the case for differential geometry in the control of single and coupled pdes the structural acoustic chamber

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1、THECASEFORDIFFERENTIALGEOMETRYINTHECONTROLOFSINGLEANDCOUPLEDPDES:THESTRUCTURALACOUSTICCHAMBERR.GULLIVER,I.LASIECKAy,W.LITTMAN,ANDR.TRIGGIANIyContentsPartI:TheneedfordierentialgeometryinthecontrolofPDEs.Itsmainfeatures........................30Informaloverviewofsomecontroltheoryconceptsa

2、ndterminology...........................41Introduction:Aimandscopeofthepresentpaper.Qualita-tivestatementofresults.Twoapproaches..........111.1LinearsinglePDEs,orcoupledsystemsofPDEs,withvariablecoecientsand/ordenedoncurvedsurfaces.........................111.2Twogeometricapproaches.....

3、..........121.3SummaryofmainfeaturesofdierentialgeometricmethodsinthecontrolofPDEs............131.4GeometricmethodsandcontrolofPDEs:Atwo-wayinteraction.......................141.5SystemsofcoupledPDEs.Nonlinearproblems...152AbenchmarkPDE-controlproblemadvocatingabroaduseofdierentialgeomet

4、ry:Thestructuralacousticproblem..162.1Variousstructuralacousticsmodelswithconstantco-ecientsandwith at exiblewall..........172.2Morerealisticmodels:Variablecoecientsandcurv-edwalls.........................192.3Conclusion.......................20PartII:Riemanniangeometricassumptions.Overv

5、iewofresults203BasicgeometricassumptionfortheenergymethodapproachintheRiemannianmetric...................223.1StrictlyconvexfunctionsandcoerciveeldsintheRiemannianmetricforhyperbolicandnon-hyperbolic(Petrowski)-problems.................223.2Useofassumption(H.1).GeneralstatementofCar-leman

6、estimatesandoverview.............24SchoolofMathematics,VincentHall,UniversityofMinnesota,Minneapolis,MN55455yDepartmentofMathematics,KerchofHall,UniversityofVirginia,Charlottesville,VA22904.ResearchpartiallysupportedbytheNationalScienceFoundationunderGrantDMS-0104305andbytheArmyResearchO

7、ceunderGrantDAAD19-02-1-0179.12R.GULLIVERETAL.4Surveyofgeometricresultsandmethodsfortheconstructionofstrictlyconvexfunctionsandcoerciveelds........314.1Theneedforageometricalperspective........314.1.1EuclideanSpace...............324.1.2TheHessianComparisonThe