Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Subseries in Control
93
Ionut¸ Munteanu
Boundary
Stabilization
of Parabolic
Equations
Progress in Nonlinear Differential Equations
and Their Applications
PNLDE Subseries in Control
Volume 93
Editors
Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France
Editorial Board
ViorelBarbu,FacultateadeMatematică,Universitatea“AlexandruIoanCuza”din,
Iaşi, Romania
Piermarco Cannarsa, Department of Mathematics, University of Rome “Tor
Vergata”, Roma, Italy
Karl Kunisch, Institute of Mathematics and Scientific Computing, University of
Graz, Graz, Austria
Gilles Lebeau, Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis,
Nice, France
Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai, China
Shige Peng, Institute of Mathematics, Shandong University, Jinan, China
EduardoSontag,DepartmentofElectricalandComputerEngineering,Northeastern
University, Boston, MA, USA
EnriqueZuazua,DepartamentodeMatemáticas,UniversidadAutónomadeMadrid,
Madrid, Spain
More information about this series at http://www.springer.com/series/15137
ţ
Ionu Munteanu
Boundary Stabilization
of Parabolic Equations
Ionuţ Munteanu
Faculty of Mathematics
Alexandru IoanCuzaUniversity
Iaşi,Romania
ISSN 1421-1750 ISSN 2374-0280 (electronic)
Progressin Nonlinear Differential EquationsandTheir Applications
PNLDESubseries inControl
ISBN978-3-030-11098-7 ISBN978-3-030-11099-4 (eBook)
https://doi.org/10.1007/978-3-030-11099-4
LibraryofCongressControlNumber:2018966441
Mathematics Subject Classification (2010): 35K05, 93D15, 93B52, 93C20, 47F05, 60H15, 35R09,
35Q30,35Q92
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To my beloved daughter, Anastasia
Preface
In recent years, many researchers have been working on designing stabilizers in
different technological areas suchas surface designfor controlling aircraft, voltage
regulators in electronics, camera stabilizers, chemical substances to prevent
unwanted change in the state of another substance, different types offood preser-
vatives,medicalprocessesforpreventingshockinsickorinjuredpeople,andmood
stabilizers, among many others.
Inthisbook,wewilltreatthissubjectfromamathematicalpointofview.More
exactly, we will consider different models from different fields such as fluid flows
modeledbytheNavier–Stokesequations;electricallyconductedfluidflowsmodeled
by the magnetohydrodynamic equations; phase separation modeled by the Cahn–
Hilliardequations;differentcasesofsemilinearheatequationsarisingfrombiology,
chemistry,orpopulationdynamicsaswellastheirstochasticversions.Thenwewill
address the problem of boundary stabilization associated with these models. All
thesemodelscanbecombinedundertherubricofabstractparabolic-likeequations,
namelyequationswhoselinearpartsaregeneratedbyanalyticC -semigroups.That
0
is why, in Chap. 2, we consider the boundary stabilization problem associated to
abstract parabolic-like equations and develop an algorithm to design proportional-
type boundary feedback stabilizers, offinite-dimensional structure, expressed in a
verysimpleform,thatareeasytomanipulateinnumericalsimulations.Itshouldbe
emphasized that no rigorous stabilization theory is possible without a unique con-
tinuation theory for the eigenfunctions of the linear operator obtained from the
linearization of the equation around the target solution. So, once a model, such as
thoseabove,canbeformulatedinaparabolicabstractform,theboundarystabilizing
controldesignmethodcanbeapplied,providedthatauniquecontinuationproperty
oftheeigenfunctions isestablished. Thisprovides thepowerofthiscontroldesign
technique;namely,itcanbeappliedtoawiderangeofmodels.Butitrequiresthat
we prove a priori a unique continuation result that relies onsome advancedresults
andtechniquesinvolvingboththetheoryofparabolic-likeequationsandfunctional
analysis.
vii
viii Preface
Wementionthatintheliterature,therearealsoothernotableresultsconcerning
the boundary stabilization of parabolic equations, and though we mention some
basicreferencesandofferabriefpresentationofothersignificantworksinthefield,
wehavenotpresentedthemindetail.Weconfineourselvestotheproportional-type
feedbackdesignonly,whichisbasedonthespectraldecompositionofalinearized
system in stable and unstable systems, thereby omitting other important results in
theliterature.Thisbookwaswrittenwiththegoalofpresentingindetailnewresults
related to an algorithm for the design of proportional-type feedback forms, which
enabledustoobtainsomeofthefirstresultsinareassuchasboundarystabilization
of the Cahn–Hilliard system, and trajectories for the semilinear heat equation and
even for stochastic partial differential equations. These ideas are still being devel-
oped, andone might expect inthe future toobtain other spectacular achievements.
Besides stabilization, the robustness of stabilizable feedback under stochastic
perturbations is also discussed. The form of the feedback is based on the eigen-
functions of the linear operator, and we have tried to use a minimal set of them.
The reader is assumed to have a basic knowledge of linear functional analysis,
linear algebra, probability theory, and the general theory of elliptic, parabolic, and
stochastic equations. Most ofthis isreviewed inChap.1. The material includedin
this book (excepting the comments on the references) represents the original con-
tribution of the author and his coworkers.
TheauthorisindebtedtoProf.ViorelBarbuforsuggestingtous,fiveyearsago,
that we develop some of his own earlier ideas on constructing proportional-type
feedbackforms,whichledtotheconceptionofthisentirebook.Weareindebtedto
him as well for encouraging us to write this book and for useful discussions,
pertinent observations and suggestions, andunstinting supportand guidance inthe
writing of this book. Many thanks go to Hanbing Liu, and special thanks to my
parents for their love and support. Also, the author is indebted to Mrs. Elena
Mocanu, from the Institute of Mathematics Iaşi, who assisted in the typesetting of
this text.
Iaşi, Romania Ionuţ Munteanu
August 2018
Contents
1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notation and Theoretical Results. . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Stabilization of Abstract Parabolic Equations. . . . . . . . . . . . . . . . . . 19
2.1 Presentation of the Abstract Model . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Design of the Boundary Stabilizer . . . . . . . . . . . . . . . . . . . . 25
2.2.1 The Case of Mutually Distinct Unstable Eigenvalues. . . . . 26
2.2.2 The Semisimple Eigenvalues Case . . . . . . . . . . . . . . . . . . 37
2.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Stabilization of Periodic Flows in a Channel. . . . . . . . . . . . . . . . . . . 49
3.1 Presentation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 The Stabilization Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 The Feedback Law and the Stability of the System . . . . . . 63
3.3 Design of a Riccati-Based Feedback . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 Stabilization of the Magnetohydrodynamics Equations
in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 The Magnetohydrodynamics Equations of an Incompressible
Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 The Stabilizing Proportional Feedback. . . . . . . . . . . . . . . . . . . . . 86
4.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Stabilization of the Cahn–Hilliard System . . . . . . . . . . . . . . . . . . . . 93
5.1 Presentation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.1 Stabilization of the Linearized System . . . . . . . . . . . . . . . 96
5.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
ix
x Contents
6 Stabilization of Equations with Delays . . . . . . . . . . . . . . . . . . . . . . . 109
6.1 Presentation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Stability of the Linearized System . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Feedback Stabilization of the Nonlinear System (6.1) . . . . . . . . . . 120
6.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Stabilization of Stochastic Equations. . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1 Robustness in the Presence of Noise Perturbation
of the Boundary Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Stabilization of the Stochastic Heat Equation on a Rod. . . . . . . . . 136
7.2.1 Mild Formulation of the Solution and Proof
of the Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 Stabilization of the Stochastic Burgers Equation. . . . . . . . . . . . . . 150
7.4 Stabilization by Discrete-Time Feedback Control . . . . . . . . . . . . . 162
7.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8 Stabilization of Unsteady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.1 Presentation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2 The Stabilization Result and Applications . . . . . . . . . . . . . . . . . . 172
8.2.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9 Internal Stabilization of Abstract Parabolic Systems . . . . . . . . . . . . 187
9.1 Presentation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.2 Stabilization of the Full Nonlinear Equation (9.9). . . . . . . . . . . . . 195
9.3 The Design of a Real Stabilizing Feedback Controller . . . . . . . . . 203
9.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
References.... .... .... .... ..... .... .... .... .... .... ..... .... 207
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 213
