Dirichlet_boundary_stabilization_of_the_wave_equation.1

《Dirichlet_boundary_stabilization_of_the_wave_equation.1》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、AsymptoticAnalysis30(2002)117130117IOSPressDirichletboundarystabilizationofthewaveequationKaisAmmariInstitutÉlieCartan,DépartementdeMathématiques,UniversitédeNancyI,F-54506Vandoeuvre-lès-Nancycedex,FranceE-mail:ammari@iecn.u-nancy.frAbstract.Weconsiderthestabilizationproblemf

2、orawaveequation.Inthecasewhenthegeometriccontrolassumption,see[3],isnotsatisfied,weprovethattheenergyofsystemdecaywithalogarithmicrateforallinitialdatainthedomainoftheinfinitesimalgeneratorofevolutionequation.Thetechniqueusedconsistsindeducingthedecayestimatefromanobservability

3、inequalityfortheassociatedundampedproblem,viasharpregularityresults.Keywords:boundarystabilization,Dirichlettypeboundaryfeedback,waveequation1.IntroductionandmainresultsLetΩ⊂Rn,n
2,beanopenboundeddomainwithasufficientlysmoothboundary∂Ω=Γ0∪Γ1,whereΓ0,Γ1aredisjointpartsoftheboun

4、daryrelativelyopenin∂Ω,int(Γ0)=∅.Weconsiderthewaveequation:∂2u−Du=0,Ω×(0,+∞),(1.1)∂t2∂
u=Gu
,Γ×(0,+∞),(1.2)0∂νu=0,Γ1×(0,+∞),(1.3)0∂u1u(x,0)=u(x),(x,0)=u(x),Ω,(1.4)∂twhereνistheunitnormalvectorof∂ΩpointingtowardstheexteriorofΩandG=(−D)−1:H−1(Ω)→H01(Ω).Theproblem(1.1)(1.4)isw

5、ell-posedinL2(Ω)×H−1(Ω),i.e.,forall(u0,u1)∈L2(Ω)×H−1(Ω),Eqs(1.1)(1.4)admitsauniquesolution

u∈C[0,∞);L2(Ω)∩C1[0,∞);H−1(Ω).Moreover,ifwemultiplied1.1byG(∂u/∂t)weobtainthefollowingenergyestimate:2
012∂uu,uL2(Ω)×H−1(Ω)−u(t),∂t(t)L2(Ω)×H−1(Ω)t2∂[G(∂u/∂s)]=2

6、(x,s)dΓ0ds,∀t>0.(1.5)0Γ0∂ν0921-7134/02/$8.00Ó2002IOSPress.Allrightsreserved118K.Ammari/DirichletboundarystabilizationofthewaveequationLetφbethesolutionofthefollowingproblem:∂2φ−Dφ=0,Ω×(0,∞),(1.6)∂t2φ=0,∂Ω×(0,∞),(1.7)0∂φ1φ(x,0)=φ(x),(x,0)=φ(x),Ω.(1.8)∂tDenote0IAd=,D0with

7、D(A)=(u,v)∈L2(Ω)×H−1(Ω)

8、(v,Du)∈L2(Ω)×H−1(Ω),d2∂[Gv]u

9、∂Ω∈L(∂Ω),u

10、Γ0=,u

11、Γ1=0.∂νIf(u,v)∈D(Ad)wedenote222(u,v)D(A)=(u,v)L2(Ω)×H−1(Ω)+(v,Du)L2(Ω)×H−1(Ω).dThefollowingholds:Proposition1.1.SupposethatthereexistsT0>0suchthatthesolutionφof(1.6)(1.8)satisfyforallT>T0T

12、2∂φ
012dΓ0dt
Cφ,φH1(Ω)×L2(Ω),(1.9)0Γ0∂ν0whereCisapositivec