Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and application to controllability

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x)$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.

In the continuous framework, we refer to [FI96] and [LR95] who proved such a controllability result by means of a global/local Carleman observability estimates in the case the diffusion coefficient c is smooth. The authors of [BDL07] produced this controllability result in the case of a discontinuous coefficient in the one-dimensional case later extended to arbitrary dimension by [LR10]. Additionally, a result of controllability in the case of a coefficient with bounded variation (BV) was shown in [FCZ02,L07].
The authors of [LZ98] show that uniform controllability holds in the one-dimensional case with constant diffusion coefficient c and for a constant step size finite-difference scheme. Here, "uniform" is meant with respect to the discretization parameter h. The situation becomes more complex in higher dimension. In fact, a counter-example to null-controllability due to O. Kavian is provided in [Zua06] for a finite-difference discretization scheme for the heat equation in a square.
In recent works, by means of discrete Carleman estimate, the authors of [BHL10a], [BHL10b] and [BL12] obtained weak observability inequalities in the case of a smooth diffusion coefficient c(x). Such observability estimates are charaterized by an additional term that vanishes exponentially fast. Morever, also with a constant diffusion coffiencient c, under the assumption that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, a uniform observability property of semi-discrete approximations for System (1.1) is achieved in L 2 [LT06]. Besides that, such a result continues to hold even with the condition that the degree of unboundedness of control operator is greater than 1/2 [N12].
In the case of a non-smooth coefficient, our aim is to investigate the uniform controllability of System (1.1) after discretization. It is well known that controllability and observability are dual aspects of the same problem. We shall therefore focus on uniform observability which is shown to hold when the observability constant of the finite dimensional approximation systems does not depend on the step-size h.
In the present paper we prove a Carleman estimate for system (1.1) in the case of: • the heat equation in one space dimension; • a piecewise C 1 coefficient c with jumps at a finite number of points in Ω; • a finite-difference discretization in space.
The main idea of the proof is combination of the derivation of a discrete Carleman estimate as in [BHL10a,BL12] and tecniques of [BDL07] for operators with discontinuous coefficients in the one-dimensional case. A similar question in n-dimensional case, n ≥ 2, remains open, to our knowledge. When considering a discontinuous coefficient c the parabolic problem (1.1) can be understood as a transmission problem. For instance, assume that c exhibits a jump at a ∈ Ω. Then we write , 1) , c∂ x y| a + = c∂ x y| a − , y| a + = y| a − , y| ∂Ω = 0, and y| t=0 = y 0 .
The second line is thus a transmission condition implying the continuity of the solution and of the flux at x = a. When one gives a finite-difference version of this transmission problem, a similar condition can be given for the continuity of the solution. Yet, for the flux, it is only achieved up to a consistent term. In what follows, in the finite-difference approximation, we shall in fact write y(a − ) = y(a + ) = y n+1 , (c d Dy) n+ 3 2 − (c d Dy) n+ 1 2 = h D (c d Dy) n+1 , (the discrete notation will be given below). Note that the flux condition converges to the continuous one if h → 0, h being the discretization parameter. This difference between the continuous and the discrete case will be the source of several technical points.
An important point in the proof of Carleman estimate is the construction of a suitable weight function ψ whose gradient does not vanish in the complement of the observation region. The weight function is chosen smooth in the case of a smooth diffusion coefficient c(x). In general, the technique to construct such a function is based on Morse functions (see some details in [FI96]). In one space dimension, this construction is in fact straightforward. In the case of a discontinuous diffusion coefficient, authors of [BDL07] introduced an ad hoc transmission condition on the weight function: its derivative exhibits jumps at the singular points of the coefficient. In this paper, we construct a weight function based on these techniques in the one-dimentional discrete case.
From the semi-discrete Carleman we obtain, we give an observability inequality for semi-discrete parabolic problems with potential. As compared to the result in continuous case [BDL07] the observability estimate we state here is weak because of an additional term that describes the obstruction to the null-controllability. This term is exponentially small in agreement with the results obtained in [BHL10a,BHL10b] in the smooth coefficient case. A precise statement is given in Section 6.
Finally, the observability inequality allows one to obtain controllability results for semi-discrete parabolic with semi-linear terms. In continuous case, this was achieved in [BDL07]. Taking advantage of one-dimensional situation, the results we state are uniform with respect to the discretization parameter h (see Section 6).
In this paper, we shall address to some families of non uniform meshes, that will be precisely defined in Section 1.2. We introduce the following notation [ρ 1 ⋆] a = ρ 1 (a + ) − ρ 1 (a − ), (1.3) [⋆ρ 2 ] a = ρ 2 (n + 3 2 ) − ρ 2 (n + 1 2 ), (1.4) [ρ 1 ⋆ ρ 2 ] a = ρ 1 (a + )ρ 2 (n + 3 2 ) − ρ 1 (a − )ρ 2 (n + 1 2 ). (1.5) We follow some notation of [BHL10a] for discrete functions in the onedimensional case. 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 ′ And for u ∈ C M we define Ω u : For u ∈ C M we define As above, for u ∈ C M , we define Ω u : In particular we define the following L 2 inner product on C M (resp. C M ) For some u ∈ C M , we shall need to associate boundary conditions u ∂M = {u 0 , u n+m+2 }. 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+m+2 = 0, in short u ∂M = 0. We can define translation operators τ ± , a difference operator D and an averaging operator as the map C M∪∂M → C M given by We also define, on the dual mesh, translation operators τ ± , a difference operatorD and an averaging operator as the map C M → C M given by

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 with a to be kept fixed in what follows and chosen such that a ∈ (0, 1) ∩ Q, i.e a = p q with p, q ∈ N * . Clearly, we have q > p. We impose the function ϑ to be affine on [a − δ, a + δ] ϑ| [a−δ,a+δ] (for some δ > 0).
Given r ∈ N * and set m = (q − p)r and n = pr. The parameter r is used to refine the mesh when increased. Set a = x n+1 = x pr+1 . The interval Ω 01 = [0, a] is then discretized with n = pr interior grid points (excluding 0 and a). The interval Ω 02 = [a, 1] is discretized with m = (q − p)r exterior grid points (excluding a and 1).
be uniform mesh of Ω 0 and M 0 be the associated dual mesh. We define a non-uniform mesh M of Ω as image of M 0 by the map ϑ, settings (1.7) The dual mesh M and the general notation are then those of the previous section.

Main results
With the notation we have introduced, a consistent finite-difference approximation of Au with homogeneous boundary condition is For a suitable weight function ϕ (to be defined below), the announced semi-discrete Carleman estimate for the operator P M = −∂ t + A M with a discontinuous diffusion coefficient c, for the non-uniform meshes we consider, is of the form , (1.8) for properly chosen functions θ = θ(t) and ϕ = ϕ(x), for all τ ≥ τ 0 (T + T 2 ), 0 < h ≤ h 0 and τ h(αT ) −1 ≤ ǫ 0 , 0 < α < T and for all u ∈ C ∞ (0, T ; C M ) satisfying the discrete transmission conditions, where τ 0 , h 0 , ǫ 0 only depend on the data. We refer to Theorem 4.1 (uniform mesh) and Theorem 5.6 (non uniform mesh) below for a precise result. The proof of this estimate will be first carried out for piecewise uniform meshes, and then adapted to the case of the non-uniform meshes we introduced in Section 1.2.
From the semi-discrete Carleman estimate we obtain allows we deduce following weak observability estimate for any q solution to the adjoint system A precise statement is given in Section 6. Moreover, from the weak observability estimate given above we obtain a controllability result for the linear operator P M . This result can be extended to classes of semi-linear equations We shall state controllability results with a control that satisfies Thanks to one space dimension the size of the control function is uniform with respect to the discretization parameter h.

Sketch of proof of the Carleman estimate
The main idea of the proof lays in the combination of the derivation of a discrete Carleman estimate as in [BHL10a,BL12] and techniques used in [BDL07] to achieve such estimates for operators with discontinuous coefficients in the one-dimensional case.
We set v = e −sϕ u yielding e sϕ P e −sϕ v = e sϕ f 1 in Q ′ 0 if P u = f 1 We obtain g = Av + Bv in Q ′ 0 , with A and iB 'essentially' selfadjoint. We write g 2 L 2 = Av 2 L 2 + Bv 2 L 2 + 2(Av, Bv) L 2 and the main part of the proof is dedicated to computing the inner product (Av, Bv) L 2 (Q ′ 0 ) , involving (discrete) integration by parts.
We proceed with these computations separately in each domain Ω 01 , Ω 02 . As in [BL12] we obtain terms involving boundary points x = 0 and . In our case we obtain additional terms involving the jump point a such as v(a), ∂ t v(a), v n+ 1 2 ,ṽ n+ 3 2 , (Dv) n+ 1 2 , (Dv) n+ 3 2 . Main difficulties of our work come from dealing with these new terms. To reduce the number of terms to control, we find relations among connecting these various values at jump point allowing to focus our computations on terms only involving v(a), ∂ t v(a) and (Dv) n+ 1 2 . Those relations are stated in Lemma 3.17. In the limit h → 0 they give back the transmission conditions for the function v = e −sϕ u used crucial way in [BDL07]. The idea of this technique comes from a similar technique shown in continuos case by [BDL07].
The discrete setting could allow computation on the whole Ω. Yet such computation would yield constant that would depend on discrete derivatives of the diffusions coefficient, yielding non-uniformity with respect to the discretization parameter h. This explains why we resort to working on both Ω 0 and Ω 1 separately and deal with the interface terms that appear. As in [BDL07] the weight function is chosen to obtain positive contributions for these terms.
Sketch of proof Theorem 1. We compute the inner product (Av, Bv) in a series of terms and collect them together in an estimate (see Lemma 4.4-Lemma 4.12). In that estimate, we need to tackle two parts: volume integrals, integrals involving boundary points and the jump point. Volume integrals and boundary terms are dealt with similar to [BL12]. Terms at the jump point require special case.

Treatment of terms the jump point
• Terms at jump point involving ∂ t v : when treating the term Y 13 we obtain a positive integral of (∂ t v(a)) 2 in the LHS of the estimate as shown in Lemma 4.15. We keep this term in the LHS of the estimate.
• Other terms: We collect together the terms at the jump point that already exist in the continuous case. As in [BDL07] we obtain a quadratic form because of the choice of the weight function (jump of its slope). This allows us to obtain positive two integrals involving v 2 (a), (Dv) 2 n+ 1 2 in the LHS of our estimate (see Lemma 4.14).
• The remaining terms at the jump point are placed in the RHS of estimate. After that, we apply Young's inequality to them (as shown in Lemma 4.16) and they then can be absorded by the positive integrals involving v 2 (a), (Dv) 2 n+ 1 2 , (∂ t v(a)) 2 in the LHS of estimate as described above.

Outline
In section 2, we construct the weight functions to be used in the Carleman estimate. In section 3 we have gathered some preliminary discrete calculus results and we present how transmission conditions can be expressed in the discretization scheme. Section 4 is devoted to the proof the semidiscrete parabolic Carleman estimate in the case of a discontinuous diffusion cofficient for piecewise uniform meshes in the one-dimensional case. To ease the reading, a large number of proofs of intermediate estimates have been provided in Appendix A. This result is then extended to non-uniform meshes in Section 5. Finally, in Section 6, as consequences of the Carleman estimate, we present the weak observability estimate and associated some controllability results.

Weight functions
We shall first introduce a particular type of weight functions, which are constructed through the following lemma.
Lemma 2.1. LetΩ 1 ,Ω 2 be a smooth open and connected neighborhoods of intervals Ω 1 , Ω 2 of R and let ω ⊂ Ω 2 be a non-empty open set. Then, there exists a function ψ ∈ C(Ω) such that in Ω 1 and the function ψ satisfies the following trace properties, for some with the matrix A defined by A = a 11 a 12 a 21 a 22 , Remark 2.2. Here we choose a weight function that yields an observation in the region ω ⊂ Ω 2 in the Carleman estimate of Section 4. This choice is of course arbitrary.
Proof. We refer to Lemma 1.1 in [BDL07] for a similar proof.
Choosing a function ψ, as in the previous lemma, for λ > 0 and K > ψ ∞ , we define the following weight functions (2.1) and min [0,T ] θ ≥ T −2 . We note that For the Carleman estimate and the observation/control results we choose here to treat the case of an distributed-observation in ω ⊂ Ω. The weight function is of the form r = e sϕ with ϕ = e λψ , with ψ fulfilling the following assumption. Construction of such a weight function is classical (see e.g [FI96]).
where V ∂Ω is a sufficiently small neighborhood of ∂Ω inΩ, in which the outward unit normal n to Ω is extended from ∂Ω.
3 Some preliminary discrete calculus results for uniform meshes Here, to prepare for Section 4, we only consider constant-step discretizations, i.e., h i+ 1 2 = h, i = 0, . . . , n + m + 1. We use here the following notation: Ω 0 = (0, 1), Ω 01 = (0, a), Ω 02 = (a, 1), This section aims to provide calculus rules for discrete operators such as D i ,D i 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 on continuous functions. For a function f defined on Ω 0 we set Remark 3.1. To iterate averaging symbols we shall sometimes write Af = f , and thus A 2 f =f.

Discrete calculus formulae
We present calculus results for finite-difference operators that were defined in the introductory section. Proofs can be found in Appendix of [BHL10a] in the one-dimension case.
Lemma 3.2. Let the functions f 1 and f 2 be continuously defined in a neighborhood ofΩ. We have: 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 )f 2 +f 1 D(f 2 ),D(g 1 g 2 ) =D(g 1 )ḡ 2 +ḡ 1D (g 2 ).
Lemma 3.3. Let the functions f 1 and f 2 be continuously defined in a neighborhood ofΩ. We have: 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 Some of the following properties can be extended in such a manner to discrete functions. We shall not always write it explicitly.
Averaging a function twice gives the following formula.
Lemma 3.4. Let the function f be continuously defined over R. We then have DDf.
The following proposition covers discrete integrations by parts and related formula.
Proposition 3.5. Let f ∈ C M∪∂M and g ∈ C M . We have the following formulae: Lemma 3.6. Let f be a smooth function defined in a neighborhood ofΩ.
We have
Lemma 3.7. Let α, β ∈ N, i=1,2. We have The same expressions hold with r and ρ interchanged and with s changed into -s.
A proof is given in [BHL10a, proof of Lemma 3.7] in the time independent case. Additionally, we provide a result below to the time-dependent case whose proof is refered to [BL12, proof of Lemma 2.8]. Note that the With Leibniz formula we have the following estimates The proofs of the following properties can be found in Appendix A of [BHL10a].
The same estimates hold with ρ i and r i interchanged.
Proposition 3.14. Provided 0 < τ h(max [0,T ] θ) ≤ K and σ is bounded, we have The same estimates hold with r i and ρ i interchanged.

Transmission conditions
We consider here discrete version of the transmission conditions (TC) at the point a.
Remark 3.16. These transmission conditions provide the continuity for u and the discrete flux at the singular point of coefficient up to a consistent factor.
From these conditions, we obtain the following lemma whose proof is given in Appendix A Lemma 3.17. For the parameter λ chosen sufficiently large and sh sufficiently small and with u = ρv we have Furthermore, we have For simplicity, (3.1) can be written in form

Carleman estimate for uniform meshes
In this section, we prove a Carleman estimate in case of picewise uniform meshes, i.e, constant-step discretizations in each subinterval (0, a) and (a, 1). The case of non-uniform meshes is treated in Section 5.
We let ω 0 ⊂ Ω 02 be a nonempty open subset. We set the operator P M to be P The Carleman weight function is of the form r = e sϕ with ϕ = e λψ − e λK where ψ satisfies the properties listed in Section 2 in the domain Ω 0 . Here, to treat the semi-discrete case, we use the enlarged neighborhoodsΩ 01 ,Ω 02 of Ω 01 , Ω 02 as introduced in Lemma 2.1. This allows one to apply multiple discrete operators such as D and A on the weight functions. In particular, we take ψ such that ∂ x ψ ≥ 0 in V 0 and ∂ x ψ ≤ 0 in V 1 where V 0 and V 1 are neighborhoods of 0 and 1 respectively. This then yields on ∂Ω 0 Theorem 4.1. Let ω 0 ⊂ Ω 02 be a non-empty open set and we set f := D(c d Du). For the parameter λ > 1 sufficiently large, there exists C, τ 0 ≥ 1, h 0 > 0, ǫ 0 > 0, depending on ω 0 so that the following estimate holds Remark 4.2. Observation was chosen in Ω 02 here. This is an arbitrary choice (see Remark 2.2).
Proof. We set f 1 : . At first, we shall work with the function v = ru, i.e., u = ρv, that satisfies We write: as derived in [BL12]. Equation (4.3) now reads Av + Bv = g and we write Av 2 First we need an estimation of g 2 . The proof can be adapted from [BHL10a]. (4.5) Most of the remaining of the proof will be dedicated to computing the inner product (Av, Bv) . The proofs of the following lemmata are provided in Appendix A.
Lemma 4.5 (Estimate of I 12 ). For τ h(max [0,T ] θ) ≤ K, the term I 12 is of the following form where δ 12 ,δ 12 are of the form s λφ(a)O(1) + O λ,K (sh) 2 and Lemma 4.6 (Estimate of I 13 ). There exists ǫ 1 (λ) > 0 such that, for 0 < τ h(max [0,T ] θ) ≤ ǫ 1 (λ), the term I 13 can be estimated from below in following way: Lemma 4.7 (Estimate of I 21 ). For τ h(max [0,T ] θ) ≤ K, the term I 21 can be estimated as Lemma 4.8 (Estimate of I 22 ). For sh ≤ K, we have and Lemma 4.9 (Estimate of I 23 ). For τ h(max [0,T ] θ) ≤ K, the term I 23 can be estimated from below in the following way Lemma 4.11 (Estimate of I 32 ). [BL12] For τ h(max [0,T ] θ) ≤ K, the term I 32 can be estimated from below in the following way Lemma 4.12 (Estimate of I 33 ). [BL12, proof of Lemma 3.9] For τ h(max [0,T ] θ) ≤ K, the term I 33 can be estimated from below in the following way Continuation of the proof of Theorem 4.1. Collecting the terms we have obtained in the previous lemmata, from (4.4) and (4.
With the following lemma, we may in fact ignore the term Y in the previous inequality.
Recalling that ▽ψ ≥ C > 0 in Ω\ω 0 we may thus write Lemma 4.14. With the function ψ satisfing the properties of Lemma 2.1 and for with α 0 as given in Lemma 2.1 and where with r 0 as given in Lemma 3.17 and For a proof see Appendix A.
For a proof see Appendix A.
If we choose λ 2 ≥ λ 1 sufficiently large, then for λ = λ 2 (fixed for the rest of the proof) and 0 < τ h(max [0,T ] θ) ≤ ǫ 3 , from (4.6) and Lemma 4.14 and Lemma 4.15, we can thus achieve the following inequality where Z = µ r + µ 1 with µ r and µ 1 are given as in Lemma 4.14 and where By using the Young's inequality, we estimate in turn all the terms of Y , Z and the two terms at the RHS of (4.7) through the following Lemma whose proof can be found in Appendix A Futhermore, we can estimate the term in X 12 as follows by Lemma 3.3 and as Ω ′ We can now choose ǫ 4 and h 0 sufficiently small, with 0 < ǫ 4 ≤ ǫ 3 (λ 2 ), 0 < h 0 ≤ h 1 (λ 2 ), and τ 2 ≥ 1 sufficiently large, such that for τ ≥ τ 2 (T + T 2 ), 0 < h ≤ h 0 , and τ h(max [0,T ] θ) ≤ ǫ 4 , from (4.7) and Lemma 4.16 we get (4.8) where we used that (Dv) 2 ≤ Ch −2 ((τ + v) 2 +(τ − v) 2 ) and the last three terms whose integral taken on domain Q 0 come from the term in X 12 , X 13 and X 23 respectively. As τ ≥ τ 2 (T + T 2 ) then s ≥ τ 2 > 0 and furthermore we observe that .
We then add the following terms T 0 hs 3 v 2 n+1 and T 0 hs −1 (∂ t v(a)) 2 on both the right hand side and the left hand side of (4.8). This allows us to change the domain of integration from Q ′ 0 to Q 0 for the discrete integrals on the primal mesh.
No additional term is required for discrete integrals on the dual mesh. For sh sufficiently small and s ≥ 1 sufficiently large, these terms at the right hand side are then absorbed by the terms at the left hand side. More precisely, with 0 < ǫ 0 ≤ ǫ 4 sufficiently small and for τ ≥ τ 2 (T + T 2 ), 0 < h ≤ h 0 , and 0 < τ h(max [0,T ] θ) ≤ ǫ 0 we thus obtain . (4.9) Now we shall estimate the term sO λ, It follows that, as sh is bounded Similarly, we treat the term sO λ,K (1)v 2 (a) |t=0 as Therefore, (4.9) can be written as We next remove the volume norm s θ ≤ 1 T α , we see that a sufficient condition for τ h max To finish the proof, we need to express all the terms in the estimate above in terms of the original function u. We can proceed exactly as in the end of proof of Theorem 4.1 in [BHL10a].

Carleman estimates for regular non uniform meshes
In this section we focus on extending the above result to the class of non piecewise uniform meshes introduced in Section 1.2. We choose a function ϑ satisfying (1.6) and further ϑ| [a−δ,a+δ] is chosen affine (for some δ > 0 to remain fixed in the sequel). The way we proceed here is similar to what is done in [BHL10a]. In this framework, we shall prove a non-uniform Carleman estimate for the parabolic operator P M = −∂ t + A M on the mesh M by using the result on uniform meshes of Section 4.
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 These two discrete functions are connected to the geometry of the primal and dual meshes M and M and we have We introduce some notation. To any u ∈ C M∪∂M , we associate the discrete function denoted by Q M0 M u ∈ C M0∪∂M0 defined on the uniform mesh M 0 which takes the same values as u at the corresponding nodes. More precisely, if u = n+m+1 i=1 The operators Q M0 M and Q M0 M are invertible and we denote by Q M M0 and Q M M0 their respective inverses. We give commutation properties between these operators and discrete-difference operators through the following Lemmata whose proofs can be found in [BHL10a]. 1. For any u ∈ C M∪∂M and any v ∈ C M , we have Futhermore, the same inequalities hold by replacing Ω by ω and Ω 0 by ω 0 , respectively.
For any continuous function f defined on Ω (resp. on Ω 0 ) we denote by In particular, for u ∈ C M∪∂M we have Moreover, by making use of Taylor formulae we get the following result Lemma 5.5. With ζ defined as in Lemma 5.1 we have Proof. From the definition of ζ, Q M0 M and D acting on C M0 ,D acting on C M0 we have By using Taylor formulae we write Thus we have From (5.1) we obtainD which proves the first result. Next, we proceed with the second result in the same manner as above. We have
Proof. We set w = Q M0 M u defined on the uniform mesh M 0 . By using Lemma 5.2 we have We observe that the right-hand side of (5.3) is a semi-discrete parabolic operator of the form by using Lemma 3.3 and Lemma 5.5. Thus, the operator P M0 can be written in form as Moreover, using Lemma 3.2 we havē We set P M0 From the properties ofν and ξ d it follows that First, we shall obtain a Carleman estimate for P M0 0 . Then we shall deduce a Carleman estimate for the operator Now, we consider the function ψ • ϑ : (t, x) → ψ t, ϑ(x) . By using the properties listed in Lemma 2.1 and (1.6), we shall see that ψ • ϑ is a suitable weight function associated to the control domain ω 0 = ϑ −1 (w) in Ω 0 , i.e., that ψ•ϑ satisfies Lemma 2.1 for the domaims Ω 0 and ω 0 .
The important property to checking is the trace property. The remaining properties are left to the reader. We set (recall that ϑ| [a−δ,a+δ] is an affine function). It follows that We can see that (Bw, w) = (Aw, w) ≥ α 0 w 2 . This means that ψ • ϑ satisfies the trace property.
We thus obtain which allows one to absorb by the term in the LHS of Carleman estimate by choosing τ sufficiently large. Futhermore, by using the previous Lemmata 5.1 -5.4 and considering each term in (5.7) separately, we see that we have the following estimates • For the first term in LHS of (5.7) , and a similar inequality holds for θ 3 2 e τ θϕ0 w 2 L 2 (Q0) .
• For the second term of LHS of (5.7) we use Lemma 5.2 and Lemma 5.3 as follows The proof is complete.

Controllability results
The Carleman estimate proved in the previous Section allows to give observability estimate that yields results of controllability to the trajectories for classes of semi-linear heat equations.
Proposition 6.1. There exists positive constants C 0 , C 1 and C 2 such that for all T > 0 and all potential fucntion a, under the condition h ≤ min(h 0 , h 1 ) with any solution of (6.2) satisfies Remark 6.2. In comparision the observability inequality in continuous case which performed in [BDL07], we find that the observability inequality obtained here is weak since there is an additional term depending upon h at right-hand-side of inequality (6.3).
From the result of Proposition 6.1 we deduce the following controllability result for system (6.1). Proposition 6.3. There exists positive constants C 1 , C 2 , C 3 and for T > 0 a map L T,a : for all initial data y 0 ∈ R M , there exists a semi-discrete control function v given by v = L a (y 0 ) such that the solution to (6.1) satisfies Note that the final state is of size e −C/h |y 0 | L 2 (Ω) . The proof of these proposition are given in [BL12].

The semilinear case
We consider the following semilinear semi-discrete control problem where ω ⊂ Ω. The function G : R → R is assumed of the form x ∈ R, (6.5) with g Lipschitz continuous. Here, we consider the function g in two cases: g ∈ L ∞ (R) and the more general case as |g(x)| ≤ K ln r (e + |x|), x ∈ R, with 0 ≤ r < 3 2 (6.6) The results of semi-discrete parabolic with potential above allows one to obtain controllability results for parabolic equation with semi-linear terms whose proofs are given in [BL12] Theorem 6.4. We assume that g ∈ L ∞ (R) and c satisfies (1.2). There exists positive constants C 0 , C 1 such that for all T > 0 and h chosen sufficiently small, for all initial data y 0 ∈ R M , there exists a semi-discrete control function v with v L 2 (Q) ≤ C |y 0 | L 2 (Ω) such that the solution to the semi-linear parabolic equation (6.4) satisfies Theorem 6.5. Let Ω = (0, 1), c satisfy (1.2) and G satisfy (6.5) -(6.6). There exists C 0 such that, for T > 0 and M > 0, there exists positive constants C, h 0 such that for 0 < h ≤ h 0 and for all initial data y 0 ∈ R M satisfying |y 0 | L 2 (Ω) ≤ M there exists a semi-discrete control function v such that the solution to the semi-linear parabolic equation where C = C(T, M ).
Observe that the constants are uniform with respect to discretization parameter h.

A Proofs of Lemma 3.17 and intermediate results in Section 4
A.1 Proof of Lemma 3.17 We have We write From (A.1) we thus write Then Moreover, as r∂ρ = −λsφ∂ψ = sO λ (1) we have From that, we can write As L = 1 + O λ,K (sh) = 0 (see below) then we read We thus have By using Proposition 3.10 we find For sh sufficiently small we have L −1 = 1 + O λ,K (sh) and then we obtain By using Proposition 3.14, Lemma 3.8 and Lemma 3.6 yield where sh sufficiently small and It follows that we have Furthermore, we can write (A.2) in the simple form which yields the conclusion.
A.2 Proof of Lemma 4.4 By using Lemma 3.2 in each domain Ω 01 , Ω 02 , we have We then apply a discrete integration by parts (Proposition 3.5) in each domain Ω 01 , Ω 02 with ∂Ω 01 = {0, a} and ∂Ω 02 = {a, 1} for the first two terms and we obtain Moreover, by Lemma 3.3 and Proposition 3.5 in each domain Ω 01 , Ω 02 we obtain Similarly, we have Thus where X 11 = Q ′ 0 ν 11 (Dv) 2 with ν 11 of the form sλφO(1) + sO λ,K (sh) and A.3 Proof of Lemma 4.5 We set q = rρcφ ′′ . By using a discrete integrations by parts (Proposition 3.5) and Lemma 3.2 in each domain Ω 01 , Ω 02 we have Provided sh ≤ K we have Note that the proof and the use of Lemma A.2 are carried out in each domain Ω 01 , Ω 02 independently.
We then obtain We thus write I 21 A.6 Proof of Lemma 4.8 We set q = c 2 r(DDρ)φ ′′ and by Lemma 3.4 we haveṽ = v + h 2D Dv/4 in each domain Ω 01 , Ω 02 . It follows that Applying a discrete integration by parts (Proposition 3.5) and Lemma 3.2 in each domain Ω 01 , Ω 02 yield as v| ∂Ω0 = 0.
In each domain Ω 01 , Ω 02 , we have φ ′′ = O λ (1) and from Proposition 3.13 we have q = s 2 O λ,K (1) and Dq = s 2 O λ,K (1). We thus obtain Note that the proof and use of above Lemma A.4 are done in each domain Ω 01 , Ω 02 separately.
Note that the proof and use of Lemma A.5 are done in each domain Ω 01 , Ω 02 separately.
Moreover, we observe that h 2 (ṽ n+ 3 2 ). By an integration by parts w.r.t t and Lemma 3.2 in each domain Ω 01 , Ω 02 we findQ By means of Lemma 3.2 and a discrete intergration by parts in space (Proposition 3.5) in each domain Ω 01 , Ω 02 we see that Note that all above terms are done in each domain Ω 01 , Ω 02 separately. We thus obtain Applying the Young's inequality and using that |∂ tṽ | 2 ≤|∂ t v| 2 in each domain Ω 01 , Ω 02 , we havē (1) |∂ t v| 2 . By using Proposition 3.5, Lemma A.6 in each domain Ω 01 , Ω 02 separately yield

Collecting (A.3), (A.4) and (A.5) we obtain
where X 23 and Y 23 are as given in the statement of Lemma 4.9.
A.8 Proof of Lemma 4.10 By means of a discrete integration by parts (Proposition 3.5) in each domain Ω 01 , Ω 02 separately, we get We have ϕcrDρv=ϕcrDρṽ + h 2 4 D(ϕcrDρ)Dv in each domain Ω 01 , Ω 02 . It follows that by using a discrete integration by parts in each domain Ω 01 , Ω 02 separately and as v| ∂Ω = 0. By using the Lipschitz continuity and Proposition 3.13 we get The proof is done in each domain Ω 01 , Ω 02 separately. Note that max t ∂ t θ = T θ 2 . It thus follows that We thus write Y (1,21) 21 as follows: (A.8) Moreover, the term Y (2,1) 11 is given by We estimate as We thus obtain Y (2,1) 11 Y (2,1) 11 and µ 1 can be written as We can write Moreover, by using Lemma 3.17, we obtain Moreover, we have: We thus write µ as where µ r can be written as We have thus achieved with u(t, a) = (c d Dv) n+ 1 2 , sλφ(a)v n+1 t and the symmetric matrix A defined in Lemma 2.1. From the choice made for the weight function β in Lemma 2.1 we find that: A.11 Proof of Lemma 4.15 By using Lemma 3.17 we have where J 1 , J 2 and J 3 are given as in Lemma 3.17.