R E S E A R C H
Open Access
Global robust exponential synchronization of
BAM recurrent FNNs with infinite distributed
delays and diffusion terms on time scales
Kaihong Zhao
**Correspondence:
[email protected] Department of Applied
Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, People’s Republic of China
Abstract
In this article, the global robust exponential synchronization of reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) with infinite distributed delays on time scales is investigated. Applied Lyapunov functional and inequality skills, some sufficient criteria are established to guarantee the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs with infinite distributed delays on time scales. One example is given to illustrate the effectiveness of our results.
Keywords: globally robust exponential synchronization; reaction-diffusion BAM recurrent FNNs; infinite distributed delays; Lyapunov functional; time scales
1 Introduction
The study on the artificial neural networks has attracted much attention because of their potential applications such as signal processing, image processing, pattern classification, quadratic optimization, associative memory, moving object speed detection,etc. Many kinds of models of neural networks have been proposed by some famous scholars. One of these important neural network models is the bidirectional associative memory (BAM) neural network models, which were first introduced by Kosko [–]. It is a special class of recurrent neural networks that can store bipolar vector pairs. The BAM neural network is composed of neurons arranged in two layers, the X-layer and the Y-layer. The neurons in one layer are fully interconnected to the neurons in the other layer. Through iterations of forward and backward information flows between the two layers, it performs a two-way associative search for stored bipolar vector pairs and generalize the single-layer auto-associative Hebbian correlation to a two-layer pattern-matched heteroauto-associative circuits. Therefore, this class of networks possesses good application prospects in some fields such as pattern recognition, signal and image process, artificial intelligence []. In general, ar-tificial neural networks have complex dynamical behaviors such as stability, synchroniza-tion, periodic or almost periodic solutions, invariant sets and attractors, and so forth. We can refer to [–] and the references cited therein. Therefore, the analysis of dynamical behaviors for neural networks is a necessary step for practical design of neural networks. As one of the famous neural network models, it has attracted many attention in the past two decades [–] since the BAM model was proposed by Kosko. The dynamical be-haviors such as uniqueness, global asymptotic stability, exponential stability and invariant
sets and attractors of the equilibrium point or periodic solutions were investigated for BAM neural networks with different types of time delays (see [–, ]).
Synchronization has attracted much attention after it was proposed by Carrolet al.[, ]. The principle of drive-response synchronization is this: the driver system sends a signal through a channel to the responder system, which uses this signal to synchronize itself with the driver. Namely, the response system is influenced by the behavior of the drive system, but the drive system is independent of the response one. In recent years, many results concerning a synchronization problem of time lag neural networks have been investigated in the literature [, , –, , , , ].
As is well known, both in biological and man-made neural networks, strictly speaking, diffusion effects cannot be avoided when electrons are moving in asymmetric electromag-netic fields, so we must consider that the activations vary in space as well as in time. Many researchers have studied the dynamical properties of continuous time reaction-diffusion neural networks (see, for example, [, , , , , , , , ]).
However, in mathematical modeling of real world problems, we will encounter some other inconveniences such as complexity and 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 LB Yang proposed the fuzzy CNNs (FCNNs) [] 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 their template input and/or output besides the sum of product operation. FCNNs are very a useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. Therefore, it is nec-essary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks. To the best of our knowledge, few authors have considered the synchronization of reaction-diffusion recurrent fuzzy neural networks with delays and Dirichlet boundary conditions on time scales which is a challenging and important problem in theory and application. Therefore, in this paper, we will investigate the global robust exponential syn-chronization of delayed reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) on time scales as follows:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u
i (t,x) =
l
k=∂∂xk(aik
∂ui
∂xk) –biui(t,x) +
m
j=cijfj(vj(t–τ,x)) +Ii
+nj=pijFj(uj(t–τ,x)) +
n
j=rij
+∞
kij(s)Fj(uj(t–s,x))s + nj=qijFj(uj(t–τ,x)) + nj=wij
+∞
kij(s)Fj(uj(t–s,x))s +nj=dijμj+
n
j=Sijμj+ nj=Tijμj,
v
j (t,x) =
l
k=
∂ ∂xk(ξjk
∂vj
∂xk) –ηjvj(t,x) +
n
i=ζjigi(ui(t–τ,x)) +Jj
+mi=λjiGi(vi(t–τ,x)) +
m
i=ρji
+∞
κji(s)Gi(vi(t–s,x))s + mi=πjiGi(vi(t–τ,x)) + mi=σji
+∞
κji(s)Gi(vi(t–s,x))s
+mi=hjiνi+
m
i=Mjiνi+ mi=Njiνi,
(.)
subject to the following initial conditions
ui(s,x) =φi(s,x), (s,x)∈[–τ, ]T×,
and Dirichlet boundary conditions
ui(t,x) = , (t,x)∈[,∞)T×∂,
vj(t,x) = , (t,x)∈[,∞)T×∂,
(.)
wherei= , , . . . ,n;j= , , . . . ,m.T⊂Ris a time scale andT∩[, +∞)[, +∞)Tis un-bounded andT∩[–τ, ][τ, ]T=φ.τ> is constant time delay.x= (x,x, . . . ,xl)T∈
⊂Rland={x= (x,x, . . . ,xl)T:|xi|<li,i= , , . . . ,l}is a bounded compact set with
smooth boundary ∂ in space Rl. u= (u
,u, . . . ,un)T∈Rn, v= (v,v, . . . ,vm)T ∈Rm.
ui(t,x) andvj(t,x) are the state of theith neurons and thejth neurons at timetand in
spacex, respectively.I= (I,I, . . . ,In)T∈RnandJ= (J,J, . . . ,Jm)T∈Rmare constant
in-put vectors. The smooth functions aik> andξjk > correspond to the transmission
diffusion operators along with theith neurons and thejth neurons, respectively.bi> ,
ηj> ,μj,νi,cij,pij, rij, qij, wij,dij,Sij,Tij, ζji, λji, ρji, πji,σji,hji, Mji,Mjiare constants.
biandηjdenote the rate with which theith neurons andjth neurons will reset their
po-tential to the resting state in isolation when disconnected from the network and external inputs, respectively.cij,pij,rij, qij,wij,dij,Sij,Tij, ζji,λji,ρji,πji, σji, hji,Mji,Mji denote
the connection weights.fj(·) (j= , , . . . ,m) andgi(·) (i= , , . . . ,n) denote the activation
function of the jth neurons of Y-layer on theith neurons of X-layer and theith neu-rons of X-layer on thejth neurons of Y-layer at timetand in spacex, respectively.Fj(·)
(j= , , . . . ,n) denotes the fuzzy activation function of thejth neurons on theith neu-rons inside of X-layer.Gi(·) (i= , , . . . ,m) denotes the fuzzy activation function of the
ith neurons on thejth neurons inside of Y-layer.μj(j= , , . . . ,n) denotes the bias of the
jth neurons on theith neurons inside of X-layer.νi(i= , , . . . ,m) denotes the bias of the
ith neurons on thejth neurons inside of Y-layer., denote the fuzzy AND and fuzzy OR operations, respectively. φ(t,x) = (φ(t,x),φ(t,x), . . . ,φn(t,x))T: [–τ, ]T×→Rn,
ϕ(t,x) = (ϕ(t,x),ϕ(t,x), . . . ,ϕm(t,x))T: [–τ, ]T×→Rmare rd-continuous with respect tot∈[–τ, ]Tand continuous with respect tox∈.
In order to investigate the global robust exponential synchronization for system (.)-(.), the quantitiesbi,aik,cij,pij,rij,qij,wij,ηj,ξjk,ζji,λji,ρji,πjiandσjimay be considered
as intervals as follows: <bi≤bi<∞,aik≤aik≤aik,|cij| ≤ |cij| ≤ |cij|,|pij| ≤ |pij| ≤ |pij|,
|rij| ≤ |rij| ≤ |rij|,|qij| ≤ |qij| ≤ |qij|,|wij| ≤ |wij| ≤ |wij|, <ηj≤ηj<∞,ξjk ≤ξjk ≤ξjk,
|ζ
ji| ≤ |ζji| ≤ |ζji|,|λji| ≤ |λji| ≤ |λji|,|ρji| ≤ |ρji| ≤ |ρji|,|πji| ≤ |πji| ≤ |πji|,|σji| ≤ |σji| ≤
|σji|.
Take the time scaleT=R(real number set), then system (.)-(.) can be changed into the following continuous case (.)-(.):
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ui(t,x)
∂t =
l
k=
∂ ∂xk(aik
∂ui
∂xk) –biui(t,x) +
m
j=cijfj(vj(t–τ,x)) +Ii
+nj=pijFj(uj(t–τ,x)) +
n
j=rij
+∞
kij(s)Fj(uj(t–s,x)) ds
+ nj=qijFj(uj(t–τ,x)) + nj=wij
+∞
kij(s)Fj(uj(t–s,x)) ds +nj=dijμj+
n
j=Sijμj+ nj=Tijμj,
∂vj(t,x)
∂t =
l k=
∂ ∂xk(ξjk
∂vj
∂xk) –ηjvj(t,x) +
n
i=ζjigi(ui(t–τ,x)) +Jj
+mi=λjiGi(vi(t–τ,x)) +
m
i=ρji
+∞
κji(s)Gi(vi(t–s,x)) ds + mi=πjiGi(vi(t–τ,x)) + mi=σji
+∞
κji(s)Gi(vi(t–s,x)) ds +mi=hjiνi+
m
i=Mjiνi+ mi=Njiνi,
subject to the following initial conditions
ui(s,x) =φi(s,x), (s,x)∈[–τ, ]×,
vj(s,x) =ϕj(s,x), (s,x)∈[–τ, ]×,
(.)
and Dirichlet boundary conditions
ui(t,x) = , (t,x)∈[,∞)×∂,
vj(t,x) = , (t,x)∈[,∞)×∂.
(.)
Take the time scaleT=Z(integer number set), then system (.)-(.) can be changed into the following discrete case (.)-(.):
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
tui(t,x) =
l k=
∂ ∂xk(aik
∂ui
∂xk) –biui(t,x) +
m
j=cijfj(vj(t–τ,x)) +Ii
+nj=pijFj(uj(t–τ,x)) +
n
j=rij
∞
s=kij(s)Fj(uj(t–s,x))
+ nj=qijFj(uj(t–τ,x)) + nj=wij
∞
s=kij(s)Fj(uj(t–s,x))
+nj=dijμj+
n
j=Sijμj+ nj=Tijμj,
tvj(t,x) =
l k=
∂ ∂xk(ξjk
∂vj
∂xk) –ηjvj(t,x) +
n
i=ζjigi(ui(t–τ,x)) +Jj
+mi=λjiGi(vi(t–τ,x)) +
m
i=ρji
∞
s=κji(s)Gi(vi(t–s,x))
+ mi=πjiGi(vi(t–τ,x)) + mi=σji
∞
s=κji(s)Gi(vi(t–s,x))
+mi=hjiνi+
m
i=Mjiνi+ mi=Njiνi,
(.)
subject to the following initial conditions
ui(s,x) =φi(s,x), (s,x)∈ {–τ, –τ+ , . . . , –, –, } ×,
vj(s,x) =ϕj(s,x), (s,x)∈ {–τ, –τ+ , . . . , –, –, } ×,
(.)
and Dirichlet boundary conditions
ui(t,x) = , (t,x)∈Z+×∂,
vj(t,x) = , (t,x)∈Z+×∂,
(.)
where t∈Z, τ is a positive integer, Z+={, , , . . .}, tui(t,x)ui(t+ ,x) –ui(t,x),
tvj(t,x)vj(t+ ,x) –vj(t,x).
If we chooseT=R, thenσ(t) =t,μ(t) = . In this case, system (.)-(.) is the contin-uous reaction-diffusion BAM recurrent FNNs (.)-(.). IfT=Z, thenμ(t) = , system (.)-(.) is the discrete difference reaction-diffusion BAM recurrent FNNs (.)-(.). In this paper, we study the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs (.)-(.), which unify both the continuous case and the discrete difference case. What is more, system (.)-(.) is a good model for handling many prob-lems such as predator-prey forecast or optimizing of goods output.
2 Preliminaries
In this section, we first recall some basic definitions and lemmas on time scales which are used in what follows.
LetTbe a nonempty closed subset (time scale) ofR. The forward and backward jump operatorsρ,σ:T→Tand the graininessμ:→R+are defined, respectively, by
σ(t) =inf{s∈T:s>t},
ρ(t) =sup{s∈T:s<t},
μ(t) =σ(t) –t.
A pointt∈Tis called left-dense ift>infTandρ(t) =t, left-scattered ifρ(t) <t, right-dense ift<supTandσ(t) =t, and right-scattered ifσ(t) >t. IfThas a left-scattered maxi-mumm, thenTk=T\ {m}, otherwiseTk=T. IfThas a right-scattered minimumm, then
Tk=T\ {m}, otherwiseTk=T.
Definition .([]) A functionf:T→Ris called regulated provided its right-hand side limits exist (finite) at all right-hand side points inTand its left-hand side limits exist (finite) at all left-hand side points inT.
Definition .([]) A functionf :T→Ris called rd-continuous provided it is con-tinuous at right-dense point in Tand its left-hand side limits exist (finite) at left-dense points inT. The set of rd-continuous functionf :T→Rwill be denoted byCrd=Crd(T) =
Crd(T,R).
Definition .([]) Assumef:T→Randt∈Tk. Then we definef(t) to be the number
(if it exists) with the property that given any> there exists a neighborhoodUoft(i.e.,
U= (t–,t+)∩Tfor some> ) such that fσ(t)–f(s)–f(t)σ(t) –s<σ(t) –s
for alls∈U. We callf(t) the delta (or Hilger) derivative off att. The set of functions f :T→Rthat is a differentiable and whose derivative is rd-continuous is denoted byC
rd=
Crd(T) =Crd(R,T).
Iff is continuous, thenf is rd-continuous. Iff is rd-continuous, thenf is regulated. Iff
is delta differentiable att, thenf is continuous att.
Lemma .([]) Let f be regulated,then there exists a function F which is delta differen-tiable with region of differentiation D such that F(t) =f(t)for all t∈D.
Definition .([]) Assume thatf :T→Ris a regulated function. Any functionFas in Lemma . is called a-antiderivative off. We define the indefinite integral of a regulated functionf by
whereCis an arbitrary constant andFis a-antiderivative off. We define the Cauchy integral byabf(s)s=F(b) –F(a) for alla,b∈T.
A functionF:T→Ris called an antiderivative off :T→RprovidedF(t) =f(t) for
allt∈Tk.
Lemma .([]) If a,b∈T,α,β∈Rand f,g∈C(T,R),then
(i) ab[αf(t) +βg(t)]t=αabf(t)t+βabg(t)t, (ii) iff(t)≥for alla≤t≤b,thenabf(t)t≥,
(iii) if|f(t)| ≤g(t)on[a,b){t∈T:a≤t≤b},then|abf(t)t| ≤abg(t)t.
A functionp:T→Ris called regressive if +μ(t)p(t)= for allt∈Tk. The set of all
regressive and rd-continuous functionsf :T→Rwill be denoted byR=R(T) =R(T,R). We define the setR+of all positively regressive elements ofRbyR+=R+(T,R) ={p∈
R: +μ(t)p(t) > for allt∈T}. Ifpis a regressive function, then the generalized expo-nential functionep(t,s) is defined byep(t,s) =exp{
t
sξμ(τ)(p(τ))τ}for alls,t∈T, with the
cylinder transformation
ξh(z) =
Log (+hz)
h , ifh= ,
z, ifh= .
Letp,q:T→Rbe two regressive functions, we define
p⊕q=p+q+μpq,
p= – p
+μp,
pq=p⊕p(q).
Ifp∈R+, thenp∈R+.
The generalized exponential function has the following properties.
Lemma .([]) Assume that p,q:T→Rare two regressive functions,then
(i) ep(σ(t),s) = ( +μ(t)p(t))ep(t,s);
(ii) /ep(t,s) =ep(t,s);
(iii) ep(t,s) = /ep(s,t) =ep(s,t);
(iv) ep(t,s)ep(s,r) =ep(t,r);
(v) [ep(t,s)]=p(t)ep(t,s);
(vi) [ep(c,·)]= –p[ep(c,·)]σ for allc∈T;
(vii) (d/dz)[ez(t,s)] = [
t
s /( +μ(τ)z)τ]ez(t,s).
Lemma .([]) Assume that f,g:T→Rare delta differentiable at t∈Tk.Then
(fg)(t) =f(t)g(t) +fσ(t)g(t) =g(t)f(t) +gσ(t)f(t).
Lemma .([]) For each t∈T,let N be a neighborhood of t.Then,for V ∈Crd(T,R+),
that
u(t)
Vσ(t)–V(t) –μ(t)f(t)<D+V(t) + for each s∈N,s>t,
whereμ(t) =σ(t) –s.If t is right-scattered and V(t)is continuous at t,this reduces to D+V(t) = V(σ(t))–V(t)
σ(t)–t .
Next, we introduce the Banach space which is suitable for system (.)-(.).
Let={x= (x,x, . . . ,xl)T:|xi|<li,i= , , . . . ,l}be an open bounded domain inRlwith
smooth boundary∂. LetCrd(T×,Rn+m) be the set consisting of all the vector function
y(t,x) = (y(t,x),y(t,x), . . . ,yn+m(t,x))T which is rd-continuous with respect tot∈Tand
continuous with respect tox∈. For everyt∈Tandx∈, we define the setCTt ={y(t,·) :
y∈C(,Rn+m)}. ThenCt
Tis a Banach space with the normy(t,·)= (in=+myi(t,·))/, whereyi(t,·)= (
|yi(t,x)|
dx)/. LetC
rd([–τ, ]T×,Rn+m) consist of all functions
f(t,x) which map [–τ, ]T× into Rn+m andf(t,x) is rd-continuous with respect to
t∈[–τ, ]T and continuous with respect tox∈. For everyt∈[–τ, ]Tandx∈, we define the setCt[–τ,]T={u(t,·) :u∈C(,Rn+m)}. ThenCt
[–τ,]Tis a Banach space equipped
with the normψ= ( n+m
i= φi)/, whereψ(t,x) = (ψ(t,x),ψ(t,x), . . . ,ψn+m(t,x))T∈
Ct
[–τ,]T,ψi(t,·)= (
|ψi(·,x)|
τdx)/,|ψi(·,x)|τ =sups∈[–τ,]T|ψi(s,x)|.
In order to achieve the global robust exponential synchronization, the following system (.)-(.) is the controlled slave system corresponding to the master system (.)-(.):
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˜ u
i (t,x) =
l k=
∂ ∂xk(aik
∂˜ui
∂xk) –biu˜i(t,x) +
m
j=cijfj(v˜j(t–τ,x)) +Ii
+nj=pijFj(u˜j(t–τ,x)) +
n
j=rij
+∞
kij(s)Fj(u˜j(t–s,x))s
+ nj=qijFj(u˜j(t–τ,x)) + nj=wij
+∞
kij(s)Fj(u˜j(t–s,x))s +nj=dijμj+
n
j=Sijμj+ nj=Tijμj+miEi(t,x),
˜ v
j (t,x) =
l
k=
∂ ∂xk(ξjk
∂˜vj
∂xk) –ηjv˜j(t,x) +
n
i=ζjigi(u˜i(t–τ,x)) +Jj
+mi=λjiGi(v˜i(t–τ,x)) +
m
i=ρji
+∞
κji(s)Gi(v˜i(t–s,x))s + mi=πjiGi(v˜i(t–τ,x)) + mi=σji
+∞
κji(s)Gi(v˜i(t–s,x))s
+mi=hjiνi+
m
i=Mjiνi+ mi=Njiνi+mn+jEn+j(t,x),
(.)
subject to the following initial conditions
˜
ui(s,x) =φ˜i(s,x), (s,x)∈[–τ, ]T×,
˜
vj(s,x) =ϕ˜j(s,x), (s,x)∈[–τ, ]T×, (.)
and Dirichlet boundary conditions
˜
ui(t,x) = , (t,x)∈[,∞)T×∂,
˜
vj(t,x) = , (t,x)∈[,∞)T×∂,
(.)
where Ei(t,x) = u˜i(t,x) –ui(t,x) (i = , , . . . ,n) and En+j(t,x) =v˜n+j(t,x) –vn+j(t,x) (j=
, , . . . ,m) are error functions.mk> (k= , , . . . ,n+m) is a constant error weighting
co-efficient.u˜(t,x) = (u˜(t,x),u˜(t,x), . . . ,u˜n(t,x))T∈Crd(T×,Rn),v˜(t,x) = (v˜(t,x),˜v(t,x), . . . ,v˜m(t,x))T∈Crd(T×,Rm),φ˜(t,x) = (φ˜(t,x),φ˜(t,x), . . . ,φ˜n(t,x))T∈C([–τ, ]×,Rn),
˜
From (.)-(.) and (.)-(.), we obtain the error system (.)-(.) as follows: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
E
i (t,x) =
l
k=∂∂xk(aik
∂Ei
∂xk) + (mi–bi)Ei(t,x) +mj=cij[fj(v˜j(t–τ,x)) –fj(vj(t–τ,x))]
+nj=pij[Fj(u˜j(t–τ,x)) –Fj(uj(t–τ,x))]
+nj=rij
+∞
kij(s)[Fj(u˜j(t–s,x)) –Fj(uj(t–s,x))]s
+ nj=qij[Fj(u˜j(t–τ,x)) –Fj(uj(t–τ,x))]
+ nj=wij
+∞
kij(s)[Fj(u˜j(t–s,x)) –Fj(uj(t–s,x))]s,
E
n+j(t,x) =
l
k=∂∂xk(ξjk
∂En+j
∂xk ) + (mn+j–ηj)En+j(t,x) +ni=ζji[gi(u˜i(t–τ,x)) –gi(ui(t–τ,x))]
+mi=λji[Gi(v˜i(t–τ,x)) –Gi(vi(t–τ,x))]
+mi=ρji
+∞
κji(s)[Gi(v˜i(t–s,x)) –Gi(vi(t–s,x))]s
+ mi=πji[Gi(˜vi(t–τ,x)) –Gi(vi(t–τ,x))]
+ mi=σji
+∞
κji(s)[Gi(v˜i(t–s,x)) –Gi(vi(t–s,x))]s,
(.)
subject to the following initial conditions
Ei(s,x) =φ˜i(s,x) –φi(s,x), (s,x)∈[–τ, ]T×,
En+j(s,x) =ϕ˜j(s,x) –ϕj(s,x), (s,x)∈[–τ, ]T×,
(.)
and Dirichlet boundary conditions
Ei(t,x) = , (t,x)∈[,∞)T×∂,
En+j(t,x) = , (t,x)∈[,∞)T×∂.
(.)
The following definition is significant to study the global robust exponential synchro-nization of coupled neural networks (.)-(.) and (.)-(.).
Definition . Let y(t,x) = (u(t,x),u(t,x), . . . ,un(t,x),v(t,x),v(t,x), . . . ,vm(t,x))T ∈
Rn+m andy˜(t,x) = (u˜
(t,x),u˜(t,x), . . . ,u˜n(t,x),v˜(t,x),v˜(t,x), . . . ,v˜m(t,x))T∈Rn+m be the
solution vectors of system (.)-(.) and its controlled slave system (.)-(.), respec-tively.E(t,x) = (E(t,x),E(t,x), . . . ,En+m(t,x))T∈Rn+mis the error vector. Then the
cou-pled systems (.)-(.) and (.)-(.) are said to be globally exponentially synchronized if there exists a controlled input vectorz(t,x) = (mE(t,x),mE(t,x), . . . ,mn+mEn+m(t,x))T
and a positive constantα∈R+andM≥ such that E(t,·)=y˜(t,·) –y(t,·)≤Meα(t, ), t∈[,∞)T,
whereαis called the degree of exponential synchronization on time scales.
3 Main results
In this section, we will consider the global robust exponential synchronization of coupled systems (.)-(.) and (.)-(.). At first, we need to introduce some useful lemmas.
Lemma . ([]) Let be a cube |xi|<li (i= , , . . . ,l) and assume that h(x) is a
h(x)|∂= .Then
Throughout this paper, we always assume that:
(H) The neurons activationfj,Fi,giandGjare Lipschitz continuous, that is, there exist
positive constantsαj,βi,γiandδjsuch that|fj(ξ) –fj(η)| ≤αj|ξ–η|,|Fi(ξ) –Fi(η)| ≤βi|ξ–
η|,|gi(ξ) –gi(η)| ≤γi|ξ–η|,|Gj(ξ) –Gj(η)| ≤δj|ξ–η|for anyξ,η∈R,i= , , . . . ,n;j=
, , . . . ,m.
(H) The delay kernelskij,κji: [, +∞)T→[, +∞) (i= , , . . . ,n;j= , , . . . ,m) are
real-valued non-negative rd-continuous functions and satisfy the following conditions:
(H) The following conditions are always satisfied:
globally robustly exponentially synchronous with the master system(.)-(.).
+
and
By applying Lemma ., (.)-(.), conditions (H)-(H) and the Hölder inequality, and noting the robustness of parameter intervals, we get
+ Similar to the arguments of (.), we obtain
En+j(t,·)
If the first inequality of condition (H) holds, there exists one positive numberς> (may be sufficiently small) such that
–
Now we consider the functions
whereθ(zi) =
Similar to the above arguments of (.)-(.), we can always choose << such that for
+
m
=
δj
|ρj|+|σj|
+∞
κj(s)e⊕(s, )s+
n
i=
αj|cij|e⊕(τ, )
+
max{e⊕(σ(t), ),e(θ()–)μ(t)R(t)Ei(t,·)
(t, )}θ()μ(t)R(t)
e⊕(σ(t), ) ≤. (.)
Take the Lyapunov functionalV(t) as follows:
V(t) =Vt,E(t)=V(t) +V(t), (.)
where
V(t) =
n
i=
e⊕(t, )Ei(t,·)
+e(θ()–)μ(t)Q(t)Ei(t,·)(t, )
+
m
j=
αj|cij|
t t–τ
e⊕σ(s+τ), En+j(s,·)s
+
n
ν=
βν
|piν|+|qiν|
t t–τ
e⊕
σ(s+τ), Eν(s,·)s
+
n
ν=
βν
|riν|+|wiν|
+∞
kiν(s)
t
t–s
e⊕σ(s+r), Eν(r,·)r
s
,
V(t) =
m
j=
e⊕(t, )En+j(t,·)+e(θ()–)μ(t)R(t)En+j(t,·)(t, )
+
n
i=
γi|ζji|
t t–τ
e⊕
σ(s+τ), Ei(s,·)
s
+
m
=
δ
|λj|+|πj| t t–τ
e⊕σ(s+τ), En+(s,·)
s
+
m
=
δ
|ρj|+|σj|
+∞
κj(s)
t
t–s
e⊕σ(s+r), En+(r,·)
r
s
.
Calculating D+V
(t) along (.) associated with (.) and noting that (d/dz)[ez(t,s)] =
[st/( +μ(τ)z)τ]ez(t,s) > if and only ifz∈R+(that is,ez(t,s) is increasing with respect
tozif and only ifz∈R+), we have
D+V(t) =
n
i=
(⊕)e⊕(t, )Ei(t,·)+e⊕σ(t), Ei(t,·)
+θ() – μ(t)Q(t)Ei(t,·)
e(θ()–)μ(t)Q(t)Ei(t,·)(t, ) +
m
j=
αj|cij|e⊕
σ(t+τ), En+j(t,·)
–
m
j=
αj|cij|e⊕
σ(t), En+j(t–τ,·)
×maxe⊕
By applying (.), we can similarly calculateD+V
+
+
which implies that
E(t,·)≤Me(t, ). (.)
Obviously,M> . According to Definition ., we conclude that the controlled slave sys-tem (.)-(.) is globally robustly exponentially synchronous with the master syssys-tem
(.)-(.) on the time scale [, +∞)T. The proof is complete.
When the time scaleT=RandT=Z, we will obtain the following two important corol-laries.
Corollary . Assume that the following(H)-(H)hold.Then the master system(.) -(.)and its controlled slave system are globally robustly exponentially synchronous.
(H) The neurons activationfj,Fi,giandGjare Lipschitz continuous, that is, there exist
η|,|gi(ξ) –gi(η)| ≤γi|ξ–η|,|Gj(ξ) –Gj(η)| ≤δj|ξ–η|for anyξ,η∈R,i= , , . . . ,n;j=
, , . . . ,m.
(H) The delay kernelskij,κji: [, +∞)→[, +∞) (i= , , . . . ,n;j= , , . . . ,m) are
real-valued non-negative continuous functions and satisfy the following conditions:
∞
kij(s) ds= ,
∞
skij(s) ds<∞,
∞
κji(s) ds= ,
∞
sκji(s) ds<∞
and there exist constantsω> ,ω> such that
∞
kij(s)esωds<∞,
∞
κji(s)esωds<∞.
(H) The following conditions are always satisfied:
–
l
k= aik
lk + (mi–bi) +
m
j=
αj|cij|+ n
ν=
βν
|piν|+|qiν|+|riν|+|wiν|
+
n
ν=
βi
|pνi|+|qνi|
eτ+
n
ν=
βi
|rνi|+|wνi|
+∞
kνi(s)esds
+
m
j=
γi|ζji|eτ < , i= , , . . . ,n;
–
l
k= ξ
jk
l
k
+ (mn+j–ηj) + n
i=
γi|ζji|+ m
=
δ
|λj|+|πj|+|ρj|+|σj|
+
m
=
δj
|λj|+|πj|
eτ+
m
=
δj
|ρj|+|σj|
+∞
κj(s)esds
+
n
i=
αj|cij|eτ< , j= , , . . . ,m.
Corollary . Assume that the following(H)-(H)hold. Then the master system(.) -(.)and its controlled slave system are globally robustly exponentially synchronous.
(H) The neurons activationfj,Fi,giandGjare Lipschitz continuous, that is, there exist
positive constantsαj,βi,γiandδjsuch that|fj(ξ) –fj(η)| ≤αj|ξ–η|,|Fi(ξ) –Fi(η)| ≤βi|ξ–
η|,|gi(ξ) –gi(η)| ≤γi|ξ–η|,|Gj(ξ) –Gj(η)| ≤δj|ξ–η|for anyξ,η∈R,i= , , . . . ,n;j=
, , . . . ,m.
(H) The delay kernelskij,κji:Z+→[, +∞) (i= , , . . . ,n;j= , , . . . ,m) are real-valued
non-negative rd-continuous functions and satisfy the following conditions:
∞
s=
kij(s) = ,
∞
s=
skij(s) <∞,
∞
s=
κji(s) = ,
∞
s=
sκji(s) <∞,
and there exist constantsω> ,ω> such that
∞
s=
kij(s)( +ω)s<∞,
∞
s=
(H) The following conditions are always satisfied:
4 Illustrative example
Consider the following reaction-diffusion BAM recurrent FNNs on time scales:
⎧
subject to the following initial conditions
ui(s,x) =φi(s,x), (s,x)∈[–τ, ]T×,
vj(s,x) =ϕj(s,x), (s,x)∈[–τ, ]T×, (.)
and Dirichlet boundary conditions
+
j=
γ|ζj|τ≈–. < ,
–
k= ξ
k
l
k
+ (m–η) +
i=
γi|ζi|+
=
δ
|λ|+|π|+|ρ|+|σ|
+
=
δ
|λ|+|π|
τ +
=
δ
|ρ|+|σ| +∞
s=
κ(s)s
+
i=
α|ci|τ≈–. < ,
–
k= ξ
k
lk + (m–η) +
i=
γi|ζi|+
=
δ
|λ|+|π|+|ρ|+|σ|
+
=
δ
|λ|+|π|
τ+
=
δ
|ρ|+|σ| +∞
s=
κ(s)s
+
i=
α|ci|τ≈–. < .
Thus, conditions (H)-(H) are satisfied. It follows from Theorem . that the master sys-tem (.)-(.) and its controlled slave syssys-tem are globally robustly exponentially synchro-nized.
Competing interests
The author declares to have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
The author would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (No. 11161025; No. 11326101), Yunnan Province natural scientific research fund project (No. 2011FZ058).
Received: 3 September 2014 Accepted: 28 November 2014 Published:16 Dec 2014
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