Stability and Boundary Stabilization of 1-D Hyperbolic Systems

317 Pages·2016·4.63 MB·English

by  Georges Bastin, Jean-Michel Coron (auth.)

Progress in Nonlinear Differential Equations
and Their Applications
Subseries in Control
88
Georges Bastin
Jean-Michel Coron
Stability and
Boundary
Stabilization
of 1-D Hyperbolic
Systems

Progress in Nonlinear Differential
Equations and Their Applications:
Subseries in Control
Volume 88
Editor
Jean-MichelCoron,UniversitéPierreetMarieCurie,Paris,France
EditorialBoard
ViorelBarbu,FacultateadeMatematicaL,Universitatea"AlexandruIoanCuza"dinIas¸i,Romania
PiermarcoCannarsa,DepartmentofMathematics,UniversityofRome"TorVergata",Italy
KarlKunisch,InstituteofMathematicsandScientificComputing,UniversityofGraz,Austria
GillesLebeau,LaboratoireJ.A.Dieudonné,UniversitédeNiceSophia-Antipolis,France
TatsienLi,SchoolofMathematicalSciences,FudanUniversity,China
ShigePeng,InstituteofMathematics,ShandongUniversity,China
EduardoSontag,DepartmentofMathematics,RutgersUniversity,USA
EnriqueZuazua,DepartamentodeMatemáticas,UniversidadAutónomadeMadrid,Spain
Moreinformationaboutthisseriesathttp://www.springer.com/series/15137
Georges Bastin (cid:129) Jean-Michel Coron
Stability and Boundary
Stabilization of 1-D
Hyperbolic Systems
GeorgesBastin Jean-MichelCoron
MathematicalEngineering,ICTEAM LaboratoireJacques-LouisLions
UniversitécatholiquedeLouvain UniversitéPierreetMarieCurie
Louvain-la-Neuve,Belgium ParisCedex,France
ISSN1421-1750 ISSN2374-0280 (electronic)
ProgressinNonlinearDifferentialEquationsandTheirApplications
ISBN978-3-319-32060-1 ISBN978-3-319-32062-5 (eBook)
DOI10.1007/978-3-319-32062-5
LibraryofCongressControlNumber:2016946174
MathematicsSubjectClassification(2010):35L,35L-50,35L-60,35L-65,93C,93C-20,93D,93D-05,
93D-15,93D-20
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Preface
THETRANSPORT of electrical energy, the flow of fluids in open channels or in
gas pipelines, the light propagation in optical fibres, the motion of chemicals
inplugflowreactors,thebloodflowinthevesselsofmammalians,theroadtraffic,
the propagation of age-dependent epidemics and the chromatography are typical
examples of processes that may be represented by hyperbolic partial differential
equations (PDEs). In all these applications, described in Chapter 1, the dynamics
are usefullyrepresented by one-dimensional hyperbolic balance laws although the
naturaldynamicsarethreedimensional,becausethedominantphenomenaevolvein
one privileged coordinate dimension, while the phenomena in the other directions
arenegligible.
From an engineering perspective, for hyperbolic systems as well as for all
dynamical systems, the stability of the steady states is a fundamental issue. This
bookisthereforeentirelydevotedtothe(exponential)stabilityofthesteadystates
of one-dimensional systems of conservation and balance laws considered over a
finitespaceinterval,i.e.,wherethespatial‘domain’ofthePDEisanintervalofthe
realline.
The definition of exponential stability is intuitively simple: starting from an
arbitraryinitialcondition,thesystemtimetrajectoryhastoexponentiallyconverge
in spatial norm to the steady state (globally for linear systems and locally for
nonlinear systems). Behind the apparent simplicity of this definition, the stability
analysis is however quite challenging. First it is because this definition is not so
easily translated intopractical tests of stability.Secondly, itisbecause the various
functionnormsthatcanbeusedtomeasurethedeviationwithrespecttothesteady
statearenotnecessarilyequivalentandmaythereforegiverisetodifferentstability
tests.
Asamatteroffact,theexponentialstabilityofsteadystatescloselydependson
the so-called dissipativity of the boundary conditions which, in many instances, is
a natural physical property of the system. In this book, one of the main tasks is
thereforetoderivesimplepracticaltestsforcheckingiftheboundaryconditionsare
dissipative.
v
vi Preface
Linear systemsofconservationlawsarethesimplestcase.Theyareconsidered
inChapters2and3.Forthosesystems,asforsystemsoflinearordinarydifferential
equations, a (necessary and sufficient) test is to verify that the poles (i.e., the
roots of the characteristic equation) have negative real parts. Unfortunately, this
test is not very practical and, in addition, not very useful because it is not
robust with respect to small variations of the system dynamics. In Chapter 3, we
show how a robust (necessary and sufficient) dissipativity test can be derived by
using a Lyapunov stability approach, which guarantees the existence of globally
exponentiallyconvergingsolutionsforanyLp-norm.
Thesituationismuchmoreintricatefornonlinearsystemsofconservationlaws
which are considered in Chapter 4. Indeed for those systems, it is well known
that the trajectories of the system may become discontinuous in finite time even
for smooth initial conditions that are close to the steady state. Fortunately, if
the boundary conditions are dissipative and if the smooth initial conditions are
sufficiently close to the steady state, it is shown in this chapter that the system
trajectoriesareguaranteedtoremainsmoothforalltimeandthattheyexponentially
converge locally to the steady state. Surprisingly enough, due to the nonlinearity
of the system, even for smooth solutions, it appears that the exponential stability
strongly depends on the considered norm. In particular, using again a Lyapunov
approach, it is shown that the dissipativity test of linear systems holds also in the
nonlinearcasefortheH2-norm,whileitisnecessarytouseamoreconservativetest
fortheexponentialstabilityfortheC1-norm.
In Chapters 5 and 6, we move to hyperbolic systems of linear and nonlinear
balancelaws.Thepresenceofthesourcetermsintheequationsbringsabigaddi-
tional difficulty for the stability analysis. In fact the tests for dissipative boundary
conditionsofconservationlawsaredirectlyextendabletobalancelawsonlyifthe
source terms themselves have appropriate dissipativity properties. Otherwise, as it
isshowninChapter5,itisonlyknown(throughthespecialcaseofsystemsoftwo
balance laws) that there are intrinsic limitations to the system stabilizability with
localcontrols.
There are also many engineering applications where the dissipativity of the
boundaryconditions,andconsequentlythestability,isobtainedbyusingboundary
feedbackcontrolwithactuatorsandsensorslocatedattheboundaries.Thecontrol
may be implemented with the goal of stabilization when the system is physically
unstable or simply because boundary feedback control is required to achieve an
efficient regulation with disturbance attenuation. Obviously, the challenge in that
case is to design the boundary control devices in order to have a good control
performance with dissipative boundary conditions. This issue is illustrated in
Chapters2and5byinvestigatingindetailtheboundaryproportional-integraloutput
feedbackcontrolofso-calleddensity-flowsystems.MoreoverChapter7addresses
the boundary stabilization of hyperbolic systems of balance laws by full-state
feedback and by dynamic output feedback in observer-controller form, using the
backstepping method. Numerous other practical examples of boundary feedback
controlarealsopresentedintheotherchapters.
Preface vii
Finally,inthelastchapter(Chapter8),wepresentadetailedcasestudydevotedto
thecontrolofnavigableriverswhentheriverflowisdescribedbyhyperbolicSaint-
Venant shallow water equations. The goal is to emphasize the main technological
features that may occur in real-life applications of boundary feedback control of
hyperbolic systems of balance laws. The issue is presented through the specific
applicationofthecontroloftheMeuseRiverinWallonia(southofBelgium).
Inouropinion,thebookcouldhaveadualaudience.Inonehand,mathematicians
interested in applications of control of 1-D hyperbolic PDEs may find the book
a valuable resource to learn about applications and state-of-the-art control design.
On the other hand, engineers (including graduate and postgraduate students) who
wanttolearnthetheorybehind1-Dhyperbolicequationsmayalsofindthebookan
interestingresource.Thebookrequiresacertainlevelofmathematicsbackground
which may be slightly intimidating. There is however no need to read the book in
a linear fashion from the front cover to the back. For example, people concerned
primarilywithapplicationsmayskiptheveryfirstSection1.1onfirstreadingand
start directly with their favorite examples in Chapter 1, referring to the definitions
of Section 1.1 only when necessary. Chapter 2 is basic to an understanding of a
large part of the remainder of the book, but many practical or theoretical sections
in the subsequent chapters can be omitted on first reading without problem. The
book presents many examples that serve to clarify the theory and to emphasize
thepracticalapplicabilityoftheresults.Manyexamplesarecontinuationofearlier
examples so that a specific problem may be developed over several chapters of
the book. Although many references are quoted in the book, our bibliography is
certainly not complete. The fact that a particular publication is mentioned simply
meansthatithasbeenusedbyusasasourcematerialorthatrelatedmaterialcanbe
foundinit.
Louvain-la-Neuve,Belgium GeorgesBastin
Paris,France Jean-MichelCoron
February2016

Acknowledgements
The material of this book has been developed over the last fifteen years. We want
to thank all those who, in one way or another, contributed to this work. We are
especially grateful to Fatiha Alabau, Fabio Ancona, Brigitte d’Andrea-Novel,
Alexandre Bayen, Gildas Besançon, Michel Dehaen, Michel De Wan, Ababacar
Diagne,PhilippeDierickx,MalikDrici,SylvainErvedoza,DidierGeorges,Olivier
Glass, Martin Gugat, Jonathan de Halleux, Laurie Haustenne, Bertrand Haut,
Michael Herty, Thierry Horsin, Long Hu, Miroslav Krstic, Pierre-Olivier Lamare,
Günter Leugering, Xavier Litrico, Luc Moens, Hoai-Minh Nguyen, Guillaume
Olive,VincentPerrollaz,BenedettoPiccoli,ChristophePrieur,ValérieDosSantos
Martins, Catherine Simon, Paul Suvarov, Simona Oana Tamasoiu, Ying Tang,
Alain Vande Wouwer, Paul Van Dooren, Rafael Vazquez, Zhiqiang Wang and
JosephWinkin.
During the preparation of this book, we have benefited from the support of
the ERC advanced grant 266907 (CPDENL, European 7th Research Framework
Programme (FP7)) and of the Belgian Programme on Inter-university Attraction
Poles(IAPVII/19)whicharealsogratefullyacknowledged.Theimplementationof
the Meuse regulation reported in Chapter 8 is carried out by the Walloon region,
SiemensandtheUniversityofLouvain.
ix