Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

ScienceDirect

Ann. I. H. Poincaré – AN 31 (2014) 1035–1078

www.elsevier.com/locate/anihpc

Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete

parabolic equations

Franck Boyer

, Jérôme Le Rousseau

aAix-Marseille Université, Laboratoire d’Analyse Topologie Probabilités (LATP), CNRS UMR 7353, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France

bUniversité d’Orléans, Laboratoire de Mathématiques – Analyse, Probabilités, Modélisation – Orléans (MAPMO), CNRS UMR 7349, Fédération Denis Poisson, CNRS FR 2964, B.P. 6759, 45067 Orléans cedex 2, France

Received 22 August 2012; received in revised form 13 May 2013; accepted 29 July 2013 Available online 12 September 2013

Abstract

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:35K10; 35K58; 65M06; 93B05; 93B07

Keywords:Parabolic operator; Semi-discrete Carleman estimates; Observability; Null controllability; Semi-linear equations

1. Introduction and notation

Letd 1, L1, . . . , Ld be positive real numbers, andΩ =

1id(0, Li). We setx=(x1, . . . , xd)∈Ω. With ωΩwe consider the following parabolic problem in(0, T )×Ω, withT >0,

∂_ty− ∇x·(Γ∇xy)=1ωv in(0, T )×Ω, y_|∂Ω=0, and y_|t=0=y₀, (1.1) where the diagonal diffusion tensorΓ (x)=Diag(γ1(x), . . . , γd(x))withγ_i(x) >0 satisfies

reg(Γ )^def= ess sup

x∈Ω i=1,...,d

γ_i(x)+ 1

γi(x)+∇xγ_i(x)

<+∞. (1.2)

* Corresponding author.

E-mail addresses:fboyer@latp.univ-mrs.fr(F. Boyer),jlr@univ-orleans.fr(J. Le Rousseau).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.07.011

The distributed null controllability problem consists in findingv∈L²((0, T )×Ω)such thaty(T )=0. This problem was solved in the 90’s by G. Lebeau and L. Robbiano[18]and A. Fursikov and O.Yu. Imanuvilov[12]. By a duality argument the null controllability result for(1.1)is equivalent to having the following observability inequality

q(0)²

L²(Ω)C_obs² q²_L2((0,T )×ω), (1.3)

for anyqsolution to(∂t+ ∇x·(Γ∇xy))q=0 andq_|∂Ω=0.

Let us consider the elliptic operator onΩgiven by A= −∇x·(Γ∇x)= −

1id

∂_xi(γ_i∂_xi)

with homogeneous Dirichlet boundary conditions on∂Ω. We shall introduce a finite-difference approximation of the operatorA. For a meshMthat we shall describe below, associated with a discretization steph, the discrete operator will be denoted byA^M. It will act on a finite dimensional spaceR^M, of dimension|M|, and will be selfadjoint for the standard inner product inR^M. Our main result is the derivation of a Carleman estimate for the operators∂_t±A^M, i.e., a weighted energy estimate with a localized observation term, which is uniform with respect to the discretization parameterh. The weight function is of exponential type.

There is a vast literature on Carleman estimates going back from the original work of T. Carleman[7]and the sem- inal work of L. Hörmander[13](see also[14, Chapter 8]and[15, Chapter 28]). These estimates were first introduced for the purpose of proving and quantifying unique continuation (see[22]for manifold references). In more recent years, the field of applications of Carleman estimates has gone beyond the original domain they had been introduced for. They are also used in the study of inverse problems and control theory for PDEs. For an introduction to Carle- man estimates and their applications to controllability of parabolic equations, as we shall use them here, we refer for instance to[9]and[17]. For applications of these estimates to inverse problems we refer to[20].

From the semi-discrete Carleman estimates we obtain, we deduce an observation inequality for the operator

∂_t−A^M+a, whereais a bounded potential function:

qh(0)²

L²(Ω)C_obs²

qh²_L2((0,T )×ω)+e^{−C1/ h}qh(T )²

L²(Ω)

, (1.4)

forq_h(semi-discrete) solution to(∂_t−A^M+a)q_h=0. Special care is placed in the estimation of the observability constantC_obs, in particular in its dependency upona∞. It should be emphasized at this point thatC_obsdepends on the geometrical setting,i.e., on Ω,ω, the operator diffusion matrix Γ, and on the potential functiona. YetCobs is not dependent on the discretization parameterh. The observability inequality(1.4)is weak as compared to that one can obtain in the continuous case; compare with(1.3). Here, there is an additional term in the right hand-side of the inequality. In fact, because of the presence of this term we shall speak of a relaxedobservability inequality. Earlier work[2,3]showed that this term cannot be removed and is connected to an obstruction to the null controllability of the semi-discrete problem in space dimension greater than two, as pointed out by a counter-example due to O. Kavian (see e.g. the review article [21]). Still, by duality, the relaxed observability estimate we derive is equivalent to a controllability result. Because of the aforementioned counter-example we do not exactly achieve null controllability at the discrete level, yet we reach a small target, which size goes to zero exponentially as the mesh sizeh→0. We speak ofh-null controllability. This notion should not be confused with approximate controllability: the size of the neighborhood of zero reached by the solution of the parabolic equation at the final time t=T is not fixed; it is a function of the discretization step. The study of relaxed observability estimates for dicretized parabolic equations was initiated by[19]. We refer to[6]for a review.

The dependency of the observability constant with respect to the norma_∞allows one to tackle controllability questions for parabolic equations with semi-linear terms, in particular for some cases of super-linear terms. In the continuous case, this was achieved in[1,11]. To our knowledge, in the discrete case this question was only discussed in[10]. Here, we shall consider such questions in the case of semi-discretized equations and we shall be interested in provingh-null controllability results as well as associated relaxed observability inequalities. Some of the results we give are uniform with respect to the discretization parameter:h-null controllability is achieved with a (semi-discrete) control function whoseL²-norm is bounded uniformly inh.

Fig. 1. Notation for the boundaries in the 2D case.

1.1. Discrete settings and notation

Precise statements of the results we obtain require the introduction of the settings we shall work with.

For 1id,i∈N, we setΩ_i=

1jd j=i

(0, Lj). ForT >0 we introduce

Q=(0, T )×Ω, Qi=(0, T )×Ωi, 1id.

We also set boundaries as (seeFig. 1)

∂_i⁻Ω=

1j <i

[0, Lj] × {0} ×

i<jd

[0, Lj], ∂_i⁺Ω=

1j <i

[0, Lj] × {L_i} ×

i<jd

[0, Lj],

∂_iΩ=∂_i⁺Ω∪∂_i⁻Ω, ∂Ω=

1id

∂_iΩ.

We shall use uniform meshes,i.e., meshes with constant discretization steps in each direction. The introduction of more general meshes is possible. We refer to[3] for some families of regular non-uniform meshes that one can consider.

The notation we introduce will allow us to use a formalism as close as possible to the continuous case, in particular for norms and integrations.Then most of the computations we carry out can be read in a very intuitive manner, which will ease the reading of the article. Most of the discrete formalism will then be hidden in the subsequent sections. The notation below is however necessary for a complete and precise reading of the proofs.

We shall use the notationJa, bK= [a, b] ∩N.

1.1.1. Primal mesh

Fori∈J1, dKandN_i∈N^∗, we seth_i=L_i/(N_i+1)andx_i,j=j h_i,j∈J0, Ni+1K, which gives 0=x_i,0< x_i,1<· · ·< x_i,Ni< x_i,Ni+1=L_i.

We introduce the following set of indices, N:=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK .

Fork=(k1, . . . , kd)∈Nwe setxk=(x1,k₁, . . . , xd,k_d)∈Ω. We refer to this discretization as to the primal mesh M:= {xk; k∈N}, with|M| :=

i∈J1,dK

N_i.

We seth=maxi∈J1,dKhi and we impose the following condition on the meshes that we consider: there existsC >0 such that

C⁻¹hh_iCh, i∈J1, dK. (1.5)

Fig. 2. Primal and dual meshes in the 2D case.

1.1.2. Boundary of the primal mesh

To introduce boundary conditions in theith direction and related trace operators (see Section1.1.6) we set∂_iN=

∂_i⁻N∪∂_i⁺Nwith

∂_i⁻N=

k=(k₁, . . . , k_d); k_j∈J1, NjK, j∈J1, dK, j=i, k_i=0 ,

∂_i⁺N=

k=(k₁, . . . , k_d); k_j∈J1, NjK, j∈J1, dK, j=i, k_i=N_i+1 , and

∂N=

i∈J1,dK

∂_iN, ∂M= {xk; k∈∂N}, ∂_i^±M=

xk; k∈∂_i^±N .

1.1.3. Dual meshes

We will need to operate discrete derivatives on functions defined on the primal mesh (see Section1.1.5). It is easily seen that these derivatives are naturally associated to another set of staggered meshes, called dual meshes. In fact there will be two kinds of such meshes: the ones associated to a first-order discrete derivation and the ones associated to a second-order discrete derivation. Let us define precisely these new meshes (seeFig. 2).

Fori∈J1, dKandN_i∈N^∗, we setx_i,j =j h_i forj∈J0, NiK+¹₂, which gives 0=x_i,0< x_i,1

2

< x_i,1< x_i,1+1 2

<· · ·< x_i,Ni< x_i,N

i+¹₂ < x_i,Ni+1=L_i. Fori∈J1, dK, we introduce a second type of sets of indices

Nⁱ:=

k=(k₁, . . . , k_d); k_j∈J1, NjK, j∈J1, dK, j=i,andk_i∈J0, NiK+1 2

. Forj∈J1, dK,j =i, we also set∂_jNⁱ=∂_j⁻Nⁱ∪∂_j⁺Nⁱ with

∂_j⁻Nⁱ=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, k_i∈J0, NiK+1

2, andk_j=0

,

∂_j⁺Nⁱ=

k=(k1, . . . , kd); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, ki∈J0, NiK+1

2, andkj=Nj+1

, and∂Nⁱ=

j∈J1,dK j=i

∂_jNⁱ. We moreover introduce∂_iNⁱ=∂_i⁻Nⁱ∪∂_i⁺Nⁱwith

∂_i⁻Nⁱ=

k=(k1, . . . , kd); kj∈J1, NjK, j∈J1, dK, j=i, ki=1 2

,

∂_i⁺Nⁱ=

k=(k1, . . . , kd); kj∈J1, NjK, j∈J1, dK, j=i, ki=Ni+1 2

.

Remark that∂_iNⁱ⊂Nⁱ whereas∂_jNⁱ⊂Nⁱ forj=i.

Fori, j∈J1, dK,i=j, we introduce a third type of sets of indices N^ij:=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j andk_i∈J0, NiK+1

2, k_j∈J0, NjK+1 2

. Forl∈J1, dK,l=i,l=j, we also set∂_lN^ij=∂_l⁻N^ij∪∂_l⁺N^ij with

∂_l⁻N^ij=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, i=l, k_i∈J0, NiK+1

2, k_j∈J0, NjK+1

2, andk_l=0

,

∂_l⁺N^ij=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, i=l, k_i∈J0, NiK+1

2, k_j∈J0, NjK+1

2, andk_l=N_l+1

, and∂N^ij=

l∈J1,dK l=i,l=j

∂_lN^ij. Moreover we set∂_iN^ij=∂_i⁻N^ij∪∂_i⁺N^ij with

∂_i⁻N^ij=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, k_i=1

2, k_j∈J0, NjK+1 2

,

∂_i⁺N^ij=

k=(k₁, . . . , k_d); k_i∈J1, NiK, i∈J1, dK, i=i, i=j, k_i=N_i+1

2, k_j∈J0, NjK+1 2

. Fork=(k1, . . . , kd)∈Nⁱ or∂Nⁱ(resp.N^ijor∂N^ij) we also setxk=(x1,k₁, . . . , xd,k_d), which gives the following dual meshes

Mⁱ:=

xk; k∈Nⁱ

, ∂Mⁱ:=

xk; k∈∂Nⁱ

, ∂_j^±Mⁱ:=

xk; k∈∂_j^±Nⁱ , resp.M^ij:=

xk; k∈N^ij

, ∂M^ij:=

xk; k∈∂N^ij

, ∂_l^±M^ij:=

xk; k∈∂_l^±N^ij . 1.1.4. Discrete functions

We denote byR^M(resp.R^MiorR^Mij) the sets of discrete functions defined onM(resp.MⁱorM^ij) respectively.

Ifu∈R^M(resp.R^Mi orR^Mij), we denote byu_k its value corresponding toxkfork∈N(resp.k∈Nⁱ ork∈N^ij).

Foru∈R^Mwe define u^M=

k∈N

1bku_k∈L^∞(Ω), withb_k=

i∈J1,dK

[x_i,k

i−¹2, x_i,k

i+¹2],k∈N. (1.6)

Since no confusion is possible, by abuse of notation we shall often writeuin place ofu^M. Foru∈R^Mwe define

Ω

u:=

Ω

u^M(x) dx=

k∈N

|b_k|u_k, where|b_k| =

i∈J1,dK

h_i. For someu∈R^M, we shall need to associate boundary values

u^∂M= {uk; k∈∂N},

i.e., the values ofuat the pointxk∈∂M. The set of such extended discrete functions is denoted byR^M∪∂M. Homoge- neous Dirichlet boundary conditions then consist in the choiceu_k=0 fork∈∂N, in shortu^∂M=0 or evenu_|∂Ω=0 by abuse of notation (see also Section1.1.6below).

Similarly, foru∈R^Mi(resp.R^Mij) we shall associate the following boundary values u^∂Mi=

u_k; k∈∂Nⁱ resp.u^∂Mij =

u_k; k∈∂N^ij .

The set of such extended discrete functions is denoted byR^Mi∪∂Mi (resp.R^Mij∪∂Mij).

Foru∈R^Mi (resp.R^Mij) we define u^Mi=

k∈Nⁱ

1_bⁱ

k

uk∈L^∞(Ω) withbⁱ_k=

l∈J1,dK

[x_l,k

l−¹₂, x_l,k

l+¹₂],k∈Nⁱ,

resp.u^Mij =

k∈N^ij

1_b^ij

k

uk∈L^∞(Ω) withb^ij_k =

l∈J1,dK

[x_l,k

l−¹₂, x_l,k

l+¹₂], k∈N^ij

.

As above, foru∈R^Mi (resp.R^Mij), we define

Ω

u:=

Ω

u^Mi(x) dx=

k∈Nⁱ

bⁱ_kuk, wherebⁱ_k=

l∈J1,dK

hl,

resp.

Ω

u:=

Ω

u^Mij(x) dx=

k∈N^ij

b^ij_kuk, whereb^ij_k=

l∈J1,dK

hl

.

Remark 1.1. Above, the definitions of b_k,bⁱ_k, andb^ij_k look similar. They are however different as each time the multi-indexk=(k₁, . . . , k_d)is chosen in a different set:N,Nⁱ andN^ij respectively.

In particular we define the followingL²-inner product onR^M(resp.R^Mi orR^Mij) u, vL²(Ω)=

Ω

uv=

Ω

u^M(x)v^M(x) dx,

resp.u, v_L²_(Ω)=

Ω

uv=

Ω

u^Mi(x)v^Mi(x) dx,

oru, vL²(Ω)=

Ω

uv=

Ω

u^Mij(x)v^Mij(x) dx

. (1.7)

The associated norms will be denoted by|u|L²(Ω).

For semi-discrete functionu(t )inR^M(resp.R^Mi orR^Mij),t∈(0, T ), we shall write

Qu dt=_T

0

Ωu(t ) dt, and we define the followingL²-norm

u(t )²

L²(Q)= T

0 Ω

u(t )2

dt.

Endowing the space of semi-discrete functionsL²(0, T;R^M)(resp.L²(0, T;R^Mi)orL²(0, T;R^Mij)) with this norm yields a Hilbert space.

Definition of a space of semi-discrete functions likeL^∞(0, T ,R^M)(resp.L^∞(0, T;R^Mi)orL^∞(0, T;R^Mij)) can be done similarly with the norm

u(t )

L^∞(Q)=ess sup

t∈(0,T )

sup

k∈N

u_k(t ). We shall also use mixed norms of the form

u(t )

L^∞0,T;L²(Ω)=ess sup

t∈(0,T )

u(t )

L²(Ω).

Similarly we shall use such norms for spaces of semi-discrete functions defined on (or restricted to)(0, T )×ω.

1.1.5. Translation, difference and average operators

Leti, j∈J1, dK,j=i. We define the following translations for indices:

τ_i^±:Nⁱ

resp.N^ij

→N∪∂_i^±N

resp.N^j∪∂_i^±N^j , k→τ_i^±k,

with τ_i^±k

l=

k_l ifl=i, k_l±¹₂ ifl=i.

Translations operators mappingR^M∪∂M→R^Mi andR^Mj∪∂Mj →R^Mij are then given by τ_i^±u

k=u_(τ±

i k), k∈Nⁱ

resp.N^ij .

A first-order difference operatorDiand an averaging operatorAi are then given by (D_iu)_k=(h_i)⁻¹

τ_i⁺u

k− τ_i⁻u

k

, k∈Nⁱ

resp.N^ij , (A_iu)_k= ˜uⁱ_k=1

2 τ_i⁺u

k+ τ_i⁻u

k

, k∈Nⁱ

resp.N^ij . Both mapR^M∪∂MintoR^MiandR^Mj∪∂Mj intoR^Mij.

We also define the following translations for indices:

τ^±_i :N

resp.N^j

→Nⁱ

resp.N^ij , k→τ^±_i k,

with τ^±_i k

l=

kl ifl=i, kl±¹₂ ifl=i.

Translations operators mappingR^Mi→R^MandR^Mij →R^Mj are then given by τ^±_i v

k=v_(τ±

ik), k∈N

resp.N^j .

A first-order difference operatorDi and an averaging operatorAi are then given by (D_iv)_k=(h_i)⁻¹

τ⁺_i v

k− τ⁻_i v

k

, k∈N

resp.N^j , (A_iv)_k=vⁱ_k=1

2 τ⁺_i v

k+ τ⁻_i v

k

, k∈N

resp.N^j . Both mapR^Mi intoR^MandR^Mij intoR^Mj.

1.1.6. Traces

Leti∈J1, dK. Foru∈R^M∪∂M(resp.R^Mj∪∂Mj,j =i), its trace on∂_i⁺Ω, corresponds tok∈∂_i⁺N(resp.∂_i⁺N^j), i.e.,k_i=N_i+1 in our discretization and will be denoted byu_|ki=Ni+1or simplyu_Ni+1. Similarly its trace on∂_i⁻Ω, corresponds tok∈∂_i⁻N(resp.∂_i⁻N^j),i.e.,k_i=0 and will be denoted byu_|ki=0or simplyu₀. The latter notation will be used if no confusion is possible, that is if the context indicates that the trace is taken on∂_i⁻Ω.

By abuse of notation, we shall also use∂_iΩ,i∈J1, dK, to denote the boundaries ofΩ in the discrete setting. For homogeneous Dirichlet boundary condition we shall write

v_|∂Ω=0 ⇔ v_|∂iΩ =0, i∈J1, dK

⇔ v_|ki=0=v_|ki=Ni+1=0, i∈J1, dK.

We set

R^M∪∂M0=

v∈R^M∪∂M; v_|ki=0=v_|ki=Ni+1=0, i∈J1, dK .

Forv∈R^Mi∪∂Mi (resp.R^Mij∪∂Mij,j=i), its trace on∂_i⁺Ω, corresponds tok∈∂_i⁺Nⁱ (resp.∂_i⁺N^ij),i.e.,k_i= N_i+¹₂in our discretization and will be denoted byv_|k

i=Ni+¹₂ or simplyv_N

i+¹₂. Similarly its trace on∂_i⁻Ω, corresponds tok∈∂_i⁻Nⁱ (resp.∂_i⁻N^ij),i.e.,k_i=¹₂and will be denoted byv_|k

i=¹2 or simplyv1

2. The latter notation will be used if no confusion is possible, if the context indicates that the trace is taken on∂_i⁻Ω.

For such functionsu∈R^M∪∂M(resp.R^Mj∪∂Mj,j=i) we can then define surface integrals of the type

∂_i⁺Ω

u_|∂+

i Ω=

Ωi

u_|ki=Ni+1=

k∈∂_i⁺N (resp.k∈∂_i⁺N^j)

|∂_ib_k|u_k, where|∂_ib_k| =

l∈J1,dK l=i

h_l, k∈∂_i⁺N

resp.∂_i⁺N^j ,

and forv∈R^Mi∪∂Mi(resp.R^Mij∪∂Mij,j=i)

∂_i⁺Ω

v_|∂+

i Ω=

Ω_i

v_|k

i=N_i+¹2 =

k∈∂_i⁺Nⁱ (resp.k∈∂_i⁺N^ij)

∂_ibⁱ_kv_k, where∂_ibⁱ_k=

l∈J1,dK l=i

h_l, k∈∂_i⁺Nⁱ

resp.∂_i⁺N^ij .

Observe that ifk∈∂_i⁺N(resp.∂_i⁺N^j) andk∈∂_i⁺Nⁱ (resp.∂_i⁺N^ij) withkl=k_lforl=ithen|∂ibk| = |∂ibⁱ_k|. We thus have

∂_i⁺Ω

v_|∂+

i Ω=

Ωi

v_|k

i=Ni+¹2 =

Ωi

τ⁻_i v

|k_i=N_i+1=

∂_i⁺Ω

τ⁻_i v

|∂_i⁺Ω

whereτ⁻_i v∈R^M∪∂i−M(resp.R^{Mj∪∂i−Mj}) with the translation operatorτ⁻_i defined in Section1.1.5. It is then natural to define the following integrals

Ωi

u_Ni+1v_N

i+¹₂ =

Ωi

u_|ki=Ni+1v_|k

i=Ni+¹₂ =

Ωi

u τ⁻_i v

|ki=Ni+1=

∂_i⁺Ω

u τ⁻_i v

|∂_i⁺Ω.

Such trace integrals will appear when applying discrete integrations by parts in the following sections.

Similar definitions and considerations can be made for integrals over∂_i⁻Ω.

Foru∈R^M∪∂M(resp.R^Mj∪∂Mj,j=i) we can then introduce the followingL²-norm for the trace on∂_iΩ:

|u|²_L2(∂_iΩ)= |u_|∂_iΩ|²_L2(∂_iΩ)=

Ω_i

(u_|k_i=N_i+1)²+

Ω_i

(u_|k_i=0)².

Forv∈R^Mi∪∂Mi (resp.R^Mij∪∂Mij,j=i) we can then introduce the followingL²-norm for the trace on∂_iΩ:

|v|²_L2(∂_iΩ)= |v_|∂iΩ|²_L2(∂_iΩ)=

Ω_i

(u_|k

i=N_i+¹2)²+

Ω_i

(u_|k

i=¹2)². 1.1.7. Sampling of continuous functions

A continuous functionf defined on Ω can be sampled on the primal meshf^M= {f (xk); k∈N}, which we identify to

f^M=

k∈N

1bkf_k, f_k=f (xk),k∈N,

withb_kas defined in(1.6). We also set f^∂M=

f (xk); k∈∂N

, f^M∪∂M=

f (xk); k∈N∪∂N .

The functionf can also be sampled on the dual meshes, e.g.Mⁱ,f^Mi= {f (xk); k∈Nⁱ}which we identify to f^Mi=

k∈Nⁱ

1_bⁱ

k

fk, fk=f (xk),k∈Nⁱ

with similar definitions forf^∂Mi,f^Mi∪∂Mi and sampling on the meshesM^ij,M^ij∪∂M^ij.

In the sequel, we shall use the symbolf for both the continuous function and its sampling on the primal or dual meshes. In fact, from the context, one will be able to deduce the appropriate sampling. For example, withudefined on the primal mesh,M, in the following expression,D_i(γ D_iu), it is clear that the functionγ is sampled on the dual meshMⁱ asDiuis defined on this mesh and the operatorDiacts on functions defined on this mesh.

To evaluate the action of multiple iterations of discrete operators, e.g.D_i, D_i, A_i, A_i on a continuous function we may require the function to be defined in a neighborhood ofΩ. This will be the case here of the diffusion coefficients in the elliptic operator and the Carleman weight function we shall introduce. For a functionf defined on a neighborhood ofΩ we set

τ_i^±f (x):=f

x±h_i 2ei

, ei=(δ_i1, . . . , δ_id), D_if:=(h_i)⁻¹

τ_i⁺−τ_i⁻

f, A_if = ˆfⁱ=1 2

τ_i⁺+τ_i⁻ f.

For a function f continuously defined in a neighborhood ofΩ, the discrete function Dif is in fact equal to^Dif sampled on the dual mesh, Mⁱ, and D_if is equal to D_if sampled on the primal mesh, M. We shall use similar meanings for averaging symbols,f˜,f, and for more general combinations: for instance, ifi=j,D_jfⁱ,D_iD_jfⁱ, D_iD_jfⁱ will be respectively the functionsD_jfⁱ sampled onM^ij,D_iD_jfⁱ sampled onM, andD_iD_jfⁱ sampled on M^j.

1.2. Statement of the main results

With the notation we have introduced, the usual consistent finite-difference approximation of the elliptic operator Awith homogeneous Dirichlet boundary conditions is

A^Mu= −

i∈J1,dK

D_i(γ_iD_iu), (1.8)

for u∈R^M∪∂M satisfyingu_|∂Ω=u^∂M=0. Recall that, in each term,γ_i is the sampling of the given continuous diffusion coefficientγi on the dual meshMⁱ, so that for anyu∈R^M∪∂Mandk∈N, we have

(A^Mu)_(k)= −

i∈J1,dK

h⁻_i ² γ_i(x_τ+

i(k)) τ⁺_i τ_i⁺u

(k)−u_(k)

−γ_i(x_τ− i(k))

u_(k)− τ⁻_i τ_i⁻u

(k)

.

In 2D, this operator is nothing but the standard 5-point discretization. Note however that other consistent choices of discretization ofγ_i on the dual meshes are possible, such as the averaging on the dual meshMⁱ of the sampling ofγ_i on the primal mesh.

The semi-discrete forward and backward parabolic operators are then given byP_±^M=∂_t±A^M. 1.2.1. Carleman estimate

For the Carleman estimate and observation and control results we choose here to treat the case of a distributed observation in ωΩ. The weight function is of the form r=e^sϕ withϕ=e^λψ, withψ fulfilling the following assumption. Construction of such a weight function is classical (see e.g.[12]).