On the numerical computation of controls for the D heat equation

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2010

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01 novembre 2010

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English

Poids de l'ouvrage

10 Mo

On the numerical computation of controls for the 1-D heat equation ARNAUD MÜNCH Laboratoire de Mathématiques de Clermont-Ferrand Université Blaise Pascal, France Nov. 2010, IHP supported by the project CONUM ANR-07-JC-183284 Arnaud Münch Exact Controllability / Heat Equation / Numerics

norm assuming

additional references

heat equation

enrique fernandez-cara

ly ?

pablo pedregal

norm

enrique zuazua

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Publié par

pefav

Publié le

01 novembre 2010

Nombre de lectures

Langue

English

Poids de l'ouvrage

10 Mo

nüMduanrAlaoltrontCacExch/noiemuNscir

ARNAUDMÜNCH

Laboratoire de Mathématiques de Clermont-Ferrand Université Blaise Pascal, France arnaud.munch@math.univ-bpclermont.fr

On the numerical computation of controls for the 1-D heat equation

libi/HtytEeaatqu

Nov. 2010, IHP

supported by the project CONUM ANR-07-JC-183284

etatsmeltnemobPrEtuqH/ae/nuNtaoitroltConlitylabiMduanrAcaxEhcnürime

ω⊂(0,1),a∈C1([0,1],R∗+),y0∈L2(0,1),QT= (0,1)×(0,T),qT=ω×(0,T) 8>Ly≡yt−(a(x)yx)x=v1ω,(x,t)∈QT <>:yy((xx,,t0))==0y,0(x),(x,t)∈ {0,1x×}∈((00,,1T)).

We introduce the linear manifold

C(y0,T) ={(y,v) :v∈L2(qT),ysolves (1) and satisﬁesy(T,∙) =0}.

(1)

∀y0∈L2(0,1),T>0 andv∈L2(qT),y∈C0([0,T];L2(0,1))∩L2(0,T;H01(0,1)).

non empty (see FSRKIUOV-IVVILOMANU’96, RONAIBBO-LUAEBE’95).

The goal is to compute numerically some elements ofC(y0,T), i.e. compute some controls for the heat equation

utliOen

4- Without dual variable via a variational approach (with PABLOPLGAREED)

3- Transmutation method : from wave to heat (with ERNQIEUZUZAAU)

2- Change of norm : framework of Fursikov-Imanuvilov’96 (with ENIRUQEFNDEZREAN-CARA)

1- Ill-posedness for the control of minimalL2-norm (the "HUM control")

5- Conclusions / Additional references

umericsitauN/noeH/yqEtaabllitilCoctrontEhaxüMcnandurA

rAortnballtilieH/yudnancMüxahECoct

(y,v)i∈nCf(y0,T)J(v,y) =21kvk2L2(qT)

PARTI Control of minimalL2(0,1)-norm assuming thata(x) =a0>0

(P)

cisuatiatEqumeron/N

ytilaeH/lortibalNun/rimequtEioatsc

Figure:

(0,1)-norm of the HUM control with respect to time

L(01)

y()xs=ni=1-Tx)(π2,0.=(-ωk→t-)8.0k)t,∙(vin2]Arn[0,TnühcuaMdCtnoxEca

L(01)

Figure:

L2the HUM control with respect to time: Zoom near-norm of T

recivk→t-)8.2k)t,∙(1-T=)-πx,0.2(0ω=yis(nx(=)taeHauqEnoitmuN/ntCollroilaby/itrAanduüMcnEhaxtcin[0.92T,T]

/HealityatiotEquCtnoxEcaalibrtlo

whereφsolves the backward system (Lφ?=φ0≡ −ΣφT0−(=(a0(,xT))φx×)x∂=Ω,0φQ(TT,∙)(==0,φTT)×Ω.Ω,

(yv)∈inCf(y0T)J(y,v) =−φiTn∈fHJ?(φT),J?(φT) =12ZqTφ2dxdt+Zφ(0,∙)y0dx Ω

iremuN/n

The Hilbert spaceHis deﬁned as the completion ofD(0,1)with respect to the norm

kφTkH=„ZqTφ2(t,x t«1/2 )dxd.

From theobservability inequality

C(T, ω)kφ(0,∙)k2L2(Ω)≤ kφTkH2∀φT∈L2(Ω),

Since it is difﬁcult to construct pairs(v,y)∈ C(y0,T)(a fortioriminimizing sequences forJ! ), it is by now standard to consider the corresponding dual :

sceonH.TheHUMcontrJi?cseocrviQTonrn.AdMauchünsiloevigvybnωXφ=or2nlLmaniMicmnortlo

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