Fatiha Alabau-Boussouira

Matthieu Léautaud
email: leautaud@ann.jussieu.fr

Indirect controllability of locally coupled wave-type systems and applications

Keywords: Observability, controllability, wave equation, hyperbolic systems, parabolic systems, geometric conditions

We consider symmetric systems of two wave-type equations only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. We deduce similar nullcontrollability results in any positive time for parabolic systems and Schrödinger-type systems under the same geometric conditions on the coupling and the control regions. This includes several examples in which these two regions have an empty intersection.

Introduction 1.Motivation

During the last decade, the controllability properties of coupled parabolic equations like

       ∂ t u 1 -∆ c u 1 + au 1 + pu 2 = bf in (0, T ) × Ω, ∂ t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 )| t=0 = (u 0 1 , u 0 2 ) in Ω,
(1.1) have been intensively studied. Here, the coefficients p and b are smooth non-negative functions on the bounded domain Ω and -∆ c is a second order selfadjoint elliptic operator. The null-controllability problem under view is the following: given a time T > 0 and initial data, is it possible to find a control function f so that the state (u 1 , u 2 ) is driven to zero in time T ? It has been proved in [dT00, AKBD06, GBPG06, Léa10] with different methods that System (1.1) is null-controllable as soon as {p > 0} ∩ {b > 0} = ∅. In these works, the case {p > 0} ∩ {b > 0} = ∅ has been left as an open problem. However, Kavian and de Teresa [START_REF] Kavian | Unique continuation principle for systems of parabolic equations[END_REF] have proved for a cascade system (i.e. without the term pu 2 in the first equation of (1.1)) that approximate controllability holds. The natural question is then whether or not null-controllability (which is a stronger property) still holds in the case {p > 0} ∩ {b > 0} = ∅: Question 1: is System (1.1) null-controllable in the case {p > 0} ∩ {b > 0} = ∅?

The second Problem under interest here is the boundary controllability of systems like

       ∂ t u 1 -∆ c u 1 + au 1 + pu 2 = 0 in (0, T ) × Ω, ∂ t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = b ∂ f, u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 )| t=0 = (u 0 1 , u 0 2 ) in Ω, (1.2)
where b ∂ is a smooth function on ∂Ω. The recent work [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF] studies such systems in one space dimension and with constant coupling coefficients. The cases of higher space dimensions and varying coupling coefficients (and in particular when the coefficients vanish in a neighborhood of the boundary) are to our knowledge completely open. The second question under interest is then: Question 2: is System (1.2) null-controllable for non-constant coupling coefficients p? Is System (1.2) null-controllable if the dimension of Ω is larger than one?

Concerning these two open problems, it seems that the parabolic theory and associated tools encounter for the moment some essential difficulties.

On the other hand, it is known from [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF] that controllability properties can be transferred from hyperbolic equations to parabolic ones. And it seems, at least for boundary control problems, that the theory for coupled hyperbolic equations of the type

       ∂ 2 t u 1 -∆ c u 1 + au 1 + pu 2 = bf in (0, T ) × Ω, ∂ 2 t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 )| t=0 = (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) in Ω, (1.3) and        ∂ 2 t u 1 -∆ c u 1 + au 1 + pu 2 = 0 in (0, T ) × Ω, ∂ 2 t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = b ∂ f , u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 )| t=0 = (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) in Ω,
(1.4) is better understood (see [START_REF] Alabau-Boussouira | Indirect boundary observability of a weakly coupled wave system[END_REF][START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]), even less studied. In the case of varying coefficients and several space-dimensions, the associated stabilization problem is addressed in [AB99, [START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled systems[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF]. In particular in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF], a polynomial stability result is proved for solutions of the system

   ∂ 2 t u 1 -∆ c u 1 + au 1 + pu 2 + b∂ t u 1 = 0 in R + × Ω, ∂ 2 t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in R + × Ω, u 1 = u 2 = 0 on R + × ∂Ω, (1.5)
in some cases where {p > 0} ∩ {b > 0} = ∅. This gives hope to prove some null-controllability results for (1.3) in the same situations.

In the present work, we answer Questions 1 and 2 for hyperbolic problems (like (1.3) and (1.4)), improving the results of [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF]. Then, we deduce a (partial) solution to the two open questions raised above for parabolic systems. Indeed, we prove that Systems (1.1)-(1.3) are null-controllable (in appropriate spaces) as soon as {p > 0} and {b > 0} both satisfy the Geometric Control Condition (recalled below) and p L ∞ (Ω) satisfies a smallness assumption. This contains several examples with {p > 0} ∩ {b > 0} = ∅, and partially answer to the first question. We prove as well that similar controllability results hold for the boundary control problems (1.2) and (1.4), which partially answers to the second question. Of course, the geometric conditions needed here are essential (and even sharp) for coupled waves, but inappropriate for parabolic equation. However, this is a first step towards a better understanding of these types of systems. In one space dimension in particular, the geometric conditions are reduced to a non-emptiness condition and are hence optimal for parabolic systems as well.

In the end of the present introduction, we state our main results for wave/heat/Shrödinger-type Systems. In Section 2, we introduce an abstract setting adapted to second order (in time) control problems. Then, in Section 3, we present the tools used in the proof of the main theorem, together with a key lemma: an observability inequality for an equation with a right hand-side (for which we give another proof in Appendix 7). Section 4 is devoted to the proof the observability of hyperbolic systems in the abstract setting, and controllability is deduced in Section 5. Finally, in Section 6, we come back to the applications to wave/heat/Shrödinger-type Systems. The results of this paper were announced in [START_REF] Alabau-Boussouira | Indirect controllability of locally coupled systems under geometric conditions[END_REF].

Main results

Let Ω be a bounded domain in R n with smooth (say C ∞ ) boundary (or a smooth connected compact Riemannian manifold with or without boundary) and ∆ c = div(c∇) a (negative) elliptic operator (or the Laplace Beltrami operator with respect to the Riemannian metric) on Ω. Here, c denotes a smooth (say C ∞ ) positive symmetric matrix i.e. in particular C -1 0 |ξ| 2 ≤ c(x)ξ • ξ ≤ C 0 |ξ| 2 for some C 0 > 0, for all x ∈ Ω, ξ ∈ R n . We consider the more general first order (in time) control problem

       e iθ ∂ t u 1 -∆ c u 1 + au 1 + pu 2 = bf in (0, T ) × Ω, e iθ ∂ t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 )| t=0 = (u 0 1 , u 0 2 ) in Ω, (1.6) with θ ∈ [-π/2, π/2],
including Schrödinger-type systems for θ = ±π/2 and diffusion-type systems for θ ∈ (-π/2, π/2). In particular we recover System (1.1) when taking (1.6) for θ = 0. We also consider System (1.3), consisting in a wave-type system, with only one control force. In these systems, a = a(x), p = p(x) and b = b(x) are smooth real-valued functions on Ω and f is the control function, that can act on the system.

We shall also consider the same systems controlled from the boundary through the (smooth) real-valued function b ∂ :

       e iθ ∂ t u 1 -∆ c u 1 + au 1 + pu 2 = 0 in (0, T ) × Ω, e iθ ∂ t u 2 -∆ c u 2 + au 2 + pu 1 = 0 in (0, T ) × Ω, u 1 = b ∂ f , u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 )| t=0 = (u 0 1 , u 0 2 ) in Ω, (1.7)
which includes System (1.2) when θ = 0. We first notice that, on the space L 2 (Ω) 2 endowed with the natural inner product

(U, V ) L 2 (Ω)×L 2 (Ω) = (u 1 , v 1 ) L 2 (Ω) + (u 2 , v 2 ) L 2 (Ω) , U = (u 1 , u 2 ), V = (v 1 , v 2 ),
the operator

A p = -∆ c + a p p -∆ c + a , with domain D(A p ) = H 2 (Ω) ∩ H 1 0 (Ω) 2 , is selfadjoint. As a consequence, for f ∈ L 2 ((0, T ) × Ω), the Cauchy problem (1.6), resp. (1.3), is well-posed in L 2 (Ω) 2 , resp. H 1 0 (Ω) 2 × L 2 (Ω)
2 , in the sense of semigroup theory. Then, taking f ∈ L 2 ((0, T ) × ∂Ω) the initial-boundary value problem (1.7), resp.

(1.4), is well-posed in

H -1 (Ω) 2 , resp. L 2 (Ω) 2 × H -1 (Ω)
2 , in the sense of transposition solution (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF][START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]).

The strategy we adopt here is to prove some controllability results for the hyperbolic systems (1.3) and (1.4), extending the two-levels energy method introduced in [AB03]. Then, using transmutation techniques, we deduce controllability properties of (1.6) and (1.7).

An important remark to make before addressing the controllability problem for the hyperbolic systems (1.3)-(1.4) is concerned with the regularity of solutions. If one takes for system (1.3

) (resp. (1.4)) initial data (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ H 1 0 (Ω) × H 2 ∩ H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) (resp. (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ L 2 (Ω) × H 1 0 (Ω) × H -1 (Ω) × L 2 (Ω)), and a control function f ∈ L 2 ((0, T ) × Ω) (resp. f ∈ L 2 ((0, T ) × ∂Ω)), then the state (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 ) remains in the space H 1 0 (Ω) × H 2 ∩ H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) (resp. L 2 (Ω) × H 1 0 (Ω) × H -1 (Ω) × L 2 (Ω)
) for all time. We recall that for Systems (1.3) and (1.4), the nullcontrollability is equivalent to the exact controllability. As a consequence, it is not possible, taking for instance zero as initial data to reach any target state in

H 1 0 (Ω) 2 × L 2 (Ω) 2 (resp. L 2 (Ω) 2 × H -1 (Ω) 2 ).
The controllability question for (1.3)-(1.4) hence becomes: starting from rest at time t = 0, is it possible to reach all

H 1 0 (Ω) × H 2 ∩ H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) (resp. L 2 (Ω) × H 1 0 (Ω) × H -1 (Ω) × L 2 (Ω)) in time t = T sufficiently large?
To state our results, we recall the classical Geometric Control Conditions GCC (resp. GCC ∂ ). We recall that GCC was introduced in [RT74] for manifolds without boundaries, in [START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF] for domains with boundaries and that GCC ∂ was introduced in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. From these works, it is known that, making the generic assumption that ∂Ω has no contact of infinite order with its tangents, GCC (resp. GCC ∂ ) is a necessary and sufficient condition for the internal (resp. boundary) observability and controllability of one wave equation (see also [START_REF] Burq | Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes[END_REF]).

Definition 1.1 (GCC (resp. GCC ∂ )). Let ω ⊂ Ω (resp. Γ ⊂ ∂Ω) and T > 0. We say that the couple (ω, T ) satisfies GCC (resp. (Γ, T ) satisfies GCC ∂ ) if every generalized geodesic (i.e. ray of geometric optics) traveling at speed one in Ω meets ω (resp. meets Γ on a non-diffractive point) in a time t < T . We say that ω satisfies GCC (resp. Γ satisfies GCC ∂ ) if there exists T > 0 such that (ω, T ) satisfies GCC (resp. (Γ, T ) satisfies GCC ∂ ).

We shall make the following key assumptions.

Assumption 1.2. (i) We have (-∆ c + a)u, u L 2 (Ω) ≥ λ 0 u 2 L 2 (Ω) , for some λ 0 > 0, for all u ∈ H 2 (Ω) ∩ H 1 0 (Ω) . (ii) We have p ≥ 0 on Ω, {p > 0} ⊃ ω p
for some open subset ω p ⊂ Ω and we set p

+ := p L ∞ (Ω) . (iii) We have b ≥ 0 on Ω, {b > 0} ⊃ ω b (resp. b ∂ ≥ 0 on ∂Ω and {b ∂ > 0} ⊃ Γ b ) for some open subset ω b ⊂ Ω (resp. Γ b ⊂ ∂Ω).
Note that in the case where c = Id and a = 0, the best constant λ 0 is the smallest eigenvalue of the Laplace operator on Ω with Dirichlet boundary conditions. We also have the identity λ 0 = 1/C 2 P , where C P is the Poincaré's constant of Ω.

We shall also require that the operator A p satisfies, for some constant C > 0,

A p (v 1 , v 2 ), (v 1 , v 2 ) L 2 (Ω)×L 2 (Ω) ≥ C v 1 2 H 1 0 (Ω) + v 2 2 H 1 0 (Ω)
for all (v 1 , v 2 ) ∈ D(A p ). This is the case when assuming p + < λ 0 .

In the case of coupled wave equations, our main result can be formulated as follows.

Theorem 1.3 (Wave-type systems). Suppose that (i) holds, that ω p satisfies GCC and that ω b (resp. Γ b ) satisfies GCC (resp. GCC ∂ ). Then, for all b (resp. b ∂ ) satisfying (iii), there exists a constant p * > 0 (depending only on the geometry of Ω and on b, resp. b ∂ ) such that for all p + < p * , there exists a time T * > 0 such that for all T > T * , all p satisfying (ii), and all initial data

(u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ H 1 0 (Ω) × H 2 ∩ H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) (resp. (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ L 2 (Ω) × H 1 0 (Ω) × H -1 (Ω) × L 2 (Ω)), there exists a control function f ∈ L 2 ((0, T )×Ω) (resp. f ∈ L 2 ((0, T )×∂Ω)) such that the solution of (1.3) (resp. (1.4)) satisfies (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 )| t=T = 0.
Another way to formulate this result is to say that, under the assumptions of Theorem 1.3, the reachable set at time T > T * with zero initial data is exactly

H 1 0 (Ω) × H 2 ∩ H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) in the case of L 2 internal control and L 2 (Ω) × H 1 0 (Ω) × H -1 (Ω) × L 2 (Ω)
in the case of L 2 boundary control. Some comments should be made about this result. First this is a generalization of the work [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] where the coupling coefficients considered have to satisfy p(x) ≥ C > 0 for all x ∈ Ω. The geometric situations covered by Theorem 1.3 are richer, and include in particular several examples of coupling and control regions that do not intersect (see Figure 1). Second, we do not know if the coercivity assumption (i) for -∆ c + a and the smallness assumption on p + are only technical and inherent to the method we use here. Moreover, the control time T * we obtain depends on all the parameters of the system, and not only the sets ω p , ω b and Γ b (as it is the case for a single wave equation). This feature does not seem to be very natural. Finally, the fact that we consider twice the same elliptic operator ∆ c is a key point in our proof and it is likely that this result does not hold for waves with different speeds (see [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] for results with different speeds and different operators). Similarly, the fact that p ≥ 0 (see Assumption(ii)) is very important here. It is possible that Theorem 1.3 does not work if the sign of p varies. As a consequence of Theorem 1.3 and using transmutation techniques (due to [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF][START_REF] Miller | Controllability cost of conservative systems: resolvent condition and transmutation[END_REF] for heattype equations and to [START_REF] Phung | Observability and control of Schrödinger equations[END_REF][START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF] for Schrödinger-type equations), we can now state the associated results for Systems (1.6) and (1.7).

Corollary 1.4 (Heat-type systems). Suppose that (i) holds, that ω p satisfies GCC and that ω b (resp. Γ b ) satisfies GCC (resp. GCC ∂ ). Then, for all b (resp. b ∂ ) satisfying (iii), there exists a constant p * > 0 (depending only on the geometry of Ω and on b, resp. b ∂ ) such that for all p + < p * , for all T > 0, θ ∈ (-π/2, π/2), for all p satisfying (ii) and all initial data

(u 0 1 , u 0 2 ) ∈ L 2 (Ω) 2 (resp. (u 0 1 , u 0 2 ) ∈ H -1 (Ω) 2 ),
there exists a control function f ∈ L 2 ((0, T ) × Ω) (resp. f ∈ L 2 ((0, T ) × ∂Ω)) such that the solution of (1.6) (resp. (1.7)) satisfies (u 1 , u 2 )| t=T = 0.

To our knowledge, this corollary gives the first controllability result for coupled parabolic symmetric equations when the coupling region ω p and the control region ω b do not intersect. Moreover, this seems to be also the first positive result for boundary control of parabolic symmetric systems in several space dimensions or with variable coupling coefficients. A one-dimensional controllability result has been obtained in [START_REF] Rosier | Exact controllability of a cascade system of conservative equations[END_REF] by a different method [START_REF] Dáger | Insensitizing controls for the 1-D wave equation[END_REF] for systems of two parabolic equations in cascade, that is when the coupling term vanishes in the uncontrolled equation.

Note that the spaces for which the controllability result of Corollary 1.4 holds are symmetric. This is due to the smoothing effect of parabolic equations. The proof of this result is given in Section 6.2.

The geometric conditions in this theorem are not sharp for parabolic equations. This leads us to think that the same result still holds under the only conditions ω p = ∅ (i.e. p is not the null function) and ω b = ∅ (i.e. b is not the null function). This remains an open problem. Note that our control result holds for any T > 0, which is natural for parabolic equations.

Concerning coupled Schrödinger equations, we have the following result.

Corollary 1.5 (Schrödinger-type systems). Suppose that (i) holds, that ω p satisfies GCC and that ω b (resp. Γ b ) satisfies GCC (resp. GCC ∂ ). Then, for all b (resp. b ∂ ) satisfying (iii), there exists a constant p * > 0 (depending only on the geometry of Ω and on b, resp. b ∂ ) such that for all p + < p * , for all T > 0, all p satisfying (iii) and all initial data

(u 0 1 , u 0 2 ) ∈ L 2 (Ω) × H 1 0 (Ω) (resp. (u 0 1 , u 0 2 ) ∈ H -1 (Ω) × L 2 (Ω))
, there exists a control function f ∈ L 2 ((0, T ) × Ω) (resp. f ∈ L 2 ((0, T ) × ∂Ω)) such that the solution of (1.6) (resp. (1.7)) with θ = ±π/2 satisfies (u 1 , u 2 )| t=T = 0.

The proof of this result is given in Section 6.3. Since there is no smoothing effect in this case, we still obtain a controllability result in asymmetric spaces here: the uncontrolled variable u 2 has to be more regular than the other one. This shows that the attainable set from zero for a L 2 internal control (resp.

L 2 boundary control) contains L 2 (Ω) × H 1 0 (Ω) (resp. H -1 (Ω) × L 2 (Ω)). Whether or not a general target in L 2 (Ω) 2 (resp. H -1 (Ω)
2 ) is reachable for (1.6) (resp. (1.7)) with θ = ±π/2 remains open.

Note finally that the geometric conditions GCC and GCC ∂ are not necessary in the case of Schrödinger equations but are not far from being optimal. The most general control result for a single Schrödinger equation [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF] is that GCC (resp. GCC ∂ ) implies null-controllability in any positive time. However, in some cases (see [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF][START_REF] Burq | Geometric control in the presence of a black box[END_REF]), these conditions are not necessary. Here, we do not recover these properties (since our result is deduced from a controllability result for waves). It would be interesting to prove such a result with weaker geometric conditions in particular situations (see [START_REF] Rosier | Exact controllability of a cascade system of conservative equations[END_REF] in the case of a square). Controllability results have been obtained in [START_REF] Rosier | Exact controllability of a cascade system of conservative equations[END_REF] with a different method based on [START_REF] Dáger | Insensitizing controls for the 1-D wave equation[END_REF] in the periodic case for cascade systems of two Schrödinger equations. Note also that our control result holds for any T > 0, which is natural for Schrödinger equations [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF] (and which is not the case in the results of [START_REF] Rosier | Exact controllability of a cascade system of conservative equations[END_REF]).

Remark 1.6. Different boundary conditions (like Neumann or Fourier boundary conditions) can also be addressed with the same techniques since we use the observability inequality for a single wave equation as a black box. In the work [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], the authors prove this observability inequality with all these boundary conditions (all compatible with the Melrose-Sjöstrand theorem of propagation of singularities).

Remark 1.7. Similarly, our results also hold (in the case of Dirichlet boundary conditions) for a boundary ∂Ω and a metric c having a limited smoothness (i.e. C 3 for ∂Ω and C 2 for c) according to the article [START_REF] Burq | Contrôlabilité exacte des ondes dans des ouverts peu réguliers[END_REF]. Note as well that we supposed that the coefficients a, p and b are smooth. It is sufficient that a and p preserve the regularity of u 1 and u 2 and b (resp. b ∂ ) that of the control function f . For instance, one can take

b ∈ L ∞ (Ω) (resp. b ∂ ∈ L ∞ (∂Ω)) and a, p ∈ W 2,∞ (Ω).
Remark 1.8. We could also replace the operator A p in all the systems studied here, by the operator

A δ = -∆ c + a δp p -∆ c + a , for δ > 0.
This is what we did in [START_REF] Alabau-Boussouira | Indirect controllability of locally coupled systems under geometric conditions[END_REF]. This operator is selfadjoint on L 2 (Ω) 2 endowed with the

inner product (U, V ) δ = (u 1 , v 1 ) L 2 (Ω) + δ(u 2 , v 2 ) L 2 (Ω)
. The controllability results obtained in this case (for all equations, as well as in the abstract setting) hold for all (δ, p + ) such that √ δp + < p * . Such results in this setting seem more general since they allow to consider large p + or large δ (provided that the other is small enough). For these choices of δ, the systems obtained are "less symmetric" than the ones for δ = 1.

However, we can pass from the system with δ to the system without δ with a change of variables. Suppose that (z 1 , z 2 ) is the solution of the system

       ∂ 2 t z 1 -∆ c z 1 + az 1 + δpz 2 = bf in (0, T ) × Ω, ∂ 2 t z 2 -∆ c z 2 + az 2 + pz 1 = 0 in (0, T ) × Ω, z 1 = z 2 = 0 on (0, T ) × ∂Ω, (z 1 , z 2 , ∂ t z 1 , ∂ t z 2 )| t=0 = (z 0 1 , z 0 2 , z 1 1 , z 1 2 ) in Ω,
then, setting u 1 = z 1 and u 2 = √ δz 2 , the new variable (u 1 , u 2 ) satisfies the fully symmetric system

       ∂ 2 t u 1 -∆ c u 1 + au 1 + √ δpu 2 = bf in (0, T ) × Ω, ∂ 2 t u 2 -∆ c u 2 + au 2 + √ δpu 1 = 0 in (0, T ) × Ω, u 1 = u 2 = 0 on (0, T ) × ∂Ω, (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 )| t=0 = (z 0 1 , √ δz 0 2 , z 1 1 , √ δz 1 2 ) in Ω,
to which Theorem 1.3 applies (for √ δp + < p * ).

Abstract setting

In this section, we describe the abstract setting in which we prove Theorem 1.3 for Systems (1.3) and (1.4), and define the appropriate spaces and operators. Let H be a Hilbert space and (A, D(A)) a selfadjoint positive operator on H with compact resolvent. We denote by (•, •) H the inner product on H and • H the associated norm. For k ∈ N, we set

H k = D(A k 2 ) endowed with the inner product (•, •) H k = (A k 2 •, A k 2 •) H and associated norm • H k = A k 2 • H . We define H -k as the dual space of H k with respect to the pivot space H = H 0 . We write •, • H k ,H -k = (A k 2 •, A -k 2 •) H the duality product between H k and H -k , and • H -k = A -k 2 • H is the norm on H -k .
The operator A can be extended to an isomorphism from H k to H k-2 for any k ≤ 1, still denoted by A. According to the properties of the operator A, the injection H k ֒→ H k-1 is dense and compact for any k ∈ Z.

We denote by λ 0 > 0 the largest constant satisfying

v 2 H1 ≥ λ 0 v 2 H for all v ∈ H 1 , (2.1)
that is, the smallest eigenvalue of the selfadjoint positive operator A. Note that we also have, for all α ≥ 0,

A -α L(H) = λ -α 0 .
In this abstract setting, we shall denote ϕ ′ the derivative with respect to time of a function ϕ : R → H k , for some k ∈ Z. In the following, as in [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], we shall make use of the different energy levels

e k (ϕ(t)) = 1 2 ϕ(t) 2 H k + ϕ ′ (t) 2 H k-1 , k ∈ Z
which are all preserved through time if ϕ is a solution of ϕ ′′ +Aϕ = 0. Moreover, the coercivity assumption (2.1) yields, for all

k ∈ Z, v ∈ H k , v 2 H k ≥ λ 0 v 2 H k-1 and e k (v) ≥ λ 0 e k-1 (v). (2.2)
We consider that the coupling operator P is bounded on H and denote by P * is its adjoint, p + := P L(H) = P * L(H) . Before addressing the control problem, let us introduce the adjoint system

   v ′′ 1 + Av 1 + P v 2 = 0, v ′′ 2 + Av 2 + P * v 1 = 0, (v 1 , v 2 , v ′ 1 , v ′ 2 )| t=0 = (v 0 1 , v 0 2 , v 1 1 , v 1 2 ), (2.3)
which shall stand for our observation system. This system can be recast as a first order differential equation

V ′ = A P V, V(0) = V 0 , (2.4)
where

A P = 0 Id -A P 0 , A P = A P P * A , V = (v 1 , v 2 ), V = (V, V ′ ) = (v 1 , v 2 , v ′ 1 , v ′ 2 ).
Note that the operator A P is selfadjoint on the space H ×H endowed with the inner product V,

Ṽ H×H = (v 1 , ṽ1 ) H + (v 2 , ṽ2 ) H . Using (2.2) with k = 1, we obtain A P V, V H×H = (Av 1 , v 1 ) H + (Av 2 , v 2 ) H + 2(P v 2 , v 1 ) H ≥ 1 - p + λ 0 v 1 2 H1 + v 2 2 H1 . (2.5)
As a consequence, we shall suppose that p + < λ 0 , so that A P is coercive. Under this assumption, A

1 2 P V, A 1 2 P Ṽ
H×H defines an inner product on (H 1 ) 2 , equivalent to the natural one. Assuming that P, P * ∈ L(H k ) and writing

H k = (H k ) 2 × H 2 k-1 , k ∈ Z, the operator A P is an isomorphism from H k to H k-1 and is skewadjoint on H k , equipped with the inner product (U, V ), ( Ũ , Ṽ ) H k = (A k 2 P U, A k 2 P Ũ ) H×H + (A k-1 2 P V, A k-1 2 P Ṽ ) H×H .
Note that this is an inner product according to the coercivity assumption for A P , which is equivalent to the natural inner product of H k . Hence, A P generates a group e tA P on H k , and the homogeneous problem (2.3) is well-posed in these spaces. An important feature of solutions

V(t) of System (2.3) is that all energies E k (V(t)) = 1/2 V(t) 2 H k , k ∈ Z,
are positive and preserved through time.

Main results: admissibility, observability and controllability

For System (2.3), now studied in H 1 , we shall observe only the state of the first component, i.e. (u 1 , u ′ 1 ), and hence define an observation operator B * ∈ L(H 2 × H, Y ), where Y is a Hilbert space, standing for our observation space. This definition is sufficiently general to take into account both the boundary observation problem (taking B * ∈ L(H 2 , Y )) and the internal observation problem (taking B * ∈ L(H, Y )). We assume that B * is an admissible observation for one equation:

    
For all T > 0 there exists a constant C > 0, such that all solutions ϕ of ϕ

′′ + Aϕ = f ∈ L 2 (0, T ; H) satisfy T 0 B * (ϕ, ϕ ′ ) 2 Y dt ≤ C e 1 (ϕ(0)) + e 1 (ϕ(T )) + T 0 e 1 (ϕ(t))dt + T 0 f 2 H dt . (A1)
Under this assumption, we have the following lemma.

Lemma 2.1 (Admissibility). The operator B * is an admissible observation for (2.3). More precisely, for all p + < λ0 √ 2 and all T > 0, there exists a constant C > 0, such that all the solutions of (2.3) satisfy

T 0 B * (v 1 , v ′ 1 )(t) 2 Y dt ≤ C e 1 (v 1 (0)) + e 0 (v 2 (0)) . (2.6)
Note that only the e 0 energy level of the second component v 2 is necessary in this admissibility estimate. Hence, we cannot hope to observe the whole H 1 energy of V and the best observability we can expect only involves e 0 (v 2 ). Our aim is now to prove this inverse inequality of (2.6). For this, we have to suppose some additional assumptions on the operators P and B * . Let us first precise Assumption (A2), related with the operator P :

We have P v 2 H ≤ p + (P v, v) H and there exists an operator Π P ∈ L(H), Π P L(H) = 1, and a number p -> 0 such that

(P v, v) H ≥ p -Π P v 2 H ∀v ∈ H. (A2)
Note that p -≤ p + = P L(H) and that (A2) implies that the operators P and P * are non-negative.

Next, we shall suppose that a single equation is observable both by B * and by Π P in sufficiently large time:

   There exist T B , T P > 0 such that all solutions ϕ of ϕ ′′ + Aϕ = 0 satisfy e 1 (ϕ(0

)) ≤ C B (T ) T 0 B * (ϕ, ϕ ′ ) 2 Y for all T > T B , e 1 (ϕ(0)) ≤ C P (T ) T 0 Π P ϕ ′ 2
H dt for all T > T P .

(A3)

In the context of Theorem 1.3, these observability assumptions are satisfied as soon as ω p and ω b satisfy GCC (resp. Γ b satisfies GCC ∂ ). We can now state our main result, i.e. an observability inequality.

Theorem 2.2 (Observability). Suppose that Assumptions (A1)-(A3) hold. Then there exists a constant p * such that for all p + < p * , there exists a time T * such that for all T > T * there exists C > 0, such that for all V 0 ∈ H 1 , the solution V(t) = e tA P V 0 of (2.3) satisfies

e 1 (v 1 (0)) + e 0 (v 2 (0)) ≤ C T 0 B * (v 1 , v ′ 1 )(t) 2 Y dt.
(2.7)

Note that the constants p * and T * can be given explicitely in terms of the different parameters of the system. In particular, T * ≥ max{T B , T P }, and p * depends only on λ 0 , on the time T B and on the observability constant C B (T * B ) (given in Assumption (A3)) for some T * B > T B . See Proposition 4.7 and Lemma 3.3 below for more precision.

Applying the Hilbert Uniqueness Method (HUM) of [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], we deduce now controllability results for the adjoint system. In this context, we have to define more precisely the observation operator. We shall treat two cases: First,

B * (v 1 , v ′ 1 ) = B * v ′ 1 with B * ∈ L(H, Y ), corresponding to internal observability (with Y = L 2 (Ω)), and second B * (v 1 , v ′ 1 ) = B * v 1 with B * ∈ L(H 2 , Y
), corresponding to boundary observability (with Y = L 2 (∂Ω)). In both cases, we define the control operator B as the adjoint of B * , and the control problem reads, for a control function f taking its values in Y ,

   u ′′ 1 + Au 1 + P u 2 = Bf, u ′′ 2 + Au 2 + P * u 1 = 0, (u 1 , u 2 , u ′ 1 , u ′ 2 )| t=0 = (u 0 1 , u 0 2 , u 1 1 , u 1 2 ).
(2.8) This is an abstract version of (1.3)-(1.4). Note that under this form, System (2.8) not only contains (1.3)-(1.4), but also locally coupled systems of plate equations, with a distributed or a boundary control.

First case: B * (v 1 , v ′ 1 ) = B * v ′ 1 with B * ∈ L(H, Y ).
In this case, B ∈ L(Y, H) and the control problem (2.8) is well-posed in H 1 for f ∈ L 2 (0, T ; Y ). Note that, as in the concrete setting, it also preserves the space H 1 × H 2 × H × H 1 through time as soon as P ∈ L(H 1 ). There is thus no hope to control in whole H 1 . In this setting, we first deduce from (2.7) the following other observability estimate for solutions W of (2.3) in H 0 : e 0 (w

1 (0)) + e -1 (w 2 (0)) ≤ C T 0 B * w 1 (t) 2
Y dt. The internal control result of Theorem 1.3 is then a direct consequence of the HUM since Assumptions (A1)-(A3) are satisfied in this application.

Theorem 2.3 (Controllability). Suppose that Assumptions (A1)-(A3) hold. Then, there exists a constant p * such that for all p + < p * , there exists a time T * such that for all T > T * and U 0 ∈ H 1 × H 2 × H × H 1 , there exists a control function f ∈ L 2 (0, T ; Y ) such that the solution U(t) of (2.8) satisfies U(T ) = 0.

To prove this theorem, we first have to deduce from the observability estimate (2.7) the following other observability estimate for all W = e tA P W 0 (solution of (2.3) in H 0 ): e 0 (w

1 (0)) + e -1 (w 2 (0)) ≤ C T 0 B * w 1 (t) 2 Y dt.
Then, Theorem 2.3 is a direct consequence of the HUM.

Second case:

B * (v 1 , v ′ 1 ) = B * v 1 with B * ∈ L(H 2 , Y ).
As a consequence of the admissibility inequality (2.6), System (2.8) is well-posed in H 0 in the sense of transposition solutions. Moreover, System (2.8) also preserves the space H × H 1 × H -1 × H, and there is no hope to control in whole H 0 . In this setting, the boundary control result of Theorem 1.3 is a direct consequence of the HUM and Theorem 2.2 since Assumptions (A1)-(A3) are satisfied in this application.

Theorem 2.4 (Controllability). Suppose that Assumptions (A1)-(A3) hold. Then, there exists a constant p * such that for all p + < p * , there exists a time T * such that for all T > T * and

U 0 ∈ H × H 1 × H -1 × H, there exists a control function f ∈ L 2 (0, T ; Y ) such that the solution U(t) of (2.8) satisfies U(T ) = 0.
Theorem 1.3 is a consequence of Theorem 2.3 in the case of an internal control and of Theorem 2.4 in the case of a boundary control since Assumptions (A1)-(A3) are satisfied in this application. Theorem 2.3 and 2.4 are proved in Section 5.

Some remarks

Let us make some remarks about these results and their proofs.

First notice that System (2.8) is reversible in time, so that the concepts of null-controllability and controllability from zero are equivalent.

A consequence of Lemma 2.1 and Theorem 2.2 is that we here describe exactly the attainable set of the control system (2.8) (this was not considered in [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]). More precisely, starting for zero initial data, we prove that for T > T * the attainable set is

H 1 × H 2 × H × H 1 in the first case and H × H 1 × H -1 × H in the second case.
Let us briefly describe the method of the proof of Theorem 2.2 which is inspired by the "two energy levels" method of [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF]. Everything here is based on energy estimates, considering the H 1 energy of v 1 and the H 0 energy of v 2 , and comes from multiplier methods. There are three main ingredients in our proof.

Our first ingredient is an observability inequality for a single wave-type equation with a right handside (see Lemma 3.3). Such observability inequality used to be proved with multiplier techniques [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF], and thus, under too strong (and not optimal) geometric conditions. Here, we prove such inequalities as a consequence of usual observability inequalities. We also prove for wave equations that such an observability inequality with optimal geometric conditions for an equation with a right hand-side is very natural (see Appendix 7). This improvement can in fact be used in several works using multiplier conditions, replacing them with optimal geometric conditions ; in particular, the stabilization results of [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] now hold with GCC.

Our second main ingredient is the energy estimate obtained by multiplying the first line of (2.3) by v 2 and the second one by v 1 and taking the difference of the two equalities. This coupling inequality allows us to estimate the "localized" energy of the unobserved componenent v 2 by the energy of the observed one v 1 (see Lemma 4.1 below).

Finally to conclude the proof, we use in a crucial way the conservation of the H k -energy of the solution V. This implies roughly that the integral on (0, T ) of the energy is increasing linearly with respect to T , for T sufficiently large.

If the operator B is bounded, we proved in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] (under similar assumptions) a polynomial decay result for the E 1 (U(t)) energy of the system

   u ′′ 1 + Au 1 + P u 2 + BB * u ′ 1 = 0, u ′′ 2 + Au 2 + P * u 1 = 0, (u 1 , u 2 , u ′ 1 , u ′ 2 )| t=0 = U 0 ∈ H 1 × H 1 × H × H.
The observability estimates for a single equation with a right hand-side we prove here (Lemma 3.3) also improve the results of [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] in the case B bounded. Now, these results also hold under optimal geometric conditions for waves. We have the following proposition, where -∆ D denotes the Laplace operator with Dirichlet boundary conditions.

Proposition 2.5. Suppose that Assumption 1.2 holds, and that ω b and ω p satisfy GCC. Then there exists p * > 0 such that for all 0 < p + < p * , the solution

U = (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 ) of (1.5) satisfies for n ∈ N, for any initial data U 0 = (u 1 , u 2 , ∂ t u 1 , ∂ t u 2 )| t=0 ∈ D (-∆ D ) n+1 2 2 × D (-∆ D ) n 2 2 , E 1 (U(t)) ≤ C n t n n i=0 E 1 (U (i) (0)) ∀t > 0. Besides, if U 0 ∈ (H 1 0 ) 2 × (L 2 ) 2 , then E 1 (U(t)
) converges to zero as t goes to infinity.

The method of energy estimates we use here has several advantages and drawbacks. The main advantage is that it furnishes a systematic method for both internal and boundary controllability problems for a large class of second order in time equations. We only have to check if an observability inequality is known for a single equation, and the results can directly be transferred to systems.

However, we have here to make several assumptions: symmetry of the system, coercivity of the elliptic operator, smallness of the coupling coefficients, large control time... We do not know precisely which assumptions are really needed and which ones are unnecessary. We here provide a general a priori analysis of such coupled models. A more precise analysis (for instance for wave systems) remains to be done.

Finally, note that these exact controllability results for abstract second order hyperbolic equations yield null-controllability results for heat or Schrödinger type systems in the abstract setting as well. However, for the sake of clarity, we do not state these results in an abstract setting but only for heat or Schrödinger systems (see Sections 6.2 and 6.3).

3 Two energy levels and two key lemmata

Two energy levels

In the following sections, when proving Lemma 2.1 and Theorem 2.2, we shall use two different energy levels. Let us consider V a H 1 solution of (2.4). We regularize the state V once by setting

W = A -1 P V, i.e.        w ′ 1 = v 1 , w ′ 2 = v 2 , Aw 1 + P w 2 = -v ′ 1 , Aw 2 + P * w 1 = -v ′ 2 .
(3.1) Note that this system has a unique solution (w 1 , w 2 ), since the operator A P is coercive, that also satisfies

w ′′ 1 + Aw 1 + P w 2 = 0, w ′′ 2 + Aw 2 + P * w 1 = 0. (3.2)
Now, the idea, is that, for p + /λ 0 sufficiently small, the e 0 energy of v 2 is almost equivalent to the e 1 energy of w 2 . And we shall see that the e 1 energy level is more practical to handle. This is summarized in the first three identities of the following proposition. The last two identities are technical estimates, used at some points of the proof of the observability inequality.

Proposition 3.1. We have the following energy estimates for all

V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) in H 1 and W = (w 1 , w 2 , w ′ 1 , w ′ 2 ) defined by (3.1): 1 - p + λ 0 [e 1 (v 1 ) + e 1 (v 2 )] ≤ E 1 (V) ≤ 1 + p + λ 0 [e 1 (v 1 ) + e 1 (v 2 )] , (3.3) 1 - p + λ 0 [e 1 (w 1 ) + e 1 (w 2 )] ≤ E 1 (W) ≤ 1 + p + λ 0 [e 1 (w 1 ) + e 1 (w 2 )] , (3.4) 1 2 - p + λ 0 2 (e 1 (w 1 ) + e 1 (w 2 )) ≤ e 0 (v 1 ) + e 0 (v 2 ) ≤ 2 + 2 p + λ 0 2
(e 1 (w 1 ) + e 1 (w 2 )) , (3.5)

e 0 (v 2 ) ≤ 2 1 + (p + ) 2 λ 2 0 2 e 1 (w 2 ) + 2 (p + ) 2 λ 2 0 e 0 (v 1 ), (3.6) e 1 (v 1 ) ≥ λ 0 2 e 1 (w 1 ) - (p + ) 2 λ 0 e 1 (w 2 ). (3.7)
As a consequence of this lemma, assuming that p + λ0 < 1 √ 2 , we see that the energies e 1 (v 1 ) + e 1 (v 2 ), e 1 (w 1 ) + e 1 (w 2 ) and e 0 (v 1 ) + e 0 (v 2 ) are almost preserved through time for (v 1 , v 2 ) solutions of (2.3).

Proof. First recall that E 1 is defined by

2E 1 (V) = v ′ 1 2 H + v ′ 2 2 H + (Av 1 , v 1 ) H + (Av 2 , v 2 ) H + 2(P v 2 , v 1 ) H .
Then, using the fact that 2|

(P v 2 , v 1 ) H | ≤ p + λ0 A 1 2 v 1 2 H + A 1 2 v 2 2 H , we have 2E 1 (V) ≥ v ′ 1 2 H + v ′ 2 2 H + 1 - p + λ 0 A 1 2 v 1 2 H + A 1 2 v 2 2 H ≥ 2 1 - p + λ 0 (e 1 (v 1 ) + e 1 (v 2 )) , together with 2E 1 (V) ≤ v ′ 1 2 H + v ′ 2 2 H + 1 + p + λ 0 A 1 2 v 1 2 H + A 1 2 v 2 2 H ≤ 2 1 + p + λ 0 (e 1 (v 1 ) + e 1 (v 2 )) ,
and (3.3) is proved. Since (3.3) holds for all V ∈ H 1 , it also holds for W, which gives (3.4). Now, applying A -1 2 to the last two lines of System (3.1) gives

       w ′ 1 = v 1 , w ′ 2 = v 2 , A 1 2 w 1 + A -1 2 P w 2 = -A -1 2 v ′ 1 , A 1 2 w 2 + A -1 2 P * w 1 = -A -1 2 v ′ 2 .
(3.8)

Since we have A -1 2 P w 2 H ≤ p + λ0 A 1 2 w 2 H and A -1 2 P * w 1 H ≤ p + λ0 A 1 2 w 1 H , System (3.8) yields 2 (e 0 (v 1 ) + e 0 (v 2 )) = v 1 2 H + v 2 2 H + A -1 2 v ′ 1 2 H + A -1 2 v ′ 2 2 H ≤ w ′ 1 2 H + w ′ 2 2 H + 2 A 1 2 w 1 2 H + 2 p + λ 0 2 A 1 2 w 2 2 H + 2 A 1 2 w 2 2 H + 2 p + λ 0 2 A 1 2 w 1 2 H ≤ 2 + 2 p + λ 0 2
2 (e 1 (w 1 ) + e 1 (w 2 )) , together with 2 (e 0 (v 1 ) + e 0 (v 2 )) ≥ w ′ 1 2

H + w ′ 2 2 H + 1 2 A 1 2 w 1 2 H - p + λ 0 2 A 1 2 w 2 2 H + 1 2 A 1 2 w 2 2 H - p + λ 0 2 A 1 2 w 1 2 H ≥ 1 2 - p + λ 0 2
2 (e 1 (w 1 ) + e 1 (w 2 )) , and (3.5) is proved. To prove (3.6), we also use the last equation of (3.8), which gives

A -1 2 v ′ 2 H ≤ p + λ 0 A 1 2 w 1 H + A 1 2 w 2 H .
Using the third equation of (3.8) to eliminate A 1 2 w 1 in this estimate, we obtain

A -1 2 v ′ 2 H ≤ p + λ 0 A -1 2 P w 2 H + p + λ 0 A -1 2 v ′ 1 H + A 1 2 w 2 H ≤ 1 + (p + ) 2 λ 2 0 A 1 2 w 2 H + p + λ 0 A -1 2 v ′ 1 H .
Hence, with the second equation of (3.8), we have

A -1 2 v ′ 2 2 H + v 2 2 H ≤ 2 1 + (p + ) 2 λ 2 0 2 A 1 2 w 2 2 H + 2 (p + ) 2 λ 2 0 A -1 2 v ′ 1 2 H + w ′ 2 2 H ,
which concludes the proof of (3.6). To prove (3.7), we first notice that the third line of (3.8) gives

A 1 2 w 1 2 H ≤ 2 A -1 2 P w 2 2 H + 2 A -1 2 v ′ 1 2 H ≤ 2λ -2 0 (p + ) 2 A 1 2 w 2 2 H + 2λ -1 0 v ′ 1 2 H .
Hence, using the first line of (3.8), we obtain

2e 1 (v 1 ) = A 1 2 w ′ 1 2 H + v ′ 1 2 H ≥ λ 0 w ′ 1 2 H + λ 0 2 A 1 2 w 1 2 H - (p + ) 2 λ 0 A 1 2 w 2 2 H ≥ λ 0 e 1 (w 1 ) - 2(p + ) 2 λ 0 e 1 (w 2 ),
which yields (3.7), and concludes the proof of the lemma.

Two key lemmata

In this section, we prove Lemma 2.1 together with a key observability inequality for a classical "wave-type" equation with a right hand-side. For both proofs, we shall use the classical well-posedness properties of the equation ϕ ′′ + Aϕ = f that we recall in the following lemma.

Lemma 3.2. For any k ∈ Z, there exists C > 0 such that for all (ϕ 0 , ϕ 1 ) ∈ H k × H k-1 and f ∈ L 1 (R + ; H k-1 ) the equation ϕ ′′ + Aϕ = f has a unique solution, satisfying for all T > 0,

e k (ϕ(T )) ≤ C e k (ϕ(0)) + f 2 L 1 (0,T ;H k-1 ) . (3.9)
Note that in this energy inequality, the constant C does not depend on the time T .

Proof of Lemma 2.1: admissibility

Here, we prove that Assumption (A1) implies the amissibility inequality (2.6) for the whole system.

Proof of Lemma 2.1. We suppose that (v 1 , v 2 ) satisfies System (2.3). In particular, we have,

v ′′ 1 + Av 1 = -P v 2 .
As a consequence of Assumption (A1), we have for all T > 0,

T 0 B * (v 1 , v ′ 1 ) 2 Y dt ≤ C(T ) e 1 (v 1 (0)) + e 1 (v 1 (T )) + T 0 e 1 (v 1 (t))dt + T 0 P v 2 (t) 2 H dt . (3.10)
Then, the energy estimate (3.9) for k = 1, the Cauchy-Schwarz inequality and the boundedness of P on H yield

e 1 (v 1 (T )) ≤ C e 1 (v 1 (0)) + P v 2 2 L 1 (0,T ;H) ≤ C e 1 (v 1 (0)) + T (p + ) 2 v 2 2 L 2 (0,T ;H) , (3.11)
together with

T 0 e 1 (v 1 (t))dt ≤ C T e 1 (v 1 (0)) + T 2 (p + ) 2 v 2 2 L 2 (0,T ;H) .
(3.12)

Now, according to (3.1), (3.4) and since p + < λ 0 , we note that v 2 2

L 2 (0,T ;H) = w ′ 2 2 L 2 (0,T ;H) ≤ 2 1-p + /λ0 T 0 E 1 (W(t))dt.
Since W is a solution of (3.2), its energy is preserved through time, so that T 0 E 1 (W(t))dt = T E 1 (W(0)). Using inequalities (3.4) and (3.5) (i.e. the equivalence of the different energies), we obtain, for all p

+ < λ0 √ 2 , v 2 2 L 2 (0,T ;H) ≤ 2T 1 + p + λ0 1 -p + λ0 [e 1 (w 1 (0)) + e 1 (w 2 (0))] ≤ 2T 1 + p + λ0 1 -p + λ0 1 2 - p + λ 0 2 -1 [e 0 (v 1 (0)) + e 0 (v 2 (0))] . (3.13)
We recall that (2.2) yields e 0 (v 1 (0)) ≤ λ -1 0 e 1 (v 1 (0)). Finally, combining (3.10)-(3.13), we obtain

T 0 B * (v 1 , v ′ 1 ) 2 Y dt ≤ C(T, λ 0 , p + ) (e 1 (v 1 (0)) + e 0 (v 2 (0))) , with C(T, λ 0 , p + ) = C(T ) 1 + C(1 + T ) + max(1, 1/λ 0 )(p + ) 2 2T (1 + CT + CT 2 ) 1+ p + λ 0 1-p + λ 0 1 2 -p + λ0 2 -1 ,
and the admissibility of B * is proved.

Proof of an observability inequality with a right hand-side

Here, we prove the following lemma.

Lemma 3.3. Suppose that Assumptions (A1) and (A3) hold. Then, for all T * B > T B and T * P > T P , there exist constants K B , K P > 0 such that for any solution ϕ of ϕ ′′ + Aϕ = f ∈ L 2 (0, T ; H), we have

T 0 e 1 (ϕ(t))dt ≤ K B T 0 B * (ϕ, ϕ ′ ) 2 Y + T 0 f (t) 2 H dt for all T ≥ T * B , (3.14) and T 0 e 1 (ϕ(t))dt ≤ K P T 0 Π P ϕ ′ 2 H dt + T 0 f (t) 2
H dt for all T ≥ T * P .

(3.15)

Note that the important feature is that the observability constants K B , K P here do not depend on the time T as T → +∞. This property is important for the sequel. It is linked to the invariance properties of the system with respect to translation in time. More precisely, K B only depends on T * B and on C B (T * B ) for some T * B > T B , where T B and C B are defined in Assumption (A3). Such observability inequalities are proved with multiplier techniques for the wave equation or the plate equation in [START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled systems[END_REF][START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF][START_REF] Alabau-Boussouira | Indirect controllability of locally coupled systems under geometric conditions[END_REF]. It is then often used to perform energy estimates. Here, proving (3.14) and (3.15) as a consequence of the associated observability inequality for the free equation (Assumption (A3)) and an admissibility assumption (A1) has several advantages. In particular, we can use as a black-box the different observability inequalities obtained for the different equations (i.e. for instance [START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for waves) and we hence obtain results with the optimal geometric conditions.

However, for the sake of completeness, we also prove Lemma 3.3 in a simple case for the wave equation in a direct way in Appendix 7, with optimal geometric conditions. This shows that the norms used here are the natural ones.

Proof of Lemma 3.3. We here only prove (3.14). The proof of (3.15) is simpler since the observation operator Π P is bounded (and we thus do not need an admissibility assumption).

To prove the first inequality of (3.14), we first split a solution of ϕ ′′ + Aϕ = f into ϕ = φ + ψ, where φ and ψ satisfy φ ′′ + Aφ = f, (φ(0), φ ′ (0)) = (0, 0), and

ψ ′′ + Aψ = 0, (ψ(0), ψ ′ (0)) = (ϕ(0), ϕ(0)).
We have

T 0 e 1 (ϕ(t))dt ≤ 2 T 0 e 1 (φ(t))dt + 2 T 0 e 1 (ψ(t))dt, (3.16)
and we provide upper bounds for both integrals on the right hand-side. The energy estimate (3.9) applied to φ gives, for all t > 0, e 1 (φ(t)) ≤ C f 2 L 1 (0,t;H) ≤ Ct f 2 L 2 (0,t;H) .

(3.17)

Then, the observability Assumption (A3) can be applied to ψ, which gives, for all t > 0 and T > T B ,

e 1 (ψ(t)) = e 1 (ψ(0)) ≤ C(T ) T 0 B * (ψ, ψ ′ ) 2 Y dt. (3.18)
Integrating (3.17) and (3.18) for t ∈ (0, T ), and using (3.16), we now have, for all T > T B ,

T 0 e 1 (ϕ(t))dt ≤ 2CT 2 T 0 f 2 H dt + 2C(T )T T 0 B * (ψ, ψ ′ ) 2 Y dt. (3.19)
To obtain the observation on ϕ instead of ψ in the right hand-side, we write

T 0 B * (ψ, ψ ′ ) 2 Y dt ≤ 2 T 0 B * (ϕ, ϕ ′ ) 2 Y dt + 2 T 0 B * (φ, φ ′ ) 2 Y dt. (3.20)
Then, using the admissibility Assumption (A1) for φ, we obtain

T 0 B * (φ, φ ′ ) 2 Y dt ≤ C(T ) e 1 (φ(T )) + T 0 e 1 (φ(t))dt + T 0 f 2 H dt ≤ C(T ) T + T 2 + 1 T 0 f 2 H dt, (
T 0 e 1 (ϕ(t))dt ≤ D(T ) T 0 f 2 H dt + T 0 B * (ϕ, ϕ ′ ) 2 Y dt .
Now, we explain how this constant can be uniformly bounded for large times. For all T * B > T B , we set D * (T ) = sup T * B ≤t≤T D(t), so that the application T → D * (T ) is nondecreasing on [T * B , +∞). Since the equation ϕ ′′ + Aϕ = f is invariant under time translations, the last inequality yields, for all

T 2 ≥ T 1 + T * B , T2 T1 e 1 (ϕ(t))dt ≤ D * (T 2 -T 1 ) T2 T1 f 2 H dt + T2 T1 B * (ϕ, ϕ ′ ) 2 Y dt . (3.22)
For T ≥ T * B , there exists an integer k 0 ≥ 1 such that T ∈ [k 0 T * B , (k 0 + 1)T * B ). Assume first that k 0 ≥ 2, then we have,

T 0 e 1 (ϕ(t))dt = k0-2 k=0 (k+1)T * B kT * B e 1 (ϕ(t))dt + T (k0-1)T * B e 1 (ϕ(t))dt
In each of these integrals the time interval is larger than T * B so that we can apply (3.22). This yields, for all

T ≥ 2T * B T 0 e 1 (ϕ(t))dt ≤ D * (T * B ) k0-2 k=0 (k+1)T * B kT * B f 2 H + B * (ϕ, ϕ ′ ) 2 Y dt + D * (T -(k 0 -1)T * B ) T (k0-1)T * B f 2 H + B * (ϕ, ϕ ′ ) 2 Y dt ≤ D * (2T * B ) T 0 f 2 H + B * (ϕ, ϕ ′ ) 2 Y dt, since T ∈ [k 0 T * B , (k 0 + 1)T * B
) and D * is nondecreasing. This inequality is also true for T * B ≤ T ≤ 2T * B , that is in the case k 0 = 1. This concludes the proof of (3.14), taking K B = D * (2T * B ). The proof of (3.15) is similar.

Proof of Theorem 2.2

In this section, we shall often use the notation A B, meaning that there exists a universal numerical constant C > 0 (depending on none of the parameters of the system) such that A ≤ CB.

The coupling Lemma

In this section, we give the link between v 1 and v 2 that we shall use in the sequel.

Lemma 4.1. Let V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 be solution of (2.3) and W = (w 1 , w 2 , w ′ 1 , w ′
2 ) be defined by (3.1). Then, for all T ≥ 0, we have

T 0 (P v 2 , v 2 ) H dt ≤ T 0 (P v 1 , v 1 ) H dt + 2λ 1 2 0 1 + (p + ) 2 λ 2 0 2 [e 1 (w 2 (T )) + e 1 (w 2 (0))] + 1 λ 1 2 0 1 + 2 (p + ) 2 λ 2 0 [e 1 (v 1 (T )) + e 1 (v 1 (0))] , (4.1) and T 0 (P w 2 , w 2 ) H dt ≤ T 0 (P w 1 , w 1 ) H dt + 1 λ 1 2 0 1 + 2(p + ) 2 λ 2 0 [e 1 (w 2 (T )) + e 1 (w 2 (0))] + 2 λ 3 2 0 [e 1 (v 1 (T )) + e 1 (v 1 (0))] . (4.2)
Proof. Since V is a solution of (2.3), we have

T 0 (v ′′ 1 + Av 1 + P v 2 , v 2 ) H -(v ′′ 2 + Av 2 + P * v 1 , v 1 ) H dt = 0. (4.3)
We first notice that (Av 1 , v 2 ) H -(Av 2 , v 1 ) H = 0 since A is selfadjoint, and

T 0 (v ′′ 1 , v 2 ) H -(v ′′ 2 , v 1 ) H dt = [(v ′ 1 , v 2 ) H -(v ′ 2 , v 1 ) H ] T 0 ≤ 1 2 1 ε v 2 2 H + ε v ′ 1 2 H + 1 ε v ′ 2 2 H-1 + ε v 1 2 H1 (t = 0) + 1 2 1 ε v 2 2 H + ε v ′ 1 2 H + 1 ε v ′ 2 2 H-1 + ε v 1 2 H1 (t = T ) ≤ 1 ε [e 0 (v 2 (T )) + e 0 (v 2 (0))] + ε [e 1 (v 1 (T )) + e 1 (v 1 (0))] ,
for all ε > 0 and T ≥ 0. Once having isolated the term

T 0 (P v 2 , v 2 ) H dt in (4.3), this yields T 0 (P v 2 , v 2 ) H dt ≤ T 0 (P v 1 , v 1 ) H dt + 1 ε [e 0 (v 2 (T )) + e 0 (v 2 (0))] + ε [e 1 (v 1 (T )) + e 1 (v 1 (0))] .
Using (3.6) in this expression, we now have for all ε > 0 and T ≥ 0,

T 0 (P v 2 , v 2 ) H dt ≤ T 0 (P v 1 , v 1 ) H dt + 1 ε 2 1 + (p + ) 2 λ 2 0 2 e 1 (w 2 (T )) + 2 (p + ) 2 λ 2 0 e 0 (v 1 (T )) +2 1 + (p + ) 2 λ 2 0 2 e 1 (w 2 (0)) + 2 (p + ) 2 λ 2 0 e 0 (v 1 (0)) + ε [e 1 (v 1 (T )) + e 1 (v 1 (0))] ≤ T 0 (P v 1 , v 1 ) H dt + 2 ε 1 + (p + ) 2 λ 2 0 2 [e 1 (w 2 (T )) + e 1 (w 2 (0))] + ε + 2 (p + ) 2 ελ 3 0 [e 1 (v 1 (T )) + e 1 (v 1 (0))] ,
since e 0 (v 1 ) ≤ λ -1 0 e 1 (v 1 ). We then set ε = λ

-1 2 0
and Estimate (4.1) is proved.

Since W is a solution of (3.2), we also have

T 0 (w ′′ 1 + Aw 1 + P w 2 , w 2 ) H -(w ′′ 2 + Aw 2 + P * w 1 , w 1 ) H dt = 0.
Following the same procedure, and recalling that w ′ 1 = v 1 , we obtain,

T 0 (P w 2 , w 2 ) H dt ≤ T 0 (P w 1 , w 1 ) H dt + T 0 (w ′′ 1 , w 2 ) H -(w ′′ 2 , w 1 ) H dt ≤ T 0 (P w 1 , w 1 ) H dt + [(v 1 , w 2 ) H -(w ′ 2 , w 1 ) H ] T 0 . (4.4)
Next, we estimate

|(v 1 , w 2 ) H | + |(w ′ 2 , w 1 ) H | ≤ 1 2 1 ε w 2 2 H + ε v 1 2 H + ε w ′ 2 2 H + 1 ε w 1 2 H , (4.5)
and notice that System (3.1) yields w 1 = -A -1 v ′ 1 -A -1 P w 2 , and hence

w 1 H ≤ λ -1 0 v ′ 1 H + p + λ -3 2 0 A 1 2 w 2 H .
This, together with (4.5) gives

|(v 1 , w 2 ) H | + |(w ′ 2 , w 1 ) H | ≤ 1 2 1 ελ 0 A 1 2 w 2 2 H + ε λ 0 A 1 2 v 1 2 H + ε w ′ 2 2 H + 2 ελ 2 0 v ′ 1 2 H + 2(p + ) 2 ελ 3 0 A 1 2 w 2 2 H . Taking ε = λ -1 2 0 , we obtain |(v 1 , w 2 ) H | + |(w ′ 2 , w 1 ) H | ≤ 1 λ 1 2 0 1 + 2(p + ) 2 λ 2 0 e 1 (w 2 ) + 2 λ 3 2 0 e 1 (v 1 ),
which, together with (4.4) yields Estimate (4.2), and concludes the proof of the lemma.

A first series of estimates

Note that until now, we did not assume that p + /λ 0 is small, except for the coercivity assumption on A P and the equivalence of the different energies in (3.4)-(3.5) (used in the proof of the Admissibility Lemma 2.1). Using the coupling relation (4.1), we now prove a first series of estimates, that will be made more precise later.

Lemma 4.2. For all T ≥ 0, all

(p + ) 2 λ 2 0 ≤ 1 2 , all V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 solution of (2.3) and W = (w 1 , w 2 , w ′ 1 , w ′
2 ) defined by (3.1), we have the following estimates

e 1 (w 2 (T )) + e 1 (w 2 (0)) λ -1 0 e 1 (v 1 (0)) + p + λ 3 2 0 T 0 e 1 (v 1 (t))dt + e 1 (w 2 (0)) + p + λ 1 2 0 T 0 w 1 2 H dt, (4.6)
e 1 (v 1 (T )) + e 1 (v 1 (0)) e 1 (v 1 (0)) + p + λ 1 2 0 T 0 e 1 (v 1 (t))dt + λ 0 e 1 (w 2 (0)) + p + λ 3 2 0 T 0 w 1 2 H dt, (4.7)
T 0

(P v 2 , v 2 ) H dt 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2 0 e 1 (w 2 (0)) + p + λ 0 T 0 w 1 2 H dt. (4.8)
Proof. Taking the inner product of the first equation of (2.3) by v ′ 1 , we obtain the following

H 1 -dissipation relation for v 1 : d dt e 1 (v 1 ) = -(P v 2 , v ′ 1 ) H
Integrated on the time interval (0, T ), this yields,

e 1 (v 1 (T )) + e 1 (v 1 (0)) = 2e 1 (v 1 (0)) - T 0 (P v 2 , v ′ 1 ) H dt ≤ 2e 1 (v 1 (0)) + p + 2ελ 1 2 0 T 0 v ′ 1 2 H dt + ελ 1 2 0 2 T 0 (P v 2 , v 2 ) H dt,
after having used Assumption (A2) on P and the Young inequality. Using the coupling relation (4.1) of Lemma 4.1 in this estimate gives

e 1 (v 1 (T )) + e 1 (v 1 (0)) ≤ 2e 1 (v 1 (0)) + p + 2ελ 1 2 0 T 0 v ′ 1 2 H dt + ελ 1 2 0 p + 2 T 0 v 1 2 H dt + ελ 0 1 + (p + ) 2 λ 2 0 2 [e 1 (w 2 (T )) + e 1 (w 2 (0))] + ε 2 1 + 2 (p + ) 2 λ 2 0 [e 1 (v 1 (T )) + e 1 (v 1 (0))] .
We obtain for all (p + ) 2 λ 2 0 ≤ 1 2 and ε sufficiently small,

e 1 (v 1 (T )) + e 1 (v 1 (0)) e 1 (v 1 (0)) + p + λ 1 2 0 T 0 v ′ 1 2 H dt + λ 1 2 0 p + T 0 v 1 2 H dt + λ 0 [e 1 (w 2 (T )) + e 1 (w 2 (0))] . (4.9)
Putting this back into (4.1), we also have for all

(p + ) 2 λ 2 0 ≤ 1 2 , T 0 (P v 2 , v 2 ) H dt T 0 (P v 1 , v 1 ) H dt + λ 1 2 0 [e 1 (w 2 (T )) + e 1 (w 2 (0))] + 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 v ′ 1 2 H dt + p + T 0 v 1 2 H dt 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2 0 [e 1 (w 2 (T )) + e 1 (w 2 (0))] . (4.10)
Now, we take the inner product of the second equation of System (3.2) on W by w ′ 2 = v 2 . We obtain the following H 1 -dissipation relation for w 2 :

d dt e 1 (w 2 ) = -(P * w 1 , w ′ 2 ) H = -(w 1 , P v 2 ) H
Integrated on the time interval (0, T ), this yields,

e 1 (w 2 (T )) + e 1 (w 2 (0)) = 2e 1 (w 2 (0)) - T 0 (w 1 , P v 2 ) H dt ≤ 2e 1 (w 2 (0)) + p + λ 1 2 0 2ε T 0 w 1 2 H dt + ε 2λ 1 2 0 T 0 (P v 2 , v 2 ) H dt,
for all ε > 0, after having used Assumption (A2) on P and the Young inequality. Using (4.10) in this last inequality, we obtain, for all ε > 0,

e 1 (w 2 (T )) + e 1 (w 2 (0)) e 1 (w 2 (0)) + p + λ 1 2 0 ε T 0 w 1 2 H dt + ε λ 0 e 1 (v 1 (0)) + p + ε λ 3 2 0 T 0 e 1 (v 1 (t))dt + ε [e 1 (w 2 (T )) + e 1 (w 2 (0))] .
Taking ε sufficiently small, this yields

[e 1 (w 2 (T )) + e 1 (w 2 (0))] e 1 (w 2 (0)) + p + λ 1 2 0 T 0 w 1 2 H dt + 1 λ 0 e 1 (v 1 (0)) + p + λ 3 2 0 T 0 e 1 (v 1 (t))dt.
When using this estimate in (4.9) and (4.10), we obtain

e 1 (v 1 (T )) + e 1 (v 1 (0)) e 1 (v 1 (0)) + p + λ 1 2 0 T 0 e 1 (v 1 (t))dt + λ 0 e 1 (w 2 (0)) + p + λ 3 2 0 T 0 w 1 2 H dt,
and

T 0 (P v 2 , v 2 ) H dt 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2 0 e 1 (w 2 (0)) + p + λ 0 T 0 w 1 2 H dt.
These three inequalities yield the result of the lemma.

A second series of estimates

Using the weak coupling relation (4.2), we now eliminate the terms with T 0 w 1 2 H dt in Estimates (4.6)-(4.8) of the previous section.

Lemma 4.3. There exists η > 0 such that for all T ≥ 0,

all p + λ0 ≤ η, all V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 solution of (2.
(P v 2 , v 2 ) H dt 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2
0 e 1 (w 2 (0)). (4.12)

Proof. Using Estimates (4.6) and (4.7) in relation (4.2), together with p + λ0 ≤ 1 2 , we have

T 0 (P w 2 , w 2 ) H dt p + T 0 w 1 2 H dt + 1 λ 1 2 0 1 + 2(p + ) 2 λ 2 0 1 λ 0 e 1 (v 1 (0)) + p + λ 3 2 0 T 0 e 1 (v 1 (t))dt + e 1 (w 2 (0)) + p + λ 1 2 0 T 0 w 1 2 H dt + 1 λ 3 2 0 e 1 (v 1 (0)) + p + λ 1 2 0 T 0 e 1 (v 1 (t))dt + λ 0 e 1 (w 2 (0)) + p + λ 3 2 0 T 0 w 1 2 H dt p + T 0 w 1 2 H dt + 1 λ 3 2 0 e 1 (v 1 (0)) + p + λ 2 0 T 0 e 1 (v 1 (t))dt + 1 λ 1 2 0 e 1 (w 2 (0)). (4.13)
Now, we want to eliminate the term with

T 0 w 1 2
H dt in these estimates. According to the third line of (3.1), we have w 1 = -A -1 v ′ 1 -A -1 P w 2 , so that

w 1 2 H ≤ 2 A -1 v ′ 1 2 H + 2 A -1 P w 2 2 H ≤ 2λ -2 0 v ′ 1 2
H + 2λ -2 0 p + (P w 2 , w 2 ) H after having used Assumption (A2) on the operator P . Integrating this estimate on (0, T ) and using (4.13) yields

T 0 w 1 2 H dt 1 λ 2 0 T 0 v ′ 1 2 H + (p + ) 2 λ 2 0 T 0 w 1 2 H dt + p + λ 7 2 0 e 1 (v 1 (0)) + (p + ) 2 λ 4 0 T 0 e 1 (v 1 (t))dt + p + λ 5 2 0 e 1 (w 2 (0)).
Now, we suppose that p + λ0 ≤ η, with η sufficiently small. This yields

T 0 w 1 2 H dt 1 λ 2 0 T 0 v ′ 1 2 H + p + λ 7 2 0 e 1 (v 1 (0)) + (p + ) 2 λ 4 0 T 0 e 1 (v 1 (t))dt + p + λ 5 2 0 e 1 (w 2 (0)) 1 λ 2 0 T 0 e 1 (v 1 (t))dt + p + λ 7 2 0 e 1 (v 1 (0)) + p + λ 5 2 0 e 1 (w 2 (0)),
which is exactly (4.11). Finally, using this inequality in (4.8), we obtain for all p + λ0 ≤ η, T 0

(P v 2 , v 2 ) H dt 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2
0 e 1 (w 2 (0))

+ p + λ 0 1 λ 2 0 T 0 e 1 (v 1 (t))dt + p + λ 7 2 0 e 1 (v 1 (0)) + p + λ 5 2 0 e 1 (w 2 (0)) 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2
0 e 1 (w 2 (0)).

This yields (4.12), and concludes the proof of the lemma.

Using the two estimates of this lemma, together with the observability inequality with a right hand-side for the operator Π P , given in Lemma 3.3, we are now able to prove the following lemma.

Lemma 4.4. Assume the hypotheses of Lemma 3.3 and that (A2) holds. Then, there exists η > 0 such that for all T * P > T P , there exists a constant K P such that for all T > T * P and p + /λ 0 ≤ η, for all

V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 solution of (2.
3) and W = (w 1 , w 2 , w ′ 1 , w ′ 2 ) defined by (3.1), we have

T 0 e 1 (w 2 (t))dt K P 1 p -+ 1 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2 0 e 1 (w 2 (0)) . (4.14)
Proof. First, we Apply (3.15) to w 2 for some T * P > T P , which, according to (3.2), satisfies w ′′ 2 + Aw 2 = -P * w 1 . We have, for all T > T * P ,

T 0 e 1 (w 2 (t))dt ≤ K P T 0 Π P w ′ 2 2 H dt + T 0 P * w 1 2 H dt .
Using Assumption (A2) on P together with the fact that w ′ 2 = v 2 , this yields

T 0 e 1 (w 2 (t))dt ≤ K P 1 p - T 0 (P v 2 , v 2 ) H dt + (p + ) 2 T 0 w 1 2 H dt .
Combining this inequality with Estimates (4.11)-(4.12) of Lemma 4.3, we obtain, for all T > T * P and p + /λ 0 ≤ η,

T 0 e 1 (w 2 (t))dt K P 1 p - 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2
0 e 1 (w 2 (0))

+ K P (p + ) 2 1 λ 2 0 T 0 e 1 (v 1 (t))dt + p + λ 7 2 0 e 1 (v 1 (0)) + p + λ 5 2 0 e 1 (w 2 (0)) K P λ 1 2 0 1 p -+ 1 e 1 (v 1 (0)) + K P p + λ 0 1 p -+ 1 T 0 e 1 (v 1 (t))dt + K P λ 1 2 0 1 p -+ 1 e 1 (w 2 (0)),
which concludes the proof of the lemma.

Lemma 4.5 (Almost conservation of the energy). For all T ≥ 0,

all p + λ0 ≤ 1 √ 2 , all V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 solution of (2.3) and W = (w 1 , w 2 , w ′ 1 , w ′
2 ) defined by (3.1), we have

T 0 λ -1 0 e 1 (v 1 (t)) + e 1 (w 2 (t)) dt ≥ T 2 1 + p + λ 0 -1 1 - p + λ 0 [e 1 (w 1 (0)) + e 1 (w 2 (0))] . (4.15)
Proof. First, as a consequence of (3.7) of Proposition 3.1, we have

λ -1 0 e 1 (v 1 ) + e 1 (w 2 ) ≥ 1 2 e 1 (w 1 ) + 1 - (p + ) 2 λ 2 0 e 1 (w 2 ) ≥ 1 2 e 1 (w 1 ) + e 1 (w 2 ) ,
as soon as (p + ) 2 λ 2 0 ≤ 1 2 . Integrating this inequality on the interval (0, T ), and using Identity (3.4) together with the the conservation of the energy E 1 (W), we obtain

T 0 λ -1 0 e 1 (v 1 (t)) + e 1 (w 2 (t)) dt ≥ 1 2 T 0 e 1 (w 1 (t)) + e 1 (w 2 (t)) dt ≥ 1 + p + λ 0 -1 1 2 T 0 E 1 (W(t))dt ≥ 1 + p + λ 0 -1 T 2 E 1 (W(0)) ≥ T 2 1 + p + λ 0 -1 1 - p + λ 0 [e 1 (w 1 (0)) + e 1 (w 2 (0))] ,
which yields (4.15), and concludes the proof of the lemma.

Lemma 4.6 (Lower bound for T 0 e 1 (v 1 (t))dt). There exists C > 0 and η > 0 such that for all T ≥ 0, all

p + λ0 ≤ η, all V = (v 1 , v 2 , v ′ 1 , v ′ 2 ) ∈ H 1 solution of (2.
3) and W = (w 1 , w 2 , w ′ 1 , w ′ 2 ) defined by (3.1), we have

1 + T p + λ 1 2 0 T 0 e 1 (v 1 (t))dt T e 1 (v 1 (0)) -T λ 0 e 1 (w 2 (0)). (4.16)
Proof. Taking the inner product of the first equation of (2.3) with v ′ 1 gives

d dt e 1 (v 1 ) = -(P v 2 , v ′ 1 ) H .
For 0 < t < T , we integrate this identity on the interval (0, t) and obtain, for all ε > 0,

e 1 (v 1 (t)) ≥ e 1 (v 1 (0)) - t 0 P v 2 H v ′ 1 H ds ≥ e 1 (v 1 (0)) - ε 2 T 0 P v 2 2 H dt - 1 2ε T 0 v ′ 1 2 H dt.
Using now Assumption (A2) and integrating the last inequality on the interval (0, T ), this gives, for all ε > 0,

T 0 e 1 (v 1 (t))dt ≥ T e 1 (v 1 (0)) - T εp + 2 T 0 (P v 2 , v 2 ) H dt - T ε T 0 e 1 (v 1 (t))dt.
This, together with (4.12) yields, for some constant C > 0 and for all ε > 0,

T 0 e 1 (v 1 (t))dt ≥ T e 1 (v 1 (0)) - CT εp + λ 1 2 0 e 1 (v 1 (0)) - CT ε(p + ) 2 λ 0 + T ε T 0 e 1 (v 1 (t))dt -CT εp + λ 1 2 0 e 1 (w 2 (0)). Now we choose ε = λ 1 2 0 2Cp + , so that we have T 0 e 1 (v 1 (t))dt ≥ T 2 e 1 (v 1 (0)) - T p + λ 1 2 0 2C + 1 2 T 0 e 1 (v 1 (t))dt - T λ 0 2 e 1 (w 2 (0)),
and the lemma is proved.

End of the proof of Theorem 2.2

In this section, we conclude the proof of Theorem 2.2. Using the key estimates of the preceding sections, we prove in fact the following more precise proposition.

Proposition 4.7. Suppose that Assumptions (A1)-(A3) hold. Then, there exists η, γ > 0 such that for all p + < p * := min{ηλ 0 , γ λ0 K B }, there exists a time T * ≥ max{T * B , T * P } (depending on p + , p -, λ 0 , K B , K P ) such that for all T > T * there exists C > 0, such that for all V 0 ∈ H 1 , the solution V(t) = e tA P V 0 of (2.3) satisfies

e 1 (v 1 (0)) + e 0 (v 2 (0)) ≤ C T 0 B * (v 1 , v ′ 1 )(t) 2 Y dt.
A numerical inspection of the proof shows that one can take for instance η = 1 5 and γ = 1 50 . Proof. We proceed as in [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF] and use balance of energies. First, we use the observability inequality for a single equation with a right hand-side, given by Lemma 3.3. Since v 1 is a solution of v ′′ 1 + Av 1 + P v 2 = 0 from (2.3), Assumptions (A1) and (A3) and Estimate (3.14) yield, for all T ≥ T * B ,

T 0 e 1 (v 1 (t))dt ≤ K B T 0 B * (v 1 , v ′ 1 ) 2 Y dt + T 0 P v 2 2 H dt .
According to Assumption (A2), this gives

K B T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ T 0 e 1 (v 1 (t))dt -K B p + T 0 (P v 2 , v 2 ) H dt ≥ ε T 0 e 1 (v 1 (t)) + λ 0 e 1 (w 2 (t)) dt + (1 -ε) T 0 e 1 (v 1 (t))dt -ελ 0 T 0 e 1 (w 2 (t))dt -K B p + T 0 (P v 2 , v 2 ) H dt,
for some ε ∈ (0, 1), to be chosen later on. In this expression, we replace T 0 (P v 2 , v 2 ) H dt by Estimate (4.12) given in Lemma 4.3 and T 0 e 1 (w 2 (t))dt by Estimate (4.14) given in Lemma 4.4. We obtain, for some constant C 0 > 0 all T ≥ max{T * B , T * P },

K B T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ ε T 0 e 1 (v 1 (t)) + λ 0 e 1 (w 2 (t)) dt + (1 -ε) T 0 e 1 (v 1 (t))dt -ελ 0 C 0 K P 1 p -+ 1 1 λ 1 2 0 e 1 (v 1 (0)) + p + λ 0 T 0 e 1 (v 1 (t))dt + λ 1 2 0 e 1 (w 2 (0)) -C 0 K B p + λ 1 2 0 e 1 (v 1 (0)) + K B (p + ) 2 λ 0 T 0 e 1 (v 1 (t))dt + K B p + λ 1 2 0 e 1 (w 2 (0)) .
This can be rewritten under the form

K B T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ ε T 0 e 1 (v 1 (t)) + λ 0 e 1 (w 2 (t)) dt + L 1 (ε) T 0 e 1 (v 1 (t))dt -L 2 (ε)e 1 (v 1 (0)) -L 3 (ε)e 1 (w 2 (0)), (4.17) with          L 1 (ε) = 1 -ε -εC 0 K P p + 1 p -+ 1 -C 0 K B (p + ) 2 λ0 , L 2 (ε) = εC 0 K P λ 1 2 0 1 p -+ 1 + C 0 K B p + λ 1 2 0 , L 3 (ε) = εC 0 K P λ 3 2 0 1 p -+ 1 + C 0 K B p + λ 1 2 0 .
Taking p + such that C 0

K B (p + ) 2 λ0
< 1 and ε sufficiently small, we obtain L 1 (ε) > 0. In Inequality (4.17), we replace the two terms integrated on (0, T ) by their estimates (4.15) given in Lemma 4.5 (almost conservation of the energy) and (4.16) of Lemma 4.6. We obtain, for some

C 1 > 0, for p + λ0 ≤ η, (p + ) 2 < λ0 C0K B , K B T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ ελ 0 T 3 [e 1 (w 1 (0)) + e 1 (w 2 (0))] -L 2 (ε)e 1 (v 1 (0)) -L 3 (ε)e 1 (w 2 (0)) + C 1 L 1 (ε) 1 + T p + λ 1 2 0 -1 [T e 1 (v 1 (0)) -T λ 0 e 1 (w 2 (0))] .
This yields

K B T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ M 1 (ε)e 1 (v 1 (0)) + M 2 (ε)e 1 (w 2 (0)) + ελ 0 T 3 e 1 (w 1 (0)), (4.18) with          M 1 (ε) = C 1 L 1 (ε) 1 + T p + λ 1 2 0 -1 T -L 2 (ε), M 2 (ε) = ελ 0 T 3 -C 1 L 1 (ε)T λ 0 1 + T p + λ 1 2 0 -1 -L 3 (ε).
Now, it remains to check that these coefficients are positive for a suitable choice of ε (small) and T (large).

The coefficients M 1 (ε) and M 2 (ε) are positive if and only if we have

     T C 1 L 1 (ε) -L2(ε)p + λ 1 2 0 -L 2 (ε) > 0, T 2 ε 3 λ 1 2 0 p + + T ελ0 3 -C 1 L 1 (ε)λ 0 -L3(ε)p + λ 1 2 0 -L 3 (ε) > 0. (4.19)
The first condition of (4.19) is satisfied for large T if

1 -ε -εC 0 1 + 1 C 1 K P p + 1 p -+ 1 -C 0 1 + 1 C 1 K B (p + ) 2 λ 0 > 0,
i.e. as soon as

   C 0 1 + 1 C1 K B (p + ) 2 λ0
< 1, and

ε 1 + C 0 1 + 1 C1 K P p + 1 p -+ 1 < 1 -C 0 1 + 1 C1 K B (p + ) 2 λ0
. This is the case when taking p + ≤ min{ηλ 0 , γ λ0 K B } for some constant γ > 0, and ε sufficiently small. Then, the second condition of (4.19) is always satisfied for large T since ε > 0. Hence, for this choice of p + , ε sufficiently small, and T sufficiently large, we obtain from (4.18) the existence of a constant C > 0 such that

C T 0 B * (v 1 , v ′ 1 ) 2 Y ≥ e 1 (v 1 (0)) + e 1 (w 2 (0)).
This concludes the proof of Proposition 4.7 (and hence, that of Theorem 2.2), since e 0 (v 2 (0)) can be estimated by e 1 (w 2 (0)) and e 1 (v 1 (0)) according to (3.6).

From observability to controllability

In this Section, we prove that the observability inequality of Theorem 2.2 (or equivalently Proposition 4.7) implies the controllability results of Theorems 2.3 and 2.4. This is done classically with the use of the Hilbert Uniqueness Method (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]), that we shall follow here.

According to (5.3), we have (skipping the time dependence) 2{e 0 (z 1 ) + λ 0 e -1 (z 2 )} = A -1 2 z ′ 1 2

H + z 1 2 H + λ 0 A -1 z ′ 2 2 H + λ 0 A -1 2 z 2 2 H = A 1 2 v 1 + A -1 2 P v 2 2 H + v ′ 1 2 H + λ 0 v 2 + A -1 P * v 1 2 H + λ 0 A -1 2 v ′ 2 2 H ≤ 2 A 1 2 v 1 2 H + 2 A -1 2 P v 2 2 H + v ′ 1 2 H + 2λ 0 v 2 2 H + 2λ 0 A -1 P * v 1 2 H + λ 0 A -1 2 v ′ 2 2 H ≤ 2 1 + (p + ) 2 λ -2 0 A 1 2 v 1 2 H + v ′ 1 2 H + 2 λ 0 + (p + ) 2 λ -1 0 v 2 2 H + λ 0 A -1 2 v ′ 2 2
H ≤ 2C{e 1 (v 1 ) + λ 0 e 0 (v 2 )}, which proves the right inequality of (5.5). We also have 2{e 1 (v 1 ) + λ 0 e 0 (v 2 )} = v ′ 1 2

H + A 1 2 v 1 2 H + λ 0 A -1 2 v ′ 2 2 H + λ 0 v 2 2 H = z 1 2 H + A -1 2 z ′ 1 + A -1 2 P v 2 2 H + λ 0 A -1 2 z 2 2 H + λ 0 A -1 P * v 1 + A -1 z ′ 2 2 H ≤ z 1 2 H + 2 A -1 2 z ′ 1 2 H + 2 A -1 2 P v 2 2 H + λ 0 A -1 2 z 2 2 H + 2λ 0 A -1 P * v 1 2 H + 2λ 0 A -1 z ′ 2 2 H
≤ 4{e 0 (z 1 ) + λ 0 e -1 (z 2 )} + 2(p + ) 2 λ -1

0 v 2 2 H + 2(p + ) 2 λ -2 0 A 1 2 v 1 2 H .
For (p + ) 2 λ -2 0 < 1/2, the last two terms in this inequality can be absorbed in the left hand side, yielding e 1 (v 1 ) + λ 0 e 0 (v 2 ) ≤ C{e 0 (z 1 ) + λ 0 e -1 (z 2 )} This proves (5.5) and concludes the proof of the lemma.

To prove Theorem 2.3 with the HUM, we shall also make use of the following lemma.

Lemma 5.2. Let (u 1 , u 2 , u ′ 1 , u ′ 2 ) be the solution of (2.8) associated with (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ H 1 ×H 2 ×H ×H 1 and f ∈ L 2 (0, T ; Y ) and (z 1 , z 2 , z ′ 1 , z ′ 2 ) the solution of (5.1) associated with

(z T 1 , z T 2 , zT 1 , zT 2 ) ∈ H × H -1 × H -1 × H -2 .
Then, we have Proof. It suffices to prove (5.6) for regular data. The general case can be deduced with a density argument. We take the inner product of the first line of (2.8) with z 1 and the second line of (2.8) with z 2 and integrate on (0, T ). Summing the two identities, we obtain After two integrations by parts, using the selfadjointness of A together with (5.1), we have

(u ′ 1 , z 1 ) H -u 1 , z ′ 1 H1,H-1 + u ′ 2 , z 2
T 0 (f, B * z 1 ) Y dt = [(u ′ 1 , z 1 ) H -(u 1 , z ′ 1 ) H + (u ′ 2 , z 2 ) H -(u 2 , z ′ 2 ) H ] T 0 ,
which direcly yields (5.6) for smooth solutions. We conclude the proof of the lemma with a density argument.

With Lemmata 5.1 and 5.2, we can now prove Theorem 2.3, following [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF].

Second case: boundary control. Let T > 0. On the time interval (0, T 2 ), we set f = 0. Hence, the initial data (u 0 1 , u 0 2 ) ∈ H -1 (Ω) 2 are driven to some (u 1 , u 2 )| t= T 2 ∈ H 1 0 (Ω) 2 ⊂ L 2 (Ω) × H -1 (Ω).

As a consequence of Theorem 1.3, combined with [Mil06, Theorem 3.4] there exists a control function f ∈ L 2 (( T 2 , T ) × ∂Ω) such that (u 1 , u 2 )| t=T = 0.

Note that we could have taken the initial data in less regular spaces, provided that the coefficients a, p are smooth enough.

Control of Schrödinger systems

The proof of Corollary 1.5 is the same as that of Corollary 1.4, except that the Schrödinger equation does not enjoy smoothing properties. Hence, Corollary 1.5 is a direct consequence of Theorem 1.3, combined with [Mil05, Theorem 3.1].

7 Appendix: a direct proof of Lemma 3.3: Observability for a wave equation with a right hand-side

In this section, we provide a direct proof of Lemma 3.3 for a wave equation in a very simple setting.

For this, we suppose that (Ω, g) is a compact connected Riemannian manifold without boundary, and we closely follow the proofs of [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF][START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. This shows in particular that the observability inequality for equations with a right hand-side (3.14)-(3.15) are indeed the natural energy estimates in the spaces we consider. In the following, ∆ denotes the (negative) Laplace-Beltrami operator on Ω for the metric g, and P = P (t, x, ∂ t , ∂ x ) = ∂ 2 t -∆ + 1 denotes the d'Alembert operator on R × Ω. Its principal symbol is given by p(t, x, τ, η) = -τ 2 + |η| 2

x for (t, x, τ, η) ∈ R × Ω × R × T *

x Ω ⊂ T * (R × Ω), where |η| 2

x = g x (η, η) denotes the Riemannian norm in the cotangent space of Ω at x. Proof. We first remark that it is sufficient to prove (7.1) with a time depending constant C b = C(T ). The time invariance property of the equation P v = f then yields the desired result.

The proof relies on a compactness-uniqueness method. In a first step, we prove the following weaker energy estimate

T 0 v 2 H 1 (Ω) + ∂ t v 2 L 2 (Ω) dt ≤ C(T ) T 0 ω |∂ t v| 2 dx dt + T 0 f 2 L 2 (Ω) dt + T 0 v 2 L 2 (Ω) dt , (7.2)
in which a compact term has been added in the right hand-side. In a second step we use a uniqueness argument to get rid of this additional term. We define the two following vector spaces: E = F = {v ∈ H 1 ((0, T ) × Ω); P v ∈ L 2 ((0, T ) × Ω)}, endowed with the norms v 2 E = v 2 H 1 ((0,T )×Ω) + P v 2 L 2 ((0,T )×Ω) , v 2 F = v 2 L 2 ((0,T )×Ω) + ∂ t v 2 L 2 ((0,T )×ω) + P v 2 L 2 ((0,T )×Ω) .

We first remark that E is a Hilbert space for the norm • E and that we have E ⊂ F with • F ≤ • E . If we prove that the space F is complete, the Banach isomorphism theorem then yields the inverse inequality:

Figure 1 :

 1 Figure 1: Examples of open sets (Ω, ω, O) such that ω and O both satisfy GCC in Ω, with ω ∩ O = ∅: in the case (a), Ω is the flat torus (or the square), in the case (b), Ω is the disk.

  3) and W = (w 1 , w 2 , w ′ 1 , w ′2 ) defined by (3.1), we have the estimates

  H1,H-1u 2 , z ′ 2 H2,H-2 T 0 = (f, B * z 1 ) L 2 (0,T ;Y )(5.6)

T 0 ( 0 (u ′′ 1 + 0 (u ′′ 2 +

 00102 Bf, z 1 ) H dt = T Au 1 + P u 2 ), z 1 H dt + T Au 2 + P * u 1 ), z 2 H dt

Lemma 7. 1 .

 1 Suppose that the couple (ω b , T b ) satisfies GCC. Then, there exists a constant C b > 0 such that for all T ≥ T b and v ∈ H 1 ((0, T ) × Ω) solution of P v = f ∈ L 2 ((0, T ) × Ω), we haveT 0 v 2 H 1 (Ω) + ∂ t v 2 L 2 (Ω) dt ≤ C b T 0 ω|∂ t v| 2 dx dt +

and L. Miller for discussions on these two articles. The first author would like to thank the Fondation des Sciences Mathématiques de Paris, the organizers of the IHP trimester on control of PDE's and the Laboratoire MAPMO for their support. The second author wishes to thank O. Glass and J. Le Rousseau for very fruitful discussions and encouragements. Both authors were partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01 and the GDRE CONEDP (CNRS/INDAM/UP).

First case: B

In this case, B ∈ L(Y, H) and the control problem (2.8) is well-posed in H 1 for f ∈ L 2 (0, T ; Y ). Note that, as in the concrete setting, it also preserves the space H 1 × H 2 × H × H 1 through time as soon as P ∈ L(H 1 ). There is thus no hope to control in whole H 1 . Note that a direct application of the HUM, at the H 1 × H × H × H -1 energy level for the adjoint variable V would yield a controllability result in H × H -1 × H 1 × H with a control Bf ∈ L 2 (0, T ; H -1 ) (and we would have to suppose that B ∈ L(Y, H -1 )). Since we want the control result to hold at a more regular level, we study an adjoint problem in a less regular space.

Lemma 5.1. For any

(5.1)

is well-posed backward in time and the unique solution satisfies Z ∈ C 0 ([0, T ];

Moreover suppose that Assumptions (A1)-(A3) hold and that p + < p * = min{ηλ 0 , γ λ0 K B } (where η and γ are given by Proposition 4.7). Then, for all T > T * (given by Proposition 4.7), the solution Z satisfies for some constant C > 1 the estimates

Of course, we prove this lemma as a consequence of (2.6) and (2.7) for more regular functions. We thus regularize Z, setting V := A -1 P Z, end then apply to V the results of the previous sections. Proof. First, we know that A P generates a group on H -1 . Hence, for

, and, in particular, z 2 ∈ C 0 ([0, T ]; H -1 ). Then, z 1 is solution of the first line of (5.1) with final data in H ×H -1 and a right hand side -P z 2 ∈ C 0 ([0, T ]; H -1 ). Hence, the solution (z 1 , z ′ 1 ) (which is unique in

). Since Z in C 0 ([0, T ]; H -1 ), this concludes the first part of the lemma.

Second, we set

(5.3) Now, let us only consider the smooth solutions, i.e. Z ∈ C 0 ([0, T ]; H 0 ), which yields V ∈ C 0 ([0, T ]; H 1 ). For these solutions, Theorem 2.2 and Lemma 2.1 yield

(5.4)

We notice that B * v ′ 1 = B * z 1 , so that in order to prove (5.2), it only remains to show the existence of a constant C > 1 such that

(5.5)

and the linear form

is the the solution of (5.1) associated with the final data

The linear form L is continuous on X by definition and, according to (5.2), the bilinear form a is both continuous and coercive on X × X as soon as T > T * . The Lax-Milgram theorem then yields for T > T * the existence (and uniqueness) of

According to (5.6), this yields (u

2 ) and f . This concludes the proof of Theorem 2.3.

Second case:

In this setting, we directly apply the HUM in the space H × H 1 × H -1 × H for the control problem, and thus in H 1 × H × H × H -1 for the observation problem. There is no need of regularizing our observation system and observability inequality. This means that the adjoint problem of the control problem (2.8) is directly System (2.3), for which we proved the admissibility inequality (2.6) and the observability inequality (2.7). Recall that in this case, the control operator B is in L(Y, H -2 ), which is not sufficient for (2.8) to be well-posed in the classical sense in H 0 for a control function f ∈ L 2 (0, T ; Y ). Nevertheless, as a consequence of the admissibility inequality (2.6), System (2.8) is well-posed in H 0 in the sense of transposition solutions (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]). Moreover, these solutions remain in H × H 1 × H -1 × H for all time if the initial data are in this space, and there is no hope to control in whole H 0 .

In this setting, the boundary control result of Theorem 1.3 is a direct consequence of the HUM and Theorem 2.2 since Assumptions (A1)-(A3) are satisfied in this application.

2 ) be the transposition solution of (2.8) associated with

The proof of this lemma is exactly the same as the one of Lemma 5.2. We can now sketch the proof of Theorem 2.4, which follows that of Theorem 2.3.

and the linear form

is the the solution of (2.3) associated with the final data

The linear form L is continuous on X and, according to the admissibility inequality (2.6) and the observability inequality (2.7), the bilinear form a is both continuous and coercive on X × X as soon as T > T * . We conclude the proof as the one of Theorem 2.3, using the Lax-Milgram Theorem.

Applications

In this section, we explain how Theorem 1.3 and Corollaries 1.4 and 1.5 can be deduced from the abstract results.

Control of wave systems: Proof of Theorem 1.3

Here, we prove Theorem 1.3. We only have to explain how the situation of this theorem can be put in the abstract setting. Here, H = L 2 (Ω) with usual inner product. For A, we take the operator -∆ c + a with domain H 2 (Ω)∩H 1 0 (Ω), which, according to Assumption 1.2-(i) is coercive. Hence

For the operator P we take the multiplication in L 2 (Ω) by the bounded function p, and the operator Π P needed in Assumption (A2) is the multiplication in L 2 (Ω) by the characteristic function 1 ωp . According to Assumption 1.2-(ii), ω p satisfies GCC, so that the observability inequality of [START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] directly implies the second part of Assumption (A3).

First case: internal control. The observation space here is Y = L 2 (Ω) and the observation operator B * is the multiplication in L 2 (Ω) by the bounded (real) function b. In this case, the operator B * is bounded and we have B = B * . Since B * is bounded, the admissibility assumption (A1) is directly satisfied. Finally, according to Assumption 1.2-(iii), ω b satisfies GCC, so that the observability inequality of [START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] directly implies the first part of Assumption (A3).

All the assumptions of Theorem 2.3 are then satisfied, so that it implies Theorem 1.3 in the internal control case.

Second case: boundary control. The observation space here is Y = L 2 (∂Ω) and the observation operator B * is defined on H 2 (Ω) ∩ H 1 0 (Ω) by

where n denotes the outward normal to ∂Ω. Hence, in this case B * ∈ L(H 2 (Ω)∩H 1 0 (Ω); L 2 (∂Ω)). The fact that this observation is admissible is a well-known hidden regularity result, see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] or [TW09, Section 7.1]. As a consequence, the admissibility assumption (A1) is satisfied. The control operator B is defined in this case as the Dirichlet map for which we refer to [TW09, Section 10.6]. The duality identity (5.7) shows in fact that it corresponds to a boundary control, i.e. to Problem (1.4). Finally, according to Assumption 1.2-(iii), Γ b satisfies GCC ∂ , so that the observability inequality of [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] directly implies the first part of Assumption (A3).

All the assumptions of Theorem 2.4 are then satisfied, so that it implies Theorem 1.3 in the boundary control case.

Control of diffusive systems

Here, we prove Corollary 1.4. Our control strategy consists in first regularizing the initial data (thanks to the natural smoothing effect of the heat equation), and then apply a transmutation argument.

First case: internal control. Let T > 0. On the time interval (0, T 2 ), we set f = 0. Hence, the initial data (u 0 1 , u 0 2 ) ∈ L 2 (Ω) 2 are driven to some (u

As a consequence of Theorem 1.3, combined with [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF]Theorem 3.4] there exists a control function f ∈ L 2 (( T 2 , T ) × Ω) such that (u 1 , u 2 )| t=T = 0.

• E ≤ C • F , which implies (7.2). Let us consider a Cauchy sequence (v k ) k∈N of F . Since L 2 ((0, T ) × Ω) and L 2 ((0, T ) × ω) are complete, there exist v ∈ L 2 ((0, T ) × Ω), w ∈ L 2 ((0, T ) × ω) and f ∈ L 2 ((0, T ) × Ω) such that

Since ∂ t v k → ∂ t v and P v k → P v in D ′ ((0, T ) × Ω), we also have ∂ t v| ω = w ∈ L 2 ((0, T ) × ω) and P v = f ∈ L 2 ((0, T ) × Ω). The first order differential operator ∂ t is microlocally elliptic on Char(P ) = {ρ ∈ T * (R × Ω), p(ρ) = 0}, so that v ∈ H 1 ((0, T ) × ω). As a consequence, v satisfies

(7.3) Now, pick any point ρ = (t, x, τ, η) ∈ T * (R × Ω) \ 0 such that t ∈ (0, T ). If ρ / ∈ Char(P ), then P is elliptic of order two at ρ and the first equation of (7.3) yields that v ∈ H 2 microlocally at ρ. If ρ ∈ Char(P ), we denote by Γ = {γ(s), s ∈ (-S -, S + )} the maximal bicharacteristic curve of P satisfying γ(0) = ρ. Since the couple (ω, T ) satisfies the geometric control condition, there exists s * ∈ (-S -, S + ) such that π(γ(s * )) ∈ (0, T ) × ω, where π : T * (R × Ω) → R × Ω denotes the natural projection. The second line of (7.3) implies that v ∈ H 1 , microlocally at γ(s * ). Hörmander's theorem on propagation of singularities [Tay81, Chapter 6, Theorem 2.1] (see also [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]Theorem 26.1.1]) yields that v ∈ H 1 microlocally at ρ since v satisfies P v ∈ L 2 . Note that the L 2 regularity for f is the natural one, required by the propagation theorem.

Finally we obtain v ∈ H 1 ((0, T ) × Ω). Hence, the Cauchy sequence (v k ) k∈N of F converges towards v ∈ F , and F is complete. The Banach isomorphism theorem gives the existence of a constant C > 0 (depending on T ) such that • E ≤ C • F , which implies (7.2). Now, we must get rid of the additional term v 2 L 2 ((0,T )×Ω) in (7.2). For this, we prove that there exists a constant C > 0 such that for all v ∈ E satisfying P v = f ∈ L 2 ((0, T ) × ω), we have

L 2 ((0,T )×Ω) ≤ C ∂ t v 2 L 2 ((0,T )×ω) + f 2 L 2 ((0,T )×Ω) .

(7.4)

We suppose that this inequality is false. Then, there exists a sequence (v k ) k∈N , such that v k L 2 ((0,T )×Ω) = 1, ∂ t v k | ω → 0 in L 2 ((0, T ) × ω), P v k → 0 in L 2 ((0, T ) × Ω).

Hence, the energy estimate (7.2) gives v k H 1 ((0,T )×Ω) ≤ C uniformly, so that we can extract a subsequence (also denoted v k ) that converges in L 2 ((0, T ) × Ω). Calling v ∈ L 2 ((0, T ) × Ω) its limit, we have      v L 2 ((0,T )×Ω) = 1 ∂ t v| ω = 0 on (0, T ) × ω P v = 0 in (0, T ) × Ω (7.5) Once again, the propagation of regularity [Tay81, Chapter 6, Theorem 2.1] together with GCC gives (for instance) v ∈ H 2 ((0, T ) × Ω). According to a uniqueness result [START_REF] Robbiano | Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques[END_REF], the last two lines of (7.5) yield v = 0 on (0, T ) × Ω, which contradicts the first line of (7.5). This gives (7.4), and concludes the proof of the lemma.