Applied Mathematics, 2011, 2, 1323-1326
doi:10.4236/am.2011.211185 Published Online November 2011 (http://www.SciRP.org/journal/am)
A New Uniformly Ultimate Boundedness Criterion for
Discrete-Time Nonlinear Systems
Zhishuai Ding1, Guifang Cheng1,2
1Department of Mat hem at i cs, Zhengzhou Uni versi t y , Zhengzhou, China
2Physical Engineering College, Zhengzhou University, Zhengzhou, China
Received August 11, 2011; revised September 27, 2011; aceepted O ct o ber 5, 2011
Abstract
A new type criterion of globally uniformly ultimate boundedness for discrete-time nonlinear systems is in-
troduced. In classical Lyapunov theory about globally uniformly ultimate boundedness, Lyapunov function
is assumed to be positive definite and its difference at the every latter moment and the former moment is
negative definite. In this paper the condition of difference of Lyapunov function is relaxed. Under the re-
laxed condition, the result of this paper can be considered as the extension of the classical Lyapunov theory
about uniformly ultimate boundedness.
Keywords: Uniformly Ultimate Boundedness, Lyapunov Function, Discrete-Time Nonlinear Systems
1. Introduction
Discrete-time systems of difference equations have attra-
cted considerable attention. A remarkable book [1] by
Agarwal makes the general survey on theory of diffe-
rence equation and introduces a detailed account of the
application of difference equations. Stability analysis is
one of the main issues in the area of control systems the-
ory.The classical Lyapunov theory [2] is an important
tool to discuss the stability and stabilization problems of
dynamical systems. A candidate Lyapunov function is
required to satisfy positive definite, and its difference be
negative definite or semi-negative definite at the every
latter moment and the former moment [3-4]. For dis-
crete-time nonlinear systems, the relation of smooth Lya-
punov function and asymptotical stability is presented in
paper [5]. In paper [6], new results on the existence of
Lyapunov function are presented for discrete-time sys-
tems described by difference inclusions. Some new cha-
racterizations of uniform global asymptotic stability for
nonlinear timevarying discrete-time systems with or with-
out an output-dominant perturbation are proposed on the
basis of the detectability for the reduced limiting systems
associated with the original system in paper [7]. A new
asymptotic stability criterion for nonlinear time-varying
differential equations is demonstrated in paper [8] by
Aeyels and Peuteman. The Lyapunov function need not
be differentiable, and not even be Lipschitz continuous.
Under the relaxed condition, a new asymptotic stability
criterion is introduced.
Uniform boundness and uniformly ultimate bounded-
ness is an indispensable part of stability problems [1,2].
In paper [9], Aeyels, Peuteman and Sepulchre transform
the problem of uniform boundedness and uniform ulti-
mate boundedness for nonautonomous continuous sys-
tems to time-invariant frozen systems and introduced
some important results. Bu and Mu [10] extend those
results and present Lyapu nov theorems of uniform boun -
dedness and uniform ultimate boundedness for nonau-
tonomous homogeneous systems. Paper [11] by Cheng,
Mu and Ding discusses the problem of uniformly ulti-
mate boundedness of nonautonomous nonlinear systems
with discontinuous right-hand sides and gives some re-
sults based on differential inclusions and Filippov solu-
tions. Under arbitrary switching laws, a continuous state
feedback control [12] scheme is proposed in order to
guarantee uniformly ultimate boundedness of every sys-
tem response within an arbitrary small neighborhood of
the zero state.
In the research area about uniformly ultimate bound-
edness, the condition of Lyapunov function usually con-
sidered is differentiable or Lipschitz continuous and re-
gular. In this paper, based on [8] given by Aeyels and
Peuteman, we relax the assumption condition of Lya-
punov function. We don’t suppose the difference of Lya-
punov function at the every latter moment and the former

Z. S. DING ET AL. 1324
moment is negative definite but suppose there exists a
finite integer such that Lyapunov function satisfies cer-
tain condition. Our object is to provide a new uniformly
ultimate boundedness criterion for discrete-time nonlin-
ear systems.
Without loss of generality, denotes the n-dimen-
sional Euclidean space, the notation
n
R
 is used to de-
note the Euclidean 2-norm of a vector.
Z
denotes the set
of integers. Continuous function ()

 is said to be a
function of class
K
if, it is strictly increasing and
(0) 0

. A closed ball B

is denoted by


Bxx


.
The rest of the paper is organized as follows. Mathe-
matical preliminary is stated in Section 2. A new type
criterion of uniformly ultimate boundedness for nonlin-
ear discrete-time systems is proposed in Section 3. A
brief conclusion is provided to summarize the paper in
Section 4. Finally, acknowledgements are given in the
final section.
2. Mathematical Preliminary of the Problem
Consider the systems
(1) ((),)
x
kfxk k (1)
where () n
x
kR is the state vector at time instant ,
. We assume that k
kZ:nn
f
RZ R
kZ
(
satisfies
for all and is globally Lipschitz
continuous. Without loss of generality, let be the
Lipschitz constant. Thus the existence and uniqueness of
the solution of system (1) is satisfied.
(0,fk) 0L
)
x
k denotes the
solution with the initial value 00
()
x
kx.
Firstly the definitions of globally uniform bounded-
ness and uniformly ultimate boundedness for discrete-
time nonlinear systems (1) arepresented.
Definition 2.1. The origin of system (1) is said to be
globally uniformly bounded if, for any positive constant
, there is (independent of ), such
that when
0a() 0bba 0 k
0
()
x
ka, there holds
0
() ,
x
kbkk.
Definition 2.2. The origin of system (1) is said to be
globally uniformly ultimately bounded if, there exist
positive constants r, for all 0

, there is
(,) 0KK r

 (independent of ), such that when
0
k
0
()xk

, there holds
0
() ,(,)
x
krkkKr   .
Lemma 2.3. ([5]) For all 0

, and choose the
closed ball . Then for any finite integer ,
there exists
n
BR

00 m

 such that for all 00
(,)
x
kBZ
 

,
there holds
00
(),[ ,]
x
kkkk   
3. Main Result
A new type globally uniformly ultimate boundedness
criterion for nonlinear discrete-time systems is proposed
in this section.
Theorem 3.1. There exists a Lyapunov function
which satisfies: :n WR ZR
1) there exist two functions 1

, 2

of class ,
such that K





12
() (),()

x
kWxkk xk  ; (2)
2) for all 0

, choose
L
m
e 

, there exists


21
1
0()





 , while for all x

, there exists a
function 3

of class
K
, and a finite integer ,
such that 0m



 
3
(),(), ()Wxkmkm Wxkkxk   

. (3)
Then the origin of system (1) is globally uniformly ul-
timate bounded.
Proof: Choose arbitrary initial value 00 (,)
x
kBZ



,
there are only two cases:
(I) 002 (,) ()Wx k



; (II) 00 2 (,) ()Wx k

 .
Case I: When (I) holds, then we have

 
00 002
(),(), (WxkmkmWxkk)

  ;
and

 
00 002
(2),2(),(WxkmkmWxkk)

  .
By iterative approach, for all , there holds
0
lZ



 
00 002
(),(), (WxklmklmWxk k)

  .
From (2), we have
 

11
01221
()xk lm

 




.
Hence for all , there exists a constant
0
kk0 kZ


,
such that 0 kk mKm

, and


0
()( 1)xkxkK m



 
.
By Lemma 2.1, for all , there has
0
kk
() Lm xk e



.
Case II: When (II) holds, Let





*00 2
sup,( )KkWxksmksm

 , (4)
where 0
s
k

,0 kZ


.
We claim *
K

 .
Contradiction If not, there must hold *
K

 , and
for all 0
kZ


, there has


00 2
(),Wxkkmk km()

 .
Then

 
20020
()( ),( )Wx kkmkkmx kkm    , m
.

Z. S. DING ET AL.1325
which implies
00
();
x
kkm kZ 

.
By (4) and iteration, we can get the following ine-
qualities



00
00
30 3
((1)),(1)
(),
() ();

 

 
Wxkkmk km
Wxk kmkkm
xk km





00
00
30 3
(),
((1)),(1)
((1)) ();




 
Wxkkmk km
Wxk k mkk m
xkkm




00 0
30 3
(), (),
() (), 

 
WxkTmk TmWxkk
k
0
then





00
00 3
(1),(1)
(),( 1)(). 
 

Wxkkmkkm
Wxkkk
When , clearly K


00
(1), (1)Wxk k mk k m , which contra-
dicts to the positive definite property of .

(),Wxkk
Then th ere exists a n integer *0
K
Z 

2
m
, when ,
we have
*
Kk
00
(),Wxkkmk k()

 
*
;
when , we have kK

00 2
(),(Wxkkmk km)

 .
Consequently
11
01221
()((( )))xkkm

 


  .
Furthermore, we can estimate *
K
. By iteration, there
has
 

**
00
*
00 3
(1),(1)
(),( 1)()
WxkKmk Km
WxkkK

 


i.e.

 
*
**
00 00
3
22
3
(),( 1),( 1)1
()
()()) 1.
()

 





K
WxkkWxkKmkK m

Then choose
** 22
3
() ())
(,)1(, )(,)
()
KK
 
where *(,)K


is independent of .
0
t
Therefore
11
01221
() ((()))xk kT

 



.
So in view of Lemma 2, for all *
0(,)kkK

  ,
() Lm xk e




.
The proof is completed.
4. Conclusions
We conclude with a brief discussion. In this paper, an
extensive Lyapunov theorem of uniform ultimately boun-
dedness is presented. In the classical Lyapunov theory
about uniform ultimately boundedness, the difference of
Lypunov function at the every latter moment and the
former moment is negative definite. Here only need to
exist an integer such that the condition 2) of Theo-
rem 3.1 is satisfied. When , the condition 2) be-
comes the condition in classical Lyapunov theory. Thus
theorem 3.1 is less restrictive than that in the classical
Lyapunov theory.
m1m
5. Acknowledgements
This work is supported by the Natural Science Founda-
tion of China under the contracts no. 60874006 and
no.10826078.
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*
K




 
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