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1、具有边界流的p-Laplace方程最优控制的存在性和稳定性Existenceandstabilityoftheoptimalcontroltop-Laplacianwithconvectiveboundarycondition??PeidongLei??andHangGaoSchoolofMathematics&Statistics,NortheastNormalUniversity,Changchun130024,People’sRepublicofChinaAbstract.Inthispaper,weareconcerned
2、withtheoptimalcontroloftheevolutionaryp-Laplacianwiththeconvectiveboundaryconditiontakingtheheattransfercoefficientasthecontrol.WetakeasourcostfunctionaltheLp-normofthedifferencebetweenthetemperatureattainedandthedesiredtemperature.Theexistenceandthestabilityoftheoptimalc
3、ontrolthatminimizesthecostfunctionalareproved.Keywords.Optimalcontrol,convectiveboundarycondition,nonlineardiffusion,degen-eracy.AMSSubjectClassification.35K,93B1IntroductionandmainresultsInthispaper,wediscusstheoptimalboundarycontrolgovernedbythefollowingnon-linearheatco
4、nductionsystem??????????yt+div~J+λy=0,inQT,~J·~n=1∑uy,onΓT,y(x,0)=y0(x),in??,(1.1)where??isaboundeddomainofRNwithboundaryofclassC1,QT=??×(0,T),ΓT=????×(0,T),∑??????isafixedpartialboundarywithpositivemeasure,namely,
5、∑
6、>0(
7、·
8、denotesthemeasureofaset),1∑standsforthecharact
9、eristicfunctionoftheset∑,λ≥0isaconstant,~ndenotestheunitouternormaltotheboundary????,and~J=??
10、??y
11、p??2??y,p>2(1.2)istheheatflux.Inthesystem(1.1),y0istheinitialtemperature,y0∈L∞(??)andy0≥0.Thefunctionurepresentstheheattransfercoefficient,whichistakenasourcontrolandactso
12、nthesystemthroughthepartialboundary∑.??TheresearchwassupportedbytheKeyProjectofChineseMinistryofEducation(No.108046),andbyGrantNENU-STC07007.1//.paper.eduThroughoutthispaper,thecostfunctionalischosenasJ(u)=1p{β∫∫QT
13、y??Zd
14、pdxdt+γ∫∫ΓT
15、u
16、pdσdt},(1.3)whereyisthecorrespondingt
17、emperaturedistributionsatisfying(1.1)andZd∈L∞(QT)isadesiredtemperaturedistribution.Asin[11],thecoefficientsβandγareperunitcostsassociatedwithfailingtoachievethedesiredtemperaturedistributionandwithimposingaheattransfercoefficientdifferentfromzero.Accordingtodifferentreque
18、sts,βandγcantakedifferentvalues.Physicallyreasonableheattransfercoefficientsarenonnegativeandbou
