability, not necessary for stability. Indeed, we use MATLAB LMI toolbox to solve (10)–(12) in Theorem 4 and obtain tmin=-0.0794<0, which implies being feasible. Further, extracting the datum results in that q_=0.9803, q¯=1.8381, and qi=1.3671, for i=1,2. Thereby, we can conclude from Theorem 4 that the null solution for this fuzzy dynamic equation is globally asymptotically stable.There are some interesting comparisons between [2, Corollary 15] and our Theorem 4 (see Table 1).Table 1
Comparisons between [2, Corollary 15] and Theorem 4.
[2, Corollary 15]λ=0.001
λ
=
1
λ
=
10
Theorem4
t
min
>0
>0
>0
<0
Feasibility
Unfeasible
Unfeasible
Unfeasible
Feasible
Stability
Unknown
Unknown
Unknown
Global asymptotical stabilityRemark 7.
Since we consider the role of the nonlinearp-Laplace diffusion item in the stability criterion, Example 2 illustrates that our Theorem 4 can judge what [2, Corollary 15] cannot do.


## Conclusions

In this paper, the global asymptotical stability criterion of the nonlinearp-Laplace fuzzy T-S dynamical equations with infinite delay was derived by the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space W01,p(Ω). The p-Laplace diffusion item plays its role in our stability criterion while the stability criteria obtained in many previous literatures did not contain the diffusion terms. In fact, when p=2,2-Laplace is the linear Laplace, and there are many papers (see, e.g., [25–29]) in which the Laplace diffusion item plays its role in their stability criteria, for the linear Laplace PDEs can be considered in the special Hilbert space H1(Ω) that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces. However, the nonlinear p-Laplace (p>1, p≠2) brings great difficulties, for the nonlinear p-Laplace PDEs should be considered in the frame of Sobolev space W1,p(Ω) that is only a reflexive Banach space. Indeed, owing to the great difficulties, the authors only provide in [4] the stability criterion in which the nonlinear p-Laplace items play roles in the case of 1<p<2. Now in this paper, we present the stability criterion in which the nonlinear p-Laplace items play roles in the case of p>2 or p>1. Moreover, numerical example shows the effectiveness of the proposed methods. Since the non-linear p-Laplace dynamical equations have many physics and engineering background, including the famous Cohen-Grossberg neural networks, a further profound study is very interesting in mathematical theories, methods, and even practice. Up to now, we do not know how the p-Laplace (p>1) diffusion item plays a role in p-stability criteria. This problem remains open and challenging.



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