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
- (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
is continuous at every right-dense point ,
exists and is finite at every left-dense point .
where is the th column of .
where is the identity matrix.
where , is called the matrix exponential function and it is denoted by .
3. Stability of Linear Dynamic Systems
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).
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 .
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 .
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 .
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 .
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.
Equation (3.2) is stable if and only if all solutions of (3.2) are bounded for all .
It follows that all solutions of (3.2) are bounded.
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.
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 .
where is a matrix exponential function of (3.2).
Hence, (3.2) is strongly stable.
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 ). (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  is related to strong stability, and we obtain their equivalence for (3.2) as a consequence of Theorem 3.7.
where denotes the conjugate transpose of , is stable.
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.
since . This implies that (3.2) is reducible to zero.
Then, we have . Thus, is a matrix exponential function of (3.2). Since and are bounded for all , the proof is complete.
If (3.2) is stable and reducible on a time scale with the constant graininess, then it is uniformly stable.
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 , respectively.
It does not hold in general that every stable linear homogeneous system with constant coefficient matrix on a time scale is uniformly stable.
If (3.11) is stable and with the eigenvalues ( ) of , then it is restrictively stable.
it is clear that is bounded for all . The proof is complete.
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.
- (ii)There exists a positive constant such that(3.23)
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.
for all .
where and .
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]).
for all .
The regressiveness of in (3.30) is preserved by the Lyapunov transformation in the following lemma.
Suppose that is the transformation matrix for all . Then is regressive if and only if is also regressive.
This completes the proof.
Suppose that is a Lyapunov transformation. Then (3.2) is strongly stable if and only if (3.27) is strongly stable.
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.
If (3.2) is strongly stable, then (3.28) is uniformly stable.
where . Hence, (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 .
where . If for all , then (4.1) is strongly stable.
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)
respectively. Thus, we obtain the following results for (4.9) and .
If , then (4.9) is uniformly stable but not strongly stable.
If , then (4.9) is strongly stable but not asymptotically stable.
If with and , then (4.9) is neither asymptotically stable nor strongly stable. However, goes to zero as .
If with , then (4.9) is neither asymptotically stable nor strongly stable.
If with , then (4.9) is unbounded and is oscillatory.
If , then (4.9) is bounded and goes to zero as .
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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