On the numerical computation of controls for the D heat equation

On the numerical computation of controls for the D heat equation

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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
Ajouté le 01 novembre 2010
Nombre de lectures 37
Langue English
Signaler un abus
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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
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ω(0,1),aC1([0,1],R+),y0L2(0,1),QT= (0,1)×(0,T),qT=ω×(0,T) 8>Lyyt(a(x)yx)x=v1ω,(x,t)QT <>:yy((xx,,t0))==0y,0(x),(x,t)∈ {0,1x×}((00,,1T)).
cs
We introduce the linear manifold
C(y0,T) ={(y,v) :vL2(qT),ysolves (1) and satisfiesy(T,) =0}.
(1)
y0L2(0,1),T>0 andvL2(qT),yC0([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
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rAortnballtilieH/yudnancMüxahECoct
(y,v)inCf(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
L2
Figure:
(0,1)-norm of the HUM control with respect to time
L(01)
0
y()xs=ni=1-Tx)(π2,0.=(-ωkt-)8.0k)t,(vin2]Arn[0,TnühcuaMdCtnoxEca
L(01)
0
s
Figure:
L2the HUM control with respect to time: Zoom near-norm of T
recivkt-)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,
(yv)inCf(y0T)J(y,v) =φiTnfHJ?(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)≤ kφTkH2φTL2(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