Stabilization of the wave equation on d

Description

Niveau: Secondaire, Lycée, Terminale
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedba ks Serge Ni aise, Julie Valein Université de Valen iennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956 Institut des S ien es et Te hniques of Valen iennes F-59313 - Valen iennes Cedex 9 Fran e Julie.Valein,Serge.Ni aiseuniv-valen iennes.fr O tober 24, 2006 Abstra t In this paper we onsider the wave equation on 1-d networks with a delay term in the boundary and/or transmission onditions. We rst show the well posedness of the problem and the de ay of an appropriate energy. We give a ne essary and su ient ondition that guarantees the de ay to zero of the energy. We further give su ient onditions that lead to exponential or polynomial stability of the solution. Some examples are also given. 1 Introdu tion/Notations Time delay ee ts arise in many pra ti al problems, see for instan e [20, 28, 1? for biologi al, ele tri al engineering, or me hani al appli ations. Furthermore it is well known that they an indu e some instabilities [17, 18, 19, 30, 25?, or on the ontrary improve the performan e of the system [28, 1?.

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Subjects

• Documents
• Education
• Secondary scrool

Informations

• Published by: profil-zyan-2012
• Reads: 33
• Language: English

Exrait

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ZN ljX ˜∂φ ∂φj j˜<φ, φ> = dx.V
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2∂ uj
(Δu) = ∀j = 1,···,N,u∈X.j 2∂x
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ρ ∈ (0, 1) t > 0

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 j∈E v u (v,t) = 0 ∀v∈D,t> 0,j v ∂ujv (v,t) = 0 ∀v∈N,t> 0, ∂njv ∂u z (0, t) = (v, t) ∀v∈V , t> 0, v c∂t (0) ∂u (1) u(t = 0) =u , (t = 0) =u , ∂t 0z (ρ, 0) =f (−τ ρ) ∀v∈V , 0<ρ< 1.v v cv
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j=1
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(v) =−(α w(v)+α z (1))∀v∈V ;v c1 2∂nj
j∈EvX
∂uj c(v) = 0∀v∈V \V ;int int∂nj
j∈Ev
∂ujv(v) = 0∀v∈N ; z (0) =w(v)∀v∈V },v c∂njv
V Vc c
2 2 VcH :=V ×L (R)×L (0, 1) ,
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∂x ∂x0 0z z˜ j=1 v∈Vc
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⊤(f, g,h) ∈H D(A),
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∂x ∂x0 0z h j=1 v∈Vc
⊤(u, w, z) ∈D(A).
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ert
um
b
b
of
no
taking
tak
Hilb
is
the
that

with
,
all
as
ab
manner
v
same
orthogonalit
and

the
then
w
to
namely
dense
in
domain
tro
is
of
where
ts
the
elemen
n
that
Since

2.1
e
er
Let
No
Pro
e
of.
w
.
all
W
to
rst
orthogonal
e
e
dense
w
and
e
see
y
6
Inw = 0 z = 0,
ZN ljX ∂u ∂fj j
dx = 0,∀(u, 0, 0)∈D(A).
∂x ∂x0j=1
(u, 0, 0)∈D(A) u∈D(A).
D(A) V < ., . >V
f = 0
(v) (v)
α ≤α ,∀v∈V .c2 1
A C0
H
vξ
(v) (v) (v)(v)τ α ≤ξ ≤τ (2α −α ),∀v∈V .v v c2 1 2
H
   * + Z Zu u˜ N l 1jX X∂u ∂u˜j j (v)   w , w˜ = ( +w w˜ )dx+ ξ ( z (ρ)z˜ (ρ)dρ)j j v v
∂x ∂x0 0z z˜ j=1 v∈VcH
H
U ∈ H0
U ∈C([0, +∞),H) U ∈D(A)0
1U ∈C([0, +∞), D(A))∩C ([0, +∞),H).
A
⊤A U = (u,w, z) ∈D(A)
   * +
w u
   (AU, U) = Δu , w
1 ∂zv−( ) zvτ ∂ρv H
Z ZN l 1j 2X X∂w ∂u ∂ u 1 ∂zj j j v(v)
= ( + w )dx+ ξ ( − (ρ)z (ρ)dρ).j v2∂x ∂x ∂x τ ∂ρv0 0j=1 v∈Vc
b
exist

ts
e

w
,
this
then
pro
These
in
Since
ositiv
it

is
ose
w
.
that
to
wn
inner
if
or
kno
ose
and
if
only
ourselv
if
(4).
on
no

(3).
pro
e
inner
of
wing
usual
in
alen
ell

k
-semi-
dense
.
is
purp
that

(equipp


real

e
op
Mor


erator
that
solution
w
Pro
(4)
of.
that
By
no
Lumer-Phillips'
to
theorem,
the
it
tro
suces
w
to

sho
inner
w
the
that
t
unique
equiv
is
is
dissipativ
pro
e
This
and
group
maximal
sho
monotone.
F
W
that
e
ose,
rst
e
pro
ho
v
By
e
p
that
e
a
n
is
over,
dissipativ
um
e.
ers
T
will
ak
e
e
to
exists
h
e
(5)
ther
W
,

the
Under
generates
e
a
w
wing
follo
o
supp
w
Let
es
to
oblem
that
that
us
datum
.
initial
and
an

pr
w
),
in
w
obtain
inner
w
the
Then
or
But
the

pro
F
e
2.2
easily
Theorem
.
ed
7
withZN l 2X j ∂ uj
(AU, U) = (−wj 2∂x0j=1
ZN 12 (v)X X∂ u ∂u ξ ∂zj j l vj+ w )dx+ [w ] − ( (ρ)z (ρ)dρ)j j v02∂x ∂x τ ∂ρv 0j=1 v∈Vc
ZN 1(v)X X∂u ξ ∂zj l vj= [w ] − ( (ρ)z (ρ)dρ).j v0∂x τ ∂ρv 0j=1 v∈Vc
Z 1
∂z 1v 2 2(ρ)z (ρ)dρ = (z (1)−z (0)).v v v∂ρ 20
⊤(u,w, z) ∈
D(A)
NX XX
∂u l ∂uj j j
[w ] = w (v) (v)j j0∂x ∂nj
j=1 v∈Vj∈EvX X X X
∂u ∂u ∂uj j jv v= w (v) (v)+ w (v) (v)+ w (v) (v)j j jv v∂n ∂n ∂nj j jv v
v∈V j∈E v∈D v∈Nc vX X
∂uj+ w (v) (v)j ∂nj
cv∈V \V j∈Eint vintX X X X
∂u ∂uj j= ( (v))w (v)+ ( (v))w (v)j j∂nj ∂nj
cv∈V j∈E v∈V \V j∈Ec v int vintX
(v) (v)
= −(α w(v)+α z (1))z (0)v v1 2
v∈VcX
(v) (v)2= − (α z (0) +α z (1)z (0)).v v v1 2
v∈Vc
X X (v)(v) (v) ξ 2 22(AU, U) = − (α z (0) +α z (1)z (0))− (z (1)−z (0))v v v1 2 v v2τv
v∈V v∈Vc cX (v) (v)(v) (v)ξ ξ2 2= − [(α − )z (0) + z (1)+α z (1)z (0)].v v v1 v 22τ 2τv v
v∈Vc
(v) (v)
α α(v) 2 2 2 2−α z (1)z (0)≤ z (1)+ z (0)v v2 v v
2 2
(v) (v)(v) (v)X ξ α ξ α(v) 2 2 2 2(AU, U)≤− [(α − − )z (0) +( − )z (1)]v1 v2τ 2 2τ 2v vv∈Vc
b
w
Again
8
b
y-Sc
e
therefore
By
to
Moreo
By
parts
e
in
v
e
ha
parts,
e
v
w
e
arz's
v
,
and
y
leads
b
y
satised
tegration

an
inequalit
obtain
y
ha
h
w
w
prop
y
yield
tegrating
the
in
y
These
b
er
erties
oundary/transmission
b
Cauc
h(v) (v)(v) (v)(v) ξ α ξ α (v) (v)2 2α − − ≥ 0 − ≥ 0 α α1 2τ 2 2τ 2 1 2v v
(AU, U)≤ 0 A
A λI−A
λ> 0
⊤ ⊤(f, g,h) ∈H U = (u,w, z) ∈D(A)
   
u f
   w g(λI−A) =
z h

 λu −w =f ∀j∈{1,...,N},j j j
2∂ ujλw − =g ∀j∈{1,...,N},j 2 j∂x 1 ∂z vλz + =h ∀v∈V .v v cτ ∂ρv
u
j∈{1,...,N},
1w :=λu −f ∈H (0,l )j j j j
w (v) =λu (v)−f (v) = 0 v∈Dj j jv v v
z w(v) =z (0). v∈V , zv c v
1 ∂zv
λz + =hv v
τ ∂ρv
z (0) =w(v) =λu(v)−f(v).v
zv
Z ρ
−λτ ρ −λτ ρ −λτ ρ λτ σv v v vz (ρ) =λu(v)e −f(v)e +τ e e h (σ)dσ.v v v
0
u
z w
R 1−λτ −λτ −λτ λτ σv v v vz (1) = λu(v)e −f(v)e +τ e e h (σ)dσv v v0
−λτ 0v= λu(v)e +z (v)v
R 10 −λτ −λτ λτ σv v vz (v) = −f(v)e +τ e e h (σ)dσv vv 0
f h
u uj
2∂ uj2λ u − =g +λf .j j j2∂x
determine
e
Indeed,
w
maximal
and
n
nd
and

e
e
and
w
since
erties,
m
prop
that
appropriate
tial
the
no
with
eness
found
is
is
dep

It
that
and
means
W
This
e
(7)
i.
tly
that
Supp
the
ose
pro
that
with
w
.
e
the
ha
ws
v
xed
e
b
found
only
alen
This
with
to
y
By
b
then
en
satisfy
giv
that
explicitly
for
is
is
the
e.,
appropriate
monotone,
Therefore
is
regularit
equation
y
dieren
.
satises
Then
v
for
w
all
us
equiv
for
or
Let
(6)
of
of
dissipativ
solution
then
for
that
ok
sho
lo
a
e
real
W
um
.
er
Let
ending
w
on
e
and
ha
.
v
remains

nd
oundary
.
b
(7)
the
(8),
and

where
ust
e
e
(8)
in
.
with
some
b
v
ecause
ha
satisfy
particular
and

surjectiv
for
(5).
e
.
Note
9
.φj
Z ZN N Nl lj 2 jX X X∂ u ∂u ∂φ ∂uj j j j l2 2 j(λ u − )φ dx = (λ u φ + )dx− [ φ ]j j j j j 02∂x ∂x ∂x ∂x0 0j=1 j=1 j=1
ZN ljX XX∂u ∂φ ∂uj j j2= (λ u φ + )dx− (v)φ (v).j j j
∂x ∂x ∂nj0j=1 v∈Vj∈Ev
⊤(u, w, z) D(A)
XX X X X
∂u ∂u ∂uj j jv(v)φ (v) = (v)φ (v)+ (v)φ (v)j j jv∂n ∂n ∂nj j jv
v∈Vj∈E v∈V j∈E v∈Dv c vX X X
∂u ∂uj jv+ (v)φ (v)+ (v)φ (v)j jv∂n ∂nj jv
cv∈N v∈V \V j∈Eint vintX X
∂uj= ( (v))φ(v)
∂nj
v∈V j∈Ec vX
(v) (v)
= − (α w (v)+α z (1))φ(v).j v1 2
v∈Vc
z (1)v
NX XRlj ∂u ∂φ (v) (v)2 j j −λτv(λ u φ + )dx + (α +α e )λu(v)φ(v)j j 1 20 ∂x ∂x
j=1 v∈Vc
NXRlj
= (g +λf )φ dxj j j0
j=1X
(v) (v) 0+ (α f(v)−α z (v))φ(v), ∀φ∈V.1 2 v
v∈Vc
u ∈ V
NY
V φ∈ D(0, l )⊂ Vj
j=1
u
2∂ uj2 ′λ u − =g +λf D (0,l ) ∀j = 1,···,N.j j j j2∂x
N NY Y
2 2u ∈ H (0, l ) u ∈ V ∩ H (0, l )j j
j=1 j=1
X X
∂u (v) (v) (v) (v)j −λτ 0v[ (v)+(α +α e )λu(v)+(α z (v)−α f(v))]φ(v)v∂n 1 2 2 1j
v∈V j∈Ec v X X X
∂u∂uj j
=− (v)φ (v)− ( (v))φ (v),∀φ∈V.j j∂n ∂nj j
cv∈N v∈V \V j∈Evint int
in
e
side
in
in
w
lemma,
for
and
expression

e
b
v
parts,
o
unique
ab
at
the
e
Using
(9)
e
the
v
implies
ha
y
e
b
satises
y
w
has
then
.
,
Coming
,
(9)
to

elong
is
b
of
ust
left-hand
m
ecause
that


that
the
Lax-Milgram's
using
w
But
y
obtain
tegration
e
using
nd
solution
b
a
This
problem
in
then
b
e
b
the
arriv
tegrating
This
problem
to
and
tit
and
this
(9)
b
on
space
,
tegrating
test
e
w
in
parts,
function
y
a

y
e
y
w
iden
If
Multiplying
.

10
k