Robust Stability Analysis of Fuzzy Neural Network with Delays

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

𝑥̇_𝑖(𝑡)=−𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝜔_𝑖𝑗𝑔_𝑗(𝑥_𝑗(𝑡))+𝐼_𝑖,𝑖=1,2,…,𝑛, (1.1) where 𝑔_𝑗 denotes the activation function of the 𝑗th neuron; 𝑥_𝑖 is the state of the 𝑖th neuron; 𝐼_𝑖 is the external input imposed on the 𝑖th neuron; 𝜔_𝑖𝑗 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

𝑥̇_𝑖(𝑡)=−𝑥_𝑖(𝑡)+𝑔_𝑖(^𝑛_𝑗=1𝜔_𝑖𝑗𝑥_𝑗(𝑡)+𝐼_𝑖),𝑖=1,2,…,𝑛. (1.2)

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:

𝑢̇_𝑖(𝑡)=−𝑐_𝑖(𝜆)𝑢_𝑖(𝑡)+𝑔_𝑖(^𝑛_𝑗=1⁰_{−𝜏(𝜆)}𝑢_𝑗(𝑡+𝜃)𝑑𝜔_𝑖𝑗(𝜃,𝜆)+𝐼_𝑖)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_𝑗+(𝑡))_𝑛𝑗=1𝛼_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_𝑗(𝑡−𝜏_𝑗(𝑡)))+^𝑛_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_𝑗(𝑡−𝜏_𝑗(𝑡))),𝑖=1,2,…,𝑛, (1.3) where 𝛼_𝑖𝑗(𝜆) and 𝛽_𝑖𝑗(𝜆) are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, respectively. 𝑎_𝑖𝑗(𝜆) are elements of feedback template. 𝑢_𝑖(𝑡) stands for state of the 𝑖th neurons. 𝜏_𝑗(𝑡) is the transmission delay and 𝑓_𝑗(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)𝑎_𝑖𝑗(𝜆),𝛼_𝑖𝑗(𝜆), and 𝛽_i𝑗(𝜆) are bounded in Ξ(𝑖,𝑗=1,2,…,𝑛).(𝐻₁)inf_𝜆∈Ξ𝑐_𝑖(𝜆)>0,0≤𝜏(𝜆)≤𝜏 and 𝜔_𝑖𝑗(𝜃,𝜆)(𝑖,𝑗=1,2,…,𝑛) are nondecreasing bounded variation function on [−𝜏(𝜆),0] with 𝜔_𝑖𝑗(𝜃,𝜆)>0, and ∫⁰_{−𝜏(𝜆)}𝑢_𝑗(𝑡+𝜃)𝑑𝜔_𝑖𝑗(𝜃,𝜆) is Lebesgue-Stieltjes integral. 𝐼=(𝐼₁,𝐼₂,…,𝐼_𝑛)^𝑇 is a constant vector which denotes an external input.(𝐻₂)𝑔_𝑖(⋅),𝑖=1,2,…,𝑛 are second-order differentiable, bounded, and Lipschitz continuous. There exist positive constants 𝐿_𝑖 and 𝐵_𝑖 such that |𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦)|≤𝐿_𝑖|𝑥−𝑦| and |𝑔_𝑖(𝑥)|≤𝐵_𝑖 for any 𝑥,𝑦∈𝑅.(𝐻₃) The activation functions 𝑓_𝑖(𝑢(𝑡)) with 𝑓_𝑖(0)=0 bounded and Lipschitz continuous; that is, there are some numbers 𝜇_𝑖>0 and 𝑙_𝑖>0 such that |𝑓_𝑖(𝑢)|≤𝜇_𝑖 and |𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)|≤𝑙_𝑖|𝑢−𝑣| for any 𝑢,𝑣∈𝑅,𝑖=1,2,…,𝑛.(𝐻₄) Functions 𝜏_𝑗(𝑡),𝑗=1,2,…,𝑛 are nonnegative, bounded, and continuously differentiable defined on 𝑅₊ and 0≤𝜏_𝑗(𝑡)≤𝜏(𝜆).

The 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 𝐶([−𝜏(𝜆),0],𝑅^𝑛) the set of all real-valued continuous mappings from [−𝜏(𝜆),0] to 𝑅^𝑛 equipped with supremum norm ‖⋅‖_∞ defined by

‖𝜙‖=max_{1≤𝑖≤𝑛}sup_{−𝜏(𝜆)<𝑡≤0}|𝜙_𝑖(𝑡)|, (2.1) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇∈𝐶([−𝜏(𝜆),0],𝑅^𝑛). Denote by 𝑢(𝑡,𝜙,𝜆) the solution of FNNs (1.3) with initial condition 𝜙∈𝐶([−𝜏(𝜆),0],𝑅^𝑛).

Definition 2.1. A vector 𝑢^∗(𝜆)=(𝑢^∗₁(𝜆),𝑢^∗₂(𝜆),…,𝑢^∗_𝑛(𝜆))^𝑇 is said to be an equilibrium point of FNNs (1.3) if for each 𝑖=1,2,…,𝑛, one has 𝑐_𝑖(𝜆)𝑢^∗_𝑖(𝜆)=𝑔_𝑖(^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)𝑢^∗_𝑗(𝜆)+𝐼_𝑖)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_𝑗(𝑢^∗_𝑗+(𝜆))_𝑛𝑗=1𝛼_𝑖𝑗(𝜆)𝑓_𝑗(𝑢^∗_𝑗(𝜆))+^𝑛_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_𝑗(𝑢^∗_𝑗(𝜆)),𝑖=1,2,…,𝑛, (2.2) where 𝜔_𝑖𝑗(𝜆)=∶∫⁰_{−𝜏(𝜆)}𝑑𝜔_𝑖𝑗(𝜃,𝜆). Denote by Ω the set of all equilibrium points of FNNs (1.3).

Definition 2.2. Let 𝑢^∗(𝜆)∈Ω.𝑢^∗(𝜆) is said to be a locally robust attractive equilibrium point if for any given 𝜆∈Ξ, there is a neighborhood 𝑌_𝜆(𝑢^∗(𝜆))⊂𝐶([−𝜏(𝜆),0],𝑅^𝑛) such that 𝜙∈𝑌_𝜆(𝑢^∗(𝜆)) implies that lim_𝑡→∞‖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. Let 𝐷,𝐷 be subsets of 𝑅^𝑛 and let 𝑢(𝑡,𝜙,𝜆) be a solution of FNNs (1.3) with 𝜙∈𝐶([−𝜏(𝜆),0],R^𝑛).(i)For any given 𝜆∈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 given 𝜆∈Ξ, if 𝜙(𝜃)∈𝐷 for all 𝜃∈[−𝜏(𝜆),0] implies that 𝑢(𝑡,𝜙,𝜆) converges to 𝑢^∗(𝜆), then 𝐷 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]). Let 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇 and 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇 be two states of (1.3); then one has |^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗(𝑢_𝑗)−^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗(𝑣_𝑗)|≤^𝑛_𝑗=1|𝛼_𝑖𝑗||𝑓_𝑗(𝑢_𝑗)−𝑓_𝑗(𝑣_𝑗|)|,_𝑛𝑗=1𝛼_𝑖𝑗𝑓_𝑗(𝑢_𝑗)−^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗(𝑣_𝑗)|≤^𝑛_𝑗=1|𝛼_𝑖𝑗||𝑓_𝑗(𝑢_𝑗)−𝑓_𝑗(𝑣_𝑗)|. (2.3)

Lemma 2.5. Let 𝑢(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 𝐻=∶𝐻₁×𝐻₂×⋯×𝐻_𝑛,𝐻_𝑖𝐵=[−^𝑖+𝑀_𝑖inf_𝜆∈Ξ𝑐_𝑖,𝐵(𝜆)^𝑖+𝑀_𝑖inf_𝜆∈Ξ𝑐_𝑖(𝜆)],𝑖=1,2…,𝑛. (2.4)

Proof. By (1.3) and Lemma 2.4, we have d⁺d𝑡|𝑢_𝑖(𝑡)|≤−inf_𝜆∈Ξ𝑐_𝑖(𝜆)|𝑢_𝑖(𝑡)|+𝐵_𝑖+𝑀_𝑖, (2.5) where 𝑀_𝑖=𝑛max_{1≤𝑗≤𝑛}{𝜇_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)},𝑖=1,2…,𝑛. (2.6) By using differential inequality, we have for 𝑡≥𝜎, |𝑢_𝑖(𝑡)|≤exp((𝜎−𝑡)inf_𝜆∈Ξ𝑐_𝑖(𝜆))[|𝑢_𝑖𝐵(𝜎)|−^𝑖+𝑀_𝑖inf_𝜆∈Ξ𝑐_𝑖𝐵(𝜆)]+^𝑖+𝑀_𝑖inf_𝜆∈Ξ𝑐_𝑖(𝜆),𝑖=1,2…,𝑛, (2.7) which leads to the uniform boundedness of 𝑢(𝑡). Furthermore, given any |𝑢_𝑖(𝜎)|≤(𝐵_𝑖+𝑀_𝑖)/inf_𝜆∈Ξ𝑐_𝑖(𝜆),𝑖=1,2…,𝑛, we get for all 𝑡≥𝜎, |𝑢_𝑖𝐵(𝑡)|≤^𝑖+𝑀_𝑖inf_𝜆∈Ξ𝑐_𝑖.(𝜆) (2.8) Hence 𝐻 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 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. Let 𝑢^∗(𝜆)=(𝑢^∗₁(𝜆),𝑢^∗₂(𝜆),…,𝑢^∗_𝑛(𝜆))^𝑇∈Ω. If there exist positive constants 𝛽_𝑖(𝑖=1,2,…,𝑛) such that for each 𝑖=1,2,…,𝑛^𝑛_𝑗=1𝛽_𝑗(sup_𝜆∈Ξ{𝜔_𝑗𝑖(𝜆)|𝑔̇_𝑗(𝜅_𝑗(𝜆))|+𝑙_𝑗(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)})<𝛽_𝑖inf_𝜆∈Ξ𝑐_𝑖(𝜆), (3.1) where 𝜅_𝑖(𝜆)=∑^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)𝑢^∗_𝑗(𝜆)+𝐼_𝑖, then one has the following. (1)𝑢^∗(𝜆)∈Ω/Ω₀, that is, 𝑢^∗(𝜆) is locally robust stable. (2)Let 𝑅=∶2min_𝑖∈𝑁+{𝛽_𝑖inf_𝜆∈Ξ𝑐_𝑖(𝜆)∑^𝑛_𝑘=1∑^𝑛_𝑗=1𝛽_𝑗max_𝜁∈𝑅|𝑔̈_𝑗(𝜁)|sup_𝜆∈Ξ(𝜔_𝑗𝑖(𝜆)𝜔_𝑗𝑘−∑(𝜆))_𝑛𝑗=1𝛽_𝑗(sup_𝜆∈Ξ{𝜔_𝑗𝑖(𝜆)|𝑔̇_𝑗(𝜅_𝑗(𝜆))|+𝑙_𝑗(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)})∑^𝑛_𝑘=1∑^𝑛_𝑗=1𝛽_𝑗max_𝜁∈𝑅|𝑔̈_𝑗(𝜁)|sup_𝜆∈Ξ(𝜔_𝑗𝑖(𝜆)𝜔_𝑗𝑘(𝜆))}. (3.2) Then every solution 𝑢(𝑡,𝜙,𝜆) of FNNs (1.3) with 𝜙∈𝑂(𝑢^∗(𝜆)) satisfies lim_𝑡→+∞‖𝑢(𝑡,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞=0, (3.3) where 𝑂(𝑢^∗(𝜆))={𝜙∈𝐶([−𝜏(𝜆),0],𝑅^𝑛)∶‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)}. (3.4)(3)The open set _{𝑢∗(𝜆)∈Ω}𝐵(𝑢^∗(𝜆))=∶{𝑢∈𝑅^𝑛∶‖𝑢−𝑢^∗(𝜆)‖_∞<𝑅∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)} (3.5) is an attracting domain of Ω,and𝐵(𝑢^∗(𝜆)) is an attracting domain of 𝑢^∗(𝜆).

The proof of Theorem 3.1 relies on the following lemma.

Lemma 3.2. Let 𝑢^∗(𝜆)=(𝑢^∗₁(𝜆),𝑢^∗₂(𝜆),…,𝑢^∗_𝑛(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],𝑅^𝑛). Let 𝑉(𝑡)=^𝑛_𝑖=1𝛽_𝑖|𝑢_𝑖(𝑡,𝜙,𝜆)−𝑢^∗_𝑖(𝜆)|, (3.6) where 𝛽_𝑖 is given by (3.1). Then one has the following. (𝐴₁) If ‖𝑢_𝜎(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖<𝑅 for some 𝜎≥0, then 𝐷⁺𝑉(𝜎)<0.(𝐴₂) If ‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅/∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖) and sup_{𝜎−𝜏≤𝑠≤𝜎}𝑉(𝑠)≤sup_{−𝜏≤𝑠≤0}𝑉(𝑠) for some 𝜎≥0, then ‖𝑢_𝜎(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖<𝑅.(𝐴₃) If ‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅/(∑^𝑛_𝑖=1𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖), then 𝐷⁺𝑉(𝑡)<0 for all 𝑡≥0.

Proof. Under transformation 𝑦(𝑡)=𝑢(𝑡,𝜙,𝜆)−𝑢^∗(𝜆), we get that 𝑑⁺|𝑦_𝑖(𝑡)|𝑑𝑡≤−𝑐_𝑖(𝜆)|𝑦_𝑖(𝑡)|+^𝑛_𝑗=1|𝑎_𝑖𝑗(𝜆)||𝑦_𝑗(𝑡)|+^𝑛_𝑗=1(|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)|𝑦_𝑗(𝑡−𝜏_𝑗+(𝑡))|_𝑛𝑗=1|𝑔̇_𝑖(𝜅_𝑖(𝜆))|⁰_{−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗+|𝑔̈(𝜃,𝜆)_𝑖(𝜁_𝑖)|2(^𝑛_𝑗=1⁰_{−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗(𝜃,𝜆))², (3.7) due to 𝑔_𝑖(^𝑛_𝑗=1⁰_{−𝜏(𝜆)}𝑢_𝑗(𝑡+𝜃)𝑑𝜔_𝑖𝑗(𝜃,𝜆)+𝐼_𝑖)−𝑔_𝑖(^𝑛_𝑗=1⁰_{−𝜏(𝜆)}𝑢^∗_𝑗(𝜆)𝑑𝜔_𝑖𝑗(𝜃,𝜆)+𝐼_𝑖)=𝑔̇_𝑖(^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)𝑢^∗_𝑗(𝜆)+𝐼_𝑖)^𝑛_𝑗=1⁰_{−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗+|𝑔̈(𝜃,𝜆)_𝑖(𝜁_𝑖)|2(^𝑛_𝑗=1⁰_{−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗(𝜃,𝜆))², (3.8) where 𝜁_𝑖 lies between ∑^𝑛_𝑗=1∫⁰_{−𝜏(𝜆)}𝑢_𝑗(𝑡+𝜃)𝑑𝜔_𝑖𝑗(𝜃,𝜆)+𝐼_𝑖 and ∑^𝑛_𝑗=1∫⁰_{−𝜏(𝜆)}𝑢^∗_𝑗(𝜆)𝑑𝜔_𝑖𝑗(𝜃,𝜆)+𝐼_𝑖. From (3.7), we can derive that 𝑑⁺𝑉(𝑡)≤𝑑𝑡^𝑛_𝑖=1𝛽_𝑖{−inf_𝜆∈Ξ𝑐_𝑖(𝜆)|𝑦_𝑖(𝑡)|+^𝑛_𝑗=1𝑙_𝑗|𝑎_𝑖𝑗(𝜆)||𝑦_𝑗+(𝑡)|_𝑛𝑗=1𝑙_𝑗(|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)|𝑦_𝑗(𝑡−𝜏_𝑗𝑔̇(𝑡))|+[|_𝑖(𝜅_𝑖|𝑔̈(𝜆))|+^𝑖(𝜁_𝑖)|2^𝑛_𝑗=1⁰_{−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗×(𝜃,𝜆)]_{𝑛𝑗=10−𝜏(𝜆)}|𝑦_𝑗(𝑡+𝜃)|𝑑𝜔_𝑖𝑗≤(𝜃,𝜆)}_𝑛𝑖=1𝛽_𝑖{−inf_𝜆∈Ξ𝑐_𝑖(𝜆)|𝑦_𝑖(𝑡)|+^𝑛_𝑗=1𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗𝑔̇(𝑠)|+[|_𝑖(𝜅_𝑖|𝑔̈(𝜆))|+^𝑖(𝜁_𝑖)|2^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗(𝑠)|]×^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗≤(𝑠)|}_𝑛𝑖=1𝛽_i{−inf_𝜆∈Ξ𝑐_𝑖(𝜆)|𝑦_𝑖(𝑡)|+^𝑛_𝑗=1𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗𝑔̇(𝑠)|+[|_𝑖(𝜅_𝑖|𝑔̈(𝜆))|+^𝑖(𝜁_𝑖)|2^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗(𝑠)|]×^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑗≤(𝑠)|}_𝑛𝑖=1{−𝛽_𝑖inf_𝜆∈Ξ𝑐_𝑖(𝜆)+^𝑛_𝑗=1𝛽_𝑗𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗+(𝜆)|)_𝑛𝑗=1𝛽_𝑗𝜔_𝑗𝑖(𝜆)[|𝑔̇_𝑗(𝜅_𝑗|𝑔̈(𝜆))|+^𝑗(𝜁_𝑗)|2^𝑛_𝑗=1𝜔_𝑗𝑘(𝜆)sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑘(𝑠)|]}sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑖(𝑠)|. (3.9)(1)(2)(3)(4)(5)(6)(7)(8)(9) As ‖𝑢_𝜎(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞<𝑅, we have for each 𝑖=1,2,…,𝑛,sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑖(𝑠)|<𝑅, which imply that 𝐷⁺𝑉(𝜎)<0.
Since min_{1≤𝑖≤𝑛}{𝛽_𝑖}‖𝑢_𝜎(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞≤sup_{𝜎−𝜏≤𝑠≤𝜏}𝑉(𝑠) and sup_{−𝜏≤𝑠≤0}𝑉(𝑠)=^𝑛_𝑖=1𝛽_𝑖{sup_{−𝜏≤𝑠≤0}|𝑢_𝑖(𝑠,𝜙,𝜆)−𝑢^∗_𝑖(𝜆)|}≤^𝑛_𝑖=1𝛽_𝑖‖𝜙−𝑢^∗(𝜆)‖_∞, (3.10) we have ‖𝑢_𝜎(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞≤∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅.
Since ‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅/∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)<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 sup_{𝑡0−𝜏≤𝑠≤𝑡0}𝑉(𝑠)≤sup_{−𝜏≤𝑠≤𝑡0}𝑉(𝑠). By using (𝐴₂), we get that ‖𝑢_𝑡0(⋅,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 Theorem 3.1.

Proof. Let 𝑢(𝑡,𝜙,𝜆) be an arbitrary solution of FNNs (1.3) other than 𝑢^∗(𝜆) and satisfy ‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅/∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖). It follows from (𝐴₃) that 𝐷⁺𝑉(𝑡)<0 for all 𝑡≥0, that is, sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠)≤sup_{−𝜏≤𝑠≤0}𝑉(𝑠) for all 𝑡≥0. Together with (𝐴₂) we get ‖𝑢_𝑡(⋅,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞<𝑅 for all 𝑡≥0. Take 𝜒_𝑖=𝛽_𝑖inf_𝜆∈Ξ𝑐_𝑖−(𝜆)_𝑛𝑗=1𝛽_𝑗(sup_𝜆∈Ξ{𝜔_𝑗𝑖(𝜆)|𝑔̇_𝑗(𝜅_𝑗(𝜆))|}+𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗𝜂(𝜆)|)),_𝑖=^𝑛𝑛_𝑘=1𝑗=1𝛽_𝑗max_𝜁∈𝑅|𝑔̈_𝑗(𝜁)|sup_𝜆∈Ξ(𝜔_𝑗𝑖(𝜆)𝜔_𝑗𝑘(𝜆))𝑅. (3.11) It is obvious that 𝜒_𝑖−𝜂_𝑖>0 for each 𝑖=1,2,…,𝑛. From (3.9) we have 𝐷⁺𝑉(𝑡)<−min_{1≤𝑖≤𝑛}{𝜒_𝑖−𝜂_𝑖}^𝑛_𝑖=1sup_{𝑡−𝜏≤𝑠≤𝑡}|𝑦_𝑖(𝑠)|≤−min_{1≤𝑖≤𝑛}{𝜒_𝑖−𝜂_𝑖}^𝑛_𝑖=1|𝑦_𝑖(𝑠)|. (3.12) By integrating both sides of above inequality from 0 to 𝑡, we have 𝑉(𝑡)+min_{1≤𝑖≤𝑛}{𝜒_𝑖−𝜂_𝑖}^𝑡₀^𝑛_𝑖=1|𝑦_𝑖(𝑠)|≤𝑉(0). (3.13) It follows that limsup_𝑡→∞min_{1≤𝑖≤𝑛}{𝜒_𝑖−𝜂_𝑖}^𝑡₀^𝑛_𝑖=1|𝑦_𝑖(𝑠)|≤𝑉(0)<∞. (3.14) 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 lim_𝑡→∞∑^𝑛_𝑖=1|𝑢_𝑖(𝑡,𝜙,𝜆)−𝑢^∗_𝑖(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 ‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅∑^𝑛_𝑖=1𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖. (3.15) From (2), we get lim_𝑡→∞‖𝑢(𝑡,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞=0. Hence 𝐵(𝑢^∗(𝜆)) is an attracting domain of 𝑢^∗(𝜆). Consequently, the open set ∪_{𝑢∗(𝜆)∈Ω}𝐵(𝑢^∗(𝜆)) is an attracting domain of Ω. The proof is complete.

Corollary 3.3. Let 𝑢^∗(𝜆)=(𝑢^∗₁(𝜆),𝑢^∗₂(𝜆),…,𝑢^∗_𝑛(𝜆))^𝑇∈Ω. If there exist positive constants 𝛽_𝑖(𝑖=1,2,…,𝑛) such that for each 𝑖=1,2,…,𝑛^𝑛_𝑗=1𝛽_𝑗({𝜔_𝑗𝑖|𝑔̇_𝑗(𝜅_𝑗)|}+𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|))<𝛽_𝑖𝑐_𝑖, (3.16) where 𝜅_𝑖=∑^𝑛_𝑗=1𝜔_𝑖𝑗𝑢^∗_𝑗(𝜆)+𝐼_𝑖, then one has the following.(1)𝑢^∗(𝜆)∈Ω/Ω₀, that is, 𝑢^∗(𝜆) is locally asymptotically stable. (2)Let 𝑅=∶2min_𝑖∈𝑁+{𝛽_𝑖𝑐_𝑖∑^𝑛_𝑘=1∑^𝑛_𝑗=1𝛽_𝑗max_𝜁∈𝑅|𝑔̈_𝑗(𝜁)|𝜔_𝑗𝑖𝜔_𝑗𝑘−∑^𝑛_𝑗=1𝛽_𝑗({𝜔_𝑗𝑖|𝑔̇_𝑗(𝜅_𝑗)|}+𝑙_𝑗sup_𝜆∈Ξ(|𝑎_𝑖𝑗(𝜆)|+|𝛼_𝑖𝑗(𝜆)|+|𝛽_𝑖𝑗(𝜆)|))∑^𝑛_𝑘=1∑^𝑛_𝑗=1𝛽_𝑗max_𝜁∈𝑅|𝑔̈_𝑗(𝜁)|𝜔_𝑗𝑖𝜔_𝑗𝑘}. (3.17) Then every solution 𝑢(𝑡,𝜙,𝜆) of FNNs (1.3) with 𝜙∈𝑂(𝑢^∗(𝜆)) satisfies lim_𝑡→+∞‖𝑢(𝑡,𝜙,𝜆)−𝑢^∗(𝜆)‖_∞=0, (3.18) where 𝑂(𝑢^∗(𝜆))={𝜙∈𝐶([−𝜏,0],𝑅^𝑛)∶‖𝜙−𝑢^∗(𝜆)‖_∞<𝑅∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)}. (3.19)(3)The open set _{𝑢∗(𝜆)∈Ω}𝐵(𝑢^∗(𝜆))=∶{𝑢∈𝑅^𝑛∶‖𝑢−𝑢^∗(𝜆)‖_∞<𝑅∑^𝑛_𝑖=1(𝛽_𝑖/min_{1≤𝑖≤𝑛}𝛽_𝑖)} (3.20) 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

𝜔_𝑖𝑗𝜔(𝜃,𝜆)=^𝑖𝑗(𝜆),𝜃=0,𝜏(𝜆)≡𝜏.0,−𝜏≤𝜃<0, (4.1) Then fuzzy neural network with two neurons can be modeled by

𝑢̇_𝑖(𝑡)=−𝑐_𝑖(𝜆)𝑢_𝑖(𝑡)+𝑔_𝑖(²_𝑗=1𝑢_𝑗(𝑡)𝜔_𝑖𝑗(𝜆)+𝐼_𝑖)+²_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_j+(𝑡))_2𝑗=1𝛼_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_𝑗(𝑡−𝜏_𝑗(𝑡)))+²_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_𝑗(𝑢_𝑗(𝑡−𝜏_𝑗(𝑡))),𝑖=1,2. (4.2) Take

𝑐₁(𝜆)=tanh(4−2sin𝜆),𝜔₁₁(𝜆)=4.02−2sin𝜆,𝜔₁₂𝑐(𝜆)=0.02,₂(𝜆)=tanh(2.3−cos𝜆),𝜔₂₁(𝜆)=0.01,𝜔₂₂𝑔(𝜆)=2.31−cos𝜆,₁(𝜉)=𝑔₂𝜋(𝜉)=tanh𝜉,Ξ=[0,2],𝐼₁=−0.02,𝐼₂𝜏=−0.01,_𝑗2(𝑡)=𝜏arctan𝜋𝑓𝑡,𝑗=1,2,₁(𝜉)=𝑓₂(𝜉)=sin𝜋𝜉,𝑎_𝑖𝑗(𝜆)=𝛼_𝑖𝑗(𝜆)=𝛽_𝑖𝑗(𝜆)=−sin𝜆100,𝑖,𝑗=1,2. (4.3)

It is easy to check that (H₁)–(H₅) hold and 𝐿_𝑖=𝐵_𝑖=𝜇_𝑖=𝑙_𝑖=1 for 𝑖=1,2. We can check that

²_𝑖=1(sup_{𝜆∈[0,𝜋/2]}𝜔_𝑖1(𝜆)𝐿_𝑖+𝑙₁sup_{𝜆∈[0,𝜋/2]}(|𝑎_𝑖1(𝜆)|+|𝛼_𝑖1(𝜆)|+|𝛽_𝑖1(𝜆)|))=4.1>inf_{𝜆∈[0,𝜋/2]}𝑐₁(𝜆)=tanh2. (4.4)

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 sup_{𝜆∈[0,𝜋/2]}|𝑔̇₁(𝜅₁(𝜆))|=0.0680,sup_{𝜆∈[0,𝜋/2]}|𝑔̇₂(𝜅₂(𝜆))|=0.0386. Taking 𝛽₁=𝛽₂=1, we get

²_𝑗=1(sup_{𝜆∈[0,𝜋/2]}𝜔_1𝑗(𝜆)|𝑔̇_𝑗(𝜅_𝑗(𝜆))|+𝑙_𝑗sup_{𝜆∈[0,𝜋/2]}(|𝑎_1𝑗(𝜆)|+|𝛼_1𝑗(𝜆)|+|𝛽_1𝑗(𝜆)|))<0.04<tanh2=inf_{𝜆∈[0,𝜋/2]}𝑐₁(𝜆),_2𝑗=1(sup_{𝜆∈[0,𝜋/2]}𝜔_2𝑗(𝜆)|𝑔̇_𝑗(𝜅_𝑗(𝜆))|+𝑙_𝑗sup_{𝜆∈[0,𝜋/2]}(|𝑎_2𝑗(𝜆)|+|𝛼_2𝑗(𝜆)|+|𝛽_2𝑗(𝜆)|))<0.04<tanh1.3=inf_{𝜆∈[0,𝜋/2]}𝑐₂(𝜆). (4.5) Similarly, we can check that (3.1) holds for 𝑂_𝑘(𝑘=1,2). Therefore, from Theorem 3.1, the four equilibrium points 𝑂_𝑘(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.

Acknowledgment

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