Introduction The problem The result convex case The proof Non convex case and corollaries

Matthieu Léautaud

Description

Introduction The problem The result (convex case) The proof Non convex case and corollaries On the uniform controllability of viscous perturbations of scalar conservation laws Matthieu Leautaud Laboratoire Jacques Louis Lions, UPMC Paris 6 Workshop Control of parabolic equations and systems, applications to fluids, Control of Partial and Differential Equations and Applications Trimester November, 16. 2010

• coron-guerrero


• applications trimester


• upmc paris


• fursikov imanuvilov


• ?g ??l2


• conservation laws


• fattorini-russell


• c1 ?

Sujets

• Paul Léautaud
• Conservation law

Informations

• Publié par: pefav
• Nombre de lectures: 23
• Langue: English

Extrait

nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)hTerpofooNnocvnxeOntheuniformcontrollabilityofviscous
perturbationsofscalarconservationlaws

MatthieuLe´autaud

LaboratoireJacquesLouisLions,UPMCParis6
[e-mail]

acesaWorkshopControlofparabolicequationsandsystems,applicationstofluids,
ControlofPartialandDifferentialEquationsandApplicationsTrimester
November,16.2010

dnocorllraeis
nIrtoudcitnohTerpIntroduction

oTheproblem

lbmehTerTheresult(convexcase)

Theproof

Firststep

Secondstep

selutc(noevxaces)Outline

Nonconvexcaseandcorollaries

hTerpofooNnocvnxeacesnadocorllraeis
nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)hTerpofoIntroduction:anexample

Coron-Guerrero’05

oNnocn
u
t
+
Mu
x
−
ε
u
xx
=0in(0
,
T
)
×
(0
,
L
)
,
u
|
t
=0
=
u
0
in(0
,
L
)
,
u
|
x
=0
=
g
(
t
)and
u
|
x
=
L
=
0
in(0
,
T
)
.

evxcTheorem(Fattorini-Russell’71,FursikovImanuvilov’96)
∀
T
>
0
,
∀
u
0
∈
L
2
(0
,
L
)
,
∃
g
=
g
ε
∈
L
2
(0
,
T
)
satisfying

εk
g
k
L
2
(0
,
T
)
≤
C
(
T
,
L
,
M
,
ε
)
k
u
0
k
L
2
(0
,
L
)
,

s.t.u
|
t
=
T
=0
.

Moreover,C
(
T
,
L
,
M
,
ε
)=
C
0
(
T
,
L
,
M
)exp
C
1
(
T
ε
,
L
,
M
)
.

saenadocorllraeis
nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)hTerpofoIntroduction:anexample

Coron-Guerrero’05

However,forthelimitproblem(
ε
=0)for
M
>
0:


u
t
+
Mu
x
=0in(0
,
T
)
×
(0
,
L
)
,
u
|
t
=0
=
u
0
in(0
,
L
)
,
u
|
x
=0
=
g
(
t
)in(0
,
T
)
.

oNnocvnxeacesnadocTheorem
SupposethatM
>
0
,then,
∀
T
>
ML
,
∀
u
0
∈
L
2
(0
,
L
)
,taking
g
=0
impliesu
|
t
=
T
=0
.(and
k
g
k
L
2
=0
)

orllraeis
nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)hTerpofoIntroduction:anexample

Coron-Guerrero’05

Theorem(Coron-Guerrero’05)

oNnocvnxeSupposethatM
>
0
.
(i)
IfT
<
ML
,then
∃
C
0
,
C
1
>
0
s.t.forallcontrol
g
=
g
ε
,

C1k
g
ε
k
L
2
(0
,
T
)
≥
C
0
exp
ε
k
u
0
k
L
2
(0
,
L
)

(ii)
IfT
>
4
.
3
ML
,then
∃
C
0
,
C
1
>
0
and
∃
g
=
g
ε
s.t.

Tthreeu(ffiorrtslaplraTtC1k
g
ε
k
L
2
(0
,
T
)
≤
C
0
exp
−
ε
k
u
0
k
L
2
(0
,
L
)

o>ftLMeh)oCor-nuGreerorocjnceuter:(ii)hsuolacdesahnodlcdroloalirse
Theproblem

(3)

)2(

(1)

u
|
x
=0
=
g
0
(
t
)and
u
|
x
=
L
=
g
L
(
t
)

in(0
,
T
)
.

u
|
t
=0
=
u
0
in(0
,
L
)
,

u
t
+[
f
(
u
)]
x
−
ε
u
xx
=0in(0
,
T
)
×
(0
,
L
)
,

Semilinearparabolicequation

.Tu=T=t|usefisitas)2(,)3(,)1(fonoitulosehttahtosεLg=Lgdnaε0g=0g?∃,Tutegratadna,0>T,0unevig:melborpytiliballortnoCseirallorocdnaesacxevnocnoNfoorpehT)esacxevnoc(tluserehTmelborpehTnoitcudortnI
nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)TTheproblem

Semilinearparabolicequation

ehrpofooNnu
t
+[
f
(
u
)]
x
−
ε
u
xx
=0in(0
,
T
)
×
(0
,
L
)
,

u
|
t
=0
=
u
0
in(0
,
L
)
,

ocnu
|
x
=0
=
g
0
(
t
)and
u
|
x
=
L
=
g
L
(
t
)in(0
,
T
)
.

evxacesnadocr(1)

)2(

(3)

Controllabilityproblem:given
u
0
,
T
>
0,andatarget
u
T
,
∃
?
g
0
=
g
0
ε
and
g
L
=
g
L
ε
sothatthesolutionof(1),(3),(2)satisfies
u
|
t
=
T
=
u
T
.

loalirse
foorpehT)esacxevnoc(tluserehTmelborpehTnoitcudortnITheproblem

,)1(fonoitulosehttahtos

(Globalexact)controllabilityproblem:given
u
0
,
T
>
0,anda
target
u
T
,
∃
?
g
0
=
g
0
ε
and
g
L
=
g
L
ε
sothatthesolutionof(1),
(3),(2)satisfies
u
|
t
=
T
=
u
T
.

.εCpxe+0→ε∼”tsoc“:snoitauqecilobaraprofsmelborpytiliballortnoctcaxelacollausU.C≤)T,0(∞LkεLgk+)T,0(∞Lkε0gk.e.i,+0→εsa”tsoc“dednuobahtiwtiod:melborpytiliballortnoc)tcaxelabolg(mrofinUseirallorocdnaesacxevnocnoN
.εCpxe+0→ε∼”tsoc“:snoitauqecilobaraprofsmelborpytiliballortnoctcaxelacollausUTheproblem

k
g
0
ε
k
L
∞
(0
,
T
)
+
k
g
L
ε
k
L
∞
(0
,
T
)
≤
C
.

Uniform
(globalexact)controllabilityproblem:doitwitha
bounded“cost”as
ε
→
0
+
,i.e.

(Globalexact)controllabilityproblem:given
u
0
,
T
>
0,anda
target
u
T
,
∃
?
g
0
=
g
0
ε
and
g
L
=
g
L
ε
sothatthesolutionof(1),
(3),(2)satisfies
u
|
t
=
T
=
u
T
.

seirallorocdnaesacxevnocnoNfoorpehT)esacxevnoc(tluserehTmelborpehTnoitcudortnI
nIrtoudcitnohTerpboelmhTeerustlc(noevxaces)TTheproblem

ehrpofooNnocvnxeacesnad(Globalexact)controllabilityproblem:given
u
0
,
T
>
0,anda
target
u
T
,
∃
?
g
0
=
g
0
ε
and
g
L
=
g
L
ε
sothatthesolutionof(1),
(3),(2)satisfies
u
|
t
=
T
=
u
T
.

Uniform
(globalexact)controllabilityproblem:doitwitha
bounded“cost”as
ε
→
0
+
,i.e.

k
g
0
ε
k
L
∞
(0
,
T
)
+
k
g
L
ε
k
L
∞
(0
,
T
)
≤
C
.

Usuallocalexactcontrollabilityproblemsforparabolicequations:

C“cost”
∼
ε
→
0
+
exp
.
ε

ocorllraeis