Asymptoti ontrol of haos for a partial dierential equation
K.-D. PHUNG
kim-dang.phungmla.ens-ahan.fr
.We ontrol asymptotially the haoti behaviour of a innite-dimensional dynamial
system generated by a rst-order linear partial dierential equation desribed in [1℄.
Abstrat
1
Introdution and main result
For n 2 N nf0g , let Xn be the spae of funtions dened by
n
Xn = v 2 C n [0; 1℄ ; v (0) = v 0 (0) = = v (n) (0) = 0
o
,
and equipped with the topology of uniform onvergene with derivatives of order
norm
n
kvk = xmax
v
2 ;
( )
[0 1℄
n and with the
(x) .
Let fSt gt0 be the semiow on the spae Xn generated by the rst-order partial dierential equation
ut + xux = u for (x; t) 2 [0; 1℄ [0; +1) ,
(1.1)
where 2 R and with initial ondition u (; t = 0) in Xn
for x 2 [0; 1℄ .
u (x; 0) = v (x)
Then
St v (x) = u (x; t) = et v xe t .
In [1℄, the authors prove that if n, then for all v 2 Xn , lim kSt v k = 0 and that if > n, then
t!+1
for almost every v 2 Xn , the trajetory starting from v is strongly turbulent. Our motivation is to
ontrol the semiow fSt gt0 when > n.
Let f = f (x; t) 2 C ([0; +1) ; Xn ) denote the additive ontrol ation. Introduing f loally in spaetime variables into the innite-dimensional dynami system (1.1) yields
ut (x; t) + xux (x; t) = u (x; t) + f (x; t) (x)[0;2"℄ 1 (t)[0;T ℄
(1.2)
for (x; t) 2 [0; 1℄ [0; +1) and with initial ondition u (; t = 0) in Xn
for x 2 [0; 1℄ ,
u (x; 0) = v (x)
(1.3)
where T > 0, " 2 (0; 1=2), 1 (x)[0 ℄ is the harateristi funtion on [0; R℄ for some R > 0, i.e.
1 (x)[0 ℄ = 1 if x 2 [0; R℄, 1 (x)[0 ℄ = 0 if x 2= [0; R℄, and (x)[0;R℄ denotes a smooth funtion C 1 (R)
suh that (x)[0;R℄ = 1 if x 2 [ R=2; R=2℄, (x)[0;R℄ = 0 if x 2= [ R; R℄.
;R
;R
;R
1
The ontrol problem is to nd a ontrol funtion f suh that the solution u of system (1.2) satises
lim ku (; t)k = 0 .
t!+1
The main result of this note is as follows.
For all v 2 Xn the ontrol funtion f given by
1 t
e v xe t
f (x; t) =
T
implies that the solution u of the innite-dimensional dynamial system (1.2)-(1.3) satises
lim ku (; t)k = 0 .
Theorem .-
t!+1
2
Proof of Theorem
Consider the rst-order partial dierential equation with seond member F = F (x; t) 2 C ([0; T ℄ ; Xn )
ut + xux = u + F for (x; t) 2 [0; 1℄ [0; T ℄ ,
with initial ondition u (; t = 0) in Xn
u (x; 0) = v (x) for x 2 [0; 1℄ .
Then
u (x; t) = St v (x) +
and
Z t
0
St s F (x; s) ds = et v xe t +
u (x; T ) 1 (x)[0;"℄ = eT v xe
T
1 (x)
+
[0;"℄
If F (x; t) = T1 et v (xe t ) (x)[0;2"℄ then F xe (T
and
e s F xe (T s) ; s 1 (x)[0;"℄ = T1 v
= T1 v
Z T
0
Z t
0
e s F xe (T s) ; s
s) ; s =
1
T
e(t s) F xe (t s) ; s ds
1 (x)
!
[0;"℄
ds
.
es v xe (T s) e s xe (T s) [0;2"℄ ,
xe T xe (T s) [0;2"℄ 1 (x)[0;"℄
xe T 1 (x)[0;"℄ ,
beause when (x; s) 2 [0; "℄ (0; T ), xe (T s) 2 [0; "℄. We onlude that u (x; T ) = 0 for all x 2 [0; "℄.
Now, onsider the rst-order partial dierential equation for times t 2 (T; +1)
ut + xux = u for (x; t) 2 [0; 1℄ (T; +1) ,
with initial ondition u (; t = T ) in Xn
u (x; T ) = w (x) for x 2 [0; 1℄ ,
suh that w (x) = 0 for x 2 [0; "℄. Then
u (x; t) = e(t T ) w xe (t T )
and onsequently, for all (x; t) 2 [0; 1℄ ln e" ; +1 , xe (t T ) 2 [0; "℄ and u (x; t) = 0.
T
This ompletes the proof.
Referenes
[1℄ J. Myjak and R. Rudniki, Stability versus haos for a partial dierential equation. Chaos, Solitons
and Fratals 14 (2002) 607-612.
2