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

Abstract

By the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space 𝑊₀^1,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.

1. Introduction

Very recently, time-delay 𝑝-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 𝑊₀^1,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 𝜇_𝑗1 and ⋯𝜔_𝑠(𝑡) is 𝜇_𝑗𝑠,  THEN𝜕𝑣Γ()𝜕𝑡=∇⋅𝑡,𝑥,𝑣∘∇_𝑝𝑣(𝑣)×𝐵(𝑣)−𝐶−𝐴_𝑗𝑓(𝑣)+𝐷_𝑗𝑔(𝑣((𝑡),𝑣()(),()∈(]𝑣()𝑡−𝜏,𝑥))𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−∞,0×Ω,𝑡,𝑥=0∈𝑅^𝑛,(𝑡,𝑥)∈𝑅×𝜕Ω, (1) where Ω is an arbitrary open bounded subset in 𝑅^𝑛. 𝜔_𝑘(𝑡)(𝑘=1,2,…,𝑠) is the premise variable, 𝜇_𝑗𝑘(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),…,𝑣_𝑛(𝑡,𝑥))^𝑇∈𝑅^𝑛, where 𝑣_𝑖(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)=(Γ_𝑖𝑗(𝑡,𝑥,𝑣))_𝑛×𝑚 with each Γ_𝑖𝑗(𝑡,𝑥,𝑣)⩾0, and Γ_𝑖𝑗(𝑡,𝑥,𝑣) is diffusion operator. Γ(𝑡,𝑥,𝑣)∘∇_𝑝𝑣=(Γ_𝑖𝑘(𝑡,𝑥,𝑣)|∇𝑣_𝑖|^𝑝−2(𝜕𝑣_𝑖/𝜕𝑥_𝑘))_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),…,𝑎_𝑛(𝑣_𝑛)) and vector 𝐵(𝑣)=(𝑏₁(𝑣₁),𝑏₂(𝑣₂),…,𝑏_𝑛(𝑣_𝑛))^𝑇, where 𝑎_𝑖(𝑣_𝑖) and 𝑏_𝑖(𝑣_𝑖) represent an amplification function at time 𝑡 and an appropriately behaved function at time 𝑡. 𝐶_𝑗=(𝑐^(𝑗)_𝑖𝑘)_𝑛×𝑛 and 𝐷_𝑗=(𝑑^(𝑗)_𝑖𝑘)_𝑛×𝑛 are connection matrices. The time-varying delays are 𝜏(𝑡)∈[0,+∞). 𝑓(𝑣)=(𝑓₁(𝑣₁),𝑓₂(𝑣₂),…,𝑓_𝑛(𝑣_𝑛))^𝑇 and 𝑔(𝑣(𝑡−𝜏(𝑡)))=(𝑔₁(𝑣₁(𝑡−𝜏(𝑡))),𝑔₂(𝑣₂(𝑡−𝜏(𝑡))),…,𝑔_𝑛(𝑣_𝑛(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: 𝜕𝑣Γ()𝜕𝑡=∇⋅𝑡,𝑥,𝑣∘∇_𝑝𝑣−𝐴(𝑣)𝐵(𝑣)−^𝑟_𝑗=1ℎ_𝑗(𝜔(𝑡×𝐶))_𝑗𝑓(𝑣)+𝐷_𝑗𝑔×(𝑣((𝑡),𝑣()(),()∈(]𝑣()𝑡−𝜏,𝑥))𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−∞,0×Ω,𝑡,𝑥=0∈𝑅^𝑛,(𝑡,𝑥)∈𝑅×𝜕Ω, (2) where 𝜔(𝑡)=[𝜔₁(𝑡),𝜔₂(𝑡),…,𝜔_𝑠(𝑡)], ℎ_𝑗(𝜔(𝑡))=(𝑤_𝑗(𝜔(𝑡)))/(∑^𝑟_𝑘=1𝑤_𝑘(𝜔(𝑡))), 𝑤_𝑗(𝜔(𝑡))∶𝑅^𝑠→[0,1](𝑗=1,2,…,𝑟) is the membership function of the system with respect to the fuzzy rule 𝑗. ℎ_𝑗 can be regarded as the normalized weight of each IF-THEN rule, satisfying ℎ_𝑗(𝜔(𝑡))⩾0 and ∑^𝑟_𝑗=1ℎ_𝑗(𝜔(𝑡))=1.

For convenience’s sake, we need to introduce some standard notations.(i)Denote |𝐶|=(|𝑐_𝑖𝑗|)_𝑛×𝑛 for any matrix 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛.(ii)Denote |𝑣(𝑡,𝑥)|=(|𝑣₁(𝑡,𝑥)|,|𝑣₂(𝑡,𝑥)|,…,|𝑣_𝑛(𝑡,𝑥)|)^𝑇 for any 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥),𝑣₂(𝑡,𝑥),…,𝑣_𝑛(𝑡,𝑥))^𝑇.(iii)𝑢⩾𝑣 if 𝑢_𝑖⩾𝑣_𝑖 for all 𝑖=1,2,…,𝑛, where 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇, 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇∈𝑅^𝑛.

In addition, we also introduce the following standard notations similarly as [2, (iii)–(X)]: 𝑄=𝑞_{𝑖𝑗𝑛×𝑛}>0(<0),𝑄=𝑞_{𝑖𝑗𝑛×𝑛}(),𝑄⩾0⩽0₁⩾𝑄₂𝑄₁⩽𝑄₂,𝑄₁>𝑄₂𝑄₁<𝑄₂,andtheidentitymatrix𝐼. (3)

Throughout this paper, we assume the following.(H1)There exist positive definite matrices 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) and 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) such that 0<𝑎_𝑖⩽𝑎_𝑖(𝑟)⩽𝑎_𝑖,𝑟∈𝑅,𝑖=1,2,…,𝑛. (4)(H2)There exists a positive definite matrix 𝐵=diag(𝑏₁,𝑏₂,…,𝑏_𝑛) such that 𝑏_𝑖(0)=0 and 𝑏_𝑖(𝑟)𝑟^𝑝−1⩾𝑏_𝑖,0≠𝑟∈𝑅,𝑖=1,2,…,𝑛. (5)(H3)There exist positive definite matrices 𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛) and 𝐺=diag(𝐺₁,𝐺₂,…,𝐺_𝑛) such that 𝑓_𝑖(𝑟)⩽𝐹_𝑖|𝑟|^𝑝−1,𝑔_𝑖(𝑟)⩽𝐺_𝑖|𝑟|^𝑝−1,𝑟∈𝑅,𝑖=1,2,…,𝑛. (6)

From (H1)–(H3), we know that 𝑓(0)=𝑔(0)=0∈𝑅^𝑛, and 𝑣=0 is an equilibrium of (2).

Lemma 1. For any 𝑎,𝑏∈(0,+∞), One has 𝑎^𝑝−1𝑏⩽𝑝−1𝑝𝑎^𝑝+𝑏^𝑝𝑝. (7)

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 space 𝑊₀^1,𝑝(Ω), which is the completion of 𝐶^∞₀(Ω) with respect to the norm ‖𝜂‖=(∫_Ω|∇𝜂(𝑥)|^𝑝𝑑𝑥)^1/𝑝. Denote by 𝜆₁ the first eigenvalue of −Δ_𝑝 in Sobolev space 𝑊₀^1,𝑝(Ω), where 𝜆₁=min{∫_Ω|∇𝜂(𝑥)|^𝑝𝑑𝑥∶𝜂(𝑥)∈𝑊₀^1,𝑝(Ω),∫_Ω|𝜂(𝑥)|^𝑝𝑑𝑥=1} (see [22–24] for details).

Lemma 2. Let 𝑄=diag(𝑞₁,𝑞₂,…,𝑞_𝑛) be a positive definite matrix, and let 𝑣 be a solution of the fuzzy system (2). Then One has _Ω𝑣^𝑇𝑄∇⋅Γ(𝑡,𝑥,𝑣)∘∇_𝑝𝑣𝑑𝑥⩽−𝜆₁𝑞Γ‖𝑣‖^𝑝_𝑝, (8) where Γ=min_𝑖,𝑘(inf_{𝑡,𝑥,𝑣}Γ_𝑖𝑘(𝑡,𝑥,𝑣)), ‖𝑣‖^𝑝_𝑝=∑^𝑛_𝑖=1∫_Ω|𝑣_𝑖|^𝑝𝑑𝑥, and 𝑞 is a positive scalar, satisfying 𝑄>𝑞𝐼.

Proof. Since 𝑣 is a solution of system (2), it follows by Gauss formula and the Dirichlet boundary condition that _Ω𝑣^𝑇𝑄∇⋅Γ(𝑡,𝑥,𝑣)∘∇_𝑝𝑣=𝑑𝑥_Ω𝑣^𝑇𝑄^𝑚_𝑘=1𝜕𝜕𝑥_𝑘Γ_1𝑘∇𝑣₁^𝑝−2𝜕𝑣₁𝜕𝑥_𝑘,…,_𝑚𝑘=1𝜕𝜕𝑥_𝑘Γ_𝑛𝑘∇𝑣_𝑛^𝑝−2𝜕𝑣_𝑛𝜕𝑥_𝑘^𝑇𝑑𝑥=−^𝑚𝑛_{𝑘=1𝑗=1Ω}𝑞_𝑗Γ_𝑗𝑘∇𝑣_𝑗^𝑝−2𝜕𝑣_𝑗𝜕𝑥_𝑘²𝑑𝑥⩽−𝜆₁𝑞Γ‖𝑣‖^𝑝_𝑝. (9)

Remark 3. Lemma 2 actually generalizes the conclusion of [7, Lemma 2.1] and [25, Lemma 2.4] from Hilbert space 𝐻¹₀(Ω) to Banach space 𝑊₀^1,𝑝(Ω). Particularly in the case of Ω=(0,𝑇)⊂𝑅¹ or 𝑊₀^1,𝑝(0,𝑇), the first eigenvalue 𝜆₁=((2/𝑇)∫^{(𝑝−1)1/𝑝}₀(𝑑𝑡/(1−(𝑡^𝑝/(𝑝−1)))^1/𝑝))^𝑝 (see, e.g., [22]).

Theorem 4. Suppose that 𝑝=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,…,𝑞_𝑛) and two positive scalars 𝑞, 𝑞 such that the following inequalities hold: 𝜆₁Γ+𝜆_min𝐴𝐵𝑞>𝑛𝜆_max𝐴𝜆_max𝐹×^𝑟_𝑗=1𝑐_𝑗+𝑛𝑝+𝑛𝑝−1()𝑝×𝜆1−𝜏_max𝐴𝜆_max𝐺^𝑟_𝑗=1𝑑_𝑗𝑞𝑞,_𝑖>𝑞𝑞,𝑖∈𝒩,_𝑖<𝑞,𝑖∈𝒩, (10)(11)(12) then the null solution of (2) is globally asymptotically stable, where 𝒩={1,2,…,𝑛}, 𝜏^′(𝑡)⩽𝜏<1, |𝑐_𝑗|=max_𝑖,𝑘|𝑐^(𝑗)_𝑖𝑘| for the matrix 𝐶_𝑗=(𝑐^(𝑗)_𝑖𝑘)_𝑛×𝑛, and |𝑑_𝑗|=max_𝑖,𝑘|𝑑^(𝑗)_𝑖𝑘| for the matrix 𝐷_𝑗=(𝑑^(𝑗)_𝑖𝑘)_𝑛×𝑛.

Proof. Define the Lyapunov-Krasovskii functional as follows: 𝑉(𝑡)=𝑉₁(𝑡)+1𝑉1−𝜏²(𝑡), (13) where 𝑉₁(𝑡)=_Ω𝑣^𝑇𝑉𝑄𝑣𝑑𝑥,₂(𝑡)=2𝑛𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝×^𝑟𝑛_𝑗=1𝑘=1𝑑_𝑗^𝑡_{(𝑡)𝑡−𝜏Ω}𝑣_𝑘(𝑠,𝑥)^𝑝𝑑𝑠𝑑𝑥. (14)
Evaluating the time derivation of 𝑉₁(𝑡) along the trajectory of the fuzzy system (2), we can get by Lemma 2𝑉^′₁(𝑡)⩽−2𝜆₁𝑞Γ‖𝑣‖^𝑝_𝑝−2_Ω𝑣^𝑇𝑄𝐴(𝑣)𝐵(𝑣)𝑑𝑥+2^𝑟_𝑗=1ℎ_𝑗(𝜔(𝑡))_Ω𝑣^𝑇𝑄𝐴(𝑣)𝐶_𝑗𝑓(𝑣)+𝐴(𝑣)𝐷_𝑗𝑔𝑣_𝜏𝑑𝑥⩽−2𝜆₁𝑞Γ‖𝑣‖^𝑝_𝑝−2_Ω𝑣^𝑇𝑄𝐴(𝑣)𝐵(𝑣)𝑑𝑥+2^𝑟_𝑗=1Ω𝑣^𝑇𝑄𝐴𝐶_𝑗𝑓(𝑣)+𝐴𝐷_𝑗𝑔𝑣_𝜏𝑑𝑥, (15) where we denote 𝑣_𝜏=𝑣(𝑡−𝜏(𝑡),𝑥) for convenience.
Then we can get by (H1), (H2) and the restrictive conditions on the parameter 𝑝_Ω𝑣^𝑇𝑄𝐴(𝑣)𝐵(𝑣)𝑑𝑥⩾𝑞_Ω𝑣^𝑇𝐴𝐵𝑣₁^𝑝−1,𝑣₂^𝑝−1,…,𝑣_𝑛^𝑝−1𝑇𝑑𝑥⩾𝑞𝜆_min𝐴𝐵‖𝑣‖^𝑝_𝑝. (16) It follows by (H3) and Lemma 1 that _Ω𝑣^𝑇𝑄𝐴𝐶_𝑗𝑓(𝑣⩽)𝑑𝑥_Ω𝑣^𝑇𝑄𝐴𝐶_𝑗𝐹𝑣₁^𝑝−1,𝑣₂^𝑝−1,…,𝑣_𝑛^𝑝−1𝑇=𝑑𝑥𝑞^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑣_𝑖𝑎_𝑖𝑐^(𝑗)_𝑖𝑘𝐹_𝑘𝑣_𝑘^𝑝−1⩽𝑑𝑥𝑞^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑎_𝑖𝑐^(𝑗)_𝑖𝑘𝐹_𝑘𝑝−1𝑝𝑣_𝑘^𝑝+𝑣_𝑖^𝑝𝑝=𝑑𝑥𝑞𝑝−1𝑝^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑎_𝑖𝑐^(𝑗)_𝑖𝑘𝐹_𝑘𝑣_𝑘^𝑝+𝑑𝑥𝑞1𝑝^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑎_𝑖𝑐^(𝑗)_𝑖𝑘𝐹_𝑘𝑣_𝑖^𝑝⩽𝑑𝑥𝑞𝑐_𝑗𝜆_max𝐴𝜆_max𝐹×𝑝−1𝑝^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑣_𝑘^𝑝1𝑑𝑥+𝑝^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑣_𝑖^𝑝𝑑𝑥=𝑛𝑞𝑐_𝑗𝜆_max𝐴𝜆_max𝐹×𝑝−1𝑝^𝑛_𝑘=1Ω𝑣_𝑘^𝑝1𝑑𝑥+𝑝^𝑛_𝑖=1Ω𝑣_𝑖^𝑝𝑑𝑥=𝑛𝑞𝑐_𝑗𝜆_max𝐴𝜆_max𝐹‖𝑣‖^𝑝_𝑝. (17)
Similarly, _Ω𝑣^𝑇𝑄𝐴𝐷_𝑗𝑔𝑣_𝜏⩽𝑑𝑥𝑞^𝑛𝑛_{𝑘=1𝑖=1Ω}𝑎_𝑖𝑑^(𝑗)_𝑖𝑘𝐺_𝑘×𝑝−1𝑝𝑣_𝑘(𝑡−𝜏(𝑡),𝑥)^𝑝+𝑣_𝑖(𝑡,𝑥)^𝑝𝑝𝑑𝑥⩽𝑛𝑞𝑑_𝑗𝜆_max𝐴𝜆_max𝐺×𝑝−1𝑝^𝑛_𝑘=1Ω𝑣_𝑘(𝑡−𝜏(𝑡),𝑥)^𝑝1𝑑𝑥+𝑝‖𝑣‖^𝑝_𝑝. (18)
Combining (15)–(18) results in 𝑉^′₁(𝑡)⩽−2𝜆₁Γ𝑞+𝑞𝜆_min𝐴𝐵−𝑛𝑞𝜆_max𝐴𝜆_max𝐹×^𝑟_𝑗=1𝑐_𝑗−1𝑝𝑛𝑞𝜆_max𝐴𝜆_max𝐺^𝑟_𝑗=1𝑑_𝑗×‖𝑣‖^𝑝_𝑝+2𝑛𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝×^𝑟𝑛_𝑗=1𝑘=1𝑑_𝑗Ω𝑣_𝑘(𝑡−𝜏(𝑡),𝑥)^𝑝𝑑𝑥. (19)
On the other hand, 𝑉^′₂(𝑡)=2𝑛𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝‖𝑣‖^𝑝_𝑝^𝑟_𝑗=1𝑑_𝑗−1−𝜏^′(𝑡)2𝑛𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝×^𝑟𝑛_𝑗=1𝑘=1𝑑_𝑗Ω𝑣_𝑘(𝑡−𝜏(𝑡),𝑥)^𝑝𝑑𝑥⩽2𝑛𝑞𝑑_𝑗𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝‖𝑣‖^𝑝_𝑝^𝑟_𝑗=1𝑑_𝑗−(1−𝜏)2𝑛𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝×^𝑟𝑛_𝑗=1𝑘=1𝑑_𝑗Ω𝑣_𝑘(𝑡−𝜏(𝑡),𝑥)^𝑝𝑑𝑥. (20)
So we conclude from (10) that 𝑉^′(𝑡)⩽−2𝜆₁𝑞Γ+𝑞𝜆_min𝐴𝐵−𝑛𝑞𝜆_max𝐴𝜆_max𝐹×^𝑟_𝑗=1𝑐_𝑗−1𝑝𝑛𝑞𝜆_max𝐴𝜆_max𝐺^𝑟_𝑗=1𝑑_𝑗−1𝑛1−𝜏𝑞𝜆_max𝐴𝜆_max𝐺𝑝−1𝑝^𝑟_𝑗=1𝑑_𝑗×‖𝑣‖^𝑝_𝑝⩽0. (21) 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 𝑊^1,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 when 𝑝=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 the 𝑝-Laplace fuzzy T-S dynamic equations as follows.
Fuzzy Rule  1.
IF  𝜔₁(𝑡) is 1/𝑒^{−2𝜔1(𝑡)},  THEN𝜕𝑣Γ()𝜕𝑡=∇⋅𝑡,𝑥,𝑣∘∇_𝑝𝑣−𝐴(𝑣)𝐵(𝑣)−𝐶₁𝑓(𝑣)+𝐷₁𝑔(𝑣((𝑡),𝑣()(),()∈(]𝑣()𝑡−𝜏,𝑥))𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−∞,0×Ω,𝑡,𝑥=0∈𝑅²,(𝑡,𝑥)∈𝑅×𝜕Ω. (22)
Fuzzy Rule  2.
IF  𝜔₂(𝑡) is 1−(1/𝑒^{−2𝜔1(𝑡)}),  THEN𝜕𝑣Γ()𝜕𝑡=∇⋅𝑡,𝑥,𝑣∘∇_𝑝𝑣(𝑣)×𝐵(𝑣)−𝐶−𝐴₂𝑓(𝑣)+𝐷₂𝑔(𝑣((𝑡),𝑣()(),()∈(]𝑣()𝑡−𝜏,𝑥))𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−∞,0×Ω,𝑡,𝑥=0∈𝑅²,(𝑡,𝑥)∈𝑅×𝜕Ω, (23) 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)∫^{(𝑝−1)1/𝑝}₀(𝑑𝑡/(1−(𝑡^𝑝/(𝑝−1)))^1/𝑝))^𝑝=0.7915 (see Remark 3). The initial value function is presented as follows: 𝜙(𝑠,𝑥)=0.251−cos5𝜋𝑥²sin³⁹⁸𝑥²−0.25𝑒^−100𝑠0.2cos²4𝜋𝑥²sin⁵⁰⁸𝑥²−0.55𝑒^−100𝑠. (24)
Let 𝜏(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)=(2𝑣₁^1/3,1.9𝑣₂^1/3)^𝑇, and 𝑓(𝑣)=𝑔(𝑣)=(0.1𝑣₁^1/3,0.2𝑣₂^1/3)^𝑇, and correspondingly we assume 𝐴=,0.7000.6𝐴=,0.9000.88𝐵=2001.9,𝐹=0.1000.2=𝐺. (25) In addition, 𝑟=2, and 𝐶₁=−0.20.30.280.3=𝐷₁,𝐶₂=0.290.180.2−0.1=𝐷₂,Γ(𝑡,𝑥,𝑣)=.0.0080.0070.0060.006 (26) Hence, Γ=0.006, |𝑐₁|=|𝑑₁|=0.3, |𝑐₂|=|𝑑₂|=0.29, 𝜆_min(𝐴𝐵)=1.1400, 𝜆_max𝐴=0.9, and 𝜆_max𝐹=𝜆_max𝐺=0.2. Then we can use MATLAB LMI toolbox to solve the inequalities (10)–(12), and obtain 𝑡_min=−0.5569<0 which implies feasible. Further, extracting the datum shows 𝑞=19.7994, 𝑞=31.1620, 𝑞_𝑖=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 variable 𝑣1(𝑡,𝑥).

Figure 2 

The state variable 𝑣2(𝑡,𝑥).

Example 2. To compare our Theorem 4 with [2, Corollary 15], we may consider the following 𝑝-Laplace dynamic equations. 𝜕𝑣Γ()𝜕𝑡=∇⋅𝑡,𝑥,𝑣∘∇₃𝑣(𝑣𝐵(𝑣)−(𝑣)(𝑣(,𝑣()(),()∈(]𝑣()−𝐴)𝐶𝑓+𝐷𝑔𝑡−30,𝑥))𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−∞,0×Ω,𝑡,𝑥=0∈𝑅²,(𝑡,𝑥)∈𝑅×𝜕Ω, (27) 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 𝐴=,0.7000.6,()=.𝐴=0.9000.88𝐵=0.2000.1,𝐹=0.1000.2=𝐺,𝐶=−0.20.30.30.3=𝐷,Γ𝑡,𝑥,𝑣0.080.070.060.06 (28)

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 𝑡_min=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 𝑡_min=2.1479×10⁻²⁰>0 and 𝑡_min=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 𝑡_min=−0.0794<0, which implies being feasible. Further, extracting the datum results in that 𝑞=0.9803, 𝑞=1.8381, and 𝑞_𝑖=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.

Remark 7. Since we consider the role of the nonlinear 𝑝-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 nonlinear 𝑝-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 𝑊₀^1,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 𝑊^1,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.

Acknowledgments

The authors would like to thank the referees for their valuable suggestions. This work was supported by the National Basic Research Program of China (2010CB732501), by the Scientific Research Fund of Science Technology Department of Sichuan Province (2010JY0057, 2012JYZ010), and by the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349, 08ZB002).