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Stability of Linear Dynamic Systems on Time Scales
Advances in Difference Equations volume 2008, Article number: 670203 (2008)
Abstract
We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples.
1. Introduction
Continuous and discrete dynamical systems have a number of significant differences mainly due to the topological fact that in one case the time scale , real numbers, and the corresponding trajectories are connected while in other case
, integers, they are not. The correct way of dealing with this duality is to provide separate proofs. All investigations on the two time scales show that much of the analysis is analogous but, at the same time, usually additional assumptions are needed in the discrete case in order to overcome the topological deficiency of lacking connectedness. Thus, we need to establish a theory that allows us to handle systematically both time scales simultaneously. To create the desired theory requires to setup a certain structure of
which is to play the role of the time scale generalizing
and
. Furthermore, an operation on the space of functions from
to the state space has to be defined generalizing the differential and difference operations. This work was initiated by Hilger [1] in the name of "calculus on measure chains or time scales."
In this paper, we examine the various types of stability-stability, uniform stability, asymptotic stability, strong stability, restrictive stability, and so forth, for the solutions of linear dynamic systems on time scales and give two examples.
2. Preliminaries on Dynamic Systems
We mention without proof several foundational definitions and results in the calculus on time scales from an excellent introductory text by Bohner and Peterson [2]. A time scale is a nonempty closed subset of
, and the forward jump operator
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ1_HTML.gif)
(supplemented by ), while the graininess
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ2_HTML.gif)
If has a left-scattered maximum
, then
and otherwise
. A function
is called differentiable at
, with (delta) derivative
if given
there exists a neighborhood
of
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ3_HTML.gif)
where .
Some basic properties of delta derivatives are given in the following [3–5]:
-
(i)
If
is differentiable at
, then
(2.4) -
(ii)
If both
and
are differentiable at
, then the product
is also differentiable at
with
(2.5)
A function is said to be rd-continuous (denoted by
if
-
(i)
is continuous at every right-dense point
,
-
(i)
exists and is finite at every left-dense point
.
A function is called an antiderivative of
on
if it is differentiable on
and satisfies
for
. In this case, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ6_HTML.gif)
where .
The norm of an matrix
is defined to be
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ7_HTML.gif)
where is the
th column of
.
Let be the set of all
matrices over
. The class of all rd-continuous and regressive functions
is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ8_HTML.gif)
Here, a matrix-valued function is called regressive provided:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ9_HTML.gif)
where is the identity matrix.
Definition 2.1.
Let . The unique matrix-valued solution of the IVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ10_HTML.gif)
where , is called the matrix exponential function and it is denoted by
.
3. Stability of Linear Dynamic Systems
We consider the dynamic system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ11_HTML.gif)
where with
and
is the delta derivative of
with respect to
. We assume that the solutions of (3.1) exist and are unique for
, and
is unbounded above.
We give the definitions about the various types of stability for the solutions of (3.1).
Definition 3.1.
The solution of (3.1) is said to be stable if, for each
, there exists a
such that, for any solution
of (3.1), the inequality
implies
for all
.
Definition 3.2.
The solution of (3.1) is said to be uniformly stable if, for each
, there exists a
such that, for any solution
of (3.1), the inequalities
and
imply
for all
.
Definition 3.3.
The solution of (3.1) is said to be asymptotically stable if it is stable and there exists a
such that
implies
as
.
The following notion of strong stability is due to Ascoli [6].
Definition 3.4.
The solution of (3.1) is said to be strongly stable if, for each
, there exists a
such that, for any solution
of (3.1), the inequalities
and
imply
for all
.
For the other types of stability, that is, -stability, we refer to [7].
We note that the stability of any solution of (3.1) is closely related to the stability of the null solution of the corresponding variational equation. Therefore, we will discuss the stability of linear dynamic system.
We consider the linear homogeneous dynamic system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ12_HTML.gif)
where .
It follows that any solution of the linear dynamic system is (uniformly, strongly, asymptotically) stable if and only if the same holds for the zero solution of (3.2). We say that (3.2) is (uniformly, strongly, asymptotically stable) stable if so is the null solution of (3.2). See [8].
Firstly, we show that the stability for solutions of (3.2) is equivalent to the boundedness.
Theorem 3.5.
Equation (3.2) is stable if and only if all solutions of (3.2) are bounded for all .
Proof.
Suppose that (3.2) is stable. Since the trivial solution is stable, given any
, there exists a
such that
implies
. Note that
for all
. Now, let
be a vector of length
in the
th direction for
. Then,
, where
is the
th column of
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ13_HTML.gif)
Consequently, for any solution of (3.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ14_HTML.gif)
It follows that all solutions of (3.2) are bounded.
For the converse, we note that all solutions of (3.2) are bounded if and only if there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ15_HTML.gif)
It follows from that (3.2) is stable. This completes the proof.
In [9, Theorem 2.1], DaCunha obtained the following characterization of uniform stability by means of the operator norm. It is not difficult to prove this result by using the maximum norm.
Theorem 3.6.
Equation (3.2) is uniformly stable if and only if there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ16_HTML.gif)
for all with
.
The following is the characterization of strong stability for linear dynamic system (3.2). Note that its continuous version was presented in [10].
Theorem 3.7.
Equation (3.2) is strongly stable if and only if there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ17_HTML.gif)
where is a matrix exponential function of (3.2).
Proof.
Suppose that (3.7) holds. For any given , we can choose
such that for any
,
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ18_HTML.gif)
Hence, (3.2) is strongly stable.
Conversely, if (3.2) is strongly stable, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ19_HTML.gif)
whenever and
holds. Since
is arbitrary, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ20_HTML.gif)
where . It is clear that
, and hence
, is independent of
and
as well as of
. Putting
and
, we obtain the result.
Example 3.8 (see [8]). (i) The system is strongly stable, but it is not asymptotically stable.
(ii) The system with
is asymptotically stable, but it is not strongly stable.
Restrictive stability in [10] is related to strong stability, and we obtain their equivalence for (3.2) as a consequence of Theorem 3.7.
Definition 3.9.
System (3.2) is said to be restrictively stable if it is stable and its adjoint system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ21_HTML.gif)
where denotes the conjugate transpose of
, is stable.
Remark 3.10.
We note that (3.2) is strongly stable if and only if it is restrictively stable.
Definition 3.11.
System (3.2) is said to be reducible (or reducible to zero), if there exists which is bounded together with its inverse
on
such that
is a constant (or zero) matrix on
. Here,
, and the set of all functions
that are differentiable and whose derivative is rd-continuous is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ22_HTML.gif)
Theorem 3.12.
System (3.2) is restrictively stable if and only if it is reducible to zero.
Proof.
Let be a matrix exponential function of (3.2). Suppose that (3.2) is restrictively stable. Then, there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ23_HTML.gif)
by means of Theorem 3.7. Consider the transformation . Then, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ24_HTML.gif)
Hence, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ25_HTML.gif)
since . This implies that (3.2) is reducible to zero.
For the converse, suppose that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ26_HTML.gif)
Then, we have . Thus,
is a matrix exponential function of (3.2). Since
and
are bounded for all
, the proof is complete.
Theorem 3.13.
If (3.2) is stable and reducible on a time scale with the constant graininess, then it is uniformly stable.
Proof.
Since (3.2) is reducible, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ27_HTML.gif)
where by the transformation
. Let
be a matrix exponential function of (3.2). The stability of (3.2) implies the boundednesss of
. Let
, where
is a matrix exponential function of (3.17). Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ28_HTML.gif)
and hence the boundedness of implies the boundedness of
since
is bounded. Thus, (3.17) is stable and, in fact, is uniformly stable. Hence, it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ29_HTML.gif)
for some positive constant and all
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ30_HTML.gif)
for some positive constant and all
. Consequently, (3.2) is uniformly stable.
The continuous versions of Theorems 3.12 and 3.13 are presented in (3.9.v) and (3.9.vi) in [10], respectively.
Remark 3.14.
It does not hold in general that every stable linear homogeneous system with constant coefficient matrix on a time scale is uniformly stable.
Corollary 3.15.
If (3.11) is stable and with the eigenvalues
(
) of
, then it is restrictively stable.
Proof.
It follows from the stability of (3.2) that is bounded for all
. Furthermore, by Liouville's formula [11], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ31_HTML.gif)
where is a positive constant. Thus, from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ32_HTML.gif)
it is clear that is bounded for all
. The proof is complete.
Remark 3.16.
Pötzsche et al. [12] proved a necessary and sufficient condition for the exponential stability of time-variant linear systems on time scales in terms of the eigenvalues of the system matrix. They used a representation formula for the transition matrix of Jordan reducible systems in the regressive case.
Remark 3.17.
In summary, the following assertions are all equivalent [13, Theorem 4.2].
-
(i)
System (3.2) is strongly stable.
-
(ii)
There exists a positive constant
such that
(3.23) -
(iii)
Adjoint system (3.11) of (3.2) is strongly stable.
-
(iv)
System (3.2) is restrictively stable.
-
(v)
System (3.2) is reducible to zero.
It is widely known that the stability characteristics of a nonautonomous linear system of differential or difference equations can be characterized completely by a corresponding autonomous linear system by the Lyapunov transformation. DaCunha and Davis in [14] gave a definition of the Lyapunov transformation as follows.
Let . The Lyapunov transformation is an invertible matrix-valued function
with the property that, for some positive
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ34_HTML.gif)
for all .
Remark 3.18.
Note that the boundedness of the coefficient matrices is not preserved by the Lyapunov transformation in the case of the time scales with right-dense point [13]. This can be seen by considering the time scale
and the Lyapunov transformation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ35_HTML.gif)
It shows that the coefficient matrices and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ36_HTML.gif)
where and
.
Now, we consider the linear dynamic system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ37_HTML.gif)
and its perturbed system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ38_HTML.gif)
where .
The following theorem means that the strong stability for the system (3.27) is equivalent to that of (3.2).
Lemma 3.19 (see [14, Theorem 3.8]).
Suppose that is invertible for all
, and
is regressive. Then, the transformation matrix for the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ39_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ40_HTML.gif)
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ41_HTML.gif)
for all .
The regressiveness of in (3.30) is preserved by the Lyapunov transformation in the following lemma.
Lemma 3.20.
Suppose that
is the transformation matrix for all
. Then
is regressive if and only if
is also regressive.
Proof.
We see that for every right-scattered point , the following identity holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ42_HTML.gif)
This completes the proof.
Theorem 3.21.
Suppose that is a Lyapunov transformation. Then (3.2) is strongly stable if and only if (3.27) is strongly stable.
Proof.
Suppose that (3.2) is strongly stable. Then, there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ43_HTML.gif)
By using Lemma 3.19, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ44_HTML.gif)
Hence, (3.27) is strongly stable.
The converse holds similarly.
If we assume that the perturbing term is absolutely integrable, then we obtain the uniform stability for the perturbed system (3.28) when system (3.2) is strongly stable.
Theorem 3.22.
Suppose that is a Lyapunov transformation and there exists a
such that for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ45_HTML.gif)
If (3.2) is strongly stable, then (3.28) is uniformly stable.
Proof.
It follows from Theorem 3.21 that (3.27) is strongly stable. Then, there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ46_HTML.gif)
For any and
, the solution
of (3.28) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ47_HTML.gif)
By taking the norms of both sides of (3.37), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ48_HTML.gif)
In view of Gronwall's inequality [15], we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ49_HTML.gif)
for all . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ50_HTML.gif)
where . Hence, (3.28) is uniformly stable.
4. Examples
In this section, we give two examples about the various types of stability for solutions of linear dynamic systems on time scales in [16].
Example 4.1.
To illustrate Theorem 3.7, we consider the linear dynamic system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ51_HTML.gif)
where . If
for all
, then (4.1) is strongly stable.
Remark 4.2.
We give some remarks about Example 4.1.
-
(1)
If
, then
of linear differential system
is given by
(4.2) -
(2)
If
with the positive constant
for all
, then
of linear difference system
(4.3)is given by
(4.4) -
(3)
If
with the constant
, then
of
-difference system
(4.5)is given by
(4.6) -
(4)
If
and
, then
of linear dynamic system
(4.7)is given by
(4.8)
Example 4.3.
We consider the linear dynamic system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ59_HTML.gif)
where . If
for all
, then the matrix exponential function
of (4.9) is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ60_HTML.gif)
We see that the generalized exponential function is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F670203/MediaObjects/13662_2007_Article_1146_Equ61_HTML.gif)
respectively. Thus, we obtain the following results for (4.9) and .
-
(1)
If
, then (4.9) is uniformly stable but not strongly stable.
-
(2)
If
, then (4.9) is strongly stable but not asymptotically stable.
-
(3)
If
with
and
, then (4.9) is neither asymptotically stable nor strongly stable. However,
goes to zero as
.
-
(4)
If
with
, then (4.9) is neither asymptotically stable nor strongly stable.
-
(5)
If
with
, then (4.9) is unbounded and
is oscillatory.
-
(6)
If
, then (4.9) is bounded and
goes to zero as
.
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Acknowledgments
The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper. This work was supported by the Korea Research Foundation Grant founded by the Korea Government (MOEHRD) (KRF-2005-070-C00015).
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Choi, S.K., Im, D.M. & Koo, N. Stability of Linear Dynamic Systems on Time Scales. Adv Differ Equ 2008, 670203 (2008). https://doi.org/10.1155/2008/670203
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DOI: https://doi.org/10.1155/2008/670203