Éphane St

Chiraz Gerbi

Amina Kassem
email: shiraz.kassem@hotmail.com

Ali Mortada
email: aminamortada2010@hotmail.com

Wehbe

EXACT CONTROLLABILITY AND STABILIZATION OF LOCALLY COUPLED WAVE EQUATIONS : THEORETICAL RESULTS

Keywords: 2010 Mathematics Subject Classification. 35L10, 35B40, 93D15, 90D20 Coupled wave equations, internal damping, exact controllability

.

y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.2) u = y = 0 on Γ × R * + , (1.3)
where a > 0 constant, b ∈ C 0 (Ω, R) and ρ(x, u t ) is a non linear damping. Using an approach based on multiplier techniques, weighted nonlinear inequalities and the optimal-weight convexity method (developed in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]), the authors established an explicit energy decay formula in terms of the behavior of the nonlinear feedback close to the origin. Their results are obtained in the case when the following three conditions are satisfied: the waves propagate at the same speed (a = 1), the coupling coefficient b(x) is small and positive (0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ⋆ ] where b ⋆ is a constant depending on Ω and on the control region) and both the coupling and the damping regions satisfying an appropriated geometric conditions named Piecewise Multipliers Geometric Conditions (introduced in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF], recalled below in Definition 2 and denoted by PMGC in short). In their work, the case where the waves are not assumed to be propagated with equal speeds (a is not necessarily equal to 1) and/or the coupling coefficient b(x) is not assumed to be positive and small has been left as an open problem even when the damping term ρ is linear with respect to the second variable. Recently, C. Kassem et al. in [START_REF] Kassem | Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions[END_REF], answered this important open question by studying the stabilization of the following linear system:

u tt -a∆u + c(x)u t + b(x)y t = 0 in Ω × R * + , (1.4) y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.5) u = y = 0 on Γ × R * + , (1.6)
in the case where the waves propagate with equal or different speeds and the coupling coefficient is not assumed to be positive and small. Indeed, they distinguished two cases. The first one is when the waves propagate at the same speed (i.e. a = 1), but unlike the works of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], the coupling coefficient function b is not necessarily assumed to be positive and small. In this case, under the condition that the coupling region and the damping region have non empty intersection satisfying the PMGC conditions, they established an exponential energy decay rate for weak initial data. On the contrary (i.e. a = 1 ) they first proved the lack of the exponential stability of the system. However, under the same geometric condition, an optimal energy decay rate of type 1 t was established for smooth initial data. Notice that, the PMGC conditions is much more restrictive than the Geometric Control Condition (introduced in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], recalled below in Definition 1 and, denoted by GCC in short). The natural question is then whether or not stabilization and the exact controllability still hold in the case where the coupling region and the damping region have non empty intersection satisfying the GCC condition? The aim of this paper is to investigate the exact controllability of the following system:

u tt -a∆u + b(x)y t = c(x)v t
in Ω × R * + , (1.7)

y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.8) u = y = 0 on Γ × R * + , (1.9)
with the following initial data (1.10) u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y 1 , x ∈ Ω, under appropriate geometric conditions. Here, a > 0 constant, b ∈ C 0 (Ω, R), c ∈ C 0 (Ω, R + ) and v is an appropriate control. The idea is to use a result of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the observability of the homogeneous system associated to (1.7)-(1.9) is equivalent to the exponential stability of system (1.4)- (1.6) in an appropriate Hilbert space. So, we provide a complete analysis for the exponential stability of system (1.4)- (1.6) in different Hilbert spaces. First, when the waves propagate at the same speed (i.e., a = 1), under the condition that the coupling region is included in the damping region and satisfies the so-called Geometric Control Condition (GCC in Short), we establish the exponential stability of system (1.4)- (1.6). Consequently, an observability inequality of the solution of the homogeneous system associated to (1.7)-(1.9) in the space

H 1 0 (Ω) × L 2 (Ω) 2
is established. This leads, by the HUM method introduced by Lions in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations[END_REF], to the exact controllability of system (1.7)-(1.9) in the space

H -1 (Ω) × L 2 (Ω) 2 .
Noting that, the geometric situations covered here are richer than those considered in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] and [START_REF] Kassem | Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions[END_REF]. Furthermore, on the contrary when the waves propagate at different speeds, (i.e., a = 1), we establish the exponential stability of system (1.4)- (1.6) in the space

H 1 0 (Ω) × L 2 (Ω) × L 2 (Ω) × H -1
(Ω) provided that the damping region satisfies the PMGC condition while the coupling region includes in the damping region and satisfying the GCC conditions. Consequently, an observability inequality of the solution of the homogeneous system associated to (1.7)-(1.9) is established. This leads, by the HUM method, to the exact controllability of system (1.7)-(1.9) in the space

L 2 (Ω) × H -1 (Ω) × H 1 0 (Ω) × L 2 (Ω).
1.2. Literature. Since the work of J. L. Lions in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations[END_REF], the observability and controllability of coupled wave equations have been studied by an intensive number of publications. In [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations[END_REF], J. L. Lions studied the complete and partial observability and controllability of coupled systems of either hyperbolic-hyperbolic type or hyperbolic-parabolic type. These results assume that the coupling parameter is sufficiently small. In [START_REF] Alabau | Observabilité frontière indirecte de systèmes faiblement couplés[END_REF] and [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], F. Alabau studied the indirect boundary observability of an abstract system of two weakly coupled second order evolution equations where the coupling coefficient is strictly positive in the whole domain. In particular, using a piecewise multiplier method, she proved that, for a sufficiently large time T , the observation of the trace of the normal derivative of the first component of the solution on a part of the boundary allows us to get back a weakened energy of the initial data. Consequently, using Hilbert Uniqueness Method, she proved that the system is exactly controllable for small coupling parameter by means of one boundary control. Noting that, the situation where the waves propagate with different speeds is not covered. Later, the indirect boundary controllability of a system of two weakly coupled one-dimensional wave equations has been studied by Z. Liu and B. Rao in [START_REF] Liu | A spectral approach to the indirect boundary control of a system of weakly coupled wave equations[END_REF]. Using the non harmonic analysis, they established several weak observability inequalities which depend on the ratio of the wave propagation speeds and proved the indirect exact controllability. The null controllability of the reaction diffusion System has been studied by F. Ammar-Khodja et al. in [START_REF] Ammar-Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF], by deriving an observability estimate for the linearized problem. The exact controllability of a system of weakly coupled wave equations with an internal locally control acted on only one equation has been studied by A. Wehbe and W. Youssef in [START_REF] Wehbe | Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées[END_REF] and [START_REF] Wehbe | Indirect locally internal observability of weakly coupled wave equations[END_REF]. They showed that, for sufficiently large time, the observation of the velocity of the first component of the solution on a neighborhood of a part of the boundary allows us to get back a weakened energy of initial data of the second component, this if the coupling parameter is sufficiently small, but non-vanishing and by the HUM method, they proved that the total system is exactly controllable. F. Alabau and M. Léautaud in [START_REF] Alabau-Boussouira | Indirect controllability of locally coupled wave-type systems and applications[END_REF], considered a symmetric systems of two wave-type equation, where only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. They obtained an internal and a boundary controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition.

1.3. Description of the paper. This paper is organized as follows: In section (2), first, we show that the system (1.4)-(1.6) can be reformulated into a first order evolution equation and we deduce the well posedness property of the problem by the semigroup approach. Second, by using Theorem 2.2 of [START_REF] Kassem | Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions[END_REF], we show that our problem is strongly stable without geometric conditions. In section 3, we show the exponential decay rate of system (1.4)-(1.6) when the coupling region b is a subset of the damping region c and satisfies the geometric control condition GCC. After that, we show that our system is exactly controllable by using Proposition 2 of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]. In section 4, we show the exponential decay rate of system (1.4)-(1.6) in the weak energy space provided that the damping region satisfies the PMGC condition while the coupling region is a subset of the damping region and satisfies the GCC condition.

Well posedeness and strong stability

Let us define the energy space H = H 1 0 (Ω) × L 2 (Ω)

2
equipped with the following inner product and norm, respectively : for all U = (u, v, y, z), U = ( u, v, y, z) ∈ H,

(U, U ) H = a Ω (∇u • ∇ u)dx + Ω v v dx + Ω (∇y • ∇ y)dx + Ω z zdx, U H = (U, U ) H ,
Let (u, u t , y, y t ) be a regular solution of the system (1.4)- (1.6). Its associated energy is defined by

E(t) = 1 2 Ω |u t | 2 + a|∇u| 2 + |y t | 2 + |∇y| 2 dx. A straightforward computations gives E ′ (t) = - Ω c(x)|u t | 2 dx ≤ 0. Consequently, system (1.4)- (1.6
) is dissipative in the sense that its energy is non-increasing with respect to t. Setting U = (u, u t , y, y t ), system (1.4)-(1.6) may be recast as:

U t = AU, in (0, +∞), U (0) = (u 0 , u 1 , y 0 , y 1 ),
where the unbounded operator A : D(A) ⊂ H → H is given by:

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω)
AU = ( v, a∆u -bz -cv, z, ∆y + bv ), ∀ U = (u, v, y, z) ∈ D(A). (2.2)
Note that due to the fact that c(x) ≥ 0, the operator A is dissipative in H. And, by applying the Lax-Milgam Theroem, it is easy to prove that the operator A is maximal in H i.e. R(I -A) = H. Consequently, it generates a C 0 -semigroup of contractions (e tA ) t≥0 . So, system (1.4)-(1.6) is wellposed in H.

We need now to study the asymptotic behavior of E(t). For this aim, we suppose that there exists a non empty open ω c+ ⊂ Ω satisfying the following condition

{x ∈ Ω : c(x) > 0} ⊃ ω c+ . (LH1)
On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open

ω b ⊂ Ω such that {x ∈ Ω : b(x) = 0} ⊃ ω b . (LH2)
If ω = ω c+ ∩ ω b = ∅ and condition (LH1) holds, then system (1.4)-(1.6) is strongly stable using Theorem 2.2 in [START_REF] Kassem | Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions[END_REF], i.e. lim t→+∞ e tA (u 0 , u 1 , y 0 , y 1 ) H = 0 ∀(u 0 , u 1 , y 0 , y 1 ) ∈ H.

3.

Exponential stability and exact controllability in the case a = 1 3.1. Exponential stability. This subsection is devoted to study the exponential stability of system (1.4)-(1.6) in the case when the waves propagate at the same speed, i.e., a = 1 under an appropriate geometric conditions. Before we state our results, we recall the Geometric Control Conditions GCC introduced by Rauch and Taylor in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] for manifolds without boundaries and by Bardos, Lebeau and Rauch in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for domains with boundaries and the Piecewise Multipliers Geometric Condition introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF].

Definition 1. We say that a subset ω of Ω satisfies the GCC if every ray of the geometrical optics starting at any point x ∈ Ω at t = 0 enters the region ω in finite time T.

Definition 2. We say that ω satisfies the Piecewise Multipliers Geometric Condition (PMGC in short) if there exist Ω j ⊂ Ω having Lipschitz boundary Γ j = ∂Ω j and x j ∈ R N , j = 1, ..., J such that Ω j ∩Ω i = ∅ for j = i and ω contains a neighborhood in Ω of the set ∪ J j=1 γ j (x j )∪ Ω \ ∪ J j=1 Ω j where γ j (x j ) = {x ∈ Γ j : (xx j ) • ν j (x) > 0} and ν j is the outward unit normal vector to Γ j . Remark 1. The PMGC is the generalization of the Multipliers Geometric Condition (MGC in short) introduced by Lions in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations[END_REF], saying that ω contains a neighborhood in Ω of the set {x ∈ Γ : (xx 0 ) • ν(x) > 0}, for some x 0 ∈ R N , where ν is the outward unit normal vector to Γ = ∂Ω. Now, we are in position to state our first main result by the following theorem : Theorem 3.1. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c+ satisfies the geometric control conditions GCC and that b, c ∈ W 1,∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ H the energy of the system (1.4)- (1.6) satisfies the following decay rate:

(3.1) E(t) ≤ M e -θt E(0), ∀t > 0.
Remark 2. The geometric situations covered by Theorem 3.1 are richer than those considered in [START_REF] Kassem | Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions[END_REF] and [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF]. Indeed, in the previous references, the authors consider the PMGC geometric conditions that are more restrictive than GCC. On the other hand, unlike the results in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], we have no restriction in Theorem 3.1 on the upper bound and the sign of the coupling function coefficient b. This theorem is then a generalization in the linear case of the result of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] where the coupling coefficient considered have to satisfy 0

≤ b(x) ≤ b 0 , b 0 ∈ (0, b ⋆ ]
where b ⋆ is a constant depending on Ω and on the control region.

In order to prove Theorem 3.1, we apply a result of Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]. A C 0 -semigroup of contraction (e tA ) t 0 in a Hilbert space H is exponentially stable if and only if the two following hypotheses are fulfilled:

iR ⊆ ρ(A) (H1) lim sup β ∈R,|β|→+∞ (iβI -A) -1 L(H) < ∞ (H2)
Since the resolvent of A is compact and 0 ∈ ρ(A), then from the fact that our system is strongly stable, we deduce that condition (H1) is satisfied. We now prove that condition (H2) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence β n with β n → +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A) such that U n H = 1 and (3.2) lim n→∞ (iβ n I -A)U n H = 0. (3.3)
Next, detailing equation (3.3), we get

iβ n u n -v n = f 1 n → 0 in H 1 0 (Ω), (3.4) iβ n v n -∆u n + bz n + cv n = g 1 n → 0 in L 2 (Ω), (3.5) iβ n y n -z n = f 2 n → 0 in H 1 0 (Ω), (3.6) iβ n z n -∆y n -bv n = g 2 n → 0 in L 2 (Ω). (3.7)
Eliminating v n and z n from the previous system, we obtain the following system (3.8)

β 2 n u n + ∆u n -iβ n by n -iβ n cu n = -g 1 n -bf 2 n -iβ n f 1 n -cf 1 n , (3.9)
β 2 n y n + ∆y n + iβ n bu n = -iβ n f 2 n + bf 1 n -g 2 n .
On the other side, we notice that v n and z n are uniformly bounded in L 2 (Ω). It follows, from equations (3.4) and (3.6), that (3.10)

Ω |y n | 2 dx = O(1) β 2 n and Ω |u n | 2 dx = O(1) β 2 n .
For clarity, we divide the proof into several Lemmas.

Lemma 3.2. The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7
) satisfies the following estimates

(3.11) Ω c|β n u n | 2 dx = o(1) and ωc + |β n u n | 2 dx = o(1).
Proof. First, since U n is uniformly bounded in H, then from (3.3), we get

(3.12) Re iβ n U n 2 H -(AU n , U n ) H = Ω c(x)|v n | 2 dx = o(1).
Under condition (LH1), it follows that (3.13)

ωc + |v n | 2 dx = o(1).
Then, using equations (3.12) and (3.4), we get

(3.14) Ω c|β n u n | 2 dx = o(1)
.

Consequently, we have

ωc + |β n u n | 2 dx = o(1).
The proof is thus complete. Proof. Multiplying equation (3.8) by cu n , integrating by parts and using the fact that u n = 0 on Γ, we get

Ω c|β n u n | 2 dx - Ω c|∇u n | 2 dx - Ω (∇c • ∇u n )u n dx -i Ω β n by n cu n dx -i Ω β n cu n u n dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )cu n dx. (3.16) Using the fact that f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and β n u n is uniformly bounded in L 2 (Ω), we obtain (3.17) Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )cu n dx = o(1).
Using the fact that ∇u n , β n y n , β n u n are uniformly bounded in L 2 (Ω) and u n = o(1), we get

(3.18) - Ω (∇c • ∇u n )u n dx -i Ω β n by n cu n dx -i Ω β n cu n u n dx = o(1).
Inserting (3.17) and (3.18) into (3.16), we get

(3.19) Ω c|β n u n | 2 dx - Ω c|∇u n | 2 dx = o(1).
Finally, using estimation (3.11) in (3.19), we deduce

ωc + |∇u n | 2 dx = o(1).
The proof is thus complete.

Lemma 3.4. The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimate (3.20) ω b |∇y n | 2 dx = o(1).
Proof. The proof contains three points. i) First, multiplying equation (3.8) by 1 βn ∆y n , then using Green's formula and the fact that

u n = f 1 n = 0 on Γ, we obtain - Ω β n (∇u n • ∇y n )dx + 1 β n Ω ∆u n ∆y n dx + i Ω (∇b • ∇y n )y n dx + i Ω b|∇y n | 2 dx + i Ω (∇c • ∇y n )u n dx + i Ω c(∇u n • ∇y n )dx (3.21) = Ω (-g 1 n -bf 2 n -cf 1 n ) 1 β n ∆y n dx + i Ω (∇f 1 n • ∇y n )dx.
As f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and the fact that 1 βn ∆y n , ∇y n are uniformly bounded in L 2 (Ω), we have

(3.22) Ω (-g 1 n -bf 2 n -cf 1 n ) 1 β n ∆y n dx + i Ω (∇f 1 n • ∇y n )dx = o(1).
Using the fact that ∇y n is uniformly bounded in

L 2 (Ω), u n L 2 (Ω) = o(1), y n L 2 (Ω) = o(1)
and using the estimation (3.15), we get

(3.23) i Ω (∇b • ∇y n )y n dx + i Ω (∇c • ∇y n )u n dx + i Ω c(∇u n • ∇y n )dx = o(1)
.

Inserting now (3.22) and (3.23) into (3.21), we get

(3.24) - Ω β n (∇u n • ∇y n )dx + 1 β n Ω ∆u n ∆y n dx + i Ω b|∇y n | 2 dx = o(1).
ii) Similarly, multiplying equation (3.9) by 1 βn ∆u n , then using Green's formula and the fact that

y n = f 2 n = 0 on Γ, we obtain - Ω β n (∇y n • ∇u n )dx + 1 β n Ω ∆y n ∆u n dx -i Ω (∇b • ∇u n )u n dx -i Ω b|∇u n | 2 dx = Ω (bf 1 n -g 2 n ) 1 β n ∆u n dx + i Ω (∇f 2 n • ∇u n )dx. (3.25) Using the fact that f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 2 n converges to zero in L 2
(Ω) and the fact that 1 βn ∆u n , ∇u n are uniformly bounded in L 2 (Ω), we get

(3.26) Ω (bf 1 n -g 2 n ) 1 β n ∆u n dx + i Ω (∇f 2 n • ∇u n )dx = o(1).
Also, using the fact that ∇u n is uniformly bounded in L 2 (Ω), u n L 2 (Ω) = o(1), we have

(3.27) -i Ω (∇b • ∇u n )u n dx = o(1).
Inserting (3.26) and (3.27) into (3.25), we get

(3.28) - Ω β n (∇y n • ∇u n )dx + 1 β n Ω ∆y n ∆u n dx -i Ω b|∇u n | 2 dx = o(1).
iii) Finally, by combining (3.24) and(3.28) and taking the imaginary part, we obtain

(3.29) Ω b|∇y n | 2 dx = Ω b|∇u n | 2 dx + o(1).
Since ω b ⊂ ω c + , it follows from (3.15) and (3.29) that

ω b |∇y n | 2 dx = o(1).
The proof is thus complete. Proof. Multiplying equation (3.9) by by n . Then using Green's formula and the fact that y n = 0 on Γ, we obtain

Ω b|β n y n | 2 dx - Ω b|∇y n | 2 dx - Ω (∇b • ∇y n )y n dx + i Ω b 2 β n u n y n dx = Ω (-iβ n f 2 n + bf 1 n -g 2 n )by n dx. (3.31) As f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 2 n converges to zero in L 2 (Ω) and β n y n is uniformly bounded in L 2 (Ω), we get (3.32) Ω (-iβ n f 2 n + bf 1 n -g 2 n )by n dx = o(1).
Using the fact that β n u n and ∇y n are uniformly bounded in L 2 (Ω) and y n L 2 (Ω) = o(1), we get

(3.33) Ω (∇b • ∇y n )y n + i Ω b 2 β n u n y n dx = o(1).
Inserting (3.32), (3.33) into (3.31), we obtain

Ω b|β n y n | 2 dx - Ω b|∇y n | 2 dx = o(1).
Using the estimation (3.20) in the previous equation, we get

Ω b|β n y n | 2 dx = o(1).
This yields

ω b |β n y n | 2 dx = o(1).
The proof is thus complete.

Lemma 3.6. Let f n be a bounded sequence in L 2 (Ω). Then the solution

φ n ∈ H 1 0 (Ω) ∩ H 2 (Ω) of the following system (3.34) β 2 n φ n + ∆φ n -ibβ n φ n = f n in Ω, φ n = 0 on Γ,
verifies the following estimate

(3.35) Ω (|β n φ n | 2 + |∇φ n | 2 )dx ≤ C Ω |f n | 2 dx,
where C is a constant independent of n.

Proof. Consider the following wave equation

(3.36) φ tt -∆φ + bφ t = 0 in Ω, φ = 0 on Γ.
System (3.36) is wellposed in the space H = H 1 0 (Ω) × L 2 (Ω) and since ω b verifies GCC condition then it is exponentially stable (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]). Therefore, following Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], we deduce that the resolvent of its corresponding operator

A aux : D(A aux ) -→ H 1 0 (Ω) × L 2 (Ω) defined by D(A aux ) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) and A aux (φ, φ) = ( φ, ∆φ -b φ)
is uniformly bounded on the imaginary axis. On the other hand, system (3.34) can be rewritten in the form:

(3.37) iβ n φ n -φn = 0, iβ n φn -∆φ n + b φn = -f n . So, (3.38) iβ n -A aux φ n φn = 0 -f n .
Equivalently,

(3.39) φ n φn = iβ n -A aux -1 0 -f n .
This yields

(φ n , φn ) 2 H ≤ (iβ n -A aux ) -1 2 L(H) (0, -f n ) 2 H ≤ C Ω |f n | 2 dx, (3.40)
where C is a constant independent of n. Consequently, we deduce

Ω (|β n φ n | 2 + |∇φ n | 2 )dx ≤ C Ω |f n | 2 dx.
The proof is thus complete. n φ n where φ n is a solution of (3.34). Then using Green's formula and the fact that u n = φ n = 0 on Γ, we obtain

(3.42) Ω β 2 n u n (β 2 n φ n + ∆φ n )dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx -i Ω β n f 1 n β 2 n φ n dx.
Substituting the first equation of system (3.34) into the first term of (3.42), we get

(3.43) Ω |β n u n | 2 dx -i Ω β 2 n φ n bβ n u n dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx -i Ω β n f 1 n β 2 n φ n dx.
As f 1 n , f 2 n converge to zero in H 1 0 (Ω) and β 2 n φ n is uniformly bounded in L 2 (Ω) due to (3.35), we get

(3.44) Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx = o(1).
From the first equation of (3.34), we have β 2 n φ n = u n -∆φ nibβ n φ n . Consequently, we have

-i Ω β n f 1 n β 2 n ϕ n dx = -i Ω β n f 1 n (u n -∆φ n -ibβ n φ n )dx = -i Ω β n f 1 n u n dx -i Ω β n (∇φ n • ∇f 1 n )dx - Ω bf 1 n β 2 n φ n , (3.45) which yields (3.46) -i Ω β n f 1 n β 2 n ϕ n dx = o(
Ω |β n u n | 2 dx --i Ω β 2 n φ n bβ n u n dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = o(1).
Finally, using estimations (3.11), (3.30) and the fact that β 2 n φ n is uniformly bounded in L 2 (Ω) into the previous equation, we obtain (3.48)

Ω |β n u n | 2 dx = o(1).
The proof is thus complete. Proof. Taking f n = y n in Lemma 3.6. Multiplying equation (3.9) by β 2 n φ n where φ n is a solution of (3.34). Then using Green's formula and the fact that y n = φ n = 0 on Γ, we obtain

(3.50) Ω β 2 n y n (β 2 n φ n + ∆φ n )dx + i Ω bβ n u n β 2 n φ n dx = -i Ω β n f 2 n β 2 n φ n + Ω (bf 1 n -g 2 n )β 2 n φ n dx.
Then, substituting the first equation of problem (3.34) into the first term of (3.50), we get

(3.51) Ω |β n y n | 2 dx -i Ω bβ 2 n φ n β n y n dx + i Ω bβ n u n β 2 n φ n dx = -i Ω β n f 2 n β 2 n φ n + Ω (bf 1 n -g 2 n )β 2 n φ n dx. Since β 2 n φ n is uniformly bounded in L 2 (Ω), f 1 n converges to zero in H 1 0 (Ω) and g 2 n converges to zero in L 2 (Ω), we have (3.52) Ω (-bf 1 n -g 2 n )β 2 n φ n dx = o(1).
Moreover, using the first equation of problem (3.34) and integrating by parts yields

-i Ω β n f 2 n β 2 n φ n dx = -i Ω (y n -∆φ n -ibβ n φ n )β n f 2 n dx = -i Ω f 2 n β n y n dx -i Ω β n (∇φ n .∇f 2 n )dx - Ω bf 2 n β 2 n φ n dx. ( 3
Ω |β n y n | 2 dx -i Ω bβ 2 n φ n β n y n dx + i Ω bβ n u n β 2 n φ n dx = o(1).
Finally, using (3.30), (3.41) and the fact that β 2 n φ n is uniformly bounded in L 2 (Ω), we deduce

Ω |β n y n | 2 dx = o(1).
The proof is thus complete. Proof. Multiplying equation (3.8) by u n , applying Green's formula and using the fact that u n = 0 on Γ, we get

(3.57) Ω |β n u n | 2 dx - Ω |∇u n | 2 dx -i Ω β n by n u n dx -i Ω β n cu n u n dx = o(1).
Using the fact that β n u n , β n y n are uniformly bounded in L 2 (Ω), u n = o(1) and the estimation (3.41) in (3.57), we obtain (3.58)

Ω |∇u n | 2 dx = o(1).
Similarly, multiplying equation (3.9) by y n and applying Green's formula and using the fact that y n = 0 on Γ, we get

(3.59) Ω |β n y n | 2 dx - Ω |∇y n | 2 dx + i Ω β n bu n y n dx = o(1).
Using the fact that β n u n is uniformly bounded in L 2 (Ω),

y n = o(1)

3.2.

Observability and exact controllability. First, we consider the following homogeneous system associated to (1.7)-(1.9) for a = 1 by:

ψ tt -∆ψ + b(x)ϕ t = 0 in Ω × R + , (3.61) ϕ tt -∆ϕ -b(x)ψ t = 0 in Ω × R + , (3.62) ψ = ϕ = 0 on Γ × R + , (3.63) ψ(•, 0) = ψ 0 , ψ t (•, 0) = ψ 1 , ϕ(•, 0) = ϕ 0 , ϕ t (•, 0) = ϕ 1 in Ω. (3.64)
Let Φ = (ψ, ψ t , ϕ, ϕ t ) be a regular solution of system (3.61)-(3.63), its associated total energy is given by:

(3.65) E(t) = 1 2 Ω |ψ t | 2 + |∇ψ| 2 + |ϕ t | 2 + |∇ϕ| 2 dx.
A direct computation gives (3.66) d dt E(t) = 0.

Thus, system (3.61)-(3.63) is conservative in the sense that its energy E(t) is constant. It is also wellposed and admits a unique solution in the energy space H. Now, we establish the direct and indirect inequality given by the following theorem:

Theorem 3.10. Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c+ satisfies the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω). Then there exists a time T 0 such that for all T > T 0 , there exist two constants M 1 > 0, M 2 > 0 such that the solution of system (3.61)-(3.63) satisfies the following observability inequalities:

(3.67) M 1 Φ 0 2 H ≤ T 0 Ω c(x) |ψ t | 2 dxdt ≤ M 2 Φ 0 2 H ,
for all Φ 0 = (ψ 0 , ψ 1 , ϕ 0 , ϕ 1 ) ∈ H.

Proof. The direct inequality follows from the definition of the total energy for all T > 0. While the proof of the inverse inequality is a direct consequence of Proposition 2 of Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the exponentially stability (3.1) implies the existence of a time T 0 > 0 such that for all T > T 0 there exist two constants M 1 > 0 and M 2 > 0 such that (3.67) holds. The proof is thus complete. Now, we are ready to study the exact controllability of a system (1.7)-(1.9) by using the HUM method. First, thanks to the direct inequality, the solution of the system of equations (1.7), (1.8), (1.9) can be obtained as usual by the method of transposition (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations[END_REF] and [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]). Let v 0 ∈ L 2 (0, T ; L 2 (w c+ )), we choose the control

(3.68) v(t) = - d dt v 0 (t) ∈ [H 1 (0, T ; L 2 (ω c+ ))] ′ ,
where the derivative d dt is not taken in the sense of distributions but in the sense of the duality H 1 (0, T ; L 2 (ω c+ )) and its dual [H 1 (0, T ; L 2 (ω c+ ))] ′ , i.e.,

- T 0 d dt v 1 (t)µ(t)dt = T 0 v 1 (t) d dt µ(t)dt, ∀µ ∈ H 1 (0, T ; L 2 (ω c+ ).
Then we have the followig result:

Theorem 3.11. Let T > 0 and a = 1. Assume that conditions (LH1) and (LH1) hold. Assume also that ω b ⊂ ωc + satisfies the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω). Given

U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ (L 2 (Ω) × H -1 (Ω)) 2 , v = - d dt v 0 ∈ [H 1 (0, T ; L 2 (ω c+ ))] ′ ,
the controlled system (1.7)-(1.9) has a unique weak solution

U = (u, u t , y, y t ) ∈ C 0 ([0, T ], (L 2 (Ω) × H -1 (Ω)) 2 ).
Proof. Let (ψ, ψ t , ϕ, ϕ t ) be the solution of (3.61)-(3.63) associated to Φ 0 = (ψ 0 , ψ 1 , ϕ 0 , ϕ 1 ). Multiplying the first equation of (1.7)-(1.9) by ψ and the second by ϕ and integrating by parts, we obtain

(3.69)                      Ω y t (T )ϕ(T )dx + Ω u t (T )ψ(T )dx - Ω y(T )ϕ t (T )dx - Ω u(T )ψ t (T )dx - Ω bu(T )ϕ(T )dx + Ω by(T )ψ(T )dx = Ω y t (0)ϕ(0)dx + Ω u t (0)ψ(0)dx - Ω ϕ t (0)y(0)dx - Ω ψ t (0)u(0)dx - Ω bu(0)ϕ(0)dx + Ω by(0)ψ(0)dx + T 0 Ω c(x)v(t)ψdxdt.
Note that

H ′ = (H -1 (Ω) × L 2 (Ω)) 2 . Then we have (3.70)    (u t (T, x), -u(T, x), y t (T, x), -y(T, x)), Φ(T ) H ′ ×H = (u 1 , -u 0 , y 1 , -y 0 ), Φ 0 H ′ ×H + T 0 Ω cv(t)ψdxdt = L(Φ 0 ).
Using the direct observability inequality (3.67), we deduce that

(3.71) L L(H,R) ≤ v 0 L 2 (0,T ;L 2 (ωc + )) + U 0 H ′ .
Using the Riesz representation theorem, there exists an element Z(x, t) ∈ H ′ solution of

(3.72) L(Φ 0 ) = Z, Φ 0 H ′ ×H , ∀Φ 0 ∈ H.
Then, define the weak solution U (x, t) of system (1.7)-(1.9) by U (x, t) = Z(x, t). The proof is thus complete.

Next, we consider the indirect locally internal exact controllability problem: For given T > 0 (sufficiently large) and initial data U 0 , does there exists a suitable control v that brings back the solution to equilibrium at time T , that is such the solution of (1.7)-(1.9) satisfies u(T ) = u t (T ) = y(T ) = y t (T ) = 0. Indeed, applying the HUM method, we obtain the following result.

Theorem 3.12. Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c+ satisfies the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω). For every T > M 1 , where M 1 is given in Theorem 3.10 and for every

U 0 ∈ (L 2 (Ω) × H -1 (Ω)) 2 ,
there exists a control

v(t) ∈ [H 1 (0, T ; L 2 (ω c+ ))] ′ ,
such that the solution of the controlled system (1.7)-(1.9) satisfies

u(T ) = u t (T ) = y(T ) = y t (T ) = 0.
Proof. We will apply the HUM method. Thanks to the indirect observability inequalities (3.67), we consider the seminorm defined by

Φ 0 2 H = T 0 ω b |ψ t | 2 dxdt,
where Φ = (ψ, ψ t , ϕ, ϕ t ) designate the solution of the homogeneous problem (3.61)-(3.63). Take the control v = d dt ψ t . Now, we solve the following time reverse problem:

(3.73)    ζ tt -∆ζ + bχ t = c d dt ψ t in (0, T ) × Ω, χ tt -∆χ -bζ t = 0 in (0, T ) × Ω, χ(T ) = χ t (T ) = ζ(T ) = ζ t (T ) = 0.
By Theorem 3.11, the system (3.73) admits a solution

Ψ(x, t) = (ζ, ζ t , χ, χ t , ) ∈ C 0 ([0, T ], H ′ ).
We define the linear operator Λ by:

Λ : H = (H 1 0 (Ω) × L 2 (Ω)) 2 → (H -1 (Ω) × L 2 (Ω)) 2 ,
where

ΛΦ 0 = (ζ t (0), -ζ(0), χ t (0), -χ(0)) ∀ Φ 0 ∈ (H 1 0 (Ω) × L 2 (Ω)
). In addition, we define the following linear form

(3.74) ΛΦ 0 , Φ0 = T 0 ωc ψ t ψt dxdt = (Φ 0 , Φ0 ) H , ∀ Φ0 ∈ H,
where (., .) H is the scalar product associated to the norm . H . Using Cauchy-Schwarz in (3.74) , we deduce that

(3.75) | ΛΦ 0 , Φ0 H×H ′ | Φ 0 H Φ0 H , ∀ Φ 0 , Φ0 ∈ H.
In particular, we have

| ΛΦ 0 , Φ 0 H×H ′ | = Φ 0 2 H ∀ Φ 0 ∈ H.
Then the inverse inequality in Theorem 3.10 implies that the operator Λ is coercive and continuous over H. Thanks to the Lax-Milgram theorem, we have Λ is an isomorphism from H into H ′ . In particular, for every

U 0 ∈ (L 2 (Ω) × H -1 (Ω)) 2 , there exists a solution Φ 0 ∈ H, such that Λ(Φ 0 ) = -U 0 = (ζ t (0), -ζ(0), χ t (0), -χ(0)).
It follows from the uniqueness of the solution of problem (3.73) that

U = Ψ.
Consequently, we have

u(T ) = u t (T ) = y(T ) = y t (T ) = 0.
The proof is thus complete.

4. Exponential stability and exact controllability in the case a = 1 4.1. Exponential stability in the weak energy space. The aim of this subsection is to show the exponential stability of system (1.4)-(1.6) in a weak energy space in the case when the waves do not propagate with same speed, i.e., a = 1. For this sake, we define the weak energy space

D = H 1 0 (Ω) × L 2 (Ω) × L 2 (Ω) × H -1
(Ω) equipped with the scalar product : for all U = (u, v, y, z) ∈ D and Ũ = (ũ, ṽ, ỹ, z) ∈ D,

(U, Ũ ) = Ω (a∇u.∇ũ + vṽ + y ỹ + (-∆) -1/2 z(-∆) -1/2 z)dx.
Next, we define the unbounded linear operator

A d : D(A d ) ⊂ D → D by A d U = ( v, a∆u -bz -cv, z, ∆y + bv ), D(A d ) = (H 1 0 (Ω) ∩ H 2 (Ω)) × H 1 0 (Ω) × H 1 0 (Ω) × L 2 (Ω) , ∀ U = (u, v, y, z) ∈ D(A d ).
We define the partial energy associated to a solution U = (u, u t , y, y t ) of (1.4)-(1.6) by

e 1 (t) = 1 2 a ∇u 2 L 2 (Ω) + u t 2 L 2 (Ω) .
We define also the weakened partial energy by

ẽ2 (t) = 1 2 y t 2 H -1 (Ω) + y 2 L 2 (Ω)
and the total mixed energy by E m (t) = e 1 (t) + ẽ2 (t).

In order to study the exponential decay rate, we need to assume that ω c+ satisfies the geometric conditions PMGC. Then there exist ε > 0, subsets Ω j ⊂ Ω, j = 1, ..., J, with Lipschitz boundary Γ j = ∂Ω j and points

x j ∈ R N such that Ω i ∩ Ω j = ∅ if i = j and ω + c ⊃ N ǫ ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j ∩ Ω with N ǫ (O) = {x ∈ R N : d(x, O) < ε} where O ⊂ R N , γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0}
where ν j is the outward unit normal vector to Γ j and that ω b satisfies the GCC condition and

ω b ⊂ Ω \ ∪ J j=1 Ω j . (LH3)
Now, we are ready to establish the following main theorem of this section:

Theorem 4.1. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω c+ satisfies the geometric conditions PMGC, ω b satisfies GCC condition and (LH3) and b, c ∈ L ∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ D the energy of system (1.4)- (1.6) satisfies the following decay rate:

(4.1) E m (t) ≤ M e -θt E m (0), ∀t > 0.
In order to prove the above theorem, we apply the same strategy using Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]. A C 0 -semigroup of contraction (e tA ) t 0 in a Hilbert space H is exponentially stable if and only if

iR ⊆ ρ(A d ) and (H1) lim sup β ∈R,|β|→+∞ (iβI -A d ) -1 L(D) < ∞ (H2)
Condition (H1) was already proved. We now prove that condition (H2) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence β n with β n → +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A d ) such that U n D = 1 and (4.2) lim n→∞ (iβ n I -A d )U n D = 0 . (4.3)
Next, detailing equation (4.3), we get

iβ n u n -v n = f 1 n → 0 in H 1 0 (Ω), (4.4) iβ n v n -a∆u n + bz n + cv n = g 1 n → 0 in L 2 (Ω), (4.5) iβ n y n -z n = f 2 n → 0 in L 2 (Ω), (4.6) iβ n z n -∆y n -bv n = g 2 n → 0 in H -1 (Ω). (4.7)
Eliminating v n and z n from the previous system, we obtain the following system (4.8)

β 2 n u n + a∆u n -iβ n by n -iβ n cu n = -g 1 n -bf 2 n -iβ n f 1 n -cf 1 n in L 2 (Ω), (4.9)
β 2 n y n + ∆y n + iβ n bu n = -iβ n f 2 n + bf 1 n -g 2 n in H -1 (Ω).
From (4.2), we have ∇u n , v n and y n are uniformly bounded in L 2 (Ω) and z n is uniformly bounded in H -1 (Ω). Using now (4.2) and (4.4), we deduce that β n u n is uniformly bounded in L 2 (Ω). In addition, using (4.2) and (4.6), we deduce that β n y n is uniformly bounded in H -1 (Ω). More precisely,

u n L 2 (Ω) = O(1) β n = o(1) and y n H -1 = O(1) β n = o(1). Lemma 4.2. The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.4)-(4.7
) satisfies the following estimates

(4.10) Ω c|β n u n | 2 dx = o (1) and ωc
+ |β n u n | 2 dx = o(1).
Proof. First, since U n is uniformly bounded in D and using (

Re

iβ n U n 2 -(A d U n , U n ) = Ω c(x)|v n | 2 dx = o(1).
Next, using equations (4.11) and (4.4), we get (4.12)

Ω c|β n u n | 2 dx = o(1).
Under condition (LH1), it follows

ωc + |β n u n | 2 dx = o(1).
The proof is thus complete. Now as ω c + satisfies the PMGC condition, let the reals 0 < ε 1 < ε 2 < ε and define Proof. First, multiplying equation (4.8) by ηū n . Then, using Green's formula and the fact that u n = 0 on Γ, we obtain (4.14)

Q i = N εi ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j , i = 1, 2. Since R N \ ω c + ∩ Q 2 = ∅, we can construct a function η ∈ C ∞ 0 (Ω) such that η(x) = 0 if x ∈ Ω \ ω c+ , 0 ≤ η(x) ≤ 1, η(x) = 1 if x ∈ Q 2 . Lemma 4.3. The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.
         Ω η|β n u n | 2 dx -a Ω η | ∇u n | 2 dx -a Ω u n (∇η • ∇u n )dx -i β n Ω bηy n u n dx -iβ n Ω cη|u n | 2 dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )ηu n dx.
As f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and β n u n is uniformly bounded in L 2 (Ω), we get (4.15)

Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )ηu n dx = o(1).
Using the fact that ∇u n , y n are uniformly bounded in L 2 (Ω), u n L 2 (Ω) = o(1) and estimation (4.10), we will have (4.16)

Ω η|β n u n | 2 dx -a Ω u n (∇η • ∇u n )dx -i β n Ω bηy n u n dx -iβ n Ω cη|u n | 2 dx = o(1).
Finally, inserting (4.15) and (4.16) into (4.14), we deduce

Ω η | ∇u n | 2 dx = o(1) and Q2∩Ω | ∇u n | 2 dx = o(1).
The proof is thus complete. Proof. The proof contains two steps.

Step 1. (Boundedness of 1 βn ∇y n ). Multiplying equation (4.9) by 1

β 2 n y n , we obtain Ω |y n | 2 dx+ < ∆y n , 1 β 2 n y n > H -1 (Ω)×H 1 0 (Ω) = -i Ω 1 β n f 2 n y n dx + Ω bf 1 n 1 β 2 n y n dx (4.18) -< g 2 n , 1 β 2 n y n > H -1 (Ω)×H
∇y n β n 2 L 2 (Ω) ≤ y n 2 L 2 (Ω) + 1 2 g 2 n 2 H -1 (Ω) + o(1).
It follows, from the uniform boundedness of y n in L 2 (Ω) and g 2 n in H -1 (Ω), that (4.20)

∇y n β n 2 = O(1).
Step 2. (Main asymptotic estimation). Multiplying equation (4.8) by η 1 βn y n . Later, using Green's formula and the fact that y n = 0 on Γ, we get (4.21)

         Ω ηβ n u n y n dx -a Ω 1 β n η(∇u n • ∇y n )dx -a Ω 1 β n (∇η • ∇u n )y n dx -i Ω bη|y n | 2 dx -i Ω cu n ηy n dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n ) η β n y n dx.
Next, using the definition of η and equations (4. Using the fact that f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and y n is uniformly bounded in L 2 (Ω), we will have (4.24)

Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n ) η β n y n dx = o(1).
Finally, inserting (4.22)-(4.24) into (4.21), we get

Ω bη|y n | 2 dx = o(1).
It follows, from condition (LH3), that

ω b |y n | 2 dx = o(1).
The proof is thus complete. Proof. Noting that ω b satisfies the GCC condition, so we can taking f n = y n in Lemma 3.6.

Multiplying equation (4.9) by φ n . Then, we have

Ω β 2 n φ n y n dx-< ∆y n , φ n > H -1 (Ω)×H 1 0 (Ω) +i Ω β n bu n φ n dx = -i Ω β n f 2 n φ n dx + Ω bf 1 n φ n dx (4.26) -< g 2 n , φ n > H -1 (Ω)×H 1 0 (Ω) .
Using the fact that φ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) and y n ∈ H 1 0 (Ω), then we have

(4.27) -< ∆y n , φ n > H -1 (Ω)×H 1 0 (Ω) = Ω y n ∆φ n dx.
It follows, from the first equation of (3.34) and (4.26), that

Ω |y n | 2 dx = i Ω bβ n φ n y n dx -i Ω β n bu n φ n dx (4.28) -i Ω β n f 2 n φ n dx + Ω bf 1 n φ n dx-< g 2 n , φ n > H -1 (Ω)×H 1 0 (Ω) .
Using the fact that β n φ n is uniformly bounded in L 2 (Ω), f 1 n converges to zero in H 1 0 (Ω), f 2 n converges to zero in L 2 (Ω) , g 2 n converges to zero in H -1 , (4.17 Proof. Multiplying equation (4.9) by (-∆) -1 y n , then integrating by parts and using the fact that y n = 0 on Γ, we get

Ω |β n (-∆) -1/2 y n | 2 dx = Ω |y n | 2 dx -i Ω β n bu n (-∆) -1 y n dx -i Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx + Ω bf 1 n (-∆) -1 y n dx (4.31) -< g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω)
. Using Cauchy-Schwarz and Poincaré inequalities, we get

| Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx| ≤ (-∆) -1/2 f 2 n L 2 (Ω) β n (-∆) -1/2 y n L 2 (Ω) (4.32) ≤ c 0 f 2 n L 2 (Ω) β n y n H -1 (Ω) . It follows, from the convergence to zero of f 2 n in L 2 (Ω) and the boundedness of β n y n in H -1 (Ω), that (4.33) Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx = o(1)
.

Similarly, we have

|< g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω) | = Ω (-∆) -1/2 g 2 n (-∆) -1/2 y n dx (4.34) ≤ (-∆) -1/2 g 2 n L 2 (Ω) (-∆) -1/2 y n L 2 (Ω) ≤ g 2 n H -1 (Ω) y n H -1 (Ω) . It follows, from the convergence of g 2 n and y n to zero in H -1 (Ω), that (4.35) < g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω) = o(1)
. Note that (-∆) -1 is compact operator from L 2 to L 2 . Then (-∆) -1 y n is uniformly bounded in L 2 . Finally, using (4.10), (4.25), (4.33), (4.35) and the fact that f 1 n converges to zero in H 1 0 (Ω) into equation (4.31), we deduce

Ω |β n (-∆) -1/2 y n | 2 dx = o(1).
The proof is thus complete.

|∇u n | 2 + |β n u n | 2 )dx = o(1). Proof. Since (Ω j \ Q 2 ) ∩ Q 1 = ∅, we define the function ψ j ∈ C ∞ 0 (R N ) by: ψ j (x) = 0 if x ∈ Q 1 , 0 ψ j 1, ψ j (x) = 1 if x ∈ Ω j \ Q 2 .
For m j (x) = (xx j ), we define h j (x) = ψ j (x)m j (x). Multiplying equation (4.8) by 2(h j • ∇u n ) and integrating over Ω j , using the dissipation (4.10) and the fact that ∇u n is uniformly bounded in L 2 (Ω), we obtain

2β 2 n Ωj u n (h j • ∇u n )dx + 2a Ωj ∆u n (h j • ∇u n )dx -2i Ωj β n by n (h j • ∇u n )dx = 2 Ωj (-g 1 n -bf 2 n -cf 1 n )(h j • ∇u n )dx -2i Ωj β n f 1 n (h j • ∇u n )dx. ( 4
.37) i) Estimation of the second member of (4.37). First, using Green's formula and the fact that u n = 0 on (Γ j \ γ j ) ∩ Γ and h j = 0 on γ j , we get

-2i Ωj β n f 1 n (h j • ∇u n )dx = 2i Ωj β n u n (h j • ∇f 1 n )dx + 2i Ωj β n u n f 1 n (divh j )dx. ( 4
(-g 1 n -bf 2 n -cf 1 n )(h j .∇u n )dx = o(1).
Finally, we deduce that the second member of (4.37) is o(1).

ii) Estimation of the first integral of equation (4.37). Using Green's formula, we get

(4.41) Re 2 Ωj β 2 n u n (h j • ∇u n )dx = - Ωj (divh j )|β n u n | 2 dx + Γj (h j • ν j )|β n u n | 2 dΓ j .
Since Ψ j = 0 on γ j and u n = 0 on (Γ j \ γ j ) ∩ Γ, then we have

(4.42) Re{2 Ωj β 2 n u n (h j • ∇u n )dx} = - Ωj (divh j )|β n u n | 2 dx.
iii) Estimation of the second integral of equation (4.37). Using Green's formula, we get

Re 2a

Ωj

∆u n (h j • ∇u n ) = -2aRe N i,k=1 Ωj ∂ i h k j ∂ i u n ∂ k u n dx + (4.43) a Ωj (divh j )|∇u n | 2 dx -a Γj (h j • ν j )|∇u n | 2 dΓ j + 2aRe Γj ∂ νj u n (h j • ∇u n )dΓ j .
According to the choice of ψ j , only the boundary terms over (Γ j \ γ j ) ∩ Γ are non vanishing in (4.43). But on this part of the boundary u n = 0, and consequently Thus, summing over j and using the fact that ψ j = 1 on Ω j \ Q 2 , we get Thus, system (4.52)-(4.54) is conservative in the sense that its energy E m (t) is constant. It is also wellposed and admits a unique solution in the energy space D. Now, we establish the direct and indirect inequality given by the following theorem:

∇u n = (∂ ν u n ) • ν = (∂ νj u n )ν j . Then, we have (4.44) -a Γj (h j • ν j )|∇u n | 2 dΓ j + 2aRe Γj (∂ νj u n )(h j • ∇u n )dΓ j = a (Γj \γj )∩Γ (ψ j m j • ν j )|∂ νj u n | 2 dΓ j 0.
Theorem 4.8. Let 0 < a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω c+ satisfies the PMGC, ω b satisfies GCC condition and (LH3) and b, c ∈ L ∞ (Ω). Then there exists a time T 0 such that for all T > T 0 , there exist two constants C 1 > 0, C 2 > 0 such that the solution of system (4.52)-(4.54) satisfies the following observability inequalities:

(4.58) C 1 Φ 0 2 D ≤ T 0 Ω c(x)|ψ t | 2 dxdt ≤ C 2 Φ 0 2 D dx,
for all Φ 0 = (ψ 0 , ψ 1 , φ 0 , φ 1 ) ∈ D.

Proof. The direct inequality follows from the definition of the total energy for all T > 0. While the proof of the inverse inequality is a direct consequence of Proposition 2 of Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the exponentially stability (4.1) implies the existence of a time T 0 > 0 such that for all T > T 0 there exist two constants C 1 > 0 and C 2 > 0 such that (4.58) holds.

It is well known that observality of the homogeneous system (4.52)-(4.54) implies the exact controllability of the system .

1 . Introduction 1 . 1 .

 111 Motivation and aims. Let Ω be a nonempty connected open subset of R N having a boundary Γ of class C 2 . In [6], F. Alabau et al. considered the energy decay of a system of two wave equations coupled by velocities u tt -a∆u + ρ(x, u t ) + b(x)y t = 0 in Ω × R * + , (1.1)

Lemma 3 . 3 .

 33 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimates (3.15) Ω c|∇u n | 2 dx = o(1) and ωc + |∇u n | 2 dx = o(1).

Lemma 3 . 5 .

 35 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimate (3.30) ω b |β n y n | 2 dx = o(1).

Lemma 3 . 7 .

 37 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimate (3.41) Ω |β n u n | 2 dx = o(1). Proof. Taking f n = u n in Lemma 3.6 and multiplying equation (3.8) by β 2

Lemma 3 . 8 .

 38 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimate (3.49) Ω |β n y n | 2 dx = o(1).

Lemma 3 . 9 .

 39 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.4)-(3.7) satisfies the following estimates (3.56) Ω |∇u n | 2 dx = o(1) and Ω |∇y n | 2 dx = o(1).

  4)-(4.7) satisfies the following estimates (4.13) Ω η | ∇u n | 2 dx = o(1) and Q2∩Ω | ∇u n | 2 dx = o(1).

Lemma 4 . 4 .

 44 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.4)-(4.7) satisfies the following estimate (4.17) ω b |y n | 2 dx = o(1).

Lemma 4 . 5 .

 45 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.4)-(4.7) satisfies the following estimate (4.25) Ω |y n | 2 dx = o(1).

  ) and u n = o(1) in equation (4.28), we obtainΩ |y n | 2 dx = o(1). (4.29)The proof is thus complete.

Lemma 4 . 6 .

 46 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.4)-(4.7) satisfies the following estimate (4.30) Ω |β n (-∆) -1/2 y n | 2 dx = o(1).

Lemma 4 . 7 .

 47 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (4.4)-(4.7) satisfies the following estimate (4.36) Ω\(Q2∩Ω)

Inserting ( 4 .

 4 44) into (4.43), we get Re 2aΩj ∆u n (h j • ∇u n ) ≤ -2aRe N i,k=1 Ωj ∂ i h k j ∂ i u n ∂ k u n dx + a Ωj (divh j )|∇u n | 2 dx. (4.45)iv) The main estimation. Inserting equations (4.39), (4.40), (4.42) and (4.45) into (4.37) and using the fact that ψ j = 0 on Q 1 , we getΩj \(Q1∩Ωj ) div(ψ j m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx +2i Ωj \(Q1∩Ωj )β n by n (ψ j m j • ∇u n )dx o(1).

( 4 .j=1j=1 4 . 2 .∇ψ 2 L 2 (Ω) + ψ t 2 L 2 (

 4422222 46) NΩ\(Q2∩Ω) |β n u n | 2 dx + (2 -N )a Ω\(Q2∩Ω) |∇u n | 2 dx +2Re i J j=1 Ωj \(Q1∩Ωj ) β n by n (ψ j m j • ∇u n )dx j m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx + o(1).Using (4.10) and (4.13), we deducej m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx = o(1). (4.47) Inserting (4.47) in (4.46), we obtain (4.48) N Ω\(Q2∩Ω)|β n u n | 2 dx + (2 -N )a Ω\(Q2∩Ω) |∇u n | 2 dx+ 2Re Ωj \(Q1∩Ωj ) β n by n (ψ j m j .∇u n )dxUnder condition (LH3) and the definition of ψ j , we will have2Re Ωj \(Q1∩Ωj )β n by n (ψ j m j .∇u n )dx estimation into (4.48), we getN Ω\(Q2∩Ω) |β n u n | 2 dx + (2 -N )a Ω\(Q2∩Ω) |∇u n | 2 dx ≤ o(1). (4.49)Multiplying (4.8) by (1 -N )u n . Then integrating on Ω, using Green's formula, the fact that y n and β n u n are bounded in L 2 (Ω) and the estimation (4.10), we obtain(4.50) (1 -N ) Ω |β n u n | 2 dx -(1 -N )a Ω |∇u n | 2 dx = o(1).Using (4.10) and (4.13) in (4.50), we deduce(4.51) (1 -N ) Ω\(Q2∩Ω) |β n u n | 2 dx -(1 -N )a Ω\(Q2∩Ω) |∇u n | 2 dx = o(1).Finally, combining (4.49) and (4.51), we get the following estimateΩ\(Q2∩Ω) a|∇u n | 2 + |β n u n | 2 )dx = o(1).The proof is thus complete.Proof of Theorem 3.10 It follows from (4.10) (4.13), (4.25), (4.30) and (4.36) that U n = o(1) which is a contradiction with (4.2). Consequently, condition (H2) holds and the energy of system (1.4)-(1.6) decays exponentially to zero in the weak energy space D. The proof is thus complete. Observability and exact controllability. First, we consider the following homogeneous system associated to (1.4)-(1.6) for a = 1 by:ψ tt -a∆ψ + b(x)ϕ t = 0 in Ω × R + , (4.52) ϕ tt -∆ϕb(x)ψ t = 0 in Ω × R + , (4.53) ψ = ϕ = 0 on Γ × R + , (4.54) ψ(•, 0) = ψ 0 , ψ t (•, 0) = ψ 1 , ϕ(•, 0) = ϕ 0 , ϕ t (•, 0) = ϕ 1 in Ω. (4.55)Let Φ = (ψ, ψ t , ϕ, ϕ t ) be a regular solution of system (3.61)-(3.63), its associated total energy is given byΩ) + ϕ t H -1 (Ω) + y L 2 (Ω) . A direct computation gives (4.57) d dt E m (t) = 0.

  1), because f 1 n converges to zero in H 1 0 (Ω) and β n u n , β 2 n φ n , β n ∇φ n are uniformly bounded in L 2 (Ω).

	Substituting now (3.44) and (3.46) into (3.43)
	(3.47)

  .53) Using the fact that β n y n , β 2 n φ n and β n ∇φ n are uniformly bounded in L 2 (Ω) and f 2

	zero in H 1 0 (Ω) in (3.53), we get	n converges to
	(3.54)	-i
	Inserting (3.52), (3.54) into (3.51), we obtain
	(3.55)

Ω β n f 2 n β 2 n φ n dx = o(1).

  .38) So, from the fact that f 1 n converges to zero in H 1 0 (Ω) and β n u n is uniformly bounded in L 2 (Ω),

	we obtain
	(4.39)	-2i	Ωj	β n f 1 n (h j • ∇u n )dx = o(1).
	Next, as f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and the sequence (∇u n ) is uniformly bounded in L 2 (Ω), we deduce
	(4.40)	2
		Ωj

Acknowledgments

The authors are grateful to the anonymous referees and the editor for their valuable comments and useful suggestions. The authors thanks professor Kais Ammari for their valuable discussions and comments. Amina Mortada and Chiraz Kassem would like to thank the AUF agency for its support in the framework of the PCSI project untitled Theoretical and Numerical Study of Some Mathematical Problems and Applications Ali Wehbe would like to thank the CNRS and the LAMA laboratory of Mathematics of the Université Savoie Mont Blanc for their supports.


Frédéric Mallet

Charles André

Sébastien Gérard

François Terrier

Keywords: domain model, multilevel modeling, UML proles, deep instantiation

Building a UML prole may be a tedious and error-prone process. There is no precise methodology to guide the process or to verify that all concepts have been implemented once and only once. Best practices recommend starting by gathering concepts in a technology-independent domain view before implementation. Still, the adequation between the domain view and the implementation should be veried. This paper proposes an automatic process to transform a domain model into a prolebased implementation. To reduce accidental complexity in the domain model and fully benet from advanced proling features in the generated prole, our process uses the multilevel paradigm and its deep characterization mechanisms. The value of this paradigm for the denition of UML proles is assessed and applied to the subset of a recently adopted OMG UML Prole, an excerpt of the MARTE time prole.

As a by-product, our process involves an inexpensive prole-based implementation of the multi-level paradigm within UML2 tools.

Un processus automatique pour la génération de prols UML Résumé :

Construire un prol UML peut s'avérer fastidieux. Il n'y a actuellement aucune méthode reconnue pour guider cette dénition, pour vérier que tous les concepts métiers ont été implémentés une et une seule fois et pour éviter les pièges nombreux.

Quelques contributions essayent d'éclaircir le chemin et il est généralement reconnu qu'il faut commencer par faire un modèle métier indépendant d'une technologie particulière (y compris d'UML) avant de se lancer dans des considérations spéciques à l'implantation.

Malgré tout, le passage du domaine métier à l'implantation reste dicile.

Nous proposons ici un procédé automatique pour transformer le modèle métier en un prol UML. Pour réduire la complexité inutile du domaine métier et obtenir toutefois un prol précis et avancé nous préconisons l'utilisation du paradigme multi-niveaux et de la caractérisation en profondeur. L'intérêt de ce paradigme pour la dénition de prols est évalué et son utilisation est illustrée sur un extrait simplié du prol UML MARTE, récemment adopté par l'OMG.

Notre procédé implique l'implantation dans les outils UML2 du paradigme de modélisation multi-niveaux et nous présentons cette implantation comme une contribution dérivée.

Mots-clés : modèle métier, modélisation multi-niveaux, prols UML, deep instantiation 1 Introduction

Building a prole for the Unied Modeling Language (UML) [START_REF]Object Management Group: Unied Modeling Language[END_REF] may be a tedious and errorprone process. There is no precise methodology to guide the process or to verify that all concepts have been implemented once and only once. The quality of the prole should be assessed with metrics and its adequation with the target domain should be veried.

This lack of methodology results in collections of heterogeneous proles that can exhibit both inconsistencies and overlaps. To overcome this issue, Bran Selic [2] suggests to start with a technology-independent description (a domain model) that can be used as a communication interface between the domain experts and the implementation team.

Starting from this domain model, several implementation choices are possible: using a Domain-specic language (DSL), a pure meta-modelling approach or building a prole.

Building tools for domain-specic languages can be quite expensive. The target community being generally small, there is little economic incentive for tool vendors to build such tools.

The result has often been the creation of incomplete tool suites and lack of support. For this reason, companies often turn to general-purpose languages, even if the modeling capabilities available are ill-suited to the domain requirements.

In this paper, we focus on the prole-based solution. Proles incrementally build on the general-purpose UML to dene only the new concepts required to represent domainspecic elements. This reduces the cost of building tools by reusing existing ones and limits the investment required for the learning process. The domain model is mapped onto the UML metamodel by dening stereotypes that extend existing UML concepts (metaclasses) in order to modify or rene their semantics. This two-stage process allows designers to focus on domain concepts and their relationships before dealing with language implementation issues.

However, the gap between the domain model and the prole may be particularly dicult to ll because of the potential inability of UML or, more generally, of the object-oriented paradigm to capture domain concepts.

One of the essential modeling capabilities missing in the object-oriented paradigm is the ability to model multiple levels. This failing may lead to accidental complexity [START_REF] Brooks | No silver bullet essence and accidents of software engineering[END_REF].

Well-known design patterns (like Item Description [4] or Type Object [5]) are articial workarounds to mimic multi-level scenarios with only two levels (classes and objects).

C. Atkinson and T. Kühne [START_REF] Atkinson | Reducing accidental complexity in domain models[END_REF] propose to go beyond the traditional two levels and promote a multi-level modeling paradigm that combines deep characterization with deep instantiation.

Use of the multi-level modeling paradigm is particularly relevant to dening proles. Proles are cross-level mechanisms between the metamodel and the model levels and mix multilevel concepts: model libraries elements, metaclasses representing descriptors (e.g., Classier) and metaclasses representing items (e.g., InstanceSpecication). Such concept mixtures can be found in the newly adopted UML Prole for Modeling and Analysis of Real-Time and Embedded systems (MARTE [7]), the result of signicant collaborative eort by domain experts. In MARTE, the prole is sometimes far from the domain model. Designers apply sophisticated design patterns, which make it hard to ensure that every domain concept is actually implemented in the prole.

We propose an automatic process to generate a UML prole directly from the domain model. To reach the same expressiveness as in MARTE, our domain model relies on the multi-level paradigm. Therefore, as a by-product we propose a UML prole for domain specication that oers multi-level annotations and a practical solution for implementing the deep instantiation mechanism within UML 2.x-compliant tools. This prole is used to specify a domain model, then the model is automatically transformed into an equivalent prole-based implementation.

We have used the Time subprole [START_REF] André | Modeling time(s)[END_REF] of MARTE to illustrate our approach. We begin by taking an excerpt from the Time subprole and we explain why it may seem unnecessarily complex (Section 2). In Section 3, we propose another domain model for the Time built on the multi-level paradigm. We then introduce our prole for Domain Specication and show how multi-level annotations are applied and used to transform the domain model into a prole-based implementation (Section 4). Section 5 outlines the dierences between a process that leverages the multi-level modeling paradigm and the approach followed by MARTE designers. Finally, we discuss some related works in Section 6 before concluding.

Motivation

The MARTE Time Prole

Figure 1 is an excerpt from the MARTE Time prole. Note that this paper does not elaborate on the underlying concepts of the Time prole, which are described in a previous work [START_REF] André | Modeling time(s)[END_REF].

This paper focuses on its design intents using a small number of concepts to justify our approach. The gure contains three parts: the left bottom part is a simplied description of the MARTE Time prole, the right-hand part shows libraries associated with the prole, and the upper part gathers UML metaclasses and relationships referred to by MARTE Time prole. The diagram in this latter part is extracted from UML::Classes::Kernel. Annotations (A) and (B) and annotated dependencies (A) and (C) are not part of the specications, they are used below to ease the understanding.

Prole core consists of two stereotypes (ClockType and Clock). These stereotypes provide for mechanisms to create new clocks and to put together (within a clock type) features common to several clocks.

A close look at the package Time shows that Time, and more generally MARTE proles make advanced usages of stereotypes. In the prole, the stereotype properties (tags) are not limited to primitive types. Some are typed by UML metaclasses and even by stereotypes.

The use of the metaclass Property for the attribute resolAttr avoids premature specication of the type and name of the property that models the clock resolution, since they may dier signicantly depending on the system considered. Sometimes the type may be a real number, representing the precision of the clock relative to a reference clock, sometimes it may be given as an integer or even as a mere enumeration (e.g., ne, medium, coarse). Typing attribute unit directly by the stereotype NFP::Unit instead of the metaclass EnumerationLiteral leads to a more precise characterization of what unit is. Not any enumeration literal can be used as a time unit; it has to be stereotyped by NFP::Unit thus giving additional information like conversion factor. The same eect could have been achieved typing by the metaclass EnumerationLiteral combined with a constraint. This alternative solution is often preferred because of the failure of most commercial tools to support stereotype-tostereotype associations.

Multi-level aspects in the Time Prole

In Figure 1, at least two modeling levels have been mixed together in one.

The rst, or metamodel level is easily identied by the metaclass keyword (e.g., InstanceSpecication, Class, Enumeration, EnumerationLiteral). A careful look at the existing relationships in the UML metamodel is, however, required to accurately qualify the modeling levels involved. The two relationships (A) and (B) in Figure 1, in the UML metamodel, are used to tell what belongs to classier level from what belongs to instantiation level. The relationship (A) tying an InstanceSpecication to one of its Classier shows that clocks are instances whereas clock types are types. This relationship is made concrete in the prole by the derived attribute type of stereotype Clock. type is a subset derived attribute of attribute classier from the metaclass InstanceSpecication.

The use of the metaclass Property for the attribute resolAttr is motivated by the relationship (B) between the metaclass Slot and the sub-class of StructuralFeature: the metaclass Property. Ultimately, the resolution value is given in an instance slot. The dependency relationship (C) is also a relationship between the metaclasses, but is only dened in the prole. It states that a clock type refers to the set of acceptable units (unitType), whereas a clock refers to a specic unit from among this set (e.g., s, ms). This specication is enforced by an OCL constraint.

The second modeling level corresponds to the use of model libraries, which are specic UML constructs that purposely escape classication by level. Elements from model libraries can be used at any levels, in a metamodel, a prole or a user model. This is the case of the primitive type Boolean, dened in the UML standard model library, which is used to dene whether a clock type is logical or not. However, certain of MARTE data types, those dened in TimeTypesLibrary, are intended for use in the prole and by users (e.g., TimeNatureKind, TimeStandardKind). Others, dened in TimeLibrary, should only be used at user level (e.g., TimeUnitKind) and are clearly situated below the prole level since TimeLibrary applies the prole. Using this library within the prole would result in cyclic denitions.

Applying the Time Prole

Figure 2 illustrates the use of the Time prole for two clock types. The left-hand side of Figure 2 shows a conventional usage of a prole. The clock type Chronometric is dened in the model libary TimeLibrary. It should be used to model discrete clocks, related to physical time, which are not necessarily perfect. The property resolution, whose type is Real, is selected to play the role of resolAttr. The clock type Cycle represents a discrete logical clock that uses units like processorCycle or busCycle to date event occurrences. For this second clock, there is no need for a property playing the role of resolAttr.

The chronometric clock cc1 completes the specication by selecting one specic unit (s) from among the literals dened in the enumeration TimeUnitKind. It also chooses a standard and a value for the resolution. The cycle clock p1 also selects a unit, but from a dierent enumeration, CycleUnitKind. Clocks of the same type must use compatible units (from the same enumeration). In that regard, the clock types acts as a dimension.

The right-hand side of the gure is a representation of the domain view using clabjects as dened by Atkinson and Kühne. This representation combines notational conventions from UML classes and instance specications. Clabjects have elds, to unify meta-attributes, attributes and slots, and thus atten the dierent modeling levels into one. Horizontal dashed lines serve to identify the logical modeling levels.

In the UML view, Clock and ClockType are both represented at the same level, as stereotypes. However, ClockType is a descriptor for a set of Clock (as dened by the pattern

Limitation

Such a design strategy denes clock properties in several steps. Part of the information is

given at the class level and, the rest, at the instance level. Some features are progressively rened using the relationship between Classier and InstanceSpecication. This is the case for the property resolAttr that references a user-dened property of the clock type Chronometric at the class level. The value of this property is given at the instance level in the related slot.

The same is true for the unit. At class level, a set of possible units is identied, but choice of the actual unit takes place only at the instance level. Certain features such as nature are only relevant at class level and others only relevant at instance level (e.g., standard).

To comply with UML requirements, the adopted solution involves complex, albeit neat, workarounds. The most visible drawback is that meaningful information is scattered across dierent locations, corresponding to the dierent modeling levels: stereotype, class, instance specication, and consequently, obscurs the domain information. Information is further hidden under elements that do not contribute to denition of the domain but are required by limitations of the modeling language. This is the case of the property resolAttr.

Multi-level modeling with deep instantiation

The inability of languages like UML to represent multiple classication levels in a same modeling level has already been pointed out by C. Atkinson and T. Kühne [START_REF] Atkinson | The Essence of Multilevel Metamodeling. UML -The Unied Modeling Language[END_REF][START_REF] Atkinson | Reducing accidental complexity in domain models[END_REF].

Their initial observation was that information about instantiation mechanisms was bound to one level. The most visible eect is that information carried by attributes cannot cross more than one level of instantiation. This obstacle is deemed to be a major impediment and also tends to increase the complexity of models. The abovementioned authors therefore call for attening the modeling levels through use of deep instantiation. The key concept is a potency that characterizes any model elements. The potency is an integer that denes the number of instantiations (depth) that may be applied on elements. Properties and slots are uniformly called elds. Fields of potency higher than one are meta-attributes. Fields of potency one are regular attributes and elds of potency zero are regular slots. Fields are thus made to persist throughout each of the instantiations (as opposed to the shallow instantiation). Each instantiation of a eld (property with potency annotation) decreases the value of potency by one.

Designers of the Time prole obviously had to solve similar problems. We know it for sure since some authors of this paper were part of the ProMARTE consortium in charge of dening the prole MARTE. Their solution plays on the instance of relationship between InstanceSpecication and Class to put, at the appropriate level, information concerning the single concept Clock. This results in information scattering. Deep instantiation, on the other hand, makes multiple levels explicit and oers new design opportunities. In this approach, both stereotypes Clock and ClockType represent the same concept. They need not be apart in the domain model, even though they are separate in the nal implementation.

Instead, instantiation levels are identied by potency independently from the underlying UML implementation details.

Figure 3 shows how we can start from the actual specication requirements and progressively rene it into a usable domain model on which the automatic transformation applies.

This renement goes in three steps.

First, an analysis of dierent usages and interpretations of time in various application domains related to real-time and embedded systems shows that a clock can be characterized by its nature (dense or discrete), whether it is logical or chronometric, its resolution, unit and standard. More features were actually identied, however we have selected here a representative subset to illustrate our approach. This results in the left-most part of the gure. Sometimes, the type of these features is obvious, sometimes there is no obvious common type. In such a case, like for the resolution, we can use the mechanism presented

Clock

Conceptual model

One-level solution Multi-level domain model

Figure 3: Clock denition with potency information in Section 2.1 to allow maximum exibility. In the middle-part of the gure, resolution is substituted by resolAttr of type Property, a higher order feature. Second, a careful look at the unit shows that clocks can be classied according to the kind of units they use. Therefore, we introduce the concept of unitType to represent a set of acceptable units. Since an NFP_unit is an enumeration literal, it seems natural to dene unitType as an Enumeration.

Third, it appears that part of the information is common to a collection of clocks whereas The nal description is very concise and has the same expressiveness as the original Time prole (Figure 1). The potency value makes it clear whether a property must obtain its value at the rst or the second instantiation level.

In the next section, we propose a practical means for implementing such mechanisms within a UML tool.

Our proposal

This section presents the mechanisms that have been devised for the creation of UML-based domain specic languages that supports the multi-level modeling paradigm. Our proposal is based on a three-step process. The rst step is to specify the domain model. Elements of the model have properties that are annotated with a potency. This artifact is used, in RR n°6599 a second step, to automatically derive a UML prole from the specication. Our premise is that using an automated transformation reduces the gap between the domain and the prole and ensures that every concept in the domain is actually implemented in the prole.

Application of this prole, in a third step, enables modeling of elements that comply with the domain model specication. In subsequent sub-sections follows a step-by-step illustration based on the Time prole.

Domain model specication

We have dened a UML prole (called DomainSpecication) for domain specication. It is used to annotate a UML model with information required for multi-level modeling. This enables to declare models in a way similar to the one presented in Figure 3. This prole consists of stereotype Field (see Figure 4, left part) that extends metaclass Property and carries potency information. We start with a class model and apply the stereotype Field to specify potency (package TimeRevisited in Figure 4). Potency is optional. A property without potency is considered a regular attribute and is equivalent to a potency of one.

« domainSpecification » TimeRevisited

Automated prole-based implementation of the domain model

In this second step, we use the domain specication as an artifact to build a prole-based implementation. The result is a prole with enough stereotypes to describe each instantiation level for each concept. Consequently, we automatically derive the prole shown in Figure 5.

A straightforward algorithm is used for transformation. Each class gives rise to as many stereotypes as instantiation levels. The name of each generated stereotype is derived from the name of the initial class suxed by an integer that reveals the level of instantiation. In We start by creating the stereotypes at the highest level. After each step, the potency of each eld is decreased by one. When the potency is one, an attribute is added to the stereotype with the same name and type as the related eld. If the potency becomes lower than one, the eld is discarded (and no longer used).

On Figure 5, the clock concept has been mapped onto two stereotypes. The stereotype Clock_2 carries information belonging to the rst instantiation level (formerly ClockType) and Clock_1 is the nal level (formerly Clock). The association between the stereotypes of two consecutive levels is added to unambiguously relate a deep instance (stereotyped by Clock_1) to its deep type (stereotyped by Clock_2).

Applying the generated prole

In this section, we apply the generated prole to declare the chronometric and cycle clocks (Figure 6).

Modeling of Cycle clock entails two new classes, one stereotyped by clock_2 and the other by clock_1. The property type avoids mixing Cycle clocks with Chronometric clocks. For instance, cycleClk is associated with CycleClock, not ChronometricClock.

The structure of this model is, at rst glance, similar to the model using the original MARTE constructs (Figure 2). One obvious dierence, however, appears in the metaclasses used as bases for our stereotypes. In MARTE, clocks were instance specications of classes ChronometricClock and CycleClock, themselves stereotyped by clockType. The instance specications carry information about the slots of this class and also provide information

Discussions on our approach

This section compares our proposed approach with the one followed by the MARTE designers and by prolers in general. It describes the process workow dierences, then discusses possible extensions to our approach.

Design ow comparison

Possible extensions

Currently the prole DomainSpecication is minimal. Nevertheless, new stereotypes can be introduced to guide the prole generation. Remind that, one of the original reason for building a prole was to be able to customize a model according to a particular point of interest. Compliant tools should (even if it is mostly not the case now) provide an easy support to hide or restore annotations of a given prole. Following that idea, our prole could be customized dierently to build a prole tting the design team legacy. Each team can implement its own generation rules. Explicit prole generation process and tool support allow designers to assess dierent candidate generation processes.

As possible customizations, we can allow a systematic naming convention and the selection of base metaclasses more adequate than the by-default Class. This kind of customization would allow us to generate the exact same result as in the actual MARTE specication where several metaclasses other than Class have been used (InstanceSpecication, Event, Observation, Activity. . . ). As illustrated in this paper, using generic metaclasses like Property and Class often leads to very exible solutions. However, this may also lead to solutions with little semantics. It is often preferrable to choose types and metaclasses that better represent domain concepts. For instance, we could also have dened the unit as being a Property instead of an enumeration literal. In that case, we had the constraint of being compatible with NFP_unit dened in MARTE NFP subprole. Being more general than an enumeration literal would have prevented us from using NFP_unit-specics, like its conversion factor.

Related work

The core motivation of this work is to facilitate building of UML-based domain-specic languages. In that matter, two communities confront each other, the meta-modeling and the proling communities. Light-weight solutions often involves the creation of a prole, whereas more complete solutions require the use of meta-modeling. This work is only concerned with the building of proles. However, use of multi-level paradigm simplies the domain model.

This multilevel-aware domain model could also be used by meta-modeling tools to produce more faithful code.

In this paper, and in multi-level modeling in general, we focus on the relationship in-stanceOf . Other approaches, related to meta-modeling, consider relationships in general, instanceOf being just one of them. Other relationships include association, dependencies, conformance, composition. . . .

In prole-based approaches, despite the ever increasing number of proles being built in many domains, there is little published literature available to support the process as a whole.

Fuentes and Vallecillo [START_REF] Fuentes-Fernández | An Introduction to UML Proles[END_REF] point to the need for rst dening a domain model (using UML itself as the language) to clearly delineate the domain of the problem. In a more recent paper [2], Bran Selic describes a staged development of UML proles and gives useful guidelines for mapping domain constructs to UML.

Our proposal also leverages the use of a domain model but explores multi-level modeling capabilities at this stage. Almost all the material available on multi-level modeling can be found in research eorts conducted by Kühne and Atkinson. They have studied the foundations of such modeling and proposed an implementation based on UML 1.x constructs [START_REF] Atkinson | The Essence of Multilevel Metamodeling. UML -The Unied Modeling Language[END_REF]. This work now needs to be aligned on UML 2.x. More recently, Kühne and Schreiber [START_REF] Kühne | Can programming be liberated from the two-level style: multi-level programming with DeepJava[END_REF] explored possibilities for support of deep instantiation in Java.

The context of our proposal is somewhat dierent. We assess values of deep instantiation mechanisms in the context of UML prole denitions, then demonstrate that the current UML specication already includes mechanisms for accessing the realm of multi-level modeling.

Conclusion

This paper presents an automatic process for generating a UML prole from a domain model by leveraging the use of the multi-level modeling paradigm. The process begins with the specication of concepts required to cover a specic domain. This design activity uses a prole to annotate concepts (mainly classes and properties) with multi-level information.

Properties become elds and carry potency information that indicates their intended in- We advocate for a well-dened process that could consistently be used to dene coherent proles. This process must rely, as much as possible, on automatic transformations. The use of the multi-level paradigm is not specic to proling and could also be used in metamodeling approaches. Such a paradigm deserves to be further explored and its adequation with more general approaches still need to be assessed.

Figure 1 :

 1 Figure 1: Excerpt from the MARTE Time Prole

INRIAFigure 2 :

 2 Figure 2: Examples of clocks and clock types

  another part is specic to each individual clock. Without the multi-level paradigm, this would result (as in MARTE) in splitting Clock into two parts, one to represent the clock type and the other to represent the clock instances. Using the multi-level paradigm, we simply assign the potency to features according to the level they belong to. Adequate potencies are shown in the right-most part of the gure. In the end, elds that operate at instance level (formerly Clock) should have the highest potency. Here, the highest potency is two, since we have three levels (metatypes, types, instances) as shown by the horizontal dashed line in Figure 2. Fields that operate at class level (formerly ClockType) should have a potency of one.

Figure 4 :

 4 Figure 4: Domain specication prole and its usage

INRIA

Figure 5 :

 5 Figure 5: Time prole generated from the domain specication

  like values relating to properties dened by the stereotype Clock. With the generated prole, since both Clock_2 and Clock_1 extend Class, we must express the fact that an instance« clock_1 » { unit = s, standard = UTC, type = ChronometricClock } chronometricClk

Figure 6 :

 6 Figure 6: Clock denition with the generated prole.

Figure 7 Figure 7 :

 77 Figure 7 shows comparison of the two process workows, from conceptual domain denition to prole creation.MARTE designers applied a two-stage process: domain description and manual mapping of domain concepts onto prole constructs. The second stage is a very sensitive activity, since dierent designers may use dierent design solutions to map a given concept. This makes it dicult to assess the implementation.

  INRIA stantiation level. The domain specication is then used to map elements onto equivalent prole constructs. The result is a prole-based implementation of the domain model that contains all stereotypes required to represent the concepts of the domain and their dierent instantiation levels. Application of this prole thus enables deep instantiation and modeling of elements complying with the domain model specication. The proposal is illustrated with an excerpt from the Time sub-prole part of the recently adopted MARTE Prole. Use of the multi-level modeling paradigm provides new design opportunities and enables simplications. It facilitates the domain specication by limiting implementation considerations. The domain description is more concise and claries the modeling levels. The resulting domain model gathers in a single class all information related to a given concept, whereas it was previously scattered over several classes. Our proposal entails several model transformations. They are being automated in an Eclipse environment as a plugin of the open source Papyrus 1 UML tool. This tooling support will enable generation of proles that support domain elements and include the necessary OCL rule enforcements. Assessment of the user's model should be made automated.

RR n°6599

This report is based on the INRIA research report RR-6525, which has also been