The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess

Kloeden, Peter E.

Zmorzynska, Alexandra

Advances in Difference Equations

English

Hochschulschriftenserver - Universität Frankfurt am Main

LYAPUNOV FUNCTIONS FOR LINEAR NONAUTONOMOUSDYNAMICAL EQUATIONS ON TIME SCALESPETER E. KLOEDEN AND ALEXANDRA ZMORZYNSKAReceived 25 January 2006; Revised 23 March 2006; Accepted 13 April 2006The existence of a Lyapunov function is established following a method of Yoshizawa forthe uniform exponential asymptotic stability of the zero solution of a nonautonomouslinear dynamical equation on a time scale with uniformly bounded graininess.Copyright © 2006 P. E. Kloeden and A. Zmorzynska. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is prop-erly cited.1.IntroductionLyapunov functions are a very useful tool for investigating the behaviour of dynamicalequations. They have been used now for over a century for diﬀerential equations of manytypes [15]a sw e l la sd i ﬀerence equations [1]. They were ﬁrst used in the context of timescales in [12]. See [16] for a more recent and an extensive investigation of Lyapunovfunctions on time scales.Much of the literature on Lyapunov functions, especially applications oriented, dealswith suﬃcient conditions, assuming that a Lyapunov function is known. An importanttheoretical issue with practical implications is whether or not a Lyapunov function char-acterizingaparticulardynamicalpropertyactuallyexists—suchresultsareknownasnec-essary conditions.In this paper, we establish the existence of a Lyapunov function characterising theuniform exponential asymptotic stability of the zero solution of a nonautonomous lineardynamic equationxΔ =A(t)x (1.1)on a time scale T with a bounded graininess, where the matrix-valued mapping t  → A(t)is right dense continuous (rd-continuous) on T, that is, A ∈rd(T,Rn×n).Such linear dynamical equations and their inhomogeneous variants play an impor-tant role in investigations of the dynamical behaviour, both in themselves and also asHindawi Publishing CorporationAdvances in Diﬀerence EquationsVolume 2006, Article ID 69106, Pages 1–10DOI 10.1155/ADE/2006/691062 Lyapunov functions for dynamical equationslinearizations of nonlinear systems, see, for example, [3, 7, 11]. We note that P¨ otzscheet al. [13] investigated various necessary and suﬃcient conditions for the uniform expo-nential stability of the zero solution of systems (1.1) in terms of spectral properties ratherthan Lyapunov functions.In the next section, we show the invertibility of the cylinder transformation on theHilger complex plane. This is required in our main result, which is presented and provedin the ﬁnal section.Background concepts and results on time scales are taken from Bohner and Peterson[4] (see also [2, 5, 8, 9]) and, for brevity, will not be stated explicitly here in general.2. Invertibilityof the cylinder transformationA classical result from function theory (see, e.g., [6, Pages 38–45]) says that the mainbranch of the complex logarithm Log(z): = log(|z|)+ıφ,w h e r ez =| z|eıφ with −π<φ≤π,i sw e l ld e ﬁ n e df o rz ∈ C∗ := C\{0}, but is not continuous for z ∈ C∗ with Im(z) =π.Moreover, Log(z) is a holomorphic function for z ∈ C− := C\{z ∈ R:z ≤0}.Deﬁne Ch and Zh for h>0b yCh := z ∈ C:z  =−1h , Zh := z ∈ C: −πh<Im(z) <πh . (2.1)Ch is called the Hilger complex plane. Hilger [10]a l s oi n t r o d u c e dt h ecylinder transfor-mation ξh : Ch → Zh, which is deﬁned byξh(z): =⎧⎪ ⎨⎪ ⎩1hLog(1+zh), h>0,z, h =0.(2.2)Proposition2.1(Hilger[10]). Let Tbeatimescalewithgraininessμandlet p∈rd(T,C)be regressive on T. Then the composition ξμ(p) of the cylinder transformation with μ and pis well deﬁned, rd-continuous on T, and thus Cauchy integrable.The cylinder transformation is useful for investigating, among other things, linear dy-namical equations. It is invertible when suitably restricted.Lemma 2.2. Let Ah :={z ∈ Ch :z<−1/h} for h>0 and deﬁne ξh : Ch \Ah → Zh byξh(z): =1hLog(1+zh). (2.3)Then ξh is a biholomorphism of Ch \ Ah onto Zh and its inverse ξ−1h : Zh → Ch \ Ah isgiven byξ−1h (z): =exp(zh)−1h. (2.4)Proof. Since ξh and ξ−1h are well deﬁned and continuous, it remains to prove that(1) ξh(Ch \Ah) ⊂ Zh and ξ−1h (ξh(z)) = z for all z ∈ Ch \Ah.P. E. Kloeden and A. Zmorzynska 3Let z ∈ Ch \Ah.T h e n1+zh∈ C− andξh(z) =1hLog(1+zh) =log|1+zh|h+i·Arg(1+zh)h(2.5)with −π<Arg(1+zh)<π, from which it follows that−πh<Arg(1+zh)h<πh, (2.6)hence ξh(Ch \ Ah) ⊂ Zh. Furthermore, a classical result from function theory says thatexp(Log(1+zh)) = 1+zh for z ∈ Ch \Ah and h>0 (see, e.g., [14, Pages 120 and 126]).Henceξ−1h ξh(z) =exp h·(1/h)Log(1+zh) −1h=1+zh−1h= z. (2.7)(2) ξ−1h (Zh) ⊂ Ch \Ah and ξh(ξ−1h (z)) =z for all z ∈ Zh.We assume that there exists a z ∈ Zh such that for h>0exp(zh) ≤−1h. (2.8)ApplyingEuler’sformulaweseethat(2.8)holdsforz ∈ Zh ifandonlyifIm(z) = 0,thatis,z ∈ R.Butthenacontradictionfollowsdirectlyfrom(2.8),sincetheexponentialfunctionisstrictlypositiveon R.H e nc eξ−1h (Zh) ⊂ Ch \Ah.W eha v e−π/h<Im(z)<π/hwheneverz ∈ Zh.I tt h e nf o l l o w sf r o m[ 14, Pages 120 and 126] that Log(exp(zh)) = zh and hence1hLog 1+h·exp(zh)−1h =1hLog exp(zh) =z (2.9)for z ∈ Zh and h>0.We have shown that ξh : Ch \Ah → Z h is bijective with inverse function ξ−1h : Zh \Ah.From function theory, it is known that exp is a holomorphism on C and that Log is aholomorphism on C− (see, e.g., [14, Page 64]). Hence ξh is a holomorphism on Ch \Ahand ξ−h is a holomorphism on Zh for h>0. Remark 2.3. Write C0 := C for h = 0. Then we deﬁne ξ0 := id on C and ξ−10 := id on all ofC, where id is the identity function on C.For ﬁxed c ∈ R,thecompositionoftheinversecylindertransformationwiththegrain-iness μ o fat i m es c a l eT will be needed in considering Lyapunov functions. Let c ∈ R beﬁxed and ξc : T → R be deﬁned byξc(t): =⎧⎪ ⎪ ⎨⎪ ⎪ ⎩exp μ(t)·c −1μ(t), t<σ(t),c, t = σ(t).(2.10)4 Lyapunov functions for dynamical equationsThen(i) the function t  → ξc(t)i sr d - c o n t i n u o u si na l lt ∈ T for arbitrary c ∈ R as thecomposition of the graininess function t  → μ(t), which is always rd-continuous(see [4, Example 1.59 and Theorem 1.60, Page 22]) and the continuity of f : R →R deﬁned byf (x): =⎧⎪ ⎨⎪ ⎩exp(x)−1x, x  = 0,1, x = 0;(2.11)(ii) since f (x) > 0f o ra l lx ∈ R,f o rc>0, it follows that ξc(t) > 0f o ra l lt ∈ T and forc<0, it follows that ξc(t)<0f o ra l lt ∈ T;(iii) moreover, ξc(t) = c for all right dense t ∈ T;(iv) for h>0, let T:= hZ,t h e nμ(t) = h andξc(t) =exp(h·c)h(2.12)for all t ∈ T, that is, t  →ξc(t) is a constant function.3. Existenceofa Lyapunov functionOurmainresult,thefollowingtheorem,isanecessaryconditionfortheexistenceofaLya-punov function characterizing the uniform exponential asymptotic stability of the zerosolution of a nonautonomous linear dynamical equation. It is adapted from Yoshizawa[15, Theorem 19.1] for linear ordinary diﬀerential equations.Theorem 3.1. Let T be an unbounded time scale with bounded graininess and suppose thatfor each t0 ∈ T, x0 ∈ Rn, there exists a classical solution y : T → Rn, y(t) = y(t,t0,x0) of theinitial value problem (IVP)yΔ =A(t)y, A ∈ rd(T,Rn×n), t ∈ T,y(t0) =x0, x0 ∈ Rn, t0 ∈ T,(3.1)for all t ∈ T, t ≥ t0. Furthermore, suppose that there exist constants K>0 and c>0,w h i c hcan be chosen independently of t0 such that   y t,t0,x0     ≤Ke−c(t−t0)   x0    (3.2)holds for any t0 ∈ T, x0 ∈ Rn,a n da l lt ≥ t0, that is, the zero solution is uniformly exponen-tially stable.Then there exists a function V : T×Rn → R such that(i)  x ≤V(t,x) ≤ K x  for all x ∈ Rn, t ∈ T;(ii) |V(t,x)−V(t,  x)|≤K x−   x  for any ﬁxed t ∈ T and all x,  x ∈ Rn;P. E. Kloeden and A. Zmorzynska 5(iii) forarbitraryﬁxed(t0,x0) ∈ T×Rn,deﬁnefunctionV∗ : T → R,V∗(t): =V(t,y(t)),where y(t) = y(t,t0,x0), then the upper right Dini derivative of V∗ deﬁned byVΔ∗(t): =⎧⎪ ⎪ ⎪ ⎪ ⎨⎪ ⎪ ⎪ ⎪ ⎩limh 0,h+t∈TV∗(t+h)−V∗(t)h, if t = σ(t),V∗ σ(t) −V∗(t)μ(t), if t<σ(t),(3.3)exists and the estimateVΔ∗(t) ≤ ξ−c(t)V∗(t)<0 (3.4)holds for all t ≥t0;(iv) V is continuous from the right in (t,x) ∈ T×Rn,t h a ti s ,lim(  t,  x)→(t,x),  t≥t   V   t,  x −V(t,x)    = 0. (3.5)Proof. Fix t ∈ T.T h e nw ed e ﬁ n eAt := τ ∈[0,∞):t+τ ∈ T , (3.6)since 0 ∈ At, we note that At is nonempty. Further, we deﬁne V : T×Rn → R byV(t,x): = supτ∈At   y(t+τ,t,x)   ecτ, (3.7)where y : T → Rn istheuniquesolutionof(3.1)withinitialval uey(t) = x.Equation(3.7)is well deﬁned since y exists for all initial values (t,x) ∈ T×Rn and all t +τ, τ ∈ At byassumption and, as we will see in detail, it also follows from the assumptions that V isbounded.It will be shown that V satisﬁes the properties (i)–(iv) asserted in the theorem.It is obvious from the deﬁnition of V that  x ≤V(t,x), while from (3.2)i tf o l l o w sfor all τ ∈At that   y(t+τ,t,x)    ≤Ke−c(t+τ−t) x =Ke−cτ x , (3.8)soV(t,x) = supτ∈At   y(t+τ,t,x)   ecτ ≤ supτ∈AtKe−cτ x ecτ = K x . (3.9)Thus V satisﬁes property (i).6 Lyapunov functions for dynamical equationsLet t ∈ T be ﬁxed and let x,  x ∈ Rn be arbitrary. By the triangle inequality and thesuperposition of solutions of a linear system, we have   V(t,x)−V t,  x     =         supτ∈At   y(t+τ,t,x)   ecτ − supτ∈At   y t+τ,t,  x    ecτ         =         supτ∈Atecτ    y(t+τ,t,x)   −   y t+τ,t,  x              ≤         supτ∈Atecτ    y(t+τ,t,x)− y t+τ,t,  x              ≤ supτ∈Atecτ   y t+τ,t,x−   x     =V t,x−   x ≤K   x−   x   ,(3.10)which shows that V is globally Lipschitz—continuous in x for ﬁxed t ∈ T, that is, satisﬁesproperty (ii).We will prove property (iii) next. Let (t0,x0) ∈ T×Rn be arbitrary. We will distinguishtwo cases, σ(t) = t and σ(t) >tin the proof.Suppose that σ(t) = t and let h ∈ At.T h e nVΔ∗(t) = limh 0,h∈AtV∗(t+h)−V∗(t)h, (3.11)where,bytheuniquenessofsolutionsof(3.1)(see[4,Theorem8.20,Page324]),itfollowsthatV∗(t+h) = supτ∈At+h   y t+h+τ,t+h,y t+h,t0,x0     ecτ= supτ∈At+h   y t+h+τ,t,y t,t0,x0     ecτ= supτ∈{τ∈[h,∞):t+τ∈T}   y t+τ,t,y t,t0,x0     ecτe−ch≤ supτ∈{τ∈[0,∞):t+τ∈T}   y t+τ,t,y t,t0,x0     ecτe−ch= V∗(t)e−ch,(3.12)hence we haveVΔ∗(t) ≤limh 0V∗(t)·e−ch −1h=V∗(t)·(−c) =V∗(t)·ξ−c(t), (3.13)since μ(t) =0 by assumption.Now suppose that σ(t) >t, in which caseVΔ∗(t) =V∗ σ(t) −V∗(t)μ(t). (3.14)P. E. Kloeden and A. Zmorzynska 7ButV∗ σ(t) = supτ∈Aσ(t)   y σ(t)+τ,σ(t),y σ(t),t0,x0     ecτ= supτ∈Aμ(t)+t   y μ(t)+t+τ,μ(t)+t,y μ(t)+t,t0,x0     ecτ= supτ∈{τ∈[μ(t),∞):t+τ∈T}   y t+τ,t,y t,t0,x0     ecτe−μ(t)c≤V∗(t)e−cμ(t),(3.15)soVΔ∗(t) ≤ V∗(t)·e−cμ(t) −1μ(t)= V∗(t)·ξ−c(t), (3.16)thus property (iii) follows from (3.13)a n d( 3.16).In order to show that V is continuous in the sense of (iv), let t ∈ T, x ∈ Rn be ﬁxedand choose ε>0 arbitrary. Then δ1 >0a n dδ2 >0 must be found such that   V   t,  x −V(t,x)   <ε (3.17)holds for all  t = t+ν, ν ∈At,0≤ν<δ 1 (3.18)and all   x ∈Bδ2(x), where Bδ2(x) is the open ball centered on x of radius δ2.If t ∈ T is right scattered, we can always choose a suitable δ1 > 0s u c ht h a t  t = t isthe only point satisfying condition (3.18). Then the assertion follows since V is globallyLipschitz continuous in x for ﬁxed t ∈ T.Let t ∈ T be right dense and   t = t +ν for ν ∈ At.T h e n( 3.17)w i l lf o l l o wf r o mt h einequalities   V   t,  x −V(t,x)    =   V t+ν,  x −V(t,x)    ≤   V t+ν,  x −V(t+ν,x)   +   V(t+ν,x)−V t+ν,y(t+ν,t,x)    +   V t+ν,y(t+ν,t,x) −V(t,x)   .(3.19)Now we can use the Lipschitz continuity of V in x to estimate   V t+ν,  x −V(t+ν,x)    ≤K   x−   x   . (3.20)Hence   V t+ν,x  −V(t+ν,x)    <ε3(3.21)for all x  ∈Bδ2(x)w h e nδ2 <ε/3K.8 Lyapunov functions for dynamical equationsIn addition, the Lipschitz continuity of V in x also implies that   V(t+ν,x)−V t+ν,y(t+ν,t,x)     ≤K   x− y(t+ν,t,x)   , (3.22)where, by the right continuity of y,w eh a v elimν→0,ν∈Aty(t+ν,t,x) = x, (3.23)hence there exists a δ 1 >0, such that   x− y(t+ν,t,x)    <ε3·1K(3.24)for all ν ∈At with 0 ≤ν<δ 1, which combines with (3.22)t og i v e   V(t+ν,x)−V t+ν,y(t+ν,t,x)     ≤ε3(3.25)for all ν ∈At with 0 ≤ν<δ 1.In order to estimate the last term in (3.19), we use the uniqueness of solutions of ourinitial value problem (3.1) (see [4, Theorem 8.20, Page 324]) to obtainy t+ν+τ,t+ν,y(t+ν,t,x) = y(t+ν+τ,t,x). (3.26)DeﬁneΔV :=   V t+ν,y(t+ν,t,x) −V(t,x)   , (3.27)that is,ΔV =          supτ∈At+ν   y(t+ν+τ,t,x)   ecτ − supτ∈At   y(t+τ,t,x)   ecτ         . (3.28)Rearrangement givessupτ∈At+ν   y(t+ν+τ,t,x)   ecτ = supτ∈{τ∈[ν,∞):t+τ∈T}   y(t+τ,t,x)   ecτe−cν, (3.29)so (3.28)c a nb er e w r i t t e na sΔV =          supτ∈{τ∈[ν,∞):τ+y∈T}   y(t+τ,t,x)   ecτe−cν − supτ∈{τ∈[0,∞):t+τ∈T}   y(t+τ,t,x)   ecτ         .(3.30)Deﬁne α(ν): = supτ∈{τ∈[ν,∞):t+τ∈T} y(t+τ,t,x) ecτ for ν ∈At.T h e nΔV ≤   α(ν)e−cν −α(0)   . (3.31)P. E. Kloeden and A. Zmorzynska 9Now α(ν) ≤ α(0) for all ν ∈ At,w h e r eν ≥ 0 by deﬁnition. Moreover, α(ν)i sn o n i n -creasing function in ν withlimν→0,ν∈Atα(ν) =α(0). (3.32)Hence, there exists a δ  1 >0s u c ht h a tΔV<ε / 3f o ra l lν ∈At with 0 ≤ν ≤δ  1 .Now choose δ1 := min{δ 1,δ  1 }.F o r  t = t +ν with ν ∈ At,w h e r e0≤ ν <δ 1 and   x ∈Bδ2(x), combining all of the above estimates of the terms in (3.19), then gives   V   t,  x −V(t,x)   <ε3+ε3+ε3= ε, (3.33)which proves property (iv) and completes the proof of Theorem 3.1. AcknowledgmentsPeterKloedenhasbeenpartlysupportedbytheMinisteriodeEducaci´ onyCienciaProjectMTM2005-01412 as well by the Programa de Movilidad del Profesorado UniversitarioEspa˜ nol y Extranjero, Grant SAB2004-0146 of the above ministerio.References[1] R. P. Agarwal, Diﬀerence Equations and Inequalities, Monographs and Textbooks in Pure andApplied Mathematics, vol. 155, Marcel Dekker, New York, 1992.[ 2 ]R .P .A g a r w a l ,M .B o h n e r ,D .O ’ R e g a n ,a n dA .P e t e r s o n ,Dynamic equations on time scales: asurvey, Journal of Computational and Applied Mathematics 141 (2002), no. 1-2, 1–26.[ 3 ]B .A u l b a c ha n dS .H i l g e r ,Linear dynamic processes with inhomogeneous time scale, NonlinearDynamics and Quantum Dynamical Systems (Gaussig, 1990), Math. Res., vol. 59, Akademie,Berlin, 1990, pp. 9–20.[4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales,B i r k h ¨ auser Boston, Mas-sachusetts, 2001.[5] M. Bohner and A. Peterson (eds.), Advances in Dynamic Equations on Time Scales,B i r k h ¨ auserBoston, Massachusetts, 2003.[ 6 ]J .B .C o n w a y ,Functions of One Complex Variable I, 2nd ed., Graduate Texts in Mathematics,vol. 11, Springer, New York, 1978.[7] A. D¨ oﬃnger, Theorie dynamischer Gleichungen—ein einheitlicher Zugang zur kontinuierlichenund diskreten Dynamik, Diplomarbeit, Universit¨ at Augsburg, Augsburg, 1995.[8] S. Hilger, Ein Maßkettenkalk¨ ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Dissertation,Universit¨ at W¨ urzburg, W¨ urzburg, 1988.[9] , Analysis on measure chains—a uniﬁed approach to continuous and discrete calculus,R e -sults in Mathematics 18 (1990), no. 1-2, 18–56.[10] , Special functions, Laplace and Fourier transform on measure chains, Dynamic Systemsand Applications 8 (1999), no. 3-4, 471–488.[11] S. Keller, Asymptotisches Verhalten invarianter Faserb¨ undel bei Diskretisierung und Mittelwert-bildung im Rahmen der Analysis auf Zeitskalen, Dissertation, Universit¨ at Augsburg, Augsburg,1999.[12] P. E. Kloeden and S. Hilger, The eﬀect of time granularity on the asymptotic stability of dynamicalsystems, Automation and Remote Control 55 (1994), no. 9, part 1, 1293–1298 (1995).10 Lyapunov functions for dynamical equations[13] C. P¨ otzsche, S. Siegmund, and F. Wirth, A spectral characterisation of exponential stability for lin-ear time-invariant systems on time scales, Discrete and Continuous Dynamical Systems 9 (2002),255–265.[14] R. Remmert, Funktionentheorie, Springer, Berlin, 1995.[15] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Publications of the MathematicalSociety of Japan, no. 9, The Mathematical Society of Japan, Tokyo, 1966.[16] A. Zmorzynska, Lyapunovfunktionen auf Zeitskalen, Diplomarbeit, Johann Wolfgang GoetheUniversit¨ at, Frankfurt am Main, 2004.Peter E. Kloeden: Institut f¨ ur Mathematik, Johann Wolfgang Goethe Universit¨ at,60054 Frankfurt am Main, GermanyE-mail address: kloeden@math.uni-frankfurt.deAlexandra Zmorzynska: Institut f¨ ur Mathematik, Technische Universit¨ at Berlin,10623 Berlin, GermanyE-mail address: zmorzynska@math.tu-berlin.deContemporary Mathematics and Its Applications “Contemporary Mathematics and Its Applications” is a book series of monographs, textbooks, and edited volumes in all areas of pure and applied mathematics. Authors and/or editors should send their proposals to the Series Editors directly. For general information about the series, please contact cmia.ed@hindawi.com.For more information and online orders please visit http://www.hindawi.com/books/cmia/volume-1For any inquires on how to order this title please contact books.orders@hindawi.comCMIA Book Series, Volume 1, ISBN: 977-5945-19-4 DISCRETE OSCILLATION THEORYRavi P. Agarwal, Martin Bohner, Said R. Grace, and Donal O’ReganThis  book  is  devoted  to  a  rapidly  developing branch  of  the  qualitative  theory  of  difference equations with or without delays. It presents the theory of oscillation of difference equations, exhibiting classical as well as very recent results in that area. While there are several books on difference equations and also on oscillation theory for ordinary differential equations, there is until now no book devoted solely to oscillation theory for difference equations. This book is ﬁlling the gap, and it can easily be used as an encyclopedia and reference tool for discrete oscillation theory.In  nine  chapters,  the  book  covers  a  wide  range  of subjects,  including  oscillation  theory  for  second-order linear difference equations, systems of difference equations, half-linear difference equations, nonlinear difference equations, neutral difference equations, delay difference  equations,  and  differential  equations  with piecewise constant arguments. This book summarizes almost  300  recent  research  papers  and  hence  covers all aspects of discrete oscillation theory that have been discussed  in  recent  journal  articles.  The  presented theory is illustrated with 121 examples throughout the book. Each chapter concludes with a section that is devoted to notes and bibliographical and historical remarks.The book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. Besides serving as a reference tool for researchers in difference equations, this book can also be easily used as a textbook for undergraduate or graduate classes. It is written at a level easy to understand for college students who have had courses in calculus.

A g a r w a l ,M .B o h n e r ,D .O ’ R e g a n ,a n dA .P e t e r s o n ,Dynamic equations on time scales: a survey,

A spectral characterisation of exponential stability for linear time-invariant systems on time scales,

Analysis on measure chains—a uniﬁed approach to continuous and discrete calculus,R e -sults in

Asymptotisches Verhalten invarianter Faserb¨ undel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Dissertation, Universit¨ at Augsburg,

C o n w a y

Diﬀerence Equations and Inequalities, Monographs and Textbooks

Dynamic Equations on Time Scales,B i r k h ¨ auser

Ein Maßkettenkalk¨ ul mit Anwendung auf Zentrumsmannigfaltigkeiten,

functions, Laplace and Fourier transform on measure chains,

Lyapunovfunktionen auf Zeitskalen, Diplomarbeit, Johann Wolfgang Goethe Universit¨ at, Frankfurt am Main,

oﬃnger, Theorie dynamischer Gleichungen—ein einheitlicher Zugang zur kontinuierlichen und diskreten Dynamik, Diplomarbeit, Universit¨ at Augsburg,

Stability Theory by Liapunov’s Second Method,

The eﬀect of time granularity on the asymptotic stability of dynamical systems,

u l b a c ha n dS .H i l g e r ,Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig,

Lyapunov functions for linear nonautonomous dynamical equations on time scales

Peter E. Kloeden

Alexandra Zmorzynska

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Lyapunov functions for linear nonautonomous dynamical equations on time scales

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