# Robust Stability Analysis of Fuzzy Neural Network with Delays

Authors: Kaihong Zhao; Yongkun Li
Journal: Mathematical Problems in Engineering
 (2009)

---

## 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.


---

## Body

## 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(1.1)ẋi(t)=-xi(t)+∑j=1nωijgj(xj(t))+Ii,i=1,2,…,n,
where gj denotes the activation function of the jth neuron; xi is the state of the ith neuron; Ii is the external input imposed on the ith neuron; ωij denotes the synaptic connectivity value between the ith neuron and the jth neuron; n is the number of neurons in the network. With the same notations, static neural network models can be written as(1.2)ẋi(t)=-xi(t)+gi(∑j=1nωijxj(t)+Ii),i=1,2,…,n.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:(1.3)u̇i(t)=-ci(λ)ui(t)+gi(∑j=1n∫-τ(λ)0uj(t+θ)dωij(θ,λ)+Ii)+∑j=1naij(λ)fj(uj(t))+⋀j=1nαij(λ)fj(uj(t-τj(t)))+⋁j=1nβij(λ)fj(uj(t-τj(t))),i=1,2,…,n,
where αij(λ) and βij(λ) are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, respectively. aij(λ) are elements of feedback template. ui(t) stands for state of the ith neurons. τj(t) is the transmission delay and fj(t) is the activation function. ⋀ and ∨ denote the fuzzy AND and fuzzy OR operation, respectively. λ∈Ξ⊂R 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(A)
aij(λ),αij(λ), and βij(λ) are bounded in Ξ(i,j=1,2,…,n).(H1)
infλ∈Ξci(λ)>0,0≤τ(λ)≤τ and ωij(θ,λ)(i,j=1,2,…,n) are nondecreasing bounded variation function on [-τ(λ),0] with ωij(θ,λ)>0, and ∫-τ(λ)0uj(t+θ)dωij(θ,λ) is Lebesgue-Stieltjes integral. I=(I1,I2,…,In)T is a constant vector which denotes an external input.(H2)
gi(·),i=1,2,…,n are second-order differentiable, bounded, and Lipschitz continuous. There exist positive constants Li and Bi such that |gi(x)-gi(y)|≤Li|x-y| and |gi(x)|≤Bi for any x,y∈R.(H3)
The activation functionsfi(u(t)) with fi(0)=0 bounded and Lipschitz continuous; that is, there are some numbers μi>0 and li>0 such that |fi(u)|≤μi and |fi(u)-fi(v)|≤li|u-v| for any u,v∈R,i=1,2,…,n.(H4)
Functionsτj(t),j=1,2,…,n are nonnegative, bounded, and continuously differentiable defined on R+ and 0≤τj(t)≤τ(λ).The rest of this paper is organized as follows. In Section2, 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 byC([-τ(λ),0],Rn) the set of all real-valued continuous mappings from [-τ(λ),0] to Rn equipped with supremum norm ∥·∥∞ defined by(2.1)∥ϕ∥=max1≤i≤nsup-τ(λ)<t≤0|ϕi(t)|,
where ϕ=(ϕ1,ϕ2,…,ϕn)T∈C([-τ(λ),0],Rn). Denote by u(t,ϕ,λ) the solution of FNNs (1.3) with initial condition ϕ∈C([-τ(λ),0],Rn).Definition 2.1.
A vectoru*(λ)=(u1*(λ),u2*(λ),…,un*(λ))T is said to be an equilibrium point of FNNs (1.3) if for each i=1,2,…,n, one has
(2.2)ci(λ)ui*(λ)=gi(∑j=1nω̃ij(λ)uj*(λ)+Ii)+∑j=1naij(λ)fj(uj*(λ))+⋀j=1nαij(λ)fj(uj*(λ))+⋁j=1nβij(λ)fj(uj*(λ)),i=1,2,…,n,
where ω̃ij(λ)=:∫-τ(λ)0dωij(θ,λ). Denote by Ω the set of all equilibrium points of FNNs (1.3).Definition 2.2.
Letu*(λ)∈Ω.u*(λ) is said to be a locally robust attractive equilibrium point if for any given λ∈Ξ, there is a neighborhood Yλ(u*(λ))⊂C([-τ(λ),0],Rn) such that ϕ∈Yλ(u*(λ)) implies that limt→∞∥u(t,ϕ,λ)-u*(λ)∥=0. Otherwise, u*(λ) is said not to be a locally robust attractive equilibrium point. Denote by Ω0 the set of all not locally robust attractive equilibrium points of FNNs (1.3).Definition 2.3.
LetD,D̃ be subsets of Rn and let u(t,ϕ,λ) be a solution of FNNs (1.3) with ϕ∈C([-τ(λ),0],Rn).(i)
For any givenλ∈Ξ, if u(σ,ϕ,λ)∈D for some σ≥0 implies that u(t,ϕ,λ)∈D for all t≥σ, then D is said to be an attracting domain of FNNs (1.3).(ii)
For any givenλ∈Ξ, if ϕ(θ)∈D̃ for all θ∈[-τ(λ),0] implies that u(t,ϕ,λ) converges to u*(λ), then D̃ is said to be an attracting domain of u*(λ)∈Ω.Correspondingly, the union of all attracting domains of equilibrium points of Ω is said to be an attracting domain of Ω.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]).
Letu=(u1,u2,…,un)T and v=(v1,v2,…,vn)T be two states of (1.3); then one has
(2.3)|⋀j=1nαijfj(uj)-⋀j=1nαijfj(vj)|≤∑j=1n|αij||fj(uj)-fj(vj)|,|⋁j=1nαijfj(uj)-⋁j=1nαijfj(vj)|≤∑j=1n|αij||fj(uj)-fj(vj)|.Lemma 2.5.
Letu(t) be any solution of FNNs (1.3). Then u(t) is uniformly bounded. Moreover, H is an attracting domain of FNNs (1.3), where
(2.4)H=:H1×H2×⋯×Hn,Hi=[-Bi+Miinfλ∈Ξci(λ),Bi+Miinfλ∈Ξci(λ)],i=1,2…,n.Proof.
By (1.3) and Lemma 2.4, we have
(2.5)d+dt|ui(t)|≤-infλ∈Ξci(λ)|ui(t)|+Bi+Mi,
where
(2.6)Mi=nmax1≤j≤n{μjsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|)},i=1,2…,n.
By using differential inequality, we have for t≥σ,
(2.7)|ui(t)|≤exp((σ-t)infλ∈Ξci(λ))[|ui(σ)|-Bi+Miinfλ∈Ξci(λ)]+Bi+Miinfλ∈Ξci(λ),i=1,2…,n,
which leads to the uniform boundedness of u(t). Furthermore, given any |ui(σ)|≤(Bi+Mi)/infλ∈Ξci(λ),i=1,2…,n, we get for all t≥σ,
(2.8)|ui(t)|≤Bi+Miinfλ∈Ξci(λ).
Hence H is an attracting domain of FNNs (1.3). The proof is complete.By Lemma2.4, we have the following theorem.Theorem 2.6.
All equilibrium points of FNNs (1.3) lie in the attracting domain H, that is, Ω⊂H.


## 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 Ω/Ω0 and estimate the attracting domains of these equilibrium points.Theorem 3.1.
Letu*(λ)=(u1*(λ),u2*(λ),…,un*(λ))T∈Ω. If there exist positive constants βi(i=1,2,…,n) such that for each i=1,2,…,n(3.1)∑j=1nβj(supλ∈Ξ{ω̃ji(λ)|ġj(κj(λ))|+lj(|aij(λ)|+|αij(λ)|+|βij(λ)|)})<βiinfλ∈Ξci(λ),
where κi(λ)=∑j=1nω̃ij(λ)uj*(λ)+Ii, then one has the following.  (1)
u*(λ)∈Ω/Ω0, that is, u*(λ) is locally robust stable.(2)
Let(3.2)R¯=:2mini∈N+{βiinfλ∈Ξci(λ)∑k=1n∑j=1nβjmaxζ∈R|g̈j(ζ)|supλ∈Ξ(ω̃ji(λ)ω̃jk(λ))-∑j=1nβj(supλ∈Ξ{ω̃ji(λ)|ġj(κj(λ))|+lj(|aij(λ)|+|αij(λ)|+|βij(λ)|)})∑k=1n∑j=1nβjmaxζ∈R|g̈j(ζ)|supλ∈Ξ(ω̃ji(λ)ω̃jk(λ))}.
Then every solution u(t,ϕ,λ) of FNNs (1.3) with ϕ∈O(u*(λ)) satisfies
(3.3)limt→+∞∥u(t,ϕ,λ)-u*(λ)∥∞=0,
where
(3.4)O(u*(λ))={ϕ∈C([-τ(λ),0],Rn):∥ϕ-u*(λ)∥∞<R¯∑i=1n(βi/min1≤i≤nβi)}.(3)
The open set(3.5)⋃u*(λ)∈ΩB(u*(λ))=:{u∈Rn:∥u-u*(λ)∥∞<R¯∑i=1n(βi/min1≤i≤nβi)}
is an attracting domain of Ω,andB(u*(λ)) is an attracting domain of u*(λ).The proof of Theorem3.1 relies on the following lemma.Lemma 3.2.
Letu*(λ)=(u1*(λ),u2*(λ),…,un*(λ))T∈Ω satisfying (3.1). Let u(t,ϕ,λ) be an arbitrary solution of FNNs (1.3) other than u*, where ϕ∈C([-τ(λ),0],Rn). Let
(3.6)V(t)=∑i=1nβi|ui(t,ϕ,λ)-ui*(λ)|,
where βi is given by (3.1). Then one has the following.  (A1)
If∥uσ(·,ϕ,λ)-u*(λ)∥<R¯ for some σ≥0, then D+V(σ)<0.(A2)
If∥ϕ-u*(λ)∥∞<R¯/∑i=1n(βi/min1≤i≤nβi) and supσ-τ≤s≤σV(s)≤sup-τ≤s≤0V(s) for some σ≥0, then ∥uσ(·,ϕ,λ)-u*(λ)∥<R¯.(A3)
If∥ϕ-u*(λ)∥∞<R¯/(∑i=1nβi/min1≤i≤nβi), then D+V(t)<0 for all t≥0.Proof.
Under transformationy(t)=u(t,ϕ,λ)-u*(λ), we get that
(3.7)d+|yi(t)|dt≤-ci(λ)|yi(t)|+∑j=1n|aij(λ)||yj(t)|+∑j=1n(|αij(λ)|+|βij(λ)|)|yj(t-τj(t))|+∑j=1n|ġi(κi(λ))|∫-τ(λ)0|yj(t+θ)|dωij(θ,λ)+|g̈i(ζi)|2(∑j=1n∫-τ(λ)0|yj(t+θ)|dωij(θ,λ))2,
due to
(3.8)gi(∑j=1n∫-τ(λ)0uj(t+θ)dωij(θ,λ)+Ii)-gi(∑j=1n∫-τ(λ)0uj*(λ)dωij(θ,λ)+Ii)=ġi(∑j=1nω̃ij(λ)uj*(λ)+Ii)∑j=1n∫-τ(λ)0|yj(t+θ)|dωij(θ,λ)+|g̈i(ζi)|2(∑j=1n∫-τ(λ)0|yj(t+θ)|dωij(θ,λ))2,
where ζi lies between ∑j=1n∫-τ(λ)0uj(t+θ)dωij(θ,λ)+Ii and ∑j=1n∫-τ(λ)0uj*(λ)dωij(θ,λ)+Ii. From (3.7), we can derive that
(3.9)d+V(t)dt≤∑i=1nβi{-infλ∈Ξci(λ)|yi(t)|+∑j=1nlj|aij(λ)||yj(t)|+∑j=1nlj(|αij(λ)|+|βij(λ)|)|yj(t-τj(t))|+[|ġi(κi(λ))|+|g̈i(ζi)|2∑j=1n∫-τ(λ)0|yj(t+θ)|dωij(θ,λ)]×∑j=1n∫-τ(λ)0|yj(t+θ)|dωij(θ,λ)}≤∑i=1nβi{-infλ∈Ξci(λ)|yi(t)|+∑j=1nljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|)supt-τ≤s≤t|yj(s)|+[|ġi(κi(λ))|+|g̈i(ζi)|2∑j=1nω̃ij(λ)supt-τ≤s≤t|yj(s)|]×∑j=1nω̃ij(λ)supt-τ≤s≤t|yj(s)|}≤∑i=1nβi{-infλ∈Ξci(λ)|yi(t)|+∑j=1nljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|)supt-τ≤s≤t|yj(s)|+[|ġi(κi(λ))|+|g̈i(ζi)|2∑j=1nω̃ij(λ)supt-τ≤s≤t|yj(s)|]×∑j=1nω̃ij(λ)supt-τ≤s≤t|yj(s)|}≤∑i=1n{-βiinfλ∈Ξci(λ)+∑j=1nβjljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|)+∑j=1nβjω̃ji(λ)[|ġj(κj(λ))|+|g̈j(ζj)|2∑j=1nω̃jk(λ)supt-τ≤s≤t|yk(s)|]∑j=1n}supt-τ≤s≤t|yi(s)|.
As ∥uσ(·,ϕ,λ)-u*(λ)∥∞<R¯, we have for each i=1,2,…,n,supt-τ≤s≤t|yi(s)|<R¯, which imply that D+V(σ)<0.
Sincemin1≤i≤n{βi}∥uσ(·,ϕ,λ)-u*(λ)∥∞≤supσ-τ≤s≤τV(s) and
(3.10)sup-τ≤s≤0V(s)=∑i=1nβi{sup-τ≤s≤0|ui(s,ϕ,λ)-ui*(λ)|}≤∑i=1nβi∥ϕ-u*(λ)∥∞,
we have ∥uσ(·,ϕ,λ)-u*(λ)∥∞≤∑i=1n(βi/min1≤i≤nβi)∥ϕ-u*(λ)∥∞<R¯.
Since∥ϕ-u*(λ)∥∞<R¯/∑i=1n(βi/min1≤i≤nβi)<R¯, from (A1), we know that D+V(0)<0. We assert that (A3) holds. Otherwise, there exist t0>0 such that D+V(t0)≥0 and D+V(t)<0 for all t∈[0,t0). This implies that V(t) is strictly monotonically decreasing on the interval [0,t0]. It is obvious that supt0-τ≤s≤t0V(s)≤sup-τ≤s≤t0V(s). By using (A2), we get that ∥ut0(·,ϕ,λ)-u*(λ)∥∞<R¯. From (A1),D+V(t0)<0. This leads to a contradiction. Hence D+V(t)<0 for all t≥0.Now we are in a position to complete the proof of Theorem3.1.Proof.
Letu(t,ϕ,λ) be an arbitrary solution of FNNs (1.3) other than u*(λ) and satisfy ∥ϕ-u*(λ)∥∞<R¯/∑i=1n(βi/min1≤i≤nβi). It follows from (A3) that D+V(t)<0 for all t≥0, that is, supt-τ≤s≤tV(s)≤sup-τ≤s≤0V(s) for all t≥0. Together with (A2) we get ∥ut(·,ϕ,λ)-u*(λ)∥∞<R¯ for all t≥0. Take
(3.11)χi=βiinfλ∈Ξci(λ)-∑j=1nβj(supλ∈Ξ{ω̃ji(λ)|ġj(κj(λ))|}+ljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|)),ηi=∑k=1n∑j=1nβjmaxζ∈R|g̈j(ζ)|supλ∈Ξ(ω̃ji(λ)ω̃jk(λ))R¯.
It is obvious that χi-ηi>0 for each i=1,2,…,n. From (3.9) we have
(3.12)D+V(t)<-min1≤i≤n{χi-ηi}∑i=1nsupt-τ≤s≤t|yi(s)|≤-min1≤i≤n{χi-ηi}∑i=1n|yi(s)|.
By integrating both sides of above inequality from 0 to t, we have
(3.13)V(t)+min1≤i≤n{χi-ηi}∫0t∑i=1n|yi(s)|≤V(0).
It follows that
(3.14)lim supt→∞min1≤i≤n{χi-ηi}∫0t∑i=1n|yi(s)|≤V(0)<∞.
Note that u(t,ϕ,λ) is bounded on R+ by Lemma 2.4; it follows from FNNs (1.3) that u̇ is bounded on R+. Hence |u(t,ϕ,λ)-u*(λ)| is uniformly continuous on R+. From Lemma 2.5, we get that limt→∞∑i=1n|ui(t,ϕ,λ)-ui*(λ)|=0. So the assertions of (1) and (2) hold. Let us consider an arbitrary solution u(t,ϕ,λ) of FNNs (1.3) satisfying ϕ(s)∈B(u*(λ)) for all s∈[-τ(λ),0] and some u*(λ)∈Ω. Then it is obvious that
(3.15)∥ϕ-u*(λ)∥∞<R¯∑i=1nβi/min1≤i≤nβi.
From (2), we get limt→∞∥u(t,ϕ,λ)-u*(λ)∥∞=0. Hence B(u*(λ)) is an attracting domain of u*(λ). Consequently, the open set ∪u*(λ)∈ΩB(u*(λ)) is an attracting domain of Ω. The proof is complete.Corollary 3.3.
Letu*(λ)=(u1*(λ),u2*(λ),…,un*(λ))T∈Ω. If there exist positive constants βi(i=1,2,…,n) such that for each i=1,2,…,n(3.16)∑j=1nβj({ω̃ji|ġj(κj)|}+ljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|))<βici,
where κi=∑j=1nω̃ijuj*(λ)+Ii, then one has the following.(1)
u*(λ)∈Ω/Ω0, that is, u*(λ) is locally asymptotically stable.(2)
Let(3.17)R¯=:2mini∈N+{βici∑k=1n∑j=1nβjmaxζ∈R|g̈j(ζ)|ω̃jiω̃jk-∑j=1nβj({ω̃ji|ġj(κj)|}+ljsupλ∈Ξ(|aij(λ)|+|αij(λ)|+|βij(λ)|))∑k=1n∑j=1nβjmaxζ∈R|g̈j(ζ)|ω̃jiω̃jk}.
Then every solution u(t,ϕ,λ) of FNNs (1.3) with ϕ∈O(u*(λ)) satisfies
(3.18)limt→+∞∥u(t,ϕ,λ)-u*(λ)∥∞=0,
where
(3.19)O(u*(λ))={ϕ∈C([-τ,0],Rn):∥ϕ-u*(λ)∥∞<R¯∑i=1n(βi/min1≤i≤nβi)}.(3)
The open set(3.20)⋃u*(λ)∈ΩB(u*(λ))=:{u∈Rn:∥u-u*(λ)∥∞<R¯∑i=1n(βi/min1≤i≤nβi)}
is an attracting domain of Ω,andB(u*(λ)) is an attracting domain of u*(λ).


## 4. Illustrative Example

For convenience of illustrative purpose, we only consider simple fuzzy neural network with time-varying and S-type distributed delays satisfying(4.1)ωij(θ,λ)={ωij(λ),θ=0,τ(λ)≡τ.0,-τ≤θ<0,
Then fuzzy neural network with two neurons can be modeled by(4.2)u̇i(t)=-ci(λ)ui(t)+gi(∑j=12uj(t)ωij(λ)+Ii)+∑j=12aij(λ)fj(uj(t))+⋀j=12αij(λ)fj(uj(t-τj(t)))+⋁j=12βij(λ)fj(uj(t-τj(t))),i=1,2.
Take(4.3)c1(λ)=tanh(4-2sinλ),ω11(λ)=4.02-2sinλ,ω12(λ)=0.02,c2(λ)=tanh(2.3-cosλ),ω21(λ)=0.01,ω22(λ)=2.31-cosλ,g1(ξ)=g2(ξ)=tanhξ,Ξ=[0,π2],I1=-0.02,I2=-0.01,τj(t)=τarctan2πt,j=1,2,f1(ξ)=f2(ξ)=sinπξ,aij(λ)=αij(λ)=βij(λ)=-sinλ100,i,j=1,2.It is easy to check that(H1)–(H5) hold and Li=Bi=μi=li=1 for i=1,2. We can check that(4.4)∑i=12(supλ∈[0,π/2]ωi1(λ)Li+l1supλ∈[0,π/2](|ai1(λ)|+|αi1(λ)|+|βi1(λ)|))=4.1>infλ∈[0,π/2]c1(λ)=tanh2.From simple calculations, we know that[-1.06/tanh2,1.06/tanh2]×[-1.06/tanh1.3,1.06/tanh1.3] is an attracting domain of FNNs (4.2). All equilibrium points of FNNs (4.2) lie in [-1.06/tanh2,1.06/tanh2]×[-1.06/tanh1.3,1.06/tanh1.3]. From some calculations, we have two equilibrium points O1=(1,0),O2=(0,1). For equilibrium O2=(1,1), we have κ1(λ)=4-2sinλ,κ2(λ)=2.3-cosλ and supλ∈[0,π/2]|ġ1(κ1(λ))|=0.0680,supλ∈[0,π/2]|ġ2(κ2(λ))|=0.0386. Taking β1=β2=1, we get(4.5)∑j=12(supλ∈[0,π/2]ω1j(λ)|ġj(κj(λ))|+ljsupλ∈[0,π/2](|a1j(λ)|+|α1j(λ)|+|β1j(λ)|))<0.04<tanh2=infλ∈[0,π/2]c1(λ),∑j=12(supλ∈[0,π/2]ω2j(λ)|ġj(κj(λ))|+ljsupλ∈[0,π/2](|a2j(λ)|+|α2j(λ)|+|β2j(λ)|))<0.04<tanh1.3=infλ∈[0,π/2]c2(λ).
Similarly, we can check that (3.1) holds for Ok(k=1,2). Therefore, from Theorem 3.1, the four equilibrium points Ok(k=1,2) 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.



---

*Source: 826908-2010-02-01.xml*