Electronic Journal of Differential Equations, Vol. 2007(2007), No. 50, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
BEHAVIOR OF IMPULSIVE FUZZY CELLULAR NEURAL
NETWORKS WITH DISTRIBUTED DELAYS
KELIN LI, ZUOAN LI, QIANKUN SONG
Abstract. In this paper, we investigate a generalized model of fuzzy cellular neural networks with distributed delays and impulses. By employing the
theory of topological degree, M -matrix and Lypunov functional, we find sufficient conditions for the existence, uniqueness and global exponential stability
of both the equilibrium point and the periodic solution. Two examples are
given to illustrate the results obtained here.
1. Introduction
Since cellular neural networks (CNN) was introduced by Chua and Yang in [7, 8],
many researchers have done extensive works on this subject due to their applications in classification of patterns, associative memories, image processing, quadratic
optimization, and other areas, e.g., [2, 5, 6, 18, 19, 25]. However, in mathematical modelling of real world problems, we encounter inconveniences, namely, the
complexity and the uncertainty or vagueness. In order to take vagueness into consideration, fuzzy theory is considered as a suitable setting. Based on traditional
CNN, Yang and Yang proposed the fuzzy cellular neural networks (FCNN) [22, 23],
which integrates fuzzy logic into the structure of the traditional CNN and maintains local connectedness among cells. Unlike previous CNN structures, FCNN has
fuzzy logic between its template input and/or output besides the sum of product
operation. FCNN is very useful paradigm for image processing problems, which is
a cornerstone in image processing and pattern recognition. In such applications, it
is of prime importance to ensure that the designed FCNN be stable. In [22, 23],
the authors have obtained some conditions for the existence and the global stability of the equilibrium point of FCNN without delays. In [16], Liu and Tang
have considered FCNN with either constant delays or time-varying delays, several
sufficient conditions have been obtained to ensure the existence and uniqueness of
the equilibrium point and its global exponential stability. Yuan, Cao and Deng
have given several novel criteria of exponential stability and periodic solutions for
FCNN with time-varying delays [24]. Recently, Huang has considered the stability
of FCNN with diffusion terms and time-varying delay [14], at the same time, Huang
has investigated the exponential stability of FCNN with distributed delay [13].
2000 Mathematics Subject Classification. 92B20, 34K20, 34K13.
Key words and phrases. Fuzzy cellular neural networks; impulses; distributed delays;
global exponential stability; periodic oscillatory solution.
c
2007
Texas State University - San Marcos.
Submitted October 2, 2006. Published April 5, 2007.
1
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KELIN LI, ZUOAN LI, QIANKUN SONG
EJDE-2007/50
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 such fields as medicine and biology, economics, mechanics, electronics and telecommunications, etc. Many interesting results on impulsive effect have
been gained, e.g., Refs. [1, 3, 10, 11, 15, 17, 20, 21]. As artificial electronic systems,
neural networks such as CNN, 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.
Motivated by the above discussions, in this paper, on the basis of the structure
of FCNN, we consider a class of impulsive fuzzy neural networks with distributed
delays described by the following system of integro-differential equations:
n
n
X
X
dxi (t)
ãij uj (t) + Ii (t)
aij fj (xj (t)) +
= −di xi (t) +
dt
j=1
j=1
Z t
Z t
n
n
^
_
bij
+
Kij (t − s)fj (xj (s))ds +
b̃ij
Kij (t − s)fj (xj (s))ds
+
j=1
n
^
j=1
−∞
Tij uj (t) +
j=1
n
_
Hij uj (t),
−∞
t 6= tk ,
j=1
−
∆xi (tk ) = xi (t+
k ) − xi (tk ) = ∆k (xi (tk )),
t = tk ,
(1.1)
for i = 1, 2, . . . , n. Where the fixed times tk satisfy t1 < t2 < . . . , limk→∞ tk = ∞.
The first part (called continuous part) of model (1.1) describes the continuous
processes of FCNN. n corresponds to the number of units in the neural network; xi
corresponds to the state variable; fj (xj (t)) denotes the activation function of the
jth neuron; ui and Ii (t) denote input and bias of the ith neuron, respectively. di
represents the rate with which the ith unit will reset its potential to the resting
state in isolation when disconnected from the networks and external inputs; aij
and ãij are elements of feedback template and feedforward template; bij , b̃ij are
elements of the distributed delay fuzzy feedback MIN template, the distributed
delay fuzzy feedback MAX template, Tij and Hij are elements of fuzzy feedforward
MIN template and fuzzy
MAX template, respectively; Kij corresponds
V feedforward
W
to the delay kernel.
and
denote the fuzzy AND and fuzzy OR operation,
respectively. The second part (called discrete part) of model (1.1) describes that
the evolution processes experience abrupt change of states at the moments of time tk
(called impulsive moments). ∆xi (tk ) represents impulsive perturbations of the ith
unit at time tk , and ∆k denotes the impulsive operator at time tk for k = 1, 2, . . . .
To the best of our knowledge, few authors has considered dynamical behaviors
of impulsive fuzzy neural networks with distributed delays. This paper studies
the existence, uniqueness and global exponential stability of both the equilibrium
point and the periodic solution for impulsive fuzzy neural networks with distributed
delays. Several sufficient conditions ensuring the existence, uniqueness and global
exponential stability of both the equilibrium point and the periodic solution for
impulsive fuzzy neural networks with distributed delays will be established for the
system (1.1).
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IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
3
The remainder part of this paper is organized as follows. some notations and
preliminaries are given in section 2. In section 3, several sufficient conditions will be
established ensuring model (1.1) to the existence, uniqueness and global exponential
stability of equilibrium point. The existence, uniqueness and global exponential
stability of the system (1.1) will be given in section 4. In section 5, two examples
are given to illustrate our theory.
2. Preliminaries
Throughout this paper we assume the following hypotheses:
(H1) There exist constant scalers Fi > 0 such that
|fi (x) − fi (y)| ≤ Fi |x − y|,
i = 1, 2, . . . , n
for any x, y ∈ R.
(H2) The delay kernels Kij : [0, +∞) → [0, +∞) are piecewise continuous functionsRand satisfies:
∞
(i) 0 Kij (s)ds = 1, i, j = 1, 2, . . . , n.
R∞
(ii) 0 sKij (s)ds < ∞, i, j = 1, 2, . . . , n.
(iii) There exists a positive number µ such that
Z ∞
seµs Kij (s)ds < ∞, i, j = 1, 2, . . . , n.
0
Let C = C((−∞, 0], Rn ) be the linear space of bounded and continuous functions
which map (−∞, 0] into Rn . The initial conditions associated with model (1.1) are
of the form
xi (t) = ϕi (t), −∞ < t ≤ 0
(2.1)
in which ϕi (·) are bounded continuous (i = 1, 2, . . . , n).
First, we introduce some notation and recall some basic definitions. For an
n × n matrix, |A| denotes the absolute value matrix given by |A| = (|aij |)n×n ;
A−1 denotes the inverse of A. Let A, B be two n × n matrices, A > B represents
aij > bij for all i, j = 1, 2, . . . , n. Let a vector norm kxkp (p = 1, ∞) (simply
denoted by kxk) for x ∈ Rn be defined as
kxk1 =
n
X
|xi |,
kxk∞ = max |xi |.
i=1
1≤i≤n
For ϕ ∈ C, kϕk∞ is defined as
kϕk∞ =
sup
−∞<s≤0
kϕ(s)k∞ =
sup
max |ϕi (s)|.
−∞<s≤0 1≤i≤n
Definition 2.1. A function x : (−∞, +∞) → Rn is said to be the special solution of
system (1.1) with initial condition (2.1) if the following two conditions are satisfied
(i) x is piecewise continuous with first kind discontinuity at the points tk ,
k = 1, 2, . . . . Moreover, x is left continuous at each discontinuity point.
(ii) x satisfies model (1.1) for t ≥ 0, and x(s) = ϕ(s) for s ∈ (−∞, 0].
Especially, a point x∗ ∈ Rn is called an equilibrium point of model (1.1), if x(t) = x∗
is a solution of (1.1).
Henceforth, we let x(t, ϕ) denote the special solution of (1.1) with initial condition ϕ ∈ C.
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KELIN LI, ZUOAN LI, QIANKUN SONG
EJDE-2007/50
Definition 2.2. The periodic solution x(t, ϕ) of system (1.1) is said to be globally
exponentially stable, if there exist positive constants α and M such that every
solution x(t, φ) of (2.1) satisfies
kx(t, φ) − x(t, ϕ)k∞ ≤ M kφ − ϕk∞ e−αt ,
∀t ≥ 0.
Definition 2.3 ([4]). A real matrix D = (dij )n×n is said to be a non-singular
M -matrix if aij ≤ 0, i, j = 1, 2, . . . , n, i 6= j, and all successive principal minors of
D are positive.
For the non-singular M -matrix, we have the following result.
Lemma 2.4 ([4]). Each of the following conditions is equivalent:
(i) D is a nonsingular M -matrix.
(ii) D−1 exists and D−1 is a nonnegative matrix.
(iii) The diagonal elements of D are all positive and there exists a positive vector
d such that Dd > 0 or DT d > 0.
Lemma 2.5 ([22]). Suppose y and ȳ are two state of model (1.1), then we have
n n
n
X
^
^
α
f
(ȳ
)
≤
α
f
(y
)
−
αij · fj (yj ) − fj (ȳj ),
ij j j ij j j
j=1
j=1
j=1
n n
n
X
_
_
βij fj (ȳj ) ≤
βij fj (yj ) −
βij · fj (yj ) − fj (ȳj ).
j=1
j=1
j=1
3. Global exponential stability of equilibrium point
In this section, we will give several sufficient conditions on the global exponential stability of equilibrium point for the impulsive FCNN with distributed delays.
Consider the case of model (1.1) as Ii (t) = Ii , ui (t) = ui , i = 1, 2, . . . , n, and let
Pn
Vn
Wn
I˜i = j=1 ãij uj + Ii + j=1 Tij uj + j=1 Hij uj , then model (1.1) becomes
Z t
n
n
^
X
dxi (t)
bij
aij fj (xj (t)) +
Kij (t − s)fj (xj (s))ds
= −di xi (t) +
dt
−∞
j=1
j=1
Z t
n
_
(3.1)
+
b̃ij
Kij (t − s)fj (xj (s))ds + I˜i , t 6= tk ,
j=1
−∞
−
∆xi (tk ) = xi (t+
k ) − xi (tk ) = ∆k (xi (tk )),
t = tk ,
for i = 1, 2, . . . , n.
Theorem 3.1. Under assumptions (H1), (H2), the first equation in system (3.1)
has a unique equilibrium point if D − (|A| + |B| + |B̃|)F is a nonsingular M matrix, where D = diag(d1 , d2 , . . . , dn ), A = (aij )n×n , B = (bij )n×n , B̃ = (b̃ij )n×n ,
F = diag(F1 , F2 , . . . , Fn ).
Proof. Let x∗ = (x∗1 , x∗2 , . . . , x∗n )T denote an equilibrium point of the first equation
in model (3.1). Then x∗ satisfies
di x∗i −
n
X
j=1
aij fj (x∗j ) −
n
^
j=1
bij fj (x∗j ) −
n
_
j=1
b̃ij fj (x∗j ) − I˜i = 0,
i = 1, 2, . . . , n. (3.2)
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IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
5
Let
hi (xi ) = di xi −
n
X
aij fj (xj )−
j=1
n
^
bij fj (xj )−
j=1
n
_
b̃ij fj (xj )− I˜i = 0,
i = 1, 2, . . . , n.
j=1
Obviously, the solutions of the above system are the equilibrium point of model
(3.1). Let us define homotopic mapping
H(x, λ) = λh(x) + (1 − λ)x,
where λ ∈ [0, 1], and
h(x) = (h1 (x1 ), h2 (x2 ), . . . , hn (xn ))T ,
H(x, λ) = (H1 (x1 , λ), H2 (x2 , λ), . . . , Hn (xn , λ))T ,
then for i ∈ {1, 2, . . . , n}, from (H1) and Lemma 2.5, we have
|Hi (x, λ)|
n
n
n
h
i
X
^
_
= λ di xi −
aij fj (xj ) −
bij fj (xj ) −
b̃ij fj (xj ) − I˜i + (1 − λ)xi j=1
j=1
≥ |λdi xi + (1 − λ)xi | − λ
n
X
j=1
|aij ||fj (xj )| − λ
j=1
−λ
n
X
n
X
|bij ||fj (xj )|
j=1
|b̃ij ||fj (xj )| − λ|I˜i |
j=1
≥ [1 + λ(di − 1)]|xi | − λ
n
X
|aij |Fj |xj | − λ
− λ |I˜i | +
n
X
|aij ||fj (0)| +
j=1
n
X
|bij |fj (0)| +
j=1
= [1 + λ(di − 1)]|xi | − λ
n
X
|bij |Fj |xj | − λ
j=1
j=1
h
n
X
n
X
n
X
|b̃ij |Fj |xj |
j=1
i
|b̃ij |fj (0)| .
j=1
Fj |xj | |aij | + |bij | + |b̃ij |
j=1
h
− λ |I˜i | +
n
X
i
|fj (0)| |aij | + |bij | + |b̃ij | .
j=1
Since D − (|A| + |B| + |B̃|)F is a nonsingular M -matrix, there exist constants li > 0
such that
n
X
li di − Fi
lj |aji | + |bji | + |b̃ji | > 0, i = 1, 2, . . . , n,
j=1
then, we have
n
X
li |Hi (x, λ)|
i=1
≥
n
X
i=1
li (1 − λ)|xi | + λ
n h
X
i=1
di li |xi | − li
n
X
j=1
i
Fj |xj | |aij | + |bij | + |b̃ij |
6
KELIN LI, ZUOAN LI, QIANKUN SONG
−λ
n
n
h
i
X
X
li |I˜i | +
|fj (0)| |aij | + |bij | + |b̃ij |
i=1
≥λ
n h
X
j=1
n
X
di li |xi | − li
i=1
n
X
−λ
h
li |I˜i | +
n h
X
i
Fj |xj | |aij | + |bij | + |b̃ij |
j=1
n
X
i
|fj (0)| |aij | + |bij | + |b̃ij |
i=1
=λ
EJDE-2007/50
j=1
n
i
X
l i d i − Fi
lj |aji | + |bji | + |b̃ji | |xi |
i=1
n
X
−λ
h
li |I˜i | +
j=1
n
X
i=1
i
|fj (0)| |aij | + |bij | + |b̃ij |
j=1
≥ λl0 kxk1 − λnI0 .
Define
n
n
o
X
lj |aji | + |bji | + |b̃ji | ,
l0 = min li di − Fi
1≤i≤n
j=1
n
n o
X
|fj (0)| |aij | + |bij | + |b̃ij | ,
I0 = max li |I˜i | +
1≤i≤n
j=1
and let
n(I0 + 1) .
Γ = x : kxk1 ≤
l0
Then it follows that kxk1 = n(I0 + 1)/l0 for any x ∈ ∂Γ, and
n
X
li |Hi (x, λ)| ≥ λl0
j=1
n(I0 + 1)
− λnI0 > 0,
l0
∀λ ∈ (0, 1],
that is F (x, λ) 6= 0, for any x ∈ ∂Γ, λ ∈ (0, 1]. Also, as λ = 0, H(x, λ) = id (x) =
x 6= 0, for any x ∈ ∂Γ, here, id is identity mapping. Hence, we have H(x, λ) 6= 0,
for any x ∈ ∂Γ, λ ∈ [0, 1].
From (H1), it is easy to prove deg(id , Γ, 0) = 1 thus we have from homotopy
invariance theorem [9] that
deg(h, Γ, 0) = deg(id , Γ, 0) = 1.
By the topological degree theory, we can conclude that (3.1) has at least one solution
in Γ. That is, model (3.1) has at least an equilibrium point.
Suppose y ∗ = (y1∗ , y2∗ , . . . , yn∗ )T is also an equilibrium point of model (3.1), then
we have
di x∗i −
di yi∗ −
n
X
j=1
n
X
j=1
aij fj (x∗j ) −
aij fj (yj∗ ) −
n
^
j=1
n
^
j=1
bij fj (x∗j ) −
bij fj (yj∗ ) −
n
_
j=1
n
_
j=1
b̃ij fj (x∗j ) − I˜i = 0,
b̃ij fj (yj∗ ) − I˜i = 0,
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IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
7
this implies
di (x∗i − yi∗ ) =
n
X
aij (fj (x∗j ) − fj (yj∗ )) +
j=1
n
_
b̃ij fj (x∗j ) −
j=1
n
X
n
_
n
^
bij fj (yj∗ )
j=1
b̃ij fj (yj∗ )
j=1
|aij ||fj (x∗j ) − fj (yj∗ )| + |
j=1
+|
bij fj (x∗j ) −
j=1
+
≤
n
^
n
^
bij fj (x∗j ) −
j=1
n
_
b̃ij fj (x∗j ) −
j=1
n
_
n
^
bij fj (yj∗ )|
j=1
b̃ij fj (yj∗ )|
j=1
for i = 1, 2, . . . , n. By using (H1) and Lemma 2.5, we have
di |x∗i − yi∗ | ≤
n
X
Fj |x∗j − yj∗ | |aij | + |bij | + |b̃ij | ,
i = 1, 2, . . . , n,
j=1
which can be rewritten as
(D − (|A| + |B| + |B̃|)F )(|x∗1 − y1∗ |, |x∗2 − y2∗ |, . . . , |x∗n − yn∗ |)T ≤ 0.
Since D − (|A| + |B| + |B̃|)F is a nonsingular M -matrix, (D − (|A| + |B| + |B̃|)F )−1
is a nonnegative matrix. Thus multiplying both sides of the above inequality by
(D − (|A| + |B| + |B̃|)F )−1 does not change the inequality direction, it follows that
(|x∗1 − y1∗ |, |x∗2 − y2∗ |, . . . , |x∗n − yn∗ |)T ≤ 0.
This implies that x∗ = y ∗ . Therefore, the system (3.1) has one unique equilibrium
point.
Theorem 3.2. Assume that (H1), (H2) hold, and D − (|A| + |B| + |B̃|)F is a
nonsingular M -matrix, furthermore, suppose that the impulsive operator ∆k (xi (tk ))
satisfies
∆k (xi (tk )) = −δik (xi (tk ) − x∗i ),
0 < δik < 2,
Then the equilibrium point x∗ = (x∗1 , x∗2 , . . . , x∗n )T
nentially stable.
i = 1, 2, . . . , n, k = 1, 2, . . . .
(3.3)
of system (3.1) is globally expo-
Proof. From (3.3), we have ∆k (x∗i ) = 0, and by Theorem 3.1, x∗ = (x∗1 , x∗2 , . . . , x∗n )T
is only equilibrium point of the system (3.1). Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T
be an arbitrary solution of the system (3.1). From assumption (H1) and Lemma
8
KELIN LI, ZUOAN LI, QIANKUN SONG
EJDE-2007/50
2.5, we obtain that
d+ |xi (t) − x∗i |
dt
d(xi (t) − x∗i )
dt
n
X
≤ −di |xi (t) − x∗i | +
|aij ||fj (xj (t)) − fj (x∗j )|
= sign(xi (t) − x∗i )
j=1
+|
+|
n
^
j=1
n
_
Z
t
Kij (t − s)fj (xj (s))ds −
bij
−∞
Z
t
Kij (t − s)fj (xj (s))ds −
b̃ij
−∞
j=1
n
^
j=1
n
_
Z
n
X
Kij (t − s)fj (x∗j )ds|
−∞
Z
t
Kij (t − s)fj (x∗j )ds|
b̃ij
j=1
≤ −di |xi (t) − x∗i | +
t
bij
−∞
|aij ||fj (xj (t)) − fj (x∗j )|
j=1
+|
+|
n
X
j=1
n
X
Z
t
(3.4)
Kij (t − s)|fj (xj (s)) − fj (x∗j )|ds
|bij |
−∞
Z
t
|b̃ij |
Kij (t − s)|fj (xj (s)) − fj (x∗j )|ds
−∞
j=1
≤ −di |xi (t) − x∗i | +
n
X
|aij |Fj |xj (t) − x∗j |
j=1
+
n X
Z
|bij | + |b̃ij |
t
Kij (t − s)Fj |xj (s) − x∗j |ds
−∞
j=1
= −di |xi (t) − x∗i | +
n
X
Fj |aij ||xj (t) − x∗j |
j=1
+
n
X
Z
Fj |bij | + |b̃ij |
+∞
Kij (s)|xj (t − s) − x∗j |ds
0
j=1
for t > 0, i = 1, 2, . . . , n, t 6= tk , k = 1, 2, . . . . Also,
∗
∗
∗
∗
xi (t+
k ) − xi = −δik (xi (tk ) − xi ) + xi (tk ) − xi = (1 − δik )(xi (tk ) − xi )
for i = 1, 2, . . . , n, k = 1, 2, . . . . Hence
∗
∗
∗
|xi (t+
k ) − xi | ≤ |1 − δik ||xi (tk ) − xi | ≤ |xi (tk ) − xi |
(3.5)
for i = 1, 2, . . . , n, k = 1, 2, . . . . Since D − (|A| + |B| + |B̃|)F is a nonsingular
M -matrix, there exist constants li > 0 such that
li di − Fi
n
X
lj |aji | + |bji | + |b̃ji | > 0,
j=1
i = 1, 2, . . . , n.
(3.6)
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IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
9
Now, for i = 1, 2, . . . , n, we define the functions
n
h
Z
X
h̃i (αi ) = li (di − αi ) − Fi
lj |aji | + |bji | + |b̃ji |
+∞
i
eαi s Kji (s)ds ,
0
j=1
where αi ∈ [0, +∞). Obviously, for i ∈ {1, 2, . . . , n}, h̃i (αi ) are continuous on
[0, +∞), and from (3.6), we know that h̃i (0) > 0, for i ∈ {1, 2, . . . , n}. Also, for
i ∈ {1, 2, . . . , n}, we have h̃i (αi ) → −∞ as αi → +∞. So there exists αi∗ such that
h̃i (αi∗ ) = 0, i ∈ {1, 2, . . . , n}. Let α = min{α1 , α2 , . . . , αn }, we get
n
h
Z +∞
i
X
h̃i (α) = li (di − α) − Fi
lj |aji | + |bji | + |b̃ji |
eαi s Kji (s)ds ≥ 0 (3.7)
0
j=1
for i = 1, 2, . . . , n. Let yi (t) = eαt |xi (t) − x∗i |, i = 1, 2, . . . , n. Then it follows from
(3.4) that
d+ yi (t)
d+ |xi (t) − x∗i |
= αeαt |xi (t) − x∗i | + eαt
dt
dt
n
X
|aij |Fj yj (t)
≤ −(di − α)yi (t) +
(3.8)
j=1
+
n X
Z
|bij | + |b̃ij | Fj
+∞
eαs Kij (s)yj (t − s)ds
0
j=1
for t > 0, i = 1, 2, . . . , n, t 6= tk , k = 1, 2, . . . . Also, from (3.5), we have
+
αtk
∗
αtk
yi (t+
|xi (t+
|xi (tk ) − x∗i | = yi (tk )
k)=e
k ) − xi | ≤ e
for i = 1, 2, . . . , n, k = 1, 2, . . . . Now, we construct the Lyapunov functional
n
n
h
Z +∞
Z t
i
X
X
αs
V (t) =
li yi (t) +
Fj |bij | + |b̃ij |
e Kij (s)
yj (r)dr ds . (3.9)
i=1
0
j=1
t−s
The derivative of V (t) along with the trajectories of model (3.1) is
D+ V (t)
=
≤
Z +∞
n
n
h d+ y (t) X
X
i
+
Fj |bij | + |b̃ij | yj (t)
eαs Kij (s)ds
li
dt
0
j=1
i=1
n
Z +∞
i
X
−
Fj |bij | + |b̃ij |
eαs Kij (s)yj (t − s)ds
0
j=1
n
X h
li − (di − α)yi (t) +
i=1
n
X
+
j=1
+
n
X
j=1
n
X
|aij |Fj yj (t)
j=1
Fj
Z
|bij | + |b̃ij |
+∞
eαs Kij (s)yj (t − s)ds
0
Z
Fj |bij | + |b̃ij | yj (t)
0
+∞
eαs Kij (s)ds
10
KELIN LI, ZUOAN LI, QIANKUN SONG
−
=−
+
n
X
j=1
n
X
Fj
Z
|bij | + |b̃ij |
+∞
eαs Kij (s)yj (t − s)ds
li (di − α)yi (t) +
n X
n
X
li Fj |aij |yj (t)
i=1 j=1
Z
li Fj |bij | + |b̃ij | |yj (t)
+∞
eαs Kij (s)ds
0
i=1 j=1
=
i
0
i=1
n X
n
X
n n
X
EJDE-2007/50
n
h
Z
X
− li (di − α) + Fi
lj |aji | + |bji | + |b̃ji |
i=1
+∞
eαs Kji (s)ds
io
yi (t) ≤ 0
0
j=1
for t > 0, t 6= tk , k = 1, 2, . . . . Also,
V
(t+
k)
n
n
h
Z
X
X
+
=
li yi (tk ) +
Fj |bij | + |b̃ij |
≤
i=1
n
X
h
li yi (tk ) +
i=1
j=1
n
X
Z
e Kij (s)
αs
+∞
Z
Fj |bij | + |b̃ij |
t+
k
t+
k −s
0
tk
Z
eαs Kij (s)
i
yj (r)dr ds
i
yj (r)dr ds
tk −s
0
j=1
= V (tk ),
+∞
k = 1, 2, . . . .
So, we have V (t) ≤ V (0), for all t > 0. From (3.9), we obtain
V (t) ≥ min {li }
1≤i≤n
n
X
yi (t).
(3.10)
i=1
Also,
V (0)
n
n
h
Z
X
X
=
li yi (0) +
Fj |bij | + |b̃ij |
i=1
≤ max {li }
1≤i≤n
≤ max {li }
1≤i≤n
+∞
Z
e Kij (s)
n
X
i=1
n h
X
j=1
n
X
yi (0) +
yi (0) +
i=1
i
yj (r)dr ds
−s
0
j=1
n h
X
0
αs
Fj
Z
|bij | + |b̃ij |
Fj
Z
|bij | + |b̃ij |
+∞
Z
eαs Kij (s)
0
−s
0
+∞
seαs Kij (s)ds
sup
i
yj (r) .
−∞<r≤0
0
j=1
i
yj (r)dr ds
From the above inequality and (3.10), we have
n
X
i=1
n
yi (t) ≤
max1≤i≤n {li } X h
yi (0)
min1≤i≤n {li } i=1
n
Z +∞
X
+
Fj |bij | + |b̃ij |
seαs Kij (s)ds
j=1
sup
i
yj (r) ,
−∞<r≤0
0
for t > 0. It follows from the definition of yi (t) and the above inequality that
n
X
i=1
|xi (t) − x∗i | ≤ M e−αt
sup
n
X
−∞<s≤0 i=1
|ϕi (s) − x∗i |
EJDE-2007/50
IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
11
for t > 0, where
M=
n
Z +∞
i
X
max1≤i≤n {li } h
1 + max
Fj |bij | + |b̃ij |
seαs Kij (s)ds .
1≤i≤n
min1≤i≤n {li }
0
j=1
The proof is completed.
As a direct result of Theorems 3.1, 3.2, we have the following result.
Corollary 3.3. Assume that (H1), (H2) hold, then model (3.1) has one unique
equilibrium point, if any one of the following conditions is true:
Pn (i) di > Fi j=1 |aji | + |bji | + |b̃ji | , i = 1, 2, . . . , n.
Pn
(ii) di > j=1 Fj |aij | + |bij | + |b̃ij | , i = 1, 2, . . . , n.
(iii) There exists a positive vector l = (l1 , l2 , . . . , ln )T > 0 such that
li di >
n
X
lj Fj |aij | + |bij | + |b̃ij | ,
i = 1, 2, . . . , n.
j=1
Furthermore, suppose that the impulsive operator ∆k (xi (tk )) satisfies
∆k (xi (tk )) = −δik (xi (tk ) − x∗i ),
0 < δik < 2,
i = 1, 2, . . . , n, k = 1, 2, . . . .
Then the equilibrium point x∗ = (x∗1 , x∗2 , . . . , x∗n )T of the system (3.1) is globally
exponentially stable.
Proof. In fact, any one of the conditions (i)-(iii) can assure, D − (|A| + |B| + |B̃|)F
is a nonsingular M -matrix.
4. Periodic oscillatory solution
In the section, we discuss the existence, uniqueness and global exponential stability of the periodic oscillatory solution of model (1.1). Let Ii : R → R and
ui : R → R be continuously periodic function with period ω, i.e. Ii (t + ω) = Ii (t),
ui (t + ω) = ui (t) for i = 1, 2, . . . , n. Furthermore, we assume that
(H3) There exists a positive integer q such that
tk+q = tk + ω,
δi(k+q) = δik ,
k = 1, 2, . . . , i = 1, 2, . . . , n,
−
where δik satisfy ∆k (xi (tk )) = xi (t+
k ) − xi (tk ) = −δik xi (tk ), 0 < δik < 2.
Theorem 4.1. Under hypothesis (H1)–(H3), there exists exactly one ω-periodic
solution of model (1.1) and all other solutions of model (1.1) converge exponentially
to it as t → +∞, if D − (|A| + |B| + |B̃|)F is a nonsingular M -matrix.
Proof. Let x(t, φ) = (x1 (t, φ), x2 (t, φ), . . . , xn (t, φ))T and let
x(t, ϕ) = (x1 (t, ϕ), x2 (t, ϕ), . . . , xn (t, ϕ))T
be an arbitrary pair of solutions of (1.1). Since D−(|A|+|B|+|B̃|)F is a nonsingular
M -matrix, (3.6) and (3.7) hold. Let ỹ(t) = eαt |xi (t, φ) − xi (t, ϕ)|, i = 1, 2, . . . , n,
12
KELIN LI, ZUOAN LI, QIANKUN SONG
EJDE-2007/50
we easily obtain
n
X
d+ ỹi (t)
≤ −(di − α)ỹi (t) +
|aij |Fj ỹj (t)
dt
j=1
n
Z +∞
X
+
Fj |bij | + |b̃ij |
eαs Kij (s)ỹj (t − s)ds,
0
j=1
and
(4.1)
+
+
αtk
ỹi (t+
|xi (t+
k)=e
k , φ) − xi (tk , ϕ)|
−
= eαtk |xi (t−
k , φ) − δik xi (tk , φ) − xi (tk , ϕ) + δik xi (tk , ϕ)|
= eαtk |xi (tk , φ) − δik xi (tk , φ) − xi (tk , ϕ) + δik xi (tk , ϕ)|
(4.2)
= eαtk |1 − δik ||xi (tk , φ) − xi (tk , ϕ)|
≤ eαtk |xi (tk , φ) − xi (tk , ϕ)| = ỹi (tk ).
Now, we construct the Lyapunov functional
n
n
h
Z +∞
Z
X
X
V (t) =
li ỹi (t) +
Fj |bij | + |b̃ij |
eαs Kij (s)
i=1
j=1
0
t
i
ỹj (r)dr ds .
t−s
By a minor modification of the proof of Theorem 3.2, we can easily derive
n
n
X
X
|φi (s) − ϕi (s)|
|xi (t, φ) − xi (t, ϕ)| ≤ M e−αt sup
−∞<s≤0 i=1
i=1
for t ≥ 0, where M ≥ 1 is constant, α = min1≤i≤n {αi } from (3.8). Therefore, we
have
kx(t, φ) − x(t, ϕ)k∞ ≤ M e−αt kφ − ϕk∞ .
(4.3)
Below, we prove that the system (1.1) has exactly one ω-periodic solution. For each
solution x(t, φ) of (1.1) and each t ≥ 0, we define a function xt (φ) in this fashion:
xt (φ)(s) = x(t + s, φ)
for s ∈ (−∞, 0].
From (4.3), we can choose a positive integer N such that M e−αN ω ≤ 61 .
Now, define a Poincare mapping C → C by P (ϕ) = xω (ϕ), then P N (ϕ) =
xN ω (ϕ). Let t = N ω, then
1
kP N (φ) − P N (ϕ)k∞ ≤ kφ − ϕk∞ .
6
This implies that P N is a contraction mapping, hence there exists one unique fixed
point ϕ∗ ∈ C such that P N (ϕ∗ ) = ϕ∗ .
Since P N (P (ϕ∗ )) = P (P N (ϕ∗ )) = P (ϕ∗ ), P (ϕ∗ ) ∈ C is also a fixed point of
N
P , it follows that P (ϕ∗ ) = ϕ∗ , that is xω (ϕ∗ ) = ϕ∗ .
Let x(t, ϕ∗ ) be the solution of model (1.1) through (0, ϕ∗ ), then x(t + ω, ϕ∗ ) is
also a solution of model (1.1). Obviously
xt+ω (ϕ∗ ) = xt (xω (ϕ∗ )) = xt (ϕ∗ )
for all t ≥ 0. Hence
x(t + ω, ϕ∗ ) = x(t, ϕ∗ ).
This shows that x(t, ϕ∗ ) is exactly one ω-periodic solution of model (1.1), and all
solutions of model (1.1) converge exponentially to it as t → +∞. The proof is
completed.
EJDE-2007/50
IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
13
As a direct result of Theorem 4.1, we have following corollary.
Corollary 4.2. Under hypothesis (H1)–(H3), there exists exactly one ω-periodic
solution of model (1.1) and all other solutions of model (1.1) converge exponentially
to it as t → +∞, if any one of the following conditions is true:
Pn (i) di > Fi j=1 |aji | + |bji | + |b̃ji | , i = 1, 2, . . . , n.
Pn
(ii) di > j=1 Fj |aij | + |bij | + |b̃ij | , i = 1, 2, . . . , n.
(iii) There exists a positive vector l = (l1 , l2 , . . . , ln )T > 0 such that
li di >
n
X
lj Fj |aij | + |bij | + |b̃ij | ,
i = 1, 2, . . . , n.
j=1
5. Examples
Example 5.1. Consider the model
2
2
X
X
dx1 (t)
ã1j uj + I1
a1j fj (xj (t)) +
= −d1 x1 (t) +
dt
j=1
j=1
+
2
^
Z
+
−(t−s)
b1j
e
fj (xj (s))ds +
−∞
j=1
2
^
t
T1j uj +
j=1
2
_
Z
H1j uj ,
e−(t−s) fj (xj (s))ds
−∞
j=1
2
_
t
b̃1j
t ≥ 0, t 6= tk
j=1
1
25
sin(1 + k))(x1 (tk ) − ), k = 1, 2, . . . ,
2
32
2
2
X
X
dx2 (t)
= −d2 x2 (t) +
a2j fj (xj (t)) +
ã2j uj + I2
dt
j=1
j=1
∆x1 (tk ) = −(1 +
+
2
^
j=1
+
2
^
j=1
Z
t
b2j
−(t−s)
e
fj (xj (s))ds +
−∞
T2j uj +
2
_
Z
H2j uj ,
e−(t−s) fj (xj (s))ds
−∞
j=1
2
_
t
b̃2j
t ≥ 0, t 6= tk
j=1
∆x2 (tk ) = −(1 +
1
2
cos(2k))(x2 (tk ) − ),
3
2
k = 1, 2, . . . ,
(5.1)
where 0 < t1 < t2 < . . . is a strictly increasing sequence such that limt→+∞ tk =
+∞; fi (x) = 12 (|x + 1| + |x − 1|), i = 1, 2;
1
1 1
1 1
4
0
−1
D=
, A= 4
, B = 22 21 , B̃ = 41 22 ,
0
5
−3 +1
3
3
3
3
1 2
à = 31 13 , I1 = I2 = 2, u1 = u2 = 1, T = (Tij ) = E, H = (Hij ) = E.
2
2
We can easily check that (H1) and (H2) hold, and for any x1 , x2 ∈ R, we have
|f1 (x1 ) − f2 (x2 )| ≤ |x1 − x2 |,
i = 1, 2,
14
KELIN LI, ZUOAN LI, QIANKUN SONG
EJDE-2007/50
hence F1 = F2 = 1. It follows that
D − (|A| + |B| + |B̃|)F =
3 −2
−4 3
is a nonsingular M -matrix. Also, α1k = 1 + 21 sin(1 + k), α2k = 1 + 32 cos(2k)
such that 0 < αik < 2, i = 1, 2, k = 1, 2, . . . . From Corollary 3.3, we know that
neural network model (5.1) has one unique equilibrium point, which is globally
exponentially
stable. Using MATLAB software, we can get the unique equilibrium
1 T
∗
point x = 25
32 , 2 ) .
Example 5.2. Consider the following impulsive neural network model with distributed delays
2
2
X
X
dx1 (t)
= −d1 x1 (t) +
a1j fj (xj (t)) +
ã1j uj + I1
dt
j=1
j=1
+
2
^
Z
+
e−2(t−s) fj (xj (s))ds +
−∞
j=1
2
^
t
b1j
T1j uj +
2
_
Z
H1j uj ,
e−2(t−s) fj (xj (s))ds
−∞
j=1
2
_
t
b̃1j
t ≥ 0, t 6= tk
j=1
j=1
1
sin(1 + k))(x1 (tk )), tk = 0.3 + 2(k − 1)π,
2
2
2
X
X
dx2 (t)
= −d2 x2 (t) +
a2j fj (xj (t)) +
ã2j uj + I2
dt
j=1
j=1
∆x1 (tk ) = −(1 +
+
2
^
Z
+
e−2(t−s) fj (xj (s))ds +
−∞
j=1
2
^
t
b2j
T2j uj +
j=1
2
_
Z
H2j uj ,
t
e−2(t−s) fj (xj (s))ds
b̃2j
−∞
j=1
2
_
k = 1, 2, . . . ,
t ≥ 0, t 6= tk
j=1
∆x2 (tk ) = −(1 +
2
cos(2k))(x2 (tk )),
3
tk = 0.3 + 2(k − 1)π, k = 1, 2, . . . ,
(5.2)
where 0 < t1 < t2 < . . . is a strictly increasing sequence such that limt→+∞ tk =
+∞; fi (x) = 1+e1−x , i = 1, 2;
1
1 1
4 0
−1
4
D=
, A=
, B = 22 12 ,
0 5
−3 +1
1 1
3 3 3 3
B̃ = 41 22 , Ã = 21 21 ,
3
3
2
2
I1 = I2 = 2 sin t, u1 = u2 = cos t, T = (Tij ) = E, H = (Hij ) = E. We can easily
check that (H1) and (H2) hold, and that for any x1 , x2 ∈ R, we have
|f1 (x1 ) − f2 (x2 )| ≤ |x1 − x2 |,
i = 1, 2,
hence F1 = F2 = 1. It follows that
D − (|A| + |B| + |B̃|)F =
3 −2
−4 3
EJDE-2007/50
IMPULSIVE FUZZY CELLULAR NEURAL NETWORKS
15
is a nonsingular M -matrix. Also, α1k = 1 + 12 sin(1 + k), α2k = 1 + 32 cos(2k) such
that 0 < αik < 2, i = 1, 2, k = 1, 2, . . . . From Theorem 4.1, we conclude that
there exists exactly one 2π-periodic solution of model (5.2), and all other solutions
converge exponentially to this solution as t → +∞.
Conclusions. Stability and periodic oscillatory behavior are important in the applications and theories of neural networks. By employing the theory of topological
degree, M -matrix and Lypunov functional, We have obtained some sufficient conditions ensuring the existence, uniqueness and global exponential stability of both
the equilibrium point and the periodic solution for a class of impulsive fuzzy cellular neural networks with distributed delays. It is believed that these results are
significant and useful for the design and applications of the fuzzy cellular neural
networks.
Acknowledgments. This work was supported by grant 2006A109 from the Scientific Research Fund of Sichuan Provincial Education Department.
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Kelin Li
Department of Mathematics, Sichuan University of Science & Engineering, Sichuan
643000, China
E-mail address: lkl@suse.edu.cn
Zuoan Li
Department of Mathematics, Sichuan University of Science & Engineering, Sichuan
643000, China
E-mail address: lizuoan@suse.edu.cn
Qiankun Song
Department of Mathematics, Chongqing Jiaotong University, Chongqing, 400074, China
E-mail address: qiankunsong@163.com