Dirihlet stabilization for the wave equation
Kim Dang PHUNG
1
Introdution and main results
The purpose of this note is to establish a polynomial deay rate for an unstable stabilization problem.
We onsider the Dirihlet boundary stabilization of the wave equation.
Let be a bounded onneted domain in Rn with a smooth boundary ( C 1 having no ontats
of innite order with its tangents ). Let S be a smooth hypersurfae in dividing the boundary into
two open sets
and + suh that = [ S [ + . Let be a ontinuous non negative funtion
on suh that (x) = 0 for all x 2
and (x) > 0 for all x 2 + and C 1 . We suppose that near
any point xo of S , there exists a loal hange of oordinates (y; xn ), y 2 Rn 2 , xn 2 R suh that +
is dened by xn > 0 and
p
= a (y; xn ) x+n =2 ,
a 2 C1,
a > 0,
x+n = sup (xn ; 0) .
Here, 2 (0; +1) is independent of xo 2 S . The Dirihlet boundary stabilization problem of the wave
equation is desribed by
8
<
:
t2 u u = 0 in (0; +1)
t u + n u = 0 on (0; +1)
u (; 0) = u0; t u (; 0) = u1 in .
with (u0; u1 ) 2 H = H 1 (
) L2 (
). This system is well-posed.
8 (u0 ; u1) 2 H 9!u 2 C 0 [0; +1) ; H 1 (
) \ C 1 [0; +1) ; L2 (
)
system.
solution of the above wave
Suh result is obtained by Hille-Yosida theorem applied to the unbounded operator A on the Hilbert
spae H,
A = 0 I0
D (A) = (u0 ; u1) 2 H; (u1; u0 ) 2 H; u1 + n u0 j = 0 .
Let us introdue Ho = (u0 ; u1 ) 2 H; u0 j = 0 a losed subspae of H and notie that
8 (u0 ; u1) 2 D (A) \ Ho 9! (u; tu) 2 C 0 ([0; +1) ; D (A) \ Ho ) \ C 1 ([0; +1) ; Ho )
the above wave system.
Moreover, posing E (t) =
1R
2 jt uj2 + jruj2
8 (u0 ; u1 ) 2 D (A)
the energy of this system, we get
2
p
d
E
(t) +
n u = 0 .
dt
Z
1
solution of
Also, posing E 0 (t) =
1 R 2 u2 + jr uj2
t
t
2 8 (u0; u1 ) 2 D (A)
, we have
E 0 (t1 ) E 0 (t2 ) =
p
Z t2 Z
t1
2
t n u .
The following result has been proved by G.Lebeau [ L℄:
First, reall the following denition. Let be an open subset of and T > 0. We say that ( ; T )
has the geometri ontrol property if any ray meets (0; T ) in a non-diratif point.
Theorem [ L℄.-
1)
Suppose that
2 (0; 1℄
ontrol property. Then there exist
+ ; T ) has the
and that there exists T > 0 suh that (
; > 0 suh that for every data (u0 ; u1) 2 Ho , one has
E (t) e tE (0)
2) Suppose
that
2 (1; +1).
Then for every
geometri
.
" > 0, T > 0,
there exists data
(u0 ; u1 ) 2 Ho ,
suh that
E (0) = 1 and E (T ) 1 " .
We prove the following result:
Theorem .-
ontrol property. Then there exists
2
T >0
b
+
suh that ( ; T ) has the geometri
> 0 suh that for every data (u0 ; u1) 2 H01 (
) L2 (
), one has
E (t) p1+ t E (0) .
Suppose that there exist
and
Proof of Theorem
2.1
step 1: polynomial deay for
E
0
( t)
We need the following result:
Proposition .-
ontrol property. Then there exists
T > 0 and
b
+
suh that ( ; T ) has the geometri
> 0 suh that for every data (u0; u1 ) 2 D (A) \ Ho , one has
E 0 (t) p1+ t [E (0) + E 0 (0)℄ .
Suppose that there exist
The proof of suh Proposition is redued to the existene of a onstant > 0, suh that for any initial
data (u0 ; u1 ) 2 D (A) \ Ho , and any > 0, > 0, we have
Z +T
Z +T
E 0 (t) dt [E (0) + E 0 (0)℄ + 1
2
d 0
E (t) dt
dt
!
.
To do this, we begin to study the following wave equation. Let z = t2 u, it solves
t2 z z = 0 in (0; +1)
z = t n u on (0; +1) .
Let > 0. We deompose the solution z as follows z = ' + where
8
<
:
t2 ' ' = 0 in (; + T )
' = 0 on (; + T )
' (; ) = u (; ) ; t ' (; ) = t u (; ) in ,
8
<
:
t2
= 0 in (; + T )
= t n u on (; + T )
(; ) = 0; t (; ) = 0 in .
We apply to the solution ' the program developed in the work of C.Bardos, J.Rauh and G.Lebeau
in order to get 9 > 0
k'kL2 (
(; +T )) kn 'kH 1 ( (; +T ))
kn z kH 1 ( (; +T )) + kn kH
1(
(; +T )) .
By a regularity result established by I.Lasieka, J.-L.Lions and R.Triggiani for the wave equation, we
have 9 > 0
k kL2 (
(; +T )) + kn kH 1 ( (; +T )) k kL2( (; +T ))
kz kL2 ( (; +T )) .
We onlude that
9 > 0
kz kL2 (
(; +T )) kn z kH
1(
(; +T )) + kz kL2 ( (; +T )) .
On another hand, n z = 1 t z on (; + T ) where b + whih implies that
kz kL2 (
(; +T )) kz kL2 ( (; +T ))
that is
9 > 0 8 > 0 8 (u0 ; u1 ) 2 D (A) \ Ho ,
2 u 2
t L (
(; +T )) kt n ukL2 ( (; +T ))
9 > 0
.
9 > 0 8; > 0 8 (u0 ; u1 ) 2 D (A) \ Ho ,
2
p
2
krt ukL2 (
( +T=3; +2T=3)) [E (0) + E 0 (0)℄ + 1 t n uL2 (
Now, we laim that
+ (; +T ))
.
Indeed, let 2 C01 ((; + T )) be suh that 0 1 and = 1 in [ + T=3; + 2T=3℄. As
2 t u = 2 t3 u in and t u = n u on , we get using Green formula
R
2 r
t u rt u
=
=
R
R
2 u u
2
n t t
R R
2 3t ut u +
2
t ut u
n t un u
Integrating over (; + T ), then integrating by parts, we have
R +T R
R
R
jrt uj2 = +T t2 u 20 t u + 2 t2u
.
p
p
R +T R
+ 2 n t u n u
R
R +T t2u2 + 2 kt ukL2 (
(;+T )) 0 t2 uL2 (
(; +T ))
p
p
.
+ n t u 2 +
n u 2 +
L ( (; +T ))
L ( (; +T ))
3
From whih we dedue that
Z +2T=3 Z
+T=3
9 > 0
jrt uj2 p
p
p
E 0 (0) + E (0)
n t u 2 +
.
L ( (; +T ))
This ends the proof of the proposition.
2.2
step 2: polynomial deay for
We begin to hek that
E (t)
8 (u0 ; u1 ) 2 H01 (
) L2 (
) 9! (y0 ; y1 ) 2 H (; ) H01 (
)
8
>
>
<
>
>
:
y0 = u1 in y0 = 0 on
n y0 = 0 on +
y1 = u0 in where H (; ) = v 2 H 1 (
) ; v 2 L2 (
) . In partiular, we have y1 + n y0 = 0 on .
Now, let y be the solution of
8
<
:
t2y y = 0 in (0; +1)
t y + n y = 0 on (0; +1)
y (; 0) = y0 ; t y (; 0) = y1 in .
Applying the last proposition, we obtain
R 2 2
y +
t
jrt yj2
9; d > 0
hR 2
2
d
p1+
t jy1 j + jry0 j +
R jy j2 + jry j2 .
p1+
0
1
t R jy0 j2 + jry1 j2
i
As u = t y, we get the desired estimate.
Referenes
[ L℄ G. Lebeau, Contr^ole et stabilisation hyperboliques, seminaire E.D.P. Eole Polytehnique, 1990.
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