^{1, 2}

^{1}

^{1}

^{2}

We investigate local robust stability of fuzzy neural networks (FNNs) with time-varying and S-type distributed delays. We derive some sufficient conditions for local robust stability of equilibrium points and estimate attracting domains of equilibrium points except unstable equilibrium points. Our results not only show local robust stability of equilibrium points but also allow much broader application for fuzzy neural network with or without delays. An example is given to illustrate the effectiveness of our results.

For the study of current neural network, two basic mathematical models are commonly adopted: either local field neural network models or static neural network models. The basic model of local field neural network is described as

It is well known that local field neural network not only models Hopfield-type networks [

However, in mathematical modeling of real world problems, we will encounter some other inconvenience, for example, the complexity and the uncertainty or vagueness. Fuzzy theory is considered as a more suitable setting for the sake of taking vagueness into consideration. Based on traditional cellular neural networks (CNNs), Yang and Yang proposed the fuzzy CNNs (FCNNs) [

Therefore, in this paper, we will study the local robust stability of fuzzy neural network with time-varying and S-type distributed delays:

Throughout this paper, we always assume the following

The activation functions

Functions

The rest of this paper is organized as follows. In Section

As usual, we denote by

A vector

Let

Let

For any given

For any given

For a class of differential equation with the term of fuzzy AND and fuzzy OR operation, there is the following useful inequality.

Let

Let

By (

By Lemma

All equilibrium points of FNNs (

In this section, we should investigate local robust stability of equilibrium points of FNNs (

Let

Let

The open set

The proof of Theorem

Let

If

If

If

Under transformation

Since

Since

Now we are in a position to complete the proof of Theorem

Let

Let

Let

The open set

For convenience of illustrative purpose, we only consider simple fuzzy neural network with time-varying and S-type distributed delays satisfying

It is easy to check that _{1})_{5})

From simple calculations, we know that

The above example implies that the system has multiple equilibrium points under the (relevant) assumption of monotone nondecreasing activation functions. These equilibrium points do not globally converge to the unique equilibrium point.

In this paper, we derive some sufficient conditions for local robust stability of fuzzy neural network with time-varying and S-type distributed delays and give an estimate of attracting domains of stable equilibrium points except isolated equilibrium points. Our results not only show local robust stability of equilibrium points but also allow much broader application for fuzzy neural network with or without delays. An example is given to show the effectiveness of our results.

This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.