# Attracting and Quasi-Invariant Sets of Cohen-Grossberg Neural Networks with Time Delay in the Leakage Term under Impulsive Perturbations

Authors: Guiying Chen; Linshan Wang
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

A class of impulsive Cohen-Grossberg neural networks with time delay in the leakage term is investigated. By using the method ofM-matrix and the technique of delay differential inequality, the attracting and invariant sets of the networks are obtained. The results in this paper extend and improve the earlier publications. An example is presented to illustrate the effectiveness of our conclusion.


---

## Body

## 1. Introduction

Cohen-Crossberg neural network model, which is initially proposed by Cohen and Grossberg [1] in 1983, has been found successful applications in many fields such as pattern recognition, parallel computing, associative memory, signal and image processing, and combinatorial optimization. Hence, there has been increasing interest in studying the stability and asymptotic behavior of this model with delays, impulses, and unique equilibrium, and many significant results have been obtained (see, e.g., [2–6]). However, the equilibrium point sometimes does not exist in many real physical systems, so it is an interesting subject to discuss the attracting and invariant sets of the neural networks [7, 8].On the other hand, a leakage delay, which is the time delay in the leakage term and a factor affecting the stability of the system, has attracted considerable attentions (see, e.g., [9–13]). However, to the best of our knowledge, so far there are few results on the attracting and invariant sets of the Cohen-Grossberg neural networks with leakage delay. Motivated by the above discussion, in this paper, we investigate the attracting and quasi-invariant sets of a class of impulsive Cohen-Grossberg neural networks with leakage delay. By using the method of M-matrix and the technique of delay differential inequality, the attracting and invariant sets of the addressed networks are obtained. The results in this paper extend and improve the earlier publications. An example is presented to illustrate the effectiveness of our conclusion.


## 2. Model Description and Preliminaries

LetR
n be the space of n-dimensional real column vectors, N
≜
{
1,2
,
…
,
n
}, R
+
≜
[
0
,
+
∞
), N
≜
{
1,2
,
…
}, and R
m
×
n denotes the set of m
×
n real matrices. Usually, E denotes an n
×
n unit matrix. For A
,
B
∈
R
m
×
n or A
,
B
∈
R
n, the notation A
≥
B (A
≤
B, 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 z
∈
R
n is called a positive vector if z
≥
0.Letτ
>
0; for x
(
t
)
=
(
x
1
(
t
)
,
x
2
(
t
)
,
…
,
x
n
(
t
)
)
T
:
R
→
R
n, we define
(1)
x
(
t
)
+
=
x
1
t
,
x
2
t
,
…
,
x
n
t
T
,
x
i
(
t
)
τ
=
sup
⁡
-
τ
≤
s
≤
0
x
i
t
+
s
,
x
(
t
)
τ
=
x
1
(
t
)
τ
,
x
2
(
t
)
τ
,
…
,
x
n
t
τ
T
,
x
(
t
)
τ
+
=
x
t
+
τ
.C
[
X
,
Y
] denotes the space of continuous mappings from the topological space X to the topological space Y. Particularly, let C
≜
C
[
[
-
τ
,
0
]
,
R
n
].P
C
1
≜
{
ϕ
:
[
-
τ
,
0
]
→
R
n is continuous and with continuous derivative everywhere except at finite number of point t at which ϕ
(
t
+
), ϕ
(
t
-
), ϕ
˙
(
t
+
), and ϕ
˙
(
t
-
) exist and ϕ
(
t
+
)
=
ϕ
(
t
), ϕ
˙
(
t
+
)
=
ϕ
˙
(
t
), where ϕ
˙ denotes the derivative of ϕ
}. P
C
1 is a space of piecewise right-hand continuous functions with the norm ϕ
=
su
p
-
τ
≤
s
≤
0
|
ϕ
(
s
)
|,  ϕ
∈
P
C
1, where |
·
| is a norm in R
n.P
C
[
[
t
0
,
∞
)
,
R
m
×
n
]
≜
{
ψ
:
[
t
0
,
∞
)
→
R
m
×
n
∣
ψ
(
t
) is continuous at t
≠
t
k, ψ
(
t
k
+
) and ψ
(
t
k
-
) exist, ψ
(
t
k
)
=
ψ
(
t
k
+
), for k
∈
N
}.In this paper, we consider the following Cohen-Grossberg neural networks with impulses and time delays:(2)
x
˙
i
t
=
-
α
i
t
,
x
t
β
i
x
i
t
-
σ
-
∑
j
=
1
n
a
i
j
f
j
x
j
t
-
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
+
I
i
,
t
≥
t
0
,
t
≠
t
k
,
Δ
x
i
k
=
x
i
(
t
k
+
)
-
x
i
(
t
k
-
)
=
h
i
k
x
1
t
k
-
,
x
2
t
k
-
,
…
,
x
n
t
k
-
,
k
∈
N
,
x
i
(
t
0
+
s
)
=
ϕ
i
(
s
)
,
-
τ
<
s
≤
0
,
i
=
1,2
,
…
,
n
,
where n corresponds to the number of units in a neural network; x
i
(
t
) corresponds to the state of the ith unit at time t; f
j and g
j are the activation functions of the jth unit; τ
i
j
(
t
) denotes the transmission delay and satisfies 0
≤
τ
i
j
(
t
)
≤
r (r is a constant); σ
>
0 is the leakage delay. Consider τ
=
max
⁡
{
σ
,
r
}; α
i
(
t
,
x
(
t
)
)
>
0 represents the amplification function of the ith neuron; β
i
(
x
i
(
t
)
) is the behaved function at time t. Consider (
s
)
=
(
ϕ
1
(
s
)
,
…
,
ϕ
n
(
s
)
)
T
∈
P
C
1
[
[
-
τ
,
0
]
,
R
n
]. The fixed impulsive moments t
k (k
∈
N) satisfy τ
<
t
1
<
t
2
<
⋯ and li
m
k
→
∞
t
k
=
∞.Definition 1 (see [14]).
A functionx
(
t
)
:
[
t
0
-
τ
,
∞
)
→
R
n is said to be a solution of (2) through (
t
0
,
ϕ
), if x
(
t
)
∈
P
C
[
[
t
0
,
∞
)
,
R
n
] as t
≥
t
0, and satisfies (2) with the initial condition
(3)
x
t
0
+
s
=
ϕ
s
,
s
∈
-
τ
,
0
,
ϕ
∈
P
C
1
.
Throughout the paper, we always assume that, for anyϕ
∈
P
C
1, system (2) has at least one solution through (
t
0
,
ϕ
), denoted by x
(
t
,
t
0
,
ϕ
) or x
t
(
t
0
,
ϕ
) (simply x
(
t
) and x
t if no confusion should occur), where x
t
(
t
0
,
ϕ
)
=
x
(
t
+
s
,
t
0
,
ϕ
)
∈
P
C, s
∈
[
-
τ
,
0
].Definition 2 (see [7]).
The setS
⊂
P
C
1 is called a positive invariant set of (2), if for any initial value ϕ
∈
S we have the solution x
t
(
t
0
,
ϕ
)
∈
S for t
≥
t
0.Definition 3 (see [7]).
The setS
⊂
P
C
1 is called a quasi-invariant set of (2), if there exist a matrix W
≥
0 and a vector b
≥
0 such that, for any ϕ
∈
S, there exists a vector z such that the solution x
(
t
)
=
x
t
(
t
0
,
ϕ
) of (2) satisfies [
x
(
t
)
]
τ
+
≤
W
z
+
b, t
≥
t
0, as [
ϕ
]
τ
+
≤
z. Obviously, the set S is an invariant set of (2) if W
=
E and b
=
0.Definition 4 (see [7]).
The setS
⊂
P
C
1 is called a global attracting set of (2), if for any initial value ϕ
∈
P
C
1 the solution x
t
(
t
0
,
ϕ
) converges to S as t
→
+
∞. That is,
(4)
dist
⁡
x
t
,
S
⟶
0
,
t
⟶
+
∞
,
where dist
⁡
(
φ
,
S
)
=
in
f
ψ
∈
S
dist
⁡
(
φ
,
ψ
), dist
⁡
(
φ
,
ψ
)
=
su
p
s
∈
[
-
τ
,
0
]
|
φ
(
s
)
-
ψ
(
s
)
| for φ
∈
P
C
1.Definition 5 (see [8]).
The zero solution of (2) is said to be globally exponentially stable if for any solution x
(
t
,
t
0
,
ϕ
) there exist constants λ
>
0 and κ
≥
1 such that |
x
(
t
,
t
0
,
ϕ
)
|
≤
κ
ϕ
e
-
λ
(
t
-
t
0
),  t
≥
t
0.Definition 6 (see [15]).
Let the matrixD
=
(
d
i
j
)
n
×
n have nonpositive off-diagonal elements (i.e., d
i
j
≤
0,  i
≠
j); then each of the following conditions is equivalent to the statement that D is a nonsingular M-matrix.(i)
All the leading principle minors ofD are positive.
(ii)
D
=
C
-
G and ρ
(
C
-
1
G
)
<
1, where G
≥
0 and C
=
diag
⁡
{
c
1
,
…
,
c
n
}.
(iii)
The diagonal elements ofD are all positive and there exists a positive vector d such that D
d
>
0 or D
T
d
>
0.
For a nonsingular matrixD
∈
R
n
×
n, we denote Ω
M
(
D
)
≜
z
∈
R
n
,
z
>
0
∣
D
z
>
0.Lemma 7 (see [14]).
For a nonsingularM-matrix D, Ω
M
(
D
) is nonempty, and for any z
1
,
z
2
∈
Ω
M
(
D
) we have
(5)
k
1
z
1
+
k
2
z
2
∈
Ω
M
D
∀
k
1
,
k
2
>
0
.
So Ω
M
(
D
) is a cone without conical surface in R
n. We call it an “M-cone.”Lemma 8 (see [16]).
Lett
0
≤
b
≤
+
∞ and u
(
t
)
∈
[
[
t
0
,
b
)
,
R
n
]  satisfy
(6)
D
+
u
(
t
)
+
≤
R
(
t
,
u
(
t
)
)
P
u
(
t
)
+
u
(
t
)
τ
+
+
I
^
+
Q
u
(
t
)
τ
+
+
I
^
,
t
∈
[
t
0
,
b
)
,
u
(
t
0
+
s
)
∈
P
C
,
s
∈
[
-
τ
,
0
]
,
where P
=
(
p
i
j
)
n
×
n, p
i
j
≥
0, (i
≠
j), Q
=
(
q
i
j
)
n
×
n
≥
0, I
^
=
I
^
1
,
…
,
I
^
n
T
≥
0, R
(
t
,
u
)
=
diag
⁡
(
R
1
t
,
u
,
…
,
R
n
t
,
u
), and R
i
(
t
,
u
)
∈
C
[
[
t
0
,
b
)
×
R
n
,
R
+
], i
∈
N. Suppose that -
(
P
+
Q
) is a nonsingular M-matrix.
(1) If the initial condition satisfies
(7)
u
(
t
)
+
≤
x
*
,
t
0
-
τ
≤
t
≤
t
0
,
where x
*
=
(
x
1
*
,
…
,
x
n
*
)
T
=
-
d
(
P
+
Q
)
-
1
I
^, d
≥
1, then [
u
(
t
)
]
+
≤
x
*, for t
≥
t
0.
(2) Suppose that there exist a scalar λ
>
0 and a vector z
=
(
z
1
,
…
,
z
n
)
T
>
0 such that
(8)
(
λ
E
+
P
+
Q
e
λ
H
τ
)
z
≤
0
,
where
(9)
H
=
sup
⁡
t
≥
t
0
⁡
max
⁡
(
s
,
[
u
]
+
)
∈
[
t
-
τ
,
t
]
×
(
0
,
x
*
]
⁡
R
^
s
,
u
s
<
∞
,
0
≤
R
^
t
,
u
t
≤
min
⁡
1
≤
i
≤
n
⁡
R
i
t
,
u
t
.
If the initial condition satisfies
(10)
u
(
t
)
+
≤
z
e
-
λ
∫
t
0
t
R
^
s
,
u
s
d
s
-
P
+
Q
-
1
I
^
,
t
0
-
τ
≤
t
≤
t
0
,
then
(11)
u
(
t
)
+
≤
z
e
-
λ
∫
t
0
t
R
^
s
,
u
s
d
s
-
P
+
Q
-
1
I
^
,
t
≥
t
0
.


## 3. Main Results

In this paper, we always suppose the following.(A1)
α
i
(
t
,
x
(
t
)
)
∈
C
[
[
t
0
,
∞
)
×
R
n
,
[
0
,
α
-
i
]
], where α
-
i
>
0 is a constant, i
∈
N.
(A2)
β
i
(
·
) is differentiable, and there exist constants β
i
′
,
β
-
i
>
0 such that 0
<
β
i
′
<
β
˙
i
(
t
)
<
β
-
i,  i
∈
N, for any t
∈
[
t
0
,
+
∞
).
(A3)
f
i
(
·
) and g
i
(
·
) are Lipschitz continuous; that is, there exist constants k
i and l
i such that, for any x
1
,
x
2
∈
R,
(12)
f
i
x
1
-
f
i
x
2
≤
k
i
x
1
-
x
2
,
|
g
i
(
x
1
)
-
g
i
(
x
2
)
|
≤
l
i
x
1
-
x
2
.
(A4)
-
(
P
^
+
Q
^
) is a nonsingular M-matrix, where
(13)
P
^
=
p
^
i
j
n
×
n
,
Q
^
=
q
^
i
j
n
×
n
,
p
^
i
i
=
-
β
i
′
+
a
i
i
k
i
,
p
^
i
j
=
a
i
j
k
j
i
≠
j
,
q
^
i
i
=
σ
α
-
i
β
-
i
2
+
1
+
σ
α
-
i
β
-
i
b
i
i
l
i
+
σ
α
-
i
β
-
i
a
i
i
k
i
,
q
^
i
j
=
1
+
σ
α
-
i
β
-
i
b
i
j
l
j
+
σ
α
-
i
β
-
i
a
i
j
k
j
.
(A5)
x
+
H
k
(
x
)
+
≤
Γ
k
x
+,  k
∈
N, for any x
∈
R
n, where H
k
(
·
)
=
(
h
1
k
(
·
)
,
…
,
h
n
k
(
·
)
)
T,  Γ
k
=
(
γ
i
j
(
k
)
)
n
×
n
≥
0.
(A6)
Forz
∈
Ω
M
(
-
P
^
-
Q
^
),
(14)
Γ
k
z
≤
μ
k
z
,
Γ
k
-
P
^
-
Q
^
-
1
I
*
≤
v
k
-
P
^
-
Q
^
-
1
I
*
,where I
*
=
(
I
1
*
,
I
1
*
,
…
,
I
n
*
)
T,  I
i
*
=
(
1
+
σ
α
-
i
β
-
i
)
|
β
i
(
0
)
|
+
∑
j
=
1
n
(
|
a
i
j
|
|
f
j
(
0
)
|
+
|
b
i
j
|
|
g
j
(
0
)
|
)
+
|
I
i
|
+
σ
m
i
β
-
i, and μ
k,  v
k
≥
1 satisfy
(15)
ln
⁡
μ
k
≤
λ
∫
t
k
-
1
t
k
α
^
s
,
x
s
d
s
,
v
=
∑
k
=
1
n
ln
⁡
v
k
<
∞
,
k
∈
N
,
where α
^
(
s
,
x
(
s
)
)
≜
mi
n
1
≤
i
≤
n
{
α
i
(
s
,
x
(
s
)
)
}.Theorem 9.
Assume that (A1)–(A6) hold. Then, for anyd
^
≥
1, the set S
d
^
=
{
ϕ
∈
P
C
1
∣
[
ϕ
]
τ
+
≤
d
^
(
-
P
^
-
Q
^
)
-
1
I
*
} is a quasi-invariant set of (2).Proof.
Combining with the middle value theorem, from (2) we can get
(16)
d
x
i
(
t
)
d
t
=
-
α
i
t
,
x
t
β
i
x
i
t
-
σ
-
∑
j
=
1
n
a
i
j
f
j
x
j
t
-
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
+
I
i
=
α
i
t
,
x
t
-
β
i
x
i
t
-
σ
+
β
i
x
i
t
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
-
β
i
x
i
t
+
∑
j
=
1
n
a
i
j
f
j
x
j
t
+
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
-
I
i
=
α
i
t
,
x
t
σ
β
˙
i
x
i
t
-
1
-
θ
σ
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
×
x
˙
i
t
-
1
-
θ
σ
-
β
i
x
i
t
+
∑
j
=
1
n
a
i
j
f
i
x
i
t
+
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
-
I
i
,
where 0
<
θ
<
1.
Case 1. Let us first consider t
0
≤
t
≤
t
0
+
(
1
-
θ
)
σ. In this case x
˙
i
(
t
-
(
1
-
θ
)
σ
)
=
ϕ
˙
i
(
t
-
(
1
-
θ
)
σ
)
=
ϕ
˙
i
(
s
),  s
∈
[
-
τ
,
0
]. Notice that ϕ
∈
P
C
1; there exist m
i
>
0 such that [
ϕ
˙
i
(
s
)
]
τ
≤
m
i, for all i
∈
N.
From (A1)–(A4) and (16), we have
(17)
D
+
x
i
t
=
sgn
⁡
x
i
t
d
x
i
(
t
)
d
t
≤
α
i
t
,
x
t
-
β
i
′
x
i
t
+
∑
j
=
1
n
a
i
j
k
j
x
j
t
+
∑
j
=
1
n
b
i
j
l
j
x
j
(
t
)
τ
+
β
i
0
+
∑
j
=
1
n
a
i
j
f
j
0
+
b
i
j
g
j
0
∑
j
=
1
n
a
i
j
f
j
0
+
I
i
+
σ
β
-
m
i
i
∑
j
=
1
n
a
i
j
k
j
x
j
t
≤
α
i
t
,
x
t
∑
j
=
1
n
p
^
i
j
x
j
t
+
∑
j
=
1
n
q
^
i
j
x
j
t
τ
+
1
-
θ
σ
+
I
i
*
.
Hence,
(18)
D
+
x
t
+
≤
α
t
,
x
t
P
^
x
(
t
)
+
+
Q
^
x
(
t
)
τ
+
1
-
θ
σ
+
+
I
*
,
t
0
≤
t
≤
t
0
+
1
-
θ
σ
.
Case 2. Let us consider t
≥
t
0
+
(
1
-
θ
)
σ. From (16), we get
(19)
d
x
i
(
t
)
d
t
=
-
α
i
t
,
x
t
×
σ
β
˙
i
x
i
t
-
1
-
θ
σ
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
-
I
i
×
-
α
i
x
i
t
-
1
-
θ
σ
∑
j
=
1
n
a
i
j
f
j
x
j
t
-
1
-
θ
σ
×
β
i
x
i
t
-
2
-
θ
σ
-
∑
j
=
1
n
a
i
j
f
j
x
j
t
-
1
-
θ
σ
-
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
1
-
θ
σ
-
∑
j
=
1
n
a
i
j
f
j
x
j
t
-
1
-
θ
σ
τ
i
j
t
-
1
-
θ
σ
+
I
i
-
β
i
x
i
t
+
∑
j
=
1
n
a
i
j
f
j
x
j
t
+
∑
j
=
1
n
b
i
j
g
j
x
j
t
-
τ
i
j
t
-
I
i
.
Then from (A1)–(A4), we have
(20)
D
+
x
i
t
=
sgn
⁡
x
i
t
d
x
i
(
t
)
d
t
≤
α
i
(
t
,
x
(
t
)
)
×
σ
β
-
i
∑
j
=
1
n
a
i
j
α
-
i
β
-
i
x
i
t
τ
+
1
-
θ
σ
+
β
i
0
∑
j
=
1
n
a
i
j
∑
j
=
1
n
a
i
j
+
∑
j
=
1
n
a
i
j
k
j
x
j
t
τ
+
1
-
θ
σ
+
f
j
0
x
j
t
τ
+
1
-
θ
σ
+
∑
j
=
1
n
b
i
j
l
j
x
j
t
τ
+
1
-
θ
σ
+
g
j
0
x
j
t
τ
+
1
-
θ
σ
+
I
i
∑
j
=
1
n
a
i
j
-
β
i
′
x
i
t
+
β
i
0
+
∑
j
=
1
n
a
i
j
k
j
x
j
t
+
f
j
0
+
∑
j
=
1
n
b
i
j
l
j
(
x
j
(
t
)
τ
+
(
1
-
θ
)
σ
+
g
j
0
+
I
i
*
∑
j
=
1
n
a
i
j
=
α
i
t
,
x
t
∑
j
=
1
n
p
^
i
j
x
j
t
+
∑
j
=
1
n
q
^
i
j
x
j
t
τ
+
1
-
θ
σ
+
I
i
*
.
From (17) and (20), we get
(21)
D
+
x
t
+
≤
α
t
,
x
t
P
^
x
(
t
)
+
+
Q
^
x
(
t
)
τ
+
1
-
θ
σ
+
+
I
*
,
t
∈
t
k
,
t
k
+
1
,
k
∈
N
.
For the initial conditionsx
(
t
0
+
s
)
=
ϕ
(
s
), s
∈
[
-
τ
,
0
], where ϕ
(
s
)
∈
S
d
^, we have
(22)
x
t
+
≤
-
d
^
P
^
+
Q
^
-
1
I
*
,
t
0
-
τ
≤
t
≤
t
0
.
By (21) and Lemma 8, we have
(23)
x
t
+
≤
-
d
^
P
^
+
Q
^
-
1
I
*
,
t
0
≤
t
≤
t
1
.
Suppose that for allm
=
1,2
,
…
,
k the inequalities
(24)
x
t
+
≤
-
v
0
…
v
m
-
1
d
^
P
^
+
Q
^
-
1
I
*
,
t
m
-
1
≤
t
<
t
m
,
hold, where v
0
=
1. Then, from (A5) and (A6),
(25)
x
t
k
+
=
x
(
t
k
-
)
+
H
k
(
x
(
t
k
-
)
)
+
≤
Γ
k
x
(
t
k
-
)
+
≤
Γ
k
v
0
⋯
v
k
-
1
d
^
-
P
^
-
Q
^
-
1
I
*
≤
v
0
⋯
v
k
d
^
-
P
^
-
Q
^
-
1
I
*
.
This, together with v
k
≥
1, leads to
(26)
x
(
t
)
+
≤
v
0
⋯
v
k
d
^
-
P
^
-
Q
^
-
1
I
*
,
for
t
∈
t
k
-
τ
,
t
k
.
On the other hand,
(27)
D
+
x
t
+
≤
α
t
,
x
t
P
^
x
t
+
+
Q
^
x
t
τ
+
1
-
θ
σ
+
+
v
0
⋯
v
k
I
*
,
t
≠
t
k
.
It follows from (A4), (26), (27), and Lemma 8 that
(28)
x
(
t
)
+
≤
v
0
⋯
v
k
d
^
-
P
^
-
Q
^
-
1
I
*
for
t
∈
t
k
,
t
k
+
1
.
By the induction, we can conclude that
(29)
x
(
t
)
+
≤
v
0
⋯
v
k
d
^
-
P
^
-
Q
^
-
1
I
*
,
t
∈
t
k
-
1
,
t
k
,
k
∈
N
.
From (A6),
(30)
v
0
⋯
v
k
≤
e
v
,
and we have
(31)
x
(
t
)
+
≤
e
v
d
^
-
P
^
-
Q
^
-
1
I
*
,
t
∈
t
0
,
t
k
,
k
∈
N
.
This implies that the conclusion holds and the proof is complete.Theorem 10.
Assume that (A1)–(A6) hold. Then the setS
=
ϕ
∈
P
C
1
∣
[
ϕ
]
τ
+
≤
e
v
(
-
P
^
-
Q
^
)
-
1
I
* is a global attracting set of (2).Proof.
By a similar proof of Theorem9, we can get (21) and (31).
By continuity, we can findλ
>
0 and an enough small ε
>
0 such that
(32)
λ
+
ε
E
+
P
^
+
Q
^
e
λ
H
τ
+
σ
z
<
0
,
where
(33)
H
≜
sup
⁡
t
≥
t
0
⁡
max
⁡
s
∈
t
-
τ
,
t
,
x
∈
0
,
x
*
⁡
α
^
s
,
x
s
<
∞
,
α
^
s
,
x
s
≜
min
⁡
1
≤
i
≤
n
⁡
α
i
s
,
x
s
,
x
*
=
-
e
2
v
-
P
^
-
Q
^
-
1
I
*
.
For the initial conditionsx
(
t
0
+
s
)
=
ϕ
(
s
), s
∈
[
-
τ
,
0
], where ϕ
(
s
)
∈
P
C
1, we have
(34)
x
(
t
)
+
≤
κ
0
z
,
κ
0
=
ϕ
mi
n
1
≤
i
≤
n
z
i
,
t
0
-
τ
≤
t
≤
t
0
,
and so
(35)
x
(
t
)
+
≤
κ
0
z
e
-
λ
+
ε
∫
t
0
t
α
^
s
,
x
s
d
s
-
P
^
+
Q
^
-
1
I
*
,
t
0
-
τ
≤
t
≤
t
0
.
By the induction and Lemma 8, we can conclude that
(36)
x
(
t
)
+
≤
μ
0
⋯
μ
k
κ
0
z
e
-
(
λ
+
ε
)
∫
t
0
t
α
^
(
s
,
x
(
s
)
)
d
s
+
v
0
⋯
v
k
-
P
^
-
Q
^
-
1
I
*
,
t
∈
t
k
-
1
,
t
k
,
k
∈
N
.
From (A6), we conclude that
(37)
x
(
t
)
+
≤
e
λ
(
t
1
-
t
0
)
⋯
e
λ
(
t
k
-
1
-
t
k
-
2
)
κ
0
z
e
-
(
λ
+
ε
)
∫
t
0
t
α
^
(
s
,
x
(
s
)
)
d
s
+
v
0
⋯
v
k
-
P
^
-
Q
^
-
1
I
*
≤
κ
0
z
e
-
ε
∫
t
0
t
α
^
(
s
,
x
(
s
)
)
d
s
+
e
v
-
P
^
-
Q
^
-
1
I
*
,
t
∈
t
0
,
t
k
,
k
∈
N
.
This implies that the conclusion holds and the proof is complete.Theorem 11.
Assume that (A1)–(A5) withΓ
k
=
E hold. Then S
=
ϕ
∈
P
C
1
∣
[
ϕ
]
τ
+
≤
(
-
P
^
-
Q
^
)
-
1
I
* is a positive invariant set and also a global attracting set of (2).Proof (straightforward).
Obviously, ifα
(
0
)
=
0, then x
(
t
)
=
0 is a solution of (2).Corollary 12.
Assume that (A1)–(A5) withα
i
(
t
,
x
(
t
)
)
≥
δ
>
0,  I
*
=
0 hold. If
(38)
ln
⁡
μ
k
≤
λ
t
k
-
t
k
-
1
,
where μ
k
≥
1 satisfy Γ
k
z
≤
μ
k
z,  k
∈
N, then the zero solution of (2) is globally exponentially stable.Remark 13.
Ifσ
=
0, and H
k
(
·
)
≡
0, then from Theorems 9 and 10 we can get Theorem 5.1 and Corollary 5.1 in [16].Remark 14.
Ifσ
=
0, and H
k
(
·
)
≡
0, then from Corollary 12 we can get Corollary 3.1 in [5].


## 4. Illustrative Example

Consider the Cohen-Grossberg neural networks (2) with the following parameters, activation functions, amplification functions, behaved functions, and delay functions (n
=
2
,
i
,
j
=
1,2):
(39)
a
11
=
0.2
,
a
12
=
0.1
,
a
21
=
0.2
,
a
22
=
0.3
,
b
11
=
0.2
,
b
12
=
0.3
,
b
21
=
0.2
,
b
22
=
0.5
,
J
1
=
1
2
=
J
2
,
σ
=
0.2
,
α
1
t
,
x
t
=
max
⁡
1
2
,
sin
x
1
t
,
α
2
(
t
,
x
(
t
)
)
=
max
⁡
1
2
,
sin
2
x
2
t
β
1
x
1
t
=
3
x
1
t
,
β
2
x
2
t
=
2
x
2
t
,
f
1
(
x
1
)
=
g
1
(
x
1
)
=
1
2
x
1
+
1
-
x
1
-
1
f
2
x
2
=
g
2
x
2
=
x
2
,
τ
i
j
t
=
sin
j
t
,
Γ
k
=
e
1
/
2
k
E
.Obviously,k
=
l
=
1,  H
=
1
/
2,  μ
k
=
v
k
=
e
1
/
2
k,  and v
=
1. The parameters of condition (A4) are as follows:
(40)
P
^
=
-
2.8
0.1
0.2
-
1.7
,
Q
^
=
2.04
0.24
0.16
1.12
,
-
P
^
+
Q
^
=
0.76
-
0.34
-
0.36
0.58
.We can easily observe that-
(
P
^
+
Q
^
) is a nonsingular M-matrix and
(41)
Ω
M
-
P
^
-
Q
^
=
z
1
,
z
2
T
>
0
∣
18
29
z
1
<
z
2
<
38
17
z
1
.Letz
=
(
1,1
)
T
∈
Ω
M
(
-
P
^
-
Q
^
) and λ
=
0.082 which satisfies the inequality
(42)
(
λ
E
+
P
^
+
Q
^
e
λ
H
τ
+
σ
)
z
=
-
0.223
,
-
1.123
T
<
0
.Clearly, all conditions of Theorems9 and 10 are satisfied. So S
=
ϕ
∈
P
C
1
∣
ϕ
τ
+
≤
-
P
^
-
Q
^
-
1
I
*
=
(
31.89,32.92
)
T is a quasi-invariant set and also a global attracting set of (2).



---

*Source: 491801-2015-01-14.xml*