# Exponential Stability and Periodicity of Fuzzy Delayed Reaction-Diffusion Cellular Neural Networks with Impulsive Effect

Authors: Guowei Yang; Yonggui Kao; Changhong Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper considers dynamical behaviors of a class of fuzzy
impulsive reaction-diffusion delayed cellular neural networks
(FIRDDCNNs) with time-varying periodic self-inhibitions,
interconnection weights, and inputs. By using delay differential
inequality,M-matrix theory, and analytic methods, some new
sufficient conditions ensuring global exponential stability of the
periodic FIRDDCNN model with Neumann boundary conditions are
established, and the exponential convergence rate index is
estimated. The differentiability of the time-varying delays is not
needed. An example is presented to demonstrate the efficiency and
effectiveness of the obtained results.


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## Body

## 1. Introduction

The fuzzy cellular neural networks (FCNNs) model, which combines fuzzy logic with the structure of traditional neural networks (CNNs) [1–3], has been proposed by Yang et al. [4, 5]. Unlike previous CNNs structures, the FCNNs model has fuzzy logic between its template and input and/or output besides the “sum of product” operation. Studies have shown that the FCNNs model is a very useful paradigm for image processing and pattern recognition [6–8]. These applications heavily depend on not only the dynamical analysis of equilibrium points but also on that of the periodic oscillatory solutions. In fact, the human brain is naturally in periodic oscillatory [9], and the dynamical analysis of periodic oscillatory solutions is very important in learning theory [10, 11], because learning usually requires repetition. Moreover, an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period. Stability analysis problems for FCNNs with and without delays have recently been probed; see [12–22] and the references therein. Yuan et al. [13] have investigated stability of FCNNs by linear matrix inequality approach, and several criteria have been provided for checking the periodic solutions for FCNNs with time-varying delays. Huang [14] has probed exponential stability of fuzzy cellular neural networks with distributed delay, without considering reaction-diffusion effects.Strictly speaking, reaction-diffusion effects cannot be neglected in both biological and man-made neural networks [19–32], especially when electrons are moving in noneven electromagnetic field. In [19], stability is considered for FCNNs with diffusion terms and time-varying delay. Wang and Lu [20] have probed global exponential stability of FCNNs with delays and reaction-diffusion terms. Song and Wang [21] have studied dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays without considering pulsing effects. Wang et al. [22] have discussed exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays. Zhao and Mao [30] have investigated boundedness and stability of nonautonomous cellular neural networks with reaction-diffusion terms. Zhao and Wang [31] have considered existence of periodic oscillatory solution of reaction-diffusion neural networks with delays without fuzzy logic and impulsive effect.As we all know, many practical systems in physics, biology, engineering, and information science undergo abrupt changes at certain moments of time because of impulsive inputs [33]. Impulsive differential equations and impulsive neural networks have been received much interest in recent years; see, for example, [34–42] and the references therein. Yang and Xu [36] have investigated existence and exponential stability of periodic solution for impulsive delay differential equations and applications. Li and Lu [38] have discussed global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses without reaction-diffusion. To the best of our knowledge, few authors have probed the existence and exponential stability of the periodic solutions for the FIRDDCNN model with variable coefficients, and time-varying delays. As a result of the simultaneous presence of fuzziness, pulsing effects, reaction-diffusion phenomena, periodicity, variable coefficients and delays, the dynamical behaviors of this kind of model become much more complex and have not been properly addressed, which still remain important and challenging.Motivated by the above discussion, we will establish some sufficient conditions for the existence and exponential stability of periodic solutions of this kind of FIRDDCNN model, applying delay differential inequality,M-matrix theory, and analytic methods. An example is employed to demonstrate the usefulness of the obtained results.Notations. Throughout this paper, ℝn and ℝn×m denote, respectively, the n-dimensional Euclidean space and the set of all n×m real matrices. The superscript “T” denotes matrix transposition and the notation X≥Y (resp., X>Y), where X and Y are symmetric matrices, means that X-Y is positive semidefinite (resp., positive definite). Ω={x=(x1,…,xm)T,|xi|<μ} is a bounded compact set in space ℝm with smooth boundary ∂Ω and measure mesΩ>0; Neumann boundary condition ∂ui/∂n=0 is the outer normal to ∂Ω; L2(Ω) is the space of real functions Ω which are L2 for the Lebesgue measure. It is a Banach space with the norm ∥u(t,x)∥2=(∑i=1n‍∥ui(t,x)∥22)1/2, where u(t,x)=(u1(t,x),…,un(t,x))T, ∥ui(t,x)∥2=(∫Ω‍|ui(t,x)|2dx)1/2, |u(t,x)|=(|u1(t,x)|,…,|un(t,x)|)T. For function g(x) with positive period ω, we denote g-=maxt∈[0,ω]g(t),g_=mint∈[0,ω]g(t). Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.


## 2. Preliminaries

Consider the impulsive fuzzy reaction-diffusion delayed cellular neural networks (FIRDDCNN) model:(1)∂ui(t,x)∂t=∑l=1m‍∂∂xl(Di∂ui(t,x)∂xl)-ci(t)ui(t,x)+∑j=1n‍aij(t)fj(uj(t,x))+∑j=1n‍bij(t)vj(t)+Ji(t)+⋀j=1nαij(t)gj(uj(t-τj(t),x))+⋁j=1nβij(t)gj(uj(t-τj(t),x))+⋀j=1nTij(t)vj(t)+⋁j=1nHij(t)vj(t),t≠tk,x∈Ω,ui(tk+,x)-ui(tk-,x)=Iik(ui(tk-,x)),t=tk,k∈ℤ+,x∈Ω,∂ui(t,x)∂n=0,t≥t0,x∈∂Ω,ui(t0+s,x)=ψi(s,x),-τj≤s≤0,x∈Ω,
where n≥2 is the number of neurons in the network and ui(t,x) corresponds to the state of the ith neuron at time t and in space x; D=diag(D1,D2,…,Dn) is the diffusion-matrix and Di≥0; Δ=∑k=1m‍(∂2/∂xk2) is the Laplace operator; fj(uj(t,x)) denotes the activation function of the jth unit and vj(t) the activation function of the jth unit; Ji(t) is an input at time t; ci(t)>0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs at time t; aij(t) and bij(t) are elements of feedback template and feed forward template at time t, respectively. Moreover, in model (1), αij(t),βij(t),Tij(t), and Hij(t) are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed forward MIN template, and fuzzy feed forward MAX template at time t, respectively; the symbols “⋀" and “⋁" denote the fuzzy AND and fuzzy OR operation, respectively; time-varying delay τj(t) is the transmission delay along the axon of the jth unit and satisfies 0≤τj(t)≤τj (τj is a constant); the initial condition ϕi(s,x) is bounded and continuous on [-τ,0]×Ω, where τ=max1≤j≤nτj. The fixed moments tk satisfy 0=t0<t1<t2…,limk→+∞tk=+∞,k∈ℕ. ui(tk+,x) and ui(tk-,x) denote the right-hand and left-hand limits at tk, respectively. We always assume ui(tk+,x)=ui(tk,x), for all k∈N. The initial value functions ψ(s,x) belong to PCΩ([-τ,0]×Ω;Rn). PCΩ(J×Ω,L2(Ω))={ψ:J×Ω→L2(Ω)∣ for every t∈J,ψ(t,x)∈L2(Ω); for any fixed x∈Ω,ψ(t,x) is continuous for all but at most countable points s∈J and at these points, ψ(s+,x) and ψ(s-,x) exist, ψ(s+,x)=ψ(s-,x)}, where ψ(s+,x) and ψ(s-,x) denote the right-hand and left-hand limit of the function ψ(s,x), respectively. Especially, let PCΩ=PC([-τ,0]×Ω,L2(Ω)). For any ψ(t,x)=(ψ1(t,x),…,ψn(t,x))∈PCΩ, suppose that |ψi(t,x)|τ=sup-τ<s≤0|ψi(t+s,x)| exists as a finite number and introduce the norm ∥ψ(t)∥2=(∑i=1n‍∥ψi(t)∥22)1/2, where ∥ψi(t)∥2=(∫Ω‍|ψi(t,x)|2dx)1/2.Throughout the paper, we make the following assumptions.(H1)
There exists a positive diagonal matrixF=diag(F1,F2,…,Fn), and G=diag(G1,G2,…,Gn) such that
(2)Fj=supx≠y|fj(x)-fj(y)x-y|,Gj=supx≠y|gj(x)-gj(y)x-y|for allx≠y,j=1,2,…,n.(H2)
ci(t)>0,aij(t),bij(t),αij(t),βij(t),Tij(t),Hij(t),vi(t),Ii(t), and τj(t)≥0 are periodic function with a common positive period ω for all t≥t0,i,j=1,2,…,n.(H3)
Forω>0,i=1,2,…,n, there exists q∈Z+ such that tk+ω=tk+q,  Iik(ui)=Ii(k+q)(ui) and Iik(ui(tk,x)) are Lipschitz continuous in ℝn.Definition 1.
The model in (1) is said to be globally exponentially periodic if (i) there exists one ω-periodic solution and (ii) all other solutions of the model converge exponentially to it as t→+∞.Definition 2 (see [26]).
LetC=([t-τ,t],ℝn), where τ≥0 and F(t,x,y)∈C(ℝ+×ℝn×C,ℝn). Then the function F(t,x,y)=(f1(t,x,y),f2(t,x,y),…,fn(t,x,y))T is called an M-function, if (i) for every t∈ℝ+,x∈ℝn,y(1)∈C, there holds F(t,x,y(1))≤F(t,x,y(2)), for y(1)≤y(2), where y(1)=(y1(1),…,yn(1))T and y(2)=(y1(2),…,yn(2))T; (ii) every ith element of F satisfies fi(t,x(1),y)≤fi(t,x(2),y) for any y∈C,t≥t0, where arbitrary x(1) and x(2)(x(1)≤x(2)) belong to ℝn and have the same ith component xi(1)=xi(2). Here, x(1)=(x1(1),…,xn(1))T,x(2)=(x1(2),…,xn(2))T.Definition 3 (see [26]).
A real matrixA=(aij)n×n is said to be a nonsingular M-matrix if aij≤0(i≠j; i,j=1,…,n) and all successive principal minors of A are positive.Lemma 4 (see [13]).
Letu and u* be two states of the model in (1), then we have
(3)|⋀j=1nαij(t)fj(uj)-⋀j=1nαij(t)fj(uj*)|≤∑j=1n‍|αij(t)|·|fj(uj)-fj(uj*)|,|⋁j=1nβij(t)fj(uj)-⋁j=1nβij(t)fj(uj*)|≤∑j=1n‍|βij(t)|·|fj(uj)-fj(uj*)|.Lemma 5 (see [26]).
Assume thatF(t,x,y) is an M-function, and (i) x(t)<y(t), t∈[t-τ,t0], (ii) D+y(t)>F(t,y(t),ys(t)), D+x(t)≤F(t,x(t),xs(t)), t≥t0, where xs(t)=sup-τ≤s≤0x(t+s), ys(t)=sup-τ≤s≤0y(t+s). Then x(t)<y(t), t≥t0.


## 3. Main Results and Proofs

We should first point out that, under assumptions(H1), (H2), and (H3), the FIRDDCNN model (1) has at least one ω-periodic solution of [26]. The proof of the existence of the ω-periodic solution of (1) can be carried out similar to [26, 28] by the nonlinear functional analysis methods such as topological degree and here is omitted. We will mainly discuss the uniqueness of the periodic solution and its exponential stability.Theorem 6.
Assume that(H1)–(H3) holds. Furthermore, assume that the following conditions hold(H4)
C_-A-F-(α-+β-)G is a nonsingular M-matrix.(H5)
The impulsive operatorshk(u)=u+Ik(u) is Lipschitz continuous in ℝn; that is, there exists a nonnegative diagnose matrix Γk=diag(γ1k,…,γnk) such that |hk(u)-hk(u*)|≤Γk|u-u*| for all u,u*∈ℝn, k∈N+, where |hk(u)|=(|h1k(u1)|,…,|hnk(un)|)T, Ik(u)=(I1k(u1),…,Ink(un))T.(H6)
η=supk∈N+{lnηk/(tk-tk-1)}<λ, where ηk=max1≤i≤n{1,γik},k∈N+.Then the model (1) is global exponential periodic and the exponential convergence rate index λ-η and λ can be estimated by
(4)ξi(λ-c_i)+∑j=1n‍ξj(|a-ij|Fj+eτλ(|α-ij|+|β-ij|)Gj)<0i=1,…,n,
where C_=diag(c_1,…,c_n) and ξi>0, A-=(|a-ij|)n×n, α-=(|α-ij|)n×n, β-=(|β-ij|)n×n, satisfies -ξic_i+∑j=1nξi(|a-ij|Fi+(|α-ij|+|β-ij|)Gi)<0.Proof.
For anyϕ,ψ∈PCΩ, let u(t,x,ϕ)=(u1(t,x,ϕ),…,un(t,x,ϕ))T be a periodic solution of the system (1) starting from ϕ and u(t,x,ψ)=(u1(t,x,ψ),…,un(t,x,ψ))T, a solution of the system (1) starting from ψ. Define
(5)ut(ϕ,x)=u(t+s,x,ϕ),ut(ψ,x)=u(t+s,x,ψ),s∈[-τ,0],
and we can see that ut(ϕ,x),ut(ψ,x)∈PCΩ for all t>0. Let Ui=ui(t,x,ϕ)-ui(t,x,ψ), then from (1) we get
(6)∂Ui∂t=∑l=1m‍∂∂xl(Di∂Ui∂xl)-ci(t)Ui+∑j=1n‍aij(t)×[fj(uj(t,x,ϕ))-fj(uj(t,x,ψ))]+[⋀j=1nαij(t)fj(uj(t-τj(t),x,ϕ))-⋀j=1nαij(t)fj(uj(t-τj(t),x,ψ))]+[⋁j=1nβij(t)fj(uj(t-τj(t),x,ϕ))-⋁j=1nβij(t)fj(uj(t-τj(t),x,ψ))]
for all t≠tk, x∈Ω, i=1,…,n.
Multiplying both sides of (6) by Ui and integrating it in Ω, we have
(7)12ddt∫Ω‍Ui2dx=∫Ω‍Ui∑l=1m‍∂∂xl(Di∂Ui∂xl)dx-ci(t)∫Ω‍Ui2dx+∑j=1n‍aij(t)∫Ω‍Ui×[fj(uj(t,x,ϕ)-fj(uj(t,x,ψ)))]dx+∫Ω‍Ui[⋀j=1nαij(t)fj(uj(t-τj(t),x,ϕ))-⋀j=1nαij(t)fj(uj(t-τj(t),x,ψ))]dx+∫Ω‍Ui[⋁j=1nβij(t)fj(uj(t-τj(t),x,ϕ))-⋁j=1nβij(t)fj(uj(t-τj(t),x,ψ))]dx
for t≠tk,  x∈Ω, i=1,…,n. By boundary condition and Green Formula, we can get
(8)∫Ω‍Ui∑l=1m‍∂∂xl(Di∂Ui∂xl)dx≤-Di∫Ω‍(∇Ui)2dx.
Then, from (8), (9), (H1)-(H2), Lemma 4, and the Holder inequality,
(9)ddt∥Ui∥22≤-2c_i∥Ui∥22+2∑j=1n‍|a-ij|Fj∥Ui∥2∥Uj∥2+2∑j=1n‍(|α-ij|+|β-ij|)Gj∥Ui∥2×∥uj(t-τj(t),x,ϕ)-uj(t-τj(t),x,ψ)∥2,t≠tk.
Thus,
(10)D+∥Ui∥2≤-c_i∥Ui∥2+∑j=1n‍|α-ij|Fj∥Uj∥2+∑j=1n‍(|α-ij|+|β-ij|)Gj×∥uj(t-τj(t),x,ϕ)-uj(t-τj(t),x,ψ)∥2,t≠tk
for i=1,…,n. Since C_-(A-F+(α-+β-)G) is a nonsingular M-matrix, there exists a vector ξ=(ξ1,…,ξn)T>0 such that
(11)-ξic_i+∑j=1n‍ξj(|a-ij|Fj+(|α-ij|+|β-ij|)Gj)<0.
Considering functions
(12)Ψi(y)=ξi(y-c_i)+∑j=1n‍ξj(|a-ij|Fj+eτy(|α-ij|+|β-ij|)Gj),i=1,…,n,
we know from (11) that Ψi(0)<0 and Ψi(y) is continuous. Since dΨi(y)/dy>0, Ψi(y) is strictly monotonically increasing, there exists a scalar λi>0 such that
(13)Ψi(λi)=ξi(λi-c_i)+∑j=1n‍ξj(|a-ij|Fj+eτλi(|α-ij|+|β-ij|)Gj)=0,i=1,…,n.
Choosing 0<λ<min{λ1,…,λn}, we have
(14)ξi(λi-c_i)+∑j=1n‍ξj(|a-ij|Fj+eτλi(|α-ij|+|β-ij|)Gj)<0,i=1,…,n.
That is,
(15)λξ-(C_-A-F)ξ+(α-+β-)Gξe-λt<0.
Furthermore, choose a positive scalar p large enough such that
(16)pe-λtξ>(1,1,…,1)T,t∈[-τ,0].
For any ε>0, let
(17)r(t)=pe-λt(∥ϕ-ψ∥2+ε)ξ,t0≤t<t1.
From (15)–(17), we obtain
(18)D+r(t)>-(C_-A-F)r(t)+(α-+β-)Grs(t)=:V(t,r(t),rs(t)),t0≤t<t1,
where rs(t)=(r1s(t),…,rns(t))T and ris(t)=sup-τ≤s≤0pe-λ(t+s)(∥ϕ-φ∥2+ε)ξi. It is easy to verify that V(t,r(t),rs(t)) is an M-function. It follows also from (16) and (17) that
(19)∥Ui∥2≤∥ϕ-φ∥2<pe-λtξi∥ϕ-φ∥2<ri(t),t∈[-τ,0],i=1,2,…,n.
Denote
(20)U⋄:=(∥u1(t,x,ϕ)-u1(t,x,ψ)∥2,…,∥un(t,x,ϕ)-un(t,x,ψ)∥2)T,U⋄(s):=(∥u1(t,x,ϕ)-u1(t,x,ψ)∥2(s),…,∥un(t,x,ϕ)-un(t,x,ψ)∥2(s))T,
where ∥Ui∥2(s)=sup-τ≤s≤0∥ui(t+s,x,ϕ)-ui(t+s,x,ψ)∥2, then
(21)U⋄<r(t),t∈[-τ,0].
From (10), we can obtain
(22)D+U⋄≤-(C_-A-F)U⋄+(α-+β-)GU⋄(s)=V(t,U⋄,U⋄(s)),t≠tk.
Now, it follows from (18)–(22) and Lemma 5 that
(23)U⋄<r(t)=pe-λt(∥ϕ-ψ∥2+ε)ξ,t0≤t<t1.
Letting ε→0, we have
(24)U⋄≤pξ∥ϕ-ψ∥2e-λt,t0≤t<t1.
And moreover, from (24), we get
(25)(∑i=1n‍∥Ui∥22)1/2≤p(∑i=1n‍ξi2)1/2∥ϕ-ψ∥2e-λt,t0≤t<t1.
Let M~=p(∑i=1n‍ξi2)1/2, then M~≥1. Define W(t)=∥ut(x,ϕ)-ut(x,ψ)∥2; it follows from (25) and the definitions of ut(ϕ,x) and ut(ψ,x) that
(26)W(t)=∥ut(x,ϕ)-ut(x,ψ)∥2≤M~∥ϕ-ψ∥2e-λt,t0≤t<t1.
It is easily observed that
(27)W(t)≤M~∥ϕ-ψ∥2e-λt,-τ≤t≤t0=0.
Because (26) holds, we can suppose that for l≤k inequality
(28)W(t)≤η0⋯ηl-1M~∥ϕ-ψ∥2e-λt,tl-1≤t<tl
holds, where η0=1. When l=k+1, we note (H5) that
(29)W(tk)=∥utk(x,ϕ)-utk(x,ψ)∥2=∥hk(utk-(x,ϕ))-hk(utk-(x,ψ))∥2≤ρ(Γk2)∥utk-(x,ϕ)-utk-(x,ψ)∥2=ρ(Γk2)W(tk-)≤η0⋯ηl-1ρ(Γk2)M~∥ϕ-ψ∥2e-λtk≤η0⋯ηl-1ηkρ(Γk2)M~∥ϕ-ψ∥2e-λtk,
where ρ(Γk2) is the spectral radius of Γk2. Let M=max{M~,ρ(Γk2)M~}, by (28), (29), and η≥1, we obtain
(30)W(t)≤η0⋯ηl-1ηkM∥ϕ-ψ∥2e-λt,tk-τ≤t≤tk.
Combining (10), (17), (30), and Lemma 5, we get
(31)W(t)≤η0⋯ηl-1ηkM∥ϕ-ψ∥2e-λt,tk≤t<tk+1,k∈N+.
Applying mathematical induction, we conclude that
(32)W(t)≤η0⋯ηl-1M∥ϕ-ψ∥2e-λt,tk-1≤t<tk,k∈N+.
From (H6) and (32), we have
(33)W(t)≤eηt1eη(t2-t1)⋯eη(tk-1-tk-2)×M∥ϕ-ψ∥2e-λt≤M∥ϕ-ψ∥2eηte-λt=M∥ϕ-ψ∥2e-(λ-η)t,tk-1≤t<tk,k∈N+.
This means that
(34)∥ut(x,ϕ)-ut(x,ψ)∥2≤M∥ϕ-ψ∥2e-(λ-η)t≤M∥ϕ-ψ∥2e-(λ-η)(t-τ),t≥t0,
choosing a positive integer N such that
(35)Me-(λ-η)(Nω-τ)≤16.
Define a Poincare mapping 𝔇:Γ→Γ by
(36)𝔇(ϕ)=uω(x,ϕ),
Then
(37)𝔇N(ϕ)=uNω(x,ϕ).
Setting t=Nω in (34), from (35) and (37), we have
(38)∥𝔇N(ϕ)-𝔇N(ψ)∥2≤16∥ϕ-ψ∥2,
which implies that 𝔇N is a contraction mapping. Thus, there exists a unique fixed point ϕ*∈Γ such that
(39)𝔇N(𝔇(ϕ*))=𝔇(𝔇N(ϕ*))=𝔇(ϕ*).
From (37), we know that 𝔇(ϕ*) is also a fixed point of 𝔇N, and then it follows from the uniqueness of the fixed point that
(40)𝔇(ϕ*)=ϕ*,thatis,uω(x,ϕ*)=ϕ*.
Let u(t,x,ϕ*) be a solution of the model (1), then u(t+ω,x,ϕ*) is also a solution of the model (1). Obviously,
(41)ut+ω(x,ϕ*)=ut(uω(x,ϕ*))=ut(x,ϕ*),
for all t≥t0. Hence, u(t+ω,x,ϕ*)=u(t,x,ϕ*), which shows that u(t,x,ϕ*) is exactly one ω-periodic solution of model (1). It is easy to see that all other solutions of model (1) converge to this periodic solution exponentially as t→+∞, and the exponential convergence rate index is λ-η. The proof is completed.Remark 7.
Whenci(t)=ci, aij(t)=aij, bij(t)=bij, αij(t)=αij, βij(t)=βij, Tij(t)=Tij, Hij(t)=Hij, vi(t)=vi, Ii(t)=Ii, and τt=τi(ci,aij,bij,αij,βij,Tij,Hij,vi,Ii, and τi are constants), then the model (1) is changed into(42)∂ui(t,x)∂t=∑l=1m‍∂∂xl(Di∂ui(t,x)∂xl)-ciui(t,x)+∑j=1n‍aijfj(uj(t,x))+∑j=1n‍bijvj+Ji+⋀j=1nαijgj(uj(t-τj(t),x))+⋁j=1nβijgj(uj(t-τj(t),x))+⋀j=1nTijvj+⋁j=1nHijvj,t≠tk,x∈Ω,ui(tk+,x)-ui(tk-,x)=Iik(ui(tk-,x)),t=tk,k∈ℤ+,x∈Ω,∂ui(t,x)∂n=0,t≥t0,x∈∂Ω,ui(t0+s,x)=ψi(s,x),-τj≤s≤0,x∈Ω.
For any positive constant ω≥0, we have ci(t+ω)=ci(t), aij(t+ω)=aij(t), bij(t+ω)=bij(t), αij(t+ω)=αij(t), βij(t+ω)=βij(t), Tij(t+ω)=Tij(t), Hij(t+ω)=Hij(t), vi(t+ω)=vi(t), Ii(t+ω)=Ii(t), and τi(t+ω)=τi(t) for t≥t0. Thus, the sufficient conditions in Theorem 6 are satisfied.Remark 8.
IfIk(·)=0, the model (1) is changed into
(43)∂ui(t,x)∂t=∑l=1m‍∂∂xl(Di∂ui(t,x)∂xl)-ci(t)ui(t,x)+∑j=1n‍aij(t)fj(uj(t,x))+∑j=1n‍bij(t)vj(t)+Ji(t)+⋀j=1nαij(t)gj(uj(t-τj(t),x))+⋁j=1nβij(t)gj(uj(t-τj(t),x))+⋀j=1nTij(t)vj(t)+⋁j=1nHij(t)vj(t),t≠tk,x∈Ω,∂ui(t,x)∂n=0,t≥t0,x∈∂Ω,ui(t0+s,x)=ψi(s,x),-τj≤s≤0,x∈Ω,
which has been discussed in [22]. As Song and Wang have pointed out, the model (43) is more general than some well-studied fuzzy neural networks. For example, when ci(t)>0,aij(t),bij(t),αij(t),βij(t),Tij(t),Hij(t),vi(t), and Ii(t) are all constants, the model in (43) reduces the model which has been studied by Huang [19]. Moreover, if Di=0, τi(t)=0, fi(θ)=gi(θ)=(1/2)(|θ+1|-|θ-1|), (i=1,…,n), then model (42) covers the model studied by Yang et al. [4, 5] as a special case. If Di=0 and τj(t) is assumed to be differentiable for i,j=1,2,…,n, then model (43) can be specialized to the model investigated in Liu and Tang [12] and Yuan et al. [13]. Obviously, our results are less conservative than that of the above-mentioned literature, because they do not consider impulsive effects.


## 4. Numerical Examples

Example 9.
Consider a two-neuron FIRDDCNN model:(44)∂ui(t,x)∂t=∑l=1m‍∂∂xl(Di∂ui(t,x)∂xl)-ci(t)ui(t,x)+∑j=12‍aij(t)fj(uj(t,x))+∑j=12‍bij(t)vj(t)+Ji(t)+⋀j=12αij(t)gj(uj(t-τj(t),x))+⋁j=12βij(t)gj(uj(t-τj(t),x))+⋀j=12Tij(t)vj(t)+⋁j=12Hij(t)vj(t),t≠tk,x∈Ω,ui(tk+,x)=(1-γik)ui(tk-,x),t=tk,k∈ℤ+,x∈Ω,∂ui(t,x)∂n=0,t≥t0,x∈∂Ω,ui(t0+s,x)=ψi(s,x),-τj≤s≤0,x∈Ω,
where i=1,2.c1(t)=26, c2(t)=20.8, a11(t)=-1-cos(t), a12(t)=1+cos(t), a21(t)=1+sin(t), a22(t)=-1-sin(t), D1=8, D2=4, ∂ui(t,x)/∂n=0(t≥t0,x=0,2π), γ1k=0.4, γ2k=0.2, ψ1(·)=ψ1(·)=5, b11(t)=b21(t)=cos(t), b12(t)=b22(t)=-cos(t), J1(t)=J2(t)=1, H11(t)=H21(t)=sin(t), H12(t)=H22(t)=-1+sin(t), T11(t)=T21(t)=-sin(t), T12(t)=T22(t)=2+sin(t), τ1(t)=τ2(t)=1, fj(uj)=uj(t,x)(j=1,2), gj(uj(t-1,x))=uj(t-1,x)e-uj(t-1,x)(j=1,2), α11(t)=-12.8, α21(t)=α12(t)=-1+cos(t), α22(t)=-10, β11(t)=12.8, β12(t)=-1+sin(t)=β21(t), β22(t)=10, vj(t)=sin(t). We assume that there exists q=6 such that tk+2π=tk+q. Obviously, f1,f2,g1, and g2 satisfy the assumption (H1) with F1=F2=G1=G2=1 and (H2) and (H3) are satisfied with a common positive period 2π(45)C_-(A-+α-+β-)F=[250045]
is a nonsingular M-matrix. The conditions of Theorem 6 are satisfied, hence there exists exactly one 2ω-periodic solution of the model and all other solutions of the model converge exponentially to it as t→+∞. Furthermore, the exponential converging index can be calculated as λ=0.021, because here ηk=1 and η=0. The simulation results are shown in Figures 1, 2, 3, and 4, respectively.Figure 1
State responseu1(t,x) of model (44) without impulsive effects.Figure 2
State responseu1(t,x) of model (44) with impulsive effects.Figure 3
State responseu2(t,x) of model (44) without impulsive effects.Figure 4
State responseu2(t,x) of model (44) with impulsive effects.


## 5. Conclusions

In this paper, periodicity and global exponential stability of a class of FIRDDCNN model with variable both coefficients and delays have been investigated. By using Halanay’s delay differential inequality,M-matrix theory, and analytic methods, some new sufficient conditions have been established to guarantee the existence, uniqueness, and global exponential stability of the periodic solution. Moreover, the exponential convergence rate index can be estimated. An example and its simulation have been given to show the effectiveness of the obtained results. In particular, the differentiability of the time-varying delays has been removed. The dynamic behaviors of fuzzy neural networks with the property of exponential periodicity are of great importance in many areas such as learning systems.



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*Source: 645262-2013-03-20.xml*