On the numerical computation of controls for the D heat equation

Blaise Pascal

Description

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

Sujets

• Pascal
• Heat equation
• Enrique Zuazua
• mathématiques
• mathématique

Informations

• Publié par: pefav
• Publié le: 01 novembre 2010
• Nombre de lectures: 37
• Langue: English
• Poids de l'ouvrage: 10 Mo

Extrait

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A RNAUD M ÜNCH

Laboratoire de Mathématiques de Clermont-Ferrand Université Blaise Pascal, France [e-mail]

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

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Nov. 2010, IHP

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

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ω⊂(0,1),a∈C 1 ([0,1],R ∗ + ),y 0 ∈L 2 (0,1),Q T = (0,1)×(0,T),q T =ω×(0,T) 8 > Ly≡y t −(a(x)y x ) x =v1 ω ,(x,t)∈Q T < > : y y ( ( x x , , t 0) ) = =0 y , 0 (x), (x,t)∈ {0,1 x ×} ∈ ( ( 0 0, , 1 T ) ) .

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We introduce the linear manifold

C(y 0 ,T) ={(y,v) :v∈L 2 (q T ),ysolves (1) and satisfiesy(T,∙) =0}.

(1)

∀y 0 ∈L 2 (0,1),T>0 andv∈L 2 (q T ),y∈C 0 ([0,T];L 2 (0,1))∩L 2 (0,T;H 01 (0,1)).

non empty (see F SRKIUOV -I VVILOMANU ’96, R ONAIBBO -L UAEBE ’95).

The goal is to compute numerically some elements ofC(y 0 ,T), i.e. compute some controls for the heat equation

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4- Without dual variable via a variational approach (with P ABLO P LGAREED )

3- Transmutation method : from wave to heat (with E RNQIEU Z UZAAU )

2- Change of norm : framework of Fursikov-Imanuvilov’96 (with E NIRUQE F NDEZREAN -C ARA )

1- Ill-posedness for the control of minimalL 2 -norm (the "HUM control")

5- Conclusions / Additional references

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(y,v) i ∈ n C f (y 0 ,T) J(v,y) = 2 1 kvk 2 L 2 (q T )

P ART I Control of minimalL 2 (0,1)-norm assuming thata(x) =a 0 >0

(P)

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L 2

Figure:

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

L(01)

0

y()xs=ni=1-Tx)(π2,0.=(-ωk→t-)8.0k)t,∙(vin 2 ] Arn [0,T nühcuaMdCtnoxEca

L(01)

0

s

Figure:

L 2 the HUM control with respect to time: Zoom near-norm of T

reci vk→t-)8. 2 k)t,∙(1-T=)-πx,0.2(0ω=yis(nx(=) taeHauqEnoitmuN/ntCollroilaby/itrAanduüMcnEhaxtc in[0.92T,T]

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whereφsolves the backward system ( L φ ? = φ 0 ≡ − Σ φ T 0 −( =( a 0 ( , x T ) ) φ x × ) x ∂ = Ω, 0 φ Q (T T ,∙) ( = =0, φ T T )× Ω. Ω,

(yv)∈ in C f (y 0 T) J(y,v) =− φ i T n ∈ f H J ? (φ T ),J ? (φ T ) = 1 2 Z q T φ 2 dxdt+ Z φ(0,∙)y 0 dx Ω

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

kφ T k H = „Z q T φ 2 (t,x t « 1/2 )dxd.

From theobservability inequality

C(T, ω)kφ(0,∙)k 2 L 2 (Ω) ≤ kφ T k H 2 ∀φ T ∈L 2 (Ω),

Since it is difficult to construct pairs(v,y)∈ C(y 0 ,T)(a fortioriminimizing sequences forJ! ), it is by now standard to consider the corresponding dual :

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