Arthur Bottois
email: arthur.bottois@uca.fr

Nicolae Cîndea
email: nicolae.cindea@uca.fr

Arnaud Münch
email: arnaud.munch@uca.fr

Uniform observability of the one-dimensional wave equation for non-cylindrical domains. Application to the control's support optimization

Uniform observability of the one-dimensional wave equation for non-cylindrical domains. Application to the control's support optimization

Arthur Bottois, Nicolae Cindea, Arnaud Munch

Introduction

This work is concerned with the distributed controllability of the one-dimensional wave equation. We define the space domain Ω = (0, 1), the controllability time T > 0 and the space-time domain Q T = Ω × (0, T ), with Σ T = ∂Ω × (0, T ). Moreover, in the sequel we shall denote by L = ∂ 2 t -∂ 2

x the one-dimensional wave operator.

The controllability problem for the one-dimensional wave equation reads as follows: for a given control domain q ⊂ Q T , for every initial datum (y 0 , y 1 ) ∈ V := H 1 0 (Ω) × L 2 (Ω), find a control v ∈ L 2 (q) such that the corresponding solution of the wave equation

     Ly = v1 q in Q T , y = 0
on Σ T , (y, y t )(•, 0) = (y 0 , y 1 ) in Ω [START_REF] Brouwer | Spectra of graphs[END_REF] satisfies (y, y t )(•, T ) = (0, 0) in Ω.

(

) 2
The application 1 q denotes the characteristic function of q. We recall that for every (y 0 , y 1 ) ∈ V and v ∈ L 2 (q), there exists a unique solution y to [START_REF] Brouwer | Spectra of graphs[END_REF] with the regularity y ∈ C([0, T ]; H 1 0 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) (see for instance [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]).

In the cylindrical case, i.e. when q = ω × (0, T ), with ω ⊂ Ω an open non empty interval, the exact controllability of (1) holds for controllability time T greater than a critical time T * , related to the measure of the set Ω \ ω. In the non-cylindrical case, the controllability of [START_REF] Brouwer | Spectra of graphs[END_REF] has been established in [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF] if the control domain q satisfies the usual geometric optics condition (we also refer to [START_REF] Rousseau | Geometric control condition for the wave equation with a time-dependent observation domain[END_REF][START_REF] Shao | On Carleman and observability estimates for wave equations on time-dependent domains[END_REF] for results in any dimension). We recall that a domain q verifies such condition if every characteristic line of the wave equation, starting from a point of Ω × {0} and following the laws of geometric optics when reflected on the boundary Σ T , meets the domain q.

In both cylindrical and non-cylindrical cases, the controllability of (1) can be proven by the Hilbert uniqueness method (HUM) introduced by J.-L. Lions [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]. The main idea of this method is to obtain the controllability as a consequence of an observability inequality for the adjoint problem associated to [START_REF] Brouwer | Spectra of graphs[END_REF]: there exists a constant C obs (q) > 0 such that

(ϕ 0 , ϕ 1 ) 2 W ≤ C obs (q) ϕ 2 L 2 (q) , ∀(ϕ 0 , ϕ 1 ) ∈ W := L 2 (Ω) × H -1 (Ω), ( 3
)
where ϕ is the solution of the following homogeneous wave equation

     Lϕ = 0 in Q T , ϕ = 0
on Σ T , (ϕ, ϕ t )(•, 0) = (ϕ 0 , ϕ 1 ) in Ω. [START_REF] Chung | Spectral graph theory[END_REF] We recall that for every (ϕ 0 , ϕ 1 ) ∈ W, there exists a unique solution ϕ to (4) (defined in the sense of transposition) with the regularity ϕ ∈ C([0, T ]; L 2 (Ω)) ∩ C 1 ([0, T ]; H -1 (Ω)) (see, for instance, [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]). We emphasize that the observability constant C obs appearing in (3) depends on the observation domain q.

According to the HUM method, the control of minimal L 2 (q)-norm is obtained as the restriction to q of the solution ϕ of (4) corresponding to the initial datum (ϕ 0 , ϕ 1 ) which minimize the functional J (ϕ 0 , ϕ 1 ) = 1 2 q ϕ 2 -ϕ 1 , y 0 H -1 (Ω),H 1 0 (Ω) + ϕ 0 , y 1 L 2 (Ω) ∀(ϕ 0 , ϕ 1 ) ∈ W.

(

) 5
The existence and uniqueness of the minimum of the functional J over the space W are mainly consequences of the observability inequality [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF].

In the first part of this work, we provide a class of observation domains, based on the geometric condition considered in [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF]Proposition 2.1], for which the observability constant C obs in (3) is uniformly bounded. More precisely, for every ε > 0 small enough, we define the ε-interior of q by q ε = (x, t) ∈ q; d((x, t), ∂q) > ε , [START_REF] Coron | Mathematical Surveys and Monographs[END_REF] and the admissible set of control domains by

Q ε ad = q ⊂ Q T ;
q open and q ε verifies the geometric optics condition . [START_REF] Cui | Exact controllability for a one-dimensional wave equation in non-cylindrical domains[END_REF] We prove the following uniform observability inequality: there exists a constant C ε obs such that for every q ∈ Q ε ad , (ϕ 0 , ϕ 1 ) 2 W ≤ C ε obs ϕ 2 L 2 (q) , ∀(ϕ 0 , ϕ 1 ) ∈ W, [START_REF] Haak | Exact observability of a 1-dimensional wave equation on a noncylindrical domain[END_REF] where ϕ is the solution of (4) associated to the initial datum (ϕ 0 , ϕ 1 ). This uniform property then allows, in a second part, to analyze the problem of the optimal distribution of the control domain q. Precisely, we consider controls acting on a horizontal neighborhood of a regular curve: for a given half-weight δ 0 > 0, we define the domain associated to the curve γ : (0, T ) → Ω by

q γ = (x, t) ∈ Q T ; |x -γ(t)| < δ 0 . ( 9
)
The curves γ are chosen in the following set

G ad = γ ∈ W 1,∞ (0, T ); γ L ∞ (0,T ) ≤ M, δ 0 ≤ γ ≤ 1 -δ 0 (10)
consisting of uniformly Lipschitz functions of fixed constant M > 0. For T ≥ 2 and ε > 0 small enough, the class {q γ ; γ ∈ G ad } is a subset of Q ε ad . The optimization problem we shall consider reads as follows: for a given initial datum (y 0 , y 1 ) ∈ V, solve inf

γ∈G ad v 2 L 2 (qγ ) , ( 11
)
where v is the control of minimal L 2 (q γ )-norm distributed over q γ ⊂ Q T . Controllability of partial differential equations by means of moving controls, although less studied than the cylindrical case, becomes more and more popular in the literature. One of the first contributions for the wave equation is due to Khapalov [START_REF] Khapalov | Controllability of the wave equation with moving point control[END_REF]. The author proved an observability inequality for the one-dimensional wave equation with moving pointwise observation. This time-dependent observation allows to avoid the issue of strategic observation points and to get uniform controllability. More recently, the works [START_REF] Castro | Exact controllability of the 1-D wave equation from a moving interior point[END_REF][START_REF] Cui | Exact controllability for a one-dimensional wave equation in non-cylindrical domains[END_REF][START_REF] Haak | Exact observability of a 1-dimensional wave equation on a noncylindrical domain[END_REF] addressed the controllability for the one-dimensional case. The controllability of the one-dimensional wave equation is proved for controls acting on an interior curve and on a moving boundary, by using d'Alembert's formula and the multiplier method respectively. For the N -dimensional case, in [START_REF] Liu | Rapid exact controllability of the wave equation by controls distributed on a time-variant subdomain[END_REF] the authors employ the multiplier method to prove that the wave equation is controllable using a control acting on a time-dependent domain q, under the hypothesis that this domain covers the whole space domain before the control time T . Under similar hypotheses, we also mention the work [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF] where the control of the damped wave equation y tt -y xx -εy txx = 0 defined on the 1D torus is obtained in a non-cylindrical case. Because of the presence of an essential spectrum, such property does not hold true in the cylindrical case. Moreover, assuming the standard geometric optics condition, the observability inequality has been obtained in [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF] in dimension one by way of d'Alembert's formula, and extended in [START_REF] Rousseau | Geometric control condition for the wave equation with a time-dependent observation domain[END_REF] to the multi-dimensional case using microlocal analysis. It is also worth mentioning the obtention of Carleman type inequality for general hyperbolic equations in [START_REF] Shao | On Carleman and observability estimates for wave equations on time-dependent domains[END_REF].

On the other hand, in the cylindrical situation, the uniform observability property for the wave equation with respect to the observation domain is addressed in [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF]. For T ≥ 2, the author proves, using Fourier series, a uniform observability inequality for domains of the form q = ω × (0, T ), with ω ⊂ Ω an open set of fixed length. The uniform property is then employed to analyze the optimal position of the support of the corresponding null control. This problem of the optimal shape and position of the support is also numerically investigated in [START_REF] Münch | Optimal design of the support of the control for the 2-D wave equation: a numerical method[END_REF][START_REF]Optimal location of the support of the control for the 1-D wave equation: numerical investigations[END_REF] for the one and two dimensional wave equation. In a similar context, we also mention [START_REF] Humbert | Observability properties of the homogeneous wave equation on a closed manifold[END_REF] and the references therein.

This paper is organized as follows. In Section 2 we prove the uniform observability inequality (8) on Q ε ad and its variant on the subset G ad . This is achieved by defining an appropriate decomposition of the observation domains in Q ε ad , and by using d'Alembert's formula. The proof also relies on arguments from graph theory. Then, in Section 3, following arguments from [START_REF] Henrot | Variation et optimisation de formes[END_REF][START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF], we analyze a variant of the extremal problem [START_REF] Humbert | Observability properties of the homogeneous wave equation on a closed manifold[END_REF]. Introducing a C 1 -regularization of the support q γ , we prove that the underlying cost is continuous over G ad for the L ∞ (0, T )-norm, and admits at least one local minimum. Section 4 is concerned with numerical experiments. Minimization sequence for the regularized cost are constructed using a gradient method: each iteration requires the computation of a null control, performed using the space-time formulation developed in [START_REF] Cîndea | A mixed formulation for the direct approximation of the control of minimal L 2 -norm for linear type wave equations[END_REF] and used in [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF], and well-suited to the description of the non-cylindrical domains, where the control acts.

Uniform observability with respect to the domain of observation

We prove in this section the uniform observability inequality [START_REF] Haak | Exact observability of a 1-dimensional wave equation on a noncylindrical domain[END_REF] with respect to the domain of observation. Precisely, we prove the following equivalent result for regular data in V.

Theorem 1. Let T > 0 and let ε > 0 be a small enough fixed parameter such that the set Q ε ad defined by [START_REF] Cui | Exact controllability for a one-dimensional wave equation in non-cylindrical domains[END_REF] is non-empty. There exists a constant C ε obs > 0 such that for every q ∈ Q ε ad , the following inequality holds

(ϕ 0 , ϕ 1 ) 2 V ≤ C ε obs ϕ t 2 L 2 (q) , ∀(ϕ 0 , ϕ 1 ) ∈ V, ( 12
)
where ϕ is the solution of the wave equation (4) associated to the initial datum (ϕ 0 , ϕ 1 ).

In the remaining part of this section, we assume that the hypotheses of Theorem 1 are satisfied.

Some notations and technical lemmas

We first introduce some notations and state some preliminary lemmas.

Let N > 0 be an integer and κ N = 1/N . We denote S N = (x N i ) 0≤i≤N a regular subdivision of Ω in N intervals, i.e. for every i ∈ {0, . . . , N }, we set x N i = i/N . Associated to the functions ϕ 0 ∈ H 1 0 (Ω) and ϕ 1 ∈ L 2 (Ω), we define the continuous function ϕ N 0 affine on intervals [x N i-1 , x N i ] of the subdivision S N and the function ϕ N 1 constant on intervals of S N :

ϕ N 0 (x) = N i=1 ϕ 0 (x N i ) x -x N i-1 κ N + ϕ 0 (x N i-1 )
x

N i -x κ N 1 [x N i-1 ,x N i ] (x), (13)
ϕ N 1 (x) = N i=1 β N i 1 [x N i-1 ,x N i ] (x), with β N i = 1 κ N x N i x N i-1 ϕ 1 . ( 14
)
We also denote by (ϕ N 0 ) ∈ L 2 (Ω) the "derivative" of ϕ N 0 :

(ϕ N 0 ) (x) = N i=1 α N i 1 [x N i-1 ,x N i ] (x), with α N i = ϕ 0 (x N i ) -ϕ 0 (x N i-1 ) κ N . ( 15
)
Using that ϕ 0 ∈ H 1 0 (Ω), we then easily check that

ϕ N 0 2 H 1 0 (Ω) = 1 N N i=1 (α N i ) 2 , ϕ N 1 2 L 2 (Ω) = 1 N N i=1 (β N i ) 2 and N i=1 α N i = 0. ( 16
)
In order to use d'Alembert's formula for the solution of the wave equation ( 4) associated to initial datum (ϕ N 0 , ϕ N 1 ) of the form ( 13)-( 14), we need to extend these functions to odd functions to [-1, 1] and then by 2-periodicity to R. In this respect, we first extend the definition of x N i to i ∈ Z by putting x N i = i/N for every i ∈ Z, and then denote by I N i for every i ∈ Z * the following interval:

I N i =    [x N i-1 , x N i ] if i > 0, [x N i , x N i+1 ] if i < 0. (17)
Similarly, we extend α N i and β N i for every i ∈ Z * as follows: if i ∈ {-N, . . . , -1}, we set

α N i = α N -i and β N i = -β N -i ; if |i| > N
, the definitions of α N i and β N i are a little more complex:

α N i = α N j N (i) , β N i = β N j N (i) ,
with j N (i) defined for every integer i ≥ 1 by

j N (i) = (i -1) mod (2N ) + 1 if (i -1) mod (2N ) < N, (i -1) mod (2N ) -2N if (i -1) mod (2N ) ≥ N, ( 18
)
and j

N (i) = -j N (-i) if i ≤ -1.
Remark that for every i ∈ Z * , we have j N (i) ∈ I N with the set

I N given by I N = {-N, . . . , -1, 1, . . . , N }. ( 19
)
For i, j ∈ I N , we also define

γ N i = α N i + β N i . ( 20
)
We extend the functions ϕ N 0 and ϕ N 1 to odd functions on [-1, 1] and by 2-periodicity to R. Then, using the notations above, we obtain

(ϕ N 0 ) (x) = i∈Z * α N i 1 I N i (x), ϕ N 1 (x) = i∈Z * β N i 1 I N i (x), ∀x ∈ R. (21)
Furthermore, from d'Alembert's formula, the solution ϕ N of (4) associated to the initial datum (ϕ N 0 , ϕ N 1 ) is given as follows:

ϕ N (x, t) = 1 2 ϕ N 0 (x + t) + ϕ N 0 (x -t) + 1 2 x+t x-t ϕ N 1 , ∀(x, t) ∈ Q T . ( 22
)
Taking the derivative with respect to t and replacing the expressions [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF] in the above equation, we deduce that for all (x, t) ∈ Q T , we have

ϕ N t (x, t) = 1 2 (ϕ N 0 ) (x + t) -(ϕ N 0 ) (x -t) + ϕ N 1 (x + t) + ϕ N 1 (x -t) = 1 2 i∈Z * (α N i + β N i )1 I N i (x + t) -(α N i -β N i )1 I N i (x -t) = 1 2 i∈Z * j∈Z * (α N i + β N i -α N j + β N j )1 I N i (x + t)1 I N j (x -t). ( 23
)
Using the properties of the function j N defined in (18), we deduce that for i, j ∈ Z * ,

α N i + β N i -α N j + β N j = α N j N (i) + β N j N (i) -α N j N (j) + β N j N (j) = (α N j N (i) + β N j N (i) ) -(α N -j N (j) + β N -j N (j) ) = γ N j N (i) -γ N -j N (j) . ( 24
)
In view of the expression (23), we introduce the following definition.

Definition 1.

For every i, j ∈ Z * , the elementary square of indices (i, j) associated to the subdivision S N is defined as the following closed set of R 2 :

C N (i,j) = (x, t) ∈ R 2 such that x + t ∈ I N i and x -t ∈ I N j , ( 25
)
where for every i ∈ Z * , the interval I N i is given by [START_REF] Mohar | The Laplacian spectrum of graphs[END_REF]. We denote by C N = {C N (i,j) ; i, j ∈ Z * } the set of all the elementary squares associated to the subdivision S N of Ω. It is easy to see that

R 2 = i,j∈Z * C N (i,j) .
Figure 1 illustrates the way the elementary squares are indexed, using elementary squares associated to the subdivision S 4 of Ω.

T = 2 q ε R 8 (q)
q Figure 2: Cover R 8 (q) of q ε , for ε = 0.15.

Remark 1.

For every i, j ∈ Z * , the coordinates of the center of the elementary square C N (i,j) associated to the subdivision S N are given by

   x N (i,j) = m N i +m N j 2 , t N (i,j) = m N i -m N j 2 , with m N i =    x N i-1 +x N i 2 if i > 0, x N i +x N i+1 2 if i < 0. ( 26
)
The area of every elementary square

C N (i,j) ∈ C N is given by |C N (i,j) | = 1 2N 2 .
Notice that for every i, j ∈ Z * with |i|, |j| > 1, the elementary squares having one side in common with the elementary square C N (i,j) are C N (i±1,j) and C N (i,j±1) .

Definition 2. For every q ∈ Q ε ad , we denote by C N (q) and C N (Q T ) the sets of the elementary squares in C N with their interior included in q and Q T respectively:

C N (q) = C N (i,j) ∈ C N ; • C N (i,j) ⊂ q , C N (Q T ) = C N (i,j) ∈ C N ; • C N (i,j) ⊂ Q T . ( 27
)
If N is large enough, the sets C N (q) and C N (Q T ) are non-empty. We also define R N (q) the union of the elementary squares in C N (q):

R N (q) = • C N (i,j) ∈C N (q) C N (i,j) . ( 28
)
With these notations, we can now prove the following lemma.

Lemma 1. Let N > 1/ε be a fixed integer. For every q ∈ Q ε ad , the set C N (i,j) ∈C N (q) C N (i,j) is a cover of q ε given by [START_REF] Coron | Mathematical Surveys and Monographs[END_REF]. Moreover, the set R N (q) defined by (28) satisfies q ε ⊂ R N (q) ⊂ q.

Proof. Let X ∈ q ε . Using the definition of q ε , we have X ∈ q and d(X, ∂q) > ε. Since R 2 is covered by squares in C N , there exists

C N (i,j) ∈ C N such that X ∈ C N (i,j) . Moreover, since diam(C N (i,j) ) = κ N , we have C N (i,j) ⊂ B(X, κ N ). Let Y ∈ B(X, κ N ). Then, for every Z ∈ R 2 \ q, it holds that d(Y, Z) ≥ |d(Y, X) -d(X, Z)| > ε -κ N > 0. Consequently, d(Y, R 2 \ q) > 0, which implies Y ∈ q. Therefore, C N (i,j) ⊂ B(X, κ N ) ⊂ q and, finally, C N (i,j) ∈ C N (q).
Figure 2 illustrates Lemma 1 in the case of the cylindrical observation domain q = ( 5 16 , 11 16 ) × (0, 2), for ε = 0.15 and N = 8. In order to write several expressions in a simpler form, we use the following graph theory framework. Definition 3. Let q ∈ Q ε ad an observation domain. We define the weighted graph G N (q) as follows:

• I N given by ( 19) is the set of vertices;

• for every i ∈ I N , the degree of the vertex i is given by:

d N i = Card C N (k,-l) ∈ C N (q); i ∈ {j N (k), j N (l)} ;
• for every i, j ∈ I N , the weight of the edge linking the vertices i and j is

w N i,j = w N j,i = Card C N (k,-l) ∈ C N (q); {i, j} = {j N (k), j N (l)} .
Definition 4. Let q ∈ Q ε ad and let i, j ∈ I N be two vertices of the graph G N (q). We say that there is a path in G N (q) from i to j and we denote i N ∼ j if the vertices i and j are in the same connected component of G N (q). In particular, if w N i,j = 0, then i N ∼ j.

We then recall the definition of the Laplacian matrix associated to a graph.

Definition 5. Let q ∈ Q ε ad . The Laplacian matrix associated to the graph G N (q) (see Definition 3) is the symmetric positive matrix A N (q) ∈ M 2N (R) defined by

A N (q) =             d N -N • • • -w N -N,-1 -w N -N,1 • • • -w N -N,N . . . . . . . . . . . . . . . -w N -1,-N • • • d N -1 -w N -1,1 • • • -w N -1,N -w N 1,-N • • • -w N 1,-1 d N 1 • • • -w N 1,N . . . . . . . . . . . . . . . -w N N,-N • • • -w N N,-1 -w N N,1 • • • d N N             2N ×2N . ( 29
)
Remark 2. Remark that for every q ∈ Q ε ad , the graph G N (q) has no loop, i.e. w N i,i = 0 for every i ∈ I N . Indeed, the elementary squares

C N (k,-l) such that j N (k) = j N (l) = i have their centers x N (k,-l) ∈ Z and, consequently, cannot be in Q T .
Remark also that the Laplacian matrix A N (q) of the graph G N (q) verifies the following property (see [START_REF] Brouwer | Spectra of graphs[END_REF][START_REF] Chung | Spectral graph theory[END_REF]

) : for every η = (η -N , . . . , η -1 , η 1 , . . . , η N ) ∈ R 2N , η T A N (q)η = i∈I N d N i η 2 i - i,j∈I N w N i,j η i η j = C N (i,j) ∈C N (q) (η j N (i) -η -j N (j) ) 2 . ( 30
)
From now on, we consider that the assumption of Lemma 1 holds true, i.e. we take N > 1/ε. More precisely, we fix N the smallest integer strictly greater than ε -1 . Lemma 2. Let q ∈ Q ε ad , so q ε verifies the usual geometric optics condition. Then the associated graph G N (q) is connected.

Proof. Let i ∈ {1, . . . , N -1}. We denote by D + i the support of the characteristic line "x + t = x N i ", starting from x N i in the direction of decreasing x and following the rules of geometric optics for its reflexion on Σ T . Since q ε satisfies the geometric optics condition, there exists (x * , t * ) ∈ q ε ∩ D + i . From Lemma 1, we have q ε ⊂ R N (q), so (x * , t * ) belongs to the common side of two elementary squares in C N (q):

C N (k,l) and C N (k+1,l) with j N (k) = i or C N (l,k) and C N (l,k-1) with j N (k) = -i .
Therefore i N ∼ i + 1 and, so, the vertices {1, . . . , N } are in the same connected component. We denote by D - i the support of the characteristic line "x -t = x N i ", starting from x N i in the direction of increasing x and following the rules of geometric optics for its reflexion on Σ T . Since q ε satisfies the geometric optics condition, there exists (x * , t * ) ∈ q ε ∩ D - i . From Lemma 1, we have q ε ⊂ R N (q), so (x * , t * ) belongs to the common side of two elementary squares in C N (q):

C N (l,k) and C N (l,k+1) with j N (k) = i or C N (k,l) and C N (k-1,l) with j N (k) = -i .
Therefore -i N ∼ -i-1 and, so, the vertices {-N, . . . , -1} are in the same connected component. In order to finish the proof, it remains to show that the vertices N and -N belong to the same connected component. We denote by D + N the support of the characteristic line "x+t = x N N ", starting from x N N in the direction of decreasing x and following the rules of geometric optics for its reflexion on Σ T . Since q ε satisfies the geometric optics condition, there exists (x * , t * ) ∈ q ε ∩ D + N . From Lemma 1, we have q ε ⊂ R N (q), so (x * , t * ) belongs to the common side of two elementary squares in C N (q):

C N (k,l) and C N (k+1,l) with j N (k) = N or C N (l,k) and C N (l,k-1) with j N (k) = -N .
Hence, N N ∼ -N .

Remark 3. A well known graph theory result (see, for instance, [1, Proposition 1.3.7]) states that the graph G

N (q) is connected if and only if dim(ker(A N (q))) = 1. Moreover, if G N (q) is connected, then ker(A N (q)) = Vect(1 2N )
, where 1 2N is the vector in R 2N with all its component equal to 1. Let us denote λ N (q) > 0 the smallest non-zero eigenvalue of the matrix A N (q). This eigenvalue is known in graph theory as the algebraic connectivity of the graph. We also define λ N by λ N = min

q∈Q ε ad λ N (q) > 0. ( 31
)
Note that since the set {G N (q); q ∈ Q ε ad } has a finite number of elements, λ N is well defined. Definition 6. For every p ∈ N * , we denote by C p N (q) the set formed by the elementary squares associated to the subdivision S pN having their interior in R N (q):

C p N (q) = C pN (i,j) ∈ C pN ; • C pN (i,j) ⊂ R N (q) . ( 32
)
We then define the graph G p N (q) following Definition 3, substituting N by pN , and substituting C N (q) by C p N (q) in the definitions of the vertex degrees and the edge weights. Finally, we denote A p N (q) ∈ M 2pN (R) the Laplacian matrix associated to the graph G p N (q). This matrix has the following block form:

A p N (q) =             d N -N pI p • • • -w N -N,-1 J p -w N -N,1 J p • • • -w N -N,N J p . . . . . . . . . . . . . . . -w N -1,-N J p • • • d N -1 pI p -w N -1,1 J p • • • -w N -1,N J p -w N 1,-N J p • • • -w N 1,-1 J p d N 1 pI p • • • -w N 1,N J p . . . . . . . . . . . . . . . -w N N,-N J p • • • -w N N,-1 J p -w N N,1 J p • • • d N N pI p             2pN ×2pN , ( 33
)
where I p , J p ∈ M p (R) are respectively the identity matrix and the matrix with all its elements equal to 1. Moreover, for every η = (η -pN , . . . , η -1 , η 1 , . . . , η pN ) ∈ R 2pN , we have

η T A p N (q)η = C N (i,j) ∈C N (q) i ∈J p i j ∈J p j (η j pN (i ) -η -j pN (j ) ) 2 , ( 34
)
with J p i defined in (40).

For any p ∈ N * , the following lemma makes the link between the spectrum of the Laplacian matrix A p N (q) (see Definition 6) and the spectrum of the Laplacian matrix A N (q) (see Definition 5). Lemma 3. Let p ∈ N * . The spectrum of the Laplacian matrix 1 p A p N (q) (see (33)) is composed of the spectrum of the Laplacian matrix A N (q) (see (29)), and the diagonal elements of A N (q) repeated p -1 times. Moreover, dim(ker(A p N (q))) = 1 and ker(A p N (q)) = Vect(1 2pN ).

Proof. Let η = (η -pN , . . . , η -1 , η 1 , . . . , η pN ) ∈ R 2pN . For every i ∈ I N , we denote by Γ i = (η i ) i ∈J p i ∈ R p and we group these vectors in the matrix

Γ = (Γ -N | • • • |Γ -1 |Γ 1 | • • • |Γ N ) ∈ M p,2N (R).
In view of (33), it follows that

η T A p N (q)η = p i∈I N d N i Γ T i Γ i - i,j∈I N w N i,j Γ T i J p Γ j .
Since J p is a real symmetric matrix, there exists an orthonormal basis

(b k ) 1≤k≤p of R p diago- nalizing J p . Let us denote b 1 = 1 √ p 1 p .
Then, there exists a diagonal matrix D ∈ M p (R) and a unitary matrix Q ∈ M p (R) such that J p = QDQ T . These matrices have the following form:

D =       p 0 • • • 0 0 0 • • • 0 . . . . . . . . . . . . 0 0 • • • 0       p×p and Q = (b 1 | • • • |b p ) p×p .
We also define the matrix U = Q T Γ ∈ M p,2N (R), and denote respectively

U k,. = (b T k Γ i ) i∈I N ∈ R 2N and U .,i = (b T k Γ i ) 1≤k≤p ∈ R p
the rows and the columns of U . Then, for i, j ∈ I N , we have

Γ T i Γ i = U T .,i U .,i = p k=1 U 2 k,i and Γ T i J p Γ j = U T .,i DU .,j = pU 1,i U 1,j .
The spectrum of the matrix 1 p A p N (q) can now be computed from

1 p η T A p N (q)η = i∈I N d N i U 2 1,i - i,j∈I N w N i,j U 1,i U 1,j + p k=2 i∈I N d N i U 2 k,i = U T 1,. A N (q)U 1,. + p k=2 U T k,. Diag(A N (q))U k,. .
Indeed, the expression above shows that A p N (q) is unitarily similar to the block diagonal matrix

      A N (q) Diag(A N (q)) . . . Diag(A N (q))       2pN ×2pN
.

Proof of Theorem 1

We are now in position, using all these notations and results, to prove the Theorem 1.

Proof of Theorem 1. We prove the theorem in three steps.

Step 1. Let N be the smallest integer strictly greater than ε -1 . The first step is to prove an observability inequality for the function ϕ N given by ( 22) and associated to the initial datum ( 13)- [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF].

In view of ( 23) and (24), remark that the function (ϕ N t ) 2 is constant on each elementary square

C N (i,j) in C N : (ϕ N t ) 2 | C N (i,j) = 1 4 (γ N j N (i) -γ N -j N (j) ) 2 , ( 35
)
where γ N j N (i) is given by (20) and j N by [START_REF] Münch | Optimal design of the support of the control for the 2-D wave equation: a numerical method[END_REF]. Using the definition (28) of the set R N (q), we can minorate the L 2 -norm of ϕ N t restricted to q as follows:

q (ϕ N t ) 2 ≥ R N (q) (ϕ N t ) 2 = C N (i,j) ∈C N (q) C N (i,j) (ϕ N t ) 2 = 1 8N 2 C N (i,j) ∈C N (q) (γ N j N (i) -γ N -j N (j) ) 2 . ( 36
)
For the last equality, we used that the area of every elementary square in C N is 1 2N 2 . Combining (36) with the relation (30) in Remark 2, we obtain

q (ϕ N t ) 2 ≥ 1 8N 2 (γ N ) T A N (q)γ N , ( 37
) with γ N = (γ N -N , . . . , γ N -1 , γ N 1 , . . . , γ N N ) ∈ R 2N and γ N i
given by [START_REF]Numerical estimations of the cost of boundary controls for the equation y t -εy xx + M y x = 0 with respect to ε[END_REF].

It is easy to see that γ N ∈ ker(A N (q)) ⊥ . Indeed, applying Lemma 2, the graph G N (q) is connected. Then, from Remark 3, we have ker(A N (q)) = Vect(1 2N ) and, since α N i = α N -i and

β N i = -β N -i , the vector γ N verifies (γ N ) T 1 2N = i∈I N γ N i = 2 N i=1 α N i = 0.
Then, using λ N defined in (31), it follows that

(γ N ) T A N (q)γ N ≥ λ N i∈I N (γ N i ) 2 = 2λ N N i=1 (α N i ) 2 + (β N i ) 2 = 2N λ N ϕ N 0 2 H 1 0 (Ω) + ϕ N 1 2 L 2 (Ω) . ( 38
)
From ( 37) and (38), we deduce the following observability inequality :

(ϕ N 0 , ϕ N 1 ) 2 V ≤ 4N λ N ϕ N t 2 L 2 (q) , ( 39
)
where the constant 4N λ N is independent of the domain q and the initial datum (ϕ 0 , ϕ 1 ).

Step 2. For any p ∈ N * , the second step of the proof consists in obtaining a uniform observability inequality for an initial datum (ϕ pN 0 , ϕ pN 1 ) of the form ( 13)-( 14). More precisely, we aim to obtain a uniform inequality with respect to the domain q ∈ Q ε ad and the integer p ∈ N * .

Let p ∈ N * . As in the first step of the proof, we easily see that (ϕ pN t ) 2 is constant on every elementary square C pN (i ,j ) ∈ C pN :

(ϕ pN t ) 2 | C pN (i ,j ) = 1 4 (γ pN j pN (i ) -γ pN -j pN (j ) ) 2 ,
where ϕ pN is the solution of (4) associated to the initial datum (ϕ pN 0 , ϕ pN 1 ) given by ( 13)-( 14), and γ pN j pN (i ) is defined by [START_REF]Numerical estimations of the cost of boundary controls for the equation y t -εy xx + M y x = 0 with respect to ε[END_REF]. For any i ∈ Z * , we define the set J p i by

J p i =    {p(i -1) + 1, . . . , pi} if i > 0, {pi, . . . , p(i + 1) -1} if i < 0. (40)
Then, remark that every elementary square C N (i,j) ∈ C N is the union of p 2 elementary squares in C pN , or more precisely that

C N (i,j) = i ∈J p i j ∈J p j C pN (i ,j ) , ∀i, j ∈ Z * .
Using the above expression in the evaluation of the L 2 (q)-norm of ϕ pN , we have

q (ϕ pN t ) 2 ≥ R N (q) (ϕ pN t ) 2 = C N (i,j) ∈C N (q) C N (i,j) (ϕ pN t ) 2 = C N (i,j) ∈C N (q) i ∈J p i j ∈J p j C pN (i ,j ) (ϕ pN t ) 2 = 1 8p 2 N 2 C N (i,j) ∈C N (q) i ∈J p i j ∈J p j (γ pN j pN (i ) -γ pN -j pN (j ) ) 2 = 1 8p 2 N 2 (γ pN ) T A p N (q)γ pN . ( 41
)
Since the graph G N (q) is a connected graph, the degree d N i of every vertex i ∈ I N verifies d N i ≥ 1. Applying Lemma 3, the smallest non-zero eigenvalue λ p N (q) of 1 p A p N (q) verifies λ p N (q) = min(λ N (q), min

i∈I N d N i ) ≥ min(λ N , 1) > 0, (42)
so we set λ N = min(λ N , 1). The vector γ pN = (γ pN i ) i ∈I pN belongs to ker(A p N (q)) ⊥ . Indeed,

(γ pN ) T 1 2pN = i ∈I pN γ pN i = 2 pN i =1 α pN i = 0. It follows that 1 p (γ pN ) T A p N (q)γ pN ≥ λ N i ∈I pN (γ pN i ) 2 = 2 λ N pN i =1 (α pN i ) 2 + (β pN i ) 2 = 2pN λ N ϕ pN 0 2 H 1 0 (Ω) + ϕ pN 1 2 L 2 (Ω) .
Consequently, combining the above relation with (41), we obtain the following observability inequality

(ϕ pN 0 , ϕ pN 1 ) 2 V ≤ 4N λ N ϕ pN t 2 L 2 (q) , ∀p ∈ N (43)
with the observability constant 4N λ N independent of the domain q, the initial datum (ϕ 0 , ϕ 1 ) and the integer p.

Step 3. In order to finish the proof, we pass to the limit when p → ∞ in the observability inequality (43). It is easy to see that when p → ∞, we have the convergences

ϕ pN 0 → ϕ 0 in H 1 0 (Ω) and ϕ pN 1 → ϕ 1 in L 2 (Ω).
Moreover, since the solution ϕ of the wave equation ( 4) depends continuously on its initial condition (ϕ 0 , ϕ 1 ) ∈ V , we can write

ϕ pN t → ϕ t in L 2 (0, T ; L 2 (Ω)).
Finally, passing to the limit in (43), we get

(ϕ 0 , ϕ 1 ) 2 V ≤ max 4N, 4N λ N ϕ t 2 L 2 (q) , ∀(ϕ 0 , ϕ 1 ) ∈ V
which concludes the proof with C ε obs = max 4N, 4N λ N . We recall that N depends on ε by the condition N > 1/ε. Remark 4. Let q ⊂ Q T be a finite union of open sets. If q verifies the usual geometric optics condition, there exists ε > 0 small enough such that q ε still verifies the geometric optics condition. We then set N = 1/ε + 1. The associated graph G N (q) being connected, there exists a relation (see, for instance, [START_REF] Mohar | The Laplacian spectrum of graphs[END_REF]) between the algebraic connectivity λ N (q), the number of vertices N V and the diameter D G of the graph. More exactly, this relation is λ N (q) ≥ 4 N V D G . Since in our case N V = 2N and D G ≤ 2N , we deduce that λ N (q) ≥ 1 N 2 and therefore that C obs (q) ≤ 4N 3 . In the worst situation, we can have an observability constant of order 1/ε 3 . Therefore, if we consider ε as a measure of the "thickness" of the observation domain q, we find the estimation of the observability constant given in [21, Proposition 2.1].

One explicit example

We illustrate in this section the proof of Theorem 1 on a simple example for which the observation domain q depicted in Figure 3 (colored in red) is well adapted to the subdivision S 4 . The study of this example is also the opportunity to develop a method for the computation of the observability constant for observation domains which are exactly the union of elementary squares associated to a given subdivision S N , for a fixed integer N > 0.

We start by enumerating the elementary squares composing the observation domain q. In Table 1, we list, for i, j ∈ Z * , the elementary squares C 4 (i,j) included in q and the values of the indices j 4 (i) and -j 4 (j), allowing to compute the Laplacian matrix A 4 (q) associated to the corresponding graph G 4 (q). C 4 (i,j)

j 4 (i) -j 4 (j) C 4 (i,j) j 4 (i) -j 4 (j) C 4 (i,j) j 4 (i) -j 4 (j) C 4 (2,1) 2 -1 C 4 (6,-1) -3 1 C 4 (9,-4) 1 4 C 4 (2,-1) 2 1 C 4 (7,1) -2 -1 C 4 (8,-5) -1 -4 C 4 (3,1) 3 -1 C 4 (7,-1) -2 1 C 4 (9,-5) 1 -4 C 4 (3,-1) 3 1 C 4 (8,-1) -1 1 C 4 (8,-6) -1 -3 C 4 (4,1) 4 -1 C 4 (8,-2) -1 2 C 4 (9,-6) 1 -3 C 4 (4,-1) 4 1 C 4 (9,-2) 1 2 C 4 (8,-7) -1 -2 C 4 (5,1) -4 -1 C 4 (8,-3) -1 3 C 4 (9,-7) 1 -2 C 4 (5,-1) -4 1 C 4 (9,-3) 1 3 C 4 (6,1) -3 -1 C 4 (8,-4) -1 4
Table 1: Elementary squares associated to S 4 and belonging to C 4 (q).

The Laplacian matrix associated to the graph G 4 (q) is given by

A 4 (q) =                 4 0 0 -2 -2 0 0 0 0 4 0 -2 -2 0 0 0 0 0 4 -2 -2 0 0 0 -2 -2 -2 13 -1 -2 -2 -2 -2 -2 -2 -1 13 -2 -2 -2 0 0 0 -2 -2 4 0 0 0 0 0 -2 -2 0 4 0 0 0 0 -2 -2 0 0 4                 8×8 .
The spectrum of A 4 (q) can be explicitly computed:

Sp(A 4 (q)) = {0, 4, 4, 4, 4, 4, 14, 16}. It confirms that the kernel of A 4 (q) is one-dimensional -therefore G 4 (q) is connected -and implies that the smallest non-zero eigenvalue of A 4 (q) is λ 4 (q) = 4. If we replace the subdivision S 4 by the subdivision S 4p for any p ∈ N * , then the Laplacian matrix associated to the graph G p 4 (q) is the following one:

A p 4 (q) =                 4pI p 0 p 0 p -2J p -2J p 0 p 0 p 0 p 0 p 4pI p 0 p -2J p -2J p 0 p 0 p 0 p 0 p 0 p 4pI p -2J p -2J p 0 p 0 p 0 p -2J p -2J p -2J p 13pI p -J p -2J p -2J p -2J p -2J p -2J p -2J p -J p 13pI p -2J p -2J p -2J p 0 p 0 p 0 p -2J p -2J p 4pI p 0 p 0 p 0 p 0 p 0 p -2J p -2J p 0 p 4pI p 0 p 0 p 0 p 0 p -2J p -2J p 0 p 0 p 4pI p                 8p×8p .
According to Lemma 3, the smallest non-zero eigenvalue of 1 p A p 4 (q) is given by λ p 4 (q) = min(λ 4 (q), min

i∈I 4 d 4 i ) = min(4, 4) = 4.
Consequently, the observability constant associated to the observation domain q depicted in Figure 3 is given by

C obs (q) = 4 • 4 λ p 4 (q) = 4.

A corollary

We show in this section a uniform observability inequality for the observation domains q γ defined in [START_REF] Hecht | New development in Freefem++[END_REF], with γ ∈ G ad , which will be used in the next section.

Corollary 1. Let T ≥ 2.

There exists a constant C obs > 0 such that for every γ ∈ G ad ,

(ϕ 0 , ϕ 1 ) 2 W ≤ C obs ϕ 2 L 2 (qγ ) , ∀(ϕ 0 , ϕ 1 ) ∈ W, ( 44
)
where ϕ is the solution of the homogeneous wave equation (4) associated to the initial condition (ϕ 0 , ϕ 1 ).

Proof. We show that for any ε > 0 small enough,

{q γ ; γ ∈ G ad } ⊂ Q ε ad . Let γ ∈ G ad . We introduce the sets Γ ± = (γ(t) ± δ 0 , t); t ∈ [0, T ] , Γ ± = (γ(t) ± δ 0 2 , t); t ∈ [0, T ] and Q ε T = Ω × (ε, T -ε). γ being a M -Lipschitz curve, we can show that d( Γ ± , Γ ± ) ≥ δ 0 2 √ M 2 + 1 .
Then, for ε <

δ 0 2 √ M 2 +1 , we have q γ ∩ Q ε T ⊂ q ε γ
, with the observation domain q γ defined as in ( 9) with a half-width of δ 0 /2. The domain q γ ∩ Q ε T verifies the geometric optics condition because δ 0 ≤ γ ≤ 1 -δ 0 and T -2ε ≥ 2(1 -δ 0 ). Consequently, q ε γ also verifies the geometric optics condition and q γ ∈ Q ε ad . We conclude the proof by noticing that the constant

δ 0 2 √ M 2 +1 is independent of the choice of γ.

Optimization of the shape of the control domain

In this section, we study the problem of finding the optimal shape and position of the control domain, for a given initial condition (y 0 , y 1 ) ∈ V.

Existence of an optimal domain

In order to show a well-posedness result, we consider a variant of the optimal problem [START_REF] Humbert | Observability properties of the homogeneous wave equation on a closed manifold[END_REF] and replace the characteristic function 1 q in (1) by a more regular function in space. More precisely, we fix δ ∈ (0, δ 0 ) and, for every γ ∈ G ad , we define χ γ (x, t) = χ(x -γ(t)), with χ : R → [0, 1] a C 1 even function such that

χ(x) =      1 if x ∈ (-δ 0 + δ, δ 0 -δ), 0 if x / ∈ (-δ 0 , δ 0 ), ∈ (0, 1) otherwise. ( 45
)
In the sequel, we will also use the function χ γ defined by χ γ (x, t) = χ (x -γ(t)). In this new setting, the HUM control now lives in the weighted space

L 2 χ (q γ ) := L 2 (q γ ; χ γ ) = v : q γ → R; qγ v 2 χ γ < +∞ .
Moreover, we can adapt the uniform observability inequality given in Corollary 1. For T ≥ 2, there exists a constant C obs > 0 such that for every γ ∈ G ad ,

(ϕ 0 , ϕ 1 ) 2 W ≤ C obs ϕ 2 L 2 χ (qγ ) , ∀(ϕ 0 , ϕ 1 ) ∈ W, ( 46
)
where ϕ is the solution of ( 4) associated to the initial condition (ϕ 0 , ϕ 1 ). Then, our optimization problem reads as follows: for a given initial datum (y 0 , y 1 ) ∈ V, solve inf

γ∈G ad J(γ) = v 2 L 2 χ (qγ ) = qγ ϕ 2 χ γ , ( 47
)
where v is the control of minimal L 2 χ -norm distributed over q γ ⊂ Q T , and ϕ is the associated adjoint state such that v = ϕ| qγ . This adjoint state can be characterized using the HUM method, it is the solution of (4) associated to the minimum (ϕ 0 , ϕ 1 ) of the conjugate functional

J γ (ϕ 0 , ϕ 1 ) = 1 2 qγ ϕ 2 χ γ -ϕ 1 , y 0 H -1 (Ω),H 1 0 (Ω) + ϕ 0 , y 1 L 2 (Ω) , ∀(ϕ 0 , ϕ 1 ) ∈ W. ( 48
)
To show the well-posedness of (47), we follow the steps of [21, Theorem 2.1]. We start with a convergence result on the function χ γ .

Lemma 4. Let (γ n ) n≥0 ⊂ G ad and γ ∈ G ad . If γ n → γ in L ∞ (0, T ), then χ γn → χ γ in L ∞ (Q T ).
Proof. It is a direct consequence of the Taylor's inequality applied to χ. Indeed, this inequality gives

χ γn -χ γ L ∞ (Q T ) ≤ χ L ∞ (R) γ n -γ L ∞ (0,T ) .
We then have that the following continuity result.

Proposition 1. The cost J is continuous over G ad for the norm L ∞ (0, T ).

Proof. Let (γ n ) n≥0 ⊂ G ad and γ ∈ G ad such that γ n → γ in L ∞ (0, T ) as n → ∞.
For any n ∈ N, we denote (ϕ n 0 , ϕ n 1 ) ∈ W the minimum of J γn , and ϕ n the corresponding solution of (4). Using the uniform observability inequality (46) and the optimality condition of J γn , it follows that

(ϕ n 0 , ϕ n 1 ) 2 W ≤ C obs qγ n (ϕ n ) 2 χ γn = C obs ϕ n 1 , y 0 H -1 ,H 1 0 -ϕ n 0 , y 1 L 2 ≤ C obs (ϕ n 0 , ϕ n 1 ) W (y 0 , y 1 ) V
leading to the uniform bound (ϕ n 0 , ϕ n 1 ) W ≤ C obs (y 0 , y 1 ) V . Consequently, there exist two functions ϕ 0 ∈ L 2 (Ω) and ϕ 1 ∈ H -1 (Ω) such that, up to a subsequence, as n → ∞, we have

ϕ n 0 ϕ 0 weakly in L 2 (Ω) and ϕ n 1 ϕ 1 weakly in H -1 (Ω).
From the continuous dependence of the solution of the wave equation with respect to the initial condition, it follows

ϕ n ϕ weakly in L 2 (0, T ; L 2 (Ω)),
where ϕ is the solution of (4) associated to (ϕ 0 , ϕ 1 ). Let ψ ∈ L 2 (0, T ; L 2 (Ω)). We then have

Q T ψϕ n χ γn = Q T ψϕ n χ γ + Q T ψϕ n (χ γn -χ γ ) → Q T ψϕχ γ .
Indeed, we can take the weak limit in the first term because ψχ γ ∈ L 2 (0, T ; L 2 (Ω)). Using Lemma 4 and the boundedness of (ϕ n ) n≥0 in L 2 (0, T ; L 2 (Ω)), the second term converges to 0 because

Q T ψϕ n (χ γn -χ γ ) ≤ ψ L 2 (L 2 ) ϕ n L 2 (L 2 ) χ γn -χ γ L ∞ (Q T ) .
Consequently, we obtain the convergence

ϕ n χ γn ϕχ γ weakly in L 2 (0, T ; L 2 (Ω)).
Let now (ψ 0 , ψ 1 ) ∈ W and ψ the corresponding solution of (4). Taking the weak limit in the optimality condition

qγ n ψϕ n χ γn = ψ 1 , y 0 H -1 ,H 1 0 - Ω ψ 0 y 1 , we find qγ ψϕχ γ = ψ 1 , y 0 H -1 ,H 1 0 - Ω ψ 0 y 1 .
This means that (ϕ 0 , ϕ 1 ) is the minimum of J γ . Besides, we remark that this property uniquely characterizes the weak limit of any subsequence of (ϕ n 0 , ϕ n 1 ). This implies that the whole sequence (ϕ n 0 , ϕ n 1 ) weakly converges. The continuity of J is finally obtain by taking the weak limit in the optimality condition

qγ n (ϕ n ) 2 χ γn = ϕ n 1 , y 0 H -1 ,H 1 0 - Ω ϕ n 0 y 1 → ϕ 1 , y 0 H -1 ,H 1 0 - Ω ϕ 0 y 1 = qγ ϕ 2 χ γ .
The continuity of J then allows to show that the extremal problem (47) is well-posed.

Proposition 2. The cost J reaches its minimum over G ad .

Proof. The cost J being bounded by below, there exists a minimizing sequence (γ n ) n≥0 ⊂ G ad . By definition of G ad , this sequence is bounded in W 1,∞ (0, T ). By the generalized Rellich theorem, W 1,∞ (0, T ) is compactly embedded in L ∞ (0, T ). Consequently, there exists a curve γ ∈ L ∞ (0, T ) such that, up to a subsequence, γ n → γ in L ∞ (0, T ). From the definition of G ad , all the curves γ n are M -Lipschitzian, with M independent of n. So, taking the pointwise limit in the expressions

|γ n (t) -γ n (s)| ≤ M |t -s|, ∀t, s ∈ [0, T ], δ 0 ≤ γ n (t) ≤ 1 -δ 0 , ∀t ∈ [0, T ],
we notice that γ ∈ G ad . Finally, using Proposition 1, we obtain J(γ n ) → J(γ) = inf G ad J which means that γ is a minimum of J over G ad .

First directional derivative of the cost

We now give the expression of the directional derivative of the cost J.

Definition 7. Let γ, γ ∈ W 1,∞ (0, T ), with δ 0 ≤ γ ≤ 1 -δ 0 .
The perturbation γ is said admissible if and only if for any η > 0 small enough, the perturbed curve

γ η = γ + ηγ verifies δ 0 ≤ γ η ≤ 1 -δ 0 . Lemma 5. Let χ ∈ C 2 (R) and γ, γ ∈ W 1,∞ (0, T ), with δ 0 ≤ γ ≤ 1 -δ 0 .
For any η > 0, we define the perturbed curve γ η = γ + ηγ. Taking η → 0, we then have

χ γη -χ γ η → -γχ γ in L ∞ (Q T ).
Proof. It is a direct consequence of the Taylor's inequality applied to χ. Indeed, this inequality gives

χ γη -χ γ + ηγχ γ L ∞ (Q T ) ≤ η 2 2 χ L ∞ (R) γ 2 L ∞ (0,T ) . Proposition 3. Let χ ∈ C 2 (R) and γ, γ ∈ W 1,∞ (0, T ), with δ 0 ≤ γ ≤ 1 -δ 0 .
For any η > 0, we define the perturbed curve γ η = γ + ηγ. If γ is an admissible perturbation, then the directional derivative of J at γ in the direction γ, denoted by dJ(γ; γ), reads

dJ(γ; γ) = lim η→0 J(γ η ) -J(γ) η = T 0 γ Ω ϕ 2 χ γ , ( 49
)
where ϕ is the solution of (4) associated to the minimum (ϕ 0 , ϕ 1 ) of J γ .

Proof. For η > 0 small enough, we denote (ϕ η 0 , ϕ η 1 ) the minimum of J γη , and ϕ η the corresponding solution of (4). Likewise, we denote (ϕ 0 , ϕ 1 ) the minimum of J γ , and ϕ the corresponding solution of (4). Using the optimality conditions of J γη and J γ , we can write

J(γ η ) -J(γ) = qγ η (ϕ η ) 2 χ γη - qγ ϕ 2 χ γ = ϕ η 1 , y 0 H -1 ,H 1 0 - Ω ϕ η 0 y 1 -ϕ 1 , y 0 H -1 ,H 1 0 - Ω ϕ 0 y 1 = qγ ϕ η ϕχ γ - qγ η ϕϕ η χ γη = - Q T ϕ η ϕ(χ γη -χ γ ).
Arguying as in the proof of Proposition 1, we can show that ϕ η ϕ weakly in L 2 (0, T ; L 2 (Ω)). As a result, we have

J(γ η ) -J(γ) η = Q T ϕ η ϕγχ γ - Q T ϕ η ϕ χ γη -χ γ η + γχ γ → Q T ϕ 2 γχ γ = T 0 γ Ω ϕ 2 χ γ .
Indeed, we can take the weak limit in the first term because ϕγχ γ ∈ L 2 (0, T ; L 2 (Ω)). Using Lemma 5 and the boundedness of (ϕ η ) η>0 in L 2 (0, T ; L 2 (Ω)), the second term converges to 0 because

Q T ϕ η ϕ χ γη -χ γ η + γχ γ ≤ ϕ η L 2 (L 2 ) ϕ L 2 (L 2 ) χ γη -χ γ η + γχ γ L ∞ (Q T )
.

Remark 5. We emphasize that the directional derivative does not depend on the solution of an adjoint problem. This is due to the fact that we minimize with respect to the curve γ over controls of minimal L 2 (q γ )-norm. We refer to the proof of [START_REF] Münch | Optimal design of the support of the control for the 2-D wave equation: a numerical method[END_REF]Theorem 2.3] for more details.

Regularization and Gradient algorithm

At the practical level, in order to solve the optimal problem (47) numerically, we need to handle the Lipschitz constraint included in G ad . In this respect, we add a regularizing term to the cost J in order to keep the derivative of γ uniformly bounded. The optimization problem is now the following one: for > 0 fixed, solve min γ∈W 1,∞ (0,T )

δ 0 ≤γ≤1-δ 0 J (γ) = J(γ) + 2 γ 2 L 2 (0,T ) . ( 50
)
The regularization parameter , which can be compared to the Lipschitz constant M in the definition of G ad , controls the speed of variation of the curves γ ∈ W 1,∞ (0, T ). We fix γ ∈ W 1,∞ (0, T ) such that δ 0 ≤ γ ≤ 1 -δ 0 . Using Proposition 3, for any admissible perturbation γ ∈ W 1,∞ (0, T ), a direct calculation provides the expression of the directional derivative of J dJ (γ; γ) = dJ(γ; γ)

+ T 0 γ γ = j γ , γ L 2 (0,T ) + γ , γ L 2 (0,T ) , ( 51
) with j γ (t) = Ω ϕ 2 (x, t)χ γ (x, t) dx, ∀t ∈ [0, T ]. ( 52
)
In the expression of j γ , the function ϕ is the solution of (4) associated to the minimum (ϕ 0 , ϕ 1 ) of J γ . Consequently, a minimizing sequence (γ n ) n∈N for J is defined as follows:

γ 0 given in H 1 (0, T ), γ n+1 = P [δ 0 ,1-δ 0 ] (γ n -ρj γn ), for n ≥ 0, (53)
where P [δ 0 ,1-δ 0 ] is the pointwise projection in the interval [δ 0 , 1 -δ 0 ], ρ > 0 a descent step and j γn ∈ H 1 (0, T ) is the solution of the variational formulation j γn , γ L 2 (0,T ) + j γn , γ L 2 (0,T ) = j γn , γ L 2 (0,T ) + γ n , γ L 2 (0,T ) , ∀ γ ∈ H 1 (0, T ), (54) so that dJ (γ n ; j γn ) = j γn 2 L 2 (0,T ) + j γn 2 L 2 (0,T ) ≥ 0.

Numerical experiments

Before to present some numerical experiments, let us briefly mention some aspects of the resolution of the underlying discretized problem.

• The discretization of the curve γ is performed as follows. For any fixed integer N > 0, we denote δt = T /N and define the uniform subdivision

{t i } i=0,••• ,N of [0, T ] such that t i = iδt.
We then approximate the curve γ in the space of dimension N + 1

P δt 1 = γ ∈ C([0, T ]); γ| [t i-1 ,t i ] affine, ∀i ∈ {1, . . . , N } .
For any γ ∈ P δt

1 , γ = N i=0 γ i L δt i , with (γ i ) 0≤i≤N ∈ Ω N +1
where (L δt i ) 0≤i≤N is the usual Lagrange basis. Consequently, γ is defined by the N + 1 points (γ i , t i ) ∈ Ω × [0, T ]. The knowledge of the initial curve γ 0 ∈ P δt 1 such that δ 0 ≤ γ 0 ≤ 1 -δ 0 determines such points and then a triangular mesh of Q T . At each iteration n ≥ 0, these points are updated along the x-axis according to the pointwise time descent direction j γn ∈ H 1 (0, T ) (see (54)) as follows:

x n+1 i = P [δ 0 ,1-δ 0 ] x n i -ρ j γn (t n i ) , t n+1 i = t n i ∀i = 0, • • • , N + 1.
and T = 2, = 10 -2 , ρ = 10 -4 . We initialize the descent algorithm with the following three initial curves:

γ 1 0 (t) = 2 5 , γ 2 0 (t) = 3 5 , γ 3 0 (t) = 1 2 + 1 10 cos π t T , for t ∈ (0, T ). ( 57
)
The corresponding initial and optimal domains are depicted in Figure 4 together with typical space-time meshes. The numbers of iterations until convergence, the values of the functional J ε evaluated at the optimal curve γ opt and the performance indices of γ opt are listed in Table 2. given by (57).

Initial
In Figure 4, we observe that the optimal domain computed by the algorithm depends on the initial domain chosen. This indicates that our functional J ε does have several local minima. Moreover, one can show that, among the cylindrical domains, there are two optimal values, x 0 = 1/4 and x 0 = 3/4, leading to J ε (x 0 ) ≈ 46.94. These values correspond to the extrema of the function sin(2πx) in [0, 1]. The simulations associated with the initial curves γ 1 0 and γ 2 0 are in agreement with this result. On the other hand, the worst cylindrical domain corresponds to x 0 = 1/2 (see Figure 6-Left).

Eventually, the adjoint states ϕ (from which we obtain the control v = ϕ| q n γ ) computed for the optimal domains in Figure 4-Bottom, are displayed in Figure 5.

• We now consider the initial datum (y 0 , y 1 ) given by y 0 (x) = (10x -4) 2 (10x -6) 2 1 [0.4,0.6] (x), y 1 (x) = y 0 (x), for x ∈ (0, 1).

(EX2)

This initial condition, plotted in Figure 7, generates a travelling wave, as can be seen in For T = 2, = 10 -2 and ρ = 10 -4 , we initialize the descent algorithm with the curve γ 0 ≡ 1/2. The convergence is reached after 68 iterations and the optimal cost is J (γ opt ) ≈ 48.70. Moreover, the minimal cost for cylindrical domains is min x 0 J (x 0 ) ≈ 179.22 leading to a performance index Π(γ opt ) ≈ 72.83%. The non-cylindrical setup is in that case much more efficient that the cylindrical one. It is due to the fact that the domains we consider can follow very closely the propagation of the travelling wave. This can be noticed in Figure 8, where we display the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and controlled solutions over the optimal domain.

The evolution of the cost J n and the derivative dJ n with respect to n are displayed in Figure 9. Figure 6-Right depicts the values of the functional J for the constant curves γ ≡ x 0 used to determine the best cylindrical domain and highlights the low variation of the cost with respect to the position of such domains.

• We now consider the initial datum (y 0 , y 1 ) given by y 0 (x) = (10x -4) 2 (10x -6) 2 1 [0.4,0.6] (x), y 1 (x) = 0, for x ∈ (0, 1).

(EX3)

This initial condition generates two travelling waves going in opposite directions, as can be seen in Figure 10.3. For T = 2, = 10 -2 and ρ = 10 -4 , we initialize the algorithm with the initial curve Figure 4: (EX1) -Initial (top) and optimal (bottom) control domains for the initial curves (γ i 0 ) i∈{1,2,3} given by (57) (from left to right).

γ 0 ≡ 1/2. The convergence is observed after 111 iterations leading to J (γ opt ) ≈ 41.02. Moreover, the minimal cost for cylindrical domains is min x 0 J (x 0 ) ≈ 85.08, so that the performance index is Π(γ opt ) ≈ 51.79%. Once again, our non-cylindrical setup is much more efficient than the cylindrical one. It is still due to the fact that the domains we consider can follow the propagation of the travelling waves, one after the other. This can be noticed in Figure 10, where we display the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and the controlled wave over the optimal domain. In order to show the influence of the controllability time on the optimal domain, for = 10 -2 and γ 0 ≡ 1/2, we use the descent algorithm with T = 1 and ρ = 2.5 × 10 -5 , initialized with the curve γ 0 = 1/2. Remark that the corresponding domain do satisfies the geometric optic condition. The convergence is observed after 213 iterations and the optimal cost is J (γ opt ) ≈ 94.78. Moreover, the minimal cost for cylindrical domains is min x 0 J (x 0 ) ≈ 183.98, so that the performance index is Π(γ opt ) ≈ 48.48%. Here, we mention that for x 0 ≤ 0.25 or x 0 ≥ 0.75, the cylindrical domain associated to x 0 does not verify the geometric optics condition needed to ensure controllability. This highlights the necessity to use non-cylindrical domains. Compared to the simulation for T = 2, the optimal cost increases by a factor around 2.3. Figure 11 displays the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and controlled wave over the optimal domain. We remark that the projection of the optimal domain on the x-axis covers the whole domain Ω, in contrast with the domain associated with T = 2.

Figure 11: (EX3) -From left to right, optimal control domain, isovalues of the corresponding adjoint state ϕ, isovalues of the uncontrolled and controlled wave over the optimal domain, for T = 1, for the initial curve γ 0 ≡ 1/2.

• Eventually, in order to highlight the influence of the regularization parameter on the optimal domain, we now consider the initial datum (y 0 , y 1 ) given by

y 0 (x) =      3x if 0 ≤ x ≤ 1/3, 3(1 -2x) if 1/3 ≤ x ≤ 2/3, -3(1 -x) if 2/3 ≤ x ≤ 1, y 1 (x) = 0,
for x ∈ (0, 1). (EX4)

For T = 2 and ρ = 10 -5 , we initialize the descent algorithm with the curve γ ≡ 1/2 and consider = 10 -2 and = 0. The numbers of iterations until convergence, the values of the functional J evaluated at the optimal curve γ opt and the performance indices of γ opt are listed in Table 3. For the initial datum (EX4), the minimal cost for cylindrical domains is min x 0 J (x 0 ) ≈ 47.71. -26.51% 9.38% Table 3: (EX4) -Number of iterations, optimal value of the functional J and performance index, for ∈ {0, 10 -2 }, for the initial curve γ 0 ≡ 1/2.

In Figure 12, we clearly see the regularizing effect of and the need of regularization in this case, as the optimal domain obtained when ε = 0 is very oscillating.

Iterative approximation of the observability constant

In this last part, we formally describe and use an algorithm allowing to approximate the observability constant appearing in (3), associated to any domain q ⊂ Q T . The algorithm is based on the following characterization: C obs (q) = sup where Λ q and R are respectively the control operator associated to the domain q and the duality operator between the space W and V:

Λ q : V → W y 0 → ϕ 0 , R : W → V (ϕ 0 , ϕ 1 ) → ((-∂ 2 x ) -1 ϕ 1 , -ϕ 0 ) . ( 59
)
In the definition of Λ q , ϕ 0 ∈ W is the minimum of the functional J (cf. ( 5)) associated to y 0 ∈ V. The characterization (58) can be obtained by following the steps of [20, Section 2] and [START_REF] Coron | Mathematical Surveys and Monographs[END_REF]Remark 2.98]. The main consequence of this characterization is that C obs (q) can be viewed as the largest eigenvalue of the operator RΛ q in V. Consequently, we can formally adapt the power iteration method to our infinite-dimensional setting. The algorithm reads as follows.

Let y 0 0 ∈ V be given such that y 0 0 V = 1. For n ≥ 0, using the space-time finite element method described in [3, Section 3-4], we compute ϕ n 0 = Λ q y n 0 then set z n 0 = R ϕ n 0 and y n+1 0 = z n 0 / z n 0 V . We finally have C obs (q) = lim n→∞ z n 0 V while y n 0 converges in V to the most expensive initial datum to control. For the control domain of Figure 3, this algorithm (after an appropriate spacetime), initialized with y 0 0 = K(x(1 -x), 0) -K such that y 0 0 V = 1-, produces the following sequence { z n 0 V } n≥0 = {2.6895, 3.829, 3.981, 3.994, 3.997, • • • } converging toward the value 4, in agreement with the result of Section 2.3 based on a graph argument. The most expensive initial datum to control is displayed in Figure 13. Remark that the initial datum solution of (58) is not unique.

Conclusion and perspectives

Making use of the d'Alembert formulae for the solutions of the one dimensional wave equation, we have shown a uniform observability inequality with respect to the class of non cylindrical domains satisfying the geometric option condition. The proof based on arguments from graph theory allows notably to relate the value of the observability constant to the spectrum of the Laplacian matrix, defined in term of the graph of any domain q ⊂ Q T . The uniform observability property then allows to consider and analyze the problem of the control's optimal support associated to fixed initial conditions. For simplicity, the optimization is made over connected domains defined by regular curves. As expected, the optimal domains (approximated within a space-time finite element method) are closely related to the travelling waves generated by the initial conditions. This work may be extended in several directions. First, the characterization of the observability constant in term of a computable eigenvalue problem in Section 4.2 may allow to consider the optimization of such constant with respect to the domain of observation, i.e. inf q∈Q ε ad C obs (q). Moreover, from an approximation point of view, we may also consider more general domains (than connected ones) and use, for instance, a level set method to describe the geometry (as done in [START_REF] Münch | Optimal design of the support of the control for the 2-D wave equation: a numerical method[END_REF]). Eventually, this work may be adapted to the case of controls supported on single curves of Q T , using the uniform observability property given in [START_REF] Castro | Exact controllability of the 1-D wave equation from a moving interior point[END_REF].

The extension of this work to the N -dimensional case studied in [START_REF] Rousseau | Geometric control condition for the wave equation with a time-dependent observation domain[END_REF][START_REF] Shao | On Carleman and observability estimates for wave equations on time-dependent domains[END_REF] is also a challenge.

Figure 1 :

 1 Figure 1: Some elementary squares in C 4 .

Figure 3 :

 3 Figure 3: Observation domain q adapted to S 4 .

Figure 5 :

 5 Figure 5: (EX1) -Isovalues of the adjoint states ϕ computed for the optimal domains obtained for the initial curves (γ i 0 ) i∈{1,2,3} given by (57) (from left to right).

Figure 6 :

 6 Figure 6: Values of J for constant curves γ ≡ x 0 (•), for the initial data (EX1) (left) and (EX2) (right). The dashed line (--) represents the value of J (γ opt ), for the initial curves 0 ≡ 2/5 (left) and γ 0 ≡ 1/2 (right).

Figure 7 :

 7 Figure 7: Initial datum (y 0 , y 1 ) defined in (EX2).

Figure 8 :

 8 Figure 8: (EX2) -From left to right, optimal control domain, isovalues of the corresponding adjoint state ϕ, isovalues of the uncontrolled and controlled wave over the optimal domain, for the initial curve γ 0 ≡ 1/2.

Figure 9 :

 9 Figure 9: (EX2) -Evolution of the cost J n (left) and the derivative dJ n (right) for the initial curve γ 0 ≡ 1/2.

Figure 10 :

 10 Figure 10: (EX3) -From left to right, optimal control domain, isovalues of the corresponding adjoint state ϕ, isovalues of the uncontrolled and controlled wave over the optimal domain, for T = 2, for the initial curve γ 0 ≡ 1/2.

2 V.Figure 12 :

 212 Figure 12: (EX4) -Optimal control domain and isovalues of the adjoint state for = 10 -2 (left), = 0 (right), for the initial curve γ 0 ≡ 1/2.

Figure 13 :

 13 Figure 13: Most expensive initial data y 0 (left) and y 1 (right) to be controlled.

Table 2 :

 2 (EX1) -Number of iterations, optimal value of the functional J ε and performance index, for the initial curves (γ i 0 ) i∈{1,2,3}

	curve	γ 1 0	γ 2 0	γ 3 0
	Number of iterations	33	33	84
	J (γ opt )	47.09	47.09	47.93
	Π(γ opt )	-0.32% -0.32% -2.11%

We emphasize in particular that a remeshing of Q T is performed at each iteration n according to the set of points (x n i , t n i ) i=0,••• ,N +1 . • Each iteration of the algorithm requires the numerical approximation of the control of minimal L 2 (q γn ) for the initial data (y 0 , y 1 ). We use the space-time method described in [START_REF] Castro | Controllability of the linear one-dimensional wave equation with inner moving forces[END_REF] which is very well-adapted to the description of γ embedded in a space-time mesh of Q T . The minimization of the conjugate functional J γn (see (48)) with respect to (ϕ 0 , ϕ 1 ) ∈ V is replaced by the search of the unique saddle-point of the Lagrangian L : Z × L 2 (0, T ; H 1 0 (Ω)) → R defined by

The corresponding mixed formulation is solved with a conformal space-time finite element method while a direct method is used to invert the discrete matrix. The interesting feature of the method for which the adaptation of the mesh is very simple to handle with, is that only a small part of the matrix -corresponding to the term ϕ 2 L 2 (qγ n ) -is modified from two consecutive iterations n and n + 1.

Numerical illustrations

We discuss several experiments performed with FreeFEM (see [START_REF] Hecht | New development in Freefem++[END_REF]) for various initial data and control domains. We notably use an UMFPACK type solver. We fix δ 0 = 0.15 and δ = δ 0 /4. Moreover, according to (45), we define the function χ ∈ C 2 (R) in [δ 0 -δ, δ 0 ] as the unique polynomial of degree 5 such that χ(δ

Concerning the stopping criterion for the descent algorithm, we observed that the usual one based on the relative quantity |J n -J n-1 |/J 0 is inefficient because too noisy. This is due to the uncertainty on the numerical computation of the adjoint state ϕ n and the perturbation j γn . Consequently, in order to better capture the variations of the sequence (J n ) n∈N , we replace J n and J n-1 by the right and left p-point average respectively leading to the stopping criterion

In the sequel, we fix p = 10 and η = 10 -3 . Furthermore, in order to measure the gain obtained by using non-cylindrical control domains rather than cylindrical ones, we define a performance index associated to each optimal curves γ opt ; identifying any constant curve γ ≡ x 0 with its value x 0 ∈ [δ 0 , 1 -δ 0 ], we compute the minimal cost min x 0 J (x 0 ) for cylindrical domains. We then define the performance index of γ opt by

In the sequel, in practice, the minimum of J with respect to x 0 is searched among 13 distincts values equi-distributed between 0.2 and 0.8.

• We first consider the regular initial datum (y 0 , y 1 ) given by y 0 (x) = sin(2πx), y 1 (x) = 0, for x ∈ (0, 1). (EX1)