Mobile point sensors and actuators in the controllability theory of partial differential equations

235 Pages·2017·2.783 MB·English

by  Khapalov, Alexander Y

Alexander Y. Khapalov
Mobile Point Sensors
and Actuators in the
Controllability Theory
of Partial Diff erential
Equations
Mobile Point Sensors and Actuators in the
Controllability Theory of Partial Differential
Equations
Alexander Y. Khapalov
Mobile Point Sensors
and Actuators in the
Controllability Theory of
Partial Differential Equations
123
AlexanderY.Khapalov
DepartmentofMathematics
andStatistics
WashingtonStateUniversity
Pullman,WA,USA
ISBN978-3-319-60413-8 ISBN978-3-319-60414-5 (eBook)
DOI10.1007/978-3-319-60414-5
LibraryofCongressControlNumber:2017943185
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ToDasha,ElenaandIrina
Foreword
Many mathematical models of controlled distributed-parameter processes choose
to employ locally distributed actuators and sensors to describe the admissible
governing actions and the methods of collection of available data. An intrinsic
feature of such “devices” is that they act in an open set of a given spatial domain
and,thus,are infinite-dimensionalateverymomentoftime.Currently,thereexists
a comprehensive mathematical controllability theory dealing exactly with such
actuators(orso-calledcontrols)andsensors.
Thegoalofthismonographistopresentaconcisecontrollabilitytheoryofpartial
differential equations in the case when they are equipped with actuators and/or
sensors that are finite-dimensional at every moment of time. Typical examples
here could be the point “devices,” static or mobile, that are designed to act at
isolatedspatialpoints.Suchfinite-dimensionaldesignsofcontrollingactionsseem
to be better suited for many real-world applications. On the other hand, for the
samereason,theirmathematicalcapabilitiesareprincipallyreducedrelativetotheir
infinite-dimensionalcounterparts,andthusrequireaspecialconsideration.
The subjects of interest in this monograph are the issues of controllability,
observability and stabilizability for parabolic and hyperbolic partial differential
equations,andsomerelatedappliedquestions,suchastheproblemoflocalization
ofunknownpollutionsourcesbasedontheinformationobtainedfrompointsensors,
arisinginenvironmentalmonitoring.
Pullman,WA,USA AlexanderY.Khapalov
February2017
vii
Preface
Themodernmathematicalcontroltheoryemergedinthe1950sasaresultofdemand
from numerous real-world applications, particularly in space engineering. Since
then it has significantly developed both in engineering (including bioengineering)
andappliedmathematicscommunities.
A typical control problem studies the evolution processes that can be affected
by a certain parameter, called “control.” The goal of such problems is to steer the
process at hand from a given initial state to a desirable target state by selecting
acontrolparameterwithinthesetofavailableoptions.Ifthisispossible,thenone
usuallywishestoachievetheabovewhileoptimizingagivencriterion(afunctional).
Thissetuprepresentswhatiscalledanoptimalcontrolproblem.
In its turn, the controllability theory focuses on the steering capabilities of the
controlledevolutionprocesses.Namely,givenanyinitialstate,itstudiestherichness
of the range of the mapping: control ! the state of the process at some moment
of time. The controllability theory was initially developed in the 1960s for linear
ordinarydifferentialequations,governedbyadditivecontrols.
Themodernmathematicalcontrollabilitytheoryoflinearandsemilinearpartial
differentialequationswithadditivecontrolsemergedabout50yearsago.Sincethen,
manypowerfulmathematicalmethodshavebeenintroducedand/oradoptedtodeal
with a wide variety of applied problems along the so-called duality approach and
theHilbertuniquenessmethod,thetechniquesofharmonicandnonharmonicanal-
ysis, unique continuation, the multiplier method, Carleman estimates, microlocal
analysis,andothers(werefertotheBibliographyforfurtherreferences).
Thereadercannotethatthetypicalmathematicalmodelsconsideredwithinthis
theorymakeuseofeitherboundaryorinternallocallydistributedadditivecontrols
or sensors to describe respectively the effect of external controlling actions on the
processathandorthemethodstocollectavailabledata.Theintrinsiccharacteristic
of such controls and sensors is that they act in their “full-dimensional capacity,”
namely,eitherinanopenpartoftheboundaryofasystem’sspatialdomainorinan
opensetwithinthisdomainand,thus,are infinite-dimensionalateverymomentof
time. Namely, if the spatial dimensionality of the original system is n, the actions
ix
x Preface
oftheformer“devices”arerepresentedbythefunctionsofn(cid:2)1spatialvariables,
whilethelatter “devices” arerepresented bythefunctions of nspatialvariables at
every moment of time. An example of such controls might be a source/sink in a
heat/masstransferprocessplacedinsidethespatialdomainoronitsboundary.
Thegoalofthismonographistopresentaconcisecontrollabilitytheoryofpartial
differentialequationsincaseswhentheyareequippedwithactuatorsand/orsensors
that are finite-dimensional at every moment of time. Typical examples here could
be the point “devices,” static or mobile, that are designed to act at isolated spatial
points. Such finite-dimensional designs of controlling actions seem to be better
suited for many real-world applications. On the other hand, for the same reason,
their mathematical capabilities are principally reduced relative to their infinite-
dimensionalcounterparts,andthusrequirespecialconsideration.
The subjects of interest in this monograph are the issues of controllability,
observabilityandstabilizabilityfortheparabolicandhyperbolicpartialdifferential
equations,andsomerelatedappliedquestions,suchastheproblemoflocalization
ofunknownpollutionsourcesbasedontheinformationobtainedfrompointsensors,
arisinginenvironmentalmonitoring.
Thismonographisbasedonresearchconductedbytheauthorin[36–58]inthe
areaofcontrollabilitytheoryofpartialdifferentialequations.
Pullman,WA,USA AlexanderY.Khapalov
February2017
Acknowledgments
The author’s research presented in this monograph was supported in part by the
NSFGrantsECCS9312745,DMS10007981andSimonsFoundationawardnumber
317297.
TheauthorexpressesspecialthankstoDariaKhapalovaforthedrawingsonthe
titlepageandinChap.11.
xi
Contents
1 Introduction................................................................. 1
1.1 ControllingPDEs:WhyPointControlsandSensors?.............. 1
1.2 ObservabilityandControllability:MethodologyofDuality ....... 2
1.2.1 ControllabilityProblem...................................... 2
1.2.2 ObservabilityProblem....................................... 4
1.3 DegenerateSensorsandControls:ChallengesandProposed
MethodologytoAddressThem...................................... 5
1.3.1 ObservabilityProblemwithStaticPointSensors:
PrincipalDifficulties......................................... 6
PartI Observability and Controllability of Linear Parabolic
EquationsbyMeansofDegenerateSensorsandControls
2 ContinuousObservabilityoftheHeatEquationUnderaSingle
MobilePointSensor........................................................ 13
2.1 Introduction........................................................... 13
2.2 AuxiliaryResults..................................................... 15
2.3 CŒ";(cid:2)(cid:3)-ContinuousObservabilityinFinite-Dimensional
Subspaces............................................................. 16
2.4 ProofofTheorem2.1andSomeCorollaries........................ 21
2.5 ExplicitObservationCurvesin1-DCase........................... 24
2.6 ObservabilitywithMobileDiscrete-TimePointSensors........... 25
2.7 DualApproximateControllability................................... 26
3 Continuous Observability of Second-Order Parabolic
EquationsUnderDegenerateMobileSensors........................... 29
3.1 Introduction........................................................... 29
3.2 PreliminaryResults .................................................. 30
3.3 MobilePointEstimate ............................................... 31
3.4 NonsmoothCase ..................................................... 35
3.5 ObservabilityandControllability.................................... 37
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