Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 405939, 10 pages
doi:10.1155/2012/405939
Research Article
Stability of the Stochastic
Reaction-Diffusion Neural Network with
Time-Varying Delays and p-Laplacian
Pan Qingfei,1 Zhang Zifang,2 and Huang Jingchang3
1
College of Civil Engineering and Architecture, Sanming University, Sanming 365004, China
Department of Mathematics and Physics, Huaihai Institute of Technology, Lianyungang 222005, China
3
Science and Technology on Microsystems Laboratory, Shanghai Institute of MicroSystems and Information
Technology, CAS, Shanghai 200050, China
2
Correspondence should be addressed to Pan Qingfei, pqf101@yahoo.com.cn
Received 22 October 2012; Accepted 9 November 2012
Academic Editor: Maoan Han
Copyright q 2012 Pan Qingfei et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The main aim of this paper is to discuss moment exponential stability for a stochastic reactiondiffusion neural network with time-varying delays and p-Laplacian. Using the Itô formula, a delay
differential inequality and the characteristics of the neural network, the algebraic conditions for the
moment exponential stability of the nonconstant equilibrium solution are derived. An example is
also given for illustration.
1. Introduction
In many neural networks, time delays cannot be avoided. For example, in electronic neural
networks, time delays will be present due to the finite switching speed of amplifies. In
fact, time delays are often encountered in various engineering, biological, and economical
systems. On the other hand, when designing a neural network to solve a problem such as
optimization or pattern recognition, we need foremost to guarantee that the neural networks
model is globally asymptotically stable. However, the existence of time delay frequently
causes oscillation, divergence, or instability in neural networks. In recent years, the stability
of neural networks with delays or without delays has become a topic of great theoretical and
practical importance see 1–16.
The stability of neural networks which depicted by partial differential equations
was studied in 6, 7. Stochastic differential equations were employed to research the
stability of neural networks in 8–11, while 12, 13 used stochastic partial differential
2
Journal of Applied Mathematics
equations to analysis this question. In 15, the authors studied almost exponential stability
for a stochastic recurrent neural network with time-varying delays. In addition, moment
exponential stability for a stochastic reaction-diffusion neural network with time-varying
delays is discussed in 16.
In this paper, we consider the stochastic reaction-diffusion neural network with timevarying delays and p-Laplacian as follows:
⎤
⎡
m
n
∂
∂u
i
− ai uui Ii Tij gj uj t − τj t, x ⎦dt
dui t, x ⎣
|∇ui |p−2
∂xk
∂xk
j1
k1
m
σil ui t, xdWl t,
1.1
i 1, 2, . . . , n, t > t0 , x ∈ Ω,
l1
∂ui
:
∂n
∂ui
∂ui
,...,
∂x1
∂xm
ui t0 s, x φi s, x,
T
0,
1.2
i 1, 2, . . . , n, t ≥ t0 , x ∈ ∂Ω,
−τi t0 ≤ s ≤ 0, 0 ≤ τi t ≤ τi , i 1, 2, . . . , n, x ∈ Ω.
1.3
In 1.1, ∇ui ∂ui /∂x1 , . . . , ∂ui /∂xm T , p ≥ 2 is a common number. Ω ⊆ Rm is a bounded
convex domain with smooth boundary ∂Ω and measure mes Ω > 0. n denotes the numbers
of neurons in the neural network, ui t, x corresponds to the state of the ith neurons at time
t and in space x, the ai u is an amplification function. Ii is output. gj uj t − τj , x denotes
the output of the jth neuron at time t − τj and in space x, namely, activation function which
shows how neurons respond to each other. Wt W1 t, . . . , Wm tT is an m-dimensional
Brownian motion which is defined on a complete probability space S, F, P with a natural
filtration {Ft }t≥t0 i.e., Ft σ{Ws : t0 ≤ s ≤ t}. σu σ1 u1 , . . . , σn un T , σi ui σi1 ui , . . . , σim ui . σij ui denotes the intensity of the stochastic perturbation. Functions
σij ui and gi are subject to certain conditions to be specified later. T : Tij n×n is a real
constant matrix and represents weight of the neuron interconnections, namely, Tij denotes
the strength of jth neuron on the ith neuron at time t − τj and in space x, and τj ∈ 0, τ
corresponds to axonal signal transmission delay.
2. Definitions and Lemmas
Throughout this paper, unless otherwise specified, let | · | denote Euclidean norm. Define that
|x|p ni1 |xi |2 p/2 and |x|p ni1 |xi |p where x x1 , . . . , xn T ∈ Rn . Denote by C−τ, 0 ×
Ω; Rn the family of continuous functions ϕ from −τ, 0 × Ω to Rn . For every t ≥ t0 and p ≥ 2,
p
denote by LFt −τ, 0×Ω; Rn the family of all Ft -measurable C−τ, 0×Ω; Rn valued random
p
p
variables such that φLp sup−τ≤θ≤0 Eφθp < ∞, where φθp Ω |φθ, x|p dx1/p ,
Eφ denotes the expectation of random variable φ.
Definition 2.1. The ut, x u1 t, x, . . . , un t, xT is called a solution of problem 1.1–1.3
if it satisfies following conditions 1, 2, and 3:
1 ut, x adapts {Ft }t≥t0 ;
2 for T ∈ Rt0 : {t ∈ R : t ≥ t0 }, ut, x ∈ Ct0 , T ×Ω, Rn , and Emaxx∈Ω
|∇ut, x|p dt < ∞, where ∇ux, t ∂u/∂x1 , . . . , ∂u/∂xn ;
T
t0
|ut, x|p Journal of Applied Mathematics
3
3 for T ∈ Rt0 , t ∈ t0 , T , it holds that
⎧
t ⎨
m
∂
p−2 ∂ui
ui t, xdx φi t0 , xdx − ai uui Ii
|∇ui |
∂xk
Ω
Ω
Ω t0 ⎩ k1 ∂xk
⎫
n
⎬
Tij gj uj ξ − τj ξ, x
dξ dx
⎭
j1
m t
l1
Ω
2.1
σil ui ξ, xdWl ξdx,
t0
i 1, . . . , n,
ui t0 s, x φi s, x,
t, x ∈ t0 , T × Ω,
P a.s.,
−τi ≤ s ≤ 0, i 1, . . . , n, P a.s.
Definition 2.2. The u u∗ x is called a nonconstant equilibrium solution of problem 1.1–
1.3 if and only if u u∗ x satisfies 1.1 and 1.2.
Definition 2.3. The nonconstant equilibrium solution u∗ x of 1.1 about the given norm · Ω
is called exponential stability in pth moment, if there are constants M > 0, δ > 0 for every
stochastic field solution ut, x of problem 1.1–1.3 such that
Eut, x − u∗ Ω ≤ Me−δt−t0 ,
2.2
1
lim sup log Eut, x − u0 Ω ≤ −δ.
t→∞ t
2.3
namely,
The constant −δ on the right hand side in 2.3 is called Lyapunov exponent of every solution
of problem 1.1–1.3 converging on equilibrium about norm · Ω .
In order to obtain pth moment exponential stability for a nonconstant equilibrium
solution of problem 1.1–1.3, we need the following lemmas.
Lemma 2.4 see 17. Let P pij n×n and Q qij n×n be two real matrices. The continuous
function ui t ≥ 0 satisfies the delay differential inequalities
D ui t ≤
n
pij uj t qij uj t − τj t ,
0 ≤ τi t ≤ τ, i 1, . . . , n.
2.4
j1
j and qij ≥ 0 i, j 1, 2, . . . , n and −P Q is an M-matrix, then there are constants
If pij > 0 for i /
ki > 0, α > 0 such that
⎞
n ui t ≤ ki ⎝ φj ⎠ exp−αt − t0 ,
⎛
i 1, . . . , n, t ≥ t0 ,
2.5
j1
where φ φ1 , . . . , φ1 are initial functions. D is right-hand upper derivate. · represents a norm.
4
Journal of Applied Mathematics
Lemma 2.5 see 10. Let p > 2, then there are positive constants ep n and dp n for any x x1 , . . . , xn T ∈ Rn such that
ep n
n
i1
p/2
|xi |
2
p/2
n
n
p
2
≤
.
|xi | ≤ dp n
|xi |
i1
2.6
i1
Remark 2.6. If p 2, Lemma 2.5 also holds with ep n dp n 1.
Suppose that σij ui , ai u, and gi are Lipschitz continuous such that the following
conditions hold:
H1 |σi v1 − σi v2 σi v1 − σi v2 T | ≤ 2λi |v1 − v2 |2 , i 1, 2, . . . , n,
H2 |gi v1 − gi v2 | ≤ ci |v1 − v2 |, i 1, 2, . . . , n,
H3 u − vai uu − ai vv ≥ ai |u − v|2 , for all u, v ∈ R, i 1, 2, . . . , n,
where ci , ai , and λi 1 ≤ i ≤ n are positive constants.
3. Main Result
Set ut, x u1 t, x, . . . , un t, xT as a solution of the problem 1.1–1.3 and u∗ x u∗1 x, . . . , u∗n xT as a nonconstant equilibrium solution of the problem 1.1–1.3.
Theorem 3.1. Let p ≥ 2 and H1–H3 hold. Assume that there are positive constants ε1 , . . . , εn
such that the matrix Mp −pji qij n×n : P T Q is an M-matrix, where
pij −pδij ai p p − 1 δij λi p − 1 Tij cj εi ,
1−p
qij Tij cj εi ,
δii 1, δij 0 i /
j , i, j 1, . . . , n,
i, j 1, . . . , n,
3.1
then the nonconstant equilibrium solution of problem 1.1–1.3 about Lp norm is exponential
stability in pth moment, that is, there are constants C1 > 0 and α > 0, for any t0 ∈ R and any
p
φ ∈ LFt such that
0
p
E ut, x − u∗ p ≤ C1 e−αt−t0 ,
t ≥ t0 .
3.2
Proof. Set Vi ut, x |ui t, x − u∗i x|p . For every t ≥ t0 and dt > 0, by means of Itô formula
and H3, one has that
⎧
!
m
⎨
∂ui
∂
∂ ∗ p−2 ∂u∗i
p−2
∇ui
dVi u pui − u∗i −
ui − u∗i
|∇ui |p−2
⎩ k1 ∂xk
∂xk
∂xk
∂xk
− ai uui − ai u∗ u∗i
⎫
n
⎬
Tij gj uj t − τj , x − gj u∗j
dt
⎭
j1
Journal of Applied Mathematics
5
p−2
T
1 dt
p p − 1 ui − u∗i sgn ui − u∗i σi ui − σi u∗i
σi ui − σi u∗i
2
p−2 pui − u∗i ui − u∗i σi ui − σi u∗i dWt
!
m p−2 ∂
∂ui
∂ ∗ p−2 ∂u∗i
∇ui
≤ p ui − u∗i −
dt
ui − u∗i
|∇ui |p−2
∂xk
∂xk
∂xk
∂xk
k1
p
−pai p p − 1 λi ui − u∗i dt
p
n Tij cj ui − u∗ p−2 ui − u∗ uj t − τj , x − u∗ dt
i
i
j
j1
p−2 pui − u∗i ui − u∗i σi ui − σi u∗i dWt.
3.3
Both sides of Inequality 3.3 are integrated about x over Ω. Set Vi u One has that
m ui − u∗ p−2 ui − u∗
dVi u ≤ p
k1
i
Ω
Ω
Ω
p
Vi udx ui −u∗i p .
!
∂ ∗ p−2 ∂u∗i
∂
p−2 ∂ui
∇ui
−
dx dt
|∇ui |
∂xk
∂xk
∂xk
∂xk
i
−pai p p − 1 λi
ui − u∗ p dx dt
i
n ui − u∗ p−2 ui − u∗ Tij cj uj t − τj , x − u∗ dx dt
p
p
i
Ω
j1
i
j
m ui − u∗ p−2 ui − u∗ σil ui − σil u∗ dx dWl t.
i
Ω
l1
i
i
3.4
p−2
Set |∇ui |p−2 ∂ui /∂xk − |∇u∗i |p−2 ∂u∗i /∂xk m
∂ui /∂x1 − |∇u∗i |p−2 ∂u∗i /∂x1 , . . . ,
k1 : |∇ui |
∗ p−2
∗
p−2
|∇ui | ∂ui /∂xm − |∇ui | ∂ui /∂xm . By 1.2, one has that
m ui − u∗ p−2 ui − u∗
k1
i
Ω
i
!
∂ ∗ p−2 ∂u∗i
∂
p−2 ∂ui
∇ui
−
dx
|∇ui |
∂xk
∂xk
∂xk
∂xk
∗ p−2 ∂u∗i m
p−2 ∂ui
∗ p−2
∗
u i − ui
− ∇ui
dx
ui − ui ∇ · |∇ui |
∂xk
∂xk k1
Ω
∗ p−2 ∂u∗i m
p−2 ∂u
i
p−2
∗
∗
dx
∇ · ui − ui − ∇ui ui − ui |∇ui |
∂xk
∂xk k1
Ω
−
Ω
|∇ui |
p−2
∂ui ∗ p−2 ∂u∗i
− ∇ui
∂xk
∂xk
m
k1
p−2 ∇ ui − u∗i ui − u∗i dx
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Journal of Applied Mathematics
∗ p−2 ∂u∗i m
p−2 ∂ui
∗ p−2
∗
ui − ui
·n
ds
− ∇ui
ui − ui |∇ui |
∂xk
∂xk k1
∂Ω
∗ p−2 ∂u∗i m
p−2 ∂ui
∗ p−2
∗
−
∇ ui − ui · |∇ui |
− ∇ui
dx
p − 1 u i − ui
∂xk
∂xk k1
Ω
m p−2
∂ui ∗ p−2 ∂u∗i
∂ui ∂u∗i
−
− ∇ui
−
·
dx
p − 1 ui − u∗i |∇ui |p−2
∂xk
∂xk
∂xk ∂xk
k1 Ω
≤ 0,
3.5
where n
is unit outer cotangent vector on ∂Ω. By 3.4, 3.5, H1, and Young’s inequality,
one has that
m p−2
∂ui ∗ p−2 ∂u∗i
∂ui ∂u∗i
− ∇ui
−
·
dx
p − 1 ui − u∗i |∇ui |p−2
∂xk
∂xk
∂xk ∂xk
k1 Ω
p
−pai p p − 1 λi ui t − u∗i p dt
dVi u ≤ −
p
n ui − u∗ p−2 ui − u∗ Tij cj uj t − τj , x − u∗ dx dt
j1
i
G
i
j
m ui − u∗ p−2 ui − u∗ σil ui − σil u∗ dx dWl t
p
l1
≤
i
G
i
3.6
i
n
n
p
p
pij ui t − u∗i p dt qij uj t − τj , x − u∗j dt
j1
p
p
j1
m ui − u∗ p−2 ui − u∗ σil ui − σil u∗ dx dWl t,
l1
i
G
i
i
where pij and qij are defined by 3.1.
For Δt > 0, both sides of 3.6 are integrated about t from t to t Δt, then both sides of
3.6 are calculated expectation. By the properties of Brownian motion, one has that
"
"
p #
p #
E ui t Δt − u∗i p − E ui t − u∗i p
≤
n
j1
$
% tΔt
pij E
t
&
% tΔt
&'
p
p
∗
∗
ui s − u ds qij E
uj s − τj s − uj ds .
i p
t
p
3.7
Journal of Applied Mathematics
7
tΔt
tΔt
p
p
Since the integrals t
Eui s − u∗i p ds and t
Euj s − τj − u∗i p ds are finite, by Fubini
theorem 18 and 3.7, one obtain that
tΔt "
n
"
"
#
# p #
∗ p
∗ p
pij
E ui s − u∗i p ds
E ui t Δt − ui p − E ui t − ui p ≤
j1
n
t
tΔt
qij
t
j1
p !
E uj s − τj s − u∗j ds.
3.8
p
p
Set vi t Eui t − u∗i p . Both sides of Inequality 3.8 are divided by Δt, let Δt → 0,
one has the following inequality:
D vi t ≤
n
pij vi t qij vj t − τj t .
3.9
j1
By Lemma 2.4, there are positive constants Ki , α such that
n p !
"
p #
∗ E
−
u
E ui t − u∗i p ≤ Ki j exp−αt − t0 ,
φj
p
j1
i 1, . . . , n, t ≥ t0 ,
3.10
where φj is initial value. Set K max{Ki : 1 ≤ i ≤ n}, then
⎤
⎡
n n p
p !
∗ E
E⎣ uj − u∗j ⎦ ≤ nK −
u
φ
j
j exp−αt − t0 ,
p
j1
p
j1
t ≥ t0 .
3.11
t ≥ t0 .
3.12
By 3.11 and Lemma 2.5, one obtains that
"
E u −
p
u∗ p
#
≤
ep−1 ndp nnK
n p !
∗ E φj − uj exp−αt − t0 ,
p
j1
In order to prove Theorem 3.1, we need the following lemma.
Lemma 3.2. The nonconstant equilibrium solution of the problem 1.1–1.3, u∗ x satisfies
p
Eu∗ p 0.
Proof. Set Fi u∗ x |u∗i x|p . Similar to 3.8 in proof of Theorem 3.1, one has that
n
j1
tΔt "
tΔt !
n
∗ p #
p
pij
E ui p ds qij
E u∗j ds ≥ 0.
t
j1
t
p
3.13
8
Journal of Applied Mathematics
By 3.13 and the assumption that −pji qij n×n : P T Q is an M-matrix, one obtains that
−
n
pji qij
p !
E u∗j Δt ≤ 0,
p
j1
p
i 1, 2, . . . , n.
3.14
p
Because of Eu∗j p ≥ 0 and Δt > 0, one has that Eu∗ p 0.
We continue the proof of Theorem 3.1 as the following.
By Lemma 3.2, one has that
p !
E φj − u∗ j ≤ M,
p
3.15
where M > 0 is a common number. We derive every solution of problem 1.1–1.3 such that
"
#
1
p
lim sup log E u − u∗ p ≤ −α,
t→∞ t
3.16
then a nonconstant equilibrium solution of problem 1.1–1.3 about Lp norm is exponential
stability in pth moment. The proof of Theorem 3.1 is complete.
In order to illustrate the application of the theorem, we give an example.
Example 3.3. Discuss the stochastic reaction-diffusion neural network with time-varying
delays and p-Laplacian as the following:
⎤
⎡
2
2
∂
∂u
1
− a1 u1 I1 T1j gj uj t − τj t, x ⎦dt
du1 t, x ⎣
|∇u1 |p−2
∂x
∂x
k
k
j1
k1
3.17
σ1 u1 t, xdWt,
⎤
⎡
2
2
∂
∂u
2
− a2 u2 I2 T2j gj uj t − τj t, x ⎦dt
du2 t, x ⎣
|∇u2 |p−2
∂x
∂x
k
k
j1
k1
3.18
σ2 u2 t, xdWt,
∂ui
:
∂n
∂ui ∂ui
,
∂x1 ∂x2
ui t0 s, x φi s, x,
T
0,
i 1, 2, t ≥ t0 ≥ 0, x ∈ ∂Ω,
−τi t0 ≤ s ≤ 0, 0 ≤ τi t ≤ τi , i 1, 2, x ∈ Ω,
3.19
3.20
Journal of Applied Mathematics
9
where
a1 1.2,
a2 1.8,
T11 1.5,
σ1 u1 0.2 u1 − u∗1 ,
T12 0.5,
T21 0.6,
σ2 u2 0.3 u2 − u∗2 ,
T22 1,
g1 u1 t − τ1 t, x 0.3u1 t − τ1 t, x,
3.21
g2 u2 t − τ2 t, x 0.2u2 t − τ2 t, x cosu2 t − τ2 t, x.
Set u∗ x u∗1 x, u∗1 xT as a nonconstant equilibrium solution of 3.17 and 3.18. One
can derive that
g1 v1 − g1 v2 ≤ 0.3|v1 − v2 |, c1 0.3;
g2 v1 − g2 v2 ≤ 0.4|v1 − v2 |, c2 0.4;
σ1 v1 − σ1 v2 σ1 v1 − σ1 v2 T ≤ 0.04|v1 − v2 |, λ1 0.02;
σ2 v1 − σ2 v2 σ2 v1 − σ2 v2 T ≤ 0.09|v1 − v2 |, λ2 0.045.
3.22
Taking εi 1 i 1, 2 and p 3, one has that
M3 2.58 −0.56
,
−0.58 1.53
3.23
and M3 is an M-matrix. The nonconstant equilibrium solution of 3.17 and 3.18 about L3
norm is exponential stability in the 3rd moment.
Remark 3.4. The Theorem 3.1 extends the correlative results in 12, 13, 16 to the situation
related to the p-Laplacian.
Acknowledgments
The authors thank the reviewers for their constructive comments.This work is supported by
the National Science Foundation of China no. 10971240.
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The Scientific
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International Journal of
Differential Equations
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International Journal of
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Mathematical Physics
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Journal of
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International
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Optimization
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