Franck Boyer
email: fboyer@latp.univ-mrs.fr

AND Florence Hubert
email: fhubert@latp.univ-mrs.fr

AND J Ér

Ôme Le Rousseau

J Le Rousseau

DISCRETE CARLEMAN ESTIMATES FOR ELLIPTIC OPERATORS AND UNIFORM CONTROLLABILITY OF SEMI-DISCRETIZED PARABOLIC EQUATIONS *

Keywords: Elliptic operator, discrete Carleman estimate, spectral inequality, parabolic equation, semi-discrete scheme, uniform controllability / observability. AMS subject classifications. 35K05 -65M06 -93B05 -93B07 -93B40

We derive a semi-discrete two-dimensional elliptic global Carleman estimate, in which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of smoothly varying meshes. As a consequence of the Carleman estimate, we derive a partial spectral inequality of the form of that proved by G. Lebeau and L. Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the spectrum. Moreover, with the control function that we construct, the L 2 norm of the final state converges to zero super-algebraically as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.

Introduction and settings.

Let Ω, ω be connected non-empty bounded open subsets of R n with ω ⋐ Ω. We consider the following parabolic problem in (0, T ) × Ω, with T > 0, ∂ t y -∇ x • (γ∇ x y) = 1 ω v in (0, T ) × Ω, y| ∂Ω = 0, and y| t=0 = y 0 , (

where the diffusion coefficient γ = γ(x) > 0 satisfies reg(γ)

def = sup x∈Ω γ(x) + 1 γ(x) + |∇ x γ(x)| < +∞. (1.2)
G. Lebeau and L. Robbiano proved in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] the null controllability of system (1.1), i.e., for all y 0 ∈ L 2 (Ω), there exists v ∈ L 2 ((0, T ) × Ω), such that y(T ) = 0 and v L 2 ((0,T )×Ω) ≤ C|y 0 | L 2 (Ω) , where C > 0 only depends on Ω, ω, γ and T . They in fact constructed the control function v semi-explicitly. This construction is based on the following spectral inequality.

Theorem 1.1 ([LR95, JL99, LZ98a]). Let (φ k ) k∈N * be a set of L 2 (Ω)-orthonormal eigenfunctions of the operator A := -∇ x •(γ∇ x ) with homogeneous Dirichlet boundary conditions, and (µ k ) k∈N * be the set of the associated eigenvalues (with finite multiplicities) sorted in a non-decreasing sequence. There exists C > 0 such that for all consider the following system,

∂ t y h + A M y h = 1 ω v h , y h | ∂Ω = 0, y| t=0 = y h 0 ,
where A M is a discrete approximation of A for a mesh M with step-size h to be precisely introduced below. We prove that there exists a control function v h , with v h L 2 ((0,T )×ω) ≤ C|y h 0 | L 2 (Ω) , C > 0 independent of h, such that the frequencies of the controlled solution y h associated to the lower part of the spectrum vanish at the final time T . We furthermore prove that

|y h (T )| L 2 (Ω) ≤ Ce -C/h 2 |y h 0 | L 2 (Ω) . (1.3)
This should not be considered as an approximate controllability result and should rather be compared with the result obtained in [START_REF] Labbé | Uniform controllability of semidiscrete approximations of parabolic control systems[END_REF], where they proved (in a somewhat more general framework) a result of the form (1.3) with e -C/h 2 replaced by h α , for some explicit exponent α > 0. See also the observability estimate (1.10) below. Note that in the sequel we shall drop the subscript h, in the case of discrete function, as in y h or v h , for the sake of concision.

As mentioned above, we chose to restrict ourselves in one space dimension since additional technicalities are needed for the multidimensional case. This issue will be developed in future work [START_REF] Boyer | Discrete carleman estimates for elliptic operators in arbitrary dimension and applications[END_REF]. With the discrete partial Lebeau-Robbiano inequality we prove here, the full discrete problem can also be addressed [START_REF] Boyer | On the approximation of the nullcontrollability problem for parabolic equations[END_REF][START_REF] Boyer | Uniform null-controllability properties for space/time-discretized parabolic equations[END_REF].

A challenging question lays in the derivation of uniform discrete parabolic global Carleman estimates. In the continuous case, global parabolic Carleman estimates were introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF] and they in particular lead to the null controllability of linear and semi-linear parabolic equations [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. Like in the elliptic case that we treat here, we cannot hope to obtain such estimates, in the discrete parabolic case, with an arbitrary large parameter.

1.1. Discrete settings. As mentioned above we restrict our analysis of semidiscrete parabolic operators to one dimension in space. Let us consider the elliptic operator on Ω = (a, b) given by A = -∂ x (γ∂ x ) with homogeneous Dirichlet boundary conditions and γ satisfying (1.2).

We introduce finite difference approximations of the operator A. Let a = x 0 < x 1 < • • • < x N < x N +1 = b, see Figure 1.1. We refer to this discretization as to the primal mesh M := {x i ; i = 1, . . . , N }. We set |M| := N . We set h i+ 1 2 = x i+1 -x i and x i+ 1 2 = (x i+1 + x i )/2, i = 0, . . . , N , and h = max 0≤i≤N h i+ 1 2 . We call M := {x i+ 1 2 ; i = 0, . . . , N } the dual mesh and we set h i = x i+ 1 2 -x i-1 2 = (h i+ 1 2 + h i-1 2 )/2, i = 1, . . . , N .

x i+ 1 2 a x 0 x 1 b x N+1 x N h i h i-1 2 x i-1 2 x i x i-1
x 1 2 Fig. 1.1. Notation for the mesh geometry

In the present article, we shall only consider some families of regular non uniform meshes, that will be precisely defined in Section 1.2. Note that the extension of our results to more general mesh families does not seem to be straightforward.

We denote by C M and C M the sets of discrete functions defined on M and M respectively. If u ∈ C M (resp. C M ), we denote by u i (resp. u i+ 1 2 ) its value corresponding to x i (resp. x i+ 1 2 ). For u ∈ C M we define

u M = N i=1 1 [x i-1 2 ,x i+ 1 2 ] u i ∈ L ∞ (Ω).
Since no confusion is possible, by abuse of notation we shall often write u in place of u M . For u ∈ C M we define Ω u := Ω u M (x)dx = N i=1 h i u i . For u ∈ C M we define

u M = N i=0 1 [xi,xi+1] u i+ 1 2 .
As above, for u ∈ C M , we define Ω u := Ω u M (x)dx = N i=0 h i+ 1 2 u i+ 1 2 . Similarly, with Q = (0, T ) × Ω, and u(t) in C M or C M for all t ∈ (0, T ), we shall write Q u dt = T 0 Ω u(t) dt. In particular we define the following L 2 inner product on C M (resp.

C M ) (u, v) L 2 = Ω u M (x)(v M (x)) * dx, resp. (u, v) L 2 = Ω u M (x)(v M (x)) * dx.
(1.4)

For some u ∈ C M , we shall need to associate boundary conditions u ∂M = {u 0 , u N +1 }. The set of such extended discrete functions is denoted by C M∪∂M . Homogeneous Dirichlet boundary conditions then consist in the choice u 0 = u N +1 = 0, in short u ∂M = 0. We can now define translation operators τ ± , a difference operator D and an averaging operator as the maps C M∪∂M → C M given by (τ + u) i+ 1 2 := u i+1 , (τ -u) i+ 1 2 := u i , i = 0, . . . , N, (Du) i+ 1 2 := 1 h i+ 1 2 (τ + u -τ -u) i+ 1 2 , ũ := 1 2 (τ + + τ -)u.

(1.5)

We also define, on the dual mesh, translations operators τ ± , a difference operator D and an averaging operator as the maps C M → C M given by (τ + u) i := u i+ 1 2 , (τ -u) i := u i-1 2 , i = 1, . . . , N, (Du) i := 1 h i (τ + u -τ -u) i , u := 1 2 (τ + + τ -)u.

(1.6)

Note that there is no need for boundary conditions here.

A continuous function f defined in a neighborhood of Ω can be sampled on the primal mesh f M = {f (x 1 ), . . . , f (x N )} which we identify to

f M = N i=1 1 [x i-1 2 ,x i+ 1 2 ] f i , f i = f (x i ), i = 1, . . . , N.
We also set The function f can also be sampled on the dual mesh f M = {f (x 1 2 ), . . . , f (x N + 1 2 )} which we identify to

f ∂M = {f (x 0 ), f (x N +1 )}, f M∪∂M = {f (x 0 ), f (x 1 ), . . . , f (x N ), f (x N +1 )}.
f M = N i=0 1 [xi,xi+1] f i+ 1 2 , f i+ 1 2 = f (x i+ 1 2 ), i = 0, . . . , N.
In the sequel, we shall often use f for both the continuous function and its discretization on the primal mesh, i.e., f M∪∂M . We shall write f d for the sampling f M of f on the dual mesh. In fact we shall write Df := Df M∪∂M and Df d := Df M , with similar conventions for compositions of the discrete operators we defined above. See also Remark 3.1 for conventions concerning the action of discrete operators on continuous functions.

Throughout the article, a volume norm, i.e., over an open subset of Q = (0, T )×Ω, will be denoted by . ; a surface norm will be denoted by |.|. Note that we shall use the same norm signs for continuous, semi-discrete and discrete norms over volumes and surfaces. For a semi-discrete function u on Q, i.e., with u(t) ∈ C M or C M for all t ∈ (0, T ), we thus set

u 2 L 2 (Q) = T 0 Ω |u(t)| 2 dt.
1.2. Regular families of non-uniform meshes. In this paper, we address non uniform meshes that are obtained as the smooth image of an uniform grid.

More precisely, let Ω 0 =]0, 1[ and let ϑ : R → R be an increasing map such that

ϑ ∈ C ∞ , ϑ(Ω 0 ) = Ω, inf Ω0 ϑ ′ > 0. (1.7)
Given an integer N , let M 0 = (ih ⋆ ) 1≤i≤N , with h ⋆ = 1 N +1 be a uniform mesh of Ω 0 and M 0 the dual mesh. We define a non-uniform mesh M of Ω as the image of M 0 by the map ϑ, setting

x i = ϑ(ih ⋆ ), ∀i ∈ {0, . . . , N + 1}.
(1.8)

The dual mesh M, and the general notation are those of the previous section. We give in Figure 1.2 an example of such a family of non-uniform meshes and the map ϑ that we used to construct those meshes.

1.3. Statement of the main results. With the notation we have introduced, a consistent finite difference approximation of Au with homogeneous boundary conditions is A M u = -D(γ d Du) for u ∈ C M∪∂M satisfying u ∂M = 0. Recall that γ d is the sampling of the given continuous diffusion coefficient γ on the dual mesh M, so that for any u ∈ C M∪∂M we have

(A M u) i = - γ(x i+ 1 2 ) ui+1-ui h i+ 1 2 -γ(x i-1 2 ) ui-ui-1 h i-1 2 h i , i = 1, ..., N.
Note however that other consistent choices of discretization of γ are possible, such as γ, i.e. the averaging on the dual mesh of the sampling of γ on the primal mesh.

Remark 1.2. Note that the discretization we have introduced can also be viewed as a finite volume approximation of the problem on the dual mesh.

For a suitable weight function ϕ, the announced semi-discrete Carleman estimate for the operator P M = -∂ 2 t + A M on (0, T * ) × Ω, for the non-uniform meshes we consider, is of the form

s 3 e sϕ u 2 L 2 (Q) + s e sϕ ∂ t u 2 L 2 (Q) + s e sϕ d Du 2 L 2 (Q) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (Ω) + se 2sϕ(T * ) |∂ t u(T * , .)| 2 L 2 (Ω) + s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ C e sϕ P M u 2 L 2 (Q) + se 2sϕ d (T * ) |Du(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) ,
for any s ≥ s 0 , and any h ≤ h 0 such that sh ≤ ε 0 , and any u satisfying u| {0}×Ω = 0, u| (0,T * )×∂Ω = 0, where s 0 , h 0 and ε 0 only depend on the data (see Theorem 5.5). The proof of this estimate will be first carried out for uniform meshes, and then adapted to the case of non-uniform meshes we introduced in Section 1.2. Note that the discrete operator A M is selfadjoint with respect to the L 2 inner product on C M introduced in (1.4). We denote by φ M a set of discrete L 2 orthonormal eigenfunctions, φ j ∈ C M , 1 ≤ j ≤ |M|, of the operator A M , and by µ M = {µ j , 1 ≤ j ≤ |M|} the set of the associated eigenvalues sorted in a non-decreasing sequence.

The announced partial Lebeau-Robbiano spectral inequality for the lower part of the spectrum reads

µ k ∈µ M µ k ≤µ |α k | 2 = Ω µ k ∈µ M µ k ≤µ α k φ k 2 ≤ Ce C √ µ ω µ k ∈µ M µ k ≤µ α k φ k 2 , ∀(α k ) 1≤k≤|M| ⊂ C.
for C > 0 only depending on (Ω, ω, γ, ϑ) and for µh 2 and h sufficiently small. (see Theorem 6.1 for details).

Remark 1.3. The inequality we have obtained only concerns a constant portion of the discrete spectrum. It is however quasi-optimal by the following argument. Observe indeed that the map

(α k ) 1≤k≤M ∈ C M → 1≤k≤M α k φ k (x j ) xj∈ω ∈ C Nω , where N ω = #(M ∩ ω), is never injective if M > N ω .
The maximal number of eigenfunctions we could possibly have in such an inequality is then of the order of |ω|N |Ω| .

Since we can prove the asymptotic behavior µ k ∼ Ck 2 , we are clearly restricted to the condition µh 2 ≤ C |ω| 2 |Ω| 2 . We show here that the discrete Lebeau-Robbiano inequality holds for µh 2 ≤ ε 0 but we do not know if the ε 0 we obtain is optimal.

We introduce the following finite dimensional spaces

E j = Span{φ k ; 1 ≤ µ k ≤ 2 2j } ⊂ C M , j ∈ N,
and denote by Π Ej the L 2 orthogonal projection onto E j . The controllability result we can deduce from the above results is the following.

Theorem 1.4. Let T > 0 and ϑ satisfying (1.7). There exist h 0 > 0, C T > 0 and C 1 , C 2 , C 3 > 0 such that for all meshes M defined by (1.8), with 0 < h ≤ h 0 , and all initial data y 0 ∈ C M , there exists a semi-discrete control function v such that the solution to

∂ t y -D(γ d Dy) = 1 ω v, y ∂M = 0, y| t=0 = y 0 .
(1.9)

satisfies Π E j M y(T ) = 0, for j M = max{j; 2 2j ≤ C 1 /h 2 }, with v L 2 (Q) ≤ C T |y 0 | L 2 (Ω)
and furthermore |y(T

)| L 2 (Ω) ≤ C 2 e -C3/h 2 |y 0 | L 2 (Ω) .
The different constants h 0 , C j , j = 1, 2, 3, appearing in the statement of the theorem will be made more explicit in the main text.

Remark 1.5. Here the highest mode we are able to control uniformly satisfies µ k ≤ ε 1 /h 2 . In fact for some d 1 > 0 and d 2 > 0, for all 1 ≤ k ≤ N we have

d 1 k 2 ≤ µ k ≤ d 2 k 2 . It follows that we can treat any mode that satisfies d 2 k 2 ≤ ε 1 /h 2 ≤ CN 2 , or rather k ≤ C ′ N .
The result of Theorem 1.4 thus states the null controllability of a constant portion of the discrete spectrum. Furthermore, note that for h sufficiently small the error made for the remainder of the spectrum goes to zero super-algebraically.

The (relaxed) observability estimate we then obtain is of the form

|q(0)| L 2 (Ω) ≤ C T T 0 ω |q(t)| 2 dt 1 2 + Ce -C/h 2 |q(T )| L 2 (Ω) .
(1.10) for any q solution to the adjoint system of system (1.9) (see Corollary 7.5 for details).

1.4. Outline. In Section 2, in the continuous case, we present an alternative method to prove the Lebeau-Robbiano spectral inequality. A large part of the article is dedicated to the extension of this approach to the discrete case. In Section 3 we have gathered preliminary discrete calculus results. To ease the reading most of the proofs have been placed in Appendix A. Section 4 is devoted to the proof of the semi-discrete elliptic Carleman estimate for uniform meshes. Again, to ease the reading, a large number of proofs of intermediate estimates have been placed in Appendix B. This result is then extended to non-uniform meshes in Section 5. In Section 6, with such a Carleman estimate at hand, we derive a partial discrete Lebeau-Robbiano spectral inequality. Finally, in Section 7, as an application, we prove the controllability result of Theorem 1.4.

1.5. Additional notation. We shall denote by z * the complex conjugate of z ∈ C. In the sequel, C will denote a generic constant independent of h, whose value may change from line to line. As usual, we shall denote by O(1) a bounded function. We shall denote by O µ (1) a function that depends on a parameter µ and is bounded once µ is fixed. The notation C µ will denote a constant whose value depends on the parameter µ.

We sometimes use multi-indices. We say that α is a multi-index if α = (α 1 , . . . , α n ) ∈ N n . For α and β multi-indices ξ ∈ R n then write

|α| = α 1 + • • • + α n , ∂ α = ∂ α1 x1 • • • ∂ αn xn , ξ α = ξ α1 1 • • • ξ αn n , β ≤ α, if β 1 ≤ α 1 , . . . , β n ≤ α n , α β = α1 β1 • • • αn βn if β ≤ α.
2. The continuous case. Let Ω be a bounded open subset of R n with C 2 boundary. Let ω be a nonempty open subset of Ω such that ω ⋐ Ω. Let T * > 0 and Q = (0, T * ) × Ω. We shall use the notation ∇ = (∂ t , ∇ x ) t here and we denote by n the outward unit normal to Q on ∂Q and by n x the outward unit normal to Ω on ∂Ω. We consider the operator A = -∇ x • (γ∇ x ) defined on Ω with domain D(A) = H 2 (Ω) ∩ H 1 0 (Ω) (homogeneous Dirichlet boundary conditions). The Lebeau-Robbiano spectral inequality of Theorem 1.1 measures the loss of orthogonality of the eigenfunctions (φ k ) k∈N * , when restricted to ω. It yields the null controllability of the associated parabolic equation through a semi-explicit construction of the control function, which makes use of the natural parabolic exponential decay of the solution (see e.g. [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF][START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional laplacian[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF]). Other applications can be found in [START_REF] Jerison | Chicago Lectures in Mathematics, ch. Nodal sets of sums of eigenfunctions[END_REF].

In this section we give a proof of the Lebeau-Robbiano inequality that differs from the original proof provided in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. Specifically, the proof in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] relies on an interpolation inequality, itself based on local Carleman estimates. Here, we do not rely on such an interpolation inequality and use a global Carleman estimate instead. The alternative method we propose will be used in the sequel for the discrete version A M of the operator A.

From the regularity of the boundary we may choose a function ψ that satisfies the following property. We enlarge the open set Ω to a larger open set Ω as this will be needed for the discrete case in the following sections.

Assumption 2.1. Let Ω be a smooth open and connected neighborhood of Ω in R n and set Q = (0, T * ) × Ω. The function ψ is in C 2 ( Q, R) and satisfies, for some c > 0,

|∇ψ| ≥ c and ψ > 0 in Q, ∂ nx ψ(t, x) < 0 in (0, T * ) × V ∂Ω , ∂ t ψ ≥ c on {0} × (Ω \ ω), ∇ x ψ = 0 and ∂ t ψ ≤ -c on {T * } × Ω,
where V ∂Ω is a sufficiently small neighborhood of ∂Ω in Ω, in which the outward unit normal n x to Ω is extended from ∂Ω.

Such a function can be obtained by following the technique of [START_REF] Fursikov | Controllability of evolution equations[END_REF], i.e., making use of Morse functions and the associated approximation theorem [START_REF] Aubin | Applied Non Linear Analysis[END_REF]. Some details of the construction of ψ are given in Appendix C.

With such a function ψ, we define the weight function ϕ := e λψ . We denote by ϕ(T * ) the constant value taken by ϕ over {T * } × Ω. We have the following global Carleman estimate for the elliptic operator P = -∂ 2 t + A.

Theorem 2.2. For λ ≥ 1 sufficiently large, there exist C > 0 and s 0 ≥ 1, both depending on Ω, ω, T * , and reg(γ), such that

s 3 e sϕ u 2 L 2 (Q) + s e sϕ ∇u 2 L 2 (Q) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (Ω)
(2.1)

+ se 2sϕ(T * ) |∂ t u(T * , .)| 2 L 2 (Ω) + s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ C e sϕ P u 2 L 2 (Q) + se 2sϕ(T * ) |∇ x u(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) ,
for s ≥ s 0 , and for all u ∈ H 2 (Q), satisfying u| {0}×Ω = 0, u| (0,T * )×∂Ω = 0.

Remark 2.3. Note that we do not impose any boundary condition for u on {T * }×Ω. The proof of the Carleman estimate can be found in Appendix 3.A of [Le 07]. Note also that letting the step size h go to zero in the discrete Carleman estimate of Theorem 4.1 below yields a proof for Theorem 2.2.

With this global Carleman estimate we can now prove the Lebeau-Robbiano inequality.

Proof of Theorem 1.1. We set u(t, x) = µj ≤µ α j sinh( √ µj t) √ µj φ j (x). We observe that u satisfies P u = 0, u| {0}×Ω = 0 and u| (0,T * )×∂Ω = 0. Simply keeping the fifth term in the l.h.s. of (2.1) we have

s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ C se 2sϕ(T * ) |∇ x u(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) ,
for all s ≥ s 0 > 0. We note that

|u(T * , .)| 2 L 2 (Ω) = µj ≤µ α j sinh(T * √ µ j ) √ µ j 2 ≥ T * 2 µj ≤µ |α j | 2 ,
since the eigenfunctions (φ k ) k∈N are chosen orthonormal in L 2 (recall that the L 2 inner product is defined in (1.4)). We furthermore note that |e sϕ(0,.) ∂ t u(0, .)| L 2 (ω) ≤ e s sup ϕ(0,.) |∂ t u(0, .)| L 2 (ω) = e s sup ϕ(0,.) µj ≤µ α j φ j (x)

L 2 (ω)
, where the supremum is taken for x ∈ Ω. The result will thus follow if we prove

1 2 s 2 |u(T * , .)| 2 L 2 (Ω) ≥ C|∇ x u(T * , .)| 2 L 2 (Ω) (2.2) for s ≥ C √ µ. We write |u(T * , .)| 2 L 2 (Ω) = µj ≤µ α j sinh(T * √ µ j ) √ µ j 2 ≥ 1 µ µj ≤µ α j sinh(T * √ µ j ) 2 ,
and

|∇ x u(T * , .)| 2 L 2 (Ω) ≤ 1 γ min ∇ x u(T * , .), γ∇ x u(T * , .) L 2 (Ω) ≤ 1 γ min µj ≤µ α j sinh(T * √ µ j ) 2 , since the functions (∇ x φ k ) k∈N satisfy ∇ x φ k , γ∇ x φ l L 2 (Ω) = µ k δ kl , k, l ∈ N.
We thus see that condition (2.2) is fulfilled for s 2 ≥ Cµ.

3. Some preliminary discrete calculus results. Here, to prepare for Section 4, we only consider constant-step discretizations, i.e., h i+ 1 2 = h, i = 0, . . . , N . This section aims to provide calculus rules for discrete operators such as D, D and also to provide estimates for the successive applications of such operators on the weight functions. To avoid cumbersome notation we introduce the following continuous difference and averaging operators. For a function f defined on R we set

τ + f (x) := f (x + h/2), τ -f (x) := f (x -h/2), Df := (τ + -τ -)f /h, f = (τ + + τ -)f /2.
Remark 3.1. To iterate averaging symbols we shall sometimes write Af = f , and thus A 2 f = f . Discrete versions of the results we give below will be natural; with the notation given in the introduction, for a function f continuously defined on R, the discrete function Df is in fact Df sampled on the dual mesh, M, and Df d is Df sampled on the primal mesh, M. We shall use similar meanings for averaging symbols, f , f (see (1.5) and (1.6)), and for more general combinations: for instance DDf will be the function DDf sampled on M.

3.1. Discrete calculus formulae. We provide calculus results for the finitedifference operators that were defined in the introductory section.

Lemma 3.2. Let the functions f 1 and f 2 be continuously defined over R. We have

D(f 1 f 2 ) = D(f 1 ) f2 + f1 D(f 2 ).
Note that the immediate translation of the proposition to discrete functions

f 1 , f 2 ∈ C M , and g 1 , g 2 ∈ C M is D(f 1 f 2 ) = D(f 1 ) f2 + f1 D(f 2 ), D(g 1 g 2 ) = D(g 1 ) g 2 + g 1 D(g 2 ).
Proof. We have

D(f 1 f 2 )(x) = h -1 (f 1 f 2 )(x + h/2) -h -1 (f 1 f 2 )(x -h/2) = (Df 1 )(x)(τ + f 2 )(x) + (τ -f 1 )(x)(Df 2 )(x).
For symmetry reasons we also have

D(f 1 f 2 ) = D(f 1 )τ -(f 2 )+τ + (f 1 )D(f 2 )
. Averaging the two equations we obtain the result.

Lemma 3.3. Let the functions f 1 and f 2 be continuously defined over R. We then have

f 1 f 2 = f1 f2 + h 2 4 D(f 1 )D(f 2 ).
Note that the immediate translation of the proposition to discrete functions

f 1 , f 2 ∈ C M , g 1 , g 2 ∈ C M is f 1 f 2 = f1 f2 + h 2 4 D(f 1 )D(f 2 ), g 1 g 2 = g 1 g 2 + h 2 4 D(g 1 )D(g 2 ).
Some of the following properties can be extended in such a manner to discrete functions. We shall not always write it explicitly.

Proof. We have

4 f1 f2 = (τ + f 1 + τ -f 1 )(τ + f 2 + τ -f 2 ) = 2τ + (f 1 f 2 ) + 2τ -(f 1 f 2 ) + τ + f 1 (τ -f 2 -τ + f 2 ) + τ -f 1 (τ + f 2 -τ -f 2 ) = 4 f 1 f 2 -h 2 (Df 1 )(Df 2 ).
Averaging a function twice gives the following formula. Lemma 3.4. Let the function f be continuously defined over R. We have

A 2 f := f = f + h 2 4 DDf.
Proof. We have

4 f = ((τ + ) 2 f + (τ -) 2 f + 2f ) = 4f + ((τ + ) 2 f + (τ -) 2 f -2f ) = 4f + h(τ + (Df ) -τ -(Df )) = 4f + h 2 DDf.
The following proposition covers discrete integrations by parts and related formulae.

Proposition 3.5. Let f ∈ C M∪∂M and g ∈ C M . We have the following formulae:

Ω f (τ + g) = Ω (τ -f )g -hf 0 g 1 2 , Ω f (τ -g) = Ω (τ + f )g -hf N +1 g N + 1 2 , Ω f (Dg) = - Ω (Df )g + f N +1 g N + 1 2 -f 0 g 1 2 , Ω f g = Ω f g - h 2 f N +1 g N + 1 2 - h 2 f 0 g 1 2 .
Lemma 3.6. Let f be a smooth function on R. We have

τ ± f = f ± h 2 1 0 ∂ x f (. ± σh/2) dσ, A j f = f + C j h 2 1 -1 (1 -|σ|) ∂ 2 x f (. + l j σh) dσ, D j f = ∂ j x f + C ′ j h 2 1 -1 (1 -|σ|) j+1 ∂ j+2 x f (. + l j σh) dσ, j = 1, 2, l 1 = 1 2 , l 2 = 1.
Proof. The results follow from Taylor formulae,

f (x + y) = n-1 j=0 y j j! f (j) (x) + y n 1 0 (1 -σ) n-1 (n -1)! f (n) (x + σy) dσ,
at order n = 1 for the first result, order n = 2 for the second one and orders n = 3 and 4 for the last one.

3.2. Calculus results related to the weight functions. We now provide some technical lemmata related to discrete operations performed on the Carleman weight functions that is of the form e sϕ with ψ ∈ C k , with k sufficiently large. For concision, we set r = e sϕ and ρ = r -1 . The positive parameters s and h will be large and small respectively and we are particularly interested in the dependence on s, h and λ in the following basic estimates.

We assume s ≥ 1 and λ ≥ 1. We shall use multi-indices of the form α = (α t , α x ) ∈ N 2 . The proofs can be found in Appendix A.

Lemma 3.7. Let α and β be multi-indices. We have

∂ β (r∂ α ρ) =|α| |β| (-sϕ) |α| λ |α+β| (∇ψ) α+β (3.1) + |α||β|(sϕ) |α| λ |α+β|-1 O(1) + s |α|-1 |α|(|α| -1)O λ (1) = O λ (s |α| ). Let σ ∈ [-1, 1]. We have ∂ β (r(x)(∂ α ρ)(x + σh)) = O λ (s |α| (1 + (sh) |β| )) e O λ (sh) . (3.2) Provided sh ≤ K we have ∂ β (r(x)(∂ α ρ)(x + σh)) = O λ,K (s |α| ).
The same expressions hold with r and ρ interchanged and with s changed into -s.

With Leibniz formula we have the following estimate. Corollary 3.8. Let α, β and δ be multi-indices. We have

∂ δ (r 2 (∂ α ρ)∂ β ρ) =|α + β| |δ| (-sϕ) |α+β| λ |α+β+δ| (∇ψ) α+β+δ + |δ||α + β|(sϕ) |α+β| λ |α+β+δ|-1 O(1) + s |α+β|-1 (|α|(|α| -1) + |β|(|β| -1))O λ (1) = O λ (s |α+β| ).
Proposition 3.9. Let α be a multi-index. Provided sh ≤ K, we have

rτ ± ∂ α ρ = r∂ α ρ + s |α| O λ,K (sh) = s |α| O λ,K (1)
,

rA j ∂ α ρ = r∂ α ρ + s |α| O λ,K ((sh) 2 ) = s |α| O λ,K (1), j = 1, 2, rA j Dρ = r∂ x ρ + sO λ,K ((sh) 2 ) = sO λ,K (1), j = 0, 1 rD 2 ρ = r∂ 2 x ρ + s 2 O λ,K ((sh) 2 ) = s 2 O λ,K (1).
The same estimates hold with ρ and r interchanged.

Lemma 3.10. Let α, β be multi-indices and k = 1, 2, j = 1, 2. Provided sh ≤ K, we have

D k (∂ β (r∂ α ρ)) = ∂ k x ∂ β (r∂ α ρ) + h 2 O λ,K (s |α| ), A j ∂ β (r∂ α ρ) = ∂ β (r∂ α ρ) + h 2 O λ,K (s |α| ). Let σ ∈ [-1, 1], we have D k ∂ β (r(x)∂ α ρ(x + σh)) = O λ,K (s |α| ).
The same expressions hold with r and ρ interchanged.

Lemma 3.11. Let α, β, δ be multi-indices and k = 1, 2, j = 1, 2. Provided sh ≤ K, we have

A j ∂ δ (r 2 (∂ α ρ)∂ β ρ) = ∂ δ (r 2 (∂ α ρ)∂ β ρ) + h 2 O λ,K (s |α|+|β| ) = O λ,K (s |α|+|β| ), D k ∂ δ (r 2 (∂ α ρ)∂ β ρ) = ∂ k x (∂ δ (r 2 (∂ α ρ)∂ β ρ)) + h 2 O λ,K (s |α|+|β| ) = O λ,K (s |α|+|β| ). Let σ, σ ′ ∈ [-1, 1]. We have A j ∂ δ r(x) 2 (∂ α ρ(x + σh))∂ β ρ(x + σ ′ h) = O λ,K (s |α|+|β| ), D k ∂ δ r(x) 2 (∂ α ρ(x + σh))∂ β ρ(x + σ ′ h) = O λ,K (s |α|+|β| ).
The same expressions hold with r and ρ interchanged.

Proposition 3.12.

Let α be a multi-index. For k = 0, 1, 2, j = 0, 1, 2, and for sh ≤ K, we have

D k A j ∂ α (r Dρ) = ∂ k x ∂ α (r∂ x ρ) + sO λ,K ((sh) 2 ) = sO λ,K (1), D k (rD 2 ρ) = ∂ k x (r∂ 2 x ρ) + s 2 O λ,K ((sh) 2 ) = s 2 O λ,K (1), D k (rA 2 ρ) = O λ,K ((sh) 2 ).
The same expressions hold with r and ρ interchanged.

Proposition 3.13. Let α and β be multi-indices and k = 0, 1, 2, j = 0, 1, 2 Provided sh ≤ K we have

A j D k ∂ β (r 2 (∂ α ρ) Dρ) = ∂ k x ∂ β (r 2 (∂ α ρ)∂ x ρ) + s |α|+1 O λ,K ((sh) 2 ) = s |α|+1 O λ,K (1), A j D k ∂ β (r 2 (∂ α ρ)A 2 ρ) = ∂ k x ∂ β (r(∂ α ρ)) + s |α| O λ,K ((sh) 2 ) = s |α| O λ,K (1), A j D k ∂ β (r 2 (∂ α ρ)D 2 ρ) = ∂ k x ∂ β (r 2 (∂ α ρ)∂ 2 x ρ) + s |α|+2 O λ,K ((sh) 2 ) = s |α|+2 O λ,K (1), A j D k ∂ α (r 2 DρD 2 ρ) = ∂ k x ∂ α (r 2 (∂ x ρ)∂ 2 x ρ) + s 3 O λ,K ((sh) 2 ) = s 3 O λ,K (1), A j D k ∂ α (r 2 Dρ A 2 ρ) = ∂ k x ∂ α (r∂ x ρ) + sO λ,K ((sh) 2 ) = sO λ,K (1)
.

Remark 3.14. We set D 2 := ((τ + ) 2 -(τ -) 2 )/2h = AD and A 2 := ((τ + ) 2 + (τ -) 2 )/2. We see that the results in the previous Lemmata and Propositions are preserved when we replace some of the D by D 2 and some of the A by A 2 .

4. A semi-discrete elliptic Carleman estimate for uniform meshes. Here we consider constant-step discretizations. The case of non-uniform meshes is treated in the following section.

For any uniform mesh M, let ξ 1 ∈ R M and ξ 2 ∈ R M be two positive discrete functions. We denote by reg(ξ) the following quantity reg(ξ) = max sup

M ξ 1 + 1 ξ 1 , sup M ξ 2 + 1 ξ 2 , sup M |Dξ 1 |, sup M |Dξ 2 | .
Hence, reg(ξ) measures the boundedness of ξ 1 and ξ 2 and of their discrete derivatives as well as the distance to zero of ξ 1 and ξ 2 . We extend ξ 1 and ξ 2 to piecewise affine functions in the neighborhood Ω of Ω on the dual and the primal meshes respectively. Continuous versions of the previous properties are then satisfied. We also call ξ 1 and ξ 2 the two piecewise affine functions. Note that ξ 2,d gives the discrete function ξ 2 we started from.

We let ω ⋐ Ω be a nonempty open subset. We set the operator P M to be

P M = -ξ 1 ∂ 2 t + D(ξ 2,d D)
, continuous in the variable t ∈ (0, T * ), with T * > 0, and discrete in the variable x ∈ Ω.

The Carleman weight function is of the form r = e sϕ with ϕ = e λψ , where ψ ∈ C k ( Q), with k ∈ N sufficiently large, satisfies Assumption 2.1. Here, to treat the semi-discrete case, we shall use the enlarged neighborhood Ω of Ω introduced in Assumption 2.1. This will allow multiple actions of discrete operators such as D and A on the weight functions. In particular we take ψ such that ∂ x ψ ≥ 0 in (0, T * ) × V a and ∂ x ψ ≥ 0 in (0, T * )×V b where V a and V b are neighborhoods of a and b respectively. This then yields (rDρ) 0 ≤ 0, (rDρ) N +1 ≥ 0.

(4.1)

We recall that ρ = r -1 . We introduce the following notation

∇ ξ f = (ξ 1 2 1 ∂ t f, ξ 1 2 2 ∂ x f ) t , ∆ ξ f = ξ 1 ∂ 2 t f + ξ 2 ∂ 2 x f.
We prove the following semi-discrete Carleman estimate. The function u denotes a function that is continuously defined and regular (C 2 ) w.r.t. t and discrete w.r.t. x. Theorem 4.1. Let reg 0 > 0 be given. For the parameter λ ≥ 1 sufficiently large, there exist C, s 0 ≥ 1, h 0 > 0, ε 0 > 0, depending on ω, T * , reg 0 , such that for any ξ = (ξ 1 , ξ 2 ) with reg(ξ) ≤ reg 0 , we have

s 3 e sϕ u 2 L 2 (Q) + s e sϕ ∂ t u 2 L 2 (Q) + s e sϕ d Du 2 L 2 (Q) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (Ω) + se 2sϕ(T * ) |∂ t u(T * , .)| 2 L 2 (Ω) + s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ C e sϕ P M u 2 L 2 (Q) + se 2sϕ(T * ) |Du(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) , (4.2)
for all s ≥ s 0 , 0 < h ≤ h 0 and sh ≤ ε 0 , and

u ∈ C 2 ([0, T ], C M∪∂M ), satisfying u| {0}×Ω = 0, u| (0,T * )×∂Ω = 0.
The proof of some of the lemmata below can be found in Appendix B.

Proof. We set f := -P M u. At first, we shall work with the function v = ru, i.e., u = ρv, that satisfies

r ξ 1 ∂ 2 t (ρv) + D(ξ 2,d D(ρv)) = rf. (4.3) By Lemma 3.2, we have ∂ 2 t (ρv) = (∂ 2 t ρ)v + 2(∂ t ρ)∂ t v + ρ∂ 2 t v and D(ξ 2,d D(ρv)) = (D(ξ 2,d Dρ)) ṽ + ξ 2,d Dρ Dṽ + (D ρ) ξ 2,d Dv + ρD(ξ 2,d Dv), since rρ = 1. By Lemma 3.3 we have ξ 2,d Dv = ξ 2,d Dv + h 4 (Dξ 2,d )(τ + Dv -τ -Dv), ξ 2,d Dρ = ξ 2,d Dρ + h 2 4 (Dξ 2,d )(DDρ), D(ξ 2,d Dρ) = (Dξ 2,d )Dρ + ξ 2,d DDρ. Equation (4.3) thus reads Av + B 1 v = g ′ where Av = ξ 1 ∂ 2 t v + r ρ D(ξ 2,d Dv) A1v + ξ 1 r(∂ 2 t ρ) v + ξ 2 r(DDρ) ṽ A2v , B 1 v = 2ξ 1 r(∂ t ρ)∂ t v + 2rDρ ξ 2 Dv, g ′ = rf - h 4 rDρ(Dξ 2,d )(τ + Dv -τ -Dv) - h 2 4 (Dξ 2,d )r(DDρ)Dv -hO(1)rDρ Dv -r(Dξ 2,d )Dρ + hO(1)r(DDρ) ṽ,
since D w = Dw, for any function w and since ξ 2,d -ξ 2 ∞ ≤ Ch. Following [START_REF] Fursikov | Controllability of evolution equations[END_REF] we now set

Bv = 2ξ 1 r(∂ t ρ)∂ t v + 2rDρ ξ 2 Dv B1v -2s(∆ ξ ϕ)v B2v , g = g ′ -2s(∆ ξ ϕ)v.
Equation (4.3) now reads Av + Bv = g and we write

Av 2 L 2 (Q) + Bv 2 L 2 (Q) + 2 Re (Av, Bv) L 2 (Q) = g 2 L 2 (Q) . (4.4)
We shall need the following estimation of g L 2 (Q) .

Lemma 4.2 (Estimate of the r.h.s.). For sh ≤ K we have

g 2 L 2 (Q) ≤ C λ,K rf 2 L 2 (Q) + s 2 v 2 L 2 (Q) + (sh) 2 Dv 2 L 2 (Q) . (4.5)
Most of the remaining of the proof will be dedicated to computing the innerproduct Re (Av, Bv) L 2 (Q) . Developing the inner-product Re (Av, Bv) L 2 (Q) , we set

I ij = Re (A i v, B j v) L 2 (Q) .
Note that all the estimates depend on reg 0 , which is a bound of the regularity measure reg(ξ) of ξ 1 and ξ 2 .

I a 11 = -sλ 2 Q ξ 1 ϕ|∇ ξ ψ| 2 |∂ t v| 2 dt -sλ 2 Q (ξ 2 ϕ|∇ ξ ψ| 2 ) d |Dv| 2 dt -sλ Ω ξ 2 1 ϕ(∂ t ψ) |∂ t v| 2 T * 0 + sλ Ω (ξ 1 ξ 2 ϕ∂ t ψ) d (T * ) |Dv| 2 (T * ),
and

Y 11 = T * 0 (1 + O λ,K ((sh) 2 )) (ξ 2 ξ 2,d rDρ) N +1 |Dv| 2 N + 1 2 -(ξ 2 ξ 2,d rDρ) 0 |Dv| 2 1 2 dt, X 11 = Q β 11 |∂ t v| 2 dt + Q ν 11 |Dv| 2 dt + Re Q α (1) 11 ∂ t v Dv * dt, J 11 = Ω δ 11 |Dv| 2 (T * ), W 11 = Q γ 11 |D∂ t v| 2 dt,
where

γ 11 = 1 2 h 2 sλ 2 (ξ 1 ϕ|∇ ξ ψ| 2 ) d + h 2 sλϕ d O(1) + hO λ,K ((sh) 2 ), δ 11 = sO λ,K (sh)
,

β 11 = sλϕO(1) + sO λ,K (sh) + hO λ (1), ν 11 = sλϕ d O(1) + sO λ,K (sh), α (1)
11 = sλϕ d O(1) + sO λ,K (sh).

Lemma 4.4 (Estimate of I 12 ). For sh ≤ K, the term I 12 is of the following form

I 12 = 2sλ 2 Q ξ 1 ϕ|∇ ξ ψ| 2 |∂ t v| 2 dt + 2sλ 2 Q (ϕξ 2 |∇ ξ ψ| 2 ) d |Dv| 2 dt -X 12 -J 12 , with
X 12 = Q β 12 |∂ t v| 2 dt + Q ν 12 |Dv| 2 dt + Q µ 12 |v| 2 dt + Re Q α (1) 12 ṽ * Dv dt, J 12 = Re Ω (α (2) 12 v * ∂ t v)(T * ) + Ω η 12 |v| 2 (T * ),
where

β 12 = sλϕO(1), ν 12 = sλϕO(1) + sO λ,K (h + (sh) 2 ), µ 12 = sO λ,K (1), α (1)
12 = sO λ,K (1), α

(2) 12 = sO λ,K (1) and η 12 = sO λ (1).

Lemma 4.5 (Estimate of I 21 ). For sh ≤ K, the term I 21 can be estimated as

I 21 ≥3s 3 λ 4 Q ϕ 3 |∇ ξ ψ| 4 |v| 2 dt -(sλ) 3 Ω ξ 1 (ϕ 3 (∂ t ψ)|∇ ξ ψ| 2 )(T * ) |v| 2 (T * ) + Y 21 + W 21 -X 21 -J 21 ,
with

W 21 = Q γ 21 |D∂ t v| 2 dt, Y 21 = T * 0 O λ,K ((sh) 2 )(rDρ) 0 |Dv| 2 1 2 dt + T * 0 O λ,K ((sh) 2 )(rDρ) N +1 |Dv| 2 N + 1 2 dt, X 21 = Q µ 21 |v| 2 dt + Q ν 21 |Dv| 2 dt, J 21 = Ω η 21 |v| 2 (T * ) + Ω δ 21 |Dv| 2 (T * ),
where

γ 21 = hO(sh), µ 21 = (sλϕ) 3 O(1) + s 2 O λ (1) + s 3 O λ,K ((sh) 2 ), ν 21 = sO λ,K ((sh) 2 ), η 21 = s 3 O λ,K ((sh) 2 ) + s 2 O λ,K (1)
, and δ 21 = sO λ,K ((sh) 2 ).

Lemma 4.6 (Estimate of I 22 ). For sh ≤ K, the term I 22 is of the following form

I 22 = -2s 3 λ 4 Q ϕ 3 |∇ ξ ψ| 4 |v| 2 dt -X 22 , X 22 = Q µ 22 |v| 2 dt + Q ν 22 |Dv| 2 dt,
where µ 22 = (sλϕ) 3 O(1) + s 2 O λ,K (1) + s 3 O λ,K ((sh) 2 ) and ν 22 = sO λ,K ((sh) 2 ).

Continuation of the proof of Theorem 4.1. Collecting the terms we have obtained in the previous lemmata, from (4.4) we obtain, for sh ≤ K,

2sλ 2 Q ξ 1 ϕ|∇ ξ ψ| 2 |∂ t v| 2 dt+2sλ 2 Q (ϕξ 2 |∇ ξ ψ| 2 ) d |Dv| 2 dt+2s 3 λ 4 Q ϕ 3 |∇ ξ ψ| 4 |v| 2 dt -2sλ Ω ξ 2 1 ϕ(∂ t ψ) |∂ t v| 2 T * 0 + 2sλ Ω (ξ 1 ξ 2 ϕ∂ t ψ) d (T * ) |Dv| 2 (T * ) -2(sλ) 3 Ω ξ 1 (ϕ 3 (∂ t ψ)|∇ ξ ψ| 2 )(T * ) |v| 2 (T * ) + W + Y ≤ C λ,K rf 2 L 2 (Q) + X + J, (4.6) where X = X 11 + X 12 + X 21 + X 22 + C λ,K s 2 v 2 L 2 (Q) + C λ,K (sh) 2 Dv 2 L 2 (Q) , J = J 11 + J 12 + J 21 , W = W 11 + W 21 and Y = Y 11 + Y 21 .
With the following lemma, we may in fact ignore the terms W and Y .

Lemma 4.7. Let sh ≤ K. There exists λ 1 ≥ 1, and ε 1 (λ) > 0 such that for λ ≥ λ 1 and 0 < sh ≤ ε 1 (λ), we have W ≥ 0 and Y ≥ 0.

Lemma 4.8. We have

Re Q α (1) 11 ∂ t v Dv * dt ≤ Q β (2) 11 |∂ t v| 2 dt + Q ν (2) 11 |Dv| 2 dt, Re Q α (1) 12 ṽ * Dv dt ≤ Cs 2 Q O λ,K (1)|v| 2 dt + C Q |Dv| 2 dt, Re Ω (α (2) 12 v * ∂ t v)(T * ) ≤ C Ω |∂ t v| 2 (T * ) + s 2 Ω O λ,K (1)|v| 2 (T * ), with β (2) 11 = sλϕO(1) + sO λ,K (sh), ν (2) 11 = sλϕ d O(1) + sO λ,K (sh).
Recalling the properties satisfied by ψ listed in Assumption 2.1, if we choose λ 2 ≥ λ 1 sufficiently large, then for λ = λ 2 (fixed for the rest of the proof) and sh ≤ ε 1 (λ 2 ), from (4.6) and Lemmata 4.7 and 4.8, we obtain

s 3 v 2 L 2 (Q) + s ∂ t v 2 L 2 (Q) + s Dv 2 L 2 (Q) (4.7) + s|∂ t v(0, .)| 2 L 2 (Ω) + s|∂ t v(T * , .)| 2 L 2 (Ω) + s 3 |v(T * , .)| 2 L 2 (Ω) ≤ C λ2,K rf 2 L 2 (Q) + s|Dv(T * , .)| 2 L 2 (Ω) + s|∂ t v(0, .)| 2 L 2 (ω) + X + J, where X = Q µ 1 |v| 2 dt + Q β 1 |∂ t v| 2 dt + Q ν 1 |Dv| 2 dt, J = Ω η 1 |v| 2 (T * ) + Ω α 1 |∂ t v| 2 (T * ) + Ω δ 1 |Dv| 2 (T * ),
with

µ 1 = s 2 O λ2,K (1) + s 3 O λ2,K ((sh) 2 ), β 1 = sO λ2,K (sh)
,

ν 1 = sO λ2,K (h + sh) + O λ2,K (1), η 1 = s 2 O λ2,K (1) + s 3 O λ2,K ((sh) 2 ), α 1 = sO λ2,K ((sh) 2 ) + C, δ 1 = sO λ2,K,η (1).
We can now choose ε 0 and h 0 sufficiently small, with 0 < ε 0 ≤ ε 1 (λ 2 ), and s 0 ≥ 1 sufficiently large, such that for s ≥ s 0 , 0 < h ≤ h 0 , and sh ≤ ε 0 , we obtain

s 3 v 2 L 2 (Q) + s ∂ t v 2 L 2 (Q) + s Dv 2 L 2 (Q) (4.8) + s|∂ t v(0, .)| 2 L 2 (Ω) + s|∂ t v(T * , .)| 2 L 2 (Ω) + s 3 |v(T * , .)| 2 L 2 (Ω) ≤ C λ2,K,ε0,s0 rf 2 L 2 (Q) + s|Dv(T * , .)| 2 L 2 (Ω) + s|∂ t v(0, .)| 2 L 2 (ω) .
We now proceed with using back the unknown function u in the estimates. In fact we have the following lemma. Lemma 4.9. For sh ≤ K we have

r d Du 2 L 2 (Q) ≤ C λ,K s 2 v 2 L 2 (Q) + Dv 2 L 2 (Q) , r∂ t u 2 L 2 (Q) ≤ C λ,K s 2 v 2 L 2 (Q) + ∂ t v 2 L 2 (Q) , |r∂ t u(T * , .)| 2 L 2 (Ω) ≤ C λ,K s 2 |v(T * , .)| 2 L 2 (Ω) + |∂ t v(T * , .)| 2 L 2 (Ω) .
Since ϕ(T * ) = Cst by the properties of ψ (see Assumption 2.1) and because of the zero-boundary condition imposed on u at t = 0 we have

|∂ t v(0, .)| 2 L 2 (Ω) = |r∂ t u(0, .)| 2 L 2 (Ω) , |∂ t v(0, .)| 2 L 2 (ω) = |r∂ t u(0, .)| 2 L 2 (ω) , |Dv(T * , .)| 2 L 2 (Ω) = r(T * ) 2 |Du(T * , .)| 2 L 2 (Ω) .
We hence obtained the desired Carleman estimate from (4.8) and Lemma 4.9.

Remark 4.10. Note that the term W in (4.6), that we proved to be non-negative, has no counterpart in the continuous case.

5. Carleman estimates for regular non uniform meshes. We present in this section a way to extend the above results to the class of non uniform meshes introduced in Section 1.2, see also Figure 1.1. We chose a function ϑ satisfying (1.7) to remain fixed in the sequel.

By using first-order Taylor formulae we obtain the following result.

Lemma 5.1. Let us define ζ ∈ R M and ζ ∈ R M as follows

ζ i+ 1 2 = h i+ 1 2 h ⋆ , ∀i ∈ {0, . . . , N }, ζ i = h i h ⋆ , ∀i ∈ {1, . . . , N }.
These two discrete functions are connected to the geometry of the primal and dual meshes M and M, and we have

0 < inf Ω0 ϑ ′ ≤ ζ i+ 1 2 ≤ sup Ω0 ϑ ′ , ∀i ∈ {0, . . . , N }, 0 < inf Ω0 ϑ ′ ≤ ζ i ≤ sup Ω0 ϑ ′ , ∀i ∈ {1, . . . , N }. |Dζ| L ∞ (Ω) ≤ ϑ ′′ L ∞ inf Ω0 ϑ ′ , |Dζ| L ∞ (Ω) ≤ ϑ ′′ L ∞ inf Ω0 ϑ ′ .
We aim to prove uniform Carleman estimates in this framework by using the result on uniform meshes of Section 4. To any u ∈ C M∪∂M , we associate the discrete function denoted by Q M 0 M u ∈ C M 0 ∪∂M 0 defined on the uniform mesh M 0 which takes the same values as u at the corresponding nodes. More precisely, if u

= N i=1 1 [x i-1 2 ,x i+ 1 2 ] u i , we let Q M 0 M u = N i=1 1 [(i-1 2 )h ⋆ ,(i+ 1 2 )h ⋆ ] u i , and (Q M 0 M u) 0 = u 0 , (Q M 0 M u) N +1 = u N +1 . Similarly, for any u ∈ C M , u = N i=0 1 [xi,xi+1] u i+ 1 2 , we set Q M 0 M u = N i=0 1 [ih ⋆ ,(i+1)h ⋆ ] u i+ 1 2 . The operators Q M 0 M and Q M 0 M
are invertible and we denote by Q M M 0 and Q M M 0 their respective inverses. Let us now give commutation properties between these operators and discrete difference operators. To lighten notation we shall use the same symbols D (resp. D) for the difference operators acting on C M 0 ∪∂M 0 and C M∪∂M (resp. on C M 0 and C M ).

Lemma 5.2.

1. For any u ∈ C M∪∂M and any v ∈ C M , we have

D(Q M 0 M u) = Q M 0 M (ζDu), D(Q M 0 M v) = Q M 0 M (ζDv).
2. For any u ∈ C M∪∂M we have

D(γ d Du) = (ζ) -1 Q M M 0 D Q M 0 M γ d ζ D(Q M 0 M u) .
Proof. Let 0 ≤ i ≤ N . On the one hand, by the definitions of Q M 0 M and D acting on C M 0 ∪∂M 0 , we have

(D(Q M 0 M u)) i+ 1 2 = (Q M 0 M u) i+1 -(Q M 0 M u) i h ⋆ = u i+1 -u i h ⋆ .
On the other hand, by the definitions of ζ, Q M 0 M , and D acting on C M∪∂M we have

(Q M 0 M (ζDu)) i+ 1 2 = (ζDu) i+ 1 2 = h i+ 1 2 h ⋆ u i+1 -u i h i+ 1 2 = u i+1 -u i h ⋆ ,
which proves the first result. The other statements can be proven in a similar manner.

Lemma 5.3. For any u ∈ C M and any v ∈ C M we have

(sup Ω0 ϑ ′ ) -1 |u| 2 L 2 (Ω) ≤ |Q M 0 M u| 2 L 2 (Ω0) ≤ (inf Ω0 ϑ ′ ) -1 |u| 2 L 2 (Ω) , (sup Ω0 ϑ ′ ) -1 |v| 2 L 2 (Ω) ≤ |Q M 0 M v| 2 L 2 (Ω0) ≤ (inf Ω0 ϑ ′ ) -1 |v| 2 L 2 (Ω) .
Furthermore, the same inequalities hold by replacing Ω by ω and Ω 0 by ω 0 , respectively.

Proof. By definition of Q M 0 M and of the discrete norms, we have

|Q M 0 M u| 2 L 2 (Ω0) = N i=1 h ⋆ |u i | 2 = N i=1 1 ζ i h i |u i | 2 , and |u| 2 L 2 (Ω) = N i=1 h i |u i | 2 ,
so that the first property follows from Lemma 5.1. The property for v is proved similarly.

To avoid any ambiguity we introduce the following notation. For any continuous function f defined on Ω (resp. on Ω 0 ) we denote by

Π M f = (f (x i )) 0≤i≤N +1 ∈ C M∪∂M the sampling of f on M (resp. Π M 0 f = (f (ih ⋆ )) 0≤i≤N +1 ∈ C M 0 ∪∂M 0 the sampling of f on M 0 ).
Lemma 5.4. Let f be a continuous function defined on Ω. We have

Q M 0 M Π M f = Π M 0 (f • ϑ).
In particular, for any u ∈ C M∪∂M we have

Q M 0 M (Π M f )u = Π M 0 (f • ϑ)(Q M 0 M u).
We can now prove the following discrete Carleman estimate for our elliptic operator

P M = -∂ 2 t -D(γ d D•)
on the mesh M. Theorem 5.5. Let ϑ satisfy (1.7) and ψ be a weight function satisfying assumption 2.1 for the observation domain ω. For the parameter λ ≥ 1 sufficiently large, there exist C, s 0 ≥ 1, h 0 > 0, ε 0 > 0, depending on ω, T * , ϑ, reg(γ), such that for any mesh M obtained from ϑ by (1.8), we have

s 3 e sϕ u 2 L 2 (Q) + s e sϕ ∂ t u 2 L 2 (Q) + s e sϕ d Du 2 L 2 (Q) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (Ω) + se 2sϕ(T * ) |∂ t u(T * , .)| 2 L 2 (Ω) + s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ C e sϕ P M u 2 L 2 (Q) + se 2sϕ(T * ) |Du(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) , (5.1)
for all s ≥ s 0 , 0 < h ≤ h 0 and sh ≤ ε 0 , and u ∈ C 2 ([0, T ], C M∪∂M ), satisfying u| {0}×Ω = 0, u| (0,T * )×∂Ω = 0. Proof. We set w = Q M 0 M u defined on the uniform mesh M 0 . By using Lemma 5.2, we have

Q M 0 M ζP M u = Q M 0 M ζ ∂ 2 t w -D Q M 0 M γ d ζ Dw .
(5.2)

We see that the right-hand side of (5.2) is a semi-discrete elliptic operator of the form

P M 0 = ξ 1 ∂ 2 t -D(ξ 2 D•) applied to w,
where

ξ 1 = Q M 0 M ζ, ξ 2 = Q M 0 M γ d ζ .
(5.3) By using assumption 2.1 and (1.7), we now observe that , the function ψ • ϑ : (t, x) → ψ(t, ϑ(x)) is a suitable weight function associated to the control domain ω 0 = ϑ -1 (ω) in Ω 0 , i.e. that ψ • ϑ satisfies assumption 2.1 for the domains Ω 0 and ω 0 . In Theorem 4.1, we have obtained a discrete uniform Carleman estimate for P M 0 and the weight function ψ • ϑ on the uniform mesh M 0 . We can now deduce the same result on the non-uniform mesh M.

Firstly, we observe that there exists C ϑ,γ such that we have reg(ξ) ≤ C ϑ,γ uniformly with respect to h ⋆ , with ξ = (ξ 1 , ξ 2 ) as defined in (5.3). Then, choosing reg 0 = C ϑ,γ in the proof of Theorem 4.1, we see that estimate (4.8) holds

s 3 e sϕ•ϑ w 2 L 2 (Q0) + s ∂ t (e sϕ•ϑ w) 2 L 2 (Q0) + s D(e s(ϕ•ϑ) w) 2 L 2 (Q0) + s|e sϕ•ϑ(0,.) ∂ t w(0, .)| 2 L 2 (Ω0) + s|∂ t (e 2sϕ w)(T * , .)| 2 L 2 (Ω0) + s 3 e 2sϕ(T * ) |w(T * , .)| 2 L 2 (Ω0)
≤ C e sϕ•ϑ P M 0 w 2 L 2 (Q0) +se 2sϕ(T * ) |Dw(T * , .)| 2 L 2 (Ω0) +s|e sϕ•ϑ(0,.) ∂ t w(0, .)| 2 L 2 (ω0) , (5.4) and the constant C is uniform in h ⋆ for s sufficiently large and with sh ⋆ ≤ ε 0 , for ε 0 sufficiently small. Note that, setting ε0 = (inf Ω0 ϑ ′ )ε 0 , we see that the condition sh ≤ ε0 on the size of the non-uniform mesh M implies the condition sh ⋆ ≤ ε 0 for the uniform mesh M 0 . Secondly, by using the previous lemmata 5.1, 5.2, 5.3, 5.4 and considering each term above separately, we see that we have the following estimates.

• For the third term in the l.h.s. of (5.4)

D(e s(ϕ•ϑ) w) 2 L 2 (Q0) = D(e s(ϕ•ϑ) Q M 0 M u) 2 L 2 (Q0) = DQ M 0 M (e sϕ u) 2 L 2 (Q0) = Q M 0 M ζD(e sϕ u) 2 L 2 (Q0) ≥ (sup Ω0 ϑ ′ ) -1 ζD(e sϕ u) 2 L 2 (Q) ≥ (sup Ω0 ϑ ′ ) -1 (inf Ω0 ϑ ′ ) 2 D(e sϕ u) 2 L 2 (Q) .
• For any α ∈ {0, 1}, we have

∂ α t (e sϕ•ϑ w) 2 L 2 (Q0) = Q M 0 M ∂ α t (e sϕ u) 2 L 2 (Q0) ≥ (sup Ω0 ϑ ′ ) -1 ∂ α t (e sϕ u) 2 L 2 (Q) ,
and similar inequalities hold for the other terms in the l.h.s. of (5.4). • By using (5.2) and (5.3) we have

e sϕ•ϑ P M 0 w 2 L 2 (Q0) = e sϕ•ϑ Q M 0 M ζP M u ) 2 L 2 (Q0) = Q M 0 M e sϕ ζP M u 2 L 2 (Q0) ≤ (inf Ω0 ϑ ′ ) -1 e sϕ ζP M u 2 L 2 (Q) ≤ (sup Ω0 ϑ ′ ) 2 (inf Ω0 ϑ ′ ) -1 e sϕ P M u 2 L 2 (Q) .
• Finally, since ϑ(ω 0 ) = ω, we have

|e sϕ•ϑ(0,.) ∂ t w(0, .)| 2 L 2 (ω0) = |Q M 0 M (e sϕ(0,•) ∂ t u(0, •))| 2 L 2 (ω0) ≤ (inf Ω0 ϑ ′ ) -1 |e sϕ(0,•) ∂ t u(0, •)| 2 L 2 (ω) .
The proof is complete.

6. A partial discrete Lebeau-Robbiano spectral inequality. In this section, with the Carleman estimate we just proved, we obtain a Lebeau-Robbiano type spectral inequality for the lower part of the spectrum of the operator A M . The constant we shall obtain in this inequality is in fact uniform w.r.t. to the step size of the chosen mesh M.

We recall that we denote by φ M a set of discrete orthonormal eigenfunctions, φ j ∈ C M , 1 ≤ j ≤ |M|, of the operator A M with homogeneous Dirichlet boundary conditions, and by µ M the set of the associated eigenvalues sorted in a non-decreasing sequence, µ j , 1 ≤ j ≤ |M|. Theorem 6.1 (Partial discrete Lebeau-Robbiano inequality). Let ϑ satisfying (1.7). There exist C > 0, ε 1 > 0 and h 0 such that, for any mesh M obtained from ϑ by (1.8) such that h ≤ h 0 , for all 0 < µ ≤ ε 1 /h 2 , we have

µ k ∈µ M µ k ≤µ |α k | 2 = Ω µ k ∈µ M µ k ≤µ α k φ k 2 ≤ Ce C √ µ ω µ k ∈µ M µ k ≤µ α k φ k 2 , ∀(α k ) 1≤k≤|M| ⊂ C.
Proof. We adapt the proof presented in Section 2. We introduce the following semi-discrete function u

(t) = µ k ∈µ M α k sinh( √ µ k t) √ µ k
φ k , which satisfies the boundary conditions listed in the discrete Carleman estimate of Theorem 5.5 and P M u = -∂ 2 t u+ A M u = 0. For some K > 0, s 0 > 0, h 0 > 0 and ε 0 > 0, uniform w.r.t. M, we thus have

s 3 e 2sϕ(T * ) |u(T * , .)| 2 L 2 (Ω) ≤ K se 2sϕ d (T * ) |Du(T * , .)| 2 L 2 (Ω) + s|e sϕ(0,.) ∂ t u(0, .)| 2 L 2 (ω) , for s ≥ s 0 , 0 < h ≤ h 0 and sh ≤ ε 0 . As in the proof of Theorem 1.1 it suffices to obtain 1 2 s 2 |u(T * )| 2 L 2 (Ω) ≥ K|Du(T * )| 2 L 2 (Ω)
. In fact we have

|u(T * )| 2 L 2 (Ω) ≥ 1 µ µ k ≤µ |α k sinh(T * √ µ k )| 2 , |Du(T * )| 2 L 2 (Ω) ≤ 1 γ min µ k ≤µ |α k sinh(T * √ µ k )| 2 , since the discrete functions Dφ k , 1 ≤ j ≤ |M|, satisfy Ω γ d Dφ k Dφ k = δ jk µ k .
It thus suffices to have s 2 /(2µ) ≥ K/γ min . Since sh ≤ ε 0 , this can be made possible if µ ≤ γ min ε 2 0 /(2Kh 2 ). The result follows with ε 1 = γ min ε 2 0 /2K. 7. Uniform controllability of the lower part of the spectrum. Proof of Theorem 1.4. Let ϑ satisfy (1.7) and M be a mesh defined by (1.8) such that h ≤ h 0 . We set µ M max = ε 1 /h 2 , with h 0 and ε 1 given by Theorem 6.1. Let j M = max{j; 2 2j ≤ µ M max }. We recall the following notation from the introduction

E j = Span{φ k ; µ k ≤ 2 2j } ⊂ C M , j ∈ N,
and denote by Π Ej the L 2 (Ω)-orthogonal projection onto E j .

Lemma 7.1. There exists C ≥ 0 such that, for j ≤ j M and S > 0, the semidiscrete solution q in C ∞ ([0, S], E j ) to the adjoint parabolic system

     -∂ t q + A M q = 0 in (0, S) × Ω, q = 0 on (0, S) × ∂Ω, q(S) = q F ∈ E j , (7.1)
satisfies the following observability estimate

|q(0)| 2 L 2 (Ω) ≤ Ce C2 j S S 0 ω |q(t)| 2 dt. Proof. If q(0) = µ k ≤2 2j b k φ k . Then q(t) = µ k ≤2 2j α k (t)φ k with α k (t) = b k e µ k t . Parabolic dissipation and Theorem 6.1, since 2 2j ≤ ε 1 /h 2 , then yield S|q(0)| 2 L 2 (Ω) ≤ S 0 |q(t)| 2 L 2 (Ω) dt = S 0 | µ k ≤2 2j α k (t)φ k | 2 L 2 (Ω) dt ≤ Ce C2 j S 0 ω | µ k ≤2 2j α k (t)φ k | 2 dt = Ce C2 j S 0 ω |q(t)| 2 dt.
We now consider the following partial control problem

     ∂ t y + A M y = Π Ej (1 ω v) in (0, S) × Ω, y = 0 on (0, S) × ∂Ω, y(0) = y 0 ∈ E j in Ω. (7.2)
With the previous observability result we have the following lemma.

Lemma 7.2. There exists C > 0, such that for j ≤ j M , there exists a control function w ∈ L 2 ((0, S) × Ω) that brings the solution to system (7.2) to zero at time S, and which satisfies

w L 2 ((0,S)×Ω) ≤ CS -1 2 e C2 j |y 0 | L 2 (Ω) .
We shall denote by V j (y 0 , a, S) such a control when working on the time interval (a, a + S) instead.

We now present the iterative construction of the control function. We write [0, T /2] = j∈N [a j , a j+1 ], with a 0 = 0, a j+1 = a j + 2T j , for j ∈ N and T j = K2 -jρ with ρ ∈ (0, 1) and the constant K is such that 2 ∞ j=0 T j = T /2. The control function is defined as follows, for 0 ≤ j ≤ j M , if t ∈ (a j , a j + T j ], v(t) = V j (Π Ej y(a j ), a j , T j ) and y(t) = S(t -a j )y(a j ) + t aj S(t -s)v(s)ds, if t ∈ (a j + T j , a j+1 ], v(t) = 0 and y(t) = S(t -a j -T j )y(a j + T j ), and v(t) = 0 for t ∈ [a j M +1 , T ] where S(t) denote the semi-group S(t) = e -tA M . In particular, S(t) (L 2 ,L 2 ) ≤ 1. This choice of the control function v in the time interval [a j , a j + T j ], j ≤ j M , implies

|y(a j + T j )| L 2 (Ω) ≤ (1 + Ce C2 j )|y(a j )| L 2 (Ω)
, and Π Ej y(a j + T j ) = 0.

During the passive period, t ∈ [a j +T j , a j+1 ], there is an exponential decrease of the L 2 norm, |y(a j+1 )| L 2 (Ω) ≤ e -2 2j Tj |y(a j + T j )| L 2 (Ω) , and from the value of T k introduced above we thus obtain

|y(a j+1 )| L 2 (Ω) ≤ e C2 j -K2 j(2-ρ) |y(a j )| L 2 (Ω) , which gives |y(a j+1 )| L 2 (Ω) ≤ e P j k=0 (C2 k -K2 k(2-ρ) ) |y 0 | L 2 (Ω) . With ρ ∈ (0, 1), there exists C > 0, such that |y(a j+1 )| L 2 (Ω) ≤ Ce -C2 j(2-ρ) |y 0 | L 2 (Ω) , 0 ≤ j ≤ j M . (7.3) Since 2 2(j M +1) ≥ ε 1 /h 2 = µ M max it furthermore follows that y(a j M +1 ) L 2 (Ω) ≤ Ce -C/h (2-ρ) y 0 L 2 (Ω) .
The constant C depends only on the map ϑ defining the mesh M but not on the mesh size h.

Concerning the L 2 norm of v we have v 2 L 2 (Q) = 0≤j≤j M v 2
L 2 ((aj ,aj+Tj )×Ω) . From Lemma 7.2 and Estimate (7.3) we deduce

v 2 L 2 ((0,T )×Ω) ≤ CT -1 0 e 2C + 1≤j≤j M CT -1 j e C2 j e -C2 (j-1)(2-ρ) |y 0 | 2 L 2 (Ω) .
Hence, arguing as above there exists 0 < C T < ∞, independent of h, depending only on ϑ, such that

v L 2 ((0,T )×Ω) ≤ C T |y 0 | L 2 (Ω) . Since v(t) = 0 for t ∈ [a j M +T j M , T ] and since 2 2(j M +1) ≥ ε 1 /h 2 = µ M max it furthermore follows that |y(T )| L 2 (Ω) ≤ Ce -(C/h) 2 |y 0 | L 2 (Ω) ,
as Π E j M y(a j M + T j M ) = 0. This concludes the proof of Theorem 1.4.

Remark 7.3. If we chose to directly control in the space E j M based on the partial observability result of Lemma 7.1, instead of the Lebeau-Robbiano construction of the control function we have done here, we would obtain a L 2 norm of the control that diverges to +∞ as h goes to zero. The Lebeau-Robbiano construction, making use of the natural parabolic exponential decay, is a key point to obtain a uniform bound for the L 2 norm of the control.

With the null controllability result we have obtained in E j M in Theorem 1.4, we have the following observability result which improves upon Lemma 7.1.

Corollary 7.4. For j ≤ j M and S > 0, the semi-discrete solution q in C ∞ ([0, S], E j ) to system (7.1) satisfies the following uniform observability estimate

|q(0)| L 2 (Ω) ≤ C T S 0 ω |q(t)| 2 dt 1 2 .
Finally, in the spirit of the work of [START_REF] Labbé | Uniform controllability of semidiscrete approximations of parabolic control systems[END_REF] the controllability result we have obtained yields the following relaxed observability estimate Corollary 7.5. There exist C T > 0 and C > 0 depending on Ω, ω, T , and ϑ, such that the semi-discrete solution

q in C ∞ ([0, T ], C M ) to      -∂ t q + A M q = 0 in (0, T ) × Ω, q = 0 on (0, T ) × ∂Ω, q(T ) = q F ∈ C M , in the case h ≤ h 0 , satisfies |q(0)| L 2 (Ω) ≤ C T T 0 ω |q(t)| 2 dt 1 2 + Ce -C/h 2 |q F | L 2 (Ω) .
Using this observability inequality, we can now provide some constructive way to compute a suitable semi-discrete control function. To this end, let h → φ(h) ∈ R + be a function which tends to zero when h goes to 0 and such that e -C/h 2 /φ(h) → 0. We have the following result.

Theorem 7.6. Let C T , C and h 0 being the same as in Corollary 7.5.

For any mesh M obtained from ϑ by (1.8) such that h ≤ h 0 , and any y 0 ∈ C M , we consider the functional q F ∈ C M → J M (q F ) defined by

J M (q F ) = 1 2 T 0 |q(t)| 2 L 2 (ω) dt + φ(h) 2 |q F | 2 L 2 (Ω) + (y 0 , q(0)) L 2 (Ω) ,
where t → q(t) is the solution to the adjoint problem -∂ t q + A M q = 0 with final data q(T ) = q F . This functional J M has a unique mimiser denoted by q F,opt ∈ C M . This minimiser produces a solution q opt of the adjoint problem such that, if we define the control function v(t) = 1 ω q(t) then we have:

• The cost of the control is bounded as follows

T 0 |v(t)| 2 L 2 (ω) dt ≤ (C 2 T + φ(h))|y 0 | 2 L 2 (Ω) .
• The controlled solution y to (1.9) is such that

|y(T )| L 2 (Ω) ≤ φ(h) C T + φ(h) |y 0 | L 2 (Ω) .
The proof of this result is not written here as it can be done along the lines of the proofs given for instance in [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF][START_REF] Labbé | Uniform controllability of semidiscrete approximations of parabolic control systems[END_REF]. Some further details will be given in [START_REF] Boyer | Uniform null-controllability properties for space/time-discretized parabolic equations[END_REF], in connection with its fully-discrete counterpart.

Let us give some final remarks: 1. In practice, the functional J M is quadratic, strictly convex and coercive. Hence, the computation of q F,opt can be performed by using a conjugate gradient algorithm. 2. The same result holds with φ(h) = Ce -C/h 2 . Such a choice can be however quite unconvenient in practice as we do not know in general the value of the constant C and since e -C/h 2 is very likely to be smaller than machine precision for reasonable values of h. 3. A natural choice for φ is φ(h) = h β with β > 0 as large as desired. Minimizing J M we then obtain a control family that is uniformly bounded with respect to h and such that the final state y(T ) tends to zero like h β/2 . Note that the numerical scheme defined by the semi-discrete operator ∂ t -D(γ d D) provides at most second-order accuracy for the computation of smooth solutions of the parabolic problem under study. A natural choice is then φ(h) = h 4 . In fact, a choice of a smaller value for φ(h) only results in a larger number of iterations of the conjugate gradient algorithm to achieve convergence. 4. As the semi-discrete controls we have obtained are bounded in L 2 , then, up to a subsequence, these semi-discrete controls converge towards a function v ∈ L 2 ((0, T )×ω) that actually drives the solution of the continuous parabolic problem to zero at time T . 5. In addition to space discretization, a time discretization can also be carried out (implicit Euler scheme or more general θ-schemes). One can then observe the convergence of the fully-discrete control function to the semidiscrete control function as the time step goes to zero. See [START_REF] Boyer | Uniform null-controllability properties for space/time-discretized parabolic equations[END_REF] for details, in particular for error estimates.

Appendix A. Proofs of some technical results in Section 3.

A.1. Proof of Lemma 3.7. For a multi-index δ, by induction we have

∂ δ ϕ = λ |δ| ∇ψ δ ϕ + |δ|(|δ| -1)λ |δ|-1 ϕO(1).
To prove (3.1) we proceed by induction on |α| + |β|. The result holds for |α| = 0 and any β, and we assume it also holds in the case |α|+|β| = n. In the case |α|+|β| = n+1, with |α| ≥ 1, we write α = α ′ + α ′′ with |α ′′ | = 1 and we have

∂ β (r∂ α ρ) = -s∂ β (r∂ α ′ ((∂ α ′′ ϕ) ρ)) = -s∂ β δ ′ +δ ′′ =α ′ α ′ δ ′ (∂ δ ′′ +α ′′ ϕ) r∂ δ ′ ρ = -s δ ′ +δ ′′ =α ′ β ′ +β ′′ =β α ′ δ ′ β β ′ (∂ δ ′′ +β ′′ +α ′′ ϕ) ∂ β ′ (r∂ δ ′ ρ).
Using the inductive hypothesis we see that the largest power in s in obtained by picking δ ′ = α ′ and δ ′′ = 0 in the previous sum. The remainder is of the form (|α| -1)s |α|-1 O(1). The terms we selected lead to -s

β ′ +β ′′ =β β β ′ λ |β ′′ | ϕ(λ∇ψ β ′′ +α ′′ + |β ′′ |O(1)) |α ′ | |β ′ | (-sϕ) |α ′ | λ |β ′ +α ′ | ∇ψ α ′ +β ′ + |α ′ ||β ′ |(sϕ) |α ′ | λ |α ′ +β ′ |-1 O(1) + |α ′ |(|α ′ | -1)s |α ′ |-1 O λ (1) ,
which follows as

β ′ +β ′′ =β β β ′ |α ′ | |β ′ | (-sϕ) |α| λ |β|+|α| ∇ψ β+α + |β|(sϕ) |α| λ |α|+|β|-1 O(1) + (|α| -1)s |α|-1 O λ (1),
which concludes the proof of the first result, since

β ′ +β ′′ =β β β ′ |α ′ | |β ′ | = |α| |β| .
The same proof applies to ∂ β (ρ∂ α r).

For (3.2), we first consider the case |α| = 0. We set ν(x, σh) := r(x)ρ(x + σh) and simply have ν(x, σh) = e s(ϕ(x)-ϕ(x+σh)) = e O λ (sh) , by a first-order Taylor formula, which gives the result in the case |β| = 0. For |β| ≥ 1, we observe that ∂ β ν(x, σh) is a linear combination of terms of the form sh) , i.e., the result in the case |α| = 0.

s k ∂ β1 (ϕ(x) -ϕ(x + σh)) • • • ∂ β k (ϕ(x) -ϕ(x + σh))ν(x, σh), 1 ≤ k ≤ |β|, β 1 + • • • + β k = β, which gives ∂ β ν(x, σh) = O λ ((sh) |β| )e O λ (
Next, for |α| ≥ 1, we write r(x)(∂ α ρ)(x + σh) = σh)µ α (x + σh), where we have set µ α := r∂ α ρ. By (3.1), this yields

∂ β (r(x)(∂ α ρ)(x + σh)) = β ′ +β ′′ =β β β ′ (∂ β ′ ν(x, σh)) (∂ β ′′ µ α (x + σh)) = O λ (s |α| (1 + (sh) |β| ))e O λ (sh) .
A.2. Proof of Proposition 3.9. We recall that rρ = 1. By Lemma 3.6 we have τ

+ ∂ α ρ(x) = ∂ α ρ(x) + Chρ(x) 1 0 r(x)∂ x ∂ α ρ(x + σh/2) dσ, which by Lemma 3.7 yields rτ + ∂ α ρ = r∂ α ρ + s |α| O λ (sh)e O λ (sh) = s |α| O λ,K (1)
. The proof is the same for rτ -∂ α ρ. For rDρ, rA∂ α ρ = r ∂ α ρ, rA 2 ∂ α ρ = r ∂ α ρ, and rDDρ we proceed similarly, exploiting the formula in Lemma 3.6 and then applying the result of Lemma 3.7, e.g.,

Dρ(x) = ∂ x ρ(x) + Ch 2 ρ(x) 1 -1 (1 -|σ|) 2 r(x)(∂ 3 x ρ)(x + σh/2) dσ = ∂ x ρ(x) + sρ(x)O λ,K ((sh) 2 ) = sr(x)O λ,K (1).
Noting that ADρ = Dρ(x) = (2h) -1 (ρ(x + h) -ρ(x -h)) we proceed as we did for Dr.

A.3. Proof of Lemma 3.10. By Lemma 3.6, we write

D(∂ β (r∂ α ρ))(x) = ∂ x ∂ β (r∂ α ρ)(x) + Ch 2 1 -1 (1 -|σ|) 2 ∂ 3 x ∂ β (r∂ α ρ)(x + σh/2) dσ.
By Lemma 3.7 we have

∂ 3 x ∂ β (r∂ α ρ) = O λ (s |α|
), which yields the first result in the case k = 1. For the case k = 2, we proceed similarly, making use of the last formula listed in Lemma 3.6. For the averaging cases, we make use of the second formula in Lemma 3.6.

As in the proof of Lemma 3.7 we set ν(x, σh) := r(x)ρ(x + σh). We have

D∂ β ′ ν(x, σh) = 1 2 1 -1 (∂ x ∂ β ′ ν)(x + σ ′ h/2, σh) dσ ′ = O λ,K (1), |β ′ | ≤ |β|, (A.1)
for sh ≤ K by Lemma 3.7. Next, with µ α = r∂ α ρ, we write r(x)∂ α ρ(x + σh) = ν(x, σh)µ α (x + σh), which gives D∂ β (r(x)∂ α ρ(x + σh)) as a linear combination of terms of the form

A(∂ β ′ ν(., σh)) D(∂ β ′′ µ α (. + σh)) + D(∂ β ′ ν(., σh)) A(∂ β ′′ µ α (. + σh)), β ′ + β ′′ = β,
by the continuous and discrete Leibniz rules (Lemma 3.2). By the first part and Lemma 3.7 we have D(∂ β ′′ µ α (x + σh)) = O λ,K (s |α| ). By Lemma 3.7, ∂ β ′ ν(x, σh) = O λ,K (1) and ∂ β ′′ µ α (x + σh) = O λ,K (s |α| ). The last result hence follows from (A.1). We proceed in a similar way for the case k = 2.

A.4. Proof of Lemma 3.11. For the first two results, we proceed as in Lemma 3.10 and use Corollary 3.8.

For the last results we use the continuous and discrete Leibniz rules (Lemma 3.2) and Lemma 3.10.

A.5. Proof of Proposition 3.12. Taylor formulae yield

Dρ(x) = ρ(x + h) -ρ(x -h) 2h = ∂ x ρ(x) + Ch 2 1 -1 (1 -|σ|) 2 ∂ 3 x ρ(x + σh) dσ, (A.2)
which in turn gives

D k A j ∂ α (r Dρ))(x) = D k A j ∂ α (r∂ x ρ)(x) + Ch 2 1 -1 (1 -|σ|) 2 D k A j ∂ α (r(x)∂ 3 x ρ(x + σh)) dσ,
and the first result follows by Lemma 3.10 (and Lemma 3.7 for the second equality). Next, from Lemma 3.6, we write

D k (rDDρ)(x) = D k (r∂ 2 x ρ)(x) + Ch 2 1 -1 (1 -|σ|) 3 D k (r(x)∂ 4 x ρ(x + σh)) dσ,
and the third result follows as above. For D k (rA 2 ρ) we use the formula for A 2 ρ given in Lemma 3.6 and proceed as above.

A.6. Proof of Proposition 3.13. From (A.2) we write

A j D k ∂ β r 2 (∂ α ρ) Dρ (x) =A j D k ∂ β r 2 (∂ α ρ)∂ x ρ (x) + Ch 2 1 -1 (1 -|σ|) 2 A j D k ∂ β r 2 (∂ α ρ)∂ 3 x ρ(. + σh) (x) dσ,
and we conclude with Lemma 3.11. For the next two results we use the formulae listed in Lemma 3.6 and proceed as above. From Lemma 3.6, equation (A.2), and by Lemma 3.11 we have

A j D k ∂ α (r 2 DρD 2 ρ) = A j D k ∂ α (r 2 (∂ x ρ)∂ 2 x ρ) + Ch 2 1 -1 (1 -|σ|) 2 A j D k ∂ α (r 2 ∂ 3 x ρ(. + σh)∂ 2 x ρ) dσ + C ′ h 2 1 -1 (1 -|σ|) 3 A j D k ∂ α (r 2 (∂ x ρ)∂ 4 x ρ(. + σh) dσ + CC ′ h 4 [-1,1] 2 (1 -|σ|) 2 (1 -|σ ′ |) 3 A j D k ∂ α (r 2 ∂ 3 x ρ(. + σh)∂ 4 x ρ(. + σ ′ h)) dσ dσ ′ = ∂ k x ∂ α (r 2 (∂ x ρ)∂ 2 x ρ) + s 3 O λ,K ((sh) 2 ).
The last result follows similarly.

Appendix B. Proofs of intermediate results in Section 4.

In this section, the calculus results of section 3 will be used and multiple averaging and difference operators will act on the weight functions and the coefficients ξ 1 and ξ 2 . In the discrete setting, this in fact requires additional discretization points outside the meshes. This can be done quite naturaly since the weight functions and the coefficients are sufficiently smooth in a neighborhood of Ω.

We shall also use the notation D 2 and A 2 introduced in Remark 3.14 and denote by D 2 f (resp. A 2 f ) their respective actions on C M (with extended boundary conditions).

B.1. Proof of Lemma 4.2. By Propositions 3.5 and 3.9, we have

|rDρ(τ + Dv)| 2 L 2 (Ω) = |τ -(rDρ)Dv| 2 L 2 (Ω) -h(rDρ) 2 0 |Dv| 2 1 2 ≤ C λ,K s 2 |Dv| 2 L 2 (Ω) (B.1)
Similarly we have

|rDρ(τ -Dv)| 2 L 2 (Ω) ≤ C λ,K s 2 |Dv| 2 L 2 (Ω) . (B.2)
We also observe that

|r(DDρ)Dv| 2 L 2 (Ω) ≤ Ω (rDDρ) 2 (Dv) 2 = Ω (rDDρ) 2 ) (Dv) 2 - h 2 (r(DDρ)) 2 0 |Dv| 2 1 2 - h 2 (r(DDρ)) 2 N +1 |Dv| 2 N + 1 2 ,
which, by Proposition 3.9, yields

|r(DDρ)Dv| 2 L 2 (Ω) ≤ C λ,K s 4 |Dv| 2 L 2 (Ω) . (B.3)
We also find

|rDρ Dv| 2 L 2 (Ω) ≤ C λ,K s 2 |Dv| 2 L 2 (Ω) . (B.4)
We note that

|ṽ| 2 L 2 (Ω) ≤ Ω |ṽ| 2 = |ṽ| 2 L 2 (Ω) - h 2 (|ṽ| 2 1 2 + |ṽ| 2 N + 1 2 ) ≤ Ω |v| 2 = |v| 2 L 2 (Ω) , (B.5)
by Proposition 3.5 and since v ∂M = 0. Since Dξ 2d is bounded by reg 0 , by Proposition 3.9 and (B.5), we thus have

|(r(Dξ 2d )Dρ + hO(1)r(DDρ))ṽ| 2 L 2 (Ω) ≤ C λ,K s 2 (1 + (sh) 2 )|v| 2 L 2 (Ω) . (
= Q 1 + Q 2 + Q 3 + Q 4 with Q 1 = 2 Re Q ξ 2 1 r(∂ t ρ) (∂ 2 t v)∂ t v * dt, Q 2 = 2 Re Q ξ 1 ξ 2 rDρ (∂ 2 t v)Dv * dt, Q 3 = 2 Re Q ξ 1 r 2 (∂ t ρ)ρ D(ξ 2d Dv)∂ t v * dt, Q 4 = 2 Re Q ξ 2 r 2 ρ Dρ D(ξ 2d Dv)Dv * dt.
Computation of Q 4 . By Lemmata 3.2 and 3.3, We have

Q 4 = - Q D(ξ 2 ξ 2 d r 2 ρ Dρ) |Dv| 2 dt + 2 Q ξ 2 (Dξ 2 d )r 2 ρ Dρ |Dv| 2 dt + T * 0 (ξ 2 ξ 2 d r 2 ρ Dρ) N +1 |Dv| 2 N + 1 2 -(ξ 2 ξ 2 d r 2 ρ Dρ) 0 |Dv| 2 1 2 dt,
by a discrete integration by parts (Proposition 3.5).

Lemma B.3. Provided sh ≤ K, we have

D(ξ 2 ξ 2d r 2 ρ Dρ) = -sλ 2 (ξ 2 2 ϕ(∂ x ψ) 2 ) d + sλϕ d O(1) + sO λ,K (sh), ξ 2 (Dξ 2 d )r 2 ρ Dρ = sλϕO(1) + sO λ,K ((sh) 2 ).
Since r ρ = 1 + O λ,K ((sh) 2 ) by Proposition 3.9, and since |Dv| 2 ≤ |Dv| 2 , it follows that we have

Q 4 ≥sλ 2 Q (ξ 2 2 ϕ(∂ x ψ) 2 ) d |Dv| 2 dt + Q ν (4) 11 |Dv| 2 dt (B.10) + T * 0 (1 + O λ,K ((sh) 2 )) (ξ 2 ξ 2d rDρ) N +1 |Dv| 2 N + 1 2 -(ξ 2 ξ 2d rDρ) 0 |Dv| 2 1 2 dt,
where ν

D(ξ 2 ξ 2d r 2 ρ Dρ) = D(ξ 2 ξ 2d ) r 2 ρ Dρ + ξ 2 ξ 2d D(r 2 ρ Dρ) = O(1)(r∂ x ρ + sO λ,K ((sh) 2 )) d + (ξ 2 2 d + hO(1))(∂ x (r∂ x ρ) + sO λ,K ((sh) 2 )) d ,
and the first result follows from Lemma 3.7. The second result follows from Lemma 3.7 and Proposition 3.9.

Gathering

I b 11 = -sλ 2 h 2 2 Q (ϕξ 2 1 (∂ t ψ) 2 ) d |D∂ t v| 2 dt + 2sλ 2 T * 0 Ω ξ 2 1 ϕ(∂ t ψ) 2 |∂ t v| 2 + Ω (ξ 2 2 ϕ(∂ x ψ) 2 ) d |Dv| 2 + 2 Re Ω (ξ 1 ξ 2 ϕ(∂ t ψ)(∂ x ψ)) d ∂ t v Dv * dt.
Note that the first term in I b 11 comes from the fact that we added exactly the opposite term in W 11 in order to ensure that W ≥ 0 (see Lemma 4.7 and its proof). We conclude the proof of Lemma 4.3 with the following lemma.

Lemma B.4. Provided sh ≤ K, we have

I b 11 ≥ Q O λ (sh) |∂ t v| 2 dt. Proof. We write I b 11 = 2sλ 2 Q ϕ d ξ 1 (∂ t ψ) d ∂ t v + (ξ 2 ∂ x ψ) d Dv 2 dt + 2sλ 2 T * 0 L(t) dt ≥ 2sλ 2 T * 0 L(t) dt, with
L(t) = Ω ϕξ 2 1 (∂ t ψ) 2 |∂ t v| 2 - Ω (ϕξ 2 1 (∂ t ψ) 2 ) d | ∂ t v| 2 - h 2 4 Ω (ξ 2 1 ϕ(∂ t ψ) 2 ) d |D∂ t v| 2 dt = Ω ξ 2 1 ϕ(∂ t ψ) 2 |∂ t v| 2 - Ω (ξ 2 1 ϕ(∂ t ψ) 2 ) d |∂ t v| 2 = Ω ξ 2 1 ϕ(∂ t ψ) 2 -(ξ 2 1 ϕ(∂ t ψ) 2 ) d |∂ t v| 2 ,
by Lemma 3.3 and Proposition 3.5 as v ∂M = 0. We conclude since ξ

2 1 ϕ(∂ t ψ) 2 - (ξ 2 1 ϕ(∂ t ψ) 2 ) d = hO λ (
1) by Lemma 3.6. B.3. Proof of Lemma 4.4. From the forms of A 1 v and B 2 v we have

I 12 = Q 1 + Q 2 with Q 1 = -2s Re Q ξ 1 (∆ ξ ϕ) v * ∂ 2 t v dt and Q 2 = -2s Re Q r ρ (∆ ξ ϕ) v * D(ξ 2d Dv) dt.
With an integration by parts w.r.t.

t we obtain Q 1 = 2s Q ξ 1 (∆ ξ ϕ) |∂ t v| 2 dt + R 1 , where R 1 = 2s Re Q ξ 1 ∂ t (∆ ξ ϕ) v * ∂ t v dt -2s Re Ω ξ 1 (∆ ξ ϕ)(T * )v * (T * )∂ t v(T * ) = s Q O λ (1) |v| 2 dt + s Ω O λ (1) |v| 2 (T * ) + Re Ω O λ (1)v * (T * )∂ t v(T * ), using 2 Re v * ∂ t v = ∂ t |v| 2
, and an additional integration by parts w.r.

t. t, since ξ 1 ∂ t (∆ ξ ϕ) = O λ (1), ξ 1 ∂ 2 t (∆ ξ ϕ) = O λ (1) and ξ 1 ∆ ξ ϕ(T * ) = O λ (1)
. For concision we now set q = r ρ (∆ ξ ϕ). For the term Q 2 , a discrete integration by parts gives

Q 2 = 2s Q qξ 2 d |Dv| 2 dt + 2s Re Q (Dq)ξ 2 d ṽ * Dv dt.
Since by Proposition 3.9 we have q = ∆ ξ ϕ + O λ,K ((sh) 2 ), then

q = (∆ ξ ϕ) d + O λ,K (h + (sh) 2 ) as ∆ ξ ϕ = (∆ ξ ϕ) d + O λ (h) since reg(ξ) ≤ reg 0 . We note also that Dq = D(r ρ) ∆ ξ ϕ + (r ρ)D(∆ ξ ϕ) = O λ,K (1)
, by Propositions 3.9 and 3.12. We thus obtain

Q 2 = 2s Q (ξ 2 ∆ ξ ϕ) d |Dv| 2 dt + R 2 , with R 2 = s Q O λ,K (h + (sh) 2 )|Dv| 2 dt + s Re Q O λ,K (1)ṽ * Dv dt.
Observing that

∆ ξ ϕ = λ 2 |∇ ξ ψ| 2 ϕ + λϕO(1), (B.11)
by Lemma 3.7, we obtain the desired result.

B.4. Proof of Lemma 4.5. From the forms of A 2 v and B 1 v we have

I 21 = Q 1 + Q 2 + Q 3 + Q 4 with Q 1 = 2 Re Q ξ 2 1 r 2 (∂ 2 t ρ)(∂ t ρ) v∂ t v * dt, Q 2 = 2 Re Q ξ 1 r 2 (∂ 2 t ρ)Dρ ξ 2 vDv * dt, Q 3 = 2 Re Q ξ 1 ξ 2 r 2 (DDρ) (∂ t ρ) ṽ∂ t v * dt, Q 4 = 2 Re Q ξ 2 2 r 2 (DDρ) Dρ ṽ Dv * dt.
Computation of Q 1 . We set q 1 = ξ 2 1 r 2 (∂ 2 t ρ)(∂ t ρ). With an integration by parts, we have (1) 21 = s 2 O λ (1). Computation of Q 2 . We set q 2 = ξ 1 ξ 2 r 2 (∂ 2 t ρ)Dρ. We have Proof of Lemma B.7. From Proposition 3.13 we have q 3 = ξ 1 ξ 2 r 2 (∂ 2

Q 1 = Q q 1 ∂ t |v| 2 dt = - Q (∂ t q 1 )|v| 2 dt + Ω q 1 (T * )|v| 2 (T * ) = 3s 3 λ 4 Q ξ 2 1 ϕ 3 (∂ t ψ)
Q 2 = 2 Re Q q 2 v Dv * dt = Q q2 D|v| 2 dt + h 2 2 Q Dq 2 |Dv| 2 dt = - Q Dq 2 |v| 2 dt + h 2 2 Q Dq 2 |Dv| 2 dt,
x ρ)∂ t ρ + s 3 O λ,K ((sh) 2 ), ∂ t q 3 = ξ 1 ξ 2 ∂ t (r 2 (∂ 2 x ρ)∂ t ρ) + s 3 O λ,K ((sh) 2 ). Iterating the averaging procedure we obtain similar estimates for q3 and ∂ t q3 (sampled on the primal mesh) and we conclude with Corollary 3.8.

Computation of Q 4 . We set q 4 = ξ 2 2 r 2 (DDρ) Dρ. Observing that Dv * = Dṽ * , we have

Q 4 = Q q 4 D|ṽ| 2 dt = - Q (Dq 4 )|ṽ| 2 dt Q (1) 4 + T * 0 ((q 4 ) N +1 |ṽ N + 1 2 | 2 -(q 4 ) 0 |ṽ 1 2 | 2 ) dt Q (2) 4
, by Lemma 3.2 and Proposition 3.5. We note that ṽ 1 2 = h 2 (Dv) 1 2 and ṽN+ 1 2 = -h 2 (Dv) N + 1 2 . By Proposition 3.9 we have q 4 = s 2 O λ,K (1)rDρ. It follows that We have ∆ ξ ϕ = O λ (1) and thus from Lemma B.9 we have q = s 2 O λ,K (1). The same estimate naturally holds for q. With the following lemma we conclude the proof.

Lemma B.10. Provided sh ≤ K, we have h 2 DDq = s(sh)O λ,K (1).

Proof. We set p = ξ 2 (∆ ξ ϕ) and observe that p ∞ = O λ (1), Dp ∞ = O λ (1), and hDDp ∞ = O λ (1). We thus have h 2 DDq = h 2 (DDp) rDDρ + 2h 2 Dp D(rDDρ) + h 2 p (DD(rDDρ)) = (h + h 2 )s 2 O λ,K (1), by Propositions 3.9 and 3.12. B.6. Proof of Lemma 4.7. We have W = Q p|D∂ t v| 2 dt with p = 1 2 h 2 sλ 2 (ξ 1 ϕ|∇ ξ ψ| 2 ) d + h 2 sλϕ d O(1) + hO(sh) + hO λ,K ((sh) 2 ).

Since |∇ ξ ψ| ≥ C > 0, we see that for λ sufficiently large, the first term above dominates the second and third terms for any h, s, so that we obtain p ≥ h 2 s(C -C ′ sh) and thus W ≥ 0 for sh sufficiently small. Next, since reg(ξ) ≤ reg 0 , we see that

Y = T * 0 q N +1 |Dv| 2 N + 1 2 -q 0 |Dv| 2 1 2
dt, with q = (1 + O λ,K ((sh) 2 ))ξ 2 ξ 2d rDρ.

By (4.1) we have Y ≥ 0 for sh sufficiently small. The proof of the second result is similar, yet simpler. We have r∂ t u = ∂ t v + r(∂ t ρ)u, which implies

|r∂ t u| 2 L 2 (Ω) ≤ C λ,K (|∂ t v| 2 L 2 (Ω) + s 2 |v| 2 L 2 (Ω) ).
The last result follows the same.

Appendix C. On the construction of the Carleman weight function. We describe here the succession of arguments used in the construction of the Carleman weight function ψ. Its regularity class is C k ( Q) for a certain k ∈ N prescribed in advance. Note however that the set Ω itself needs to be of class C k .

We first start with a function φ 1 (t) ∈ C ∞ ([0, T * ]) such that ∂ t φ 1 (0) ≥ C > 0, ∂ t φ 1 (T * ) ≤ -C < 0, and φ 1 (0) = φ 1 (T * ) = 0, and φ 1 (t) > 0 if t ∈ (0, T * ). We also choose φ 2 (x) ∈ C k ( Ω) such that φ 2 ≥ C > 0 and ∂ nx φ 2 ≤ -C ′ < 0 in V ∂Ω , which can be achieved by choosing the neighborhood V ∂Ω sufficiently small. We next set φ(t, x) = φ 1 (t)φ 2 (x). This function satisfies the desired properties listed in

Fig. 1. 2 .

 2 Fig. 1.2. Some non-uniform meshes for N = 10, 20, 40 (left) and the corresponding map ϑ (right).

  Lemma 4.3 (Estimate of I 11 ). For sh ≤ K we have I 11 ≥ I a 11 + W 11 + Y 11 -X 11 -J 11 , with

  sλϕ d O(1) + sO λ,K (sh). Proof of Lemma B.3. By Lemma 3.3, and Proposition 3.13 we write

  of the different terms. The results obtained in (B.7)-(B.10) yield I 11 ≥ I a 11 + I b 11 + W 11 + Y 11 -X11 -J 11 , where I a 11 , W 11 , Y 11 , and J 11 are as given in the statement of Lemma 4.3, X11 has the same form as X 11 in the statement of Lemma 4.3, and

  4 |v| 2 dt -(sλ) 3 Ω ξ 2 1 (ϕ∂ t ψ) 3 (T * )|v| 2 (T * ) sλϕ) 3 O(1) + s 2 O λ(1) and η

by

  Proposition 3.5 and Lemmata 3.2 and 3.3, using v ∂M = 0. Lemma B.5. Provided sh ≤ K, we have Dq 2 = s 3 O λ,K (1) andD q2 = Dq 2 = -3(sϕ) 3 λ 4 ξ 1 (∂ t ψ) 2 ξ 2 (∂ x ψ) 2 + (sλϕ) 3 O(1) + s 2 O λ (1) + s 3 O λ,K ((sh) 2 ).It follows thatQ 2 = 3s 3 λ 4 Q ϕ 3 ξ 1 (∂ t ψ) 2 ξ 2 (∂ x ψ) 2 |v| 2 dt + sλϕ) 3 O(1) + s 2 O λ (1) + s 3 O λ,K ((sh) 2 ) and ν (2) 21 = sO λ,K ((sh) 2 ).Proof of Lemma B.5. We writeDq 2 = D(ξ 1 ξ 2 ) r 2 (∂ 2 t ρ)Dρ + ξ 1 ξ 2 D(r 2 (∂ 2 t ρ)Dρ) = O(1)((r 2 (∂ 2 t ρ)∂ x ρ) d + s 3 O λ,K ((sh) 2 )) + ((ξ 1 ξ 2 ) d + hO(1))((∂ x (r 2 (∂ 2 t ρ)∂ x ρ)) d + s 3 O λ,K ((sh) 2 )),by Lemmata 3.2 and 3.13. Sincer 2 (∂ 2 t ρ)∂ x ρ = -(sλϕ) 3 (∂ t ψ) 2 (∂ x ψ) + s 2 O λ (1), ∂ x (r 2 (∂ 2 t ρ)∂ x ρ) = -3(sϕ) 3 λ 4 (∂ t ψ) 2 (∂ x ψ) 2 + s 2 O λ (1) + (sλϕ) 3 O(1) sλϕ) 3 O(1) + s 2 O λ (1) + s 3 O λ,K ((sh) 2 ), η(3)21 = s 3 O λ,K ((sh) 2 ) + s 2 O λ,K(1), ν(3) 21 = sO λ,K ((sh) 2 ), and δ(3) 21 = sO λ,K ((sh) 2 ), γ 21 = hO(sh).Proof of Lemma B.6. We haveDq 3 = D(ξ 1 ξ 2 ) r 2 (DDρ)(∂ t ρ) + ξ 1 ξ 2 D(r 2 (DDρ)(∂ t ρ)) = s 3 O λ,K(1), by Proposition 3.13 and Corollary 3.8, since D(ξ 1 ξ 2 ) is bounded.

QO 2 +Q

 2 λ,K (1)(rDρ) 0 |Dv| 2 1 O λ,K (1)(rDρ) N +1 |Dv| 2 Dq 4 |v| 2 dt + h 2 4 Q (Dq 4 )|Dv| 2 dt.Lemma B.8. Provided sh ≤ K, we have Dq 4 = s 3 O λ,K (1) andDq 4 = -s 3 λ 4 ϕ 3 ξ 2 2 (∂ x ψ) 4 + (sλϕ) 3 O(1) + s 2 O λ,K(1) + s 3 O λ,K ((sh) 2 ).

B. 7 .

 7 Proof of Lemma 4.9. By Lemma 3.2 we have r d Du = ṽr d Dρ + r d ρDv, which by Proposition 3.9, yields|r d Du| 2 L 2 (Ω) ≤ C λ,K |ṽr d Dρ| 2 L 2 (Ω) + |Dv| 2 L 2 (Ω) ,We observe that|ṽr r Dρ| 2 L 2 (Ω) = Ω |ṽ| 2 (r d Dρ) 2 ≤ Ω |v| 2 (r d Dρ) 2 = Ω |v| 2 (r d Dρ) 2 = s 2 Ω O λ,K (1)|v| 2 ,since v ∂M = 0 and by Proposition 3.9, which yields the first result.

  B.6) Similarly, since Dξ 2d and ∆ ξ ϕ are bounded, estimates (B.1)-(B.4) and (B.6) yield the result, after an integration w.r.t. t. B.2. Proof of Lemma 4.3. From the forms of A 1 v and B 1 v we have I 11

  By Lemma 3.3, we now writeq3 = ξ 1 ξ 2 r 2 (∂ 2 x ρ)∂ t ρ + h 2 4 D(ξ 1 ξ 2 ) D r 2 (∂ 2 x ρ)∂ t ρ + s 3 O λ,K ((sh) 2 ) = ((ξξ 2 ) d + hO(1))((r 2 (∂ 2 x ρ)∂ t ρ) d + s 3 h 2 O λ,K (1)) + h 2 O(1)O λ,K (s 3 ) + s 3 O λ,K ((sh) 2 ) = (ξ 1 ξ 2 r 2 (∂ 2 x ρ)∂ t ρ) d + s 3 O λ,K ((sh) 2 ) + s 2 O λ,K (1), (B.18)by Lemma 3.11 and Corollary 3.8. Similarly, we find∂ t q3 = (ξ 1 ξ 2 ∂ t (r 2 (∂ 2 x ρ)∂ t ρ)) d + s 3 O λ,K ((sh) 2 ) + s 2 O λ,K(1). (B.19)

Acknowledgments: The authors thank Assia Benabdallah for discussions on the subject of the present article.

The three authors were partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01.

Computation of Q 1 . With 2 Re(∂ 2 t v)∂ t v * = ∂ t |∂ t v| 2 , an integration by parts yields

11 |∂ t v| 2 dt, (B.7) with β

(1) 11 = sλϕO(1), by Lemma 3.7.

Computation of Q 2 . Since v| t=0 = 0, an integration by parts yields

The last term, Q

(3) 2 , vanishes since ψ| t=T * = Cst. Since v ∂M = 0, by Proposition 3.5 and Lemma 3.3 we have

and, after an integration by parts w.r.t. t, we have

Since 2 Re ∂ t v ∂ t Dv * = D|∂ t v| 2 by Lemma 3.2, a discrete integration by parts (Proposition 3.5) yields

If follows that

11 = sλϕO(1) + sO λ,K (sh), α

(2,1) 11

Proof of Lemma B.1. By Lemma 3.2 and Proposition 3.12, we have

which yields the second result by Lemma 3.7. We note that D(ξ

and we proceed as above. The other results follow similarly.

Computation of Q 3 . With a discrete integration by parts (Proposition 3.5) and Lemma 3.2, followed by an integration by parts w.r.t. t, we have

The proof follows from Lemma 3.7, Propositions 3.9 and 3.13. We thus have

where α

(3)

11 = sλϕ d O(1) + sO λ,K ((sh) 2 ), and δ

(3) 11 = sO λ,K ((sh) 2 ). by Corollary 3.8, we have

and the first result follows. We note that Dq 2 = D 2 q 2 (see Remark 3.14). We have

Using Remark 3.14, proceeding as above we obtain the second result.

Computation of Q 3 . We set q 3 = ξ 1 ξ 2 r 2 (DDρ) (∂ t ρ). By Proposition 3.5 and Lemma 3.3, we then have

.

By Young inequalities, we have the following estimate

since |ṽ| 2 ≤ |v| 2 and then exploiting Proposition 3.5 and v ∂M = 0. Next, with an integration by parts, we see that

by Lemma 3.3 and Proposition 3.5, using v ∂M = 0.

Lemma B.7. We have

We have thus obtained,

We have thus obtained

where

21 = sO λ,K ((sh) 2 ). Proof of Lemma B.8. By Proposition 3.13 we have

). Arguing as we did in the proof of Lemma B.5, we find that a similar estimate (sampled on the primal mesh) holds for Dq 4 . conclude with Corollary 3.8.

Collecting the estimates of Q j , j = 1, 2, 3, 4, we have obtained in (B.12), (B.13), (B.17), and (B.20), we conclude the proof of Lemma 4.5. B.5. Proof of Lemma 4.6. From the forms of A 2 v and B 2 v we have

By Lemma 3.4 we have ṽ = v + h 2 DDv/4 which gives

We first work on the expressions Q 1 and Q ′ 2 . Lemma B.9. Provided sh ≤ K we have ξ 1 r∂ 2

). The proof follows by Proposition 3.9 and Lemma 3.7.

Using (B.11), we have

We now turn to the term Q ′′ 2 . For concision we set q := rξ 2 (DDρ)(∆ ξ ϕ). Since v ∂M = 0, discrete integrations by parts give

Assumption 2.1 on the boundaries (0, T * )×∂Ω (and in its neighborhood (0, T * )×V ∂Ω ), {0} × (Ω \ ω) and {T * } × Ω. We choose y 0 in {0} × ω. We enlarge Q in a small neighborhood of y 0 which leaves ∂Q unchanged outside of {0} × ω. We call Q this extension of Q and we extend the function φ to Q in a C k manner.

The function φ exhibits only one critical point points in Q. It can be pulled back to the interior of Q \ Q by composing φ with a diffeomorphism (see [START_REF] Fursikov | Controllability of evolution equations[END_REF] for the construction of such a diffeomorphism). The resulting function is the weight function ψ and it satisfies all the properties listed in Assumption 2.1.