Stability of Linear Dynamic Systems on Time Scales
© Sung Kyu Choi et al. 2008
Received: 31 May 2007
Accepted: 16 March 2008
Published: 19 March 2008
We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples.
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  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
3. Stability of Linear Dynamic Systems
We give the definitions about the various types of stability for the solutions of (3.1).
The following notion of strong stability is due to Ascoli .
For the other types of stability, that is, -stability, we refer to .
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.
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 .
Firstly, we show that the stability for solutions of (3.2) is equivalent to the boundedness.
It follows that all solutions of (3.2) are bounded.
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.
The following is the characterization of strong stability for linear dynamic system (3.2). Note that its continuous version was presented in .
Hence, (3.2) is strongly stable.
Example 3.8 (see ). (i) The system is strongly stable, but it is not asymptotically stable.
Restrictive stability in  is related to strong stability, and we obtain their equivalence for (3.2) as a consequence of Theorem 3.7.
We note that (3.2) is strongly stable if and only if it is restrictively stable.
System (3.2) is restrictively stable if and only if it is reducible to zero.
The continuous versions of Theorems 3.12 and 3.13 are presented in (3.9.v) and (3.9.vi) in , respectively.
Pötzsche et al.  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.
In summary, the following assertions are all equivalent [13, Theorem 4.2].
System (3.2) is strongly stable.
Adjoint system (3.11) of (3.2) is strongly stable.
System (3.2) is restrictively stable.
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  gave a definition of the Lyapunov transformation as follows.
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]).
This completes the proof.
Hence, (3.27) is strongly stable.
The converse holds similarly.
If (3.2) is strongly stable, then (3.28) is uniformly stable.
In this section, we give two examples about the various types of stability for solutions of linear dynamic systems on time scales in .
We give some remarks about Example 4.1.
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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