Boundary Value Problems
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Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions
Boundary Value Problems 2012, 2012:2
doi:10.1186/1687-2770-2012-2
WeiYuan Zhang (ahzwy@163.com) JunMin Li (jmli@mail.xidian.edu.cn)
ISSN 1687-2770
Article type Research
Submission date
25 October 2011
Acceptance date
13 January 2012
Publication date
13 January 2012
Article URL http://www.boundaryvalueproblems.com/content/2012/1/2
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Global exponential synchronization of delayed BAM neural networks
with reaction–diffusion terms and the Neumann boundary conditions
WeiYuan Zhang*1,2 and JunMin Li1
1School of Science, Xidian University, Shaan Xi Xi’an 710071, P.R. China
2Institute of Maths and Applied Mathematics, Xianyang Normal University, Xianyang,
ShaanXi 712000, P.R. China
*Corresponding author: ahzwy@163.com
Email address:
JML: jmli@mail.xidian.edu.cn
Abstract
In this article, a delay-differential equation modeling a bidirectional associative memory
(BAM) neural networks (NNs) with reaction-diffusion terms is investigated. A feedback
control law is derived to achieve the state global exponential synchronization of two
identical BAM NNs with reaction-diffusion terms by constructing a suitable Lyapunov
functional, using the drive-response approach and some inequality technique. A novel
global exponential synchronization criterion is given in terms of inequalities, which can
1
be checked easily. A numerical example is provided to demonstrate the effectiveness of
the proposed results.
Keywords: neural networks;
reaction–diffusion; delays; global
exponential
synchronization; Lyapunov functional.
1.
Introduction
Aihara et al. [1] firstly proposed chaotic neural network (NN) models to simulate the
chaotic behavior of biological neurons. Consequently, chaotic NNs have drawn
considerable attention and have successfully been applied in combinational optimization,
secure communication, information science, and so on [2–4]. Since NNs related to
bidirectional associative memory (BAM) have been proposed by Kosko [5], the BAM
NNs have been one of the most interesting research topics and extensively studied
because of its potential applications in pattern recognition, etc. Hence, the study of the
stability and periodic oscillatory solution of BAM with delays has raised considerable
interest in recent years, see for example [6–12] and the references cited therein.
Strictly speaking, diffusion effects cannot be avoided in the NNs when electrons
are moving in asymmetric electromagnetic fields. Therefore, we must consider that the
activations vary in space as well as in time. In [13–27], the authors have considered
various dynamical behaviors such as
the stability, periodic oscillation, and
synchronization of NNs with diffusion terms, which are expressed by partial differential
equations. For instance, the authors of [16] discuss the impulsive control and
2
synchronization for a class of delayed reaction-diffusion NNs with the Dirichlet boundary
conditions in terms of p-norm. In [25], the synchronization scheme is discussed for a
class of delayed NNs with reaction-diffusion terms. In [26], an adaptive synchronization
controller is derived to achieve the exponential synchronization of the drive-response
structure of NNs with reaction-diffusion terms. Meanwhile, although the models of
delayed feedback with discrete delays are good approximation in simple circuits
consisting of a small number of cells, NNs usually have a spatial extent due to the
presence of a multitude of parallel pathways with a variety of axon sizes and lengths.
Thus, there is a distribution of conduction velocities along these pathways and a
distribution of propagation delays. Therefore, the models with discrete and continuously
distributed delays are more appropriate.
To the best of the authors’ knowledge, global exponential synchronization is
seldom reported for the class of delayed BAM NNs with reaction-diffusion terms. In the
theory of partial differential equations, Poincaré integral inequality is often utilized in the
deduction of diffusion operator [28]. In this article, the problem of global exponential
synchronization is investigated for the class of BAM NNs with time-varying and
distributed delays and reaction-diffusion terms by using Poincaré integral inequality,
Young inequality technique, and Lyapunov method, which are very important in theories
and applications and also are a very challenging problem. Several sufficient conditions
are in the form of a few algebraic inequalities, which are very convenient to verify.
3
2. Model description and preliminaries
In this article, a class of delayed BAM NNs with reaction–diffusion terms is described as
follows
l
i
,
=
−
)
∑
D ik
( p u t x i i
(
)
u ∂ t ∂
k
1 =
∂ x ∂ k
u ∂ i x ∂ k
n
n
n
t
x
k
t
I
,
,
+
+
−
+
−
+
( v t x ,
)
( ) t
(
) s f
) v s x ds ,
(
( ) t
b f ji
j
j
% b f ji
j
θ ji
ji
ji
j
j
i
∑
∑
∑ ∫ b
(
( v t j
(
)
)
)
(
)
−∞
j
j
j
1 =
1 =
1 =
l
m
v
∂
v ∂
j
j
D
t x ,
,
=
−
+
(
)
(
)
∗ jk
j
ij
i
∑
∑
( d g u t x i
)
( q v j
)
t
∂
∂ x ∂
∂
k
i
1 =
1 =
k
x k
m
m
t
x
t
J
,
,
,
+
−
+
−
+
( ) t
(
)
( ) t
ij
i
τ ij
ij
k ij
) ( s g u s x ds i
i
j
∑ %
(
)
)
∑ ∫ d
( ( d g u t i
)
−∞
i
i
1 =
1 =
(1)
T
x
,...,
=
where
,l
∈ Ω ⊂ (cid:2) Ω is a compact set with smooth boundary ∂Ω and
(
)
x l
x x , 1
2
T
T
m
n
(cid:2)
(cid:2)
u
u
v
,
,...,
,
,...,
,
=
∈
=
∈
l(cid:2) ;
0
mesΩ > in space
(
(
)
(
)
) iu t x and
m
v n
u u , 1
2
v v , 1 2
,
(
)
jv t x represent the states of the ith neurons and the jth neurons at time t and in space
x , respectively.
,
,
are known constants denoting the synaptic
% b b b , ji
ji
ji
% ijd
d d , and ,ij ij
connection strengths between the neurons, respectively;
jf and
ig denote the activation
functions of the neurons and the signal propagation functions, respectively;
jJ
iI and
denote the external inputs on the ith and jth neurons, respectively;
jq are
ip and
differentiable real functions with positive derivatives defining the neuron charging time,
and
tθ represent continuous time-varying discrete delays,
respectively;
( )
( ) tτ ij
ji
0
respectively;
jkD∗ ≥ stand for the transmission diffusion coefficient along
ikD ≥ and 0
i
m
k
l
j
the ith and jth neurons, respectively.
and
1, 2,...,
,
1, 2,...,
1, 2,...,
n .
=
=
=
4
System (1) is supplemented with the following boundary conditions and initial
values
v ∂
v ∂
v ∂
v ∂
j
j
j
,
,
,...,
0,
,...,
0
: =
=
: =
=
t
x
,
(2)
0,
≥
∈ ∂Ω ,
u ∂ i n ∂
j n ∂
T
T
u ∂ i x ∂ l
x ∂ l
u ∂ i x ∂ 1
u ∂ i x ∂ 2
x ∂ 1
x ∂ 2
,
,
s x ,
s x ,
,
, 0
=
=
s x ∈ −∞ × Ω , .
(3)
)
(
) s x v ,
(
)
(
)
(
)
(
]
( u s x i
ϕ ui
j
ϕ vj
i
m
j
n
for any
and
where n is the outer normal vector of
1, 2,...,
1, 2,...,
,
=
=
T
C
are bounded and continuous, where
∂Ω ,
,...,
,...,
=
=
∈
ϕ
)
( ϕ ϕ ϕ ϕ , um vn
u
v 1
1
ϕ u ϕ v
m
(cid:2)
m n +
(cid:2)
C
.
It is the Banach space of continuous
, : ϕ
=
=
→
n
(cid:2)
( (
ϕ u ϕ v
] , 0 −∞ × ] , 0 −∞ ×
| ϕ ϕ
m n+(cid:2)
functions which map
into
with the topology of uniform converge for the
,0 ,0
] ]
( −∞ ( −∞
norm
m
n
r
r
dx
dx
r
,
2.
=
=
+
≥
ϕ
ϕ ui
ϕ vj
∑
∑
∫
∫
Ω
Ω
0
0
sup s −∞≤ ≤
sup s −∞≤ ≤
i
j
1 =
1 =
ϕ u ϕ v
Throughout this article, we assume that the following conditions are made.
(A1) The functions
, θ are piecewise-continuous of class
1C on the closure of
( ) t
( ) t
τ ij
ji
each continuity subinterval and satisfy
0
,0
1,
1,
≤
≤
≤
≤
≤
≤
<
( ) t
( ) t
( ) t
( ) t
τ ij
τ ij
θ ji
& , θ τ ij ji
& µ θ < ji
τ
µ θ
,
τ
=
=
ij
{ } , τ θ
{ θ ji
}
j n
j n
max i m ,1 ≤ ≤
1 ≤ ≤
max i m ,1 ≤ ≤
1 ≤ ≤
0,
≥
with some constants
for all
0t ≥ .
0,
0, τ θ>
>
τ ij
0, θ≥ ji
5
1C on the closure
(A2) The functions
( )
( ) ip ⋅ and
jq ⋅ are piecewise-continuous of class
of each continuity subinterval and satisfy
0,
0,
=
>
=
( ζ
)
( ) 0
a i
′ p i
p i
inf (cid:2) ζ ∈
c
q
q
0,
0.
=
>
=
( ζ
)
( ) 0
j
′ j
j
inf (cid:2) ζ ∈
(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist
g
f
positive constants
,η η ∈ (cid:2)
jL and
iL such that for all
1
2
f
f
L
g
g
−
≤
− η η
−
≤
− η η
)
)
)
)
j
j
f j
i
i
g L i
( η 1
( η 2
2 ,
1
( η 1
( η 2
2 .
1
: 0,
,
i
(A4) The delay kernels
are
1, 2,...,
m j ,
n 1, 2,..., )
=
=
( ) K s K s ,
)
)
( ) [
[ 0, ∞ → ∞ (
ij
ji
real-valued non-negative continuous functions that satisfy the following conditions
+∞
+∞
1,
1,
=
=
(i)
( ) K s ds
( ) K s ds
ji
ji
∫
∫
0
0
+∞
+∞
, < ∞
, < ∞
(ii)
( ) sK s ds
( ) sK s ds
ji
ij
∫
∫
0
0
(iii) There exist a positive µ such that
+∞
+∞
s µ
s µ
, < ∞
. < ∞
( )
( )
se K s ds ji
se K s ds ij
∫
∫
0
0
We consider system (1) as the drive system. The response system is described by
the following equations
l
n
∂
∂
)
)
,
=
−
+
)
% ( v t x ,
)
D ik
% ( p u t x i i
b f ji
j
j
∑
∑
(
)
(
)
% ( u t x , i t ∂
k
j
1 =
1 =
∂ x ∂ k
% ( u t x , i x ∂ k
n
n
t
x
k
t
I
,
t x ,
,
+
−
+
−
+
+
( ) t
(
) s f
% ) v s x ds ,
(
( ) t
(
)
% b f ji
j
θ ji
ji
ji
j
j
i
σ i
∑
∑ ∫ b
( % v t j
(
)
)
(
)
−∞
j
j
1 =
1 =
6
l
m
∂
∂
)
)
j
D
,
,
=
−
+
)
(
)
∗ jk
j
ij
i
∑
∑
( % d g u t x i
)
( % ( q v t x j
)
t
% ( v t x , ∂
k
i
1 =
1 =
∂ x ∂ k
% ( v t x , j x ∂ k
m
m
t
x
k
t
J
,
,
t x ,
,
+
−
+
−
+
+
( ) t
(
)
( ) t
(
)
% i
τ ij
ij
ij
% ( ) s g u s x ds i
i
j
ϑ j
∑
(
)
∑ ∫ d
)
% ( ( d g u t ij i
)
−∞
i
i
1 =
1 =
(4)
,
,
where
and
denote the external control inputs that will be appropriately
( t xσ
)
( t xϑ
)
i
j
T
% u
,...,
t x ,
=
designed for a certain control objective. We denote
,
% ( u t x ,
)
)
(
)
m
% ( u t x , 1
(
)
T
T
,...,
,
t x ,
t x ,
,...,
t x ,
=
=
and
% ( v t x ,
)
)
)
( , σ
)
(
)
(
)
% ( v t x n
σ m
% ( v t x , 1
(
)
( σ 1
)
T
t x ,
t x ,
,...,
t x ,
=
.
( ϑ
)
(
)
(
)
ϑ n
( ϑ 1
)
The boundary and initial conditions of system (4) are
% v ∂
% v ∂
% v ∂
% v ∂
j
j
j
,
,
,...,
0,
,...,
0
: =
=
: =
=
t
x
,
(5)
0,
≥
∈ ∂Ω ,
% u ∂ i n ∂
j n ∂
T
T
% u ∂ i x ∂ l
x ∂ l
% u ∂ i x ∂ 1
% u ∂ i x ∂ 2
x ∂ 1
x ∂ 2
,
,
s x ,
s x ,
, 0
=
=
and
s x ∈ −∞ × Ω , ,
(6)
)
(
% ) s x v ,
(
)
(
)
, (
)
(
]
% ( u s x i
ψ ui
j
ψ v j
T
C
where
.
,...,
,...,
ψ
=
=
∈
)
( , ψ ψ ψ ψ % % v n u m
% u
% v
1
1
ψ % u ψ % v
Definition 1. Drive-response systems (1) and (4) are said to be globally exponentially
synchronized, if there are control inputs
,
, and
2
r ≥ , further there exist
( ,t xσ
)
( ,t xϑ
)
constants
0α> and
1β≥ such that
−
+
−
( u t x ,
)
% ( u t x ,
)
( v t x ,
)
% ( v t x ,
)
s x ,
s x ,
s x ,
s x ,
, for all
t αβ 2 − e
≤
+
0t ≥ ,
(
)
(
)
(
)
(
)
ϕ u
− ψ % u
ϕ v
− ψ % v
(
)
7
m
r
−
=
−
,
in
which
( u t x ,
)
% ( u t x ,
)
)
(
( u t x , i
% ) u t x dx , i
∑∫
Ω
i
1 =
n
r
,
,
,
2
,
,
−
=
−
≥
,
and
and
( v t x ,
)
% ( v t x ,
)
( v t x
)
% ) v t x dx r
(
) ( u t x v t x ,
(
)
j
j
(
)
∑∫
Ω
j
1 =
%
,
,
are the solutions of drive-response systems (1) and (4) satisfying
% ) ( u t x v t x ,
(
)
(
)
boundary conditions and initial conditions (2), (3) and (5), (6), respectively.
m(cid:2) with a
Lemma 1. [21] (Poincaré integral inequality). Let Ω be a bounded domain of
2C by Ω .
smooth boundary ∂Ω of class
( ) u x is a real-valued function belonging to
)
1
0.
Then
=
(
) 0H Ω and
| ∂Ω
( u x ∂ n ∂
2
2
dx
dx
,
≤
∇
( ) u x
( ) u x
∫
∫
1 λ 1
Ω
Ω
which
1λ is the lowest positive eigenvalue of the Neumann boundary problem
x
x
x
,
=
, ∈ Ω
)
( λϕ 1
(7)
x
0,
=
. ∈ ∂Ω
| ∂Ω
−∆ ∂
( ) ϕ ( ) u x n ∂
3. Main results
From the definition of synchronization, we can define the synchronization error signal
T
t x ,
,
,
,...,
t x ,
,
=
−
=
−
=
,
( e t x ,
)
)
(
)
)
)
)
(
)
( v t x
)
% ( v t x
)
( e t x , i
( u t x , i
% ( u t x , i
, ω j
j
j
e m
( e t x , 1
(
)
T
t x ,
t x ,
,...,
t x ,
=
and
. Thus, error dynamics between systems (1) and (4)
( ω
)
(
)
(
)
ω n
( ω 1
)
can be expressed by
8
l
n
∂
∂
)
)
,
t x ,
=
−
+
)
(
)
D ik
% ( p e t x i i
% b f ji
j
∑
∑
(
)
( ω j
)
( e t x , i t ∂
k
j
1 =
1 =
∂ x ∂ k
( e t x , i x ∂ k
n
n
t
x
k
t
,
t x ,
,
+
−
+
−
−
( ) t
(
% ) s f
(
(
)
% % b f ji
j
j
ji
ji
j
σ i
∑
∑ ∫ b
( ω j
)
) ) s x ds ,
( ( t ω θ ji
)
−∞
j
j
1 =
1 =
l
m
t x ,
t x ,
)
)
%
D
% q
t x ,
,
=
−
+
(
)
(
)
∗ jk
j
ij
i
∑
∑
( d g e t x i
)
( ω j
(
)
)
( ∂ ω j t ∂
∂ ω j ∂
k
i
1 =
1 =
∂ x ∂ k
( x k
(8)
m
m
t
%
x
,
,
t x ,
,
+
−
+
−
−
( ) t
)
(
)
ij
i
τ ij
ij
( k t ij
) ( s g e s x ds i
i
ϑ j
∑
(
)
∑ ∫ d
)
% % ( ( d g e t i
)
−∞
(
)
i
i
1 =
1 =
% f
f
f
% v
where
,
t x ,
,
t x ,
,
,
,
,
=
−
=
−
(
)
( v t x
)
(
)
)
)
)
j
j
j
j
j
% ( g e t x i
i
( g u t x i
i
% ( g u t x i
i
(
)
(
)
(
)
( ω j
)
(
)
(
)
% q
.
,
,
,
,
t x ,
t x ,
t x ,
=
−
=
−
)
)
)
(
)
(
)
(
)
% ( p e t x i i
( p u t x i i
% ( p u t x i i
j
j
j
(
)
(
)
(
)
( ω j
)
( q v j
)
( % q v j
)
The control inputs strategy with state feedback are designed as follows:
m
n
t x ,
,
t x ,
t x ,
µ
=
=
i
n
,
.
1, 2,...,
m j ,
1, 2,...,
=
=
(
)
)
(
)
(
)
σ i
( e t x ik k
, ϑ j
ρ ω jk k
∑
∑
k
k
1 =
1 =
that is,
t x ,
t x ,
t x ,
µ
=
=
,
(9)
( σ
)
( e t x ,
)
( , ϑ
)
( ρω
)
where
and
are the controller gain matrices.
( = µ µ
( = ρ ρ
)ik m m
×
)jk n n
×
The global exponential synchronization of systems (1) and (4) can be solved if
the controller matrices µ and ρ are suitably designed. We have the following result.
Theorem 1. Under the assumptions (A1)–(A4), drive-response systems (1) and (4) are in
global
exponential
synchronization,
if
there
exist
i
0,
0
0
1, 2,...,
,
2,
>
=
+
≥
(
) n m r
iw
ijγ >
jiβ > such that the controller gain matrices µ
and ρ in (9) satisfy
9
n
n
m
1 −
r
r 1 −
rn
r
r
r
2
−
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
w i
r rna i
µ ii
r a i
r a i
− β ji
r a i
r a i
r 1 − rna D λ i i 1
∑
∑
∑
j
j
k
i k
1 =
1 =
1, = ≠
n
m
r
r
r
r
r
r
r
r
w m d
d
n
0
+
+
<
+
ij
g L i
ij
r γ ij
g L i
µ ki
w k
m j +
% d ij
+∑
∑
(
)
(
(
rg ) L i
) )
(
1
τ e −
j
k
i k
= 1
= ≠ 1,
τµ
and
m
n
m
1 −
r
r 1 −
w
rmc
r
c
r
c
r
c
2
−
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
r 1 − rmc D j
r j
rm c ρ jj
r j
r j
r j
− r γ j ij
∗ λ j 1
m j +
∑
∑
∑
i
k
i
1 =
1, =
j k ≠
1 =
m
n
r
r
r
r
r
r
r
r
r
b
L
% b
L
b
L
m
w
0,
(10)
+
+
+
+
<
w n i
ji
f j
ji
% f j
ji
r β ji
f j
ρ kj
k m +
∑
∑
(
)
(
(
)
(
) )
1
θ e −
i
k
1 =
1, =
j k ≠
µ θ
g
f
i
m
j
n
and
are Lipschitz
constants,
in which
1, 2,...,
,
1, 2,...,
,
=
=
jL
iL
D
,
=
=
D i
D D , ik
∗ j
∗ jk
1λ is the lowest positive eigenvalue of problem (7).
min k l ≤ ≤ 1
min k l ≤ ≤ 1
Proof. If (10) holds, we can always choose a positive number
0δ > (may be very small)
such that
n
n
m
1 −
r
r 1 −
rn
r
r
r
2
−
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
w i
r rna i
µ ii
r a i
r a i
− β ji
r a i
r a i
r 1 − rna D λ i i 1
∑
∑
∑
j
j
k
i k
1 =
1 =
1, = ≠
n
m
r
r
r
r
r
r
r
r
w m d
d
n
0
+
+
δ + <
+
ij
g L i
ij
r γ ij
g L i
µ ki
w k
m j +
% d ij
+∑
∑
(
)
(
(
rg ) L i
) )
(
1
τ e −
j
k
i k
= 1
= ≠ 1,
τµ
and
m
n
m
1 −
r
r 1 −
w
rmc
r
c
r
c
r
c
2
−
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
r 1 − rmc D j
r j
rm c ρ jj
r j
r j
r j
− r γ j ij
∗ λ j 1
m j +
∑
∑
∑
i
k
i
1 =
1, =
j k ≠
1 =
m
n
r
r
r
r
r
r
r
r
r
b
L
% b
L
b
L
m
w
0,
+
+
+
+
+ < δ
w n i
ji
f j
ji
f j
ji
r β ji
f j
ρ kj
k m +
∑
∑
(
)
(
)
(
) )
1
θ e −
i
k
1,
1 =
=
j k ≠
µ θ
(11)
10
i
where
1, 2,...,
m j ,
1, 2,...,
n .
=
=
Let us consider functions
n
m
r
r
2
−
+
−
+
−
−
(
) 1
(
) 1
r µ − 1 rn a ii i
r a i
∑
∑ r a i
i
w i
r rna i
r 1 λ− rna D i i 1
)* ( iF x =
−
j
k
i k
= 1
= ≠ 1,
n
n
r
+∞
r
r
1 −
r
r 1 −
w m d
r
k
2
+
−
+
(
) 1
( ) s ds
ij
g L i
m j +
+∑
− β ji
r a i
ji
r ∗ x na i i
∑
(
)
∫
0
(
j
j
1 =
= 1
m
r
+∞
r
r
r
2
r
∗ x s i
d
e
k
+
+
( ) s ds
ij
r γ ij
g L i
ij
µ ki
w k
% d ij
+ ∑ n
(
)
(
rg ) L i
∫
0
)
1
τ e −
k
i k
= ≠ 1,
τµ
and
m
n
1 −
w
rmc
r
c
r
2
−
−
−
+
−
+
−
(
) 1
(
) 1
r 1 − rmc D j
r j
rm c ρ jj
r j
r j
r j
∗ λ j 1
m j +
j
∑
∑ c
) ( G y∗ = j
i
k
1,
1 =
=
j k ≠
m
m
r
r
+∞
r
1 −
r
r 1 −
b
L
r
c
k
2
+
−
+
(
) 1
( ) s ds
w n i
ji
f j
+∑
− r γ j ij
ij
∗ y mc j
r j
∑
(
)
∫
0
(
i
i
1 =
1 =
n
r
+∞
r
r
r
r
2
r
∗ y s j
% b
L
b
L
e
k
w
(12)
,
+
+
ji
f j
ji
r β ji
f j
ji
ρ kj
k m +
∑
(
)
(
)
∫
0
1
θ e −
k
1,
=
j k ≠
( ) s ds m +
µ θ
,
i
where
1, 2,...,
m j ,
1, 2,...,
n .
=
=
)
[ * x y ∈ +∞ , 0, i
* j
From (12) and (A4), we derive
0,
0;
,
δ< − <
δ< − <
and
are continuous for
( )0
( )0
)
iF
jG
[ * x y ∈ +∞ . 0, i
* j
i
( iF x∗
)
( G y∗ j j
)
Moreover,
G y∗ → +∞ as
jy∗ → +∞ , thus there exist
ix∗ → +∞ and
iF x∗ → +∞ as
i
j
j
(
)
(
)
0,
,
constants
[
) jε ν ∈ +∞ such that
i
n
m
r
r
2
−
+
−
+
−
−
(
) 1
(
) 1
) ( iF ε =
i
r 1 µ − rn a ii i
r a i
∑
∑ r a i
w i
r rna i
r 1 λ− rna D i i 1
−
j
k
i k
1 =
1, = ≠
11
n
n
r
+∞
r
r
1 −
r
r 1 −
w m d
r
k
+
−
+
(
) 1
( ) s ds
ij
g L i
m j +
+∑
− β ji
r a i
ji
2 ε i
r na i
∑
(
)
∫
0
(
j
j
1 =
1 =
m
r
+∞
r
r
r
s
r
2 ε i
d
e
k
n
0
+
+
=
+
( ) s ds
ij
r γ ij
g L i
ij
µ ki
w k
% d ij
∑
(
)
(
rg ) L i
∫
0
)
1
τ e −
k
i k
1, = ≠
τµ
and
m
n
rm c
r
c
r
2
−
+
−
+
−
w
rmc
−
(
) 1
(
) 1
r 1 ρ − jj j
r j
r j
∑
∑ c
j
m j
r rmc D j
r j
1 ∗ λ− j 1
) ( G ν = j
( + −
i
k
1 =
1, =
j k ≠
m
m
+∞
r
r
r
r
r
r 1 −
r
c
k
b
L
+
+
(
( ) s ds
− r γ j ij
ij
w n i
ji
f j
+∑
j
− ∑ ) 1
(
)
)1 jmcν − 2
∫
0
(
i
i
1 =
1 =
n
r
+∞
r
r
r
r
s
r
2 ν j
% b
L
b
L
e
k
m
w
(13)
0.
+
+
+
=
( ) s ds
ji
f j
ji
r β ji
f j
ji
ρ kj
k m +
∑
(
)
(
)
∫
0
1
θ e −
k
1,
=
j k ≠
µ θ
By using
obviously, we get
,
α
=
{ , ε ν i j
}
j n
min i m ,1
1 ≤ ≤
≤ ≤
n
m
r
r
2
−
+
−
+
−
−
−
(
) 1
(
) 1
) ( iF α =
r 1 µ − rn a ii i
r a i
∑
∑ r a i
w i
r rna i
r 1 λ− rna D i i 1
(
j
k
i k
1 =
1, = ≠
n
n
+∞
r
r
r
1 −
r
r 1 −
w m d
r
k
)
+
−
+
2 α
(
) 1
( ) s ds
ij
g L i
m j +
− β ji
r a i
ji
r na i
+∑
∑
(
)
∫
0
(
j
j
1 =
1 =
m
r
+∞
r
r
r
r
s 2 α
d
e
k
n
0
+
+
≤
+
( ) s ds
ij
r γ ij
g L i
ij
µ ki
w k
% d ij
∑
(
)
(
rg ) L i
∫
0
)
1
τ e −
k
i k
1, = ≠
τµ
and
m
n
rm c
r
c
r
2
−
+
−
+
−
w
rmc
−
)
(
) 1
(
) 1
( jG α =
r 1 ρ − jj j
r j
r j
∑
∑ c
m j
r rmc D j
r j
1 ∗ λ− j 1
( + −
i
k
1 =
1, =
j k ≠
m
m
+∞
r
r
r
r
r
r 1 −
r
c
k
b
L
+
+
(
( ) s ds
w n i
ji
f j
− r γ j ij
ij
+∑
− ∑ ) 1
(
)
)1 jmcα − 2
∫
0
(
i
i
1 =
1 =
12
n
r
+∞
r
r
r
r
r
s 2 α
% b
L
b
L
e
k
w
0.
(14)
+
+
+
≤
( ) s ds m
ji
f j
ji
r β ji
f j
ji
ρ kj
k m +
∑
(
)
(
)
∫
0
)
1
θ e −
k
1,
=
j k ≠
µ θ
,
Multiplying both sides of the first equation of (8) by
(
)
ie t x and integrating over Ω
yields
l
2
2
)
=
−
(
)
)
(
) e t x dx , i
( e t x , i
D ik
′ p i
( ξ i
) e t x dx , i
∑
∫
∫
∫
Ω
Ω
Ω
d dt
1 2
k
1 =
∂ x ∂ k
( e t x , ∂ i x ∂ k
n
n
,
(15)
+
+
−
% % ( ) b e t x f ,
( ) t
% ( ) b e t x f ,
(
ji
i
j
ji
i
j
j
∑
∑
( ω j
) ) t x dx ,
∫
∫
dx ( ( t ω θ ji
) ) x dx
Ω
Ω
j
j
1 =
1 =
n
m
t
k
t
,
+
−
−
µ
)
(
% ) s f
(
)
(
)
( b e t x i
ji
ji
j
( e t x , i
e t x dx . , ik k
∑
∑
( ω j
) ) s x dsdx ,
∫
∫
∫
Ω
−∞
Ω
j
k
1 =
1 =
It is easy to calculate by the Neumann boundary conditions (2) that
l
l
)
)
=
∇
)
)
( e t x , i
D ik
( e t x , i
D ik
∑
∑
∫
∫
Ω
Ω
k
k
1 =
1 =
∂ x ∂ k
( e t x , ∂ i x ∂ k
( e t x , ∂ i x ∂ k
dx
dx
(16)
2
2
l
l
l
)
)
)
dx
dx
dx
=
−
= −
(
)
e t x D , i ik
D ik
D ik
∑
∑
∑
∫
∫
∫
∂Ω
Ω
k
k
k
1 =
1 =
( e t x , ∂ i x ∂ k
( e t x , ∂ i x ∂ k
( e t x , ∂ i x ∂ k
1 = Ω
Moreover, from Lemma 1, we can derive
2
2
l
l
2
)
)
dx
dx
,
.
−
≤ −
≤ −
(17)
(
)
D ik
D i
D e t x i
λ i 1
∑
∑
∫
∫
2
k
k
1 =
1 =
( e t x , ∂ i x ∂ k
( e t x , ∂ i x ∂ k
Ω
Ω
From (13)–(17), (A2), and (A3), we obtain that
2
2
2
2
2
,
≤ −
−
(
(
(
) e t x dx , i
) e t x dx , i
a i
) e t x dx i
D λ i 1
∫
∫
∫
Ω
Ω
Ω
d dt
n
n
% f
x
dx
,
2
,
,
2
+
−
+
( b e t x L
)
(
) t x dx ,
)
( ) t
ji
i
f j
ω j
% ( b e t x i
ji
j
j
∑
∑
)
∫
∫
( ( t ω θ ji
)
Ω
Ω
j
j
1 =
1 =
m
n
t
k
t
% f
dsdx
b
,
s x ,
2
2
−
−
+
(
)
(
)
)
(
)
ji
( ) s e t x i
j
( e t x , i
µ ik
e t x dx . , k
ji
∑
∑
( ω j
)
∫
∫
∫
Ω
Ω
−∞
k
j
1 =
1 =
(18)
13
,
Multiplying both sides of the second equation of (8) by
, similarly, we also have
)
( t xω j
2
D
dx
t x ,
c 2
2 ≤ −
−
2 ) t x dx ,
(
(
)
2 ) t x dx ,
(
ω j
j
ω j
1
∫
∫
Ω
Ω
d dt
∫ ∗ λ ω j j Ω
m
m
x
,
2
,
(19)
2 +
+
−
(
)
(
) t x dx ,
( ) t
(
) t x dx ,
g d L e t x i
ij
i
ω j
i
τ ij
ω j
∑
∑
)
∫
∫
% ( ( % d g e t i ij
)
Ω
Ω
i
i
1 =
1 =
m
n
t
%
d
k
t
,
2
t x ,
2 +
−
−
(
(
)
(
) t x dsdx ,
(
)
(
) t x dx . ,
ij
ij
i
ω j
ρ ω jk k
ω j
∑
∑
( ) s g e s x i
)
∫
∫
∫
Ω
−∞
Ω
i
k
1 =
1 =
Consider the following Lyapunov functional
m
n
r
r
t
r
r
t 2 α
2 αξ
1 −
( ) V t
)
)
i
( e t x , i
ji
j
∑
∑
( ( , ω ξ j
)
∫
∫
t
Ω
( ) t
θ − ji
θ e −
i
j
1 =
1 =
n
r
t
+∞
r
r
) ( s 2 α ξ +
e % b n e % f x d ξ = + r w na i 1 µ θ
( ) s
)
ji
r β ji
ji
j
∑
( ( , ω ξ j
)
∫
∫
0
t s −
j
1 =
n
m
t
r
r
r
t 2 α
2 αξ
1 −
b n k e % f x + d ds dx ξ
(
)
( , ξ
)
r j
m j +
∑
∑
)
∫
∫
t
Ω
( ) t
− τ ij
τ e −
j
i
1 =
1 =
m
t
+∞
r
) ( s 2 + α ξ
r d m
w mc e e x t x , d ξ + + ω j % r d m ij % ( g e i i 1 µ τ
( ) s
( , ξ
)
ij
r r γ ij
ij
∑
)
∫
∫
0
t s −
i
1 =
k e x (20) + % ( g e i i d ds dx . ξ
m
Its upper Dini-derivative along the solution to system (8) can be calculated as
r
r
1 −
)
t 2 α
1 −
1 −
( ) + D V t
)
)
i
( e t x , i
r t 2 α e na i
( e t x , i
∑∫
Ω
( e t x , i t ∂
i
1 =
n
n
r
r
r
+∞
r
r
r
t 2 α
t 2 α
s 2 α
∂ e 2 α ≤ +
(
)
( ) s
(
)
ji
j
ji
r β ji
ji
j
∑
∑
( ω j
)
( ω j
)
∫
0
j
j
1 =
1 =
n
r
r
( ) t
r
( t 2 − α θ ji
)
e % b n % f e b n e k % f ds t x , t x , + + 1 r w rna i θ e − µ θ
( ) t
ji
j
j
∑
)
( 1
) ( ) t e
( ( t ω θ ji
)
θ e −
j
1 =
n
r
+∞
r
r
t 2 α
% b n % f x , − − − & θ ji 1 µ θ
( ) s
(
)
ji
r β ji
ji
j
∑
( ω j
)
∫
0
j
1 =
n
e b n k % f t s x , − − ds dx
r
r
1 −
)
t 2 α
1 −
1 −
t 2 α e mc
(
)
(
)
r j
r j
m j +
∑∫
Ω
( ω j t ∂
j
1 =
14
t x , ∂ w rmc e t x , t x , + + 2 α ω j ω j
m
r
r
t 2 α
)
i
∑
(
)
τ e −
i
1 =
m
r
( ) t
r
( t 2 − α τ ij
)
e , + % ( g e t x i % r d m ij 1 µ τ
( ) t
( ) t
i
∑
( 1
)
)
( ( % g e t i
)
τ e −
i
1 =
m
+∞
r
t 2 α
s 2 α
e x , − − − % r d m ij & τ ij τ ij 1 µ τ (21)
r d m
(
)
ij
r r γ ij
i
i
∑
( ( ) s g e t x
)
∫
0
i
1 =
m
+∞
r
t 2 α
% e e ds , + k ij
r d m
)
ij
r r γ ij
ij
i
∑
)
∫
0
i
1 =
e k s x , − − % ( ( ) ( s g e t i ds dx
m
m
n
t 2 α
1 −
From (21) and Young inequality, we can conclude
e
r
2
≤
−
−
+
−
(
r 1 α − na 2 + i
r a i
( ) + D V t
(
) 1
i
r i
r rna i
r a i
w rna D λ i 1
∑
(
∫
∑
Ω
i
j
k
i k
1 =
1 =
1, = ≠
n
n
t
r
r
r
1 −
r
r 1 −
r + − ∑ ) 1
rn
r
k
t
−
+
−
−
(
) 1
(
) s ds
ij
g L i
m j +
µ ii
r a i
− β ji
r a i
ji
∑
(
)
∫
−∞
(
j
j
1 =
1 =
m
+∞
r
r
r
r
r
s 2 α e
r n
+
+
w m d +∑
( w e t x dx ,
)
d ij
r γ ij
k s L ds ij
g i
µ ki
k
i
∑
( )(
)
∫
(
rg ) L i
0
+ % d ij
)
τ e −
k
i k 1, = ≠
τµ
m
n
n
t 2 α
1
w
rmc
+
−
−
(
) 1
(
) 1
r 1 ρ − jj j
r j
r j
∑
∑ c
r rmc D j
r j
1 ∗ λ− j 1
m j +
(
∑∫ e
Ω
j
i
k
1 =
1 =
1, =
j k ≠
m
m
t
r
r
r
r
r
r 1 −
rm c r c r 2 − + − + −
b
L
+
(
) 1
(
) s ds
w n i
ji
f j
− r γ j ij
+∑
∑
(
)
)1 jmcα − 2
∫
−∞
(
i
i
1 =
1 =
n
r
+∞
r
r
r
r
r
r
s 2 α
L
b
L
e
k
w
dx
% b
t x ,
+
+
+
( ) s ds m
(
)
f j
ji
r β ji
f j
ji
ρ kj
ω j
ji
k m +
∑
(
)
(
)
∫
0
)
1
θ e −
k
1, =
j k ≠
µ θ
r c t + − − k ij
(22)
V
t
≤
+ D V t
From (10), we can conclude
0 ≥
( ) V t
( )0 ,
( ) 0, ≤
15
and so (23)
m
r
1 −
V
x
0,
=
( ) 0
(
)
i
e i
Ω
∑∫ r w na i
i
1 =
n
r
r
0
r
Since
)
ji
j
∑
( ( , ω ξ j
)
( ) t
θ e −
j
1 =
∫ µ − θ ji θ
n
r
0
+∞
r
r
) ( s 2 + α ξ
b
n
k
e
% f
x
d ds dx ξ
+
( ) s
)
ji
r β ji
ji
j
∑
( ( , ω ξ j
)
∫
∫
s
0
−
j
1 =
n
m
0
r
r
r
% b n % f x d ξ + 1
)
( , ξ
)
r j
( 1 xω− j
m j +
∑
)
Ω
( ) t
∑∫ w
τ e −
i
j
1 =
1 =
∫ µ − τ ij τ
m
0
+∞
r
) ( s 2 + α ξ
r d m
mc 0, x + d ξ + % r d m ij % ( g e i i 1
( ) s
( , ξ
)
ij
r r γ ij
ij
∑
)
∫
∫
s
0
−
d ds dx ξ
i
1 =
m
+∞
r
r
r
k e x + % ( g e i i
s 2 dsα
≤
+
( ) s se
ij
r γ ij
ij
m j +
{ } w i
∑
{
}
( g d m L i
)
∫
0
{1
max i m ≤ ≤
i
1 =
m
r
r
r
w
s x ,
s x ,
+
(
)
(
)
ij
ϕ u
− ψ u
m j +
∑ % d
{
}
(
r ) g L m i
max j n 1 ≤ ≤
max j n 1 ≤ ≤
1
i
1 =
τ e τ − µ τ
n
+∞
r
r
s 2 α
b
se
k
w k max j n 1 ≤ ≤ max j n 1 ≤ ≤
( ) s ds
m j
{ } w i
ji
f j
r β ji
ji
∑
{
}
( r n L
)
∫
0
max i m 1 ≤ ≤
max i m 1 ≤ ≤
+ + w + max j n ≤ ≤
{1
j
1 =
n
r
r
r
+
(
)
(
)
{ } w i
ji
∑ % b
( r g n L j
)
max i m 1 ≤ ≤
max i m 1 ≤ ≤
1
j
1 =
θ e θ − µ θ
s x , s x , ϕ v ψ− v
(24)
t 2 α
e
t
t x ,
,
0.
+
≤
≥
( e t x ,
)
( ω
)
( ) V t
w i
(
)
(
)
min i m n 1 ≤ ≤ +
Noting that
(25)
16
Let
m
+∞
r
r
r
s 2 dsα
( ) s se
{ } w i
ij
r γ ij
ij
m j +
∑
{
}
( g d m L i
)
∫
0
{
i
1 =
m
r
r
w
,
+
ij
m j +
∑ % d
{
}
(
rg ) L m i
max j n 1 ≤ ≤
max j n 1 ≤ ≤
1
i
1 =
τ e τ µ − τ
n
+∞
r
r
s 2 α
w
b
se
k
+
( ) s ds
{ } w i
ji
f j
r β ji
ji
m j +
∑
{
}
( r n L
)
∫
0
max j n 1 ≤ ≤
max i m 1 ≤ ≤
max i m 1 ≤ ≤
j
1 =
n
r
r
+
{ } w i
ji
∑ % b
( r g n L j
)
max i m 1 ≤ ≤
max i m 1 ≤ ≤
1
j
1 =
θ e θ − µ θ
.
{ } w i
min i m n 1 ≤ ≤ +
w k β = + max max i m 1 ≤ ≤ max j n 1 ≤ ≤ max j n 1 ≤ ≤
1.β≥
Clearly,
t 2 α
−
It follows that
( e t x ,
)
( ω
)
(
)
(
)
(
)
(
)
(
)
(26) t x , s x , s x , s x , s x , . e β + ≤ + ϕ u − ψ % u ϕ v − ψ % v
0,
t ≥
1β≥ is a constant. This implies that drive-response systems (1) and
for any where
(4) are globally exponentially synchronized. This completes the proof of Theorem 1.
Remark 1. In Theorem 1, the Poincaré integral inequality is used firstly. This is a very
important step. Thus, the derived sufficient condition includes diffusion terms. We note
that, in the proof in the previous articles [24–26], a negative integral term with gradient is
left out in their deduction. This leads to those criteria that are irrelevant to the diffusion
term. Therefore, Theorem 1 is essentially new and more effectiveness than those
obtained.
Remark 2. It is noted that we construct a novel Lyapunov functional here as defined in
17
(20) since the considered model contains time-varying and distributed delays and
reaction-diffusion terms. We can see that the results and research method obtained in this
article can also be extended to many other types of NNs with reaction-diffusion terms,
e.g., the cellular NNs, cohen-grossberg NNs, etc.
Remark 3. In our result, the effects of the reaction-diffusion terms on the
synchronization are considered. Furthermore, we note a very interesting fact, that is, as
long as diffusion coefficients in the system are large enough, then condition (10) can
always satisfy. This shows that a large enough diffusion coefficient may always make the
system globally exponentially synchronous.
Some famous NN models are a special case of model (1). In system (1), ignoring
the role of reaction-diffusion, then system (1) will degenerate into the following delayed
n
n
BAM NNs
j
( ) v t j
j
( ) tθ ji
∑ %
∑
( ( ) p u t i i
)
(
)
( v t j
)
(
)
j
j
1 =
1 =
n
t
= − + + − & u i b f ji b f ji
(
) s f
( ) t
ji
ji
j
j
i
(
) ( ) s ds
∑ ∫ b
−∞
j
1 =
m
m
& v
= −
+
+
−
j
j
j
ij
i
ij
i
( ) tτ ij
∑ %
∑
( ( ) d g u t i
)
( ( ) q v t
)
)
( ( d g u t i
)
i
i
1 =
1 =
m
t
k
t
J
+
−
+
k t v I , + − +
(
( )
( ) t
ij
ij
) s g u s ds i
i
j
(
)
∑ ∫ d
−∞
i
1 =
(27)
n
n
and the corresponding response system (4) will become the following form
( ) t
j
j
∑
∑
)
(
)
( % v t j
)
(
)
j
j
1 =
1 =
n
t
= − + + − &% ( ) u t i b f ji % ( ) v t j % b f ji θ ji % ( ( ) p u t i i
(
) s f
( ) t
( ) t
ji
j
j
i
ji
(
) ( ) s ds
∑ ∫ b
−∞
j
1 =
18
k t % v I , − + + + σ i
m
m
= −
+
+
−
( ) t
&% ( ) v t j
j
j
% i
ij
% i
τ ij
∑
∑
( ( ) d g u t i
)
( % ( ) q v t
)
)
% ( ( d g u t ij i
)
i
i
1 =
1 =
m
t
k
t
J
.
+
−
+
+
(
( )
( ) t
( ) t
ij
ij
% ) s g u s ds i
i
j
ϑ j
(
)
∑ ∫ d
−∞
i
1 =
=
−
=
−
(28)
( ) t
( ) e t i
( ) u t i
% ( ) u t i
, ω j
( ) v t j
% ( ) v t j
Define the synchronization error signal , then
n
n
the error dynamics between systems (27) and (28) can be expressed by
( ) t
( ) t
j
j
j
∑
∑
)
( ω j
)
)
( ( t ω θ ji
)
j
j
1 =
1 =
n
t
= − + + − & ( ) e t i % b f ji % % b f ji % ( ( ) p e t i i
(
( ) t
ji
ji
j
( ω j
) ( ) s ds
∑ ∫ b
−∞
j
1 =
m
m
%
%
% q
= −
+
+
−
( ) t
( ) t
( ) t
& ω j
j
ij
i
i
τ ij
∑
∑
( ( ) d g e t i
)
( ω j
)
)
% ( ( d g e t ij i
)
i
i
1 =
1 =
m
t
%
t
,
−
−
+
(
( )
( ) t
k ij
) s g e s ds i
i
ϑ j
ij
(
)
∑ ∫ d
−∞
i
1 =
k t , + − − % ) s f σ i
(29)
m
n
µ
=
=
i
n
We consider the following control inputs strategy
1, 2,...,
m j ,
1, 2,...,
=
=
( ) t
( ) t
( ) t
σ i
( ) e t ik k
, ϑ j
ρ ω jk k
∑
∑
k
k
1 =
1 =
, . (30)
As a consequence of Theorem 1, we have the following result:
Corollary 1. Under the assumptions (A1)–(A4), drive-response systems (27) and (28) are
i
0,
0
0
1, 2,...,
,
2,
>
=
+
≥
(
) n m r
iw
ijγ >
jiβ > such that the controller gain matrices µ
in global exponential synchronization, if there exist
n
n
m
1 −
r
r 1 −
rn
r
r
r
2
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
w i
r rna i
µ ii
r a i
r a i
− β ji
r a i
r a i
∑
∑
∑
j
j
k
i k
1 =
1 =
1, = ≠
19
and ρ in (9) satisfy
n
m
r
r
r
r
r
r
r
r
ij
g L i
ij
r γ ij
g L i
m j +
∑
(
)
(
(
rg ) L i
) )
(
τ e −
j
k
i k
1 =
1, = ≠
τµ
w m d d n 0 + + < + µ ki w k % d ij +∑ 1
m
n
m
1 −
r
r 1 −
w
rmc
r
c
r
c
r
c
2
−
−
+
−
+
−
+
−
(
) 1
(
) 1
(
) 1
r j
rm c ρ jj
r j
r j
r j
− r γ j ij
m j +
∑
∑
∑
i
k
i
1 =
1, =
j k ≠
1 =
m
n
r
r
r
r
r
r
r
r
r
and
ji
f j
ji
% f j
ji
r β ji
f j
k m +
∑
∑
(
)
(
(
)
(
) )
θ e −
i
k
1 =
1, =
j k ≠
g
f
i
m
j
n
b L % b L b L m w 0, (31) + + + + < w n i ρ kj 1 µ θ
1, 2,...,
,
1, 2,...,
,
=
=
jL and
iL are Lipschitz constants.
in which
4.
Illustration example
To illustrate the effectiveness of our criterion, we give the following example.
Example
1.
T
2
| 0
0.2 ,
kπ
Ω =
<
<
=
⊂ (cid:2)
)
x k
x x , 1 2
{ (
} 1, 2
2
n
n
Consider the following system on
( ) t
)
( v t x
)
j
( a u t x i i
j
j
∑ %
∑
( v t j
)
(
)
(
)
j
j
1 =
1 =
n
t
x , + − , , = − + b f ji θ ji b f ji u ∂ i t ∂ u ∂ i 2 x ∂
(
) s f
(
ji
j
j
i
ji
(
) ) s x ds ,
∑ ∫ b
−∞
j
1 =
2
m
m
k t v I , − + +
j
x
,
+
−
( ) t
)
(
)
ij
i
τ ij
( c v t x j
j
ij
i
∑ %
∑
( d g u t x i
)
)
( ( d g u t i
)
i
i
1 =
1 =
m
t
k
t
J
,
,
−
+
+
v ∂ ∂ , , − + = v j 2 t ∂ x ∂
(
)
ij
) ( s g u s x ds i
i
j
ij
)
(
∑ ∫ d
−∞
i
1 =
(32)
20
and
2
2
2
∂
)
)
x
,
,
,
=
−
+
+
−
)
% ( v t x
)
( ) t
% ( a u t x i i
b f ji
j
j
% b f ji
j
θ ji
∑
∑
(
)
( % v t j
)
(
)
% ( u t x , ∂ i t ∂
% ( u t x , i 2 x ∂
j
j
1 =
1 =
2
2
t
b
k
t
% v
I
,
,
+
−
+
+
µ
(
) s f
(
( ) t
( e t x
)
ji
ji
j
j
i
ik k
∑
∑
(
) ) s x ds ,
∫
−∞
k
j
1 =
1 =
2
2
2
)
)
j
)
(
)
( ) t
( c v t x j
j
ij
i
∑
∑
)
)
)
i
i
1 =
1 =
2
2
t
, , ∂ % x , , , = − + + − % i τ ij % ( d g u t x i % ( ( d g u t ij i % ( v t x ∂ t ∂ % ( v t x j 2 x ∂ (33)
(
)
( ) t
(
)
ij
i
j
∑
∑
)
(
∫
−∞
k
i
1 =
1 =
n m r
k
te−
2,
,t
= =
=
=
=
f
d t J , t x , , + − + + k ij % ) ( s g u s x ds i ρ ω jk k
1
η
η
=
+ + −
( ) t
( ) t
( ) gη =
( ) η
ji
k ij
j
i
) 1 ,
(
1 2
where
i
,
1, 2
j =
f j
f j
g L i
g L i
1
d
L L 1, . ln 2. 1, 2, 2, 5, λ τ θ = = = = = = = = = = = 1, a 1 a 2 c 1 c 2
0.2,
0.6,
0.5,
=
=
=
d 11
d 12
21
21
22
θµ µ=
τ
0.8,
1,
0.2,
0.5,
0.4.
= −
=
=
=
d d 0.2, 0.5, 1, 0.5, 0.2, = = = = = d 11 d 12
d = 22
b 11
b 12
b 21
b 22
0.5, 0.6, 1, 0.8, = = = = − b 11 b 12 b 21 b 22
0.5, 0.3, 0.7, 0.1, 0.6, 2, 1, 0.4 . = = = = = = = = µ 11 µ 12 µ 21 µ 22 ρ 11 ρ 12 ρ 21 ρ 22
1
3
11
12
21
By simple calculation with 1, = = = β β β β = = w w = 2 w w = 4 = 22 1,
11
12
21
22 1
2
2
1 −
r 1 −
and = γ γ γ γ = = = , we get
(
) 1
(
) 1
(
) 1
− β j
r rna 1
1 − λ 1
r rna 1
r a 11 1
r a 1
r a 1
r 1
r a 1
∑
∑
j
j
1 =
1 =
2
r
r
r
r
r
r
n
12.04 0,
+
+
= −
<
+
j
j
j
r m d 1
g L 1
d 1
r γ 1
g L 1
µ 12
∑
(
)
(
)
(
j
1 =
2
2
1 −
r 1 −
rn r r r 2 µ − − − + − + − + −
(
) 1
(
) 1
(
) 1
− β j
r rna 2
1 − λ 1
r rna 2
r a 22 2
r a 2
r a 2
r 2
r a 2
∑
∑
j
j
1 =
1 =
2
r
r
r
r
r
r
r m d
rn r r r 2 µ − − − + − + − + −
j
j
j
2
g L 2
2
r γ 2
g L 2
∑
(
)
(
) )
(
j
1 =
21
d n 22.12 0, + + = − < + µ 21
2
2
1 −
r
r 1 −
r
r
2 +
−
+
−
−
−
−
(
) 1
(
) 1
(
r c 1
r c 1
− r c γ i 1 1
r rmc 1
1 − λ 1
r rmc 1
r rm c ρ 11 1
∑
i
i
1 =
1 =
2
r
r
r
r
r
r
r
r + − ∑ ) 1
f L 1
r β i 1
f L 1
∑
(
)
(
)
(
i
1 =
m n + + = − 9.8 0 < + b i 1 b i 1 ρ 12
2
2
1 −
r 1 −
r
r
and
2 +
−
+
−
−
−
−
(
) 1
(
) 1
(
r c 2
r c 2
− r c γ i 2
r 2
r rmc 2
1 − λ 1
r rmc 2
r rm c ρ 22 2
∑
i
i
1 =
1 =
2
r
r
r
r
r
r
r
r + − ∑ ) 1
i
i
f L 2
r β i 2
f L 2
∑
(
)
(
)
(
i
1 =
n m + + + = − 9.4 0, < b 2 b 2 ρ 21
Hence, it follows from Theorem 1 that (32) and (33) are globally exponentially
synchronized.
5.
Conclusions
In this article, global exponential synchronization has been considered for a class of
BAM NNs with time-varying and distributed delays and reaction–diffusion terms. We
have established a new sufficient condition which includes the diffusion coefficients by
constructing the suitable Lyapunov functional, introducing many real parameters and
applying inequality techniques. From condition (10) in Theorem 1, we see that diffusion
coefficients directly affect the synchronization behavior of the delayed BAM NNs with
reaction–diffusion terms. In comparison with previous literature, diffusion effects are
taken into account in our models. A numerical example has been given to show the
22
effectiveness of the obtained results.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WZ designed and performed all the steps of proof in this research and also wrote the
paper. JL participated in the design of the study and suggest many good ideas that made
this paper possible and helped to draft the first manuscript. All authors read and approved
the final manuscript.
Acknowledgments
This study was partially supported by the National Natural Science Foundation of China
under Grant No. 60974139 and partially supported by the Fundamental Research Funds
for the Central Universities under Grant No. 72103676, the Natural Science Foundation
of Shannxi Province, China under Grant No. 2010JQ1013, and the Special research
projects in Shannxi Province Department of Education under Grant No. 2010JK896.
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