- Research Article
- Open access
- Published:
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
(supplemented by 
), while the graininess
is given by
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 
,
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)
is differentiable at 
, then 
and 
are differentiable at 
is also differentiable at 
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 .
is continuous at every right-dense point 
,
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
where 
.
The norm of an 
matrix 
is defined to be
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
Here, a matrix-valued function 
is called regressive provided:
where 
is the identity matrix.
Definition 2.1.
Let 
. The unique matrix-valued solution of the IVP
where 
, is called the matrix exponential function and it is denoted by 
.
3. Stability of Linear Dynamic Systems
We consider the dynamic system
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
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
Consequently, for any solution 
of (3.2),
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
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
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
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
Hence, (3.2) is strongly stable.
Conversely, if (3.2) is strongly stable, then we have
whenever 
and 
holds. Since 
is arbitrary, we have
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
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
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
by means of Theorem 3.7. Consider the transformation 
. Then, it follows that
Hence, we obtain
since 
. This implies that (3.2) is reducible to zero.
For the converse, suppose that there exists 
such that
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
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
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
for some positive constant 
and all 
. Therefore,
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
where 
is a positive constant. Thus, from
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.
such that 
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 
,
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:
It shows that the coefficient matrices 
and 
satisfy
where 
and 
.
Now, we consider the linear dynamic system
and its perturbed system
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
where
is given by
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:
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
By using Lemma 3.19, we have
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 
:
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
For any 
and 
, the solution 
of (3.28) satisfies
By taking the norms of both sides of (3.37), we have
In view of Gronwall's inequality [15], we obtain
for all 
. Thus
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
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)
, then 
of linear differential system 
is given by 
with the positive constant 
for all 
, then 
of linear difference system 
with the constant 
, then 
of 
-difference system 
and 
, then 
of linear dynamic system 
Example 4.3.
We consider the linear dynamic system
where 
. If 
for all 
, then the matrix exponential function 
of (4.9) is given by
We see that the generalized exponential function 
is given by
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 
.
, then (4.9) is uniformly stable but not strongly stable.
, then (4.9) is strongly stable but not asymptotically stable.
with 
and 
, then (4.9) is neither asymptotically stable nor strongly stable. However, 
goes to zero as 
.
with 
, then (4.9) is neither asymptotically stable nor strongly stable.
with 
, then (4.9) is unbounded and 
is oscillatory.
, then (4.9) is bounded and 
goes to zero as 
.References
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990, 18(1-2):18-56.
Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Agarwal R, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002, 141(1-2):1-26. 10.1016/S0377-0427(01)00432-0
Bohner M, Hering R: Perturbations of dynamic equations. Journal of Difference Equations and Applications 2002, 8(4):295-305. 10.1080/1026190290017360
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996.
Ascoli G: Osservazioni sopra alcune questioni di stabilità . I. Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1950, 9: 129-134.
Choi SK, Koo N, Im DM:
-stability for linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006, 324(1):707-720. 10.1016/j.jmaa.2005.12.046Rama Mohana Rao M: Ordinary Differential Equations. Edward Arnold, London, UK; 1981.
DaCunha JJ: Stability for time varying linear dynamic systems on time scales. Journal of Computational and Applied Mathematics 2005, 176(2):381-410. 10.1016/j.cam.2004.07.026
Cesari L: Asymtotic Behavior and Stability Properties in Ordinary Differential Equations. 2nd edition. Springer, Berlin, Germany; 1963.
Cormani V: Liouville's formula on time scales. Dynamic Systems and Applications 2003, 12(1-2):79-86.
Pötzsche C, Siegmund S, Wirth F: A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical Systems 2003, 9(5):1223-1241.
Aulbach B, Pötzsche C: Reducibility of linear dynamic equations on measure chains. Journal of Computational and Applied Mathematics 2002, 141(1-2):101-115. 10.1016/S0377-0427(01)00438-1
DaCunha JJ, Davis JM: Periodic linear systems: Lyapunov transformations and a unified Floquet theory for time scales. preprint, 2005
Agarwal P, Bohner M, Peterson A: Inequalities on time scales: a survey. Mathematical Inequalities & Applications 2001, 4(4):535-557.
Choi SK, Koo N: On the stability of linear dynamic systems on time scales. to appear in Journal of Difference Equations and Applications
-stability for linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006, 324(1):707-720. 10.1016/j.jmaa.2005.12.046Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/670203