Tính ổn định mũ và phổ nhị phân mũ của phương trình vi phân đại số

The spectral theory for ODEs was further developed throughout the 20th century, and concepts such as Bohl exponents, exponential dichotomy also \vell-known as Sacker-Sell spectra were i

TRƯỜNG ĐẠI HỌC KHOA HỌC T ự NHIÊN

h iệu quả vào cá c bài toán c ô n g n gh ệ và kỹ thuật trong cá c dự án c ô n g n g h iệp ở

cá c nước tiên tiến , v í dụ như các bài toán đ iều k h iển tố i ưu, bài toán m ô p h ỏng

m ạch đ iện tử, m ổ p h ỏn g hệ c ơ h ọc nh iều vật và m ộ t s ố bài toán tính toán k h oa

h ọ c khác.

T ại khoa T oán — Cơ — Tin h ọ c, trường Đ ạ i h ọ c K h o a h ọ c Tự n h iên , Đ H Q G

H N , từ c u ố i những năm 9 0 , m ột nh óm n g h iên cứu vể p h ư ơn g trình v i phân đại s ố

đã được hình thành (G S T SK H Phạm K ỳ A n h , G S T S N g u y ễ n Hữu Dư,

PG S.TS V ũ H o àn g L in h , TS Lê C ôn g L ợi) T rong 5 n ă m vừa qua ch ú n g tôi đã

trì m ột sem in ar về phư ơng trình vi phân và tính toán k h o a h ọ c N s o à i m ục tiêu

ch ín h là đạt được các kết quả k h oa h ọ c có chất lư ợ n g, ch ú n g tỏi c ũ n g hướng tới

v iệc bổi dưỡng, đ ào tạo các sinh v iên , h ọ c viên ca o h ọ c , và lớp cán bộ trẻ có năng lực trong lĩnh vực T oán h ọ c tính toán và T oán ứng d ụ ng thành những cán

bộ k h oa h ọ c c ó c h u y ên m ô n tốt, đảm nhận được c ô n g tác đ ào tạo và n g h iên cứu khoa h ọ c, đ ổ n g thời đ ó n g g ó p vào v iệ c n g h iên cứu lý th u yết p h ư ơ n g trình vi phân đại số.

- N ộ i d u n g : P h ư ơ n g tr ìn h vi p h â n đ ạ i số c ấ p 1 có d ạ n g t ổ ng q u á t:

trong đ ó m a trận Jacob i củ a F theo biến thứ nhất được g iả thiết là su y b iến D ạn g

tu yến tính củ a (1 ) c ó thể v iết như sau:

2 Báo c á o tại X e m in a liê n trường, V iện T oán h ọ c, 9 /2 0 0 8 và tại X e m in a cùa

P h òn g n g h iên cứu K h oa h ọc T ính toán và K ỹ thuật, Đ ại h ọ c C a liío m ia Santa Barbara, H o a K ỳ , 4 /2 0 0 9 N g ư ờ i báo cáo: V ũ H o à n g L in h , tên b á o

cáo: S p e c t r a l i n t e r v a l s f o r D A E s a n d t h e ir n u m e r i c a ì a p p r o x i m a t i o n

Đ à o tạ o đ ạ i h ọ c v à s a u đ ạ i h ọ c: 3 luận văn đ ạ i h ọ c , 2 luận văn c a o h ọ c đ ã b ả o

v ệ , 1 N C S ( n ă m t h ứ n h ấ t )

/ Tình hình kinh phí của đê tài (hoặc dự án).

K inh p h í 2 0 triệu đ ổ n g đã ch i v ào các m ụ c như sau:

1 V ật tư văn phòng: l.OOO.OOOđ

2 T h ô n g tin liên lạc: l.OOO.OOOđ

2 ABSTRACT

a Project’s title E x p o n e n t i a l stabili tv a nd e x p o n e n t i a l d i c h o t o m y s p e c t r u m for d i f f e r e n t i a l - a l g e b r a i c e q u a t i o n s

C ode: Q T -0 8 -0 2

b Project’s supervisor Dr V u H o a n g Linh

c Project’s members Prof.D r N g u y e n Huu D u, Tran Q u o c T uan, Le H uy

H oan g, N g u y e n Thi Y e n , V u Thi V an, Le The Sac, L e D ieu H u o n g , N g u y e n Thu Hau

d Objective and content of the project.

In the project w e c o n sid e r the d ifferen tial eq u ation o f g en era l form

w here the Jacob ian o f íu n ctio n F w r.t the first variable is su p p o sed to be singular T he linear variant o f sy ste m (1 ) is g iv e n as

The m ain o b je c tiv e s o f the research are as fo llo w s

1 E x p o n en tia l sta b ility , B ohl ex p o n en ts, e x p o n en tia l d ic h o to m y , S ack er-S ell spectral intervals and their properties.

2 N u m e rica l m eth o d s for c a lc u la tin s the spectral in tervals.

3 E x te n sio n to n o n lin ea r D A E s o f the form (1 ) vvhen th ey are su b jected to linearization alo n g a trajectory.

e Main results of the projects.

P u b l i c a t i o n s (in j o u r n a l s a n d c o n f e r e n c e p r o c ee d i n g s ) :

1 V H L inh, V M eh rm an n , L y a p u n ov, B ohl, and S a ck er-S ell spectral

intervals for d iffer e n tia l-a lg e b r a ic eq u ation s, 4 0 p a g e s (a cce p te d for

3 PHẦN CHÍNH CỦA BÁO CÁO:

kh oản g thời gian 25 năm trở lại đây M ột sô trường phái n g h iê n cứu tiêu biểu

đã được hình thành ở M ỹ (G ear, P etzold , C am p b ell, R h e in b o ld ), Đ ứ c (M aerz,

K u n k el, M eh n n a n n , L u b ich ), T hụy Sỹ (H airer), N g a (B o ja rin cev ,

C h isty a k o v ), w N h iể u bộ ch ư ơn g trình phần m ềm đã được x â y d ự n s và áp dụng h iệu quả vào c á c bài toán c ô n g nghệ và k ỹ thuật trong cá c dự án c ô n c

n g h iệp ở cá c nước tiên tiến , v í dụ như các bài toán đ iều k h iển tối ưu, bài toán

in ô p h ỏng m ạch đ iện tử, m ô p h ỏn g hệ cơ h ọc n h iều vật và m ộ t s ố bài toán tính toán k h oa h ọ c khác.

Phương trình vi phân đại s ố cấp 1 c ó dạng tổn g quát

trong đó m a trận J a co b i củ a F th eo b iến thứ nhất được g iả thiết là su y b iến

D ạ n g tu yến tính củ a (1 ) c ó thể v iết như sau

V í dụ, khi tu yến tính h ó a hệ (1 ) d ọc theo m ột lời g iả i riên g X nào đ ó, c h ú n s

ta nhận được hệ d ạn g (2 ) K hi E (t) = I, hệ (2 ) trở thành h ệ p hư ơng trình vi phân thường quen th u ộ c và đã được khảo sát, n g h iên cứu trong su ốt n h iều th ế

k ỷ qua.

N ăm 1 8 9 2 , trong lu ận án tiến s ĩ n ỗ i tiến g củ a m ìn h , nhà toán h ọc N g a

L yap u n ov đã đặt n ền m ó n g ch o lý thuyết ổn định củ a PT V P M ột trong những khái n iệm quan trọng m à ó n g đưa ra c ó tên g ọ i s ố m ũ đặc trung, sau này được g ọ i là s ố m ũ L y a p u n o v , nhằm đặc trưng ch o tốc đ ộ tãng trưởng của lời g iả i và c ó thể sử d ụ n g đ ể k h ảo sát tính ổn định L ý th u yết L yap u n ov đã được tiếp tục k h ảo sát và m ở rộng trong suốt th ế kỷ 2 0 với sự đ ó n g g ó p củ a

B oh l, Perron, M ilio n c h ik o v , O se le d e ts, Sacker, S ell, w T rong thời gian gần đây, D ie c i và V an V le c k (2 0 0 2 ) đã khảo sát bài toán s ố m ũ đặc trưng và phổ

từ g ó c đ ộ toán h ọc tinh toán và đưa ra m ột s ố thuật toán tính toán cá c k h o ả n s phổ c h o PT V P C ơ sở toán h ọ c cũ n g như các bài toán liên quan như đánh g iá sai số , k ỹ thuật cà i đặt h iệu quả đã được dẫn dắt ch i tiết.

M ục tiêu và n ộ i d u n g ch ín h củ a đ ề tài là n g h iên cứu tính ổn định củ a hệ (2 )

th ôn g qua v iệ c k h ảo sát cá c s ố m ũ đặc trưng và phổ củ a h ệ, m ở rộng các kết quả từ P T V P thường sa n g P T V P Đ S Đ ổ n g thời, ch ú n g tòi cũ n g đưa ra m ột sỏ

M ố i liên hệ g iữ a s ố m ũ L y a p u n o v củ a hệ V P Đ S và s ố m ũ cù a P T V P thư ờns cãn bản;

M ố i liên hệ giữ a s ố m ũ L y a p u n o v và tính ch ín h qui củ a hệ V P Đ S với s ố mũ

T rong phần tiếp th eo ch ú n g tôi đã g iả i q u yết các câu h ỏ i sau:

Đ ư a ra khái n iệm s ố m ũ B oh l đặc trưng ch o tốc đ ộ tăn g trưởng đểu của lời

g iả i và cá c tính chất c ơ bản củ a chúng;

C húng tôi cũ n g đã trình b ày khái n iệm nhị phân m ũ và định n g h ĩa phổ nhị phân m ũ K h ảo sát c á c tính chất củ a phổ nhị phân m ũ (h ay cò n g ọ i là phổ

S ack er-S ell);

M ố i liên hệ g iữ a k h o ả n g phổ S a ck er-S ell, s ố m ũ B o h l, s ố m ũ và k h o ả n g phổ

L yapunov;

M ố i liên hệ g iữ a k h o ả n g p h ổ S a ck er-S ell của hệ P T V P Đ S và h ệ liên hợp;

C u ố i cù n g ch ú n g tô i c ũ n g k h ảo sát tính ổn định củ a p h ổ S a ck er-S ell, cụ thể

ch ỉ ra rằng, dưới tác đ ộ n g củ a n h iễu chấp nhận đư ợc, p h ổ S a ck er-S ell lu ôn ổn định.

3 2 3 T í n h t oán x ấ p xỉ k h o ả n g p h ổ

C hú n g tôi đã đưa ra hai thuật toán, Q R liên tục và Q R rời rạc đ ể xấ p x ỉ các

k h o ả n g phổ C ác v í dụ m in h h ọa đã chứ ng tỏ h iệu lực củ a cá c thuật toán và

m inh họa tốt ch o cá c kết quả lý thuyết Thuật toán c ũ n g đã được x â y dự ne

ch o h ệ tu yến tính h ó a nhận được từ m ột hệ V P Đ S phi tu yến

In: Collected lectures on the preservation o f stability under discreĩization

(Fort C o llin s, c o , 2 0 0 1 ) , p a g e s 1 9 7 —2 1 8 S IA M , P h ila d elp h ia , P A , 2 0 0 2

7 A M L y a p u n o v T h e g e n e r a l p ro b lem o f the s ta b ility o f m o tio n

T r a n sla te d by A T F u lle r fro m E d o u a rd D a v a u x 's F r e n c h tr a n sla tio n ( 1 9 0 7

o f the 1 8 9 2 R u s s ia n o r ig in a l Internat J C o n tro ỉ, p a s e s 5 2 1 —7 9 0 , 1 9 9 2

8 R J S a ck er and G R S e ll A sp e ctra l th eo ry for lin e a r d iffe r e n tia l s y s t e m s

J Diff Equations, 2 7 : 3 2 0 —3 5 8 , 1 9 7 8

‘HỤ LỤC: CÁC BÀI BÁO VÀ BÁO C Á O HỘI T H Ả O

BÌA LUẬN V ÃN VÀ K H Ó A L U Ậ N

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Lyapunov Bohl and Sacker-Sell Spectral Inter\'als for

or divergence rates of nearby solutions The spectral theory for ODEs was further developed

throughout the 20th century, and concepts such as Bohl exponents, exponential dichotomy (also

\vell-known as Sacker-Sell) spectra were introduced, see [1, 19, 20, 70] ưnlike the development

of the analytic theorv, the development of numerical methods to compute Lvapunov exponents and also other spectral intervals has only recently been studied In a series of papers, see [23,

24, 26, 27, 29, 30, 31], Dieci and Van Vleck have developed algorithms for the computation of Lyapunov and Bohl exponents as \vell as Sacker-Sell spectral intervals These methods have also been analyzed concerning their sensitivity under small perturbations (stability), the relationship betvveen diíĩerent spcctra, the error analysis and eíĩìcient implementation techniques.

•T h is rosearch vvas su pp orted by Deutsche Forschungsgemeinschaft through M a t h e o n th e DFG Research

C enter “M a th em a tics for K ey Technologies" in Berlin \ ’u Hoang Linh s research vvas partiallv su p p orted by

\ ’N U ’s project Q T 08-02.

^Faculty o f M a th em atics, M echanics and Iníorm atics V ietnam N ational University 334 N guyen Trai Str

Thanh Xuan, Hanoi, VMetnam.

M nstitut fiir M ath em atik , M A 4-5, T echnische U n iversitat Berlin D-10G23 Berlin Fed Rep Germ any.

This paper is devoted to rhe generalization of some theoretical results as \vell as numerical methods from the spectral theory for ODEs ro diẩerential-algebraic equations (DAEs) In par- ticular, we are interested in the characterization of the dynamical behavior of solutions to initial value problems for linear system s of DAEs

on the half-line E = [0,oc), together with an initial condition

Here we assume th a t E A e C (I,M nxn), and / € C(H,Rn) are suíRciently sm ooth We use the

notation C(H,Rnxrl) to denote the space of continuous íunctions from I to R n xn

Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system

The fact th a t the dynamics of DAEs is constrained also requires a modiíication of most classical concepts of the qualitative theory th a t was developed for ODEs Diíĩerent stabilitv concepts for DAEs have been discussed already in [2, 42, 43, 53, 60, 62, 68, 69, 71, 72, 73, 74] Only very few papers, ho\vever, discuss the spectral theory for DAEs, see [17, 18] for results on Lyapunov expo- nents and Lyapunov regularity [57] for the concept of exponential dichotomv used in numerical solution to boundary value problems, and [16, 35] for robustness results of exponential stability and Bohl exponents All these papers use the tractability index approach as it was introduced in [37, 61] and consider linear svstem s of DAEs of tractability index 1, only Here we allow general regular DAEs of arbitrary index and we use reíormulations based on derivative arravs as well as the strangeness index concept [50] As in the ODE case there is also a close relation of the spectral theorv to the theory of adjoint equations which has recently been studied in the context of control problems in [4, 5, 6, 14, 51, 52]

In this paper, we system atically extend the classical spectral concepts (Lvapunov, Bohl, Saeker- Sell) th at were introduced for ODEs, to general linear DAEs \vith variable coeíĩicients of the form (1) We show th a t substantial diíĩerences in the theorỵ arise and th a t most statem ents in the classical ODE theory hold for DAEs only under íurther restrictions, here our results extend results

on asym ptotic stability given in [53] After deriving the concepts and analyzing the relationship between the diẩerent concepts of spectral intervals, we then derive two alternative numerical approaches to com pute the corresponding spectra

The outline of the papcr is as follows In the following section, we recall some concepts from the theory of differential-algebraic equations We discuss in detail the extension of spectral concepts from ODEs to DAEs in Section 3 The relation bet\veen the spectral characteristics of DAE systems and those of their underlying ODE systerns is investigated Furtherm ore, the stability

of the spectra vvith respect to perturbations arising in the system data is analyzed In Section 4

\ve propose numerical m ethods for com puting the Lyapunov and the Sacker-Sell (exponential dichotomy) spectral intervals and discuss implementation details as \vell as the associated error analysis In Section 5 \ve present Iiumerical examples to illustrate the theoretical results and the properties of the numerical m ethods \ \ e finish the paper \vith a sum m ary and a discussion of open problems

This paper is devoted to the generalization of some theoretical results as well as numerical nethods from the spectral theory for ODEs ro diẩerential-algebraic equations (DAEs) In par- icular we are interested in the characterization of the dynamical behavior of solutions to initial /alue problems for linear system s of DAEs

jn the half-line n = [0,oc), together with an initial condition

Here we assume th at E A € C (3 ,R nxrl), and / € C ( ĩ , R n) are sufficiently sm ooth We use the

notation C (n,R nxn) to denote the space of continuous íunctions from I to IRnx n

Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system

of DAEs

F ( t , X, ±) = 0 , t > 0, (3 )

along a particular solution [12] In this paper for the discussion of spectral intervals, we restrict

ourselves to regular DAEs, i e., we require th at (1) (or (3) locally) has a unique solution for

sufficiently smooth E , A , f (F) and appropriately chosen (consistent) initial conditions see [50

for a discussion of existence and uniqueness of solut-ion of more general nonregular DAEs

DAEs like (1) and (3) arise in constrained multibody dynamics [36], electrical circuit sim ulation [38, 39], chemical engineering [32, 33] and many other applications, in particular \vhen the dynamics

of a system is constrained or \vhen difFerent physical models are coupled together in automatically generated models [64] \Vhile DAEs provide a very convenient modeling concept m any numerical difficulties arise due to the fact th a t the dynamics is constrained to a maniíold, \vhich often is only given implicitly, see [9 41, 67] or the recent textbook [50] These difficulties are typically characterized by one of many index concepts th a t exist for DAEs, see [9, 37, 41, 50]

The fact th at the dynamics of DAEs is constrained also requires a m odiíication of most classical concepts of the qualitative theory th a t was developed for ODEs DiíTerent stability concepts for DAEs have been discussed already in [2, 42, 43, 53, 60, 62, 68, 69, 71, 72, 73, 74] Only very few papers, however, discuss the spectral theory for DAEs, see [17, 18] for results on Lyapunov expo- nents and Lyapunov regularity [57] for the concept of exponential dichotomv used in numerical solution to boundary value problems, and [16, 35] for robustness results of exponential stability and Bohl exponents All these papers use the tractability index approach as it was introduced in [37, 61] and consider linear system s of DAEs of tractability index 1, only Here vve allo\v general regular DAEs of arbitrary index and we use reformulations based on derivative arrays as well as the strangeness index concept [50] As in the ODE case there is also a close relation of the spectral theory to the theory of adjoint equations which has recently been studied in the context of control problenis in [4, 5, 6, 14, 51, 52]

In this paper, we system atically extend the classical spectral concepts (Lyapunov, Bohl, Sacker- Sell) th a t were introduced for ODEs, to general linear DAEs \vith variable coeíĩìcients of the form (1) We show th at substantial diíĩerences in the theory arise and th a t most statem ents in the classical ODE theory hold for DAEs only under further restrictions, here our results extend results

on asym ptotic stability given in [53] After deriving the concepts and analỵzing the relationship between the diíĩerent concepts of spectral intervals, \ve then derive two alternative numerical approaches to compute the corresponding spectra

The outline of the paper is as follows In the following seetion, we recall some concepts from the theory of diíĩerential-algebraic equations \Ye discuss in detail the extension of spectral concepts from ODEs to DAEs in Section 3 The relation between the spectral characteristics of DAE svstems and those of their underlying ODE systems is investigated Furthermore, the stabilitv

of the spectra vvitli respect to perturbations arising in the system data is analyzed In Section 4

\ve propose numerical methods for computing the Lyapunov and the Sacker-Sell (exponential dichotomy) spectral intervals and discuss implem entation details as \vell as the associated error analysis In Section 5 \ve present numerical examples to illustrate the theoretical results and the

properties of the numerical methods \Xe finish the paper \vith a summ ary and a discussion of

open problems

This paper is devoted to the generalization of some theoretical results as \vell as numerical nethods from the spectral theory for ODEs to differential-algebraic equations (DAEs) In par- icular vve are interested in the characterization of the dynamical behavior of solutions to initial /alue problems for linear system s of DAEs

along a particular solution [ 1 2 ] In tlìis paper for the discussion of spectral intervals, \ve restrict

Durselves to regular DAEs, i e., we require th at (1) (or (3) locally) has a unique solution for suíĩiciently srnooth E , A , f (F) and appropriately chosen (consistent) initial conditions, see [50]

for a discussion of existence and uniqueness of solution of more general nonregular DAEs

DAEs like (1) and (3) arise in constrained multibody dynamics [36], electrical circuit sim ulation

38, 39], chemical engineering [32, 33] and many other applications, in particular when the dynamics

of a system is constrained or \vhen diíĩerent physical models are coupled together in autom atically generated models [64] \Vhile DAEs provide a very convenient modeling concept, many numerical diíRcuỉties arise due to the fact th at the dynamics is constrained to a maniíold, \vhich often is only given implicitly, see [9, 41, 67] or the recent textbook [50] These difficưlties are typically characterized by one of many index concepts th a t exist for DAEs, see [9, 37, 41, 50]

The fact th at the dynamics of DAEs is constrained also requires a m odiíication of most classical concepts of the qualitative theory th a t was developed for ODEs Diíĩerent stabilitv concepts for DAEs have been discussed alreadv in [2 42, 43, 53, 60, 62, 68, 69, 71, 72, 73, 74] Only very few papers, however, discuss the spectral theory for DAEs, see [17, 18] for results on Lyapunov expo- nents and Lyapunov regularity [57] for the concept of exponential dichotomy used in numerical solution to boundary value problems, and [16, 35] for robustness results of exponential stability and Bohl exponents All these papers use the tractability index approach as it \vas introduced in

37, 61] and eonsider linear systems of DAEs of tractabilitv index 1, only Here \ve allow general regular DAEs of arbitrary index and we use reíbrmulations based on derivative arrays as well as the strangeness index concept [50] As in the ODE case there is also a close relation of the spectral theory to the theory of adjoint equations which has recently been studied in the context of control problems in [4, 5, 6, 14, 51, 52]

In this paper, we system atically extend the classical spectral concepts (Lyapunov, Bohl, Sacker- Sell) th a t were introduced for ODEs, to general linear DAEs \vith variable coeíĩìcients of the form (1) We show th a t substantial diẩerences in the theory arise and th a t most statem ents in the classical ODE theory hold for DAEs only under further restrictions, here our results extend results

OI1 asym ptotic stability given in [53] Aíter deriving the concepts and analyzing the relationship between the different concepts of spectral intervals, we then derive t\vo alternative numerical approaches to com pute the corresponding spectra

The outlinc of the paper is as follows In the following section, we recall some concepts from the theory of difFerential-algebraic equations \Ve discuss in detail the extension of spectral concepts from ODEs to DAEs in Section 3 The relation bet\veen the spectral characteristics of DAE systems and those of their underlying ODE systems is investigated Furtherm ore the stability

of the spcctra with respect to perturbations arising in the system d ata is analyzed In Section 4

\ve propose numerical m ethods for com puting the Lyapunov and the Sacker-Sell (exponential dichotomy) spectral intervals and discuss implernentation details as \vell as the associated error analysis In Section 5 \ve present numerical examples to illustrate the theoretical results and the properties of the numerical methods \Ve finish the paper \vith a summary and a discussion of open problems

This paper is devoted to the generalization of some theoretical results as \vell as numerical ĩiethods from the spectral theory for ODEs ro diíĩerential-algebraic equations (DAEs) In par- icular we are interested in the characterization of the dynamical behavior of solutions to initial /alue problems for linear systems of DAEs

along a particular solution [12] In this paper for the discussion of spectral intervals, \ve restrict

ourselves to regular DAEs, i e., we require th a t (1) (or (3) locally) has a unique solution for sufficiently smooth E , A , f (F) and appropriately chosen (consistent) initial conditions see [50]

for a discussion of existence and uniqueness of solution of more general nonregular DAEs

DAEs like (1) and (3) arise in constrained multibody dynamics [36], electrical circuit simulation [38, 39], chemical engineering [32, 33] and manv other applications, in particular when the dynamics

of a system is constrained or \vhen diíĩerent physical models are coupled together in autom atically generated niodels [64] \Vhile DAEs provide a very convenient modeling concept many numerical diíĩiculties arise due to the fact th at the dynamics is constrained to a maniíold \vhich oíten is only given implicitly, see [9, 41, 67] or the recent textbook [50] These difRculties are typically characterized by one of many index concepts th at exist for DAEs, see [9, 37, 41, 50]

The fact th at the dynamics of DAEs is constrained also requires a modiíication of most classical concepts of the qualitative theory th a t was developed for ODEs DifFerent stability concepts for DAEs have been discussed alreadv in [2, 42, 43, 53, 60, 62, 68, 69, 71 72, 73, 74] Only very few papers, ho\vever, discuss the spectral theory for DAEs, see [17, 18] for results on Lyapunov expo- nents and Lyapunov regularity [57] for the concept of exponential dichotomy used in numerical solution to boundary value problems, and (16, 35] for robustness results of exponential stability and Bohl exponents All these papers use the tractability index approach as it \vas introduced in [37, 61] and consider linear systems of DAEs of tractability index 1, only Here \ve allow general regular DAEs of arbitrary index and \ve use reformulations based on derivative arrays as well as the strangeness index concept [50] As in the ODE case there is also a close relation of the spectral theory to the theory of adjoint equations which has recently been studied in the context of control problems in [4, 5, 6, 14 51, 52]

In this paper, we system atically extend the classical spectral concepts (Lyapunov, Bohl, Saeker- Sell) th a t were introduced for ODEs, to general linear DAEs with variable coeíĩỉcients of the form (1) We sho\v th at substantial diíĩerences in the theory arise and th a t most statem ents in the classical ODE theory hold for DAEs only under further restrietions, here our results extend results

on asym ptotic stability given in [53] After deriving the concepts and analyzing the relationship between the diíĩerent concepts of spectral intervals, we then derive two alternative numerical approaches to com pute the corresponding spectra

The outline of the paper is as follo\vs In the following section \ve recall some concepts from the theory of diíĩerential-algebraic equations \Ye discuss in detail the extension of spectral concepts from ODEs to DAEs in Section 3 The relation between the spectral characteristics of DAE systems and tliosc of their underlying ODE systcms is investigated Furtherm ore the stability

of the spectra \vitli respect to perturbations arising in the system data is analyzed In Section 4

\ve propose numerical methods for computing the Lyapunov and the Sacker-Sell (exponential dichotomy) spectral intervals and discuss implementation details as well as the associated error analysis In Section 5 \ve present numerical examples to illustrate the theoretical results and the properties of the numerical methods ^^e íìnish the paper with a summary and a discussion of open problems

This paper is devoted to rhe generalization of some theoretical results as well as numerical nethods from the spectral theory for ODEs to diíĩerential-algebraic equations (DAEs) In par- icular we are interested in the characterization of the dynamical behavior of solutions to initial ,'alue problems for linear system s of DAEs

E ( t ) x = A ( t ) x + f { t ) , (1)

)n the half-line E = [0, oc), together with an initial condition

riere we assume that E A e C (I,R nxn), and / e C(]I,R n) are suASciently smooth \Ve use the

lotation C (I,R nxn) to denote the space of continuous íunctions from n to R n x n

Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system )f DAEs

F ( t , X, x) = 0, t > 0, ( 3 )

along a particular solution [12j In this paper for the discussion of spectral intervals, \ve restrict Durselves to r e g u l a r DAEs, i e., we require th at (1) (or (3) locally) has a unique solution for

suíĩiciently smooth E , A , f (F) and appropriately chosen (consistent) initial conditions see [50

for a discussion of existence and uniqueness of solution of more general nonregular DAEs

DAEs like (1) and (3) arise in constrained m ultibody dynamics [36], electrical circuit sim ulation [38, 39], chemical engineering [32, 33] and many other applications, in particular when the dynamics

of a system is constrained or \vhen diíĩerent physical models are coupled together in autom atically generated models [64] While DAEs provide a very convenient modeling concept, many numerical diíĩìculties arise due to the fact th at the dynamics is constrained to a maniíbld, which often is only given implicitly, see (9, 41, 67] or the recent textbook [50] These diíĩiculties are typically characterized by one of manv index concepts th a t exist for DAEs, see [9, 37, 41, 50]

The fact th at the dynamics of DAEs is constrained also requires a m odiíìcation of most classical concepts of the qualitative theory th a t was developed for ODEs Diíĩerent stabilitv concepts for DAEs have been discussed alreadv in [2, 42, 43, 53, 60, 62, 68, 69, 71 72, 73, 74] Only very few papers, however, discuss the spectral theory for DAEs, see [17, 18) for results on Lyapunov expo- nents and Lyapunov regularity, [57] for the concept of exponential dichotomy used in numerical solution to boundary value problems, and [16, 35] for robustness results of exponential stabilitv and Bohl exponents All these papers use the tractability index approach as it was introduced in [37, 61] and consider linear system s of DAEs of tractability index 1, only Here we allow general regular DAEs of arbitrary index and we use reíormulations based on derivative arrays as well as the strangeness index concept [50] As in the ODE case there is also a close relation of the spectral theory to the theory of adjoint equations which has recently been studied in the context of control problems in [4, 5, 6, 14, 51, 52]

In this paper, we system atically extend the classical spectral concepts (Lvapunov, Bohl, Sacker- Sell) th at were introduced for ODEs, to general linear DAEs \vith variable coefficients of the form (1) We show th a t substantial differences in the theory arise and th a t most statem ents in the classical ODE theory hold for DAEs only under further restrictions, here our results extend results

on asym ptotic stability given in [53] After deriving the concepts and analyzing the relationship

betAveen t h e d i í ĩ e r e n t c o n c e p t s o f s p e c t r a l in t e r v a l s , w e t h e n d e r i v e t w o a l t e r n a t i v e n u m e r i c a l

approaches to com pute the corresponding spectra

The outline of the papcr is SLS follo\vs In the following section, we recall some concepts from the

theory of diíĩerential-algebraic equations We discuss in detail the extension of spectral concepts from ODEs to DAEs in Section 3 The relation bet\veen the spectral characteristics of DAE systems and those of their underlying ODE systems is investigated Furtherm ore the stability

of the spectra \vith respect to perturbations arising ÌII the system d a ta is analyzed In Section 4

\ve propose numerical methods for computing the Lyapunov and the Sacker-Sell (exponential dichotomy) spectral intervals and discuss implementation details as \vell as the associated error

analysis In Section 5 \ve present numerical examples to illustrate the theoretical results and the

properties of the numerical methods \ \ e íỉnish the paper \vith a summarv and a discussion of open problems

2 A r e v ie w o f D A E t h e o r y

In this section we briefly recall some concepts from the theory of differential-algebraic equations see e.ơ [9, 37, 50, 66] \Ye follow [-50] in notation and style of presentation

D e fín itio n 1 Considtr system (1) with sufficiently smooth coefficient Ịunctions E .4 -4 / unction

solution of the initial value problem ( l ) - ( 2 ) i f X IS a soỉution o f (1) and satisỷies (2) A n in ỉtỉa l

condition (2) is called consistent if the corresponding initial value problem has at least one solution.

For the analysis as in [11 13 48 50], we use derivative arrays

Mi(t)zi = Xi(t)zt + gi(t)<

To guarantee existence and uniqueness of solutions, \ve make the following hypothesis, see 150;

with the given pair of matrix Ịunctions ( E , A ) has the following properties:

1 For all t £ I lue have rankA //i(í) = (ụ + l) n — a such that there exists a smooth matrix Ịunction Z 2 of size (ụ 4- l ) n X a and pointĩvise maximal rank satisỊying Z Ĩ M /J = 0.

2 For all t € I we have rank;42(£) = a, ivhere Á 2 = z ĩ N ự ự n 0 • • • 0jr such that there eiists a

smooth m a trix Ịunction T 2 o f size n X d d = n — a and pointivise maximal ranh satisỊying

À 2 T 2 = 0.

3 For aỉl t G I we have ĩd n kE (t)T 2 {t) = d such that there exists a smooth matrix Ịunction Z\

of size n X d and pointmse maximaỉ ranh satisỊying r a n k i ì i ^ = d with E\ = Z Ị E

Since Gram -Schm idt orthonorm alization is a continuous process, we may assume without loss of

generality th a t the columns of the m atrix functions Z\, Zi< and To in Hypothesis 2 are pointwise

orthonorm al

D e fin itio n 3 The smaỉlest possible ụ for which Hypothesis 2 hoỉds is called the strangeness index

of (1) Systems with vanishing strangeness index are called strangeness-free.

The strangeness index can be considered as a generalization of the differentiation Index as intro-

duced in [8], seo [50] for a dctailed analysis of the relationship bet\veen diíĩerenr index concepts

It has been sho\vn in [47], see also [50 th a t under som e constan t ra n k conditions every uniquely

solvable (regular) linear DAE of the form (1) with sufficiently smooth E .4 satisíìes Hypothesis 2 and th a t tliere exists a reduced system

th at is strangeness-free and ha5 the same solution as (1) where

\vith block entries

System (7) can be viewed as a different representation (remodeling) of sỵstem (1) \vhere all necessary differentiations of (1) th a t are needed to describe the solution are already represented in the model This representation avoids many of the numerical diíĩiculties that are associated \vith DAEs th at have a non-vanishing strangeness-index (differentiation index larger than 1), see '9 50]

The reduction to the form (7) can be carried out in a numerically stable way at any time instance

t, see [55, 50] and this idea can also be extended to over- and underdeterm ined systems as well as

locally to general nonlinear system s [54, 49, 50] For this reason in the follo\ving, we assume that

the DAE is given in the form (7) and for ease of notation we leave oíĩ the hats Furtherm ore a

m atrix íunction will be said nonsingular (orthogonal) if it is pointwise nonsingular (orthogonal)

In this section we generalize the classical spectral results for ODEs to DAEs \Ve refer to 26 27,

30, 44] or [58] for niore details on the theory for ODEs An essential step in the com putation of spectral intervals for linear DAEs of the form (1) is to íirst transform the system to a reduced strangeness-free form (7), which has the same solution set as (1), see (50 and then to consider the spectral results in this frame\vork This transform ation will not alter the spectral sets which will be deíìned in term s of the íundam ental solution matrices th at have not changed ưnder Hypothesis 2 this transíorm ation can ahvays be done and this reduced form can even be computed numerieally

at every tim e instance t For this reason, we may assume in the follo\ving that the system is given

in the reduced form (7), i.e we assume th a t our homogeneous DAE is already strangeness-free and has the form

and E\ € C { l R d x n) and Ao e C (I, R (n" d)xn) are of full row rank.

3.1 L y a p u n o v e x p o n e n ts a n d L y a p u n o v sp e c tr a l in ter v a ls

\Ve first discuss the concepts of Lyapunov exponents and Lyapunov spectral intervals

D e fìn itio n 4 A matrix Ịunction X £ c 1 ( Ị MnxA:), d < k < n, ỈS called íundam ental solution

matrix of (9) if each of ỉts columns is a solution to (9) and rank X ( t ) = d Ịor aỉl t > 0.

.4 / undamental solution m a tr ừ is said to be maximal if k = n and minimal if k = d re-

spectively A maximal Ịundamentaỉ matrix soỉution, denoted by X ( t , s ) , is caỉled principal if it satisfỉes the projected initial condition E( t o) ( X( t o, t o) - / ) = 0 for some to > 0.

not necessarily square and of full-rank Every fundamental soỉution matrix has exactlv d linearly

independent columns and a minimal íundam ental m atrix solution can be easilv made maximal bv

adding n — d zero columns.

3 S p e c t r a l t h e o r y for D A E s

\vhere

D eR nition 5 For a given fundamental solution matrix X of a strangeness-free DAE syỉtem of

the form (9), and Ị ot d < k < n we introduce

AỊ* = limsup — ln \ X(t )ex\ andA- = lim inf - ln ỊÀ’(í)e ,|| i = 1 '2.k.

t — oc t t — oc t

where et denotes the i-th unit vector The columns of a minỉmal /undamental solution m atns form

a normal basis if £ f=1A“ is minimal The XỴ,i = 1,2 , (ỉ, belonging to a normal basis are called

(upper) Lyapunov exponents and the intervals = 1.2, , á, are called Lyapunov spectral

D efinition 6 Suppose that u € C(3 Rn><n) and V £ C 1(E.Rnxn) are nonsingular matrii

func-tions such that V and v ~ l are bounded Then the transỊormed DAE system

Ẽ ( t ) x = Ã ( t ) ĩ , (10)

U A V — U E V and X = V ĩ is called globally kinematically equivalent to

w i t h Ẽ = U E V , À — U A V — u ty V a n a X = V X IS c a u e a giooaiiy Kinematicauy e q u iv a ie n t to

(9) and the trans/ormation is called a global kinematical equivalence transformation / / ư €

kinematical equivalence transíormation

are bounded then we call this a strono; slobal

It is clear that the Lyapunov exponents of a DAE system as \vell as the normalitỵ of a basis formed

by the columns of a íundamental solution matrix are preserved under global kinematic equivalence transĩormations

L em m a 7 Consider a strangeness-free DAE system of the form (9) vưith continuous coefficient.s

and a minimal Ịundamental soỉution matrix X Then there exist orthogonal matnx functions

uG C ( n , R nxrl) and V £ C 1( 3 , R nxr’ ) such that in the /undamental matrix equation E X = .-LY

where S\ := U Ị E V1 is nonsingular and A \ := U Ị A X1 — U Ị E \ \ Here U\.V\ are the matnx

Ịunctions consisting of the ỷỉrst d columns of u, V , respectively.

Since, by assumption, the íìrst d rows of E are of full row rank \ve have that the first d columns

of E V , given by E V1, have full column rank Thus, there exists a smooth QZ?-decomposition

\vhich proves the assertion D

The system (11) is an implicitly given ODE, since S\ is nonsingular It is called essentiallỵ

norm invariant, the Lyapunov exponents of the columns of the matrices X and R. and thereíore those of the two systems are the same

T h e o rem 8 Let z be a mimmal Ịundamental solution matrix for (9) such that the upper Lyapunov

exponents of its columns art ordered decreasingly Then there exists a nonsingular upper tnangular matnx c € Rdxd such that the columns of X(-) - Z(-)C jorm a normaỉ basis.

I" R "Ị

Proof By Lemma 7, there exists an orthogonal matrix íunction V such that V Tz = Q1 j \vith

Ri satisíying the implicit system

a normal basis as well Because the normalitv is preserved under global kinematical equivalence transformations, the proof is complete □

As in the case of ODEs it is useful to introduce the adjoint equation to (9) see also [14, 5, 51 52]

D efinition 9 The DAE system

dt

IS called th e adjoint system a s s o c i a te d w it h (9).

L em m a 10 Fundamental solution matnces X , Y of (9) and its adjoint equation (12) satisỊy the

Lagrange identity

Y T (t )E(t )X(t ) = Y t {0)E(0)X{0), t e I Let u , v e C '( I ,R rixn) deỷine a strong global kinematic equivalence for system (9) Then the adjoint of the trans/ormed DAE system (10) is strongly globally kinematically equivalent to the adjoint of (9).

ProoỊ Diíĩerentiating the product Y ( t ) TE( t ) X( t ) and using the definition of the adjoint equation

we obtain (leaving ofĩ the arguments) that

ị ( Y TE ) X + Y t E X = - Y t A X + Y t A X = 0

dt

and hence the Lagrange identity follo\vs By assumption, the matrices V T , Ư T deíìne a stronơ

global kinematic equivalence transíormation for the adjoint equation leading to the adjoint of

( 10 ) □

R e m a rk 11 In the ODE theory, the adjoint equations are easily derived from the Lagran°-e identity Nevertheless for DAEs, since a íundamental matrix solution is not necessarilv square or may be singular, the Lagrange identity does not imply the ajoint system (12) The concept of adjoint is deíìned only for some classes of DAEs That is, given a DAE, it may happen that its adjoint DAE does not exist or sometimes it is not clear at all \vhat is an adjoint system For more details on adjoint DAEs, see [5 14] and references therein

The relationship bet\veen the dynamics of a DAE system and its adjoint is more complicated than in the ODE case, except if some extra assumptions are added In order to see this and to better understand the dynamical behavior of DAEs, we apply an orthogonal change of basis to transform the system (9) into appropriate condensed íorms

T h e o rem 12 Consider the strangeness-free DAE system (9) If the pair of coefficient matrices

is sufficiently smooth then there exísts an orthogonal matnx Ịunction Q € C 1( I 3 nxnJ i'ich that

by the change of variables X = QTX the submatrix E\ is compressed i.e the transform.ed system has the Ịorrn

Furthermore, the system (13) IS still strangeness-free and thus E11 and 422 are nonsingular.

ProoỊ In order to show the existence of appropriate transíormations \ve use again the th e o r e m

on the existence of smooth Q R decompositions, see [21, Prop 2.3] and [50, Thm 3.9' If E is continuously diíĩerentiable then there exist a matrix function Q\ G c 1 (3 !Rnxd) \vith orthonormal

columns and a nonsingular Ê \\ e c 1 (3, Rdxd) S U C Ỉ 1that

Ei = Ẻ n Q Ĩ

-Since d rows of Q f pointwise form orthonormal basis in Kn and since the Gram-Schmidt process

is continuous, we can complete this basis by adding a smooth (and pointwise orthonormal ) matrix

ộ 2 e C 1( i , Rnx(n~d>) so thd.t

is pointwise orthogonal Then we have

Since we have started with a strangeness-free system, it follo\vs that the corresponding transíormed

matrix A partitioned as in (13) has a nonsingular block A 2 2 - □

R e m a rk 13 Alternatively we could have used a transíormation in Theorem 12 that compresses

the block Aỉ , thus obtaining a transformed system

■Ẻ n Cl

X = ^11 Á \0

\vith E\\ and Aoo nonsingular The proof for the condensed form (14) follo\vs analogouslv to that

of Theorem 12 by compressing the second block row of A, see also [15, Corollary 2.5] Most ofthe results that vve present below carry over directly to this system Due to the use of orthogonal transíormations, it is also clear th at the two transformed systems (13) and (14) are globallv kinematically equivalent It is important to note in addition that the form (13) generalizes the semi-explicit form which appears írequently in applications, see [9] So all the theoretical results derived for (13) apply directly to the class of semi-exlicit DAEs In this case, all conditions can

be checked directly for the original system Hovvever, for numerical computations, the form (14)

is more convenient To caleulate spectral intervals eíĩìciently, \ve prefer transíorming the DAE of general form (1) or (9) into the form (14) rather than (13)

System (13) is a strangeness-free DAE in semi-implicit form Since Q is orthogonal and since

the Euclidean norm is used, it íolloNvs that IIĩII = ||x|| Performing this transíormations allows to

separate the differential and the algebraic components of the solutions Partitioning X = [ i f ±Ị]T

appropriately, solving for the second component and substituting it into the íìrst block equation

one gets the associated underlying (implicit) ODE,

T h e o rem 14 Let Au(Á7,1.4->i) be the upper Lyapunov exponent of the matrix /unction 4v,'.4ịi.

Au( i i ) < Au( i ) < Au ( 1 + ||.4J2U 2i | | ) + \ u ( i i ) = Au ( f 1

and thus, Au(i) = Au(fi) Analogously we prove that A;( ii) < Ằl(x).

Since Au(i4j21Â2i) < 0, for any e > 0, there exists T > 0 such that

(18)

- In ^1 + i4ai IP < £ for all í > T.

\vhich implies that 1 + | ^ í o21' ^ 2 i I I < f°r all t > T As in the case of upper exponents, we

|ĩ(í)ll < v/2e£t |ĩ i ( í ) ||, t > T

Hence, we obtain that A;(x) < e + A; (xi) Since £ can be chosen arbitrarily, it follows that

the columns of the íundamental solution matrix X of (13) form a normal basis if and only if the corresponding columns of X \ form a normal basis of (15) □

R e m a rk 15 Assumption (17) ensures that the “algebraic” variable ± 2 cannot grow exponentially íaster than the “diẩerential" variable X i- Thus, the dynamics of the underlying ODE (15) essen- tially determines the dynamics of the DAE (13), see also [53] A suíRcient condition for (17) is

that A 22 is bounded or has a less than exponential growth rate This is for example the case

if there exist constants 7 > 0 and /c 6 N such that ^221^ 2 i(0 < l t k for all t G

R e m a rk 1G Alternatively, we could use (14) and the corresponding underlying ODE (16) It is easy to prove the equality for the Lyapunov exponents of (14) and those of (16) In this case such a boundedness or restriction in the grcnvth rate like (17) is not required Hovvever, a similar boundednoss condition on 4>-) in (14) will be needed, if one considers the analvsis of perturbed

or inhoniogcncous DAE systems

The next step of our analysis is the extension of the concept of Lyapunov-regularity to DAEs

D e íìn itio n 17 The DAE system (9) is said to be Lyapunov-regular if each of its Lyapunov spectral

intervals reduces to a point i.e., A| = X ị , i = 1,2,

To analyze the Lyapunov-regularity of the DAE system (9) \ve again study the transíormed semi-implicit system (13) and the underlying ODE system Since the Lyapunov exponents for a DAE system are preserved under global kinematic equivalence transíormations also t h e Lyapunov- reơularity is preserved, i e the DAE system (9) is Lyapunov-regular if and only if the semi-implicit DAE system (13) is Lyapunov-regular Thus, we immediately have the follo\ving equivalence result.

P ro p o s itio n 18 Consider the DAE system (13) and suppose that the boundedness condition (17)

holds Then, the DAE system (13) is Lyapunov-regular if and only if the underlying ODE system

(15) is Lyapunov-regular.

Unlike for ODEs, to obtain the equivalence between the Lyapunov-regularity of (9) and that

of its adjoint system we need some extra conditions

T h e o re m 19 Consider the DAE system (13) and suppose that the boundedness condition (17)

holds Assume Ịurther, that for the trans/ormed system (13) the conditíons

Au ( i i 2 - 4 J 2 ) < 0, Au ( £ n ) < 0 A“ ( £ f 11) < 0 (19)

hold If \[ are the louier Lyapunov exponents order of (9) and —fiị are the upper Lyapunov exponents of the adjoint system (12), both in increasing order, then

X \ = n Ỵ , Furthennore system (9) is regular if and only if (12) is regular and in this case we have the

Perron identity

ProoỊ VVithout loss of generalitv, we may consider the adjoint system (12) for the semi-implicit

The underlying ODE of the adjoint system is then given by

E\\ỹ\ — ~(^ĩi - ^ĩì^22 Ái2 + Eu)ỹi, (22)

which is exactly the adjoint of the underlying ODE system (15) It also follows immediately that

the Lagrange identity Ỳỵ Ẻ \ \ X \ = const holds for íundamental matrix solutions X \ Ỳ \ of (15) and its adjoint, respectively Note that if the columns of X \ form a normal basis for (15) then those of ỹ'i := Ẻ ^ X { T form a normal basis for the adjoint Hence (20) holds for the Lyapunov

spectra of the underlying ODE systems and due to the preservation of Lyapunov spectra under global kinematic equivalence, the prooí is complete □

C o ro lla ry 20 Suppose that (9) is Lyapunov-regular and the assumptions of Theorem 19 hold

Then, with A s deỷíned as in (15), the limit

-lim - / tr(Êfj1.4s(5))ds,

t - ~ x r J 0

exists and is equal to ỵy!=i ^1

-R e m a rk 21 The last two inequalities in (19) just imply Au( £ n ) = Au(£i"11j = 0 It is also important to note th at the relation bet\veen the spectral intervals of (9) and its adjoint (12) stated in Theorem 19 is invariant under strongly global kinematic equivalence transíormations

Hcnvever a crlobal kinematic equivalence transíormation that is not strong may destrov the Perron identity as the íollovving simple example demonstrates: Consider a scalar ODE and a kinematicallv equivalent implicit equation

E xam ple 22 Consider the DAE system

eati 1 = eat\ x i + x 2

for t € D, with constants a < 0 b < 0 and A € R The adjoint system is

It is easy to see that both (23) and (24) are Lvapunov-regular The only Lvapunov exponent for

(23) is A, while the onlv Lyapunov exponent for (24) is -A — a — b So, the Perron identity (20) betvveen the Lyapunov exponents does not hold if a + b ^ 0 In addition the Lvapunov exponent

of (24) is not necessarily equal to that of its underlying ODE Note th at in this example all the coeíĩicient matrices are bounded

E xam ple 23 Consiđer the systems (23) and (24) as in Example 22 but assume that a is positive,

i e the leading coeíRcient matrix is unbounded and assume that A is given by the time-varying íunction A(í) = sin(ln(í + 1)) + cos(ln(< + 1)) Then, the Lyapunov spectrum of (23) is [—1.1] and

that of the adjoint (24) is [—1 — a — b, 1 — a — 6] Neither the DAEs nor their underlving ODEs are Lyapunov-regular Hovvever, if a + b = 2, then (20) holds for the upper Lyapunov exponents

but the spectra of the DAE and its adjoint are not symmetric at all

As vve have deíìned it, Lyapunov-regularity is an asymptotic property of solutions to a DAE system Hence, the Lyapunov-regularity deíìnition presented here seems to be more natural than that based on the Perron identity (20) given in [18] Clearly, if the conditions (17) and (19) hold, then the diíĩerent definitions of Lyapunov-regularity are equivalent

The following two examples demonstrate the effect of the algebraic constraint on the dynamical behavior of solutions We stress that again in these examples the coeíRcient matrices are bounded

E x am p le 24 Consider the DAE system

Here the DAE is Lyapunov-regular but the underlying ODE is not

E x an ip le 25 The DAE system

Xi = - 3 x i + eísin(t)_íj->

0 = e-tX 2 ,

is Lvapunov-regular However its adjoint system

the algebraic equation in (9) íìom the left with a nonsingular m atrix function of size a X a may

change the boundedness of the coeíĩìcient m atrix A However, the validity of (17) is invariant un-

der this transíorm ation Using the relation bet\veen the coefficients of (9) and those of (13) which folỉows by the proof of Theorem 12, one may easily reíormulate the conditions (17) (19) in term

of the original data, i.e the coefficients of (9) Of course, the derivative of the m atrix íunction

ộ appearing in Theorem 12 \vill be involved in the reformulated conditions Furthermore since

E\ = [ Ẻ n 0 ] ộ , the assum ption on the gro\vth rate of E\ \ in (19) \vill autom atically imply the

same on the growth rate of the original coeíĩỉcient E.

In this section we have introduced the concepts of Lyapunov spectra and Lyapunov-regularity íor strangeness-free DAEs of the form (9) Since these concepts only depend on the solution of the DAE and not on the representation of the system of DAEs, whether it is in the form (9) or in the general form (1), we immediatelv ha ve all the results also for DAEs in the general form

3.2 S ta b ility o f L yap u n ov e x p o n e n ts

The analysis períormed in the last subsection changes substantiallv if the DAE (9) is subject to perturbations, i e if one studies perturbed DAEs

\vith perturbation íunctions A E(t), Ai4(í) If we allow general perturbations, then it is very diffì-

cult to analyze the behavior of the system due to the fact th at the strangeness-index may change

or the solvability of the system may be destroyed, see [10, 34, 63] The complete perturbation analysis for this case is still an o p e n p ro b le m even for constant coeíĩicient systems For this

reason we require th at the pair of perturbation íunctions ( A E , A A ) , A E , A.4 G C(E,Rnxn) are

suíRciently smooth such th at by applying a similar orthogonal transíorm ation as from (9) to (13) (but not the same), we obtain

If this is the case then we say that the perturbations are admissibỉe.

L e m m a 27 Consider a strangeness-free D A E of the form (9) and the set V of all pairs of admis-

and Ả 22 CLre still invertible and have bounded inverses I f ( A E , A.4) G V is sufficiently smalỊ then (26) remains strangeness-free.

Proof The assertion follo\vs, since for suíỉỉciently small admissible pairs of perturbations (A E A A)

the íunctions I + È ĩ i A Ê n and I + A õ ị & A 22 remain nonsingular □

If the unperturbed DAE system s corresponding to the transíorm ed system (26) has boundedly invertible blocks £"11 and A- 2 2 ' then \ve call these DAEs robustly strangtness-free In tlie folio\vinơ

we restrict ourselves to robustly strangeness-free DAE systems under admissible perturbarions

D efinition 28 The upper Lyapunov exponents Aj > > of (13) are said to be stable if for any € > 0, there exists ổ > 0 such that the conditions sup£ j|A.E(í)|| < ổ, supị ||Av4(í)Ị| < ỗ on the perturbations imply that the perturbed DAE system (26) is strangeness-free and

ỊA“ — ' “ 1 < € f o r aỉl i = 1 2 . d.

uihere the 7tu are the ordered upper Lyapunov exponents of the perturbed system (26).

The DAE system (13) and the perturbed system (26) are called asymptotically equivalent if they are strangeness-free and

lim ||A £(í)|| = lim |Ai4(í)|| = 0.

f—o c í —oo

It is clear that the stability of upper Lyapunov exponents and the asymptotic equivalence of DAE systems are invariant under strong global kinematic equivalence transíormations Since the Lyapunov exponents do not depend on the behavior of the coeíĩìcient matrices on a finite interval,

we have the following result (see also [1, Theorem 5.2.1] or [27, Theorem 3.1])

T h e o rem 29 Suppose that the DAE system (13) and the perturbed system (26) are asymptntically

equivalent Then the stability of the Lyapunov exponents of (13) implies A“ = 7 for all i —

where again the 7“ are the ordered upper Lyapunov exponents of the perturbed system

(26).

Proof Due to the asymptotic equivalence of the two systems, given an arbitrary í > 0 there

exists T > 0 (suíĩìciently large) such that

|AJ‘ - 7ỉ ‘ | < c , for all í = 1 2 d.

Since € can be chosen arbitrarily small, the prooí is complete □

As our next step we extend the concept of integral separation to DAEs

D eíìn itio n 30 A minimal Ịundamental solution matrix X for (9) (or (13)) is called integrally separated if for i = l , 2 , d — 1 there exist constants (3 > 0 and 7 > 0 such that

||.Y(s)e,I ||X (í)e1+i!| - re

for all t,s with t > s > 0 / / a DẢE system has an ìntegrally separated mimmal Ịundamtntal solution matrix, then we say it has the integral separation property.

Analogous to the result for ODEs, see e.g [26], we then have the following facts

P ro p o s itio n 31 Consider a strangeness-free DAE system of the form (13).

1 If (13) is integrally separated then the same holds for any globally kinematically equivalent system, i e also for (9).

2 1/(13) is integrally separated, then it has painuise distmct upper and pairwi.se dìstinct lower Lyapunov exponents.

3 Sưppose that À ^ À 2 l IS bounded Then, the DAE system (13) is integrally separated if and onỉy if and the underlying ODE (15) is integrally separated.

for all t, s and i = 1 ,2 d — 1, which immediatelv yields the assertion.

2 This part is immediate

3 The proof is similar to th at of Part 1 \Ve use again the estim ates (18) bet\veen the columns

of X and the corresponding columns of the íundamental solution for (15).

T h e o re m 33 Suppose t h a t the coefficient matrices in (13) are such that

I f the system (13) has d pairvũise distỉnct upper and painuise distinct lower Lyạpunov exponents and they are stable, then the system admits integral separation Conversely if there exists an integraỉly separated Ịundamentaỉ solution matrix to (13) then the system has d stable painuise distinct upper and stable pairuise distinct lower Lyapunov exponents.

Lyapunov exponents as its underlying ODE (15) The boundedness conditions (28) imply th at if

the perturbations A E and A.4 are small enough, then the underlying explicit ODE

± 1 = A i \ = £ ' 111( - 4 n — A 12 Ả 22 Ả 21 ) i \

is only aíĩected by a small p ertu rb atio n in the coeíĩicient m atrix A By invoking Part 3 of Propo-

sition 31 and the \vell-known result for ODEs [1], see also [26], the prooí for system (13) follo\vs

R e m a rk 34 Theorem 31 is stated for both the upper and the lower Lyapunov exponents But one should note t h a t although both upper and lower Lyapunov exponents are painvise distinct the Lyapunov spectral intervals mav intersect each other, see Example 40 belo\v.

Unlike the case of ODEs, the integral separation of a DAE system d o e s not aatomaticallv iniplv that of its adjoint system

T h e o rem 35 Consider a strangeness-free DAE system of the form (13) and suppose that A õ J ả 2 i

A 1 2 A 22 > Ê \\, and Ẻ n are bounded Then, the system has an integrally separated Ịundamental solution matrix iỊ and only if its adjoint (12) has an integrally separated fundamental solution matrix.

Proof The prooí follows immediately by using the structure of che íundamental solution matrices

of (13), the Lagrange identity the first statement of Theorem 19, and Part 3 of Proposition 31

□

Recall that if the columns of X i form a normal basis for (15) then the columns of >'1 :=

Ẻ^ÌTX [ T form a normal basis for the adjoint But, as the following example sho\vs, vvithout the

boundedness conditions in Theorem 35, the integral separation of the DAE (13) does not imply the integral separation of the underlying ODE (15)

E xam ple 36 Consider the DAE system

±1

Xi

õ0

Clearly, the underlying ODE

3.3 B ohl ex p o n en ts and Sacker-Sell spectrum

Another concept that can be used to describe the behavior of solutions to ordinary differential equations is that of Bohl exponents [7], see also [20] The extension of this concept to DAEs is straightforward

D efinition 37 Let X be a nontrivial solution of (9) The (upper) Bohl exponent K g(i) 0/ this

solution is the greatest lower bound of all those values p for Vũhich there exist constants Np > 0 such that

for any t > s > 0 I f such numbers p do not exist, then One sets Kg(x) = +OC.

Similarly, the ỉovoer Bohl exponent K g(i) is the least upper bound of all those values p' for uhich there exist constants N'p > 0 such that

It íollovvs directly from the deíìnition, that Lyapunov exponents and Bohl exponents are related via

K b ( x ) — ^ ( ( x ) < ^ u ( x ) 5 : k b ( x

)-Bohl exponents characterize the uniíòrm growth rate of solutions \vhile Lyapunov exponents

simply characterize the groivth rate of solutions departing from t = 0.

R em ark 38 The Bohl exponent of linear ODEs, \vhich was introcluced íìrst in [7] has been proven

to be a useíul tool in the qualitative theory and in the control of íìnite as well as iníìnite dimensional linear systems Numerous properties of Bohl exponent are discussed in [20] Though less well- known than the íamous characteristic number introduced by Lvapunov, the Bohl exponent is often preíerable, since it is stable with respect to small perturbations occurring in the coeíĩìcient matrix For this reason, the Bohl exponent was used for characterizing the robust stability of linear systems, see e g., [44 16] and the reíerences therein

It is straightforward to extend the íormulas for Bohl exponents of ODEs see e.g [20], to DAEs, i.e

KB{x) = limsup - - , KB (x) = lim inf — -

P ro p o sitio n 39 Consider the DAE system (9) and the transỊormed sỵstem (13) Then we have

the following properties of Bohl exponents.

1 Bohl exponents are invariant under global kinematical equivalence transỊormations.

2 Consider a minimal /undamental solution matnx X for (13) If A 22 Ả 21 is bounded then the Bohl intervals for the columns of X are exactly the Bohl intervals for the corresponding fundamental solution matrix X \ of the underlying ODE (15).

3 If the Bohl intervals of the columns of a minimal Ịundamental solution matrix X of (9) are

d disjoint closed intervals, then X has integrally separated columns.

ị If the columns of a Ịundamental solution matrix X of (9) are integrally separated, then the upper (or the lovơer) Bohl exponents of the columns of X columns are distinct, but the Bohl intervals may intersect each other.

)-Converselv, we have X = V H 'x hence similarly we obtain

i <' b (x ) ^ x

)-As a consequence, we have Kg( Vx) = Kg{x) The proof for the lo\ver Bohl exponents is

obtained analogously

2 The proof is similar to that of Theorem 14

3 Without loss of generality, we assume that the Bohl intervals of X are ordered decreasinglv

For the sake of simplicitv, it suổìces to consider the íìrst t\vo columns X \ and x- 2 - By the

deíinition of the Bohl exponents there exist positive constants Nị A/;, í — 1.2 such that

iV1e't‘( í- 's) ||xj(s)Ị| < ||x,(í)Ị| < A/,e'ĩ- (í-s) ||xi(s)|| for all t > s.

fo r ỉ = 1 ,2 , vvhere Kị a n d k Ỵ a re lo \v e r a n d u p p e r B o h l e x p o n e n ts fo r X ị , r e s p e c tiv e ly T h e n ,

it is easy to see that

11,(01 \\x2 (s)\\ > = ỊVị_

||xi(s)Ị| IIX2(í ) I M 2eK'i(t- s'1 M:

holds for all t > s Since the intervals [«11«“] and {k1 2,Ko} are disjoint, the positivity of (k[ — kỊị) is obvious, which veriíìes the integral separation of Xi and X 2 -

4 For the converse statement, we consider again the íìrst t\vo columns X \ and Bv the

integral separation property, there exist positive constants 3, 7 such that

Taking the limsup on both sides of the above inequality as s, t — s tend to infinity \ve obtain

ln ||xi(í)Ị| - ln ||xi(s)|| ln ||xi(í)|| - ln ||ii(s)|Ịlimsup - ; - > limsup - ;

E x am p le 40 Consider the system

±1 (í) = Ịsin(ln t) + cos(ln f)]xi ( í).

IỊxịÍí)-\vith t > 1 t h a t is an extension to DAEs of an example by Perron [65] It is easv to see that

this system is integrally separated Houever, the Bohl intervals ’- \ / 2 >/2] and ị - y / ĩ - 1 - 1’are clearly not disjoint Similarly, also the Lyapunov spectral intervals of this system [-1 1Ị and [-2,0] overlap

In order to extend the concept of exponential dichotomy to DAEs \ve first introduce shifted DAE systems.

D eíìnition 41 Consider a strangeness-free DAE of the Ịorm (9) For A € R the DAE system

is called a shifted DAE system.

By the transíòrmation of Theorem 12, the shiíted DAE transforms as

and clearly, the shifted DAE svstem inherits the strangeness-free propertv from the original DAE

In the previous subsection we have seen that minimal fundamental solution matrices are useíul

in the analysis of Lyapunov exponents Uníortunately, they do not have the semi-group property

as fundamental solutions in the ODE case have Hovvever, for strangeness-free systems it is easy

to introduce a (unique) maximal fundamental solution matrix X \vhich possesses a semi-group

property by demanding that it satisíìes the projected initial condition

see [46, 50] The strangeness index of this system is 1 It is easy to see th at the trivial solution A’ = 0 is the only maximal íundamental matrix solution but it does not satisív (33)

In the following, when it is necessary to emphasize the dependence of the íundamental solution

matrix on the initial time to, we write X(t,t ũ).

In the case of the transíormed strangeness-free system (13), a maximal íundamental solution matrix that satisíìes the corresponding projected initial condition (33) is easilv obtained as

where X\ is a íundamental solution m atrix for (15) satisfving Xi (t o to) = ỉ d■ For this maxi-

mal fundamental solution matrix A'(í, to), we introduce the generalized inverse matrix function

x ~ ( t , to) defined by

_ Ã ' r 1 ( t , _ t o ) 0

X ~ ( t J0) : =

for t > to- The matrix functions X and x ~ satisíy the follo\ving relations.

P ro p o s itio n 43 Let X as in ( 3 4 ) be the maximal fundamental solution m atrư of (13) and let

x~ be as in (35) Then for t\ > to- the following identities hold.

2 X~ { t i , t o ) X( t i , t o ) X~ ( t i , t o ) = A “ ( í i ,

ío)-3 X( t i , t o ) X~( t i , t o ) = [ À 2 2 (t 1 ) ^ 2 1 (t\) 0 ]• : - -

-4- X ( Í Ị ,t ữ ) X ( t \, ío) / 0

I -422' (^oM'21 (to) 0

Vice versa these Ịour propertie.s define the matnx X uniquely.

Furthermore X satisỷies the semigroup property i.e for to > 11 > t(j we have

X( t 2, t \ ) X( t \ , t o) = X ( t ‘ 2 to).

ProoỊ By using the íòrmulae (34) and (35), the identities 1.-4 as \vell as the semigroup propertv

are easily verified by elementary matrix calculations □

The identities 1.-2 in Proposition 43 mean that X~( t i , t o) is a reAexive generalized inverse of

X(t\,to), while the identities 3.-4 guarantee that this generalized inverse is unique, see 5 61.

In the following, for ease of notation, we use ío = 0 and the abbreviation X( t ) := X ị t 0) \Ye

then introduce the concept of exponential d ic h o to m y for DAEs as in [57]

D eíỉnition 44 The semi-implicit DAE system (13) is said to have an exponential dichotomy if

for a maximal Ịundamental solution matrix X( t ) there exists a projection matrix p £ <d and

constants a,Ị3 > 0 and K, L > 1 such that

Furthermore, we say that a general DAE system (9) has an exponential dichotomy if there exist.s

a global kinematical equivalence transỊormation that reduces (9) to the semi-irnplicit form and the reduced system has an exponential dichotomy.

For a strangeness-free DAE in the form (13), exponential dichotomy can again be eharacterized via the underlying ODE

T h eo rem 45 The DAE system (13) has an exponential dichotomy if and only if ÂTi.-i?! is

bounded and the corresponding underlying ODE (15) has an exponential dichotomy.

ProoỊ Suppose that (36) holds ưsing the structure of X and x~, \ve can revvrite (36) as

If Ằ 2 2 À 2 1 is bounded and (15) has an exponential dichotomy, then clearly the inequalities in

(37) hold □

The following íacts associated \vith exponential dichotomy of DAEs follow easilv

P ro p o s itio n 46 Consider a strangeness-free DAE of the form (13) that has an exponential di-

chotomy.

1 Euery globally kinematically equivalent system has an exponential dichotomy I e in par- ticular the exponential dichotomic property is invariant under global kinematical equivalence transỊormations.

2 If a ịundamental solution matrix X of (13) has an exponentia! dichotomy then so dots the Ịundamental solution matrix that fulfills the projected initial condition (33) at 10 = 0

Furthermore, the projection p can be chosen to be orthogonal.

ProoỊ The íìrst part simplv follows by the deíỉnition of exponential dichotomic DAE systems By

Theorem 45, to veriíy the second statement, it suíRces to consider the underlying ODE system (15) and analyze its exponential dichotomy Invoking [27, Lemma 6.1], the underlying ODE system (15)

also admits an exponential dichotomy for its (principal) matrix solution that satisíìes Xi(0) = Id

a n d th e D ro je c tio n p can be ch o se n to be o r th o g o n a l F in a lly , n o te t h a t th e íu n d a m e n ta l m a t r ix

solution for (13) constructed with this X \ , see (34), is exactly the unique íundamental solution

matrix that fulfills the projected initial condition □

After these preparations we can deíine Sacker-Sell spectra for DAEs

D eíìnition 47 The Sacker-Sell (or exponential dichotomy) spectrum of the DAE system (13) is

defined by

Es := {A € R, the shiíted DAE (32) does not have an exponential dichotomy} (38)

The complement ofĩ<s is called the resolvent set for the DAE system (13).

The Sacker-Sell spectrum of the DAE system (9) is defined as the Sacker-Sell spectrum of its transformed DAE system (13).

With these definitions we have the following properties of Sacker-Sell spectra for DAEs

Lem m a 48 Consider the DAE system (13) and suppose that A 22 A 21 is bounded Then.

1 the Sacker-Sell spectrum of the DAE system (13) is exactly the Sacker-Sell spectrum of the underlying ODE (15) It consists of at most d closed interuals.

2 the Sacker-Sell spectrum of the DAE system (9) does not depend on the choice of an orthog- onal change of basis that transỊorms it into the form (13).

ProoỊ.

1 Consider an arbitrary A ễ R By Theorem 45, the shiíted DAE system (32) with this A has

an exponential dichotomy if and only if the corresponding shifted underlying ODE system has an exponential dichotomy This implies that the resolvent set of (13) and that of (15) are exactly the same which proves the assertion Since the dimension (the size) of the underlying

ODE system (15) is d, it has been shown in [70] that the Sacker-Sell spectrum of (15) consists

of at most d closed intervals.

2 As a consequence of Proposition 46, Part 1, two globally kinematically equivalent implicit DAE svstems must possess the same Sacker-Sell spectrum Therefore, the Sacker- Sell spectrum of the DAE system (9) does not depend on the choice of a global kinematic equivalence transíormation that transíorms it into the form (13) □

semi-We also obtain the relationship of Sacker-Sell spectra and the integral separation property

T h e o ren i 49 Suppose that the Sacker-Sell spectrum of (ũ) is given bỵ d disjoint closed intervaỉs

Then the re exists a minimal fundarnental solution matnx of (9) unth integralỉy separated column.s.

ProoỊ Without loss of generality, it is suíỉìcient to consider the transíormed system (13) Bv

Theorem 45, the assumption implies that A 22 A- 2 \ is bounded Then, by Lemma 48 the Sacker-

Sell spectrum of the underlying ODE (15) consists of the same d disjoint intervals By invoking [27

Theorem 6.3], then (15) has an integrally separated íundamental solution matrix denoted by A'i- Hence, by construction (see Part 3 of Proposition 31) there exists a corresponding fundamental solution for the DAE system (13) whose columns are integrallv separated □

For the converse of this result we again need a boundedness condition

T h eo rem 50 Suppose that for the DAE system (13), A 22 A ‘ 2 \ is bounded and there exists a min- imal and integrally separated fundamental solution matĩix X Then the Sacker-Sell spectrum for (13) is given exactly by the d (not necessarily d.isjoint) Bohl intervals associated with the columns ÕfX.

ProoỊ Because of the relations bet\veen (13) and its underlving ODE (15), the veriíìcation of the

statement reduces to that for the underlying ODE system (15) Due to a theorem of Bylov (see [1 Corollary 5.3.2] or [26, Theorem 2.31]), there exists a kinematic equivalence transíormation that

tra n s fo rm s (1 5 ) in to d ia g o n a l fo r m T h e d ia g o n a liz e d s v s te m o b ta in e d in th is vvay is in te g r a lly

separated as well and the Sacker-Sell spectrum for the diagonal ODE system is exactly the set of Bohl intervals for all scalar equations corresponding to the diagonal elements, see also [58, Lemma 21] Since Bohl intervals are invariant under global kinematic equivalence transíbrmations the proof is complete □

R e m a rk 51 As is well-known already for ODEs [58], if we take an arbitrary minimal fundamental solution matrix of (13), then the set of Bohl intervals associated \vith its columns is only a subset

of the Sacker-Sell spectrum of (13) The integral separation assumption then ensures that the two sets coincide We underline th at the relation between the Bohl exponents and the Sacker-Sell spectrum of a scalar ODE [58, Lemma 21] has resulted in a more elegant interpretation of Sacker- Sell intervals compared vvith the approach based on two associated planar systems in [26 27:

C orollary 52 Consider the strangeness-free DAE system (9) Then the Lyapunov spectrum IS

contained in the Sacker-Sell spectrum i.e we have T-L c E5

ProoỊ Suppose that the columns of a íundamental solution matrix X of (9) form a normal

basis Then, by deíìnition, the Lyapunov spectrum is exactly the set of Lyapunov intervals for

the columns of X Since for an arbitrary solution X of (9), the Lyapunov interval is contained

in the Bohl interval and since the Bohl intervals are contained in the Sacker-Sell spectrum (see Remark 51), the prooí is complete □

The following well-known example example of Perron [65] shows that Lyapunov spectral inter- vals can be strict subsets of Sacker-Sell spectral intervals

E xam ple 53 Consider the ODE

where e ,e _1, and e la are continuous and bounded Let the Sacker-Sell spectrum associated with this system be given by the interval [a,/3Ị Then the Sacker-Sell spectrum of the adjoint equation

e ( t ) ý = + è( t ) ) y

given by [—/3, — q].

ProoỊ Without loss of generalitv, we assume that e(f) > 0 for a]] í € I Let the Sacker-Sell

spectrum of the adjoint equation be denoted by Ịõ,/3] Due to (58, Proposition 22], vve have

and, therefore, p = —Q The prooí that Q = —p follows analogouslv 0

Beíore vve can prove the svmmetry property for the Sacker-Sell spectra of the DAE and its adjoint, we need the follo\ving lemma

Lem m a 55 Consider an implicit ODE of the Ịorrn

with E(t) nonsingular Suppose that both E(t) and A(t) are continuous and that E ~ l A I S bounded Then there exist orthogonal matrix Ịunctions u £ C(I,Rn-n) and V € C 1(I,R n n) such that the

transỊormed matrix Ịunctions

í — [e,j] = UT E V and A = [a,jl = UTẢ V — UT E V

are both in upper triangular form.

ProoỊ We give a constructive proof We want to determine triangular matrix functions í and Ả

U ' , J , 1 > j

- W ] X , i <

j-V = j-VS(j-V). ( 4 1 ,

At t = 0, we can obtain V(0) bv using a Qi?-factorization of an initial value for the íundamental solution rnatrix X(0) associated \vith (39) say the identity matrix Since i r = S ~ l ư TA V =

V TE ~ l A V , the boundedness of E ~ 1Ả implies the boundedness of i r thus the boundedness of

5(1^), as well Thus, V can be determined as the (unique) solution to the initial value problem (41) With this, a continuous orthogonal matrix íunction u (as \vell as a triangular m atrix íunction

£) is then determined via

E V = u s ( 4 2 )

and this then uniquely determines A □

It should be noted that the computation of the triangularization in (42) can actually be im- plemented numerically by using smooth Q/?-factorizations see [21, Subsection 2.1\

Using Lemma 55, we can no\v prove the foỉlo\ving theorem

T h eo rem 56 Consider the DAE system (13) ivith continuous differentiable E \\ Suppose that

 2 2 Á 2 1 , Á 1 2 Ả 22 ĩ Ẻ u , o.nd E ĩị are bounded Then, the Sacker-Sell spectrum of (13) and that

of the adjoint system are symmetric xrith respect to the origin i e if [a„ ,5,] (1 < i < d) is an arbitrary Sacker-Sell spectral interval for (13) then [— ữ,Ị is a Sacker-Sell spectral ìnterval

for the adjoint system and vice versa.

Proof By Lemma 48, it suffices to consider the Sacker-Sell spectruni associated with the under-

lving ODEs (15) and (22) Then, using similar arguments as in [23 Lemma 1.2 the proof is straightforward Here \ve give an alternative proof By Lemma 55 there exist orthogonal matrix

íunctions U\,V\ € C ^ I , Rd'd) such that (15) is equivalently transíormed to

In this section we have introduced the concepts of Sacker-Sell (exponential diehotomv) spectra for strangeness-free DAEs of the form (9) Again, since these concepts onlv depend on the solution

of the DAE and not on the representation of the system of DAEs, \vhether it is in the form (9) or

in the general form (1), we immediately have all the results also for DAEs in the general form

3 4 S t a b ilit y o f t h e S a c k e r -S e ll s p e c t r u m

We have seen that for Lyapunov spectra of ODEs we do not ahvays have stability under small perturbations and this instability may clearly carry over to the DAE case On the other hand we

ha ve stability of Bohl spectra for ODEs and DAEs It remains to analyze the stability properties

of Sacker-Sell spectra for DAE systems of the form (9) Again we have to restrict ourselves to admissible perturbations, i e perturbations of the form (26) \ \ ’e assume that the boundedness condition (28) holds and to make our perturbation bounds quantitative, \ve assume that there

The orthogonal matrix function V then solves the diíĩerential equation