# Global Robust Attractive and Invariant Sets of Fuzzy Neural Networks with Delays and Impulses

Authors: Kaihong Zhao; Liwenjing Wang; Juqing Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated. With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.


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## Body

## 1. Introduction

The theoretical and applied studies of the current neural networks (CNNs) have been a new focus of studies worldwide because CNNs are widely applied in signal processing, image processing, pattern recognition, psychophysics, speech, perception, robotics, and so on. The scholars have introduced many classes of CNNs models such as Hopfield-type networks [1], bidirectional associative memory networks [2], cellular neural networks [3], recurrent back-propagation networks [4–6], optimization-type networks [7–9], brain-state-in-a-box-(BSB-) type networks [10, 11], and Cohen-Grossberg recurrent neural networks (CGCNNs) [12]. According to the choice of the variable for CNNs [13], two basic mathematical models of CNNs 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)x˙i(t)=-xi(t)+∑j=1n‍ωijfi(xj(t))+Ii,i=1,2,…,n,
where fj 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.It is well known that local field neural network not only models Hopfield-type networks but also models bidirectional associative memory networks and cellular neural networks. In the past few years, there has been increasing interest in studying dynamical characteristics such as stability, persistence, periodicity, robust stability of equilibrium points, and domains of attraction of local field neural network. Many deep theoretical results have been obtained for local field neural network. We can refer to [14–32] and the references cited therein.However, in mathematical modeling of real world problems, we will encounter some other inconveniences, 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),T. Yang and L.-B. Yangproposed the fuzzy CNNs (FCNNs) [33], which integrate fuzzy logic into the structure of traditional CNNs and maintain 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. In addition, many evolutionary processes in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system. Therefore, it is necessary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks with impulses. Nevertheless, to the best of our knowledge, there are few published papers considering the global robust domain of attraction for the fuzzy neural network (FNNs). Therefore, in this paper, we will study the global robust attracting set and invariant set of the following fuzzy neural networks (FNNs) with time-varying delays and impulses:
(2)u˙i(t)=-ci(λ)ui(t)+∑j=1n‍ωij(λ)fj(uj(t-τij(t)))+∑j=1n‍γij(λ)νj+Ii+⋀j=1naij(λ)fj(uj(t))+⋀j=1nαij(λ)fj(uj(t-τij(t)))+⋀j=1ncij(λ)νj+⋁j=1nbij(λ)fj(uj(t))+⋁j=1nβij(λ)fj(uj(t-τij(t)))+⋁j=1nδij(λ)νj,t≠tk,t≥0,Δui(t)=ui(t+)-ui(t-)=μik(λ)ui(t),t=tk,t≥0,ui(t)=ϕi(t),t∈[-τ(λ),0],
where γij(λ) are elements of fuzzy feed-forward template, aij(λ) and αij(λ) are elements of fuzzy feedback MIN template, bij(λ) and βij(λ) are elements of fuzzy feedback MAX template, and cij(λ) and δij(λ) are elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. ωij(λ) is the weight of connection between the ith neurons and the jth neurons. ui(t),Ii, and νi stand for state, input, and bias of the ith neurons, respectively. τij(t) is the transmission delay and fj is the activation function. ∧ and ∨ denote the fuzzy AND and fuzzy OR operation, respectively. Δui(tk) is the impulses at moments tk, and 0≤t1<t2<⋯ is a strictly increasing sequence such that limk→∞tk=+∞. gi=μik(λ)ui(t) is the impulsive function. Function ϕi is the initial function. τ(λ)>0 is a constant. λ∈Ξ⊂R is the parameter.The main purpose of this paper is to investigate the global robust attracting and invariant sets of FNNs (2). Different from [34, 35], in this paper, we will introduce a new nonlinear differential inequality, which is more effective than the linear differential inequalities for studying the asymptotic behavior of some nonlinear differential equations. Applying this new nonlinear delay differential inequality, sufficient conditions are gained for global robust attracting and invariant sets.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 (2). In Section 3, we will obtain the proof of the usefully nonlinear delay differential inequality. In Section 4, our main results will be proved by this delay differential inequality. Finally, an example is given to illustrate the effectiveness of our results in Section 5.


## 2. Preliminaries

As usual,C(X,Y) denotes the space of continuous mappings from the topological space X to the topological space Y. In particular, let C([-τ(λ),0],ℝn) denote the set of all real-valued continuous mappings from [-τ(λ),0] to ℝn equipped with supremum norm ∥·∥∞ defined by
(3)∥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 FCNNs (2) with initial condition ϕ∈C([-τ(λ),0],ℝn).LetE denote the n-dimensional unit matrix. For A,B∈ℝm×n or A,B∈ℝn,A≥B(A>B) means that each pair of the corresponding elements of A and B satisfies the inequality “≥(>)”. For any x∈ℝn,A∈ℝn×n,φ∈C([-τ,0],ℝn), we define [x]+=(|x1|,|x2|,…,|xn|)T=col{|xi|},  [A]+=(|aij|)n×n, [φ(t)]τ=([φ1(t)]τ,[φ2(t)]τ,…,[φn(t)]τ)T, [φ(t)]τ+=[[φ(t)]+]τ, [φi(t)]τ=sup-τ≤θ≤0{φi(t+θ)}. For an M-matrix D [36], we denote D∈ℳ and ΩM(D)={z∈Rn:Dz>0,z>0}. For the sake of simplicity, we denote that g-=supλ∈Ξg(λ),g_=infλ∈Ξg(λ), where g(λ) is bounded in Ξ.As usual, in the theory of impulsive differential equations, at the points of discontinuitytk,k=1,2,…, we assume that ui(tk)≡ui(tk-) and ui′(tk)≡ui′(tk-).Inspired by [37], we construct an equivalent theorem between (2) and (4). Then we establish some lemmas which are necessary in the proof of the main results.Throughout this paper, we always assume the following.(A1)
For allλ∈Ξ,0<|μik(λ)|<1 and ∑k=1∞‍μik(λ) is uniformly absolute convergence.(A2)
ci(λ)>0, ωij(λ), γij(λ), aij(λ), bij(λ), cij(λ), αij(λ), βij(λ), and δij(λ) are bounded in Ξ(i,j=1,2,…,n).(A3)
The activation functionfi(·) with fi(0)=0,i=1,2,…,n is second-order differentiable and Lipschitz continuous; that is, there exist positive constants Li such that |fi(x)|≤Li|x| for any x∈ℝ.(A4)
Functionsτij(t),i,j=1,2,…,n are nonnegative, bounded, and continuous defined on ℝ+ and 0≤τij(t)≤τ(λ).(A5)
LetD^=-(P^+Q^)∈ℳ, where P^=(p^ij)n×n,p^ii=-ci_+(|a-ii|+|b-ii|)Li, p^ij=Li∏k=1∞‍(1-|μ-ik|)-1∑j=1,j≠in‍∏k=1∞‍(1+ |μ-jk|)(|a-ij|+|b-ij|), Q^=(q^ij)n×n,q^ij=Li∏k=1∞‍(1-|μ-ik|)-1∑j=1n‍∏k=1∞‍(1+|μ-jk|)(|ω-ij|+|α-ij|+|β-ij|), and I^=col{|Ii|+∑j=1n‍(|γ-ij|+|c-ij|+|δ-ij|)|νj|}.Consider the following non-impulsive system (4):
(4)x˙i(t)=-ci(λ)xi(t)+∏0≤tk<t(1+μik)-1x˙i(t)=×[∑j=1n‍ωij(λ)fj(∏0≤tk<t(1+μjk)xj(t-τij(t)))x˙i(t)=×s+⋀j=1naij(λ)fj(∏0≤tk<t(1+μjk)xj(t))x˙i(t)=×s+⋁j=1nbij(λ)fj(∏0≤tk<t(1+μjk)xj(t))x˙i(t)=×s+⋀j=1nαij(λ)fj(∏0≤tk<t(1+μjk)xj(t-τij(t)))x˙i(t)=×s+∑j=1n‍γij(λ)νj+⋀j=1ncij(λ)νjx˙i(t)=×s+⋁j=1nβij(λ)fj(∏0≤tk<t(1+μjk)xj(t-τij(t)))x˙i(t)=×s+⋁j=1nδij(λ)νj+Ii(∏0≤tk<t‍(1+μjk)xj(t-τij(t)))],t≥0,xi(t)=ϕi(t),t∈[-τ(λ),0].
We have the following lemma, which shows that system (2) and (4) is equivalent.Lemma 1.
Assume(A1) holds, thenwe have the following.(i)
Ifxi(t) is a solution of (4), then ui(t)=∏0≤tk<t‍(1+μik)xi(t) is a solution of (2).(ii)
Ifui(t) is a solution of (2), then xi(t)=∏0≤tk<t‍(1+μik)-1ui(t) is a solution of (4).Proof.
Firstly, let us prove (i). For a givenλ∈Ξ, it is easy to see that ui(t)=∏0≤tk<t‍(1+μik(λ))xi(t) is absolutely continuous on the interval (tk,tk+1] and for any t≠tk,  k=1,2,…,
(5)u(t)=(∏0≤tk<t(1+μ1k(λ))x1(t),…,∏0≤tk<t(1+μnk(λ))xn(t))
satisfies system (2). In addition, for every tk∈{tk}k=1∞,
(6)ui(tk+)=limt→tk+∏0≤tj<t(1+μij(λ))xi(t)=∏0≤tj≤tk(1+μij(λ))xi(t),ui(tk)=∏0≤tj<tk(1+μij(λ))xi(t).
Thus, for every k=1,2,…,
(7)ui(tk+)=(1+μik(λ))ui(tk).
The proof is complete.
Next, we prove (ii). Sinceui(t)=∏0≤tk<t‍(1+μik(λ))xi(t) is absolutely continuous on the interval (tk,tk+1] and, in view of (7), it follows that, for any k=1,2,…,
(8)xi(tk+)=∏0≤tj≤tk(1+μij(λ))-1ui(tk+)=∏0≤tj<tk(1+μij(λ))-1ui(tk)=xi(tk),xi(tk-)=∏0≤tj≤tk-1(1+μij(λ))-1ui(tk-)=xi(tk),i=1,2,…,
which implies that xi(t) is continuous on [0,∞). It is easy to prove that xi(t) is absolutely continuous on [0,∞). Now, one can easily check that
(9)x(t)=(∏0≤tk<t(1+μ1k(λ))-1u1(t),…,2∏0≤tk<t(1+μnk(λ))-1un(t))
is the solution of (4). The proof is complete.Definition 2.
LetS be subsets of C([-τ(λ),0],ℝn)≜C which is independent of the parameter λ∈Ξ and let u(t,ϕ,λ) be a solution of FNNs (2) with ϕ∈C.(i)
For any givenλ∈Ξ, if for any initial value ϕ∈S implies that u(t,ϕ,λ)∈S for all t≥0, then S is said to be a robust positive invariant set of system of FNNs (2).(ii)
For any givenλ∈Ξ, if for any initial value ϕ∈S, the solution u(t,ϕ,λ)∈S converges to S as t→∞, that is, dist(u(t,ϕ,λ),S)→0 as t→∞, then S is said to be a global robust attracting set of system of FNNs (2), where dist(φ,S)=infψ∈Sdist(φ,ψ), and dist(φ,ψ)=sups∈[-τ,0]|φ(s)-ψ(s)| for φ∈C.
For a class of differential equations with the term of fuzzy AND and fuzzy OR operation, there is the following useful inequality.Lemma 3 (see [33]).
Letu=(u1,u2,…,un)T and v=(v1,v2,…,vn)T be two states of (2), then we have
(10)|⋀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)|.


## 3. Nonlinear Delay Differential Inequality

In this section, we will establish a new nonlinear delay differential inequality which will play the important role to prove our main results.Lemma 4.
Assume thatu(t)∈C([t0,∞),ℝn) satisfies
(11)d+dt[u(t)]+≤P[u(t)]++Q[u(t)]τ++I,t≥t0,u(t0+θ)=φ(θ)∈C([-τ,0],ℝn),θ∈[-τ,0],
where P=(pij)n×n and pij≥0 for i≠j,  Q=(qij)n×n≥0, I=(I1,I2,…,In)T≥0. If D=-(P+Q)∈ℳ and L=D-1I, then we have the following.(i)
For any constantd≥1, the solution u(t) of (11) satisfies
(12)[u(t)]+≤dL,t≥t0,provided that[φ]τ+≤dL.(ii)
Consider that(13)[u(t)]+≤ze-κ(t-t0)+L,t≥t0,provided that(14)[u(t)]+≤ze-κ(t-t0)+L,t∈[t0-τ,t0],wherez=col{|zi|}∈ΩM(D) and the positive constant κ is determined by the following inequality:
(15)[κE+P+Qeκτ]<0.Proof.
SinceD=-(P+Q)∈ℳ, we have D-1≥0. Let ε=D-1col{1}ϵ (ϵ>0 small enough), then ε>0. In order to prove (12), we will first prove that
(16)[u(t)]+≤dL+ε=col{x-i}=x-,∀t≥t0,
for any given initial function φ∈C([t0-τ,t0],ℝn) with [φ]τ+≤dL.
If (16) does not hold, then there exist i∈{1,2,…,n} and t1>t0 such that
(17)|ui(t1)|=x-i,[u(t)]+≤x-,fort≤t1,(18)d+dt|ui(t1)|≥0.
It follows from (11) and (17) that
(19)d+dt[u(t1)]+≤P[u(t1)]++Q[u(t1)]τ++I≤(P+Q)x-+I=-[dI+col{1}ϵ-I]≤-col{1}ϵ<0,
which contradicts the inequality (18). So (16) holds for all t≥t0. Letting ϵ→0 in (16), we have
(20)[u(t)]+≤dL,t≥t0.
The proof of part (i) is complete.
SinceL=D-1I, we have (P+Q)L+I=0. Then
(21)∑j=1n‍[pij+qij]Lj+Ii=0,i=1,2,…,n.
From (15), we can get
(22)∑j=1n‍[pij+qijeκτ]zj<-κzi,i=1,2,…,n.
In the following, we at first will prove that for any positive constant ϵ,
(23)|ui(t)|≤(1+ϵ)[zie-κ(t-t0)+Li]=wi(t),t≥t0,i=1,2,…,n.
We let
(24)℘={i∈{1,2,…,n}:|ui(t)|>wi(t)forsomet∈[t0,+∞)},ϑi=inf{t∈[t0,+∞):|ui(t)|>wi(t),i∈℘}.
If inequality (23) is not true, then ℘ is nonempty set and there must exist some integer m∈℘ such that ϑm=mini∈℘ϑi∈[t0,+∞).
Byum(t)∈([t-0,+∞),ℝ) and the inequality (23), we can get
(25)ϑm>t0,|um(ϑm)|=wm(ϑm),d+dt|um(ϑm)|≥w˙m(ϑm),(26)|ui(t)|≤wi(t),t∈[t0-τ,ϑm],i=1,2,…,n.
By applying (11) and (21)–(26), we obtain
(27)d+dt|um(ϑm)|≤∑j=1n‍(1+ϵ)zje-κ(ϑm-t0)[pmmj+qmjeκτ]-ϵIm≤∑j=1n‍[pmj+qmjeκτ](1+ϵ)zje-κ(ϑm-t0)<-κzm(1+ϵ)e-κ(ϑm-t0)=w˙m(ϑm),
which contradicts the inequality in (25). Thus the inequality (23) holds. Therefore, letting ϵ→0, we have (13). The proof is complete.By the process of proof of Lemma4, we easily derive the following theoremTheorem 5.
Under the conditions of Lemma4, then
(28)limt→+∞[u(t)]+≤L.


## 4. Main Results

In this section, we will state and prove our main results. The following lemma is very useful to prove Theorem7.Lemma 6.
Assume that0<|ak|<1,(k=1,2,…) and the series of number ∑k=1∞‍ak is absolute convergence, then the infinite products ∏k=1∞‍(1-|ak|)-1,  ∏k=1∞‍(1+ak) and ∏k=1∞‍(1+|ak|) are convergent and ∏k=1∞‍(1-|ak|)-1≥∏k=1∞‍(1+ak)-1,∏k=1∞‍(1+|ak|)≥∏k=1∞‍(1+ak).Proof.
In fact, by the assumption0<|ak|<1,(k=1,2,…), we have
(29)11-|ak|>1,11-|ak|≥11+ak>0,1+|ak|>1,1+|ak|≥1+ak≥1-|ak|>0
which imply that
(30)ln(1-|ak|)≤ln(1+ak)≤ln(1+|ak|),ln(1-|ak|)<0,ln(1+|ak|)>0.
On the other hand, since ∑k=1∞‍ak is absolute convergence, we derive that
(31)limk→∞ak=limk→∞|ak|=0.
Equations (30) and (31) give that
(32)limk→∞-ln(1-|ak|)|ak|=1,limk→∞ln(1+ak)|ak|=1,limk→∞ln(1+|ak|)|ak|=1.
According to (32) and considering that ∑k=1∞‍ak is absolute convergence, we get that the series of positive number -∑k=1∞‍ln(1-|ak|)=∑k=1∞‍ln(1-|ak|)-1,  ∑k=1∞‍ln(1+ak), and ∑k=1∞‍ln(1+|ak|) is convergent. Thus the infinite products ∏k=1∞‍(1-|ak|)-1, ∏k=1∞‍(1+ak), and ∏k=1∞‍(1+|ak|) are convergent. At the same time, combined with (29), we conclude that ∏k=1∞‍(1-|ak|)-1≥∏k=1∞‍(1+ak)-1, ∏k=1∞‍(1+|ak|)≥∏k=1∞‍(1+ak). The proof of Lemma 6 is complete.Theorem 7.
Assume that (A1)–(A5) hold, then
(33)S={ϕ∈C:[ϕ]τ(λ)+≤ΠD^-1I^}
is a robust positive invariant and global robust attracting set of FNNs (2),where
(34)Π=diag(∏k=1∞(1+|μ-1k|),∏k=1∞(1+|μ-2k|),…,∏k=1∞(1+|μ-nk|)).Proof.
Calculating the upper right derivative(d+/dt)[x(t)]+ along system (4) and by using Lemma 6, we have
(35)d+dt|xi(t)|=sgn(xi(t))x˙i(t)≤(-c_i+Li|a-ii|+Li|b-ii|)|xi(t)|≤+Li∏tk<t(1+μik)-1∑j=1,j≠in‍∏tk<t(1+μjk)≤+Li∏tk<t(1+μik)-1∏∑j=1,j≠in‍×(|a-ij|+|b-ij|)|xj(t)|≤+Li∏tk<t(1+μik)-1∑j=1n‍∏tk<t(1+μjk)≤+Li∏tk<t(1+μik)-1∏∑j=1n‍×(|ω-ij|+|a-ij|+|b-ij|)≤+Li∏tk<t(1+μik)-1∏∑j=1n‍×|xj(t-τij(t))|≤+|Ii|+∑j=1n‍(|γ-ij|+|c-ij|+|δ-ij|)|νj|≤(-c_i+Li|a-ii|+Li|b-ii|)|xi(t)|≤+Li∏tk<t(1-|μ-ik|)-1∑j=1,j≠in‍∏tk<t(1+|μ-jk|)≤+Li∏tk<t(1-|μ-ik|)-1∏∑j=1,j≠in‍×(|a-ij|+|b-ij|)|xj(t)|≤+Li∏tk<t(1-|μ-ik|)-1∑j=1n‍∏tk<t(1+|μ-jk|)≤+Li∏tk<t(1-|μ-ik|)-1∏∑j=1n‍×(|ω-ij|+|a-ij|+|b-ij|)≤+Li∏tk<t(1-|μ-ik|)-1∏∑j=1n‍×|xj(t-τij(t))|≤+|Ii|+∑j=1n‍(|γ-ij|+|c-ij|+|δ-ij|)|νj|≤(-c_i+Li|a-ii|+Li|b-ii|)|xi(t)|≤+Li∏k=1∞(1-|μ-ik|)-1∑j=1,j≠in‍∏k=1∞(1+|μ-jk|)≤+Li∏k⩽1∞(1-|μ-ik|)-1∏∑j=1,j≠in‍×!×(|a-ij|+|b-ij|)|xj(t)|≤+Li∏k=1∞(1-|μ-ik|)-1∑j=1n‍∏k=1∞(1+|μ-jk|)≤+Li∏k=1∞(1-|μ-ik|)-1∏∑j=1n‍×(|ω-ij|+|a-ij|+|b-ij|)≤+Li∏k=1∞(1-|μ-ik|)-1∏∑j=1n‍×|xj(t-τij(t))|≤+|Ii|+∑j=1n‍(|γ-ij|+|c-ij|+|δ-ij|)|νj|.
From (A5), (35) can be rewritten as follows:(36)d+dt[x(t)]+≤P^[x(t)]++Q^[x(t)]τ(λ)++I^,t≥0.
Then from the conclusion (i) of Lemma 4, we can obtain
(37)[x(t)]+≤L^1,t≥0,
provided that [ϕ]τ(λ)+≤L^1, where L^1=D^-1I^.
According to Lemma1 and (37), one has
(38)[u(t)]+=diag(∏0≤tk<t(1+μ1k),…,∏0≤tk<t(1+μnk))[x(t)]+≤diag(∏k=1∞(1+|μ-1k|),…,∏k=1∞(1+|μ-nk|))[x(t)]+=Π[x(t)]+≤ΠL^1=L^,
provided that [ϕ]τ(λ)+≤L^, where L^=ΠD^-1I^. In view of Definition 2, we get that S denoted by (33) is a robust positive invariant set of FNNs (2).
On the other hand, sinceD^∈ℳ, there exists a positive vector z=(z1,z2,…,zn)T such that
(39)D^z>0,thatis,(P^+Q^)z<0.
By using continuity, we know that there must exist a positive scalar κ(λ) such that
(40)[κ(λ)E+P^+Q^eκ(λ)τ(λ)]z<0,
where d^≥1 is a constant such that [ϕ]τ(λ)+≤d^L^.
Then by (36), (40), and (A4), all the conditions of Theorem 5 are satisfied, and we have
(41)limt→+∞[x(t)]+≤L^.
According to Lemma 1 and Definition 2, we yield that S denoted by (33) is also a global robust attracting set of FNNs (2).  The proof is complete.Theorem 8.
In addition to (A1)–(A5), further assume I^=0. Then FNNs (2) has a zero solution and the zero solution is global robust exponential stability and the exponential convergent rate equals κ which is determined by (40).


## 5. Illustrative Example

The following illustrative example will demonstrate the effectiveness of our results. Consider the following FNNs with time-varying delays and impulses:(42)u˙i(t)=-ci(λ)ui(t)+∑j=12‍ωij(λ)fj(uj(t-τij(t)))+∑j=1n‍γij(λ)νj+Ii+⋀j=12aij(λ)fj(uj(t))+⋀j=12αij(λ)fj(uj(t-τij(t)))+⋀j=1ncij(λ)νj+⋁j=12bij(λ)fj(uj(t))+⋁j=12βij(λ)fj(uj(t-τij(t)))+⋁j=1nδij(λ)νj,t≠tk,t≥0,Δui(t)=ui(t+)-ui(t-)=μik(λ)ui(t),t=tk,t≥0,ui(t)=ϕi(t),t∈[-τ(λ),0],
where i=1,2,tk∈{1,2,3,…}, Ξ=[π/4,π/2], τ(λ)=1, c1(λ)=h(8+sin(2λ)), c2(λ)=h(9+cos(2λ)),  f1(x)=f2(x)=(1/2)(|x+1|-|x-1|), τij(t)=|sin(i-j)t|, μ1k(λ)=μ2k(λ)=(-1)ksinλ/2k, aii(λ)=h(1+sin2iλ)/4, aij(λ)=((1+sin(i+j)λ)/4)(i≠j), bii(λ)=h(1+2sin(i+j)λ)/6, bij(λ)=((1+2sin(i+j)λ)/6)(i≠j),  αij(λ)=(2+sin(i+j)λ)/9, βij(λ)=(1+2sin(i+j)λ)/9,  ωij(λ)=(1+sin(i+j)λ)/6, cij(λ)=sin(i+j)λ/4,γij(λ)= -sin(i+j)λ/4, and δij(λ)=cos(i+j)λ/4, Ii=-((-1)i/4), νi=(-1)i. By the simple calculation, we obtain(43)P^=h(-711-7),Q^=h(1111),I^=(11),
where h=∏k=1∞‍(2k+1)/(2k-1). Then D^=-(P^+Q^) is a nonsingular M-matrix, and taking κ=0.1, z=(1,1)T, we get
(44)[κE+P^+Q^eκτ]z<0.
Therefore, by Theorem 7, we obtain that
(45)S={ϕ∈C:[ϕ]τ(λ)+≤ΠD^-1I^=(14,14)T}
is a robust positive invariant and global robust attracting set of FNNs (42).



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*Source: 935491-2013-03-21.xml*