Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 494067, 7 pages
http://dx.doi.org/10.1155/2013/494067
Research Article
The Asymptotic Behavior for a Class of Impulsive Delay
Differential Equations
Zhichun Yang
Department of Mathematics, Chongqing Normal University, Chongqing 400047, China
Correspondence should be addressed to Zhichun Yang; zhichy@yahoo.com.cn
Received 27 January 2013; Accepted 21 March 2013
Academic Editor: Chuangxia Huang
Copyright © 2013 Zhichun Yang. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for
determining attracting sets and attracting basin of the impulsive system are obtained by developing the properties of quasi-invariant
sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the
impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive
systems are unbounded.
1. Introduction
Impulsive delay differential equations have attracted increasing interests since time delays and impulsive effects commonly exist in many fields such as population dynamics,
automatic control, drug administration, and communication
networks [1–4]. In past two decades, its asymptotical behaviors such as stability and attractivity of the equilibrium point
or periodical solutions have been deeply studied for impulsive
functional differential equations (see, [5–18]).
However, under impulsive perturbation, the solutions
may not be attracted to an equilibrium point or periodical
trajectory but to some bounded region. In this case, it is
interesting to investigate the attracting set and attracting
basin, that is, the region attracting the solutions and the range
in which initial values vary when remaining the attractivity
for impulsive delay differential equations. In [19], Xu and
Yang first give the method to estimate global attracting set
and invariant set for impulsive delayed systems by developing
delayed differential inequalities. The techniques are further
developed to study global attractivity for some complex
impulsive systems such as impulsive neutral differential equations [20, 21] and impulsive stochastic systems [22]. But the
techniques and methods given in the existing publications are
invalid for determining locally attracting set and attracting
basin for impulsive delay differential equations.
In this paper, our objective is to mainly discuss the asymptotical behavior on (locally) attracting set and its attracting
basin for a class of impulsive delay differential equations.
Based on the quasi-invariant properties, we estimate the
existence range of attracting set and attracting basin of the
impulsive delay systems by solving algebraic equations and
employing differential inequality technique. Examples are
given to illustrate the effectiveness of our method and show
that the asymptotic behavior of the impulsive systems may be
different from one of the corresponding continuous systems.
2. Preliminaries
Let 𝑁 be the set of all positive integers, 𝑅𝑛 the space of 𝑛dimensional real column vectors, and 𝑅𝑚×𝑛 the set of 𝑚 × 𝑛
real matrices. For 𝐴, 𝐵 ∈ 𝑅𝑚×𝑛 or 𝐴, 𝐵 ∈ 𝑅𝑛 , 𝐴 ≥ 𝐵(𝐴 ≤
𝐵, 𝐴 > 𝐵, 𝐴 < 𝐵) means that each pair of corresponding
elements of 𝐴 and 𝐵 satisfies the inequality “≥(≤, >, <).” 𝑅+𝑛 =
{𝑥 ∈ 𝑅𝑛 | 𝑥 ≥ 0}, 𝐸 = (1, 1, . . . , 1)𝑇 ∈ 𝑅𝑛 , and 𝐼 denotes an
𝑛 × 𝑛 unit matrix.
Let 𝜏 > 0 and 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ be the fixed points with
lim𝑘 → ∞ 𝑡𝑘 = ∞ (called impulsive moments).
𝐶[𝑋, 𝑌] denotes the space of continuous mappings from
Δ
the topological space 𝑋 to the topological space 𝑌. Let 𝐶 =
𝐶[[−𝜏, 0], 𝑅𝑛 ] especially.
2
Abstract and Applied Analysis
Δ
Morever, PC = {𝜙 : [−𝜏, 0] → 𝑅𝑛 | 𝜙(𝑡+ ) = 𝜙(𝑡) for
𝑡 ∈ [−𝜏, 0), 𝜙(𝑡− ) exists for 𝑡 ∈ (−𝜏, 0], 𝜙(𝑡− ) = 𝜙(𝑡) for all but
at most a finite number of points 𝑡 ∈ (−𝜏, 0]}. PC is a space of
piecewise right-hand continuous functions which is a nature
extension of the phrase space 𝐶.
Δ
We define PC[[𝑡0 , ∞), 𝑅𝑛 ] = {𝜓 : [𝑡0 , ∞) → 𝑅𝑛 | 𝜓(𝑡) is
continuous at 𝑡 ≠ 𝑡𝑘 , 𝜓(𝑡𝑘+ ) and 𝜓(𝑡𝑘− ) exist, 𝜓(𝑡𝑘 ) = 𝜓(𝑡𝑘+ ), for
𝑘 ∈ 𝑁}.
For 𝑥 ∈ 𝑅𝑛 , 𝐴 ∈ 𝑅𝑛×𝑛 , 𝜙 ∈ 𝐶 or 𝜙 ∈ PC, we define
󵄨 󵄨𝑇
󵄨 󵄨 󵄨 󵄨
[𝑥]+ = (󵄨󵄨󵄨𝑥1 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑥2 󵄨󵄨󵄨 , . . . , 󵄨󵄨󵄨𝑥𝑛 󵄨󵄨󵄨) ,
󵄨 󵄨
[𝐴]+ = (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨)𝑛×𝑛 ,
+
󵄩 󵄩 󵄩 󵄩
󵄩 󵄩
[𝜙]𝜏 = (󵄩󵄩󵄩𝜙1 󵄩󵄩󵄩𝜏 , 󵄩󵄩󵄩𝜙2 󵄩󵄩󵄩𝜏 , . . . , 󵄩󵄩󵄩𝜙𝑛 󵄩󵄩󵄩𝜏 ) ,
(1)
where ‖ 𝜙𝑖 ‖𝜏 = sup𝑠∈[−𝜏,0] ‖ 𝜙𝑖 (𝑠) ‖ and ‖ ⋅ ‖ is an norm in 𝑅𝑛 .
In this paper, we will consider a impulsive delay differential equations:
𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝑓 (𝑡, 𝑥𝑡 ) ,
𝑡 ≠ 𝑡𝑘 , 𝑡 ≥ 𝑡0 ,
Δ𝑥 = 𝐵𝑥 (𝑡𝑘− ) + 𝐼𝑘 (𝑡𝑘− , 𝑥 (𝑡𝑘− )) ,
𝑘 ∈ 𝑁,
(2)
̇ denotes the right-hand derivative of 𝑥(𝑡), Δ𝑥 =
where 𝑥(𝑡)
𝑥(𝑡𝑘+ ) − 𝑥(𝑡𝑘− ), 𝑥(𝑡𝑘 ) = 𝑥(𝑡𝑘+ ), 𝐴 = diag{𝑎1 ,𝑎2 , . . . , 𝑎𝑛 }, 𝐵 =
diag{𝑏1 , 𝑏2 , . . . , 𝑏𝑛 }, 𝑓 ∈ 𝐶[[𝑡𝑘−1 , 𝑡𝑘 ) × PC, 𝑅𝑛 ], and the limit
lim(𝑡,𝜙) → (𝑡𝑘− ,𝜑) 𝑓(𝑡, 𝜙) = 𝑓(𝑡𝑘− , 𝜑) exists, 𝐼𝑘 ∈ 𝐶[[𝑡0 , ∞) ×
𝑅𝑛 , 𝑅𝑛 ], and 𝑥𝑡 ∈ PC is defined by 𝑥𝑡 (𝑠) = 𝑥(𝑡 + 𝑠), 𝑠 ∈ [−𝜏, 0].
A function 𝑥(𝑡) : [𝑡0 − 𝜏, ∞) → 𝑅𝑛 is called to be a
solution of (2) through (𝑡0 , 𝜙), if 𝑥(𝑡) ∈ PC[[𝑡0 , ∞), 𝑅𝑛 ] as
𝑡 ≥ 𝑡0 , and satisfies (2) with the initial condition
𝑥 (𝑡0 + 𝑠) = 𝜙 (𝑠) ,
𝑠 ∈ [−𝜏, 0] , 𝜙 ∈ PC.
(3)
Throughout the paper, we always assume that for any 𝜙 ∈ PC,
system (2) has at least one solution through (𝑡0 , 𝜙), denoted
by 𝑥(𝑡, 𝑡0 , 𝜙) or 𝑥𝑡 (𝑡0 , 𝜙) (simply 𝑥(𝑡) and 𝑥𝑡 if no confusion
should occur), where 𝑥𝑡 (𝑡0 , 𝜙) = 𝑥(𝑡 + 𝑠, 𝑡0 , 𝜙) ∈ PC, 𝑠 ∈
[−𝜏, 0].
In this paper, we need the following definitions involving
attracting set, attracting basin, the quasi-invariant set of
impulsive systems, and monotonous vector functions.
Definition 1. The set 𝑆 ⊂ PC is called to be an attracting set
of (2), and 𝐷 ⊂ PC is called an attraction basin of 𝑆, if for
any initial value 𝜙 ∈ 𝐷, the solution 𝑥𝑡 (𝑡0 , 𝜙) converges to 𝑆
as 𝑡 → +∞. That is,
Definition 3. Let Ω ⊂ 𝑅𝑛 . The vector function 𝐹(𝑥) : Ω →
𝑅𝑛 is called to be monotonically nondecreasing in Ω, if for
any 𝑥󸀠 , 𝑥󸀠󸀠 ∈ Ω, 𝑥󸀠 ≤ 𝑥󸀠󸀠 implies 𝐹(𝑥󸀠 ) ≤ 𝐹(𝑥󸀠󸀠 ).
3. Main Results
In this paper, we always make the following assumptions.
(𝐻1 ) There exist nonnegative constants 𝜃, 󰜚 such that 0 <
𝜃 ≤ 𝑡𝑘 − 𝑡𝑘−1 ≤ 󰜚, for 𝑘 ∈ 𝑁.
(𝐻2 ) [𝑓(𝑡, 𝜑)]+ ≤ 𝑝([𝜑]+𝜏 ) for 𝑡 ≥ 𝑡0 and 𝜑 ∈ PC, where the
vector function 𝑝(⋅) : 𝑅+𝑛 → 𝑅+𝑛 is continuous and
monotonically nondecreasing in 𝑅+𝑛 .
(𝐻3 ) [𝐼𝑘 (𝑡, 𝑥)]+ ≤ 𝑞([𝑥]+ ), for 𝑡 ≥ 𝑡0 , 𝑘 ∈ 𝑁 and 𝑥 ∈
𝑅𝑛 , where the vector function 𝑞(⋅) : 𝑅+𝑛 → 𝑅+𝑛 is
continuous and monotonically nondecreasing in 𝑅+𝑛 .
To obtain attractivity, we first give the quasi-invariant properties of (2).
Theorem 4. Assume that in addition to (𝐻1 )–(𝐻3 ), there is a
vector 𝑧∗ ≥ 0 such that
−1
𝑝 (𝑀𝑧∗ ) + 𝑊[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧∗ ) − 𝑊𝑧∗ < 0,
(5)
where 𝑊 = diag{𝑤1 , . . . , 𝑤𝑛 }, 𝑀 = diag{𝑚1 , . . . , 𝑚𝑛 }, 𝑤𝑖 >
0, 𝑚𝑖 ≥ 1, 𝑖 = 1, 2, . . . , 𝑛, are defined by
󵄨
󵄨
ln 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
󵄨
󵄨
{
{
−
, if 0 < 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨 < 1,
−𝑎
{ 𝑖
{
󰜚
𝑤𝑖 = {
{
󵄨󵄨
󵄨󵄨
{
{−𝑎 − ln 󵄨󵄨1 + 𝑏𝑖 󵄨󵄨 , if 󵄨󵄨1 + 𝑏 󵄨󵄨 ≥ 1,
𝑖
𝑖 󵄨󵄨
󵄨󵄨
(6)
{
𝜃
1
󵄨
󵄨
{ 󵄨󵄨
󵄨 , if 0 < 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨 < 1,
𝑚𝑖 = { 󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
󵄨
󵄨
if 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨 ≥ 1.
{1,
Then, the set 𝐷 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ 𝑧∗ } is a positive quasiinvariant set of (2). When 𝑀 = 𝐼 especially, 𝐷 is a positive
invariant set of (2).
Proof. Let 𝑥(𝑡) = 𝑥(𝑡, 𝑡0 , 𝜙) be a solution of (2) through
(𝑡0 , 𝜙). It is easily verified that the following formula for the
variation of parameters is valid:
𝑡
𝑥 (𝑡) = 𝐾 (𝑡, 𝑡0 ) 𝜙 (0) + ∫ 𝐾 (𝑡, 𝑠) 𝑓 (𝑠, 𝑥𝑠 ) 𝑑𝑠
𝑡0
+ ∑ 𝐾 (𝑡, 𝑡𝑘 ) 𝐼𝑘 (𝑡𝑘− , 𝑥 (𝑡𝑘− )) ,
𝑡0 <𝑡𝑘 ≤𝑡
(7)
𝑡 ≥ 𝑡0 ,
(4)
where 𝐾(𝑡, 𝑠) is the Cauchy matrix of linear impulsive system
where dist(𝜑, 𝑆) = inf 𝜓∈𝑆 dist(𝜑, 𝜓), dist(𝜑, 𝜓) = sup𝑠∈[−𝜏,0]
‖𝜑(𝑠) − 𝜓(𝑠)‖, for 𝜑 ∈ PC.
𝑡 ≠ 𝑡𝑘 ,
{
{𝑦̇ (𝑡) = 𝐴𝑦 (𝑡) ,
(8)
{
{
+
−
Δ𝑦
(𝑡
)
=
𝐵𝑦
(𝑡
)
,
𝑘
∈
𝑁.
𝑘
𝑘
{
According to the representation of the Cauchy matrix (see
page 74 [2]),
dist (𝑥𝑡 (𝑡0 , 𝜙) , 𝑆) 󳨀→ 0,
a𝑠 𝑡 󳨀→ +∞,
Definition 2. The set 𝐷 ⊂ PC is called to be a positive quasiinvariant set of (2), if there is a positive diagonal matrix
𝐿 = diag{𝑙𝑖 } such that for any initial value 𝜙 ∈ 𝐷, the solutions
𝑥𝑡 (𝑡0 , 𝜙) satisfy 𝐿𝑥𝑡 (𝑡0 , 𝜙) ∈ 𝐷, for 𝑡 ≥ 𝑡0 . When 𝐿 =
𝐼(identity matrix) especially, the set 𝐷 is called positively
invariant.
𝐾 (𝑡, 𝑠) = 𝑒𝐴(𝑡−𝑠) ∏ (𝐼 + 𝐵) ,
𝑠<𝑡𝑘 ≤𝑡
𝑡 ≥ 𝑠 ≥ 𝑡0 .
(9)
Abstract and Applied Analysis
3
Since 0 < 𝜃 ≤ 𝑡𝑘 − 𝑡𝑘−1 ≤ 󰜚, for 𝑘 ∈ 𝑁, we obtain the following
estimate:
󵄨
󵄨
∏ 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
+
[𝑥 (𝑡∗ )] ≤ 𝑒−𝑊(𝑡
𝑠<𝑡𝑘 ≤𝑡
1
󵄨󵄨
󵄨
󵄨
󵄨
(ln|1+𝑏𝑖 |/󰜚)(𝑡−𝑠)
= 󵄨󵄨
, if 0 < 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨 < 1,
󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
{
󵄨𝑒
{
{󵄨
󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
≤{
{
{󵄨
󵄨(𝑡−𝑠)/𝜃
󵄨
󵄨
= 𝑒(ln|1+𝑏𝑖 |/𝜃)(𝑡−𝑠) ,
if 󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨 ≥ 1.
{󵄨󵄨󵄨1 + 𝑏𝑖 󵄨󵄨󵄨
(10)
((𝑡−𝑠)/󰜚)−1
+
−𝑊(𝑡−𝑠)
[𝐾 (𝑡, 𝑠)] ≤ 𝑀𝑒
,
𝑡 ≥ 𝑠 ≥ 𝑡0 .
(11)
−𝑊(𝑡−𝑡0 )
[𝑥 (𝑡)] ≤ 𝑀𝑒
𝑡
+
[𝜙]𝜏
+ 𝑀∫ 𝑒
𝑡0
𝑡 ≥ 𝑡0 .
𝑡0 <𝑡𝑘 ≤𝑡
−𝑊(𝑡−𝑡𝑘 )
≤ ∑ 𝑒
𝑡0 <𝑡𝑘 ≤𝑡
∗
−𝑡𝑘 )
≤𝑡∗
−𝑡0 )
−𝑡0 )
𝑞 (𝑀𝑧)
𝑀𝑧 + 𝑀 (𝐼 − 𝑒−𝑊(𝑡
∗
−𝑡0 )
∗
−𝑡0 )
) 𝑊−1 𝑝 (𝑀𝑧)
−1
) [𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧)
𝑀𝑊−1
−𝑊𝜃
[𝐼 − 𝑒
−1
−𝑊(𝑡𝑘 −𝑡𝑘−1 )
[𝐼 − 𝑒
= 𝑀𝑧.
(18)
−𝑊𝜃 −1
] [𝐼 − 𝑒
−1
< 𝑀𝑊−1 [𝑊𝑧 − 𝑝 (𝑀𝑧) − 𝑊[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧)]
+ 𝑊−1 𝑀𝑝 (𝑀𝑧) + 𝑀[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧)
Since 𝑡𝑘 − 𝑡𝑘−1 ≥ 𝜃 > 0 and 𝑊 = diag{𝑤1 , . . . , 𝑤𝑛 } > 0, we have
= ∑ 𝑒
∗
∗
𝑝 (𝑀𝑧) 𝑑𝑠
+ 𝑊−1 𝑀𝑝 (𝑀𝑧) + 𝑀[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧)
(12)
𝑡0 <𝑡𝑘 ≤𝑡
∑ 𝑒
−𝑠)
−1
+
𝑝 ([𝑥𝑠 ]𝜏 ) 𝑑𝑠
+
−𝑊(𝑡−𝑡𝑘 )
𝑡0 <𝑡𝑘
= 𝑒−𝑊(𝑡
∗
𝑡0
+ 𝑀 ∑ 𝑒−𝑊(𝑡
< 𝑒−𝑊(𝑡
𝑡∗
+
𝑀[𝜙]𝜏 + 𝑀 ∫ 𝑒−𝑊(𝑡
−1
−𝑊(𝑡−𝑠)
+ 𝑀 ∑ 𝑒−𝑊(𝑡−𝑡𝑘 ) 𝑞 ([𝑥 (𝑡𝑘− )] ) ,
−𝑊(𝑡−𝑡𝑘 )
−𝑡0 )
× [𝑊𝑧 − 𝑝 (𝑀𝑧) − 𝑊[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑀𝑧)]
By (7) and (11) and the assumptions (𝐻2 ) and (𝐻3 ), then
+
∗
+ 𝑀 (𝐼 − 𝑒−𝑊(𝑡
In terms of the definition of 𝑀 and 𝑊,
𝑡0 <𝑡𝑘 ≤𝑡
By using (12), (13), (14), (17), 𝑊 > 0, and the monotonicity of
𝑝(⋅), 𝑞(⋅), we can get
]
This contradicts (16), and so (15) holds. Letting 𝜀 → 0, from
(15), we have for any 𝜙 ∈ 𝐷 (i.e., [𝜙]+𝜏 ≤ 𝑧∗ ),
−𝑊𝜃 −1
] [𝐼 − 𝑒
]
+
[𝑥 (𝑡, 𝑡0 , 𝜙)] ≤ 𝑀𝑧∗ ,
+
that is, [𝑀−1 𝑥 (𝑡, 𝑡0 , 𝜙)] ≤ 𝑧∗ ,
𝑡 ≥ 𝑡0 .
(19)
−𝑊𝜃 −1
= ∑ [𝑒−𝑊(𝑡−𝑡𝑘 ) − 𝑒−𝑊(𝑡−𝑡𝑘−1 ) ] [𝐼 − 𝑒
𝑡0 <𝑡𝑘 ≤𝑡
]
−1
≤ [𝐼 − 𝑒−𝑊(𝑡−𝑡0 ) ] [𝐼 − 𝑒−𝑊𝜃 ] .
(13)
From the strict inequality (5), there is an enough small
number 𝜀 > 0 such that
−𝑊𝜃 −1
𝑝 (𝑀𝑧) + 𝑊[𝐼 − 𝑒
Based on the obtained quasi-invariant set, we have the
following
Theorem 5. Let
−1
] 𝑞 (𝑀𝑧) − 𝑊𝑧 < 0,
(14)
Δ
Therefore, the set 𝐷 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ 𝑧∗ } is a positive
quasi-invariant set of (2). When 𝑀 = 𝐼 especially, 𝐷 is a
positive invariant set of (2). The proof is complete.
Δ (𝑧) = 𝑝 (𝑧) + 𝑊[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝑧) − 𝑀−1 𝑊𝑧,
𝑧 = 𝑧∗ + 𝜀𝐸 > 0.
Assume that all conditions in Theorem 4 hold. Define
In the following, we will prove that [𝜙]+𝜏 < 𝑧 implies
+
[𝑥 (𝑡)]+ = [𝑥 (𝑡, 𝑡0 , 𝜙)] < 𝑀𝑧,
𝑡 ≥ 𝑡0 .
Ω1 = {𝑧 ∈ 𝑅+𝑛 | Δ (𝑀𝑧) < 0} ,
(15)
Ω2 = {𝑧 ∈ 𝑅+𝑛 | Δ (𝑧) < 0} ,
Otherwise, from the piecewise continuity of 𝑥(𝑡), there must
be an integer 𝑖 and 𝑡∗ > 𝑡0 such that
Ω3 = {𝑧 ∈ 𝑅+𝑛 | Δ (𝑧) ≥ 0} ,
󵄨󵄨
∗ 󵄨
󵄨󵄨𝑥𝑖 (𝑡 )󵄨󵄨󵄨 ≥ 𝑚𝑖 𝑧𝑖 ,
+
[𝑥 (𝑡)] ≤ 𝑀𝑧,
(16)
∗
𝑡0 ≤ 𝑡 < 𝑡 .
𝑧 ∈ 𝑅+𝑛 .
(20)
(17)
Ω∗1
= ⋃ {𝑧 ∈
𝑧∗ ∈Ω1
𝑅+𝑛
∗
| 𝑧 ≤ 𝑧 },
Ω∗2 = ⋃ {𝑧 ∈ 𝑅+𝑛 | 𝑧 ≤ 𝑧∗ } .
𝑧∗ ∈Ω2
(21)
4
Abstract and Applied Analysis
Then, 𝑆 = {𝜙 ∈ PC | [𝜙]+𝜏 ∈ Ω∗2 ∩ Ω3 } is an attracting set of
(2) and 𝐷 = {𝜙 ∈ PC | [𝜙]+𝜏 ∈ Ω∗1 } is the attracting basin of 𝑆.
Proof. From (5) and the definitions of the above sets, then
𝑧∗ ∈ Ω1 , 𝑀𝑧∗ ∈ Ω2 , 0 ∈ Ω∗1 , 0 ∈ Ω∗2 , 0 ∈ Ω3 . Obviously,
Ω1 , Ω2 , Ω∗1 , Ω∗2 , Ω3 and Ω∗2 ∩ Ω3 are nonempty, and so the
definitions of the sets of 𝑆 and 𝐷 are valid. For any 𝜙 ∈ 𝐷,
there is a 𝑧∗ ∈ Ω1 satisfying [𝜙]+𝜏 ≤ 𝑧∗ . According to
Theorem 4, we obtain
+
+
∗
[𝑥 (𝑡)] = [𝑥 (𝑡, 𝑡0 , 𝜙)] ≤ 𝑀𝑧 ∈ Ω2 ,
∀𝑡 ≥ 𝑡0 .
(22)
𝑡
+∫
𝑡−𝑇2
𝑒−𝑊(𝑡−𝑠) 𝑀𝑝 (𝜀𝐸 + 𝜎) 𝑑𝑠
+ 𝑀 ∑ 𝑒−𝑊(𝑡−𝑡𝑘 ) 𝑞 (𝑀𝑧∗ )
𝑡0 <𝑡𝑘 ≤𝑇1
+ 𝑀 ∑ 𝑒−𝑊(𝑡−𝑡𝑘 ) 𝑞 (𝜀𝐸 + 𝜎)
𝑇1 <𝑡𝑘 ≤𝑡
≤ 𝑒−𝑊(𝑡−𝑡0 ) 𝑀𝑧∗ + 𝜀𝑀𝑝 (𝑀𝑧∗ )
+ (𝐼 − 𝑒−𝑊𝑇2 ) 𝑊−1 𝑀𝑝 (𝜀𝐸 + 𝜎) + 𝜀𝑀𝑞 (𝑀𝑧∗ )
−1
+ 𝑀 (𝐼 − 𝑒−𝑊(𝑡−𝑇1 ) ) [𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝜀𝐸 + 𝜎)
≤ 𝑒−𝑊(𝑡−𝑡0 ) 𝑀𝑧∗ + 𝜀𝑀 [𝑝 (𝑀𝑧∗ ) + 𝑞 (𝑀𝑧∗ )]
That is,
−1
Δ
+
𝜎 = lim sup [𝑥 (𝑡)] ∈
𝑡→∞
+ 𝑊−1 𝑀𝑝 (𝜀𝐸 + 𝜎) + 𝑀[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝜀𝐸 + 𝜎) .
Ω∗2 .
(23)
(26)
This implies that
Then, for any given 𝜀 > 0, there is a 𝑇1 > 𝑡0 such that
+
[𝑥 (𝑡)] ≤ 𝜀𝐸 + 𝜎,
𝜎 = lim sup[𝑥 (𝑡)]+ ≤ 𝜀𝑀 [𝑝 (𝑀𝑧∗ ) + 𝑞 (𝑀𝑧∗ )]
𝑡 → +∞
(27)
−1
𝑡 ≥ 𝑇1 .
(24)
+ 𝑊−1 𝑀𝑝 (𝜀𝐸 + 𝜎) + 𝑀[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝜀𝐸 + 𝜎) .
Letting 𝜖 → 0+ , then
In light of 𝑊 = diag{𝑤𝑖 } > 0, for the above 𝜀 > 0 and 𝑇1 , we
can find an enough large 𝑇2 > 0 such that
−1
𝜎 ≤ 𝑊−1 𝑀𝑝 (𝜎) + 𝑀[𝐼 − 𝑒−𝑊𝜃 ] 𝑞 (𝜎) .
(28)
That is, Δ(𝜎) ≥ 0 and 𝜎 ∈ Ω3 . Thus,
𝜎 ∈ Ω∗2 ∩ Ω3 .
∞
∫ 𝑒−𝑊𝑠 𝑑𝑠 ≤ 𝜀𝐼,
𝑇2
−𝑊(𝑡−𝑡𝑘 )
∑ 𝑒
≤ 𝜀𝐼,
𝑡0 <𝑡𝑘 ≤𝑇1
(25)
𝑡 > 𝑇2 .
Using (12), (13), (22), (24), and (25), we have for 𝑡 ≥ 𝜏+𝑇1 +𝑇2 ,
+ ∑ 𝑀𝑒
𝑡0 <𝑡𝑘 ≤𝑡
+
𝑞 ([𝑥 (𝑡𝑘− )] )
+
𝑡−𝑇2
𝑡0
𝑡
+∫
𝑡−𝑇2
Corollary 6. Assume that (𝐻1 )–(𝐻3 ) hold with
𝑃 = (𝑝𝑖𝑗 )𝑛×𝑛 ≥ 0,
𝑄 = (𝑞𝑖𝑗 )𝑛×𝑛 ≥ 0,
+
} 𝑒−𝑊(𝑡−𝑠) 𝑀𝑝 ([𝑥𝑠 ]𝜏 ) 𝑑𝑠
}
{
+
+ { ∑ + ∑ } 𝑀𝑒−𝑊(𝑡−𝑡𝑘 ) 𝑞 ([𝑥 (𝑡𝑘− )] )
{𝑡0 <𝑡𝑘 ≤𝑇1 𝑇1 <𝑡𝑘 ≤𝑡}
∞
≤ 𝑒−𝑊(𝑡−𝑡0 ) 𝑀𝑧∗ + ∫ 𝑒−𝑊𝑠 𝑀𝑝 (𝑀𝑧∗ ) 𝑑𝑠
𝑇2
+
𝑝 ([𝜑]𝜏 ) = 𝑃[𝜑]𝜏 + 𝜇,
𝑇
𝜇 = (𝜇1 , . . . , 𝜇𝑛 ) ≥ 0,
𝑞 ([𝑥]+ ) = 𝑄[𝑥]+ + ],
≤ 𝑒−𝑊(𝑡−𝑡0 ) 𝑀[𝜙]𝜏
+ {∫
From the above theorems, we can obtain sufficient conditions ensuring global attractivity and stability in the following
corollaries.
+
𝑡0
−𝑊(𝑡−𝑡𝑘 )
From the definition of 𝜎 and 𝑆, dist (𝑥𝑡 (𝑡0 , 𝜙), 𝑆) → 0 as 𝑡 →
+∞. The proof is complete.
+
𝑡
+
[𝑥 (𝑡)]+ ≤ 𝑒−𝑊(𝑡−𝑡0 ) 𝑀[𝜙]𝜏 + ∫ 𝑒−𝑊(𝑡−𝑠) 𝑀𝑝 ([𝑥𝑠 ]𝜏 ) 𝑑𝑠
(29)
(30)
𝑇
] = (]1 , . . . , ]𝑛 ) ≥ 0.
If the spectral radius
𝜌 (Λ) < 1,
−1
where Λ = 𝑊−1 𝑀𝑃 + 𝑀[𝐼 − 𝑒−𝜃𝑊] 𝑄, (31)
Δ
then 𝐷 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ 𝑍 = (𝐼 − Λ)−1 𝑊−1 (𝜇 + 𝑊[𝐼 −
𝑒−𝜃𝑊]−1 ])} is a positive quasi-invariant set of (2), and 𝑆 = {𝜙 ∈
PC | [𝜙]+𝜏 ≤ (𝐼 − Λ)−1 𝑊−1 𝑀(𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊]−1 ])} is a global
attracting set of (2).
Abstract and Applied Analysis
5
Proof. Since 𝑝(𝑧) = 𝑃𝑧 + 𝜇 and 𝑞(𝑧) = 𝑄𝑧 + ], we directly
calculate
−1
Δ (𝑧) = 𝑀−1 𝑊 (Λ − 𝐼) 𝑧 + 𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ],
(32)
−1
Δ (𝑀𝑧) = 𝑊 (Λ − 𝐼) 𝑧 + 𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ].
Without loss of generality, we assume that 𝜇, ] > 0. Since
𝜌(Λ) < 1, (𝐼 − Λ)−1 exists and (𝐼 − Λ)−1 ≥ 0 (see [23]), and so
𝑍 > 0. For any 𝜅 > 0, we take 𝑧∗ = (1 + 𝜅)𝑍 > 0 in Theorem 4
and verify the condition (5):
−1
Δ (𝑀𝑧∗ ) = −𝜅 (𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ]) < 0.
(33)
According to Theorem 4, when 𝜅 → 0, we deduce that 𝐷 is
a positive quasi-invariant set of (2). Furthermore, by (33),
(1 + 𝜅) 𝑍 ∈ Ω1 ,
(1 + 𝜅) 𝑀𝑍 ∈ Ω2 .
(34)
From the arbitrariness of 𝜅, we obtain Ω∗1 = Ω∗2 = 𝑅+𝑛 .
Moreover,
Ω3 = {𝑧 ∈ 𝑅+𝑛 | Δ (𝑧) ≥ 0}
−1
According to Theorems 4 and 5, we have 𝜃 = 󰜚 = 0.15, 𝐴 =
0.2, 𝐵 = −0.6, 𝑀 = 2.5, 𝑊 ≐ 5.9086, 𝑝(𝑧) = 0.2𝑧2 + 0.1,
𝑞(𝑧) = 0.1𝑧2 + 0.1, Δ(𝑧) ≐ 1.2052𝑧2 − 2.3634𝑧 + 1.1052, and
so
Ω1 = {𝑧 ∈ 𝑅+ | Δ (𝑀𝑧) < 0} = (0.3079, 0.4765) ,
Ω∗1 = [0, 0.4765] ,
Ω2 = {𝑧 ∈ 𝑅+ | Δ (𝑧) < 0} = (0.7698, 1.1913) ,
Ω∗2 = [0, 1.1913] ,
Ω3 = {𝑧 ∈ 𝑅+ | Δ (𝑧) ≥ 0} = [0, 0.7698] ∪ [1.1913, +∞) ,
Ω∗2 ∩ Ω3 = [0, 0.7698] .
(37)
Thus, 𝑆 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ 0.7698} is an attracting set of (36),
and 𝐷 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ 0.4765} is an attracting basin of 𝑆.
However, solutions of the corresponding continuous system
(i.e., Δ𝑥 = 0 in (36)) may be unbounded. Taking the initial
condition 𝜙(𝑠) = 0.2, 𝑠 ∈ [−1, 0], Figure 1 shows the different
asymptotic behavior between the solution of (36) with no
impulse and one with impulses.
= {𝑧 ∈ 𝑅+𝑛 | 𝑀−1 𝑊 (Λ − 𝐼) 𝑧 + 𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ] ≥ 0}
−1
= {𝑧 ∈ 𝑅+𝑛 | (𝐼 − Λ) 𝑧 ≤ 𝑊−1 𝑀 (𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ])}
Example 9. Consider a 2-dimensional impulsive delay system
𝑥̇ 1 (𝑡) = 𝑥1 (𝑡) + 0.5 sin (𝑥1 (𝑡 − 1)) − 0.4𝑥2 (𝑡 − 1) − 0.5,
⊂ {𝑧 ∈ 𝑅+𝑛 | 𝑧 ≤ (𝐼 − Λ)−1 𝑊−1 𝑀
𝑡 ≥ 0,
𝑥̇ 2 (𝑡) = −4𝑥2 (𝑡) − 0.5𝑥1 (𝑡 − 1) + 0.4 cos (𝑥2 (𝑡 − 1)) + 0.5,
−1
× (𝜇 + 𝑊[𝐼 − 𝑒−𝜃𝑊] ])} .
𝑡 ≠ 𝑡𝑘 ,
(35)
It follows from Theorem 5 that 𝑆󸀠 = {𝜙 ∈ PC | [𝜙]+𝜏 ∈ Ω3 } is a
global attracting set of (2) and 𝑆 is also a global attracting set
due to 𝑆󸀠 ⊂ 𝑆. The proof is complete.
Δ𝑥1 = −0.5𝑥1 (𝑡𝑘− ) + 0.1 cos (𝑥1 (𝑡𝑘− )) + 0.5 sin (𝑒𝑡𝑘 ) ,
Corollary 7. Assume that all conditions in Corollary 6 hold
with 𝜇 = ] = 0. Then, the zero solution 𝑥(𝑡) = 0 of (2) is
globally asymptotically stable.
Δ𝑥2 = 0.1𝑥2 (𝑡𝑘− ) + 0.2 sin (𝑥2 (𝑡𝑘− )) − 0.5 cos (𝑒𝑡𝑘 ) ,
4. Illustrative Examples
The following illustrative examples will demonstrate the
effectiveness of our results and also show the different
asymptotical behaviors between the impulsive system and the
corresponding continuous system.
Example 8. Consider a scalar nonlinear impulsive delay
system
𝑥̇ (𝑡) = 0.2𝑥 (𝑡) + 0.2𝑥2 (𝑡 − 1) + 0.1,
𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ 𝑁, 𝑡 ≥ 𝑡0 = 0,
Δ𝑥 = −0.6𝑥 (𝑡𝑘− ) + 0.1𝑥2 (𝑡𝑘− ) + 0.1 sin (𝑒𝑡𝑘 ) ,
𝑡𝑘 = 𝑡𝑘−1 + 0.15.
(36)
𝑡𝑘 = 0.1𝑘,
𝑘 ∈ 𝑁.
(38)
According to Corollary 6, we have 𝜃 = 󰜚 = 0.1,
𝐴 = diag{1, −4},
𝐵 = diag{−0.5, 0.1},
𝑀 = diag{2, 1},
𝑊 = diag{5.9315, 3.0469}, 𝑝(𝑧) = 𝑃𝑧 + 𝜇, 𝑞(𝑧) = 𝑄𝑧 + ],
Λ = 𝑊−1 𝑀𝑃 + 𝑀[𝐼 − 𝑒−𝜃𝑊]−1 𝑄, where
𝑃=(
0.5 0.4
),
0.5 0.4
0.5
𝜇 = ] = ( ),
0.5
𝑄=(
0.1 0
),
0 0.2
0.6156 0.1349
Λ=(
),
0.1641 0.8928
(39)
and so 𝜌(Λ) = 0.9575 < 1. Therefore, 𝐷 = {𝜙 ∈ PC |
[𝜙]+𝜏 ≤ (1.7105, 2.5228)𝑇 } is a positive quasi-invariant set of
(38), and 𝑆 = {𝜙 ∈ PC | [𝜙]+𝜏 ≤ (1.8637, 2.8720)𝑇 } is a
global attracting set of (38). Figure 2 shows the asymptotic
properties of solutions of (38) under the different initial
conditions.
6
Abstract and Applied Analysis
10
𝑥(𝑡)
0.3
0.2
𝑥(𝑡)
×103
15
5
0
−5
0.1
0
−0.1
2
0
4
6
8
12
10
−0.2
0
2
4
6
8
10
12
14
16
Time 𝑡
Time 𝑡
(a)
(b)
Figure 1: The trajectory of (36) with: (a) no impulse (i.e., Δ𝑥 = 0) and (b) impulses.
Global attracting set
4
3
2
𝑦2
1
0
−1
−2
−3
−4
−5
0
𝑦1
5
Figure 2: Global attracting set of (38).
Acknowledgments
This work is supported partially by the National Natural
Science Foundation of China under Grant nos. 10971240,
61263020, and 61004042, the Key Project of Chinese Education Ministry under Grant no. 212138, the Natural Science
Foundation of Chongqing under Grant CQ CSTC 2011BB0117,
and the Foundation of Science and Technology project of
Chongqing Education Commission under Grant KJ120630.
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Applied Mathematics
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Function Spaces
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Optimization
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Volume 2014