Dynamical Analysis of DTNN with Impulsive Effect

Abstract

We present dynamical analysis of discrete-time delayed neural networks with impulsive effect. Under impulsive effect, we derive some new criteria for the invariance and attractivity of discrete-time neural networks by using decomposition approach and delay difference inequalities. Our results improve or extend the existing ones.

Full text

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Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 201068, 12 pages
doi:10.1155/2009/201068
Research Article
Dynamical Analysis of DTNN with Impulsive Effect
Chao Chen, Zhenkun Huang, Honghua Bin, and Xiaohui Liu
School of Sciences, Jimei University, Xiamen 361021, China
Correspondence should be addressed to Chao Chen, firedoctor@163.com
Received 11 March 2009; Accepted 30 September 2009
Recommended by Yong Zhou
We present dynamical analysis of discrete-time delayed neural networks with impulsive effect.
Under impulsive effect, we derive some new criteria for the invariance and attractivity of discrete-
time neural networks by using decomposition approach and delay difference inequalities. Our
results improve or extend the existing ones.
Copyright q 2009 Chao Chen 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.
1. Introduction
As we know, the mathematical model of neural network consists of four basic components:
an input vector, a set of synaptic weights, summing function with an activation, or
transfer function, and an output. From the view point of mathematics, an artificial neural
network corresponds to a nonlinear transformation of some inputs into certain outputs.
Due to their promising potential for tasks of classification, associative memory, parallel
computation and solving optimization problems, neural networks architectures have been
extensively researched and developed ?1–25?. Most of neural models can be classified as
either continuous-time or discrete-time ones. For relative works, we can refer to ?20, 24, 26–
28?.
However, besides the delay effect, an impulsive effect likewise exists in a wide variety
of evolutionary process, in which states are changed abruptly at certain moments of time
?5, 29?. As is well known, stability is one of the major problems encountered in applications,
and has attracted considerable attention due to its important role in applications. However,
under impulsive perturbation, an equilibrium point does not exist in many physical systems,
especially, in nonlinear dynamical systems. Therefore, an interesting subject is to discuss the
invariant sets and the attracting sets of impulsive systems. Some significant progress has been
made in the techniques and methods of determining the invariant sets and attracting sets for
delaydifferenceequationswithdiscretevariables,delaydifferentialequations,andimpulsive

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2Discrete Dynamics in Nature and Society
functional differential equations ?30–37?. Unfortunately, the corresponding problems for
discrete-timeneuralnetworkswithimpulsesanddelayshavenotbeenconsidered.Motivated
by the above-mentioned papers and discussion, we here make a first attempt to arrive
at results on the invariant sets and attracting sets of discrete-time neural networks with
impulses and delays.
2. Preliminaries
In this paper, we consider the following discrete-time networks under impulsive effect:
xi?k? ? e−aihxi?k − 1? ?1 − e−aih
ai
n ?
??,
j?1
bijfj
?xj?k − τ???1 − e−aih
i ? 1,2,...,n, l ? 1,2,...,
ai
ci,k ≥ k0, k/ ?kl,
xi?kl? ? Iil
?xi
?k−
l
?2.1?
where bij,ci?i,j ? 1,2,...,n? are real constants, ai?i ? 1,2,...,n?; h, τ are positive real
numbers such that τ > 1. kl?l ? 1,2,...? is an impulsive sequence such that k1< k2< · · · <
kl< · · · and liml→∞kl? ∞.fj,Iil: R → R are real-valued functions.
By a solution of ?2.1?, we mean a piecewise continues real-valued function xi?k?
defined on the interval ?k0− τ,∞? which satisfies ?2.1? for all k ≥ k0.
In the sequel, by Φiwe will denote the set of all continuous real-valued function xi?k?
defined on the interval ?k0− τ,∞?, which satisfies the compatibility condition:
φi?k0? ? e−aihφ?k0− 1? ?
n ?
j?1
bijfj
?φj?k − τ???1 − e−aih
ai
ci.
?2.2?
By the method of steps, one can easily see that, for any given initial function φi∈ Φi,
thereexistsaunique solutionxi?k? ?i ? 1,2,...,n?of ?2.1?whichsatisfiestheinitialcondition:
xi?k? ? φi?k?,
for k ∈ ?k0− τ,k0?,
?2.3?
this function will be called the solution of the initial problem ?2.1?–?2.3?.
For convenience, we rewrite ?2.1? and ?2.3? into the following vector form
x?k? ? Ax?k − 1? ? Bf?x?k − τ?? ? C,
x?kl? ? Il
k ≥ k0, k/ ?kl,
?x?k−
l
??,
k ∈ ?k0− τ,k0?,
l ? 1,2,...,
x?k? ? φ?k?,
?2.4?
where x?k?
e−aih/ai?bij?n×n, C ? diag{??1−e−a1h?/a1?c1,??1−e−a2h?/a2?c2,...,??1−e−anh?/an?cn}, f?x? ?
?f1?x?,f2?x?,...,fn?x??T, Il?x? ? ?I1l?x?,I2l?x?,...,Inl?x??T, and φ ? ?φ1,φ2,...,φn?T∈ Φ, in
which Φ ? ?Φ1,Φ2,...,Φn?T.
In what follows, we will introduce some notations and basic definitions.
??x1?k?,x2?k?,...,xn?k??T, A
? diag{e−a1h,e−a2h,...,e−anh}, B
???1 −

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Discrete Dynamics in Nature and Society3
Let Rnbe the space of n-dimensional real column vectors and let Rm×ndenote the
set of m × n real matrices. E denotes an identical matrix with appropriate dimensions. For
A,B ∈ Rm×nor A,B ∈ Rn,A ≥ B ?A > B? means that each pair of corresponding elements of
A and B satisfies the inequality ≥ ?>?. Particularly, A is called a nonnegative matrix if A ≥ 0
and is denoted by A ∈ Rm×n
radius of A.
C?X,Y? denotes the space of continuous mappings from the topological space X to
the topological space Y.
PC?I,Rn? ? {ϕ : I → Rn| ϕ?k?? and ϕ?k−? exist for k ∈ I,ϕ?k?? ? ϕ?k? for k ∈ I and
ϕ?k−? ? ϕ?k? except for points kl∈ I}, where I ⊂ R is an interval, ϕ?k?? and ϕ?k−? denote the
right limit and left limit of function ϕ?k?, respectively. Especially, let PC ? PC??k0−τ,k0?,Rn?.
Definition 2.1. The set S ⊂ Rnis called a positive invariant set of ?2.4?, if for any initial value
φ ∈ S, one has the solution x?k? ∈ S for k ≥ k0.
Definition 2.2. The set S ⊂ Rnis called a global attracting set of ?2.4?, if for any initial value
φ ∈ PC, the solution x?k? converges to S as k → ?∞. That is,
?
and z is called a positive vector if z > 0. ρ?A? denotes the spectral
dist?x?k?,S? −→ 0,
as k −→ ?∞,
?2.5?
where dist?x,S? ? infy∈Sd?x,y?;?x,y? ? supk0−τ≤k|x?k?−y?k?|. In particular, S ? {0} is called
asymptotically stable.
Following ?33?, we split the matrices A,B into two parts, respectively,
A ? A?− A−,B ? B?− B−,C ? C?− C−
?2.6?
with a?
e−aih?/ai?bij,0}, c?
Then the first equation of ?2.4? can be rewritten as
i? max{ai,0}, a−
i? max{−ai,0}, b?
i? max{??1 − e−aih?/ai?ci,0}, c−
ij? max{?1 − e−aih/ai?bij,0}, b−
i? max{−??1 − e−aih?/ai?ci,0}.
ij? max{−??1 −
x?k? ??A?− A−?x?k − 1? ??B?− B−?f?x?k − τ?? ??C?− C−?.
?2.7?
Now take the symmetric transformation y ? −x. From ?2.7?, it follows that
x?k? ? A?x?k − 1? ? A−y?k − 1? ? B?f?x?k − τ?? ? B−g?y?k − τ??? ?C?− C−?,
y?k? ? A?y?k − 1? ? A−x?k − 1? ? B?g?y?k − τ??? B−f?x?k − τ?? ? ?C−− C??,
?2.8?
where f?−u? ? −g?u?.

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4Discrete Dynamics in Nature and Society
Set
z?k? ?
?x?k?
y?k?
?
?
,h?z?k?? ?
?f?x?k??
g?y?k??
C ?
?
,
A ?
?A?A−
A−A?
,
B ?
?B?B−
B−B?
?
,
?C?−C−
C−−C?
?
,
?2.9?
By virtue of ?2.8? and ?2.7?, we have
z?k? ? Az?k − 1? ? Bh?z?k − τ?? ? C.
?2.10?
Set Il?−v? ? −Jl?v?, in view of the impulsive part of ?2.4?, we also have x?kl? ?
Il?x?k−
l??, y?kl? ? Jl?y?k−
l??, and so we have
z?kl? ? ωl
?z?k−
l
??,l ? 1,2,...,
?2.11?
where ωl?z? ? ?Il?x?T,Jl?y?T?T.
Lemma 2.3 ?see ?34??. Suppose that M ∈ Rn×n
such that
?
and ρ?M? < 1, then there exists a positive vector z
?E − M?z > 0.
?2.12?
For M ∈ Rn×n
?
and ρ?M? < 1, one denotes
Ωρ?M? ? {z ∈ Rn| ?E − M?z > 0, z > 0}.
?2.13?
By Lemma 2.3, we have the following result.
Lemma 2.4. Ωρ?M? is nonempty, and for any scalars k1> 0,k2> 0 and vectors z1,z2∈ Ωρ?M?,
one has
k1z1? k2z2∈ Ωρ?M?.
?2.14?
Lemma 2.5. Assume that u?k? ? ?u1?k?,u2?k?,...,un?k??T∈ C??k0,∞?,Rn? satisfy
u?k? ≤ Mu?k − 1? ? Nu?k − τ? ? J,
u?θ? ∈ PC,
k ≥ k0,
θ ∈ ?k0− τ,k0?,
?2.15?
where M ? ?mij? ∈ Rn×n
If ρ?M ? N? < 1, then there exists a positive vector v ? ?v1,v2,...,vn?Tsuch that
?
,N ? ?nij? ∈ Rn×n
?
, J ∈ Rn.
u?k? ≤ ve−λ?k−k0?? ?E − M − N?−1J,k ≥ k0,
?2.16?

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Discrete Dynamics in Nature and Society5
where λ > 0 is a constant and defined as
?
E − Meλ− Neλτ?
v ≥ 0?2.17?
for the given v.
Proof. Since M,N ∈ Rn×n
p ∈ Ωρ?M ? N? such that ?E − M − N?p > 0.
Set Hi?λ? ? pi−?n
?
and ρ?M ? N? < 1, by Lemma 2.3, there exists a positive vector
j?1?mijeλ? nijeλτ?pj?i ? 1,2,...,n?, then we have
˙ Hi?λ? ? −
n ?
j?1
?
mijeλ? nijτeλτ?
pj< 0.
?2.18?
Due to
Hi?0? ? pi−
n ?
j?1
?mij? nij
?pj> 0,
?2.19?
there must exist a λ > 0, such that
n ?
j?1
?
mijeλ? nijeλτ?
pj≤ pi,i ? 1,2,...,n.
?2.20?
For u?θ? ∈ PC, θ ∈ ?k0− τ,k0?, there exists a positive constant l > 1 such that
u?θ? ≤ lpe−λ?θ−k0?? W,θ ∈ ?k0− τ,k0?,
?2.21?
where W ? ?E − M − N?−1J.
By Lemma 2.4, lp ∈ Ωρ?M ? N?. Without loss of generality, we can find a v ∈ Ωρ?M ?
N? such that
n ?
j?1
?
mijeλ? nijeλτ?
u?θ? ≤ ve−λ?θ−k0?? W,
vj≤ vi,i ? 1,2,...,n,
?2.22?
θ ∈ ?k0− τ,k0?.
?2.23?
Set u?k? ? v?k?e−λ?k−k0?? W,k ≥ k0, substituting this into ?2.15?, we have
v?k? ≤ Meλv?k − 1? ? Neλτv?k − τ?.
?2.24?
By ?2.23?, we get that
v?θ? ≤ v,θ ∈ ?k0− τ,k0?.
?2.25?