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Nonlinear Dynamics and Systems Theory, (1) (2009) 3750 Exponential Stability of Linear Time-Invariant Systems on Time Scales T.S. Doan, A. Kalauch and S. Siegmund Department of Mathematics, Dresden University of Technology 01069 Dresden, Germany Received: May 19, 2008; Revised: December 23, 2008 Abstract: Several notions of exponential stability of linear time-invariant sys- tems on arbitrary time scales are discussed. We establish a necessary and suﬃ- cient condition for the existence of uniform exponential stability. Moreover, we characterize the uniform exponential stability of a system by the spectrum of its matrix. In general, exponential stability of a system can not be characterized by the spectrum of its matrix. Keywords: time scale; linear dynamic equation; exponential stability; uniform ex- ponential stability. Mathematics Subject Classiﬁcation (2000): 34C11, 39A10, 37B55, 34D99. 1 Introduction It is well-known that exponential decay of the solution of a linear autonomous ordinary diﬀerential equation ) = Ax , t or of an autonomous diﬀerence equation +1 Ax , t , can be characterized by spectral properties of . Namely, the solutions tend to 0 exponentially as , if and only if all the eigenvalues of have negative real parts or a modulus smaller than 1, respectively. The question, which notion of stability of a linear time-invariant dynamic equation on a time scale inherits such a property, is answered partly in Potzsche et al [16]. The history of asymptotic stability of an equation on a general time scale goes back to the work of Aulbach and Hilger [2]. Although it uniﬁes the time scales or , h > 0, its assumptions are often too pessimistic since the maximal graininess is involved. For a real scalar dynamic equation, stability and instability results are obtained Corresponding author: siegmund@tu-dresden.de 2009 InforMath Publishing Group/1562-8353 (print)/1813-7385 (online)/www.e-ndst.kiev.ua 37

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38 T.S DOAN, A. KALAUCH AND S. SIEGMUND by Gard and Hoﬀacker [7]. Another way to approach to the asymptotic stability of linear dynamic equations using Lyapunov functions can be found in Hilger and Kloeden [10]. Potzsche [15, Abschnitt 2.1] provides suﬃcient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time-varying systems on time scales. Properties of exponential stability of a time-varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [3], DaCunha [5], Du and Tien [6], Hoﬀacker and Tisdell [11], Martynyuk [13] and Peterson and Raﬀoul [14]. As a thorough introduction into dynamic equations on time scales we refer to the paper by Hilger [9] or the monograph by Bohner and Peterson [4]. The paper [2] presents the theory with a focus on linear systems. time scale is a non-empty, closed subset of the reals . For the purpose of this paper we assume from now on that is unbounded from above, i.e. sup . On the graininess is deﬁned as ) := inf t < s } t. This paper is organized as follows. In Section 2 we introduce the class of systems we wish to study and deﬁne the concepts of exponential, uniform exponential, robust exponential and weak-uniform exponential stability. In Section 3 we ﬁrst provide a necessary and suﬃcient condition for the existence of a uniformly exponentially stable linear time-invariant system. We show that uniform exponential stability implies robust exponential stability. An example illustrates that robust exponential stability, in general, does not imply weak-uniform exponential stability. The uniform exponential stability and the robust exponential stability of a system are characterized by the spectrum of its matrix, respectively. In Section 4 we provide an example which indicates that, in general, exponential stability of a system is not determined by the spectrum. We intend to relate the stability of a scalar system to the stability of the according Jordan system. We arrive at the statement that weak-uniform exponential stability of a system is characterized by the spectrum of its matrix. 2 Preliminaries In the following denotes the real ( ) or the complex ( ) ﬁeld. As usual, is the space of square matrices with rows, is the identity mapping on the -dimensional space over and denotes the set of eigenvalues of a matrix Let and consider the -dimensional linear system of dynamic equations Ax. (1) Let t, } denote the transition matrix corresponding to (1), that is, t, τ, ) = t, solves the initial value problem (1) with initial condition ) = for and t, with . The classical examples for this setup are the following. Example 2.1 If we consider linear time-invariant systems of the form ) = Ax ). If , then (1) reduces to ( )) /h Ax ) or equivalently ) = [ hA ).

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 39 The subsequent notions of exponential stability (i), (ii), (iii) of system (1) are intro- duced here as in Potzsche et al [16]. Deﬁnition 2.1 (Exponential stability) Let be a time scale which is unbounded above. We call system (1) (i) exponentially stable if there exists a constant α > 0 such that for every there exists 1 with t, s k )exp( )) for s. (ii) uniformly exponentially stable if can be chosen independently of in the deﬁ- nition of exponential stability. (iii) robustly exponentially stable if there is an ε > 0 such that the exponential stability of (1) implies the exponential stability of Bx for any with k (iv) weak-uniformly exponentially stable if there exists a constant α > 0 such that for every there exists 1 with t, k for all s. Remark 2.1 (i) The diﬀerent notions of stability (i), (ii) and (iii) are partly inves- tigated in the paper by Potzsche et al [16], where examples are provided which show that exponential stability, in general, does neither imply uniform exponential stability nor robust exponential stability. (ii) The notion of weak-uniform exponential stability serves as an intermediate notion between exponential stability and uniform exponential stability. Note that weak-uniform exponential stability coincides with uniform exponential stability if we can choose a bounded function in (iv). One of the observations in this paper is the following diagram about the relations between the stability notions: uniform e. s. weak-uniform e. s. robust e.s. exponential stability 3 Uniform Exponential Stability In this section, we deal with some fundamental properties of uniform exponential sta- bility. More precisely, the existence and robustness of uniform exponential stability are investigated. As a consequence, we obtain a characterization of uniform exponential stability for a linear time-invariant system based on the spectrum of its matrix. Theorem 3.1 (Existence of a uniformly exponentially stable system) Let be a time scale which is unbounded above. Then there exists a uniformly exponentially stable system on Ax, A , x if and only if the graininess of is bounded above, i.e. there exists h > such that for all

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40 T.S DOAN, A. KALAUCH AND S. SIEGMUND Proof ) Assume that there exists such that the system Ax, (2) is uniformly exponentially stable, i.e. there exist K > 0, α > 0 such that t, s k exp( )) for all s. (3) We ﬁrst show that = 0. Indeed, suppose that = 0, then t, s ) = . Hence, we have t, s = 1 for all and the inequality (3) thus does not hold. Let be an arbitrary right scattered point, i.e. 0. Then at the point the equation (2) becomes )) Ax This implies that , t ) = and then by using (3) we have k exp( )) i.e. 1 + k K. Therefore, + 1 for every right scattered point , i.e. has bounded graininess. ) Assume that there exists h > 0 so that for all . Deﬁne Clearly, is invertible for all , i.e. is a regressive matrix. Now we will show that the system Ax, (4) is uniformly exponentially stable. Since is a regressive diagonal matrix, Hilger [9, Theorem 7.4(iii)] implies the following explicit representation of the norm of the transition matrix of (4) t, s = exp lim log exp = exp This completes the proof. From now on we only deal with a time scale with bounded graininess. In order to show the roughness of uniform exponential stability, we provide the following preparatory lemma. Lemma 3.1 Let α > be a positive number. Then for the corresponding scalar system αx the following inequality holds t, s exp( )) for all s. Proof Since α > 0 we have 1+ α > 0 for all . Hence, by Hilger [9, Theorem 7.4(iii)] we have t, s = exp lim log 1 + αu exp = exp( )) This concludes the proof.

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 41 Proposition 3.1 (Robustness of uniform exponential stability) Let be a time scale which is unbounded above and with bounded graininess. Assume that the system Ax, (5) where , is uniformly exponentially stable. Then there exists ε > such that the system Bx (6) is also uniformly exponentially stable for all with k Proof Let K > 0 and α > 0 such that t, s k exp( )) for all s. (7) The equation (6) can be rewritten as follows Ax + ( x. Using the variation of constants formula (see Bohner and Peterson [4, pp 195]) with the inhomogeneous part ) := ( t, s ) for a ﬁxed , the transition matrix of (6) is determined by t, s ) = t, s ) + t, u ))( u, s ) u, for all s. Fix and deﬁne ) = exp( )) t, s . We thus obtain the following estimate exp( )) ≤k t, s k t, u )) exp( )) ) u. This implies with (7) that exp( )) ) for all s. (8) Due to Theorem 3.1 the graininess of is bounded. Fix H > 0 such that for all . Hence, we get from (8) that exp( αH ) for all s. Applying Gronwalls inequality (see Bohner and Peterson [4, Corollary 6.7]) and with ) = 1 we obtain Ke t, s ) for all s, where exp( αH ). By virtue of Lemma 3.1 and the deﬁnition of the function ) we get t, s k exp(( )( )) for all s. (9)

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42 T.S DOAN, A. KALAUCH AND S. SIEGMUND Choose and ﬁx ε > 0 such that K exp( αH < . Now for any with k , we obtain from (9) that t, s k exp(( K exp( αH ))( )) for all s. Since K exp( αH 0, the claim follows. The robustness of uniform exponential stability of a time-varying system is also in- vestigated in DaCuhna [5, Theorem 5.1]. However, the notion of uniform exponential stability and the type of perturbation in DaCuhna [5] are diﬀerent to those here. Pre- cisely, he used the exponential functions to deﬁne uniform exponential stability. For a more details, we refer the reader to DaCuhna [5], Du and Tien [6] and the references therein. Corollary 3.1 (Uniform implies robust exponential stability) Let be a time scale which is unbounded above and with bounded graininess. Suppose that the system Ax, A (10) is uniformly exponentially stable. Then system (10) is also robustly exponentially stable. Next we construct an example which asserts that, in general, robust exponential stability does not imply weak-uniform exponential stability. As a consequence, robust exponential stability does not imply uniform exponential stability. Example 3.1 Let = 1. We deﬁne a sequence recursively by := 0 , s +1 + 4 k, k and := = 0 , k }∪{ + 3 = 0 , k , k and the time scale by the discrete set := =1 Clearly, is unbounded above and has a bounded graininess. Consider on the scalar equation x. (11) For 1 an elementary calculation yields for that + 4 k, s k, x ) = ( 2) This shows that the system (11) is not weak-uniformly exponentially stable, as a solution starting in = 1 may become arbitrarily large depending on the initial time . Now we are going to show that, on the other hand, the system (11) is robustly exponentially stable. To verify this claim we show that the perturbed system = ( 1 + (12)

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 43 is exponentially stable for all 10 10 ). Let be an arbitrary initial value and . Denote by the smallest integer such that . Now we are going to prove inductively the following estimate t, s , x | for all < t +1 and k. (13) We ﬁrst prove (13) in case . Indeed, a straightforward computation yields that t, s , x ) = if m, for = 1 , . . .k 2 + 3 if + 3 m, for = 0 , . . .k This implies with the inequality 2 + 3 | the inequality (13) in case Suppose that the inequality (13) holds for 1. We will show that this also holds for + 1. Indeed, an elementary computation gives t, s , x ) = , s , x ) if m, for = 1 , . . .n, 2 + 3 , s , x ) if + 3 m, for = 0 , . . .n. This implies with the inequality 2 + 3 | the inequality (13) in case and then the claim follows. Deﬁne ) = Φ( , t , x , log 2 and by (13) we get t, t | )exp( )) for all t. Corollary 3.2 Let be a time scale which is unbounded above and with bounded graininess. For consider the Jordan block given by := 1 0 . . . . . . The scalar equation λx (14) is uniformly exponentially stable if and only if the system (15) is uniformly exponentially stable. Proof ) Assume that (14) is uniformly stable. Hence, the equation λI is also uniformly exponentially stable. So, by virtue of Proposition 3.1 there exists ε > such that the system Bx (16)

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44 T.S DOAN, A. KALAUCH AND S. SIEGMUND is uniformly exponentially stable for all such that λI k . Deﬁne = diag(1 , , . . . , ). A straightforward computation yields that := . . . λ ε . . . Consequently, t, s ) = t, s ) for all s. (17) On the other hand, λI k . Hence, by (16) there exists K, α > 0 such that t, s k Ke for all s. This implies with (17) that t, s k for all s. Therefore, (15) is uniformly exponentially stable and it completes the proof. ) The converse direction is trivial. In the next theorem, we will show that uniform exponential stability of a linear system depends only on the eigenvalues of its matrix. Theorem 3.2 The system Ax, A is uniformly exponentially stable if and only if the system λx is uniformly exponentially stable for every Proof Without loss of generality, we deal with the norm = max i,j ij for all = ( ij . Let be the transformation such that AP diag( , J , . . . , J ), where 1 0 . . . . . . for = 1 , . . ., p, are the Jordan blocks of . Clearly, ) = , , . . . , . A straightforward compu- tation yields that t, s ) = diag( t, s , e t, s , . . . , e t, s )) Therefore, max t, s k≤k t, s k≤k max t, s This implies that (3.2) is exponentially stable if and only if the systems x, k = 1 , . . ., p, are exponentially stable. Then by virtue of Corollary 3.2 the claim follows.

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 45 Remark 3.1 The robust exponential stability depends also only on the eigenvalues of the matrix of a system. 4 Exponential Stability and Weak-uniform Exponential Stability In view of Corollary 3.2, the question arises whether the exponential stability of a time- invariant linear system could be characterized by the spectrum of its matrix. In general, this is not the case, since in the subsequent example two systems are presented whose matrices have the same spectrum, one of them is exponentially stable and the other is not. Example 4.1 There exists a time scale , which has bounded graininess such that the system (18) is exponentially stable and the system 1 1 (19) is not exponentially stable. Indeed, denote by the positive number such that (1 + ) = 2 +1 for all (20) Equivalently, := 1 + 1 + 2 +1 for all We deﬁne a discrete time scale as follows =1 where := }∪{ + 1 + 3 = 0 (21) To verify exponential stability of system (18) we show that t, 4) | for all , t (22) Indeed, let be an arbitrary but ﬁxed element in . Deﬁne := max < t A straightforward computation together with (20) yields that (4 +1 ) = (1 + )2 +1 for all (23)

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46 T.S DOAN, A. KALAUCH AND S. SIEGMUND Therefore, t, 4) = t, (4 4) t, Clearly, if = 4 +1 then (22) follows. Hence, it remains to deal with the case t < +1 By deﬁnition of , see (21), we obtain t, where = 4 + 1 + 3 for ∈{ , . . ., . This implies with (20) that t, | +1 +1 for all ∈{ , . . ., proving (22). As a consequence, system (18) is exponentially stable. It remains to show that system (19) is not exponentially stable. System (19) can be represented in the following form where = ( , x ). Denote by ( , x )) the solution of this system starting at = 4 in (1 1). A straightforward computation yields (4 +1 (4 +1 2 0 (4 (4 which gives (4 +1 ) = (2 +1 1) (4 ) + 2 +1 (4 This implies together with (4 ) = 2 that (4 +1 ) = 8 +1 + 2 +1 (4 ) for all Hence, lim (4 ) = 8. As a consequence, system (19) is not exponentially stable. Proposition 4.1 Let be a time scale which is unbounded above. For consider the Jordan block given by := 1 0 . . . . . .

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 47 The scalar equation λx (24) is weak-uniformly exponentially stable if and only if the system (25) is weak-uniformly exponentially stable. Proof Clearly, if = 0 then t, ) = 1 for all . Hence, neither the system (24) nor the system (25) is weak-uniformly exponentially stable. Therefore we are only interested in the case = 0. ) Assume that system (24) is weak-uniformly exponentially stable. Fix Hence, there exist α > 0 and 1 such that t, | exp( )) for all s. (26) To verify the assertion we construct explicit bounds for the solution of (25) with initial condition ) = for a ﬁxed with . Without loss of generality we use the norm := max {| , . . . , |} for = ( , . . . , x in our consideration. Choose and ﬁx ε > 0 such that α > d . Deﬁne j d for = 1 , . . ., d Clearly, 0 and we will prove by induction on d, . . . , 1 that there exist constants such that the -th component of the solution of (25) is exponentially bounded by | exp( )) for all τ. (27) For the assertion follows from the assumption as the -th entry of ) is a solution of (24) and hence by (26) we have t, | exp( )) for all τ, where := . Assume that the assertion (27) is shown for some index + 1 2, i.e. there exists +1 with +1 | +1 exp( +1 )) for all τ. (28) By construction, the -th component of the solution satisﬁes the equation ) = λx ) + +1 ) for Using the variation of constants formula (see Bohner and Peterson [4, pp 77]) we have the representation ) = t, t, u )) +1 ) u. Fix . Using the exponential bound of t, ) and (28) we obtain |≤| t, t, u )) || +1 exp( )) + +1 ) u, (29)

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48 T.S DOAN, A. KALAUCH AND S. SIEGMUND where ) := t, u )) exp( +1 )). Denote by < t < t the right scattered points in [ τ, t ] with 1 + | , i = 1 , . . ., n. Now we are going to estimate ) for τ, t ]. If for = 1 , . . ., n we get t, u ) = t, u )) , u t, u ))(1 + )) This implies with (26) that t, u )) | exp( )) Therefore, exp( +1 )) if ∈{ , . . . , t (30) If 6∈{ , . . ., t , we get . Applying (26) to t, u )), we obtain exp( ))exp( )))exp( +1 )) Therefore, exp exp( +1 )) for 6∈{ , . . ., t (31) Combining (30) and (31), there exists M > 0 such that exp( +1 )) for all τ, t This implies with (29) that | exp( )) + MK +1 )exp( +1 )) On the other hand, exp( )) for all . We thus obtain | MK +1 exp( )) for all τ, proving (27) with := MK +1 . As we have exponential decay of all components of the solution ), we obtain the assertion. We now construct an example which ensures that in general weak-uniform exponential stability does not imply uniform exponential stability. Example 4.2 We deﬁne a discrete time scale by =1 [0 where := { = 0 , . . ., k for all

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 49 Consider on the scalar system x. (32) We ﬁrst show that the system (32) is weak-uniformly exponentially stable. Obviously, for any with 0 we have t, = exp( )) for all (33) For an arbitrary but ﬁxed with s < 0, we are going to estimate t, for . A straightforward computation yields that t, if > τ, exp( )) if We thus obtain t, | )exp( )) for all s, where ) := max t, exp( Hence system (32) is weak-uniformly exponentially stable. On the other hand, a direct computation gives that k, 1) = (1 for all which implies that system (32) is not uniformly exponentially stable. Remark 4.1 Observe that Proposition 4.1 in combination with example 4.2 provides a negative answer to the question mentioned in the conclusion of Potzsche et al [16] whether the uniform exponential stability of system (24) is a necessary condition for the exponential stability of system (25). Moreover, the time scale in Potzsche et al [16] is assumed to have bounded graininess. This assumption is dropped in Proposition 4.1. By virtue of Proposition 4.1 in combination with an analogous argument as in the proof of Theorem 3.2 we get the following corollary to characterize weak-uniform exponential stability. Corollary 4.1 The system Ax, A is weak-uniformly exponentially stable if and only if the system λx is weak-uniformly exponentially stable for every

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50 T.S DOAN, A. KALAUCH AND S. SIEGMUND References [1] Agarwal, R.P. Diﬀerence Equations and Inequalities . Marcel Dekker Inc., New York, 1992. [2] Aulbach, B. and Hilger, S. Linear dynamic processes with inhomogeneous time scale. In: Nonlinear Dynamics and Quantum Dynamical Systems (G.A. Leonov, V. Reitmann, W. Timmermann, eds.). Mathematical Research Bd. 59, Akademie-Verlag, Berlin, 1990, 920. [3] Bohner, M. and Martynyuk, A.A. Elements of stability theory of A. M. Lyapunov for dynamic equations on time scales. Prikladnaya Mekhanika 43 (9) (2007) 327. Translation in Nonlinear Dynamics and Systems Theory (3) (2007) 225-257. [4] Bohner, M. and Peterson, A. Dynamic Equations on Time Scales . Birkhauser, Boston, 2001. [5] DaCunha, J.J. Stability for time varying linear dynamic systems on time scales. J. Comput. Appl. Math. 176 (2005) 381410. [6] Du, N.H. and Tien, L.H. On the exponential stability of dynamic equations on time scales. J. Math. Anal. Appl. 331 (2007) 11591174. [7] Gard, T. and Hoﬀacker, J. Asymptotic behavior of natural growth on time scales. Dynam. Systems Appl. 12 (2003) 131147. [8] Hilger, S. Ein Makettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten . Ph.D. the- sis, Universitat Wurzburg, 1988. [9] Hilger, S. Analysis on measure chains A uniﬁed approach to continuous and discrete calculus. Results in Mathematics 18 (1990) 1856. [10] Hilger, S. and Kloeden, P.E. Comparative time gainyness and asymptotic stability of dy- namical systems. Autom. Remote Control 55 (9) Pt.1 (1994) 12931298. [11] Hoﬀacker, J. and Tisdell, C.C. Stability and instability for dynamic equations on time scales. Comput. Math. Appl. 49 (2005) 13271334. [12] Keller, S. Asymptotisches Verhalten invarianter Faserbundel bei Diskretisierung und Mit- telwertbildung im Rahmen der Analysis auf Zeitskalen . Ph.D. thesis, Augsburger Mathema- tische Schriften 36, Winer Verlag, Augsburg, 1999. [13] Martynyuk, A.A. On exponential stability of a dynamical system on a time scale. Dokl. Akad. Nauk 421 (2008) 312317. [Russian] [14] Peterson, A. and Raﬀoul, R.F. Exponential stability of dynamic equation on time scales. Adv. Diﬀerence Equ. Appl. (2) (2005) 133144. [15] Otzsche, C.P. Langsame Faserbundel dynamischer Gleichungen auf Maketten . Ph.D. the- sis, Logos Verlag, Berlin, 2002. [16] Otzsche, C. P, Siegmund, S. and Wirth, F. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical System . Series A, (5) (2003) 12231241.

S Doan A Kalauch and S Siegmund Department of Mathematics Dresden University of Technology 01069 Dresden Germany Received May 19 2008 Revised December 23 2008 Abstract Several notions of exponential stability of linear timeinvariant sys tems on arbit ID: 24218

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Nonlinear Dynamics and Systems Theory, (1) (2009) 3750 Exponential Stability of Linear Time-Invariant Systems on Time Scales T.S. Doan, A. Kalauch and S. Siegmund Department of Mathematics, Dresden University of Technology 01069 Dresden, Germany Received: May 19, 2008; Revised: December 23, 2008 Abstract: Several notions of exponential stability of linear time-invariant sys- tems on arbitrary time scales are discussed. We establish a necessary and suﬃ- cient condition for the existence of uniform exponential stability. Moreover, we characterize the uniform exponential stability of a system by the spectrum of its matrix. In general, exponential stability of a system can not be characterized by the spectrum of its matrix. Keywords: time scale; linear dynamic equation; exponential stability; uniform ex- ponential stability. Mathematics Subject Classiﬁcation (2000): 34C11, 39A10, 37B55, 34D99. 1 Introduction It is well-known that exponential decay of the solution of a linear autonomous ordinary diﬀerential equation ) = Ax , t or of an autonomous diﬀerence equation +1 Ax , t , can be characterized by spectral properties of . Namely, the solutions tend to 0 exponentially as , if and only if all the eigenvalues of have negative real parts or a modulus smaller than 1, respectively. The question, which notion of stability of a linear time-invariant dynamic equation on a time scale inherits such a property, is answered partly in Potzsche et al [16]. The history of asymptotic stability of an equation on a general time scale goes back to the work of Aulbach and Hilger [2]. Although it uniﬁes the time scales or , h > 0, its assumptions are often too pessimistic since the maximal graininess is involved. For a real scalar dynamic equation, stability and instability results are obtained Corresponding author: siegmund@tu-dresden.de 2009 InforMath Publishing Group/1562-8353 (print)/1813-7385 (online)/www.e-ndst.kiev.ua 37

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38 T.S DOAN, A. KALAUCH AND S. SIEGMUND by Gard and Hoﬀacker [7]. Another way to approach to the asymptotic stability of linear dynamic equations using Lyapunov functions can be found in Hilger and Kloeden [10]. Potzsche [15, Abschnitt 2.1] provides suﬃcient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time-varying systems on time scales. Properties of exponential stability of a time-varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [3], DaCunha [5], Du and Tien [6], Hoﬀacker and Tisdell [11], Martynyuk [13] and Peterson and Raﬀoul [14]. As a thorough introduction into dynamic equations on time scales we refer to the paper by Hilger [9] or the monograph by Bohner and Peterson [4]. The paper [2] presents the theory with a focus on linear systems. time scale is a non-empty, closed subset of the reals . For the purpose of this paper we assume from now on that is unbounded from above, i.e. sup . On the graininess is deﬁned as ) := inf t < s } t. This paper is organized as follows. In Section 2 we introduce the class of systems we wish to study and deﬁne the concepts of exponential, uniform exponential, robust exponential and weak-uniform exponential stability. In Section 3 we ﬁrst provide a necessary and suﬃcient condition for the existence of a uniformly exponentially stable linear time-invariant system. We show that uniform exponential stability implies robust exponential stability. An example illustrates that robust exponential stability, in general, does not imply weak-uniform exponential stability. The uniform exponential stability and the robust exponential stability of a system are characterized by the spectrum of its matrix, respectively. In Section 4 we provide an example which indicates that, in general, exponential stability of a system is not determined by the spectrum. We intend to relate the stability of a scalar system to the stability of the according Jordan system. We arrive at the statement that weak-uniform exponential stability of a system is characterized by the spectrum of its matrix. 2 Preliminaries In the following denotes the real ( ) or the complex ( ) ﬁeld. As usual, is the space of square matrices with rows, is the identity mapping on the -dimensional space over and denotes the set of eigenvalues of a matrix Let and consider the -dimensional linear system of dynamic equations Ax. (1) Let t, } denote the transition matrix corresponding to (1), that is, t, τ, ) = t, solves the initial value problem (1) with initial condition ) = for and t, with . The classical examples for this setup are the following. Example 2.1 If we consider linear time-invariant systems of the form ) = Ax ). If , then (1) reduces to ( )) /h Ax ) or equivalently ) = [ hA ).

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 39 The subsequent notions of exponential stability (i), (ii), (iii) of system (1) are intro- duced here as in Potzsche et al [16]. Deﬁnition 2.1 (Exponential stability) Let be a time scale which is unbounded above. We call system (1) (i) exponentially stable if there exists a constant α > 0 such that for every there exists 1 with t, s k )exp( )) for s. (ii) uniformly exponentially stable if can be chosen independently of in the deﬁ- nition of exponential stability. (iii) robustly exponentially stable if there is an ε > 0 such that the exponential stability of (1) implies the exponential stability of Bx for any with k (iv) weak-uniformly exponentially stable if there exists a constant α > 0 such that for every there exists 1 with t, k for all s. Remark 2.1 (i) The diﬀerent notions of stability (i), (ii) and (iii) are partly inves- tigated in the paper by Potzsche et al [16], where examples are provided which show that exponential stability, in general, does neither imply uniform exponential stability nor robust exponential stability. (ii) The notion of weak-uniform exponential stability serves as an intermediate notion between exponential stability and uniform exponential stability. Note that weak-uniform exponential stability coincides with uniform exponential stability if we can choose a bounded function in (iv). One of the observations in this paper is the following diagram about the relations between the stability notions: uniform e. s. weak-uniform e. s. robust e.s. exponential stability 3 Uniform Exponential Stability In this section, we deal with some fundamental properties of uniform exponential sta- bility. More precisely, the existence and robustness of uniform exponential stability are investigated. As a consequence, we obtain a characterization of uniform exponential stability for a linear time-invariant system based on the spectrum of its matrix. Theorem 3.1 (Existence of a uniformly exponentially stable system) Let be a time scale which is unbounded above. Then there exists a uniformly exponentially stable system on Ax, A , x if and only if the graininess of is bounded above, i.e. there exists h > such that for all

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40 T.S DOAN, A. KALAUCH AND S. SIEGMUND Proof ) Assume that there exists such that the system Ax, (2) is uniformly exponentially stable, i.e. there exist K > 0, α > 0 such that t, s k exp( )) for all s. (3) We ﬁrst show that = 0. Indeed, suppose that = 0, then t, s ) = . Hence, we have t, s = 1 for all and the inequality (3) thus does not hold. Let be an arbitrary right scattered point, i.e. 0. Then at the point the equation (2) becomes )) Ax This implies that , t ) = and then by using (3) we have k exp( )) i.e. 1 + k K. Therefore, + 1 for every right scattered point , i.e. has bounded graininess. ) Assume that there exists h > 0 so that for all . Deﬁne Clearly, is invertible for all , i.e. is a regressive matrix. Now we will show that the system Ax, (4) is uniformly exponentially stable. Since is a regressive diagonal matrix, Hilger [9, Theorem 7.4(iii)] implies the following explicit representation of the norm of the transition matrix of (4) t, s = exp lim log exp = exp This completes the proof. From now on we only deal with a time scale with bounded graininess. In order to show the roughness of uniform exponential stability, we provide the following preparatory lemma. Lemma 3.1 Let α > be a positive number. Then for the corresponding scalar system αx the following inequality holds t, s exp( )) for all s. Proof Since α > 0 we have 1+ α > 0 for all . Hence, by Hilger [9, Theorem 7.4(iii)] we have t, s = exp lim log 1 + αu exp = exp( )) This concludes the proof.

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 41 Proposition 3.1 (Robustness of uniform exponential stability) Let be a time scale which is unbounded above and with bounded graininess. Assume that the system Ax, (5) where , is uniformly exponentially stable. Then there exists ε > such that the system Bx (6) is also uniformly exponentially stable for all with k Proof Let K > 0 and α > 0 such that t, s k exp( )) for all s. (7) The equation (6) can be rewritten as follows Ax + ( x. Using the variation of constants formula (see Bohner and Peterson [4, pp 195]) with the inhomogeneous part ) := ( t, s ) for a ﬁxed , the transition matrix of (6) is determined by t, s ) = t, s ) + t, u ))( u, s ) u, for all s. Fix and deﬁne ) = exp( )) t, s . We thus obtain the following estimate exp( )) ≤k t, s k t, u )) exp( )) ) u. This implies with (7) that exp( )) ) for all s. (8) Due to Theorem 3.1 the graininess of is bounded. Fix H > 0 such that for all . Hence, we get from (8) that exp( αH ) for all s. Applying Gronwalls inequality (see Bohner and Peterson [4, Corollary 6.7]) and with ) = 1 we obtain Ke t, s ) for all s, where exp( αH ). By virtue of Lemma 3.1 and the deﬁnition of the function ) we get t, s k exp(( )( )) for all s. (9)

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42 T.S DOAN, A. KALAUCH AND S. SIEGMUND Choose and ﬁx ε > 0 such that K exp( αH < . Now for any with k , we obtain from (9) that t, s k exp(( K exp( αH ))( )) for all s. Since K exp( αH 0, the claim follows. The robustness of uniform exponential stability of a time-varying system is also in- vestigated in DaCuhna [5, Theorem 5.1]. However, the notion of uniform exponential stability and the type of perturbation in DaCuhna [5] are diﬀerent to those here. Pre- cisely, he used the exponential functions to deﬁne uniform exponential stability. For a more details, we refer the reader to DaCuhna [5], Du and Tien [6] and the references therein. Corollary 3.1 (Uniform implies robust exponential stability) Let be a time scale which is unbounded above and with bounded graininess. Suppose that the system Ax, A (10) is uniformly exponentially stable. Then system (10) is also robustly exponentially stable. Next we construct an example which asserts that, in general, robust exponential stability does not imply weak-uniform exponential stability. As a consequence, robust exponential stability does not imply uniform exponential stability. Example 3.1 Let = 1. We deﬁne a sequence recursively by := 0 , s +1 + 4 k, k and := = 0 , k }∪{ + 3 = 0 , k , k and the time scale by the discrete set := =1 Clearly, is unbounded above and has a bounded graininess. Consider on the scalar equation x. (11) For 1 an elementary calculation yields for that + 4 k, s k, x ) = ( 2) This shows that the system (11) is not weak-uniformly exponentially stable, as a solution starting in = 1 may become arbitrarily large depending on the initial time . Now we are going to show that, on the other hand, the system (11) is robustly exponentially stable. To verify this claim we show that the perturbed system = ( 1 + (12)

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 43 is exponentially stable for all 10 10 ). Let be an arbitrary initial value and . Denote by the smallest integer such that . Now we are going to prove inductively the following estimate t, s , x | for all < t +1 and k. (13) We ﬁrst prove (13) in case . Indeed, a straightforward computation yields that t, s , x ) = if m, for = 1 , . . .k 2 + 3 if + 3 m, for = 0 , . . .k This implies with the inequality 2 + 3 | the inequality (13) in case Suppose that the inequality (13) holds for 1. We will show that this also holds for + 1. Indeed, an elementary computation gives t, s , x ) = , s , x ) if m, for = 1 , . . .n, 2 + 3 , s , x ) if + 3 m, for = 0 , . . .n. This implies with the inequality 2 + 3 | the inequality (13) in case and then the claim follows. Deﬁne ) = Φ( , t , x , log 2 and by (13) we get t, t | )exp( )) for all t. Corollary 3.2 Let be a time scale which is unbounded above and with bounded graininess. For consider the Jordan block given by := 1 0 . . . . . . The scalar equation λx (14) is uniformly exponentially stable if and only if the system (15) is uniformly exponentially stable. Proof ) Assume that (14) is uniformly stable. Hence, the equation λI is also uniformly exponentially stable. So, by virtue of Proposition 3.1 there exists ε > such that the system Bx (16)

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44 T.S DOAN, A. KALAUCH AND S. SIEGMUND is uniformly exponentially stable for all such that λI k . Deﬁne = diag(1 , , . . . , ). A straightforward computation yields that := . . . λ ε . . . Consequently, t, s ) = t, s ) for all s. (17) On the other hand, λI k . Hence, by (16) there exists K, α > 0 such that t, s k Ke for all s. This implies with (17) that t, s k for all s. Therefore, (15) is uniformly exponentially stable and it completes the proof. ) The converse direction is trivial. In the next theorem, we will show that uniform exponential stability of a linear system depends only on the eigenvalues of its matrix. Theorem 3.2 The system Ax, A is uniformly exponentially stable if and only if the system λx is uniformly exponentially stable for every Proof Without loss of generality, we deal with the norm = max i,j ij for all = ( ij . Let be the transformation such that AP diag( , J , . . . , J ), where 1 0 . . . . . . for = 1 , . . ., p, are the Jordan blocks of . Clearly, ) = , , . . . , . A straightforward compu- tation yields that t, s ) = diag( t, s , e t, s , . . . , e t, s )) Therefore, max t, s k≤k t, s k≤k max t, s This implies that (3.2) is exponentially stable if and only if the systems x, k = 1 , . . ., p, are exponentially stable. Then by virtue of Corollary 3.2 the claim follows.

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 45 Remark 3.1 The robust exponential stability depends also only on the eigenvalues of the matrix of a system. 4 Exponential Stability and Weak-uniform Exponential Stability In view of Corollary 3.2, the question arises whether the exponential stability of a time- invariant linear system could be characterized by the spectrum of its matrix. In general, this is not the case, since in the subsequent example two systems are presented whose matrices have the same spectrum, one of them is exponentially stable and the other is not. Example 4.1 There exists a time scale , which has bounded graininess such that the system (18) is exponentially stable and the system 1 1 (19) is not exponentially stable. Indeed, denote by the positive number such that (1 + ) = 2 +1 for all (20) Equivalently, := 1 + 1 + 2 +1 for all We deﬁne a discrete time scale as follows =1 where := }∪{ + 1 + 3 = 0 (21) To verify exponential stability of system (18) we show that t, 4) | for all , t (22) Indeed, let be an arbitrary but ﬁxed element in . Deﬁne := max < t A straightforward computation together with (20) yields that (4 +1 ) = (1 + )2 +1 for all (23)

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46 T.S DOAN, A. KALAUCH AND S. SIEGMUND Therefore, t, 4) = t, (4 4) t, Clearly, if = 4 +1 then (22) follows. Hence, it remains to deal with the case t < +1 By deﬁnition of , see (21), we obtain t, where = 4 + 1 + 3 for ∈{ , . . ., . This implies with (20) that t, | +1 +1 for all ∈{ , . . ., proving (22). As a consequence, system (18) is exponentially stable. It remains to show that system (19) is not exponentially stable. System (19) can be represented in the following form where = ( , x ). Denote by ( , x )) the solution of this system starting at = 4 in (1 1). A straightforward computation yields (4 +1 (4 +1 2 0 (4 (4 which gives (4 +1 ) = (2 +1 1) (4 ) + 2 +1 (4 This implies together with (4 ) = 2 that (4 +1 ) = 8 +1 + 2 +1 (4 ) for all Hence, lim (4 ) = 8. As a consequence, system (19) is not exponentially stable. Proposition 4.1 Let be a time scale which is unbounded above. For consider the Jordan block given by := 1 0 . . . . . .

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 47 The scalar equation λx (24) is weak-uniformly exponentially stable if and only if the system (25) is weak-uniformly exponentially stable. Proof Clearly, if = 0 then t, ) = 1 for all . Hence, neither the system (24) nor the system (25) is weak-uniformly exponentially stable. Therefore we are only interested in the case = 0. ) Assume that system (24) is weak-uniformly exponentially stable. Fix Hence, there exist α > 0 and 1 such that t, | exp( )) for all s. (26) To verify the assertion we construct explicit bounds for the solution of (25) with initial condition ) = for a ﬁxed with . Without loss of generality we use the norm := max {| , . . . , |} for = ( , . . . , x in our consideration. Choose and ﬁx ε > 0 such that α > d . Deﬁne j d for = 1 , . . ., d Clearly, 0 and we will prove by induction on d, . . . , 1 that there exist constants such that the -th component of the solution of (25) is exponentially bounded by | exp( )) for all τ. (27) For the assertion follows from the assumption as the -th entry of ) is a solution of (24) and hence by (26) we have t, | exp( )) for all τ, where := . Assume that the assertion (27) is shown for some index + 1 2, i.e. there exists +1 with +1 | +1 exp( +1 )) for all τ. (28) By construction, the -th component of the solution satisﬁes the equation ) = λx ) + +1 ) for Using the variation of constants formula (see Bohner and Peterson [4, pp 77]) we have the representation ) = t, t, u )) +1 ) u. Fix . Using the exponential bound of t, ) and (28) we obtain |≤| t, t, u )) || +1 exp( )) + +1 ) u, (29)

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48 T.S DOAN, A. KALAUCH AND S. SIEGMUND where ) := t, u )) exp( +1 )). Denote by < t < t the right scattered points in [ τ, t ] with 1 + | , i = 1 , . . ., n. Now we are going to estimate ) for τ, t ]. If for = 1 , . . ., n we get t, u ) = t, u )) , u t, u ))(1 + )) This implies with (26) that t, u )) | exp( )) Therefore, exp( +1 )) if ∈{ , . . . , t (30) If 6∈{ , . . ., t , we get . Applying (26) to t, u )), we obtain exp( ))exp( )))exp( +1 )) Therefore, exp exp( +1 )) for 6∈{ , . . ., t (31) Combining (30) and (31), there exists M > 0 such that exp( +1 )) for all τ, t This implies with (29) that | exp( )) + MK +1 )exp( +1 )) On the other hand, exp( )) for all . We thus obtain | MK +1 exp( )) for all τ, proving (27) with := MK +1 . As we have exponential decay of all components of the solution ), we obtain the assertion. We now construct an example which ensures that in general weak-uniform exponential stability does not imply uniform exponential stability. Example 4.2 We deﬁne a discrete time scale by =1 [0 where := { = 0 , . . ., k for all

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NONLINEAR DYNAMICS AND SYSTEMS THEORY, (1) (2009) 3750 49 Consider on the scalar system x. (32) We ﬁrst show that the system (32) is weak-uniformly exponentially stable. Obviously, for any with 0 we have t, = exp( )) for all (33) For an arbitrary but ﬁxed with s < 0, we are going to estimate t, for . A straightforward computation yields that t, if > τ, exp( )) if We thus obtain t, | )exp( )) for all s, where ) := max t, exp( Hence system (32) is weak-uniformly exponentially stable. On the other hand, a direct computation gives that k, 1) = (1 for all which implies that system (32) is not uniformly exponentially stable. Remark 4.1 Observe that Proposition 4.1 in combination with example 4.2 provides a negative answer to the question mentioned in the conclusion of Potzsche et al [16] whether the uniform exponential stability of system (24) is a necessary condition for the exponential stability of system (25). Moreover, the time scale in Potzsche et al [16] is assumed to have bounded graininess. This assumption is dropped in Proposition 4.1. By virtue of Proposition 4.1 in combination with an analogous argument as in the proof of Theorem 3.2 we get the following corollary to characterize weak-uniform exponential stability. Corollary 4.1 The system Ax, A is weak-uniformly exponentially stable if and only if the system λx is weak-uniformly exponentially stable for every

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50 T.S DOAN, A. KALAUCH AND S. SIEGMUND References [1] Agarwal, R.P. Diﬀerence Equations and Inequalities . Marcel Dekker Inc., New York, 1992. [2] Aulbach, B. and Hilger, S. Linear dynamic processes with inhomogeneous time scale. In: Nonlinear Dynamics and Quantum Dynamical Systems (G.A. Leonov, V. Reitmann, W. Timmermann, eds.). Mathematical Research Bd. 59, Akademie-Verlag, Berlin, 1990, 920. [3] Bohner, M. and Martynyuk, A.A. Elements of stability theory of A. M. Lyapunov for dynamic equations on time scales. Prikladnaya Mekhanika 43 (9) (2007) 327. Translation in Nonlinear Dynamics and Systems Theory (3) (2007) 225-257. [4] Bohner, M. and Peterson, A. Dynamic Equations on Time Scales . Birkhauser, Boston, 2001. [5] DaCunha, J.J. Stability for time varying linear dynamic systems on time scales. J. Comput. Appl. Math. 176 (2005) 381410. [6] Du, N.H. and Tien, L.H. On the exponential stability of dynamic equations on time scales. J. Math. Anal. Appl. 331 (2007) 11591174. [7] Gard, T. and Hoﬀacker, J. Asymptotic behavior of natural growth on time scales. Dynam. Systems Appl. 12 (2003) 131147. [8] Hilger, S. Ein Makettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten . Ph.D. the- sis, Universitat Wurzburg, 1988. [9] Hilger, S. Analysis on measure chains A uniﬁed approach to continuous and discrete calculus. Results in Mathematics 18 (1990) 1856. [10] Hilger, S. and Kloeden, P.E. Comparative time gainyness and asymptotic stability of dy- namical systems. Autom. Remote Control 55 (9) Pt.1 (1994) 12931298. [11] Hoﬀacker, J. and Tisdell, C.C. Stability and instability for dynamic equations on time scales. Comput. Math. Appl. 49 (2005) 13271334. [12] Keller, S. Asymptotisches Verhalten invarianter Faserbundel bei Diskretisierung und Mit- telwertbildung im Rahmen der Analysis auf Zeitskalen . Ph.D. thesis, Augsburger Mathema- tische Schriften 36, Winer Verlag, Augsburg, 1999. [13] Martynyuk, A.A. On exponential stability of a dynamical system on a time scale. Dokl. Akad. Nauk 421 (2008) 312317. [Russian] [14] Peterson, A. and Raﬀoul, R.F. Exponential stability of dynamic equation on time scales. Adv. Diﬀerence Equ. Appl. (2) (2005) 133144. [15] Otzsche, C.P. Langsame Faserbundel dynamischer Gleichungen auf Maketten . Ph.D. the- sis, Logos Verlag, Berlin, 2002. [16] Otzsche, C. P, Siegmund, S. and Wirth, F. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical System . Series A, (5) (2003) 12231241.

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