Open Access

Stability of Linear Dynamic Systems on Time Scales

Advances in Difference Equations20082008:670203

https://doi.org/10.1155/2008/670203

Received: 31 May 2007

Accepted: 16 March 2008

Published: 19 March 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
(2.1)
(supplemented by ), while the graininess is given by
(2.2)
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 ,
(2.3)

where .

Some basic properties of delta derivatives are given in the following [35]:

  1. (i)
    If is differentiable at , then
    (2.4)
     
  2. (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

  1. (i)

    is continuous at every right-dense point ,

     
  2. (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
(2.6)

where .

The norm of an matrix is defined to be
(2.7)

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
(2.8)
Here, a matrix-valued function is called regressive provided:
(2.9)

where is the identity matrix.

Definition 2.1.

Let . The unique matrix-valued solution of the IVP
(2.10)

where , is called the matrix exponential function and it is denoted by .

3. Stability of Linear Dynamic Systems

We consider the dynamic system
(3.1)

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
(3.2)

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
(3.3)
Consequently, for any solution of (3.2),
(3.4)

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
(3.5)

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
(3.6)

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
(3.7)

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
(3.8)

Hence, (3.2) is strongly stable.

Conversely, if (3.2) is strongly stable, then we have
(3.9)
whenever and holds. Since is arbitrary, we have
(3.10)

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
(3.11)

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
(3.12)

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
(3.13)
by means of Theorem 3.7. Consider the transformation . Then, it follows that
(3.14)
Hence, we obtain
(3.15)

since . This implies that (3.2) is reducible to zero.

For the converse, suppose that there exists such that
(3.16)

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
(3.17)
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
(3.18)
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
(3.19)
for some positive constant and all . Therefore,
(3.20)

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
(3.21)
where is a positive constant. Thus, from
(3.22)

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].

  1. (i)

    System (3.2) is strongly stable.

     
  2. (ii)
    There exists a positive constant such that
    (3.23)
     
  3. (iii)

    Adjoint system (3.11) of (3.2) is strongly stable.

     
  4. (iv)

    System (3.2) is restrictively stable.

     
  5. (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 ,
(3.24)

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:
(3.25)
It shows that the coefficient matrices and satisfy
(3.26)

where and .

Now, we consider the linear dynamic system
(3.27)
and its perturbed system
(3.28)

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
(3.29)
where
(3.30)
is given by
(3.31)

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:
(3.32)

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
(3.33)
By using Lemma 3.19, we have
(3.34)

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 :
(3.35)

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
(3.36)
For any and , the solution of (3.28) satisfies
(3.37)
By taking the norms of both sides of (3.37), we have
(3.38)
In view of Gronwall's inequality [15], we obtain
(3.39)
for all . Thus
(3.40)

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
(4.1)

where . If for all , then (4.1) is strongly stable.

Remark 4.2.

We give some remarks about Example 4.1.

  1. (1)
    If , then of linear differential system is given by
    (4.2)
     
  2. (2)
    If with the positive constant for all , then of linear difference system
    (4.3)
    is given by
    (4.4)
     
  3. (3)
    If with the constant , then of -difference system
    (4.5)
    is given by
    (4.6)
     
  4. (4)
    If and , then of linear dynamic system
    (4.7)
    is given by
    (4.8)
     

Example 4.3.

We consider the linear dynamic system
(4.9)
where . If for all , then the matrix exponential function of (4.9) is given by
(4.10)
We see that the generalized exponential function is given by
(4.11)

respectively. Thus, we obtain the following results for (4.9) and .

  1. (1)

    If , then (4.9) is uniformly stable but not strongly stable.

     
  2. (2)

    If , then (4.9) is strongly stable but not asymptotically stable.

     
  3. (3)

    If with and , then (4.9) is neither asymptotically stable nor strongly stable. However, goes to zero as .

     
  4. (4)

    If with , then (4.9) is neither asymptotically stable nor strongly stable.

     
  5. (5)

    If with , then (4.9) is unbounded and is oscillatory.

     
  6. (6)

    If , then (4.9) is bounded and goes to zero as .

     

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Chungnam National University
(2)
Department of Mathematics Education, Cheongju University

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Copyright

© Sung Kyu Choi et al. 2008

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