Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries

. The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than 1. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general. The main assumption we make on the geometry and on the evolution equa- tion itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisﬁes some necessary Kalman con- dition) with uniform bounds on the controls. To achieve this objective we actually propose an abstract result on ordinary diﬀerential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.

(Communicated by the associate editor name) Abstract. The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than 1. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general.
The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls.
To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.
where Ω is a bounded domain of R d (d ≥ 1), Γ is a non empty part of the boundary ∂Ω, α and β are the two components of the system, and v is the boundary control we are looking for. The main difficulty in the analysis of the controllability of such system comes from the fact that we only have one boundary control v to drive the two components (α, β) to 0 at the final time. The coupling terms (here the term α in the equation for β) plays a key role in the problem and it can be seen as a kind of indirect control for the second component of the system. Note that this indirect controllability issue arises even if Γ = ∂Ω.
Actually, it appears that the results for such systems may be quite different from the case of scalar equations or from the case of coupled systems with a distributed control. We refer for instance to the survey [3] for a review on that topic. In particular, it is explained in that reference that usual techniques based on Carleman estimates are useless on those problems. This is mainly because those techniques naturally give the controllability of the system when there is as many controls as components of the system (for (1) it would consist in another boundary control for β on Γ) and, in a second step, it is needed to prove that only one control is necessary. This is done, at the observability inequality level (see the discussion in Section 2.3) by removing one observation term thanks to the PDE itself. This last step cannot be done for boundary controls (or observations). This is why other approaches have to be developed.
Most of the results available up to now for such controllability problems are only proved in dimension d = 1 by using the so-called moments method. This is a quite powerful method but, unfortunately, restricted to autonomous problems in space dimension 1. In the multi-dimensional case, one of the more advanced result available in the literature is proved in [5], in the case where the geometry and the diffusion operator can be tensorized and this is also the case we shall consider in the present work. We also refer to [1] for results on similar multi-dimensional systems, yet under the geometric control condition which is not satisfied in the present study (see Figure 1). In this work, the geometric assumption we shall consider for problem (1), is as follows: Ω = Ω 1 × Ω 2 with Ω 1 = (0, a) and Ω 2 ⊂ R d−1 , and Γ = {0} × ω 2 (see  Figure 1). Then, we can rewrite the system as follows where ∆ 2 is the (d − 1)-dimensional Laplace operator in Ω 2 . The fact that the diffusion operator is split into two parts, each of them acting on different sets of variables, is crucial in the analysis. That is the reason why we shall adopt a tensor product formalism that consists essentially in identifying L 2 (Ω) to L 2 (Ω 1 ) ⊗L 2 (Ω 2 ) (see Remark 3.1 for a definition of ⊗) and in writing the two equations above in the following equivalent form where the same symbol I is used for the identity operator in L 2 (Ω 1 ) and L 2 (Ω 2 ). All the necessary notations and properties concerning tensor products will be recalled in Section 3.1. By exploiting this tensor product structure, even though the tensor product formalism was not explicitly used, it was proved in [5], that the null-controllability of (3) holds at any time T > 0.

1.2.
Passing to the discrete world. We are now interested in semi-discrete versions of the controllability result for (1) mentioned just before. To simplify the presentation in this introduction, we assume that d = 2, that Ω 1 = Ω 2 = (0, 1) and that the computation grid is made of N × N uniformly distributed points (ih, jh) 1≤i,j≤N with h = 1 N +1 . The semi-discrete system we consider is obtained by the finite difference method and reads The grid geometry is essentially the one described in Figure 2, where the control v = (v j ) j is only appearing in the first equation of the system (the one for α i,j ) and only on the boundary points represented by the symbol corresponding to the subdomain ω 2 .
At each time t, both components α h = (α i,j ) i,j ∈ R N ×N and β h = (β i,j ) i,j ∈ R N ×N of the system are now considered as elements of the tensor product R N ⊗ R N and we observe that the five-point discrete Laplace operator can be written as the tensor product A h ⊗ I + I ⊗ A h with the usual definition of the three-point discrete 4

Laplace matrix
We finally end up with the following equivalent form of our semi-discrete system where B h is a matrix that accounts for the influence of the control v h in the system through the boundary conditions in (4), the precise definition of which will be given in Section 4.1. This is the semi-discrete version of (3). For this particular system, the main result of this paper is the following (see Section 4.1 for the precise statement and definition of the norms involved).
Theorem. There exist C > 0 and h 0 > 0 such that for any h < h 0 , any time T > 0 and any initial data , and the associated solution to (5) satisfies . It is well known, see [7,19], that we cannot expect in general to achieve exactly α h (T ) = β h (T ) = 0 since the semi-discrete system may be not even approximately controllable. In this sense, achieving exponentially small targets with respect to h is an optimal result. Our aim will be to provide similar results for more general semi-discrete systems. That is the reason why, in order to formulate them more conveniently for any number of coupled equations and to ease the reading of the proofs, we shall actually gather the two components (α h , β h ) ∈ (R N ⊗ R N ) 2 into a single unknown y h ∈ R N ⊗ R N ⊗ R n , with n = 2 in the present case, in such a way that the considered system (5) will finally be written in the compact form In the sequel of this paper we shall not explicitly mention the subscript h for the notation of quantities related to the discretization process but taking into account the fact that the considered spaces, norms and operators are grid-dependent is a central point in the analysis.

1.3.
Main results and outline of the paper. Considering the previous discussion we shall analyze in this paper the controllability of parabolic systems of n components and m controls of the following tensorized form ∂ t y + A 1 ⊗ I ⊗ I y + I ⊗ A 2 ⊗ I y + I ⊗ I ⊗ Cy = B 1 ⊗ B 2 ⊗ Bv, in (0, T ), (6) BOUNDARY CONTROL OF SEMI-DISCRETE PARABOLIC SYSTEMS IN MULTI-D   5 where A i is a diffusion operator in Ω i , B i is a (boundary or distributed) control operator in Ω i , I is the n × n identity matrix, C is a n × n coupling matrix and B a n × m control matrix.
Our main aim being to analyze semi-discrete versions of (6), we shall also consider linear ordinary differential equations of the similar form (7) where the unknown y and the control v belong to a finite dimensional tensor space, A i and B i are linear operators on those space. All those objects depend, by nature, on discretization parameters, as in the example of Section 1.1. Therefore, it will be crucial to take care of all the constants in the estimates so as to obtain, at the end, controllability results for (7), that will not depend on those parameters. Remark 1.1. We have used in (6) and (7) a convention that will be used all along the paper: • Operators acting in infinite dimensional function spaces are written with calligraphic letters: A, B, I, ... • Operators (matrices) acting in finite dimensional spaces coming from discretization issues (whose dimension may be large and depends on the discretization parameters) are written with upright letters: A, B, I, ... • Matrices acting in finite dimensional spaces coming from the number of components or controls in the system (their dimension is fixed and independent of discretization parameters) are written with sans serif letters: B, C, I , ...
The outline of the paper is the following. Section 2 is dedicated to recall the main results in the controllability theory for ODEs while paying a particular attention to the discrete functional setting that will be adapted to the analysis of finite difference approximations of parabolic PDEs. In section 3, we first review the material we need concerning tensorized operators that are central in the present work, then we state the precise assumptions we need and our main abstract result (Theorem 3.1). In short, we will assume for our tensorized system (6) (in the continuous setting) or (7) (in the discrete setting): • that the associated sub-problem concerning only the first coordinate of the tensor product which is is null-controllable (or a relaxed version of this) at any time T with a precise control of the cost of the control. If we come back to our motivating example, this subsystem reads Note that this system does not depend on the control set ω 2 . • that the diffusion operator A 2 (resp. A 2 ) and the control operator B 2 (resp. B 2 ) in the other direction satisfy a suitable spectral estimate similar to the

DAMIEN ALLONSIUS AND FRANCK BOYER
Lebeau-Robbiano spectral inequality, except that we allow the inequality to hold only for a portion of the spectrum.
For system (2), this amounts to ask that the Lebeau-Robbiano spectral inequality holds, relative to the control set ω 2 , for the eigenfunctions of the operator −∆ 2 in Ω 2 with homogeneous boundary condition. The complete proof of the theorem is given in Section 3.3. It consists, following the strategy developed in [5], to implement a construction of the control similar to the one originally proposed by Lebeau and Robbiano in [15]. We split the time interval into a suitable number of subintervals whose length is carefully chosen and, on each of those subintervals, we construct a partial control (obtained by combining the two assumptions above) that is able to damp out exponentially the part of the solution corresponding to the frequencies less than some threshold. Note that, contrary to the usual construction, we are not necessary able to drive this part of the solution exactly to zero at this stage. This threshold is then increased while the construction progresses towards the final time T ; this eventually gives the expected control.
The main novelties in the present work are that : we allow relaxed controllability and spectral inequalities in our assumptions and moreover we precisely take care of the dependence of all the quantities of interest (norms, constants, ...) with respect to parameters on which the problem may depend. Those two refinements of the proof in [5] are mandatory since we want to apply this abstract result to systems obtained by semi-discretization processes.
To conclude the paper, in section 4, we precisely explain how to use the abstract formalism developed here to achieve the uniform controllability results of semidiscrete coupled parabolic systems as announced in this introduction. As another example, we also show how to deduce the result of [5] (slightly generalized to variable coefficients operators) from the present abstract result and give some insights on other possible applications.
2. Controllability for linear ODEs. Before studying discrete versions of System (6), which is the main aim of the paper, we start by introducing the main notations and results that we shall use in the sequel concerning the controllability of linear ODEs. Most of this material is already well-known, however we propose a specific point of view adapted to our needs.
2.1. Framework. Let (E, •, • 0 ) and (U, [•, •] 0 ) be two (finite dimensional) Euclidean spaces (each of them being identified with its own dual space). The corresponding norms are denoted by • 0 and • 0 . The presence of a subscript 0 in the notation is related to the fact that, in Section 2.2, those norms will be embedded in a scale of Sobolev-like norms.
We consider for the moment a general linear autonomous controlled system of the form y + Ly = Bv, on (0, T ), where L : E → E and B : U → E are two linear operators, y : [0, T ] → E is the state and v : [0, T ] → U is the control we are looking for.
In the sequel of this paper, different such systems will be considered, coming in particular from the discretization of multi-D parabolic control problems such as (7) or their reduced version (9). In particular, the spaces E, U and the operators L and B will depend on some discretization parameter h. We will be interested in properties of those systems that are uniform with respect to h, that is the reason why we will pay, in this section, a particular attention to the various constants appearing in the estimates. In section 4 we will propose a suitable framework ensuring that all those constants will be uniform with respect to h.

2.2.
Well-posedness. It is clear that (11) is well-posed for any choice of y 0 and v and that sup where L (resp. B ) is the operator norm of L : E → E (resp. B : U → E). However, in the framework we are interested in which comes from the semi-discretization in space of an evolution PDEs, those operator norms will not be bounded in general with respect to the discretization parameter. This is the consequence at the discrete level of the fact that differential operators are naturally unbounded operators in Sobolev spaces. For example if L the discrete Laplace operator on a uniform mesh of size h and B the boundary control operator as defined in section 4.1, then B and L both behave like C/h 2 . Thus, inequality (12) will not give usable estimates.
Therefore, we need to introduce adapted estimates and some kind of discrete Sobolev norms to take into account the particular geometry of the (discrete) control operators under study. To this end, we introduce D : E → E a self-adjoint definite positive operator on E (one can think of the discrete Laplace operator for instance) and we define a scale of inner products in E defined, for any s ∈ R, by We shall now define, for given s ∈ R, the two constants M s,adm , M s,cont > 0 that satisfy sup where the adjoint operators L * and B * are relative to the ambient inner product on E and U . Observe that (14) and (15) automatically holds since we consider finite dimensional spaces, and the only interesting point is the uniformity (or not) of the constants with respect to the spaces and the operators involved. Depending on the targeted application (distributed control, Dirichlet boundary control or Neumann boundary control for instance) we will need to choose a convenient value of s and of the operator D to ensure that those constants are actually uniform with respect to the discretization parameter.

DAMIEN ALLONSIUS AND FRANCK BOYER
Proof. We write the Duhamel formula then we take the inner product with any ψ ∈ E It follows that and then By (14) and (15) we deduce that Since this is valid for any ψ ∈ E, we deduce the expected estimate by the duality property (13).
Remark 2.1. During this work we will often use the following very standard duality formula that was given in the proof above 2.3. Relaxed observability inequalities. It is well-known (see [11,18] for instance) that System (11) is null controllable at time T for any initial condition y 0 if and only if there exists C > 0, such that the following observability inequality for the adjoint problem is satisfied The value of the constant C in this inequality is crucial since it appears in the measure of the control cost. It happens that, when (11) comes from a discretization of a parabolic equation with the finite difference method then the null controllability of the semi-discrete system may not hold (in particular in a multi dimensional setting, see the example given by Kavian and reported in [19]). To tackle this problem, it was proposed (in [12,8,9,7] for instance) to relax the controllability requirements by considering instead the ϕ(h)-null-controllability of (11). It consists in constructing uniformly bounded controls such that the solution y(T ) does not identically vanish but is small enough with respect to the discretization parameter h. This approach is based on the penalized HUM construction, where the penalization parameter is a given function h → ϕ(h) of the discretization parameter, given its name to this notion. Note that, the spaces E, U and the operators L and B all depend on h, in particular the dimensions of E and U may increase when h tends to zero. This is one of the main difficulty that we need to take care of in the analysis.
The ϕ(h)-null-controllability property is equivalent to a relaxed version of inequality (17) and the following Lemma 2.2, whose proof is given in appendix A, aims at establishing such an equivalence. This lemma is somehow related to [16,Lemma 3.4] or [4,Proposition 1] and is stated in a quite general framework : the constant s and the operator D can be chosen arbitrarily. Moreover using appropriate spaces F 0 and F T (which are defined below), one can show that Lemma 2.2 encompasses some already known situations (see Remark 2.3). Roughly speaking, this Lemma is about controlling the components in the final state space F T of the solution of system (24) which starts from an initial condition y 0 in the initial state space F 0 . Even though we only state it in a finite dimensional setting, it is clear that infinite dimensional versions also hold, as in the references quoted above.
Let F 0 and F T be two subspaces of E and P F0 (resp. P F T ) the orthogonal projection onto F 0 (resp. F T ) with respect to the inner product •, • −s,D .
We will denote the adjoint operators of the projectors P F0 and P F T for the inner product •, • 0,D by P * F0 and P * F T . Observe that P * F0 and P * F T are also the orthogonal projectors in (E, •, • s,D ) onto D −s F 0 and D −s F T respectively. In particular, we have P * F0 y s,D ≤ y s,D , ∀y ∈ E, does not depend on s. In particular, those projections are orthogonal for the inner product •, • 0 .
Dealing with such subspaces will be crucial in the sequel when we will apply the Lebeau-Robbiano strategy since it requires to be able to control some precise components of the solution at each step (depending on eigenspaces of A 2 ), see Section 3.3.
Lemma 2.2 (Relaxed observability inequalities and controllability). We use the above notations and assume that s, D, F 0 and F T are given. Let M obs > 0 and M rel ≥ 0 be two given numbers.
The following two propositions are equivalent.

For any
where y is the corresponding solution of (11). 2. For any q T ∈ D −s F T , the following relaxed observability inequality holds: Of course, for M rel = 0, the inequality (18) should be understood as Remark 2.3. Throughout this paper F 0 will always be equal to the whole space E. However it is worth noticing that by specifying spaces F 0 and F T , one can recover usual inequalities related to different notions of controllability. (19) is the usual relaxed inequality. If (19) holds with M rel = 0 then system (11) is null controllable.
As explained above, we cannot always expect M rel to be equal to zero when system (11) is discretized by finite differences method with a space domain of dimension greater than one. However, if this inequality holds with M 2 rel = ϕ(h) and with M obs independent of h, we recover the ϕ(h)-null-controllability notion briefly described above.
• When dim(F 0 ) = 1 and F T = E, then proving (19) amounts to drive only one given initial condition to zero (when M rel = 0) or close to zero (when M 2 rel is small, like ϕ(h) for instance). This question is tackled for instance in [7].
• The partial null-controllability consists in driving to zero only some components of the solution of a system of parabolic PDEs. It amounts to prove inequality (19) for M rel = 0 and to choose an appropriate subspace F T . This kind of controllability is studied for instance in [4] where related ϕ(h)-partialnull-controllability is also investigated.

3.1.
Notations. We introduce in this section the main notations used all along this paper. They mostly rely on usual notations and properties of tensor products (see for instance [13] for the algebraic properties of such structures, and [17] for related Euclidean/Hilbertian properties) Let (E i , •, • 0,i ), i = 1, 2 be two finite dimensional Euclidean spaces of dimensions N 1 and N 2 . The associated norms are denoted by • 0,i . Let D i be two positive definite self-adjoint operators in those spaces and for any s ∈ R, we introduce the following scalar products In the case of a finite difference approximate system, the two spaces E i have to be understood as the spaces of discrete in space (scalar) functions defined on a grid of Ω i , i = 1, 2. We will then consider the tensor product space E 1 ⊗ E 2 as a natural discretization space for functions defined on the tensor product grid of Ω. This space is equipped with the natural Euclidean structure defined by Remark 3.1. If E 1 and E 2 are infinite dimensional Hilbert spaces, the algebraic tensort product E 1 ⊗ E 2 equipped with the above inner product is only a prehilbertian space. The natural functional space to consider in that case is the completion of E 1 ⊗ E 2 , which is then an Hilbert space denoted by E 1 ⊗E 2 , see [17].
On E 1 ⊗ E 2 we consider the operator D = D 1 ⊗ I + I ⊗ D 2 , where I stands for the identity operator in E 1 and E 2 . This is a positive definite self-adjoint operator, from which we can define the following natural inner products and associated norms Note, in particular that we have Since we will be interested in vector-valued discrete functions that aim at being approximations of the solution (at any time t) of system (6), we will naturally work with the space E = E 1 ⊗ E 2 ⊗ R n , which corresponds to the fact that all the n components of the system are approximated at each grid point.
We recall that if L i is a linear operator in E i and L a linear operator in R n , the tensor product L 1 ⊗ L 2 ⊗ L is a linear operator on E defined by Let (•, •) be the Euclidean inner product in R n and |•| the associated norm. The previous definitions are naturally extended to the space E as follows and we do similar extensions for spaces E 1 ⊗ R n and E 2 ⊗ R n .
Recall that E can be identified to We now introduce two other finite dimensional Euclidean spaces (U i , [•, •] 0,i ), i = 1, 2, with the associated norms • 0,i that correspond to the control space for each subproblem.
We define U := U 1 ⊗ U 2 ⊗ R m and its inner product [•, •] 0 , whose definition on U 1 ⊗ U 2 is analogous to (20) : and is extended to U as before.
3.2. Main theorem. Let A i be a symmetric definite positive operator in E i . One can think of A i as an approximation of the continuous operator A i but the statement of our result is generic and does not explicitly make use of such an assumption. The eigenvalues of A i will be denoted by (λ i,k ) Ni k=1 and the corresponding eigenfunctions are (φ i,k ) Ni k=1 . Those form an orthonormal family for •, • 0,i . In our estimates, the following discrete Poincaré inequality for A 1 , will be needed for some M P,1 . Without loss of generality, we will assume that M P,1 ≥ 1. Note that the best possible value for M P,1 is λ but it is not sure that this value is greater than 1. Note also that the following generalized Poincaré estimate holds Our goal is to analyze the controllability properties (uniform with respect to any parameter on which the system may depend) of the following tensorized ODE where the control v belongs to L 2 (0, T ; U ) and L := A + I ⊗ I ⊗ C and A := A 1 ⊗ I ⊗ I + I ⊗ A 2 ⊗ I .
Here, the symbol I stands for the identity operator on E 1 or E 2 and the symbol I is the identity operator of R n .
To this end, we will take benefit from the tensor product structure of the system and make two main assumptions: 1. the first one concerns the vector-valued sub-control-system in E 1 defined by with 2. the second one concerns a spectral property related with the pair of operators (A 2 , B 2 ) similar to the well-known Lebeau-Robbiano inequality, excepted that it only holds for a certain portion of the spectrum of A 2 .
The main theorem of this paper is the following.
Theorem 3.1. Assume that Assumptions 1 and 2 hold. Let µ * = min(µ E 1 , µ E 2 ). There exist a M obs > 0 depending only on M obs,1 , M LR,2 , M s,cont , M s,adm and C such that for any y 0 ∈ E and any T > 0 there exists a control v ∈ L 2 (0, Note that in this theorem the constant M obs,1 is built upon the Sobolev norms associated with D 1 = A 1 and the constants M s,cont , M s,adm with the Sobolev norms associated with D = A. Those norms are thus problem dependent.
In the case where A i are discrete versions of diffusion operators, it can be tempting to use instead usual discrete Sobolev norms, that are defined by using for D 1 and D the discrete Laplace operators. However, the equivalence between those norms, for large values of s, is uniform with respect to the discretization parameter if and only if the diffusion coefficients γ i are smooth enough. In the case of non smooth diffusion coefficients, using the norms given in the theorem is mandatory.

3.3.
Proof of the main result. In this section, we will prove Theorem 3.1.

BOUNDARY CONTROL OF SEMI-DISCRETE PARABOLIC SYSTEMS IN MULTI-D 13
3.3.1. Preliminary estimates. We start with some preliminary results.
Lemma 3.2 (Norms comparison). For any s ≥ 0, and any Proof. We write each q j under the form with q k,j ∈ R n in such a way that and we immediately obtain Moreover, we have and the claim follows.

DAMIEN ALLONSIUS AND FRANCK BOYER
Let us now state some properties of the uncontrolled system ∂ t y + Ly = 0, Proposition 3.3 (Dissipation estimates).

For any
for any t and moreover, for any s ∈ R, we have 2. For any µ ≥ 0, and any y 0 ∈ E, the unique solution to (27) satisfies, for any s ∈ R, Proof.
1. The first point is just a consequence of the fact that E µ and E ⊥ µ are stable by L = A + I ⊗ I ⊗ C. A straightforward computation, using that A commutes with I ⊗ I ⊗ C shows that and the claim follows by the differential form of Gronwall's Lemma. 2. We simply observe that P µ y and P ⊥ µ y solve the same equation as y with initial conditions P µ y 0 ∈ E ⊥ 0 , P ⊥ µ y 0 ∈ E ⊥ µ . It is then enough to use the inequality of the first point to conclude.

3.3.2.
Partial controllability results. The following proposition is a partial version of Theorem 3.1. More precisely, we establish the existence of a control v that reduces the norm of the projection on E µ of the final state y(T ) as much as possible. However, the bound on the control is not uniform with respect to µ yet. and Proof. Let q T ∈ E µ and q be the solution of the backward equation Since E µ is stable by L * , we can decompose q in the following way The new variable z j (t) := q j (t)e λ2,j (T −t) satisfies the following system Thanks to Assumption 1, we can apply Lemma 2.2 in the space and therefore, coming back to the variable q j , we get and thus, Using the second inequality in Lemma 3.2, we get

DAMIEN ALLONSIUS AND FRANCK BOYER
Applying now the first inequality in Lemma 3.2 and the following inequality λ s e −2λT ≤ We shall now apply Assumption 2. To this end we choose any orthonormal basis (Ψ k ) k=1,...,K of U 1 ⊗ R n and we decompose each B * 1 ⊗ B * q j (t) in this basis, the coefficients being denoted by a k,j (t). It follows For any k ∈ {1, ..., K}, given that 0 < µ < µ E 2 , we can apply the discrete Lebeau-Robbiano inequality given by Assumption 2 to the vector ψ = λ2,j ≤µ a k,j (t)φ 2,j , to obtain λ2,j ≤µ Apply now (22) and the fact that for any k, Ψ k 0,1 = 1: Hence, Note that, by construction, E µ is stable by A so that (30) is valid for any q T ∈ F T = D −s E µ . Thus, we can apply Lemma 2.2 in the space E with D = A, F 0 = E and F T = E µ . It follows that there exists a control v ∈ L 2 (0, T ; U ) such that Finally, inequality (29) comes from Proposition 2.1 and the estimate on the control cost just proved which give Those estimates can easily be put into the expected form for a suitable choice of M part .
The following corollary contains the main idea of Lebeau and Robbiano's strategy: during the first half of a given time interval, we control the lowest frequencies then, during the second half of the time interval, we turn the control to zero to take advantage of the natural dissipation of the problem.
Proof. We apply Proposition 3.4 on the time interval (0, τ /2). We get a control v and a solution y which satisfy (28) (with T replaced by τ /2). Then, on the interval (τ /2, τ ), we set the control to zero. Thus, we have constructed the following control v(t) := v(t) for t ∈ (0, τ /2) 0 for t ∈ (τ /2, τ ), and the associated solution of the system is still denoted by y. Clearly, the L 2 norm ofv on (0, τ ) is equal to that of v on (0, τ /2). 1. Since the control is 0 on (τ /2, τ ), the dissipation properties given in Proposition 3.3 yield Then, we use (28) to get 2. By the dissipation properties given in Proposition 3.3 we get Then inequality (29) of Proposition 3.4 (with still T replaced by τ /2) leads to We combine (31) and (32) to get the result for a suitable value of M LR depending on M part .
Observe that Corollary 3.5 is slightly different from the similar result in the classical Lebeau and Robbiano's strategy. Indeed, the modes corresponding to frequencies less than µ of the final state y(τ ) are not cancelled; they still exist but are controlled by the small term e −µ E 1 τ 2 . When applying the complete strategy on the interval (0, T ), one has to make sure that the term e −µ E 1 τ /2 is smaller than e −µτ /2 . This constraint is fulfilled in Theorem 3.1 by dealing with frequencies µ smaller than µ E 1 .

3.3.3.
Conclusion of the proof of the main theorem. We can now apply Lebeau-Robbiano's strategy, with a well chosen finite number of steps, and prove Theorem 3.1.
Proof. Without loss of generality we suppose that M LR ≥ ln (2).
• During the time interval (0, T 1 ) = (0, τ 1 ), we apply a control v 1 as given by Corollary 3.5 with µ = µ 1 (which applies here since µ 1 < µ * ≤ µ E 2 ) and we get • For any index j ≤ j * , we continue this procedure by applying Corollary 3.5 on time interval (T j−1 , T j ) = (T j−1 , T j−1 + τ j ) with µ = µ j . We get a control v j that satisfies Let us focus on the estimation of the term y(T j ). First, we use that e −µ E 1 τj /2 ≤ e −µj τj /2 and the relations It follows Hence since ln(2) ≤ M LR , inequality 1 ≤ 1 τj + τ j leads to which, associated with (36), finally yields Therefore, and with (35), When j = j * , we end up with (recall that µ * < µ j * +1 = 4µ j * ) Let us now estimate the control v. Taking back (34) for j ≥ 2, combined with (37) and (35), Note that, (33) gives thatβ := β/2−1− and the last inequality came from j≥0 e − M LRβ with M obs depending on M LR but not on T .
The last step of the proof consists in applying Corollary 3.5 on the interval (T j * , T ) to recover the estimates of Theorem 3.1.

Applications.
4.1. Dirichlet boundary null control of a semi-discrete cascade system on a rectangle. In this section, we will apply the general framework introduced above to prove the φ(h) null controllability properties for semi-discrete versions of the boundary control problem of the coupled system (6) in the case of a cascade form (that is for particular B and C given below). The penalization term φ(h) will be exponentially small in h, just like in similar results obtained in the literature and quoted in the introduction section 1.1.
The results are valid for the finite difference discretization in space of the system in any dimension. However, for the simplicity of the presentation we will only state and prove our theorem in dimension d = 2. Observe that other usual techniques (based on Carleman estimates, or on moments methods as in [2,10,9,8]) do not directly apply in this setting since we are considering boundary controls for multidimensional coupled systems with less controls than components in the system.
For any function f : Ω i → R, we will use the same letter f to denote the sampling of f on the grid (f (x i,j )) j ∈ E i or the multiplication by f operator in E i which means that, for any y ∈ E i , the vector f y ∈ E i is defined by Let A i be the self-adjoint operator in E i (that can be seen as a N i × N i matrix) corresponding to the discretization of the scalar 1D operator A i by the finite difference method, which is defined for any y ∈ E i by with the usual convention that y 0 = y Ni+1 = 0. Here, we have used the notation γ i,j+1/2 := γ i (x i,j+1/2 ), for i = 1, 2 and j ∈ {0, ..., N i − 1}, for the sampling of the diffusion coefficient γ i on the dual grid of the mesh of Ω i . At some point we will also need to consider the discrete Laplace operators ∆ i defined by

BOUNDARY CONTROL OF SEMI-DISCRETE PARABOLIC SYSTEMS IN MULTI-D 23
The discretization of the vector-valued 2D operator A we will be interested in is thus given by A = A 1 ⊗ I ⊗ I + I ⊗ A 2 ⊗ I , and, if we take into account the coupling terms, the complete discretization of the operator appearing in (6) is given by We introduce now the discrete control spaces and operators associated with our boundary control problem. We set U 2 = E 2 , and to avoid confusions, we will use the notation [•, •] 0,2 instead of •, • 0,2 when dealing with objects in U 2 .
Using the convention (43), we define the discrete control operator in Ω 2 by This means that for any y ∈ E 2 , Observe that this operator is self-adjoint in E 2 , that is B * 2 = B 2 . Since the domain Ω 1 is a 1D interval, the corresponding boundary control v is in fact a scalar control that lives in the space U 1 = R. However, the discretization A 1 of the operator A 1 is built upon the assumption that we are dealing with homogeneous Dirichlet boundary condition. Therefore, in order to include non-homogeneous Dirichlet boundary condition on the left end-point of Ω 1 in the discretization, we need to add a source term in the discretization which is given by For the analysis, it will be sightly more convenient to work with this operator in the following form where r is (the sampling of) the affine map defined by x ∈ Ω 1 → 1 − x/a 1 . This is the discrete counterpart of the boundary control operator as analyzed in [18,Chapter 2].
A simple computation shows that the adjoint of B 1 is given by where ∂ l q is the discrete normal derivative of q ∈ E 1 on the left boundary of the domain defined by Note that this formula takes into account implicitly the homogeneous Dirichlet boundary condition for q.
We can now precisely write the semi-discrete control problem that we consider

DAMIEN ALLONSIUS AND FRANCK BOYER
where y(t) ∈ E = E 1 ⊗ E 2 ⊗ R n and v(t) ∈ U = R ⊗ U 2 ⊗ R. Note that, in this particular case, the control space U can be in fact identified with E 2 .
The main theorem of this section is the following. The crucial point is that all the constants appearing in the estimates (47) do not depend on the discretization parameter h. In short, we prove that we can drive the semi-discrete system (46) to a target which is exponentially small with respect to h with controls that are uniformly bounded. Up to a subsequence, this results imply the weak convergence of the semi-discrete controls towards a control of the continuous problem which leads the solution to zero, as soon as the discrete initial data converges towards the suitable initial data.
Theorem 4.1. There exist C > 0, C > 0 and h 0 > 0, depending only on γ 1 , γ 2 and ω 2 such that, for any T > 0, any mesh such that h < h 0 , and any y 0 ∈ E there exists a control v ∈ L 2 (0, where y is the corresponding solution to the semi-discrete problem (46) with control v and h = max(h 1 , h 2 ) is the space discretization parameter.
Once such a theorem is proved, even with a non explicit/constructive proof, one can produce an optimization algorithm, based on the penalized HUM approach, that is able to compute a control v satisfying (47). We can even relax the requirements by replacing the exponential factor e −C/h 2 by any more convenient φ(h), such as φ(h) = h p for some large enough integer p. Those questions are discussed in details for instance in [7] where some numerical illustrations are given.

Additional notations and properties.
In order to simplify the presentation of the following proofs we need to introduce a few more notations. For any i ∈ {1, 2}, we define γ ± i to be the translated sampling of the diffusion coefficient γ i defined by (γ ± i ) j = γ i,j±1/2 , ∀j ∈ {1, ..., N i }. We also introduce the forward and backward difference operators ∇ ± i defined, for any y ∈ E i , by For any i ∈ {1, 2}, we define (e i,j ) 1≤j≤Ni to be the canonical basis of E i (each element corresponds to a point in the 1D grid of Ω i ). At some point in the forthcoming analysis, we shall need to work with compactly supported discrete functions in order to justify some discrete integrations by parts. To this end, we introduce the subspaces E 00 . With those notations, we observe that the gradient operators defined above satisfy the duality property which is obtained by a summation by parts.

Lemma 4.2.
There exists a C > 0 depending only on inf γ i , sup γ i , γ i L ∞ , a i such that: for any i ∈ {1, 2}, and any y ∈ E i , we have Remark 4.1. By combining the above properties, we can obtain the following estimate where C depends only on γ i and a i . This implies in particular that the Poincaré inequality (23) holds with a constant M P,1 uniform with respect to the discretization parameters.
Proof. The first inequality is very classical. We recall the sketch of proof: for i ∈ {1, 2}, we first write and by the Cauchy-Schwarz inequality, we obtain . . , N i }. The claim follows by multiplying by h i and summing over j. The proof with the operator ∇ + i , is done in the same way but starting from the equality For the third estimate, a simple calculation shows that and we deduce (50) with C depending only on min Ωi (γ i ). Finally, a straightforward algebraic computation gives where C only depends on min Ωi (γ i ). By using (50), the Cauchy-Schwarz inequality and the Young inequality, we finally deduce that where C only depends on the function γ i and we conclude by using (49) (where we take the square on both sides) and (50).

4.1.3.
Proof of the semi-discrete controllability result. The proof of Theorem 4.1 consists essentially in applying the analysis above to a suitable setting. To achieve the results, it is just needed to check that all the assumptions of Theorem 3.1 are satisfied with constants that do not depend on the discretization parameter.
• In a first step, we will check that the discrete diffusion operator L and the discrete control operator B = B 1 ⊗ B 2 ⊗ B are compatible in the sense that inequalities (14) and (15) are satisfied with s = 1, and D = A, uniformly with respect to the mesh size. • In a second step, using previous results in the literature, we shall prove that the discrete Lebeau-Robbiano spectral inequality (Assumption 2) and the semidiscrete controllability of the 1D system (Assumption 1) hold. Let us start by proving the following result.
Proof. The proof of (53) is exactly the same as the one of the dissipation estimates of Proposition 3.3, for µ = 0 and s = −1, except that we deal with the adjoint matrix C * . The more intricate part is now to prove (54). This will be a combination of discrete trace estimates and of discrete elliptic regularity properties for the operator A. In that proof the fact that we consider s = 1 in the definition of the norms is crucial, this is the discrete counter-part of the fact that, in the continuous setting, we need to take H 1 0 initial data to ensure that the normal derivative of the solution of the backward heat equation belongs to L 2 .
1. First, we prove a 1D trace inequality. More precisely, we show that there exists C > 0 independent of h 1 such that: To this end, we first use (45) and the fact that ∆ 1 is self-adjoint to get It follows that, for any q ∈ E 1 ⊗ R n , where C depends only on γ 1 and Ω 1 .
From (51) (that we apply on each component of the vector-valued unknown q), we deduce that which proves (55). 2. In the second step, we prove that the inequality (55) can be extended to 2D discrete unknowns. More precisely, we show that there exists a constant C > 0 independent of h 1 and h 2 such that We define f j = e 2,j / √ h 2 , in such a way that (f 2,j ) 1≤j≤N2 is an orthonormal basis of E 2 . Note that the particular structure of B 2 implies that B 2 f 2,j = α j f 2,j with α j ∈ {0, 1} (α j = 1 if and only if the mesh point x 2,j lies in ω 2 ).
We can decompose any q ∈ E into the unique form with q j ∈ E 1 ⊗ R n for any j. We have, by orthogonality of (f 2,j ) j , Using (55) for each j, and the fact that α 2 j ≤ 1, we get and, still by orthogonality of (f 2,j ) j , we obtain the claimed inequality (56), the constant C being the same as in (55). 3. In the estimate (56) we only have the operator A 1 ⊗ I ⊗ I that appears in the right-hand side and not the complete discrete 2D elliptic operator A. The third step consists in proving a discrete elliptic regularity property that will allow us to get that there exists a constant C > 0 independent of h 1 and h 2 such that B * q 2 0 ≤ C Aq 2 0 , ∀q ∈ E.
(57) The main idea is based on the following well-known computation: for any smooth and compactly supported function f defined on R 2 , we can write Moreover, by a double integration by parts, we find We want to apply the same idea to prove roughly speaking that , which, according to (56), would prove (57).
However, because of boundary terms, it is easier to prove this inequality for compactly supported discrete functions, that is for q ∈ E 00 1 ⊗ E 00 2 . We will thus proceed by extension and truncation of the discrete functions under study.

DAMIEN ALLONSIUS AND FRANCK BOYER
Let ξ 1 : (0, a 1 ) → R be a smooth function such that ξ 1 = 1 on (0, a1 3 ), and ξ 1 = 0 on ( 2a1 3 , a 1 ), and ξ 2 : (0, a 2 ) → R be a smooth compactly supported function such that ξ 2 = 1 on ω 2 . Such a function exists thanks to the assumption (42). We introduce now the truncation operator on E 1 ⊗ E 2 , defined by T = ξ 1 ⊗ ξ 2 . The choice of ξ 1 and ξ 2 implies that 2 0 , and thus by (56) we have . The discrete function (T ⊗ I )q vanishes near all the boundaries of the domain, except the one corresponding to {x 1 = 0} because of the choice of ξ 1 . We will now introduce a symmetrization procedure that will let us work with a compactly supported discrete function.
We start by introducing the extended space in the first variable defined by which stands for discrete functions defined on a uniform discretization of (−a 1 , a 1 ) with a mesh size h 1 . If we denote by (ē 1,j ) −N1≤j≤N1 the canonical basis of E 1 , we can define the odd symmetrization operator S : E 1 → E 1 by Se 1,j =ē 1,j −ē 1,−j , ∀j ∈ {1, ..., N 1 }. With this notation, for q ∈ E 1 ⊗ E 2 , (S ⊗ I)q is the odd symmetrization of q with respect to the x 1 variable in the extended 2D domain.
We considerγ 1 to be the even extension of γ 1 to (−a 1 , a 1 ) and let A 1 be the discrete diffusion operator associated withγ 1 and defined on E 1 in the same way as in (44).
A simple computation shows that the symmetrization is compatible with the diffusion operator definitions, that is In particular, using the same notation for the norm in E 1 as for the one in E 1 , we have A 1 Sq 2 0,1 = 2 A 1 q 2 0,1 , ∀q ∈ E 1 . As a consequence, all the previous estimates lead to the following inequality where, we have setq = (S⊗I⊗I )(T⊗I )q. Observe now that, by construction of the symmetrization and truncation operators, the discrete functionq belongs to E 00 1 ⊗ E 00 2 ⊗ R n . Note that for such compactly supported discrete functions we have the algebraic identities We can now make the following computation and the double product term can be evaluated as follows, the discrete integration by parts being justified by (48) and the fact that At the end, we obtain that To conclude the proof of the final claim, we just need to show that q. Since ξ 1 and ξ 2 are smooth and γ 1 and γ 2 are bounded, we conclude, with the mean-value theorem that The conclusion follows by (49), (50), (52) and the Cauchy-Schwarz and Young inequalities. 4. The last step of the proof consists in showing (54) by using (57) and a classical energy estimate. Let ψ ∈ E and let q(t) = e −tL * ψ. We set z(t) = I ⊗ I ⊗ e tC * q(t). One can check that z satisfies ∂ t z + Az = 0, We multiply (59) by Az and integrate on (0, T ), Applying (57) to q(t) for any t ∈ (0, T ) and integrating in time this inequality, we finally get and this concludes the proof of (54).

Remark 4.2.
Note that we just proved that M adm depends on T like e CT and this is consistent with estimates of v and y(T ) given by Theorem 4.1.

Now we can prove Assumption 2
Proof of Assumption 2. In [8] the authors proved a discrete Lebeau-Robbiano inequality on quite general meshes. We translate the statement of their Theorem 6.1 in our setting.
This result exactly yields that Assumption 2 is fulfilled in this setting with Finally, we prove that Assumption 1 holds.
Proof of Assumption 1. We base our proof on the strategy developed in [2], where the semi-discretized (on a uniform mesh) boundary null-control problem in space dimension 1 with operator A 1 is tackled by applying the moments method. However, the explicit dependence in T of the control cost by some bound in e C/T was not given in that work. This is crucial in the present analysis. We can actually obtain this precised bound by using the expression of the control obtained with this method, in setting (S2) described in [2], and by using the refined bounds on biorthogonal families to exponential functions given by Theorem 1.5 of [5]. We will just describe here the new estimate that we need to adapt the results of [2] to our needs.
Remark 4.3. In [5], Theorem 1.5, hypothesis of 6 on the counting function is slightly different from item 6 given above. Indeed, in this reference the authors require the following condition : for some p, α > 0, Actually, looking carefully at the proof of Theorem 1.5, we realize that we can use and using (60), thus there exists C 1 > 0 depending only on ε (whose value has been arbitrarily set to 1/2) and γ 1 such that which gives (63) since k 0 ≤ Still using the estimates in the proof of Theorem 5.4 of [2], we get and C 1 > 0 and C 2 > 0 depending only on γ 1 . Now we can apply Theorem 3.1 with µ E 1 and µ E 2 both of the same order C h 2 . The proof of Theorem 4.1 is complete.

4.2.
Dirichlet boundary null control of a continuous n-dimensional system on a cylindrical domain of dimension d. Our aim is to show how the finite dimensional framework developed in Section 3 actually applies to the study of nullcontrollability problems of tensorized parabolic systems.
As an illustration of this statement, we shall give a short proof of the main result of [5] by using Theorem 3.1 of the present article. This is of course not surprising since our approach is directly inspired from the one developed in [5]. However, it seems to us interesting to show how our general finite dimensional framework actually encompasses already known results through a spectral projection technique. This example should convince the reader that, by using the same strategy, one can easily adapt the proof to other kinds of tensorized controlled systems like, for instance, Neumann or Robin boundary controls, or even mixed (distributed and boundary) controls, as soon as we have in hand a suitable controllability result on the associated 1D system.
Finally, since we have taken care of all the constants in the proofs, this strategy can be used to derive controllability properties that are uniform with respect to some parameters present in the problem. As an illustration, in the case of a 1D Robin boundary control problem, we can prove estimates that are uniform in the Robin parameter (see [6]). By the present technique, those result will automatically be translated to the corresponding multi-D result, generalizing the one proved in [5].
Let us recall the statement of [5, Theorem 1.3] with the notation of the present paper. We suppose given a coupling matrix C ∈ M n (R) and a control matrix B ∈ M n,m (R). Assume that the following system of n equations      ∂ t y 1 + A 1 ⊗ I y 1 + I ⊗ Cy 1 = 0, in (0, T ) × Ω 1 , is null controllable for any y 1 0 ∈ H −1 (Ω 1 )⊗R n and any time T > 0 with, in addition, the following bound: Then, for any nonempty open set ω 2 ⊂ Ω 2 , the following system of n equations: where L := A 1 ⊗ I ⊗ I + I ⊗ A 2 ⊗ I + I ⊗ I ⊗ C, is null controllable for any y 0 ∈ H −1 (Ω) ⊗ R n and any time T > 0 with, in addition, the following bound: Proof. Let y 0 ∈ H −1 (Ω) ⊗ R n , recall that, by definition, the solution y of (65) satisfies for any ψ ∈ H 1 0 (Ω) ⊗ R n , the equality Therefore, v is a null-control for this system, if and only if, it satisfies for any ψ ∈ H 1 0 (Ω) ⊗ R n . Actually, it is enough to check the previous equality for any ψ belonging to a total family of H 1 0 (Ω) ⊗ R n . Let us consider the total family of H 1 0 (Ω) made of the eigenfunctions of A 1 and A 2 . For i = 1, 2, we denote by (φ i,j , λ i,j ) j≥1 the eigenfunctions and eigenvalues of the operator A i with homogeneous Dirichlet boundary conditions. We choose them to form an orthonormal basis of L 2 (Ω i ). We will thus consider the total family of H 1 0 (Ω) defined by (φ 1,j1 ⊗ φ 2,j2 ) j1,j2 that will be tensorized with the canonical basis of R n , (e k ) k∈{1,...,n} to finally produce a total family of H 1 0 (Ω) ⊗ R n . We are thus led to find a control satisfying, ∀j 1 , j 2 ≥ 1 and ∀k ∈ {1, . . . , n}, e −T (λ1,j 1 +λ2,j 2 ) y 0 , φ 1,j1 ⊗ φ 2,j2 ⊗ (e −T C * e k ) In order to apply Theorem 3.1, we need to consider a projection of (65) on a finite dimensional space. For i = 1, 2 and J ≥ 1, we define the spaces F i,J = span (φ i,j , j ≤ J) , and the operator A i : Let P i,J be the orthogonal projection from the space H −1 (Ω i ) equipped with the inner product •, • −1,Ai onto the space F i,J .
Appendix A. Proof of Lemma 2.2.
1. ⇒ 2. For any q T ∈ D −s F T , we set q(t) = e −(T −t)L * q T and we consider the initial data y 0 := P F0 D s P * F0 q(0); by assumption there exists a control v and an associated solution y satisfying (18).
Since D s q T ∈ F T and P F T is orthogonal with respect to •, • −s,D , we deduce P * F0 (q(0)) Appendix B. Numerical Illustrations. In this section we give a numerical illustration of the boundary controllability of a 2D parabolic system of the cascade form using the HUM approach (see [7] for an introduction to this method and also the last paragraph of section 4.1.1). We consider a discretization in time of system (4) of coupled heat-like equations, with a constant diffusion coefficients equal to 0.05 in both directions, on a unit square with a uniform discretization of 50 mesh points in both directions and 100 times steps. The time horizon is T = 1, the coupling coefficient is equal to 10 and the initial condition of the controlled equation is α 0 = 0 and the initial condition of the second component is β 0 = sin(πx) sin(πy). The boundary control is acting on a part of the boundary situated on three of the edges of the square namely Observe that Theorem 4.1 ensures that the controllability properties of the discrete system holds even if the control only acts on one of those three parts. In this fully discretized framework, we make the parameter ε of the (fully-discrete version of the) penalised functional F ε (v) := 1 2 v 2 L 2 (0,T ;U ) + 1 2ε α(T ) 2 + β(T ) 2 depend on the mesh parameter h and we set ε = 0.3 × h 2 . We give two sets of simulations below for different times t ∈ {0, T /3, 2T /3, T }.
In figure 3, we set the control to zero and we plot solutions α and β. We see on figure 3d that the second component β of the system at time T is not equal to zero since the solution only decreases because of the dissipation of the heat equation. The first component α remains equal to zero since its initial condition is zero and it is not controlled.
In figure 4, however, we see how the control affects the first component of the system α so that it can drive both components of the system to zero.
In figure 5 we plot the norms of the two components t → α(t) and t → β(t) as a function of time. When no control is applied to the system (dashed lines), we observe that the norm of the component β decreases until the final time T where it is close to 0.2, whereas α remains constant equal to zero. However, when applying the control (solid lines), α does not remain constant anymore, and both components α and β eventually get close to zero. We recall that we do not exactly reach zero since we used a penalized HUM approach with a penalty term depending on the mesh size.