Exponential stability criteria for fuzzy bidirectional associative memory Cohen Grossberg neural networks with mixed delays and impulses He and Chu Advances in Difference Equations (2017) 2017 61 DOI[.]
Trang 1R E S E A R C H Open Access
Exponential stability criteria for fuzzy
bidirectional associative memory
Cohen-Grossberg neural networks with
mixed delays and impulses
Weina He and Longxian Chu*
* Correspondence:
chulongxian_pdsu@126.com
Software College, Pingdingshan
University, Pingdingshan, 467000,
PR China
Abstract
This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses By constructing
an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model The results in this paper extend and complement the previous publications An example is given to illustrate the effectiveness of our obtained results
MSC: 34C20; 34K13; 92B20 Keywords: fuzzy BAM Cohen-Grossberg neural networks; exponential stability;
mixed delays; periodic solution; impulse
1 Introduction
In recent years, considerable attention has been paid to bidirectional associative memory (BAM) Cohen-Grossberg neural networks [] due to their potential applications in var-ious fields such as neural biology, pattern recognition, classification of patterns, parallel computation and so on [–] In real life, numerous application examples appear, for ex-ample, emerging parallel/distributed architectures were explored for the digital VLSI im-plementation of adaptive bidirectional associative memory (BAM) [], Teddy and Ng [] applied a novel local learning model of the pseudo self-evolving cerebellar model articu-lation controller (PSECMAC) associative memory network to produce accurate forecasts
of ATM cash demands Chang et al [] proposed a maximum-likelihood-criterion based
on BAM networks to evaluate the similarity between a template and a matching region
Sudo et al [] proposed a novel associative memory that operated in noisy environments
and performed well in online incremental learning applying self-organizing incremental neural networks On the one hand, the existence and stability of the equilibrium point of BAM Cohen-Grossberg neural networks plays an important role in practical application
On the other hand, time delay is inevitable due to the finite switching speed of amplifiers
in the electronic implementation of analog neural networks, moreover, time delays may have important effect on the stability of neural networks and lead to periodic oscillation,
© The Author(s) 2017 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
Trang 2bifurcation, chaos and so on [, , ] Thus many interesting stability results on BAM
Cohen-Grossberg neural networks with delays have been available [–]
As is well known, numerous dynamical systems of electronic networks, biological neu-ral networks, and engineering fields often undergo abrupt change at certain moments due
to instantaneous perturbations which leads to impulsive effects [, , , –] Many
scholars [, ] think that uncertainty or vagueness often appear in mathematical
model-ing of real world problems, thus it is necessary to take vagueness into consideration Fuzzy
neural networks (FNNs) pay an important role in image processing and pattern recogni-tion [] and some results have been reported on stability and periodicity of FNNs [, –
] Here we would like to point out that most neural networks involve negative feedback
terms and do not possess amplification functions or behaved functions The model (.)
of this paper has amplifications function and behaved functions which differ from most
neural networks with negative feedback term Up to now, there are rare papers that
con-sider exponential stability of this kind of fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses
Inspired by the discussion above, in this paper, we are to consider the following fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays
and impulses,
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
˙x i (t) = ι i (x i (t))[–a i (t, x i (t)) +m
j=c ji (t)f j (y j (t – τ (t)))
+m
j=α ji (t)t
–∞K ji (t – s)f j (y j (s)) ds +m
j=T ji u j+ m
j=H ji u j
+ m
j=β ji (t)t
–∞K ji (t – s)f j (y j (s)) ds + I i (t)], t = t k , i ∈ ,
x i (t k ) = x i (t k ) – x i (t–
k ) = –γ ik x i (t–
k) +m
j=e ij (t–
k )E j (y j (t–
k – τ )), k∈ Z+,
˙y j (t) = ϑ j (y j (t))[–b j (t, y j (t)) +n
i=d ij (t)g i (x i (t – τ (t)))
+n
i=p ij (t)t
–∞N ij (t – s)g i (x i (s)) ds +n
i=S ij u i+ n
i=L ij u i
+ n
i=q ij (t)t
–∞N ij (t – s)g i (x i (s)) ds + J j (t)], t = t k , j∈ ,
y j (t k ) = y j (t k ) – y j (t–
k ) = –δ jk y i (t–
k) +n
i=h ji (t–
k )H i (x i (t–
k – τ )), k∈ Z+,
(.)
with initial conditions
x i (s) = φ i (s), s ∈ (–∞, ], i ∈ ,
y j (s) = φ i (s), s ∈ (–∞, ], j ∈ , (.) where n and m correspond to the number of neurons in X-layer and Y -layer, respectively.
x i (t) and y j (t) are the activations of the ith neuron and the jth neurons, respectively ι i(·)
and ϑ j(·) are the abstract amplification functions, ai (t, ·) and b j (t,·) stand for the rate
func-tions with which the ith neuron and jth neuron will reset its potential to the resting state
in isolation when disconnected from the network and external inputs; α ji (t), β ji (t), T ji
and H jiare elements of fuzzy feedback MIN template and fuzzy feedback MAX template,
fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in X-layer,
re-spectively; p ij (t), q ij (t), S ij and L ijare elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX
template in Y -layer, respectively;
and denote the fuzzy AND and fuzzy OR operation,
respectively; u j , u i denote external input of the ith neurons in X-layer and external input
of the jth neurons in Y -layer, respectively; I i (t) and J j (t) are external bias of Xlayer and Y -layer, respectively, f j(·) and gi(·) are signal transmission functions, Kji (t) and N ij (t) are delay
Trang 3kernels, = {, , , n}, = {, , , m}, Z+denotes the set of positive integral numbers,
the impulse times t k satisfy = t< t< t<· · · < t k<· · · , limk→∞t k=∞, φ i(·), φ j(·) ∈ C, whereC denotes real-valued continuous functions defined on (–∞, ], τ(t) is the
trans-mission delay such that ≤ τ(t) ≤ τ , τ is a positive constant, e ij (t–
k) represents impulsive
perturbations of the ith unit at time t k , h ji (t–k ) represents impulsive perturbations of the jth unit at time t k , E j (y j (t–
k )) represents impulsive perturbations of the jth unit at time t kand
y j (t–
k ) denotes impulsive perturbations of the jth unit at time t kcaused by the transmission
delays, H i (x i (t–
k )) represents impulsive perturbations of the ith unit at time t k and x i (t–
k)
de-notes impulsive perturbations of the ith unit at time t kwhich caused by the transmission delays For details, see [–]
The main purpose of this paper is to investigate the existence and global exponential sta-bility of a periodic solution of fuzzy BAM Cohen-Grossberg neural networks with mixed
delays and impulses By constructing a suitable Lyapunov function and a new differential inequality, we establish some sufficient conditions to ensure the existence and global
ex-ponential stability of a periodic solution of the model (.) The results obtained in this
paper extend and complement the previous studies in [, ] Two examples are given to
illustrate the effectiveness of our theoretical findings To the best of our knowledge, there
are very few papers that deal with this aspect Therefore we think that the study of the
fuzzy BAM Cohen-Grossberg neural networks with mixed delays and impulses has
im-portant theoretical and practical value Here we shall mention that since the existence of
amplifications function and behaved functions in model (.), thus there are some
diffi-culties in dealing with the exponential stability We will apply some inequality techniques,
meanwhile, the construction of Lyapunov function is a key issue
The remaining part of this paper is organized as follows In Section , the necessary definitions and lemmas are introduced In Section , we present some new sufficient con-ditions to ensure the existence and global exponential stability of a periodic solution of
model (.) In Section , an illustrative example is given to show the effectiveness of the
proposed method A brief conclusion is drawn in Section
2 Preliminaries
LetR denote the set of real number, Rn the n-dimensional real space equipped with the
Euclidean norm| · |, R+the set of positive numbers Denote PC(R, R+) ={φ : R → R n:
φ (t) is continuous for t = t k , φ(t+k ), φ(t k–)∈ Rn and φ(t–k ) = φ(t k)}
Throughout this paper, we make the following assumptions:
(H) For i ∈ , j ∈ , c ji (t), α ji (t), β ji (t), e ij (t), d ij (t), p ij (t), q ij (t), h ji (t), τ (t), I i (t) and J j (t) are all continuously periodic functions defined on t∈ [, ∞) with common period
ω>
(H) For i ∈ , j ∈ , there exist positive constants L f
j , L E j , L g i and L H j such that
f j (u) – f j (v) ≤ L f j |u – v|, E j (u) – E j (v) ≤ L E j |u – v|,
g i (u) – g i (v) ≤ L g i |u – v|, H i (u) – H i (v) ≤ L H j |u – v|
for all u, v∈ R
(H) For i ∈ , j ∈ , ι i(·) and ϑ j(·) are continuous and satisfy ≤ ι i ≤ ι i(·) ≤ ι i,
≤ ϑ ≤ ϑ j(·) ≤ ϑj , where ι , ι i , ϑ , ϑ jare some positive constants
Trang 4(H) For i ∈ , j ∈ , there exist continuous positive ω-periodic functions i (t) and σ j (t)
such that
a i (t, u) – a i (t, v)
u – v ≥ i (t), b j (t, u) – b j (t, v)
u – v ≥ σ j (t) for all u, v∈ R
(H) For i ∈ , j ∈ , the delay kernels K ij(·), Nji(·) ∈ C(R+,R+)are piecewise continuous
and satisfy K ij (s) ≤ ˜K(s) and N ji (s) ≤ ˜K(s) for all s ∈ R+, where ˜K (s) ∈ C(R+,R+) and integrable, satisfying∞
˜K(s)e μs ds<∞, in which the constant μ denotes
some positive number
(H) For i ∈ , j ∈ ω > , there exists q ∈ Z+such that t k + ω = t k +q and γ ik = γ i (k+q),
δ jk = δ j (k+q) , k∈ Z+
(H) For i ∈ , j ∈ , c∗
ji= maxt ∈[,ω] |c ji (t)|, α∗
ji= maxt ∈[,ω] |α ji (t)|, β∗
ji= maxt ∈[,ω] |β ji (t)|,
e∗ij= maxt ∈[,ω] |e ij (t)|, d∗
ij= maxt ∈[,ω] |d ij (t)|, p∗
ij= maxt ∈[,ω] |p ij (t)|,
q∗ij= maxt ∈[,ω] |q ij (t) |, h∗
ji= maxt ∈[,ω] |h ji (t) |,
i = mint ∈[,ω] | i (t)|,
σ
i = mint ∈[,ω] |σ j (t)|.
In this paper, we use the following norm ofRn +m:
n
i=
|x i| +
m
j=
|y j|,
s∈(–∞,]
n
i=
φ i (s) + m
j=
φ j (s)
for u = (x, x, , x n , y, y, , y m)T∈ Rn +m , φ = (φ, φ, , φ n , φ, φ, , φ m)T∈ Cn +m
Lemma .([]) Let x and y be two states of system (.) Then
n
j=
α ij (t)g j (x) –
n
j=
α ij (t)g j (y)
≤
n
j=
α ij (t) g j (x) – g j (y)
and
n
j=
β ij (t)g j (x) –
n
j=
β ij (t)g j (y)
≤
n
j=
β ij (t) g j (x) – g j (y) .
Lemma .([]) Let p, q, r and τ denote nonnegative constants and f ∈ PC(R, R+) satisfies
the scalar impulsive differential inequality
D+f (t) ≤ –pf (t) + q sup t –τ≤s≤t f (s) + rσ
k (s)f (t – s) ds, t = t k , t ≥ t,
f (t k)≤ a k f (t–
k ) + b k f (t–
k – τ ), k∈ Z+, (.)
where < σ ≤ +∞, a k , b k ∈ R, k(·) ∈ PC([, σ ], R+) satisfiesσ
k (s)e ηs ds<∞ for some
positive constant η> in this case when σ = +∞ Moreover, when σ = +∞, the
inter-val [t – σ , t] is understood to be replaced by (– ∞, t] Assume that (i) p > q + rσ
k (s) ds (ii) There exist constant M > , η > such that
n
, a k + b k e λτ
≤ Me η (t n –t ), n∈ Z+,
Trang 5where λ ∈ (, η) satisfies
λ < p – qe λτ – r
σ
k (s)e λs ds
Then
f (t) ≤ Mf (t)e –(λ–η)(t–t ), t ≥ t,
where f (t) = supt–max{σ,τ}f (s).
3 Global exponential stability of the periodic solution
In this section, we will discuss the global exponential stability of the periodic solution for (.)
Theorem . Assume that (H)-(H) hold, then there exists a unique ω-periodic solution
of system (.) which is globally exponentially stable if the following conditions are fulfilled.
(H)
min{
i , σ
j} mini ∈,j∈ {ι i , ϑ j}
maxi ∈,j∈ {ι i , ϑ j}
> max
n
i=
max
j∈ c∗ji L f j,
m
j=
max
i ∈ d∗ij L g i
+ max
n
i=
max
j∈ α
∗
ji L f j,
n
i=
max
j∈ β
∗
ji L f j,
m
j=
max
i ∈ p∗ij L g i,
m
j=
max
i ∈ q∗ij L g i
×
∞
˜K(s)ds.
(H) There exist constants M ≥ , λ ∈ (, λ)and η ∈ (, λ) such that
n
l=max{, χ l } ≤ Me ηt n for all n∈ Z+holds and
λ< min{
i , σ
j} mini ∈,j∈ {ι i , ϑ j}
maxi ∈,j∈ {ι i , ϑ j} – max
n
i=
max
j∈ c∗ji L f j,
m
j=
max
i ∈ d ij∗L g i
e λτ
– max
n
i=
max
j∈ α
∗
ji L f j,
n
i=
max
j∈ β
∗
ji L f j,
m
j=
max
i ∈ p∗ij L g i,
m
j=
max
i ∈ q∗ij L g i
×
∞
˜K(s)ds,
where
χ l=maxi ∈,j∈ {ι i , ϑ j}
mini ∈,j∈ {ι i , ϑ j} imax∈,j∈ |–γ il |, |–δ jl|+max
n
i=
max
j∈ e∗ij L E j,
m
j=
max
i ∈ h∗ji L H i
e λτ
Proof Assume that u(t) = (x(t, φ), x(t, φ), , x n (t, φ), y(t, φ), y(t, φ), , y m (t, φ))Tis
an arbitrary solution of system (.) through (t, φ, φ), where φ = (φ, φ, , φ n)T,
Trang 6φ= (φ, φ, , φ m)T Define
x i (t + ω, φ) = ϕ i, t ≤ , i ∈ , y j (t + ω, φ) = ϕ j, t ≤ , j ∈ ,
then ϕ= (ϕ, ϕ, , ϕ n)T∈ Cn , ϕ= (ϕ, ϕ, , ϕ m)T∈ Cm
Now we construct the following Lyapunov function:
V (t) =
n
i=
x i (t+ω)
x i (t)
ι i (s) ds sgn
x i (t + ω) – x i (t)
+
m
j=
y j (t+ω)
y j (t)
ϑ j (s) ds sgn
y j (t + ω) – y j (t)
It is easy to see that
min
i ∈,j∈
ι i,
ϑ j
n
i=
x i (t + ω) – x i (t) + m
j=
y j (t + ω) – y j (t)
≤ V(t) ≤ max
i ∈,j∈
ι i,
ϑ j
n
i=
x i (t + ω) – x i (t) + m
j=
y j (t + ω) – y j (t) . (.)
When t = t k , calculating the derivative of D+V (t) along the solution of (.), we have
D+V (t)≤
n
i=
D+x i (t + ω)
ι i (x i (t + ω))–
D+x i (t)
ι i (x i (t))
sgn
x i (t + ω) – x i (t)
+
m
j=
D+y j (t + ω)
ϑ j (y j (t + ω))–
D+y j (t)
ϑ j (y j (t))
sgn
y j (t + ω) – y j (t)
=
n
i=
–a i
t , x i (t + ω) + a i
t , x i (t) +
m
j=
c ji (t)f j
y j
t + ω – τ (t + ω)
–
m
j=
c ji (t)f j
y j
t – τ (t)
+
m
j=
α ji (t)
t +ω
–∞ K ji (t + ω – s)f j
y j (s) ds
–
m
j=
α ji (t)
t
–∞K ji (t – s)f j
y j (s) ds+
m
j=
β ji (t)
t +ω
–∞ K ji (t + ω – s)f j
y j (s) ds
–
m
j=
β ji (t)
t
–∞K ji (t – s)f j
y j (s) ds
sgn
x i (t + ω) – x i (t)
+
m
j=
–b j
t , y j (t + ω) + b j
t , y j (t) +
n
i=
d ij (t)g i
x i
t + ω – τ (t + ω)
–
n
i=
d ij (t)g i
x i
t – τ (t)
+
n
i=
p ij (t)
t +ω
–∞ N ij (t + ω – s)g i
x i (s) ds
–
n
p ij (t)
t
–∞N ij (t – s)g i
x i (s) ds+
n
q ij (t)
t +ω
–∞ N ij (t + ω – s)g i
x i (s) ds
Trang 7n
i=
q ij (t)
t +ω
–∞ N ij (t + ω – s)g i
x i (s) ds
sgn
y j (t + ω) – y j (t)
≤ – min
i ∈
i n
i=
x i (t + ω) – x i (t) + n
i=
m
j=
c∗ji L f j y j
t + ω – τ (t) – y j
t – τ (t) +
n
i=
m
j=
α∗ji L f j
∞
˜K(s) y j (t + ω – s) – y j (t – s) ds +
n
i=
m
j=
β ji∗L f j
∞
˜K(s) y j (t + ω – s) – y j (t – s) ds
– min
j∈ σ
j
m
j=
y j (t + ω) – y j (t) + m
j=
n
i=
d∗ij L g i x i
t + ω – τ (t) – x i
t – τ (t) +
m
j=
n
i=
p∗ij L g i
∞
˜K(s) x i (t + ω – s) – x i (t – s) ds +
m
j=
n
i=
q∗ij L g i
∞
˜K(s) x i (t + ω – s) – x i (t – s) ds
≤ – min i , σ j n
i=
x i (t + ω) – x i (t) +
m
j=
y j (t + ω) – y j (t)
+
n
i=
max
j∈ c∗ji L f j
m
j=
y j
t + ω – τ (t) – y j
t – τ (t) +
n
i=
max
j∈ α
∗
ji L f j
∞
˜K(s) y j (t + ω – s) – y j (t – s) ds +
n
i=
max
j∈ β
∗
ji L f j
∞
˜K(s) m
j=
y j (t + ω – s) – y j (t – s) ds +
m
j=
max
i ∈ d∗ij L g i
n
i=
x i
t + ω – τ (t) – x i
t – τ (t) +
m
j=
max
i ∈ p∗ij L g i
∞
˜K(s) n
i=
x i (t + ω – s) – x i (t – s) ds +
m
j=
max
i ∈ q∗ij L g i
∞
˜K(s) n
i=
x i (t + ω – s) – x i (t – s) ds
≤ – min i , σ
j
n
i=
x i (t + ω) – x i (t) +
m
j=
y j (t + ω) – y j (t)
+ max
n
i=
max
j∈ c∗ji L f j,
m
j=
max
i ∈ d ij∗L g i
×
n
i=
x i
t + ω – τ (t) – x i
t – τ (t) + m
j=
y j
t + ω – τ (t) – y j
t – τ (t)
Trang 8+ max
n
i=
max
j∈ α
∗
ji L f j,
n
i=
max
j∈ β
∗
ji L f j,
m
j=
max
i ∈ p∗ij L g i,
m
j=
max
i ∈ q∗ij L g i
×
∞
˜K(s) n
i=
x i (t + ω – s) – x i (t – s) + m
j=
y j (t + ω – s) – y j (t – s) ds.
(.)
In view of (.), it follows from (.) that
D+V (t)≤ – min i , σ j
min
i ∈,j∈ {ι i , ϑ j }V(t)
+ max
n
i=
max
j∈ c∗ji L f j,
m
j=
max
i ∈ d ij∗L g i
max
i ∈,j∈ {ι i , ϑ j }V t – τ (t)
+ max
n
i=
max
j∈ α
∗
ji L f j,
n
i=
max
j∈ β
∗
ji L f j,
m
j=
max
i ∈ p∗ij L g i,
m
j=
max
i ∈ q∗ij L g i
×
∞
˜K(s)V(t – s)ds. (.)
When t = t k, in view of (H), (H), and (.), we get
V (t k)≤ max
i ∈,j∈
ι i,
ϑ j
n
i=
x i (t k + ω) – x i (t k) + m
j=
y j (t k + ω) – y j (t k)
= max
i ∈,j∈
ι i,
ϑ j
n
i=
x i (t k +q ) – x i (t k) + m
j=
y j (t k +q ) – y j (t k)
≤ max
i ∈,j∈
ι i,
ϑ j
n
i=
| – γ ik| x i
t k–+q – x i
t–k +
n
i=
m
j=
e∗ij L E j y j
t k–+q – τ – y j
t k–– τ +
m
j=
| – δ jk| y j
t k–+q – y j
t–k + m
j=
n
i=
h∗ji L H i x i
t k–+q – τ – x i
t k–– τ
≤ max
i ∈,j∈
ι i,
ϑ j maxi ∈ | – γ ik|
n
i=
x i
t–k + ω – x i
t k– +
n
i=
max
j∈ e∗ij L E j
m
j=
y j
t k–+ ω – τ – y j
t k–– τ
+ max
j∈ | – δ jk|
m
j=
y j
t k–+ ω – y j
t k– +
n
i=
max
i ∈ h∗ji L H i
m
j=
x i
t k–+ ω – τ – x i
t k–– τ
≤ max
i ∈,j∈
ι,
ϑ
max
i ∈,j∈
| – γ ik |, | – δ jk|
Trang 9n
i=
x i
t–k + ω – x i
t k– + m
j=
y j
t k–+ ω – y j
t–k + max
n
i=
max
j∈ e∗ij L E j,
n
i=
max
i ∈ h∗ji L H i
×
n
i=
x i
t–k + ω – x i
t k– + m
j=
y j
t k–+ ω – y j
t–k
≤maxi ∈,j∈ {ι i , ϑ j}
mini ∈,j∈ {ι i , ϑ j}i ∈,j∈max
| – γ ik |, | – δ jk|V
t k–
+maxi ∈,j∈ {ι i , ϑ j}
mini ∈,j∈ {ι i , ϑ j}max
n
i=
max
j∈ e∗ij L E j,
n
i=
max
i ∈ h∗ji L H i
V
t k–– τ (.)
In view of (.)-(.) and (H)-(H), using Lemma ., we have
V (t) ≤ MV()e –(λ–η)t, t≥ , (.)
where V () = sup–∞≤s≤V (s) It follows from (.) that
n
i=
x i (t + ω) – x i (t) + m
j=
y j (t + ω) – y j (t) ≤ ν –(λ–η)t, t≥ , (.) where
ν = Mmaxi ∈,j∈ {ι i , ϑ j}
mini ∈,j∈ {ι i , ϑ j} ≥ ,
λsatisfies the condition (H) Notice that
x i (t + kω) = x i (t) +
k
l=
x i (t + lω) – x i
t + (l – )ω
, i ∈ ,
y j (t + kω) = y j (t) +
k
l=
y j (t + lω) – y j
t + (l – )ω
, j∈
In view of (.), we have
∞
l=
x i (t + lω) – x i
t + (l – )ω
= lim
k→∞
k
l=
x i (t + lω) – x i
t + (l – )ω
k→∞
k
l=
e –(λ–η)(t+(l–)ω)
–(λ–η)t
∞
Trang 10
for any given t≥ By (.), we know that limk→∞x i (t + kω) exists Similarly, we know
that limk→∞y j (t + kω) also exists.
Set (x∗(t), y∗(t)) T = (x∗(t), x∗(t), , x∗n (t), y∗(t), y∗(t), , y∗m (t)) T , where x∗i = limk→∞x i (t +
kω ), y∗j = limk→∞y j (t + kω), then (x∗(t), y∗(t)) T is a periodic function with period ω for
system (.)
Assume that system (.) has another ω-periodic solution (x∗∗(t), y∗∗(t)) Tas follows:
x∗∗(t, ψ), y∗∗(t, ψ) T=
x∗∗ (t, ψ), x∗∗ (t, ψ), , x∗∗n (t, ψ),
y∗∗ (t, ψ), y∗∗ (t, ψ), , y∗∗m (t, ψ) T,
where ψ∈ Cn , ψ∈ Cm It follows from (.) that
n
i=
x∗
i (t) – x∗∗i (t) + m
j=
y∗
j (t) – y∗∗j (t)
=
n
i=
x∗
i (t + kω) – x∗∗i (t + kω) + m
j=
y∗
j (t + kω) – y∗∗j (t + kω)
Let k → ∞, then x∗
i (t) = x∗∗i (t), y∗j (t) = y∗∗j (t), t≥ Thus we can conclude that system
(.) has a unique ω-periodic solution which is globally exponentially stable The proof of
Theorem . is complete
Remark . Li [] investigated the existence and global exponential stability of a peri-odic solution for impulsive Cohen-Grossberg-type BAM neural networks with
continu-ously distributed delays, the model in [] is not concerned with fuzzy terms Bao []
dis-cussed the existence and exponential stability of a periodic solution for BAM fuzzy
Cohen-Grossberg neural networks with mixed delays, the model in [] is not concerned with
impulsive effects Yang [] considered the periodic solution for fuzzy Cohen-Grossberg
BAM neural networks with both time-varying and distributed delays and variable
coef-ficients, the model in [] is not concerned with impulsive effect and distributed delays
Balasubramaniam et al [] analyzed the global asymptotic stability of stochastic fuzzy
cellular neural networks with multiple time-varying delays, the model in [] is not
con-cerned with impulsive effect and distributed delays, Balasubramaniam and Vembarasan
[] studied the robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses, the authors did not discuss the
exis-tence and global exponential stability of a periodic solution of neural networks and the
model in [] is also not concerned with distributed delays In this paper, we study the exponential stability for fuzzy bidirectional associative memory Cohen-Grossberg
neu-ral networks with mixed delays and impulses All the obtained results in [, , , ,
] cannot be applicable to model (.) to obtain the exponential stability of model (.)
From this viewpoint, our results on the exponential stability for fuzzy bidirectional asso-ciative memory Cohen-Grossberg neural networks with mixed delays and impulses are
essentially new and complement earlier works to some extent
... (s).3 Global exponential stability of the periodic solution
In this section, we will discuss the global exponential stability of the periodic solution for (.)
Theorem...
and satisfy K ij (s) ≤ ˜K(s) and N ji (s) ≤ ˜K(s) for all s ∈ R+, where ˜K (s) ∈ C(R+,R+) and integrable,...
some positive number
(H) For i ∈ , j ∈ ω > , there exists q ∈ Z+such that t k + ω = t k +q and γ ik = γ i