Surveys in Differential-algebraic Equations I

237 Pages·2013·1.469 MB·English

by  Ilchmann, Achim, Reis, Timo (eds.)

Differential-Algebraic Equations Forum
Editors-in-Chief
AchimIlchmann(TUIlmenau,Ilmenau,Germany)
TimoReis(UniversitätHamburg,Hamburg,Germany)
EditorialBoard
LarryBiegler(CarnegieMellonUniversity,Pittsburgh,USA)
SteveCampbell(NorthCarolinaStateUniversity,Raleigh,USA)
ClausFührer(LundsUniversitet,Lund,Sweden)
RoswithaMärz(HumboldtUniversitätzuBerlin,Berlin,Germany)
StephanTrenn(TUKaiserslautern,Kaiserslautern,Germany)
PeterKunkel(UniversitätLeipzig,Leipzig,Germany)
RicardoRiaza(UniversidadPolitécnicadeMadrid,Madrid,Spain)
VuHoangLinh(VietnamNationalUniversity,Hanoi,Vietnam)
MatthiasGerdts(UniversitätderBundeswehrMünchen,Munich,Germany)
SebastianSager(Otto-von-Guericke-UniversitätMagdeburg,Magdeburg,Germany)
SebastianSchöps(TUDarmstadt,Darmstadt,Germany)
BerndSimeon(TUKaiserslautern,Kaiserslautern,Germany)
WilSchilders(TUEindhoven,Eindhoven,Netherlands)
EvaZerz(RWTHAachen,Aachen,Germany)
Differential-Algebraic Equations Forum
Theseries“Differential-AlgebraicEquationsForum”isconcernedwithanalytical,algebraic,
controltheoreticandnumericalaspectsofdifferentialalgebraicequations(DAEs)aswellas
theirapplicationsinscienceandengineering.Itisaimedtocontainsurveyandmathematically
rigorousarticles,researchmonographsandtextbooks.ProposalsareassignedtoanAssociate
Editor,whorecommendspublicationonthebasisofadetailedandcarefulevaluationbyat
leasttworeferees.Theappraisalswillbebasedonthesubstanceandqualityoftheexposition.
Forfurthervolumes:
www.springer.com/series/11221
Achim Ilchmann (cid:2) Timo Reis
Editors
Surveys in
Differential-Algebraic
Equations I
Editors
AchimIlchmann TimoReis
InstitutfürMathematik FachbereichMathematik
TechnischeUniversitätIlmenau UniversitätHamburg
Ilmenau,Germany Hamburg,Germany
ISBN978-3-642-34927-0 ISBN978-3-642-34928-7(eBook)
DOI10.1007/978-3-642-34928-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013935149
MathematicsSubjectClassification(2010): 34A08,65L80,93B05,93D09
©Springer-VerlagBerlinHeidelberg2013
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Preface
We are pleased to present the first of three volumes of survey articles in various
fieldsofdifferential-algebraicequations(DAEs).Inthelasttwodecades,therehas
beenasubstantialresearchactivityinthetheory,applications,andcomputationsof
DAEs;ouraimistogiveanalmostcompletepictureoftheselatestdevelopments.
What are DAEs? They certainly belong to differential equations, but the termi-
nologyisnotclear.Intheirmostgeneralform,DAEsareimplicitdifferentialequa-
tions.However,thisisstilltoowideandinviewoflinearizationsandthefactthat
most research is on linear DAEs, one uses the more narrow notion of differential-
algebraic systems. This fact is reflected in the Mathematics Subject Classification
(MSC 2010), which is a taxonomy on a first, second, and third level (2-, 3-, and
5-digitclass,respectively).DAEsarementionedtwiceonlevelthree:34Ordinary
differential equations, 34A General theory, 34A09 Implicit equations, differential-
algebraic equations, and 65 Numerical analysis, 65L Ordinary differential equa-
tions,65L80Methodsfordifferential-algebraicequations.
WhatisthehistoryofDAEs?AlthoughDAEscanbetracedbackearlier,itwas
not until the 1960s that mathematicians and engineers started to thoroughly study
computational issues, mathematical theory, and applications of DAEs. There are
manyrelationshipswithmathematicaldisciplinessuchasdifferentialgeometry,al-
gebra, functional analysis, numerical analysis, stochastics, and control theory, to
mention but a few; and there are extensive applications in electric circuit theory,
chemical processes, constrained mechanics, as well as in economics. In addition
to the intrinsic mathematical interest, there are two fundamental reasons for these
advances:first,automaticmodeling,whichresultsinlargedimensionalDAEs,and
secondtheadvancementofcomputersandhencethefeasibilityofsolvingproblems
numerically. In quantitative terms, this development has lead to more than 1500
journalandconferencepapersonDAEseachyear.
Isaleveltworank,insteadofthecurrentlevelthree,forDAEsappropriate?The
MSC tries to rank the different levelshierarchically.However,terminologicaluni-
tiesfordifferentfields,sothattheycanbeaccuratelyseparatedfromeachother,do
notnecessarilyexist.Moreover,fieldsandtheirimportancevaryintime:newfields
arise,othersbecomelessimportant.OnecouldimaginethatDAEsareequallyim-
v
vi Preface
portantas,forexample,34BBoundaryvalueproblems,34GDifferentialequations
in abstract spaces, 34K Functional-differential and differential-difference equa-
tions, 34L Ordinary differential operators, to name but a few within the 34 ODEs
class.
TheimmensenumberofpapersonDAEsiscertainlynotasufficientreasonfor
any taxonomy, and the underlying methods in DAEs are very distinct: differential
geometry, distributions, and linear algebra. But today’s changing importance and
relevance of DAEs have been shown by about ten research monographs in fields
of DAEs in the last decade and, most importantly, recently the first textbooks on
the mathematical theory of DAEs have been written. This may indicate a turning
point:DAEsarebecomingafieldintheirownright,besideotherfieldsinordinary
differentialequations.
The collectionof surveyarticles in DAEs presentedin the upcomingthreevol-
umeswillincludethetopics
– Linearsystems
– Nonlinearsystems
– Solutiontheory
– Stabilitytheory
– Controltheory
– Modelreduction
– Analyticalmethods
– Differentialgeometricmethods
– Algebraicmethods
– Numericalmethods
– Coupledproblemswithpartialdifferentialequations
– StochasticDAEs
– Chemicalengineering
– Circuitmodelling
– Mechanicalengineering
This may show the depth and width of the recent progress in differential-
algebraic equations and will possibly underpin the fact that differential-algebraic
equationsareinastatewheretheyarenolongeronlyacollectionofresultsonthe
sametopic,butafieldwithintheclassofordinarydifferentialequations.
Ilmenau,Germany AchimIlchmann
Hamburg,Germany TimoReis
Contents
ControllabilityofLinearDifferential-AlgebraicSystems—ASurvey . . . 1
ThomasBergerandTimoReis
RobustStabilityofDifferential-AlgebraicEquations . . . . . . . . . . . . 63
NguyenHuuDu,VuHoangLinh,andVolkerMehrmann
DAEsinCircuitModelling:ASurvey . . . . . . . . . . . . . . . . . . . . 97
RicardoRiaza
SolutionConceptsforLinearDAEs:ASurvey . . . . . . . . . . . . . . . 137
StephanTrenn
Port-HamiltonianDifferential-AlgebraicSystems. . . . . . . . . . . . . . 173
A.J.vanderSchaft
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
vii
Controllability of Linear Differential-Algebraic
Systems—A Survey
ThomasBergerandTimoReis
Abstract Different concepts related to controllability of differential-algebraic
equations are described. The class of systems considered consists of linear
differential-algebraic equations with constant coefficients. Regularity, which is,
looselyspeaking,aconceptrelatedtoexistenceanduniquenessofsolutionsforany
inhomogeneity, is not required in this article. The concepts of impulse controlla-
bility,controllabilityatinfinity,behavioralcontrollability,andstrongandcomplete
controllabilityaredescribedanddefinedinthetimedomain.Equivalentcriteriathat
generalizetheHautustestarepresentedandproved.
Special emphasis is placed on normal forms under state space transformation
and, further, under state space, input and feedback transformations. Special forms
generalizing the Kalman decomposition and Brunovský form are presented. Con-
sequences for state feedback design and geometric interpretation of the space of
reachablestatesintermsofinvariantsubspacesareproved.
Keywords Differential-algebraicequations·Controllability·Stabilizability·
Kalmandecomposition·Canonicalform·Feedback·Hautuscriterion·Invariant
subspaces
MathematicsSubjectClassification(2010) 34A09·15A22·93B05·15A21·
93B25·93B27·93B52
ThomasBergerwassupportedbyDFGgrantIL25/9andpartiallysupportedbytheDAAD.
T.Berger(B)
InstitutfürMathematik,TechnischeUniversitätIlmenau,WeimarerStraße25,98693Ilmenau,
Germany
e-mail:[email protected]
T.Reis
FachbereichMathematik,UniversitätHamburg,Bundesstraße55,20146Hamburg,Germany
e-mail:[email protected]
A.Ilchmann,T.Reis(eds.),SurveysinDifferential-AlgebraicEquationsI, 1
Differential-AlgebraicEquationsForum,
DOI10.1007/978-3-642-34928-7_1,©Springer-VerlagBerlinHeidelberg2013
2 T.BergerandT.Reis
1 Introduction
Controllability is, roughly speaking, the property of a system that any two trajec-
tories can be concatenated by another admissible trajectory. The precise concept,
however, depends on the specific framework, as quite a number of different con-
ceptsofcontrollabilityarepresenttoday.
Since the famous work by Kalman [81–83], who introduced the notion of con-
trollability about 50 years ago, the field of mathematical control theory has been
revivedandrapidlygrowingeversince,emergingintoanimportantareainapplied
mathematics,mainlyduetoitscontributionstofieldssuchasmechanical,electrical
and chemical engineering (see e.g. [2, 47, 148]). For a good overview of standard
mathematicalcontroltheory,i.e.,involvingordinarydifferentialequations(ODEs),
anditshistoryseee.g.[70,76,77,80,138,142].
Just before mathematical control theory began to grow, Gantmacher published
his famous book [60] and therewith laid the foundations for the rediscovery of
differential-algebraicequations(DAEs),thefirstmaintheoriesofwhichhavebeen
developedbyWeierstraß[158]andKronecker[93]intermsofmatrixpencils.DAEs
havethenbeendiscoveredtobeappropriateformodelingavastvarietyofproblems
ineconomics[111],demography[37],mechanicalsystems[7,31,59,67,127,149],
multibody dynamics [55, 67, 139, 141], electrical networks [7, 36, 54, 106, 117,
134,135],fluidmechanics[7,65,106]andchemicalengineering[48,50–52,126],
which often cannot be modeled by standard ODE systems. Especially the tremen-
dous effort in numerical analysis of DAEs [10, 96, 98] is responsible for DAEs
beingnowadaysapowerfultoolformodelingandsimulationoftheaforementioned
dynamicalprocesses.
Ingeneral,DAEsareimplicitdifferentialequations,andinthesimplestcasejust
acombinationofdifferentialequationsalongwithalgebraicconstraints(fromwhich
the name DAE comes from). These algebraic constraints, however, may cause the
solutionsofinitialvalueproblemsnolongertobeunique,orsolutionsnottoexist
at all. Furthermore, when consideringinhomogeneousproblems, the inhomogene-
ityhastobe“consistent”withtheDAEinorderforsolutionstoexist.Dealingwith
theseproblemsahugesolutiontheoryforDAEshasbeendeveloped,themostim-
portantcontributionofwhichistheonebyWilkinson[159].Nowadays,therearea
lotofmonographs[31,37,38,49,66,98]andonetextbook[96],wherethewhole
theory can be looked up. A comprehensive representation of the solution theory
of general linear time-invariant DAEs, along with possible distributional solutions
basedonthetheorydevelopedin[143,144],isgivenin[22].Agoodoverviewof
DAEtheoryandahistoricalbackgroundcanalsobefoundin[99].
DAEs found its way into control theory ever since the famous book by Rosen-
brock [136], in which he developed his ideas of the description of linear systems
by polynomial system matrices. Then a rapid development followed with impor-
tant contributions of Rosenbrock himself [137] and Luenberger [107–110], not
to forget the work by Pugh et al. [131], Verghese et al. [151, 153–155], Pan-
dolfi[124,125],Cobb[42,43,45,46],Yipetal.[169]andBernard[27].Themost
important of these contributions for the development of concepts of controllabil-
ity are certainly [46, 155, 169]. Further developments were made by Lewis and