Practical Stability of Impulsive Discrete Systems with Time Delays
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 954121, 10 pages
http://dx.doi.org/10.1155/2014/954121
Research Article
Practical Stability of Impulsive Discrete Systems with
Time Delays
Liangji Sun,1 Chengyan Liu,2 and Xiaodi Li2
1
2
Department of Computer Science, Shaanxi Vocational & Technical College, Xi’an 710100, China
Department of Mathematics, Shandong Normal University, Ji’nan 250014, China
Correspondence should be addressed to Xiaodi Li;
Received 5 December 2013; Accepted 8 February 2014; Published 18 March 2014
Academic Editor: Haydar Akca
Copyright © 2014 Liangji Sun et al. 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.
The purpose of this paper is to investigate the practical stability problem for impulsive discrete systems with time delays. By
using Lyapunov functions and the Razumikhin-type technique, some criteria which guarantee the practical stability and uniformly
asymptotically practical stability of the addressed systems are provided. Finally, two examples are presented to illustrate the criteria.
1. Introduction
As we all know, in many applications, we use discrete systems
rather than continuous ones as the mathematical modeling,
for example, numerical analysis, control theory, population
models, and computer science [1–3]. Therefore, more and
more attention has been paid to the theory of discrete
systems, and some results for the stability of discrete systems
have been obtained over the past few years [4–8].
The theory of practical stability has developed into a
branch of the theory of motion stability [9]. Its notion is very
useful, since it only needs to stabilize a system into a region
of phase space. Based on this method, the desired state of a
system can be unstable only if it oscillates sufficiently near
this state. Recently, there has been a significant development
in the theory of practical stability [10–15]. Moreover, impulses
and time delays exist in many processes of dynamic systems, for example, physics, chemical technology, population
dynamics, and neural networks, and they may impact systems
seriously [16–30]. Therefore, it is necessary and important to
analyze the practical stability of impulsive discrete systems
with time delays.
In [7, 8], authors have obtained some results for asymptotic stability and exponential stability of impulsive discrete
systems with time delays. Unfortunately, there is almost no
result concerning uniformly asymptotically practical stability
of impulsive discrete systems with time delays. The purpose
of this paper is to establish some criteria which guarantee uniformly asymptotically practical stability of the addressed systems by using Lyapunov functions and the Razumikhin-type
technique. This work is organized as follows. In Section 2, we
introduce some basic definitions and notations. In Section 3,
the main results are presented. In Section 4, two examples are
discussed to illustrate the results.
2. Preliminaries
Let R+ denote the set of nonnegative real numbers, R𝑚 the
𝑚-dimensional real space equipped with the Euclidean norm
‖ ⋅ ‖, Z the set of integers, and Z+ the set of positive integers.
For any 𝑟 > 0, 𝑟 ∈ Z+ , 𝐽 ≜ {−𝑟, −𝑟 + 1, −𝑟 + 2, . . . , −1, 0},
and set 𝐶(R+ , R+ ) ≜ {𝜙 : R+ → R+ | 𝜙 is continuous}. Let
𝑆 ≜ {𝜑 : 𝐽 → R𝑚 }. Let 𝑆𝜌 ≜ {𝜑 ∈ 𝑆 : ‖𝜑‖ < 𝜌}. The norm
of 𝜑 is defined by ‖𝜑‖𝐽 = max𝑠∈𝐽 |𝜑(𝑠)|. The impulse times
𝑛𝑘 satisfy 0 < 𝑛1 < 𝑛2 < ⋅ ⋅ ⋅ < 𝑛𝑘 < ⋅ ⋅ ⋅ , 𝑛𝑘 , 𝑘 ∈ Z+ , and
lim𝑘 → +∞ 𝑛𝑘 = +∞.
Consider the following impulsive discrete systems with
time delays:
𝑥 (𝑛 + 1) = 𝑓 (𝑛, 𝑥𝑛 ) ,
𝑥 (𝑛) = {
𝑛 ≥ 𝑛0 , 𝑛 ∈ Z+ ,
𝑥 (𝑛) ,
𝑛 ≠ 𝑛𝑘 , 𝑘 ∈ Z+ ,
𝑥 (𝑛𝑘 ) + 𝐼𝑘 (𝑛𝑘 , 𝑥 (𝑛𝑘 )) , 𝑛 = 𝑛𝑘 , 𝑘 ∈ Z+ ,
𝑥𝑛0 (𝑠) = 𝜑 (𝑠) ,
𝑠 ∈ 𝐽,
(1)
2
Abstract and Applied Analysis
where 0 ≤ 𝑛0 < 𝑛1 , 𝜑 ∈ 𝑆, 𝑓 ∈ Z+ × 𝑆𝜌 → R𝑚 , 𝑓(𝑛, 0) = 0.
For each 𝑛 ≥ 𝑛0 , 𝑥𝑛 ∈ 𝑆𝜌 is defined by 𝑥𝑛 (𝑠) = 𝑥(𝑛 + 𝑠), 𝑠 ∈ 𝐽.
For each 𝑘 ∈ Z+ , 𝐼𝑘 ∈ Z+ × R𝑚 → R𝑚 , 𝐼𝑘 (𝑛, 0) = 0, and,
for any 𝜌 > 0, there exists a 𝜌1 ∈ (0, 𝜌) such that 𝑥 ∈ 𝑆(𝜌1 )
implies that 𝑥 + 𝐼𝑘 ∈ 𝑆(𝜌).
In this paper, we assume that 𝑓 and 𝐼𝑘 satisfy certain
conditions such that the solution of system (1) exists on [𝑛0 −
𝑟, +∞)∩Z+ and is unique [4]. We denote by 𝑥(𝑛) = 𝑥(𝑛, 𝑛0 , 𝜑)
the solution of system (1) with initial value 𝜑.
For convenience, we define the following classes of
functions:
𝐾 = {𝑤 ∈ 𝐶(R+ , R+ ) : 𝑤 is strictly increasing
and 𝑤(0) = 0};
𝐾1 = {𝑤 ∈ 𝐶(R+ , R+ ) : 𝑤(0) = 0 and 𝑤(𝑠) >
0 for 𝑠 > 0};
𝐾2 = {𝜓 ∈ 𝐶(R+ , R+ ) : 𝜓 is increasing and 𝜓(𝑠) <
𝑠 for 𝑠 > 0}.
Then, the system (1) with respect to (𝜆, 𝐴) is uniformly
asymptotic practically stable.
Proof. Let
𝜔 (𝑠)
,
𝑠
𝑠>0
𝑞 ≜ sup
𝜓−1 (𝑠)
)
𝑠>0
𝑠
−1
< 1.
𝑝 ≜ (inf
(4)
For any 𝑛0 ≥ 0, let 𝑥(𝑛) ≐ 𝑥(𝑛, 𝑛0 , 𝜑) be the solution of system
(1) through (𝑛0 , 𝜑), where (𝑛0 , 𝜑) ∈ Z+ × 𝑆, and ‖𝜑‖𝐽 < 𝜆. It
suffices to show that
‖𝑥‖ < 𝐴,
𝑛 ≥ 𝑛0 ,
𝑛 ∈ Z+ .
(5)
In addition, we introduce some definitions as follows.
Definition 1 (see [9]). Given two constants 𝜆 and 𝐴, 0 < 𝜆 <
𝐴. Then, the impulsive discrete system (1) with respect to
(𝜆, 𝐴) is said to be
(𝑆1 ) practically stable, if ‖𝜑‖𝐽 < 𝜆 implies ‖𝑥(𝑛)‖ < 𝐴, 𝑛 ≥
𝑛0 , 𝑛 ∈ Z+ ,
(𝑆2 ) uniformly practically stable if (𝑆1 ) holds, for every
𝑛0 ∈ Z+ ,
(𝑆3 ) asymptotically practically stable, if (𝑆1 ) holds and, for
any 𝜖 > 0, there exists 𝑇 = 𝑇(𝑛0 , 𝜖) > 0, 𝑇 ∈ Z+ , such
that ‖𝜑‖𝐽 < 𝜆 implies ‖𝑥(𝑛)‖ < 𝜖, 𝑛 ≥ 𝑛0 + 𝑇, 𝑛 ∈ Z+ ,
(𝑆4 ) uniformly asymptotically practically stable if (𝑆2 )
holds and the latter part of (𝑆3 ) holds for a constant
𝑇 = 𝑇(𝜖) > 0, 𝑇 ∈ Z+ , only dependent on 𝜖.
3. Main Results
Now, we show that
𝑉 (𝑛, 𝑥 (𝑛)) ≤ 𝜓−1 (𝑏 (𝜆)) ,
𝑛 ∈ [𝑛0 , 𝑛1 ] ∩ Z+ .
(6)
If it does not hold, then there exists a 𝑟 ∈ [𝑛0 , 𝑛1 ] ∩ Z+ , such
that 𝑉(𝑟, 𝑥(𝑟)) > 𝜓−1 (𝑏(𝜆)). Let 𝑟2 = min{𝑛 : 𝑉(𝑛, 𝑥(𝑛)) >
𝜓−1 (𝑏(𝜆)), 𝑛 ∈ [𝑛0 , 𝑛1 ] ∩ Z+ }. Since 𝑉(𝑛0 , 𝑥(𝑛0 )) ≤ 𝑏(𝜆) ≤
𝜓−1 (𝑏(𝜆)), it is clear that 𝑟2 > 𝑛0 . Let 𝑟1 = max{𝑛 :
𝑉(𝑛, 𝑥(𝑛)) ≤ 𝑏(𝜆), 𝑛 ∈ [𝑛0 , 𝑟2 ) ∩ Z+ }. Thus,
𝑉 (𝑟2 , 𝑥 (𝑟2 )) > 𝜓−1 (𝑏 (𝜆)) ,
𝑉 (𝑟1 , 𝑥 (𝑟1 )) ≤ 𝑏 (𝜆) ,
𝑏 (𝜆) < 𝑉 (𝑡, 𝑥 (𝑡)) ≤ 𝜓−1 (𝑏 (𝜆)) ,
𝑛 ∈ (𝑟1 , 𝑟2 ) ∩ Z+ .
(7)
Theorem 2. Assume that there exist functions 𝑎, 𝑏 ∈ 𝐾, 𝜔 ∈
𝐶(R+ , R+ ), 𝜓 ∈ 𝐾2 , 𝑉 : Z+ × R𝑚 → R+ , such that
(i) 0 < 𝜆 < 𝐴 are given,
(ii) 𝑎(‖𝑥‖) ≤ 𝑉(𝑛, 𝑥) ≤ 𝑏(‖𝑥‖) for (𝑛, 𝑥) ∈ Z+ × R𝑚 ,
(iii) 𝑉(𝑛𝑘 , 𝑥(𝑛𝑘 )) = 𝑉(𝑛𝑘 , 𝑥(𝑛𝑘 ) + 𝐼𝑘 (𝑛𝑘 , 𝑥(𝑛𝑘 ))) ≤
𝜓(𝑉(𝑛𝑘 , 𝑥(𝑛𝑘 )));
(iv) there is a function 𝑃(𝑠) continuous and nondecreasing
for 𝑠 ≥ 0 and satisfying 𝑃(𝑠) > 𝜓−1 (𝑠), 𝑠 > 0, such that,
for any solution 𝑥(𝑛) of system (1), 𝑃(𝑉(𝑛, 𝑥(𝑛))) ≥
𝑉(𝑛 + 𝑠, 𝑥(𝑛 + 𝑠)), 𝑠 ∈ 𝐽, implies that
Δ𝑉 (𝑛, 𝑥 (𝑛)) = 𝑉 (𝑛 + 1, 𝑥 (𝑛 + 1)) − 𝑉 (𝑛, 𝑥 (𝑛))
≤ 𝜔 (𝑉 (𝑛, 𝑥 (𝑛))) ,
Hence, we obtain
𝑉 (𝑟2 , 𝑥 (𝑟2 )) − 𝑉 (𝑟1 , 𝑥 (𝑟1 )) > 𝜓−1 (𝑏 (𝜆)) − 𝑏 (𝜆)
= 𝜓−1 (𝑏 (𝜆)) (1 −
𝑏 (𝜆)
)
𝜓−1 (𝑏 (𝜆))
≥ 𝜓−1 (𝑏 (𝜆)) (1 − 𝑝) .
(8)
(2)
By (7), we obtain that, for any 𝑛 ∈ [𝑟1 , 𝑟2 ] ∩ Z+ ,
where 𝜏 ≜ max𝑘∈Z+ {𝑛𝑘+ (...truncated)
