Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 175934, 20 pages
doi:10.1155/2012/175934
Research Article
Pth Moment Exponential Stability of Impulsive
Stochastic Neural Networks with Mixed Delays
Xiaoai Li,1 Jiezhong Zou,1 and Enwen Zhu2
1
2
School of Mathematics and Statistics, Central South University, Hunan, Changsha 410083, China
School of Mathematics and Computational Science, Changsha University of Science and Technology,
Hunan, Changsha 410004, China
Correspondence should be addressed to Enwen Zhu, engwenzhu@126.com
Received 29 June 2012; Revised 15 October 2012; Accepted 2 November 2012
Academic Editor: Dane Quinn
Copyright q 2012 Xiaoai Li 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.
This paper investigates the problem of pth moment exponential stability for a class of stochastic
neural networks with time-varying delays and distributed delays under nonlinear impulsive
perturbations. By means of Lyapunov functionals, stochastic analysis and differential inequality
technique, criteria on pth moment exponential stability of this model are derived. The results of
this paper are completely new and complement and improve some of the previously known results
Stamova and Ilarionov 2010, Zhang et al. 2005, Li 2010, Ahmed and Stamova 2008, Huang
et al. 2008, Huang et al. 2008, and Stamova 2009. An example is employed to illustrate our
feasible results.
1. Introduction
The dynamics of neural networks have drawn considerable attention in recent years due to
their extensive applications in many fields such as image processing, associative memories,
classification of patters, and optimization. Since the integration and communication delays
are unavoidably encountered in biological and artificial neural systems, it may result
in oscillation and instability. The stability analysis of delayed neural networks has been
extensively investigated by many researchers, for instance, see 1–30.
In real nervous systems, there are many stochastic perturbations that affect the stability
of neural networks. The result in Mao 24 suggested that one neural network could be
stabilized or destabilized by certain stochastic inputs. It implies that the stability analysis of
stochastic neural networks has primary significance in the design and applications of neural
networks, such as 7, 12–16, 18, 20, 22–24, 26, 27, 30.
2
Mathematical Problems in Engineering
On the other hand, it is noteworthy that the state of electronic networks is often
subjected to some phenomenon or other sudden noises. On that account, the electronic
networks will experience some abrupt changes at certain instants that in turn affect
dynamical behaviors of the systems 5, 6, 17–23, 28, 29. Therefore, it is necessary to take
both stochastic effects and impulsive perturbations into account on dynamical behaviors of
delayed neural networks 18, 20, 22, 23.
Very recently, Li et al. 22 have employed the properties of M-cone and inequality
technique to investigate the mean square exponential stability of impulsive stochastic neural
networks with bounded delays. Wu et al. 23 studied the exponential stability of the
equilibrium point of bounded discrete-time delayed dynamic systems with linear impulsive
effects by using Razumikhin theorems. To the best of authors’ knowledge, however, few
authors have considered the pth moment exponential stability of impulsive stochastic neural
networks with mixed delays.
Motivated by the discussions above, our object in this paper is to present the sufficient
conditions ensuring pth moment exponential stability for a class of stochastic neural networks
with time-varying delays and distributed delays under nonlinear impulsive perturbations
by virtue of Lyapunov method, inequality technique and Itô formula. The results obtained
in this paper generalize and improve some of the existing results 5, 8, 18, 19, 26–28.
The effectiveness and feasibility of the developed results have been shown by a numerical
example.
2. Model Description and Preliminaries
Let R denote the set of real numbers, Rn the n-dimensional real space equipped with
the Euclidean norm | · |, Z the set of nonnegative integral numbers. E· stands for the
mathematical expectation operator. L denotes the well-known L-operator given by the Itô
formula. wt w1 t, . . . , wm t is m-dimensional Brownian motion defined on a complete
probability space Ω, F, P with a natural filtration {Ft }t≥0 generated by {ws : 0 ≤ s ≤ t},
where we associate Ω with the canonical space generated by wt and denote by F the
associated σ-algebra generated by wt with the probability measure P . Let σt, x, y σil t, xi , yi n×m ∈ Rn×m , and σi t, xi , yi be ith row vector of σt, x, y.
In 5, 6, the researchers investigated the following impulsive neural networks with
time-varying delays:
ẋi t −ai xi t n
n
bij fj xj t cij gj xj t − τj t Ii ,
j1
xi tk pik x t−k ,
t/
tk ,
j1
2.1
k ∈ Z , i ∈ Λ.
The authors in 7, 26, 27 studied the stochastic recurrent neural networks with time-varying
delays:
⎛
dxi t ⎝−ai xi t n
j1
m
l1
⎞
n
bij fj xj t cij gj xj t − τj t Ii ⎠dt
j1
σil t, xi t, xi t − τi tdwl t.
2.2
Mathematical Problems in Engineering
3
In this paper, we will study the generalized stochastically perturbed neural network
model with time-varying delays and distributed delays under nonlinear impulses defined by
the state equations:
⎡
dxi t ⎣−ai xi t n
n
bij fj xj t cij gj xj t − τj t
j1
j1
⎤
t
n
dij
Kij t − s · hj xj s ds Ii ⎦dt
m
2.3
−∞
j1
σil t, xi t, xi t − τi tdwl t,
t
/ tk ,
l1
xi tk pik x t−k ,
k ∈ Z , i ∈ Λ,
where Λ {1, 2, . . . , n}, the time sequence {tk } satisfies 0 t0 < t1 < t2 < · · · < tk < tk1 · · · ,
limk → ∞ tk ∞; xt x1 t, x2 t, . . . , xn tT and xi t corresponds to the state of the
ith unit at time t; bij , cij , and dij denote the constant connection weight; τj t is the timevarying transmission delay and satisfies 0 ≤ τj t ≤ τj , 0 < ηj inft∈R {1 − τ̇j t}, for
j ∈ Λ. fj ·, gj ·, hj · denote the activation functions of the jth neuron; the delay kernel
Kij · is the real-valued nonnegative piecewise continuous functions defined on 0, ∞; n
corresponds to the numbers of units in a neural network; Ii denotes the external bias on the
ith unit; pik xtk represents the abrupt change of the state xi t at the impulsive moment tk .
System 2.3 is supplemented with initial condition given by
xi s ϕi s,
s ∈ −∞, 0, i ∈ Λ,
2.4
where ϕs ϕ1 s, ϕ2 s, . . . , ϕn sT ∈ PCBbF0 −∞, 0, Rn ˙ BPCBbF0 . Denote by PCBbF0 the
family of all bounded F0 -measurable, PC−∞, 0, Rn -value random variables ϕ, satisfying
supθ∈−∞,0 E|ϕθ|p < ∞, where PC−∞, 0, Rn {ϕ : −∞, 0 → Rn } is continuous
everywhere except at finite number of points tk , at which ϕtk and ϕt−k exist and ϕtk ϕtk .
The norms are defined by the following norms, respectively:
ϕ sup
p
1/p
n p
ϕi s
,
s∈−∞,0
xp i1
n
1/p
|xi |
p
.
2.5
i1
Throughout this paper, the following standard hypothesis are needed.
H1 Functions ai · : R → R are continuous and monotone increasing, that is, there
exist real numbers ai > 0 such that
ai u − ai v
≥ ai ,
u−v
for all u, v ∈ R, u /
v, i ∈ Λ.
2.6
4
Mathematical Problems in Engineering
H2 Functions fj , gj , and hj are Lipschitz-continuous on R with Lipschitz constants
f
Lj
g
> 0, Lj > 0, and Lhj > 0, respectively. That is,
fj u − fj v ≤ Lf |u − v|,
j
gj u − gj v ≤ Lg |u − v|,
hj u − hj v ≤ Lh |u − v|,
j
j
2.7
for all u, v ∈ R, i ∈ Λ.
H3 The delay kernels Kij : 0, ∞ → R satisfy
Kij s ≤ Ks
∞
∀i, j ∈ Λ, s ∈ 0, ∞,
Kseμ0 s ds < ∞,
2.8
0
where Ks : 0, ∞ → R is continuous and integrable, and the constant μ0 denotes some
positive number.
H4 There exist nonnegative constants ei , li such that
2
2
T
σi t, u , v − σi t, u, v σi t, u , v − σi t, u, v ≤ ei u − u li v − v ,
2.9
for all u, v, u , v ∈ R, i ∈ Λ.
H5 There exist nonnegative matrixes Pk pijk n×n such that
n
p
pik u1 , u2 , . . . , un − pik v1 , v2 , . . . , vn p ≤
pijk uj − vj ,
2.10
j1
for any u1 , u2 , . . . , un T , v1 , v2 , . . . , vn T ∈ Rn , where p ≥ 1 is an integer.
We end this section by introducing three definitions.
Definition 2.1. A constant vector x∗ x1∗ , x2∗ , . . . , xn∗ T ∈ Rn is said to be an equilibrium point
of system 2.3 if x∗ is governed by the algebraic system
n
n
n
ai xi∗ bij fj xj∗ cij gj xj∗ dij
j1
j1
j1
t
−∞
Kij t − shj xj∗ ds Ii ,
2.11
where it is assumed that impulse functions pik · satisfy pik x1∗ , x2∗ , . . . , xn∗ xi∗ for all i ∈ Λ
and σt, xi∗ , xi∗ 0.
Definition 2.2. The equilibrium x∗ x1∗ , x2∗ , . . . , xn∗ T ∈ Rn of system 2.3 is said to be pth
moment exponentially stable if there exist λ > 0 and M ≥ 1 such that
p
Ext − x∗ p ≤ Eϕ − x∗ Me−λt
for t ≥ 0,
where xt is an any solution of system 2.3 with initial value ϕ ∈ PCBbF0 −∞, 0, Rn .
2.12
Mathematical Problems in Engineering
5
Definition 2.3 Forti and Tesi, 1995 25. A map H : Rn → Rn is a homeomorphism of Rn
onto itself if H is continuous and one-to-one, and its inverse map H −1 is also continuous.
3. Main Result
For convenience, we denote that
φi pai −
p−1 n pαl,ij f pβl,ij pγl,ij g pδl,ij pξl,ij h pηl,ij ∞
bij Lj
cij Lj
dij Lj
Ksds
0
j1 l1
−
n μ pβ
j
bji pαp,ji Lf p,ji ,
i
μ
j1 i
ψi n μ j
cji pγp,ji Lg pδp,ji ,
i
μ
j1 i
p p−1
p−1 p−2
Φi φi −
ei −
li ,
2
2
μi pξp,ij h pηp,ij
dij
ζij Lj
,
μ max μi ,
i
μj
3.1
Ψi ψi p − 1 li ,
μ min μi ,
i
where μi are positive constant, αl,ij , βl,ij , γl,ij , δl,ij , ξl,ij , and ηl,ij are real numbers and satisfy
p
αl,ij 1,
p
l1
l1
βl,ij 1,
p
p
γl,ij 1,
l1
δl,ij 1,
p
ξl,ij 1,
p
ηl,ij 1.
l1
l1
l1
3.2
Lemma 3.1. If ai i 1, 2, . . . , p denote p nonnegative real numbers, then
p
a1 a2 · · · ap ≤
p
p
a1 a2 · · · ap
p
,
3.3
where p ≥ 1 denotes an integer.
A particular form of 3.3, namely,
p−1
a1 a2
p
p
p − 1 a1 a2
,
≤
p
p
for p 1, 2, 3, . . . .
Lemma 3.2. If H : Rn → Rn is a continuous function and satisfies the following conditions.
Hy for all x /
y.
1 Hx is injective on Rn , that is, Hx /
2 Hx → ∞ as x → ∞.
Then, Hx is homeomorphism of Rn .
3.4
6
Mathematical Problems in Engineering
Theorem 3.3. System 2.3 exists a unique equilibrium x∗ under the assumptions (H1)–(H3) if the
following condition is also satisfied:
∞
(H6) φi > ψi nj1 ζji 0 Ksds.
Proof. Defining a map Hx h1 x, h2 x, . . . , hn xT ∈ C0 Rn , Rn , where
t
n
n
n
hi x −ai xi bij fj xj cij gj xj dij
Kij t − shj xj ds Ii ,
j1
j1
−∞
j1
3.5
the map H is a homeomorphism on Rn if it is injective on Rn and satisfies Hx → ∞ as
x → ∞.
In the following, we will prove that Hx is a homeomorphism.
Firstly, we claim that Hx is an injective map on Rn . Otherwise, there exist xT , yT ∈
n
yT such that Hx Hy, then
R , and xT /
n
n
ai xi − ai yi bij fj xj − fj yj cij gj xj − gj yj
j1
j1
n
t
dij
j1
−∞
Kij t − s hj xj − hj yj ds.
3.6
It follows from H1–H3 that
n n bij Lf xj − yj cij Lg xj − yj ai xi − yi ≤
j
j
j1
j1
n h
t
dij Lj xj − yj
Kt − sds.
−∞
j1
Therefore,
n
p
pμi ai xi − yi i1
≤
n
n
n n f
p−1 g
p−1 pμi bij Lj xi − yi xj − yj pμi cij Lj xi − yi xj − yj i1 j1
n
n p−1 pμi dij Lhj xi − yi xj − yj i1 j1
i1 j1
t
−∞
Kt − sds
3.7
Mathematical Problems in Engineering
7
p−1
n n
pαl,ij f pβl,ij p pαp,ij f pβp,ij p
bij
xi − yi bij
xj − yj
≤
μi
Lj
Lj
i1 j1
l1
p−1
n
n pγl,ij g pδl,ij p pγp,ij g pδp,ij p
cij L
xi − yi cij xj − yj μi
Lj
j
i1 j1
l1
t
p−1
n
n pξ
pηl,ij xi − yi p
μi
Kt − sds dij l,ij Lhj
−∞
i1 j1
n
n μi
t
i1 j1
n
−∞
l1
pξ
pηp,ij xj − yj p
Kt − sdsdij p,ij Lhj
⎞
∞
n
p
μi ⎝pai − φi ψi ζji
Ksds⎠xi − yi .
⎛
i1
0
j1
3.8
From H6, it leads to a contradiction with our assumption. Therefore, Hx is an injective
map on Rn .
To demonstrate the property Hx → ∞ as x → ∞, we have
n
sgnxi pμi hi x − hi 0|xi |p−1
i1
n
⎛
sgnxi pμi |xi |p−1 ⎝−ai xi i1
n
sgnxi pμi |xi |
p−1
i1
≤−
pai μi |xi |p n
i1
j1
j1
−∞
⎛
μi ⎝pai − φi ψi i1
n
∞
ζji
j1
⎞
3.9
Ksds⎠|xi |p
0
⎞
⎛
−
⎞
n
bij fj xj cij gj xj ⎠
t
n
dij
Kij t − shj xj ds
j1
n
n
∞
n
n
μi ⎝φi − ψi − ζji
Ksds⎠|xi |p
i0
0
j1
p
≤ −ωxp ,
where ω min1≤i≤n {μi φi − ψi −
∞
j1 ζji 0
n
p
ωxp ≤ pμ
Ksds}. Then, we have
n
|hi x − hi 0||xi |p−1 .
i1
3.10
8
Mathematical Problems in Engineering
Using the Hölder inequality, we obtain
p
xp
pμ
≤
ω
n
1−1/p 1/p
n
p
,
|xi |
|hi x − hi 0|
p
i1
3.11
i1
which leads to
xp ≤
pμ Hxp H0p .
ω
3.12
From 3.12, we see that Hx → ∞ as x → ∞. Thus, the map Hx is a
homeomorphism on Rn under the sufficient condition H6, and hence it has a unique
fixed point x∗ . This fixed point is the unique solution of the system 2.3. The proof is now
complete.
To establish some sufficient conditions ensuring the pth moment exponential stability
of the equilibrium point of x∗ of system 2.3, we transform x∗ to the origin by using the
transformation yi t xi t − x∗ for i ∈ Λ. Then system 2.3 can be rewritten as the following
form:
⎡
n
n
dyi t ⎣−
ai yi t bij fj yj t cij gj yj t − τj t
j1
j1
⎤
t
n
j yj s ds⎦dt
dij
Kij t − sh
j1
m
−∞
σil t, yi t, yi t − τi tdwl t,
3.13
t/
tk ,
l1
yi tk pik y t−k ,
k ∈ Z , i ∈ Λ,
where
ai yi t ai yi t xi∗ − ai xi∗ ,
fj yj t fj yj t xj∗ − fj xj∗ ,
gj yj t − τj t gj yj t − τj t xj∗ − gj xj∗ ,
j yj s hj yj s x∗ − hj x∗ ,
h
j
j
σij t, yj t, yj t − τj t σij t, yj t xj∗ , yj t − τj t xj∗ − σij t, xj∗ , xj∗ ,
pik ytk pik ytk x∗ − pik x∗ .
In order to obtain our results, the following assumptions are necessary.
3.14
Mathematical Problems in Engineering
9
∞
H7 When ηi ≥ 1, Φi > Ψi nj1 ζji 0 Ksds for any i ∈ Λ.
∞
When 0 < ηi ≤ 1, Φi > Ψi /ηi nj1 ζji 0 Ksds for any i ∈ Λ.
H8 There exist constant λ ∈ 0, μ0 and α ∈ 0, λ, ρk maxi∈Λ {1, nj1 μj /μi pjik } ≤
eαtk −tk−1 such that
λ < Φi −
∞
n
Ψi λτ e − ζji
ηi
j1
Kse sds ≤
λs
∞
∞
0
Kseλs ds
0
3.15
Kse
μ0 s
ds.
0
Theorem 3.4. Assume that (H1)–(H5) and (H7)-(H8) hold, then system 2.3 exists a unique
equilibrium x∗ , and the equilibrium point is pth moment exponentially stable.
Proof. From H7, if 0 < ηi ≤ 1, then
φi > Φi >
n
Ψi ζji
ηi j1
∞
Ksds >
0
n
ψi ζji
ηi j1
∞
Ksds > ψi 0
∞
n
ζji
Ksds,
j1
0
3.16
if ηi ≥ 1,
φi > Φi > Ψi n
j1
∞
ζji
Ksds > ψi 0
∞
n
ζji
Ksds,
j1
0
3.17
for any i ∈ Λ.
Then regardless of cases, H6 and
n
Ψi ζji
Φi >
ηi j1
∞
Ksds
0
are satisfied. Therefore, system 2.3 exists a unique equilibrium point x∗ .
3.18
10
Mathematical Problems in Engineering
Now, we define
n
ui t.
U t, yt p
ui t, yt eλt μi yi t ui t,
3.19
i1
For t / tk , k ∈ Z , we obtain
1
LU Ut trace σ T t, yi t, yi t − τi t Uyi yi t, y σ t, yi t, yi t − τi t
2
⎞
⎛
n
n
n
Uyi ⎝−
ai yi t bij fj yj t cij gj yj t − τj t ⎠
i1
j1
n
n t
Uyi dij
−∞
i1 j1
j1
j yj s ds
Kij t − sh
n
p p−1
λ − pai ei ui t
≤
2
i1
⎧
n
n ⎨
p−1 f
peλt
μi bij Lj yi t yj t
⎩ i1 j1
"
n
n p−1 g
μi cij Lj yi t yj t − τj t i1 j1
#
p−2 2
p−1 li yi t yi t − τi t
2
⎫
n
n h
p−1 ∞
⎬
μi dij Lj yi t
Ksyj t − sds
⎭
0
i1 j1
≤
n
i1
n
p−2 2
p p−1
p p − 1 λt λ − pai ei ui t e
μi li yi t yi t − τi t
2
2
i1
p−1
n
n pαl,ij f pβl,ij p pαp,ij f pβp,ij p
bij yi t bij yj t
e
μi
Lj
Lj
λt
i1 j1
l1
p−1
n
n pγl,ij g pδl,ij p pγp,ij g pδp,ij p
λt
cij L
yi t cij yj t
e
μi
L
j
i1 j1
j
l1
p−1
∞
n
n pξl,ij pηl,ij p pξl,ij h pηp,ij p
h
dij L
yi t dij L
yj t − s ds
e
μi
Ks
j
j
λt
i1 j1
0
l1
Mathematical Problems in Engineering
⎤
⎡
p−1 n
n pαl,ij f pβl,ij pγl,ij g pδl,ij pξl,ij h pηl,ij ∞
bij ⎣
≤
Lj
cij Lj
dij Lj
Ksds ⎦ui t
i1
11
0
j1 l1
⎛
⎞
n
n μ pβ
p p−1
p−1 p−2
p,ji
j
bji pαp,ji Lf
⎠ui t
ei li ⎝λ − pai j
2
2
μ
i
i1
j1
n n
μi pξp,ij h pηp,ij ∞
dij
Lj
Kseλs uj t − sds
μ
j
0
i1 j1
⎛
⎞
n
n μ j
pδp,ji
g
pγ
p,ji
λτi t ⎝
cji e
Li
p − 1 li ⎠ui t − τi t
μ
i1
j1 i
n ∞
n
n ζij
Kseλs uj t − sds.
λ − Φi ui t Ψi eλτi t ui t − τi t i1
i1 j1
0
3.20
while Lemma 3.1 is used in the second inequality. Let
n
V t, yt U t, yt Ψi eλτi
i1
t
t−τi t
ui s
ds
1 − τ̇i ψi−1 s
3.21
∞
t
n
n ζij
Kseλs
uj rdrds,
i1 j1
0
t−s
o formula to 3.21, we can get
for t ≥ 0, where ψi s s − τi s, applying It'
n
LV LU Ψi eλτi
i1
n
n i1 j1
∞
ζij
ui t
ui t − τi t
t − τi t
−
1 − τ̇i t
1 − τ̇i ψi−1 t
Kseλs uj t − uj t − s ds
0
⎧
n ⎨
n
Ψi λτ e − ζji
Φi − λ −
≤ −
⎩
ηi
i1
j1
∞
0
3.22
⎫
⎬
Kseλs ds ui t.
⎭
From H8, we have
LV < 0.
3.23
12
Mathematical Problems in Engineering
On the other hand, we have
n
n
n
p
p
EU tk , ytk Eeλtk μi yi tk ≤ Eeλtk μi pijk yj t−k ≤ ρk EU t−k , y t−k ,
i1
i1
tk
n
EV tk , ytk EU tk , ytk E
Ψi eλτi
i1
tk −τi tk ui s
ds
1 − τ̇i ψi−1 s
⎞
⎛
∞
tk
n
n ζij
Kseλs
uj rdrds⎠
E⎝
3.25
tk −s
0
i1 j1
3.24
j1
≤ ρk EV t−k , y t−k ,
for t ∈ tk−1 , tk , k ∈ Z , by 3.23, 3.25, and H8, we have
EU t, yt ≤ EV t, yt ≤ EV tk−1 , ytk−1 ≤ ρk−1 EV t−k−1 , y t−k−1
≤ ρ0 ρ1 · · · ρk−1 EV t0 , yt0 ≤ eαt1 · · · eαtk−1 −tk−2 EV t0 , yt0 ≤ eαt EV 0, y0 ,
3.26
where ρ0 1.
On the other hand, we observe that
0
n
n
p λτi
μi yi 0 Ψi e
V 0, y0 ≤
i1
i1
−τ
ui s
ds
1 − τ̇i ψi−1 s
∞
0
n n
ζij
Kseλs
uj rdrds
i1 j1
0
−s
3.27
⎛
⎞
∞
n
n
p
τ
Ψ
i
≤
μi ⎝1 ζji
Kseλs sds eλτi ⎠ sup yi s .
ηi
−∞<s≤0
0
i1
j1
It follows that
n n p
yi sp ,
yi t
≤ Me−λ−αt sup
E
i1
−∞<s≤0 i1
3.28
for t ≥ 0, where
⎧
⎫⎞
⎛
∞
n
⎨
μ⎝
Ψi τ λτi ⎬⎠
< ∞,
1 max
1≤M
ζji
Kseλs sds e
⎭
i∈Λ ⎩
μ
ηi
0
j1
3.29
Mathematical Problems in Engineering
13
which means that
p
Ext − x∗ p ≤ Eϕ − x∗ Me−λ−αt
3.30
for t ≥ t0 .
Therefore, the equilibrium point of system 2.3 is pth moment exponentially stable.
Corollary 3.5. If p ≥ 2, under the assumptions (H1)–(H5), system 2.3 exists a unique equilibrium
point x∗ , and x∗ is pth moment exponentially stable if the following two conditions are satisfied:
(H9)
∞
n n f
bij Lf cij Lg dij Lh
pai − p − 1
Ksds
− bji Li
j
j
j
0
j1
j1
3.31
n h ∞
p p−1
g
cji Li dji Li
>
Ksds li ei .
2
0
j1
(H10) There exist constant λ ∈ 0, μ0 and α ∈ 0, λ, ρk maxi∈Λ {1,
e
αtk −tk−1 n
k
j1 μj /μi pji }
≤
such that
n n f g h ∞
f p p−1
bij Lj cij Lj dij Lj
bji Li −
ei
Ksds −
λ < pai − p − 1
2
0
j1
j1
⎞
⎛
n n h ∞
p−2 p−1
g
λτ
⎠
⎝
cji Li p − 1 li e −
dji Li
li −
−
Kseλs ds.
2
0
j1
j1
3.32
Proof. In Theorem 3.4, let αl,ij βl,ij γl,ij δl,ij ξl,ij ηl,ij 1/p, μi 1 for all i, j, l ∈ Λ. The
result is obtained directly.
Remark 3.6. In Corollary 3.5, if p 2, the conditions H9 and H10 are less weak than the
following conditions:
H9 ⎛
⎞
⎛
⎞
⎛
⎞
∞
n n n f
g
2min ai − max ⎝ bij Lj ⎠ − max⎝ cij Lj ⎠ − max⎝ dij Lhj
Ksds⎠
i∈Λ
i∈Λ
>
n
j1
i∈Λ
j1
i∈Λ
j1
0
j1
∞
n
n
f g max bji Li max cji Li max dji Lhi
Ksds max li 3.33
i∈Λ
n
j1
max ei .
i∈Λ
j1
i∈Λ
j1
i∈Λ
0
i∈Λ
14
Mathematical Problems in Engineering
H10 There exist constant λ ∈ 0, μ0 and α ∈ 0, λ, ρk max {1,
e
i∈Λ
αtk −tk−1 n
k
j1 μj /μi pji }
≤
such that
⎞
⎞
⎛
⎞
⎛
n n n f
h ∞
g
λ < 2min ai − max⎝ bij Lj ⎠ − max⎝ cij Lj ⎠ − max⎝ dij Lj
Ksds⎠
⎛
i∈Λ
i∈Λ
i∈Λ
j1
i∈Λ
j1
−
j1
0
⎞
⎛
n
n
f
g
max bji Li − max ei − ⎝ max cji Li max li ⎠eλτ
j1
i∈Λ
i∈Λ
j1
i∈Λ
i∈Λ
n
h ∞
− max dji Li
Kseλs ds,
j1
i∈Λ
0
3.34
while H9 and H10 were asked in Theorem 3.2 in 18.
Corollary 3.7. Under the assumptions H2 and H8, system 2.1 exists a unique equilibrium point
x∗ , and x∗ is globally exponentially stable if 0 < ηi ≤ 1, and the following condition is also satisfied:
n pα /p−1
f
g pγ /p−1
Lj bij ij
Lj cij ij
pai − p − 1
j1
n μ n μ j
j
bji p1−αji Lf > 1
cji p1−γji Lg .
−
i
i
μ
ηi j1 μi
j1 i
3.35
Proof. In Theorem 3.4, when p > 1, choosing αl,ij αij /p − 1, γl,ij γij /p − 1 for l 1, 2, . . . , p − 1, and dij 0, βl,ij δl,ij 1/p for all l ∈ Λ, then the result is obtained.
f
g
Remark 3.8. If 0 < ηi ≤ 1, Li Li Li , it follows from 3.35 that
n n μ n μ pα /p−1
pγ /p−1 j
j
bji p1−αji Li >
cji p1−γji Li ,
Lj bij ij
−
pai − p − 1
Lj cij ij
μ
μ
i
i
j1
j1
j1
3.36
this is less conservative than the following inequality:
⎧
⎨
⎫
n n μ pαij /p−1
pγij /p−1 p1−αji ⎬
j
bji Lj bij −
min pai − p − 1
Lj cij Li
⎭
i∈Λ ⎩
μ
i
j1
j1
⎧
⎫
n μ ⎨
⎬
j
p1−γ
ji
cji > max
Li ,
⎭
i∈Λ ⎩
μ
j1 i
while 3.37 was required in Theorem 3.1 in 5 and in the only theorem in 8.
3.37
Mathematical Problems in Engineering
15
When p 2, αij γij 1/2, μi 1, 3.37 is equal to
⎧
⎨
⎫
⎧
⎫
n
n n ⎬
⎨
⎬
cji Li ,
Lj bij Lj cij − bji Li > max
min 2ai −
⎭
⎭
i∈Λ ⎩
i∈Λ ⎩
j1
j1
j1
3.38
while 3.38 was required in Theorem 3.2 in 19.
Similar to Corollaries 3.5 and 3.7, the following results are directly gained from
Theorem 3.4.
Corollary 3.9. Under the assumptions (H2), (H4), and (H8), system 2.2 exists a unique equilibrium
point x∗ , and x∗ is pth moment exponentially stable if 0 < ηi ≤ 1, and the following condition is
satisfied:
n n μ f g f p p−1
p−1 p−2
j
bji Li −
bij Lj cij Lj −
ei −
li
pai − p − 1
μ
2
2
j1
j1 i
⎛
⎞
n 1 ⎝ μj g
cji L p − 1 li ⎠.
>
i
ηi μi j1
3.39
Especially, if p 2, then the equilibrium x∗ is exponentially stable in mean square if the following
condition is also satisfied:
2ai −
n n μ 2β
2−2βji
j
bij 2αij Lf ij cij 2γij Lg 2δij −
bji 2−2αji Lf
− ei
j
j
i
μ
j1
j1 i
⎛
⎞
n 1 ⎝ μj 2−2δji
g
2−2γ
ji
cji >
Li
li ⎠.
ηi μi j1
3.40
Remark 3.10. If 0 < ηi ≤ 1, it follows from 3.39 that
n n μ f g f p p−1
p−1 p−2
j
bji Li −
bij Lj cij Lj −
ei −
li
pai − p − 1
μ
2
2
j1
j1 i
n μj cji Lg p − 1 li .
>
i
μi j1
3.41
16
Mathematical Problems in Engineering
This is less conservative than the following inequality:
⎧
⎨
n n μ n p−1 p−2
f g f j
bji L −
bij L cij L −
min pai − p − 1
ei li j
j
i
1≤i≤n⎩
μ
2
j1
j1 i
j1
⎫
⎧
⎫
n ⎨ μj ⎬
μ
⎬
j
cji Lg ,
−
p − 1 ei > max
p
−
1
l
i
i
⎭ 1≤i≤n ⎩ μi j1
⎭
μ
μi
j1 i
n μ
j
3.42
while 3.42 was required in Theorem 3.3 in 26.
Remark 3.11. If 0 < ηi ≤ 1, it follows from 3.40 that
2ai −
n n μ 2β
2−2βji
j
bij 2αij Lf ij cij 2γij Lg 2δij −
bji 2−2αji Lf
− ei
j
j
i
μ
j1
j1 i
n μj cji 2−2γji Lg 2−2δji li ,
>
i
μi j1
3.43
this is less conservative than the following inequality:
⎧
⎨
⎫
n n μ 2αij f 2βij 2γij g 2δij
2−2αji f 2−2βji max1≤i≤n μi ⎬
j
bij L
bji cij Lj
Li
−
ei
−
min 2ai −
j
⎭
1≤i≤n⎩
μ
μi
j1
j1 i
⎧
⎫
n ⎨ μj max1≤i≤n μi ⎬
2−2δji
g
2−2γ
ji
cji Li
li ,
> max
⎭
1≤i≤n ⎩ μi
μi
j1
3.44
while 3.44 was required in Theoerm 3.3 in 27.
4. Illustrative Example
In this section, we will give an example to show that the conditions given in the previous
sections are less weak than those given in some of the earlier literatures, such as 18.
Consider the following stochastic neural networks with mixed time delays
⎛
dxi t ⎝−ai xi t 2
2
2
bij fj xj t cij gj xj t − τj t dij
j1
·
t
−∞
j1
j1
⎞
Kij t − shj xj s ds Ii ⎠dt σi t, xi t, xi t − τi tdwi t,
4.1
Mathematical Problems in Engineering
17
where t /
tk , i ∈ Λ {1, 2}, tk k, fj s gj s hj s 1/2|x 1| − |x − 1|, and Kij s e−s .
1.7
,
ai 2×1 3
0.1 0.2
dij 2×2 ,
0.2 0.3
bij
0.1 0.2
0.2 0.3
,
cij 2×2 ,
2×2
0.4 0.3
0.1 0.4
0.94
0.1x1 t − τ1 t − 1.2
.
,
σi 2×1 0.1x2 t − τ2 t − 1.4
2.5
Ii 2×1
4.2
In the following, we introduce the following nonlinear impulsive controllers:
x1 tk 0.05 sin x1 t−k − 1.2 − 0.02x2 tk 1.48, k ∈ Z ,
x2 tk − 0.03x1 tk 0.04 cos x2 t−k − 1.4 1.72, k ∈ Z .
f
4.3
g
In this case, we have Lj Lj Lhj 1, Ks e−s , ej 0, lj 0.01 for j 1, 2, and μ0 0.9.
For p 2, we can compute that
2a1 −
∞
) 2 (
2 f
b1j Lf c1j Lg d1j Lh
Ksds
− bj1 L1 1.8
j
j
j
0
j1
>
∞
)
2 (
cj1 Lg dj1 Lh
Ksds
l1 e1 0.61,
1
1
0
j1
2a2 −
j1
∞
) 2 (
2 b2j Lf c2j Lg d2j Lh
bj2 Lf 3.8
Ksds
−
j
2
j
j
0
j1
>
j1
∞
)
2 (
cj2 Lg dj2 Lh
Ksds
l2 e2 1.21,
1
1
0
j1
Pk 0.2 0.08
.
0.12 0.16
4.4
18
Mathematical Problems in Engineering
Choosing μi 1 for i 1, 2, we have ρk maxi∈Λ {1,
0.5, then
0.6 λ < 2a1 −
2
j1
pjik } 1 and λ 0.6 and α ∞
) 2 (
2 b1j Lf c1j Lg d1j Lh
bj1 Lf − e1
Ksds
−
j
j
j
1
0
j1
j1
⎡
⎤
2 2 g
h ∞
λτ
⎣
⎦
cj1 L1 l1 e −
dj1 L1
−
Kseλs ds 1.05 − 0.31e0.12 0.7,
j1
0
j1
) 2 (
2 f g h ∞
f
b2j Lj c2j Lj d2j Lj
0.6 λ < 2a2 −
Ksds − bj2 L2 − e2
0
j1
4.5
j1
⎤
⎡
∞
2 2 g
− ⎣ cj2 L2 l2 ⎦eλτ − dj2 Lh2
Kseλs ds 2.55 − 0.71e0.12 1.749,
j1
0
j1
∞
Kseλs sds 6.25 ≤
∞
0
Kseμ0 s ds 10.
0
Thus, all conditions of Theorem 3.4 in this paper are satisfied; the equilibrium solution is
exponentially stable in mean square. From above discussion, it is easy to see that
⎛
⎞
⎛
⎞
⎛
⎞
2 2 2 f
h ∞
g
Ksds⎠
2min ai − max⎝ bij Lj ⎠ − max⎝ cij Lj ⎠ − max⎝ dij Lj
i∈Λ
i∈Λ
−
i∈Λ
j1
0
j1
n
f max bji Li 1.0
j1
<
i∈Λ
j1
4.6
i∈Λ
∞
2
2
2
g max cji Li max dji Lhi
Ksds max li max ei 1.11,
j1
i∈Λ
j1
i∈Λ
0
i∈Λ
j1
i∈Λ
which implies that the condition H9 in 18 do not hold for this example. So our results are
less weaker than some previous results.
5. Conclusion
In this paper, we investigate the pth moment exponential stability for stochastic neural
networks with mixed delays under nonlinear impulsive effects. By means of Lyapunov
functionals, stochastic analysis, and differential inequality technique, some sufficient
conditions for the pth moment exponential stability of this system are derived. The results
of this paper are new, and they supplement and improve some of the previously known
results 5, 8, 18, 19, 26, 27. Moreover, examples are given to illustrate the effectiveness of our
results. Furthermore, the method given in this paper may be extended to study other neural
networks, such as the model in 29 and stochastic Cohen-Grossberg neural networks in 30,
and we can get improved results too.
Mathematical Problems in Engineering
19
Acknowledgments
The authors would like to thank the editor and the anonymous referees for their detailed
comments and valuable suggestions which considerably improved the presentation of this
paper. This work was supported in part by the National Natural Science Foundation of China
under Grants no. 11101054, Hunan Provincial Natural Science Foundation of China under
Grant no. 12jj4005, the Open Fund Project of Key Research Institute of Philosophies and
Social Sciences in Hunan Universities under Grants no. 11FEFM11 and the Scientific Research
Funds of Hunan Provincial Science and Technology Department of China under Grants no.
2012SK3096.
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