Note on the cost of the approximate controllability for the heat equation with potential

Kim Dang Phung

Description

Niveau: Supérieur, Licence, Bac+2
J. Math. Anal. Appl. 295 (2004) 527–538 Note on the cost of the approximate controllability for the heat equation with potential K.-D. Phung Received 2 September 2003 Available online 28 May 2004 Submitted by R. Triggiani Abstract We prove the approximate controllability for the heat equation with potential with a cost of order ec/? when the target is in H 10 (?) with a precision in L 2(?) norm. Also a quantification estimate of the unique continuation for initial data in L2(?) of the heat equation with potential is established. ? 2004 Elsevier Inc. All rights reserved. 1. Introduction and main results Throughout this paper, ? is a bounded domain in Rn, n 1, with a boundary ∂? of class C2, ? is a non-empty open subset of ? and T > 0 is a real number. Further, we denote ? · ? ∞ the usual norm in L∞(? ? (0, T )) and we consider a = a(x, t) a function in L∞(? ? (0, T )). In this paper we study the following heat equation with a potential a in L∞(? ?(0, T )): ? ? ? ∂tu ? ∆u + au = f · 1|? in ? ? (0, T ), u = 0 on ∂? ? (0, T

• ∂tu ?


• equation without


• let u0 ?


• estimate already


• ud ?


• e?? ·


• heat equation


• quantitative estimate


• approximate controllability

Sujets

• Heat equation

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 15
• Langue: English

Extrait

J. Math. Anal. Appl. 295 (2004) 527–538
www.elsevier.com/locate/jmaa
Note on the cost of the approximate controllability
for the heat equation with potential
K.-D. Phung
Received 2 September 2003
Available online 28 May 2004
Submitted by R. Triggiani
Abstract
We prove the approximate controllability for the heat equation with potential with a cost of order
c/ε 1 2e when the target is in H (Ω) with a precision in L (Ω) norm. Also a quantification estimate of0
2the unique continuation for initial data in L (Ω) of the heat equation with potential is established.
 2004 Elsevier Inc. All rights reserved.
1. Introduction and main results
nThroughout this paper, Ω is a bounded domain inR , n 1, with a boundary ∂Ω of
2class C , ω is a non-empty open subset of Ω andT> 0 is a real number. Further, we
∞denote · the usual norm in L (Ω × (0,T)) and we consider a = a(x,t) a function∞
∞in L (Ω × (0,T)).
∞In this paper we study the following heat equation with a potential a in L (Ω ×(0,T)):

∂ u − ∆u + au = f · 1 in Ω × (0,T), t |ω
u =0on ∂Ω × (0,T), (1.1)

u(·, 0) = u in Ω,0
2 1where f ∈ L (ω × (0,T)), u ∈ H (Ω) and 1 denotes the characteristic function of the0 |ω0
set ω.
2It is well known from [6] or [2] that we can act through f ∈ L (ω × (0,T)) when
2u ∈ L (Ω) in order to get the null controllability result u(·,T) = 0 and furthermore, the0
following estimate holds [3]: there exists a constant c > 0 such that0
E-mail address: [e-mail].
0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2004.03.059528 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538

1 2/3
f 2 exp c 1 + + T a + a u 2 . (1.2)0 ∞ ∞ 0L (ω×(0,T )) L (Ω)T
(1.2) is an explicit estimate with respect to both quantitiesT>0and a 0, of the∞
control function f and may be viewed as a measure of the cost of the null controllability
for the heat equation with potential.
Here we ask whether the following steering property for the heat equation with potential
holds when u = 0: there exist two constantsD>1and c = c(T,a )> 1 depending0 ∞
1on both quantitiesT>0and a 0 such that for allε> 0, for all u ∈ H (Ω) ,there∞ d 0
exists a suitable approximate control function f depending on ε such that
Du 1d H (Ω)
0f 2 c exp u 2 (1.3)dL (ω×(0,T )) L (Ω)ε
and

u(·,T) − u ε. (1.4)d 2L (Ω)
Our goal is to measure the cost of the approximate control function f and furthermore to
give an explicit estimate with respect to ε, T and a 0.∞
This problem has received a particular attention from [3] where it is proved that the cost
c/εof the approximate controllability for the heat equation with potential is of order e if
22 1 c/ε 1u ∈ H (Ω) ∩H (Ω) , and of order e if u ∈ H (Ω) .Butwhen a is a constant or mored d0 0
∞ ∞generally of the form a(x,t) = a (x) + a (t),where a ∈ L (Ω) and a ∈ L (0,T),the1 2 1 2√
1c/ ε 2order e if u ∈ H (Ω) ∩ H (Ω) is optimal [3, Theorem 6.2].d 0
Let us make the first observation: by simple changes of variables, we are reduced to
1check that for allε> 0, for allT> 0, for all w ∈ H (Ω) there exists a function ϑ suchd 0
that
D/εϑ 2 ce w 2 , (1.5)dL (ω×(0,T )) L (Ω)
and such that the following steering property holds:

w(·, 0) − w εw , (1.6)1d 2 d H (Ω)L (Ω) 0
where w is the solution of the following backward heat equation with potential:

∂ w + ∆w − aw = ϑ · 1 in Ω × (0,T), t |ω
w =0on ∂Ω × (0,T), (1.7)

w(·,T) =0in Ω.
Now let us consider ϕ the solution of the heat equation without control when the initial
2data ϕ ∈ L (Ω) :0

∂ ϕ − ∆ϕ + aϕ =0in Ω × (0,T), t
ϕ =0on ∂Ω × (0,T), (1.8)

ϕ(·, 0) = ϕ in Ω.0
Then by classical integrations by parts, we getK.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 529
T
ϑ(x,t)ϕ(x,t)dxdt =− w(x, 0)ϕ(x, 0)dx
ω0 Ω

= (−w(·, 0) + w )ϕ dx − w ϕ dx. (1.9)d 0 d 0
Ω Ω
Also suppose that the solution w of (1.7) exists with (1.6) and (1.5) then for allε> 0,
1for all w ∈ H (Ω) , we have using Cauchy–Schwarz inequalityd 0

D/εw ϕ dx εw 1 ϕ 2 + ce w 2 ϕ 2 . (1.10)d 0 d 0 dL (Ω) L (Ω) L (ω×(0,T ))H (Ω)
0
Ω
−1Consequently, we obtain choosing w = (−∆) ϕ thatd 0
D/εϕ −1 εϕ 2 + ce ϕ 2 , ∀ε> 0, (1.11)0 0H (Ω) L (Ω) L (ω×(0,T ))
or equivalently

ϕ 20 L (Ω)
ϕ c exp D ϕ if ϕ = 0, (1.12)2 20 0L (Ω) L (ω×(0,T ))ϕ −10 H (Ω)
or equivalently
D
ϕ −1 ϕ 2 , (1.13)0 0H (Ω) L (Ω) ϕ 0 21 L (Ω)ln 2 +
c ϕ 2L (ω×(0,T ))
where the values of the constants c = c(T,a )>1andD> 1 may changed from line∞
(1.11) to line (1.13) but not their dependence.
2(1.13) is a quantitative estimate for unique continuation for initial data in L (Ω) of
the heat equation with potential from an interior observation. This kind of logarithmic
estimate already appears in the context of the cost of approximate control and stabilization
for hyperbolic equation [5,7]. Here we will follow the strategy in [7] (see also [4]) to prove
that (1.13) implies an approximate result with an estimate (1.3) of the cost function.
The first main result of this paper is as follows.
Theorem 1. There exist two constants c ,c > 1 such that for all ε> 0, for allT> 0,1 2
∞ 1for all a ∈ L (Ω × (0,T)), for all u ∈ H (Ω) , there exists a control function f ∈d ε0
2L (ω × (0,T)) such that

D ∇ u 2d L (Ω)
f 2 E exp u 2 ,ε dL (ω×(0,T )) L (Ω)ε

u(·,T) − u ε, (1.14)d 2L (Ω)
2where u ∈ C([0,T ];L (Ω)) is the unique solution of the heat equation with potential and
control function

∂ u − ∆u + au = f · 1 in Ω × (0,T), t ε |ω
u = 0 on ∂Ω × (0,T), (1.15)

u(·, 0) = 0 in Ω,
530 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
andE> 1,D> 1 are given by

2 1c T a1 ∞D = D(T,a ) = c Te + ,∞ 1 T (1.16)2 2/3c T a2 ∞) = exp(c (1 + T a (1 + e )+ a )).E = E(T,a∞ 2 ∞ ∞
Of course we will also need to prove the estimate (1.13). Our second main result is as
follows.
Theorem 2. There exist two constants c ,c > 1 such that for all T> 0, for all a ∈1 2
∞ 2L (Ω × (0,T)), for all initial data u ∈ L (Ω) such that u = 0, the solution u of the0 0
homogeneous heat equation with potential

∂ u − ∆u + au = 0 in Ω × (0,T), t
u = 0 on ∂Ω × (0,T), (1.17)

u(·, 0) = u in Ω,0
satisfies
D
u u , (1.18)−1 20 0H (Ω) L (Ω) u 0 21 L (Ω)ln 2 + E u 2L (ω×(0,T ))
whereE> 1,D> 1 are given by

2c T a 11 ∞D = c Te + ,1 T
2 2/3c T a2 ∞E = exp(c (1 + T a (1 + e )+ a )).2 ∞ ∞
Remarks. (1) Note that E(T, 0) is a constant not depending onT> 0.
(2) An application of Theorem 1 to get a space of exact controllable target data may be
possible following [7] based on properties of Riesz basis (see [8]).
The plan of the paper is as follows. In Section 2 we prove Theorem 1 as an application
of Theorem 2. Section 3 contains the proof of Theorem 2. Finally in the last section some
comments are added.
2. Proof of Theorem 1
Theorem 1 is easily deduced from the following result.
Theorem 3. There exist two constants c ,c > 1 such that for all ε> 0, for allT> 0,1 2
∞ 1for all a ∈ L (Ω × (0,T)), for all w ∈ H (Ω) , there exists a control function ϑ ∈d ε0
2L (ω × (0,T)) such that

D
ϑ exp w ,2 2ε dL (ω×(0,T )) L (Ω)ε

w(·, 0) − w εw 1 , (2.1)d 2 dL (Ω) H (Ω)0
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 531
2where w ∈ C([0,T ];L (Ω)) is the unique solution of the backward heat equation with
potential and control function

−∂ w − ∆w + aw = Eϑ · 1 in Ω