- Research Article
- Open access
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Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales
Journal of Inequalities and Applications volume 2011, Article number: 979705 (2011)
Abstract
This paper studies the stability of the solution for a class of quasilinear implicit dynamic equations on time scales of the form
. We deal with an index concept to study the solvability and use Lyapunov functions as a tool to approach the stability problem.
1. Introduction
The stability theory of quasilinear differential-algebraic equations (DAEs for short)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ1_HTML.gif)
with being a given
-matrix function, has been an intensively discussed field in both theory and practice. This problem can be seen in many real problems, such as in electric circuits, chemical reactions, and vehicle systems. März in [1] has dealt with the question whether the zero-solution of (1.1) is asymptotically stable in the Lyapunov sense with
, with
being constant and small perturbation
.
Together with the theory of DAEs, there has been a great interest in singular difference equation (SDE) (also referred to as descriptor systems, implicit difference equations)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ2_HTML.gif)
This model appears in many practical areas, such as the Leontiev dynamic model of multisector economy, the Leslie population growth model, and singular discrete optimal control problems. On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and partial differential-algebraic equations, and so forth, which have already attracted much attention from researchers (cf. [2–4]). When , in [5], the authors considered the solvability of Cauchy problem for (1.2); the question of stability of the zero-solution of (1.2) has been considered in [6] where the nonlinear perturbation
is small and does not depend on
.
Further, in recent years, to unify the presentation of continuous and discrete analysis, a new theory was born and is more and more extensively concerned, that is, the theory of the analysis on time scales. The most popular examples of time scales are and
. Using "language" of time scales, we rewrite (1.1) and (1.2) under a unified form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ3_HTML.gif)
with in time scale
and
being the derivative operator on
. When
, (1.3) is (1.1); if
, we have a similar equation to (1.2) if it is rewritten under the form
.
The purpose of this paper is to answer the question whether results of stability for (1.1) and (1.2) can be extended and unified for the implicit dynamic equations of the form (1.3). The main tool to study the stability of this implicit dynamic equation is a generalized direct Lyapunov method, and the results of this paper can be considered as a generalization of (1.1) and (1.2).
The organization of this paper is as follows. In Section 2, we present shortly some basic notions of the analysis on time scales and give the solvability of Cauchy problem for quasilinear implicit dynamic equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ4_HTML.gif)
with small perturbation and for quasilinear implicit dynamic equations of the style
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ5_HTML.gif)
with the assumption of differentiability for . The main results of this paper are established in Section 3 where we deal with the stability of (1.5). The technique we use in this section is somewhat similar to the one in [6–8]. However, we need some improvements because of the complicated structure of every time scale.
2. Nonlinear Implicit Dynamic Equations on Time Scales
2.1. Some Basic Notations of the Theory of the Analysis on Time Scales
A time scale is a nonempty closed subset of the real numbers , and we usually denote it by the symbol
. We assume throughout that a time scale
is endowed with the topology inherited from the real numbers with the standard topology. We define the forward jump operator and the backward jump operator
by
(supplemented by
) and
(supplemented by
). The graininess
is given by
. A point
is said to be right-dense if
, right-scattered if
, left-dense if
, left-scattered if
, and isolated if
is right-scattered and left-scattered. For every
, by
, we mean the set
. The set
is defined to be
if
does not have a left-scattered maximum; otherwise, it is
without this left-scattered maximum. Let
be a function defined on
, valued in
. We say that
is delta differentiable (or simply: differentiable) at
provided there exists a vector
, called the derivative of
, such that for all
there is a neighborhood
around
with
for all
. If
is differentiable for every
, then
is said to be differentiable on
. If
, then delta derivative is
from continuous calculus; if
, the delta derivative is the forward difference,
, from discrete calculus. A function
defined on
is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of all rd-continuous functions from
to a Banach space
is denoted by
. A matrix function
from
to
is said to be regressive if
for all
, and denote
the set of regressive functions from
to
. Moreover, denote
the set of positively regressive functions from
to
, that is, the set
.
Let and let
be a rd-continuous
-matrix function and
rd-continuous function. Then, for any
, the initial value problem (IVP)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ6_HTML.gif)
has a unique solution defined on
. Further, if
is regressive, this solution exists on
.
The solution of the corresponding matrix-valued IVP ,
always exists for
, even
is not regressive. In this case,
is defined only with
(see [12, 13]) and is called the Cauchy operator of the dynamic equation (2.1). If we suppose further that
is regressive, the Cauchy operator
is defined for all
.
We now recall the chain rule for multivariable functions on time scales, this result has been proved in [14]. Let and
be continuously differentiable. Then
is delta differentiable and there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ7_HTML.gif)
where is the derivative (in the second variable of the function
) in normal meaning and
is the scalar product.
We refer to [12, 15] for more information on the analysis on time scales.
2.2. Linear Equations with Small Nonlinear Perturbation
Let be a time scale. We consider a class of nonlinear equations of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ8_HTML.gif)
The homogeneous linear implicit dynamic equations (LIDEs) associated to (2.3) are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ9_HTML.gif)
where and
is rd-continuous in
. In the case where the matrices
are invertible for every
, we can multiply both sides of (2.3) by
to obtain an ordinary dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ10_HTML.gif)
which has been well studied. If there is at least a such that
is singular, we cannot solve explicitly the leading term
. In fact, we are concerned with a so-called ill-posed problem where the solutions of Cauchy problem may exist only on a submanifold or even they do not exist. One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of the equation.
We introduce the so-called index-1 of (2.4). Suppose that rank for all
and let
such that
is an isomorphism between
and
;
. Let
be a projector onto
satisfying
. We can find such operators
and
by the following way: let matrix
possess a singular value decomposition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ11_HTML.gif)
where ,
are orthogonal matrices and
is a diagonal matrix with singular values
on its main diagonal. Since
, on the above decomposition of
, we can choose the matrix
to be in
(see [16]). Hence, by putting
and
, we obtain
and
as the requirement.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ12_HTML.gif)
and .
Under these notations, we have the following Lemma.
Lemma 2.2.
The following assertions are equivalent
(i);
(ii)the matrix is nonsingular;
(iii), for all
.
Proof.
(i)(ii) Let
and
such that
. This equation implies
. Since
and
, it follows that
. Hence,
which implies
. This means that
. Thus,
, that is, the matrix
is nonsingular.
(ii)(iii) It is obvious that
. We see that
and
. Thus,
and we have
.
Let , that is,
and
. Since
, there is a
such that
and since
,
. Therefore,
. Hence,
which follows that
. Thus,
and then
. So, we have that (iii). (iii)
(i) is obvious.
Lemma 2.2 is proved.
Lemma 2.3.
Suppose that the matrix is nonsingular. Then, there hold the following assertions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ18_HTML.gif)
Proof.
-
(1)
Noting that
, we get (2.8).
-
(2)
From
, it follows
. Thus, we have (2.9).
-
(3)
and
. This means that
is a projector onto
. From the proof of (iii), Lemma 2.2, we see that
is the projector onto
along
.
-
(4)
Since
for any
,
(2.14)Therefore,
so we have (2.11). Finally,
(2.15)Thus, we get (2.12).
-
(5)
Let
be another linear transformation from
onto
satisfying
to be an isomorphism from
onto
and
a projector onto
. Denote
. It is easy to see that
(2.16)
Therefore, . The proof of Lemma 2.3 is complete.
Definition 2.4.
The LIDE (2.4) is said to be index-1 if for all , the following conditions hold:
(i)rank ,
(ii) ker.
Now, we add the following assumptions.
Hypothesis 2.5.
-
(1)
The homogeneous LIDE (2.4) is of index-1.
-
(2)
is rd-continuous and satisfies the Lipschitz condition,
(2.17)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ23_HTML.gif)
Remark 2.6.
By the item (2.13) of Lemma 2.3, the condition (2.18) is independent from the choice of and
.
We assume further that we can choose the projector function onto
such that
for all right-dense and left-scattered
;
is differentiable at every
and
is rd-continuous. For each
, we have
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ24_HTML.gif)
and the implicit equation (2.3) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ25_HTML.gif)
Thus, we should look for solutions of (2.3) from the space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ26_HTML.gif)
Note that does not depend on the choice of the projector function since the relations
and
are true for each two projectors
and
along the space
.
We now describe shortly the decomposition technique for (2.3) as follows.
Since (2.3) has index-1 and by virtue of Lemma 2.2, we see that the matrices are nonsingular for all
. Multiplying (2.3) by
and
, respectively, it yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ27_HTML.gif)
Therefore, by using the results of Lemma 2.3, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ28_HTML.gif)
By denoting , (2.23) becomes a dynamic equation on time scale
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ29_HTML.gif)
and an algebraic relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ30_HTML.gif)
For fixed and
, we consider a mapping
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ31_HTML.gif)
We see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ32_HTML.gif)
for any . Since
,
is a contractive mapping. Hence, by the fixed point theorem, there exists a mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ33_HTML.gif)
and it is easy to see that is rd-continuous in
.
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ34_HTML.gif)
This deduces
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ35_HTML.gif)
Thus, is Lipschitz continuous with the Lipschitz constant
. Substituting
into (2.24), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ36_HTML.gif)
It is easy to see that the right-hand side of (2.31) satisfies the Lipschitz condition with the Lipschitz constant
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ37_HTML.gif)
Applying the global existence theorem (see [12]), we see that (2.31), with the initial condition has a unique solution
.
Thus, we get the following theorem.
Theorem 2.7.
Let Hypothesis 2.5 and the assumptions on the projector be satisfied. Then, (2.3) with the initial condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ38_HTML.gif)
has a unique solution. This solution is expressed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ39_HTML.gif)
where is the solution of (2.31) with
.
We now describe the solution space of the implicit dynamic equation (2.3). Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ40_HTML.gif)
Lemma 2.8.
There hold the following statements:
(i),
(ii)If for all
then
.
Proof.
(i) Let , that is,
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ41_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ42_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ43_HTML.gif)
it yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ44_HTML.gif)
Conversely, suppose that , that is, there exists
such that
. We have to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ45_HTML.gif)
or equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ46_HTML.gif)
Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ47_HTML.gif)
where we have already used a result of Lemma 2.3 that is a projector onto
. So
.
(ii) Let . Then
and
. Since
, we have
. This means that
. From the assumption
, it follows that
. The fact
implies that
. Thus
. The lemma is proved.
Remark 2.9.
-
(1)
By virtue of Lemma 2.8, we find out that the solution space
is independent from the choice of projector
and operator
.
-
(2)
Since
and
, the initial condition (2.33) is equivalent to the condition
. This implies that the initial condition is not also dependent on choice of projectors.
-
(3)
Noting that if
is a solution of (2.3) with the initial condition (2.33), then
for all
. Conversely, let
and let
,
, be the solution of (2.3) satisfying the initial condition
. We see that
. This means that there exists a solution of (2.3) passing
.
2.3. Quasilinear Implicit Dynamic Equations
Now we consider a quasilinear implicit dynamic equation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ48_HTML.gif)
with and
assumed to be continuously differentiable in the variable
and continuous in
.
Suppose that rank for all
. We keep all assumptions on the projector
and operator
stated in Section 2.2.
Equation (2.43) is said to be of index-1 if the matrix
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ49_HTML.gif)
is invertible for every and
.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ50_HTML.gif)
Further introduce the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ51_HTML.gif)
containing all solutions of (2.43). The subspace manifests its geometrical meaning
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ52_HTML.gif)
where is the tangent space of
at the point
.
Suppose that (2.43) is of index-1. Then, by Lemma 2.2, this condition is equivalent to one of the following conditions:
(1),
(2).
(3)Let be a matrix such that the matrix
is invertible (we can choose
, e.g.). From the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ53_HTML.gif)
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ54_HTML.gif)
is invertible.
Lemma 2.10.
Suppose that the bounded linear operator triplet: is given, where
are Banach spaces. Then the operator
is invertible if and only if
is invertible.
Proof .
See [17, Lemma 1].
By virtue of (2.49) and Lemma 2.10, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ55_HTML.gif)
Now we come to split (2.43). Multiplying both sides of (2.43) by and
, respectively, and putting
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ56_HTML.gif)
Consider the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ57_HTML.gif)
We see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ58_HTML.gif)
where .
Let be a vector satisfying
. Paying attention to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ59_HTML.gif)
Therefore, by (2.50) we get . This means that
is an isomorphism of
. By the implicit function theorem, equation
has a unique solution
. Moreover, the function
is continuous in
and continuously differentiable in
. Its derivative is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ60_HTML.gif)
Then, by substituting into the first equation of (2.51) we come to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ61_HTML.gif)
It is obvious that the ordinary dynamic equation (2.56) with the initial condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ62_HTML.gif)
is locally uniquely solvable and the solution of (2.43) with the initial condition (2.33) can be expressed by
.
Now suppose further that satisfies the Lipschitz condition in
and we can find a matrix
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ63_HTML.gif)
is bounded for all and
. Then, the right-hand side of (2.56) also satisfies the Lipschitz condition. Thus, from the global existence theorem (see [12]), (2.56) with the initial condition (2.57) has a unique solution defined on
.
Therefore, we have the following theorem.
Theorem 2.11.
Given an index-1 quasilinear implicit dynamic equation (2.43), then there holds the following.
-
(1)
Equation (2.43) is locally solvable, that is, for any
,
, there exists a unique solution
of (2.43), defined on
with some
,
, satisfying the initial condition (2.33).
-
(2)
Moreover, if
satisfies the Lipschitz condition in
and we can find a matrix
such that
(2.59)
is bounded, then this solution is defined on and we have the expression
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ65_HTML.gif)
where is the solution of (2.56) with
.
Remark 2.12.
-
(1)
We note that the expression
depends only on choosing the matrix
.
-
(2)
The assumption that
is bounded for a matrix function
seems to be too strong. In Section 3, we show a condition for the global solvability via Lyapunov functions.
-
(3)
If
, there exists
satisfying
. Hence,
. Therefore, by the same argument as in Section 2.2, we can prove that for every
, there is a unique solution passing through
.
3. Stability Theorems of Implicit Dynamic Equations
For the reason of our purpose, in this section we suppose that is an upper unbounded time scale, that is,
. For a fixed
, denote
.
Consider an implicit dynamic equation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ66_HTML.gif)
where and
.
First, we suppose that for each , (3.1) with the initial condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ67_HTML.gif)
has a unique solution defined on . The condition ensuring the existence of a unique solution can be refered to Section 2. We denote the solution with the initial condition (3.2) by
. Remember that we look for the solution of (3.1) in the space
. Let
for all
, which follows that (3.1) has the trivial solution
.
We mention again that . Noting that if
is the solution of (3.1) and (3.2) then
for all
.
Definition 3.1.
The trivial solution of (3.1) is said to be
(1)-stable (resp.,
-stable) if, for each
and
, there exists a positive
such that
(resp.,
) implies
for all
,
(2)-uniformly (resp.,
-uniformly) stable if it is
-stable (resp.,
-stable) and the number
mentioned in the part (1). of this definition is independent of
,
(3)-asymptotically (resp.,
-asymptotically) stable if it is stable and for each
, there exist positive
such that the inequality
(resp.,
) implies
. If
is independent of
, then the corresponding stability is
-uniformly asymptotically (
-uniformly asymptotically) stable,
(4)-uniformly globally asymptotically (resp.,
-uniformly globally asymptotically) stable if for any
there exist functions
,
such that
(resp.,
) implies
for all
and if
(resp.,
) then
for all
,
(5)P-exponentially stable if there is positive constant with
such that for every
there exists an
, the solution of (3.1) with the initial condition
satisfies
. If the constant
can be chosen independent of
, then this solution is called
-uniformly exponentially stable.
Remark 3.2.
From and
, the notions of
-stable and
-stable as well as
-asymptotically stable and
-asymptotically stable are equivalent. Therefore, in the following theorems we will omit the prefixes
and
when talking about stability and asymptotical stability. However, the concept of
-uniform stability implies
-uniform stability if the matrices
are uniformly bounded and
-uniform stability implies
-uniform stability if the matrices
are uniformly bounded.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ68_HTML.gif)
and is the domain of definition of
.
Proposition 3.3.
The trivial solution of (3.1) is
-uniformly (resp.,
-uniformly) stable if and only if there exists a function
such that for each
and any solution
of (3.1) the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ69_HTML.gif)
holds, provided (resp.,
).
Proof.
We only need to prove the proposition for the -uniformly stable case.
Sufficiency. Suppose there exists a function satisfying (3.4) for each
; we take
such that
, that is,
. If
is an arbitrary solution of (3.1) and
, then
, for all
.
Necessity. Suppose that the trivial solution of (3.1) is
-uniformly stable, that is, for each
there exists
such that for each
the inequality
implies
, for all
. For the sake of simplicity in computation, we choose
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ70_HTML.gif)
It is clear that is an increasing positive function in
. Further,
and by definition, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ71_HTML.gif)
By putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ72_HTML.gif)
it is seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ73_HTML.gif)
Let be the inverse function of
. It is clear that
also belongs to
.
For , we denote
. If
, then
by
(remember that
does not imply that
). Consider the case where
. If
, then by the relations (3.6) and (3.8) we have
. In particular,
which is a contradiction. Thus
, this implies
, provided
.
The proposition is proved.
Similarly, we have the following proposition.
Proposition 3.4.
The trivial solution of (3.1) is
-stable (resp.,
-stable) if and only if for each
and any solution
of (3.1) there exists a function
such that there holds the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ74_HTML.gif)
provided (resp.,
).
In order to use the Lyapunov function technique related to (3.1), we suppose that . By using (2.3), we can define the derivative of the function
along every solution curve as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ75_HTML.gif)
Remark 3.5.
Note that when the function is independent of
and even if the vector field associated with the implicit dynamic equation (3.1) is autonomous, the derivative
may depend on
.
Theorem 3.6.
Assume that there exist a constant and a function
being rd-continuous and a function
,
defined on
satisfying
(1) for all
and
,
(2), for any
and
.
Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable, that is, every solution with the initial condition (3.2) is defined on .
Proof.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ76_HTML.gif)
By the condition (2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ77_HTML.gif)
Therefore, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ78_HTML.gif)
From the condition (1), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ79_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ80_HTML.gif)
The last inequality says that the solution can be lengthened on
, that is, (3.1) is globally solvable.
Theorem 3.7.
Assume that there exist a function being rd-continuous and a function
,
defined on
satisfying the conditions
(1) for all
,
(2) for all
and
,
(3) for any
and
.
Assume further that (3.1) is locally solvable. Then the trivial solution of (3.1) is stable.
Proof.
By virtue of Theorem 3.6 and the conditions (2) and (3), it follows that (3.1) is globally solvable. Suppose on the contrary that the trivial solution of (3.1) is not stable. Then, there exists an
such that for all
there exists a solution
of (3.1) satisfying
and
for some
. Put
.
By the assumption that and
is rd-continuous, we can find
such that if
then
. With given
, let
be a solution of (3.1) such that
and
for some
.
Since and by the condition (3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ81_HTML.gif)
Therefore, . Further,
and by the condition (2) we have
. This is a contradiction. The theorem is proved.
Theorem 3.8.
Assume that there exist a function being rd-continuous and functions
,
defined on
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ82_HTML.gif)
satisfying the conditions
(1) uniformly in
,
(2) for all
and
,
(3) for any
and
.
Further, (3.1) is locally solvable. Then the trivial solution of (3.1) is asymptotically stable.
Proof.
Also from Theorem 3.6 and the conditions (2) and (3), it implies that (3.1) is globally solvable.
And since , the trivial solution of (3.1) is stable by Theorem 3.7. Consider a bounded solution
of (3.1). First, we show that
Assume on the contrary that
. From the condition (1), it follows that
By the condition (3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ83_HTML.gif)
as , which gets a contradiction.
Thus, Further, from the condition (3) for any
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ84_HTML.gif)
This means that is a decreasing function. Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ85_HTML.gif)
which follows that by the condition (2).
Theorem 3.9.
Suppose that there exist a function ,
defined on
, and a function
such that
(1) uniformly in
and
for all
and
,
(2) for any
and
.
Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly stable.
Proof.
The proof is similar to the one of Theorem 3.7 with a remark that since uniformly in
, we can find
such that if
then
The proof is complete.
Remark 3.10.
The conclusion of Theorem 3.9 is still true if the condition (1) is replaced by "there exist two functions,
defined on
and a function
such that
for all
and
".
We present a theorem of uniform global asymptotical stability.
Theorem 3.11.
If there exist functions ,
defined on
, and a function
satisfying
(1) for all
and
,
(2) for any
and
.
Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly globally asymptotically stable.
Proof.
Let be given. Define
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ86_HTML.gif)
( is not necessary in
).
Let be a solution of (3.1) with
. From the condition (2), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ87_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ88_HTML.gif)
Hence, for all
.
Because the trivial solution of (3.1) is -uniformly stable, we only need to show that there exists a
such that
. Assume that such a
does not exist, that is
for all
. From the condition (2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ89_HTML.gif)
Since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ90_HTML.gif)
which contradicts the definition of in (3.21). The proof is complete.
When is not differentiable, one supposes that there exists a
-differentiable projector
onto
and
is rd-continuous on
; moreover,
for all
. Let
.
We choose matrix functions such that
is an isomorphism between
and
and the matrix
is invertible. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ91_HTML.gif)
where (see (2.51)).
From now on we remain following the above assumptions on the operators whenever
is mentioned.
By the same argument as Theorem 3.6, we have the following theorem.
Theorem 3.12.
Assume that there exist a constant and a function
being rd-continuous and a function
,
defined on
satisfying
(1) for all
and
,
(2), for any
and
.
Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable.
Theorem 3.13.
Assume that (3.1) is locally solvable. Then, the trivial solution of (3.1) is stable if there exist a function
being rd-continuous and a function
,
defined on
such that
(1) for all
,
(2) for all
and
,
(3) for all
and
.
Proof.
Assume that there is a function satisfying the assertions (1), (2), and (3) but the trivial solution
of (3.1) is not stable. Then, there exist a positive
and a
such that
; there exists a solution
of (3.1) satisfying
and
, for some
. Let
. Since
, it is possible to find a
satisfying
when
. Consider the solution
satisfying
and
for a
.
From the assumption (3), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ92_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ93_HTML.gif)
We get a contradiction because when
.
The proof of the theorem is complete.
Theorem 3.14.
Assume that (3.1) is locally solvable. If there exist two functions ,
defined on
and a function
being rd-continuous such that
(1) for all
and
,
(2) for all
and
,
then the trivial solution of (3.1) is -uniformly stable.
Proof.
The proof is similar to the one of Theorem 3.9.
Theorem 3.15.
If there exist functions ,
defined on
and a function
satisfying
(1) for all
and
,
(2) for any
and
.
Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly globally asymptotically stable.
Proof.
Similarly to the proof of Theorem 3.11.
It is difficult to establish the inverse theorem for Theorems from 3.7 to 3.15, that is, if the trivial solution of (3.1) is stable, there exists a function satisfying the assertions in the above theorems. However, if the structure of the time scale
is rather simple we have the following theorem.
Theorem 3.16.
Suppose that contains no right-dense points and the trivial solution
of (3.1) is
-uniformly stable. Then, there exists a function
being rd-continuous satisfying the conditions (1), (2), and (3) of Theorem 3.13, where
is an open neighborhood of 0 in
.
Proof.
Suppose the trivial solution of (3.1) is -uniformly stable. Due to Proposition 3.3, there exist functions
such that for any solution
of (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ94_HTML.gif)
provided .
Let and
. For any
satisfying
and
, we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ95_HTML.gif)
where is the unique solution of (3.1) satisfying the initial condition
. It is seen that
is defined for all
satisfying
,
, and
.
Let . By the definition,
. From (2.60),
for all
. In particular,
. Thus,
. Hence, we have the assertion (2) of the theorem.
Due to the unique solvability of (3.1), we have with
. Therefore,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ96_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ97_HTML.gif)
The proof is complete.
Now we give an example on using Lyapunov functions to test the stability of equations. The following result finds out that the stability of a linear equation will be ensured if nonlinear perturbations are sufficiently small Lipschitz.
Consider a nonlinear equation of the form (2.3)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ98_HTML.gif)
where and
are constant matrices with ind
,
, and
satisfing the Lipschitz condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ99_HTML.gif)
where is sufficiently small. Let
be defined by (2.9) with
and
. By Theorem 2.7, we see that there exists a unique solution satisfying the condition
for any
.
Besides, also consider the homogeneous equation associated to (3.33)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ100_HTML.gif)
and suppose this equation has index-1. As in Section 2, multiplying (3.33) by we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ101_HTML.gif)
where .
Note that the general solution of (3.35) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ102_HTML.gif)
in there is denoted the set of right-scattered points of the interval
.
Denote . It is easy to show that the trivial solution
of (3.35) is
-uniformly exponentially stable if and only if
, where
is the domain of uniform exponential stability of
. On the exponential stable domain of a time scale, we can refer to [10, 18, 19]. By the definition of exponential stability, it implies that the graininess function of the time scale
is upper bounded. Let
.
We denote the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ103_HTML.gif)
and suppose . Since
, this condition implies that (3.35) is
-uniformly exponentially stable.
If , define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ104_HTML.gif)
where the matrix is supposed to be symmetric positive definite. It is clear that
is symmetric positive definite.
Since , the above series is convergent. Further, for any
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ105_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ106_HTML.gif)
Letting and paying attention to
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ107_HTML.gif)
In the case where and
is symmetric positive definite, by putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ108_HTML.gif)
we can examine easily that the matrix also satisfies (3.42),
is symmetric and positive definite.
Theorem 3.17.
Suppose that and the homogeneous equation (3.35) is of index-1 and the constant
is sufficiently small. Then, the trivial solution
of (3.33) is
-uniformly globally asymptotically stable.
Proof.
Let be a symmetric and positive definite (constant) matrix satisfying (3.42). Consider the Lyapunov function
. The derivative of
along the solution of (3.33) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ109_HTML.gif)
From the Lipschitz condition and (2.25), it is seen that where
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ110_HTML.gif)
Combining this inequality and the above appreciation, we see that when is sufficiently small there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ111_HTML.gif)
By Theorem 3.15, (3.33) is -uniformly globally asymptotically stable.
Example 3.18.
Let and consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ112_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ113_HTML.gif)
We have span
, rank
for all
. It is easy to verify that
is the canonical projector onto
,
. Let us choose
. We see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ114_HTML.gif)
Since ,
, (3.47) has index-1.
It is obvious that . Further,
. Thus, according to Theorem 2.7 for each
, (3.47) with the initial condition
has the unique solution.
It is easy to compute, ,
,
, where
, and
.
Therefore, satisfies
. Moreover, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ115_HTML.gif)
Let the Lyapunov function be .
Put , we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ116_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ117_HTML.gif)
We have for any solution of (3.47) and
(noting that
),
(+) if is right-scattered then
,
(+)if is right-dense then
, where
.
In both two cases, we have , so the trivial solution of (3.47) is
-uniformly stable by Theorem 3.14.
Note that if we let then the result is still true. Indeed, by the simple calculations we obtain
(a), where
defined by
,
(b). Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979705/MediaObjects/13660_2010_Article_2372_Equ118_HTML.gif)
Therefore, having the above result is obvious.
4. Conclusion
We have studied some criteria ensuring the stability for a class of quasilinear dynamic equations on time scales. So far, the inverse theorem of the theorems of the stability in Section 3 of this paper is still an open problem for an arbitrary time scale meanwhile it is true for discrete and continuous time scales.
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This work was done under the support of NAFOSTED no 101.02.63.09.
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Du, N., Liem, N., Chyan, C. et al. Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales. J Inequal Appl 2011, 979705 (2011). https://doi.org/10.1155/2011/979705
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DOI: https://doi.org/10.1155/2011/979705