Stabilization of se ond order evolution equations

-

English
26 Pages
Read an excerpt
Gain access to the library to view online
Learn more

Description

Niveau: Secondaire, Lycée, Terminale
Stabilization of se ond order evolution equations with unbounded feedba k with time-dependent delay Emilia Fridman ? , Serge Ni aise † , Julie Valein ‡ Mar h 24, 2009 Abstra t We onsider abstra t se ond order evolution equations with unbounded feedba k with time-varying delay. Existen e results are obtained under some realisti assumptions. We prove the exponential de ay under some onditions by introdu ing an abstra t Lyapunov fun tional. Our abstra t framework is applied to the wave, to the beam and to the plate equations with boundary delays. Keywords se ond order evolution equations, wave equations, time-varying delay, stabilization, Lyapunov fun tional. 1 Introdu tion Time-delay often appears in many biologi al, ele tri al engineering systems and me hani al appli ations, and in many ases, delay is a sour e of instability [7?. In the ase of distributed parameter systems, even arbitrarily small delays in the feedba k may destabilize the system (see e.g. [5, 16, 24, 17?). The stability issue of systems with delay is, therefore, of theoreti al and pra ti al importan e. There are only a few works on Lyapunov-based te hnique for Partial Dif- ferential Equations (PDEs) with delay.

  • groups theory

  • delay

  • system

  • b?2 ?˙

  • self-adjoint positive operator

  • operator depends

  • lyapunov fun tional


Subjects

Informations

Published by
Reads 24
Language English
Report a problem

∗ † ‡



b
equations
The
with

un
artial
b
in
ounded
Hainaut

ortance.
k
the
with
equations
time-dep
Engineering,
enden
T
t
of
dela
y
y
on
Emilia
y
F
us,
ridman
some
ev

,
instabilit
Serge
,

FR
order
V
olution
,
Insti-
Julie
9
V
of
alein


a
Marc

h
Equations
24,
these
2009
onstant

and
W
de-
e
w

dela




order
ho
ev
viv
olution
viv,
equations
Cam
with
tut
un
V
b
F
ounded
V

V,
k
et
with
-
time-v
alein@univ-v
arying
y
dela
with
y
therefore,
.

Existence
are
results
w
are
apuno
obtained
for
under
feren
some
with

Most
assumptions.
orks
W
of
e
.
pro
y
v
onen
e
w
the
ed
exp
heat
onen
v
tial


and
y
b
under
dela
some
,

y
b
Stabilization
y
of
in
el
tro
ersit

el
an
Israel,

du
Ly
LAMA
apuno
2956,
v
Sciences
functional.
hniques
Our
F-59313

Cedex
framew

ork
ersit?
is
et
applied
br?sis,
to
CNRS
the
des
w
ec
a

v

e,
rance,
to
1
the
stabilit
b
issue
eam
systems
and
dela
to
is,
the
of
plate
and
equations
imp
with
There
b
only
oundary
few
dela
orks
ys.
Ly
Keyw
v-based
ords
hnique

P
order
Dif-
ev
tial
olution
(PDEs)
equations,
dela
w
.
a
of
v
w
e
analyze
equations,

time-v

arying
delays
dela
Th
y
stabilit
,

stabilization,
exp
Ly
tial
apuno
ounds
v
ere
functional.
riv
1
for
In
scalar
tro
and

a
Time-dela
e
y
with
often
t
app
ys
ears
with
in
hlet
man
oundary
y
without
biological,
y
electrical
[25
engineering
26
systems
Stabilit
and
and

y

Sc
applications,
ol
and
Electrical
in
T
man
A
y
Univ

y
dela
T
y
A
of
69978
Univ

de
is
ersit?
a
V

et
of
Hainaut
instabilit
br?sis,
y
V,

CNRS
In
Insti-
the
des

et
of
ec
distributed
of
parameter

systems,
-
ev

en
9
arbitrarily
rance,
small

dela
Univ
ys
de
in

the
du

Cam
k
LAMA
ma
FR
y
2956,
destabilize
tut
the
Sciences
system
T
(see
hniques
e.g.
V
[5
F-59313
,
V
16
Cedex
,
F
24
Julie.V
,

17
℄H
k.kH
(.,.) . A : D(A) → HH
1/2 1/2 1/2 ′H. V :=D(A ) A . D(A )
1/2D(A ) H.
i = 1, 2 Ui
k.kUi
1/2 ′(.,.) B ∈L(U , D(A ) ).U i ii

ω¨(t)+Aω(t)+B u (t)+B u (t−τ(t)) = 0, t> 0, 1 1 2 2
ω(0) =ω , ω˙(0) =ω ,0 1
 0u (t−τ(0)) =f (t−τ(0)), 0<t<τ(0),2
t ∈ [0,∞) τ(t) > 0 ω :
[0,∞)→ H ω˙ ω u ∈1
2 2L ([0, ∞), U ) u ∈L ([−τ, ∞), U )1 2 2
τ(t)
∃d< 1,∀t> 0, τ˙(t)≤d< 1,
∃M > 0,∀t> 0, 0<τ ≤τ(t)≤M.0
2,∞∀T > 0, τ ∈W ([0, T]).
ω
where
ternal
tly
in
acting
and
in
ts
b

satises

[19
b
in
e
example).
a
of
self-adjoin
ert
t
,
p
er,
ositiv
the
e
The
op
dela
erator
represen
with
,
a


b
in
22
v
time-v
erse
apuno
in
delays
t
y

order
Let
domain.
with
trol
systems
pap
olic
Ly
parab
e
linear
b
of
and
y
of
stabilit
time-v
The
b

state
b
time
e
means
the
y
domain
the
of
of
20
the
,
in
[17
In
in
22
found
as
Denote
of
b
ounded
y
with
e
that
b
stabilit

w
y
linear
dela

t


where
with
of
the
v
dual

space
een
of
ha
equations
time-v
e
equations
v
a
a
1-d
w
the
the
stabilit
for
is
obtained
dela
b
dela
y
applied
means
is
of
the
the
is
inner
ativ
pro
and


in
state

the

,
F
assume
urther,
6
for
the
t
an
Let
functions.
dela
dela
ys
linear
has

,
metho
let
via
b
21
een
in
b
and
e
with
a
stabilit
real
ks.
Hilb
un
ert
t
space
Moreo

e
h
olution
studied
of
stands
for
the
refer
t
v
2
of
to
mo
its
of
dual
distributed
space)
dela
with
e
norm
form
and
er
inner
this
pro
aim

functional.
denoted
apuno
resp
via
ectiv
[21
ely
studied
b
b
y
v
in
ys
and
arying
[8
oundary

with
in
e
and
v
the
w
y
heat
b
the
ely
ts
ectiv
time,
resp
y
denoted
The
,
for
and
the
let
arying

y
pro
ys
inner
oundary
and
to
norm
e
with
not
space
the
ert
of
Hilb
system,
real

a
the
e
deriv
b
e
frequency
this
Let
that
ork.
h
framew
ounded

is
W
ed
e
dela

on
the
erator
system
op
describ
that
ed

b
,
y
[3,
(1)

our
space,
t
Hilb
presen
are
us
input
let
The
on,
arying
going
y
Before
systems
equation).
of
e
(2)
v
the
a
d.
w
v
the
Ly
for


,
[21
6,
of
[3
results
analyzed
the
w
particular
(3)
in
time-varying
tains
PDEs

y
h
the


ks


b
y
in
dela
dela
arying

v
equation
time-
v
with
w
problems
assume
of
(4)

ev
large

quite
y
a
the
tain


to
to
e
order
er
in
Moreo
ossible
Most
p
the
as
equations
large
deling
as
vibrations
and


with
[19

to
with
similar
y
setting
b

written
an
the

(1),
to
is
will
for
b

e
eld.
iden
tiedui
∗u (t) = B ω˙(t)i i

∗ ∗ω¨(t)+Aω(t)+B B ω˙(t)+B B ω˙(t−τ(t)) = 0, t> 0, 1 21 2
ω(0) =ω , ω˙(0) =ω ,0 1
 ∗ 0B ω˙(t−τ(0)) =f (t−τ(0)), 0<t<τ(0).2

2 2∗ ∗∃0<α< 1−d, ∀u∈V, kB uk ≤αkB uk2 1U U2 1

(5)
system
system
and
op

lo
w

our
the
us
y
ell-p
a
follo
w
dissipativ
this
giv
in
ples,
The
that

Therefore

Hence
order
w
ev
is
olution
in
equations
e
without
3
delay
whic
or
particular
with
T

are
on-
system
stant
theory
delay
3
of
the
t
[9
yp
question
e
system.
(5)
e
ha
that
v
where
e
h
b
the
een
equations.
studied
if
in
sho

(6)
and
to
[19


,
resp
y
ectiv

ely
wledge,
.
of
In
ell-p
these
to
t
ell-p
w
e
o
from
pap
the
ers,
y
the
e
exp
ariable
onen
of
tial
10
stabilit
rst
y
the
(or
of
p

olynomial
will
stabilit
sucien
y)
guaran
is
system
sho
ell-p
wn,
e
under
the
some
elop


via
heat
an
v
observ
,
abilit
y
y
system
inequalit
W
y
in
for
the
solution
W
of



onding
form

in
ativ
inputs
e

system.

In
In
our
framew

e
for
our
time-v
the
arying
the
dela
rst
y
2
,
of
this
e
metho
w
d
is

F
not
ose,
obtain
semi-
guaran
an
that

energy
tro
ys.
v
further
dela
if
.
is
w
satised,
use
exist
v
where
norm
instabilities
hnique
y
Kato
ear
,
[17

20
the
27
natural
for
is
w
w
v
osedness
equation
this

In
dela
2
Hence
e
assumption
giv

a
a
t
step,
that
under
tees

this
w
(5)
pro
w
e
osed,
exp
w
tial

y
w
the

(5)
dev
y
ed
tro
[21
an
for
Ly
1-d
v
and
Moreo
a
er
e
e

e
w
dep
ma
of
ask

this
rate
is
resp
e.
to
e
dela
w
,

particular
that
e

w
e
if
sensors.
dela
actuators



onds
rate
h
This
,
the
k
ten
the
of
en
4.
are
w
the
nish
y
pap
from
b
in

problems,

man
dieren
examples
b
tees
e
the
applied

due
Note
to
that
the
(6)
loss
not
of
there
the

time
some
translation
ma
in
app
v
(see
ariance.
,
Hence
,
w

e
the
in
a
tro
e

with
new
t

y).
Ly
this
apuno
seems
v
In
functionals
third
with
again
exp
the
onen
(6),
tial
e
terms
v
and
the
an
onen
additional

term,
of
whic
system
h
b
tak
in
e

in
appropriate
to
apuno

functional.
t
v
the
w
dep
giv
endence
the
of
endence
the
the
dela
y
y
with
with
ect
resp
the
ect
y
to
in
time.
w
F
sho
or
that
the
the
treatmen
y
t
the
of
y
other

problems
is
with

Ly
t
apuno

v
Finally

e
hnique
this
see
er
[6
y
,
in
18
5
,
t
22
where


Moreo
ork
v
b
er,
applied.

o
trary
kno
to
all
[17
exam-
,
with
19


the
the
one,
existence
new.
results
W
do
osedness
not
the
follo
W
w
aim
from
sho
stan-
that
dard
(5)
semi-group
w
theory
osed.
b
or
ecause
purp
the
w
spatial
use
op
group
erator
and
dep
idea
ends
[17
on
Let
time
in
due

to
auxiliary
the
ariable
time-v
arying∗z(ρ, t) = B ω˙(t−τ(t)ρ) ρ ∈ (0, 1) t > 0 z2

∂z ∂zτ(t) +(1−τ˙(t)ρ) = 0, 0<ρ< 1, t> 0 ∂t ∂ρ
∗z(0, t) =B ω˙(t)2 ∗ 0z(ρ, 0) =B ω˙(−τ(0)ρ) =f (−τ(0)ρ).2

∗ω¨(t)+Aω(t)+B B ω˙(t)+B z(1, t) = 0, t> 0, 1 21 ∂z ∂zτ(t) +(1−τ˙(t)ρ) = 0, t> 0, 0<ρ< 1,
∂t ∂ρ
0 ω(0) =ω , ω˙(0) =ω , z(ρ, 0) =f (−τ(0)ρ), 0<ρ< 1,0 1 ∗z(0, t) =B ω˙(t), t> 0.2
TU := (ω, ω˙, z) ,
U
T
τ˙(t)ρ−1∂zT′ ∗U = (ω˙, ω¨, z˙) = ω˙,−Aω(t)−B B ω˙(t)−B z(1, t), .1 21
τ(t) ∂ρ

′U =A(t)U
0U(0) = (ω , ω ,f (−τ(0).)),0 1
A(t)
   

∗   −Aω−B B u−B z(1)A(t) u = 1 2 ,1
τ˙(t)ρ−1 ∂zz
τ(t) ∂ρ
1 ∗ ∗D(A(t)) :={(ω, u, z)∈V×V×H ((0, 1), U ); z(0) =B u, Aω+B B u+B z(1)∈H}.2 1 22 1
A(t) t
D(A(t)) =D(A(0)), ∀t> 0.
2H =V ×H×L ((0, 1), U )2
   * + Zω ω˜ 1
1 1
    2 2u , u˜ = A ω, A ω˜ +(u, u˜) + (z(ρ), z˜(ρ)) dρ.H U2H 0z z˜
where
t
the
alen
e
equiv
dened
No
with
w,
Consequen
w
note
e
enden
in
(7)
tro
rewritten
is
in
4
Note
Hilb
the
ert
W
space
y
(5)
time
system
tion
the
ev
Therefore,
the
equation
b
ort
ma
(10)
the
i.e.
and
,
satises
transp

time
that
the
e
of
is
w
(9)
t
b
enden
t
indep
dep
is
the
wing
(8)
equipp
equa-
ed
olution
with
order
the
rst
usual
as
inner
e
pro
to

y
(11)
(5)
If
system
erator
tly
op
for
erator
.
op
that
domain
satises
follo
tro
of
then
domain
the

the{A,H,Y} A = {A(t) : t ∈ [0, T]}
T > 0 Y = D(A(0))
Y =D(A(0)) H
t∈ [0, T] A(t) H
A = {A(t) : t ∈ [0, T]} C
m t (S (s)) A(t)t s≥0
mskS (s)uk ≤Ce kuk u∈H s≥ 0t H H
∞∂ A L ([0, T], B(Y, H))t ∗
[0, T] B(Y, H)
Y H
1U ∈C([0, T],Y)∩C ([0, T],H)
Y
√ 2 2∗ ∗∃0<α≤ 1−d, ∀u∈V,kB uk ≤αkB uk ,2 U 1 U2 1
d α d< 1
∗ ∗X ={u∈V : B B u+B B u∈H} H1 21 2
D(A(0)) H.
⊤(f, g, h) ∈H D(A(0)),
   * + Zω f 1
   0 = u , g = (ω, f) +(u, g) + (z(ρ), h(ρ)) dρ,V H U2
0z h
⊤(ω, u, z) ∈D(A(0)).
⊤ω = 0 u = 0 z ∈ D((0, 1),U ) (0, 0, z) ∈2
D(A(0)),
Z 1
(z(ρ), h(ρ)) dρ = 0.U2
0
2D((0, 1), U ) L ((0, 1), U ), h = 0.2 2
triplet
or
sential
[1
d
,
w

dense
Theorem
een
2.1
W

uniqueness
Assume
system,
that
that
(i)
yp
l
ongly
al
ac
for
and
(iii)
v
holds,
,
(10)
forms
(ii)
10
,
es
is
fr
of
for
subset
into
dense
functions
a
b
where
quivalent
ates
elop
is
e
giv
and
en
to
b
(iv)
y
to
(2).
the
Note
some
that

the
d

,
hoice
oup
of
and
a
theory
is
er
p
b
ossible
of
since
set
str
has
ongly
om

asur
b
d,
y
all
(2).
of
The
of
follo
the
ontinuous
using
[10
rst
giv
theory
es
,
a
The
sucien
y
t
b

.
to
existence
obtain
results
(i):
al
Lemma
w
2.2
e
Assume
with
that
and
semigr
CD-system
oup
satises
on
[9,
of
ate
(12)
More
that
follo
ose
theorem
supp
the
us
endent

Let
e
(8).
onstants
system
om
for
ators
assumptions
op
e
ounde
v
of
o
equations
ab
t
the
e
k
the

(8)

b
to
dev
is
fr
dense
able
in
me
then
str
.
ounde
Then
for
(13)
ly
is
es-
goal

Our
e
.
e
and
sp
in
,
data
ed
is
semi-
dense
e
in
tak
initial
group
any
[9,
Pro
10
of.
23
Let

for
simplest
the
a
family
to
2.13
elongs
Theorem
pro
also
e
and
As

),
[9
and
of
and
b
is
e
l
orthogonal
for
to
sho
all
that
elemen
w
ts
get
of
,
A
for
1.9
xed
Theorem
,
see
a
of
(or
pro
t
(for
domain
results
see
namely
by
uniqueness
Since
and
gener
is

stable
precisely
with
the
gener
wing
stability
is

in
solution
giv
unique
semigr
a
(i.e.
has
of
(8)
indep
oblem
some
pr
w
Then,

.
existence
general
into
wing
5
lemma∗ω = 0 z = B u u ∈ X2
∗ T(0, u, B u) ∈ D(A(0)) (u, g) = 0 u ∈ X XH2
H g = 0
⊤0 = (ω, f) ,∀(ω, u, z) ∈D(A(0)).V
u = 0 z = 0,
⊤(ω, f) = 0, ∀(ω, 0, 0) ∈D(A(0)).V
⊤(ω, 0, 0) ∈ D(A(0)) ω ∈ D(A)
D(A) V < ., . >V
f = 0
∗ ∗ ∗ker(B ) B X ker(B )1 1 1
H D(A(0)) H
A(t) C H0
H
   * + Zω ω˜ 1
1 1
    2 2u , u˜ = A ω, A ω˜ +(u, u˜) +qτ(t) (z(ρ), z˜(ρ)) dρ,H U2H 0z z˜
t
q
1 2 1
√ ≤q≤ − √
α1−d 1−d
k.k . q 0<α≤t√
1−d
H
U ∈D(A(t))0
1U ∈C([0, +∞), D(A(t)))∩C ([0, +∞),H)
ckφk |t−s|t 2τ0≤e , ∀t, s∈ [0, T],
kφks
is
enden
es
time-dep
(12)
wing

follo
usual
the
easily

.
tro
e
in

e
Theorem
w
if
ose,
e
purp

that
to
or
pro
F
step,
solution.
This
unique
alen
a
(11)
has
assumptions
(5))
an
then
that
(and
exists
(8)
e
system
to
that
v
e
w
v
of.
pro
b
will
therefore
e
,
w
in

y
from
inner
Kato

of
to
hnique
pro

Since
norm
Under
ariable
(3),
v
(13),
the
datum
using
and
6

t
ther
and,
unique
in
But
-semigroup
and
a

generates
then
erator
orthogonalit
op
ab
the

that
yp
w
(8).
sho
e
will
(15)
e
dense
w
for
w,
that
No
,
.
y
in
a
dense
dense
is
b
then
(12).
,
new
in
pro
dense
is
is
equiv
where
t
if
the
is
inner
a

p
on
ositiv
.
e
2.4

the
t
(2),

(4),
hosen
and

for
h
initial
that
.
(14)
only
,
if
in
k


is
,
of
e
ernel
a
k
solution
the
w
(12),
obtain
y
w
b
to
As,
ourselv
2.3
By
Remark

.
is
that
y
with
e
asso
o

The
norm
that
denoted
e
b
othesis,
y
h

system
e
Pro
w
W
),
rst

that
This
y

in
on

is
inner
.
hoice
As
of
all
pro
,
is
and
p
see
ossible
w
since
and
inner
taking
the
b
with

ed
In
(equipp
b
y⊤φ = (ω, u, z) c s, t∈ [0, T]
2c c 12 2 |t−s| |t−s| 2
τ τ0 0 2kφk −kφk e = 1−e A ω +kuk t s H
H
Z 1
c |t−s| 2
τ0+q τ(t)−τ(s)e kz(ρ)k dρ.U2
0
c c|t−s| |t−s|
τ τ0 01−e ≤ 0 τ(t)−τ(s)e ≤ 0 c> 0
τ(t) =τ(s)+τ˙(a)(t−s), a∈ (s, t),
τ(t) |τ˙(a)|
≤ 1+ |t−s|.
τ(s) τ(s)
τ˙ c> 0
cτ(t) c |t−s|
τ0≤ 1+ |t−s|≤e ,
τ(s) τ0
A(t) t > 0
⊤U = (ω, u, z) ∈D(A(t))
   * +u ω
∗ −Aω−B B u−B z(1)   1 2hA(t)U, Ui = 1 , ut
τ˙(t)ρ−1 ∂z zτ(t) ∂ρ t
1 1 ∗
2 2= A u, A ω −(Aω +B B u+B z(1), u)1 21 H
H Z 1
∂z
−q (ρ), z(ρ) (1−τ˙(t)ρ)dρ.
∂ρ0 U2
∗Aω +B B u+B z(1)∈H,1 21

1 1 ∗2 2hA(t)U, Ui = A u, A ω −hAω, ui −hB B u, ui −hB z(1), ui′ 1 ′ 2 ′1t V ,V V ,V V ,V
H Z 1
∂z
−q (ρ), z(ρ) (1−τ˙(t)ρ)dρ
∂ρ0 U2
2∗ ∗= hAω, ui −hAω, ui −kB uk −(z(1), B u)′ ′ U1 2 2V ,V V ,V U1Z 1 ∂z
−q (ρ), z(ρ) (1−τ˙(t)ρ)dρ,
∂ρ0 U2
ρ
Z Z1 1 ∂z 1 ∂ 2
(ρ), z(ρ) (1−τ˙(t)ρ)dρ = kzk (1−τ˙(t)ρ)dρU2∂ρ 2∂ρ0 0U2 Z 1τ˙(t) 12 2
= kzk dρ+ kz(1)k (1−τ˙(t))U U2 22 20
1 2∗− kB uk .2 U22
translation
e
that
ha
a
w
dissipativ
b
where
e
t.
v
whic
and
up
where
b
Indeed,

obtain
ositiv
e
ounded
w
pro
,
y
.
.
in
that
parts
a
y
h
b
ak
tegrating
and
us,
therefore,
(4),
and
in
for
By
e
By
v
W
h
.
(3),
y
is
e
Moreo
dualit
e
y
to
b
a
note
for
some
xed
for
that
Since
.
.
T
Then
e
no
exists
w
is
e
there
W
p
(15).
e
es

w
pro
Indeed,
v
all
e
obtain
,
is
v
er
7
thq q2 2 2∗ ∗ ∗hA(t)U, Ui = −kB uk −(z(1), B u) − kz(1)k (1−τ˙(t))+ kB ukUt 1 U 2 2 U 2 U1 2 22 2Z 1qτ˙(t) 2
− kzk dρ.U22 0

α qα 1−d q(1−d)2 2∗hA(t)U, Ui ≤ √ + −1 kB uk + − kz(1)k +κ(t)hU,Ui ,t 1 U U t1 22 2 22 1−d
2 1/2(τ˙(t) +1)
κ(t) = .
2τ(t)

qα q(1−d)α 1−d√ + −1≤ 0 − ≤ 0 q2 2 22 1−d
hA(t)U, Ui −κ(t)hU, Ui ≤ 0,t t
˜A(t) =A(t)−κ(t)I
12 2τ¨(t)τ˙(t) τ˙(t)(τ˙(t) +1)
κ˙(t) = − [0, T]1 22τ(t)22τ(t)(τ˙(t) +1)2
T > 0
 
0
d
 0A(t)U =
dt τ¨(t)τ(t)ρ−τ˙(t)(τ˙(t)ρ−1)
z2 ρτ(t)
τ¨(t)τ(t)ρ−τ˙(t)(τ˙(t)ρ−1) [0, T]2τ(t)
d ∞˜A(t)∈L ([0, T], B(D(A(0)), H)),∗dt
[0, T] B(D(A(0)), H)
λI−A(t) t> 0 λ> 0.
T T(f, g, h) ∈H. U = (ω, u, z) ∈D(A(t))
   
ω f
   u g(λI−A(t)) =
z h
ok
for
ounded
all
(3)
(b
satises
in
(12),
to
functions
y
b
(3)
e
and
where
(4))
us
and
solution
w
b
e
inequalit
ha
This
v
and
e
e
with
Observ
of
e
.
y
Let
(4).
us
b
space
By
or
strongly
alen
essen
8
of
e

that
equiv
the
(14).
er
since
v
that
Moreo
W
e.
lo
dissipativ
for
is
(16)
is
nd
surjectiv
w
e
and
for
(18)
a
Th
xed
and
erator
y
op
is
the
of
and
from
an
measurable
y
ounded,
that
tially
Therefore
b
Y
on
whic

Let
on
(17)
ounded
that
oung's
ws
sho
means
h
no
equiv
w
tly
pro
v
λω−u =f
∗λu+Aω +B B u+B z(1) =g1 21
 1−τ˙(t)ρ ∂zλz + =h.τ(t) ∂ρ
ω
u =−f +λω∈V.
z z
1−τ˙(t)ρ∂z
λz + =h
τ(t) ∂ρ
∗ ∗ ∗z(0) = B u = −B f +λB ω. z2 2 2
Z ρ
∗ −λτ(t)ρ ∗ −λτ(t)ρ −λτ(t)ρ λτ(t)σz(ρ) =λB ωe −B fe +τ(t)e e h(σ)dσ,2 2
0
τ˙(t) = 0
λτ(t) λτ(t)
∗ ln(1−τ˙(t)ρ) ∗ ln(1−τ˙(t)ρ)τ˙(t) τ˙(t)z(ρ) = λB ωe −B fe2 2Z ρ
λτ(t) λτ(t)h(σ)ln(1−τ˙(t)ρ) − ln(1−τ˙(t)σ)τ˙(t) τ˙(t)+τ(t)e e dσ,
1−τ˙(t)σ0
ω
z u τ˙(t) = 0
∗ −λτ(t) 0z(1) =λB ωe +z ,2
R10 ∗ −λτ(t) −λτ(t) λτ(t)σz =−B fe +τ(t)e e h(σ)dσ U22 0
f h,
λτ(t) ln(1−τ˙(t))∗ 0τ˙(t)z(1) =λB ωe +z ,2
λτ(t) λτ(t) R λτ(t)ln(1−τ˙(t)) ln(1−τ˙(t)) 1 h(σ) − ln(1−τ˙(t))0 ∗ τ˙(t) τ˙(t) τ˙(t)z =−B fe +τ(t)e e dσ2 0 1−τ˙(t)σ
U f h2
ω. ω
2 ∗ ∗λ ω +Aω +λB B ω +B z(1) =g +B B f +λf,1 2 11 1
2 ∗ −λτ(t) ∗ ∗ 0λ ω +Aω +λB B ω +λe B B ω =g +B B f +λf −B z =:q,1 2 1 21 2 1
′q∈V τ˙(t) = 0
λτ(t) ln(1−τ˙(t))2 ∗ ∗ ∗ 0τ˙(t)λ ω+Aω+λB B ω+λe B B ω =g+B B f +λf−B z =:q,1 2 1 21 2 1
ust
w
of
particular,
ose
In
b
.
to
us
t
with
that
erties,
and
th
en
and
the
ha
dep
e
is
w
ending
and
e
plicitely
if
ex-
y
is
the
Therefore
(19)

remains
Then,
Supp
w
only
e
a
oundary
xed
ha
elemen
v
,
where
on
.
y
y
otherwise
regularit
and
otherwise.
v
e
b
prop
(21)
W
m
e
(19),

where
then
nd
determine
nd
.
It
Indeed
.
satises
and
the
on
appropriate
ending
dieren
is
This
of
means
elemen
that
a
where
xed

t
appropriate
(20)
is
dep
tial
only
,
(20),
if
and
equation
w
e
and,
,
(21)
y
ha
v

and
if
the
b
b
e
found
found
with
e,
satisfy
,
giv
9
By′q ∈ V τ˙(t) = 0
h.,.i φ∈V′V ,V
D E
2 ∗ −λτ(t) ∗λ ω +Aω +λB B ω +λe B B ω, φ =hq, φi .′1 21 2 V ,V′V ,V


2 ∗ −λτ(t) ∗λ ω +Aω +λB B ω +λe B B ω, φ1 21 2 ′V ,V
2 ∗ −λτ(t) ∗=λ hω, φi +hAω, φi +λ(hB B ω, φi +e hB B ω, φi )′ ′ ′ ′1 2V ,V V ,V 1 V ,V 2 V ,V
1 12 ∗ ∗ −λτ(t) ∗ ∗
2 2=λ (ω, φ) + A ω, A φ +λ((B ω, B φ) +e (B ω, B φ) )1 1 2 2H U U1 2H
ω∈V ⊂H

1 12 ∗ ∗ −λτ(t) ∗ ∗2 2λ (ω, φ) + A ω, A φ +λ((B ω, B φ) +e (B ω, B φ) )1 1 2 2H U U1 2H
=hq, φi , ∀φ∈V.′V ,V
V.