Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 523812, 12 pages
doi:10.1155/2012/523812
Research Article
LMI Approach to Stability Analysis of
Cohen-Grossberg Neural Networks with
p-Laplace Diffusion
Xiongrui Wang,1, 2 Ruofeng Rao,1, 2 and Shouming Zhong2, 3
1
Department of Mathematics, Yibin University, Yibin 644007, China
Institute of Mathematics, Yibin University, Yibin 644007, China
3
College of Applied Mathematics, University of Electronic Science and Technology of China,
Chengdu 610054, China
2
Correspondence should be addressed to Ruofeng Rao, ruofengrao@163.com
Received 1 August 2012; Revised 24 October 2012; Accepted 12 November 2012
Academic Editor: Zhilong L. Huang
Copyright q 2012 Xiongrui Wang 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 nonlinear p-Laplace diffusion p > 1 was considered in the Cohen-Grossberg neural network
CGNN, and a new linear matrix inequalities LMI criterion is obtained, which ensures the
equilibrium of CGNN is stochastically exponentially stable. Note that , if p 2, p-Laplace diffusion
is just the conventional Laplace diffusion in many previous literatures. And it is worth mentioning
that even if p 2, the new criterion improves some recent ones due to computational efficiency. In
addition, the resulting criterion has advantages over some previous ones in that both the impulsive
assumption and diffusion simulation are more natural than those of some recent literatures.
1. Introduction and Preparation
It is well known that Cohen-Grossbeg neural network CGNN was proposed by Cohen and
Grossberg 1 in 1983. Since then there have been a lot of interested results obtained in many
literatures see 2–9 due to its general applications, such as pattern recognition, image
and signal processing, optimization automatic control, and artificial intelligence. Usually,
there exist the impulsive effect and time-varying delays phenomenon in various neural
networks 3, 5–7, 10–14. Besides, diffusion effects cannot be avoided in the neural networks
when electrons are moving in asymmetric electromagnetic fields 15–18. However, diffusion
disturbance was always simulated simply by linear Laplace diffusion 15–18. Few papers
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Journal of Applied Mathematics
involved the nonlinear reaction-diffusion 19. So in this paper, we investigate the stability of
the following stochastic CGNN with nonlinear p-Laplace diffusion p > 1:
x, u ◦ ∇p u − Aut, x
dut, x ∇ · Dt,
× But, x − Cfut, x Dgut − τt, x dt
σut, xdwt, t ∈ tk , tk1 u tk , x Mu t− , x , t tk
ut0 θ, x ϕθ, x,
∂ui t, x
0,
∂ν
1.1
θ, x ∈ −τ, 0 × ∂Ω
t, x ∈ −τ, ∞ × ∂Ω,
i 1, 2, . . . , n,
where Ω is a bounded subset in Rm with smooth boundary ∂Ω, and ∂ui t, x/∂ν ∂ui t, x/∂x1 , ∂ui t, x/∂x2 , . . . , ∂ui t, x/∂xm T denotes the outward normal derivative
on ∂Ω.
Remark 1.1. If p 2, system 1.1 was studied by 5 though there is a little difference between
Dirichlet boundary condition and Neumann boundary condition. However, our impulsive
assumption utk , x Mut−k , x is more natural than that of 5, which will result in some
difference in methods.
Here, M is a diagonal matrix, Ω ∈ Rm is a bounded compact set with smooth boundary,
u u1 , u2 , . . . , un T ∈ Rn , and wt w1 t, w2 t, . . . , wn tT is a n-dimensional Brownian
motion defined on a complete probability space Ω, F, P with the natural filtration {Ft }t≥0
generated by the process {ws : 0 ≤ s ≤ t}. We associate Ω with the canonical space
generated by all {wi t} and denoted by F the associated σ-algebra generated by wt with
the probability measure p. Aut, x presents an amplification function, But, x is an
appropriately behavior function, and f and g denote the activation function. τt 0 ≤ τt ≤
τ corresponds to the transmission delays at time, and tk is called the impulsive moment
with 0 < t1 < t2 < · · · < tk < · · · and limk → ∞ tk ∞. We always assume utk , x ◦
utk , x. ∇p u ∇p u1 , . . . , ∇p un T , ∇p ui |∇ui |p−2 ∂ui /∂x1 , . . . , |∇ui |p−2 ∂ui /∂xm T , D
p−2
ik |∇ui | ∂ui /∂xk n×m is Hadamard product of matrix D
and ∇p u 20. Here,
∇p u D
ik n×m . Let Yi the diffusion parameters matrix Dt, x, u is denoted simply as D D
T
T
yi1 , . . . , yim , i 1, 2, . . . , n, and matrix Y Y1 , . . . , Yn , and we denote ∇ · Yi m
T
k1 ∂yik /∂xk , ∇ · Y ∇ · Y1 , ∇ · Y2 , . . . , ∇ · Yn . Particularly, ∇p u ∇u for the case of
p 2.
Remark 1.2. Diffusion effects always occur in the neural networks when electrons are moving
in asymmetric electromagnetic fields 15–18, and diffusion behavior is so complicated that
it cannot always be simulated by linear Laplace diffusion. So in this paper, the nonlinear
p-Laplace diffusion is considered in System 1.1.
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3
Assume, in addition, the following.
H1 Aut, x is a bounded, positive, and continuous diagonal matrix, that is, there
exist two positive diagonal matrices A and A such that 0 < A ≤ Aut, x ≤ A.
H2 But, x b1 u1 t, x, b2 u2 t, x, . . . , bn un t, xT such that there exists a
positive diagonal matrix B diagB1 , B2 , . . . , Bn satisfying bi ui t, x/ui t, x ≥
Bi for all i.
H3 There exist two positive diagonal matrices F diagF1 , F2 , . . . , Fn and G diagG1 , G2 , . . . , Gn such that
0≤
fi r
≤ Fi ,
r
0≤
gi r
≤ Gi ,
r
∀i.
1.2
H4 The null solution is the equilibrium point of system 1.1, that is, the following
conditions hold:
B0 − Cf0 − Dg0 0,
trace σ T ut, xσut, x ≤ uT t, xQut, x,
1.3
where the symmetrical matrix Q > 0.
For convenience’s sake, we introduce some standard notations
i L2 RR × Ω: the space of real Lebesgue measurable functions of R × Ω, it
is a Banach space for the 2-norm ut
2 ni1 ui t
1/2 with ui t
Ω |ui t, x|2 dx1/2 , where |ui t, x| is Euclid norm.
ii L2F0 −τ, 0 × Ω; Rn : the family of all F0 -measurable C−τ, 0 × Ω; Rn - value
random variable ξ {ξθ, x : −τ ≤ θ ≤ 0, x ∈ Ω} such that sup−τ≤θ≤0 E
ξθ
22 < ∞,
where E{·} stands for the mathematical expectation operator with respect to the
given probability measure p.
iii Q qij n×n > 0 <0: a positive negative definite symmetrical matrix, that is,
yT Qy > 0 <0 for any 0 / y ∈ Rn .
iv Q qij n×n ≥ 0 ≤0: a semipositive semi-negative definite symmetrical matrix,
that is, yT Qy ≥ 0 ≤0 for any y ∈ Rn .
Q ≤ Q:
this means Q − Q
is a semi-positive semi-negative symmetrical
v Q ≥ Q
definite matrix.
Q < Q:
this means Q − Q
is a positive negative symmetrical definite
vi Q > Q
matrix.
vii λmax Φ, λmin Φ denotes the largest and smallest eigenvalue of symmetrical matrix
Φ, respectively.
viii I: identity matrix with compatible dimension.
ix Denote |C| |cij |n×n for any matrix Cn×n ; |ut, x| |u1 |, |u2 |, . . . , |un | for any
u ∈ Rn .
Let ut, x; ϕ denote the state trajectory from the initial data ut0 θ, x; ϕ ϕθ, x on
−τ ≤ θ ≤ 0 in L2F0 −τ, 0 × Ω; Rn .
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Journal of Applied Mathematics
Definition 1.3. The null solution of impulsive system 2.2 is globally stochastically
exponentially stable in the mean square if for every ϕ ∈ L2F0 −τ, 0 × Ω; Rn , there exists
scalars β > 0 and γ > 0 such that
2 2
E u t, ϕ 2 ≤ γe−βt sup E ϕθ2 .
1.4
−τ≤θ≤0
Lemma 1.4 see 11. Let U, P be any matrices, ε > 0 is a positive number and matrix H H T > 0,
then
P T U UT P ≤ εP T HP ε−1 UT H −1 U.
1.5
Lemma 1.5 Schur complement 3. The LMI
Qt St
ST t Rt
1.6
> 0,
where matrix St and symmetrical matrices Qt and Rt depend on t, is equivalent to any one of
the following conditions:
L1 Rt > 0, Qt − StR−1 tST t > 0;
L2 Qt > 0, Rt − ST tQ−1 tST t > 0.
Lemma 1.6 see 21. Consider the following differential inequality:
D vt ≤ −avt bvtτ , t /
tk
−
− vtk ≤ ak v tk bk v tk τ ,
1.7
where vt ≥ 0, vtk τ supt−τ≤s≤t vs, vt−k τ supt−τ≤s≤t vs and vt is continuous except
tk , k 1, 2, . . ., where it has jump discontinuities. The sequence tk satisfies 0 t0 < t1 < t2 < · · · <
tk < tk1 < · · · , and limk → ∞ tk ∞. Suppose that
1 a > b ≥ 0;
2 tk − tk−1 > δτ, where δ > 1, and there exist constants γ > 0, M > 0 such that
ρ1 ρ2 · · · ρk1 ≤ Meγtk ,
1.8
where ρi max{1, ai bi eλτ }, λ > 0 is the unique solution of equation λ a − beλτ then
vt ≤ Mv0τ e−λ−γt .
1.9
In addition, if θ supk∈Z {1, ak bk eλτ }, then
vt θv0τ e−λ−lnθe
λτ
/δτt
,
t ≥ 0.
Computer simulation is shown in Figures 1, 2, 3, and 4.
1.10
Journal of Applied Mathematics
5
0.2
0.18
0.16
0.14
0.12
u1
0.1
0.08
0.06
0.04
0.02
0
0
5
1 15
10
0.5
ce:
Spa
Time: t
0
x
Figure 1: Computer simulation of the state u1 t, x .
0.18
0.16
0.14
u1 (t, 0.259)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
Figure 2: Sectional curve of the state variable u1 t, x .
2. Main Results
Theorem 2.1. If assumptions (H1)–(H4) hold, in addition, the following conditions are satisfied:
C1 there exists diagonal matrices P1 diagp11 , p12 , . . . , p1n > 0 and P2 > 0 such that
⎛
−2P1 AB F 2 P1 Q P2
T
⎜
C AP
⎝
T 1
D AP1
P1 A|C|
−I
0
⎞
P1 A|D|
⎟
0 ⎠ < 0;
−I
2.1
C2 min{λmin Θ/λmax P1 , 1 − μ} > λmax G2 /λmin P2 ≥ 0, where Θ 2P1 AB −
P1 A|C||C|T AP1 − P1 A|D||D|T AP1 − F 2 − P1 Q − P2 , where τ t ≤ μ < 1 for all t;
6
Journal of Applied Mathematics
0.02
0.018
0.016
0.014
u2
0.012
0.01
0.008
0.006
0.004
0.002
0
1
0.5
0
ce: x
5
Spa
10
15
Time: t
Figure 3: Computer simulation of the state u2 t, x .
0.02
0.018
u2 (t, 0.267)
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
5
10
15
Figure 4: Sectional curve of the state variable u2 t, x .
C3 there exists a constant δ > 1 such that infk∈Z tk − tk−1 > δτ, δ2 τ > lnρeλτ and
λ − lnρeλτ /δτ > 0, where λ > 0 is the unique solution of the equation λ a − beλτ ,
and ρ max{1, λmax MP1 M/λmin P1 eλτ }, a min{λmin Θ/λmax P1 , 1 − μ}, b λmax G2 /λmin P2 , then the null solution of system 1.1 is stochastically exponentially stable
with convergence rate 1/2λ − lnρeλτ /δτ.
Proof. First, we can get by Guass formula see 20, Lemma 2.3
Ω
x, u ◦ ∇p u dx
uT P1 ∇ · Dt,
T
m
m
∂
∂
p−2 ∂u1
p−2 ∂un
u P1
dx
D1k |∇u1 |
,...,
Dnk |∇un |
∂xk
∂xk
∂xk
∂xk
Ω
k1
k1
T
Journal of Applied Mathematics
n
m
p−2 ∂uj
∂
dx
p1j uj
Djk ∇uj
∂xk
∂xk
Ω j1
k1
m n −
Ω
k1 j1
jk ∇uj p−2
p1j D
∂uj
∂xk
7
2
dx.
2.2
Construct the Lyapunov functional as follows:
V V1 V2 ,
2.3
where
V1 u t, xP1 ut, xdx uT t, xP1 |ut, x|dx
T
Ω
V2 Ω
t
Ω
u s, xP2 us, xds dx.
T
t−τt
Then
LV1 Ω
−
m n
jk ∇uj p−2
2p1j D
k1 j1
∂uj
∂xk
2
dx
−2
2
2
Ω
Ω
Ω
Ω
uT t, xP1 Aut, xBut, xdx
uT t, xP1 Aut, xCfut, xdx
uT t, xP1 Aut, xDgut − τt, xdx
trace σ T ut, xP1 σut, x dx
≤ − 2 uT t, xP1 AB|ut, x|dx
Ω
T
u t, xP1 A|C||C|T AP1 |ut, x| f T ut, xfut, x dx
Ω
T
u t, xP1 A|D||D|T AP |ut, x| g T ut − τt, xgut − τt, x dx
Ω
Ω
uT t, xP1 Qut, xdx
2.4
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Journal of Applied Mathematics
≤ − uT t, x 2P1 AB − P1 A|C||C|T AP1 − P1 A|D||D|T AP1 − F 2 − P1 Q |ut, x|dx
Ω
Ω
uT t − τt, xG2 ut − τt, xdx
uT t, xP2 ut, x − 1 − τ t uT t − τtP2 ut − τt dx
LV2 Ω
T
≤
uT t − τtP2 ut − τtdx.
u t, xP2 |ut, x|dx μ − 1
Ω
Ω
2.5
And then we have
LV ≤ −
T
x|dx
uT t − τt, x G2 μ − 1 P2 ut − τt, xdx.
x
t,
Θ|ut,
u
Ω
Ω
2.6
Next, we use the method similar as that of 22. Since ut, x is the solution of system,
o formula
and V ut, x ∈ C2 Rm , R for all t, we can get by It
V t V tk t
LV us, xds tk
t
tk
∂V
σus, xdws.
∂u
2.7
Then we have
EV ut, x EV utk , x t
ELV s, xds,
t ∈ tk , tk1 .
2.8
tk
Thus, for small enough Δt > 0, we have
EV ut Δt, x EV utk , x tΔt
ELV s, xds,
t ∈ tk , tk1 ,
2.9
tk
and then
EV ut Δt, x − EV ut, x
tΔt
ELV s, xds
t
tΔt − uT s, xΘ|us, x|dx
≤E
Ω
t
2
u s − τs, x G μ − 1 P2 us − τs, xdx ds,
T
Ω
2.10
t ∈ tk , tk1 .
Journal of Applied Mathematics
9
Since
T
uT t − τt, x 1 − μ P2 ut − τt, xdx
u t, xΘ|ut, x|dx Ω
Ω
λmin Θ
T
uT t − τt, xP2 ut − τt, xdx
≥
u t, xP1 |ut, x|dx 1 − μ
λmax P1 Ω
Ω
!
λmin Θ , 1−μ V
≥ min
λmax P1
λmax G2
uT t − τt, xG2 ut − τt, xdx ≤
uT t − τt, xP2 ut − τt, xdx,
λmin P2 Ω
Ω
2.11
Then,
!
λmin Θ λmax G2
D EV ut, x ≤ − min
, 1 − μ EV t EV tτ ,
λmax P1
λmin P2
t ∈ tk , tk1 .
2.12
Next, we have
V tk u tk , xP1 utk , xdx T
Ω
≤
Ω
u
T
t−k , x
Ω
uT tk − τtk , xP2 utk − τtk , xdx
MP1 Mu t−k , x dx Ω
uT tk − τtk , xP2 utk − τtk , xdx
2.13
λmax MP1 M − − V tk V t k τ .
λmin P1
Now the conditions C1–C3 and Lemma 1.6 deduce
EV t ≤ ρV 0τ e−λ−lnρe
λτ
/δτt
2.14
or
2 −λ−lnρeλτ /δτt
λmax P
sup E φs 2 e
ρ
λmin P −τ≤s≤0
2
E
ut, x
≤
2.15
which together with Definition 1.3 implies the accomplishment of the proof.
Remark 2.2. The nonlinear p-Laplace diffusion p > 1 brings a great difficulties in judging
the stability. However, even if p 2, Theorem 2.1 has more computational efficiency than 15,
Theorem 3.1 due to LMI criterion.
10
Journal of Applied Mathematics
3. Examples
Consider the following impulsive CGNN:
dut, x 0
0.0005
0
1.65 0.5 sin u1
◦ ∇p u −
0
1.58 0.5 sin u2
0
0.0005
5.19u1 − 0.02u1 cos u1
0.1 −0.03
×
−
fut, x
5.18u2 − 0.01u2 cos u2
−0.03 0.1
!
0.1 −0.03
−
gut − τt, x dt σut, xdwt, t ∈ tk , tk1 −0.03 0.1
1.350 0
u tk u t− , x , t tk
0 1.350
∇·
ut0 θ, x φθ, x,
∂ui t, x
0,
∂ν
where Ω {x1 , x2 T ∈ R2 | −
F
√
θ, x ∈ −0.65, 0 × ∂Ω
t, x ∈ −τ, ∞ × ∂Ω,
3.1
2 ≤ x1 , x2 ≤
0.100 0
0 0.100
i 1, 2,
√
2}, and the corresponding matrices
G,
Q
0.0001
0
.
0
0.0001
3.2
We might as well assume that t0 0, tk − tk−1 0.525, τt 0.65, τ t ≤ μ 0.99 for all t ≥ t0 ,
and
⎛
⎞
|u1 1| − |u1 − 1|
⎜
⎟
20
⎟
fu gu ⎜
⎝ |u2 1| − |u2 − 1| ⎠,
20
x1 − cos5πxcos189 x − 0.25e−100s
, −0.65 ≤ s ≤ 0,
1 − xsin2 4πxcos201 x − 0.55e−100s
1 0
2 0
5.160 0
A
A
,
,
B
.
0 1
0 2
0 5.160
φs, x 3.3
By way of MATLAB LMI Control Toolbox, we can solve the LMI condition in C1 and get
P1 0.2392
0
,
0
0.2392
1.2291
0
P2 .
0
1.2291
3.4
Next, we will prove that such P1 and P2 make C2 and C3 hold. Indeed, by computing
directly, we can obtain a 0.01, b 0.0081, and then C2 holds. Moreover, we might as well
Journal of Applied Mathematics
11
assume δ 82, and then we have λ 0.0019, ρ 2.8235, thus C3 is satisfied. Now from
Theorem 2.1 we can compute the convergence 9.3809 ∗ 10−0.004 .
4. Conclusions
In this paper, we investigate the influence of impulse, time-delays and diffusion behaviors on
the stability of stochastic Cohen-Grossberg neural network CGNN. The LMI conditions of
stochastic exponential stability of impulsive CGNN with p-Laplace reaction-diffusion terms
was given, and an illustrate example was also given to show the effectiveness of the obtained
result. Besides, the result obtained in this paper is also valid to the Laplace reaction-diffusion
in the case of p 2 and has more computational efficiency due to the LMI approach even if
p 2 Remark 2.2.
Acknowledgments
This work was supported by the National Basic Research Program of China 2010CB732501,
by Scientific Research Fund of Science Technology Department of Sichuan Province
2011JYZ010, and by Scientific Research Fund of Sichuan Provincial Education Department
11ZA172, 12ZB349.
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