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Global Carleman inequalities for the wave
equations and applications to controllability
and inverse problems
Jean-Pierre Puel
1
1
J.-P. Puel: ([email protected]) Laboratoire de Mathématiques Appliquées, Université de Versailles Saint-Quentin, 45 avenue des Etats Unis, 78035 Versailles
Chapter 1
Presentation of the problems
Exact controllability problems for wave equations have been extensively studied in
the last fifteen years (see [13] for example) and this problem corresponds to many
applications like how to stop vibrations by a boundary action for example and many
others. We will consider here the case of wave equations with bounded potentials
and present a direct approach to this problem. Another problem which will be considered here is to retrieve an unknown potential in a wave equation from boundary
measurements. This is now an inverse problem and again it corresponds to important applications, even if a more interesting problem would be to retrieve a diffusion
coefficient or elasticity coefficients. The situation considered here constitutes one
step in this direction.
The results we present here are not new but they are given in a unified way and
the attempt is to make them rather simple and comprehensible despite of some long
computations.
The link between the two questions appears clearly from Lions’ Hilbert Uniqueness Method for example. Here we obtain results on the two problems using the
same mathematical machinery.
We will present this machinery which is the very technical but powerful mathematical tool given by global Carleman estimates in the present context of wave
equations with potentials. We will give a complete proof of these estimates for
strong solutions of wave equations, following the method developped in [8]. Then
we give applications to the two above mentionned problems. We start with well
known preliminaries on wave equations which we will need all along this study.
This text corresponds to lectures which have been given during the advanced
school “Control of Solids and Structures: Mathematical Modelling and Engineering
Applications” at the C.I.S.M. in Udine in June 2004 and to a part of D.E.A. course
at the University Pierre et Marie Curie in 2003-2004.
1
1.1
Basic properties for the wave equation
In the whole text, Ω is a bounded (regular enough) open subset of IRN , Γ denotes its
boundary and T is a strictly positive number. We will sometimes use the notation
Q = Ω × (0, T ). We will call ν the outward unit normal vector on Γ.
We will consider various kinds of wave equations and they will all be of the
following type :


∂2u



 ∂t2 − ∆u + p.u = f, in Ω × (0, T ),
(1.1.1)
u = g, on Γ × (0, T ),



∂u

 u(0) = u0 ;
(0) = u1 , in Ω.
∂t
A very important special situation is the case of homogeneous Dirichlet boundary
condition, correponding to g = 0 which is


∂2u



 ∂t2 − ∆u + p.u = f, in Ω × (0, T ),
(1.1.2)
u = 0, on Γ × (0, T ),



∂u

 u(0) = u0 ;
(0) = u1 , in Ω.
∂t
1.1.1
Existence of finite energy solutions
We recall here the basic results concerning existence of finite energy solutions for
equation (1.1.2) in the case of homogeneous Dirichlet boundary conditions (g = 0).
We can refer to [5] for a complete study of the wave equations.
Theorem 1.1.1 Suppose that p ∈ L∞ (Ω×(0, T )) . Then for every f ∈ L1 (0, T ; L2 (Ω)),
u0 ∈ H01 (Ω) and u1 ∈ L2 (Ω), there exists a unique solution u to (1.1.2) with
∂u
∈ C([0, T ]; L2 (Ω)).
∂t
u ∈ C([0, T ]; H01 (Ω)) ,
Proof. (idea)
The proof is based on an energy estimate which can be justified on an approximate equation (finite dimensional or regularized...) and which works as follows.
Multiply equation (1.1.2) by ∂u
∂t and integrate in Ω, we obtain
1d
(
2 dt
∂u
| |2 dx +
Ω ∂t
Z
∂u
|∇u| dx) +
p.u dx =
∂t
Ω
Ω
Z
Z
2
Z
f
Ω
If we define the energy by
1
E(t) = (
2
Z
Ω
|
∂u
(t)|2 dx +
∂t
2
Z
Ω
|∇u(t)|2 dx),
∂u
dx.
∂t
we have
d
∂u
∂u
E(t) ≤ ||p||L∞ (Q) |u(t)|L2 (Ω) | (t)|L2 (Ω) + |f (t)|L2 (Ω) | (t)|L2 (Ω)
dt
∂t
∂t
As Ω is bounded the Poincaré inequality says that there exists a constant C0 such
that
∀v ∈ H01 (Ω), |v|L2 (Ω) ≤ C0 |∇v|L2 (Ω) .
Therefore we have
q
√
d
E(t) ≤ C0 ||p||L∞ (Q) E(t) + 2|f (t)|L2 (Ω) E(t),
dt
so that
q
dq
C0
1
E(t) ≤
||p||L∞ (Q) E(t) + √ |f (t)|L2 (Ω) .
dt
2
2
Using Gronwall’s lemma we obtain
∀t ∈ (0, T ),
q
1
E(t) ≤ ( √
2
Z T
0
|f (s)|L2 (Ω) ds +
q
E(0))exp(
C0
||p||L∞ (Q) T ).
2
Remark 1.1.2 In fact we only need p ∈ L1 (0, T ; L∞ (Ω)).
This gives an explicit bound on the energy E(t) in terms of E(0) (which only depends
on the initial datas), the potential p and the right hand side f and this is the essential
part of the proof of Theorem 1.1.1.
We also have an existence and uniqueness result for finite energy solutions in
another context.
Theorem 1.1.3 We still suppose that p ∈ L∞ (Ω × (0, T )) . Then for every f ∈
W 1,1 (0, T ; H −1 (Ω)), u0 ∈ H01 (Ω) and u1 ∈ L2 (Ω), there exists a unique solution u
to (1.1.2) with
∂u
∈ C([0, T ]; L2 (Ω)).
∂t
u ∈ C([0, T ]; H01 (Ω)) ,
The difference in the proof comes from the term
< f (t),
∂u
(t) >
∂t
which is treated as
∂
∂f (t)
< f (t), u(t) > − <
, u(t) > .
∂t
∂t
3
1.1.2
Regularity of the normal derivative
We now recall a regularity result for solutions given by Theorem 1.1.1. This result
is known as a result of “hidden regularity” (see for example [13]).
Theorem 1.1.4 We assume the same hypotheses as in Theorem 1.1.1. Then the
solution u of (1.1.2) satisfies
∂u
∈ L2 (0, T ; L2 (Γ)).
∂ν
Moreover, the mapping
∂u
∂ν
which is well defined for regular (and dense) datas, is linear continous from
L1 (0, T ; L2 (Ω)) × H01 (Ω) × L2 (Ω) into L2 (0, T ; L2 (Γ)).
(f, u0 , u1 ) →
Proof. (idea).
It relies on the multiplier method. The computations are done in the case of
regular solutions (for a dense set of regular datas). Let m ∈ W 1,∞ (Ω; IRN ) (called
multiplier) and let us multiply (1.1.2) by m.∇u. We successively obtain (using the
summation convention for repeated indices)
Z TZ
∂2u
0
Z
=
Ω
Z
=
Ω
Z
=
Ω
Ω
∂u
(T )m.∇u(T )dx −
∂t
∂u
(T )m.∇u(T )dx −
∂t
∂u
(T )m.∇u(T )dx −
∂t
Z
∂t2
m.∇udxdt
u1 m.∇u0 dx −
Z TZ
Ω
Z
0
u1 m.∇u0 dx −
Ω
Z
u1 m.∇u0 dx +
Ω
Ω
Z Z
1 T
2
0
m.∇(|
Ω
Z Z
1 T
2
∂u ∂u
dxdt
∂t ∂t
m.∇
0
∂u 2
| )dxdt
∂t
(div m)|
Ω
∂u 2
| dxdt.
∂t
Z TZ
(−∆u)m.∇udxdt
0
Z TZ
=
0
Ω
Z TZ
=
0
Ω
Z TZ
=
0
Ω
Ω
Z TZ
∂u ∂mj ∂u
dxdt+
∂xi ∂xi ∂xj
0
Ω
Z TZ
∂u
∂ ∂u
mj
(
)dxdt−
∂xi
∂xj ∂xi
∂u ∂mj ∂u
1
dxdt +
∂xi ∂xi ∂xj
2
Z TZ
∂u ∂mj ∂u
1
dxdt−
∂xi ∂xi ∂xj
2
Z TZ
mj
0
0
Ω
(∇u.ν)(m.∇u)dσdt
0
∂
(|∇u|2 )dxdt −
∂xj
(div m)|∇u|2 dxdt−
Ω
4
1
2
Γ
Z TZ
0
Z TZ
0
|∇u.ν|2 (m.ν)dσdt
Γ
Γ
|∇u.ν|2 (m.ν)dσdt.
We now take the multiplier m as a lifting of the outward unit normal ν so that
m.ν = 1 on Γ. This is possible if Ω is regular enough. We obtain, gathering all
terms, the following estimate
1
2
Z TZ
0
Z TZ
Z T
|∇u.ν|2 dσdt ≤ C(
E(t)dt +
0
0
Γ
|p||u||∇u|dxdt +
Z TZ
Ω
0
|f ||∇u|dxdt).
Ω
This gives the desired result.
Remark 1.1.5 If f ∈ W 1,1 (0, T ; H −1 (Ω)), we still have existence and uniqueness
of a finite energy solution from Theorem 1.1.3. But in that case, we cannot obtain
any regularity result on the normal derivative ∂u
∂ν .
1.1.3
Solution obtained by transposition
We will also need a weaker notion of solution which is obtained by transposition of
the previous results. These solutions are usually called “solution by transposition”.
Let us consider a wave equation with non homogeneous Dirichlet boundary condition


∂2z



 ∂t2 − ∆z + p.z = f, in Ω × (0, T ),
(1.1.3)
z = g, on Γ × (0, T ),



∂z

 z(0) = z0 ;
(0) = z1 , in Ω.
∂t
Theorem 1.1.6 For every f ∈ L1 (0, T ; H −1 (Ω)), g ∈ L2 (0, T ; L2 (Γ)), z0 ∈ L2 (Ω)
and z1 ∈ H −1 (Ω), there exists a unique solution z of (1.1.3) with z ∈ C([0, T ]; L2 (Ω))
−1 (Ω)).
and ∂z
∂t ∈ C([0, T ]; H
Proof. .
Let h ∈ L1 (0, T ; L2 (Ω)) and let u be solution of


∂2u



 ∂t2 − ∆u + p.u = h, in Ω × (0, T ),
(1.1.4)
u = 0, on Γ × (0, T ),



∂u

 u(T ) = 0 ;
(T ) = 0, in Ω.
∂t
We know from Theorem 1.1.1 that u ∈ C([0, T ]; H01 (Ω)) and from Theorem 1.1.3
2
2
that ∂u
∂ν ∈ L (0, T ; L (Γ)).
Now let us define
Z T
L(h) =
0
∂u
< f (t), u(t) > dt −
z0 (0)dx+ < z1 , u(0) > −
∂t
Ω
Z
5
Z TZ
g
0
Γ
∂u
dσdt.
∂ν
It is immediate to see that L is a linear continuous functional on L1 (0, T ; L2 (Ω)).
Therefore there exists a unique function z ∈ L∞ (0, T ; L2 (Ω)) such that
1
Z TZ
2
∀h ∈ L (0, T ; L (Ω)),
zhdxdt = L(h)
0
Z T
=
< f (t), u(t) > dt −
0
Z
z0
Ω
Ω
∂u
(0)dx+ < z1 , u(0) > −
∂t
Z TZ
g
0
Γ
∂u
dσdt.
∂ν
It is now standard to interpret the above equation and to give a sense to the value
of z on the boundary as well as the initial values z(0) and ∂z
∂t (0) to obtain (1.1.3).
By taking first the datas f , z0 and z1 in a dense subset of regular functions, we
−1 (Ω)).
can show easily that in fact z ∈ C([0, T ]; L2 (Ω)) and then that ∂z
∂t ∈ C([0, T ]; H
This gives the result of Theorem 1.1.6.
1.2
Retrieving a potential from boundary measurements
We will consider the inverse problem consisting in retrieving a potential, depending
only on the space variable, in a wave equation from the knowledge of measurements
of the flux on a part of the boundary during a period of time (0, T ). More precisely
we consider a wave equation
(1.2.5)


∂2u



 ∂t2 − ∆u + p.u = f, in Ω × (0, T ),
u = g, on Γ × (0, T ),



∂u

 u(0) = u0 ;
(0) = u1 , in Ω,
∂t
where the potential p = p(x) is unknown (but assumed to be bounded). For every
p ∈ L∞ (Ω) (assuming enough regularity on f , g, u0 and u1 ) we suppose that (1.2.5)
has a unique solution u(p). On the other hand we know a measure of ∂u
∂ν on a set
Γ0 × (0, T ) where Γ0 is a part of the boundary Γ say
(1.2.6)
∂u
= H on Γ0 × (0, T ).
∂ν
The question is then : can we retrieve p from the measure H?
To be more precise we can ask two questions :
• Uniqueness : do we have
∂u(p)
∂u(q)
=
on Γ0 × (0, T ) ⇒ p = q ?
∂ν
∂ν
6
• Stability : do we have
||p − q||X(Ω) ≤ C||
∂u(p) ∂u(q)
−
||
∂ν
∂ν Y (Γ0 )
for suitable norms X(Ω) and Y (Γ0 )?
Of course stability implies uniqueness. We will obtain here, following the work of [9]
and [10] , stability results in a situation where we assume to know the solution u(q)
corresponding to a given potential q with enough regularity on u(q) (for example this
solution u(q) could be a result of computations) and under a geometrical hypothesis
on Γ0 together with T sufficiently large. Previous results in the same direction have
been obtained for example in [15], [17] and [11]
We can already notice that if we call
z = u(p) − u(q)
z is solution of the following wave equation :
(1.2.7)


∂2z



 ∂t2 − ∆z + p.z = (q − p)u(q), in Ω × (0, T ),
z = 0 on Γ × (0, T ),



∂z

 z(0) = 0 ;
(0) = 0, in Ω.
∂t
∂z
The problem is then to estimate (p − q) in terms of ∂ν
. Notice that the potential p
appearing in the left hand side of the equation is unknown.
In [3] the authors consider another inverse problem for retrieving a potential in
a stationnary elliptic equation from measurements on a part of the boundary. More
precisely they consider an elliptic equation
(
(1.2.8)
−∆u + p.u = f in Ω,
u = ϕ on Γ,
where p is unknown. They suppose to know the complete mapping
Λp : ϕ →
∂u
.
∂ν /Γ0
The question they address is then : If two such mappings (corresponding to two
potentials) coincide, are the potentials equal? More precisely, do we have
Λp = Λq ⇒ p = q?
7
Notice that the knowledge of Λp = Λq implies an infinite number of measurements,
∂u(q)
but in our case, the knowledge of ∂u(p)
∂ν = ∂ν on Γ0 × (0, T ) means the knowledge
of one measurement but during a time period (0, T ) so that it also means an infinite
number of measurements.
The positive result they obtain requires hypotheses which are analogous to the
ones we will assume here and the techniques involved in their proof is similar to the
one we will describe here for the wave equation. However, up to our knowledge, the
precise relation between the two problems is unknown at the moment and it would
be very interesting know if such a relation exists.
A similar question can be treated for Schrödinger equations and the answers are
similar (see [2]).
The basic tool we will use is a global Carleman inequality for the wave equation.
The obtention of such an inequality will be the subject of Chapter 2.
1.3
Exact controllability for wave equations with bounded
potentials
In this part we are given a wave equation
(1.3.9)
 2
∂ u



− ∆u + p.u = 0, in Ω × (0, T ),

2


 ∂t
u = v, on Γ0 × (0, T ),

u = 0, on (Γ − Γ0 ) × (0, T ),




∂u

 u(0) = u0 ;
(0) = u1 , in Ω,
∂t
where p ∈ L∞ (Ω × (0, T )). We know from Theorem 1.1.6 that for every u0 ∈
L2 (Ω), u1 ∈ H −1 (Ω) and for every v ∈ L2 (0, T ; L2 (Γ0 )), there exists a unique
solution u to (1.3.9) (defined by transposition) with u ∈ C([0, T ]; L2 (Ω)) and ∂u
∂t ∈
C([0, T ]; H −1 (Ω)).
The problem of exact controllability is then the following : Given any target
(z0 , z1 ) ∈ L2 (Ω) × H −1 (Ω), can we find a control v ∈ L2 (0, T ; L2 (Γ0 )) such that the
state reaches the target at time T which means
u(T ) = z0 and
∂u
(T ) = z1 .
∂t
Because of the linearity and the reversibility of (1.3.9) it is easy to see that it suffices
to treat the case when z0 = 0 and z1 = 0.
Moreover, since the work of Lions ([13]) and his development of the Hilbert
Uniqueness Method (HUM), it is well known that solving the exact controllability
8
problem is equivalent to proving an observability inequality on the adjoint equation.
Let us consider the adjoint equation
(1.3.10)


∂2ϕ



 ∂t2 − ∆ϕ + p.ϕ = 0, in Ω × (0, T ),
ϕ = 0, on Γ × (0, T ),



∂ϕ

 ϕ(0) = ϕ0 ;
(0) = ϕ1 , in Ω,
∂t
where ϕ0 ∈ H01 (Ω) and ϕ1 ∈ L2 (Ω). If the initial energy is defined by
1
E0 = (
2
Z
|ϕ1 |2 dx +
Z
Ω
|∇ϕ0 |2 dx)
Ω
the observability inequality to be proved (which is equivalent to solving the exact
controllability problem) is the following
(1.3.11)
∃C0 > 0, such that E0 ≤ C0
Z TZ
0
Γ0
|
∂ϕ 2
| dσdt.
∂ν
Using the same basic tool as for the previous inverse problem, namely a global
Carleman inequality, we will show inequality (1.3.11) under a geometrical hypothesis
on Γ0 and for T sufficiently large.
9
Chapter 2
Global Carleman inequality for
the wave equation
Carleman estimates for regular functions with compact support (which we shall call
local) go back to [4] and have been developped in [7]. Here we are interested in
proving global Carleman estimates (up to the boundary) corresponding to the wave
operator.
We consider a function v ∈ L2 (−T, T ; L2 (Ω)) such that
L0 v =
∂2v
− ∆v ∈ L2 (−T, T ; L2 (Ω)).
∂t2
We then call
Lv = L0 v + pv =
∂2v
− ∆v + pv
∂t2
and we also have
Lv ∈ L2 (−T, T ; L2 (Ω)).
∂v
We can then define in a very weak sense the traces on Γ × (−T, T ) of v and ∂ν
because ∆v ∈ H −2 (−T, T ; L2 (Ω)). We can also define the values of v and ∂v
∂t at
∂2v
2
−2
times −T and T because ∂t2 ∈ L (−T, T ; H (Ω)).
We know consider the set of functions v such that
v ∈ L2 (−T, T ; L2 (Ω)), L0 v ∈ L2 (−T, T ; L2 (Ω)), v = 0 on Γ × (−T, T ),
∂v
∂v
(−T ) =
(T ) = 0.
∂t
∂t
It is classical to show that this set contains a dense subset of “regular” functions.
v(−T ) = v(T ) = 0 ;
10
Let us now define, for x0 ∈ IRN \ Ω a first weight
(2.0.1)
φ(x, t) = |x − x0 |2 − βt2 + M0 , 0 < β < 1,
where M0 is chosen such that
∀(x, t) ∈ Ω × (−T, T ), φ(x, t) ≥ 1.
(2.0.2)
For λ > 0 we now define
(2.0.3)
ϕλ (x, t) = eλφ(x,t) .
Now for a function v defined on Ω × (−T, T ) we set for s > 0
w = esϕλ .v.
If v is regular enough let us calculate (formally)
P0 w = esϕλ L0 (e−sϕλ .w) = esϕλ L0 v.
We have
∂ −sϕλ
∂w
∂φ
(e
.w) = e−sϕλ (
− sλ ϕλ w) ;
∂t
∂t
∂t
2
∂ 2 −sϕλ
∂2φ
−sϕλ ∂ w
2 ∂φ 2
(e
.w)
=
e
(
−
sλ
(
)
ϕ
w
−
sλ
ϕλ w
λ
∂t2
∂t2
∂t
∂t2
∂φ
∂φ ∂w
+ s2 λ2 ( )2 ϕ2λ w) ;
−2sλϕλ
∂t ∂t
∂t
∂ −sϕλ
∂w
∂φ
(e
.w) = e−sϕλ (
− sλ
ϕλ w) ;
∂xi
∂xi
∂xi
∆(e−sϕλ .w) = e−sϕλ (∆w − sλ2 |∇φ|2 ϕλ w − sλ∆φϕλ w
−2sλϕλ ∇φ.∇w + s2 λ2 ϕ2λ |∇φ|2 w).
Therefore we can write
P0 w = P1 w + P2 w + R0 w
where
(2.0.4)
P1 w =
∂2w
∂φ
− ∆w + s2 λ2 ϕ2λ (( )2 − |∇φ|2 )w,
2
∂t
∂t
(2.0.5) P2 w = (M1 − 1)sλϕλ (
∂2φ
∂φ
− ∆φ)w − sλ2 ϕλ (( )2 − |∇φ|2 )w
2
∂t
∂t
∂φ ∂w
−2sλϕλ (
− ∇φ.∇w)
∂t ∂t
11
where M1 will be chosen later on such that
2β
2
< M1 <
β+N
β+N
and
∂2φ
− ∆φ)w.
∂t2
In order to formulate the result concerning the global Carleman inequality we
need another definition. For x0 ∈ IRN , we define
R0 w = −M1 sλϕλ (
(2.0.6)
Γx0 = {x ∈ Γ, ν(x).(x − x0 ) > 0}.
(2.0.7)
We now give a global Carleman inequality following essentially Imanuvilov’s method
in [8]. Other versions of global Carleman inequalities for wave equations have been
obtained by several authors, in particular [18] (see also [16]).
Theorem 2.0.1 Let us suppose that there exists x0 ∈ IRN \ Ω such that Γ0 ⊃
Γx0 . Then for every m > 0, there exists λ0 > 0 and s0 > 0 and there exists a
constant C = C(s0 , λ0 , Ω, β, x0 , m) such that for every p ∈ L∞ (Q) with |p|L∞ (Q) ≤
m, we have, for every λ ≥ λ0 and s ≥ s0 , for every v ∈ L2 (−T, T ; L2 (Ω)), L0 v ∈
∂v
L2 (−T, T ; L2 (Ω)), v = 0 on Γ × (−T, T ), v(−T ) = v(T ) = 0, ∂v
∂t (−T ) = ∂t (T ) = 0,
Z T Z
e
sλ
−T
2sϕλ
Ω
∂v
ϕλ (| |2 + |∇v|2 )dxdt + s3 λ3
∂t
Z T Z
(2.0.8)
−T
≤C
−T
−T
|P1 w|2 dxdt +
+
Z T Z
Z T Z
Ω
Z T Z
−T
Ω
e2sϕλ |Lv|2 dxdt + Csλ
Z T Z
−T
Ω
e2sϕλ ϕ3λ |v|2 dxdt
Γ0
|P2 w|2 dxdt
Ω
e2sϕλ |
∂v 2
| dσdt.
∂ν
Remark 2.0.2 The above estimate is uniform in p when p is in a bounded set of
L∞ (Q). This is important for the applications we have in mind.
The above estimate does not require any sufficiently large size for the time interval
2T .
Proof.
Let us first assume that we have proved (2.0.8) with L0 v instead of Lv in the
right hand side. Then as
|Lv|2 ≤ 2|L0 v|2 + 2|p|L∞ (Q) |v|2 ≤ 2|L0 v|2 + 2m|v|2
and as ϕλ ≥ 1, we see that by choosing s0 large enough we can absorb the term
Z T Z
e2sϕλ |v|2 dxdt
2Cm
−T
Ω
12
in the left hand side and this gives (2.0.8) with slightly different constants. Therefore
we have to prove (2.0.8) with L0 v in the right hand side instead of Lv.
We will show the inequality for a dense subset of functions v in order to justify
our calculations.
For w = esϕλ v we have
P0 w = P1 w + P2 w + R0 w = esϕλ L0 v.
Then
P1 w + P2 w = esϕλ L0 v − R0 w.
Taking norms of both sides in L2 (−T, T ; L2 (Ω)) we obtain
Z T Z
(2.0.9)
−T
Z
(|P1 w|2 + |P2 w|2 )dxdt + 2
Ω
T Z
≤2
−T
P1 wP2 wdxdt
Ω
|esϕλ L0 v − R0 w|2 dxdt
=
−T
Z T Z
Ω
Z T Z
−T
e2sϕλ |L0 v|2 dxdt + 2
Ω
Z T Z
−T
|R0 w|2 dxdt.
Ω
We now have to get a lower bound for
Z T Z
−T
P1 wP2 wdxdt
Ω
and this requires lots of very careful calculations. The dominating terms (of which
we have to keep track carefully) are the ones containing
Z T Z
sλ
−T
Ω
s3 λp
ϕλ (|
∂w 2
| + |∇w|2 )dxdt,
∂t
Z T Z
−T
Ω
ϕ3λ |w|2 dxdt
with p ≥ 3 and the boundary terms. The other terms can be shown to be negligible.
Let us compute successively the different terms we obtain.
I11
Z T Z
∂2w
∂2φ
− ∆φ)) dxdt
2
∂t2
−T Ω ∂t
Z T Z
∂2φ
∂w 2
= (1 − M1 )sλ
|
| ϕλ ( 2 − ∆φ) dxdt
∂t
−T Ω ∂t
Z T Z
∂2φ
∂2φ
(1 − M1 ) 2
∂φ
−
sλ
|w|2 ϕλ ( 2 + λ| |2 )( 2 − ∆φ) dxdt
2
∂t
∂t
∂t
−T Ω
=
((M1 − 1)sλϕλ w(
13
I12
=
Z T Z
∂2w
−T
Ω
∂t2
(−sλ2 ϕλ w(|
∂φ 2
| − |∇φ|2 )) dxdt
∂t
T
∂w 2
∂2φ
∂φ
|
= sλ
|w|2 ϕλ | 2 |2 dxdt
| ϕλ (| |2 − |∇φ|2 ) dxdt − sλ2
∂t
∂t
−T Ω ∂t
−T Ω
Z T Z
Z T Z
3
2
1
∂φ ∂ φ
sλ
∂2φ
−(2 + )sλ3
|w|2 ϕλ | |2 2 dxdt +
|w|2 ϕλ |∇φ|2 2 dxdt
2
∂t ∂t
2 −T Ω
∂t
−T Ω
Z T Z
4
sλ
∂φ
∂φ
−
|w|2 ϕλ | |2 (| |2 − |∇φ|2 ) dxdt
2 −T Ω
∂t
∂t
2
I13
Z T Z
Z
Z T Z
∂2w
Z
∂w ∂φ
− ∇w.∇φ)) dxdt
∂t ∂t
−T Ω
Z T Z
Z T Z
∂w 2
∂φ
∂w 2 ∂ 2 φ
|
| ϕλ 2 dxdt + sλ2
| ϕλ | |2 dxdt
|
= sλ
∂t
∂t
−T Ω ∂t
−T Ω ∂t
Z T Z
Z T Z
∂w 2
∂w ∂φ
|
+sλ
| ϕλ (∆φ + λ|∇φ|2 ) dxdt − 2sλ2
ϕλ ∇w.∇φ dxdt
∂t
−T Ω
−T Ω ∂t ∂t
=
∂t2
(−2sλϕλ (
∂2φ
− ∆φ)) dxdt
∂t2
−T Ω
Z T Z
∂2φ
|∇w|2 ϕλ ( 2 − ∆φ) dxdt
= −(1 − M1 )sλ
∂t
−T Ω
Z T Z
2
∂ φ
(1 − M1 ) 2
|w|2 ϕλ ( 2 − ∆φ)(∆φ + λ|∇φ|2 ) dxdt
+
sλ
2
∂t
−T Ω
Z T Z
I21
−∆w((M1 − 1)sλϕλ w(
=
Z T Z
I22
=
−T
Ω
= −sλ2
∂φ 2
| − |∇φ|2 )) dxdt
∂t
Z
∂φ
|∇w|2 ϕλ (| |2 − |∇φ|2 ) dxdt
∂t
Ω
−∆w(−sλ2 ϕλ w(|
Z T
−T
−sλ
2
Z T Z
−T
+
sλ3
2
2
Ω
|w| ϕλ
Z T Z
−2sλ3
−T
Ω
i,j=1
∂2φ
∂xi ∂xj
!2
|w|2 ϕλ (∆φ + λ|∇φ|2 )(|
Z T Z
−T
n
X
Ω
dxdt
∂φ 2
| − |∇φ|2 ) dxdt
∂t
|w|2 ϕλ D2 φ(∇φ, ∇φ) dxdt
14
Z T Z
∂w ∂φ
− ∇w.∇φ)) dxdt
∂t ∂t
−T Ω
Z T Z
Z T Z
∂2φ
2
2
|∇φ.∇w|2 ϕλ dxdt
|∇w| ϕλ ( 2 − ∆φ) dxdt + 2sλ
= sλ
∂t
−T Ω
−T Ω
Z T Z
Z T Z
∂w
∂φ
∂φ
−2sλ2
|∇w|2 ϕλ (| |2 − |∇φ|2 ) dxdt
ϕλ ∇w.∇φ dxdt + sλ2
∂t
−T Ω ∂t ∂t
−T Ω
I23
−∆w(−2sλϕλ (
=
−sλ
Z T Z
−T
Γ
|∇w.ν|2 ϕλ ∇φ.ν dσdt + 2sλ
Z T Z
−T
Ω
ϕλ D2 φ(∇w, ∇w) dxdt
∂φ 2
∂2φ
| − |∇φ|2 )((M1 − 1)sλϕλ w( 2 − ∆φ)) dxdt
∂t
∂t
−T Ω
Z T Z
2
∂ φ
∂φ
|w|2 ϕ3λ ( 2 − ∆φ)(| |2 − |∇φ|2 ) dxdt
= −(1 − M1 )s3 λ3
∂t
∂t
−T Ω
Z T Z
s2 λ2 ϕ2λ w(|
I31 =
Z T Z
I32 =
−T
Ω
3 4
= −s λ
Z T
−T
Z T Z
I33
∂φ 2
∂φ
| − |∇φ|2 )(−sλ2 ϕλ w(| |2 − |∇φ|2 )) dxdt
∂t
∂t
Z
∂φ
|w|2 ϕ3λ (| |2 − |∇φ|2 )2 dxdt
∂t
Ω
s2 λ2 ϕ2λ w(|
∂φ 2
∂w ∂φ
| − |∇φ|2 )(−2sλϕλ (
− ∇w.∇φ)) dxdt
∂t
∂t ∂t
−T Ω
Z T Z
∂2φ
∂φ
|w|2 ϕ3λ ( 2 − ∆φ)(| |2 − |∇φ|2 ) dxdt
= s3 λ3
∂t
∂t
−T Ω
Z T Z
2
∂ φ ∂φ
|w2 |ϕ3λ ( 2 | |2 + D2 φ(∇φ, ∇φ)) dxdt
+2s3 λ3
∂t ∂t
−T Ω
Z T Z
∂φ
+3s3 λ4
|w|2 ϕ3λ (| |2 − |∇φ|2 )2 dxdt
∂t
−T Ω
=
s2 λ2 ϕ2λ w(|
Summing up all terms we obtain :
Z T Z
−T
Z T Z
P1 wP2 w dxdt = 2sλ
Ω
−T
Ω
|
∂w 2 ∂ 2 φ
| ϕλ 2 dxdt
∂t
∂t
∂w 2
∂2φ
−M1 sλ
|
| ϕλ ( 2 − ∆φ) dxdt
∂t
−T Ω ∂t
Z T Z
∂w 2 ∂φ 2
∂w ∂φ
+2sλ2
ϕλ [|
| | | −2
∇w.∇φ + |∇φ.∇w|2 ] dxdt
∂t
∂t
∂t ∂t
−T Ω
Z T Z
15
Z T Z
+2sλ
−T
Ω
ϕλ D2 φ(∇w, ∇w) dxdt
Z T Z
+M1 sλ
−T
Z T Z
Ω
|∇w|2 ϕλ (
∂2φ
− ∆φ) dxdt
∂t2
∂w 2
| ∇φ.ν(x) dσdt
∂ν
−T Γ
Z T Z
∂φ
|w|2 ϕ3λ (| |2 − |∇φ|2 )2 dxdt
+2s3 λ4
∂t
−T Ω
Z T Z
∂ 2 φ ∂φ
|w2 |ϕ3λ ( 2 | |2 + D2 φ(∇φ, ∇φ)) dxdt
+2s3 λ3
∂t ∂t
−T Ω
Z T Z
∂2φ
∂φ
+M1 s3 λ3
|w|2 ϕ3λ ( 2 − ∆φ)(| |2 − |∇φ|2 ) dxdt
∂t
∂t
−T Ω
+X1
−sλ
ϕλ |
where, because we have chosen φ(x, t) ≥ 1, we have
λ ≤ ϕλ
and therefore
X1 ≤ Csλ3
Z T Z
−T
Ω
ϕ3λ |w|2
for some suitable constant C.
We first notice that
Z T Z
∂w 2 ∂φ 2
∂w ∂φ
| | | −2
∇w.∇φ + |∇φ.∇w|2 ] dxdt
∂t
∂t
∂t ∂t
−T Ω
Z T Z
∂w ∂φ
= 2sλ2
ϕλ [
− ∇w.∇φ]2 dxdt ≥ 0.
∂t
∂t
−T Ω
2sλ
2
ϕλ [|
On the other hand with the choice of M1 such that
2
2β
< M1 <
β+N
β+N
if we set
M2 = 2M1 (β + N )
we have
4β < M2 < 4.
16
Now it is easy to see that
Z T Z
−T
P1 wP2 w dxdt ≥ (M2 − 4β)sλ
Z T Z
Ω
Z T Z
+(4 − M2 )sλ
+2s3 λ3
−T
Z T Z
−T
Ω
Ω
−T
|∇w|2 ϕλ dxdt − 2sλ
|
Ω
Z T
−T
∂w 2
| ϕλ dxdt
∂t
Z
∂w 2
ϕλ |
| (x − x0 ).ν(x) dσdt
∂ν
Γ
|w2 |ϕ3λ Fλ (φ) dxdt + X1
where Fλ (φ) is defined by
M2 ∂φ 2
∂ 2 φ ∂φ
∂φ 2
| − |∇φ|2 )2 −
(| | − |∇φ|2 ) + ( 2 | |2 + 2|∇φ|2 )
∂t
2 ∂t
∂t ∂t
2
2 2 2
2
2 2
= 16λ(|x − x0 | − β t ) + 2(M2 + 4β)(|x − x0 | − β t ) + 8(1 − β)|x − x0 |2
Fλ (φ) = λ(|
As we have taken x0 6∈ Ω, we have
8(1 − β)|x − x0 |2 ≥ K > 0.
If P (X) = 16λX 2 + 2(M2 + 4β)X + K, we know that by choosing λ large enough
(λ ≥ λ0 ) we have
min P (X) ≥ P0 > 0.
X∈IR
Then we have for every x ∈ Ω and t ∈ (−T, T )
Fλ (φ)(x, t) ≥ P0
and this implies that for some positive constant C
Z T Z
−T
P1 wP2 w dxdt ≥ −2λs
Ω
Z T Z
−T
∂Ω
ϕλ |
∂w 2
| (x − x0 ).ν(x) dσdt
∂ν
Z T Z Z T Z
∂w 2
+Csλ
|
| + |∇w|2 ϕλ dxdt + Cs3 λ3
|w|2 ϕ3λ dxdt + X1 .
−T
Ω
∂t
−T
Ω
By choosing now s large enough (s ≥ s0 ) we can absorb the terms of X1 and the
term
Z
Z
T
−T
|R0 w|2 dxdt
Ω
17
in (2.0.9) which is of the same order to obtain for another positive constant C
Z T Z
−T
P1 wP2 w dxdt −
Ω
Z T Z
−T
|R0 w|2 dxdt
Ω
Z T Z
∂w 2
ϕλ |
≥ −2λs
| (x − x0 ).ν(x) dσdt
∂ν
−T ∂Ω
Z T Z Z T Z
∂w 2
|
|w|2 ϕ3λ dxdt.
+Csλ
| + |∇w|2 ϕλ dxdt + Cs3 λ3
∂t
−T Ω
−T Ω
Now using this inequality in (2.0.9), the definition of Γx0 and going back to the
variable v = e−sϕλ w this implies Carleman inequality (2.0.8) and Theorem 2.0.1.
18
Chapter 3
Application to the inverse
problem for the potential
3.1
Uniqueness and stability results
Let us recall the inverse problem for retrieving a potential from boundary measurements. We have a wave equation
(3.1.1)


∂2u



 ∂t2 − ∆u + p.u = f, in Ω × (0, T ),
u = g, on Γ × (0, T ),



∂u

 u(0) = u0 ;
(0) = u1 , in Ω,
∂t
where the potential p = p(x) is unknown and which is supposed to have a solution
u(p) for each given bounded potential p. On the other hand, we know a measure of
the flux of the solution on a part Γ0 × (0, T ) of the boundary, more precisely
∂u(p)
= H on Γ0 × (0, T ).
∂ν
We want to find a stability result which means an inequality like
||p − q||X(Ω) ≤ C||
∂u(p) ∂u(q)
−
||
∂ν
∂ν Y (Γ0 )
for suitable norms X(Ω) and Y (Γ0 ). We can notice that the mapping p → u(p) is
strongly nonlinear.
We have already noticed that if
z = u(p) − u(q)
19
z is solution of the following wave equation :
(3.1.2)


∂2z



 ∂t2 − ∆z + p.z = (q − p)u(q), in Ω × (0, T ),
z = 0, on Γ × (0, T ),



∂z

 z(0) = 0 ;
(0) = 0, in Ω.
∂t
Calling f (x) = (q(x) − p(x)) and R(x, t) = u(q)(x, t) we can consider the “linear”
problem : If y is solution of the following equation with potential p = p(x) (supposed
to be know here for the moment)
(3.1.3)


∂2y



 ∂t2 − ∆y + p(x)y = f (x)R(x, t), x ∈ Ω, t ∈ (0, T ),
y = 0, x ∈ Γ, t ∈ (0, T ),



∂y

 y(0) = 0,
(0) = 0, x ∈ Ω,
∂t
do we have an inequality like
||f ||2X(Ω) ≤ C||
∂y 2
||
?
∂ν Y (Γ0 ×(0,T ))
Moreover if we can obtain such an inequality with constants independent of p for p
in a suitable subset, then this result will apply directly to the nonlinear problem.
These questions for the wave equations have been studied by several authors among
them [11], [15], [17], and the latest results have been obtained by [9] in which the
authors also consider different types of boundary conditions.
We start with a uniqueness result.
Theorem 3.1.1 We assume that
• ∃x0 ∈ IRN \ Ω with Γ0 ⊃ Γx0 = {x ∈ Γ, (x − x0 ).ν(x) > 0}
• T > supx∈Ω |x − x0 |
• ∃a0 > 0, |u0 (x)| ≥ a0
• p, q ∈ L∞ (Ω), u(q) ∈ H 1 (0, T ; L∞ (Ω))
Then if
∂u
∂u
(p) =
(q) on Γ0 × (0, T )
∂ν
∂ν
we have
p = q.
20
We can now give the main stability result
Theorem 3.1.2 We assume the same hypotheses as before. In addition :
• we assume a priori that ||p||L∞ (Ω) ≤ m where m > 0 is given.
Then there exists C > 0 such that for every p with ||p||L∞ (Ω) ≤ m
1
∂u
∂u
||q − p||2L2 (Ω) ≤ || (q) −
(p)||2H 1 (0,T ;L2 (Γ0 )) ≤ C||q − p||2L2 (Ω) .
C
∂ν
∂ν
Theorem 3.1.2 implies Theorem 3.1.1 and both are consequences of the following
result for the linear problem.
Theorem 3.1.3 We assume that ||p||L∞ (Ω) ≤ m and
• ∃x0 ∈ IRN \ Ω with Γ0 ⊃ Γx0 = {x ∈ Γ, (x − x0 ).ν(x) > 0}
• T > supx∈Ω |x − x0 |
• R ∈ H 1 (0, T ; L∞ (Ω)) with |R(x, 0)| ≥ a0 > 0.
Then there exists C > 0 depending on p only via m such that for every solution y
of the linear problem
∂y
1
||f ||2L2 (Ω) ≤ || ||2H 1 (0,T ;L2 (Γ0 )) ≤ C||f ||2L2 (Ω) .
C
∂ν
All these results are based on a global Carleman inequality for the wave equation.
Proof of Theorem 3.1.3.
If y is solution to the linear problem take
∂y
∂R
, R0 =
.
∂t
∂t
Then if we extend y 0 and R0 on (−T, 0) in an odd way we have
y0 =
(3.1.4)
 2 0
∂ y

0
0
0



 ∂t2 − ∆y + q(x)y = f (x)R (x, t), x ∈ Ω, t ∈ (−T, T ),
y 0 = 0, x ∈ Γ, t ∈ (−T, T ),



∂y 0

 y 0 (0) = 0,
(0) = f (x)R(x, 0), x ∈ Ω.
∂t
For δ > 0 (small) take θ ∈ C0∞ (−T, T ) such that
0 ≤ θ ≤ 1, θ(t) = 1 if − T + δ ≤ t ≤ T − δ.
21
Let us set
v = θ.y 0
Then v satisfies
 2
∂ v


− ∆v + p(x)v = θ(t)f (x)R0 (x, t)


2

∂t



∂θ ∂y 0 ∂ 2 θ 0


+2
+ 2 y , x ∈ Ω, t ∈ (−T, T ),



∂t ∂t
∂t


 v = 0, x ∈ Γ, t ∈ (−T, T ),
(3.1.5)
∂v


v(0) = 0,
(0) = f (x)R(x, 0), x ∈ Ω


∂t



∂v


(−T ) = 0, x ∈ Ω
v(−T ) =



∂t



 v(T ) = ∂v (T ) = 0, x ∈ Ω.
∂t
We then apply global Carleman inequality (2.0.8) to v (with notations analogous to
the ones of the previous chapter). Choosing T > supx∈Ω |x − x0 | = ρ, we can take
β such that
ρ2
<β<1
T2
Therefore, as φ(x, t) = |x − x0 |2 − βt2 + M0 , we have for sufficiently small δ
∀t ∈ [−T, −T + δ] ∪ [T − δ, T ], φ(x, t) < M0 < φ(x, 0)
As the derivatives of θ vanish on [−T +δ, T −δ], and the energy of y 0 can be bounded
in terms of ||f ||2L2 (Ω) , we obtain
Z T Z
2sϕλ
e
−T
Ω
0
2
|Lv| dxdt ≤ C(||q||L∞ , ||R ||L2 (L∞ ) )
Z
e2sϕλ (0) |f |2 dx.
Ω
On the other hand for w = esϕλ v we have
Z 0 Z
−T
∂w
1
P1 w
dxdt =
∂t
2
Ω
Z
Ω
|
∂w
(0)|2 dx + rest terms
∂t
which gives
Z 0 Z
−T
∂w
1
P1 w
dxdt ≥
∂t
2
Ω
Z
e
Ω
2sϕλ (0)
|f |2 a20 dx
2 3
− Cs λ
Z T Z
−T
Ω
ϕ3λ |w|2 dxdt.
From Carleman inequality and using the previous estimate for Lv we obtain
22
Z T Z
−T
|P1 w|2 dxdt + s3 λ3
Z T Z
−T
Z T Z
e
−T
≤C
Z
−T
Ω
|
+sλ
≤C
Z T Z
2sϕλ
Ω
2
Z T Z
−T
Ω
Z T Z
−T
Ω
ϕ3λ |w|2 dxdt
∂w 2
| dxdt
∂t
|Lv| dxdt + Csλ
e2sϕλ (0) |f |2 dx + Csλ
Ω
Γ0
Γ0
ϕλ e2sϕλ |
ϕλ e2sϕλ |
∂v 2
| dσdt
∂ν
∂v 2
| dσdt.
∂ν
Using Cauchy-Schwarz inequality we obtain for every s ≥ s0
Z
e2sϕλ (0) |f |2 a20 dx
Ω
C
≤√
e2sϕλ (0) |f |2 dx
s Ω
√ Z T Z
∂v
ϕλ e2sϕλ | |2 dσdt.
+C sλ
∂ν
−T Γ0
Z
Taking now s large enough we absorb the first term in the right hand side and we
obtain (with different constants)
Z
2sϕ(0)
e
Ω
|f |2 a20 dx
√
≤ C sλ
Z T Z
−T
which gives the result of Theorem 3.1.3.
23
Γ0
ϕλ e2sϕλ |
∂v 2
| dσdt
∂ν
Chapter 4
Application to exact
controllability for the wave
equation with a bounded
potential
We recall the exact controllability problem for the wave equation. For any couple
of initial data (u0 , u1 ) ∈ L2 (Ω) × H −1 (Ω) we consider the wave equation
(4.0.1)
 2
∂ u



− ∆u + p.u = 0, in Ω × (0, T ),

2


 ∂t
u = v, on Γ0 × (0, T ),

u = 0, on (Γ \ Γ0 ) × (0, T ),




∂u

 u(0) = u0 ;
(0) = u1 , in Ω,
∂t
where the potential p satisfies p ∈ L∞ (Ω × (0, T )) and where v is a control taken in
L2 (0, T ; L2 (Γ0 )). We know from Theorem 1.1.6 that equation (4.0.1) has a unique
−1 (Ω)). The problem of exact
solution u ∈ C([0, T ]; L2 (Ω)) with ∂u
∂t ∈ C([0, T ]; H
controllability is then to find a control v such that at time T we have
u(T ) = 0 and
∂u
(T ) = 0.
∂t
As we already mentionned, it is by now standard, since the work of Lions (see [13]),
that this question is equivalent to proving an observability inequality (also often
called an inverse inequality) for the adjoint problem. Let us consider the adjoint
24
equation
(4.0.2)


∂2ψ



 ∂t2 − ∆ψ + p.ψ = 0, in Ω × (0, T ),
ψ = 0, on Γ × (0, T ),



∂ψ

 ψ(0) = ψ0 ;
(0) = ψ1 , in Ω,
∂t
where ψ0 ∈ H01 (Ω) and ψ1 ∈ L2 (Ω). We want to show that (under conditions on T
and Γ0 ) we have an inequality of the following type
(4.0.3)
Z TZ
∃C0 > 0, such that E0 ≤ C0
0
Γ0
|
∂ψ 2
| dσdt,
∂ν
where E0 is the initial energy namely
1
E0 = (
2
Z
2
|ψ1 | dx +
Z
|∇ψ0 |2 dx).
Ω
Ω
Theorem 4.0.4 We assume that
• ∃x0 ∈ IRN \ Ω with Γ0 ⊃ Γx0 = {x ∈ Γ, (x − x0 ).ν(x) > 0}
• T > 2 supx∈Ω |x − x0 |
Then there exists a constant C0 such that for every ψ0 ∈ H01 (Ω) and ψ1 ∈ L2 (Ω) we
have
Z TZ
∂ψ
(4.0.4)
E0 ≤ C0
| |2 dσdt.
0
Γ0 ∂ν
Proof of Theorem 4.0.4.
We know that for every ψ0 ∈ H01 (Ω) and ψ1 ∈ L2 (Ω) equation (4.0.2) has a
unique solution ψ ∈ C([0, T ]; H01 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)). Moreover if
1
E(t) = (
2
we have
Z
Ω
dE
(t) +
dt
|
∂ψ
(t)|2 dx +
∂t
Z
p(t)ψ(t)
Ω
Z
|∇ψ(t)|2 dx)
Ω
∂ψ
(t)dx = 0
∂t
so that
dE
∂ψ
∂ψ
(t) ≤ ||p(t)||L∞ (Ω) | (t)|L2 (Ω) |ψ(t)|L2 (Ω) ≤ C||p(t)||L∞ (Ω) | (t)|L2 (Ω) |∇ψ(t)|L2 (Ω)
dt
∂t
∂t
and
dE
(t) ≤ C||p(t)||L∞ (Ω) E(t).
dt
25
If
T
EM = E( ),
2
we have
T
T
EM ≤ E0 eC 2 ||p||L∞ (Q) and E0 ≤ EM eC 2 ||p||L∞ (Q) .
Let us choose
(4.0.5)
φM (x, t) = |x − x0 |2 − β(t −
T 2
) + M0 ,
2
where β is chosen such that
(4.0.6)
1>β>
4
sup |x − x0 |2
T 2 x∈Ω
and where M0 is chosen such that
∀(x, t) ∈ Ω × (−T, T ), φM (x, t) ≥ 1.
(4.0.7)
We have
∀t ∈ [0, T ], φM (x, t) ≤ φM (x,
and
φM (x,
so that
(4.0.8)
∃η > 0, ∀t ∈ [
T
)
2
T
) = |x − x0 |2 + M0 > M0
2
T
T
− η, + η], ∀x ∈ Ω, φM (x, t) ≥ M0 .
2
2
Now
T2
+ M0 .
4
Therefore, with the choice of β and the hypothesis on T
φM (x, 0) = φM (x, T ) = |x − x0 |2 − β
∃δ > 0, ∀t ∈ [0, δ] ∪ [T − δ, T ], φM (x, t) ≤ M0 .
(4.0.9)
Let us now define θδ as a function in C ∞ ([0, T ]) such that
∀t ∈ [0, T ], 0 ≤ θδ (t) ≤ 1, θδ (0) = θδ (1) = 0, ∀t ∈ [δ, T − δ], θδ (t) = 1.
We set
z(x, t) = θδ (t)ψ(x, t).
We have


∂θδ ∂ψ ∂ 2 θδ
∂2z


−
∆z
+
p(x)z
=
2
+
ψ, (x, t) ∈ Ω × (0, T ),
Lz
=


∂t2
∂t ∂t
∂t2
(4.0.10)
z = 0, (x, t) ∈ Γ × (0, T ),



∂z
∂z

 z(0) = z(T ) = 0,
(0) =
(T ) = 0, x ∈ Ω.
∂t
∂t
26
If we denote for λ > 0
ϕM (x, t) = eλφM (x,t)
we can apply Carleman’s inequality (2.0.8) to z on the interval (0, T ). There exist
s0 , λ0 and C such that for every s ≥ s0 , λ ≥ λ0 we have
Z TZ
e2sϕM ϕM ((
sλ
0
Ω
≤C
∂z 2
) + |∇z|2 )dxdt + s3 λ3
∂t
Z TZ
0
e2sϕM |Lz|2 dxdt + Csλ
Z TZ
0
Ω
e2sϕM ϕ3M |z|2 dxdt
Z TZ
0
Ω
e2sϕM |
Γ0
∂z 2
| dσdt.
∂ν
Using Poincare inequality and the fact that derivatives of θδ vanish on [δ, T − δ] we
have
Z TZ
0
Z δZ
∂z 2
) + |∇z|2 )dxdt
∂t
0
Ω
Z T Z
∂z
λM0
+C1
e2se
(( )2 + |∇z|2 )dxdt
∂t
T −δ Ω
e2sϕM |Lz|2 dxdt ≤ C1
Ω
λM0
e2se
((
≤ 2δC1 e2se
λM0
T
EM eC||p||L∞ (Q) 2 .
On the other hand, we have
Z TZ
e
0
Ω
2sϕM
∂z
ϕM (( )2 + |∇z|2 )dxdt ≥
∂t
≥ 2e2se
λM0
Z
T
2
T
2
T
2
Z
T
2
+η
+η
Z
−η
λM0
e2se
Ω
((
∂z 2
) + |∇z|2 )dxdt
∂t
λM0
E(t)dt ≥ 4ηe2se
−η
EM e−C||p||L∞ (Q) η .
Therefore we obtain for s ≥ s0 and λ ≥ λ0
λM0
4sληe2se
EM e−C||p||L∞ (Q) η ≤ 2δC1 e2se
Z T
+Csλ
0
Dividing by e2se
λM0
T
λM0
EM eC||p||L∞ (Q) 2
Z
∂z
e2sϕM | |2 dσdt
∂ν
Γ0
and choosing sλ large enough we obtain
EM ≤ C(T, p)
Z TZ
0
with
C(T, p) ∼
λM0 )
e2s(ϕM −e
Γ0
|
∂z 2
| dσdt
∂ν
C C||p||L∞ (Q) η
e
2ηsλ
27
and
sλ ∼
δ C||p||L∞ (Q) ( T +η)
2
.
e
η
We can now obtain the observability inequality (with explicit dependence of the
constants) which gives the result of Theorem 4.0.4
T
E0 ≤ C(T, p)eC||p||L∞ (Q) 2
Z TZ
0
λM0 )
e2s(ϕM −e
Γ0
|
∂ψ 2
| dσdt
∂ν
Using now the classical result of Lions’ H.U.M. we obtain as a consequence the exact
controllability result for equation (4.0.1)
Theorem 4.0.5 We assume that
• ∃x0 ∈ IRN \ Ω with Γ0 ⊃ Γx0 = {x ∈ Γ, (x − x0 ).ν(x) > 0}
• T > 2 supx∈Ω |x − x0 |
• p ∈ L∞ (Q).
Then for every (u0 , u1 ) ∈ L2 (Ω)×H −1 (Ω) there exists a control v ∈ L2 (0, T ; L2 (Γ0 ))
such that the solution u of (4.0.1) satisfies
u(T ) = 0 and
∂u
(T ) = 0.
∂t
Remark 4.0.6 Observability inequalities (or inverse inequalities) have been obtained in several contexts. The most general result has been proved in [1] using
microlocal analysis but the constants which appear in this result are not explicit. On
the other hand, observability inequalities with explicit constants have been obtained
using the multiplier method when the potential p is absent and with the same hypotheses on Γ0 and T as the ones we make here (see [13] and also [12]). The same
result is proved in a more general geometrical situation in [14] using again the multiplier method but with a non standard multiplier. When the potential p is bounded
but does not depend on the time variable, these results can be extended easily using
a compactness uniqueness argument at the price of loosing the explicit dependence
of the constants. It would be very interesting to obtain a direct result with explicit
constants, using for example other Carleman estimates, in the geometrical situation
considered in [14] but with bounded potentials p depending on both x and t.
28
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