Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces

220 Pages·2012·1.941 MB·English

by  Birgit Jacob, Hans J. Zwart (auth.)

Operator Theory: Advances and Applications
Volume 223

Founded in 1979 by Israel Gohberg

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Subseries
Linear Operators and Linear Systems
Subseries editors:
Daniel Alpay (Beer Sheva, Israel)
Birgit Jacob (Wuppertal, Germany)
André C.M. Ran (Amsterdam, The Netherlands)

Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
Ingo Witt (Göttingen, Germany)
Birgit Jacob
Hans J. Zwart
Linear Port-Hamiltonian
Systems on
Infinite-dimensional
Spaces
L Linear
O Operators and
L Linear
S Systems
Birgit Jacob Hans J. Zwart
Fachbereich C Department of Applied Mathematics
Bergische Universität Wuppertal University of Twente
Wuppertal, Germany Enschede, Netherlands
ISBN 978-3-0348-0398-4 ISBN 978-3-0348-0399-1 (eBook)
DOI 10.1007/978-3-0348-0399-1
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012940251
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Preface
The aim of this book is to give a self-contained introduction to the theory of
infinite-dimensional systems theory and its applications to port-Hamiltonian sys-
tems.
Thefieldofinfinite-dimensionalsystemstheoryhasbecomeawell-established
field within mathematics and systems theory. There are basically two approaches
to infinite-dimensional linear systems theory: an abstract functional analytical
approach and a PDE approach. There are excellent books dealing with infinite-
dimensionallinearsystemstheory,suchas(inalphabeticalorder)Bensoussan,Da
Prato, Delfour and Mitter [6], Curtain and Pritchard[9], Curtain and Zwart [10],
Fattorini [17], Luo, Guo and Morgul [40], Lasiecka and Triggiani [34, 35], Lions
[37], Lions and Magenes [38], Staffans [51], and Tucsnak and Weiss [54].
Many physical systems can be formulated using a Hamiltonian framework.
Thisclasscontainsordinaryaswellaspartialdifferentialequations.Eachsystemin
thisclasshasaHamiltonian,generallygivenbytheenergyfunction.Inthestudyof
Hamiltoniansystems it is usually assumedthat the systemdoes notinteractwith
its environment. However, for the purpose of control and for the interconnection
of two or more Hamiltonian systems it is essential to take this interaction with
the environment into account. This led to the class of port-Hamiltonian systems,
see[56,57].TheHamiltonian/energyhasbeenusedtocontrolaport-Hamiltonian
system,seee.g.[4, 7,21,43]. Forport-Hamiltoniansystemsdescribedbyordinary
differentialequationsthisapproachisverysuccessful,seethereferencesmentioned
above. Port-Hamiltonian systems described by partial differential equations is a
subject of current research, see e.g. [14, 28, 33, 41].
In this book, we combine the abstract functional analytical approach with
the more physical approach based on Hamiltonians. For a class of linear infinite-
dimensional port-Hamiltonian systems we derive easily verifiable conditions for
well-posedness and stability.
The material of this book has been developed over a series of years. Javier
Villegas[58]studiedinhisPhD-thesisaport-Hamiltonianapproachtodistributed
parameter systems. We are grateful to Javier Villegas that we could include his
resultsintothebook.Thefirstsetupofthebookwaswrittenforagraduatecourse
oncontrolofdistributedparametersystemsfortheDutchInstituteofSystemsand
Control(DISC)inthespringof2009whichwasattendedby25PhDstudents.This
v
vi Preface
materialwasadaptedfortheCIMPA-UNESCO-MarrakechSchoolonControland
Analysis for PDE in May 2009. In 2010-2011 we were the virtual lecturers of the
14thInternet Seminar on Infinite-dimensional Linear Systems Theory. More than
300 participants attended this virtual course and a wikipage was used to discuss
the material and to post typos and comments. For this course we decided to add
extra chapters on finite-dimensional systems theory, and to make the material in
the later chapters more accessible.
Weareindebtedtothehelpfrommanycolleaguesandfriends.Wearegrateful
to the participants of the DISC-course, the CIMPA-UNESCO-MarrakeschSchool
andthe14thInternetSeminarfortheirusefulcommentsandquestions.Largeparts
of the manuscript have been read by our colleagues Mikael Kurula (Twente) and
ChristianWyss(Wuppertal),whomademanyusefulcommentsforimprovements.
We gratefully acknowledge the financial support of the German Research
Foundation (DFG) in form of a Mercator visiting professorship for the second
author.
Birgit Jacob and Hans Zwart,
November 2011
Wuppertal and Twente
Contents
List of Figures xi
1 Introduction 1
1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 How to control a system? . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 State Space Representation 13
2.1 State space models . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Solutions of the state space models . . . . . . . . . . . . . . . . . . 19
2.3 Port-Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Controllability of Finite-Dimensional Systems 27
3.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Stabilizability of Finite-Dimensional Systems 39
4.1 Stability and stabilizability . . . . . . . . . . . . . . . . . . . . . . 39
4.2 The pole placement problem. . . . . . . . . . . . . . . . . . . . . . 40
4.3 Characterizationof stabilizability . . . . . . . . . . . . . . . . . . . 44
4.4 Stabilization of port-Hamiltonian systems . . . . . . . . . . . . . . 47
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Strongly Continuous Semigroups 51
5.1 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . 51
5.2 Infinitesimal generators . . . . . . . . . . . . . . . . . . . . . . . . 57
vii
viii Contents
5.3 Abstract differential equations. . . . . . . . . . . . . . . . . . . . . 61
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Contraction and Unitary Semigroups 65
6.1 Contraction semigroups . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Groups and unitary groups . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 Homogeneous Port-Hamiltonian Systems 79
7.1 Port-Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Generation of contraction semigroups. . . . . . . . . . . . . . . . . 84
7.3 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8 Stability 97
8.1 Exponential stability . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 Spectral projection and invariant subspaces . . . . . . . . . . . . . 101
8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9 Stability of Port-Hamiltonian Systems 111
9.1 Exponential stability of port-Hamiltonian systems . . . . . . . . . 111
9.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10 Inhomogeneous Abstract Differential Equations and Stabilization 123
10.1 The abstract inhomogeneous Cauchy problem . . . . . . . . . . . . 123
10.2 Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.3 Bounded perturbations of C -semigroups. . . . . . . . . . . . . . . 132
0
10.4 Exponential stabilizability . . . . . . . . . . . . . . . . . . . . . . . 133
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 140
11 Boundary Control Systems 143
11.1 Boundary control systems . . . . . . . . . . . . . . . . . . . . . . . 143
11.2 Outputs for boundary control systems . . . . . . . . . . . . . . . . 147
11.3 Port-Hamiltonian systems as boundary control systems . . . . . . 148
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Contents ix
12 Transfer Functions 157
12.1 Basic definition and properties . . . . . . . . . . . . . . . . . . . . 158
12.2 Transfer functions for port-Hamiltonian systems . . . . . . . . . . 163
12.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
12.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13 Well-posedness 171
13.1 Well-posedness for boundary control systems . . . . . . . . . . . . 171
13.2 Well-posedness for port-Hamiltonian systems . . . . . . . . . . . . 181
13.3 P H diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
1
13.4 Proof of Theorem 13.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 189
13.5 Well-posedness of the vibrating string . . . . . . . . . . . . . . . . 191
13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
13.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A Integration and Hardy Spaces 197
A.1 Integration theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.2 The Hardy spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography 209
Index 215