Table Of ContentDAE-F
Differential-Algebraic Equations Forum
Timo Reis
Sara Grundel
Sebastian Schöps Editors
Progress in
Diff erential-Algebraic
Equations II
Differential-Algebraic Equations Forum
Editors-in-Chief
AchimIlchmann(TUIlmenau,Germany)
TimoReis(UniversitätHamburg,Germany)
EditorialBoard
LarryBiegler(CarnegieMellonUniversity,Pittsburgh,USA)
StephenCampbell(NorthCarolinaStateUniversity,Raleigh,USA)
ClausFührer(LundsUniversitet,Sweden)
MatthiasGerdts(UniversitätderBundeswehrMünchen,Germany)
SaraGrundel(MaxPlanckInstituteforDynamicsofComplexDynamicalSystems,
Magdeburg,Germany)
PeterKunkel(UniversitätLeipzig,Germany)
VuHoangLinh(VietnamNationalUniversity,Hanoi,Vietnam)
RoswithaMärz(HumboldtUniversitätzuBerlin,Germany)
RicardoRiaza(UniversidadPolitécnicadeMadrid,Spain)
SebastianSager(Otto-von-Guericke-UniversitätMagdeburg,Germany)
SebastianSchöps(TUDarmstadt,Germany)
BerndSimeon(TUKaiserslautern,Germany)
StephanTrenn(UniversityofGroningen,TheNetherlands)
EvaZerz(RWTHAachen,Germany)
Differential-Algebraic Equations Forum
The series Differential-Algebraic Equations Forum is concerned with analytical,
algebraic,controltheoreticandnumericalaspectsofdifferentialalgebraicequations,
as well as their applications in science and engineering. It is aimed to contain
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Moreinformationaboutthisseriesathttp://www.springer.com/series/11221
Timo Reis (cid:129) Sara Grundel (cid:129) Sebastian Schöps
Editors
Progress in
Differential-Algebraic
Equations II
Editors
TimoReis SaraGrundel
FachbereichMathematik Max-Planck-InstitutfürDynamik
UniversitätHamburg komplexertechnischerSysteme
Hamburg,Germany Magdeburg,Sachsen-Anhalt
Germany
SebastianSchöps
TechnischeUniversitätDarmstadt
Darmstadt,Germany
ISSN2199-7497 ISSN2199-840X (electronic)
Differential-AlgebraicEquationsForum
ISBN978-3-030-53904-7 ISBN978-3-030-53905-4 (eBook)
https://doi.org/10.1007/978-3-030-53905-4
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Preface
The “9th Workshop on Descriptor Systems” took place on March 17–20,2019 in
Paderborn,Germany.Followingthetraditionoftheprecedingworkshopsorganized
by Prof. Peter C. Müller between 1992–2005 and 2013, the workshop brought
together more than 40 mathematicians and engineers from various fields, such as
numerical and functionalanalysis, control theory, mechanics and electromagnetic
fieldtheory.Theparticipantsfocusedonthetheoreticalandnumericaltreatmentof
“descriptor”systems,i.e.,differential-algebraicequations(DAEs).
Thisbookcontainstheproceedingsofthisworkshop.Itdiscussesthewiderange
ofcurrentresearchtopicsindescriptorsystems,includingmathematicalmodeling,
index analysis, stability, stabilization, well-posedness of problems, stiffness and
differenttimescales,co-simulationandsplittingmethods,andconvergenceanalysis.
Inaddition,italsopresentsapplicationsfromtheautomotiveandcircuitindustries
thatshowthatdescriptorsystemsarechallengingproblemsfromthepointofview
oftheoryandpractice.
This book is organized into three parts with the first part covering analysis.
It features a contribution by Diana Estévez Schwarz and René Lamour that
discussesorthogonaltransformationsfordecouplingofDAEswithahigherindex.
Differenttypesoftheso-calledhigher-indexcomponentswithregardtotheexplicit
and hidden constraints are characterized. This also results into a straightforward
possibility for linear DAEs to determine an orthogonally projected explicit ODE.
The second chapter consists of an article by Michael Hanke and Roswitha März
on an operator-theoretic view to linear DAEs with constant and non-constant
coefficients.Conditionsconcerningbasiccharacteristicssuchasnormalsolvability
(closed range), Fredholmness, etc. are presented. In particular, it is proven that
actually the operators having tractability index zero and one constitute the class
ofnormallysolvabledifferential-algebraicoperators.
The second part of this book covers numerical analysis and model order
reduction. It consists of five contributions: The article by Andreas Bartel and
Michael Günther contains a complete convergence theory for inter/extrapolation-
based multirate schemes for both ODEs and DAEs of index one, along with
a convergence analysis based on linking these schemes to multirate dynamic
v
vi Preface
iteration.ThesecondcontributioninthispartbyMichaelHankeandRoswithaMärz
isonoverdeterminedpolynomialleast-squarescollocationfortwo-pointboundary
value problems for higher-index DAEs. Basic properties, such as convergence
properties, of this method for initial value problems by a windowing technique
are proven. The article by Robert Altmann and Christoph Zimmer is devoted to
the construction of exponential integrators of the first and second order for the
time discretization of constrained parabolic partial-differential-algebraic systems.
Exponentialintegratorsfor unconstrainedsystems are combinedwith the solution
of certain saddle point problems in order to meet the constraints throughout the
integrationprocess,alongwithaconvergenceanalysis.Thesucceedingcontribution
by Gerd Steinebach is about an improvement of the known Rosenbrock–Wanner
method rodasp, which results in a less distinctive drop of the convergence
order. The contribution by Thanos Antoulas, Ion Victor Gosea, and Matthias
Heinkenschlosstreatsthecomputationofreduced-ordermodelsforaclassofsemi-
explicit DAEs, which includes the semi-discretized linearized Navier–Stokes and
Oseenequations,byadata-drivenLoewnerframework.
The third part is devoted to control aspects of DAEs. The article by Thomas
Berger, Lê Huy Hoàng, and Timo Reis treats adaptive control for a large class
of multiple-inputmultiple-outputDAEs by a novel funnelcontroller. To this end,
a generalization of the conceptof vector relative degree is presented. The second
contributioninthispartbyThomasBergerandLukasLanzaisaboutstateestimation
fornonlinearDAEsystemswithinputsandoutputs.Thepresentedobserverunifies
earlierapproachesandextendsthestandardLuenbergertypeobserverdesign.The
succeeding contribution by Matthias Gerdts and Björn Martens covers implicit
Euler discretization for linear-quadratic optimal control problems with index two
DAEs. The discretized problem is reformulatedsuch that an approximationof an
indexreducedproblemwithsuitablenecessaryconditionsisobtained.Undersome
additionalcircumstances,itisshownthatthecontrolsconvergewithanorderof 1
2
in the L -norm. These error estimates are further improvedwith slightly stronger
1
smoothness conditions of the problem data and switching function, which results
intoaconvergenceorderofone.
The fourth part contains four articles on applications of DAEs. The article
by Sarah-Alexa Hauschild, Nicole Marheineke, Volker Mehrmann, Jan Mohring,
Arbi Moses Badlyan, Markus Rein, and Martin Schmidt is about modeling of
district heating network in the port-Hamiltonian framework. By introducing a
model hierarchy of flow equations on the network, a thermodynamically con-
sistent port-Hamiltonian embedding of the resulting partial differential-algebraic
systemsispresented.Itisfurthershownthataspatiallydiscretizednetworkmodel
describing the advection of the internal energy density with respect to an under-
lying incompressible stationary Euler-type hydrodynamics can be considered as
a parameter-dependentfinite-dimensional port-Hamiltonian system. Moreover, an
infinite-dimensionalport-Hamiltonianformulationfor a compressible instationary
thermodynamic fluid flow in a pipe is presented. The subject of the article by
Steffen Plunder and Bernd Simeon is a coupled system composed of a linear
DAE and a linear large-scale system of ODEs, where the latter stands for the
Preface vii
dynamics of numerous identical particles. Such systems, for instance, arise in
mathematicalmodelsformuscletissuewherethemacroscopicbehaviorisgoverned
by the equations of continuum mechanics. Replacing the discrete particles by a
kinetic equation for a particle density, the mean-field limit results into a new
class of partially kinetic systems. The influence of constraints on those systems
isinvestigated.Asa mainresult,Dobrushin’sstabilityestimateforsystemsofthis
type is presented. The estimate implies convergence of the mean-field limit and
providesarigorouslinkbetweentheparticledynamicsandtheirkineticdescription.
InthearticlebyIdoiaCortesGarcia,SebastianSchöps,ChristianStrohm,andCaren
Tischendorf,a definition of generalized circuit elements which may, for example,
containadditionalinternaldegreesoffreedom,suchthatthoseelementsstillbehave
structurally like resistances, inductances, and capacitances, is presented. Several
complexexamplesdemonstratetherelevanceofthosedefinitions.Finally,thearticle
byDianaEstévezSchwarz,RenéLamour,andRoswithaMärzfocusesonaclassical
benchmarkproblemforhigher-indexDAEs,namelyaroboticarmresultingfroma
tracking problem in mechanicalengineering. The difficulty of this problem is the
appearance of certain singularities, whose thorough analysis is the subject of this
article. To this end, different methodologiesare elaborated, such as the projector-
basedanalysis ofthe derivativearrayandthe directprojector-basedDAE analysis
associated with the tractabilityindex. As a result, with both approaches,the same
kinds of singularities are identified. Some of them are obvious, but others are
unexpected.
Wewouldliketotaketheopportunitytothankalltheindividualswhocontributed
to the workshop and to this volume. Special gratitude goes to Prof. Dr. Peter
C. Müller from BU Wuppertal for encouraging us to pursue his traditional and
distinguishedworkshopseries.Wehopetoseeyouatmanyofthefutureworkshops
ondescriptorsystems!
Hamburg,Germany TimoReis
Magdeburg,Germany SaraGrundel
Darmstadt,Germany SebastianSchöps
May2020
Contents
PartI AnalysisandDecoupling
A ProjectorBased Decoupling of DAEs Obtained from the
DerivativeArray ................................................................. 3
DianaEstévezSchwarzandRenéLamour
BasicCharacteristicsofDifferential-AlgebraicOperators.................. 39
MichaelHankeandRoswithaMärz
PartII NumericalMethodsandModelOrderReduction
Inter/Extrapolation-Based Multirate Schemes: A
Dynamic-IterationPerspective ................................................. 73
AndreasBartelandMichaelGünther
Least-Squares Collocation for Higher-Index DAEs: Global
ApproachandAttemptsTowardaTime-SteppingVersion ................. 91
MichaelHankeandRoswithaMärz
ExponentialIntegratorsforSemi-linearParabolicProblemswith
LinearConstraints............................................................... 137
RobertAltmannandChristophZimmer
ImprovementofRosenbrock-WannerMethodRODASP ................... 165
GerdSteinebach
Data-DrivenModelReductionforaClassofSemi-ExplicitDAEs
UsingtheLoewnerFramework................................................. 185
AthanasiosC.Antoulas,IonVictorGosea,andMatthiasHeinkenschloss
PartIII Closed-LoopandOptimalControl
Vector Relative Degree and Funnel Control for
Differential-AlgebraicSystems................................................. 213
ThomasBerger,HuyHoàngLê,andTimoReis
ix
x Contents
ObserversforDifferential-AlgebraicSystemswithLipschitz or
MonotoneNonlinearities........................................................ 257
ThomasBergerandLukasLanza
Error Analysis for the Implicit Euler Discretization of
Linear-QuadraticControlProblemswithHigherIndexDAEsand
Bang–BangSolutions............................................................ 291
BjörnMartensandMatthiasGerdts
PartIV Applications
Port-HamiltonianModelingofDistrictHeatingNetworks.................. 333
Sarah-Alexa Hauschild, Nicole Marheineke, Volker Mehrmann,
JanMohring,ArbiMosesBadlyan,MarkusRein,andMartinSchmidt
CoupledSystemsofLinear Differential-AlgebraicandKinetic
EquationswithApplicationtothe MathematicalModelling of
MuscleTissue..................................................................... 357
SteffenPlunderandBerndSimeon
GeneralizedElementsforaStructuralAnalysisofCircuits ................ 397
Idoia Cortes Garcia, Sebastian Schöps, Christian Strohm,
andCarenTischendorf
SingularitiesoftheRoboticArmDAE......................................... 433
DianaEstévezSchwarz,RenéLamour,andRoswithaMärz
Index............................................................................... 481
