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.
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 [
Let us consider the following delayed differential equations:
Constant matrix
Below, we describe the T-S fuzzy mathematical model with time delay as follows.
Fuzzy Rule
IF
For convenience's sake, we introduce the following standard notations similarly as [
Throughout this paper, we assume
Let
From Lemma
In many previous literature,
Similarly as is [
Dynamic equation (
Before giving the main result of this paper, we need to define the matrix exponential function as follows.
For a diagonal constants matrix
From the above definition of the matrix exponential function, we are not difficult to obtain the following lemma.
Let
where
In addition, we need to define the rule on vectors in
Now, we present the main result of this paper as follows.
Assume that there exists a positive constant
To apply the fixed point theory, we firstly define a complete metric space
Let
It is not difficult to verify that the above-mentioned space
Next, we formulate and define a contraction mapping
Let
Then, for
From (
On the other hand, it follows from (
Let
Indeed, since
Next, we verify the condition (b).
Indeed, for any given
Obviously,
Finally, it is followed from (
On the other hand, it follows from
Besides,
To prove
Indeed, we may as well assume
Below, we only need to prove
In fact, it follows from
On the one hand,
On the other hand,
Now we can conclude from (
Indeed, we can similarly define the corresponding constant
On the other hand,
From (
Indeed, for any
From mathematical analysis and computation, we can derive
Similarly, we have
Combining the above three inequalities results in
Therefore,
Consider the T-S fuzzy impulsive dynamic equations as follows.
Fuzzy Rule 1:
IF
Fuzzy Rule 2:
IF
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., [
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).