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In this paper, Banach fixed point theorem is employed to derive LMI-based exponential stability of impulsive Takagi-Sugeno (T-S) fuzzy integrodifferential equations, originated from Cohen-Grossberg Neural Networks (CGNNs). As far as we know, Banach fixed point theorem is rarely employed to derive LMI criteria for T-S fuzzy CGNNs, and this inspires our present work. It is worth mentioning that the conditions on the behavior functions are weaker than those of existing results, and the formulated contraction mapping and fixed point technique are different from those of previous literature. Even a corollary of our main result improves one of existing main results due to extending linear function to nonlinear function. Besides, the LMI-based criteria are programmable for computer MATLAB LMI toolbox. Moreover, an analytical table and a numerical example are presented to illustrate the advantage, feasibility, and effectiveness of the proposed methods.

In this paper, we consider a class of integrodifferential equations, which is originated from Cohen-Grossberg Neural Networks (CGNNs). CGNNs model was proposed originally by Cohen-Grossberg in 1983 [

If (H1)–(H4) are satisfied and there exists a positive constant

For any

For any

There exist nonnegative constants

For

Denoting

All the good results and methods in existing literature, particularly in [

Due to the weaker condition on the behavior functions, both our results and methods are novelty (see below “Remark

Comparing our Theorem

Fixed point theorems | Continuity of | Differentiability of | Programmabilit | |
---|---|---|---|---|

Our Theorem | Contraction mapping theorem | Unnecessary | Unnecessary | Yes, LMI-based |

[ | Contraction mapping theorem | Yes, necessary | [ | No |

[ | Contraction mapping theorem | Yes, necessary | [ | No |

[ | Contraction mapping theorem | Yes, necessary | Yes, necessary | No |

[ | Brouwer fixed point theorem | Yes, necessary | [ | No |

[ | Contraction mapping theorem | Yes, necessary | Yes, necessary | No |

Conditions [

For convenience’s sake, we introduce the following standard notations.

Denote

Cohen-Grossberg Neural Networks (CGNNs) with discrete and distributed delays have been investigated in many papers [

Since practice has shown that fuzzy logic theory is an efficient approach to deal with the analysis and synthesis problems for complex nonlinear system, the fuzzy model is far more important than stochastic model [

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

Throughout this paper, we assume that

There is positive definite diagonal matrix

For each

There are a large number of functions

For example, we denote

Hence,

Let

Since (H4) is replaced with (A3), the methods of [

Impulsive fuzzy CGNNs (

Letting

If (A1)–(A3) hold, we can derive the following main result.

Impulsive fuzzy CGNNs (

First of all, we need to formulate integral equations equivalent to (

Denote, for convenience,

Next, we claim that System (

On one hand, we can prove that the solution of System (

Indeed, we get by (

Differentiating both sides of (

Further, let

Thus, we have proved the above claim.

On the other hand, we claim that the solution of System (

In fact, multiplying both sides of the first equation of System (

Throughout this paper, we assume that

To apply the contraction mapping theorem, we firstly define the complete metric space

Let

It is not difficult to verify that the product space

Hence, we define the mapping

Below, we are to prove that

We may firstly prove

Indeed, for

Throughout this section, we assume that

Below, we need to prove that

Indeed, for

Similarly, we have

Besides,

Next,

Obviously,

Below, we assume that

Synthesizing the above analysis results in

Finally, we claim that

Indeed, for any

Combining the above five inequalities and (

From the proof, we know that the formulated contraction mapping is different from that of existing literature involved in the fixed point technique and CGNNs models. One can also understand from Remark

In case of ignoring fuzzy factors and distributed delay (letting

Impulsive CGNNs (

As a corollary of our main result, the conditions and conclusion are better than the main result of [

Consider the T-S fuzzy impulsive discrete and distributed delays CGNNs model as follows.

From Example

Table

From Table

In many existing literatures related to Cohen-Grossberg Neural Networks, various fixed point theorems were applied to derive the stability criteria by imitating System (

The authors declare that they have no competing interests.

All authors typed, read, and approved the final manuscript.

This work was supported by National Basic Research Program of China (2010CB732501), Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), Sichuan Educational Committee Science Foundation (08ZB002, 12ZB349, and 14ZA0274), and the Initial Founding of Scientific Research for Chengdu Normal University Introduction of Talents.