Erik Burman

ARNAUD Ali Feizmohammadi

Arnaud Munch

Lauri Oksanen
email: lauri.oksanen@helsinki.fi

Spacetime

AND M Ünch

SPACETIME FINITE ELEMENT METHODS FOR CONTROL PROBLEMS SUBJECT TO THE WAVE EQUATION

whether they are published or not. The documents may come

Introduction

We consider the classical null controllability problem for the wave equation, both with distributed and boundary control. Let T > 0, and let Ω ⊂ R n , with n ≥ 2, be a connected bounded open set with smooth boundary. We write M = (0, T ) × Ω, Γ = (0, T ) × ∂Ω.

For a fixed initial state (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the distributed null control problem on M reads: find φ ∈ L 2 (M ) such that the solution u of

     u = χφ, u| Γ = 0, u| t=0 = u 0 , ∂ t u| t=0 = u 1 , (1)
satisfies u| t=T = 0, ∂ t u| t=T = 0.

(2) Here = ∂ 2 t -∆ is the wave operator, and χ is a cutoff function that localizes the control in a subset of M . More precisely, we consider a cutoff of the form

χ(t, x) = χ 0 (t)χ 2 1 (x)
EB acknowledges funding by EPSRC grants EP/P01576X/1. AF acknowledges funding by EPSRC grant EP/P01593X/1 and support from the Fields institute for research in mathematical sciences." .

AM acknowledges funding by the French government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF][START_REF] Ervedoza | Numerical approximation of exact controls for waves[END_REF][START_REF] Evans | Partial differential equations[END_REF][START_REF] French | A space-time finite element method for the wave equation[END_REF][START_REF] French | A continuous space-time finite element method for the wave equation[END_REF][START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF].

LO acknowledges funding by EPSRC grants EP/P01593X/1 and EP/R002207/1.

where χ 0 ∈ C ∞ 0 ([0, T ]) and χ 1 ∈ C ∞ (Ω) take values in [0, 1]. Our main assumption is that (A) χ = 1 on open (a, b) × ω ⊂ M satisfying the geometric control condition.

The geometric control condition means that every compressed generalized bicharacteristic intersects the set (a, b) × ω, when projected to M . We refer to [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for the rather technical definition of a compressed generalized bicharacteristic. Roughly speaking, all continuous paths on M , consisting of lightlike line segments in its interior and reflected on Γ according to Snell's law, must intersect (a, b) × ω. However, projections of compressed generalized bicharacteristics may also glide along Γ under suitable convexity.

For our main result we assume, furthermore, that (u 0 , u 1 ) ∈ H k+1 (Ω) × H k (Ω) for some k = 2, 3, . . . , and that the following compatibility conditions of order k are satisfied (∆ j u 0 )| ∂Ω = 0 for j = 0, 1, . . . , k 2 , (C1)

(∆ j u 1 )| ∂Ω = 0 for j = 0, 1, . . . , k -1 2 . (C2)
Here • is the floor function that gives the greatest integer less than or equal to its argument. We recall that the compatibility conditions guarantee that, for smooth enough φ, the solution u of (1) is in H k+1 (M ), see e.g. [START_REF] Evans | Partial differential equations[END_REF]Theorem 6,p. 412].

Under these assumptions, we show that the stabilized finite element method introduced below gives such an approximation φ h of a certain minimum norm solution φ to the control problem that

χ(φ h -φ) L 2 (M ) h q , (3)
where h > 0 is the mesh size and q ≤ k -1 is the polynomial order of the finite element space. The implicit constant in the above inequality is independent of h and the functions φ h and φ. This notation is used in the paper when confusion is not likely to arise. See Theorem 2.5 for the precise formulation. In this result both u and φ are assumed to be at least H 2 (M )-smooth, corresponding to the above constraint k ≥ 2. We prove also a weak convergence result for our method assuming only the smoothness u 1 ∈ L 2 (Ω) in the case that u 0 = 0, see Theorem 3.5. The case with general rough data is left for future work.

Let us now sketch our result in the case of the boundary null control problem of the following form: given an initial state (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), find ψ ∈ L 2 (Γ) such that the solution u of

     u = 0, u| Γ = χψ, u| t=0 = u 0 , ∂ t u| t=0 = u 1 , (4)
satisfies the final time condition [START_REF] Aranda | A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations[END_REF]. Under a geometric control condition analogous to (A), and the above regularity assumptions on the data (u 0 , u 1 ), we introduce a finite element method that converges as

χ(ψ h -ψ) L 2 (Γ) h q-1 2 , (5)
with notations analogous to (3) and q ≤ k + 1. See Theorem 4.6 for the precise formulation. Observe that, although there is a loss of order 1/2 in (5) in comparison to [START_REF] Asch | Uniformly controllable schemes for the wave equation on the unit square[END_REF], if the highest possible polynomial orders are used, the order in (5) becomes k + 1/2 versus k -1 in [START_REF] Asch | Uniformly controllable schemes for the wave equation on the unit square[END_REF]. We can also rescale the method in the distributed case to get the order q -1/2 for q ≤ k, leading to k -1/2 for the highest possible order.

1.1. Literature. This work is a contribution to the finite dimensional approximation of null controls for the linear wave equation. The seminal work is due to Glowinski and Lions in [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF] where the search of the control of minimal L 2 norm is reduced (using the Fenchel-Rockafellar duality theory) to the unconstrained minimization of the corresponding conjugate functional involving the homogeneous adjoint problem. Minimization of the discrete functional, associated with centered finite difference approximation in time and P 1 finite element method in space is discussed at length in [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF] and exhibits a lack of convergence of the approximation with respect to the discretization parameter h. This is due to spurious high frequencies discrete modes which are not exactly controllable uniformly in h.

This pathology can easily be avoided in practice by adding to the conjugate functional a regularized Tikhonov parameter, but this leads to so called approximate controls, solving the control problem only up to a small remainder term. Several cures aiming to filter out the high frequencies have been proposed and analyzed, mainly for simple geometries (1d interval, unit square in 2d, etc) with finite differences schemes. The simplest, but artificial, approach is to eliminate the highest eigenmodes of a discrete approximation of the initial condition as discussed in one space dimension in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF], and extended in [START_REF] Lissy | Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method[END_REF]. We mention spectral methods initially developed [START_REF] Bourquin | Approximation theory for the problem of exact controllability of the wave equation with boundary control[END_REF] then used in [START_REF] Lebeau | Experimental study of the HUM control operator for linear waves[END_REF]. We also mention so called bi-grid method (based on the projection of the discrete gradient on a coarse grid) proposed in [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF] and analyzed in [START_REF] Loreti | An Ingham type proof for a two-grid observability theorem[END_REF][START_REF] Ignat | Convergence of a two-grid algorithm for the control of the wave equation[END_REF] leading to convergence results. One may also design more elaborated discrete schemes avoiding spurious modes: we mention [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF] based on a mixed reformulation of the wave equation analyzed later with finite difference schemes in [START_REF] Castro | Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method[END_REF][START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF][START_REF] Asch | Uniformly controllable schemes for the wave equation on the unit square[END_REF] at the semi-discrete level and then extended in [START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF] to a full space-time discrete setting, leading to strong convergent results.

The above previous works, notably reviewed in [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF][START_REF] Ervedoza | Numerical approximation of exact controls for waves[END_REF], fall within an approach that can be called "discretize then control" as they aim to control exactly to zero a finite dimensional approximation of the wave equation. A relaxed controllability approach is analyzed in [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] using a stabilized finite element method in space and leading for smooth two and three dimensional geometries to a strong convergent approximation. The controllability requirement is imposed via appropriate penalty terms. We also mention [START_REF] Pedregal | A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems[END_REF] based on the Russel principle, extended in [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF] and [START_REF] Gunzburger | A numerical method for exact boundary controllability problems for the wave equation[END_REF][START_REF] Aranda | A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations[END_REF] for least-squares based method. One the other hand, one may also employ a "control then discretize" procedure, where the optimality system (for instance associated with the control of minimal L 2 norm ) mixing the boundary condition in time and space and involving the primal and adjoint state is discretized within a priori a space-time approximation. The well-posedness of such system is achieved by using so called global or generalized observability inequalities. Such approach avoids the numerical pathologies mentioned above and is notably well-suited for mesh adaptivity. On the other hand, the numerical analysis, within a conformal approximation is delicate since it requires to prove inf-sup stablity that is uniform with respect to h. We mention [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] where this approach has been introduced within a conformal approximation leading to convergent numerical results for the control of minimal L 2 norm. It has been extended in [START_REF] Montaner | Approximation of controls for linear wave equations: a first order mixed formulation[END_REF] where the wave equation is reformulated as a first order system, solved in the one dimensional case with a stabilized formulation allowing to bypass the inf-sup property issue. We also mention [START_REF] Cîndea | Numerical controllability of the wave equation through primal methods and Carleman estimates[END_REF] in the 1d case where the optimality system associated to cost involving both the control and the state is reformulated as a space-time elliptic problem of order four, leading to strong convergent result with respect to the discretization parameter. The present paper falls into this category and aims, in the spirit of [START_REF] Burman | Space time stabilized finite element methods for a unique continuation problem subject to the wave equation[END_REF] devoted to the dual data assimilation problem, to provide some convergent results, including rate of convergence, with respect to the discrete parameter. We mention a growing interest for space-time (finite element) methods of approximation for the wave equation, initially advocated in [START_REF] Hulbert | Space-time finite element methods for second-order hyperbolic equations[END_REF][START_REF] French | A space-time finite element method for the wave equation[END_REF][START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF] and more recently in [START_REF] Langer | Space-Time Methods[END_REF], [START_REF] Antonietti | A space-time discontinuous Galerkin method for the elastic wave equation[END_REF], [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF], [START_REF] Dörfler | Space-time discontinuous galerkin discretizations for linear first-order hyperbolic evolution systems[END_REF], [START_REF] Steinbach | A Stabilized Space-Time Finite Element Method for the Wave Equation[END_REF].

Distributed control

We recall that, assuming (A), the distributed control problem can be solved by finding u and φ such that

         u = χφ, u| x∈∂Ω = 0, u| t=0 = u 0 , ∂ t u| t=0 = u 1 , u| t=T = 0, ∂ t u| t=T = 0, φ = 0, φ| x∈∂Ω = 0. (6) Moreover, if (u 0 , u 1 ) ∈ H k+1 (Ω) × H k (Ω)
satisfies the compatibility conditions of order k, then the unique solution (u, φ) to (6) satisfies [START_REF] Burman | A stabilized nonconforming finite element method for the elliptic Cauchy problem[END_REF] and this initial data for φ satisfies the compatibility conditions of order k -1, see [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF]Theorem 5.1]. It follows that φ ∈ H k (M ), and this again implies that u ∈ H k+1 (M ). The convergence proof for our finite element method is based on the fact that the solution of ( 6) has this regularity.

φ| t=T ∈ H k (Ω), ∂ t φ| t=T ∈ H k-1 (Ω),
The control given by φ can be characterized also as the control with the minimum norm on M with respect to the weighted measure χdtdx. The fact that (6) has a unique solution follows from this characterization, however, we give a short independent proof for the convenience of the reader. Lemma 2.1. Suppose that (A) holds. Let u, φ ∈ L 2 (M ) solve (6) with u 0 = u 1 = 0. Then u = φ = 0.

Proof. The lateral boundary traces on (0, T ) × ∂Ω are well-defined due to partial hypoellipticity, see Lemma A.7 in Appendix A. Typical energy estimates, see e.g. [START_REF] Lasiecka | Nonhomogeneous boundary value problems for second order hyperbolic operators[END_REF], give

u ∈ C(0, T ; H 1 0 (Ω)) ∩ C 1 (0, T ; L 2 (Ω)), and Lemma A.3 in Appendix A implies that φ ∈ C(0, T ; L 2 (Ω)) ∩ C 1 (0, T ; H -1 (Ω)).
In particular, we may parametrize φ by φ| t=0 and ∂

t φ| t=0 . Let φ 0 j , φ 1 j ∈ C ∞ 0 (Ω) satisfy φ 0 j → φ| t=0 in L 2 (Ω) and φ 1 j → ∂ t φ| t=0 in H -1 (Ω). Write (u j , φ j ) for the solution of      u = χφ, u| x∈∂Ω = 0, u| t=0 = 0, ∂ t u| t=0 = 0,      φ = 0, φ| x∈∂Ω = 0. φ| t=0 = φ 0 j , ∂ t φ| t=0 = φ 1 j . Then (χφ j , φ j ) L 2 (M ) = ( u j , φ j ) L 2 (M ) -(u j , φ j ) L 2 (M ) = (∂ t u j | t=T , φ j | t=T ) L 2 (Ω) -(u j | t=T , ∂ t φ j | t=T ) H 1 0 ×H -1 (Ω)
. Taking the limit j → ∞ shows that φ = 0 in supp(χ). The distributed observability estimate, see Theorem A.4, implies that φ = 0 in M . It follows that also u = 0 in M .

2.1. Notations. We write (t, x) = (x 0 , x 1 , . . . , x n ) for the coordinates on R 1+n . Let g stand for the Minkowski metric on R 1+n , and denote by g(•, •) the scalar product with respect to g. The wave operator can be written as = -div grad u, where the divergence and gradient are defined with respect to g. Let K ⊂ M be an open set with piecewise smooth boundary, and let N = (N 0 , . . . , N n ) be the outward pointing unit normal vector field on ∂K, defined with respect to the Euclidean metric on R 1+n . We write

∂ ν u = N • grad v = -N 0 ∂ x 0 v + N 1 ∂ x 1 v + • • • + N n ∂ x n v.
Note that div coincides with the Euclidean divergence, and we can apply the Euclidean divergence theorem to obtain

K u v dx = K g(du, dv) dx - ∂K u∂ ν v ds, ( 8
)
where ds is the Euclidean surface measure on ∂K, and du is the spacetime differential of u, that is, the covector with the components ∂ x j u, j = 0, . . . , n.

2.2.

Discretization. Consider a family T = {T h : h > 0} where T h is a set of 1 + n-dimensional simplices forming a simplicial complex. To keep the discussion as simple as possible, we assume in this section that K∈T h = M for all h > 0. This is a restrictive assumption since we also assumed that the spatial boundary ∂Ω is smooth. We will explain later, see Remark 4.7, how this issue can be avoided by allowing the simplices adjacent to the boundary to have curved faces, fitting Ω. This fitting technique is also described in detail in the context of the boundary control problem below.

If the set ω in assumption (A) is a neighbourhood of the boundary ∂Ω then the distributed observability estimate in Theorem A.4 holds in the case of piecewise smooth ∂Ω and large enough T > 0. In particular, we can consider polyhedral Ω and then K∈T h = M is straightforward to arrange. The multiplier method can also be used to derive the distributed observability estimate for polyhedral Ω and more general observation regions ω, however, this method can not reproduce the sharp geometric control condition in the case of smooth boundary [START_REF] Miller | Escape function conditions for the observation, control, and stabilization of the wave equation[END_REF].

We assume that the family T is quasi uniform, see e.g. [START_REF] Ern | Theory and practice of finite elements[END_REF]Definition 1.140], and indexed by h = max

K∈T h diam(K).
Then we define for p ∈ N + = {1, 2, . . . } the H 1 (M )-conformal approximation space of polynomial degree p,

V p h = {u ∈ H 1 (M ) : u| Γ = 0, u| K ∈ P p (K) for all K ∈ T h }, (9)
where P p (K) denotes the set of polynomials of degree less than or equal to p on K. Occasionally we write also V h = p∈N + V p h . For any h > 0, the control problem (6) can be formulated weakly as a(u, ψ) = h 2 c(φ, ψ) + L(ψ), a(v, φ) = 0, [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] for all v, ψ ∈ C ∞ (M ) vanishing on Γ, where

a(u, ψ) = M g(hdu, hdψ) dx -h(u, h∂ ν ψ) L 2 (∂M \Γ) (11) L(ψ) = h(hu 1 , ψ| t=0 ) L 2 (Ω) -h(u 0 , h∂ t ψ| t=0 ) L 2 (Ω) ,
and c(φ, ψ) = (χφ, ψ) L 2 (M ) . Indeed, it follows from (8) that if smooth (u, φ) solves ( 6) then [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] holds for all smooth (v, ψ) vanishing on Γ.

The bilinear form a is scaled so that there is C > 0 such that for all h > 0 and u, v ∈ H 2 (M ) + V h there holds

a(u, v) ≤ C u H 2 (T h ) v H 2 (T h ) , (12)
where the broken semiclassical Sobolev norm is defined for any k ∈ N by

u 2 H k (T h ) = k j=0 K∈T h (hD) j u 2 L 2 (K) .
Here D j u is the tensor of order j that gives the jth total derivative of u. The continuity [START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF] is consequence of the following trace inequality, see e.g. [6, Eq. 10.3.9]: there is C > 0 such that for all h > 0, K ∈ T h and u ∈ H 1 (K) there holds

h 1 2 u L 2 (∂K) ≤ C( u L 2 (K) + h∇u L 2 (K) ). ( 13
)
For u ∈ H k (M ) the broken semiclassical norm u H k (T h ) reduces to the usual semiclassical norm defined by

u 2 H k h (M ) = k j=0 (hD) j u 2 L 2 (M ) .
Moreover, there is C > 0 such that for all h > 0 and u ∈ V h there holds

u H k (T h ) ≤ C u L 2 (M ) .
This is due to the discrete inverse inequality, see e.g. [START_REF] Ern | Theory and practice of finite elements[END_REF]Lemma 1.138]: there is C > 0 such that for all h > 0, K ∈ T h , p ∈ N + and u ∈ P p (K) there holds

h∇u L 2 (K) ≤ C u L 2 (K) . (14)
We will systematically use a scaling so that all the bilinear forms in the paper satisfy the bound [START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF].

Our finite element method has the form: find the critical point of the Lagrangian

L(u, φ) : V p h × V q h → R, L(u, φ) = 1 2 h 2 c(φ, φ) + L(φ) - 1 2 R(u, φ) -a(u, φ),
where, writing U = (u, ∂ t u) and U 0 = (u 0 , u 1 ), the regularization is given by

R(u, φ) = h -κ S(u) -h κ S(φ) + h -κ E(U | t=0 -U 0 ) + h -κ E(U | t=T ) (15) + h 4-κ C(φ) + 2h 2-κ ρ(u, φ), E(U 0 ) = h u 0 2 L 2 (Ω) + h hu 1 2 L 2 (Ω) , S(u) = K∈T h h 2 u 2 L 2 (K) + F ∈F h h h∂ ν u 2 L 2 (F ) , C(φ) = χφ 2 L 2 (M ) , ρ(u, φ) = - K∈T h (h 2 u, χφ) L 2 (K) ,
where κ < 2 is a fixed constant. We have R(u, φ) = 0 for a smooth solution (u, φ) to [START_REF] Brenner | The mathematical theory of finite element methods[END_REF]. Indeed,

S(u) + 2h 2 ρ(u, φ) + h 4 C(φ) = K∈T h h 2 ( u -χφ) 2 L 2 (K) + F ∈F h h h∂ ν u 2 L 2 (F ) = 0,
and also S(φ) = 0 and

E(U | t=0 -U 0 ) = E(U | t=T ) = 0.
The equation dL(u, φ) = 0 can be written as

A[(u, φ), (v, ψ)] = h -κ e(U 0 , V | t=0 ) + L(ψ) for all (v, ψ) ∈ V p h × V q h , (16)
where the bilinear form A is given by

A[(u, φ), (v, ψ)] = h -κ s(u, v) -h κ s(φ, ψ) -h 2 c(φ, ψ) + h -κ τ =0,T e(U | t=τ , V | t=τ ) + a(v, φ) + a(u, ψ) + h 4-κ c(φ, ψ) + h 2-κ ρ(v, φ) + h 2-κ ρ(u, ψ).
Here s is the bilinear form associated to the quadratic form S, and this lowercaseuppercase convention is systematically used also for other quadratic and bilinear forms in the paper. Let us emphasize that all the bilinear forms s, c, e, c and ρ satisfy the same bound (12) as a.

We define the residual norm by

| (u, φ)| 2 = h -κ S(u) + h κ S(φ) + h 2 C(φ) + h -κ τ =0,T E(U | t=τ ). ( 17
) Lemma 2.2. Suppose that (A) holds. Then | •| is a norm on V h × V h .
Proof. Suppose | (u, φ)| = 0. Then u = 0 elementwise and ∂ ν u = 0 for all internal faces. It follows that u = 0 in the weak sense. As u| Γ = 0 and E(U | t=0 ) = 0, it follows that u = 0. Similarly φ = 0 in the weak sense. As φ| Γ = 0 and C(φ) = 0, the distributed observability estimate, see Theorem A.4, implies that φ = 0. Lemma 2.3. For all sufficiently small h and all u, φ ∈ H 2 (M ) + V h there holds

| (u, φ)| 2 A[(u, φ), (u, -φ)].
Proof. By the definition of A, we have

A[(u, φ), (u, -φ)] =| (u, φ)| 2 -h 4-κ C(φ).
As κ < 2 and χ ≤ 1, h 4-κ C(φ) can be absorbed by h 2 C(φ) for small h > 0.

The previous two lemmas imply that (16) has a unique solution. Indeed, ( 16) is a square system of linear equations and the lemmas imply that (u, φ) = 0 is the only solution when the right-hand side is zero. The right-hand side being zero is equivalent with U 0 = 0. 2.3. Error estimates. Equation ( 16) defines a finite element method that is consistent in the sense that if smooth enough u and φ satisfy (6), then [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF] holds for (u, φ). This follows from the weak formulation (10) of ( 6) together with the regularization vanishing for (u, φ). In particular, if (u h , φ h ) ∈ V p h ×V q h solves ( 16) then the following Galerkin orthogonality holds

(18) A[(u -u h , φ -φ h ), (v, ψ)] = 0 for all (v, ψ) ∈ V p h × V q h . It is straightforward to see that for all u, φ, v, ψ ∈ H 2 (M ) + V h there holds A[(u, φ), (v, ψ)] -(a(v, φ) + a(u, ψ)) | (u, φ)| | (v, ψ)| . ( 19
)
We will need the following continuity estimates for a.

Lemma 2.4. For all u, φ, v, ψ ∈ H 2 (M ) + V h vanishing on Γ there holds a(v, φ) S 1 2 (φ) v H 1 (T h ) , a(u, ψ) S 1 2 (u) + τ =0,T E 1 2 (U | t=τ ) ψ H 2 (T h ) .
Proof. Recalling [START_REF] Burman | Space time stabilized finite element methods for a unique continuation problem subject to the wave equation[END_REF] we see that

a(v, φ) = M g(hdv, hdφ) dx -h(v, h∂ ν φ) L 2 (∂M \Γ) = K∈T h K vh 2 φ dx + F ∈F h h F v h∂ ν φ ds
and the first claimed estimate follows from the Cauchy-Schwarz inequality and the trace inequality [START_REF] Cîndea | Numerical controllability of the wave equation through primal methods and Carleman estimates[END_REF].

Let us now turn to the second estimate. We have

a(u, ψ) = M g(hdu, hdψ) dx -h(u, h∂ ν ψ) L 2 (∂M \Γ) = K∈T h K h 2 uψ dx + F ∈F h h F h∂ ν u ψ ds + h(h∂ ν u, ψ) L 2 (∂M \Γ) -h(u, h∂ ν ψ) L 2 (∂M \Γ) ,
and the second estimate follows.

Let us recall estimates for the Scott-Zhang interpolant i p h taking functions in H 1 (M ), that vanish on Γ, to V p h , see [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF]. For all p ∈ N + and k = 1, . . . , p + 1 there is C > 0 such that for all h > 0 and u ∈ H k (M )

u -i p h u H k (T h ) ≤ C (hD) k u L 2 (M ) . (20)
Theorem 2.5. Suppose that (A) holds. Let κ < 2, p, q ∈ N + and let (u h , φ h ) in V p h × V q h be the solution of [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF].

Let u ∈ H p+1 (M ) and φ ∈ H q+1 (M ) solve (6). Then | (u -u h , φ -φ h )| h p+1-κ 2 u H p+1 (M ) + h q+1+ κ 2 φ H q+1 (M ) . In particular, χ(φ -φ h ) L 2 (M ) h p-κ 2 u H p+1 (M ) + h q+ κ 2 φ H q+1 (M ) . (21)
Proof. We write

w = u h -u, η = φ h -φ, w h = u h -i p h u, η h = φ h -i q h φ. (22)
By Lemma 2.3 and the Galerkin orthogonality [START_REF] Micu | Approximation of the controls for the wave equation with a potential[END_REF],

| (w, η)| 2 A[(w, η), (w, -η)] = A[(w, η), (w -w h , η h -η)].
We write

w i = i p h u -u, η i = i q h φ -φ. Observing that w -w h = w i and η h -η = -η i , it follows from (19) and Lemma 2.4 that | (w, η)| | (w i , η i )| + h -κ 2 w i H 1 (T h ) + h κ 2 η i H 2 (T h ) .
Recalling the scaling in [START_REF] Dörfler | Space-time discontinuous galerkin discretizations for linear first-order hyperbolic evolution systems[END_REF] and using the bound [START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF], with a replaced by s, c and e, we see that

| (w i , η i )| h -κ 2 w i H 2 (T h ) + h κ 2 η i H 2 (T h ) .
Finally, using [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF],

h -κ 2 w i H 2 (T h ) + h κ 2 η i H 2 (T h ) h p+1-κ 2 u H p+1 (M ) + h q+1+ κ 2 φ H q+1 (M ) . Recall that if (u 0 , u 1 ) ∈ H k+1 (Ω) × H k (Ω)
satisfies the compatibility conditions of order k, then the unique solution (u, φ) to ( 6) is in H k+1 (M ) × H k (M ). Hence we can take p ≤ k and q ≤ k -1. Choosing κ = 0 and p = q ≤ k -1 leads to the convergence rate (3) stated in the introduction.

Under the assumptions of Theorem 2.5, it is possible to show that

u -u h L ∞ (0,T ;L 2 (Ω)) + ∂ t (u -u h ) L 2 (0,T ;H -1 (Ω)) h p u H p+1 (M ) + h q φ H q+1 (M ) .
We do not detail the proof here, but refer to [START_REF] Burman | Space time stabilized finite element methods for a unique continuation problem subject to the wave equation[END_REF]Theorem 4.4] for a similar analysis.

The weak norms reflect the fact that the forward problem does not enjoy the classical energy stability of the wave equation. Instead error estimates are derived using continuum estimates on a level dictated by the regularity of (u-u h ). This quantity is in H -1 (M ), and not likely in a better space, resulting in the above estimate. Continuum theory at this energy level is reviewed in an appendix below, see Remark A.2 in particular. Remark 2.6. Observe that the corresponding stability estimates for unique con-

tinuation given in [9, Theorem 2.2], [8, Theorem 1.1] are inaccurate, claiming control of ∂ t u L ∞ (0,T ;H -1 (Ω))
when the best quantity that can be controlled (as shown in appendix below, Theorem A.4, Remark A.5 and Proposition A.1) is

∂ t u L 2 (0,T ;H -1 (Ω)) + ∂ t u| t=0 H -1 (Ω) + ∂ t u| t=T H -1 (Ω) .
The results in the above references are nevertheless correct without further modifications after correction of the stability norm for the error analysis.

However, we obtain a better approximation simply by solving

     u = f, u| x∈∂Ω = 0, u| t=T = 0, ∂ t u| t=T = 0, (23)
with f = χφ h . We will detail the arguments in an abstract setting below.

Let h denote a stable discrete wave operator with vanishing initial and boundary conditions such that the following standard stability estimate holds for the solution

u h to h u h = f , | u h | E := u h L ∞ (0,T ;H 1 (Ω)) + ∂ t u h L ∞ (0,T ;L 2 (Ω)) f L 2 (M ) .
We also assume that the following optimal error estimate holds: if u is the solution to [START_REF] French | A space-time finite element method for the wave equation[END_REF], then there holds | u -u h | E h p . For a high order scheme satisfying these assumptions see for instance [START_REF] French | A continuous space-time finite element method for the wave equation[END_REF]. Let now u be the solution to [START_REF] French | A space-time finite element method for the wave equation[END_REF] with f = χφ, v h the solution to h v h = χφ and u h the solution to u h = χφ h . It then follows by the above inequalities that

| u -u h | E ≤ | u -v h | E + | v h -u h | E h p + χ(φ -φ h ) L 2 (M ) h p + h q .
Here we used the properties of the method h and Theorem 2.5.

Distributed control with limited regularity

In this section we will study the finite element method [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF] in the case that the continuum solution (u, φ) to the control problem ( 6) is in the natural energy class H 1 (M ) × L 2 (M ). We make the standing assumption that (A) holds, so that (16) has a unique solution.

Lemma 3.1. Let (u h , φ h ) ∈ V p h × V q h be the solution of (16) with κ = 0. Then h -1 h 2 φ h H -1 (M ) h 1 2 hu 1 L 2 (Ω) + | (u h , φ h )| , h -1 h 2 u h H -1 (M ) h 1 2 u 0 L 2 (Ω) + | (u h , φ h )| .
Proof. To establish the first claimed inequality, we will show for

v ∈ H 1 0 (M ) that M g(hdv, hdφ h ) dx (h 1 2 hu 1 L 2 (Ω) + | (u h , φ h )| ) h∇v L 2 (M ) .
We have

M g(hdv, hdφ h ) dx = K∈T h K vh 2 φ h dx + F ∈F h h F v h∂ ν φ h ds.
Let v h ∈ V p h be the Scott-Zhang interpolant of v, and apply the above equation with

v replaced by v -v h . Then M g(hd(v -v h ), hdφ h ) dx | (0, φ h )| h∇v L 2 (M ) .
Moreover, using ( 16)

- M g(hdv h , hdφ h ) dx = h -κ s(u h , v h ) + h -κ τ =0,T h(h∂ t u h | t=τ , h∂ t v h | t=τ ) L 2 (Ω) + h 2-κ ρ(v h , φ h ) -h -κ h(hu 1 , h∂ t v h | t=0 ) L 2 (Ω) .
Hence, using κ = 0,

M g(hdv h , hdφ h ) dx (h 1 2 hu 1 L 2 (Ω) + | (u h , φ h )| ) h∇v h L 2 (M ) . Let us now show for ψ ∈ H 1 0 (M ) M g(hdu h , hdψ) dx (h 1 2 u 0 L 2 (Ω) + | (u h , φ h )| )h ψ H 1 (M ) .
Let ψ h ∈ V p h be the Scott-Zhang interpolant of ψ. Analogously with the above, we have

M g(hdu h , hd(ψ -ψ h )) dx | (u h , 0)| h∇ψ L 2 (M ) .
Moreover,

- M g(hdu h , hdψ h ) dx = h(u h , h∂ ν ψ h ) L 2 (∂M \Γ) -h κ s(φ h , ψ h ) -h 2 c(φ h , ψ h ) + h 4-κ c(φ h , ψ h ) + h 2-κ ρ(u h , ψ h ) + h(u 0 , h∂ t ψ h | t=0 ) L 2 (Ω) ,
and the second claimed inequality follows. Lemma 3.2. Let (u h , φ h ) be the solution of (16) with κ = 0. Then

| (u h , φ h )| h 1 2 u 0 L 2 (Ω) + h u 0 H 1 0 (Ω) + h u 1 L 2 (Ω) . (24)
Proof. By Lemma 2.3 there holds

| (u h , φ h )| 2 A[(u h , φ h ), (u h , -φ h )] = e(U 0 , U h | t=0 ) -L(φ h ) ≤ h 1 2 ( u 0 L 2 (Ω) + hu 1 L 2 (Ω) )E 1 2 (U h | t=0 ) + h 2 ( u 0 H 1 0 (Ω) + u 1 L 2 (Ω) )( φ h | t=0 L 2 (Ω) + ∂ t φ h | t=0 H -1 (Ω) )
. By the distributed observability estimate, see Theorem A.4 in Appendix A,

φ h | t=0 L 2 (Ω) + ∂ t φ h | t=0 H -1 (Ω) C 1 2 (φ h ) + φ h H -1 (M ) .
Recalling that hC 1 2 (φ h ) | (0, φ h )| , and using Lemma 3.1, we obtain

h( φ| t=0 L 2 (Ω) + ∂ t φ| t=0 H -1 (Ω) ) h 1 2 hu 1 L 2 (Ω) + | (u h , φ h )| .
As also

E 1 2 (U h | t=0 ) ≤ | (u h , φ h )| , we have | (u h , φ h )| 2 (h 1 2 u 0 L 2 (Ω) + h u 0 H 1 0 (Ω) + h u 1 L 2 (Ω) )(h 1 2 hu 1 L 2 (Ω) + | (u h , φ h )| ),
leading to [START_REF] French | A continuous space-time finite element method for the wave equation[END_REF].

Lemma 3.3. Let (u h , φ h ) be the solution of ( 16) with κ = 0 and u 0 = 0. Then

u h L 2 (M ) + φ h L 2 (M ) u 1 L 2 (Ω) .
Proof. Lemma 3.1 implies

u h H -1 (M ) + φ h H -1 (M ) u 1 L 2 (Ω) .
Moreover, it follows from (24) that

C 1 2 (φ h ) u 1 L 2 (Ω) , ∂ j t u h | t=0 L 2 (Ω) h 1 2 -j u 1 L 2 (Ω) , j = 0, 1.
The bound φ h L 2 (M ) ≤ C u 1 follows from the distributed observability estimate, see Remark A.5 below. It remains to show the same bound for u h . We face the complication that the above estimates do not allow us to conclude that ∂ t u h | t=0 is bounded.

To overcome this, we will employ ũh ∈ V p h that coincides with u h on ∂M and satisfies [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF] and ( 26) below. We have

ũ H -1 (M ) u 1 L 2 (Ω) .
Indeed, for any v ∈ H 1 0 (M ) there holds, using ( 14) and ( 26),

(h 2 ũh , v) L 2 (M ) = M g(hdũ h , hdv) dx ≤ M g(hdu h , hdv) dx + h∇(ũ h -u h ) L 2 (M ) h∇v L 2 (M ) h 2 u h H -1 (M ) v H 1 0 (M ) + h 1 2 h(∂ t u h | t=0 -u 1 ) L 2 (Ω) h∇v L 2 (M ) . h 2 ( u 1 L 2 (Ω) + h 1 2 ∂ t u h | t=0 L 2 (Ω) ) ≤ h 2 u 1 L 2 (Ω) .
Moreover, using [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF],

∂ t ũh | t=0 H -1 (Ω) ≤ ∂ t ũh | t=0 -u 1 H -1 (Ω) + u 1 H -1 (Ω) h ∂ t u h | t=0 -u 1 L 2 (Ω) + u 1 H -1 (Ω) ≤ h 1 2 u 1 L 2 (Ω) + u 1 H -1 (Ω) .
Recalling that ũh coincides with u h on ∂M , we conclude that

ũh L 2 (M ) u 1 L 2 (Ω)
follows from an energy estimate, see Proposition A.1 in Appendix A. Finally, using [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF],

u h L 2 (M ) ≤ u h -ũh L 2 (M ) + ũh L 2 (M ) h 1 2 h(∂ t u h | t=0 -u 1 ) L 2 (Ω) + u 1 L 2 (Ω) u 1 L 2 (Ω) .
Lemma 3.4. Let p ∈ N + and consider a family

u h ∈ V p h , h > 0. Let u 1 ∈ L 2 (Ω). Then there is a family ũh ∈ V p h , h > 0, such that ũh | ∂M = u h | ∂M and h -1 ∂ t ũh | t=0 -u 1 H -1 (Ω) ∂ t u h | t=0 -u 1 L 2 (Ω) , (25)
u h -ũh L 2 (M ) h 1 2 h(∂ t u h | t=0 -u 1 ) L 2 (Ω) . ( 26
)
Proof. Let us consider the trace mesh at t = 0,

F h,0 = {∂K ∩ {t = 0} : K ∈ T h }.
We decompose F h,0 into a set of N h disjoint patches ∂P i , i = 1, . . . , N h , such that each patch contains several element faces but their area and diameter satisfy

h n |∂P i | h n , h diam(P i ) h.
Then we define disjoint patches P i consisting of elements of T h so that

∂P i = P i ∩ {t = 0} and that h n+1 |P i | h n+1
. Now we define the functions p i ∈ V 1 h such that supp(p i ) ⊂ P i and p i (x) = 1 for every node x in the interior of P i . We require that the patches ∂P i are large enough so that, writing

α i = ∂P i ∂ t p i | t=0 ds, β i = ∂ t p i | t=0 L 2 (∂P i ) , γ i = p i L 2 (P i ) , there holds h n-1 α i h n-1 , h n 2 -1 β i h n 2 -1 and h 1 2 (n+1) γ i h 1 2 (n+1) . We set ũh = u h + N h i=1 w i p i , w i = -α -1 i ∂P i (∂ t u h | t=0 -u 1 ) ds.
Then

∂P i (∂ t ũh | t=0 -u 1 ) ds = 0. ( 27
)
To establish [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF] we let v ∈ H 1 0 (Ω) and show that

(∂ t ũh (•, 0) -u 1 , v) L 2 (Ω) ∂ t u h (•, 0) -u 1 L 2 (Ω) h∇v L 2 (Ω) .
Let v ∈ L 2 (Ω) be equal to the average of v on each patch ∂P i , that is,

v| ∂P i = |∂P i | -1 ∂P i v ds. Now (27) implies (∂ t ũh | t=0 -u 1 , v) L 2 (Ω) = 0, and
(∂ t ũh | t=0 -u 1 , v) L 2 (Ω) = (∂ t ũh | t=0 -u 1 , v -v) L 2 (Ω) ≤ ∂ t ũh | t=0 -u 1 L 2 (Ω) v -v L 2 (Ω) ∂ t ũh | t=0 -u 1 L 2 (Ω) h∇v L 2 (Ω) .
Here we used the Poincaré inequality as stated for example in [START_REF] Ern | Finite element quasi-interpolation and best approximation[END_REF]. To establish [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF] it remains to show that

∂ t ũh | t=0 -u 1 L 2 (Ω) ∂ t u h | t=0 -u 1 L 2 (Ω) .
Using the fact that the patches ∂P i are disjoint, we have

∂ t ũh | t=0 -u 1 L 2 (Ω) ≤ ∂ t u h | t=0 -u 1 L 2 (Ω) + N h i=1 |w i | ∂ t p i | t=0 L 2 (∂P i ) .
Recalling that α i behaves like h n-1 and β i like h n 2 -1 , we obtain using the Cauchy-Schwarz inequality

|w i | ∂ t p i | t=0 L 2 (∂P i ) = α -1 i β i ∂P i (∂ t u h | t=0 -u 1 ) ds h 1-n h n 2 -1 h n 2 ∂ t u h | t=0 -u 1 L 2 (∂P i ) = ∂ t u h | t=0 -u 1 L 2 (∂P i ) .
Let us now turn to [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF]. Note that

u h -ũh 2 L 2 (M ) = N h i=1 |w i | 2 p i 2 L 2 (P i ) .
Recalling that γ 2 i behaves like h n+1 , we obtain

|w i | 2 p i 2 L 2 (P i ) = |w i | 2 γ 2 i h n+1 h 2(1-n) h n ∂ t u h | t=0 -u 1 2 L 2 (∂P i ) , leading to u h -ũh 2 L 2 (M ) h 3 ∂ t u h | t=0 -u 1 2 L 2 (Ω) .
Theorem 3.5. Suppose that u 0 = 0 and u 1 ∈ L 2 (Ω). Let (u h , φ h ) be the solution of (16) with κ = 0, and let (u, φ) be the solution of [START_REF] Brenner | The mathematical theory of finite element methods[END_REF]. Then there is a sequence h j → 0 such that (u h j , φ h j ) converges weakly to (u, φ) in L 2 (M ).

Proof. By Lemma 3.3 both u h and φ h are bounded in L 2 (M ). Thus there is a sequence h j → 0 such that (u h j , φ h j ) converges weakly to a function (u * , φ * ) in L 2 (M ). By Lemma 2.1 it is enough to show that (u * , φ * ) satisfies [START_REF] Brenner | The mathematical theory of finite element methods[END_REF].

As the embedding H -(M ) ⊂ L 2 (M ) is compact for > 0, by passing to a subsequence, we may assume that (u h j , φ h j ) → (u * , φ * ) in H -(M ). By Lemmas 3.1 and 3.2 we may further assume that ( u h j , φ h j ) → ( u * , φ * ) in H --1 (M ). For < 1/2 it follows from Lemma A.7 that

0 = (u h j | Γ , φ h j | Γ ) → (u * | Γ , φ * | Γ ).
Thus (u * , φ * ) satisfies the homogeneous lateral boundary conditions in [START_REF] Brenner | The mathematical theory of finite element methods[END_REF].

For any ψ ∈ C ∞ (M ) with ψ| Γ = 0 and any v ∈ C ∞ 0 (M ) there holds

h -2 (a(u h , ψ) -h 2 c(φ h , ψ) -L(ψ)) → 0, ( 28
)
h -2 a(v, φ h ) → 0, ( 29
)
as h → 0. Before showing ( 28)-( 29), let us show that they imply that (u * , φ * ) satisfies [START_REF] Brenner | The mathematical theory of finite element methods[END_REF]. The equation φ * = 0 follows immediately from [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]. Observe that

h -2 a(u h , ψ) = (u h , ψ) L 2 (M ) → (u * , ψ) L 2 (M ) .
It follows from ( 28) that for any ψ ∈ C ∞ (M ) vanishing on Γ there holds

(u * , ψ) L 2 (M ) = c(φ * , ψ) + (u 1 , ψ| t=0 ) L 2 (Ω) -(u 0 , ∂ t ψ| t=0 ) L 2 (Ω) . (30)
In particular, taking ψ ∈ C ∞ 0 (M ) we see that u * = χφ * . To show that (u * , φ * ) satisfies [START_REF] Brenner | The mathematical theory of finite element methods[END_REF], it remains to verify the initial and final conditions for u * . We have u * ∈ L 2 (M ) and

∂ 2 t u * = ∆u * + χφ * ∈ L 2 (0, T ; H -2 (Ω)). Now [37, Theorem 3.1, p. 19] gives u * ∈ C(0, T ; H -1 2 (Ω)), ∂ t u * ∈ C(0, T ; H -3 2 (Ω)).
Taking ψ(t, x) = ψ 0 (t)ψ 1 (x), with ψ 0 ∈ C ∞ (0, T ) and ψ 1 ∈ C ∞ 0 (Ω), we integrate by parts

(u * , ψ) L 2 (M ) = T 0 u * , ψ 1 ∂ 2 t ψ 0 dt - T 0 u * , ∆ψ 1 ψ 0 dt = T 0 (χφ * , ψ 1 ) L 2 (Ω) ψ 0 dt + [ u * , ψ 1 ∂ t ψ 0 -∂ t u * , ψ 1 ψ 0 ] t=T t=0 ,
where •, • is the pairing between distribution and test functions on Ω. Comparison with [START_REF] Hulbert | Space-time finite element methods for second-order hyperbolic equations[END_REF] shows that u * satisfies the initial and final conditions in [START_REF] Brenner | The mathematical theory of finite element methods[END_REF].

Let us now show [START_REF] Hecht | New development in Freefem++[END_REF]. Denote by ψ h the Scott-Zhang interpolant of ψ. By ( 16)

a(u h , ψ) -h 2 c(φ h , ψ) -L(ψ) = a(u h , φ -ψ h ) -h 2 c(φ h , ψ -ψ h ) -L(ψ -ψ h ) + s(φ h , ψ h ) -h 4 c(φ h , ψ h ) -h 2 ρ(u h , ψ h ).
Using the continuity of a in Lemma 2.4, the interpolation estimate [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF], and the bound [START_REF] French | A continuous space-time finite element method for the wave equation[END_REF] for the residual norm, we obtain

|a(u h , ψ -ψ h )| | (u h , φ h )| (hD) 2 ψ L 2 (M ) h u 1 L 2 (M ) (hD) 2 ψ L 2 (M ) .
Recalling that hC 1 2 (φ h ) | (0, φ h )| , we use the continuity [START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF] for c and the interpolation estimate [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] to get

h 2 |c(φ h , ψ -ψ h )| ≤ h 2 C 1 2 (φ h )C 1 2 (ψ -ψ h ) h| (0, φ h )| (hD) 2 ψ L 2 (M ) .
Using once again [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF],

|L(ψ -ψ h )| = h 2 |(u 1 , (ψ -ψ h )| t=0 ) L 2 (Ω) | h 3/2 u 1 L 2 (Ω) (hD) 2 ψ L 2 (M ) .
Turning to the first term related to regularization, we have

|s(φ h , ψ h )| ≤ S 1 2 (φ h )S 1 2 (ψ h ),
where the first factor is bounded by

| (0, φ h )| h u 1 L 2 (Ω)
, and the second satisfies

S(ψ h ) K∈T h ( h 2 (ψ h -ψ) 2 L 2 (K) + h 2 ψ 2 L 2 (K) ) + F ∈F h h h∂ ν (ψ h -ψ) 2 L 2 (F ) (hD) 2 ψ 2 L 2 (M ) . Finally, h 4 |c(φ h , ψ h )| h 3 | (0, φ h )| ψ L 2 (M ) , h 2 |ρ(u h , ψ h )| h 2 | (u h , 0)| ψ L 2 (M ) h 3 u 1 ψ L 2 (M ) ,
and (28) follows.

We turn to [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]. Denote by v h the Scott-Zhang interpolant of v. By ( 16)

a(v, φ h ) = a(v -v h , φ h ) -s(u h , v h ) + h 2 ρ(v h , φ h ).
Similarly to the bounds above, we have

|a(v -v h , φ h )| + |s(u h , v h )| h u 1 L 2 (Ω) (hD) 2 v L 2 (M ) , h 2 |ρ(v h , φ h )| h| (0, φ h )| h 2 v L 2 (M ) ,
and (29) follows. This finishes the proof that (u * , φ * ) satisfies (6).

Boundary control

Let us begin by formulating our assumptions on the cutoff function χ in (4). We consider a function of the form

χ(t, x) = χ 0 (t)χ 2 1 (x), where χ 0 ∈ C ∞ 0 ([0, T ]) and χ 1 ∈ C ∞ (Γ) take values in [0, 1]
, and suppose that (A') χ = 1 on open (a, b) × ω ⊂ Γ satisfying the geometric control condition. In the case of boundary control, the geometric control condition means that every compressed generalized bicharacteristic intersects the set (a, b) × ω, when projected to M . Moreover, the intersection must happen at a nondiffractive point and the lightlike lines must have finite order of contact with Γ. We refer again to [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for the definitions.

We let V ∈ C ∞ (Ω) and consider the boundary control problem for the following operator

P = + V. (31)
Let (u 0 , u 1 ) ∈ L 2 (Ω) × H -1 (Ω). Then the distributed control problem for P can be solved by finding (u, φ)

∈ L 2 (M ) × H 1 (M ) such that          P u = 0, u| Γ = χ∂ ν φ, u| t=0 = u 0 , ∂ t u| t=0 = u 1 , u| t=T = 0, ∂ t u| t=T = 0, P φ = 0, φ| Γ = 0. (32)
If (u 0 , u 1 ) ∈ H k+1 (Ω) × H k (Ω) satisfies the compatibility conditions of order k, then the unique solution (u, φ) to [START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF] satisfies [START_REF] Langer | Space-Time Methods[END_REF] and this initial data for φ satisfies the compatibility conditions of order k + 1, see [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF]Thëorem 5.4]. It follows that φ ∈ H k+2 (M ) and u ∈ H k+1 (M ), and the convergence proof for our finite element method is again based on this regularity.

φ| t=T ∈ H k+2 (Ω), ∂ t φ| t=T ∈ H k+1 (Ω),
Although uniqueness of the solution (u, φ) to ( 32) is implictly contained in [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF], we give a short proof. This illustrates the difference in natural regularities between the distributed and boundary control cases.

Lemma 4.1. Suppose that (A') holds. Let (u, φ) ∈ L 2 (M ) × H 1 (M ) solve ( 32) with u 0 = u 1 = 0. Then u = φ = 0.
Proof. For the convenience of the reader, we show first that

φ ∈ C(0, T ; H 1 0 (Ω)) ∩ C 1 (0, T ; L 2 (Ω)), ∂ ν φ| Γ ∈ L 2 (Γ). ( 34
)
The proof of this fact is very similar to the proof of Lemma A.3. The standard energy estimate implies that for all s ∈ (0, T ),

φ| t=0 H 1 (Ω) + ∂ t φ| t=0 L 2 (Ω) φ| t=s H 1 (Ω) + ∂ t φ| t=s L 2 (Ω) .
Integration in s gives

φ| t=0 H 1 (Ω) + ∂ t φ| t=0 L 2 (Ω) φ H 1 (M ) ,
and the regularity (34) follows now from [START_REF] Lasiecka | Nonhomogeneous boundary value problems for second order hyperbolic operators[END_REF]. It also follows from [START_REF] Lasiecka | Nonhomogeneous boundary value problems for second order hyperbolic operators[END_REF] that

u ∈ C(0, T ; L 2 (Ω)) ∩ C 1 (0, T ; H -1 (Ω)).
In the case that u and φ are smooth

0 = (P u, φ) L 2 (M ) -(u, P φ) L 2 (M ) = -(χ∂ ν φ, ∂ ν φ) L 2 (Γ) ,
and for (u, φ) ∈ L 2 (M ) × H 1 (M ) this can be justified by approximating u and φ with smooth functions as in the proof of Lemma 2.1. It follows from the boundary observability estimate, see Theorem A.6, that φ = 0 identically, and hence also u = 0 identically. Writing M h = K∈ Th K, we assume that M ⊂ M h . We define

T h = {K ∩ M : K ∈ T h }, T = {T h : h > 0},
and require that:

(T) There is C > 0 such that for all h > 0 and all K ∈ T h , letting K ∈ Th satisfy K = K ∩ M , there are balls B 1 ⊂ K and B 2 ⊃ K such that the radii r j of B j , j = 1, 2, satisfy

C -1 r 2 ≤ h ≤ Cr 1 , (35)
and that ν(y) • ρ(y) > C -1 , for all y ∈ ∂K, [START_REF] Lebeau | Experimental study of the HUM control operator for linear waves[END_REF] where ν is the outer unit normal vector of ∂K, and ρ(y) = (y -x)/|y -x| with x is the centre of B 1 .

If M was polyhedral, then we could choose T so that M h = M for all small enough h > 0. In this case (T) follows if T is quasi-uniform, see [START_REF] Ern | Theory and practice of finite elements[END_REF]Definition 1.140]. In the case of smooth Ω, we can construct T so that (T) holds for all small enough h > 0 by choosing polyhedral sets M h ⊃ M that approximate M in the sense that the Hausdorff distance between ∂M h and ∂M is of order h 1+ for some > 0, and meshing M h in a quasi-uniform manner.

We define for p ∈ N + = {1, 2, . . . } the H 1 (M )-conformal approximation space of polynomial degree p,

V p h = {u ∈ H 1 (M ) : u| K ∈ P p (K) for all K ∈ T h }, (37)
where P p (K) denotes the set of polynomials of degree less than or equal to p on K. We write also V h = p∈N + V p h . Note that, contrary to (9) no boundary condition is imposed on Γ.

The following two lemmas are proven in Appendix B. Lemma 4.2. The trace inequality (13) holds for the family T .

Lemma 4.3. There is a family of interpolation operators i

p h : H 1 (M ) → V p h satisfying u -i p h u H k (T h ) h k u H k (M ) . (38)
For any h > 0, the control problem (32) can be formulated weakly as

a(u, ψ) = -c(h -1 φ, ψ) + L(ψ), a(v, φ) = 0, (39)
for all v, ψ ∈ C ∞ (M ), where

a(u, ψ) = M g(hdu, hdψ) dx + h 2 (u, V ψ) L 2 (M ) (40) -h(u, h∂ ν ψ) L 2 (∂M ) -h(h∂ ν u, ψ) L 2 (Γ) c(φ, ψ) = h(χh∂ ν φ, h∂ ν ψ) L 2 (Γ) ,
and L is as in [START_REF] Castro | Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method[END_REF]. Indeed, it follows from (8) that if smooth (u, φ) solves ( 32) then [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] holds for all smooth (v, ψ). We emphasize that a and c are chosen here so that they satisfy the continuity estimate [START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF].

Our finite element method has the form: find the critical point of the Lagrangian

L(u, φ) : V p h × V q h → R, L(u, φ) = 1 2 c(φ, φ) + L(φ) - 1 2 R(u, φ) + a(u, φ),
where, writing U = (u, ∂ t u) and U 0 = (u 0 , u 1 ), the regularization is given by

R(u, φ) = h -κ S(u) -h κ S(φ) + h -κ E(U | t=0 -U 0 ) + h -κ E(U | t=T ) + γh -κ B(u) -h κ B(φ) + γh -κ C(φ) + 2γh -κ ρ(u, φ), B(u) = h u 2 L 2 (Γ) , C(φ) = h χh∂ ν φ 2 L 2 (Γ) , ρ(u, φ) = -h(u, χh∂ ν φ) L 2 (Γ) ,
where κ ≤ 0 and γ ∈ (0, 1) are fixed constants. Here E and S are as 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] except that in S is replaced by P . We have R(u, h -1 φ) = 0 for a smooth solution (u, φ) to [START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF]. Indeed,

B(u) + 2ρ(u, h -1 φ) + C(h -1 φ) = h u -χ∂ ν φ 2 L 2 (Γ)
= 0, and also S(u) = S(φ) = B(φ) = 0 and E(U | t=0 -U 0 ) = E(U | t=T ) = 0. The equation dL(u, φ) = 0 can be written as [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF] where the bilinear form A is now given by

A[(u, φ), (v, ψ)] = h -κ s(u, v) -h κ s(φ, ψ) -h 2 c(φ, ψ) + h -κ τ =0,T e(U | t=τ , V | t=τ ) + γb(u, v) -b(φ, ψ) -a(v, φ) -a(u, ψ) + γh -κ c(φ, ψ) + 2γh -κ ρ(v, φ) + 2γh -κ ρ(u, ψ).
We define the residual norm by

| (u, φ)| 2 = h -κ (S(u) + B(u)) + h κ (S(φ) + B(φ)) + C(φ) + h -κ τ =0,T E(U | t=τ ).
This is indeed a norm on V h × V h as can be seen by following the proof of Lemma 2.2. Observe that in this case the vanishing boundary conditions on Γ are not imposed in the spaces

V h but follow if B(u) = B(φ) = 0. Lemma 4.4. For all u, φ ∈ H 2 (M ) + V h there holds | (u, φ)| 2 A[(u, φ), (u, -φ)].

Proof. By the definition of A, we have

A[(u, φ), (u, -φ)] =| (u, φ)| 2 -γh -κ C(φ).
As κ ≤ 0, γ < 1 and χ ≤ 1, γh -κ C(φ) can be absorbed by C(φ).

The previous two lemmas imply that the finite dimensional linear system ( 16) has a unique solution, and thus defines a finite element method. 4.2. Error estimates. Equation ( 16) defines a finite element method that is consistent in the sense that if smooth enough u and φ satisfy (6), then [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF] holds for (u, φ). This follows from the weak formulation ( 10) of ( 6) together with the regularization vanishing for (u, φ). If smooth enough u and φ satisfy [START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF] and if (u h , φ h ) ∈ V p h × V q h solves ( 16) then the Galerkin orthogonality

(41) A[(u -u h , h -1 φ -φ h ), (v, ψ)] = 0 for all (v, ψ) ∈ V p h × V q h .
Analogously to the case of distributed control, this is due to (u, φ) satisfying the weak formulation [START_REF] Loreti | An Ingham type proof for a two-grid observability theorem[END_REF] and the regularization vanishing at (u, h -1 φ).

It is straightforward to see that for all u, φ, v, ψ ∈ H 2 (M ) + V h there holds

A[(u, φ), (v, ψ)] + (a(v, φ) + a(u, ψ)) | (u, φ)| | (v, ψ)| .
We will need the following analogue of Lemma 2.4. We omit the proof, this being a modification of the earlier proof. The only difference is that the boundary terms on Γ need to be kept track of. Lemma 4.5. For all u, φ, v, ψ ∈ H 2 (M ) + V h there holds

a(v, φ) S 1 2 (φ) + B(φ) v H 2 (T h ) , a(u, ψ) S 1 2 (u) + B(u) + τ =0,T E 1 2 (U | t=τ ) ψ H 2 (T h ) .
By repeating the proof of Theorem 2.5 we obtain:

Theorem 4.6. Suppose that (A') holds. Let κ ≤ 0, p, q ∈ N + and let (u h , φ h ) in V p h × V q h be the solution of [START_REF] Dumont | 4D remeshing using a space-time finite element method for elastodynamics problems[END_REF]. Let u ∈ H p+1 (M ) and φ ∈ H q+1 (M ) solve [START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF]. Then

| (u -u h , h -1 φ -φ h )| h p+1-κ 2 u H p+1 (M ) + h q+1+ κ 2 h -1 φ H q+1 (M ) . In particular, χ∂ ν (φ -hφ h ) L 2 (Γ) h p-κ 2 + 1 2 u H p+1 (M ) + h q+ κ 2 -1 2 φ H q+1 (M ) . (42)
As discussed above, if (u 0 , u 1 ) ∈ H k+1 (Ω) × H k (Ω) satisfies the compatibility conditions of order k, then the solution to [START_REF] Johnson | Discontinuous Galerkin finite element methods for second order hyperbolic problems[END_REF] satisfies

(u, φ) ∈ H k+1 (M ) × H k+2 (M ).
Hence we can take κ = 0, q ≤ k + 1 and p = q -1 in the above theorem, leading to the convergence rate (5) stated in the introduction. Remark 4.7. A finite element method for the distributed control problem can be formulated using the spaces V p h defined by [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF] and the bilinear form a in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF]. With these choices replacing V p h and a in the Lagrangian in Section 2.2, and with B(u) -B(φ) added in the regularization R(u, φ) there, we obtain a method satisfying the estimates in Theorem 2.5. This method works for smooth Ω whenever the geometric control condition (A) holds. We omit proving this, the proof being very similar with those above.

Numerical experiments

We discuss some numerical experiments performed with the Freefem++ package (see [START_REF] Hecht | New development in Freefem++[END_REF]).

We address the distributed and boundary case in the one dimensional case and emphasize the influence of the regularity of the initial condition on the rate of convergence of the finite element method with respect to the size of the discretization. We use uniform (unstructured) meshes and the cut off functions χ 0 ∈ C ∞ 0 ([0, T ]) and

χ 1 ∈ C ∞ 0 ([0, 1]) defined as follows (43) χ 0 (t) = e -1 2t e -1 2(T -t) e -1 T e -1 T , χ 1 (x) = e -1 5(x-a) e -1 5(b-x) e -2 5(b-a) e -2 5(b-a) 1 [a,b] (x)
for any 0 < a < b < 1 and T > 0. In particular, χ 0 (T /2) = 1 and χ 1 ((a + b)/2) = 1. Figure 1 depicts the function χ 0 for T = 2.5. We consider the simplest situation for which

(Ex1) (u 0 , u 1 ) = (sin(πx), 0) ∈ H k+1 (Ω) × H k (Ω) ∀k ∈ N.
Compatibility conditions (C1)-(C2) are satisfied for any j. Moreover, we use the cut-off functions χ 0 ∈ C ∞ 0 ([0, T ]) and χ 1 ∈ C ∞ 0 ([0, 1]) defined by ( 43) with T = 2, a = 0.1 and b = 0.4. The null controllability property (A) holds true for this set of data. Since explicit solutions are not available in the distributed case, we define as "exact" solution (u, φ) the one of ( 16) from a fine and structured mesh (composed of 409 000 triangles and 205 261 vertices) corresponding to h ≈ 4.41 × 10 -3 and (u h , φ h ) ∈ V p h × V q h with (p, q) = (3, 3). Figure 2-left depicts the evolution of the relative error for the variable φ with respect to the L 2 -norm err(φ, φ h , χ)

:= χ(φ -φ h ) L 2 (M ) / χφ L 2 (M )
with respect to h for various pairs of (p, q). Table 1 collects the corresponding numerical values. We observe the convergence of the approximation w.r.t. h. Moreover, the figure exhibits the influence of the space V q h used for the variable φ h while the choice of the space V p h for the variable u h has no effect on the approximation. We observe rates close to 0.5, 2 and 3 for (p, q) = (1, 1), (p, q) = (2, 2) and (p, q) = (3, 3) respectively, in agreement with Theorem 2.5. For comparison, Figure 2-right depicts the evolution of the relative error err(φ, φ h , χ) for χ 0 (t) = 1 and χ 1 (x) = 1 (a,b) (x), i.e. when no regularization of the support is introduced. Table 2 collects the corresponding numerical values. The corresponding controlled pair (u, φ) is a priori only in C([0, T ];

H 1 0 (Ω)) × C([0, T ]; L 2 (Ω))
. Thus, if we still get the convergence with respect to the parameter h, we observe that the approximation is not improved beyond the value q = 2. As before, the choice of the approximation space V p h for u h does not affect the result. The rate is also reduced: for (p, q) = (2, 2), the rate is close to 1.5. This highlights the influence of the cut off functions, including for very smooth initial conditions. Table 3 collects some L 2 norms of u h and φ h with respect to h for the pair (p, q) = (2, 2): in particular, the relative error err(u, u h , 1) associated with the controlled solution u is order of h 2.5 for h small enough.

(x) = 1 (a,b) (x). 5.2. Distributed case : initial condition in H 1 (Ω) × L 2 (Ω). We consider the initial condition (Ex2) (u 0 , u 1 ) = 4x1 (0,1/2) (x) + 4(1 -x)1 [1/2,1) (x), 0 ∈ H 1 (Ω) × H 0 (Ω)
for which the compatibility conditions (C1)-(C2) are satisfied for any j. If the cutoff functions are introduced, the controlled pair (u, φ) belongs to H 1 (M ) × H 0 (M ). h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 (p, q) = (1, 1) 9.81 × 10 -1 9.58 × 10 -1 8.81 × 10 -1 6.83 × 10 -1 4.31 × 10 -1 (p, q) = (2, 2) 4.96 × 10 -1 3.15 × 10 -1 2.09 × 10 -1 6.05 × 10 -2 1.00 × 10 -2 (p, q) = (3, 3) 2.45 × 10 -1 6.41 × 10 -2 7.93 × 10 -3 1.81 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 (p, q) = (1, 1) 9.55 × 10 -1 × 10 -1 6.74 × 10 -1 3.58 × 10 -1 1.24 × 10 -1 (p, q) = (2, 2) 1.71 × 10 -1 1.57 × 10 -2 5.20 × 10 -3 2.24 × 10 -3 7.47 × 10 -4 (p, q) = (3, 3) 2.58 × 10 -2 1.17 × 10 -3 5.28 × 10 -3 2.24 × 10 -3 7.47

× 10 -3 1.38 × 10 -4 Table 1. (Ex1); χ(φ h -φ) L 2 (M ) / χφ L 2 (M ) ; χ from (43).
× 10 -4 Table 2. (Ex1); χ(φ h -φ) L 2 (M) / χφ L 2 (M) ; χ 0 (t) = 1; χ 1 (x) = 1 (a,b) (x). h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1) 6.22 × 10 -1 3.92 × 10 -1 2.33 × 10 -1 6.45 × 10 -2 1.04 × 10 -2 err(∂ x φ, ∂ x φ h , 1) 8.11 × 10 -1 6.84 × 10 -1 4.85 × 10 -1 1.64 × 10 -1 4.57 × 10 -2 err(u, u h , 1) 4.29 × 10 -1 1.36 × 10 -1 5.00 × 10 -2 1.08 × 10 -2 1.10 × 10 -3 err(φ, φ h , χ) 4.96 × 10 -1 3.15 × 10 -1 2.09 × 10 -1 6.05 × 10 -2 1.00 × 10 -2 Table 3. (Ex1); (p, q) = (2, 2
) and χ from (43).

Theorem 2.5 does not provide a convergence rate in this case. The strong convergence is however observed: figure 3 displays the relative error wrt h for (p, q) = (1, 1), (p, q) = (2, 2) and (p, q) = (3, 3) with rates close to 1/2. h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 (p, q) = (1, 1) 9.89 × 10 -1 9.66 × 10 -1 8.99 × 10 -1 7.34 × 10 -1 5.30 × 10 -1 (p, q) = (2, 2) 5.90 × 10 -1 4.31 × 10 -1 3.38 × 10 -1 2.98 × 10 -1 2.22 × 10 -1 (p, q) = (3, 3) 3.51 × 10 -1 3.14 × 10 -1 2.28 × 10 -1 1.54

× 10 -1 8.08 × 10 -2 Table 4. (Ex2); χ(φ -φ h ) L 2 (M ) / χφ L 2 (M ) ; χ from (43).
A similar behavior is observed with the condition u 0 = 0 and u 1 = 1 (0.4,0.6) (x) in L 2 (Ω) in agreement with Theorem 3.5 5.3. Distributed case : initial condition in H 2 (Ω) × H 1 (Ω). We consider the initial condition

(Ex2b) (u 0 , u 1 ) = ρ(x) x 0 u 2 0 (t)dt, 0 ∈ H 2 (Ω) × H 1 (Ω)
χ(φ -φ h ) L 2 (M ) / χφ L 2 (M ) vs. h; χ from (43).
where u 2 0 is the initial position defined in (Ex2) and ρ ∈ C ∞ 0 (Ω) is introduced in order to preserve the compatibility conditions (C1)-(C2). Figure 3-right displays the convergence of the approximation for (p, q) = (1, 1), (p, q) = (2, 2) and (p, q) = (3, 3). Theorem 2.5 still does not provide a convergence rate in this case. However, with respect to the previous example, smaller relative error with rates close to 1 are observed for (p, q) = (2, 2) and (p, q) = (3, 3). 5.4. Boundary case: initial condition in H k+1 (Ω) × H k (Ω) for all k ∈ N. We consider again the simple situation given by the initial condition (Ex1). Compatibility conditions (C1)-(C2) are satisfied for any j. In contrast with the distributed case, explicit exact solutions are available in the boundary case when cut-off functions are not introduced. Precisely, the corresponding control of minimal L 2 (Γ) norm with Γ = (0, T = 2) × {1} is given by v(t) = 1 2 sin(π(1 -t)) = 1 2 sin(πt) leading to v 2 = 1/2. The corresponding controlled solution is given by ( 44)

u(t, x) =    1 2 u 0 (x + t) + u 0 (x -t) x + t ≤ 1, 1 2 u 0 (x -t), x + t > 1, x -t ≥ -1, 0, x -t < -1, leading to u L 2 (M ) = 1/2. The corresponding adjoint solution is given by φ(t, x) = -1 2π sin(πt) sin(πx) leading to φ L 2 (M ) = 1 2 √ 2π and ∂ x φ L 2 (M ) = 1 2 √
2 . Tables 5 and6 collects some relative errors w.r.t. h for (p, q) = (1, 2) and (p, q) = (2, 3) respectively including

err(v, u h ) := v -u h (•, 1) L 2 (0,T ) / v L 2 (0,T ) ,
and err(v, hχ∂ ν φ h ) while Figure 4-left depicts the relative error on the control w.r.t. h for several pairs of (p, q). Since compatibility conditions hold true here, the introduction of the cut off function χ = 1 is a priori useless. However, we observe that the term ∂ x φ(•, 1) is not well approximated near t = 0 and t = T . This somehow pollutes the approximation φ h of φ inside the domain (precisely along the characteristics intersecting the points (t, x) = (0, 1) and (t, x) = (T, 1) and affects the optimal rate. We observe rate close to 0.75 for (p, q) = (1, 1) and close to 1.5 otherwise. Imposing in addition φ = 0 on the boundary ∂Ω slightly improves the approximation.

h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1)
7.17 × 10 -1 1.97 × 10 -1 4.54 × 10 -2 1.53 × 10 -2 4.50 × 10 -3 err(∂ x φ, ∂ x φ h , 1) 9.21 × 10 -1 3.05 × 10 -1 9.40 × 10 -2 5.41 × 10 -2 2.47 × 10 -2 err(u, u h , 1)

1.88 × 10 -1 5.73 × 10 -2 With χ = 1, the explicit control of minimal L 2 (χ 1/2 , (0, T )) norm is not available anymore. As for the distributed controllability, we define as "exact" control the one obtained from a fine and uniform mesh (composed of 648 000 triangles and 325 261 vertices) corresponding to h ≈ 3.92 × 10 -3 and (p, q) = (2, 3). We take T large enough, precisely T = 2.5, to ensure the null controllability property of the wave equation. Figure 4-right displays the evolution of hχ∂ x φ h (•, 1) -v L 2 (0,T ) / v L 2 (0,T ) w.r.t. h. For (p, q) = (1, 1) and (p, q) = (1, 2), we observe rates close to 0.5 and 1.5 in agreement with [START_REF] Miller | Escape function conditions for the observation, control, and stabilization of the wave equation[END_REF] of Theorem 4.6 with κ = 0. For (p, q) = (2, 3), we observe a rate close to 3, which is a bit better than the value 2.5 from [START_REF] Miller | Escape function conditions for the observation, control, and stabilization of the wave equation[END_REF]. Those results also show that the boundary control can be approximated both from the quantity h∂ x φ h (•, 1) obtained from the adjoint dual variable and from the trace u h (•, 1) of the primal variable. 5.5. Boundary case: initial condition in H 1 (Ω)×H 0 (Ω). We consider the initial data (Ex2). The corresponding control of minimal L 2 (Γ) norm with Γ = (0, T = 2) × {1} (corresponding to χ ≡ 1)is given by

v(t) = 2t 1 (0,1/2) (t) + 2(1 -t)1 (1/2,3/2) (t) + 2(t -2) 1 (3/2,1) (t).
The corresponding controlled solution is explicitly known as follows: [START_REF] Münch | Constructive exact control of semilinear 1-D wave equations by a least-squares approach[END_REF]

u(t, x) = 1 2 u 0 (x -t)1 (t-x≤0) + 1 2 u 0 (x + t)1 (t+x≤1) - 1 2 u 0 (t -x)1 (t-x≥0) 1 (t-x≤1) .
The corresponding adjoint solution is given by φ(t, x) + φ(t, 1 -x) where φ is defined in [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF] with the following initial conditions (φ 0 , φ 1 ) = 0, -2x

1 (0,1/2) (x) + 2(x -1) 1 (1/2,1) (x) ∈ H 2 (Ω) × H 1 (Ω).
Compatibility conditions are satisfied. Figure 5-left displays the evolution of χ∂ x φ h (1, •) -v L 2 (0,T ) / v L 2 (0,T ) w.r.t. h for various pairs of (p, q) and χ ≡ 1. We observe a rate close to 0.75 for (p, q) = (1, 1) and close to 1.5 otherwise. Remark that a priori u ∈ H 1 (M ) and φ ∈ H 2 (M ) so that the choice (p, q) = (2, 3) does not lead to a better rate than the choice (p, q) = (1, 2). Moreover, as expected, the introduction of the cut off χ = 1 does not improve here the rate of convergence: see Figure 5-right where similar rates are observed. 5.6. Boundary case: initial condition in H 0 (Ω) × H -1 (Ω). We consider the following stiff situation given by (Ex3) (u 0 , u 1 ) = (4x1 (0,1/2) (x), 0) ∈ H 0 (Ω) × H -1 (Ω). 7.83 × 10 -1 2.53 × 10 -1 5.82 × 10 -2 1.97 × 10 -2 5.64 × 10 -3 err(∂ x φ, ∂ x φ h , 1) 1.12 × 10 0 5.03 × 10 -1 2.02 × 10 -1 1.12 × 10 -1 5.04 × 10 -2 err(u, u h , 1)

1.91 × 10 -1 4.94 × 10 -2 2.42 × 10 -2 1.17 × 10 -2 5.04 × 10 -3 err(v, u h ) 2.20 × 10 -1 6.48 × 10 -2 2.87 × 10 -2 1.16 × 10 -2 4.95 × 10 -3 err(v, hχ∂ x φ h ) 2.49 × 10 -1 6.41 × 10 -2 2.97 × 10 -2 1.40 × 10 -2 6.06 × 10 -3 and extensively discussed 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][START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF] and T = 2. The corresponding control of minimal L 2 ((0, T ) × {1}) norm is given by v

(t) = 2(1 -t)1 (1/2,3/2) (t) leading to v L 2 (0,T ) = 1/ √ 3.
The corresponding controlled solution is explicitly known as follows:

(46)

u(t, x) =    4x 0 ≤ x + t < 1 2 , 2(x -t) -1 2 < t -x < 1 2 , x + t ≥ 1 2 , 0 else, leading to u L 2 (M ) = 1/ √ 3.
The corresponding initial conditions of the adjoint solution is (φ 0 , φ 1 ) = (0, -2x

1 (0,1/2) (x)) ∈ H 1 (Ω) × H 0 (Ω) leading to (47) φ(t, x) =            -2xt 0 ≤ x + t < 1 2 , x ≥ 0, t ≥ 0, (x-t) 2 2 -1 8 1 2 ≤ x + t < 3 2 , -1 2 < x -t < 1 2 , 2(x -1)(1 -t) 3 2 ≤ x + t, -1 2 < x -t, -(x+t-2) 2 2 + 1 8 3 2 < x + t < 5 2 , -3 2 < x -t ≤ -1 2 , 2x(2 -t)
x -t ≤ -3 2 , leading to φ L 2 (M ) ≈ 9.86 × 10 2 and ∂ x φ L 2 (M ) ≈ 4.08 × 10 -1 . In particular, we check that ∂ x φ(t, x) |x=1 = 2(1 -t) 1 (1/2,3/2) (t) = v(t). Both u and φ develop singularities (where u and ∇φ are discontinuous).

Figure 6 depicts the evolution of hχ∂ x φ h (•, 1) -v L 2 (0,T ) / v L 2 (0,T ) w.r.t. h with χ ≡ 1. We observe a rate close to 0.5.

Let us also emphasize that the space-time discretization formulation is very well appropriated for mesh adaptivity. Using the V 1 h × V 2 h approximation, Figure 7-left (resp. right) depicts the mesh obtained after seven adaptative refinements based on the local values of gradient of the variable φ h (resp. u h ). Starting with a coarse mesh composed of 288 triangles and 166 vertices, the final mesh on the right is composed with 13068 triangles and 6700 vertices and leads to a relative error err(v, u h ) of the order of 10 -3 . The final mesh follows the singularities of the controlled solution starting at the point (0, 1) of discontinuity of u 0 .

(•, 1) -v L 2 (0,T ) / v L 2 (0,T ) w.r.t. h (rate ≈ 0.5).
potential V , see [START_REF] Ignat | Convergence of a two-grid algorithm for the control of the wave equation[END_REF]. Non zero potentials notably appear from linearization of nonlinear wave equations of the form u + f (u) = χv (see [START_REF] Münch | Constructive exact control of semilinear 1-D wave equations by a least-squares approach[END_REF]). Actually, we want to emphasize that this spacetime approach, based on the resolution of the optimal condition associated with the control of minimal L 2 norm is very relevant for potential with the "bad" sign for which V (t, x)u(t, x) < 0. Indeed, in this case, the usual "à la Glowinski" strategy developed in [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF] is numerically inefficient and requires adaptations, since the uncontrolled solution (used to initialize the conjugate algorithm) grows exponentially in time, leading to numerical instabilities and overflow. Recall that the observability constant behaves like e C(T,ω) A 2 L ∞ (0,T ;L n (Ω)) (see [START_REF] Zhang | Explicit observability estimate for the wave equation with potential and its application[END_REF]) and appears notably in the constant in the a priori estimate [START_REF] Miller | Escape function conditions for the observation, control, and stabilization of the wave equation[END_REF]. We consider the initial condition (Ex1), T = 2.5 and constant negative potentials V (t, x) = V < 0. Table 12 collects the relative error on the approximation of the boundary control with respect to h for several negatives values of V . In particular, for V = -40, the L ∞ norm of the corresponding uncontrolled solution is of order 10 5 . We approximate u and φ in V 1 h and V 2 h respectively and observe a rate close to 1.5. The value of V only affects the constant. We refer to [START_REF] Micu | Approximation of the controls for the wave equation with a potential[END_REF] for a semi-discrete (in space) approximation of exact boundary controls for a semi-discretized wave equation with potential, including experiments for small and potentials with good sign.

h 1.6 × 10 -1 8 × 10 -2 4 × 10 -2 2 × 10 -2
1 × 10 -2 V = -10 7.11 × 10 -1 2.43 × 10 -1 6.91 × 10 -2 2.48 × 10 -2 7.97 × 10 -3 V = -20 7.06 × 10 -1 2.81 × 10 -1 1.57 × 10 -1 6.13 × 10 -2 1.65 × 10 -2 V = -30 9.54 × 10 -1 6.91 × 10 -1 2.07 × 10 -1 7.21 × 10 -2 2.27 × 10 -2 V = -40 1.01 × 10 -1 9.44 × 10 -1 5.05 × 10 -1 1.13 × 10 -1 3.18 × 10 -2 Table 12. (Ex1) -Boundary case -(p, q) = (1, 2) -χ from [START_REF] Montaner | Approximation of controls for linear wave equations: a first order mixed formulation[END_REF];

hχ∂ x φ h (•, 1) -v L 2 (0,T ) / v L 2 (0,T ) w.r.t.
h and V ∈ {-10, -20, -30, -40}.

Conclusion

We have introduced and analyzed a spacetime finite element approximation of a controllability problem for the wave equation. Based on a non conformal 1approximation, the analysis yields error estimates for the control in the natural L 2 -norm of order h q (resp. h q-1 2 ) where q is the degree of the polynomials used to describe the adjoint variable in the distributed (resp. boundary) case. The numerical experiments performed for initial data with various regularity exhibits the efficiency method. The convergence is also observed for initial data with minimal regularity.

We emphasize that spacetime formulations are easier to implement than timemarching methods, since in particular, there is no kind of CFL condition between the time and space discretization parameters. Moreover, as shown in the numerical section, they are well-suited for mesh adaptivity (as initially discussed in [START_REF] Hulbert | Space-time finite element methods for second-order hyperbolic equations[END_REF]).

Similarly to the formulation proposed in [START_REF] Cîndea | Numerical controllability of the wave equation through primal methods and Carleman estimates[END_REF][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], the present formulation follows the "control then discretize" approach. However, contrary to [START_REF] Cîndea | Numerical controllability of the wave equation through primal methods and Carleman estimates[END_REF][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], the H 1 -formulation of the present work does not require the introduction of sophisticated finite element spaces. On the other hand, the formulation requires additional stabilized terms which are function of the jump of the gradient across the boundary of each element. The analysis is then inspired from [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] and also from [START_REF] Burman | Space time stabilized finite element methods for a unique continuation problem subject to the wave equation[END_REF] where an analogous spacetime formulation for a data assimilation is considered.

The implementation of the stabilized terms is not straightforward, in particular, in higher dimension, and is usually not available in finite element softwares. A possible way to circumvent the introduction of the gradient jump terms is to consider nonconforming approximation of the Crouzeix-Raviart type as in [START_REF] Burman | A stabilized nonconforming finite element method for the elliptic Cauchy problem[END_REF]. A penalty is then needed on the solution jump instead to control the H 1 -conformity error. Another possible way, following [START_REF] Montaner | Approximation of controls for linear wave equations: a first order mixed formulation[END_REF] devoted to the boundary case, could be to consider the controllability problem associated to a first order reformulation of the wave equation: [START_REF] Steinbach | A Stabilized Space-Time Finite Element Method for the Wave Equation[END_REF] v t -div p = 0, p t -∇v = 0.

with v := u t and p := ∇u. A H 1 conformal stabilized approximation is employed in [START_REF] Montaner | Approximation of controls for linear wave equations: a first order mixed formulation[END_REF] leading to promising numerical experiments in the one dimensional case. A rigorous numerical analysis however remains to be done.

Proof. Let χ ∈ C ∞ (R) satisfy χ(t) = 0 near t = T and χ(t) = 1 for t ∈ (0, T /2). Applying Proposition A.1 to χu backwards in time, we obtain u L ∞ (0,T /2;L 2 (Ω)) + ∂ t u L 2 (0,T /2;H -1 (Ω))

[ , χ]u H 1 ((0,T )×Ω) u L 2 ((0,T )×Ω) .

Applying [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] backwards in time on the interval (0, s) where s < T /2, we get

∂ t u| t=0 2 H -1 (Ω)
u| t=s 2 L 2 (Ω) + ∂ t u| t=s 2 H -1 (Ω) . Integration in s gives

∂ t u| t=0 2 H -1 (Ω) u 2
L 2 (0,T /2;L 2 (Ω)) + ∂ t u 2 L 2 (0,T /2;H -1 (Ω)) . We conclude that u| t=0 L 2 (Ω) + ∂ t u| t=0 H -1 (Ω) u L 2 ((0,T )×Ω) .

The claimed estimate follows from [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF].

Let us now turn the claimed continuity. Let > 0 and u j be a mollification in time that satisfies u j → u in L 2 (( , T -) × Ω). Then u j converges in

C( , T -; L 2 (Ω)) ∩ C 1 ( , T -; H -1 (Ω))
and thus u is in this space. In particular, u| t=T /2 and ∂ t u| t=T /2 are well-defined. Solving the initial value problem starting from these gives the desired conclusion. 49), φ| t=0 L 2 (Ω) + ∂ t φ| t=0 H -1 (Ω) = ψ| t=0 L 2 (Ω) + ∂ t ψ| t=0 H -1 (Ω) ψ L 2 ((0,T )×ω) ≤ φ L 2 ((0,T )×ω) + u L 2 ((0,T )×ω) φ L 2 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + φ H -1 ((0,T )×Ω) .

Remark A.5. By applying Theorem A. φ L 2 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + φ H -1 ((0,T )×Ω) , assuming that T > 0 and ω ⊂ Ω satisfy the geometric control condition. As ψ ∈ L 2 ((0, T ) × Ω), we have using P ψ = 0, ∂ 2 t ψ L 2 (0,T ;H -2 (Ω)) = ∆ψ L 2 (0,T ;H -2 (Ω)) + V ψ L 2 (0,T ;H -2 (Ω)) ψ L 2 ((0,T )×Ω) .

Now interpolation, see e.g. [37, Theorems 2.3 and 12.2], gives ∂ t ψ L 2 (0,T ;H -1 (Ω)) ψ L 2 (0,T ;L 2 (Ω)) .

Hence also f L 2 (0,T ;H -1 (Ω)) ψ L 2 (0,T ;L 2 (Ω)) , and (50), or rather its analogue backward in time, gives u| t=0 L 2 (Ω) + ∂ t u| t=0 H -1 (Ω) ψ L 2 (0,T ;L 2 (Ω)) .

But the state of u at t = 0 coincides with that of ψ, and (55) follows from the above estimate and (57). Will now show (54). Let v be the solution of ∂ ν φ H -1 ((0,T )×ω) + ∂ ν v H -1 ((0,T )×ω)

∂ ν φ H -1 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + P φ H -1 ((0,T )×Ω) .

The analogue of Remark A.5 holds also in the case of boundary observations. We need also the following classical result.

Lemma A.7 (Partial hypoellipticity). Let I ⊂ (0, T ) be a compact interval. Suppose that s ∈ R and j ∈ N satisfy s + 1/2 > j. Then

∂ j ν u H s-j-3 2 (I×∂Ω)
u H s (M ) + u H s-1 (M ) .

4. 1 .

 1 Discretization. Let us consider a family T = { Th : h > 0} where Th is a set of 1 + n-dimensional simplices forming a simplicial complex. The family T is parametrized by h = max K∈ Th diam(K).

Figure 1 . 5 . 1 .

 151 Figure 1. The C ∞ 0 ([0, T ]) function t → χ 0 (t), t ∈ [0, T ] with T = 2.5.

Figure 2 .

 2 Figure 2. (Ex1); χ(φ -φ h ) L 2 (M ) / χφ L 2 (M ) vs. h; Left: χ 0 (t) and χ 1 (x) given by (43); Right: χ 0 (t) = 1 and χ 1 (x) = 1 (a,b) (x).

  h

Figure 3 .

 3 Figure 3. (Ex2) and (Ex2b) ; χ(φ -φ h ) L 2 (M ) / χφ L 2 (M ) vs. h; χ from (43).

Figure 4 .

 4 Figure 4. (Ex1) -Boundary case -hχ∂ x φ h (1, •)-v L 2 (0,T ) / v L 2 (0,T )vs. h with χ = 1 (left) and χ = χ 0 from (43) (right).

h 1 .

 1 57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1)

Figure 5 .

 5 Figure 5. (Ex2) -Boundary case -h∂ x χφ h (1, •)-v L 2 (0,T ) / v L 2 (0,T )vs. h with χ = 1 (left) and χ = χ 0 from (43) (right).

  h

Figure 7 .

 7 Figure 7. (Ex3); Locally refine space-time meshes with respect to φ h (left) and u h (right). (p, q) = (1, 2)

Theorem A. 4 (

 4 Distributed observability estimate). Let T > 0 and let an open set ω ⊂ Ω satisfy the geometric control condition. Thenφ| t=0 L 2 (Ω) + ∂ t φ| t=0 H -1 (Ω) φ L 2 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + φ H -1 ((0,T )×Ω) .Proof. It is classical thatψ| t=0 L 2 (Ω) + ∂ t ψ| t=0 H -1 (Ω) ψ L 2 ((0,T )×ω) u| x∈∂Ω = φ| x∈∂Ω , u| t=0 = 0, ∂ t u| t=0 = 0,and define ψ = φ -u. Then using (53) and (

  [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] to the function (t, x) → φ(T -t, x) we obtain the following variantφ| t=T L 2 (Ω) + ∂ t φ| t=T H -1 (Ω) φ L 2 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + φ H -1 ((0,T )×Ω) ,and by combining Proposition A.1 and Theorem A.4, we getφ L ∞ (0,T ;L 2 (Ω)) + ∂ t φ L 2 (0,T ;H -1 (Ω))

Theorem A. 6 (

 6 Boundary observability estimate). Let T > 0 and let an open set ω ⊂ ∂Ω satisfy the geometric control condition. Let V ∈ C ∞ (Ω) and define P by[START_REF] Ignat | Convergence of a two-grid algorithm for the control of the wave equation[END_REF]. Thenφ| t=0 L 2 (Ω) + ∂ t φ| t=0 H -1 (Ω) (54) ∂ ν φ H -1 ((0,T )×ω) + φ L 2 ((0,T )×∂Ω) + P φ H -1 ((0,T )×Ω) . Proof. It is well-known that ψ| t=0 L 2 (Ω) + ∂ t ψ| t=0 H -1 (Ω) ∂ ν ψ H -1 ((0,T )×ω) (55)for solutions ψ of P ψ = 0, ψ| x∈∂Ω = 0. (56) However, we did not find this exact formulation in the literature, and give a short proof for the convenience of the reader. It follows from(3.11) of the classical paper[START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] thatψ L 2 ((0,T )×Ω) ∂ ν ψ H -1 ((0,T )×ω) ,(57)since the space of invisible solutions is empty in our case due to unique continuation. Let τ ∈ C ∞ (R) satisfy τ (t) = 1 near t = 0 and τ (t) = 0 near t = T . Writing u = τ ψ and f = ∂ t τ ∂ t ψ + ∂ 2 t τ ψ, there holds      P u = f, u| x∈∂Ω = 0, u| t>T = 0.

  P φ v| x∈∂Ω = φ| x∈∂Ω v| t<0 = 0. Then ψ = φ -v solves (56), and (55) and (49) implyφ| t=0 L 2 (Ω) + ∂ t φ| t=0 H -1 (Ω) = ψ| t=0 L 2 (Ω) + ∂ t ψ| t=0 H -1 (Ω)

  1.92 × 10 -2 9.29 × 10 -3 4.09 × 10 -3 err(v, u h ) 2.16 × 10 -1 5.83 × 10 -2 1.94 × 10 -2 7.21 × 10 -3 3.10 × 10 -3 err(v, hχ∂ x φ h ) 2.71 × 10 -1 8.41 × 10 -2 2.79 × 10 -2 1.22 × 10 -2 5.31 × 10 -3

Table 5 .

 5 (Ex1) -Boundary case -(p, q) = (1, 2) -χ ≡ 1.h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1) 4.09 × 10 -1 1.15 × 10 -1 2.82 × 10 -2 9.29 × 10 -3 2.81 × 10 -3 err(∂ x φ, ∂ x φ h ,1) 6.73 × 10 -1 2.10 × 10 -1 6.13 × 10 -2 2.91 × 10 -2 1.24 × 10 -2 err(u, u h , 1) 7.29 × 10 -2 2.81 × 10 -2 9.02 × 10 -3 3.45 × 10 -3 1.28 × 10 -3 err(v, u h ) 1.00 × 10 -1 3.74 × 10 -2 1.13 × 10 -2 4.16 × 10 -3 1.43 × 10 -3 err(v, hχ∂ x φ h ) 1.05 × 10 -1 3.89 × 10 -2 1.23 × 10 -2 4.61 × 10 -3 1.65 × 10 -3

Table 6 .

 6 (Ex1) -Boundary case -(p, q) = (2, 3) -χ ≡ 1.

Table 7 .

 7 (Ex2) -Boundary case -(p, q) = (1, 2); χ ≡ 1 h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1) 4.90 × 10 -1 1.35 × 10 -1 3.28 × 10 -2 1.07 × 10 -2 3.41 × 10 -3 err(∂ x φ, ∂ x φ h , 1) 9.79 × 10 -1 3.75 × 10 -1 1.35 × 10 -1 6.76 × 10 -2 3.21 × 10 -2 err(u, u h , 1) 6.70 × 10 -2 2.60 × 10 -2 9.34 × 10 -3 3.78 × 10 -3 1.44 × 10 -3 err(v, u h ) 8.61 × 10 -2 3.25 × 10 -2 1.10 × 10 -2 4.19 × 10 -3 1.59 × 10 -3 err(v, hχ∂ x φ h ) 8.57 × 10 -2 3.23 × 10 -2 1.10 × 10 -2 4.45 × 10 -3 1.71 × 10 -3

Table 8 .

 8 (Ex2) -Boundary case -(p, q) = (2, 3); χ ≡ 1

  1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1) 6.38 × 10 -1 4.35 × 10 -1 2.85 × 10 -1 1.84 × 10 -1 1.05 × 10 -1 err(∂ x φ, ∂ x φ h , 1) 8.38 × 10 -1 6.23 × 10 -1 4.85 × 10 -1 3.97 × 10 -1 3.17 × 10 -1 err(u, u h , 1) 5.78 × 10 -1 4.40 × 10 -1 3.59 × 10 -1 2.93 × 10 -1 2.27 × 10 -1 err(v, u h ) 7.67 × 10 -1 5.69 × 10 -1 5.04 × 10 -1 4.05 × 10 -1 3.15 × 10 -1 err(v, hχ∂ x φ h ) 8.41 × 10 -1 6.47 × 10 -1 5.08 × 10 -1 4.09 × 10 -1 3.16 × 10 -1

Table 9 .

 9 (Ex3); (p, q) = (1, 1) -χ ≡ 1 -Boundary case 5.7. Boundary case: the wave equation with a potential. To end these numerical illustrations, we report some results for the wave equation with non vanishing h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1)1.62 × 10 0 6.33 × 10 -1 2.72 × 10 -1 1.45 × 10 -1 7.36 × 10 -2 err(∂ x φ, ∂ x φ h , 1) 2.33 × 10 0 1.52 × 10 0 1.22 × 10 0 1.08 × 10 0 1.05 × 10 0 err(u, u h , 1)3.93 × 10 -1 3.00 × 10 -1 2.27 × 10 -1 1.74 × 10 -1 1.30 × 10 -1 err(v, u h ) 5.03 × 10 -1 3.43 × 10 -1 2.41 × 10 -1 1.89 × 10 -1 1.48 × 10 -1 err(v, hχ∂ x φ h ) 4.73 × 10 -1 3.41 × 10 -1 2.64 × 10 -1 2.08 × 10 -1 1.60 × 10 -1

Table 10 .

 10 (Ex3); (p, q) = (1, 2) -χ ≡ 1 -Boundary case h 1.57 × 10 -1 8.22 × 10 -2 4.03 × 10 -2 2.29 × 10 -2 1.25 × 10 -2 err(φ, φ h , 1) 1.25 × 10 0 5.01 × 10 -1 2.02 × 10 -1 9.58 × 10 -2 4.52 × 10 -2 err(∂ x φ, ∂ x φ h , 1) 3.44 × 10 0 2.34 × 10 0 1.88 × 10 0 1.57 × 10 0 1.45 × 10 0 err(u, u h , 1) 3.12 × 10 -1 2.22 × 10 -1 1.60 × 10 -1 1.24 × 10 -1 9.01 × 10 -2 err(v, u h ) 3.52 × 10 -1 2.36 × 10 -1 1.71 × 10 -1 1.30 × 10 -1 9.64 × 10 -2 err(v, hχ∂ x φ h ) 3.40 × 10 -1 2.36 × 10 -1 1.66 × 10 -1 1.37 × 10 -1 9.56 × 10 -2

Table 11 .

 11 (Ex3); (p, q) = (2, 3) -χ ≡ 1 -Boundary case

	10	0
	10	-1
		10 -2	10 -1
	Figure 6. (Ex3); hχ∂ x φ h

Appendix A. Continuum estimates Proposition A.1 (Energy estimate). There holds u L ∞ (0,T ;L 2 (Ω)) + ∂ t u L 2 (0,T ;H -1 (Ω)) + ∂ ν u H -1 ((0,T )×∂Ω) [START_REF] Zhang | Explicit observability estimate for the wave equation with potential and its application[END_REF] u| t=0 L 2 (Ω) + ∂ t u| t=0 H -1 (Ω) + u L 2 ((0,T )×∂Ω) + u -1 ((0,T )×Ω) .

Proof. The estimate u L ∞ (0,T ;L 2 (Ω)) + ∂ t u L ∞ (0,T ;H -1 (Ω)) + ∂ ν u H -1 ((0,T )×∂Ω) [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF]

follows from [START_REF] Lasiecka | Nonhomogeneous boundary value problems for second order hyperbolic operators[END_REF]Theorem 2.3], see also Remark 2.2 there. Thus it is enough to consider the equation

and show that its solution satisfies

Let us use the shorthand notations M = (0, T ) × Ω and (t, x) = (x 0 , . . . , x n ). We recall that for any f ∈ H -1 (M ) there are

and that

where the infimum is taken over all f j ∈ L 2 (M ) satisfying (52), see e.g. [START_REF] Evans | Partial differential equations[END_REF]Theorem 1,p. 299]. Thus it is enough to show that

where u satisfies (51) with f replaced by ∂ x j f j when j ≥ 0 and by f -1 when j = -1.

The cases j = -1 and j > 0 are contained in [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF].

Let us consider the case j = 0. We denote by v the solution of (51

Moreover,

Proof. To get a contradiction, we suppose that

for solutions u of (51). Let f ∈ L 2 (0, T ), and denote by u and v the solutions of (51) with sources ∂ t f and f , respectively. Then u = ∂ t v and

If u = 0 and u| x∈∂Ω = 0 then the norm on the left-hand side of (49) controls the L ∞ (0, T ; H -1 (Ω)) norm of ∂ t u. In fact, we have:

Proof. We define a norm u X by the right-hand side of the claimed inequality, and set X = {u ∈ H s (M ) : u X < ∞}. It follows from the closed graph theorem that X is a Banach space. In normal coordinates of R × ∂Ω, there holds = ∂ 2 ν + A, where A is a differential operator in the tangential directions to R × ∂Ω, with coefficients depending on all the variables, see e.g. [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]Corollary C.5.3].

We will use the spaces H(m,s) (I × Ω), defined on p. 478 of [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], in the boundary normal coordinates, and use the shorthand notation H (m,s) for them. Here m measures Sobolev smoothness in all the variables and s additional smoothness in the tangential variables. However, s can be also negative, corresponding to a loss of smoothness in tangential directions.

Let u ∈ X. It follows from [29, Theorem B.2.9] that u ∈ H (m,r) when m + r ≤ s -1 and m ≤ s + 1. In particular, u ∈ H (s+1,-2) and the closed graph theorem implies

Moreover, using the assumption s + 1/2 > j, [29, Theorem B.2.7] implies

Appendix B. Estimates for meshes fitted to the boundary Proof of Lemma 4.2. Let u ∈ C ∞ (K). Let h > 0, K ∈ T h and consider spherical coordinates (r, θ) ∈ (0, ∞) × S n centered at x where x is as in (T). It follows from (36) that K is star-shaped with respect to x. In particular, there is R : S n → (0, ∞) such that

As ∂K is piecewise smooth, it follows from (36) that R is piecewise smooth. Applying [START_REF] Evans | Partial differential equations[END_REF]Theorem 6,p. 713] in a piecewise manner, we see that

where dθ is the canonical volume measure on the unit sphere S n . It follows from [START_REF] Rousseau | Geometric control condition for the wave equation with a time-dependent observation domain[END_REF] that h R(θ). Hence, using [START_REF] Lebeau | Experimental study of the HUM control operator for linear waves[END_REF] we have

Proof of Lemma 4.2. We choose an extension û of u so that û

h , and the now classical result [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF] says that the analogue û

of ( 38) holds. This implies (38) since T h is obtained from Th via a restriction.