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Global Exponential Stability of Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales
Journal of Inequalities and Applications volume 2009, Article number: 491268 (2009)
Abstract
Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional method, and some analysis techniques, sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg bidirectional associative memory (BAM) neural networks with distributed delays and impulses on time scales. This is the first time applying the time-scale calculus theory to unify the discrete-time and continuous-time Cohen-Grossberg BAM neural network with impulses under the same framework.
1. Introduction
In the recent years, bidirectional associative memory (BAM) neural networks and Cohen-Grossberg neural networks (CGNNs) with their various generalizations have attracted the attention of many mathematicians, physicists, and computer scientists (see [1–17]) due to their wide range of applications in, for example, pattern recognition, associative memory, and combinatorial optimization. Particularly, as discussed in [18–20], in the hardware implementation of the neural networks, when communication and response of neurons happens time delays may occur. Actually, time delays are known to be a possible source of instability in many real-world systems in engineering, biology, and so forth. (see, e.g., [21] and references therein). However, besides delay effect, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving fields such as medicine and biology, economics, mechanics, electronics, and telecommunications. As artificial electronic systems, neural networks such as Hopfield neural networks, bidirectional neural networks, and recurrent neural networks often are subject to impulsive perturbations which can affect dynamical behaviors of the systems just as time delays. Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks.
As is well known, both continuous and discrete systems are very important in implementation and applications. However, it is troublesome to study the stability for continuous and discrete systems, respectively. Therefore, it is worth studying a new method, such as the time-scale theory, which can unify the continuous and discrete situations.
Motivated by the above discussions, the objective of this paper is to study the global exponential stability of the following Cohen-Grossberg bidirectional associative memory networks with impulses and time delays on time scales:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ1_HTML.gif)
where is a time scale;
are continuous,
,
are the states of the
th neuron from the neural field
and the
th neuron from the neural field
at time
, respectively;
denote the activation functions of the
th neuron from
and the
th neuron from
, respectively;
and
are constants, which denote the external inputs on the
th neuron from
and the
th neuron from
, respectively;
and
correspond to the transmission delays;
and
represent amplification functions;
and
are appropriately behaved functions such that the solutions of system (1.1) remain bounded;
and
denote the connection strengths which correspond to the neuronal gains associated with the neuronal activations;
and
denote the external inputs. For each interval
of
, we denote that by
are the impulses at moments
, and
represent the right and left limits of
and
in the sense of time scales;
is a strictly increasing sequence.
The system (1.1) is supplement with initial values given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ2_HTML.gif)
where ,
are continuous real-valued functions defined on their respective domains.
As usual in the theory of impulsive differential equations, at the points of discontinuity of the solution
we assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ3_HTML.gif)
for .
The organization of the rest of this paper is as follows. In Section 2, we introduce some notations and definitions, and state some preliminary results which are needed in later sections. In Section 3, by means of homeomorphism theory, we study the existence and uniqueness of the equilibrium point of system (1.1). In Section 4, by constructing a suitable Lyapunov function, we establish the exponential stability of the equilibrium of (1.1). In Section 5, we present an example to illustrate the feasibility and effectiveness of our results obtained in previous sections.
2. Preliminaries
In this section, we will cite some definitions and lemmas which will be used in the proofs of our main results.
Let be a nonempty closed subset (time scale) of
. The forward and backward jump operators
and the graininess
:
are defined, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ4_HTML.gif)
A point is called left dense if
and
, left scattered if
, right dense if
and
, and right scattered if
. If
has a left-scattered maximum
, then
; otherwise
. If
has a right-scattered minimum
, then
; otherwise
.
A function is right dense continuous provided that it is continuous at right dense point in
and its left-side limits exist at left-dense points in
. If
is continuous at each right dense point and each left-dense point, then
is said to be a continuous function on
. The set of continuous functions
will be denoted by
.
For and
, we define the delta derivative of
to be the number (if it exists) with the property that for a given
, there exists a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ5_HTML.gif)
for all
If is continuous, then
is right dense continuous, and
is delta differentiable at
, then
is continuous at
.
Let be right dense continuous. If
, then we define the delta integral by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ6_HTML.gif)
Definition 2.1 (see [22]).
For each , let
be a neighborhood of
, then, for
, define
to mean that, given
, there exists a right neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ7_HTML.gif)
for each ,
, where
. If
is rd and
is continuous at
, this reduce to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ8_HTML.gif)
Definition 2.2 (see [23]).
If , and
is right dense continuous on
, then we define the improper integral by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ9_HTML.gif)
provided that this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges.
A function is called regressive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ10_HTML.gif)
for all .
If is regressive function, then the generalized exponential function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ11_HTML.gif)
with the cylinder transformation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ12_HTML.gif)
Let be two regressive functions, then we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ13_HTML.gif)
Then the generalized exponential function has the following properties.
Lemma 2.3 (see [24]).
Assume that are two regressive functions, then
(i) and
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii).
Definition 2.4.
The equilibrium point of system(1.1) is said to beexponentially stable if there exists a positive constant
such that for every
, there exists
such that the solution
of (1.1) with initial value
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ14_HTML.gif)
Lemma 2.5 (see [25]).
If satisfies the following conditions:
(i) is injective on
,
(ii) as
,
then is a homeomorphism of
onto itself.
For , we define the norm as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ15_HTML.gif)
Throughout this paper, we assume that
(), and satisfy
;
()the activation functions and there exist positive constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ16_HTML.gif)
for all ;
() and there exist positive constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ17_HTML.gif)
()the kernels and
defined on
are nonnegative continuous integral functions such that
,
3. Existence and Uniqueness of the Equilibrium
In this section, using homeomorphism theory, we will study the existence and uniqueness of the equilibrium point of system (1.1).
An equilibrium point of (1.1) is a constant vector which satisfies the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ18_HTML.gif)
where the impulsive jumps satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ19_HTML.gif)
From the assumptions and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ20_HTML.gif)
Noting that if exist and activation functions
and
are bounded, then the existence of an equilibrium point of system (1.1) is easily obtained from Brouwer's fixed point theorem. We can refer to [2–8].
Theorem 3.1.
Assume that and
hold. Suppose further that for each
, the following inequalities are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ21_HTML.gif)
Then there exists a unique equilibrium point of system (1.1).
Proof.
Consider a mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ22_HTML.gif)
where . First, we want to show that
is an injective mapping on
. By contradiction, suppose that there exists a distinct
such that
, where
and
. Then it follows from (3.5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ23_HTML.gif)
In view of -
and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ24_HTML.gif)
Thus, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ25_HTML.gif)
It follows from (3.4) and (3.8) that and
,
That is
, which leads to a contradiction. Therefore,
is an injective on
.
Then we will prove is a homeomorphism on
. For convenience, we let
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ26_HTML.gif)
We assert that as
. Otherwise there is a sequence
such that
and
is bounded as
, where
. Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ27_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ28_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ29_HTML.gif)
It follows from (3.11) and (3.12) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ30_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ31_HTML.gif)
That is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ32_HTML.gif)
which contradicts our choice of . Hence,
satisfies
as
. By Lemma 2.5,
is a homeomorphism on
and there exists a unique point
such that
. From the definition of
, we know that
is the unique equilibrium point of (1.1).
4. Global Exponential Stability of the Equilibrium
In this section, we will construct some suitable Lyapunov functions to derive the sufficient conditions which ensure that the equilibrium of (1.1) is globally exponentially stable.
Theorem 4.1.
Assume that ()–(
) hold, suppose further that
()for each , the following inequalities are satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ33_HTML.gif)
()the impulsive operators and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ34_HTML.gif)
Then the unique equilibrium point of system (1.1) is globally exponentially stable.
Proof.
According to Theorem 3.1, we know that (1.1) has a unique equilibrium point . In view of
, it is easy to see that
and
. Suppose that
is an arbitrary solution of (1.1). Let
, then system (1.1) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ35_HTML.gif)
where, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ36_HTML.gif)
Also, forall ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ37_HTML.gif)
Hence by and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ39_HTML.gif)
Also, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ40_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ41_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ42_HTML.gif)
Let and
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ43_HTML.gif)
respectively, where . By
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ44_HTML.gif)
Since are continuous on
and
, as
, there exist
such that
and
, for
, for
. By choosing
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ45_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ47_HTML.gif)
where . For
it follows from (4.6)–(4.15), we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ48_HTML.gif)
Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ49_HTML.gif)
Define a Lyapunov function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ50_HTML.gif)
And we note that for
and
. Calculating the
-derivatives of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ51_HTML.gif)
Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ52_HTML.gif)
It follows that for
and hence, for
, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ53_HTML.gif)
In view of (4.14)-(4.15) and the previous inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ55_HTML.gif)
The proof is complete.
5. An Example
In this section, we give an example to illustrate our results.
Consider the following Cohen-Grossberg BAM neural networks system with distributed delays and impulses:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491268/MediaObjects/13660_2009_Article_1962_Equ56_HTML.gif)
where . A simple computation shows that
. It is easy to check that all conditions of Theorems 3.1 and 4.1 are satisfied. Hence, (5.1) has a unique equilibrium point, which is globally exponentially stable.
6. Conclusion
Using the time-scale calculus theory, the homeomorphism theory and the Lyapunov functional method, some sufficient conditions are obtained to ensure the existence and the global exponential stability of the unique equilibrium point of Cohen-Grossberg BAM neural networks with distributed delays and impulses on time scales. This is the first time applying the time-scale calculus theory to unify and improve impulsive Cohen-Grossberg BAM neural networks with distributed delays on time scales under the same framework. The sufficient conditions we obtained can easily be checked in practice by simple algebraic methods.
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Acknowledgments
This work was supported by the National Natural Sciences Foundation of People's Republic of China and the Natural Sciences Foundation of Yunnan Province under Grant 04Y239A.
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Li, Y., Hua, Y. & Fei, Y. Global Exponential Stability of Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales. J Inequal Appl 2009, 491268 (2009). https://doi.org/10.1155/2009/491268
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DOI: https://doi.org/10.1155/2009/491268