Research Article | Open Access

Ruofeng Rao, Zhilin Pu, Shouming Zhong, Jialin Huang, "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", *Journal of Applied Mathematics*, vol. 2013, Article ID 940845, 8 pages, 2013. https://doi.org/10.1155/2013/940845

# 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

**Academic Editor:**Reinaldo Martinez Palhares

#### Abstract

By the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space , 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 -Laplace () 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 () 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 -Laplace diffusion () 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 -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 -Laplace partial differential equations with Dirichlet boundary value by the way of Lyapunov-Krasovskii functional approach and some variational methods in the Sobolev space . And in the obtained criterion, the nonlinear -Laplace diffusion item plays a positive role.

#### 2. Model Description and Preliminaries

Let us consider a class of fuzzy Takagi-Sugeno (T-S) -Laplace partial differential equations described as follows.

*Fuzzy Rule* .

**IF** is and is , **THEN**
where is an arbitrary open bounded subset in . is the premise variable, is the fuzzy set that is characterized by membership function. And is the number of the **IF-THEN** rules; 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 , where is the state variable of the th neuron and the th neuron at time and in space variable . Matrix with each , and is diffusion operator. denotes the Hadamard product of matrix and (see [1–6] for details). Matrix and vector , where and represent an amplification function at time and an appropriately behaved function at time . and are connection matrices. The time-varying delays are . and 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:
where , , is the membership function of the system with respect to the fuzzy rule . can be regarded as the normalized weight of each **IF-THEN** rule, satisfying and .

For convenience’s sake, we need to introduce some standard notations.(i)Denote for any matrix .(ii)Denote for any .(iii) if for all , where , .

In addition, we also introduce the following standard notations similarly as [2, (iii)–(X)]:

Throughout this paper, we assume the following.(H1)There exist positive definite matrices and such that (H2)There exists a positive definite matrix such that and (H3)There exist positive definite matrices and such that

From (H1)–(H3), we know that , and is an equilibrium of (2).

Lemma 1. *For any , One has
*

#### 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 , which is the completion of with respect to the norm . Denote by the first eigenvalue of in Sobolev space , where (see [22–24] for details).

Lemma 2. *Let be a positive definite matrix, and let be a solution of the fuzzy system (2). Then One has
**
where , , and is a positive scalar, satisfying .*

*Proof. *Since is a solution of system (2), it follows by Gauss formula and the Dirichlet boundary condition that

*Remark 3. *Lemma 2 actually generalizes the conclusion of [7, Lemma 2.1] and [25, Lemma 2.4] from Hilbert space to Banach space . Particularly in the case of or , the first eigenvalue (see, e.g., [22]).

Theorem 4. *Suppose that , where is an even number while is an odd number. If, in addition, there exist a positive definite matrix and two positive scalars , such that the following inequalities hold:
**
then the null solution of (2) is globally asymptotically stable, where , , for the matrix , and for the matrix .*

*Proof. *Define the Lyapunov-Krasovskii functional as follows:
where

Evaluating the time derivation of along the trajectory of the fuzzy system (2), we can get by Lemma 2
where we denote for convenience.

Then we can get by (H1), (H2) and the restrictive conditions on the parameter
It follows by (H3) and Lemma 1 that

Similarly,

Combining (15)–(18) results in

On the other hand,

So we conclude from (10) that
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 -Laplace () 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 -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 that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces. However, the nonlinear -Laplace (, ) brings great difficulties, for the nonlinear -Laplace PDEs should be considered in the frame of Sobolev space 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 -Laplace items play roles in the case of . However, in this paper, the nonlinear -Laplace diffusion terms play a positive role in our Theorem 4 for the case of or , 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 , 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 -Laplace fuzzy T-S dynamic equations as follows.*Fuzzy Rule 1*.**IF** is , **THEN***Fuzzy Rule 2*.**IF** is , **THEN**
where , , , and then the first eigenvalue (see Remark 3). The initial value function is presented as follows:

Let , and then . Besides, , , and , and correspondingly we assume
In addition, , and
Hence, , , , , , and . Then we can use MATLAB LMI toolbox to solve the inequalities (10)–(12), and obtain which implies feasible. Further, extracting the datum shows , , , . 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).

*Example 2. *To compare our Theorem 4 with [2, Corollary 15], we may consider the following -Laplace dynamic equations.
where , , , and then the first eigenvalue (see Remark 5). Let , and then . In addition, is the number of the **IF-THEN**. Assume

Below we shall employ [2, Corollary 15] and our Theorem 4 to judge the stability of System (27), respectively.

In [2, Corollary 15], letting , we use MATLAB LMI toolbox to solve the LMI conditions [2, ] and obtain , which implies that these LMI constraints are not strictly feasible. Again, let and in [2, Corollary 15], respectively. And then and , 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 , which implies being feasible. Further, extracting the datum results in that , , and , for . 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).

*Remark 7. *Since we consider the role of the nonlinear -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 -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 . The -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 -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 that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces. However, the nonlinear -Laplace (, ) brings great difficulties, for the nonlinear -Laplace PDEs should be considered in the frame of Sobolev space 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 -Laplace items play roles in the case of . Now in this paper, we present the stability criterion in which the nonlinear -Laplace items play roles in the case of or . Moreover, numerical example shows the effectiveness of the proposed methods. Since the non-linear -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 -Laplace () diffusion item plays a role in -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).

#### References

- Q. F. Pan, Z. F. Zhang, and J. C. Huang, “Stability of the stochastic reaction-diffusion neural network with time-varying delays and $p$-Laplacian,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 405939, 10 pages, 2012. View at: Google Scholar | MathSciNet - R. F. Rao, X. R. Wang, S. M. Zhong, and Z. L. Pu, “LMI approach to exponential stability and almost sure exponential stability for stochastic fuzzy Markovian-jumping Cohen-Grossberg neural networks with nonlinear $p$-Laplace diffusion,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 396903, 21 pages, 2013. View at: Google Scholar | MathSciNet - R. F. Rao and Z. Pu, “Stability analysis for impulsive stochastic fuzzy $p$-Laplace dynamic equations under Neumann or Dirichlet boundary condition,”
*Boundary Value Problems*, vol. 2013, article 133, 2013. View at: Publisher Site | Google Scholar | MathSciNet - R. F. Rao, S. M. Zhong, and X. R. Wang, “Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear reaction-diffusion,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 1, pp. 258–273, 2014. View at: Google Scholar - R. F. Rao, S. M. Zhong, and X. R. Wang, “Delay-dependent exponential stability for Marko-vian jumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partially known transition rates via a differential inequality,”
*Advances in Difference Equations*, vol. 2013, article 183, 2013. View at: Google Scholar - X. R. Wang, R. F. Rao, and S. M. Zhong, “LMI approach to stability analysis of Cohen-Grossberg neural networks with $p$-Laplace diffusion,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 523812, 12 pages, 2012. View at: Google Scholar | MathSciNet - X. Zhang, S. Wu, and K. Li, “Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1524–1532, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Liang and L. S. Wang, “Exponential stability for a class of stochastic reaction-diffusion Hopfield neural networks with delays,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 693163, 12 pages, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. T. Zhang, “Asymptotic stability of impulsive reaction-diffusion cellular neural networks with time-varying delays,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 501891, 17 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Salem, “Invariant regions and global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions,”
*Journal of Applied Mathematics*, vol. 2007, Article ID 12375, 15 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. J. Higham and T. Sardar, “Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay,”
*Applied Numerical Mathematics*, vol. 18, no. 1–3, pp. 155–173, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. K. Baranwal, R. K. Pandey, M. P. Tripathi, and O. P. Singh, “An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on a new iterative method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 10, pp. 3906–3921, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Meral and M. Tezer-Sezgin, “The comparison between the DRBEM and DQM solution of nonlinear reaction-diffusion equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 10, pp. 3990–4005, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,”
*Fuzzy Sets and Systems*, vol. 45, no. 2, pp. 135–156, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,”
*IEEE Transactions on Systems, Man, and Cybernetics*, vol. 15, pp. 116–132, 1985. View at: Google Scholar - R. F. Rao and Z. Pu, “LMI-based stability criterion of impulsive T-S fuzzy dynamic equations via fixed point theory,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 261353, 9 pages, 2013. View at: Publisher Site | Google Scholar - Q. K. Song and J. D. Cao, “Global dissipativity on uncertain discrete-time neural networks with time-varying delays,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 810408, 19 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. K. Song, Z. D. Wang, and J. L. Liang, “Analysis on passivity and passification of T-S fuzzy systems with time-varying delays,”
*Journal of Intelligent & Fuzzy Systems*, vol. 24, no. 1, pp. 21–30, 2013. View at: Google Scholar | MathSciNet - Q. K. Song, “Stochastic dissipativity analysis on discrete-time neural networks with time-varying delays,”
*Neurocomputing*, vol. 74, no. 5, pp. 838–845, 2011. View at: Google Scholar - Q. K. Song and Z. D. Wang, “New results on passivity analysis of uncertain neural networks with time-varying delays,”
*International Journal of Computer Mathematics*, vol. 87, no. 1–3, pp. 668–678, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. K. Song and J. Y. Zhang, “Novel results on mesh stability for a class of vehicle following system with time delays,” in
*Advances in Neural Networks—ISNN 2012*, vol. 7367 of*Lecture Notes in Computer Science*, pp. 336–342, 2012. View at: Google Scholar - R. F. Rao and X. R. Wang, “Infinitely many solutions for the resonant quasi-linear equation without Landesman-Lazer conditions,”
*Acta Mathematica Scientia A*, vol. 32, no. 4, pp. 744–752, 2012. View at: Google Scholar | MathSciNet - P. Drábek and R. Manásevich, “On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian,”
*Differential and Integral Equations*, vol. 12, no. 6, pp. 773–788, 1999. View at: Google Scholar | MathSciNet - P. Lindqvise, “On the equation div $({|\nabla u|}^{p-2}\nabla u)+\lambda {|u|}^{p-2}u=0$,”
*Proceedings of the American Mathematical Society*, vol. 109, pp. 159–164, 1990. View at: Google Scholar - J. Pan and S. M. Zhong, “Dynamic analysis of stochastic reaction-diffusion Cohen-Grossberg neural networks with delays,”
*Advances in Difference Equations*, vol. 2009, Article ID 410823, 18 pages, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - C. H. Zhou, H. Y. Zhang, H. B. Zhang, and C. Y. Dang, “Global exponential stability of impulsive fuzzy Cohen-Grossberg neural networks with mixed delays and reaction-diffusion terms,”
*Neurocomputing*, vol. 91, pp. 67–76, 2012. View at: Google Scholar - C. H. Wang, Y. G. Kao, and G. W. Yang, “Exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays,”
*Neurocomputing*, vol. 89, pp. 55–63, 2012. View at: Google Scholar - J. Pan and S. M. Zhong, “Dynamical behaviors of impulsive reaction-diffusion Cohen-Grossberg neural network with delays,”
*Neurocomputing*, vol. 73, pp. 1344–1351, 2010. View at: Google Scholar - J. Pan, X. Z. Liu, and S. M. Zhong, “Stability criteria for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays,”
*Mathematical and Computer Modelling*, vol. 51, no. 9-10, pp. 1037–1050, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Ruofeng Rao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.