About the controllability of two parabolic equations with first order terms in the coupling

Luz De Teresa In Collaboration With A. Benabdallah, M. Cristofol, P. Gaitan

Description

About the controllability of two parabolic equations with first order terms in the coupling. Luz DE TERESA in collaboration with A. Benabdallah, M. Cristofol, P. Gaitan IHP-CPDEA:Control of systems Paris; November 2010 LdeT Controllability of parabolic systems

• coupled heat


• a22 ·


• ihp-cpdea


• every initial


• a11 y1


• finite dimensional


• kalman condition

Sujets

• Dimension (vector space)

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

Extrait

About the controllability of two parabolic equations with rst order terms in the coupling.

ERESA

Luz DE T in collaboration with A. Benabdallah, M. Cristofol, P. Gaitan

IHP-CPDEA:Control of systems

Paris; November 2010

LdeTControllabilityofparabolicsystems

uocfdelptaehauqentColorotions Ωn T i Ω T in onΣ T   Ωin obarapfoytilibal

witha ij ∈L ∞ (Ω T )andA ij ∈L ∞ (Ω T ) n forij=12. We say that system (1) isnull controllableif for every initial data 0 (y 10 y 2 )∈L 2 (Ω)there existsf∈L 2 (ω T )such that the corresponding solution of (1) satises

y 1 (T) =y 2 (T) =0

LetΩ T = Ω×(0T),ω⊂Ω. We consider (1)  ∂ t y 1 = Δy 1 +a 11 y 1 +a 12 y 2 +A 11  ∇y 1 +A 12  ∇y 2 +fχ ω   y ∂ 1 t ( y  2 t= Δ= )y y 2 2 ( + t a ) 21 = y 1 0 +a 22 y 2 +A 21  ∇y 1 +A 22  ∇y 2   y 1 (0) =y 10 ()y 2 (0) =y 20 ()

msteyscslitrolTConLde

Y ′ =AY+Bu

T HEOREM (K ALMAN RANK CONDITION )

A∈ L(R n ),B∈ L(R m R n ).

Kalman operator K:=  BAB A n−1 B  ∈ L(R nm R n )

ms

is controllable at timeT>0if and only if rank[A|B] =rank  BAB A n−1 B  =n

ystelics

Ker(K ∗ ) ={0}

Kalman condition is equivalent to

itinFEroweDO:kalonamfrimedsien litylabiraboofpaLedrtloCTno

Y=   D C   A=   − 0 0 µ P

Kalman condition :

Ld

0 −ν 0

ν6=µ

eTControl

1    − 1 λ B= 

albilityofparabolicsyste

0 0 1 V

sm

 

Control

of

the

heat

equation

LdeT

Controllability

of

parabolic

systems