Observability for parabolic equations

Description

Observability for parabolic equations from a measurable set in time Kim Dang PHUNG , joint work Gengsheng Wang Université d'Orléans. 27 September 2011, Orléans, GDRE CONEDP

• original lebeau-robbiano


• kim dang


• given y0 ?


• heat equation


• lebeau-robbiano strategy


• lebeau

Sujets

• Heat equation
• Lebeau

Informations

• Publié par: pefav
• Nombre de lectures: 50
• Langue: English

Extrait

Orl?ans,ObservabilitWywfo27r,parkrabUniversit?olicerequationsCONfromjointaomeasurableGengshengsetangind'Orl?ans.tSeptembime2011,KimGDREDangEDPPHUNGnΩ R ω⊂ Ω T > 0
2y ∈L (Ω) (y,f)0

∂ y−Δy = 1 f Ω×(0,T)t |ω y = 0 ∂Ω×(0,T)
y(·,0) =y Ω0
 y(·,T) = 0 Ω
kfk ≤cky k2 20L (Ω×(0,T)) L (Ω)
,satisfyingov,.Robbiano,GivenAlso,G.inA.thereNote,EXAisinHEAeau,T(1995)inursiEQUAImanuvilov,TION(1996)LetfobCTinCONTROL,F,e.,LebaL.ORCPDEd.othFsmokoundedO.mainLecture,Serieso.THEOKparolicrabonequationsb
∂ u−Δu = 0 Ω×(0,T) t
u = 0 ∂Ω×(0,T)
 2u(·,0)∈L (Ω)
Z Z T
2 2ku(·,T)k ≤c |u(x,t)| dxdt2L (Ω)
ω 0
,ALENinTL,EQUIVon,.Y 1−α α
kwk ≤C kwk kwk1 2 1H (Ω×(γ,T−γ)) L (ω×(0,T)) H (Ω×(0,T))

2 2for any w∈H (Ω×(0,T)), ∂ +Δ w = 0 and w = 0 .|∂Ωt
=⇒
=⇒
inniteLEBEAyORIGINALTEGYrnitetH?lderfoControllabilitU-ROBBIANOinSTRAandObservabilitdimypellipticObservationye=⇒
=⇒
LEBEASTRA(1998)TEGYeau,H?lder.tG.ypZuazua,eDCDSObservationestimateU-ROBBIANOObservabilitfoyrLebellipticE.OFARMASHORTCUT.L.Miller,Sum(2010)ofeigenfunctions(0,T) E |E|> 0
measurablefoArtime,payrabaolicrequationsbang-bangIaI.setReplaceinywIsimilaI.meAIMI&TiPLANoptimalI.controlbn∞ q ∞a∈L (0,T;L (Ω)) q≥ 2 b∈L (Ω×(0,T))

∂ u−Δu+au+b·∇u = 0 Ω×(0,T) t
u = 0 ∂Ω×(0,T)
 2u(·,0)∈L (Ω)
Ω ω⊂ Ω E⊂ (0,T) |E|> 0
Z Z
ku(·,T)k ≤c |u(x,t)|dxdt2L (Ω)
ω E
,OBSERVandconvex,andNEW.,,,,oninABILITYINEQUALITY 1−α αC
‘ku(·,‘)k ≤Ce ku(·,‘)k ku(·,0)k2 2 2L (Ω) L (ω) L (Ω)
=⇒
Z Z
ku(·,T)k ≤c |u(x,t)|dxdt2L (Ω)
ω E
ARABOLICObservabilitoneH?lderytObservationSTRAFtimesetinfromameasurablePeointyppTEGYrORfointime→ Ω
→ ε> 0
E⊂ (0,T) |E|> 0
→ {‘ }j j≥1
→ z > 1
ε z
fI.StepStepedSteprameters:IolationI.rameterUsageChoiceofquencesetointsoffrompstaositiveomeIasure,theofandProopptimeinequaliininterpointdomainandr-shapObservationparameterrpaconvex,IyI.tofConstructionpaofoneasehere 1−α αC
‘ku(·,‘)k ≤Ce ku(·,‘)k ku(·,0)k2 2 2L (Ω) L (ω) L (Ω)
=⇒
1 C
‘ku(·,‘)k ≤ Ce ku(·,‘)k +εku(·,0)k2 1 2L (Ω) L (ω) L (Ω)γε
∀ε> 0
pointfrominatimeObservation