Control Theory of Infinite-Dimensional Systems

201 Pages·2020·2.406 MB·English

by  Kerner, Joachim, Laasri, Hafida, Mugnolo, Delio

Operator Theory
Advances and Applications
277
Joachim Kerner
Hafida Laasri
Delio Mugnolo
Editors
Control Theory
of Infinite-Dimensional
Systems

Operator Theory: Advances and Applications
Volume 277
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Joachim Kerner • Hafida Laasri • Delio Mugnolo
Editors
Control Theory
of Infinite-Dimensional
Systems
Editors
Joachim Kerner Hafida Laasri
Lehrgebiet Analysis Lehrgebiet Analysis
Fernuniversität in Hagen Fernuniversität in Hagen
Hagen, Germany Hagen, Germany

Delio Mugnolo
Lehrgebiet Analysis
Fernuniversität in Hagen
Hagen, Germany

ISSN 0255-0156 ISSN 2296-4878 (electronic)
Operator Theory: Advances and Applications
ISSN 2504-3609 ISSN 2504-3617 (electronic)
Linear Operators and Linear Systems
ISBN 978-3-030-35897-6 ISBN 978-3-030-35898-3 (eBook)
https://doi.org/10.1007/978-3-030-35898-3

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Preface
The workshop on \Control theory of in(cid:12)nite-dimensional systems" was held Jan-
uary 10{12, 2018, in Hagen and attracted from three continents more than 30
researchers who are active in the (cid:12)elds of control theory, operator theory, and
systems theory. The program included 11 plenary talks by distinguished senior
researchersand11contributedtalks;weareparticularlyproudofthemanyyoung
scientists from Africa and Europe who attended our meeting.
Held on the premises of the FernUniversit(cid:127)at in Hagen, the workshop’s goal
was to bring together leading international experts and young scientists in the
(cid:12)eldofcontroltheoryofin(cid:12)nite-dimensionalsystemsforamutuallybene(cid:12)cialex-
changeofnewideasandresults.Thecontributionstoourmeetingcoveredabroad
spectrum of topics and re(cid:13)ected various aspects of control theory, including well-
posedness,controllability,andoptimalcontrolproblems,aswellasthestabilityof
linearandnonlinearsystems.Moreover,the(cid:12)eldsofpartialdi(cid:11)erentialequations,
semigrouptheory,mathematicalphysics,graphandnetworktheory,aswellasnu-
merical analysis, were addressed. By publishing this book, it is our great pleasure
tocollectoutstandingcontributionsthatsharenovelresultsinthesehighlyactive
areas of research.
We wish to gratefully acknowledge the (cid:12)nancial support of the FernUniver-
sit(cid:127)atinHagenandthatofoursponsor,theGesellschaftderFreundederFernUni-
versit(cid:127)at e.V.
Wuppertal, Germany Ha(cid:12)da Laasri
Hagen, Germany Joachim Kerner
Delio Mugnolo
October 2019
v
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
B. Augner
Well-posedness and stability
for interconnection structures of port-Hamiltonian type . . . . . . . 1
O. Post and J. Simmer
A distance between operators acting in different Hilbert spaces
and operator convergence . . . . . . . . . . . . . . . . . . . . . . . 53
J. Schmid
Infinite-time admissibility under compact perturbations . . . . . . . 73
F.L. Schwenninger
Input-to-state stability for parabolic boundary control:
linear and semilinear systems . . . . . . . . . . . . . . . . . . . . . 83
M. Egidi, I. Nakić, A. Seelmann, M. Täufer, M. Tautenhahn,
and I. Veselić
Null-controllability and control cost estimates for the heat equation
on unbounded and large bounded domains . . . . . . . . . . . . . . 117
C. Wyss
Dichotomous Hamiltonians and Riccati equations
for systems with unbounded control and observation operators . . . 158
vii
Well-posedness and stability
for interconnection structures
of port-Hamiltonian type
Björn Augner
Abstract. We consider networks of infinite-dimensional port-Hamiltonian
systems S on 1D spatial domains. These subsystems of port-Hamiltonian
i
typeareinterconnectedviaboundarycontrolandobservationandareallowed
tobeofdistinctport-HamiltonianordersN ∈N.Well-posednessandstability
i
resultsforport-HamiltoniansystemsoffixedorderN ∈Naretherebygener-
alisedtonetworksofsuch.Theabstracttheoryisappliedtosomeparticular
modelexamples.
Mathematics Subject Classification (2010). Primary: 93D15, 35B35.
Secondary:35G46,37L15,47B44,47D06.
Keywords.Infinite-dimensionalport-Hamiltoniansystems,networksofPDE,
feedbackinterconnection,contractionsemigroups,stabilityanalysis.
1. Introduction
A port-based modelling and analysis initially had been introduced in the 1960s
to treat complex, multiphysics systems within a unified mathematical framework
[23].Eachofthesesubsystems,mayitbeofmechanical,electricalorthermaltype
etc. is described by its inner dynamics, usually by a system of ODEs or PDEs,
on the one hand, and ports, which enable the interconnection with other subsys-
tems, on the other hand. For port-Hamiltonian systems the notion of an energy
has been highlighted, similar to classical Hamiltonian systems. In contrast to the
latter,however,theport-Hamiltonianformulationallowsbesidesconservative,i.e.
energy preserving, elements also for dissipative, i.e. energy dissipating, elements,
e.g.frictionallossesinmechanicalsystemsorenergyconversioninresistorswithin
ThisworkhasbeenpartlysupportedbyDeutscheForschungsgemeinschaft(GrantJA735/8-1).
© Springer Nature Switzerland AG 2020 1
J. Kerner et al. (eds.), Control Theory of Infinite-Dimensional Systems, Operator Theory:
Advances and Applications 277, https://doi.org/10.1007/978-3-030-35898-3_1
2 B. Augner
an electric circuit, where the energy leaves the system in form of heat while the
latter is not included in the model.
Forthedescriptionandanalysisofport-Hamiltoniansystemsinageometrical
way, in [24] the concept of a Dirac structure had been introduced into the the-
oryofport-Hamiltoniansystems.TheseDiracstructureshavetheveryconvenient
property that (suitable) interconnections of Dirac structures again give a Dirac
structure (of higher dimension). The underlying models for the physical systems
uptothe2000shadbeenprimarilyfinite-dimensional,i.e.theinnerdynamicsofthe
subsystems interconnected via ports had usually been described by ODEs. Prob-
ablywiththearticle[25]firstattemptsweremadetoextendthedevelopedfinite-
dimensional theory of port-Hamiltonian systems to infinite-dimensional models,
i.e.PDEs,andtherebyfillinginthegapbetweenresultsonfinite-dimensionalsys-
tems and infinite-dimensional port-Hamiltonian systems. E.g. first in [14], it has
been demonstrated that for linear infinite-dimensional port-Hamiltonian systems
on an interval, i.e. evolution equations of the form
∂x XN ∂k(Hx)
= P (t,ζ), t≥0, ζ ∈(0,l)
∂t k ∂ζk
k=0
withx(t,·)∈L (0,l;Kd)(whereK =RorK=C)andforsuitableP ∈Kd×dand
2 k
H : [0,1] → Kd×d, those boundary conditions (or, in a rather systems theoretic
interpretation:linearclosurerelations)thatleadtogenerationofabounded(even
contractive, when L (0,l;Kd) is equipped with an appropriate energy norm) C -
2 0
semigroupcanbecharacterised:Crucialisthedissipativity(w.r.t.theenergyinner
product), which can be checked solely via a matrix criterion on the boundary
conditions[13],[14].Nextstepsthenhavebeensufficientconditionsforasymptotic
or uniform exponential stability of the system [26, 27, 5]. Then followed efforts to
generalise these results to PDE–ODE-systems, i.e. feedback control via a finite-
dimensional linear control system [18, 5], and non-linear boundary feedback [22,
19,3].Here,wewanttopushforwardintoadifferentdirectionandinasortreturn
to the beginnings of port-Hamiltonian modelling: What happens, if we consider
anetworkofinfinite-dimensionalport-Hamiltoniansubsystemsinsteadofasingle
one, where the subsystems, just in the spirit of port-based modelling, are coupled
viaboundarycontrolandobservationofthedistinctport-Hamiltoniansubsystems?
Towhatextenddotheresultsonwell-posedness(inthesenseofsemigrouptheory)
and stability extend to this network case? For special classes of PDE, especially
the wave equation and several beam models, such an analysis is not new by any
means,seee.g.[28,12,15,8]and[17].Also,notethatthecompositionofboundary
controlsystemshasalreadybeenstudiede.g.in[1]forcouplingviastaticKirchhoff
laws.Concerningthenotionofwell-posednessnotethatinthismanuscriptalways
well-posednessinthesenseofsemigrouptheoryismeant.Forthemorerestrictive
notion of well-posedness also including input, output and feed-through maps, let
us refer to the work of Staffans, Tucsnak, Weiss, etc., see e.g. the monograph [21]
or the recent conference paper [29].
Interconnection structures of port-Hamiltonian type 3
Before giving an outline of the organisation of this paper, let us emphasise
that for systems with constant Hamiltonian energy densities Hi :[0,li]→Kdi×di,
already alternative approaches to well-posedness and stability are well-known. In
particular,inthatcaseitisoftenpossibletodetermine(inananalyticalwayorvia
sufficiently good numerical approximation) the eigenvalues of the total system up
to sufficient accuracy, and derive conclusions on well-posedness and stability. For
non-constantH suchanapproachisnotthateasilyaccessible,inparticularthere
i
aresituations,inwhichstabilitypropertiesofaport-Hamiltoniansystemarevery
sensitivetomultiplicativeperturbationbyH ,seee.g.[9]foranastonishingcounter
i
example. Therefore, we deem the port-Hamiltonian approach as a legitimate way
to describe and analyse such systems.
This manuscript is structured as follows. Section 2 serves as an introduction
to the (mainly standard) notation we use throughout this paper, we recall some
basic facts on (strongly continuous) semigroup theory, and the notion of a port-
Hamiltonian system is introduced. In Section 3, we recall previous results on the
well-posedness and stability of infinite-dimensional linear port-Hamiltonian sys-
tems on a one-dimensional domain. We do this with the background of particular
interconnectionschemeswhichhavebeenconsidereduptonow,andalsocomment
onsomeofthetechniquesusedtoprovethecorrespondingresults.Thesubsequent
Sections 4, 5 and 6 constitute the main sections of this paper: First, in Section 4
we provide the general well-posedness result for multi-port Hamiltonian systems
interconnected in a dissipative way: As for single port-Hamiltonian systems, a
dissipative linear closure relation is already enough to have existence of unique
(strong) solutions for all initial data, and the solution depends continuously on
theinitialdatum,i.e.theinitialdatumtosolutionmapisgivenbyastronglycon-
tinuous semigroup of linear operators. Secondly, the focus lies on asymptotic and
exponential stability for closed loop port-Hamiltonian systems, which we investi-
gateinSection5underadditionalstructuralconstraints,e.g.theport-Hamiltonian
systemsbeingseriallyinterconnectedinachain.Then,Section6isdevotedtosys-
temsconsistingthemselvesofsystemsofport-Hamiltoniansystemsagainwhichfor
complexstructuresofthetotalsystemmightbeahelpfulpointofviewforstability
considerations. We illustrate the results of the preceding sections by networks of
first order port-Hamiltonian systems and of Euler–Bernoulli beam type. Finally,
in Section 8 we rephrase the main aspects of this paper and comment on further
openorrelatedproblems.Afterthat,sometechnicalresultsontheEuler–Bernoulli
beam equation are collected in an appendix.