- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 261353, 9 pages

http://dx.doi.org/10.1155/2013/261353

## LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

^{1}Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China^{2}College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China

Received 27 June 2013; Accepted 6 August 2013

Academic Editor: Chuanzhi Bai

Copyright © 2013 Ruofeng Rao and Zhilin Pu. 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.

#### Abstract

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.

#### 1. Introduction

In this paper, we consider a class of delayed impulsive differential equations, which admits some biomathematics, physics, and engineering backgrounds, including the famous cellular neural networks proposed by Chua and Yang in 1988 [1, 2]. In practice, both time delays and impulse are unavoidable and may cause undesirable dynamic network behaviors such as oscillation and instability. So the stability analysis for delayed impulsive neural networks has become a topic of great theoretic and practical importance in recent years [3–25]. However, the research skill of the above literature is mainly based on Lyapunov theory. And there are many difficulties in applications of corresponding theory to the specific problems [26–32]. Recently, Burton and other authors have applied fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results [11, 26–42]. For example, in [11], the authors used Leray-Schauders fixed point theorem to obtain the stability criteria of neural networks. Besides, the contraction-mapping theory is also an important fixed point theory in studying the stability of dynamics equations (see, e.g., [11, 33, 39–42]). On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear system [20–25]. In practice, the fuzzy model is far more important than stochastic model. 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. To the best of our knowledge, few authors have used the fixed point theorem to study the stability of Takagi-Sugeno fuzzy differential equations with impulses. In addition, the LMI-based stability criterion of neural networks has never been investigated or obtained via any of the fixed point theories. Such a situation motivates our present study. Motivated by the above related literature [3–9, 11, 26–42], we will not only apply the fixed point theory to study the impulsive Takagi-Sugeno fuzzy dynamics equations but also try to obtain the LMI-based stability criterion by applying the contraction-mapping theory. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. In the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods. Finally, a conclusion is given in the final chapter.

#### 2. Preliminaries

Let us consider the following delayed differential equations: equipped with the impulsive condition and the initial condition where . Functions , , , and time delays for all . The fixed impulsive moments satisfy with . , and stand for the right-hand and left-hand limits of at time , respectively. We always assume , for all . Similarly as in [33], we assume in this paper that .

Constant matrix is a positive definite diagonal matrix, and both and are matrices with dimension. It is well gknown that the above equation admits its practical implications. For example, it can serve as a model of impulsive cellular neural networks with time-varying delays. The parameter denotes the connection weight of the th neuron on the th neuron at time . And the parameter represents the connection strength of the th neuron on the th neuron at time . The constant represents the rate with which the th neuron will reset its potential to the resting state when disconnected from the network and external inputs. is the activation function of the th neuron at time , and represents the activation function of the th neuron at time .

Below, we describe the T-S fuzzy mathematical model with time delay as follows.

Fuzzy Rule :

IF is and is , THEN where is the premise variable, is the fuzzy set that is characterized by membership function, is the number of the IF-THEN rules, and is the number of the premise variables. By the way of a standard fuzzy inference method, the system (4) is 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 introduce the following standard notations similarly as [10, (iii)–(X)]: In addition, we denote for any matrix and for any . Denote the finite set .

Throughout this paper, we assume(A1) is locally Lipschitz continuous, and there exists a positive constant such that for all at which is differentiable;(A2) is locally Lipschitz continuous, and there exists a positive constant such that for all at which is differentiable;(A3) is locally Lipschitz continuous, and there exists a positive constant such that for all at which is differentiable.

Lemma 1 (see [25]). *Let be locally Lipschitz continuous. For any given , there exists an element in the union such that
**
where denotes the segment connecting and .*

*Remark 2. *From Lemma 1, (A1)–(A3), and [10, equation (27)], we can similarly derive
where matrices , , and .

*Remark 3. *In many previous literature, are always assumed to be globally Lipschtiz continuous. However, the most common function is not globally Lipschtiz continuous in . Note that we extend the functions from global Lipschtiz continuous functions to locally Lipschtiz continuous functions. Obviously, is a local Lipschitz continuous function.

Similarly as is [11, Definition 2.2], the exponential stability is defined as follows.

*Definition 4. *Dynamic equation (5) is said to be exponentially stable if, for any initial condition , there exists a pair of positive constants and such that
where the norm

#### 3. Main Result

Before giving the main result of this paper, we need to define the matrix exponential function as follows.

*Definition 5. *For a diagonal constants matrix , we denote the matrix exponential function for all .

From the above definition of the matrix exponential function, we are not difficult to obtain the following lemma.

Lemma 6. *Let be a diagonal constants matrix, and let be the matrix exponential function of . Then, we have*(1)*,*(2)*,**where , and each is a constant.*

In addition, we need to define the rule on vectors in as follows.

*Definition 7. * if for all , where .

Now, we present the main result of this paper as follows.

Theorem 8. *Assume that there exists a positive constant such that . In addition, there exists a constant such that
**
where is the inverse matrix of . Then, the impulsive fuzzy dynamic equation (5) is exponentially stable in the mean square.*

*Proof. *To apply the fixed point theory, we firstly define a complete metric space as follows.

Let be the space consisting of functions , satisfying that(a) is continuous on ,(b) and exist, and for all ,(c) for ,(d) as , where is a positive constant, satisfying .It is not difficult to verify that the above-mentioned space is a complete metric space if it is equipped with the following metric:
where , and .*Remark 9*. Here, we consider the above-defined metric, which is different from those of some previous related literature (see, e.g., [33]) so that the LMI-based stability criterion in this paper may be obtained expediently.

Next, we formulate and define a contraction mapping , which may be divided into three steps.*Step 1*. Formulating the mapping.

Let be a solution of the fuzzy equation (5).

Then, for , we have
Further, we get by the integral nature
where is the vector to be determined.

From (2) and (3) or , it is not difficult to conclude . And, hence,

On the other hand, it follows from (14) that

Let ; then we can derive from the two equations above that
So, we may define the mapping on the space as follows:
on , and on .*Step 2*. We claim that for any . That is, satisfies the conditions (a)–(d) of .

Indeed, since on , the condition (c) is satisfied. It is obvious from (17) that is continuous on and . And then the condition (a) is satisfied.

Next, we verify the condition (b).

Indeed, for any given , we can get from (17)
where

Obviously, and as . In addition, letting , we have . Letting , we get by (18)
So the condition (b) is satisfied.

Finally, it is followed from (17) that
Obviously, as .

On the other hand, it follows from that, for any given , there exists a corresponding constant such that for all , where
In addition, (A3) derives
Obviously,

Besides,

To prove , we only need to prove
for the other cases can be similarly proved.

Indeed, we may as well assume and . Then, we have
which implies that (26) holds. And, hence, .

Below, we only need to prove

In fact, it follows from that, for any given , there exists a corresponding constant such that for all . Then, we get by (A1)

On the one hand,
where .

On the other hand,
where .

Now we can conclude from (29)–(31) that
Similarly, we can prove

Indeed, we can similarly define the corresponding constant for any given , satisfying for all . Then, we have
Similarly, we can prove
where .

On the other hand,
where .

From (34)–(36), we can conclude that (33) holds. Hence, the condition (d) is satisfied.*Step 3*. Below, we only need to prove that is a contraction mapping.

Indeed, for any , we estimate , where

From mathematical analysis and computation, we can derive

Similarly, we have

Combining the above three inequalities results in
and hence

Therefore, is a contraction mapping such that there exists the fixed point of in , which implies that is the solution for the the impulsive fuzzy dynamic equation (5), satisfying as . So the proof is completed.

#### 4. Numerical Example

*Example 1. *Consider the T-S fuzzy impulsive dynamic equations as follows.

Fuzzy Rule 1:

IF is , THEN

Fuzzy Rule 2:

IF is , THEN
where
Let . Then we can use Matlab LMI toolbox to solve the LMI condition (10) and obtain which implies it is feasible (see [10, Remark 29(3)] for details). Further, extracting the datum shows
which means . Thereby, we can conclude from Theorem 8 that this impulsive fuzzy dynamic equation is exponentially stable in the mean square.

#### 5. Conclusion

By formulating a contraction mapping and the matrix exponential function, the author applies linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. It is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. The LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. A numerical example is presented to illustrate the effectiveness of the proposed methods. In the end of this paper, we have to point out that there are still many difficulties in obtaining the LMI-based stability criteria for some other dynamics equations, such as Cohen-Grossberg neural networks (see, e.g., [11, 19]) and other neural networks. These problems remain open and challenging.

#### Acknowledgments

This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2010JY0057, 2012JYZ010) and the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).

#### References

- L. O. Chua and L. Yang, “Cellular neural networks: theory,”
*IEEE Transactions on Circuits and Systems*, vol. 35, no. 10, pp. 1257–1272, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. O. Chua and L. Yang, “Cellular neural networks: applications,”
*IEEE Transactions on Circuits and Systems*, vol. 35, no. 10, pp. 1273–1290, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - D. Xu, X. Wang, and Z. Yang, “Further results on existence-uniqueness for stochastic functional differential equations,”
*Science China Mathematics*, vol. 56, no. 6, pp. 1169–1180, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - D. Xu and S. Long, “Attracting and quasi-invariant sets of non-autonomous neural networks with delays,”
*Neurocomputing*, vol. 77, no. 1, pp. 222–228, 2012. View at Publisher · View at Google Scholar · View at Scopus - D. Xu and L. Xu, “New results for studying a certain class of nonlinear delay differential systems,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 7, pp. 1641–1645, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - D. Xu and X. Wang, “A new nonlinear integro-differential inequality and its application,”
*Applied Mathematics Letters*, vol. 22, no. 11, pp. 1721–1726, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Xu and Z. Yang, “Attracting and invariant sets for a class of impulsive functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 2, pp. 1036–1044, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Xu, Z. Yang, and Y. Huang, “Existence-uniqueness and continuation theorems for stochastic functional differential equations,”
*Journal of Differential Equations*, vol. 245, no. 6, pp. 1681–1703, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Xu and H. Zhao, “Invariant set and attractivity of nonlinear differential equations with delays,”
*Applied Mathematics Letters*, vol. 15, no. 3, pp. 321–325, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Rao, X. Wang, S. Zhong, and Z. 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 Publisher · View at Google Scholar · View at MathSciNet - X. Yang, C. Huang, D. Zhang, and Y. Long, “Dynamics of Cohen-Grossberg neural networks with mixed delays and impulses,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 432341, 14 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Wan and J. Sun, “Global asymptotic stability of Cohen-Grossberg neural network with continuously distributed delays,”
*Physics Letters A*, vol. 342, no. 4, pp. 331–340, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. F. Rao, S. M. Zhong, and X. R. Wang, “Stochastic stability criteria with LMI conditions forMarkovian jumping impulsive BAM neural networks with mode-dependent time-varying delaysand nonlinear reaction-diffusion,”
*Communications in Nonlinear Science and Numerical Simulation*, 2013. View at Publisher · View at Google Scholar - Q. Song and J. Zhang, “Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 2, pp. 500–510, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Song and J. Cao, “Impulsive effects on stability of fuzzy Cohen-Grossberg neural networks with time-varying delays,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 37, no. 3, pp. 733–741, 2007. View at Publisher · View at Google Scholar · View at Scopus - Q. Song and Z. Wang, “Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays,”
*Physica A*, vol. 387, no. 13, pp. 3314–3326, 2008. View at Publisher · View at Google Scholar · View at Scopus - C. Bai, “Stability analysis of Cohen-Grossberg BAM neural networks with delays and impulses,”
*Chaos, Solitons & Fractals*, vol. 35, no. 2, pp. 263–267, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Li and D. Y. Xu, “Exponential
*p*-stability of stochastic recurrent neural networks with mixeddelays and Markovian switching,”*Neurocomputing*, vol. 103, pp. 239–246, 2013. View at Publisher · View at Google Scholar - B. Li and D. Xu, “Existence and exponential stability of periodic solution for impulsive Cohen-Grossberg neural networks with time-varying delays,”
*Applied Mathematics and Computation*, vol. 219, no. 5, pp. 2506–2520, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Song, Z. Wang, and J. 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 · View at MathSciNet - P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, “Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 4, pp. 2109–2129, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Muralisankar, N. Gopalakrishnan, and P. Balasubramaniam, “An LMI approach for global robust dissipativity analysis of T-S fuzzy neural networks with interval time-varying delays,”
*Expert Systems with Applications*, vol. 39, no. 3, pp. 3345–3355, 2012. View at Publisher · View at Google Scholar · View at Scopus - R. F. Rao and Z. L. 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 · View at Google Scholar - Z. L. Pu and R. F. Rao, “Stability analysis of Takagi-Sugeno fuzzy Markovian jumping PDEswith p-Laplace under zero-boundary condition,” to appear
*Journal of Applied Mathmatics*. View at Google Scholar - P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, “Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays,”
*Expert Systems with Applications*, vol. 39, no. 1, pp. 472–481, 2012. View at Publisher · View at Google Scholar · View at Scopus - J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 1, pp. 431–440, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Luo, “Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 342, no. 2, pp. 753–760, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Luo, “Stability of stochastic partial differential equations with infinite delays,”
*Journal of Computational and Applied Mathematics*, vol. 222, no. 2, pp. 364–371, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Luo and T. Taniguchi, “Fixed points and stability of stochastic neutral partial differential equations with infinite delays,”
*Stochastic Analysis and Applications*, vol. 27, no. 6, pp. 1163–1173, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Sakthivel and J. Luo, “Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,”
*Journal of Mathematical Analysis and Applications*, vol. 356, no. 1, pp. 1–6, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Sakthivel and J. Luo, “Asymptotic stability of nonlinear impulsive stochastic differential equations,”
*Statistics & Probability Letters*, vol. 79, no. 9, pp. 1219–1223, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Luo, “Fixed points and exponential stability for stochastic Volterra-Levin equations,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 934–940, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Lai and Y. Zhang, “Fixed point and asymptotic analysis of cellular neural networks,”
*Journal of Applied Mathematics*, Article ID 689845, 12 pages, 2012. View at Google Scholar · View at MathSciNet - T. A. Burton,
*Stability by Fixed Point Theory for Functional Differential Equations*, Dover, Mineola, NY, USA, 2006. View at MathSciNet - R. F. Rao, S. M. Zhong, and X. R. Wang, “Delay-dependent exponential stability for Markovianjumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partiallyknown transition rates via a differential inequality,”
*Advances in Difference Equations*, vol. 2013, article 183, 2013. View at Publisher · View at Google Scholar - T. A. Burton, “Fixed points, stability, and exact linearization,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 61, no. 5, pp. 857–870, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Jin and J. Luo, “Fixed points and stability in neutral differential equations with variable delays,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 3, pp. 909–918, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. N. Raffoul, “Stability in neutral nonlinear differential equations with functional delays using fixed-point theory,”
*Mathematical and Computer Modelling*, vol. 40, no. 7-8, pp. 691–700, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wu, N. j. Huang, and C. W. Zhao, “Fixed points and stability in neutral stochastic differential equations with variable delays,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 407352, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Zhang, “Fixed points and stability in differential equations with variable delays,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 63, no. 5–7, pp. e233–e242, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. T. Zhang and Q. Luo, “Global exponential stability of impulsive cellular neural networks withtime-varying delays via fixed point theory,”
*Advances in Difference Equations*, vol. 2013, article 23, 2013. View at Google Scholar - L. Ding, X. Li, and Z. Li, “Fixed points and stability in nonlinear equations with variable delays,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 195916, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet