Mobile point sensors and actuators in the controllability theory of partial differential equations PDF
Preview 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 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 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 xiii
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