RESEARCH Open Access Global exponential synchronization ......RESEARCH Open Access Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and

RESEARCH Open Access

Global exponential synchronization of delayedBAM neural networks with reaction-diffusionterms and the Neumann boundary conditionsWeiYuan Zhang1,2* and JunMin Li1

* Correspondence: [email protected] of Science, XidianUniversity, Shaan Xi Xi’an 710071,P.R. ChinaFull list of author information isavailable at the end of the article

Abstract

In this article, a delay-differential equation modeling a bidirectional associativememory (BAM) neural networks (NNs) with reaction-diffusion terms is investigated. Afeedback control law is derived to achieve the state global exponentialsynchronization of two identical BAM NNs with reaction-diffusion terms byconstructing a suitable Lyapunov functional, using the drive-response approach andsome inequality technique. A novel global exponential synchronization criterion isgiven in terms of inequalities, which can be checked easily. A numerical example isprovided to demonstrate the effectiveness of the proposed results.

Keywords: neural networks, reaction-diffusion, delays, global exponential synchroni-zation, Lyapunov functional

1. IntroductionAihara et al. [1] firstly proposed chaotic neural network (NN) models to simulate the

chaotic behavior of biological neurons. Consequently, chaotic NNs have drawn consid-

erable 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 synchroniza-

tion of NNs with diffusion terms, which are expressed by partial differential equations.

For instance, the authors of [16] discuss the impulsive control and 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 con-

troller is derived to achieve the exponential synchronization of the drive-response

structure of NNs with reaction-diffusion terms. Meanwhile, although the models of

Zhang and Li Boundary Value Problems 2012, 2012:2http://www.boundaryvalueproblems.com/content/2012/1/2

© 2012 Zhang and Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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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 pre-

sence 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 distri-

bution 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 the-

ory 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 dis-

tributed delays and reaction-diffusion terms by using Poincaré integral inequality,

Young inequality technique, and Lyapunov method, which are very important in the-

ories and applications and also are a very challenging problem. Several sufficient condi-

tions are in the form of a few algebraic inequalities, which are very convenient to

verify.

2. Model description and preliminariesIn this article, a class of delayed BAM NNs with reaction-diffusion terms is described

as follows

∂ui∂t

=l∑

k=1

∂

∂xk

(Dik

∂ui∂xk

)− pi (ui (t, x))

+n∑j=1

bjifj(vj (t, x)

)+

n∑j=1

b̃jifj(vj

(t − θji (t) , x

))+

n∑j=1

b̄ji

∫ t−∞

kji (t − s) fj(vj (s, x)

)ds + Ii (t) ,

∂vj∂t

=l∑

k=1

∂

∂xk

(D∗jk

∂vj∂xk

)− qj

(vj (t, x)

)+

m∑i=1

dijgi (ui (t, x))

+m∑i=1

d̃ijgi(ui

(t − τij (t) , x

))+

m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi (ui (s, x)) ds + Jj (t) ,

(1)

where x = (x1, x2 ,..,xl)T Î Ω ⊂ ℝl, Ω is a compact set with smooth boundary ∂Ω and

mesΩ > 0 in space ℝl; u = (u1,u2,...,um)T Î ℝm, (v1,v2,...,vn)

T Î ℝn, ui(t,x) and vj(t,x) andrepresent the states of the ith neurons and the jth neurons at time t and in space x,

respectively. bji, b̃ji, b̄ji,dij, d̄ij, and d̃ij are known constants denoting the synaptic connec-

tion strengths between the neurons, respectively; fi and gi denote the activation func-

tions of the neurons and the signal propagation functions, respectively; Ii and Ji denote

the external inputs on the ith and jth neurons, respectively; pi and qj are differentiable

real functions with positive derivatives defining the neuron charging time, respectively;

τij(t) and θji(t) represent continuous time-varying discrete delays, respectively; Dik ≥ 0

and D∗jk ≥ 0 stand for the transmission diffusion coefficient along the ith and jth neu-rons, respectively. i = 1, 2, ..., m, k = 1, 2, l and j = 1, 2,..., n.

System (1) is supplemented with the following boundary conditions and initial values

∂ui∂n̄

:=(

∂ui∂x1

,∂ui∂x2

, ...,∂ui∂xl

)T= 0,

∂vj∂n̄

:=(

∂vj∂x1

,∂vj∂x2

, ...,∂vj∂xl

)T= 0, t ≥ 0, x ∈ ∂�, (2)

ui (s, x) = ϕui (s, x) , vj (s, x) = ϕvj (s, x) , (s, x) ∈ (−∞, 0] × �. (3)


Page 2 of 14

for any i = 1,2,..., m and j = 1,2,..., n where n̄ is the outer normal vector of

∂Ω,ϕ =(

ϕuϕv

)= (ϕu1, ...,ϕum,ϕv1, ...,ϕvn)T ∈ C are bounded and continuous, where

C ={ϕ|ϕ =

(ϕuϕv

),ϕ :

((−∞, 0] × Rm(−∞, 0] × Rn

)→ Rm+n

}. It is the Banach space of continuous

functions which map

((−∞, 0](−∞, 0]

)into ℝm+n with the topology of uniform converge for the

norm

‖ϕ‖ =∥∥∥∥(

ϕuϕv

)∥∥∥∥ = sup−∞≤s≤0[∫

�

m∑i=1

|ϕui|rdx]+ sup

−∞≤s≤0

⎡⎣∫

�

n∑j=1

∣∣ϕvj∣∣rdx⎤⎦ , r ≥ 2.

Throughout this article, we assume that the following conditions are made.

(A1) The functions τij(t), θji(t) are piecewise-continuous of class C1 on the closure of

each continuity subinterval and satisfy

0 ≤ τij (t) ≤ τij, 0 ≤ θji (t) ≤ θji, τ̇ij (t) ≤ μτ < 1, θ̇ji (t) ≤ μθ < 1,τ = max

1≤i≤m,1≤j≤n{τij

}, θ = max

1≤i≤m,1≤j≤n{θji

},

with some constants τij ≥ 0, θji ≥ 0, τ > 0, θ >0, for all t ≥ 0.

(A2) The functions pi (·)and qj(·) are piecewise-continuous of class C1 on the closure

of each continuity subinterval and satisfy

ai = infζ∈R

pi ′ (ζ ) > 0, pi (0) = 0,

cj = infζ∈R

qj′ (ζ ) > 0, qj (0) = 0.

(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist

positive constants Lfj and Lgi such that for all h1, h2 Î ℝ∣∣fj (η1) − fj (η2)∣∣ ≤ Lfj |η1 − η2| , ∣∣gi (η1) − gi (η2)∣∣ ≤ Lgi |η1 − η2| .

(A4) The delay kernels Kji (s) , K̄ij (s) : [0,∞) → [0,∞) ,(i = 1, 2,...,m, j = 1, 2,...,n) arereal-valued non-negative continuous functions that satisfy the following conditions

(i)∫ +∞0 Kji (s) ds = 1,

∫ +∞0 K̄ji (s) ds = 1,

(ii)∫ +∞0 sKji (s) ds < ∞,

∫ +∞0 sK̄ij (s) ds < ∞,

(iii)There exist a positive μ such that∫ +∞0

seμsKji (s) ds < ∞,∫ +∞0

seμsK̄ij (s) ds < ∞.

We consider system (1) as the drive system. The response system is described by the

following equations

∂ũi (t, x)∂t

=l∑

k=1

∂

∂xk

(Dik

∂ũi (t, x)∂xk

)− pi

(ũi (t, x)

)+

n∑j=1

bjifj(ṽj (t, x)

)

+n∑j=1

b̃jifj(ṽj

(t − θji (t) , x

))+

n∑j=1

b̄ji

∫ t−∞

kji (t − s) fj(ṽj (s, x)

)ds + Ii (t) + σi (t, x) ,

∂ ṽj (t, x)∂t

=l∑

k=1

∂

∂xk

(D∗jk

∂ ṽj (t, x)∂xk

)− qj

(ṽj (t, x)

)+

m∑i=1

dijgi(ũi (t, x)

)

+m∑i=1

d̃ijgi(ũi

(t − τij (t) , x

))+

m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi(ũi (s, x)

)ds + Jj (t) + ϑj (t, x) ,

(4)


Page 3 of 14

where si (t,x) and ϑj(t,x) denote the external control inputs that will be appropriately

designed for a certain control objective. We denote ũ (t, x) =(ũ1 (t, x) , ..., ũm (t, x)

)T,ṽ (t, x) =

(ṽ1 (t, x) , ..., ṽn (t, x)

)T , σ (t, x) = (σ1 (t, x) , ..., σm (t, x))T and ϑ(t,x) = (ϑ1(t,x),...,ϑn(t,x))

T.

The boundary and initial conditions of system (4) are

∂ũi∂n̄

:=(

∂ũi∂x1

,∂ũi∂x2

, ...,∂ũi∂xl

)T= 0,

∂ ṽj∂n̄

:=(

∂ ṽj∂x1

,∂ ṽj∂x2

, ...,∂ ṽj∂xl

)T= 0 t ≥ 0, x ∈ ∂�, (5)

and

ũi (s, x) = ψui (s, x) , ṽj (s, x) = ψvj (s, x) , (s, x) ∈ (−∞, 0] × �, (6)

where ψ =(

ψũψṽ

)=

(ψũ1, ...,ψũm,ψṽ1, ...,ψṽn

)T ∈ C.Definition 1. Drive-response systems (1) and (4) are said to be globally exponentially

synchronized, if there are control inputs s(t,x), ϑ(t,x), and r ≥ 2, further there existconstants a > 0 and b ≥ 1 such that∥∥u (t, x) − ũ (t, x)∥∥ + ∥∥v (t, x) − ṽ (t, x)∥∥

≤ βe−2αt (‖ϕu (s, x) − ψũ (s, x)‖ + ‖ϕv (s, x) − ψṽ (s, x)‖), for all t ≥ 0,

in which∥∥u (t, x) − ũ (t, x)∥∥ = ∫

�

m∑i=1

∣∣ui (t, x) − ũi (t, x)∣∣rdx,∥∥v (t, x) − ṽ (t, x)∥∥ = ∫

�

n∑j=1

∣∣vj (t, x) − ṽj (t, x)∣∣rdx, r ≥ 2, and (u(t,x), v(t,x)) and(ũ (t, x) , ṽ (t, x)

)are the solutions of drive-response systems (1) and (4) satisfying

boundary conditions and initial conditions (2), (3) and (5), (6), respectively.

Lemma 1. [21] (Poincaré integral inequality). Let Ω be a bounded domain of ℝm

with a smooth boundary ∂Ω of class C2 by Ω. u(x) is a real-valued function belonging

to H10 (�) and∂u (x)

∂n̄|∂� = 0. Then

∫�

|u (x)|2dx ≤ 1λ1

∫�

|∇u (x)|2dx,

which l1 is the lowest positive eigenvalue of the Neumann boundary problem⎧⎨⎩

−�ϕ (x) = λ1ϕ (x) , x ∈ �,∂u (x)

∂n̄|∂� = 0, x ∈ ∂�. (7)

3. Main resultsFrom the definition of synchronization, we can define the synchronization error signal

ei (t, x) = ui (t, x) − ũi (t, x) ,ωj (t, x) = vj (t, x) − ṽj (t, x), e(t,x) = (e1(t,x),...,em(t,x))T, andω(t,x) = (ω1(t,x),..., ωn(t,x))

T . Thus, error dynamics between systems (1) and (4) can be

expressed by


Page 4 of 14

∂ei (t, x)∂t

=l∑

k=1

∂

∂xk

(Dik

∂ei (t, x)∂xk

)− p̃i (ei (t, x)) +

n∑j=1

bjif̃j(ωj (t, x)

)

+n∑j=1

b̃ji f̃j(ωj

(t − θji (t) , x

))+

n∑j=1

b̄ji

∫ t−∞

kji (t − s) f̃j(ωj (s, x)

)ds − σi (t, x) ,

∂ωj (t, x)∂t

=l∑

k=1

∂

∂xk

(D∗jk

∂ωj (t, x)∂xk

)− q̃j

(ωj (t, x)

)+

m∑i=1

(dijg̃i (ei (t, x))

)

+m∑i=1

(d̃ijg̃i

(ei

(t − τij (t) , x

)))+

m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) g̃i (ei (s, x)) ds − ϑj (t, x) ,

(8)

where f̃j(ωj (t, x)

)= fj

(vj (t, x)

) − fj (ṽj (t, x)) , g̃i (ei (t, x)) = gi (ui (t, x)) − gi (ũi (t, x)),p̃i (ei (t, x)) = pi (ui (t, x)) − pi

(ũi (t, x)

), q̃j

(ωj (t, x)

)= qj

(vj (t, x)

) − qj (ṽj (t, x)).The control inputs strategy with state feedback are designed as follows:

σi (t, x) =m∑k=1

μikek (t, x) ,ϑj (t, x) =n∑

k=1

ρjkωk (t, x) , i = 1, 2, ...,m, j = 1, 2, ...,n.

that is,

σ (t, x) = μe (t, x) ,ϑ (t, x) = ρω (t, x) , (9)

where μ = (μik)m×m and ρ =(ρjk

)n×n are the controller gain matrices.

The global exponential synchronization of systems (1) and (4) can be solved if the

controller matrices μ and r 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 wi > 0(i = 1,2,..., n+m), r ≥ 2,

gij > 0, bji > 0 such that the controller gain matrices μ and r in (9) satisfy

wi

⎛⎜⎝−rnar−1i Diλ1 − rnari − rnμiiar−1i + 2 (r − 1)

n∑j=1

ari + (r − 1)n∑j=1

ariβ−

rr − 1

ji + (r − 1)m∑

k=1,i�=kari

⎞⎟⎠

+n∑j=1

wm+jmr(∣∣dij∣∣r(Lgi )r + ∣∣∣d̃ij∣∣∣r eτ1 − μτ

(Lgi

)r+∣∣∣d̄ij∣∣∣rγ rij(Lgi )r) + nr

m∑k=1,i�=k

|μki|rwk < 0

and

wm+j

⎛⎜⎝−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj + (r − 1)

m∑i=1

crj γ−

rr − 1

ij

⎞⎟⎠

+m∑i=1

winr(∣∣bji∣∣r(Lfj )r + ∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj

)r+∣∣∣b̄ji∣∣∣rβ rji(Lfj)r) +mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m < 0,(10)

in which i = 1, 2, ..., m, j = 1, 2,..., n, Lfj and Lgi are Lipschitz constants,

Di = min1≤k≤l

Dik,D∗j = min1≤k≤l

D∗jk,l1 is the lowest positive eigenvalue of problem (7).

Proof. If (10) holds, we can always choose a positive number δ > 0 (may be very

small) such that

wi

⎛⎜⎝−rnar−1i Diλ1 − rnari − rnμiiar−1i + 2 (r − 1)

n∑j=1

ari + (r − 1)n∑j=1

ariβ−

rr − 1

ji + (r − 1)m∑

k=1,i�=kari

⎞⎟⎠

+n∑j=1


(Lgi


m∑k=1,i�=k

|μki|rwk + δ < 0


Page 5 of 14

and

wm+j

⎛⎜⎝−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj + (r − 1)

m∑i=1

crj γ−

rr − 1

ij

⎞⎟⎠

+m∑i=1

winr

⎛⎝∣∣bji∣∣r(Lfj )r + ∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj

)r+∣∣∣b̄ji∣∣∣rβ rji(Lfj )r) +mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m + δ < 0,(11)

where i = 1, 2,..., m, j = 1, 2,..., n.

Let us consider functions

Fi(x∗i

)= wi

[−rnar−1i Diλ1 − rnari − rnμiiar−1i + 2 (r − 1)n∑j=1

ari + (r − 1)m∑

k=1,i�=kari

+ (r − 1)n∑j=1

ariβ−

rr − 1

ji

∫ +∞0

kji (s) ds + 2x∗i nar−1i

⎤⎥⎦ + n∑

j=1

wm+jmr(∣∣dij∣∣r(Lgi )r

+∣∣∣d̃ij∣∣∣r eτ1 − μτ

(Lgi

)r+∣∣∣d̄ij∣∣∣rγ rij(Lgi )r

∫ +∞0

e2x∗i sk̄ij (s) ds

)+ nr

m∑k=1,i�=k

|μki|rwk

and

Gj(y∗j

)= wm+j

[−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj

+ (r − 1)m∑i=1

crjγ−

r

r − 1ij

∫ +∞0

k̄ij (s) ds + 2y∗j mcr−1j

⎤⎥⎦ + m∑

i=1

winr(∣∣bji∣∣r(Lfj )r

+∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj

)r+∣∣∣b̄ji∣∣∣rβ rji(Lfj )r

∫ +∞0

e2y∗j skji (s) ds

)+mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m,

(12)

where x∗i , y∗j ∈ [0, +∞) ,i = 1, 2, ..., m, j = 1, 2, ..., n.

From (12) and (A4), we derive

Fi(0) < -δ < 0, Gj(0) < -δ

and

Gj(νj)= wm+j

(−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj

+ (r − 1)m∑i=1

crjγ−

rr − 1

ij

∫ +∞0

k̄ij (s) ds +2νjmcr−1j

)+

m∑i=1


+∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj


∫ +∞0

e2νj skji (s) ds)+mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m = 0.(13)

By using α = min1≤i≤m,1≤j≤n

{εi, νj

}, obviously, we get

Fi (α) = wi(−rnar−1i Diλ1 − rnari − rnμiiar−1i + 2 (r − 1)

n∑j=1

ari + (r − 1)m∑

k=1,i�=kari

+ (r − 1)n∑j=1

ariβ−

rr − 1

ji

∫ +∞0

kji (s) ds + 2αnar−1i ) +

n∑j=1


+∣∣∣d̃ij∣∣∣r eτ1 − μτ

(Lgi

)r+∣∣∣d̄ij∣∣∣rγ rij(Lgi )r

∫ +∞0

e2αsk̄ij (s) ds)+ nr

m∑k=1,i�=k

|μki|rwk ≤ 0

and

Gj (α) = wm+j(−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj

+ (r − 1)m∑i=1

crj γ−

rr − 1

ij

∫ +∞0

k̄ij (s) ds +2αmcr−1j

)+

m∑i=1


+∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj

)r+∣∣∣b̄ji∣∣∣rβ rji(Lfj)r

∫ +∞0

e2αskji (s) ds)+mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m ≤ 0.(14)

Multiplying both sides of the first equation of (8) by ei (t,x) and integrating over Ω yields

12ddt

∫�

ei (t, x)2dx =∫

�

l∑k=1

ei (t, x)∂

∂xk

(Dik

∂ei (t, x)∂xk

)dx − p′ i (ξi)

∫�

ei(t, x)2dx

+∫

�

n∑j=1

bjiei (t, x) f̃j(ωj (t, x)

)dx +

n∑j=1

∫�

b̃jiei (t, x) f̃j(ωj

(t − θji (t) , x

))dx

+n∑j=1

∫�

b̄jiei (t, x)∫ t

−∞kji (t − s) f̃j

(ωj (s, x)

)dsdx −

∫�

m∑k=1

ei (t, x) μikek (t, x) dx.

(15)

It is easy to calculate by the Neumann boundary conditions (2) that

∫�

l∑k=1

ei (t, x)∂

∂xk

(Dik

∂ei (t, x)∂xk

)dx =

∫�

l∑k=1

ei (t, x)∇(Dik

∂ei (t, x)∂xk

)dx

=∫

∂�

l∑k=1

ei (t, x)Dik∂ei (t, x)

∂xkdx −

∫�

l∑k=1

Dik

(∂ei (t, x)

∂xk

)2dx = −

l∑k=1

∫�

Dik

(∂ei (t, x)

∂xk

)2dx

(16)

Moreover, from Lemma 1, we can derive

−l∑

k=1

∫�

Dik

(∂ei (t, x)

∂xk

)2dx ≤ −

l∑k=1

∫�

Di

(∂ei (t, x)

∂xk

)2dx ≤ −Diλ1 ‖ei (t, x)‖22 . (17)


Page 7 of 14

From (13)-(17), (A2), and (A3), we obtain that

ddt

∫�

|ei (t, x)|2dx ≤ −2Diλ1∫

�

|ei (t, x)|2dx − 2ai∫

�

|ei (t, x)|2dx

+2∫

�

n∑j=1

∣∣bji∣∣ |ei (t, x)| Lfj ∣∣ωj (t, x)∣∣dx + 2n∑j=1

∫�

∣∣∣b̃ji∣∣∣ |ei (t, x)| ∣∣∣f̃j (ωj (t − θji (t) , x))∣∣∣ dx+2

n∑j=1

∫�

∣∣∣b̄ji∣∣∣∫ t

−∞kji (t − s) |ei (t, x)|

∣∣∣f̃j (ωj (s, x))∣∣∣ dsdx − 2∫

�

m∑k=1

|ei (t, x)|μik |ek (t, x)| dx.

(18)

Multiplying both sides of the second equation of (8) by ωj (t,x), similarly, we also have

ddt

∫�

∣∣ωj (t, x)∣∣2dx ≤ −2D∗j λ1∫�

∣∣ωj (t, x)∣∣2dx − 2cj∫

�

∣∣ωj (t, x)∣∣2dx+2

∫�

m∑i=1

∣∣dij∣∣ Lgi |ei (t, x)| ∣∣ωj (t, x)∣∣dx + 2m∑i=1

∫�

∣∣∣d̃ij∣∣∣ ∣∣g̃i (ei (t − τij (t) , x))∣∣ ∣∣ωj (t, x)∣∣ dx+2

m∑i=1

∫�

∣∣∣d̄ij∣∣∣∫ t

−∞k̄ij (t − s)

∣∣g̃i (ei (s, x))∣∣ ∣∣ωj (t, x)∣∣ dsdx − 2∫

�

n∑k=1

ρjk |ωk (t, x)|∣∣ωj (t, x)∣∣dx.

(19)

Consider the following Lyapunov functional

V (t) =∫

�

m∑i=1

wi[nar−1i |ei (t, x)|re2αt +

n∑j=1

∣∣∣b̃ji∣∣∣rnr eθ1 − μθ∫ tt−θji(t)

e2αξ∣∣∣f̃j (ωj (ξ , x))∣∣∣rdξ

+n∑j=1

∣∣∣b̄ji∣∣∣rnrβ rji∫ +∞0

kji (s)∫ tt−s

e2α(s+ξ )∣∣∣f̃j (ωj (ξ , x))∣∣∣rdξds

⎤⎦ dx

+∫

�

n∑j=1

wm+j[mcr−1j

∣∣ωj (t, x)∣∣re2αt + m∑i=1

∣∣∣d̃ij∣∣∣rmr eτ1 − μτ∫ tt−τij(t)

e2αξ∣∣g̃i (ei (ξ , x))∣∣rdξ

+m∑i=1

∣∣∣d̄ij∣∣∣rmrγ rij∫ +∞0

k̄ij (s)∫ tt−s

e2α(s+ξ )∣∣g̃i (ei (ξ , x))∣∣rdξds

]dx.

(20)

Its upper Dini-derivative along the solution to system (8) can be calculated as

D+V (t) ≤∫

�

m∑i=1

wi

[rnar−1i |ei (t, x)|r−1

∂ |ei (t, x)|∂t

e2αt + 2αe2αtnar−1i |ei (t, x)|r

+e2αtn∑j=1

∣∣∣b̃ji∣∣∣rnr eθ1 − μθ∣∣∣f̃j (ωj (t, x))∣∣∣r + e2αt n∑

j=1


e2αskji (s)∣∣∣f̃j (ωj (t, x))∣∣∣rds

−n∑j=1

∣∣∣b̃ji∣∣∣rnr eθ1 − μθ(1 − θ̇ji (t)

)e2α(t−θji(t))

∣∣∣f̃j (ωj (t − θji (t) , x))∣∣∣r

−e2αtn∑j=1


kji (s)∣∣∣f̃j (ωj (t − s, x))∣∣∣rds

⎤⎦ dx

+∫

�

n∑j=1

wm+j

[rmcr−1j

∣∣ωj (t, x)∣∣r−1 ∂∣∣ωj (t, x)∣∣

∂te2αt + 2αe2αtmcr−1j

∣∣ωj (t, x)∣∣r

+e2αtm∑i=1

∣∣∣d̃ij∣∣∣rmr eτ1 − μτ∣∣g̃i (ei (t, x))∣∣r

−m∑i=1

∣∣∣d̃ij∣∣∣rmr eτ1 − μτ e2α(t−τij(t))(1 − τ̇ij (t)

)∣∣g̃i (ei (t − τij (t) , x))∣∣r

+e2αtm∑i=1


e2αsk̄ij (s)∣∣g̃i (ei (t, x))∣∣rds

−e2αtm∑i=1


k̄ij (s)∣∣g̃i (ei (t − s, x))∣∣rds

]dx

(21)


Page 8 of 14

From (21) and Young inequality, we can conclude

D+V (t) ≤∫

�

e2αtm∑i=1

[wi

(−rnar−1i Diλ1 − rnari + 2 (r − 1)n∑j=1

ari + 2αnar−1i + (r − 1)

m∑k=1,i�=k

ari

−rnμiiar−1i + (r − 1)n∑j=1

ariβ−

rr − 1

ji

∫ t−∞

kji (t − s) ds

⎞⎟⎠ + n∑

j=1


+∣∣∣d̃ij∣∣∣r eτ1 − μτ

(Lgi

)r+∣∣∣d̄ij∣∣∣rγ rij

∫ +∞0

e2αsk̄ij (s)(Lgi

)rds

)+ nr

m∑k=1,i�=k

|μki|rwk⎤⎦ |ei (t, x)|rdx

+∫

�

e2αtn∑j=1

[wm+j

(−rmcr−1j D∗j λ1 − rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj

+ (r − 1)m∑i=1

crj γ−

rr − 1

ij

∫ t−∞

k̄ij (t − s) ds +2αmcr−1j)+

m∑i=1


+∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj


∫ +∞0

e2αskji (s) ds)+mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m⎤⎦ ∣∣ωj (t, x)∣∣rdx

(22)

From (10), we can conclude

D+V (t) ≤ 0, and so V (t) ≤ V (0) , t ≥ 0 (23)

Since

V (0) =∫

�

m∑i=1

wi[nar−1i |ei (0, x)|r

+n∑j=1

∣∣∣b̃ji∣∣∣rnr eθ1 − μθ∫ 0

−θji(t)

∣∣∣f̃j (ωj (ξ , x))∣∣∣rdξ

+n∑j=1


kji (s)∫ 0

−se2α(s+ξ )

∣∣∣f̃j (ωj (ξ , x))∣∣∣rdξds⎤⎦ dx

+∫

�

n∑j=1

wm+j[mcr−1j

∣∣ωj (0, x)∣∣r + m∑i=1

∣∣∣d̃ij∣∣∣rmr eτ1 − μτ∫ 0

−τij(t)

∣∣g̃i (ei (ξ , x))∣∣rdξ+

m∑i=1


k̄ij (s)∫ 0

−se2α(s+ξ )

∣∣g̃i (ei (ξ , x))∣∣rdξds]dx

≤{max1≤i≤m

{wi} + max1≤j≤n

{wm+j

}max1≤j≤n

[m∑i=1

∣∣∣d̄ij∣∣∣rmr(Lgi )rγ rij∫ +∞0

k̄ij (s) se2αsds

]

+ max1≤j≤n

{wm+j

}max1≤j≤n

[m∑i=1

∣∣∣d̃ij∣∣∣r(Lgi )rmr eτ τ1 − μτ]}

‖ϕu (s, x) − ψu (s, x)‖r

+{max1≤j≤n

{wm+j

}+ max

1≤i≤m{wi} max

1≤i≤m

⎡⎣ n∑

j=1

∣∣∣b̄ji∣∣∣rnr(Lfj )rβ rji∫ +∞0

se2αskji (s)ds

⎤⎦

+ max1≤i≤m

{wi} max1≤i≤m

⎡⎣ n∑

j=1

∣∣∣b̃ji∣∣∣rnr(Lgj )r eθ θ1 − μθ⎤⎦⎫⎬⎭ ‖ϕv (s, x) − ψv (s, x)‖r

(24)

Noting that

e2αt(

min1≤i≤m+n

wi

)(‖e (t, x)‖ + ‖ω (t, x)‖) ≤ V (t) , t ≥ 0. (25)


Page 9 of 14

Let

β = max{max1≤i≤m

{wi}+ max1≤j≤n

{wm+j

}max1≤j≤n

[m∑i=1

∣∣∣d̄ij∣∣∣rmr(Lgi )rγ rij∫ +∞0

k̄ij (s) se2αsds

]

+ max1≤j≤n

{wm+j

}max1≤j≤n

[m∑i=1

∣∣∣d̃ij∣∣∣r(Lgi )rmr eτ τ1 − μτ],

max1≤j≤n

{wm+j

}+ max

1≤i≤m{wi} max

1≤i≤m

⎡⎣ n∑

j=1

∣∣∣b̄ji∣∣∣rnr(Lfj )rβ rji∫ +∞0

se2αskji (s)ds

⎤⎦

+ max1≤i≤m

{wi} max1≤i≤m

[n∑j=1

∣∣∣b̃ji∣∣∣rnr(Lgj )r eθ θ1 − μθ]}/

min1≤i≤m+n

{wi}.

Clearly, b ≥ 1.It follows that

‖e (t, x)‖ + ‖ω (t, x)‖ ≤ βe−2αt (‖ϕu (s, x) − ψũ (s, x)‖ + ‖ϕv (s, x) − ψṽ (s, x)‖) . (26)

for any t ≥ 0 where b ≥ 1 is a constant. This implies that drive-response systems (1)and (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 irrele-

vant to the diffusion term. Therefore, Theorem 1 is essentially new and more effective-

ness than those obtained.

Remark 2. It is noted that we construct a novel Lyapunov functional here as defined

in (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 synchroni-

zation 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 sys-

tem 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 BAM NNs

u̇i = −pi (ui (t)) +n∑j=1

bjifj(vj (t)

)+

n∑j=1

b̃jifj(vj

(t − θji (t)

))

+n∑j=1

b̄ji

∫ t−∞

kji (t − s) fj(vj (s)

)ds + Ii (t) ,

v̇j = −qj(vj (t)

)+

m∑i=1

dijgi (ui (t)) +m∑i=1

d̃ijgi(ui

(t − τij (t)

))

+m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi (ui (s)) ds + Jj (t)

(27)


Page 10 of 14

and the corresponding response system (4) will become the following form

˙̃ui (t) = −pi(ũi (t)

)+

n∑j=1

bjifj(ṽj (t)

)+

n∑j=1

b̃jifj(ṽj

(t − θji (t)

))

+n∑j=1

b̄ji

∫ t−∞

kji (t − s) fj(ṽj (s)

)ds + Ii (t) + σi (t) ,

˙̃vj (t) = −qj(ṽj (t)

)+

m∑i=1

dijgi(ũi (t)

)+

m∑i=1

d̃ijgi(ũi

(t − τij (t)

))

+m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi(ũi (s)

)ds + Jj (t) + ϑj (t) .

(28)

Define the synchronization error signal ei (t) = ui (t) − ũi (t) ,ωj (t) = vj (t) − ṽj (t),then the error dynamics between systems (27) and (28) can be expressed by

ėi (t) = −p̃i (ei (t)) +n∑j=1

bjif̃j(ωj (t)

)+

n∑j=1

b̃ji f̃j(ωj

(t − θji (t)

))

+n∑j=1

b̄ji

∫ t−∞

kji (t − s) f̃j(ωj (s)

)ds − σi (t) ,

ω̇j (t) = −q̃j(ωj (t)

)+

m∑i=1

dijg̃i (ei (t)) +m∑i=1

d̃ijg̃i(ei

(t − τij (t)

))

+m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) g̃i (ei (s)) ds − ϑj (t) ,

(29)

We consider the following control inputs strategy

σi (t) =m∑k=1

μikek (t) ,ϑj (t) =n∑

k=1

ρjkωk (t) , i = 1, 2, ...,m, j = 1, 2, ..., n. (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 in global exponential synchronization, if there exist wi > 0 (i = 1, 2,...,n+m), r ≥ 2,

gij > 0, bji > 0 such that the controller gain matrices μ and r in (9) satisfy

wi

⎛⎜⎝−rnari − rnμiiar−1i + 2 (r − 1)

n∑j=1

ari + (r − 1)n∑j=1

ariβ−

rr − 1

ji + (r − 1)m∑

k=1,i�=kari

⎞⎟⎠

+n∑j=1


(Lgi


m∑k=1,i�=k

|μki|rwk < 0

and

wm+j

⎛⎜⎝−rmcrj − rmρjjcr−1j + 2 (r − 1)

m∑i=1

crj + (r − 1)n∑

k=1,j�=kcrj + (r − 1)

m∑i=1

crj γ−

rr − 1

ij

⎞⎟⎠

+m∑i=1

winr(∣∣bji∣∣r(Lfj )r + ∣∣∣b̃ji∣∣∣r eθ1 − μθ

(Lfj

)r+∣∣∣b̄ji∣∣∣rβ rji(Lfj)r) +mr

n∑k=1,j�=k

∣∣ρkj∣∣rwk+m < 0,(31)


Page 11 of 14

in which i = 1, 2,..., m, j = 1, 2,..., n, Lfj and Lgi are Lipschitz constants.

4. Illustration exampleTo illustrate the effectiveness of our criterion, we give the following example.

Example 1. Consider the following system on

∂ui∂t

=∂2ui∂x2

− aiui (t, x) +n∑j=1

bjifj(vj (t, x)

)+

n∑j=1

b̃jifj(vj

(t − θji (t) , x

))

+n∑j=1

b̄ji

∫ t−∞


)ds + Ii,

∂vj∂t

=∂2vj∂x2

− cjvj (t, x) +m∑i=1

dijgi (ui (t, x)) +m∑i=1

d̃ijgi(ui

(t − τij (t) , x

))

+m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi (ui (s, x)) ds + Jj,

∂ui∂t

=∂2ui∂x2

− aiui (t, x) +n∑j=1

bjifj(vj (t, x)

)+

n∑j=1

b̃jifj(vj

(t − θji (t) , x

))

+n∑j=1

b̄ji

∫ t−∞


)ds + Ii,

∂vj∂t

=∂2vj∂x2

− cjvj (t, x) +m∑i=1

dijgi (ui (t, x)) +m∑i=1

d̃ijgi(ui

(t − τij (t) , x

))

+m∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi (ui (s, x)) ds + Jj,

(32)

and

∂ũi (t, x)∂t

=∂2ũi (t, x)

∂x2− aiũi (t, x) +

2∑j=1

bjifj(ṽj (t, x)

)+

2∑j=1

b̃jifj(ṽj

(t − θji (t) , x

))

+2∑j=1

b̄ji

∫ t−∞

kji (t − s) fj(ṽj (s, x)

)ds + Ii (t) +

2∑k=1

μikek (t, x) ,

∂ ṽj (t, x)

∂t=

∂2ṽj (t, x)

∂x2− cjṽj (t, x) +

2∑i=1

dijgi(ũi (t, x)

)+

2∑i=1

d̃ijgi(ũi

(t − τij (t) , x

))

+2∑i=1

d̄ij

∫ t−∞

k̄ij (t − s) gi(ũi (s, x)

)ds + Jj (t) +

2∑k=1

ρjkωk (t, x) ,

(33)

where n = m = r = 2, kji (t) = k̄ij (t) = te−t , fj (η) = gi (η) =12

(|η + 1| + |η − 1|) ,

Lfj = Lf̄j = L

gi = L

ḡi = 1, i, j = 1, 2. λ1 = 5, τ = θ = ln 2. a1 = 1, a2 = 2, c1 = 2, c2 = 1,

μτ = μθ = 0.2, d11 = 0.5, d12 = 1, d21 = 0.5, d22 = 0.2, d̄11 = 0.2, d̄12 = 0.6, d̄21 = 0.5,

d̄22 = 0.8, b11 = 0.5, b12 = 0.6, b21 = 1, b22 = −0.8, b̄11 = −1, b̄12 = 0.2, b̄21 = 0.5, b̄22 = 0.4.


Page 12 of 14

μ11 = 0.5,μ12 = 0.3,μ21 = 0.7,μ22 = 0.1, ρ11 = 0.6,ρ12 = 2,ρ21 = 1,ρ22 = 0.4.

By simple calculation with w1 = w2 = w3 = w4 = 1, b11 = b12 = b21 = b22 = 1,and g11 = g12 = g21 = g22 = 1, we get

−rnar−11 λ1 − rnar1 − rnμ11ar−11 + 2 (r − 1)2∑j=1

ar1 + (r − 1)2∑j=1

ar1β−

rr − 1

j1 + (r − 1) ar1

+2∑j=1

mr(∣∣d1j∣∣r(Lg1)r + ∣∣∣d̄1j∣∣∣rγ r1j(Lḡ1)r

⎞⎠ + nr|μ12|r = −12.04 < 0,

−rnar−12 λ1 − rnar2 − rnμ22ar−12 + 2 (r − 1)2∑j=1

ar2 + (r − 1)2∑j=1

ar2β−

rr − 1

j2 + (r − 1) ar2

+2∑j=1

mr(∣∣d2j∣∣r(Lg2)r +∣∣∣d̄2j∣∣∣rγ r2j(Lḡ2)r) + nr|μ21|r = −22.12 < 0,

−rmcr−11 λ1 − rmcr1 − rmρ11cr−11 + 2 (r − 1)2∑i=1

cr1 + (r − 1) cr1 + (r − 1)2∑i=1

cr1γ−

r

r − 1i1

+2∑i=1

nr(|b1i|r

(Lf1

)r+∣∣∣b̄1i∣∣∣rβ r1i(Lf̄1)r

)+mr|ρ12|r = −9.8 < 0

and

−rmcr−12 λ1 − rmcr2 − rmρ22cr−12 + 2 (r − 1)2∑i=1

cr2 + (r − 1) cr2 + (r − 1)2∑i=1

cr2γ−

rr − 1

i2

+2∑i=1

nr(|b2i|r

(Lf2

)r+∣∣∣b̄2i∣∣∣rβ r2i(Lf̄2)r

)+mr|ρ21|r = −9.4 < 0,

Hence, it follows from Theorem 1 that (32) and (33) are globally exponentially

synchronized.

5. ConclusionsIn 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 diffu-

sion 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 effectiveness of the obtained results.

AcknowledgementsThis study was partially supported by the National Natural Science Foundation of China under Grant No. 60974139and 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 researchprojects in Shannxi Province Department of Education under Grant No. 2010JK896.

Author details1School 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


Page 13 of 14

Authors’ contributionsWZ designed and performed all the steps of proof in this research and also wrote the paper. JL participated in thedesign of the study and suggest many good ideas that made this paper possible and helped to draft the firstmanuscript. All authors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 25 October 2011 Accepted: 13 January 2012 Published: 13 January 2012

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http://www.ncbi.nlm.nih.gov/pubmed/11152205?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11152205?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/18244446?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/18334360?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/18334360?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/19932996?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/19932996?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/19923046?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/19923046?dopt=Abstract

Abstract1. Introduction2. Model description and preliminaries3. Main results4. Illustration example5. ConclusionsAcknowledgementsAuthor detailsAuthors' contributionsCompeting interestsReferences

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