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

TRƯỜNG ĐẠI HỌC KHOA HỌC Tự NHIÊN
TÊN ĐỂ TÀI:
TÍNH ỔN ĐỊNH MŨ VÀ PHổ NHỊ PHÂN MŨ CỦA
PHƯƠNG TRÌNH VI PHÂN ĐẠI s ố
MÃ SỐ: QT-08-02
CHỦ TRÌ ĐỂ TÀI:
PGS.TS. VŨ HOÀNG LINH
CÁC CÁN BỘ THAM GIA:
G S .T S . N g u y ễ n H ữ u D ư , T h S . N g u y ễ n Q u ố c T u ấ n ,
T h S . L ê H u y H o à n g , C N . V ũ T h ị V â n , T h S . N g u y ễ n T h ị Y ế n ,
C N . L ê T h ê S ắ c , C N . N g u y ễ n D iệ u H ư ơ n g , C N . N g u y ễ n T h u H ậ u
HÀ NỘI - 2008
2
MỤC LỤC
1. Báo cáo tóm tắt 4
2. Tóm tắt bằng tiếng Anh 7
3. Phần chính của báo cáo 9
4. Phụ lục 14
(Bài báo và báo cáo hội thảo, bìa luận vãn và khóa luận)
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1. BÁO CÁO TÓM TẮT
a. Tên đê tài, mã số.
Tính ổn định mũ và phổ nhị phân mũ của phương trình vi phân đại sỏ
Mã sô: Q T -08-02
b. Chủ trì đề tài.
PGS.TS. Vũ Hoàng Linh
c. Các cán bộ tham gia.
GS.TS. Nguyễn Hữu Dư, ThS. Nguyễn Q uốc Tuấn, ThS. Lê Huy H càng, CN. Vũ
T hị V â n , ThS . N g u y ễ n T h ị Y ế n , C N . L ê T h ế s ắ c , C N . N g u y ễ n D iệu H ư ơ n g , C N .
N guyễn Thu Hậu
d. M ục tiêu và nội dung nghiên cứu.

- M ục tiêu: L ý th u y ế t địn h tính và lờ i g iả i số của p h ư ơ n g trìn h vi phân đ ại s ố
được các nhà nghiên cứu lý thuyết và ứng dụng trên thế giới đặc biệt quan tâm
trong khoảng thời gian 25 năm trở lại đây. M ột số trường phái nghiên cứu tiêu
biểu đã được hình thành ở Mỹ (Gear, Petzold, Cam pbell, R heinbold), Đức
(M aerz, Kunkel, M ehrm ann, Lubich), Thụy Sỹ (H airer), N ga (Bojarincev,
C h is ty a k o v ), w . N h iề u b ộ c h ư ơ n g trìn h p h ầ n m ề m đ ã đ ư ợ c x â y d ự n g v à áp d ụ n g
hiệu quả vào các bài toán công nghệ và kỹ thuật trong các dự án côn g nghiệp ở
các nước tiên tiến, ví dụ như các bài toán điều khiển tối ưu, bài toán m ô phỏng
mạch điện tử, m ổ phỏng hệ cơ học nhiều vật và m ột số bài toán tính toán khoa
học khác.
Tại khoa Toán — Cơ — Tin học, trường Đại học K hoa học Tự nhiên, ĐH Q G
H N, từ cu ối những năm 90, một nhóm nghiên cứu vể phương trình vi phân đại số
đã được hình thành (GS.TSKH. Phạm Kỳ Anh, GS.TS. N guyễn Hữu Dư,
PGS.TS. Vũ H oàng Linh, TS. Lê Công Lợi). Trong 5 năm vừa qua chúng tôi đã
th ự c h iệ n 2 đ ề tài c ấ p Đ H Q G v ề lĩn h vực n à y . C á c k ế t q u ả đ ã đ ư ợ c trìn h b à y tạ i
nhiều hội nghị khoa học trong và ngoài nước. Hơn 20 bài báo khoa học đã được
c ô n g b ố , tro n g đ ó n h iề u b à i b á o đ ư ợ c đ ă n g ở các tạ p c h í q u ố c tế c ó u y tín n h ư J.
D ijferentiaỉ Equations, A pplied N itm erical M athem atics, Systems & Control
Letters, IMA J. M athem atical C ontrol and InỊormation, J. D ifference Equations
Applic., J. M atlì. Analysis Applic., Advances in D iffercnce Equ., vv. Đ ể tiếp cận
các hướng nghiên cứu hiện đại trên thế giới, từ nhiều năm nay ch úng tôi đã duy
4
trì một seminar về phương trình vi phân và tính toán khoa học. N so à i mục tiêu
chính là đạt được các kết quả khoa học có chất lượng, chúng tỏi cũng hướng tới
việc bổi dưỡng, đào tạo các sinh viên, học viên cao học, và lớp cán bộ trẻ có
năng lực trong lĩnh vực Toán học tính toán và Toán ứng dụng thành những cán
bộ khoa học có chuyên m ôn tốt, đảm nhận được cô ng tác đào tạo và nghiên cứu
khoa học, đổng thời đóng góp vào việc nghiên cứu lý thuyết phương trình vi
phân đại số.
- Nội dung: P h ư ơng trình vi p hâ n đại số cấ p 1 có d ạ n g tổng quát:

F (x \x,t)= 0 , (1)
trong đó ma trận Jacobi của F theo biến thứ nhất được giả thiết là suy biến. Dạng
tuyến tính của (1) có thể viết như sau:
E (t)x’(t)+A (t)x(t)=q (t). (2)
N ội dung nghiên cứu của đề tài gồm các vấn đề chính như sau:
1. K h á i n iệm ổ n đ ịn h m ũ , s ố m ũ B o h l, n h ị p h â n m ũ v à p h ổ n h ị p h â n m ũ c ủ a
hệ (2). Các tính chất của chúng.
2. Phư ơ ng p háp tính toán xấp x ỉ p h ổ nhị phân m ũ c h o h ệ (2).
3. M ở rộ n g k ế t q u ả n ó i tr ê n c h o h ệ (1 ) kh i tu y ế n tín h h ó a ( 1 ) d ọ c th e o m ột
q u ỹ đ ạ o lờ i g iả i c h o trư ớ c .
e. Các kết quả đạt được.
Bài báo khoa học (công bô ở tạp chí và kỷ yếu hội thảo khoa học):
1. V .H . L in h , V . M ehrm a n n , L yapu n ov, B o h l, an d S a c k er-S e ll spectral
intervals for differential-algebraic equations, 40 trang (đã nhận đănc
tr o n g J . Dynamics Differential Equations, 2 0 0 8 )
Báo cáo tại hội nghị khoa học:
1. H ội nghị Toán học Toàn quốc lần thứ 7, 4 -8/8/2 0 0 8 , Q ui N hơn. Người
b áo cáo: V .H . L inh, tên b á o cá o: Exponential stability an d robu st stability
of differential-algebraic equations (b á o c á o m ờ i tiể u b a n P h ư ơ n g trìn h vi
phân)
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2. Báo cáo tại X em ina liên trường, V iện Toán học, 9 /2 008 và tại X em ina cùa
Phòng nghiên cứu Khoa học Tính toán và Kỹ thuật, Đại học C aliíom ia.
Santa Barbara, Hoa K ỳ, 4 /200 9. Người báo cáo: V ũ Hoàng Linh, tên báo
cáo: Spectral intervals fo r D A Es and their num ericaì approxim ation.
Đ ào tạo đại học và sau đại học: 3 luận văn đ ại 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 th ứ n h ấ t)
/.
Tình hình kinh phí của đê tài (hoặc dự án).
Kinh phí 20 triệu đổng đã chi vào các m ục như sau:

1. Vật tư
văn phòng:
l.OOO.OOOđ
2. Thông tin liên lạc: l.OOO.OOOđ
3. H ộ i nghị: l.OOO.OOOđ
4 . C ô n g tác phí: 3 .0 0 0 .0 0 0 đ
5. T h u ê m ướn: 1 2 .0 0 0 .0 0 0 đ
6. C h i p hí n g h iệ p vụ c h u y ê n m ôn: 2 .0 0 0 .0 0 0 đ
KHOA QUẢN LÝ CHỦ TRÌ ĐỂ TÀI
(K ý và ghi rõ họ tén) (Ký và ghi rõ họ tên)
TS. Lê Minh Hà PGS.TS. VD Hoàng Linh
TRƯỜNG ĐẠI HỌC KHO A HỌC T ự NHIÊN
t^rtO MlỆu I KwÓi»U
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2. ABSTRACT
a. Project’s title.

Exponential stabilitv and exponential dichotomy spectrum
for differential-algebraic equations
Code: Q T -08-02
b. Project’s supervisor.

Dr. Vu Hoang Linh
c. Project’s members.

Prof.Dr. N guyen Huu Du, Tran Q uoc Tuan, Le Huy
Hoang, N guyen Thi Y en, Vu Thi Van, Le The Sac, Le D ieu Huong, N guyen
Thu Hau
d. Objective and content of the project.
In the project we consider the differential equation o f general form

where the Jacobian o f íunction F w.r.t. the first variable is supposed to be
singular. The linear variant o f system (1) is given as
The main objectives o f the research are as follows
1. Exponential stability, Bohl exponents, exponential d ichotom y, Sacker-Sell
spectral intervals and their properties.
2. Num erical m ethods for calculatins the spectral intervals.
3. Extension to nonlinear D A E s o f the form (1) vvhen they are subjected to
linearization along a trajectory.
e. Main results of the projects.
Publications (in journals and conference proceedings):
1. V.H . Linh, V . Mehrmann, Lyapunov, Bohl, and Sacker-Sell spectral
intervals for differential-algebraic equations, 40 pages (accepted for
publication in J. D ynam ics Differential Equations, 2 008 )
Lecture at conference and \vorkshop:
2. The 7 ,h N ational C onsress 011 M athem atics, 4 -8 /8 /2 0 0 8 , Qui Nhơn.
Speaker: V .H . Linh, Title: Exponentiaì stabiỉity and robust stability of
F (x \x,t)= 0 ,
(1)
(2)
differential-alạebraic equations ( In vited talk at S e ssion D iffe rential
Equations)
3. Seminar at H anoi Insititute o f M athem atics, 9 /2008 and at C S E research
group, U n iv e rsity o f C a lifom ia , S anta B arbara, H o a K ỳ , 4 /2 0 0 9 . Speaker: V u
Hoang L inh, title: Spectral intervals for D AEs and their numericaỉ
approxim ation.
Education and training: 3 B .S c. th ese s, 2 M .S c. th e ses, 1 P h .D . stu dent
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3. PHẦN CHÍNH CỦA BÁO CÁO:
TÍNH ỔN ĐỊNH MŨ VÀ PHổ NHỊ PHÂN MŨ CỦA
PHƯƠNG TRÌNH VI PHÂN ĐẠI số

3.1 Giới thiệu:
Lý thuyết định tính và lời giải số của phương trình vi phân đại số được các
nhà nghiên cứu lý thuyết và ứng dụng trên thế giới đặc biệt quan tàm trons
khoảng thời gian 25 năm trở lại đây. Một sô trường phái nghiên cứu tiêu biểu
đã được hình thành ở M ỹ (Gear, Petzold, Campbell, R heinb old), Đ ức (M aerz,
Kunkel, M ehnnann, Lubich), Thụy Sỹ (Hairer), N ga (Bojarincev,
Chistyakov), w . N hiểu bộ chương trình phần mềm đã được xây dựns và áp
dụng hiệu quả vào các bài toán công nghệ và kỹ thuật trong các dự án cô nc
nghiệp ở các nước tiên tiến, ví dụ như các bài toán điều khiển tối ưu, bài toán
inô phỏng m ạch điện tử, m ô phỏng hệ cơ học nhiều vật và một số bài toán
tính toán khoa học khác.
Phương trình vi phân đại số cấp 1 có dạng tổng quát
F (x’,x,t)= 0, (1)
trong đó ma trận Jacobi của F theo biến thứ nhất được giả thiết là suy biến.
Dạng tuyến tính của (1) có thể viết như sau
E (t)x ’(t)+A (t)x(t)= q(t). (2)
V í dụ, khi tuyến tính hóa hệ (1) dọc theo một lời giải riêng X nào đó, chú n s
ta nhận được hệ dạng (2). Khi E(t) = I, hệ (2) trở thành hệ phương trình vi
phân thường quen thuộc và đã được khảo sát, nghiên cứu trong suốt nhiều thế
kỷ qua.
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Năm 1892, trong luận án tiến sĩ nỗi tiếng của m ình, nhà toán học Nga
Lyapunov đã đặt nền m óng cho lý thuyết ổn định của PTVP. Một trong
những khái niệm quan trọng m à óng đưa ra có tên gọi số mũ đặc trung, sau
này được gọi là số mũ Lyapunov, nhằm đặc trưng cho tốc độ tãng trưởng của
lời giải và có thể sử dụng để khảo sát tính ổn định. Lý thuyết Lyapunov đã
được tiếp tục khảo sát và mở rộng trong suốt thế kỷ 20 với sự đóng góp của
Bohl, Perron, M ilion ch ikov, O seledets, Sacker, Sell, w . Trong thời gian gần
đây, D ieci và Van V leck (2002 ) đã khảo sát bài toán số mũ đặc trưng và phổ
từ g óc độ toán học tinh toán và đưa ra một số thuật toán tính toán các khoảns

phổ cho PTVP. Cơ sở toán học cũng như các bài toán liên quan như đánh giá
sai số, kỹ thuật cài đặt hiệu quả đã được dẫn dắt chi tiết.
M ục tiêu và nội dung chính của đề tài là nghiên cứu tính ổn định của hệ (2)
thông qua việc khảo sát các số mũ đặc trưng và phổ của hệ, m ở rộng các kết
quả từ PTVP thường sang PTV PĐS. Đ ổng thời, chúng tòi cũng đưa ra một sỏ
cách tiếp cận để tính xấp xỉ các khoảng phổ.
3.2 Các kết quả chính
Bước quan trọng của nghiên cứu là chúng tôi giả thiết hệ đã được đưa về
dạng thu gọn không có tính chất lạ, dựa trên lý thuyết chỉ số của Kunkel và
M ehrmann (200 6). Sau đó, bằng các phép biến đổi trực giao, chúng tôi nhận
được các hệ động học tương đương (tức là có các tính chất dộng học như tính
ổn định giống hệ ban đầu) có dạng thưa, từ đó rút ra được phương trình vi
phân thường căn bản.
3.2.1 Sô mũ Lyapunov và khoảng phổ Lvapunov
Trước hết chúng tôi đưa ra khái niệm ma trận nghiệm cơ bản cho PTVPĐS và
định nghĩa số mũ L y a p u n o v của hệ, tính chinh qui (theo Lyapunov) của một
hệ. Sau đó chúng tôi đã khảo sát và đưa ra kết quả cho các câu hỏi sau:
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M ối liên hệ giữa số mũ Lyapunov của hệ V PĐS và số mũ cùa PTVP thườns
cãn bản;
M ối liên hệ giữa số m ũ Lyapunov và tính chính qui của hệ V P Đ S với số mũ
và tính chính qui của hệ liên hợp;
Các ví dụ cho thấy sự khác biệt giữa lý thuyết PTV P thường và lý thuyết
PTVPĐS;
Khái niệm ổn định của số mũ và điều kiện để các số m ũ L yapunov của một
hệ V PĐS là ổn định.
3.2.2 Sô mũ Bohl và phổ Sacker-Sell
Trong phần tiếp theo chúng tôi đã giải quyết các câu hỏi sau:
Đưa ra khái niệm số mũ Bohl đặc trưng cho tốc độ tăng trưởng đểu của lời
giải và các tính chất cơ bản của chúng;

Chúng tôi cũng đã trình bày khái niệm nhị phân mũ và định nghĩa phổ nhị
phân mũ. Khảo sát các tính chất của phổ nhị phân mũ (hay còn gọi là phổ
Sacker-Sell);
M ối liên hệ giữa khoảng phổ Sacker-Sell, số mũ Bohl, số mũ và khoảng phổ
Lyapunov;
M ối liên hệ giữa khoảng phổ Sacker-Sell của hệ PTV PĐS và hệ liên hợp;
Cuối cùng chúng tôi cũng khảo sát tính ổn định của phổ Sacker-Sell, cụ thể
chỉ ra rằng, dưới tác động của nhiễu chấp nhận được, phổ Sacker-Sell luôn ổn
định.
3.2.3 Tính toán xấp xỉ khoảng phổ
Chúng tôi đã đưa ra hai thuật toán, QR liên tục và QR rời rạc để xấp xỉ các
khoảng phổ. Các ví dụ minh họa đã chứng tỏ hiệu lực của các thuật toán và
minh họa tốt cho các kết quả lý thuyết. Thuật toán cũ ng đã được xây dựne
cho hệ tuyến tính hóa nhận được từ một hệ VPĐS phi tuyến.
3.3 Kết luận
Đ ề tài đã m ở rộng lý thuyết phổ từ PTV P thường sang PTV PĐS. Một số điểm
k h á c b iệt c ă n b ả n g iữ a h a i lớ p p h ư ơ n g trìn h c ũ n g đ ã đ ư ợ c c h ỉ ra. N h ữ n g c á c h
tiếp c ậ n b ư ớ c đ ầ u c h o v iệ c tín h to á n s ố k h o ả n g p h ổ đ ã đ ư ợ c x â y d ự n g c h o
PT VPĐ S tuyến tính và phi tuyến. M ột số bài toán m ở cần giải quyết trong
thời gian tới như khảo sát m ối liên hệ giữa tính ổn định của một nghiệm cùa
PT VPĐ S phi tuyến với tính ổn định của hệ tuyến tính hóa, khảo sát sai số của
thuật toán, xem xét việc áp dụng phân tích trơn SVD như đã thưc hiện với
PTVP thường.
3.4 Tài liệu tham khảo
1. W .A . C o p p el. D icho tom ies in Stabiỉity Th eoiy. S p r in ger -V e rla ơ , N e \v
York, N Y , 1978.
2. J.L . D a le c k ii an d M .G . K rein . Stability o f solutions o f differentiaỉ
equations in Banach space. A m eric a n M a them a tic a l S o c ie ty , P r o v id en ce, R I.
1974 .
3. p . K u n k el and V . M ehrm a n n . Differential-Al%ebraic Equations. Anaìysis

and N um erical Solution. E M S P u b lishin g H o u se, Z u ric h , S w itze rla n d , 2 0 0 6 .
4 . L . D ieci and E. s . V a n V lec k . L yapu n o v and o th er sp ectra: a su rvey.
In: Collected lectures on the preservation of stability under discreĩization
(Fort C ollins, c o , 2 0 0 1 ), pages 197—218. SIAM, Philadelphia, PA, 2002.
5. L . D ie c i and E. s . V a n V le c k . L y a p u n o v spec tral in ter va ls: th eo ry and
com putation. SI AM J. Numer. Anal., 40:5 1 6 —54 2, 2002.
6 . V .H . L inh and V . M eh r m an n . S pectr al inter v als fo r d iffer e n tia l a lg ebra ic
e q u a tio n s and th eir n u m e rical ap p roxim a tio n s. Preprint 4 02 , D F G Research
Center Matheon, Berlin, Germany, 2007. url: ttp://w w w .m atheon.de/
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7. A .M . L y a p u n o v . The general problem o f the sta bility o f m otion
Translated by A .T. Fuller from Edouard D avaux's French translation (1907
o f the 1892 R u ssian original. Internat. J. Controỉ, p a ses 5 2 1 —79 0 , 1992.
8. R.J. Sacker and G .R . Sell. A spectral theory for linear d ifferential system s.
J . Diff. Equations, 2 7 :3 2 0 —3 5 8 , 1978.
‘HỤ LỤC: CÁC BÀI BÁO VÀ BÁO CÁO HỘI THẢO
BÌA LUẬN VÃN VÀ KHÓA LUẬN
14
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"George R Sell" <>
JDDE 1271
ờ: ThứTƯ, 19 tháng 11, 2008 21:50

ửi: "George R Sell" <>/"Annette Triner" <annette.triner@springer-
sbm.com>
OURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
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m ail v nu.ed u.vn/vvebm ail/src/printer_friendly_bottom php'7passed _ent_id =0& m ailbo 1/11 2 009
Lyapunov. Bohl and Sacker-Sell Spectral Inter\'als for
Diíĩerential-Algebraic Equations *
Vu H oang Linh t Volker M eh rm ann *
J a n u a ry 12, 2009
Abstract
L ya p unov and exp on e ntia l d ich otom y sp ectral theory is exten ded from ordin ar y differen-
tial eq u atio ns (O D Es) to non auton om ou s diíĩerential-algebraic eq ua tion s (D A E s). B y using
orthogo nal changes of varia b les, the original D A E system is transíb r m ed in to app ro p ria te con-
den sed íorn is, for w h ich c on cep ts su ch as Lyapu nov ex pon e nts , B ohl ex p o n en ts, exp one ntial
dich o to m y and spectra l interv als of various kin ds can be an alyz ed via the re s u ltin g un derlyin g
O D E . Sorne essential diíĩer en ces betw een the spectra l th eory for O D E s and th at for D A Es
are p oin ted o ut. It is also discussed how nunierical m eth ods for com p uting the sp ec tral in-
te rv a ls associa ted w ith L yapun ov and Sacker-Sell (exp on ential d ich otom y ) can be ex ten de d
ĩroin th ose m etho ds proposed for O D Es. S o m e nu m erical exa m ples are presented to illu strate
the theo re tica ỉ results.
K e y w o r d s : diíĩeren tia l-algeb ra ic eq uations, stran g en ess in dex, Lyapu nov ex p on en t, Bohl
ex po nen t, Sacker-Seỉl sp ectru m , ex pon entia l d ichotom y, sp ectral in terval, srnoo th Q R fac-
torizatio n, co n tinu ous Q R a lgorith m , d iscrete Q R algorithm , k in em atic eq u iva lence, S tek lov
íu nctio n
AMS(MOS) subject classification: 65L07, 65L80. 3-ỈD08. 3-4D09
1 Introduction
More than a century ago, fundamental concepts and results for the stability theorv of ordinary
diẩerential equations were presented in Lyapunov’s íamous thesis [59]. One of the most important
notions, the so-called Lyapunov exponent (or Lyapunov characteristic number), has proved verv
useíul in studying growth rates of solutions to linear ODEs. In the nonlinear case, by linearizing
along a particular solution, Lyapunov exponents also give information about the convergence
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.
•This rosearch vvas supported by Deutsche Forschungsgemeinschaft. through M a th eon . the DFG Research
Center “M athematics for Key Technologies" in Berlin. \ ’u Hoang Linh s research vvas partiallv supported by
\ ’N U ’s project Q T 08-02.
^Faculty of Mathem atics, M echanics and Iníormatics. Vietnam National University. 334. Nguyen Trai Str
Thanh Xuan, Hanoi, VMetnam.
M nstitut fiir M athem atik, MA 4-5, Technische Universitat Berlin. D-10G23 Berlin. Fed Rep Germany.
1
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 systems of DAEs
E(t)x = A(t)x + f(t), (1)
on the half-line E = [0,oc), together with an initial condition
x(0) = xo. (2)
Here we assume that E .A e C(I,M nxn), and / € C(H,Rn) are suíRciently smooth. We use the
notation C(H,Rnxrl) to denote the space of continuous íunctions from I to R nxn.
Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system
of DAEs
F{t,x,x) = 0, t> 0. (3)
along a particular solution [12]. In this paper for the discussion of spectral intervals, \ve restrict
ourselves to r eg u l a r DAEs, i. e., we require that (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 engincering [32, 33] and many other applications, in partieular when the dynamics
of a system is constrained or \vhen different physical niodels are coupled together in automatieally
generated modcls [64]. \Yhile DAEs provide a very convenient modeling concept. many numerical
difficulties arise due to the fact that the dynamics is constrained to a maniíold, \vhich oíten is
only given implicitly, see [9, 41, 67] or the recent textbook [50]. These diffieulties are typically
characterized by one of niany index concepts that exist for DAEs, see [9, 37, 41, 50].
The fact that the dynamics of DAEs is constrained also requires a modiíication of most classical
concepts of the qualitative theory that 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 svstems 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 systematically extend the classical spectral concepts (Lvapunov, Bohl, Saeker-
Sell) that were introduced for ODEs, to general linear DAEs \vith variable coeíĩicients of the form
(1). We show that substantial diíĩerences in the theorỵ arise and that most statements in the
classical ODE theory hold for DAEs only under íurther restrictions, here our results extend results
on asymptotic 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 compute 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. Furthermore, the stability

of the spectra vvith 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 \vell as the associated error
analysis. In Section 5 \ve present Iiumerical examples to illustrate the theoretical results and the
properties of the numerical methods. \\e finish the paper \vith a summary and a discussion of
open problems.
2
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 systems of DAEs
E(t)± = A(t)x + /(í), (1)
jn the half-line n = [0,oc), together with an initial condition
x(0) = XQ. (2)
Here we assume that E.A € C(3,Rnxrl), and / € C(ĩ,Rn) are sufficiently smooth. We use the
notation C(n,Rnxn) to denote the space of continuous íunctions from I to IRnxn.
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 that (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 simulation
[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. many numerical
difficulties arise due to the fact that 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 that exist for DAEs, see [9, 37, 41, 50].

The fact that the dynamics of DAEs is constrained also requires a modiíication of most classical
concepts of the qualitative theory that 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 systems 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 systematically extend the classical spectral concepts (Lyapunov, Bohl, Sacker-
Sell) that were introduced for ODEs, to general linear DAEs \vith variable coeíĩìcients of the form
(1). We show that substantial diíĩerences in the theory arise and th at most statem ents in the
classical ODE theory hold for DAEs only under further restrictions, here our results extend results
on asymptotic 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 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. \Xe finish the paper \vith a summary and a discussion of
open problems.
2

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 systems of DAEs
E(t)x = A(t)x + f{t)< (1)
)n the half-line II = [0, oc), together with an initial condition
x(0) = XQ. (2)
rỉere we assume that E.A € C(2, Mnxn), and / € C(I,R n) are suíĩìciently smooth. \Ve use the
ìotation C(H, Rrixn) to denote the space of continuous íunctions from II to Rnxn.
Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system
DÍ DAEs
F(t, X, x) = 0, t > 0. (3 )
along a particular solution [12]. In tlìis paper for the discussion of spectral intervals, \ve restrict
Durselves to regular DAEs, i. e., we require that (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 simulation
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 automatically
generated models [64]. \Vhile DAEs provide a very convenient modeling concept, many numerical
diíRcuỉties arise due to the fact that 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 that exist for DAEs, see [9, 37, 41, 50].
The fact that the dynamics of DAEs is constrained also requires a modiíication of most classical
concepts of the qualitative theory that 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 systematically extend the classical spectral concepts (Lyapunov, Bohl, Sacker-
Sell) that were introduced for ODEs, to general linear DAEs \vith variable coeíĩìcients of the form
(1). We show that substantial diẩerences in the theory arise and that most statements in the
classical ODE theory hold for DAEs only under further restrictions, here our results extend results
OI1 asymptotic 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 compute 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. Furthermore. the stability
of the spcctra with 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 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.
2
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
E (t)x = A(t)x + f(t). (1)
'jn the half-line II = [0,oc), together with an initial condition
x(0) = Xo- (2)

Here we assume that E.A € C (I,R nxn), and / e C(I, Rn) are sufficiently smooth. \Ye use the
notation C(II, Rnxri) to denote the space of continuous functions from II to RnYn.
Linear systems of the form (1) occur when one linearizes a general implicit nonlinear system
of DAEs
F(t, X, x) = 0, t > 0, (3)
along a particular solution [12]. In this paper for the discussion of spectral intervals, \ve restrict
ourselves to regular DAEs, i. e., we require that (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 automatically
generated niodels [64]. \Vhile DAEs provide a very convenient modeling concept. many numerical
diíĩiculties arise due to the fact that 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 that exist for DAEs, see [9, 37, 41, 50].
The fact that the dynamics of DAEs is constrained also requires a modiíication of most classical
concepts of the qualitative theory that 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 systematically extend the classical spectral concepts (Lyapunov, Bohl, Saeker-
Sell) that were introduced for ODEs, to general linear DAEs with variable coeíĩỉcients of the form

(1). We sho\v that substantial diíĩerences in the theory arise and that most statements in the
classical ODE theory hold for DAEs only under further restrietions, here our results extend results
on asymptotic 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 compute 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. Furthermore. 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.
2
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 systems of DAEs
E (t)x = A(t)x + f{t), (1)
)n the half-line E = [0, oc), together with an initial condition
x(0)=xo. (2)
riere we assume that E.A e C(I,Rnxn), and / e C(]I,Rn) are suASciently smooth. \Ve use the
lotation C (I,R nxn) to denote the space of continuous íunctions from n to Rnxn.
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 reg u l a r DAEs, i. e., we require that (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 multibody dynamics [36], electrical circuit simulation
[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 automatically
generated models [64]. While DAEs provide a very convenient modeling concept, many numerical
diíĩìculties arise due to the fact that 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 that exist for DAEs, see [9, 37, 41, 50].
The fact that the dynamics of DAEs is constrained also requires a modiíìcation of most classical
concepts of the qualitative theory that 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 systems 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 systematically extend the classical spectral concepts (Lvapunov, Bohl, Sacker-
Sell) that were introduced for ODEs, to general linear DAEs \vith variable coefficients of the form
(1). We show that substantial differences in the theory arise and that most statem ents in the
classical ODE theory hold for DAEs only under further restrictions, here our results extend results
on asymptotic stability given in [53]. After deriving the concepts and analyzing the relationship
betAveen the diíĩerent concepts of spectral intervals, we then derive two alternative numerical
approaches to compute 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. Furthermore. the stability
of the spectra \vith respect to perturbations arising ÌII 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 \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
2 A review of DAE theory
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 efínition 1 Considtr system (1) with sufficiently smooth coefficient Ịunctions E. .4. -4 / unction
X : I Rn is called a solution of (1) if X e C l (I,R n) and X satisỊìes (1) pointĩiise. It IS caỉled a
solution of the initial value problem (l)-(2 ) if X IS a soỉution of (1) and. satisỷies (2). An inỉtỉal
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)<
\vhere
(4)
(Mthi = õ Ei
vj/_ i j = 0

í.
í V V = J A(i) for i = 0 , , l, j = 0,
i)i,j ì
Q
othenvise.
(5)
j = 0

ế,
(9i)i=f{i\ i = 0

ể,
using the convention that (l) = 0 for i < 0, j < 0 or j > i. In more detail. \ve have
E
" .4
0
■•
0
"
Ẻ - A E
.4
0
•
• 0
Mc =
Ẻ - 2À 2Ẻ - A E
,Ne =
À 0
•
• 0
1
ta
1
^ ■ '
1
• • • ÍẺ - A E _
. Ảiè)

0
•
• 0

_
(6)
To guarantee existence and uniqueness of solutions, \ve make the following hypothesis, see 150;.
H y p o th e s is 2 There exist integers ỊJL. a. and d such that the inỷlated pair a.ssociated
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 ZĨM/J = 0.
2. For all t € I we have rank;42(£) = a, ivhere Á
2
= zĩN ựựn 0 • • • 0jr such that there eiists a
smooth matrix Ịunction T2 of 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.nkE(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-Schmidt orthonormalization is a continuous process, we may assume without loss of
generality that the columns of the matrix functions Z\, Zi< and To in Hypothesis 2 are pointwise
orthonormal.
D efinition 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 . that under some constant rank conditions. every uniquely
solvable (regular) linear DAE of the form (1) with sufficiently smooth E. .4 satisíìes Hypothesis 2
and that tliere exists a reduced system
Ẻ(t)x = Â{t)x + f{t).
<)
3
that is strangeness-free and ha5 the same solution as (1). where
\vith block entries
ế 1 = z Ị e , à 1 = z Ị a . À2 = ZĨNli[In 0 0 ]T.
(S)
System (7) can be viewed as a different representation (remodeling) of sỵstem (1). \vhere all
necessary differentiations of (1) that 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 that 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 underdetermined systems as well as
locally to general nonlinear systems [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. Furthermore. a
matrix 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 computation 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 transformation will not alter the spectral sets which will be
deíìned in terms of the íundamental solution matrices that have not changed. ưnder Hypothesis 2
this transíormation can ahvays be done and this reduced form can even be computed numerieally
at every time 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 that our homogeneous DAE is already strangeness-free
and has the form
and E\ € C{lRdxn) and Ao e C(I, R (n" d)xn) are of full row rank.
3.1 Lyapunov exp onents and Lyapunov spectral intervals
\Ve first discuss the concepts of Lyapunov exponents and Lyapunov spectral intervals.
D efìnition 4 A matrix Ịunction X £ c 1 ( Ị MnxA:), d < k < n, ỈS called íundamental 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 matrừ 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(to)(X(to,to) - /) = 0. for some to > 0.
A major difference between ODEs and DAEs is that íundamental solution matrices for DAEs are
not necessarily square and of full-rank. Every fundamental soỉution matrix has exactlv d linearly
independent columns and a minimal íundamental matrix solution can be easilv made maximal bv
adding n — d zero columns.
3 Spectral theory for DAEs
E(t)x = A(t)x, t G I,
(9)
\vhere
4
DeRnition 5 For a given fundamental solution matrix X of a strangeness-free DAE syỉtem of
the form (9), and Ịot d < k < n. we introduce
1 t 1
AỊ* = limsup — ln \X(t)ex\ and A- = lim inf - ln ỊÀ’(í)e,||. i = 1. '2
.
k.
t — oc t t — o c t
where et denotes the i-th unit vector. The columns of a minỉmal /undamental solution matns 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
in te rva ls . The set of the Lyapunov spectral interưals IS called the L v ap u n o v sp ec tru m of (9).

Definition 6 Suppose that u € C(3. Rn><n) and V £ C1(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)
UAV — UEV and X = V ĩ is called. globally kinematically equivalent towith Ẽ = UE V , À — UAV — u ty V ana X = V X IS cauea giooaiiy Kinematicauy eq uiva ie nt to
(9) and the trans/ormation is called a global kinematical equivalence transformation. // ư €
c1 (I, Rnxn) and, Ịurthermore. also u and ư~ ’"■» '•'•II rrUKoi
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.
Lem ma 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
u G C(n,Rnxrl) and V £ C 1(3,Rnxr’) such that in the /undamental matrix equation EX = LY
R "*
associated with (9). the change of vanables X = VR, with R =
0
and Ri e c\l?.d <d).
and the multiplication of both sides of the system from the leỷt with UT leads to the system
S\ R\ = Ả\ R\, (11)
where S\ := UỊEV1 is nonsingular and A\ := UỊAX1 — U ỊE\\. Here. U\.V\ are the matnx
Ịunctions consisting of the ỷỉrst d columns of u, V, respectively.
\vhere
ProoỊ. Since a smooth and full column rank matrix íunction has a smooth ọ/ỉ-decomposition. see
Ri
0
R 1 is nonsingular. By substituting X = VR into the íundamental matrix equation E X = ,4A'.
vve obtain
[25, Prop. 2.3], there exists an orthogonal matrix íunction V such that X = VR =
EV

■ Ri ■
= (AV - EV)
■ R 1 ■
0
\ /
0
Since, by assumption, the íìrst d. rows of E are of full row rank. \ve have that the first. d columns
of EV, given by EV1, have full column rank. Thus, there exists a smooth QZ?-decomposition
E \\ = u
£i
0
where ư is orthogonal and í\ is nonsingular. Looking at the leading dxd block in the transíormed
equation, we arrive at
SiRi = [ưỊA\\ - UỊE\\\RX,
\vhich proves the assertion. D
The system (11) is an implicitly given ODE, since S\ is nonsingular. It is called essentiallỵ
underlying implicit ODE system of (9). Since orthonormal changes of basis keep the Euclidean
norm invariant, the Lyapunov exponents of the columns of the matrices X and R. and thereíore
those of the two systems are the same.
5
Theorem 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 VTz = Q1 j \vith
Ri satisíying the implicit system
£\R\ = -Ri
.
or equivalently, satisfying the explicit ODE svstem
R\ — £ J 1 -4i R\ ■

Here £\,Ai are deíìned as in Lemma 7. Note that the Lyapunov exponents of z are exactly the
Lyapunov exponents of R\. Due to Lyapunov's theorem on the construction of a normal basis
for ODEs (see [59]), there exists an upper triangular nonsingular matrix c G Rdxd such that the
columns of R\C form a normal basis of (11). This implies that the columns of RC = VTz c form
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].
Definition 9 The DAE system
ị(E Ty) = -A Ty. or ET (t)ỷ = ~[ẢT (t) + E T ten. (12)
dt
IS called. the adjoint system associated with (9).
Lemma 10 Fundamental solution matnces X ,Y of (9) and its adjoint equation (12) satisỊy the
Lagrange identity
YT(t)E(t)X(t) = Yt{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 + Yt EX = - Yt AX + Yt AX = 0
dt
and hence the Lagrange identity follo\vs. By assumption, the matrices VT,ƯT deíìne a stronơ
global kinematic equivalence transíormation for the adjoint equation leading to the adjoint of
( 10). □
R em ark 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.
6
Theorem 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
Ị Ẻn
0
Cì
0
0
X —
Ả21 A
20
X, te I.
(13 ì
Furthermore, the system (13) IS still strangeness-free and thus E 11 and .422 are nonsingular.
ProoỊ. In order to show the existence of appropriate transíormations. \ve use again the th eor e m
on the existence of smooth QR 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 c1 (3, Rdxd) SUCỈ1 that
Ei = ẺnQĨ-
Since d rows of Qf 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 C1(i,Rnx(n~d>) so thd.t
Q:=[ Qi Ộ2 ]
is pointwise orthogonal. Then. we have

'11
0 ] 0 7
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
22
- □
R em ark 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
0
0
0
Ả 22
t G I
(14;
\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 of
the results that vve present below carry over directly to this system. Due to the use of orthogonal
transíormations, it is also clear that 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,
£n.ri = -4s.fi. (15)
\vhcre -4S := .411 - .4i2À7o ^ 2 1 denotes the Schur complement. For (14), the associated uiiderlyino-
implicit ODE system is
£11X1 = ^ gj
respectively. The following result extends the asyrnptotic stability results of [53] in terms of
Lyapunov exponents.
7