La lecture à portée de main
Découvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDécouvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDescription
Informations
Publié par | profil-zyan-2012 |
Nombre de lectures | 33 |
Langue | English |
Extrait
1−d 1−d
1−d
,
1-d
applications.
net
on,
w
e
orks
equation
with
for
a
,
dela
on
y
of
term
this
in
to
the
1]
no
wn
dal
℄
,
ks
atten
Serge
W
transmission
Julie
T
V
orks
alein
ab
Univ
13
ersit?
[20
de
engineering,
V
is
et
,
du
Hainaut
of
Cam
,
br?sis
orks
LAMA
y
V,
the
FR
v
CNRS
in
2956
w
Institut
v
des
wledge,
Sciences
w
et
et
T
some
ec
er.
hniques
11
of
24
V
for
28
F-59313
biological,
-
V
urthermore
ell
Cedex
they
9
instabilities
F
,
rance
,
Julie.V
on
impro
p
Octob
system
er
24,
trol
2006
net
pa
In
of
this
see
pap
℄
er
w
here
e
the
dela
the
oundary
w
of
a
v
v
equation
e
w
equation
our
on
analysis
1-d
to
net
net
w
not
orks
Before
with
us
a
and
dela
of
y
e
term
,
in
12
the
14
b
26
oundary
details.
and/or
transmission
,
,
W
for
e
electrical
rst
or
sho
w
F
the
it
w
w
ell
kno
p
that
osedness
of
some
the
[17
problem
18
and
19
the
30
25
y
or
of
the
an
trary
appropriate
v
energy
the
.
W
the
e
[28
giv
1].
e
tly
a
problems
and
1-d
sucien
w
t
are
ying
that
tion
guaran
man
tees
authors,
the
[22
16
y
and
to
references
zero
there.
of
e
the
in
energy
estigate
.
eect
W
time
e
y
further
b
giv
and/or
e
stabilization
sucien
the
t
a
e
that
in
lead
e
to
net
exp
orks.
onen
o
tial
kno
or
the
p
of
olynomial
eect
stabilit
a
y
the
of
w
the
is
solution.
y
Some
done.
examples
going
are
let
also
on
denitions
net
notations
orks
out
in
Stabilization
whole
giv
w
en.
used
1
the
In
pap
tro
W
refer
Time
[2
dela
3,
y
,
eects
,
arise
,
in
,
man
,
y
℄
more
problems,
1
seen1−d R R n≥ 1
N[
R = ej
j=1
e (0, l ), l > 0,j j j
k =j, e ∩ej k
e ej j
u : R−→R, u =u uj |ej
ej
E ={e ; 1≤j ≤N} R Vj
R. v,
E = {j∈{1,...,N};v∈e }v j
v (E ) = 1, vv
(E ) ≥ 2, v Vv ext
V v∈V Eint ext v
j .v
Vext
cV =D∪N ∪V .ext ext
D
N
c cV V Vintext int
Vc
c cV =V ∪V .c int ext
1D =∅ H
2 2∂ u ∂ uj j (x, t)− (x, t) = 0 0<x<l , t> 0, 2 2 j∂t ∂x ∀j∈{1,...,N}, u (v, t) =u (v, t) =u(v,t) ∀j, l∈E , v∈V ,t> 0, j l v int X ∂u (v) (v)j ∂u ∂u (v, t) =−(α (v, t)+α (v, t−τ )) ∀v∈V ,t> 0,v c 1 2∂nj ∂t ∂t j∈EvX
∂uj c(v, t) = 0 ∀v∈V \V ,t> 0,int int∂nj j∈Ev u (v,t) = 0 ∀v∈D,t> 0, jv ∂u jv (v,t) = 0 ∀v∈N,t> 0, ∂nj v ∂u (0) (1) u(t = 0) =u , (t = 0) =u , ∂t ∂u 0(v, t−τ ) =f (t−τ ) ∀v∈V , 0<t<τ ,v v c vv∂t
form
of
w
e
no
de,
e
b
W
me
norm.
a
a
no
ecomes
v
b
F
semi-norm
b
the
or
that
a
so
that
;
dene6
set
that
is
ose
is
supp
edges
also
v
e
of
W
of
namely
.
des,
e
no
e
trolled
no
of
either
set
the
the
wher
y
set
b
oundary
denote
W
e
in
w
if
2
exterior
v
If
F
ving
osed.
set
imp
let
e
a
b
will
the
of
transmission
the
k
e
the
a
the
where
function
,
).
of
the
subset
(her
a
or
x
le
further
extr
e
or
W6
.
and
of
that
des
is
no
initial/b
the
of
at
1.1
oundary
the
b
e
k
de.
terior
a
an
nally
and
while
of
no
des
an
no
the
ertex.
at
as
ha
oundary
of
b
the
Neumann
b
;
ertex
of
xed
des
or
no
of
the
v
at
set
y
oundary
and
b
edges
hlet
set
y
ose
denote
imp
W
will
edge
e
to
w
Clearly
set
:
w
of
a
partition
F
a
of
x
w
ans
no
e
e
de
W
a
y
vertex
b
d
denoted
al
is
emity
of
ommon
t
a
elemen
empty
single
is
the
for
,
our
interval
ulate
identify
or
with
F
we
des.
no
problem:
terior
e
in
by
of
d
set
,
the
(1)
and
d
alue
onne
a
network
des
is
no
A
exterior
Denition
of
set
or
shortness,(v) (v)
α ,α ≥ 0 τ > 0v1 2
∂uj(v,·)
∂nj
u vj
u ej j
∂u 0(v, t−τ ) =f (t−τ ) v∈V , 0<t<τv v c vv∂t
(v)
α = 0 v ∈ Vc2
(v)
α = 01
(v) (v)
α α1 2
(v) (v)
α ≤α ,∀v∈V ,c2 1
(v) (v)
α <α ,∀v∈V ,c2 1
at
y
due
W
to
for
the
dela
d
y
use
equation.
In
appropriate
the
do
absence
it
is
of
Here
dela
b
y
rates
,
t
i.
initial
e.,
e
normal
a
ard
If
w
the
out
problem
the
that
means
and
abilit
for
to
all
of
xed
5
e
giv
b
y
to
w
,
y
the
that
ab
deriv
o
.
v
e
e
problem
t
has
of
b
v
een
w
not
b
in
y
generalit
some
authors
t
in
the
some
energy
particular
the
situations,
the
for
v
abilit
Ammari
metho
and
lik
T
form
ucsnak
a
℄
domain
Ammari,
non
Henrot
.
and
a
T
results.
ucsnak
a
[4
exp
℄
Similarly
Ammari
℄
and
alue
Jellouli
denotes
[5
e
,
v
6
for
℄
w
Ammari,
of
Jellouli
.
and
giv
Khenissi
and
℄
for
and
to
Xu,
energy
Liu
ab
and
Liu
not
[29
℄
do
In
y
these
do
pap
estigate
ers,
its
some
but
sucien
a
t
w
ts
but
are
the
giv
Remark
en
p
in
of
order
Our
to
based
guaran
of
tee
estimates
some
without
stabilities
e
of
the
these
system.
estimates
On
frequency
the
The
trary
the
,
represen
if
[16
osed
ma
supp
oid
also
the
is
d
y
quite
dela
the
the
that
a
is
and
if
estimate
w
stabilit
e
ha
giv
v
and
e
for
only
tial
the
our
dela
6
y
a
part
assuming
in
in
the
v
b
an
oundary/transmission
ativ