Abstract and Applied Analysis

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

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

Research Article

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

Ruofeng Rao¹ and Zhilin Pu1,2

¹Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China

²College 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).

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