# A Class of Vector Lyapunov Functions for Stability Analysis of Nonlinear Impulsive Differential Systems

Authors: Zhang Qunli
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

A novel and effective approach to stability of the solutions of nonlinear systems with impulsive effect is considered. The investigations are carried out by means of a class of vector Lyapunov functions and differential inequalities for piecewise continuous functions.
Simulation examples are given to illustrate the presented results.


---

## Body

## 1. Introduction

There has been a significant development in the theory of the impulsive differential systems within the past 20 years (see [1–17]). The efficient applications of impulsive differential systems require the finding of criteria for stability of their solutions.It is well known that the more complex the considered dynamical system is, the more difficult it is to find a Lyapunov function. This leads to employing several Lyapunov functions, which is called to be a vector Lyapunov function, for which each component provides information about a part of the dynamics. Then some of the complex aspects can be reduced to the connected dynamical systems and less rigid requirements can be satisfied. Hence, the corresponding theory, namely, the method of vector Lyapunov functions, offers a very flexible process (see [1–5, 11, 13, 18]).In this paper, without assuming Lipschitz conditions on these nonlinear functions, which play important roles in the considered systems, as needed in most other papers, we will use a class of vector Lyapunov functions, by means of the comparison principle (see [1, 2, 13]), to study global exponential stability of the solution of nonlinear impulsive differential systems.


## 2. Statement of the Problem

LetR
n be the n-dimensional Euclidean space with norm ∥
x
∥
=
x
T
x
=
〈
x
,
x
〉
=
(
∑
i
=
1
n
‍
x
i
2
)
1
/
2, x
=
(
x
1
,
x
2
,
…
,
x
n
)
T, let Ω be a bounded domain in R
m containing the origin, and 〈
u
,
v
〉
=
u
1
v
1
+
u
2
v
2
+
⋯
+
u
n
v
n, where u
=
(
u
1
,
u
2
,
…
,
u
n
)
T, v
=
(
v
1
,
v
2
,
…
,
v
n
)
T, and R
+
=
[
0
,
+
∞
).Lett
0
∈
R
+. Consider the impulsive functional differential system
(1)
x
˙
(
t
)
=
f
(
t
,
x
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
Δ
x
=
x
(
t
k
+
0
)
-
x
(
t
k
)
=
I
k
x
(
t
k
)
,
(
t
=
t
k
>
t
0
)
,
where f
:
(
t
0
,
+
∞
)
×
Ω
→
R
n, I
k
:
Ω
→
Ω, k
=
1,2
,
3
,
…, t
0
<
t
1
<
t
2
<
⋯, and lim
⁡
k
→
+
∞
t
k
=
+
∞.Definition 1 (see [12, 14, 15, 19]).
System (1) is called to be exponentially stable on a neighborhood Ω of the equilibrium point, if there exist constants μ
>
0, M
>
0 such that
(2)
∥
x
(
t
)
∥
≤
M
e
-
μ
t
∥
x
0
∥
,
(
t
≥
0
)
,
where x
(
t
) is any solution of (1) initiated from x
(
t
0
)
=
x
0.We introduce a kind of partial ordering defined in the following natural way. Foru
,
v
∈
R
n, u
≥
v
(
u
>
v
) if and only if u
j
≥
v
j
(
u
j
>
v
j
) for any j
=
1,2
,
…
,
n.Definition 2 (see [1, 2, 13]).
The functionΨ
:
Ω
→
R
n is called monotonically increasing function in Ω if Ψ
(
u
)
>
Ψ
(
v
) for u
>
v and Ψ
(
u
)
≥
Ψ
(
v
) for u
≥
v, u
,
v
∈
Ω.Definition 3 (see [1, 2, 13]).
The functionF
:
R
+
×
Ω
→
R
n is called quasi-monotonically increasing function in R
+
×
Ω if, for any two points (
t
,
u
) and (
t
,
v
) of R
+
×
Ω and for any i
∈
1,2
,
…
,
n, the inequality F
i
(
t
,
u
)
≥
F
i
(
t
,
v
) holds always when u
i
=
v
i and u
≥
v, that is, if, for any t
∈
R
+ fixed and any i
∈
1,2
,
…
,
n, the function F
i
(
t
,
u
) is nondecreasing with respect to (
u
1
,
…
,
u
i
-
1
,
u
i
+
1
,
…
,
u
n
).In the further considerations we will use the classν
0 of piecewise continuous auxiliary functions V
:
[
t
0
,
+
∞
)
×
Ω
→
R
+
m which are analogues of Lyapunov functions (see [20]).We denote byκ the class of all continuous and strictly increasing functions a
:
R
+
→
R
+ such that a
(
0
)
=
0.We are interested in focusing on the possibility of applying the comparison method to the stability of the zero solution of system (1). Together with the system (1), we will also consider the following comparison system:
(3)
u
˙
(
t
)
=
g
(
t
,
u
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
Δ
u
=
u
(
t
k
+
0
)
-
u
(
t
k
)
=
B
k
u
(
t
k
)
,
(
t
=
t
k
>
t
0
)
,
where g
:
R
+
×
S
(
ρ
)
→
R
m, B
k
:
S
(
ρ
)
→
R
m, k
=
1,2
,
3
,
…, t
0
<
t
1
<
t
2
<
⋯, lim
⁡
k
→
+
∞
t
k
=
+
∞, and S
(
ρ
)
=
{
x
∈
R
n
:
∥
x
∥
<
ρ
} is an open subset of R
m.Lemma 4 (see [1, 2, 13]).
Let the following conditions be fulfilled:(1)
V
:
R
+
×
S
(
ρ
)
→
R
+
m
,
V
∈
ν
0,
(4)
D
+
V
(
t
,
x
)
≤
g
(
t
,
V
(
t
,
x
)
)
,
t
≠
t
k
,
(
t
,
x
)
∈
R
+
×
S
(
ρ
)
,
where g
:
R
+
×
R
+
m
→
R
+
m is continuous, quasi-monotonically increasing in (
t
k
-
1
,
t
k
]
×
R
m and g
(
t
,
0
)
=
0 for t
∈
R
+ and D
+
V
(
t
,
x
)
=
lim
⁡
h
→
0
+
sup
⁡
⁡
(
1
/
h
)
[
V
(
t
+
h
,
x
+
h
f
(
t
,
x
)
)
-
V
(
t
,
x
)
];
(2)
there exists aρ
0 which satisfies when x
0
∈
S
(
ρ
0
), u
+
B
k
(
u
) are monotonically increasing in S
(
ρ
) and B
k
(
0
)
=
0;
(3)
there exist the functionsa
(
·
)
,
b
(
·
)
∈
κ which satisfy b
(
∥
x
∥
)
≤
L
(
t
,
x
)
≤
a
(
∥
x
∥
) for all (
t
,
x
)
∈
R
+
×
S
(
ρ
), where L
(
t
,
x
)
=
∑
i
=
1
m
‍
V
i
(
t
,
x
)
,
V
(
t
,
x
)
=
(
V
1
(
t
,
x
)
,
V
2
(
t
,
x
)
,
…
,
V
m
(
t
,
x
)
)
T.
If the zero solution of system (3) is stable, then the zero solution of system (1) is stable with respect to x
(
t
).


## 3. Theory Results of Impulsive Stability

### 3.1. Basic Results

Consider the following system(5)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
(
t
,
0,0
)
=
G
(
t
,
0,0
)
=
0.We will say that condition (A) is satisfied if the following conditions hold.(A) The functionsF
(
t
,
x
(
t
)
,
y
(
t
)
), G
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exists a function h
(
t
), and the condition
(6)
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
≤
h
(
t
)
holds for any α
,
β.We will say that condition (B) is satisfied if the following conditions hold.(B) The functionsF
i
(
t
,
x
(
t
)
,
y
(
t
)
), G
i
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exist functions h
j
i
(
t
), and the condition
(7)
(
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
×
(
α
j
x
(
t
)
+
β
j
y
(
t
)
)
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T

)
×
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
-
1
≤
h
j
i
(
t
)
holds for any α
j
,
β
j
,
α
i
,
β
i.We will say that condition (C) is satisfied if the following conditions hold.(C) The functionsF
i
(
t
,
x
(
t
)
,
y
(
t
)
), G
i
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exist functions h
j
i
(
t
), and the condition
(8)
(
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
×
(
α
j
x
(
t
)
+
β
j
y
(
t
)
)
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T

)
×
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
∥
α
i
x
(
t
-
τ
)
+
β
i
y
(
t
-
τ
)
∥
)
-
1
≤
s
j
i
(
t
)
holds for any α
j
,
β
j
,
α
i
,
β
i.Theorem 5.
If the operatorsF
,
G in system (5) satisfy condition (A) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(9)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
∀
t
≥
0
.Proof.
For allt
≥
0 and r
≥
0, we have
(10)
d
α
e
r
t
x
(
t
)
d
t
=
α
r
e
r
t
x
(
t
)
+
α
e
r
t
x
˙
(
t
)
d
α
e
r
t
x
(
t
)
d
t
=
α
r
e
r
t
x
(
t
)
+
α
e
r
t
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
d
β
e
r
t
y
(
t
)
d
t
=
β
r
e
r
t
y
(
t
)
+
β
e
r
t
y
˙
(
t
)
d
α
e
r
t
x
(
t
)
d
t
=
β
r
e
r
t
y
(
t
)
+
β
e
r
t
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
d
α
e
r
t
x
(
t
)
d
t
+
d
β
e
r
t
y
(
t
)
d
t
=
d
(
α
e
r
t
x
(
t
)
+
β
e
r
t
y
(
t
)
)
d
t
=
d
[
e
r
t
(
α
x
(
t
)
+
β
y
(
t
)
)
]
d
t
=
e
r
t
[
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
=
e
r
t
w
+
α
r
x
(
t
)
+
β
r
y
(
t
)
]
,
e
r
t
(
α
x
(
t
)
+
β
y
(
t
)
)
=
e
r
s
(
α
x
(
s
)
+
β
y
(
s
)
)
+
∫
s
t
‍
e
r
u
[
α
F
(
u
,
x
(
u
)
,
y
(
u
)
)
+
β
G
(
u
,
x
(
u
)
,
y
(
u
)
)
+
∫
s
t
‍
e
r
u
w
+
α
r
x
(
u
)
+
β
r
y
(
u
)
]
d
u
,
e
r
t
∥
α
x
(
t
)
+
β
y
(
t
)
∥
-
e
r
s
∥
α
x
(
s
)
+
β
y
(
s
)
∥
≤
∫
s
t
‍
e
r
u
∥
α
F
(
u
,
x
(
u
)
,
y
(
u
)
)
+
β
G
(
u
,
x
(
u
)
,
y
(
u
)
)
≤
∫
s
t
‍
e
r
u
w
+
α
r
x
(
u
)
+
β
r
y
(
u
)
∥
d
u
.
The derivative with respect to time t is
(11)
e
r
t
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
+
r
e
r
t
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
e
r
t
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
≤
e
r
t
w
e
r
t
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
.
That is,
(12)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
≤
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
.
It is easy to prove that ∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥ is monotonically decreasing on r; thus the limit
(13)
lim
⁡
r
→
+
∞
(
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
lim
⁡
r
→
+
∞
w
+
α
r
x
(
t
)
+
β
r
y
(
t
)
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
exists. From the norm ∥
x
∥
=
x
T
x
=
〈
x
,
x
〉
=
(
∑
i
=
1
n
‍
x
i
2
)
1
/
2 and condition (A), we obtain
(14)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
[
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
α
r
x
(
t
)
≤
ew
+
β
r
y
(
t
)
)
T
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
r
y
(
t
)
)
T
+
β
r
y
)
T
+
α
r
x
(
t
)
+
β
r
y
(
t
)
)
]
1
/
2
-
r
(
(
α
x
(
t
)
+
β
y
(
t
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
)
1
/
2
=
q
(
t
,
α
,
β
)
(
p
(
t
,
α
,
β
)
)
1
/
2
+
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
q
(
t
,
α
,
β
)
/
r
(
p
(
t
,
α
,
β
)
)
1
/
2
/
r
+
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
lim
⁡
r
→
+
∞
q
(
t
,
α
,
β
)
/
r
(
p
(
t
,
α
,
β
)
)
1
/
2
/
r
+
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)

+

β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)

+

β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)

+

β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)

+

β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
×
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
,
where q
(
t
,
α
,
β
)
=
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
2
+
2
r
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
) and p
(
t
,
α
,
β
)
=
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
2
+
2
r
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
+
r
2
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2.
Namely,(15)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
.Corollary 6.
Whenx
(
t
)
∈
R
n, y
(
t
)
∈
R
m, and n
>
m, let
(16)
y
(
t
)
⟶
(
y
(
t
)
0
n
-
m
)
=
y
~
(
t
)
,
G
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
G
(
t
,
x
(
t
)
,
y
(
t
)
)
0
n
-
m
)
=
G
~
(
t
,
x
(
t
)
,
y
~
(
t
)
)
,
and (
α
F
(
t
,
x
(
t
)
,
y
~
(
t
)
)
+
β
G
~
(
t
,
x
(
t
)
,
y
~
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
~
(
t
)
)
/
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
2
≤
h
(
t
) holds for any α
,
β, where 0
n
-
m
=
(
0,0
,
…
,
0
)
T
∈
R
n
-
m. Then one obtains a similar result with Theorem 5:
(17)
(
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
∀
t
≥
0
.Corollary 7.
Whenx
(
t
)
∈
R
n, y
(
t
)
∈
R
m, and n
>
m, let
(18)
x
(
t
)
⟶
(
x
(
t
)
0
1
)
=
x
~
(
t
)
,
y
(
t
)
⟶
(
y
(
t
)
0
2
)
=
y
~
(
t
)
,
F
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
F
(
t
,
x
(
t
)
,
y
(
t
)
)
0
1
)
=
F
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
,
G
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
G
(
t
,
x
(
t
)
,
y
(
t
)
)
0
2
)
=
G
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
,
and (
α
F
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
+
β
G
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
)
T
(
α
x
~
(
t
)
+
β
y
~
(
t
)
)
/
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
2
≤
h
(
t
) holds for any α
,
β, where 0
1
=
(
0,0
,
…
,
0
)
T
∈
R
m, 0
2
=
(
0,0
,
…
,
0
)
T
∈
R
n. Then one obtains a similar result with Theorem 5:
(19)
(
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
∀
t
≥
0
.Remark 8.
Whenα
=
1, β
=
-
1, and F
=
G, the constant
(20)
sup
⁡
x
,
y
∈
Ω
,
x
≠
y
(
F
(
x
(
t
)
)
-
F
(
y
(
t
)
)
)
T
(
x
(
t
)
-
y
(
t
)
)
∥
x
(
t
)
-
y
(
t
)
∥
2
is called the nonlinear measure [21–26]. But the condition
(21)
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
≤
h
(
t
)
is weaker than the nonlinear measure.When system (5) is written as
(22)
x
˙
(
t
)
=
F
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
F
2
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
G
2
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2, we have the following conclusion.Corollary 9.
If the operatorsF
1
,
F
2
,
G
1
,
G
2 in system (22) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(23)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
e
h
12
≤
h
11
(
t
)
∀
t
≥
0
.When system (5) is written as
(24)
x
˙
(
t
)
=
∑
i
=
1
s
‍
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
∑
i
=
1
s
‍
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2
,
…
,
s, we have the following conclusion.Corollary 10.
If the operatorsF
i
,
G
i,  i
=
1,2
,
…
,
s, in system (24) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(25)
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
)
t
′
≤
∑
i
=
1
s
‍
h
j
i
(
t
)
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
∀
t
≥
0
.Corollary 11.
If the operatorsF
i
,
G
i, i
=
1,2
,
…
,
s, in system (24) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω and the constant σ satisfy
(26)
(
e
σ
t
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
)
t
′
≤
∑
i
=
1
s
‍
h
j
i
(
t
)
(
e
σ
t
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
(
∥
α
j
x
(
t
)
+
β
j
(
t
)
∥
)
t
′
∑
i
=
1
s
‍
h
j
(
t
)
(
e
σ
t
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
∀
t
≥
0
.When system (5) is
(27)
x
˙
(
t
)
=
F
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
F
2
(
t
,
x
(
t
-
τ
)
,
y
(
t
-
τ
)
)
,
y
˙
(
t
)
=
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
G
2
(
t
,
x
(
t
-
τ
)
,
y
(
t
-
τ
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2, and τ
>
0 is the time delay, we have the following conclusion.Corollary 12.
If the operatorsF
1
,
F
2
,
G
1
,
G
2 in system (27) satisfy conditions (B) and (C) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(28)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
s
11
(
t
)
∥
α
1
x
(
t
-
τ
)
+
β
1
y
(
t
-
τ
)
∥
≤
e
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
∀
t
≥
0
.When system (5) is written as
(29)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
z
˙
(
t
)
=
H
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, z
(
t
)
∈
R
m, and F
(
t
,
0,0
)
=
G
(
t
,
0,0
)
=
H
(
t
,
0,0
)
=
0, we have the following conclusion.Corollary 13.
If the operatorsF
,
G
,
H in system (29) satisfy the condition
(30)
(
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
)
)
×
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
2
)
-
1
≤
l
(
t
)
for any α
,
β
,
η and x
,
y
,
z
∈
Ω, then x
(
t
)
,
y
(
t
), and z
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω, and z
(
t
0
)
=
z
0
∈
Ω satisfy
(31)
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
)
t
′
≤
l
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
)
t
′
∥
l
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
∀
t
≥
0
.


### 3.2. Impulsive Stability

Consider the impulsive differential system(32)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
Δ
(
x
(
t
k
)
,
y
(
t
k
)
)
T
=
(
x
(
t
k
+
0
)
-
x
(
t
k
)
,
y
(
t
k
+
0
)
-
y
(
t
k
)
)
T
=
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
,
(
t
=
t
k
>
t
0
)
,
where F
,
G
:
(
t
0
,
+
∞
)
×
Ω
×
Ω
→
R
m, I
k
:
Ω
×
Ω
→
Ω
×
Ω, k
=
1,2
,
3
,
…
,
t
0
<
t
1
<
t
2
<
⋯, and lim
⁡
k
→
+
∞
t
k
=
+
∞.Theorem 14.
If the operatorsF
,
G in system (32) and the constants α
i
,
β
i, i
=
1,2, satisfy condition (B) for any x
,
y
∈
Ω and the matrix (
α
1
β
1
α
2
β
2
) is invertible,
(33)
(
h
11
(
t
)
h
12
(
t
)
h
21
(
t
)
h
22
(
t
)
)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
)
is continuous, quasi-monotonically increasing in (
t
k
-
1
,
t
k
]
×
R
m,
(34)
∥
(
α
i
β
i
)
(
(
x
(
t
)
y
(
t
)
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
)
∥
are monotonically increasing, i
=
1,2, and I
k
(
0,0
)
=
(
0,0
)
T. If the zero solution of system (3) is stable, u
(
t
)
=
(
u
1
(
t
)
u
2
(
t
)
), u
i
(
t
)
=
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥, and i
=
1,2, B
k
u
(
t
)
=
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
∥
,
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
∥
)
T
-
u
(
t
), then the zero solution of system (32) is also stable.Proof.
From Corollary10, we can obtain
(35)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
11
(
t
)
e
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
≤
h
1
(
t
)
∀
t
≥
0
,
(
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
)
t
′
≤
h
21
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
22
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
21
(
t
)
e
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
22
(
t
)
≤
h
2
(
t
)
∀
t
≥
0
.
It is easy to verify that all the conditions in the lemma are satisfied when V
=
(
u
1
(
t
)
,
u
2
(
t
)
)
T. So it is true that the zero solution of system (32) is stable.Remark 15.
Imitating Theorem14 and combining the above 7 corollaries, respectively, we can obtain similar results with Theorem 14, respectively.


## 3.1. Basic Results

Consider the following system(5)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
(
t
,
0,0
)
=
G
(
t
,
0,0
)
=
0.We will say that condition (A) is satisfied if the following conditions hold.(A) The functionsF
(
t
,
x
(
t
)
,
y
(
t
)
), G
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exists a function h
(
t
), and the condition
(6)
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
≤
h
(
t
)
holds for any α
,
β.We will say that condition (B) is satisfied if the following conditions hold.(B) The functionsF
i
(
t
,
x
(
t
)
,
y
(
t
)
), G
i
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exist functions h
j
i
(
t
), and the condition
(7)
(
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
×
(
α
j
x
(
t
)
+
β
j
y
(
t
)
)
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T

)
×
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
-
1
≤
h
j
i
(
t
)
holds for any α
j
,
β
j
,
α
i
,
β
i.We will say that condition (C) is satisfied if the following conditions hold.(C) The functionsF
i
(
t
,
x
(
t
)
,
y
(
t
)
), G
i
(
t
,
x
(
t
)
,
y
(
t
)
) are continuous in (
t
k
-
1
,
t
k
]
×
R
m
×
R
m. There exist functions h
j
i
(
t
), and the condition
(8)
(
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
×
(
α
j
x
(
t
)
+
β
j
y
(
t
)
)
(
α
j
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
j
G
i
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T

)
×
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
∥
α
i
x
(
t
-
τ
)
+
β
i
y
(
t
-
τ
)
∥
)
-
1
≤
s
j
i
(
t
)
holds for any α
j
,
β
j
,
α
i
,
β
i.Theorem 5.
If the operatorsF
,
G in system (5) satisfy condition (A) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(9)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
∀
t
≥
0
.Proof.
For allt
≥
0 and r
≥
0, we have
(10)
d
α
e
r
t
x
(
t
)
d
t
=
α
r
e
r
t
x
(
t
)
+
α
e
r
t
x
˙
(
t
)
d
α
e
r
t
x
(
t
)
d
t
=
α
r
e
r
t
x
(
t
)
+
α
e
r
t
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
d
β
e
r
t
y
(
t
)
d
t
=
β
r
e
r
t
y
(
t
)
+
β
e
r
t
y
˙
(
t
)
d
α
e
r
t
x
(
t
)
d
t
=
β
r
e
r
t
y
(
t
)
+
β
e
r
t
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
d
α
e
r
t
x
(
t
)
d
t
+
d
β
e
r
t
y
(
t
)
d
t
=
d
(
α
e
r
t
x
(
t
)
+
β
e
r
t
y
(
t
)
)
d
t
=
d
[
e
r
t
(
α
x
(
t
)
+
β
y
(
t
)
)
]
d
t
=
e
r
t
[
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
=
e
r
t
w
+
α
r
x
(
t
)
+
β
r
y
(
t
)
]
,
e
r
t
(
α
x
(
t
)
+
β
y
(
t
)
)
=
e
r
s
(
α
x
(
s
)
+
β
y
(
s
)
)
+
∫
s
t
‍
e
r
u
[
α
F
(
u
,
x
(
u
)
,
y
(
u
)
)
+
β
G
(
u
,
x
(
u
)
,
y
(
u
)
)
+
∫
s
t
‍
e
r
u
w
+
α
r
x
(
u
)
+
β
r
y
(
u
)
]
d
u
,
e
r
t
∥
α
x
(
t
)
+
β
y
(
t
)
∥
-
e
r
s
∥
α
x
(
s
)
+
β
y
(
s
)
∥
≤
∫
s
t
‍
e
r
u
∥
α
F
(
u
,
x
(
u
)
,
y
(
u
)
)
+
β
G
(
u
,
x
(
u
)
,
y
(
u
)
)
≤
∫
s
t
‍
e
r
u
w
+
α
r
x
(
u
)
+
β
r
y
(
u
)
∥
d
u
.
The derivative with respect to time t is
(11)
e
r
t
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
+
r
e
r
t
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
e
r
t
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
≤
e
r
t
w
e
r
t
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
.
That is,
(12)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
≤
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
.
It is easy to prove that ∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
α
r
x
(
t
)
+
β
r
y
(
t
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥ is monotonically decreasing on r; thus the limit
(13)
lim
⁡
r
→
+
∞
(
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
lim
⁡
r
→
+
∞
w
+
α
r
x
(
t
)
+
β
r
y
(
t
)
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
-
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
exists. From the norm ∥
x
∥
=
x
T
x
=
〈
x
,
x
〉
=
(
∑
i
=
1
n
‍
x
i
2
)
1
/
2 and condition (A), we obtain
(14)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
[
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
α
r
x
(
t
)
≤
ew
+
β
r
y
(
t
)
)
T
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
r
y
(
t
)
)
T
+
β
r
y
)
T
+
α
r
x
(
t
)
+
β
r
y
(
t
)
)
]
1
/
2
-
r
(
(
α
x
(
t
)
+
β
y
(
t
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
)
1
/
2
=
q
(
t
,
α
,
β
)
(
p
(
t
,
α
,
β
)
)
1
/
2
+
r
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
q
(
t
,
α
,
β
)
/
r
(
p
(
t
,
α
,
β
)
)
1
/
2
/
r
+
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
lim
⁡
r
→
+
∞
q
(
t
,
α
,
β
)
/
r
(
p
(
t
,
α
,
β
)
)
1
/
2
/
r
+
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)

+

β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)

+

β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
=
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)

+

β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)

+

β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
×
∥
α
x
(
t
)
+
β
y
(
t
)
∥
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
,
where q
(
t
,
α
,
β
)
=
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
2
+
2
r
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
) and p
(
t
,
α
,
β
)
=
∥
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
∥
2
+
2
r
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
+
r
2
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2.
Namely,(15)
(
∥
α
x
(
t
)
+
β
y
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
.Corollary 6.
Whenx
(
t
)
∈
R
n, y
(
t
)
∈
R
m, and n
>
m, let
(16)
y
(
t
)
⟶
(
y
(
t
)
0
n
-
m
)
=
y
~
(
t
)
,
G
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
G
(
t
,
x
(
t
)
,
y
(
t
)
)
0
n
-
m
)
=
G
~
(
t
,
x
(
t
)
,
y
~
(
t
)
)
,
and (
α
F
(
t
,
x
(
t
)
,
y
~
(
t
)
)
+
β
G
~
(
t
,
x
(
t
)
,
y
~
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
~
(
t
)
)
/
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
2
≤
h
(
t
) holds for any α
,
β, where 0
n
-
m
=
(
0,0
,
…
,
0
)
T
∈
R
n
-
m. Then one obtains a similar result with Theorem 5:
(17)
(
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
(
t
)
+
β
y
~
(
t
)
∥
∀
t
≥
0
.Corollary 7.
Whenx
(
t
)
∈
R
n, y
(
t
)
∈
R
m, and n
>
m, let
(18)
x
(
t
)
⟶
(
x
(
t
)
0
1
)
=
x
~
(
t
)
,
y
(
t
)
⟶
(
y
(
t
)
0
2
)
=
y
~
(
t
)
,
F
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
F
(
t
,
x
(
t
)
,
y
(
t
)
)
0
1
)
=
F
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
,
G
(
t
,
x
(
t
)
,
y
(
t
)
)
⟶
(
G
(
t
,
x
(
t
)
,
y
(
t
)
)
0
2
)
=
G
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
,
and (
α
F
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
+
β
G
~
(
t
,
x
~
(
t
)
,
y
~
(
t
)
)
)
T
(
α
x
~
(
t
)
+
β
y
~
(
t
)
)
/
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
2
≤
h
(
t
) holds for any α
,
β, where 0
1
=
(
0,0
,
…
,
0
)
T
∈
R
m, 0
2
=
(
0,0
,
…
,
0
)
T
∈
R
n. Then one obtains a similar result with Theorem 5:
(19)
(
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
)
t
′
≤
h
(
t
)
∥
α
x
~
(
t
)
+
β
y
~
(
t
)
∥
∀
t
≥
0
.Remark 8.
Whenα
=
1, β
=
-
1, and F
=
G, the constant
(20)
sup
⁡
x
,
y
∈
Ω
,
x
≠
y
(
F
(
x
(
t
)
)
-
F
(
y
(
t
)
)
)
T
(
x
(
t
)
-
y
(
t
)
)
∥
x
(
t
)
-
y
(
t
)
∥
2
is called the nonlinear measure [21–26]. But the condition
(21)
(
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
)
∥
α
x
(
t
)
+
β
y
(
t
)
∥
2
≤
h
(
t
)
is weaker than the nonlinear measure.When system (5) is written as
(22)
x
˙
(
t
)
=
F
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
F
2
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
G
2
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2, we have the following conclusion.Corollary 9.
If the operatorsF
1
,
F
2
,
G
1
,
G
2 in system (22) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(23)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
e
h
12
≤
h
11
(
t
)
∀
t
≥
0
.When system (5) is written as
(24)
x
˙
(
t
)
=
∑
i
=
1
s
‍
F
i
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
∑
i
=
1
s
‍
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, and F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2
,
…
,
s, we have the following conclusion.Corollary 10.
If the operatorsF
i
,
G
i,  i
=
1,2
,
…
,
s, in system (24) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(25)
(
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
)
t
′
≤
∑
i
=
1
s
‍
h
j
i
(
t
)
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
∀
t
≥
0
.Corollary 11.
If the operatorsF
i
,
G
i, i
=
1,2
,
…
,
s, in system (24) satisfy condition (B) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω and the constant σ satisfy
(26)
(
e
σ
t
∥
α
j
x
(
t
)
+
β
j
y
(
t
)
∥
)
t
′
≤
∑
i
=
1
s
‍
h
j
i
(
t
)
(
e
σ
t
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
(
∥
α
j
x
(
t
)
+
β
j
(
t
)
∥
)
t
′
∑
i
=
1
s
‍
h
j
(
t
)
(
e
σ
t
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥
)
∀
t
≥
0
.When system (5) is
(27)
x
˙
(
t
)
=
F
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
F
2
(
t
,
x
(
t
-
τ
)
,
y
(
t
-
τ
)
)
,
y
˙
(
t
)
=
G
1
(
t
,
x
(
t
)
,
y
(
t
)
)
+
G
2
(
t
,
x
(
t
-
τ
)
,
y
(
t
-
τ
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, F
i
(
t
,
0,0
)
=
G
i
(
t
,
0,0
)
=
0, i
=
1,2, and τ
>
0 is the time delay, we have the following conclusion.Corollary 12.
If the operatorsF
1
,
F
2
,
G
1
,
G
2 in system (27) satisfy conditions (B) and (C) for any x
,
y
∈
Ω, then x
(
t
) and y
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω satisfy
(28)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
s
11
(
t
)
∥
α
1
x
(
t
-
τ
)
+
β
1
y
(
t
-
τ
)
∥
≤
e
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
∀
t
≥
0
.When system (5) is written as
(29)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
z
˙
(
t
)
=
H
(
t
,
x
(
t
)
,
y
(
t
)
)
,
where x
(
t
)
∈
R
m, y
(
t
)
∈
R
m, z
(
t
)
∈
R
m, and F
(
t
,
0,0
)
=
G
(
t
,
0,0
)
=
H
(
t
,
0,0
)
=
0, we have the following conclusion.Corollary 13.
If the operatorsF
,
G
,
H in system (29) satisfy the condition
(30)
(
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
α
F
(
t
,
x
(
t
)
,
y
(
t
)
)
+
β
G
(
t
,
x
(
t
)
,
y
(
t
)
)
+
η
H
(
t
,
x
(
t
)
,
y
(
t
)
)
)
T
(
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
)
)
×
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
2
)
-
1
≤
l
(
t
)
for any α
,
β
,
η and x
,
y
,
z
∈
Ω, then x
(
t
)
,
y
(
t
), and z
(
t
) initiated from x
(
t
0
)
=
x
0
∈
Ω, y
(
t
0
)
=
y
0
∈
Ω, and z
(
t
0
)
=
z
0
∈
Ω satisfy
(31)
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
)
t
′
≤
l
(
t
)
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
(
∥
α
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
)
t
′
∥
l
x
(
t
)
+
β
y
(
t
)
+
η
z
(
t
)
∥
∀
t
≥
0
.


## 3.2. Impulsive Stability

Consider the impulsive differential system(32)
x
˙
(
t
)
=
F
(
t
,
x
(
t
)
,
y
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
y
˙
(
t
)
=
G
(
t
,
x
(
t
)
,
y
(
t
)
)
,
(
t
≥
t
0
,
t
≠
t
k
,
k
=
1,2
,
3
,
…
)
Δ
(
x
(
t
k
)
,
y
(
t
k
)
)
T
=
(
x
(
t
k
+
0
)
-
x
(
t
k
)
,
y
(
t
k
+
0
)
-
y
(
t
k
)
)
T
=
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
,
(
t
=
t
k
>
t
0
)
,
where F
,
G
:
(
t
0
,
+
∞
)
×
Ω
×
Ω
→
R
m, I
k
:
Ω
×
Ω
→
Ω
×
Ω, k
=
1,2
,
3
,
…
,
t
0
<
t
1
<
t
2
<
⋯, and lim
⁡
k
→
+
∞
t
k
=
+
∞.Theorem 14.
If the operatorsF
,
G in system (32) and the constants α
i
,
β
i, i
=
1,2, satisfy condition (B) for any x
,
y
∈
Ω and the matrix (
α
1
β
1
α
2
β
2
) is invertible,
(33)
(
h
11
(
t
)
h
12
(
t
)
h
21
(
t
)
h
22
(
t
)
)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
)
is continuous, quasi-monotonically increasing in (
t
k
-
1
,
t
k
]
×
R
m,
(34)
∥
(
α
i
β
i
)
(
(
x
(
t
)
y
(
t
)
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
)
∥
are monotonically increasing, i
=
1,2, and I
k
(
0,0
)
=
(
0,0
)
T. If the zero solution of system (3) is stable, u
(
t
)
=
(
u
1
(
t
)
u
2
(
t
)
), u
i
(
t
)
=
∥
α
i
x
(
t
)
+
β
i
y
(
t
)
∥, and i
=
1,2, B
k
u
(
t
)
=
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
∥
,
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
+
I
k
(
x
(
t
k
)
,
y
(
t
k
)
)
∥
)
T
-
u
(
t
), then the zero solution of system (32) is also stable.Proof.
From Corollary10, we can obtain
(35)
(
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
)
t
′
≤
h
11
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
11
(
t
)
e
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
12
(
t
)
≤
h
1
(
t
)
∀
t
≥
0
,
(
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
)
t
′
≤
h
21
(
t
)
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
22
(
t
)
∥
α
2
x
(
t
)
+
β
2
y
(
t
)
∥
≤
h
21
(
t
)
e
∥
α
1
x
(
t
)
+
β
1
y
(
t
)
∥
+
h
22
(
t
)
≤
h
2
(
t
)
∀
t
≥
0
.
It is easy to verify that all the conditions in the lemma are satisfied when V
=
(
u
1
(
t
)
,
u
2
(
t
)
)
T. So it is true that the zero solution of system (32) is stable.Remark 15.
Imitating Theorem14 and combining the above 7 corollaries, respectively, we can obtain similar results with Theorem 14, respectively.


## 4. Example

Example 1.
We consider the system [3, 13]
(36)
d
x
d
t
=
e
-
t
x
+
y
sin
⁡
t
-
(
x
3
+
x
y
2
)
sin
2
⁡
t
,
t
≠
t
k
,
d
y
d
t
=
x
sin
⁡
t
+
e
-
t
y
-
(
x
2
y
+
y
3
)
sin
2
⁡
t
,
t
≠
t
k
,
Δ
x
=
a
x
+
b
y
,
Δ
y
=
b
x
+
a
y
,
t
=
t
k
,
where 2
a
=
1
+
c
1
+
1
+
c
2
-
2, 2
b
=
1
+
c
1
-
1
+
c
2, -
1
<
c
1
≤
0, and -
1
<
c
2
≤
0. Consider
(37)
α
F
(
t
,
x
,
y
)
+
β
G
(
t
,
x
,
y
)
=
α
e
-
t
x
+
α
y
sin
⁡
t
-
α
(
x
3
+
x
y
2
)
sin
2
⁡
t
+
β
x
sin
⁡
t
+
β
e
-
t
y
-
β
(
x
2
y
+
y
3
)
sin
2
⁡
t
=
e
-
t
(
α
x
+
β
y
)
+
(
α
y
+
β
x
)
sin
⁡
t
-
(
α
x
+
β
y
)
(
x
2
+
y
2
)
sin
2
⁡
t
,
〈
α
F
(
t
,
x
,
y
)
+
β
G
(
t
,
x
,
y
)
,
α
x
+
β
y
〉
=
e
-
t
(
α
x
+
β
y
)
2
+
(
α
y
+
β
x
)
(
α
x
+
β
y
)
sin
⁡
t
-
(
α
x
+
β
y
)
2
(
x
2
+
y
2
)
sin
2
⁡
t
,
(
α
F
(
t
,
x
,
y
)
+
β
G
(
t
,
x
,
y
)
)
T
(
α
x
+
β
y
)
∥
α
x
+
β
y
∥
≤
e
-
t
(
α
x
+
β
y
)
2
∥
α
x
+
β
y
∥
2
∥
α
x
+
β
y
∥
+
(
α
y
+
β
x
)
(
α
x
+
β
y
)
sin
⁡
t
∥
α
x
+
β
y
∥
∥
α
y
+
β
x
∥
∥
α
y
+
β
x
∥
≤
e
-
t
∥
α
x
+
β
y
∥
+
∥
α
y
+
β
x
∥
|
sin
⁡
t
|
.
So (
∥
α
x
+
β
y
∥
)
t
′
≤
e
-
t
∥
α
x
+
β
y
∥
+
∥
α
y
+
β
x
∥
|
sin
⁡
t
|, t
≠
t
k.
Similarly,(
∥
α
y
+
β
x
∥
)
t
′
≤
e
-
t
∥
α
y
+
β
x
∥
+
∥
α
x
+
β
y
∥
|
sin
⁡
t
|,  t
≠
t
k.
Whenα
x
+
β
y
=
φ
(
α
y
+
β
x
) for all x
,
y, we can obtain φ
=
±
1. That is, α
=
β
≠
0 and α
=
-
β
≠
0, respectively. Then we get the comparison system
(38)
d
u
1
d
t
=
e
-
t
u
1
+
|
sin
⁡
t
|
u
2
,
t
≠
t
k
,
d
u
2
d
t
=
|
sin
⁡
t
|
u
1
+
e
-
t
u
2
,
t
≠
t
k
,
u
1
(
t
k
+
)
=
|
(
x
+
a
x
+
b
y
)
+
(
y
+
b
x
+
a
y
)
|
=
1
+
c
1
u
1
,
t
=
t
k
,
u
2
(
t
k
+
)
=
|
(
y
+
b
x
+
a
y
)
-
(
x
+
a
x
+
b
y
)
|
=
1
+
c
2
u
2
,
t
=
t
k
.
Whenα
x
+
β
y
≠
φ
(
α
y
+
β
x
) for all x
,
y, we can obtain the comparison system
(39)
d
u
1
d
t
=
e
-
t
u
1
+
|
sin
⁡
t
|
u
2
,
t
≠
t
k
,
d
u
2
d
t
=
|
sin
⁡
t
|
u
1
+
e
-
t
u
2
,
t
≠
t
k
,
u
1
(
t
k
+
)
=
|
1
+
a
|
u
1
+
|
b
|
u
2
,
t
=
t
k
,
u
2
(
t
k
+
)
=
|
b
|
u
1
+
|
1
+
a
|
u
2
,
t
=
t
k
.
We know that when the constantsα
,
β satisfy the condition in Theorem 14, u
1
,
u
2 in the systems (38) and (39) are stable by stability theory of impulsive differential systems. So it is true that the zero solution of system (36) is stable. Taking c
1
=
-
0.2, c
2
=
-
0.5, and initial conditions (
x
,
y
)
T
=
(
1,2
)
T, the stability simulations of the states x
(
t
)
,
y
(
t
) of system (36) are as shown in Figures 1 and 2, respectively.Figure 1
Stability ofx
(
t
) of system (36).Figure 2
Stability ofy
(
t
) of system (36).Example 2.
We consider the system [3]
(40)
d
x
d
t
=
-
2
x
(
y
2
-
(
2
+
sin
⁡
t
)
x
)
2
-
x
cos
⁡
⁡
t
2
+
sin
⁡
t
,
t
≠
t
k
,
d
y
d
t
=
-
y
(
y
2
-
(
2
+
sin
⁡
t
)
x
)
2
,
t
≠
t
k
,
Δ
x
=
(
-
1
2
)
x
,
Δ
y
=
(
1
2
-
1
)
,
t
=
t
k
.
Combining Corollary11 and imitating Example 1, we can obtain the comparison system
(41)
d
u
1
d
t
=
(
σ
+
|
cos
⁡
⁡
t
2
+
sin
⁡
t
|
)
u
1
,
t
≠
t
k
,
d
u
2
d
t
=
σ
u
2
,
t
≠
t
k
,
u
1
(
t
k
+
)
=
(
1
2
)
u
1
,
u
2
(
t
k
+
)
=
(
1
2
)
u
2
,
t
=
t
k
,
where u
1
=
e
σ
t
|
α
x
|, u
2
=
e
σ
t
|
β
y
|, and 0
≤
σ
<
1. When the constants α
,
β satisfy the condition in Theorem 14, it is true that the zero solution of system (40) is stable. Taking initial conditions (
x
,
y
)
T
=
(
1,2
)
T, the stability simulations of the states x
(
t
)
,
y
(
t
) of system (40) are as shown in Figures 3 and 4, respectively.Figure 3
Stability ofx
(
t
) of system (40).Figure 4
Stability ofy
(
t
) of system (40).Remark 16.
Comparing Examples1 and 2 in this paper with Examples 2 and 4 in the literature [3, 13] or [3], the vector Lyapunov functions V
(
t
,
x
,
y
)
=
(
|
α
x
+
β
y
|
|
β
x
+
α
y
|
), V
(
t
,
x
,
y
)
=
(
e
σ
t
|
α
x
|
e
σ
t
|
β
y
|
) in this paper are simpler than the vector Lyapunov functions V
(
t
,
x
,
y
)
=
(
(
x
+
y
)
2
(
x
-
y
)
2
), V
(
t
,
x
,
y
)
=
(
y
2
-
(
2
+
sin
⁡
t
)
x
)
2 in [3, 13] or [3], respectively, and the approach is easy to operate.Example 3.
We consider the system(42)
d
x
d
t
=
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
y
+
e
-
t
x
+
y
(
t
-
τ
)
sin
⁡
t
-
(
x
3
+
x
y
2
)
sin
2
⁡
t
,
t
≠
t
k
,
d
y
d
t
=
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
x
+
e
-
t
y
+
x
(
t
-
τ
)
sin
⁡
t
-
(
x
2
y
+
y
3
)
sin
2
⁡
t
,
t
≠
t
k
,
Δ
x
=
a
x
+
b
y
,
Δ
y
=
b
x
+
a
y
,
t
=
t
k
,
where 2
a
=
1
+
c
1
+
1
+
c
2
-
2, 2
b
=
1
+
c
1
-
1
+
c
2, -
1
<
c
1
≤
0, and -
1
<
c
2
≤
0.
Combining Corollary12 and imitating Example 1, we can obtain the comparison system
(43)
d
u
1
d
t
=
|
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
|
u
2
+
e
-
t
u
1
+
|
sin
⁡
t
|
u
2
(
t
-
τ
)
,
=
|
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
|
u
2
+
e
e
-
t
e
-
t
u
1
w
+
|
sin
⁡
t
|
t
≠
t
k
,
d
u
2
d
t
=
|
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
|
u
1
+
e
-
t
u
2
+
|
sin
⁡
t
|
u
1
(
t
-
τ
)
,
=
|
sin
⁡
(
log
⁡
⁡
(
t
+
1
)
)
ww
u
1
+
ee
e
-
t
e
-
t
+
|
sin
⁡
t
|
t
≠
t
k
,
u
1
(
t
k
+
)
=
|
1
+
a
|
u
1
+
|
b
|
u
2
,
u
2
(
t
k
+
)
=
|
b
|
u
1
+
|
1
+
a
|
u
2
,
=
|
1
+
a
|
u
1
+
|
b
|
u
2
,
e
e
-
t
u
2
(
t
k
+
)
we
=
|
b
|
u
1
t
=
t
k
,
where u
1
=
|
α
x
+
β
y
|, u
2
=
|
α
y
+
β
x
|. When the constants α
,
β satisfy the condition in Theorem 14, it is true that the zero solution of system (42) is stable. Taking c
1
=
-
0.8, c
2
=
-
0.9, τ
=
1, and initial conditions (
x
,
y
)
T
=
(
1,2
)
T, the stability simulations of the states x
(
t
)
,
y
(
t
) of system (42) are as shown in Figures 5 and 6, respectively.Figure 5
Stability ofx
(
t
) of system (42).Figure 6
Stability ofy
(
t
) of system (42).Example 4.
We consider the system(44)
d
x
d
t
=
-
2
x
(
y
2
-
(
2
+
sin
⁡
t
)
z
)
2
-
x
cos
⁡
⁡
t
2
+
sin
⁡
t
,
t
≠
t
k
,
d
y
d
t
=
e
-
t
y
+
z
sin
⁡
t
-
(
y
3
+
y
z
2
)
sin
2
⁡
t
,
t
≠
t
k
,
d
z
d
t
=
y
sin
⁡
t
+
e
-
t
z
-
(
y
2
z
+
z
3
)
sin
2
⁡
t
,
t
≠
t
k
,
Δ
x
=
(
-
1
2
)
x
,
Δ
y
=
a
y
+
b
z
,
Δ
z
=
b
y
+
a
z
,
Δ
x
=
(
-
1
2
)
x
x
,
Δ
y
=
a
y
+
b
z
,
Δ
z
=
z
t
=
t
k
,
where 2
a
=
1
+
c
1
+
1
+
c
2
-
3, 2
b
=
1
+
c
1
-
1
+
c
2, -
1
<
c
1
≤
0, and -
1
<
c
2
≤
0.
Combining Corollary6 or Corollary 13 and imitating Example 1, we can obtain the comparison system
(45)
d
u
1
d
t
=
|
cos
⁡
⁡
t
2
+
sin
⁡
t
|
u
1
,
t
≠
t
k
,
d
u
2
d
t
=
|
e
-
t
+
sin
⁡
t
|
u
2
,
t
≠
t
k
,
u
1
(
t
k
+
)
=
(
1
2
)
|
α
|
u
1
,
u
2
(
t
k
+
)
=
∥
(
1
+
a
b
b
1
+
a
)
∥
u
2
,
=
(
1
2
)
|
α
|
u
1
,
u
2
(
t
k
+
)
h
=
∥
(
1
+
a
b
b
1
+
a
)
∥
u
2
t
=
t
k
,
where u
1
=
∥
α
(
x
0
)
∥, u
2
=
∥
β
(
y
z
)
∥, or the comparison system
(46)
d
u
1
d
t
=
|
cos
⁡
⁡
t
2
+
sin
⁡
t
|
u
1
,
t
≠
t
k
,
d
u
2
d
t
=
e
-
t
u
2
+
|
sin
⁡
t
|
u
3
,
t
≠
t
k
,
d
u
3
d
t
=
|
sin
⁡
t
|
u
2
+
e
-
t
u
3
,
t
≠
t
k
,
u
1
(
t
k
+
)
=
(
1
2
)
u
1
,
u
2
(
t
k
+
)
=
|
1
+
a
|
u
2
+
|
b
|
u
3
,
u
3
(
t
k
+
)
=
|
b
|
u
2
+
|
1
+
a
|
u
3
,
=
u
1
(
t
k
+
)
|
b
|
u
2
|
1
+
a
|
t
=
t
k
,
where u
1
=
|
γ
x
|, u
2
=
|
α
y
+
β
z
|, and u
3
=
|
α
z
+
β
y
|. When the constants α
,
β, and γ satisfy the conditions in Theorem 14 or Remark 15, it is true that the zero solution of system (44) is stable. Taking c
1
=
-
0.2, c
2
=
-
0.5, and initial conditions (
x
,
y
,
z
)
T
=
(
1,2
,
0.8
)
T, the stability simulations of the states x
(
t
)
,
y
(
t
), and z
(
t
) of system (44) are as shown in Figures 7, 8, and 9, respectively.Figure 7
Stability ofx
(
t
) of system (44).Figure 8
Stability ofy
(
t
) of system (44).Figure 9
Stability ofz
(
t
) of system (44).


## 5. Conclusion

This letter, which uses a class of vector Lyapunov functions, has studied the issue on impulsive stability of nonlinear systems and improves the existing results. Numerical simulations have verified the effectiveness of the method.



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*Source: 649012-2014-05-05.xml*