Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 501891, 17 pages
doi:10.1155/2012/501891
Research Article
Asymptotic Stability of Impulsive
Reaction-Diffusion Cellular Neural Networks with
Time-Varying Delays
Yutian Zhang
College of Mathematics & Physics, Nanjing University of Information Science & Technology,
Nanjing 21004, China
Correspondence should be addressed to Yutian Zhang, ytzhang81@163.com
Received 27 June 2011; Accepted 13 September 2011
Academic Editor: E. S. Van Vleck
Copyright q 2012 Yutian Zhang. 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 work addresses the asymptotic stability for a class of impulsive cellular neural networks
with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of
Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov
functions, we summarize some new and concise sufficient conditions ensuring the global
exponential asymptotic stability of the equilibrium point. The provided stability criteria are
applicable to Dirichlet boundary condition and showed to be dependent on all of the reactiondiffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial
variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained
results.
1. Introduction
Cellular neural networks CNNs, proposed by Chua and Yang in 1988 1, 2, have been the
focus of a number of investigations due to their potential applications in various fields such
as optimization, linear and nonlinear programming, associative memory, pattern recognition,
and computer vision 3–7. Moreover, on the ground that time delays are unavoidably
encountered for the finite switching speed of neurons and amplifiers in implementation of
neural networks, it was followed by the introduction of the delayed cellular neural networks
DCNNs so as to solve some dynamic image processing and pattern recognition problems.
Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors
such as stability, convergence, and oscillatory 8, 9. Particularly, stability analysis has been
a major concern in the designs and applications of the CNNs and DCNNs. The stability of
CNNs and DCNNs is a subject of current interest, and considerable theoretical efforts have
been put into this topic with many good results reported see, e.g., 10–13.
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Journal of Applied Mathematics
With reference to neural networks, however, it is noteworthy that the state of electronic
networks is actually subject to instantaneous perturbations more often than not. On this
account, the networks experience abrupt change at certain instants which may be caused by a
switching phenomenon, frequency change, or other sudden noise; that is, the networks often
exhibit impulsive effects 14, 15. For instance, according to Arbib 16 and Haykin 17, when
a stimulus from the body or the external environment is received by receptors, the electrical
impulses will be conveyed to the neural net and impulsive effects arise naturally in the net.
As a consequence, in the past few years, scientists have become increasingly interested in the
influence that impulses may have on the CNNs and DCNNs and a large number of stability
criteria have been derived see, e.g. 18–22.
In reality, besides impulsive effects, diffusion effects are also nonignorable since
diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields.
As such, the model of neural networks with both impulses and reaction-diffusion should
be more accurate to describe the evolutionary process of the systems in question, and it is
necessary to consider the effects of both diffusion and impulses on the stability of CNNs and
DCNNs.
In the past years, there have been a few theoretical contributions to the stability
of CNNs and DCNNs with impulses and diffusion. For instance, Qiu 23 formulated a
mathematical model of impulsive neural networks with time-varying delays and reactiondiffusion terms described by impulsive partial differential equations and studied, via delay
impulsive differential inequality, the problem of global exponential stability with some
stability criteria presented. Remarkably, all of the obtained stability criteria in 23 are
independent of the diffusion. In 2008, Li and song 24 investigated a class of impulsive
Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By
establishing a delay inequality with impulsive initial conditions and M-matrix theory, some
sufficient conditions ensuring global exponential stability of the equilibrium points are given.
Analogous to 23, the proposed stability criteria in 24 are also independent of the diffusion.
More recently Pan et al. 25 investigated a class of impulsive Cohen-Grossberg neural
networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay
impulsive differential inequality quoted in 23, several sufficient conditions are exploited
ensuring global exponential stability of the equilibrium points. Especially, different from
23, 24, the estimate of the exponential convergence rate depends on reaction-diffusion in
25.
In this paper, unlike the methods of impulsive differential inequalities and Poincaré
inequality used in 25, we attempt to adopt the new techniques of the impulsive integral
inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the
problem of global exponential asymptotic stability for impulsive cellular neural networks
with time-varying delays and reaction-diffusion terms. Different from the existing research,
we find, besides the reaction-diffusion coefficients, the dimension of the space and the
boundary of the spatial variables do influence the stability.
The rest of the paper is organized as follows. In Section 2, the model of impulsive
delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary
condition is outlined, and some facts and lemmas are introduced for later reference. By the
new agency of the impulsive integral inequality of Gronwall-Bellman type as well as HardySobolev inequality, we discuss the global exponential asymptotic stability and develop some
new criteria in Section 3. To conclude, two illustrative examples are given to verify the
effectiveness of our results in Section 4.
Journal of Applied Mathematics
3
2. Preliminaries
Let Rn denote the n-dimensional Euclidean space, and Ω ⊂ Rm is a bounded open set
containing the origin. The boundary of Ω is smooth and mes Ω > 0. Let R 0, ∞ and
t0 ∈ R .
We consider the following impulsive neural networks with time delays and reactiondiffusion terms:
m
n
n
∂
∂ui t, x
∂ui t, x Dis
− ai ui t, x bij fj uj t, x cij fj uj t − τj t, x
∂t
∂xs
∂xs
s
1
j
1
j
1
t ≥ t0 ,
t/
tk ,
ui tk 0, x ui tk , x Pik ui tk , x
x ∈ Ω,
i 1, 2, . . . , n,
k 1, 2, . . . ,
2.1
x ∈ Ω, k 1, 2, . . . , i 1, 2, . . . , n,
2.2
where n corresponds to the numbers of units in a neural network; x x1 , . . . , xm T ∈ Ω,
ui t, x, denotes the state of the ith neuron at time t and in space x; smooth functions Dis Dis t, x, u ≥ 0 represent transmission diffusion operators of the ith unit; activation functions
fj uj t, x stand for the output of the jth unit at time t and in space x; bij , cij , ai are constants:
bij indicates the strength of the jth unit on the ith unit at time t and in space x, cij denotes the
strength of the jth unit on the ith unit at time t − τj t and in space x, where τj t corresponds
to the transmission delay along the axon of the jth unit and satisfies 0 ≤ τj t ≤ τ τ const
•
as well as τ j t < 1 − 1/h h > 0, and ai > 0 represents the rate with which the ith unit
will reset its potential to the resting state in isolation when disconnected from the network
and external inputs at time t and in space x. The fixed moments tk k 1, 2, . . . are called
impulsive moments satisfying 0 ≤ t0 < t1 < t2 < · · · and limk → ∞ tk ∞; ui tk 0, x and
ui tk − 0, x represent the right-hand and left-hand limit of ui t, x at time tk and in space x,
respectively. Pik ui tk , x stands for the abrupt change of ui t, x at impulsive moment tk and
in space x.
Denote by ut, x ut, x; t0 , ϕ, u ∈ Rn the solution of system 2.1-2.2, satisfying the
initial condition
u s, x; t0 , ϕ ϕs, x,
t0 − τ ≤ s ≤ t0 , x ∈ Ω
2.3
t ≥ t0 , x ∈ ∂Ω,
2.4
and Dirichlet boundary condition
u t, x; t0 , ϕ 0,
where the vector-valued function ϕs, x ϕ1 s, x, . . . , ϕn s, xT is such that Ω ni
1
ϕ2i s, xdx is bounded on t0 − τ, t0 and ϕi s, x i 1, 2, . . . , n is first-order continuous
differentiable as to s on t0 − τ, t0 .
The solution ut, x ut, x; t0 , ϕ u1 t, x; t0 , ϕ, . . . , un t, x; t0 , ϕT of problems
2.5–2.8 is, for the time variable t, a piecewise continuous function with the first kind
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Journal of Applied Mathematics
discontinuity at the points tk k 1, 2, . . ., where it is continuous from the left, that is the
following relations are true:
ui tk − 0, x ui tk , x,
ui tk 0, x ui tk , x Pik ui tk , x.
2.5
Throughout this paper, the norm of ut, x; t0 , ϕ is governed by
⎛
⎞1/2
n u t, x; t0 , ϕ ⎝
u2i t, x; t0 , ϕ dx⎠ .
Ω
i
1
2.6
Ω
Before moving on, we introduce two hypotheses as follows.
H1 Activation function fj uj t, x satisfies fi 0 0, and there exists constant li > 0
such that |fi y1 − fi y2 | ≤ li |y1 − y2 | holds for all y1 , y2 ∈ R and i 1, 2, . . . , n.
H2 The functions Pik ui tk , x are continuous on R and Pik 0 0, i 1, 2, . . . , n, k 1, 2, . . . .
According to H1 and H2, it is easy to see that problems 2.5–2.8 admits an
equilibrium point u 0.
Definition 2.1. The equilibrium point u 0 of problems 2.5–2.8 is said to be globally
exponentially stable if there exist constants κ > 0 and M ≥ 1 such that
u t, x; t0 , ϕ ≤ Mϕ e−κt−t0 ,
Ω
Ω
2
where ϕΩ supt0 −τ≤s≤t0
n i
1 Ω
t ≥ t0 ,
2.7
ϕ2i s, xdx.
Lemma 2.2 Gronwall-Bellman-type impulsive integral inequality 26. Assume that
(A1) the sequence {tk } satisfies 0 ≤ t0 < t1 < t2 < · · · , with limk → ∞ tk ∞,
(A2) q ∈ P C1 R , R and qt is left-continuous at tk , k 1, 2, . . .,
(A3) p ∈ CR , R and for k 1, 2, . . .
qt ≤ c t
t0
psqsds ηk qtk ,
t ≥ t0 ,
2.8
t ≥ t0 .
2.9
t0 <tk <t
where ηk ≥ 0 and c const. Then,
t
psds ,
qt ≤ c
1 ηk exp
t0 <tk <t
t0
Journal of Applied Mathematics
5
Lemma 2.3 Hardy-Sobolev inequality 27. Let Ω ⊂ Rm (m ≥ 3) be a bounded open set
containing the origin and u ∈ H 1 Ω {ω | ω ∈ L2 Ω, Di ω ∂ω/∂xi ∈ L2 Ω, 1 ≤ i ≤ m}.
Then there exists a positive constant Cm Cm Ω such that
m − 22
4
u2
Ω
dx ≤
|x|2
Ω
|∇u|2 dx Cm
2.10
u2 dσ.
∂Ω
Lemma 2.4. If a > 0 and b > 0, then ab ≤ 1/εa2 εb2 holds for any ε > 0.
3. Main Results
Theorem 3.1. Provided that
2
1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m
s
1 xs < β.
In addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0. Denote
Dm − 22 /2β χ,
2 Pik ui tk , x −θik ui tk , x, 0 ≤ θik ≤ 2,
3 there exists a constant γ satisfying γ λ hρeγτ > 0 as well as λ hρeγτ < 0, where
λ maxi
1,...,n −χ − 2ai nj
1 bij2 cij2 ρ, ρ n maxi
1,...,n li 2 ,
then, the equilibrium point u 0 of problems (2.5–2.8) is globally exponentially stable with
convergence rate – λ hρeγτ /2.
Proof. Multiplying both sides of 2.1 by ui t, x and integrating with respect to spatial
variable x on Ω, we get
d
Ω
m ui 2 t, xdx
∂
∂ui t, x
2
ui t, x
u2i t, xdx
Dis
dx − 2ai
dt
∂x
∂x
s
s
Ω
s
1 Ω
n
ui t, xfj uj t, x dx
2 bij
Ω
j
1
n
2 cij
j
1
Ω
ui t, xfj uj t − τj t, x dx
tk , k 1, 2, . . . .
t ≥ t0 , t /
3.1
Regarding the right-hand part of 3.1, the first term becomes by using Green formula,
Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.1
m m ∂
∂ui t, x
∂ui t, x 2
ui t, x
Dis
dx
Dis
dx −2
2
∂xs
∂xs
∂xs
s
1 Ω
s
1 Ω
Dm − 22
≤−
2
u2i t, x
Ω
|x|2
Dm − 22
dx ≤ −
2β
3.2
Ω
u2i t, xdx −χ
Ω
u2i t, xdx.
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Journal of Applied Mathematics
Moreover, we derive from H1 that
n
2 bij
j
1
n bij
ui t, xfj uj t, x dx ≤ 2
|ui t, x|fj uj t, x dx
Ω
Ω
j
1
n ≤2
Ω
j
1
≤
lj bij |ui t, x|uj t, xdx
n Ω
bij2 u2i t, x lj2 u2i t, x dx,
j
1
n
n 2 cij
ui t, xfj uj t − τj t, x dx ≤ 2 cij |ui t, x|fj uj t − τj t, x dx
j
1
Ω
3.3
Ω
j
1
n ≤2
lj cij |ui t, x|uj t − τj t, x dx
Ω
j
1
≤
n Ω
j
1
cij2 u2i t, x lj2 u2i t − τj t, x dx.
Consequently, substituting 2.10–3.14 into 3.1 produces
d Ω u2i t, xdx
≤ −χ
u2i t, xdx − 2ai
u2i t, xdx
dt
Ω
Ω
n bij2 u2i t, x lj2 u2i t, x dx
j
1
3.4
Ω
n j
1
Ω
cij2 u2i t, xlj2 u2i t−τj t, x dx
for t ≥ t0 , t /
tk , k 1, 2, . . . .
We define a Lyapunov function Vi t as Vi t Ω u2i t, xdx. It is easy to find that
Vi t is a piecewise continuous function with points of discontinuity of the first kind tk k 1, 2, . . ., where it is continuous from the left, that is, Vi tk − 0 Vi tk k 1, 2, . . .. In
addition, due to Vi t0 0 ≤ Vi t0 and the following estimate derived from condition 2 of
Theorem 3.1
u2i tk 0, x −θik ui tk , x ui tk , x2 1 − θik 2 u2i tk , x ≤ u2i tk , x
k 1, 2, . . .,
3.5
we have
Vi tk 0 ≤ Vi tk ,
k 0, 1, 2, . . . .
3.6
Journal of Applied Mathematics
7
holds for t tk k 0, 1, 2, . . .. Put t ∈ tk , tk1 , k 0, 1, 2, . . . . Then for the derivative
dVi t/dt of Vi with respect to problems 2.5–2.8, it results from 3.4 that
dVi t
≤ −χ
dt
Ω
u2i t, xdx − 2ai
n j
1
Ω
Ω
u2i t, xdx n j
1
2
cij2 u2i t, x lj2 uj t − τj t, x
Ω
bij2 u2i t, x lj2 u2i t, x dx
⎛
dx ≤ ⎝−χ−2ai bij2 n
j
1
n
n
max li2
Vj t max li 2
Vj t − τj t
i
1,...,n
n
i
1,...,n
j
1
⎞
cij2 ⎠Vi t
j
1
t ∈ tk , tk1 , k 0, 1, 2, . . . .
j
1
3.7
Choose V t of the form V t n
i
1 Vi t.
From 3.7, one then reads
⎛
⎞
⎞
⎛
n dV t ⎝
2
2
2
bij cij ⎠ n max li ⎠V t
≤
max ⎝−χ − 2ai i
1,...,n
i
1,...,n
dt
j
1
n
n max li 2
Vj t − τj
i
1,...,n
λV t ρ
3.8
j
1
n
Vj t − τj t
t ∈ tk , tk1 , k 0, 1, 2, . . . .
j
1
Construct V ∗ t eγt−t0 V t, where γ satisfies γ λ hρeγτ > 0 and λ hρeγτ < 0.
Evidently, V ∗ t is also a piecewise continuous function with points of discontinuity of the
first kind tk k 1, 2, . . ., in which it is continuous from the left, that is V ∗ tk − 0 V ∗ tk k 1, 2, . . .. Moreover, at t tk k 0, 1, 2, . . ., we find by use of 3.6
V ∗ tk 0 ≤ V ∗ tk ,
k 0, 1, 2, . . . .
3.9
Set t ∈ tk , tk1 , k 0, 1, 2, . . .. By virtue of 3.8, one has
⎞
⎛
n
dV ∗ t
γt−t0 γt−t0 dV t
γt−t0 γe
≤ γe
V t e
V t ⎝λV t ρ
Vj t − τj t ⎠eγt−t0 dt
dt
j
1
n
γ λ V ∗ t ρeγt−t0 Vj t − τj t
t ∈ tk , tk1 , k 0, 1, 2, . . . .
j
1
3.10
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Journal of Applied Mathematics
Choose small enough ε > 0. Integrating 3.10 from tk ε to t gives
∗
∗
V t ≤ V tk ε γ λ
t
γs−t0 ρe
tk ε
t
V ∗ sds
tk ε
3.11
n
Vj s−τj s ds,
t ∈ tk , tk1 , k 0, 1, 2, . . .
j
1
which yields after letting ε → 0 in 3.11
V ∗ t ≤ V ∗ tk 0 γ λ
t
V ∗ sds
tk
t
γs−t0 ρe
n
Vj s−τj s ds, t ∈ tk , tk1 , k 0, 1, 2, . . . .
tk
3.12
j
1
We now proceed to estimate the value of V ∗ t at t tk1 , k 0, 1, 2, . . . . For small
enough ε > 0, we put t tk1 − ε. Now an application of 3.12 leads to, for k 0, 1, 2, . . .,
∗
∗
V tk1 − ε ≤ V tk 0 γ λ
tk1 −ε
∗
V sds tk1 −ε
tk
ρeγs−t0 tk
n
Vj s − τj s ds.
j
1
3.13
If we let ε → 0 in 3.13, there results
∗
∗
V tk1 − 0 ≤ V tk 0 γ λ
tk1
V ∗ sds
tk
tk1
γs−t0 ρe
tk
n
Vj s − τj s ds,
3.14
k 0, 1, 2, . . . .
j
1
Note that V ∗ tk1 − 0 V ∗ tk1 is applicable for k 0, 1, 2, . . . . Thus,
V ∗ tk1 ≤ V ∗ tk 0 γ λ
tk1
V ∗ sds tk1
tk
tk
ρeγs−t0 n
Vj s − τj s ds
3.15
j
1
holds for k 0, 1, 2, . . . . By synthesizing 3.12 and 3.15, we then arrive at
∗
∗
V t ≤ V tk 0 γ λ
t
V ∗ sds
tk
γs−t0 ρe
tk
t
n
Vj s − τj s ds
j
1
3.16
t ∈ tk , tk1 , k 0, 1, 2, . . . .
Journal of Applied Mathematics
9
This, together with 3.9, results in
t
V ∗ t ≤ V ∗ tk γ λ
V ∗ sds t
tk
n
ρeγs−t0 Vj s−τj s ds
tk
3.17
j
1
for t ∈ tk , tk1 , k 0, 1, 2, . . . .
•
Recalling assumptions that 0 ≤ τj t ≤ τ and τ j t < 1 − 1/h h > 0, we have
t
ρeγs−t0 tk
n t−τj t
n
Vj s − τj s ds j
1
j
1
≤ hρe
tk −τj tk γτ
ρeγθτj s−t0 Vj θ
n t−τj t
j
1
γθ−t0 e
tk −τj tk 1
•
1 − τ j s
dθ
3.18
Vj θdθ.
Hence,
V ∗ t ≤ V ∗ tk γ λ
t
V ∗ sds hρeγτ
tk
n t−τj t
tk −τj tk j
1
eγs−t0 Vj sds
3.19
t ∈ tk , tk1 , k 0, 1, 2, . . . .
By induction argument, we reach
∗
∗
V tk ≤ V tk−1 γ λ
tk
∗
V sds hρe
γτ
tk−1
n tk −τj tk j
1
tk−1 −τj tk−1 eγs−t0 Vj sds,
..
.
V ∗ t2 ≤ V ∗ t1 γ λ
t2
V ∗ sds hρeγτ
t1
V ∗ t1 ≤ V ∗ t0 γ λ
t1
n t2 −τj t2 t1 −τj t1 j
1
V ∗ sds hρeγτ
t0
n t1 −τj t1 t0 −τj t0 j
1
eγs−t0 Vj sds,
eγs−t0 Vj sds.
Therefore,
∗
∗
V t ≤ V t0 γ λ
t
∗
V sds hρe
γτ
t0
∗
≤ V t0 γ λ
t
t0
n t−τj t
j
1
∗
V sds hρe
γτ
t0 −τj t0 n t
j
1
t0 −τj t0 eγs−t0 Vj sds
eγs−t0 Vj sds
3.20
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Journal of Applied Mathematics
V ∗ t0 γ λ hρeγτ
t
V ∗ sds
t0
hρeγτ
n t0
j
1
t0 −τj t0 eγs−t0 Vj sds t ∈ tk , tk1 , k 0, 1, 2, . . . .
3.21
Since
hρeγτ
n t0
j
1
t0 −τj t0 hρeγτ
t0
t0 −τ
eγs−t0 Vj sds ≤ hρeγτ
n t0
j
1
⎛
⎝
Vj sds
3.22
⎞
n j
1
t0 −τ
Ω
2
ϕ2j s, xdx⎠ds ≤ τhρeγτ ϕΩ ,
we claim
2 V ∗ t ≤ V ∗ t0 τhρeγτ ϕΩ γ λ hρeγτ
t
V ∗ sds
t ∈ tk , tk1 , k 0, 1, 2 . . . .
t0
3.23
According to Lemma 2.2, we claim
∗
V t ≤
2
V t0 τhρe ϕ Ω exp γ λ hρeγτ t − t0 ,
∗
γτ t ≥ t0
3.24
which reduces to
u t, x; t0 , ϕ ≤ 1 τhρeγτ ϕ exp
Ω
Ω
λ hρeγτ
t − t0 ,
2
t ≥ t0 .
3.25
This completes the proof.
Remark 3.2. According to the conditions of Theorem 3.1, we see that the reaction-diffusion
term do influence the stability of problem 2.5–2.8. Moreover, besides the reactiondiffusion coefficients, the dimension of the space and the boundary of spatial variables have
also an effect on the stability of the equilibrium point u 0.
Theorem 3.3. Providing that
2
1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exist constants β such that |x|2 m
s
1 xs < β,
in addition, there exists constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote
Dm − 22 /2β χ,
√
√
2 Pik ui tk , x −θik ui tk , x, 1 − 1 α ≤ θik ≤ 1 1 α, α ≥ 0,
3 infk
1,2... tk − tk−1 > μ,
Journal of Applied Mathematics
11
4 there exists constant γ which satisfies γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0,
where λ maxi
1,...,n −χ − 2ai nj
1 bij2 cij2 ρ and ρ n maxi
1,...,n li 2 , then,
the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with
convergence rate –1/2λ hρeγτ ln1 α/μ.
Proof. Define a Lyapunov function V of the form V t ni
1 Vi t, where Vi t Ω u2i t, xdx.
Obviously, V t is a piecewise continuous function with points of discontinuity of the first
kind tk , k 1, 2, . . ., where it is continuous from the left, that is, V1 tk −0 V1 tk k 1, 2, . . ..
Furthermore, for t tk k 0, 1, 2, . . ., it follows from condition 2 of Theorem 3.3 that
u2i tk 0, x − u2i tk , x 1 − θik 2 u2i tk , x − u2i tk , x ≤ αu2i tk , x.
3.26
V tk 0 ≤ αV tk V tk , k 0, 1, 2, . . . .
3.27
Thereby,
Construct another Lyapunov function defined by V ∗ t eγt−t0 V t, where γ satisfies
γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0. Then, V ∗ t is also a piecewise continuous
function with points of discontinuity of the first kind tk , k 1, 2, . . ., where it is continuous
from the left, that is V ∗ tk − 0 V ∗ tk k 1, 2, . . .. And for t tk k 0, 1, 2, . . ., it results
from 3.27 that
V ∗ tk 0 ≤ αV ∗ tk V ∗ tk ,
k 0, 1, 2, . . . .
3.28
Set t ∈ tk , tk1 , k 0, 1, 2, . . . . Following the same procedure as in Theorem 3.1, we
get
∗
∗
V t ≤ V tk 0 γ λ
t
V ∗ sds hρeγτ
tk
×
n t−τj t
tk −τj tk j
1
eγθ−t0 Vj θdθ
t ∈ tk , tk1 , k 0, 1, 2, . . . .
3.29
The relations 3.28 and 3.29 yield
V ∗ t − V ∗ tk ≤ αV ∗ tk γ λ
t
V ∗ sds hρeγτ
tk
×
n t−τj t
j
1
tk −τj tk eγθ−t0 Vj θdθ
t ∈ tk , tk1 , k 0, 1, 2, . . . .
3.30
12
Journal of Applied Mathematics
By induction argument, we arrive at
∗
∗
∗
V tk − V tk−1 ≤ αV tk−1 γ λ
tk
∗
V sds hρe
γτ
tk−1
n tk −τj tk j
1
tk−1 −τj tk−1 eγθ−t0 Vj θdθ,
..
.
∗
∗
∗
t2
V t2 − V t1 ≤ αV t1 γ λ
∗
V sds hρe
γτ
n t2 −τj t2 t1
t1
V ∗ t1 − V ∗ t0 ≤ αV ∗ t0 γ λ
j
1
V ∗ sds hρeγτ
t1 −τj t1 n t1 −τj t1 t0
j
1
t0 −τj t0 eγθ−t0 Vj θdθ,
eγθ−t0 Vj θdθ.
3.31
Hence,
V ∗ t − V ∗ t0 ≤ αV ∗ t0 γ λ
t
V ∗ sds
t0
hρeγτ
n t−τj t
j
1
t0 −τj t0 eγθ−t0 Vj θdθ α
V tk t0 <tk <t
∗
≤ αV t0 γ λ hρe
γτ
t
V ∗ sds
t0
n t0
hρeγτ
j
1
t0 −τj t0 eγθ−t0 Vj θdθ α
V tk t ∈ tk , tk1 , k 0, 1, 2, . . . .
t0 <tk <t
3.32
2
t
Introducing hρeγτ nj
1 t00 −τj t0 eγθ−t0 Vj θdθ ≤ τhρeγτ ϕΩ as shown in the proof of
Theorem 3.1 into 3.32, the expression becomes
2 V ∗ t − V ∗ t0 ≤ αV ∗ t0 τhρeγτ ϕΩ γ λ hρeγτ
α
V tk t
V ∗ sds
t0
3.33
t ∈ tk , tk1 , k 0, 1, 2 . . . .
t0 <tk <t
It then results from Lemma 2.2 that
V ∗ t ≤
2
α 1V ∗ t0 τhρeγτ ϕΩ
2
α 1V ∗ t0 τhρeγτ ϕ
Ω
1 α exp γ λ hρeγτ t − t0 t0 <tk <t
1 αk exp γ λ hρeγτ t − t0 ,
3.34
t ≥ t0 .
Journal of Applied Mathematics
13
On the other hand, since infk
1,2,... tk − tk−1 > μ, one has k < tk − t0 /μ. Thereby,
1 αk < exp
ln1 α
tk − t0 μ
< exp
ln1 α
t − t0 .
μ
3.35
And 3.34 can be rewritten as
V ∗ t ≤
2
ln1 α
γ λ hρeγτ α 1V ∗ t0 τhρeγτ ϕΩ exp
t − t0 μ
3.36
which implies
1
ln1 α
γτ
γτ ϕ exp
u t, x; t0 , ϕ ≤
α
1
τhρe
λ
hρe
−
t
,
t
0
Ω
Ω
2
μ
t ≥ t0 .
3.37
The proof is completed.
Remark 3.4. Theorem 3.1 is in fact the special case of Theorem 3.3 by choosing α 0. Due to
Lemma 2.4, we know
n
n ui t, xf uj t, x dx ≤
2 bij
j
1
n
2 cij
j
1
Ω
Ω
ui t, xf uj t − τj , x dx ≤
j
1
Ω
n j
1
ε1 bij2 u2i t, x
Ω
ε2 cij2 u2i t, x lj2
ε1
u2i t, x
lj2
ε2
dx,
u2i t − τj , x dx
3.38
hold for any ε1 , ε2 > 0.
In the sequel, we follow the same procedures as in Theorems 3.1 and 3.3 to find the
following theorems.
Theorem 3.5. Provided that
2
1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m
s
1 xs < β.
in addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote
Dm − 22 /2β χ,
2 Pik ui tk , x −θik ui tk , x, 0 ≤ θik ≤ 2,
3 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ < 0,
where λ maxi
1,...,n −χ − 2ai nj
1 ε1 bij2 ε2 cij2 n/ε1 maxi
1,...,n li2 and ρ n/ε2 maxi
1,...,n li2 , then, the equilibrium point u 0 of problem (2.5–2.8) is globally
exponentially stable with convergence rate –λ hρeγτ /2.
Theorem 3.6. Assume that
2
1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m
s
1 xs < β;
In addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote
Dm − 22 /2β χ,
14
Journal of Applied Mathematics
2 Pik ui tk , x −θik ui tk , x, 1 −
√
1 α ≤ θik ≤ 1 √
1 α, α ≥ 0,
3 infk
1,2,... tk − tk−1 > μ,
4 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0, where λ maxi
1,...,n −χ − 2ai nj
1 ε1 bij2 ε2 cij2 n/ε1 maxi
1,...,n li2 and ρ n/ε2 maxi
1,...,n li2 .
Then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with
convergence rate −1/2λ hρeγτ ln1 α/μ.
2
< α−1/2 and α ≥ 1,
Further, on the condition that |Pik ui tk , x| ≤ θik |ui tk , x|, where θik
we obtain, for t tk (k 1, 2, . . .),
u2i tk 0, x − u2i tk , x Pik ui tk , x ui tk , x2 − u2i tk , x
≤ 2ui tk , x2 2Pik ui tk , x2 − u2i tk , x
≤ 2 2θ2ik ui tk , x2 − u2i tk , x ≤ αu2i tk , x.
3.39
Identical with the proof of Theorem 3.3, we present the theorem as follows.
Theorem 3.7. Assume that
2
1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m
s
1 xs < β.
in addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0. Denote
Dm − 22 /2β χ,
2
2 |Pik ui tk , x| ≤ θik |ui tk , x|, where θik
≤ α − 1/2 and α ≥ 1,
3 infk
1,2,... tk − tk−1 > μ,
4 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0, where λ maxi
1,...,n −χ − 2ai nj
1 ε1 b2 ij ε2 cij2 n/ε1 maxi
1,...,n li 2 and ρ n/ε2 maxi
1,...,n li 2 .
Then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with
convergence rate −1/2λ hρeγτ ln1 α/μ.
Remark 3.8. Different from Theorems 3.1–3.6, the impulsive part in Theorem 3.7 could be
nonlinear and this will be of more applicability. Actually, Theorems 3.1–3.6 can be regarded
as the special cases of Theorem 3.7.
4. Examples
Example 4.1. Consider the following impulsive reaction-diffusion delayed neural network:
m
n
n
∂ui t, x ∂
∂ui t, x
Dis
− ai ui t, x bij fj uj t, x cij fj uj t − τj t, x
∂t
∂xs
∂xs
s
1
j
1
j
1
t ≥ 0,
t/
tk ,
x ∈ Ω,
k 1, 2, . . . ,
i 1, . . . , n
4.1
Journal of Applied Mathematics
15
with the impulsive effects characterized by
u1 tk 0, x u1 tk , x 1.343u1 tk , x,
u2 tk 0, x u2 tk , x 1.343u2 tk , x
k 1, 2, . . . ,
x∈Ω
4.2
and initial condition 2.3 and Dirichlet condition 2.4,
n 2, m 4, Ω 1.2 where
T 4
2
1.3
2.3
2.5
3.1
{x1 , . . . , x4 | i
1 xi < 4}, a1 a2 6.5, Dis 2×4 1.8 3.2 2.7 3.4 , bij 2×2 −0.23
−0.1 3 ,
−0.1 −0.2 •
cij 2×2 0.1 −0.3 , fj uj 1/4|uj 1| − |uj − 1|, 0 ≤ τj t ≤ 0.5, and τ j t < 0. For
β 4 and D 1.2, we compute χ 0.6. This, together with the chosen li 1/2, yields
⎛
λ max ⎝−χ − 2ai i
1,...,n
n ⎞
bij2 cij2 ⎠ n max li2 −4,
j
1
i
1,...,n
1
ρ n max li2 .
i
1,...,n
2
4.3
By selecting γ 2.6, τ 0.5 and h 1, we estimate
1
γ λ hρeγτ 2.6 − 4 e1.3 > 0,
2
1
λ hρeγτ −4 e1.3 < 0.
2
4.4
According to Theorem 3.1, we therefore conclude that the system in Example 4.1 is
globally exponential stable.
Example 4.2. Consider the following impulsive reaction-diffusion delayed neural network:
m
n
n
∂ui t, x ∂
∂ui t, x
Dis
− ai ui t, x bij fj uj t, x cij fj uj t − τj t, x
∂t
∂x
∂x
s
s
s
1
j
1
j
1
t ≥ 0,
t/
tk ,
x ∈ Ω,
k 1, 2, . . . ,
i 1, . . . , n
4.5
with the impulsive effects featured by
u1 tk 0, x u1 tk , x arctan0.5u1 tk , x,
u2 tk 0, x u2 tk , x arctan0.5u2 tk , x
k 1, 2, . . . , x ∈ Ω
4.6
and initial condition 2.3 and Dirichlet condition 2.4,
n 2, m 4, Ω where
−0.23 1.3 ,
2.3 2.5 3.1 , b {x1 , . . . , x4 T | 4i
1 xi2 < 4}, a1 a2 6.5, Dis 2×4 1.2
ij 2×2 −0.1 3
1.8 3.2 2.7 3.4
−0.1 −0.2 •
τ
cij 2×2 u
1/4|u
1|
−
|u
−
1|,
0
≤
τ
t
≤
0.5,
t
<
0, and
,
f
j
j
j
j
j
j
0.1 −0.3
16
Journal of Applied Mathematics
infk
1,2,... tk − tk−1 > 1. For β 4 and D 1.2, we computeχ 0.6. This, together with
li 1/2 and ε1 ε2 1, yields
1
n
ρ
max li2 ,
ε2 i
1,...,n
2
⎛
λ max ⎝−χ − 2ai n i
1,...,n
j
1
⎞
n
ε1 bij2 ε2 cij2 ⎠ max li2 −4.
ε1 i
1,...,n
4.7
Select α 1.5 by setting θik 0.5. Hence, we compute by letting μ 1, γ 3, τ 0.5,
and h 1 that
1
γ λ hρeγτ 3 − 4 e1.5 > 0,
2
λ hρeγτ 1
ln1 α
−4 e1.5 ln 2.5 < 0.
μ
2
4.8
It is then concluded from Theorem 3.7 that the system in Example 4.2 is globally
exponentially stable.
Acknowledgment
The work is supported by National Natural Science Foundation of China under Grant
60904028.
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