Ioan Florin Bugariu
email: florinbugariu86@yahoo.com

Sorin Micu
email: sdmicu@yahoo.com.

Ionel Rovent ¸a
email: ionelroventa@yahoo.com

Approximation of the controls for the beam equation with vanishing viscosity

Keywords: beam equation, control approximation, vanishing viscosity, moment problem, biorthogonals. Mathematical subject codes: 93B05, 58J45, 65N06, 30E05

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.

Introduction

The controlled transversal vibrations of a 1-D beam with hinged boundary conditions are modelled by the following equation (1.1)

           u ′′ (t, x) + u xxxx (t, x) = 0
(t, x) ∈ (0, T ) × (0, 1) u(t, 0) = u(t, 1) = u xx (t, 0) = 0 t ∈ (0, T ) u xx (t, 1) = v(t) t ∈ (0, T ) u(0, x) = u 0 (x)

x ∈ (0, 1) u ′ (0, x) = u 1 (x)

x ∈ (0, 1),

where by ′ we denote the derivative in time. In (1.1) the vector u u ′ represents the state and v is the control acting on the extremity x = 1 of the beam. Given T > 0 we say that (1.1) is null-controllable in time T if, for every initial data u 0 u 1 ∈ H := H 1 0 (0, 1) × H -1 (0, 1), there exists a control v ∈ L 2 (0, T ) such that the corresponding solution of (1.1) verifies (1.2) u(T, x) = u ′ (T, x) = 0 (x ∈ (0, 1)).

for any ϕ 0 ϕ 1 ∈ H, ϕ ϕ ′ being the solution of the following homogeneous adjoint equation (1.4)

      
ϕ ′′ (t, x) + ϕ xxxx (t, x) = 0 (t, x) ∈ (0, T ) × (0, 1) ϕ(t, 0) = ϕ(t, 1) = ϕ xx (t, 0) = ϕ xx (t, 1) = 0 t ∈ (0, T ) ϕ(T, x) = ϕ 0 (x)

x ∈ (0, 1) ϕ ′ (T, x) = ϕ 1 (x)

x ∈ (0, 1).

Boundary observability inequality (1.3) can be proved by using Fourier series or multipliers techniques (see, for instance, [START_REF] Komornik | Fourier Series in Control Theory[END_REF][START_REF] Lions | Controlabilité exacte, stabilisation et perturbations des systèmes distribués[END_REF]). Consequently, equation (1.1) is null-controllable in any time T .

The main objective of this work is to study the controllability properties of the semi-discrete space approximation of (1.1). For each N ∈ N * we consider an equidistant partition of the interval (0, 1), x 0 = 0 < x 1 = h < ... < x j = jh < ... < x N +1 = 1, where the mesh-size is h = 1 N +1 . To discretize the boundary conditions of the problem using centered finite differences, we also introduce two external points x -1 = x 0 -h and x N +2 = x N +1 + h. The semi-discretization by finite differences of (1.1) is then given by the following system (1.5)

       u ′′ j (t) + uj+2(t)-4uj+1(t)+6uj (t)-4uj-1(t)+uj-2(t) h 4 = 0 1 ≤ j ≤ N, t ∈ (0, T ) u 0 (t) = 0, u N +1 (t) = 0 t ∈ (0, T ) u -1 (t) = -u 1 (t), u N +2 (t) = h 2 v h (t) -u N (t) t ∈ (0, T ) u j (0) = u 0 j (x), u ′ j (0) = u 1 j (x) 1 ≤ j ≤ N.
System (1.5) consists of N linear equation with N unknowns u 1 , u 2 , . . . , u N . The quantities u j (t) approximate u(t, x j ), the solution of (1.1), if

u 0 j u 1 j 1≤j≤N
is an approximation of the initial data of (1.1). For instance, in our numerical examples in which the initial data are sufficiently regular, we shall choose (1.6) u 0 j = u 0 (jh), u 1 j = u 1 (jh) (1 ≤ j ≤ N ).

We consider the following controllability property for (1.5): given T > 0 and u 0 j u 1 j 1≤j≤N

∈ C 2N , we look for a control v h ∈ L 2 (0, T ) such that the corresponding solution

u j u ′ j 1≤j≤N
of (1.5) verifies

(1.7) u j (T ) = u ′ j (T ) = 0 (1 ≤ j ≤ N ).
If this property is verified for every initial data u 0 j u 1 j 1≤j≤N

∈ C 2N , we say that (1.5) is null-controllable in time T .

It is not difficult to show that the null-controllability property for (1.5) holds in any time T > 0. However, as it has been proved in [START_REF] Leon | Boundary controllability of the finite-difference space semi-discretizations of the beam equation[END_REF], the corresponding observability inequality is not uniform in h. More precisely, if we denote by • 1,-1 the discrete version of the norm • H (see Section 3, formula (3.7)), it can be shown that, for any h > 0, there exists a constant C = C(T, h) such that (1.8) ϕ j ϕ ′ j 1≤j≤N

(0) = 0 1 ≤ j ≤ N, t ∈ (0, T ) ϕ 0 (t) = 0, ϕ N +1 (t) = 0 t ∈ (0, T ) ϕ -1 (t) = -ϕ 1 (t), ϕ N +2 (t) = -ϕ N (t) t ∈ (0, T ) ϕ j (T ) = ϕ 0 j (x), ϕ ′ j (T ) = ϕ 1 j (x) 1 ≤ j ≤ N, but (1.10) lim h→0 sup (ϕ, ϕ ′ ) solution of (1.9)

ϕ j ϕ ′ j (0) 2 1,-1 T 0 ϕ N (t) h 2 dt = ∞.
Note that (1.8) is the discrete version of the observability inequality (1.3) and (1.10) shows that the constant C(T, h) explodes as h tends to zero. In this situation we say that (1.9) is not uniformly (with respect to h) observable or, equivalently, system (1.5) is not uniformly controllable.

As in the continuous case, inequality (1.8) ensures the null-controllability property of (1.5) in time T for any fixed h > 0. On the other hand, (1.10) implies that there exist initial data u 0 u 1 ∈ H such that any sequence of discretes controls (v h ) h>0 of (1.5) diverges in L 2 (0, T ) as h tends to zero, when the initial data are discretized as in (1.6). In order to obtain a uniform observability inequality, two possibilities have been proposed and analyzed in [START_REF] Leon | Boundary controllability of the finite-difference space semi-discretizations of the beam equation[END_REF]:

(a) the class of solutions of (1.9) has been restricted to a space in which the high frequencies have been filtered out. Under this assumption, the corresponding observability inequality becomes uniform and, consequently, the projection of the solution of (1.5) over this filtered space is controlled to zero uniformly;

(b) the observed quantity in the right side of (1.8) has been reinforced by introducing an extra term. This shows that an additional boundary control, which vanishes in limit, makes the system uniformly controllable.

The above analysis shows that the spurious high frequencies introduced by the discretization process are responsible for the nonuniform controllability properties of (1.5) and their elimination or diminution can guarantee the convergence of the scheme. For more general uniform controllability results, by using filtered spaces, the interested reader is referred to [START_REF] Ervedoza | Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes[END_REF][START_REF] Ervedoza | Uniformly exponentially stable approximations for a class of damped systems[END_REF][START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF].

Our paper studies a third possibility to restore the uniform controllability property which is more natural and avoids the necessity of working with projections or with an additional resources consuming control. To this aim, we introduce in the discrete equation (1.5) a numerical viscosity which vanishes in the limit. Since this term damps out the high frequencies which are responsible for (1.10), we can expect that it will also help us to restore the desired uniform observability inequality and to improve the convergence properties of the discrete controls. More precisely, in this paper we study the null-controllability of the following different discretization of (1.1) given by (1.11)

         u ′′ j (t) + uj+2(t)-4uj+1(t)+6uj (t)-4uj-1(t)+uj-2(t) h 4 -ε u ′ j+1 (t)-2u ′ j (t)+u ′ j-1 (t) h 2 = 0 1 ≤ j ≤ N, t ∈ (0, T ) u 0 (t) = 0, u N +1 (t) = 0 t ∈ (0, T ) u -1 (t) = -u 1 (t), u N +2 (t) = h 2 v h (t) -u N (t) t ∈ (0, T ) u j (0) = u 0 j (x), u ′ j (0) = u 1 j (x) 1 ≤ j ≤ N. 3
Note that, if ε = 0, the system (1.11) coincides with the system (1.5). If ε > 0, the ratio ε

u ′ j+1 (t)-2u ′ j (t)+u ′ j-1 (t) h 2
represents a viscous term which damps out the spurious high frequencies. The parameter ε depends on the step size h and verifies (1.12) lim h→0 ε(h) = 0.

Hence, scheme (1.11) represents in fact (1.5) with an added numerical vanishing viscosity. Note that the parameter ε should be chosen small, in order to preserve the convergence and the accuracy of the numerical scheme, but also sufficiently large to improve the observability properties of the system. Our analysis allows to obtain the minimal range of the parameter ε answering to both desiderates. Indeed, we shall prove that

(1.13) ε ≥ h 2 2T ln 1 h
is a necessary condition for the uniform observability inequality. Moreover, our main result ensures, for each T > 0, the existence of two positive constants h 0 and c 0 such that, for any h ∈ (0, h 0 ) and ε ∈ c 0 h 2 ln 1 h , h , system (1.11) is uniformly controllable in time T (see Theorem 6.1 below). This result is obtained by constructing and evaluating some particular controls corresponding to a single eigenmode initial data.

The artificial viscosity is a common tool in many numerical schemes. Let us mention here only the works [START_REF] Ervedoza | Spectral conditions for admissibility and observability of Schrödinger systems: Applications to finite element discretizations[END_REF][START_REF] Münch | Uniform stabilization of a viscous numerical approximation for a locally damped wave equation[END_REF][START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations and application to LQR problems[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semidiscretization of a locally damped wave equation via an artificial numerical viscosity[END_REF][START_REF] Tébou | Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation[END_REF][START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] where the uniform stabilization of discrete hyperbolic equations is achieved and [START_REF] Ignat | Numerical dispersive schemes for the nonlinear Schrödinger equation[END_REF] where uniform Strichartz's estimates for the discrete Schrödinger equation are obtained by using this method.

The vanishing viscosity method has been used in the context of control problems for the wave equation in [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation with vanishing viscosity[END_REF]. As in the case of the semi-discrete beam equation (1.11), uniform observability inequality for the wave equation has been obtained by adding a viscous term of the same type with ε = h. Although the main ideas are similar, many technical details and final results are different. Indeed, in the present article an important aim is to obtain a uniform controllability result in arbitrarily small time. This is not possible in the context of the wave equation but it is perfectly realistic for the beam equation. However, in order to achieve it, additional work (like the construction of two different biorthogonals in Section 5) is required. Also, we manage to prove the uniform controllability result with any viscosity parameter ε greater than c 0 h 2 ln 1 h which, according to (1.13), belongs to the lowest allowed range of values. The corresponding result for the semi-discrete wave equation is still an open problem.

The numerical viscosity method can be easily implemented for more general domains and boundary conditions. However, how to prove a uniform controllability result in these new settings in which the spectrum is not known explicitly and/or the control depends both on time and space variables is a challenging and still open problem.

The rest of the paper is organized as follows. In Section 2 we recall some well-known facts concerning the controllability properties of the continuous beam equation. In Section 3 we introduce the semi-discrete model, we discuss its corresponding controllability properties and we present a spectral analysis. Section 4 is devoted to some remarks concerning the discrete observability inequality and the transformation of our control problem into a moment problem. Section 5 gives the construction of a biorthogonal sequence and its main properties. These results are used in Section 6 to prove the main result of uniform controllability of the scheme (1.11). Section 7 is devoted to some numerical experiments based on this discrete scheme.

The continuous problem

In this section we recall some well-known properties concerning the boundary null-controllability problem for the continuous 1-D linear beam equation and we characterize the control by using the moment theory. More precisely, given any T > 0 and any initial data u 0 u 1 ∈ H := H 1 0 (0, 1) × H -1 (0, 1) we look for a control v ∈ L 2 (0, T ) such that the solution of the beam equation

(2.1)            u ′′ (t, x) + u xxxx (t, x) = 0 (t, x) ∈ (0, T ) × (0, 1) u(t, 0) = u(t, 1) = u xx (t, 0) = 0 t ∈ (0, T ) u xx (t, 1) = v(t) t ∈ (0, T ) u(0, x) = u 0 (x) x ∈ (0, 1) u ′ (0, x) = u 1 (x)
x ∈ (0, 1),

verifies (2.2) u(T, x) = u ′ (T, x) = 0 (x ∈ (0, 1)).
If for any initial data

u 0 u 1 ∈ H there exists a control v ∈ L 2 (0, T ) for (2.1)-(2.
2), we say that problem (2.1) is null-controllable in time T > 0. We mention that, in this context, the null-controllability property is equivalent to the exact controllability property for which the target may be any state from the space H, not only the origin.

As we have mentioned above, we intend to write the control problem of the beam equation in terms of a moment problem. To do that we need the following variational result.

Lemma 2.1. Let T > 0 and the initial data

u 0 u 1 ∈ H. The function v ∈ L 2 (0, T ) is a control which drives
to zero the solution of (2.1) in time T if, and only if, the following relation holds

(2.3) T 0 v(t)ϕ x (t, 1) dt = -u 1 , ϕ(0, • ) D + u 0 , ϕ ′ (0, • ) D ϕ 0 ϕ 1 ∈ H ,
where ϕ ϕ ′ ∈ H is the solution of the following adjoint backward problem

(2.4)        ϕ ′′ (t, x) + ϕ xxxx (t, x) = 0 (t, x) ∈ (0, T ) × (0, 1) ϕ(t, 0) = ϕ(t, 1) = ϕ xx (t, 0) = ϕ xx (t, 1) = 0 t ∈ (0, T ) ϕ(T, x) = ϕ 0 (x) x ∈ (0, 1) ϕ ′ (T, x) = ϕ 1 (x)
x ∈ (0, 1), and , D denotes the duality product between the spaces H 1 0 (0, 1) and H -1 (0, 1).

Proof: If we multiply in (2.1) by ϕ and we integrate by parts over (0, T ) × (0, 1) we obtain that v ∈ L 2 (0, T ) is a null-control for (2.1) if and only if it verifies (2.3).

By denoting W = ϕ ϕ ′ , equation (2.4) is equivalent with (2.5)    W ′ + AW = 0 W (0) = W 0 = ϕ 0 ϕ 1 ,
where A : D(A) ⊂ H → H is the unbounded, maximal and monotone operator in H given by

D(A) = ϕ 0 ϕ 1 ∈ H 3 (0, 1) ∩ H 1 0 (0, 1) × H 1 0 (0, 1) : ϕ 0 xx (t, 0) = ϕ 0 xx (t, 1) = 0 , A = 0 -I ∂ 4 x 0 .
Moreover, the eigenvalues of A are given by (2.6)

η n = i sgn(n) n 2 π 2 (n ∈ Z * ),
and the corresponding eigenvectors are

(2.7) Φ n = 1 sgn(n) nπ 1 -η n sin(nπx) (n ∈ Z * ).
We note that the sequence (Φ n ) 1≤|n|≤N forms an orthonormal basis in H. The following lemma transforms the control problem into a moment problem by using the Fourier expansion of the solution of (2.1). We recall that the moment problems have been, from the very beginning, one of the most successful method used to study controllability properties (see [START_REF] Avdonin | Families of exponentials. The method of moments in controllability problems for distributed parameter systems[END_REF][START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Komornik | Fourier Series in Control Theory[END_REF][START_REF] Tucsnak | Observation and Control for Operator Semigroups, Birkhuser Advanced Texts[END_REF]

u 0 u 1 = n∈Z * a 0 n Φ n ,
there exists a control v ∈ L 2 (0, T ) such that (2.9)

T 0 v (t) e tη n dt = (-1) n i a 0 n (n ∈ Z * ).
Proof: Since (Φ n ) n∈Z * given by (2.7) is a basis in H, from Lemma 2.1 it follows that (2.1) is null-controllable if, and only if, there exists v ∈ L 2 (0, T ) such that (2.3) holds for any initial data which is an eigenfunction of the operator A. If ϕ 0 ϕ 1 = Φ n , the corresponding solution of (2.5) is given by ϕ ϕ ′ (t) = e ηn(t-T ) Φ n . In this case, for any initial data u 0 u 1 of the form (2.8) we obtain that

-u 1 , ϕ(0, • ) D + u 0 , ϕ ′ (0, • ) D = i sgn(n) a 0 n e -T η n ,
and

T 0 v(t)ϕ x (t, 1) dt = (-1) n sgn(n) e -T η n T 0 v (t) e tη n dt.
The last two relations give (2.9) and the proof ends.

The discrete control problem

In this section we study a finite-difference space discretization of equation (2.1). In order to do this let us consider N ∈ N * , a step h = 1 N +1 and a equidistant mesh of the interval (0, 1), 0 = x 0 < x 1 < . . . < x N < x N +1 = 1 with x j = jh for any j ∈ [0, N + 1]. To discretize the boundary conditions of the problem using centered finite differences, we also introduce two external points x -1 = x 0 -h and x N +2 = x N +1 + h. As we said before in this paper we study the null-controllability of the following discretization of (2.1)

(3.1)          u ′′ j (t) + uj+2(t)-4uj+1(t)+6uj (t)-4uj-1(t)+uj-2(t) h 4 -ε u ′ j+1 (t)-2u ′ j (t)+u ′ j-1 (t) h 2 = 0 1 ≤ j ≤ N, t ∈ (0, T ) u 0 (t) = 0, u N +1 (t) = 0 t ∈ (0, T ) u -1 (t) = -u 1 (t), u N +2 (t) = h 2 v h (t) -u N (t) t ∈ (0, T ) u j (0) = u 0 j (x), u ′ j (0) = u 1 j (x) 1 ≤ j ≤ N,
where ε is a nonnegative parameter which depends on the mesh size h and vanishes as h goes to zero. Later on, we shall choose conveniently the parameter ε in order to guarantee the uniform controllability, with respect to h, of scheme (3.1). Now, let us write (3.1) as an abstract Cauchy form by using the matrices A h , B h ∈ M N ×N (R) given by

A h = 1 h 2            2 -1 0 0 . . . 0 0 -1 2 -1 0 . . . 0 0 0 -1 2 -1 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 . . . -1 2 -1 0 0 0 . . . 0 -1 2 -1 0 0 . . . 0 0 -1 2            and B h = A 2 h .
If we denote by

U 0 h =      u 0 1 u 0 2 . . . u 0 N      , U 1 h =      u 1 1 u 1 2 . . . u 1 N      , U h (t) =      u 1 (t) u 2 (t) . . . u N (t)      and F h (t) = 1 h 2        0 0 . . . 0 -v h (t)       
, then system (3.1) may be written vectorially as follows:

(3.2) U ′′ h (t) + B h U h (t) + εA h U ′ h (t) = F h (t) t ∈ (0, T ) U h (0) = U 0 h , U ′ h (0) = U 1 h .
We say that (3.2) is null-controllable in time T > 0 if for any initial data

U 0 h U 1 h ∈ C 2N , we can find a control v h ∈ L 2 (0, T ) such that the corresponding solution U h U ′ h of (3.2) verifies (3.3) U h (T ) = U ′ h (T ) = 0.
Let us consider in C N the canonical inner product

(3.4) f, g = h N k=1 f k g k f = (f k ) 1≤k≤N ∈ C N , g = (g k ) 1≤k≤N ∈ C N .
Moreover, we introduce two additional discrete inner products

(3.5) (f, g) 1 = A h f, g f = (f k ) 1≤k≤N ∈ C N , g = (g k ) 1≤k≤N ∈ C N ,
with the corresponding norm • 1 and

(3.6) (f, g) -1 = A -1 h f, g f = (f k ) 1≤k≤N ∈ C N , g = (g k ) 1≤k≤N ∈ C N ,
with the corresponding norm • -1 .

Finally, we define the discrete inner product in C 2N

(3.7) (f 1 , f 2 ), (g 1 , g 2 ) 1,-1 = (f 1 , g 1 ) 1 + (f 2 , g 2 ) -1 f 1 , f 2 , g 1 , g 2 ∈ C N with the corresponding norm • 1,-1 .
The controllability properties of (3.2) are directly related to the properties of the corresponding homogeneous "adjoint" backward problem:

(3.8) W ′′ h (t) + B h W h (t) -εA h W ′ h (t) = 0 t ∈ (0, T ) W h (T ) = W 0 h , W ′ h (T ) = W 1 h ,
where the initial data

W 0 h W 1 h ∈ C 2N are
U 0 h U 1 h ∈ C 2N there exists v h ∈ L 2 (0, T ) which verifies (3.9) T 0 v h (t) W hN (t) h dt = U 1 h , W h (0) -U 0 h , W ′ h (0) -εA h W h (0) W 0 h W 1 h ∈ C 2N , where W h W ′ h
is the solution of (3.8).

Proof: If we multiply in (3.2) by W h and we integrate in time we obtain immediately that v h ∈ L 2 (0, T ) is a control which drives to zero the solution of (3.2) if, and only if, (3.9) takes place for any

W 0 h W 1 h ∈ C 2N . Now, if we put Z = W h (t) W ′ h (t)
, then system (3.8) has the following equivalent vectorial form

(3.10) -Z ′ + A h Z = 0 Z(T ) = Z T ,
where the operator A h is given by

A h = 0 -I A 2 h -εA h .
We pass to study the eigenvalues and eigenfunctions of the matriceal operator corresponding to (3.10). We begin with the operator A h , for which it is known that (see, for instance, [START_REF] Isaakson | Analysis of Numerical Methods[END_REF]) the eigenvalues are given by (3.11)

µ n = 4 h 2 sin 2 nπh 2 (1 ≤ n ≤ N ),
with the corresponding eigenvectors (3.12)

φ n h = (sin(knhπ)) 1≤k≤N ∈ R N (1 ≤ n ≤ N ).
We have the following result.

Lemma 3.2. The eigenvalues of the operator A h are given by

(3.13) λ n = µ |n| ε + i sgn(n) √ 4 -ε 2 2 (1 ≤ |n| ≤ N ),
and the corresponding eigenvectors are

(3.14) Φ n h = 1 √ µ |n| 1 -λ n φ |n| h (1 ≤ |n| ≤ N ),
where φ |n| h are given by (3.12). Moreover, the vectors (Φ n h ) 1≤|n|≤N form an orthonormal basis in C 2N with respect to the discrete inner product (3.7).

Proof:

The eigenvalues and the eigenvectors can be compute immediately. As for the last part, which is also an explicit computation, the reader is referred to [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-D wave equation[END_REF]Proposition 3.1].

Remark 3.1. From (3.13) it follows that

(3.15) |λ n | = µ |n| (1 ≤ |n| ≤ N ).
Hence, the eigenvalues λ n are obtained from the purely imaginary values i µ |n| by a simple rotation of angle sgn(n) arctan

√ 4-ε ε in the complex plane.
In what follows, we analyze more closely the eigenvalues of the operator A h and, more precisely, the distance between them. Firstly, we introduce the sets:

F = {n ∈ Z * : 1 ≤ |n| ≤ M and N -M + 1 ≤ |n| ≤ N } , G = {n ∈ Z * : M + 1 ≤ |n| ≤ N -M } .
Let γ > 0 be fixed and define the following quantity

(3.16) M = γ 4 + 1.
In the sequel we shall suppose that N is sufficiently large such that M < N 2 or, equivalently,

(3.17) h < h 1 0 = 2 6 + γ .
Since γ is a positive number independent of N and h, this can be done without any loos of generality. The following result shows that we have a uniform gap γ, at least for a part of the spectrum of A h .

Lemma 3.3. For M given by (3.16) we have that

(3.18) |λ n+1 | -|λ n | > γ (M ≤ |n| ≤ N -M + 1).
Proof: We remark that

(3.19) |λ n+1 | -|λ n | = 4 h 2 sin πh 2 sin (2n + 1)πh 2 .
Taking into account that sin (2n+1)πh 2 is an increasing function on [1, N/2] and a decreasing one on (N/2, 1/h), it is sufficient to verify that we have a gap γ only in the extremities. Indeed, for n = M we have that

|λ M +1 | -|λ M | = 4 h 2 sin πh 2 sin (2M + 1)πh 2 ≥ 4(2M + 1) ≥ γ,
and for n = N -M + 1 we deduce that

|λ N -M +2 | -|λ N -M +1 | = 4 h 2 sin πh 2 sin π - (2(N -M + 1) + 1)πh 2 ≥ 4(2M -1) ≥ γ,
which ends the proof of our lemma.

Remark 3.2. Relation (3.18) shows that the distance (the "gap") between two consecutive eigenvalues of the family (λ n ) n∈G is greater than γ. Since γ can be taken arbitrarily large, we obtain that this distance may be as large as we want. Also, note that the eigenvalues from the complementary set, (λ n ) n∈F , form a finite family with a number of elements depending only of γ.

The moment problem and the observability inequality

As mentioned at the beginning, our aim is to prove the uniform controllability property of (3.2) or, equivalently, to show that a uniform observability inequality for the corresponding adjoint system (3.8) holds. In order to fulfil our objectives, we transform first the control problem in an equivalent moment problem (see Theorem 4.1 below) and we provide a solution for it by using a biorthogonal sequence to the family of complex exponential functions e λnt 1≤|n|≤N . In the second part of this section we show how can we estimate the norm of this solution by using the observability inequality for (3.8) (see Proposition 4.1 below). This analysis allows us to provide a lower bound for the parameter ε as a function of the step size h, which represents a necessary condition for the uniform controllability property of (3.2).

Let us begin with the following theorem which transforms the null-controllability problem into a moment problem.

Theorem 4.1. Let T > 0. System (3.2) is null-controllable in time T if,

and only if, for any initial data

U 0 h U 1 h ∈ C 2N of the form (4.1) U 0 h U 1 h = 1≤|n|≤N a 0 nh Φ n h , there exists a control v h ∈ L 2 (0, T ) such that (4.2) T 0 v h (t)e λnt dt = (-1) n 1 2 cos nπh 2 i 4 -ε 2 a 0 nh -ε sgn(n)a 0 nh + ε sgn(n)a 0 -nh (1 ≤ |n| ≤ N ).
Proof: The proof of this theorem follows immediately from Lemma 3.1. Indeed, since according to Lemma 3.2

(Φ n h ) n∈Z * is a basis in C 2N
, it is sufficient to verify (3.9) for

W 0 h W 1 h = Φ n h , 1 ≤ |n| ≤ N .
In this case, the corresponding solution of (3.8) will be

W h W ′ h (t) = e (t-T )λn Φ n h , which gives us that (4.3) T 0 v h (t) h W hN (t) dt = (-1) n+1 sin(nπh) h √ µ |n| e -T λn T 0 v h (t) e tλn dt, (4.4
) U 1 h , W h (0) -U 0 h , W ′ h (0) -εA h W h (0) = - e -T λn 2 i sgn(n) 4 -ε 2 a 0 nh -εa 0 nh + εa 0 -nh .
From (4.3) and (4.4) it follows that (4.2) takes place and the proof finishes.

The notion of biorthogonality is very useful in the study of moment problems like (4.2). We recall that a sequence

(θ m ) 1≤|m|≤N ⊂ L 2 -T 2 , T 2 is biorthogonal to the family of exponential functions e λnt 1≤|n|≤N in L 2 -T 2 , T 2 if (4.5) T 2 -T 2 θ m (t)e λnt dt = δ mn (1 ≤ |m|, |n| ≤ N ) .
It is easy to see from (4.2) that, if (θ m ) 1≤|m|≤N is a biorthogonal sequence to the family of exponential functions e λnt 1≤|n|≤N in L 2 -T 2 , T 2 , then a solution v h of (4.2) will be given by

(4.6) v h (t) = N |n|=1 (-1) n e -λn T 2 2 cos nπh 2 i 4 -ε 2 a 0 nh -ε sgn(n)a 0 nh + ε sgn(n)a 0 -nh θ n t - T 2 (t ∈ (0, T )).
Now, the main task is to show that there exists a biorthogonal sequence (θ m ) 1≤|m|≤N to the family e λnt 1≤|n|≤N in L 2 -T 2 , T 2 and to evaluate its L 2 -norm. This allows us to estimate the norm of v h from (4.6) and to study the convergence properties of the family (v h ) h>0 .

Let us define

(4.7) C(T, h) = sup   W 0 h W 1 h   ∈C 2N W h W ′ h (0) 2 1,-1 T 0 W hN (t) h 2 dt , where W h W ′ h ∈ C 2N denotes the solution of (3.8) with initial data W 0 h W 1 h
. We have the following result.

Proposition 4.1. The constant C(T, h) defined by (4.7) is finite and, for each

U 0 h U 1 h ∈ C 2N , there exists a solution v h ∈ L 2 -T 2 , T 2 of (4.
2) with the property that

(4.8) v h L 2 ≤ (1 + ε) C(T, h) U 0 h U 1 h 1,-1
.

Proof: Let us first show that C(T, h) is finite. By taking the initial data of the form

W 0 h W 1 h = N |n|=1 b 0 nh Φ n h ,
we obtain that (4.9) Both quantities from the right hand sides in (4.9) and (4.10) are norms in C 2N . Hence, they are equivalent and the finiteness of the constant C(T, h) is proved. Now, since C(T, h) < ∞, we deduce from (4.9) and (4.10) that the following weighted Ingham type inequality takes place

W h W ′ h (0) 2 1,-1 = N |n|=1 b 0 nh 2 e -2T ℜ(λn) , (4.10)
(4.11) N |n|=1 b 0 nh 2 e -T ℜ(λn) cos 2 nπh 2 ≤ C(T, h) T 2 -T 2 N |n|=1 b 0 nh e tλn 2 dt,
for every finite sequence ( b0 nh ) 1≤|n|≤N .

On the other hand, by taking into account that we are dealing with a finite family of distinct exponential functions, it follows that there exists a unique biorthogonal sequence (θ m ) 1≤|m|≤N to the family e λnt 1≤|n|≤N which belongs to Span e λnt 1≤|n|≤N . This biorthogonal sequence is constructed by using the Projection Theorem (see, for instance, [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF][START_REF] Hansen | Bounds on Functions Biorthogonal to Sets of Complex Exponentials; Control of Dumped Elastic Systems[END_REF]). Moreover, from (4.11), we obtain that (4.12)

N |m|=1 α m θ m 2 L 2 (-T 2 , T 2 ) ≤ C(T, h) N |m|=1 |α m | 2 e T |ℜ(λm)| cos 2 mπh 2 ,
for any finite sequence (α m ) 1≤|m|≤N . It follows that v h defined by (4.6) is a solution of (4.2) which verifies (4.8).

Now, we pass to analyze the behavior of the observability constant C(T, h) from (4.7) with respect to h. According to (4.8), if the constant C(T, h) is bounded in h (which is equivalent to say that (3.8) is uniformly observable), then the family (v h ) h>0 is bounded in L 2 (0, T ). This represents a key condition for the convergence of our numerical scheme. In [START_REF] Leon | Boundary controllability of the finite-difference space semi-discretizations of the beam equation[END_REF] it was proved that, for ε = 0, the observability constant C(T, h) is not uniform in h. More precisely, in this particular case

(4.13) C(T, h) ≥ 1 T cos 2 N πh 2 ≥ 4 π 2 T h 2 .
When ε > 0 is sufficiently large, due to the viscous term, we can obtain a uniform estimate of the observability constant. This is a difficult problem which will be proved at the end of this paper (see Theorem 5.4 below). However, it is easy to show that at least some particular initial data can be observed uniformly, if ε is large enough.

Proposition 4.2. Let T > 0. We have that

(4.14) C(T, h) = sup   W 0 h W 1 h   ∈C 2N W h W ′ h (0) 2 1,-1 T 0 W hN (t) h 2 dt ≥ 16ε π 2 h 4 e 4T ε h 2 -1 . Proof: Let us consider the initial data W 0 h W 1 h = Φ n h .
Since the corresponding solution of (3.8) is given by W h W ′ h (t) = e (t-T )λn Φ n h , we deduce from (4.9) and (4.10) that (4.15)

W h W ′ h (0) 2 1,-1 T 0 W hN (t) h 2 dt = 1 cos 2 nπh 2 εµ |n| e T εµ |n| -1 .
The proof is completed by taking into account that the function x e T x -1 is decreasing in (0, ∞) and µ n < 4 h 2 . Remark 4.1. From Proposition 4.2 it follows that a necessary condition for the uniformity of the observability

constant C(T, h) is that the constant εh -4 e 4T ε h 2 -1 -1
is bounded as h tends to zero. From this we deduce that a necessary condition for the uniformity of the observability constant C(T, h) is

(4.16) ε > 1 2T h 2 ln 1 h .
On the other hand the proof of Proposition 4.2 shows that the solutions of (3.8) with initial data

W 0 h W 1 h = Φ n h , 1 ≤ |n| ≤ N , are uniformly observable for any (4.17) ε > 1 T h 2 ln 1 h .
Finally, note that (4.16) implies that the real parts of the large eigenvalues λ n are not uniformly bounded in h. Consequently, in order to obtain a uniform controllability result for (3.2), the perturbation introduced by the viscosity term should be unbounded.

Construction of the biorthogonal sequence

This section is devoted to construct and evaluate a biorthogonal sequence to the family Λ = e λnt 1≤|n|≤N in L 2 -T 2 , T 2 , where (λ n ) 1≤|n|≤N are given by (3.13). Since e λnt 1≤|n|≤N is a finite family of exponential functions, it follows immediately that it has infinitely many biorthogonal sequences in L 2 -T 2 , T 2 . However we are interested not only on their existence, but also on their dependence of the parameters h and ε. Therefore, we shall construct an explicit biorthogonal and we shall evaluate its L 2 -norm. In order to succeed in our purpose we need to treat separately the biorthogonal sequences for the families Λ 1 = e λnt n∈F and Λ 2 = e λnt n∈G . This fact is motivated by the different behavior of the eigenvalues λ n from these two families. Indeed, the exponents of Λ 2 have a gap at least γ (see Lemma 3.3 and Remark 3.2) whereas those of Λ 1 are not so well separated. On the other hand, the cardinality of these two families is different: Λ 2 has a number of elements which goes to infinity when h tends to zero whereas Λ 1 has a number of elements depending only on γ.

Firstly, we present some notations and results that will be used along this section. For any n ∈ Z * such that |n| ∈ (M, N -M ] we introduce the following quantities (5.1)

k n = |λ n | γ , (5.2) k n = k n if |λ n | -γk n ≤ γ 2 k n + 1 if γ(k n + 1) -|λ n | ≤ γ 2 . Lemma 5.1.
We have the following properties:

(5.3) kn γ -µ |n| ≤ γ 2 (|n| ∈ (M, N -M ]) , (5.4) kM+1 γ -µ M ≥ γ 2 , (5.5) µ N -M +1 -kN-M γ ≥ γ 2 .
Proof: From (3.15), by taking into account (5.1) and (5.2) we obtain immediately (5.3). For (5.4) and (5.5) we obtain, by using (3.18), that

kM+1 γ -µ M = µ M +1 -µ M + kM+1 γ -µ M +1 ≥ µ M +1 -µ M - γ 2 ≥ γ 2 ,
and, respectively,

µ N -M +1 -kN-M γ = µ N -M +1 -µ N -M -kN-M γ + µ N -M ≥ µ N -M +1 -µ N -M - γ 2 ≥ γ 2 ,
which ends the proof of this lemma.

In the sequel, C > 0 denotes a generic constant which may changes from one row to another but it is always independent of h, ε and m.

A biorthogonal sequence to the family Λ 1

In this subsection we construct a biorthogonal sequence to the family Λ 1 = e λnt n∈F . Let us consider the following product (5.6)

P 1 m (z) = M |j|=1 j =m z -iλ j iλ m -iλ j N |j|=N -M +1 j =m z -iλ j iλ m -iλ j (z ∈ C, m ∈ F )
Firstly, we remark that (5.7)

P 1 m (iλ n ) = δ mn (m, n ∈ F ).
In the following proposition we obtain some estimates for the product P 1 m . Proposition 5.1. Let γ > 0. There exists h 1 0 > 0 such that for any h ∈ 0, h 1 0 and ε ∈ (0, h) the function P 1 m defined by (5.6) has the following property

(5.8) |P 1 m (z)| ≤          C(1 + |z -iλ m |) 2M -1 1 + |z-iλm| N 2 2M (1 ≤ |m| ≤ M, z ∈ C) C(1 + |z -iλ m |) 2M 1 + |z-iλm| N 2
2M -1

(N -M + 1 ≤ |m| ≤ N, z ∈ C) ,
where C and ω are two positive constants independent of h, ε and m.

Proof: Let h 1 0 be the constant given by (3.17). In order to evaluate P 1 m we remark that (5.9)

|P 1 m (z)| = M |n|=1 n =m z -iλ n i(λ m -λ n ) N |n|=N -M +1 n =m z -iλ n i(λ m -λ n ) ≤ ≤ M |n|=1 n =m 1 + |z -iλ m | |λ m -λ n | N |n|=N -M +1 n =m 1 + |z -iλ m | |λ m -λ n | .
By taking into account that, for 1 ≤ |m| ≤ M we have

λ m -λ n >    C 1 ≤ |n| ≤ M, n = m CN 2 N -M + 1 ≤ |n| ≤ N,
and, for N -M + 1 ≤ |m| ≤ N we have

λ m -λ n >    CN 2 1 ≤ |n| ≤ M C N -M + 1 ≤ |n| ≤ N, n = m,
then from (5.9) we obtain immediately (5.8).

We are now in position to construct a biorthogonal sequence to the family of exponential functions Λ 1 . This is accomplished in the following theorem.

Theorem 5.1. Let T, γ > 0. There exists h 1 0 , c0 > 0 such that for any h ∈ (0, h 1 0 ) and ε ∈ (c 0 h 2 ln 1 h , h) there exists a biorthogonal sequence (ψ m ) m∈F to the family of exponential functions Λ 1 in L 2 -T 4 , T 4 with the following properties

(5.10) ψ m L 2 (-T 4 , T 4 ) ≤ C 1 e T 8 |ℜ(λm)| (m ∈ F ), (5.11) | ψ 1 (x)(x -iλ 1 )| ≤ C 1 (x ∈ R),
(5.12)

| ψ 1 (iλ k )(iλ k -iλ 1 )| ≥ C 2 (k ∈ G) ,
where C 1 and C 2 are two positive constants independent of h, ε and m.

Proof: Let h 1 0 be the constant given by (3.17) and let P 1 m (z) be the product from (5.6). For each m ∈ F we define the entire function

(5.13) E m (z) = P 1 m (z) sin (z -iλ m + iσ)δ (z -iλ m + iσ)δ 2M   sin z-iλm N 2 + iσ δ z-iλm N 2 + iσ δ   2M M 1 m (z) iσδ sin(iσδ) 4M , where δ = T 16M and σ > 2π 2 . Let (5.14) ψ m (t) = 1 2π R E m (x)e ixt dx.
From (5.8) we deduce that E m is an entire function of exponential type T 4 . Nextly, we evaluate E m on the real axis, by considering the following two cases.

Case 1. x -iλ m ≤ 2σ + 1. In this case, from the continuity of M m , we have that

|M 1 m (x)| ≤ C.
Moreover, from (5.8) we obtain that |P 1 m (x)| ≤ C, which finally gives us the following estimate

(5.15) |E m (x)| ≤ C x -iλ m ≤ 2σ + 1 . Case 2.
x -iλ m > 2σ + 1. In this case, for ε < h we obtain that

|M 1 m (x)| ≤ C e 2M δ|ℜ(λm)| e 2M δ N 2 |ℜ(λm)| |(x -iλ m + iσ)δ| 2M x-iλm N 2 + iσ δ 2M ≤ C e 2M δ|ℜ(λm)| (|x -iλ m | + 1) 2M |x-iλm| N 2 + 1 2M .
From estimates (5.8) we deduce that

(5.16) |E m (x)| ≤        Ce 2M δ|ℜ(λm)| |x-iλm|+1 , 1 ≤ |m| ≤ M Ce 2M δ|ℜ(λm)| |x-iλm| N 2 +1 , N -M + 1 ≤ |m| ≤ N x -iλ m > 2σ + 1 .
From (5.15), (5.16) and the fact that δ = T 16M it follows that (5.17)

R |E m (x)| 2 dx ≤ Ce T 4 |ℜ(λm)| .
Now, by taking into account the properties of E m and by applying Paley-Wiener Theorem [32, Theorem 18, Section 2.4], we deduce that (ψ m ) m∈F ⊂ L 2 -T 4 , T 4 . Moreover, from the inverse Fourier transform property and the fact that E m (iλ n ) = δ mn , for any m, n ∈ F , we obtain that (ψ m ) m∈F is a biorthogonal sequence to (e λnt ) n∈F in L 2 -T 4 , T 4 . Finally, from Plancherel's Theorem we deduce that (5.10) holds.

Relation (5.11) follows from the estimates we have obtained for E 1 on the real axis. Indeed, from (5.16) we have that

| ψ 1 (x)(x -iλ 1 )| ≤ C(1 + |x -iλ 1 |) 2M -1 1 + |x -iλ 1 | N 2 2M e 2M δ|ℜ(λ1)| |x -iλ 1 | (|x -iλ 1 | + 1) 2M |x-iλ1| N 2 + 1 2M ≤ C.
It remains to show that (5.12) takes place. For any k ∈ G, we have that

|P 1 1 (iλ k )| 2 = M |n|=1 n =1 λ k -λ n λ 1 -λ n 2 N |n|=N -M +1 λ k -λ n λ 1 -λ n 2 = = λ k -λ -1 λ 1 -λ -1 2 M n=2 (µ 2 k -µ 2 n ) 2 -ε 2 µ k µ n (µ k -µ n ) (µ 2 n -µ 2 1 ) 2 -ε 2 µ n µ 1 (µ n -µ 1 ) N n=N -M +1 (µ 2 n -µ 2 k ) 2 -ε 2 µ n µ k (µ n -µ k ) (µ 2 n -µ 2 1 ) 2 -ε 2 µ n µ 1 (µ n -µ 1 ) ≥ ≥ Ck 4 1 - ε 2 2 2M M n=2 (µ 2 k -µ 2 n ) 2 (µ 2 n -µ 2 1 ) 2 N n=N -M +1 (µ 2 n -µ 2 k ) 2 (µ 2 n -µ 2 1 ) 2 ≥ ≥ C k 4 N 8M M n=2 (µ 2 k -µ 2 n ) 2 N n=N -M +1 (µ 2 n -µ 2 k ) 2 .
Thus, we obtain that (5.18)

|P 1 1 (iλ k )| 2 ≥    Ck 4 µ 4M -4 k |k| ≤ N 2 CN -4 |k| > N 2 .
We pass now to evaluate M 1 (iλ k ). Firstly, we remark that iλ k -iλ1

N 2 ≤ µ k +µ1 N 2 ≤ 2π 2 , which, by choosing σ > 2π 2 , leads to (5.19) sin iλ k -iλ1 N 2 + iσ δ iλ k -iλ1 N 2 + iσ δ 2M ≥ C.
Secondly, since sin (iλ k -iλ 1 + iσ)δ

(iλ k -iλ 1 + iσ)δ 4M ≥   e -δ(ℜ(λ k -λ1)+σ) + e δ(ℜ(λ k -λ1)+σ) -2 4δ 2 (ℜ(λ k -λ 1 ) + σ) 2 + (ℑ(λ k -λ 1 )) 2   2M ,
we obtain the following estimates.

If |k| ≤ N 2 then we have

(5.20) sin (iλ k -iλ 1 + iσ)δ (iλ k -iλ 1 + iσ)δ 4M ≥ (δσ) 4M k 8M . If |k| > N 2 then we have (5.21) sin (iλ k -iλ 1 + iσ)δ (iλ k -iλ 1 + iσ)δ 4M ≥ C e 2M δℜ(λ k ) (2δℑ(λ k )) 4M ≥ C e 2M δεN 2 N 8M ≥ C,
where the last inequality takes place if we choose the positive constant c0 ≥ 64M T and ε > c0 h 2 ln 1 h .

In conclusion, from (5.18), (5. [START_REF] Komornik | Fourier Series in Control Theory[END_REF]), (5.20) and (5.21) we have that

(5.22) | ψ 1 (iλ k )(iλ k -iλ 1 )| 2 ≥      Ck 4 µ 4M -4 k 1 k 8M |λ k -λ 1 | 2 ≥ C |k| ≤ N 2 CN -4 C|λ k -λ 1 | 2 ≥ C |k| > N 2 .
This last result completes the proof of the theorem.

Remark 5.1. For any finite sequence (a n ) n∈N * we deduce from (5.10) that the following inequalities hold

T 4 -T 4   M |m|=1 + N |m|=N -M +1   a m ψ m (t) 2 dt ≤ 2 T 4 -T 4 M |m|=1 a m ψ m (t)dt 2 + 2 T 4 -T 4 N |m|=N -M +1 a m ψ m (t)dt 2 ≤ ≤ 4(M + 1) T 4 -T 4   M |m|=1 + N |m|=N -M +1   |a m | 2 |ψ m (t)| 2 dt ≤ 4(M + 1)C   M |m|=1 + N |m|=N -M +1   |a m | 2 e T 4 |ℜ(λm)| .
Therefore, in the hypothesis of Theorem 5.1, the following inequality takes place for any sequence (a n ) n∈F

(5.23)

T 4 -T 4 m∈F a m ψ m (t) 2 dt ≤ C(T ) m∈F |a m | 2 e T 4 |ℜ(λm)| ,
where C(T ) is a positive constant depending only of T .

A biorthogonal sequence to the family Λ 2

In this subsection we construct a biorthogonal sequences for the family Λ 2 = e λnt n∈G . Firstly, we construct an entire function of exponential type, as a Weierstrass product, and we evaluate it on the real axis. Secondly, by taking into account the estimates for the Weierstrass product on the real axis, we construct another entire function, called multiplier, such that the product between this two functions is an entire function of arbitrarily small exponential type bounded on the real axis. This construction allows us to obtain a biorthogonal element ζ m , through the inverse Fourier transform. Such a method was used for the first time by Paley and Wiener [START_REF] Paley | Fourier Transforms in Complex Domains[END_REF] and, in the context of controllability problems, by Fattorini and Russell [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]. The main difficulties in our analysis is to obtain good estimates for the behavior of the Weierstrass product on the real axis and to construct an appropriate multiplier.

A Weierstrass product

The aim of this section is to construct an entire function as a Weierstrass product and to study its properties.

Let γ > 0. For z ∈ C, we define the following functions:

(5.24) R m (z) = ∞ |k|=1 k = km 1 - z iν k (m ∈ G) , (5.25) P 2 m (z) = R m (z) R m (iλ m ) (m ∈ G) ,
where (5.26)

ν k =    ikγ k ∈ N * and k = k n for M < n ≤ N -M λ n M < n ≤ N -M and k = k n ν -k -k ∈ N * .
Note that in the product R m are included not only the eigenvalues (λ n ) n∈G which, since we want to construct a biorthogonal to the family Λ 1 , are the natural candidates. The presence of the extended family (ν k ) k∈Z * is justified by our desire to have a product R m as close as possible to the function sin(zπ i/γ) zπ i/γ which is a function of small exponential type π γ and an appropriate behavior on the real axis.

Lemma 5.2. Let γ > 0. There exists h 1 0 > 0 such that for any h ∈ 0, h 1 0 and ε ∈ (0, h) the following relation takes place

(5.27) R m (iλ m ) ≥ C exp - h √ ε |ℜ(λ m )| (1 ≤ |m| ≤ N ),
where C is a positive constant independent of h, ε and m.

Proof: Let h 1 0 be the constant given by (3.17). For each m ∈ F ∪ G we have the following estimates

R m (iλ m ) 2 = λ -m -λ m λ -m ∞ k=1 k = k |m| ν k -λ m ν k ν k -λ m ν k 2 ≥ ∞ k=1 k = k |m| |ν k | 2 -2ℜ(ν k )λ m + λ 2 m |ν k | 2 2 = = sin iπ|λm| γ iπ|λm| γ 2 N -M n=M +1 k2 n γ 2 k2 n γ 2 + λ 2 m 2 N -M n=M +1 n =|m| µ 2 n + λ 2 m µ 2 n 2 N -M n=M +1 n =|m| |λ n | 2 -εµ n λ m + λ 2 m |λ n | 2 + λ 2 m 2 = = sin iπ|λm| γ iπ|λm| γ 2 N -M n=M +1 k2 n γ 2 µ 2 n 2 Rm,1 N -M n=M +1 n =|m| µ 2 n + λ 2 m k2 n γ 2 + λ 2 m 2 Rm,2 N -M n=M +1 n =|m| |λ n | 2 -εµ n λ m + λ 2 m |λ n | 2 + λ 2 m 2 Rm,3
.

In order to evaluate R m,1 we use (5.3) and we deduce that

(5.28) R m,1 = N -M n=M +1 k2 n γ 2 µ 2 n 2 ≥ N -M n=M +1 1 - γ µ n 4 ≥ C.
For R m,2 we use again (5.3) and the fact that kn γ -µ |m| > 1 2 |µ n -µ |m| |, so we deduce that

(5.29) R m,2 ≥ N -M n=M +1 n =|m| 1 - |µ 2 n -k2 n γ 2 | | k2 n γ 2 + λ 2 m | 2 ≥ C N -M n=M +1 n =|m| 1 - 2γ |µ n -µ |m| | 2 ≥ C.
For the product R m,3 , by using that (

µ n + µ |m| ) 2 -ε 2 µ n µ |m| > 1 -ε 2 (µ n + µ |m| ) 2 we obtain the following estimates R m,3 = N -M n=M +1 n =|m| |λ n | 2 -εµ n λ m + λ 2 m |λ n | 2 + λ 2 m 2 ≥ N -M n=M +1 n =|m| (µ n -µ |m| ) 2 (µ n + µ |m| ) 2 -ε 2 µ n µ |m| (µ 2 n -µ 2 |m| ) 2 + ε 2 µ 2 n µ 2 |m| ≥ ≥ 1 - ε 2 2 1 h N -M n=M +1 n =|m| 1 - ε 2 µ 2 n µ 2 |m| (µ 2 n -µ 2 |m| ) 2 + ε 2 µ 2 n µ 2 |m| .
Taking into account the following inequalities

(5.30) |µ m -µ n | ≥ |m 2 -n 2 | 1 ≤ m ≤ N 2 , 1 ≤ n ≤ N , (5.31) |µ m -µ n | ≥ 4|m -n| 2 N ≥ m > N 2 , 1 ≤ n ≤ N ,
we treat separately the following two cases:

Case 1. If |m| ∈ 1, N 2 we obtain that R m,3 ≥ 1 2 m-1 n=M +1 1 - Cε 2 m 4 (m 2 -n 2 ) 2 N -M n=m+1 1 - Cε 2 m 4 (n 2 -m 2 ) 2 ≥ ≥ C exp m-1 2 ln 1 - Cε 2 m 2 (m -t) 2 dt + N -1 m+1 ln 1 - Cε 2 m 2 (t -m) 2 dt ≥ C. Case 2. If |m| ∈ N 2 , N we obtain that R m,3 ≥ 1 2 N -M n=M +1 n =|m| 1 - ε 2 µ 2 n µ 2 |m| 16(m -n) 4 (µ n + µ |m| ) 2 + ε 2 µ 2 n µ 2 |m| ≥ ≥ 1 2 |m| 1+C √ ε n=1 1 - Cε 2 n 4 (|m| -n) 4 N n= |m| 1-C √ ε 1 - Cε 2 m 4 (n -|m|) 4 |m| 1-C √ ε n= |m| 1+C √ ε n =|m| C(|m| -n) 4 ε 2 n 4 ≥ ≥ 1 2 exp |m| 1+C √ ε +1 2 ln 1 - ε 2 t 4 C(|m| -t) 4 dt + N -1 |m| 1-C √ ε -1 ln 1 - ε 2 m 4 C(t -|m|) 4 dt -C √ ε|m| ≥ ≥ 1 2 exp(-C √ ε|m|).
The previous analysis allows us to deduce that

(5.32) R m,3 ≥ C exp -C √ ε|m| (1 ≤ |m| ≤ N ).
Hence, by using (5.28), (5.29) and (5.32) we obtain that

(5.33) |R m (iλ m )| 2 ≥ exp(-C √ εm) ≥ C exp - h √ ε |ℜ(λ m )| (1 ≤ |m| ≤ N ) ,
and the proof of the lemma finishes. Now, we prove some elementary properties of the product P 2 m .

Proposition 5.2. Let γ > 0. There exists h 1 0 > 0 such that for any h ∈ 0, h 1 0 and ε ∈ (0, h), if (ν k ) k∈Z * is defined by (5.26) then the product P 2 m , which is defined by (5.25), is an entire function with the following properties

(5.34) P 2 m (z) ≤ C exp √ 2π γ |z| + ε|z| + h √ ε |ℜ(λ m )| (m ∈ G, z ∈ C), (5.35) P 2 m (iλ n ) = δ mn (m, n ∈ G),
where C is a positive constant independent of h, ε and m.

Proof: Let h 1 0 be the constant given by (3.17). For any z ∈ C, we have that

R m (z) ≤ exp Nz k=1 ln 1 - z 2 |ν k | 2 + 2izℜ 1 ν k + ∞ k=Nz+1 ln 1 - z 2 |ν k | 2 + 2izℜ 1 ν k = = exp(A(z) + B(z)),
where N z is defined by

N z = max k ≥ 1 : 2zℜ 1 ν k ≤ |z| 2 |ν k | 2 .
From (5.1) and (5.26) we deduce that |ν k | ≥ γk and consequently

A(z) ≤ Nz k=1 ln 1 + 2|z| 2 |ν k | 2 ≤ Nz k=1 ln 1 + 2|z| 2 γ 2 k 2 ≤ ∞ 0 ln 1 + 2|z| 2 γ 2 s 2 ds = √ 2π γ |z|.
Moreover, since

k≥1 ℜ 1 ν k ≤ n≥1 ℜ 1 λ n = n≥1 ε 2µ n ≤ ε 4 ,
we have that

B(z) ≤ ∞ k=Nz+1 ln 1 + 4|z| ℜ 1 ν k ≤ ∞ k=Nz+1 4|z| ℜ 1 ν k ≤ ε|z|.
Thus, we obtain that

(5.36) R m (z) ≤ exp √ 2π γ |z| + ε|z| (z ∈ C).
Therefore, by using (5.27), (5.36) and Lemma 5.2 we obtain (5.34). Property (5.35) is obvious.

The main result of this section is given in the following proposition which gives the estimate for P 2 m on the real axis.

Proposition 5.3. Let γ > 0. There exists h 1 0 > 0 such that for any h ∈ 0, h 1 0 and ε ∈ (0, h) the function P 2 m , defined by (5.25), has the following property on the real axis

(5.37) |P 2 m (x)| ≤ C exp ωϕ(x) + h √ ε |ℜ(λ m )| (m ∈ G, x ∈ R),
where ϕ is given by

(5.38) ϕ(x) = ε|x|,
and C and ω are two positive constants independent of h, ε and m.

Proof: Let h 1 0 be the constant given by (3.17). In order to prove (5.37), we observe that

|R m (x)| 2 ≤ |λ m | 2 |x -λ m | 2 sin πx γ πx γ 2 R m,11 (x)R m,12 (x)R m,13 (x), where R m,11 (x) = N -M n=M +1 kn γ µ n 4 , R m,12 (x) = N -M n=M +1 x 2 -µ 2 n x 2 -k2 n γ 2 2 , R m,13 (x) = N -M n=M +1 1 + ε 2 x 2 µ 2 n |x 2 -µ 2 n | 2 .
By using (5.3) we obtain that

(5.39) R m,11 (x) ≤ N -M n=M +1 µ n + γ µ n 4 ≤ C(γ).
In order to estimate R m,12 (x) we analyze the following three cases.

1. If 4 h 2 sin 2 M πh 2 ≤ |x| ≤ 4 h 2 sin 2 (N -M +1)πh 2 let v ∈ [M, N -M + 1] such that x = 4 h 2 sin 2 vπh 2 . We have that R m,12 (x) = [v] n=M +1 x 2 -µ 2 n x 2 -k2 n γ 2 2 N -M n=[v]+1 µ 2 n -x 2 k2 n γ 2 -x 2 2 ≤ ≤ x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 [v]-1 n=M +1 x 2 -µ 2 n x 2 -(µ n + γ 2 ) 2 2 N -M n=[v]+2 µ 2 n -x 2 (µ n -γ 2 ) 2 -x 2 2 ≤ ≤ x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 [v]-1 n=M +1 x -µ n x -(µ n + γ 2 ) 2 N -M n=[v]+2 µ n + x (µ n -γ 2 ) + x 2 ≤ ≤ x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 exp   [v]-1 n=M +1 2γ x -(µ n + γ 2 ) + N -M n=[v]+2 2γ (µ n -γ 2 ) + x   ≤ CC v ,
where

C v = x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 . 2. If |x| > 4 h 2 sin 2 (N -M +1)πh 2
, from the monotony of the function R m,12 we obtain that

R m,12 (x) ≤ N -M n=M +1   16 h 4 sin 4 (N -M +1)πh 2 -µ 2 n 16 h 4 sin 4 (N -M +1)πh 2 -k2 n γ 2   2 ≤ N -M n=M +1   16 h 4 sin 4 (N -M +1)πh 2 -µ 2 n 16 h 4 sin 4 (N -M +1)πh 2 -(µ n + γ 2 ) 2   2
and the problem is reduced to case 1.

If |x|

< 4 h 2 sin 2 M πh 2
, by taking again into consideration the monotony of R m,12 we deduce that

R m,12 (x) ≤ N -M n=M +1 µ 2 n -x 2 (µ n -γ 2 ) 2 -x 2 2 ≤ N -M n=M +1 µ 2 n -16 h 4 sin 4 M πh 2 (µ n -γ 2 ) 2 -16 h 4 sin 4 M πh 2 2
and the problem is reduced again to the case 1.

Thus, we obtain that

(5.40) R m,12 (x) ≤ CC v (x ∈ R).
In order to evaluate R m,13 (x) we consider the following cases.

Case 1. If |x| ≤ 4 h 2 there exists v ∈ 0, 1 h such that x = 4 h 2 sin 2 vπh 2 and we have R m,13 (x) = 1 + ε 2 x 2 µ 2 [v] (x 2 -µ 2 [v] ) 2 1 + ε 2 x 2 µ 2 [v]+1 (µ 2 [v]+1 -x 2 ) 2 Cv [v]-1 n=M +1 1 + ε 2 x 2 µ 2 n (x 2 -µ 2 n ) 2 N -M n=[v]+2 1 + ε 2 x 2 µ 2 n (µ 2 n -x 2 ) 2 .
To be more precisely we study first the case when v ≤ N 2 . We have that

R m,13 (x) = C v [v]-1 n=M +1 1 + ε 2 x 2 µ 2 n (x 2 -µ 2 n ) 2 N -M n=[v]+2 1 + ε 2 x 2 µ 2 n (µ 2 n -x 2 ) 2 ≤ ≤ C v [v]-1 n=M +1 1 + ε 2 µ 2 n (x -µ n ) 2 N -M n=[v]+2 1 + ε 2 x ( √ µ n - √ x) 2 = = C v [v]-1 n=M +1 1 + ε 2 sin 4 nπh 2 (sin 2 vπh 2 -sin 2 nπh 2 ) 2 N -M n=[v]+2 1 + ε 2 sin 2 vπh 2 (sin nπh 2 -sin vπh 2 ) 2 = = C v [v]-1 n=M +1   1 + ε 2 sin 4 nπh 2 sin 2 (v-n)πh 2 sin 2 (v+n)πh 2   N -M n=[v]+2   1 + ε 2 sin 2 vπh 2 4 sin 2 (n-v)πh 4 cos 2 (n+v)πh 4   ≤ ≤ C v [v]-1 n=M +1 1 + Cε 2 n 4 (v -n) 2 (v + n) 2 N -M n=[v]+2 1 + Cε 2 v 2 (n -v) 2 ≤ ≤ C v exp [v]-1 M +1 ln 1 + Cε 2 s 2 (v -s) 2 ds + N -M [v]+2 ln 1 + Cε 2 v 2 (s -v) 2 ds ≤ ≤ C v exp(Cεv) < C v exp(C ε|x|).
Secondly, we study the case v > N 2 . We have that

R m,13 (x) ≤ C v [v]-1 n=1 1 + ε 2 µ 2 n (x -µ n ) 2 N n=[v]+2 1 + ε 2 x 2 (µ n -x) 2 = = C v [v]-1 n=1   1 + ε 2 sin 4 nπh 2 sin 2 (v-n)πh 2 sin 2 (v+n)πh 2   N n=[v]+2   1 + ε 2 sin 4 vπh 2 sin 2 (n-v)πh 2 sin 2 (v+n)πh 2   ≤ ≤ C v [v]-1 n=1   1 + ε 2 sin 4 nπh 2 sin 4 (v-n)πh 2   N n=[v]+2   1 + ε 2 sin 4 vπh 2 sin 4 (n-v)πh 2   ≤ ≤ C v exp v 0 ln 1 + 2ε 2 t 4 (v -t) 4 dt I1 + N v+1 ln 1 + 2ε 2 v 4 (t -v) 4 dt I2 .
For I 1 we have that

I 1 = 8ε 2 v v 0 t 3 (v -t) 4 + 2ε 2 t 4 dt = 8ε 2 v   v 1+ 4 √ 2 √ ε 0 t 3 (v -t) 4 dt + 1 2ε 2 v v 1+ 4 √ 2 √ ε 1 t dt   ≤ C ε|x|.
For I 2 we obtain that

I 2 = (t -s) ln 1 + 2ε 2 v 4 (t -v) 4 N v+1 + 8ε 2 v 4 N v+1 1 (t -v) 4 + 2ε 2 v 4 dt ≤ ≤ (N -s) ln 1 + 2ε 2 v 4 (N -v) 4 + 8ε 2 v 4 (1+ 4 √ 2 √ ε)v v+1 1 2ε 2 v 4 dt + N (1+ 4 √ 2 √ ε)v (t -s) -4 dt ≤ C ε|x|.
R m,13 (x) = N -M n=M +1 1 + ε 2 x 2 µ 2 n (x 2 -µ 2 n ) 2 ≤ N -M n=M +1 1 + Cε 2 µ 2 n ( 4 h 2 -µ n ) 2 ≤ exp C ε|x| ,
where the last inequality follows from (5.41).

Thus, we obtain that

(5.42) |R m,13 (x)| ≤ C v exp C ε|x| (x ∈ R).
From (5.39), (5.40) and ( 5.42) we deduce that

(5.43) |R m (x)| 2 ≤ CC v C v sin πx γ πx γ 2 |λ m | 2 |x -λ m | 2 exp(C ε |x|) (x ∈ R).
Note that, there exists a constant C > 0 independent of h, ε and m such that (5.44)

|λ m | 2 |x -λ m | 2 sin πx γ πx γ 2 1 + ε 2 x 2 µ 2 [v] (x 2 -µ 2 [v] ) 2 1 + ε 2 x 2 µ 2 [v]+1 (µ 2 [v]+1 -x 2 ) 2 x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 ≤ C.
Indeed, the terms |λm| 2 |x-λm| 2 ,

x 2 -µ 2 [v] x 2 -k2 [v] γ 2 2 , µ 2 [v]+1 -x 2 k2 [v]+1 γ 2 -x 2 2 , 1 + ε 2 x 2 µ 2 [v] (x 2 -µ 2 [v] ) 2 and 1 + ε 2 x 2 µ 2 [v]+1 (µ 2 [v]+1 -x 2 ) 2
explodes as

x goes to k [v] γ, k [v]+1 γ, µ [v]
and µ [v]+1 respectively, but not in the same time. Thus, depending of which goes to zero, we combine it with the sine function,

, so that we obtain a bounded function.

Hence, from (5.43) and (5.44) it follows that

(5.45) |R m (x)| ≤ C exp(ωϕ(x)) (x ∈ R),
where C and ω are two positive constants independent of h, ε and m.

By taking into account (5.27) and (5.45) it follows that (5.37) holds which ends the proof of the proposition.

A multiplier

The aim of this section is to construct a multiplier, an entire function capable of compensating for the growth on the real axis of the product P 2 m given by (5.37). We begin with a general result (see also [START_REF] Bugariu | A numerical method with singular perturbation to approximate the controls of the heat equation[END_REF]Proposition 4]).

Proposition 5.4. Let ϕ : [0, ∞) → [0, ∞) be a continuous, increasing and onto function, r ∈ N * and define the sequences (a n ) n∈N * and (n m ) 1≤|m|≤N by

(5.46) ϕ(ea n ) = n + r, (5.47
)

n m = 1 ϕ(e|λ m |) < r [ϕ(e|λ m |)] -r + 1 ϕ(e|λ m |) ≥ r,
where λ m are the eigenvalues defined by (3.13). Moreover, suppose that the sequence (a n ) n≥1 verifies

(I1) n≥nm 1 a n ≤ L 2 < ∞, (I2) n≥nm 1 a 2 n ≤ D 1 + |ℜ(λ m )| |λ m | 2 ,
where L 2 and D are two positive constants. Then, for each m ∈ Z * such that |m| ≤ N , there exists M m : C → C with the following properties:

1. M m is an entire function of exponential type L 2 , 2. M m (x) ≤ exp (-ϕ(|x|) + ϕ(e|λ m |) + r + 1) for all x ∈ R, 3. M m (iλ m ) ≥ exp - D 6 (1 + |ℜ(λ m )|) .
Proof: By adapting an idea from [START_REF] Ingham | A note on Fourier transform[END_REF] (see also [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF]), we define the function M m : C → C as follows

(5.48) M m (z) = ∞ n=nm sin z an z an .
M m is an entire function of exponential type. Indeed, this is a consequence of (I1) and the following estimate

∞ n=nm sin z an z an ≤ ∞ n=nm e | z an | = e |z| ∞ n=nm 1 a n ≤ e L|z| .
To prove the second property of M m we have to treat separately the following two cases:

Case 1. |x| ≤ ea nm . In this case it follows easily from the monotony of ϕ, (5.46) and (5.47) that

M m (x) = ∞ n=nm sin x an x an ≤ 1 ≤ exp(-ϕ(ea nm ) + ϕ(e|λ m |) + r + 1) ≤ exp(-ϕ(|x|) + ϕ(e|λ m |) + r + 1). Case 2. |x| > ea nm . If we denote by n x = ϕ(e|x|) -r, we deduce that | M m (x)| ≤ nx j=nm a j x = exp nx j=nm ln a j x ≤ exp - |x| an m [ϕ(eu)] -r -n m + 1 u du .
Since a nm < |x| e and n m -1 ≤ ϕ(e|λ m |), we obtain that

| M m (x)| ≤ exp - |x| |x| e [ϕ(|x|)] -r -ϕ(e|λ m |) u du ≤ exp (-ϕ(|x|) + ϕ(e|λ m |) + r + 1) ,
and the second property of M m is proved. Now, we pass to study the last property of M m . We notice that

a nm = 1 e ϕ -1 (n m + r) ≥ 1 e ϕ -1 (ϕ(e|λ m |) -r + r) = |λ m |, which implies that |λ m | a n ≤ 1 (n ≥ n m ).
Thus, by taking into account the inequality

sin x x ≥ 1 - 1 6 x 2 (|x| ≤ 1),
and (I2), we deduce that

| M m (iλ m )| = ∞ n=nm sin iλm an iλm an ≥ ∞ n=nm sin λm an λm an ≥ ∞ n=nm 1 - 1 6 |λ m | 2 a 2 n = = exp ∞ n=nm ln 1 - 1 6 |λ m | 2 a 2 n ≥ exp - |λ m | 2 6 ∞ n=nm 1 a 2 n ≥ exp - D 6 (1 + |ℜ(λ m )|) .
This gives the third property of M m and completes the proof of the proposition.

To obtain our multiplier function, we use the previous result with the particular function ϕ given by (5.38).

Proposition 5.5. Let ε ∈ (0, 1), ϕ be the function defined by (5.38) and r ∈ N * . For each m ∈ Z * such that |m| ≤ N , there exists a function M m,ε : C → C with the following properties 1. M m,ε is an entire function of exponential type at most e ε r .

|M

m,ε (x)| ≤ C 1 exp -ϕ(|x|) + 2 e r |ℜ(λ m )| for all x ∈ R. 3. M m,ε (iλ m ) ≥ C 2 exp -2 e r |ℜ(λ m )| ,
where C 1 and C 2 are two positive constants independent of h, ε and m.

Proof: Let (a n ) n≥1 and n m be given by (5.46) and (5.47), respectively. Also, let M m,ε be given by (5.48).

From Proposition 5.4 and the estimate

1 e n≥nm 1 a n = ε n≥nm 1 (n + r) 2 ≤ ε ∞ 0 (t + r) -2 dt = ε r ,
we obtain that M m,ε is an entire function of exponential type at most e ε r .

On the other hand, the following estimates

1 e 2 n≥nm 1 a 2 n = ε 2 n≥nm (n + r) -4 ≤ ε 2 ∞ nm-1 (t + r) -4 dt = ε 2 3(n m -1 + r) 3 ,
allows us to prove the last two properties of M m,ε . Indeed, we have the following two cases.

|) + r + 1) ≤ exp(-ϕ(|x|) + 2r + 1) ≤ C(r) exp(-ϕ(|x|)) (x ∈ R),
where C(r) is a positive constant depending of r, but independent of h, ε and m.

Furthermore, since

n≥nm 1 a 2 n ≤ e 2 ε 2 3r 3 ≤ e 2 ε 2 3r 3 2|λ m | ε 1 + |ℜ(λ m )| |λ m | 2 ≤ 2 e r 1 + |ℜ(λ m )| |λ m | 2 ,
we obtain from Proposition 5.4 that the last property of M m,ε is verified, too.

ϕ(e|λ m |) = 2 e ε |λ m | |ℜ(λ m )| ≤ 2 e r |ℜ(λ m )|.
For the last property of M m,ε , we remark that

n≥nm 1 a 2 n ≤ e 2 ε 2 3(ϕ(e|λ m |)) 3 2|λ m | ε 1 + |ℜ(λ m )| |λ m | 2 ≤ 2 e r 1 + |ℜ(λ m )| |λ m | 2
and we apply once again Proposition 5.4. This concludes the proof of the Proposition.

The desired biorthogonal sequence to the family Λ 2

In this subsection we construct two biorthogonal sequences to the family of exponential functions Λ 2 = (e λnt ) n∈G .

Theorem 5.2. Let T > 0 and γ > 32π

√ 2 T
. There exists h 0 > 0 such that, for any h ∈ (0, h 0 ) and ε ∈

1 T h 2 ln 1 h , h , there exists a sequence (ζ m ) m∈G ⊂ L ∞ -T 8 , T 8 biorthogonal to the family Λ 2 in L 2 -T 8 , T 8
with the following properties:

(5.49) ζ m L ∞ (-T 8 , T 8 ) ≤ C 3 e T 8 |ℜ(λm)| (m ∈ G) , (5.50) ζ M +1 (x)(x -iλ M +1 ) ≤ C 3 (x ∈ R), (5.51) ζ M +1 (iλ m )(iλ m -iλ M +1 ) ≥ C 4 e -T 8 |ℜ(λm)| (m ∈ F ) ,
where C 3 and C 4 are two positive constants independent of h, ε and m.

Proof: Let ω be the constant from Proposition 5.3 and let us consider Let P2 m and M m,ε given by (5.25) and (5.48), respectively. For each m ∈ G we define the entire function

(5.53) Ψ m (z) = P 2 m (z) M m,ε (z) M m,ε (iλ m ) [ω]+1 sin(δ(z -iλ m )) δ(z -iλ m ) Let (5.54) ζ m (t) = 1 2π R Ψ m (x)e ixt dx.
Let h 1 0 be the constant given by (3.17). We remark, at this point, that there exists h 0 ∈ (0, h 1 0 ) such that, for any h ∈ (0, h 0 ) and ε ∈ 1 T h 2 ln 1 h , h , the following relations take place (5.55) (64 + T )ε 64 < T 16 ,

(5.56) h √ ε < T 16 .
From Propositions 5.2 and 5.5 and relations (5.52) and (5.55) it results that Ψ m is an entire function of exponential type at most T 8 . Moreover, from the estimates of P if and M m,ε on the real axis given by Propositions 5.3 and 5.5 and relations (5.52) and (5.56), we obtain that (5.57)

R |Ψ m (x)|dx ≤ Ce h √ ε + 2e([ω]+1) r |ℜ(λm)| R sin(δ(x -iλ m )) δ(x -iλ m ) 2 dx ≤ ≤ C δ e h √ ε + 2e([ω]+1) r +2δ |ℜ(λm)| R sin t t 2 dt ≤ Ce T 8 |ℜ(λm)| .
Nextly, Paley-Wiener Theorem implies that (ζ m ) m∈G ⊂ L 2 -T 8 , T 8 . By taking into account that Ψ m (iλ n ) = δ mn , for any m, n ∈ G, the inverse Fourier transform property gives us that (ζ m ) m∈G is a biorthogonal sequence to (e λnt ) n∈G in L 2 -T 8 , T 8 . Finally, from Plancherel's Theorem we deduce that (5.49) holds.

Furthermore, from the estimates of P if and M m,ε on the real axis given by Propositions 5.3 and 5.5, it follows that (5.50) takes place for any x ∈ R. Indeed, we have that

ζ M +1 (x)(x -iλ M +1 ) ≤ Ce h √ ε + 2e([ω]+1) r |ℜ(λ M +1 )| sin(δ(x -iλ M +1 )) δ(x -iλ M +1 ) 2 |x -iλ M +1 | ≤ C.
It remains to show that (5.51) takes place. Firstly, we remark that Proposition 5.2 implies that Secondly, from (5.27) it follows that (5.61)

|R(iλ m )| ≥ Ce -h √ ε |ℜ(λm)| (m ∈ F ).
Finally, to complete the proof of (5.51) we notice that

(5.62) sin(δ(iλ m -iλ M +1 )) δ(iλ m -iλ M +1 ) 2 (iλ m -iλ M +1 ) ≥ e -δℜ(λm-λ M +1 ) + e δℜ(λm-λ M +1 ) 2 4δ 2 |λ m -λ M +1 | ≥ C (m ∈ F ).
Thus, from (5.52), (5.56), (5.58), (5.59), (5.60), (5.61) and (5.62) it follows that (5.51) holds, fact that concludes the proof of this theorem.

To construct a biorthogonal sequence to the family of exponential functions (e λnt ) n∈G with better properties we have to work in a slightly larger time interval. Let a > 0 and k a =

k a (x) = 1 √ 2π R k a (t)e -itx dt = 2π a 2 χ a (x) χ a (x) = 4 a 2 sin 2 ( xa 2 ) x 2 .
For any m ∈ G we define ρ m (x) = e ixℑ(λm) k a (x), so that supp(ρ m ) ⊂ [-a, a]. Now, we have all the ingredients to define a new biorthogonal sequence in L2 -T 4 , T 4 . For any m ∈ G we define (5.64)

ξ m = 1 2π ρ m (iλ m ) ζ m * ρ m ,
where ζ m is given by (5.54). We have the following result which shows the properties of this new biorthogonal sequence.

Theorem 5.3. Let T > 0. There exists a positive constant h 0 such that, for any h ∈ (0, h 0 ) and ε ∈ 1 T h 2 ln 1 h , h , the family (ξ m ) m∈G , defined by (5.64) with a = T 8 , is biorthogonal to

Λ 2 in L 2 -T 4 , T 4
and has the property that, for any finite sequence (α m ) m , the following inequality holds and let h 0 be the constant given by Theorem 5.2. Since

T 4 -T 4 ξ m (t)e λnt dt = √ 2π ξ m (iλ n ) = ζ m (iλ n ) ρ m (iλ n ) ρ m (iλ m ) = δ mn ,
it follows that the sequence (ξ m ) m∈G given by (5.64) is biorthogonal to the family Λ 2 in L 2 -T 4 , T 4 .

In order to prove (5.65) we remark that Now, we are able to prove (5.65). We have that

ρ m (iλ m ) = 1 √ 2π R ρ m (t)e tλm dt = 1 √ 2π R e itℑ(λm
T 4 -T 4 m∈G α m ξ m (t) 2 dt = 2π R m∈G α m ξ m (x) 2 dx = 2π R m∈G α m ζ m (x) ρ m (x) √ 2π ρ m (iλ m ) ≤ R m∈G |α m | | ρ m (iλ m )| ζ m L ∞ (R) | ρ m (x)| 2 dx ≤ R m∈G |α m | ζ m L ∞ (-T 8 , T 8 ) | ρ m (x)| 2 dx = = R m∈G |α m | ζ m L ∞ (-T 8 , T 8 ) | k T 8 (x -ℑ(λ m ))| 2 dx ≤ ≤ T 8 -T 8 k 2 T 8 (t) m∈G |α m | ζ m L ∞ (-T 8 , T 8 ) e itℑ(λm) 2 dt ≤ 128π T 2 T 8 -T 8 m∈G |α m | ζ m L ∞ (-T 8 , T 8 ) e itℑ(λm) 2 dt.
Since for any m ∈ G we have that

|ℑ(λ m+1 ) -ℑ(λ m )| ≥ √ 4-ε 2 2 γ and γ > 32π √ 2 T
, from Ingham's inequality (see [START_REF] Ingham | Some trigonometric inequalities with applications to the theory of series[END_REF]), it follows that there exists a positive constant C, such that (5.68)

T 8 -T 8 m∈G |α m | ζ m L ∞ (-T 8 , T 8 ) e iℑ(λm)t 2 dt ≤ C m∈G |α m | 2 ζ m 2 L ∞ (-T 8 , T 8 ) .
Thus, from (5.49) and (5.68) it follows that (5.65) takes place. Moreover, relations (5.66) and (5.67) follows immediately from (5.64), (5.50), and (5.51). This completes our proof.

A biorthogonal sequence to the family Λ

In this subsection we use the sequences (ψ m ) m∈F and (ξ m ) m∈G constructed in Subsections 5.1 and 5.2.3 in order to obtain a biorthogonal sequence (θ m ) 1≤|m|≤N to the entire family Λ = (e tλn ) 1≤|n|≤N in L 2 -T 2 , T 2 . The following theorem gives us the desired biorthogonal sequence. and let h 0 , c0 be the constants given by Theorems 5.2 and 5.1. We choose c 0 (T ) = c0 where c0 is the constant from Theorem 5.1. Let (ψ m ) m∈F and (ξ m ) m∈G be the biorthogonals constructed in Theorems 5.1 and 5.3, respectively.

For any z ∈ C and m ∈ Z * such that |m| ≤ N we define the entire function

(5.70) Θ m (z) =              ψ m (z) ξ M +1 (z)(z -iλ M +1 ) ξ M +1 (iλ m )(iλ m -iλ M +1 ) m ∈ F ξ m (z) ψ 1 (z)(z -iλ 1 ) ψ 1 (iλ m )(iλ m -iλ 1 ) m ∈ G.
By taking into account that Theorems 5.1 and 5.3 implies that ψ m and ξ m are two entire functions of exponential type at most T 4 we deduce that Θ m is an entire function of exponential type at most T 2 . Moreover, from (5.17) and (5.57) it results that Θ m is L 2 on the real axis, and by applying Paley-Wiener Theorem we deduce that there exists (θ

m ) 1≤|m|≤N ⊂ L 2 -T 2 , T 2 such that (5.71) Θ m (x) = T 2 -T 2 θ m (t)e -ixt dt.
From (5.71) and the fact that

(5.72) Θ m (iλ n ) = δ mn (1 ≤ |m|, |n| ≤ N ),
we deduce that (θ m ) 1≤|m|≤N is a biorthogonal sequence to the family Λ in L 2 -T 2 , T 2 .

Moreover, by using Plancherel's Theorem, (5.11), (5.66), (5.71) and by denoting

β m =          α m ξ M +1 (iλ m )(iλ m -iλ M +1 ) m ∈ F α m ψ 1 (iλ m )(iλ m -iλ 1 ) m ∈ G,
we obtain that (5.73) Finally, from (5.12), (5.67) and (5.74) we have that (5.69) holds and the proof of the theorem ends.

T 2 -T 2 N |m|=1 α m θ m (t) 2 dt ≤ 2   R m∈F α m Θ m (x) 2 dx + R m∈G α m Θ m (x) 2 dx   ≤ ≤ C   R m∈F β m ψ m (x) 2 dx + R m∈G β m ξ m (x)
Remark 5.2. Note that, from the proof of Theorems 5.1 and 5.4, we can deduce that c0 ∼ γ T and c 0 ∼ 1 T 2 when T → 0.

Controllability results

In this section we use the biorthogonal sequence constructed in Theorem 5.4 in order to obtain a uniformly bounded family of controls for the perturbed problem (3.2) with the property that any weak limit of it is a control for the continuous problem (2.1). We present the following uniformly boundedness result.

Theorem 6.1. Let T > 0. There exist h 0 , c 0 > 0 such that for any h ∈ (0, h 0 ), ε ∈ c 0 h 2 ln 1 h , h and any initial data

U 0 h U 1 h ∈ C 2N
of the form (4.1) such that there exists a constant C > 0 independent of h and ε with the property

(6.1) (a 0 nh ) n ℓ ∞ < C,
there exists a control v h ∈ L 2 (0, T ) for problem (3.2) such that the family (v h ) h is uniformly bounded in L 2 (0, T ).

Proof: Let γ > max{ 32π √ 2 T
, 1} and let h 0 , c 0 be the constants given by Theorems 5.2 and 5.4.

Let (θ m ) 1≤|m|≤N be the biorthogonal sequence to the family e λnt 1≤|n|≤N given by Theorem 5.4. For any initial data

U 0 h U 1 h ∈ C 2N and t ∈ (0, T ) we construct the function v h ∈ L 2 (0, T ) as follows (6.2) v h (t) = 1≤|n|≤N (-1) n 2 cos nπh 2 i 4 -ε 2 a 0 nh -ε sgn(n)a 0 nh + ε sgn(n)a 0 -nh e -λn T 2 θ n t - T 2 .
Taking into account the biorthogonality properties of (θ m ) m it follows immediately that v h satisfy (4.2) and it is a control for (3.2). To estimate the norm of v h we analyze the right hand side of (6.2). By using the formula (6.2), the biorthogonal estimate (5.69) from Theorem 5.4 and hypothesis (6.1) we obtain that

T 0 |v h (t)| 2 dt ≤ C N |n|=1 1 cos nπh 2 2 e -T 2 |ℜ(λn)| ≤ C,
where C is a positive constant independent of h, ε and m. Note that, the last inequality takes place since e -T 4 |ℜ(λ N )| < h 2 , which is an easy consequence of the fact that the proof of Theorem 5.4 implies that c 0 ≥ 16 T . The proof of the theorem is complete. Now, we have all the ingredients needed to prove our main result. Theorem 6.2. Let T > 0 and u 0 u 1 ∈ H given by (2.8). There exist h 0 , c 0 > 0 such that for any for any h ∈ (0, h 0 ), ε ∈ c 0 h 2 ln 1 h , h and any initial data

U 0 h U 1 h ∈ C 2N of the form (4.1)
with the property

(6.3) (a 0 nh ) n * ⇀ (a 0 n ) n in ℓ ∞ when h → 0,
there exists a family of exact controls (v h ) h ⊂ L 2 (0, T ) for problem (3.1) which converges to a null-control for (2.1) in L 2 (0, T ).

Proof: Let γ > max{ 32π √ 2 T
, 1} and let h 0 , c 0 be the constants given by Theorems 5.2 and 5.4.

Let us remark that (6.3) implies (6.1). Since from Theorem 6.1 we obtain a family of controls (v h ) h uniformly bounded in L 2 (0, T ), it follows that there exists a subfamily, denoted in the same way, which converges weakly to a function v from L 2 (0, T ). Moreover, by taking into account that v h is a control for (3.2), it results from Theorem 4.1 that (6.4)

T 0 v h (t)e tλn dt = (-1) n 2 cos nπh 2 i 4 -ε 2 a 0 nh -ε sgn(n)a 0 nh + ε sgn(n)a 0 -nh (n ∈ Z * , |n| ≤ N ).
Now, by passing to the limit as h and ε tend to zero in the above expression, we deduce that (6.5)

T 0 v(t)e tη n dt = (-1) n i a 0 n (n ∈ Z * ),
fact that, by taking into account Lemma 2.2, concludes our proof.

Remark 6.1. The usual discretization by points leads to a convergence property of the Fourier coefficients sequence that depends on the regularity of u 0 u 1 . Indeed, it is not difficult to prove that, if u 0 and u 1 are continuous and piecewise one time derivable with continuous derivative, then (6.3) holds.

U 0 h U 1 h = u 0 (jh) u 1 (jh) 1≤j≤N , 0

Numerical results

In this section we present three numerical experiments to approximate the minimum L 2 -norm controls of (2.1) corresponding to three different initial conditions u 0 u 1 . All of them are based on the discrete scheme (3.1). In this way we shall be able to illustrate the convergence of the scheme analyzed in this paper and to highlight its effectiveness, particularly in the case of initial data with low regularity. We recall that the minimum L 2norm controls are usually called HUM controls and can be obtained by minimizing a quadratic cost functional depending on the solutions of the adjoint problem (1.4). The interested reader is referred to [START_REF] Lions | Controlabilité exacte, stabilisation et perturbations des systèmes distribués[END_REF] for a detailed analysis of these controls. An algorithm based on the conjugate gradient method applied to a discrete version of the already mentioned cost functional is used to approximate the controls. This algorithm is similar to those introduced in [START_REF] Carthel | On exact and approximate Boundary Controllability for the heat equation: A numerical approach[END_REF][START_REF] Glowinski | Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation[END_REF][START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods[END_REF][START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF] in the context of the heat and wave equations. More precisely, by considering an approximation U 0 h U 1 h of u 0 u 1 , we minimize the functional J : C 2N → C defined by (7.1) The function ϑ has the role to improve the numerical approximations of the controls, avoiding the incompatibility between the initial data and the nonhomogeneous term at the origin. other space of controllable initial conditions, we expect the viscosity to be truly necessary for the convergence of the scheme in this example.

J(W 0 h , W 1 h ) = 1 2 T 0 ϑ(t) W h,N (t) h 2 dt -U 1 h , W h (0) + U 0 h , W ′ h (0) -εA h W h (0) ,
In Figure 2 we present the first four iterations of the conjugate gradient method with ε = 0 and ε = h, respectively (N = 100 in both simulations) and Figure 3 shows the behavior of the residual in both cases. We note that only in the presence of the viscosity the process converges. In this case, the low regularity of initial data makes the viscosity necessary to restore the convergence property.

Figure 4 shows the approximations of the control and the controlled solution in the viscosity case.

Example 3

In this example we take T = 3 and the initial data to be controlled are the following u 0 (x) = 1 -|2 -|4x -1|| u 1 (x) = 0 (x ∈ (0, 1)). As in the previous two examples, the initial data belong to H but not to D(A). Once again we are in the limit case of non smooth initial data in which we expect to need viscosity for the convergence of the scheme. Figure 5 shows the behavior of the residual when ε = h, ε = 4h 1.5 , ε = 5h 2 ln(1/h) and ε = 0 whereas N = 200. We note that, indeed, the low regularity of initial data makes the process convergent only in the presence of the viscosity. The approximation of the control and solution is depicted in Figure 6 for ε = h and N = 400. Moreover, table 2 exhibits the convergence of the algorithm with different values of ε: 5h 2 ln 1 h , 4h 3/2 and h.

In conclusion, these numerical experiments illustrate the lack of convergence of the algorithm if ε = 0 and the initial data are not smooth enough (for instance, if they belong to H). On the other hand, the results show the effectiveness of the viscosity method in all these cases. Hence, the numerical experiments confirm the convergence property proved in Theorem 6.2. Moreover, we have observed that, while a larger viscosity parameter ε helps the convergence of the scheme in the non smooth case, it also produces a slower convergence rate in the regular case. Hence, the amount of dissipation introduced in the system through the parameter ε should be decided by taking into account the regularity of initial data to be controlled.

1 )

 1 n+1 sin(nπh)e -T λn h √ µ |n| e tλn 2 dt.

Case 1 .

 1 If ϕ(e|λ m |) < r, it follows from (5.47) that n m = 1 and from the definition of ϕ that |λ m | < r 2 e ε . In this case the second property of M m,ε follows immediately from Proposition 5.4 by noting that exp(-ϕ(|x|) + ϕ(e|λ m

Case 2 .

 2 If ϕ(e|λ m |) ≥ r, it follows that n m = [ϕ(e|λ m |)] -r + 1 and |λ m | ≥ r 2 e ε . Hence, the second property of M m,ε follows from Proposition 5.4 by noting that

( 5 .

 5 58) |R(iλ M +1 )| ≤ C(γ)e √ 2π γ +ε |iλM+1| , and Proposition 5.5 gives us (5.59) |M M +1,ε (iλ M +1 )| ≤ Ce eε([ω]+1) r |iλ M +1 | , and (5.60) |M M +1,ε (iλ m )| ≥ C 2 e -2([ω]+1)e r |ℜ(λm)| (m ∈ F ).

√ 2π a 2

 2 (χ a * χ a ), where χ a represents the characteristic function χ [-a/2,a/2] . Evidently supp(k a ) ⊂ [-a, a]. Also, we have (5.63)

2 dt ≤ C 5 m∈G |α m | 2 e T 4

 254 |ℜ(λm)| .Moreover, we have that(5.66) ξ M +1 (x)(x -iλ M +1 ) ≤ C 5 (x ∈ R), (5.67) ξ M +1 (iλ m )(iλ m -iλ M +1 ) ≥ C 6 e -T 8 |ℜ(λm)| (m ∈ F ),where C 5 and C 6 are two positive constants independent of h, ε and m.Proof: Let us chose a = T 8 , γ > 32π √ 2 T

) k T 8 (t)e tλm dt = k T 8 (

 88

Theorem 5 . 4 .T 2

 542 Let T > 0. There exist h 0 , c 0 > 0 such that, for any h ∈ (0, h 0 ) and ε ∈ c 0 h 2 ln 1 h , h there exists a biorthogonal sequence (θ m ) m∈Z * |m|≤N to the family (e λnt ) n∈Z * |n|≤N in L 2 -T 2 , T 2 such that, for any finite sequence (α m ) m , we have that |ℜ(λm)| , where C(T ) is a positive constant. Proof: Let δ and r as in (5.52) such that (5.55) and (5.56) take place. Let γ > 32π √ 2 T

5

 5 applying again Plancherel's Theorem and by taking into account (5.23) and (5.65), we deduce that (

Figure 1 :

 1 Figure 1: Example 1 -Two approximations of the control v h : with ε = 0 (left) and with ε = h (right).

1 h.

 1 In (7.1) ϑ ∈ C ∞ [0, T ] is a cut-off regular function with the property that there exists a positive number ǫ < T 2 such that supp(ϑ) ⊂ (ǫ/2, T -ǫ/2), 0 ≤ ϑ(t) ≤ 1 for all t ∈ [0, T ], ϑ(t) ≥ 1/2 for all t ∈ [ǫ, T -ǫ].

1 hFigure 2 :

 12 Figure 2: Example 2 -The first four iterations of the conjugate gradient method for the approximation of v h with N = 100 and ε = 0 (up) or ε = h (down).

Figure 3 :

 3 Figure 3: Example 2 -Error evolution in the conjugate gradient method with ε = 0 and ε = h.

Figure 4 :

 4 Figure 4: Example 2 -Controlled solution and the approximation of the control with N = 100 and ε = h.

Figure 5 :

 5 Figure 5: Example 3 -Error evolution in the conjugate gradient method with four different values of ε.

  2 with ε = h 0.5376 1.1518 1.6301 1.9209 2.099 v h L 2 with ε = 4h 1.5 0.7635 1.5728 1.9908 2.2008 2.3175 v h L 2 with ε = 5h 2 log(1/h) 0.9746 1.7655 2.1201 2.2929 2.3819 Table 2: Example 3 -Numerical results for v h L 2 obtained with different values of the parameters ε and N .

Figure 6 :

 6 Figure 6: Example 3 -Controlled solution and the approximation of the control with N = 400 and ε = h.

  ).

	Lemma 2.2. Problem (2.1) is null-controllable in time T > 0 if, and only if, for any initial data	u 0 u 1	∈ H
	with the Fourier expansions
	(2.8)

  given. Indeed, we have the following discrete version of Lemma 2.1.

	Lemma 3.1. Given T > 0, system (3.2) is null-controllable in time T if, and only if, for any initial data

  Therefore, in the case |x| ≤ 4 h 2 we obtain that (5.41) |R m,13 (x)| ≤ C v exp(C ε|x|). < |x| we reduce our estimate to the first case. Indeed, we have that

	Case 2. If 4 h 2

(z ∈ C).

dx ≤

Acknowledgments

The first and the third authors were partially supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS UEFISCDI. The second author was partially supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS UEFIS-CDI and by Grant MTM2011-29306 funded by MICINN (Spain).

respect to the weight ϑ. The weak limit of the family ( v h ) h>0 is the minimal L 2 -norm (with respect to the weight ϑ) control of (2.1).

In the algorithm several beam equations have to be solved. This is done by using the Newmark Method with parameters γ = 0.5 and β = 0.25 (see [START_REF] Hughes | The finite element method[END_REF]). The iterative conjugate gradient algorithm is initialized by 0 0 and we assume that the convergence is achieved when the relative residual is lower than 10 -6 .

Example 1

In this example we take T = 1.7 and the initial data to be controlled are given by

Note that this particular initial condition belongs to D(A), where A is the operator from (2.5). In Figure 1 two approximations of the control are presented for N = 100 and two different values of the viscosity parameter: ε = 0 and ε = h. In this case, the initial data to be controlled being regular, there are no significant differences between these two results. In fact, formula (6.2) can be used to show that sufficiently smooth initial data can be uniformly controlled, even if ε = 0. To prove this result, estimates of the biorthogonals sequences to the family of complex exponentials e i sgn(n) µ |n| t 1≤|n|≤N are needed. This study does not concern the viscosity approach and it will be presented in a future paper. Note that, in Example 1, all the Fourier coefficients of the initial data to be controlled, except a ±1h , are equal to zero.

From the L 2 -norms of the approximations presented in Table 1 we can appreciate the convergence of the algorithm with four different values of ε: 0, h 2 ln 1 h , h 3/2 and h. We note that the convergence rate is becoming slightly better for smaller values of ε. This shows that the viscous perturbation does affect the velocity of convergence of the numerical scheme and, in order to have better approximations, the parameter ε should be chosen as small as possible.

Example 2

In this example we take T = 2.3 and the following initial data to be controlled

Note that in this case the initial data belong to H = H 1 0 (0, 1) × H -1 (0, 1) but not to D(A). The space H is the largest space of initial data which can be controlled with L 2 -controls. As shown in Proposition 4.2, the observability constant is affected by the badly approximated high frequencies and tends to infinity if the viscosity parameter ε is not sufficiently large. Since in the space H the high frequencies are more present than in any