#### Abstract

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.

#### 1. Introduction

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

where denotes the activation function of the th neuron; is the state of the th neuron; is the external input imposed on the th neuron; denotes the synaptic connectivity value between the th neuron and the th neuron; is the number of neurons in the network. With the same notations, static neural network models can be written as

It is well known that local field neural network not only models Hopfield-type networks [1] but also models bidirectional associative memory networks [2] and cellular neural networks [3]. Many deep theoretical results have been obtained for local field neural network; we can refer to [4–12] and references cited therein. Meanwhile static neural network has a great potential of applications. It not only includes the recurrent back-propagation network [13–15] but also includes other extensively studied neural network such as the optimization type network introduced in [16–18] and the brain-state-in-a-box (BSB) type network [19, 20]. In the past few years, there has been increasing interest in studying dynamical characteristics such as stability, persistence, periodicity, local robust stability of equilibrium points, and domains of attraction of local field neural network (see[21–25]).

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) [26], which integrates fuzzy logic into the structure of traditional CNNs and maintains local connectedness among cells. Unlike previous CNNs structures, FCNNs have fuzzy logic between its template input and/or output besides the sum of product operation. FCNNs are very useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. Therefor, it is necessary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks. Nevertheless, to the best of our knowledge, there are few published papers considering the local robust stability of equilibrium points and domain of attraction for the fuzzy neural network (FNNs).

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

where and are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, respectively. are elements of feedback template. stands for state of the th neurons. is the transmission delay and is the activation function. and denote the fuzzy AND and fuzzy OR operation, respectively. is the parameter. The main purpose of this paper is to investigate local robust stability of equilibrium points of FNNs (1.3). Sufficient conditions are gained for local robust stability of equilibrium points. Meanwhile, the attracting domains of equilibrium points are also estimated.

Throughout this paper, we always assume the following

(), and are bounded in () and are nondecreasing bounded variation function on with and is Lebesgue-Stieltjes integral. is a constant vector which denotes an external input.() are second-order differentiable, bounded, and Lipschitz continuous. There exist positive constants and such that and for any () The activation functions with bounded and Lipschitz continuous; that is, there are some numbers and such that and for any () Functions are nonnegative, bounded, and continuously differentiable defined on andThe rest of this paper is organized as follows. In Section 2, we will give some basic definitions and basic results about the attracting domains of FNNs (1.3). In Section 3, we discuss the local robust stability of equilibrium points of FNNs (1.3). In Section 4, an example is given to illustrate the effectiveness of our results. Finally, we make a conclusion in Section 5.

#### 2. Preliminaries

As usual, we denote by the set of all real-valued continuous mappings from to equipped with supremum norm defined by

where Denote by the solution of FNNs (1.3) with initial condition

*Definition 2.1. *A vector is said to be an equilibrium point of FNNs (1.3) if for each , one has
where Denote by the set of all equilibrium points of FNNs (1.3).

*Definition 2.2. *Let is said to be a locally robust attractive equilibrium point if for any given , there is a neighborhood such that implies that Otherwise, is said not to be a locally robust attractive equilibrium point. Denote by the set of all not locally robust attractive equilibrium points of FNNs (1.3).

*Definition 2.3. *Let be subsets of and let be a solution of FNNs (1.3) with (i)For any given if for some implies that for all then D is said to be an attracting domain of FNNs (1.3).(ii)For any given if for all implies that converges to then is said to be an attracting domain of Correspondingly, the union of all attracting domains of equilibrium points of is said to be an attracting domain of

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

Lemma 2.4 ([26]). *Let and be two states of (1.3); then one has
*

Lemma 2.5. *Let be any solution of FNNs (1.3). Then is uniformly bounded. Moreover, is an attracting domain of FNNs (1.3), where
*

*Proof. *By (1.3) and Lemma 2.4, we have
where
By using differential inequality, we have for ,
which leads to the uniform boundedness of Furthermore, given any we get for all ,
Hence is an attracting domain of FNNs (1.3). The proof is complete.

By Lemma 2.4, we have the following theorem.

Theorem 2.6. *All equilibrium points of FNNs (1.3) lie in the attracting domain that is, *

#### 3. Local Robust Stability of Equilibrium Points

In this section, we should investigate local robust stability of equilibrium points of FNNs (1.3). We derive some sufficient conditions to guarantee local robust stable of equilibrium points in and estimate the attracting domains of these equilibrium points.

Theorem 3.1. *Let If there exist positive constants such that for each **
where then one has the following. *()* that is, is locally robust stable. *()*Let **
Then every solution of FNNs (1.3) with satisfies
**
where
*()*The open set **
is an attracting domain of is an attracting domain of *

The proof of Theorem 3.1 relies on the following lemma.

Lemma 3.2. *Let satisfying (3.1). Let be an arbitrary solution of FNNs (1.3) other than , where Let
**
where is given by (3.1). Then one has the following. *()* If for some then *()* If and for some then *()* If then for all *

*Proof. *
Under transformation we get that
due to
where lies between and From (3.7), we can derive that
As we have for each which imply that

Since and
we have

Since from , we know that We assert that holds. Otherwise, there exist such that and for all This implies that is strictly monotonically decreasing on the interval It is obvious that By using we get that From This leads to a contradiction. Hence for all

Now we are in a position to complete the proof of Theorem 3.1.

*Proof. *Let be an arbitrary solution of FNNs (1.3) other than and satisfy It follows from that for all that is, for all Together with we get for all Take
It is obvious that for each From (3.9) we have
By integrating both sides of above inequality from 0 to , we have
It follows that
Note that is bounded on by Lemma 2.4; it follows from FNNs (1.3) that is bounded on Hence is uniformly continuous on From Lemma 2.5, we get that So the assertions of (1) and (2) hold. Let us consider an arbitrary solution of FNNs (1.3) satisfying for all and some Then it is obvious that
From (2), we get Hence is an attracting domain of Consequently, the open set is an attracting domain of The proof is complete.

Corollary 3.3. *Let If there exist positive constants such that for each **
where then one has the following.*()* that is, is locally asymptotically stable. *()*Let **
Then every solution of FNNs (1.3) with satisfies
**
where
*()*The open set **
is an attracting domain of is an attracting domain of *

#### 4. Illustrative Example

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

Then fuzzy neural network with two neurons can be modeled by

Take

It is easy to check that *(H _{1})*–

*(H*hold and for We can check that

_{5})From simple calculations, we know that is an attracting domain of FNNs (4.2). All equilibrium points of FNNs (4.2) lie in From some calculations, we have two equilibrium points For equilibrium we have and Taking we get

Similarly, we can check that (3.1) holds for Therefore, from Theorem 3.1, the four equilibrium points are locally robust stable and their convergent radius is 0.04.

*Remark 4.1. *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.

#### 5. Conclusions

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.

#### Acknowledgment

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