Sorin Micu

Takéo Takahashi

Local controllability to stationary trajectories of a one-dimensional simplified model arising in turbulence

Keywords: Controllability to trajectories, viscous Burgers equation, Series of exponentials, Observability, Biorthogonal 2010 Mathematics Subject Classification. 93C20, 93B05, 93B07, 35Q53, 35P20

Riesz basis 16 4 The biorthogonal family 18 5 Observability result 25 6 The nonlinear problem 28

Introduction

In [START_REF] Fernández-Cara | Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence[END_REF], the authors tackle the local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence. More precisely, they consider the following control problem

           ∂ t w + (w • ∇)w -ν( ∇w 2
L 2 (Ω) )∆w + ∇q = uχ ω in (0, T ) × Ω, div w = 0 in (0, T ) × Ω, w = 0 on (0, T ) × ∂Ω w(0, •) = w 0 in Ω, where ν : R + → [ν 0 , ∞) is a C 1 function, with ν 0 > 0 and with bounded derivatives. Following the method used for the controllability of the Navier-Stokes system (see [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]), they show a Carleman estimate for the linearized system and, with a fixed point argument, they obtain the local controllability to the null state (w(T, •) ≡ 0). In Section 6 of [START_REF] Fernández-Cara | Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence[END_REF], it is mentioned that at the contrary to the Navier-Stokes system, it is not clear how to obtain the local controllability to the trajectories. Indeed, in the linearized system, one has to deal with non-local terms.

In [START_REF] Fernández-Cara | Null controllability of linear heat and wave equations with nonlocal spatial terms[END_REF], a partial answer to the above question is given: the authors consider the linear heat and the linear wave system and show that one can recover the controllability properties of these systems if some non local terms are added. More precisely, their results are obtained for any dimension in space, but they need that the nonlocal integral terms are analytic in space.

Unhappily, in the above paper, the proof is given through a contradiction argument and in particular, it is not clear how to keep the cost of the linear heat equation in order to tackle nonlinear problems. Our aim is here to study the controllability of a Burgers system with variable viscosity. This can be seen as a simplified model for the turbulence model considered in [START_REF] Fernández-Cara | Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence[END_REF]. More precisely, our system writes as (1.1)

     ∂ t w -ν( ∂ x w 2
L 2 (0,π) )∂ xx w + w∂ x w = f S + uχ ω in (0, T ) × (0, π), w(t, 0) = w(t, π) = 0 t ∈ (0, T ), w(0, •) = w 0 := y 0 + w S in (0, π),

where ω is a non empty open interval of (0, π). In what follows, we assume that ν : R + → [ν 0 , ∞) is a C 2 function, with ν 0 > 0. In (1.1) the function f S depends only on x and belongs to L 2 (0, π).

Our aim in this paper is to obtain the local exact controllability to the stationary trajectories. More precisely, w S and f S in the above system are related through the stationary equations:

(1.2) -ν( ∂ x w S 2 L 2 (0,π) )∂ xx w S + w S ∂ x w S = f S in (0, T ) × (0, π), w S (0) = w S (π) = 0.

We can write the above problem as a null controllability problem by setting y := w -w S .

Then y satisfies the following nonlinear heat-type system: The system (1.4) is transformed into

         ∂ t y -ν S ∂ xx y -µ
(1.6)          ∂ t z -∂ xx z + π 0 K(ξ, • )z(ξ) dξ + pz = vχ ω , z(t, 0) = z(t, π) = 0, z(0, •) = z 0 , p = 1 2 ∂ x w S + 1 4 (w S ) 2 ,
K(ξ, x) = a(ξ)e W (ξ) a(x)e -W (x) .

Let us define the unbounded operator (D(A), A) in L 2 (0, π) as follows (1.7)

D(A) = H 2 (0, π) ∩ H 1 0 (0, π), Az = -∂ xx z + π 0 K(ξ, • )z(ξ) dξ + pz (z ∈ D(A)),
where p ∈ L ∞ (0, π) and the kernel K ∈ L 2 (0, π) 2 . With this notation, system (1.6) can be equivalently written as follows (1.8) ∂ t z(t) + Az(t) = vχ ω , z(0) = z 0 .

The adjoint (D(A * ), A * ) of the operator (D(A), A) is given by (1.9)

D(A * ) = H 2 (0, π) ∩ H 1 0 (0, π), A * w = -∂ xx w + π 0 K( • , ξ)w(ξ) dξ + qw (w ∈ D(A * )),
where K ∈ L 2 (0, π) 2 and q ∈ L ∞ (0, π) are given by K(x, ξ) = K(x, ξ), q = p.

Also, we introduce the following "adjoint problem":

(1.10) ∂ t ϕ(t) + A * ϕ(t) = 0,

ϕ(0) = ϕ 0 .
The following result is classic in control theory (see [START_REF] Dolecki | A general theory of observation and control[END_REF] or Theorem 11.2.1 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]).

Theorem 1.1. Let T > 0 and let ω ⊂ [0, π] be a non empty open set. For each z 0 ∈ L 2 (0, π) there exists v ∈ L 2 ((0, T ) × ω) such that the solution z of (1.8) vanishes at T , if and only if there exists a constant C = C(T ) such that the following inequality holds

(1.11) ϕ(T ) 2 L 2 (0,π) C T 0 ω |ϕ(t, x)| 2 dx dt,
for any ϕ 0 ∈ L 2 (0, π) and ϕ solution of (1.10). Moreover, if (1.11) holds, then there exists a control v ∈ L 2 (0, T ) which verifies

(1.12) v 2 L 2 (0,T ) C z 0 2 L 2 (0,π) (z 0 ∈ L 2 (0, π)).
A key result of this paper is the following theorem concerning the observability inequality (1.11) which also estimates the behavior of the constant C as T tends to zero.

Theorem 1.2. Let ω ⊂ [0, π] be a non empty open set. Suppose that the kernel K is degenerate, i.e. there exist two functions α, β ∈ L 2 (0, π) such that K(x, ξ) = α(x)β(ξ) for each (x, ξ) ∈ [0, π] 2
and suppose that α is not identically zero in ω. Then, there exist three positive constants T 0 , M 0 and ς such that, for any T ∈ (0, T 0 ) and ϕ 0 ∈ L 2 (0, π), the corresponding solution ϕ of equation (1.10) verifies the observability inequality

(1.13) ϕ(T ) 2 L 2 (0,π) M 0 exp ς T T 0 ω |ϕ(t, x)| 2 dx dt.
To prove Theorem 1.2 we adopt the following strategy:

• We show that there exists a Riesz basis of L 2 (0, π) formed by generalized eigenvectors of the operator (D(A * ), A * ).

In order to do this we analyze in Section 2 the high part of the spectrum of (D(A * ), A * ), we localize the sufficiently large eigenvalues (λ n ) n N and we show that the corresponding eigenvectors (ψ n ) n N are geometrically simple and quadratically close to the orthonormal

sequence (φ n ) n N = 2 π sin(nx) n N : n N ψ n -φ n 2 L 2 (0,π) < ∞.
From [9, Theorem 1] we know that there exist a number N H N and generalized eigenvectors (

ψ n ) 1 n N H -1 of the operator (D(A * ), A * ) such that ( ψ n ) 1 n N H -1 ∪(ψ n ) n N H forms a Riesz basis B of L 2 (0, π).
The set of the eigenvalues corresponding to the generalized eigenvectors from B will be denoted by Σ = (λ n )

1 n N L ∪ (λ n ) n N H .
We recall that (f n ) n 1 is a Riesz basis of a Hilbert space H if it is complete in H and there exist two positive constants c 1 and c 2 such that the following inequalities are verified

(1.14) c 1 n 1 |a n | 2 n 1 a n f n 2 H c 2 n 1 |a n | 2 ,
for any finite sequence of scalars (a n ) n 1 . For equivalent definitions and properties of Riesz basis the interested reader is referred to [14, Ch. 1, Sec. 8].

• By expanding the solution ϕ of (1.10) in the Riesz basis constructed above, we reduce the proof of inequality (1.13) to obtain and evaluate the norm of a biorthogonal sequence (θ λ,j ) λ∈Σ,0 j η-1 to the family of functions Λ = t j e -λt λ∈Σ,

0 j η-1 in L 2 -T 2 , T 2
, where η 1 is the maximal dimension of the root linear space corresponding to the low eigenvalues (λ n ) 1 n N L . This is done in Theorem 4.8, by adapting some ideas from [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF] and [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF]. We recall that, given a sequence (f n ) n 1 in the Hilbert space H endowed with the inner product ( • , • ), (g n ) n 1 is a biorthogonal family to (f n ) n 1 in H if the following relations are verified (1.15) (f m , g n ) H = δ mn (n, m 1).

From Theorem 1.2, we can obtain the main result of this paper concerning the local controllability of (1.1) to the stationary states:

Theorem 1.3. Assume that f S ∈ L 2 (0, π) such that (1.2) admits a solution w S ∈ H 2 (0, π) ∩ H 1 0 (0, π), w S = 0.
Let ω ⊂ [0, π] be a non empty open set such that ∂ xx w S is not identically zero in ω and let T > 0.

Then there exists c 0 > 0 such that for any w 0 ∈ H 1 0 (0, π) with

w 0 -w S H 1 0 (0,π) c 0 ,
there exists a control u ∈ L 2 (0, T ; L 2 (0, π)) such that the solution w of (1.1) satisfies

w(T ) = w S .
In the above result, we assume that (1.2) admits a solution for some f S . Note that for an arbitrary f S ∈ L 2 (0, π), (1.2) may not have a solution. Nevertheless, it is easy to construct an infinite number of solutions of (1.2), even with the restriction that ∂ xx w S does not cancel in (0, π).

Notice that our controllability result holds for initial data in H 1 0 (0, π). This is due to the fact that the solution w of (1.1) in Theorem 1.3 should be a strong solution. We obtain such a solution by a fixed point argument and we need in particular that w 0 ∈ H 1 0 (0, π) and that w S ∈ H 2 (0, π)∩H 1 0 (0, π) to handle the nonlinear terms. Let us remark that even the well-posedness for weak solutions of (1.1) (without control) is not an easy issue. Indeed, the compactness or completeness methods can not be applied directly with a regularity such as w ∈ L ∞ (0, T ; L 2 (0, π))∩ L 2 (0, T ; H 1 0 (0, π)) because of the nonlinear term ν( ∂ x w 2 L 2 (0,π) ). The outline of the paper is the following: in Section 2, we describe the main spectral properties of the linear operator A * defined by (1.9). In Section 3, we give the spectral decomposition of the solutions of the linearized system (1.10) in terms of a Riesz basis formed from generalized eigenvectors of A * . Section 4 is devoted to the construction and the evaluation of a biorthogonal family to the set of functions Λ = t j e -λt λ∈Σ, 0 j η-1 . This allows us to prove in Section 5 the observability inequality given by Theorem 1.2. Finally, Section 6 is devoted to the proof of our main result Theorem 1.3.

Spectral analysis

The aim of this section is to give a complete description of the high part of the spectrum of the operator (D(A * ), A * ). First of all we have the following quite general result.

Theorem 2.1. The operator (D(A * ), A * ) defined by (1.7) is an unbounded linear operator in L 2 (0, π) with compact resolvent. Its spectrum σ(A * ) consists of a sequence of isolated complex eigenvalues. To each eigenvalue λ ∈ σ(A * ) corresponds a finite dimensional root linear space G λ (the space of generalized eigenvectors). Moreover, if D 0 = K L 2 ((0,π) 2 ) + q L ∞ (0,π) , then we have that

(2.1) σ(A * ) = {λ ∈ C : | (λ)| D 0 , (λ) -D 0 } .
Proof. The fact that (D(A * ), A * ) has compact resolvent follows from the compact embedding of H 2 (0, π) into L 2 (0, π). The classical spectral theory of compact operators ensures that the spectrum σ(A * ) consists of isolated complex eigenvalues and the root linear space G λ corresponding to each eigenvalue λ is of finite dimension.

Let us now show that (2.1) holds. Given λ ∈ σ(A * ) there exists a function u ∈ D(A * ) with u L 2 (0,π) = 1 such that

(2.2) -∂ xx u + π 0 K( • , ξ)u(ξ) dξ + qu = λu.
Multiplying (2.2) by u and integrating by parts we obtain that (2.3)

π 0 |∂ x u(x)| 2 dx + π 0 π 0 K(x, ξ)u(ξ)u(x) dξ dx + π 0 q(x)|u(x)| 2 dx = λ.
By taking the imaginary part of (2.3) we get that

| (λ)| = π 0 π 0 K(x, ξ)u(ξ)u(x) dξ dx + π 0 q(x)|u(x)| 2 dx K L 2 ((0,π) 2 ) + q L ∞ (0,π) ,
and the first relation in (2.1) follows. By taking the real part of (2.3) we get that

(λ) = π 0 |∂ x u(x)| 2 dx + π 0 π 0 K(x, ξ)u(ξ)u(x) dξ dx + π 0 q(x)|u(x)| 2 dx -K L 2 ((0,π) 2 ) + q L ∞ (0,π) ,
and the second relation in (2.1) follows. The proof of the theorem is complete.

We pass to localize and describe the high eigenvalues of the operator (D(A * ), A * ). In the sequel, given a ∈ C and r 0, we denote by B(a, r) the ball in the complex plane of center a and radius r. We localize the eigenvalues of the unbounded operator (D(A * ), A * ) by using a strategy similar to the one presented in [START_REF] Pöschel | Inverse spectral theory[END_REF]Chapter 2] which combines the shooting method and the Rouché Theorem. Similar ideas have been employed to analyze the spectrum of several differential operators (see, for instance, [START_REF] Benaddi | Energy decay rate of wave equations with indefinite damping[END_REF][START_REF] Cox | The rate at which energy decays in a damped string[END_REF][START_REF] Zhang | Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential[END_REF]). In order to do that let us define the map

(2.4) G : C → C, G(µ) = v(π),
where v is the unique solution of the initial value problem (2.5)

   -∂ xx v(x) -µv(x) = 0 x ∈ (0, π) v(0) = 0 ∂ x v(0) = 1.
We have the following immediate result.

Proposition 2.2. The value µ ∈ C is a root of the function G given by (2.4) if and only if it is an eigenvalue of the one dimensional Laplace operator (D( A), A),

(2.6) D( A) = H 2 (0, π) ∩ H 1 0 (0, π), Au = -∂ xx u, i.e. there exists n ∈ N * such that µ = n 2 . Now, given D > 0 and M > D 2 , let us set (2.7) ∆ M,D = {z ∈ C : (z) M, | (z)| D}.
Remark 2.3. For our future computations we need to give some estimates of √ µ in the case when

µ ∈ ∆ M,D . If µ ∈ ∆ M,D
, it is easy to see that

(2.8) ( √ µ) √ M , | ( √ µ)| D 2 √ M .
Notice that, here and in the sequel, √ µ represents the principal branch of the square root function

such that √ µ ∈ R + , if µ ∈ R + .
We define the map (2.9)

F : ∆ D,M → C, F (µ) = z(π),
where z is the unique solution of the initial value problem (2.10)

       -∂ xx z(x) -µz(x) + π 0 K(x, ξ)z(ξ) dξ + q(x)z(x) = 0 x ∈ (0, π) z(0) = 0 ∂ x z(0) = 1.
Given D > 0, let us show that there exists M > D 2 such that F is well-defined in ∆ M,D , i.e. equation (2.10) has a unique solution z ∈ H 2 (0, π) for each µ ∈ ∆ M,D . We have the following result.

Proposition 2.4. Let us set

(2.11) C K,q := 2πe π K L 2 ((0,π) 2 ) + q L ∞ (0,π) = 2πe π D 0 .
Given D > 0, K ∈ L 2 ((0, π) 2 ) and q ∈ L ∞ (0, π), let M > 0 be such that

(2.12) max {D, C K,q } < √ M .
Then, for each µ ∈ ∆ M,D , the integro-differential equation (2.10) has a unique solution z ∈ H 2 (0, π) which verifies the following variation of constants formula

z(x) = 1 √ µ sin( √ µx) + 1 √ µ x 0 sin( √ µ(x -s)) π 0 K(s, ξ)z(ξ) dξ + q(s)z(s) ds (x ∈ (0, π)), (2.13)
and the following estimate

(2.14) z L ∞ (0,π) 2 |µ| e π 2 .
Proof. Given f ∈ L 2 (0, π) and µ ∈ ∆ M,D , let us consider the nonhomogeneous equation

(2.15)    -∂ xx v(x) -µv(x) = f (x) x ∈ (0, π) v(0) = 0 ∂ x v(0) = 1,
and remark that its unique solution v ∈ H 2 (0, π) is given by the following formula

(2.16) v(x) = 1 √ µ sin( √ µx) - 1 √ µ x 0 sin( √ µ(x -s))f (s)ds (x ∈ (0, π)).
With this in mind, we define the map (2.17)

z ∈ L 2 (0, π) → Lz ∈ L 2 (0, π), Lz(x) = 1 √ µ sin( √ µx) + 1 √ µ x 0 sin( √ µ(x -s)) π 0 K(s, ξ)z(ξ) dξ + q(s)z(s) ds.
We remark that z verifies (2.13) if and only if it is a fixed point of the operator L. We show that, by choosing M as in (2.12), L becomes a contraction in L 2 (0, π) for each µ ∈ ∆ M,D . Indeed, given z 1 , z 2 ∈ L 2 (0, π) and taking into account (2.8), we obtain that

Lz 1 -Lz 2 L 2 (0,π) π |µ| e π| ( √ µ)| K L 2 ((0,π) 2 ) + q L ∞ (0,π) z 1 -z 2 L 2 (0,π) π M e πD 2 √ M K L 2 ((0,π) 2 ) + q L ∞ (0,π) z 1 -z 2 L 2 (0,π) . (2.18) Since (2.12) implies that (2.19) π M e πD 2 √ M K L 2 ((0,π) 2 ) + q L ∞ (0,π) 1 4π < 1, from (2.18) it follows that L is a contraction in L 2 (0, π) for each µ ∈ ∆ M,D .
Hence, in this case, there exists a unique z ∈ L 2 (0, π) which verifies (2.13).

It is easy to see that, if z ∈ L 2 (0, π) verifies (2.13), with K ∈ L 2 ((0, π) 2 ) and q ∈ L ∞ (0, T ), then in fact z ∈ H 2 (0, π) and it is the unique solution of (2.10). Hence, it remains to prove (2.14). We have that, for each x ∈ (0, π), the following estimate holds

|z(x)| 1 |µ| e π| ( √ µ)| + π |µ| e π| ( √ µ)| K L 2 ((0,π) 2 ) + q L ∞ (0,π) z L ∞ (0,π) . (2.20)
Since (2.12) implies that, for any µ ∈ ∆ M,D , we have

π |µ| e π| ( √ µ)| K L 2 ((0,π) 2 ) + q L ∞ (0,π) π √ M e πD 2 √ M K L 2 ((0,π) 2 ) + q L ∞ (0,π) 1 2 ,
from (2.20) and (2.8) we deduce that (2.14) holds true. The proof of the proposition is now complete.

Remark 2.5. If |µ| is not sufficiently large, equation (2.10) may have multiple solution. For instance, in the case q ≡ 0, we have the following result.

Proposition 2.6. Assume that µ ∈ C * and let α, β ∈ L 2 (0, π) be such that

(2.21) √ µ = π 0 β(x) x 0 sin( √ µ(x -s))α(s) ds dx,
where √ µ is a square root of µ. Then for any ϑ ∈ C, w given by

(2.22) w(x) = ϑ √ µ x 0 sin( √ µ(x -s))α(s) ds, is a solution of (2.23)    -∂ xx w(x) + π 0 K(x, ξ)w(ξ) dξ = µw(x) x ∈ (0, π) w(0) = ∂ x w(0) = 0 with K(x, ξ) = α(x)β(ξ) (degenerate kernel).
In particular, for any µ ∈ C * , there exists K ∈ L 2 ((0, π) 2 ) such that (2.23) has nontrivial solutions. Proof of Proposition 2.6. If ϑ denotes the number π 0 β(ξ)w(ξ) dξ, we have that the solution w of (2.23) is given by (2.22). Thus, ϑ is solution of

(2.24) ϑ = ϑ √ µ π 0 β(x) x 0 sin( √ µ(x -s))α(s) ds dx.
From (2.24) we deduce that there exists two possibilities:

1. Relation (2.21) does not hold. In this case ϑ = 0 and the unique solution of (2.23) is the trivial one.

2. Relation (2.21) holds. In this case any w given by (2.22) with ϑ ∈ C is a solution of (2.23) and the space of solutions of (2.23) is of dimension one.

We notice that, given any µ ∈ C * and α ∈ L 2 (0, π), α ≡ 0, there exists β ∈ L 2 (0, π) such that (2.21) holds.

Remark 2.8. We can show that, in the case of a degenerate kernel K(x, ξ) = α(x)β(ξ) with α, β ∈ L 2 (0, π), the geometric multiplicity of any eigenvalue λ of the operator (D(A * ), A * ) is at most two. Indeed, let λ ∈ C be an eigenvalue of the operator (D(A * ), A * ), i.e. there exists

ϕ ∈ D(A * ) with ϕ L 2 (0,π) = 1 such that (2.25)    -∂ xx ϕ(x) + π 0 K(x, ξ)ϕ(ξ) dξ + q(x)ϕ(x) = λϕ(x) x ∈ (0, π) ϕ(0) = ϕ(π) = 0.
Also, let φ 1 and φ 2 be the unique solutions of the equations

(2.26) -∂ xx φ 1 (x) + q(x)φ 1 (x) = λφ 1 (x) x ∈ (0, π) φ 1 (0) = 1, ∂ x φ 1 (0) = 0, and
(2.27) -∂ xx φ 2 (x) + q(x)φ 2 (x) = λφ 2 (x) x ∈ (0, π) φ 2 (0) = 0, ∂ x φ 2 (0) = 1,
respectively. Any solution of the non homogeneous problem

(2.28) -∂ xx φ(x) + q(x)ϕ(x) = λφ(x) + f (x) x ∈ (0, π) φ(0) = a, ∂ x φ(0) = b,
can be written as follows

(2.29) φ(x) = aφ 1 (x) + bφ 2 (x) - x 0 H(x, s)f (s) ds (x ∈ (0, π)),
where

H(x, s) = φ 1 (s)φ 2 (x) -φ 1 (x)φ 2 (s) φ 1 (s)∂ x φ 2 (s) -∂ x φ 1 (s)φ 2 (s)
.

From this formula, we obtain that any eigenfunction ϕ verifying (2.25) is of the following form

(2.30) ϕ(x) = ∂ x ϕ(0)φ 2 (x) + ϑ x 0 H(x, s)α(s) ds (x ∈ (0, π)),
where ϑ = π 0 β(ξ)ϕ(ξ) dξ. By taking into account the above considerations we have that the following two cases are possible for the eigenspace

E λ = {v ∈ D(A * ) : A * v = λv} corresponding to λ: 1. If 1 = π 0 β(x)
x 0 H(x, s)α(s) ds dx, then the eigenspace E λ is one dimensional and it is generated by the unique solution of the equation H(π, s)α(s) ds = 0 and φ 2 (π) = 0 (i.e. λ is an eigenvalue of the operator -∂ xx + q) then the space E λ is two dimensional and it is generated by the functions

(2.31)    -∂ xx w(x) + π 0 K(x, ξ)w(ξ) dξ + q(x)w(x) = λw(x) x ∈ (0, π) w(0) = 0, ∂ x w(0) = 1 2. If 1 = π 0 β(x)
φ 2 (x), x 0 H(x, s)α(s) ds. (b) Otherwise, the eigenspace E λ is one dimensional.
The following immediate property of the function F is very important in the sequel and justifies the introduction of F . Proposition 2.9. Assume that (2.12) holds true. Then the value µ ∈ ∆ M,D is a root of the function F if and only if it is an eigenvalue of the operator (D(A * ), A * ).

According to Proposition 2.9, if we want to find the eigenvalues of the operator (D(A * ), A * ) from ∆ M,D , it is sufficient to look for the roots of F in this domain. We shall localize the large roots of F by using the Rouché Theorem. Firstly, we have to prove the following three lemmas.

Lemma 2.10. Given D > 0 and M satisfying (2.12), the following estimate holds

(2.32) |F (µ) -G(µ)| C K,q |µ| (µ ∈ ∆ M,D ).
Proof. We have that

F (µ) -G(µ) = z(π) -v(π) := w(π),
where w is the unique solution of (2.33)

   -∂ xx w(x) -µw(x) + π 0 K(x, ξ)z(ξ) dξ + q(x)z(x) = 0 x ∈ (0, π) w(0) = ∂ x w(0) = 0.
Since the solution of (2.33) is given by

w(x) = 1 √ µ x 0 sin( √ µ(x -s)) π 0 K(s, ξ)z(ξ) dξ + q(s)z(s) ds (x ∈ (0, π)), it follows that (2.34) |w(x)| π |µ| e π| ( √ µ)| K L 2 ((0,π) 2 ) + q L ∞ (0,π) z L ∞ (0,π) (x ∈ (0, π)).
From (2.14) and (2.34) it follows that

|w(π)| 2π |µ| e πD 2 √ M + π 2 K L 2 ((0,π) 2 ) + q L ∞ (0,π) .
Since M > D 2 , the above inequality implies that (2.32) holds.

Given δ > 0, for each n ∈ N, we define the circles

(2.35) Γ n (δ) = z ∈ C : |z -n 2 | = δ = ∂B(n 2 , δ).
Also, we define the set

(2.36) S(δ) = z ∈ ∆ M,D : |z -n 2 | δ, ∀n 0 .
Lemma 2.11. Given δ > 0, D > 0 and M satisfying (2.12), the function G defined by (2.4) verifies the following inequality

(2.37) |G(µ)| 2δ 3|µ| µ ∈ S(δ), |µ| 1 2 .
Proof. We have that G(µ) = v(π) = 1 √ µ sin(π √ µ) and the problem is reduced to evaluate sin(π √ µ)

in S(δ). Let µ ∈ S(δ) and let us set that

√ µ = a + ib with b ∈ R and a ∈ R + . If n a ∈ N is a number which verifies (2.38) |a -n a | 1 2 ,
we have that

| sin( √ µπ)| 2 = | sin((a + ib)π)| 2 = 1 4 e 2bπ + e -2bπ -2 cos(2aπ) b 2 π 2 + sin 2 (aπ) = b 2 π 2 + sin 2 ((a -n a )π). (2.39)
Inequality (2.38), together with the facts that µ ∈ ∆ M,D (see (2.8)) and M > D 2 , implies that

| √ µ -n a | 2 = |a -n a | 2 + |b| 2 1 4 + D 2 √ M 2 1 2
.

Since, for | √ µ| 1 √ 2 , we have that | √ µ + n a | | √ µ -n a | + 2| √ µ| 1 √ 2 + 2| √ µ| 3| √ µ|, from (2
.39) and (2.38) we deduce that, for any µ ∈ S(δ), we have that

| sin( √ µπ)| 2 4b 2 + 4(a -n a ) 2 = 4 | √ µ -n a | 2 4δ 2 √ µ + n a 2 4δ 2 9 |µ| .
From the last inequality it follows immediately that (2.37

) is verified if | √ µ| 1 √
2 and the proof ends.

Lemma 2.12. Given D > 0 and M satisfying (2.12), the functions G : C → C and F : ∆ M,D → C, given by (2.4) and (2.9) respectively, are analytic in their domains of definition.

Proof. Since G(µ) = 1 √ µ sin( √ µπ), it follows immediately that G is an entire function. Let us prove the property for F . We recall that, according to Proposition 2.4, for each µ ∈ ∆ M,D , there exists a unique solution z = z( • , µ) ∈ H 2 (0, π) of equation (2.13). Let us define the Picard's recurrent sequence of functions (2.40)

                   z 0 (x, µ) = 1 √ µ sin( √ µx), z n+1 (x, µ) = z 0 (x, µ) + x 0 sin( √ µ(x -s)) √ µ π 0 K(s, ξ)z n (ξ, µ) dξ + q(s)z n (s, µ) ds, (n 0).
We have that

(2.41) (z n+1 -z n )( • , µ) L ∞ (0,π) π |µ| e | ( √ µ)|π K L 2 ((0,π) 2 ) + q L ∞ (0,π) (z n -z n-1 )( • , µ) L ∞ (0,π) .
By taking into account (2.8) and (2.12), it follows from (2.41) that

(2.42) (z n+1 -z n )( • , µ) L ∞ (0,π) 1 2 (z n -z n-1 )( • , µ) L ∞ (0,π) .
Hence, the series n 0 (z n+1 -z n )( • , µ) converges in C[0, π] and since

z 0 ( • , µ) L ∞ (0,π) 1 √ M e π| √ µ| 1 √ M e π 2 ,
this convergence is uniformly in µ ∈ ∆ M,D . Since

k n=0 (z n+1 -z n )( • , µ) = z k+1 ( • , µ) -z 0 ( • , µ),
it follows that the sequence (z n ( • , µ)) n 0 converges in C[0, π] as n tends to infinity, uniformly in µ ∈ ∆ M,D . In fact, the limit of the sequence (z n ( • , µ)) n 0 is the unique solution z( • , µ) of (2.13) encountered in Proposition 2.4. It follows that the sequence (z n (π, µ)) n 0 converges to z(π, µ) as n tends to infinity, uniformly in µ ∈ ∆ M,D . Since each z n depends analytically of µ in ∆ M,D , it follows that F (µ) = z(π, µ) is analytic in ∆ M,D and the proof of the lemma is now complete.

We have now all the ingredients needed to apply the Rouché Theorem and to localize the zeros of F from the region ∆ M,D . In the sequel • denotes the floor function.

Theorem 2.13. There exists δ > 0, such that for any D > δ and N max{ 2δ , D + 1}, there exists M satisfying (2.12) and having the following property: for each n N + 1, the function F has a unique root λ n inside the circle Γ n (δ) defined by (2.35). Moreover, the roots of the function

F in ∆ M,D are exactly (λ n ) n N +1 .
Proof. We recall that C K,q is defined by (2.11). We choose δ =

3C K,q 2
+ 1 and let D > δ. From the choice of N we deduce that (N + 1) 2 -δ > N 2 + δ and we can take M such that

(N + 1) 2 -δ > M > N 2 + δ.
From the above relations we have, in particular, that M satisfies (2.12) and Γ n (δ) ⊂ ∆ M,D

(n N + 1).

From Lemmas 2.10, 2.11 and the choice of δ it follows that, for each n N + 1, we have that

(2.43) |F (µ) -G(µ)| C K,q |µ| < 2δ 3|µ| |G(µ)| (µ ∈ Γ n (δ)).
Moreover, Lemma 2.12 ensures that F and G are analytic functions in ∆ M,D . From the Rouché Theorem we deduce that, for each n N + 1, there exists a simple root λ n of F inside the contour Γ n (δ) ⊂ ∆ M,D . On the other hand, by using estimate (2.37) from Lemma 2.11 it follows that

(2.44) |F (µ) -G(µ)| < |G(µ)| (µ ∈ S(δ)).
This implies that there are no other roots of the function F in ∆ M,D .

We pass to prove some properties of the associated eigenvectors.

Lemma 2.14. Under the hypothesis of Theorem 2.13, for each n N + 1, let φ n (x) = 2 π sin(nx) and ψ n (x) = 2 π nz(x, λ n ), where z( • , λ n ) is the solution of (2.10) with µ = λ n . The following properties are verified 1. There exist two positive constants ρ 1 and ρ 2 such that

(2.45) ρ 1 ψ n L 2 (0,π) ρ 2 (n N + 1);
2. There exists a positive constant ρ 3 such that

(2.46) ψ n -φ n L 2 (0,π) ρ 3 n (n N + 1).
3. If ω is a non empty open subset of [0, π], then there exists a positive constant ρ 4 such that

(2.47) ω |ψ n (x)| 2 dx ρ 4 (n N + 1).
Proof. We remark that φ n is the solution of the equation

(2.48)      -∂ xx φ n (x) -n 2 φ n (x) = 0 x ∈ (0, π) φ n (0) = φ n (π) = 0 ∂ x φ n (0) = 2 π n,
whereas ψ n is the solution of the equation

(2.49)          -∂ xx ψ n (x) -λ n ψ n (x) + π 0 K(x, ξ)ψ n (ξ) dξ + q(x)ψ n (x) = 0 x ∈ (0, π) ψ n (0) = ψ n (π) = 0 ∂ x ψ n (0) = 2 π n.
To prove (2.45), we use (2.49) to deduce that

ψ n (x) = φ n (x) + 1 n x 0 sin(n(x -s)) (n 2 -λ n )ψ n (s) + π 0 K(s, ξ)ψ n (ξ) dξ + q(s)ψ n (s) ds. (2.50)
From (2.50) and the fact that |n 2 -λ n | δ it follows that

ψ n L 2 (0,π) -φ n L 2 (0,π) π n δ + K L 2 ((0,π) 2 ) + q L ∞ (0,π) ψ n L 2 (0,π) .
From the last estimate, by using that φ n L 2 (0,π) = 1 and that, for n max{N + 1, 4πδ},

π δ + K L 2 ((0,π) 2 ) + q L ∞ (0,π) < 2πδ n 2 ,
we obtain that (2.45) holds with

ρ 1 = min 2 3 , min N +1 n 4πδ ψ n L 2 (0,π) , ρ 2 = max 2, max N +1 n 4πδ ψ n L 2 (0,π) .
Let us pass to prove (2.46). If we put ζ n = ψ n -φ n , it follows that ζ n verifies the following relation

ζ n (x) = 1 n x 0 sin(n(x -s)) π 0 K(s, ξ)ψ n (ξ) dξ + q(s)ψ n (s) + (n 2 -λ n )ψ n (s) ds. (2.51)
Since, according to Theorem 2.13, |λ n -n 2 | δ, we deduce from (2.51) that

ζ n L 2 (0,π) π n K L 2 ((0,π) 2 ) + q L ∞ (0,π) + δ ψ n L 2 (0,π) . (2.52)
From (2.52) we deduce, by taking into account the second inequality in (2.45), that (2.46) holds true with ρ 3 = 2πδρ 2 .

On the other hand, from (2.46), we deduce that

ω |ψ n (x)| 2 dx 1 π ω | sin(nx)| 2 dx - ω ψ n (x) - 2 π sin(nx) 2 dx 1 π ω | sin(nx)| 2 dx - ρ 2 3 n 2 ,
from which we deduce that there exists N 1 N + 1 and a constant ρ > 0 such that (2.53)

ω |ψ n (x)| 2 dx ρ (n N 1 ).
Relation (2.53) follows with

ρ 3 = min ρ , min N +1 n N1 ω |ψ n (x)| 2 dx
and the proof of the lemma is complete.

To solve our control problem we shall also need the following unique continuation principle for the solutions of equation (2.10).

Lemma 2.15. Suppose that the kernel K is degenerate, i.e. there exist two functions α, β ∈ L 2 (0, π) Since α is not identically zero in (a, b) it follows that π 0 β(ξ)z(ξ) dξ = 0. Let x 0 ∈ (a, b) and notice that z(x 0 ) = ∂ x z(x 0 ) = 0. From the uniqueness of solutions of the equation

such that K(x, ξ) = α(x)β(ξ) for each (x, ξ) ∈ [0, π] 2 . Let µ ∈ C and let z ∈ H 2 (0, π) be any solution of equation (2.54) -∂ xx z(x) + π 0 K(x, ξ)z(ξ) dξ + q(x)z(x) = µz(x) (x ∈ (0, π)),
-∂ xx z(x) + (q(x) -µ)z(x) = 0 z(x 0 ) = ∂ x z(x 0 ) = 0,
in (0, x 0 ) and (x 0 , π), we deduce that z is identically zero in [0, π] and the proof ends.

Remark 2.16. The degeneracy hypothesis of the kernel K is verified in the case of our model (1.6). It seems that, for kernels that are not degenerate, additional conditions should be imposed in order to ensure that the unique continuation property holds. For instance, in [START_REF] Fernández-Cara | Null controllability of linear heat and wave equations with nonlocal spatial terms[END_REF] an assumption of analyticity of the kernel is used in order to solve a multidimensional control problem.

Remark 2.17. The hypothesis that α is not identically zero in (a, b) is necessary since it is easy to construct in this case a nonzero function z verifying (2.54) which vanishes in (a, b).

Let us summarize the most important results concerning the high part of the spectrum of the operator (D(A * ), A * ) in the following theorem.

Theorem 2.18. There exist N ∈ N and δ > 0 such that, for each n N + 1, σ(A * ) ∩ B(n 2 , δ) is reduced to one element that we denote by λ n and

(2.55) σ(A * ) \ B(0, (N + 1) 2 -δ) = {λ n , n N + 1}.
Moreover, for any n N + 1, the eigenvalue λ n is geometrically simple and there exists an eigenvector ψ n of A * corresponding to λ n such that

(2.56) ψ n -φ n L 2 (0,π) ρ n ,
where φ n (x) = 2 π sin(nx) and ρ is a positive constant independent of n.

Proof. Let D 0 > 0 be the constant from Theorem 2.1. Also, let D, M , N ∈ N and δ > 0 be given by Theorem 2.13 with D D 0 . According to Proposition 2.9 and Theorem 2.13 there exists a a unique eigenvalue λ n of the operator (D(A * ), A * ) in each ball B(n 2 , δ) for n N + 1 and there are no others eigenvalues of this operator in ∆ M,D . By taking into account relations (2.1) from Theorem 2.1, we obtain that (2.55) is verified, too. Due to the uniqueness of solutions to problem (2.10) proved in Proposition 2.4, if n N + 1, to each λ n corresponds a one dimensional eigenspace generated by the function ψ n defined in Lemma 2.14. Finally, estimate (2.56) for ψ n follows from (2.46) and the proof of the theorem is complete.

Riesz basis

The analysis of the eigenfunctions (ψ n ) n N +1 in the previous section and a result from [9, Theorem 1] allow us to obtain a Riesz basis of L 2 (0, π). Theorem 3.1. Let N be the entire number and (ψ n ) n N +1 be the eigenvectors given by Theorem 2.18. There exist N H , N L ∈ N, with N H > max{N + 1, N L } and a Riesz basis B of L 2 (0, π) such that

(3.1) B = (ψ k ) k N H ∪ ( ψ k ) 1 k N H -1 ,
where

( ψ k ) 1 k N H -1 are generalized eigenvectors of the operator (D(A * ), A * ) corresponding to N L different eigenvalues of A * , λ 1 , . . . , λ N L , satisfying (3.2) |λ n | < |λ N H | (1 n N L ).
Proof. The result follows immediately from [9, Theorem 1], by taking into account that (D(A * ), A * ) is a densely defined operator with compact resolvent in L 2 (0, π), (φ n ) n 1 is an orthonormal basis of L 2 (0, π) and (ψ k ) k N +1 verify (2.56).

Let us use Theorem 3.1 in order to expand the functions from L 2 (0, π) and to write the solutions of (1.10). Firstly, let us denote the family of distinct eigenvalues corresponding to all the eigenvectors (ψ k ) k N H and generalized eigenvectors ( ψ k ) 1 k N H -1 from Theorem 3.1 by

(3.3) Σ = Σ L ∪ Σ H = (λ k ) 1 k N L ∪ (λ k ) k N H ⊂ C.
We recall that, according to Theorem 2.18, if k N H , the element ψ k from the basis B generates the one dimensional eigenspace corresponding to λ k . On the other hand, for any 1 k N L , let us consider a Jordan basis of the root space of A * associated to the eigenvalue λ k :

(3.4) ψ k,j,m 1 k N L , 1 j d k , 0 m r k,j -1.
More precisely, for a given k, {ψ k,j,m } is a family of generalized eigenvectors of A * associated with the eigenvalue λ k with the following property

ψ k,j,0 ∈ ker(A * -λ k I), (A * -λ k I)ψ k,j,m = -ψ k,j,m-1 1 m r k,j -1, 1 j d k ,
and the family in (3.4) is linearly independent. Note that ker(A * -λ k I) = Span{ψ k,j,0 , 1 j d k }.

We remark that not all the generalized eigenvectors (ψ k,j,m ) 1 k N L , 1 j d k , 0 m r k,j -1 belong necessarily to the Riesz basis B given by Theorem 3.1. However, in order to simplify the notation, we shall write that any function ϕ 0 ∈ L 2 (0, π) can be uniquely written as

(3.5) ϕ 0 = N L k=1 d k j=1 r k,j -1 m=0 a k,j,m ψ k,j,m + ∞ k=N H a k ψ k ,
where in (3.5) we shall consider that a k,j,m = 0 if ψ k,j,m / ∈ B. Now, if ϕ 0 ∈ L 2 (0, π) is given by (3.5), then the corresponding solution ϕ of (1.10) can be written under the following form

(3.6) ϕ(t) = N L k=1 d k j=1 r k,j -1 m=0 a k,j,m m s=0 t s s! ψ k,j,m-s e -λ k t + ∞ k=N H a k ψ k e -λ k t .
We shall need to rewrite the solution ϕ in a slightly different way. Let us set for k = 1, . . . , N L , ( Then we can rewrite (3.6) as

(3.9) ϕ(t) = N L k=1 R k -1 s=0 t s e -λ k t   R k -s-1 p=0 R k -p-1 m=s j∈I k,p 1 s! a k,j,m ψ k,j,m-s   + ∞ k=N H a k ψ k e -λ k t .
For any k ∈ {1, . . . , N L }, the coefficient corresponding to

t R k -1 e -λ k t is j∈I k,0 1 (R k -1)! a k,j,R k -1 ψ k,j,0 , the coefficient corresponding to t R k -2 e -λ k t is j∈I k,0 ∪I k,1 1 (R k -2)! a k,j,R k -2 ψ k,j,0 + j∈I k,0 1 (R k -2)! a k,j,R k -1 ψ k,j,1 ,
and so on. We shall use these expressions of the solution ϕ of (1.10) in order to prove our observability inequality for this equation. Note that in (3.9) appears the family of functions

t s e -λ k t 1 k N L , 0 s R k -1 ∪ e -λ k t k N H .
Given T > 0, we need to construct and estimate a biorthogonal family in L 2 (0, T ) to this family.

In fact, we can obtain a slightly more general result and we start with the family

Λ = t s e -λt λ∈Σ, 0 s η-1 ,
where η 1 is a given integer. The following section is devoted to the construction and evaluation of a biorthogonal sequence to Λ in L 2 (0, T ). Let us remark that such a construction has been already done in similar contexts by [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF]. However, since we need to estimate the dependence of the norm of the biorthogonal sequence with respect to small T , we shall adopt a different and more explicit approach which is closer to the one used by [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF], even though in [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF] the exponents λ are purely real and η = 1.

The biorthogonal family

Let Σ be defined by (3.3) and η ∈ N * . Before starting the construction of a biorthogonal sequence to the family Λ = t s e -λt λ∈Σ, 0 s η-1 , for reader's convenience, let us enumerate some simple properties of the set Σ which will be used in this section.

(4.1) |λ -λ | γ (λ, λ ∈ Σ, λ = λ ).
P2) There exists a constant δ ∈ (0, N H ) such that

(4.2) |λ n -n 2 | δ (n N H ), (4.3) | (λ N H ) -(λ n )| δ (1 n N L ).
P3) For each 1 n N L we have that

(4.4) |λ n | < |λ N H |.
P4) The sequence (λ n ) n N H verifies the following properties

(a) | (λ n ) -n 2 | C, n N H , (b) (λ n+1 ) -(λ n ) δ, n N H . P5) We have that (4.5) λ∈Σ H 1 |λ| < ∞.
Proof. The existence of γ > 0 verifying (4.1) follows from the localization of the elements of Σ H given by Theorem 2.18 and the fact that Σ L has only a finite number of different eigenvalues. On the other hand, with δ > 0 given by Theorem 2.13, both (4.2) and (4.3) are verified. Indeed, since 2δ

< N + 1 < N H , we deduce that dist (Γ N H (δ), Γ n (δ)) δ (1 n N L ),
which gives (4.3). Property (4.4) follows from Theorem 3.1 and the last properties are direct consequences of Theorem 2.18.

For each λ ∈ Σ, let us consider the function Φ λ : C → C defined by

(4.6) Φ λ (z) = λ ∈Σ\{λ} 1 - z λ -λ .
Some of the basic properties of Φ λ are described in the following lemma.

Lemma 4.2. The function Φ λ is an entire function of arbitrarily small exponential type. Moreover, there exist three positive constants B 0 , C 1 and C 2 , depending at most upon γ, N L , N H and δ, such that we have, uniformly with respect to λ,

(4.7) |Φ λ (z)| C 1 e π √ |z| (1 + |z|) B0 (z ∈ C), (4.8) Φ λ (ix -λ) C 2 e π √ |x| (1 + | (λ)| + |x|) B0 (x ∈ R).
Proof. Firstly, by using (4.1), we deduce that

|Φ λ (z)| λ ∈Σ\{λ} 1 + |z| |λ -λ| λ ∈Σ L \{λ} 1 + |z| γ λ ∈Σ H \{λ} 1 + |z| |λ -λ| max 1, 1 γ N L (1 + |z|) N L λ ∈Σ H \{λ} 1 + |z| | (λ ) -(λ)| . Now,
Φ λ (ix -λ) = λ ∈Σ\{λ} λ -ix λ -λ λ ∈Σ L \{λ} |λ + ix| γ λ ∈Σ H \{λ} λ + ix λ -λ 1 + 1 γ N L (max {1, |λ N H |}) N L (1 + |x|) N L λ ∈Σ H 1 + |x| |λ | λ ∈Σ H \{λ} |λ | |λ -λ| . (4.9)
Let us evaluate each of the last two products in (4.9). For the first product we remark that (4.10) ln

λ ∈Σ H 1 + |x| |λ | = k N H ln 1 + |x| k 2 S1 + k N H ln |λ k | + |x| k 2 + |x| S2 + k N H ln k 2 |λ k | S3 .
Now, from (4.2), we have that there exists a positive constant C such that

S 3 k N H ln |λ k | + |k 2 -λ k | |λ k | k N H ln 1 + δ |λ k | k N H δ |λ k | C, S 2 = k N H ln 1 + |λ k | -k 2 k 2 + |x| k N H ln 1 + δ k 2 + |x| k N H δ k 2 C, S 1
k 1 ln 1 + |x| k 2 = ln    k 1   1 - i |x| 2 k 2       = ln   sin i π |x| i π |x|   ln Ce π √ |x| .
The above inequalities for (S i ) 1 i 3 , together with (4.10), imply that there exists C > 0 such that (4.11)

λ ∈Σ H 1 + |x| |λ | Ce π √ |x| .
We pass now to evaluate the second product in (4.9). We have that

λ ∈Σ H \{λ} λ -λ λ λ ∈Σ H \{λ} | (λ ) -(λ)| | (λ )| P1 λ ∈Σ H \{λ} | (λ )| |λ | P2 .
We evaluate each of the products above. For the product P 1 we use property P 4) and the same arguments as in [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF]Lemma 4.1]. We deduce that there exist two positive constants B 0 > 0 and C > 0 such that (P 1 )

-1 C (1 + | (λ)|) B0 .
For the second product, by using (4.2), we deduce that there exists C > 0 such that

(P 2 ) 2 = λ ∈Σ H \{λ} |λ | 2 -( (λ )) 2 |λ | 2 k N H 1 - δ 2 (k 2 -δ) 2 C.
Hence, the last inequalities for (P i ) 1 i 2 imply that there exist B 0 , C > 0 such that (4.12)

λ ∈Σ H \{λ} λ λ -λ C (1 + | (λ)|) B0 .
From (4.9), (4.11) and (4.12) it follows that (4.8) holds and the proof of the lemma is complete.

Remark 4.3. Estimate (4.7) implies that Φ λ is a function of order not exceeding 1 2 and a type not exceeding π if or order 1 2 . We recall that an entire function of exponential type τ ∈ (0, ∞) is a function of order not exceeding 1 and a type not exceeding τ if of order 1 (see [START_REF] Boas | Entire functions[END_REF]Page 8]).

Let us define the C

∞ (R) function (4.13) σ ν := exp -ν 1-t 2 if |t| < 1 0 if |t| 1,
where ν is a positive constant and put c ν = 1/ σ ν L 1 . Given β > 0, we introduce the function

(4.14) H β (z) = c ν 1 -1 σ ν (t)e -iβtz dt.
The properties of H β we are interested in are listed in the following lemma.

Lemma 4.4. Let β > 0, > 0 and set ν = (π + ) 2 /β. The function H β defined by (4.14) is an entire function of exponential type β which verifies the following properties:

H β (0) = 1, (4.15) |H β (z)| exp(β|y|) (z = x + iy, x, y ∈ R), (4.16) |H β (x)| C 3 √ ν + 1 exp 3ν/4 -(π + /2) |x| (x ∈ R), (4.17) |H β (iy)| 1 11 √ ν + 1 exp β|y|/(2 √ ν + 1) (y ∈ R) , (4.18) |H β (x + iy)| C 4 √ ν + 1 exp β|y|/(2 √ ν + 1) y ∈ R, |x| π 4β , (4.19)
where C 3 and C 4 are two positive constants, independent of , ν and β.

Proof. From (4.14) we deduce that H β , being the Fourier transform of a function in L 2 (-β, β), is an entire function of exponential type β. Properties (4.15)-(4.18) are proved in [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF]Lemma 4.3] (see, also, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Lemma 9.2.3]). It remains to show estimate (4.19). We have that

|H β (x + iy)| c ν 1 -1 σ ν (t)e βty cos(βtx) dt . Since 0 β|x| π 4 , it follows that √ 2 2 cos(βtx) 1,
from which we deduce that (4.20)

|H β (x + iy)| c ν √ 2 2 1 -1 σ ν (t)e βty dt = √ 2 2 H β (iy).
By using (4.20) and (4.18) we deduce immediately that (4.19) holds and the proof of the lemma is complete.

For each λ ∈ Σ, let us now define the entire function

(4.21) Q λ (z) = Φ λ (iz -λ) H β (z) H β (-iλ) .
The properties of the function Q λ we are interested in are given by the following lemma.

Lemma 4.5. For each λ ∈ Σ, the function Q λ defined by (4.21) has the following properties:

1. For each λ ∈ Σ we have that Q λ (-iλ ) = 1 if λ = λ 0 if λ = λ .
2. Q λ is an entire function of exponential type β.

3. There exists β 0 > 0 such that, for each β ∈ (0, β 0 ], the following estimate holds

(4.22) |Q λ (x)| C 5 (1 + | (λ)| + |x|) B0 exp κ β - 2 |x| -β| (λ)|/(2 √ ν + 1) (x ∈ R),
where the two positive constants C 5 and κ are independent of β ∈ (0, β 0 ].

Proof

H β (-iλ) C 4 √ ν + 1 exp β| (λ)|/(2 √ ν + 1) .
From (4.8), (4.17) and (4.23) we obtain that, for any x ∈ R, we have

|Q λ (x)| = Φ λ (ix -λ) H β (x) H β (-iλ) C 2 C 3 C 4 (ν + 1)(1 + | (λ)| + |x|) B0 exp 3ν/4 -/2 |x| -β| (λ)|/(2 √ ν + 1) . (4.24) Since ν = (π+ ) 2 β
, from (4.24) we deduce that the following estimate holds true for any β ∈ (0,

β 0 ] |Q λ (x)| C 5 β (1 + | (λ)| + |x|) B0 exp 3(π + ) 2 4β - 2 |x| -β| (λ)|/(2 √ ν + 1) (x ∈ R),
where C 5 is a positive constant independent of β. The last inequality gives (4.22) by taking κ = 3(π+ ) 2

We need to modify our function Q λ given by (4.21) in order to be able to add conditions on the values of the derivatives. Lemma 4.6. Let λ ∈ Σ and η ∈ N * . If Q λ is the function defined by (4.21) then, for each 0 j η -1, there exists a polynomial function p λ,j of degree less than or equal to η -1 such that the following properties are verified

(4.25) ∀k η -1, (Q η λ p λ,j ) (k) (-iλ ) = 1 if λ = λ and j = k 0 otherwise.
Moreover, there exists a positive constant C 6 independent of λ and β ∈ (0, β 0 ), such that

(4.26) |p λ,j (z)| C 6 β η 2 /2 1 + z + iλ η e η 2 β| (λ)| (z ∈ C).
Proof. We shall explicitly construct and evaluate the polynomials p λ,j for each 0 j η -1. First, we define

(4.27) p λ,η-1 (z) = 1 (η -1)! (z + iλ) η-1 .
We remark that relations (4.25) are verified in the case j = η -1 and we have that

(4.28) |p λ,η-1 (x)| 1 (η -1)! |z + iλ| η-1 .
Generally, for 0 j η -1, we define

(4.29) p λ,j (z) = η-1 k=0 a λ,j,k k! (z + iλ) k ,
where the values (a λ,j,k ) 0 k η-1 are given recursively as follows (4.30)

                 a λ,j,k = 0 (0 k < j) a λ,j,j = 1, a λ,j,k = - k-j s=1 C s k (Q η λ ) (s) (-iλ)a λ,j,k-s (j + 1 k η -1).
Notice that (4.29) implies that a λ,j,k = p (k) λ,j (-iλ). From (4.30) it follows immediately that Q η λ p λ,j verifies (4.25). In order to prove (4.26), let us evaluate (Q η λ ) (s) (-iλ) and show that there exists a constant C > 0 such that

(4.31) (Q η λ ) (s) (-iλ) C β η/2 e βη| (λ)| (0 s η -1).
Indeed, by using the Cauchy formula, we have that

(Q η λ ) (s) (-iλ) = s! 2π i Γ R Q η λ (z) (z + iλ) s+1 dz ,
where R ∈ (0, γ), Γ R = {z ∈ C : |z + iλ| = R} and the contour integral is taken counter-clockwise.

Let us evaluate Q λ (z) on Γ R . From (4.21), (4.7), (4.16) and (4.19) it follows that, for any z ∈ Γ R of the form z = -iλ + Re iυ , υ ∈ [0, 2π], we have that

|Q λ (z)| = Φ λ (iRe iυ ) H β (-iλ + Re iυ ) H β (-iλ) C 1 C 4 (1 + R) B0 √ ν + 1e π √ R+Rβ+β| (λ)| ,
and (4.31) is proved. Now, from (4.30) and (4.31), we immediately deduce that

p (k) λ,j (-iλ) C β η 2 /2 e βη(k-j)| (λ)| (k j),
which, by taking into account (4.29), implies that

(4.32) |p λ,j (z)| C β η 2 /2 |z + iλ| j η-1-j s=0 1 (s + j)! |z + iλ| s e βηs| (λ)| (z ∈ C).
Estimate (4.32) shows that (4.26) holds and the proof of the lemma is complete.

For each λ ∈ Σ and j ∈ {0, 1, ..., η -1}, let us define the function

(4.33) G λ,j (z) = Q η λ (z)p λ,j (z) (z ∈ C),
where Q λ is given by (4.21) and p λ,j are the polynomials functions from Lemma 4.6. The following theorem studies the main properties of the function G λ,j .

Theorem 4.7. For each λ ∈ Σ and j ∈ {0, 1, ..., η -1}, the function G λ,j defined by (4.33) has the following properties:

1. For each λ ∈ Σ we have that

(4.34) (G λ,j ) (k) (-iλ ) = 1 if λ = λ and j = k 0 otherwise. (1 k η -1) .
2. G λ,j is an entire function of exponential type ηβ.

3. There exists β 0 > 0 such that, for each β ∈ (0, β 0 ], the following estimate holds

(4.35) G λ,j L 2 (R) C 7 | (λ)| + 1 exp κ β + η 2 β| (λ)| ,
where the two positive constants C 7 and κ are independent of λ, j and β ∈ (0, β 0 ].

Proof. Properties (4.34) are consequences of (4.25). Since the polynomials p λ,j have degree less than η and Q λ is an entire function of exponential type β, it follows that G λ,j is an entire function of exponential type ηβ.

|G λ,j (x)| 2 = |Q η λ (x)p λ,j (x)| 2 (C 5 C 6 ) 2 β η 2 (1 + | (λ)| + |x|) 2B0 exp 2κ β - |x| -β| (λ)|/ √ ν + 1 1 + x + iλ 2η e 2η 2 β| (λ)| (C 5 C 6 ) 2 β η 2 exp 2κ β + 2η 2 β| (λ)| c (1 + | (λ)| + |x|) 2B0+2η exp - |x| -β| (λ)|/ √ ν + 1 .
Now, we have that

(1 + | (λ)| + |x|) 2B0+2η exp - |x| -β| (λ)|/ √ ν + 1 (1 + | (λ)|) 2B0+2η exp -β| (λ)|/ √ ν + 1 (1 + |x|) 2B0+2η exp - |x| C √ ν + 1 β 2B0+2η 1 (1 + | (λ)|) 2 (1 + |x|) 2B0+2η exp - |x| ,
where we have used in the last inequality that (1 + s) 2B0+2η+2 e -s C = C(η, B 0 ) for any s 0. The conclusion follows by taking into account that

R (1 + |x|) 2B0+2η exp - |x| dx C = C(η, B 0 , )
and, for any κ > κ,

1 β η 2 exp 2κ β + 2η 2 β| (λ)| √ ν + 1 β 2B0+2η C exp 2κ β + 2η 2 β| (λ)| .
Now we have all the ingredients needed to construct our biorthogonal family. For each λ ∈ Σ and j ∈ {0, 1, ..., η -1}, we define the function θ λ,j as follows

(4.36) θ λ,j (t) = (-i) j 2π R G λ,j (x)e ixt dx.
The following result is a consequence of the properties of the function G λ,j proved in Theorem 4.7.

Theorem 4.8. Let Σ be the family of eigenvalues given by (3.3) and let η ∈ N * . There exists T 0 > 0 such that, for any T ∈ (0, T 0 ), the family of functions Λ = t j e -λt λ∈Σ, 0 j η-1 has a biorthogonal (θ λ,j ) λ∈Σ, 0 j η-1 in L 2 -T 2 , T 2 with the property that

(4.37) θ λ,j L 2 (-T 2 , T 2 ) c | (λ)| + 1 exp 2η 2 κ T + T 2 | (λ)| ,
where the two positive constants c and κ are independent of λ, j and T.

Proof. Let us chose T 0 = min {2ηβ 0 , 1}, T ∈ (0, T 0 ) and β = T 2η 2 . The Paley-Wiener Theorem and Theorem 4.7 imply that (4.36) defines a function θ λ,j which belongs to L 2 -T 2 , T 2 . Moreover, since

G λ,j (z) = 1 (-i) j T 2 -T 2 θ λ,j (t)e -izt dt (z ∈ C),
relations (4.34) imply that (θ λ,j ) λ∈Σ, 0 j η-1 is a biorthogonal family to Λ in L 2 -T 2 , T 2 . Since the Plancherel formula implies that

√ 2π θ λ,j L 2 (-T 2 , T 2 ) = G λ,j L 2 (
λ∈Σ, 0 j η-1 |a λ,j | 2 e -2T (λ) c exp 4η 2 κ T T 0 λ∈Σ, 0 j η-1
a λ,j t j e -λt 2 dt for any finite sequence of complex numbers (a λ,j ) λ∈Σ, 0 j η-1 .

Proof. Let (θ λ,j ) λ∈Σ, 0 j η-1 be the biorthogonal to Λ in

L 2 -T 2 , T 2
given by Theorem 4.8. It follows that there exists a biorthogonal θ λ,j λ∈Σ, 0 j η-1

to Λ in L 2 (0, T ) such that (4.39) θ λ,j L 2 (0,T ) c | (λ)| + 1 exp 2η 2 κ T + T | (λ)| .
Indeed, θ λ,j λ∈Σ, 0 j η-1 is given by the following formulas

(4.40) θ λ,j t + T 2 =   θ λ,j (t) + η-1 s=j+1 l j,s θ λ,s (t)   e T 2 λ t ∈ - T 2 , T 2
,
where the coefficients l j,s can be obtained recursively (4.41)

     l j,j+1 = -C j j+1 T 2 j , l j,s = -C j s T 2 j - s-j-1 r=1 C r s T 2 r l j,s-r , j + 2 s η -1.
From (4.37), (4.40) and (4.41) it follows that (4.39) holds with a constant c depending only on T 0 and η.

Let us now pass to prove (4.38). By using the orthogonality properties of (θ λ,j ) λ∈Σ, 0 j η-1 and Cauchy inequality we obtain that λ∈Σ, 0 j η-1

|a λ,j | 2 e -2T (λ) = = T 0   λ∈Σ, 0 j η-1 a λ,j e -2T (λ) θ λ,j (t)     λ∈Σ, 0 j η-1 a λ,j t j e -λt   dt    T 0 λ∈Σ, 0 j η-1 a λ,j e -2T (λ) θ λ,j (t) 2 dt    1/2    T 0 λ∈Σ, 0 j η-1 λ,j t j e -λt 2 dt    1/2   λ∈Σ, 0 j η-1 |a λ,j | 2 e -2T (λ)   1/2   λ∈Σ, 0 j η-1 e -2T (λ) θ λ,j 2 L 2 (0,T )   1/2    T 0 λ∈Σ, 0 j η-1 a λ,j t j e -λt 2 dt    1/2
.

By taking into account estimates (4.39) and since (4.5) implies that

λ∈Σ 1 (| (λ)| + 1) 2 < ∞,
it follows that (4.38) holds and the proof ends.

Observability result

Now, we have all the ingredients needed to prove the observability inequality (1.13) for the solutions of the adjoint equation (1.10).

Proof of Theorem 1.2: Let T 0 be given by Theorem 4.8 and let T ∈ (0, T 0 ). We recall that the set ψ k given by (3.5) then the corresponding solution ϕ of (1.10) is given by (3.6). Consequently, we have that

ϕ(T ) 2 L 2 (0,π) M 1   ∞ k=N H |a k | 2 e -2 (λ k )T + N L k=1 d k j=1 a k,j,r k,j -1 2 + a k,j,r k,j -2 + T 1! a k,j,r k,j -1 2 +... + a k,j,0 + T 1! a k,j,1 + T 2 2! a k,j,2 + ... + T r k,j -1 (r k,j -1)! a k,j,r k,j -1 2 e -2 (λ k )T . (5.1)
Since T < T 0 1, from (5.1) we deduce that

(5.2) ϕ(T ) 2 L 2 (0,π) M 1   N L k=1 d k j=1 r k,j -1 m=0 |a k,j,m | 2 e -2 (λ k )T + ∞ k=N H |a k | 2 e -2 (λ k )T   ,
where M 1 is a positive constant independent of T . Let us now evaluate the right hand side of (1.13). To this aim, we use the solution ϕ(t) of (1.10) written in the form (3.9). From (4.38) we obtain that

(5.3) T 0 ω |ϕ(t, x)| 2 dx dt = ω T 0 |ϕ(t, x)| 2 dt dx ce -4η 2 κ T N L k=1 e -2T (λ k ) R k -1 s=0 ω |S k,s (x)| 2 dx + ∞ k=N H |a k | 2 e -2T (λ k ) ω |ψ k (x)| 2 dx , where (5.4) S k,s (x) = R k -s-1 p=0 R k -p-1 m=s j∈I k,p 1 s! a k,j,m ψ k,j,m-s (x).
For each 1 k N L , let S k be the root space corresponding to the eigenvalues λ k of dimension N P (k) = (P2) For each 1 k N L , the map (5.6)

C N P (k) (a k,j,m ) 1 j d k , 0 m r kj -1 → R k -1 s=0 ω |S k,s (x)| 2 dx 1/2 ∈ R + , is a norm in C N P (k) .
Property (P 2) is a direct consequence of definition (5.4) and property (P 1). To prove property (P 1) we use the unique continuation principle given by Lemma 2.15. Notice that the properties of our kernel K allow us to apply this Lemma. Since

S k,R k -1 (x) = 1 (R k -1)! j∈I k,0
a k,j,0 ψ k,j,0 (x), and ψ k,j,0 ∈ ker(A * -λ k I), j ∈ I k,0 , from Lemma 2.15 we deduce that (5.7)

a k,j,R k -1 = 0 (j ∈ I k,0 ). Since S k,R k -2 (x) = 1 (R k -2)!   j∈I k,0 a k,j,R k -2 ψ k,j,0 (x) + j∈I k,0 a k,j,R k -1 ψ k,j,1 (x) + j∈I k,1 a k,j,R k -2 ψ k,j,0 (x)   ,
and ψ k,j,0 ∈ ker(A * -λ k I), j ∈ I k,0 ∪ I k,1 , from (5.7) and Lemma 2.15 we deduce that

(5.8) a k,j,R k -2 = 0 (j ∈ I k,0 ∪ I k,1
).

If we suppose that, for some 0 q R k -1, we have that (5.9)

a k,j,l = 0 (j ∈ ∪ R k -2-q s=0 I k,s , q + 1 l R k -1), then it follows that S k,q (x) = 1 q! R k -q-1 p=0 j∈I k,p
a k,j,q ψ k,j,0 (x).

Since ψ k,j,0 ∈ ker(A * -λ k I) for j ∈ ∪ R k -q-1 p=0 I k,p , by using again Lemma 2.15 we deduce that (5.10)

a k,j,q = 0 (j ∈ ∪ R k -1-q l=0 I k,l ).
Hence, we have proved by induction that (P 1) holds true. From (P 2) we deduce that, for each 1 k N L , there exists a constant c k > 0 such that, From (5.3), (5.12) and (5.13) it follows that where M 0 and ς are the constants in (1.13).

As stated in Theorem 1.1, for any z 0 ∈ X, there exists v ∈ L 2 (0, T ; U ) such that the solution of (1.8) satisfies z(T ) = 0 and v L 2 (0,T ;U ) γ(T ) z 0 X .

By using this result, one can handle the controllability of a system similar to (1.8) but with righthand side (see [START_REF] Liu | Single input controllability of a simplified fluidstructure interaction model[END_REF]). We introduce here some notation in order to state such a result. Let r ∈ (1, 2) be a constant. We consider ς 0 , ς 1 , ς * such that (6.1)

ς 1 1 r 2 - 1 4 > ς r -1 , ς 0 = ς 1 r 2 - ς r -1 , ς 0 > ς * > ς 1 4 > 0.
We define three functions in (0, T ) by (6. In order to do this we use the controllability of (1.8) from Theorem 1.2 and the fact that ρ F , ρ 0 and ρ satisfy the following relations ρ 0 (t) = ρ F (r 2 (t -T ) + T )γ((r -1)(T -t)) t ∈ T 1 -1 r 2 , T ,

ρ 0 ρ , ρ F ρ , ρ ρ 0 ρ 2 ∈ L ∞ (0, T ).
Consequently, applying Proposition 2.3 and Proposition 2.8 of [START_REF] Liu | Single input controllability of a simplified fluidstructure interaction model[END_REF], we deduce that for every z 0 ∈ D((-A) 1 2 ) and for every f ∈ F, there exists v ∈ U such that the solution z of (6.5) satisfies z ρ ∈ L 2 (0, T ; D(A)) ∩ H 1 (0, T ; X) ∩ C([0, T ]; D((-A)

2 )).

Moreover, there exists a positive constant C such that (6.6) z ρ L 2 (0,T ;D(A))∩H 1 (0,T ;X)∩C([0,T ];D((-A) C z 0 D((-A) 1/2 ) + f F .

Using the change of variables (1.5) and definition (1.7) of A, we deduce that for every y 0 ∈ H 1 0 (0, π), and for every f ∈ F, there exists u ∈ U such that the solution y of (6.7) C y 0 H 1 0 (0,π) + f F .

         ∂ t y -ν S ∂
Notice that (6.8) implies, in particular, that (6.9) y(T ) = 0.

In order to prove Theorem 1.3, we only need to show that the mapping

N : f ∈ F → F (y) ∈ F,
where F is given by (1.3), is well-defined and admits a fixed point.

In order to do this, we first notice that (6.1) and (6.2) yield (6.10)

ρ 4 ρ F C.
Using this inequality, relation (6.8) and standard properties of Sobolev spaces we find that (6.11) F (y) F C y 0 H 1 0 (0,π) + f F 2 1 + y 0 H 1 0 (0,π) + f F

2

.

Let us consider f 1 and f 2 in F. Assume that y 1 and y 2 are the solutions of (6.7) associated with f 1 and f 2 , respectively, with the initial condition y 0 and satisfying (6.8). Then some calculation and (6.10) imply (6.12)

F (y 1 ) -F (y 2 ) F C f 1 -f 2 F y 0 H 1 0 (0,π) + f 1 F + f 2 F + y 0 H 1 0 (0,π) + f 1 F + f 2 F 3 .
Estimates (6.11)-(6.12) imply that we can use the Banach fixed point theorem on a ball of F of radius y 0 H 1 0 (0,π) . If y 0 H 1 0 (0,π) is small enough, then the above estimates yield that F is a contraction and this completes the proof of the theorem.

π 0 ( 0 (∂

 00 ∂ x w S )(∂ x y) dx ∂ xx w S + w S ∂ x y + y∂ x w S = F + uχ ω , y(t, 0) = y(t, π) = 0, y(0, •) = y 0 , and(1.3) F = -y∂ x y + ν( ∂ x y + ∂ x w S 2 L 2 (0,π) ) -ν( ∂ x w S 2 L 2 (0,π) ) ∂ xx y + ν( ∂ x y + ∂ x w S 2 L 2 (0,π) ) -ν( ∂ x w S 2 L 2 (0,π) ) -2ν ( ∂ x w S 2 L 2 (0,π) ) π ∂ x w S )(∂ x y) dx ∂ xx w S , with ν S := ν( ∂ x w S 2 L 2 (0,π) ) > 0 and µ = 2ν ( ∂ x w S 2 L 2 (0,π) ). By integrating by parts and linearization we are lead to study(1.4) t y -∂ xx y + µ π 0 ay dx a + w S ∂ x y + y∂ x w S = uχ ω , y(t, 0) = y(t, π) = 0, y(0, •) = y 0 ,where a is smooth function and µ ∈ R. We have assumed ν S = 1 to simplify. x) := y(t, x)e -W (x) .Then standard calculation yields∂ t y -∂ xx y + µ π 0 ay dx a + w S ∂ x y + y∂ x w S = e W ∂ t z -∂ xx z -2W (∂ x z) -W z -(W ) 2 z + µ π 0ae W z dx ae -W + w S (∂ x z + W z) + z∂ x w S .

Remark 2 . 7 .

 27 If µ = 0, we can replace (2.21) s)α(s) ds dx and (2.22) by w(x) = ϑ x 0 (x -s)α(s) ds.

x 0 H

 0 (x, s)α(s) ds dx, then we have the following possibilities (a) If

π 0 φ 2

 02 (s)β(s) ds = 0, π 0

  with the property that there exists a non empty open interval (a, b) ⊂ [0, π] such that z vanishes in (a, b) and α is not identically zero in (a, b). Then z is identically zero in [0, π]. Proof. By taking into account the particular form of the kernel K and equation (2.54) verified by z, we obtain that α(x) π 0 β(ξ)z(ξ) dξ = 0 (x ∈ (a, b)).

  ) R k := max{r k,j , j = 1, . . . , d k }, and (3.8) I k,p := {j ∈ {1, . . . , d k } ; r k,j = R k -p} (0 p R k -1).

Lemma 4 . 1 .

 41 The eigenvalues of the operator (D(A * ), A * ) from Σ verify the following properties: P1) There exists γ > 0 such that

  d k j=1 r kj . For each 1 k N L , the following two properties hold true: (P1) If S k,s| ω ≡ 0 for 0 s R k -1, then (5.5) a k,j,m = 0 (1 j d k , 0 m r kj -1).

2 .- 1 m=0

 21 From(5.11), by taking c = min 1 k N L c k , we deduce that (5.12)N L k=1 e -2T (λ k ) R k -1 s=0 ω |S k,s (x)| 2 dx c |a k,j,m | 2 e -2T (λ k ) .Using (2.47), we have that(5.13) ∞ k=N H |a k | 2 e -2T (λ k ) ω |ψ k (x)| 2 dx c ∞ k=N H |a k | 2 e -2T (λ k ) .

- 1 m=0 2 6 2 x

 122 |a k,j,m | 2 e -2T (λ k ) + ∞ k=N H |a k | 2 e -2T (λ k )where M 2 = c min{c , c } is a positive constant independent of T . By taking into account (5.2) and (5.14), we deduce that (1.13) holds with M 0 = M1 M2 and ς = 4η 2 κ which concludes the proof of the theorem. The nonlinear problemThe aim of this section is to provide the proof of Theorem 1.3. Without loss of generality, we may suppose that T < T 0 , where T 0 is given in Theorem 1.2. Let us consider thatK(ξ, x) = ∂ xx w S (ξ)e W (ξ) ∂ xx w S (x)e -W (x) ,with W (x) = 1 0 w S ds. Since ∂ xx w S ≡ 0 in ω, the hypotheses Theorem 1.2 are satisfied and (1.13) holds true.Let X = L 2 (0, π) and U = L 2 (ω) and let us denote γ : (0, ∞) → [0, ∞), t → γ(t) := M 0 exp ς t ,

1 2

 1 2) ρ F (t) = exp -ς 1 T -t , ρ 0 (t) = M 0 exp -ς 0 T -t , ρ(t) = exp -ς * T -t ,so that ρ F , ρ 0 and ρ are continuous, decreasing and ρ F (T ) = ρ 0 (T ) = ρ(T ) = 0. We associate to the functions ρ F and ρ 0 the Hilbert spaces F and U defined byF = f ∈ L 2 (0, T ; X) f ρ F ∈ L 2 (0, T ; X) , (6.3) U = u ∈ L 2 (0, T ; U ) u ρ 0 ∈ L 2 (0, T ; U ) . (6.4)By supposing that z 0 ∈ D((-A) ) and f ∈ F, we are now able to obtain a controllability result for the nonhomogeneous equation (6.5) ∂ t z(t) + Az(t) = f + vχ ω , z(0) = z 0 .

xx y -µ π 0 ( 1 0

 01 ∂ x w S )(∂ x y) dx ∂ xx w S + w S ∂ x y + y∂ x w S = f + uχ ω , y(t, 0) = y(t, π) = 0, y(0, •) = y 0 , satisfies y ρ ∈ L 2 (0, T ; H 2 (0, π)) ∩ H 1 (0, T ; L 2 (0, π)) ∩ C([0, T ]; H 1 0 (0, π)),with the estimate (6.8) y ρ L 2 (0,T ;H 2 (0,π))∩H 1 (0,T ;L 2 (0,π))∩C([0,T ];H (0,π))

  . From definitions (4.21) of the function Q λ , (4.14) of the function H β and (4.6) of the function Φ λ we deduce immediately that the first property of Q λ holds true. Moreover, by taking into account estimate (4.7) for Φ λ and the fact that, according to Lemma 4.4, H β is an entire function of exponential type β, it follows that Q λ verifies the second property, too.

	To prove (4.22), let us first recall that, according to (2.1), | (λ)|	D 0 for each λ ∈ Σ. By
	taking β 0 = π 4D0 we deduce from (4.19) that, for any β β 0 , the following estimate holds
	(4.23)

+ 1.

k N L ∪ (ψ k ) k N H is a Riesz basis in L

(0, π) (see Theorem

3.1) and, ifϕ 0 ∈ L 2 (0, π) is

Acknowledgments. Part of this work was done when S.Micu was visiting Université de Lorraine and he would like to thank the members of Institut Élie Cartan de Lorraine for their kindness and warm hospitality. T. Takahashi was partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010).


Palazzo Mancini

Mara Cantoni

Molecular Dynamics Simulation of Methanol-Water Mixture

Keywords: Molecular dynamics simulation, Einstein equation, Diffusion coefficient, Hydrogen bond

In this study some properties of the methanol-water mixture such as diffusivity, density, viscosity, and hydrogen bonding were calculated at different temperatures and atmospheric pressure using molecular dynamics simulations (MDS). The results were compared with the available experimental data as well as some theoretical models; overall indicating a good agreement. This shows the useful and effective application of MDS for determination of physical properties.

1-Introduction

As molecular simulations are considered useful tools for the interpretation of experimental data, two methods namely Monte Carlo (MC) and Molecular Dynamics (MD), can be applied. This allows us to determine macroscopic properties by evaluating exactly a theoretical model of molecular behavior using a computer program. [START_REF] Monticella | The MARTINI coarse grained force field: extension to proteins[END_REF][START_REF] Sadus | Molecular simulation of fluid, theory, algorithms and object-orientation[END_REF] Molecular simulation considers small size systems, at a typical scale of a few nanometers, and determines their behaviour from a careful computation of the interactions between their components. These computations take much more computer time than classical thermodynamic models (i. e. from a few hours to several weeks of computing time, depending on system size). As such, molecular simulation is now considered in the chemical industry as capable of filling the existing gap between experimental data and engineering models in various circumstances such as unknown chemicals, extreme conditions of temperature or pressure, and toxic compounds. The first part of this article provides a brief description of Molecular dynamic simulation i and how it can be analyzed for approving experimental data. In the second part, different properties of the system will be calculated by molecular dynamic simulation. Results will then be compared with available experimental data and theoretical models.

2-Molecular dynamics method 2.1-What is molecular dynamics method?

Molecular dynamics simulation involves integration of Newtonís equation of motion for each atom in an ensemble in order to generate information on the variation in the position, velocity, and acceleration of each particle as a function of time. The equations solved are [START_REF] Nobakht | Numerical study of diodicity mechanism in different Tesla-type microvalves[END_REF]: covalently bonded atoms and the last term describes the non-bonded interactions (van der Waals and electrostatic). The constants k b , k Ë , and k ˆ are force constants for the covalent bonds, bond angles, and dihedrals, respectively, rij are the distances between atoms i and j, Aij and Bij are the Lenard Jones parameters, and q is a partial charge [START_REF] Nobakht | Numerical study of diodicity mechanism in different Tesla-type microvalves[END_REF]. To calculate the properties of a real system, it is convenient to define an ensemble. An ensemble can be regarded as an imaginary collection of a very large number of systems in different quantum states with common macroscopic attributes. The ensemble average of any dynamic property (A) can be obtained from the relationship:

f i m i a i (1) A   A i p i i (4) f  U r i (2)
Where Ai is the value of A in quantum state i, pi represents the probability of observing the i-th state, and the angled brackets denote an

Where fi is the force exerted on the i th atom, mi is its mass, and ai is its acceleration. The force acting on each atom is obtained from a potential function U, such as that depicted below [START_REF] Nobakht | Numerical study of diodicity mechanism in different Tesla-type microvalves[END_REF]: ensemble average. The time averaged properties of the real system are related to the ensemble average by invoking the following assumption:

k b  ij ij k   ijk ijk A t   A (5) U  ij (r 2  r eq ) 2    angle ijk (  eq ) 2  2
The above equivalence of the time ñ average  dihedrals k    1cos(n( eq ))   and ensemble ñ average is called the Ergodic hypothesis. Form of pi is determined by the macroscopic properties that are common to

A ij  B ij   q i q i  
all of the systems of the ensemble.

 12 6    ij   r ij r ij 4 0 r ij    (3)

2.2-Calculation of physical properties

Prediction of transport properties of fluids of Where the first three terms reflect intramolecular interactions between industrial interest is one of the main motivations to perform MD simulations [START_REF] Ungerer | Molecular simulation of the thermophysical properties of fluids: From understanding toward quantitative predictions[END_REF].

T i k  
As it is not possible to observe individual atoms or molecules directly, various models are used to describe and/or predict the Also shear viscosity can be calculated using the following equation:

 properties of a system with molecular V dynamic methods. On the other hand, it is possible to calculate macroscopic properties

   B 0  xy (t) xy (0) dt (8)
of the mixtures by calculating the interaction

Where  xy is the pressure tensor of the of the molecules. The partial density is one components. kB is the Boltzman constant and macroscopic property that could be calculated using molecular dynamics simulation. Also transport properties like the diffusion coefficient and shear viscosity can be computed directly from an MD simulation too [START_REF] Krishna | The Maxwell-Stefan approach to mass transfer[END_REF]. In an equilibrium molecular dynamics (EMD) simulation, the selfdiffusion coefficient D S α of component α is computed by taking the slope of the mean square displacement (MSD) at a long period of time:

T is the temperature of the system.

2.3-GROMACS Software

GROMACS is an acronym for GROningen MAchine for Chemical Simulation. It is the latest release of a versatile and very well known optimized package for molecular simulation. The package is a collection of programs and libraries for the simulation of molecular dynamics and the subsequent analysis of trajectory data. Although it is primarily targeted at biological molecules (6) with complex bonded interactions, the very 2dN

D S  1 lim N  (r  (t)  r  (0)) 2
  t  i i i 1
effective implementation of nonbonded force calculations makes GROMACS suitable for Where Nα is the number of molecules of component α, d is the spatial dimension of the system, t is the time, and r α i is the center of mass of molecule i of component α. Equivalently, D S α is given by the time integral of the velocity autocorrelation function: all kinds of molecular dynamics simulations based on pair potentials. Apart from normal potential functions like Lenard-Jones, Buckingham and Coulomb, it is possible to use arbitrary forms of interactions with spline-interpolated tables [START_REF] Lindahl | GROMACS 3.0: a package for molecular simulation and trajectory analysis[END_REF]. The GROMACS molecular dynamics code contains several algorithms that have been

 N D S  1   v  (t)v  (0) (7)
developed in the Berendsen group. The leap-

dN 0 i i i1
frog integrator is used in two versions, for molecular dynamics and stochastic dynamics Where v  is the center of mass -velocity of [START_REF] Van Gunsteren | A leap frog-algorithm for stochastic dynamics[END_REF]. molecule i of component α. Equation ( 6) is known as the Einstein equation and equation [START_REF] Lindahl | GROMACS 3.0: a package for molecular simulation and trajectory analysis[END_REF] is often referred to as the Green-Kubo relation. [START_REF] Dubbeldam | Recent developments in the molecular modeling of diffusion in nanoporous materials[END_REF]

Result and Discussion

3.1-System definition

In Gromacs software, the system is defined by its size and shape, the number and types of molecules it contains, and the coordinates and velocities of all atoms. Velocities may also be generated from a Maxwellian distribution [START_REF] Spoel | GROMACS: Fast, Flexible, and Free[END_REF]. In this study MD simulation was used to examine some physical and transport properties of the mixture of methanol ñ water. This system consists of 216 molecules of water and 216 molecules of methanol. Simulations are carried out using a fully atomistic system with the maximum size 4.6 * 2.3 *2.3 nm. The box is shown in Fig. 1. Gromacs software was used for the MD calculations. Simulation was performed using GROMOS 96 [START_REF] Gunsteren | Computer Simulation of molecular dynamics methodology, applied and perspectives in chemistry[END_REF][START_REF] Gunsteren | The GROMOS 96 manual and user guide[END_REF] force field for 200 ps. MD simulations are performed at atmospheric pressure with four different temperatures 273, 283, 288, 293 k at the isothermal-isobaric ensemble (NPT). Table 1 shows the Lenard-Jones parameters and partial charge for methanol molecule. Water molecules are modeled as SPC water model with their related parameters [START_REF] Gunsteren | Computer Simulation of molecular dynamics methodology, applied and perspectives in chemistry[END_REF].

3.2-Diffusion coefficient of the methanol in the mixture

As the first step, diffusion coefficient of methanol molecules in a mixture of methanol-water has been studied by MD method. For this reason mean square displacement (MSD) of methanol molecules in the mixture was calculated by Gromacs software. Fig. 2 shows MSD values versus time for methanol molecules diffusing in the methanol-water mixture at four temperatures 273, 283, 288 and 293 k. These graphs show that the behavior of MSD versus time for 0<t<200 ps is linear, so Einstein equation can describe the diffusion behavior of methanol. Calculated Diffusion coefficients of methanol in the mixture by Einstein equation are shown in Table 1 and Fig. 3. Wilke-Chang Eq.

Calculated Diffusion Coefficient

RMSD (nm^2)

A

To evaluate the calculated results, the diffusion coefficient of this mixture was calculated by Wilke-Chang equation [START_REF] Treybal | Mass transfer phenomena[END_REF]. Results are shown in Table 1 andFig 3: 12%, while this is about 10% between experimental value and Eq.9.

3.3-Calculation of density

In the second step, the density of methanol-(117.3 10 18 )(M ) 1 / 2 T water mixture was calculated at different

D AB  B V 0.6 (9)
temperatures and different points in the box.

The results are shown in Table 3 and Fig. 4 and5. The comparison between experimental The available experimental data for diffusion coefficient of methanol in water are shown in Table 2 [START_REF] Perry | Perry's chemical engineer's handbook[END_REF]. The averaged error between MD results and experimental value is about [START_REF] Perry | Perry's chemical engineer's handbook[END_REF] and calculated results shows a good agreement between these data and the averaged error is less than 1%.

3.4-Shear viscosity

One of the important properties of the mixtures is viscosity. It is possible to calculate this macroscopic property of the mixture by calculating the interaction of the molecules in microscopic scale. Here the viscosity of the methanol-water mixture was calculated by MD simulation and the results are compared with experimental data.

As it was shown in Table 4 and Fig. 7, viscosity is decreased by increasing temperature, the same as experimental value.

MD EXP.

Calculated and Experimental Density

3.5-Hydrogen Bond

Many thermodynamical properties (e.g., melting and boiling point) of the mixture depend on the strength of the hydrogen bonds (H-bond) formed among molecules. The H-bond is a special attractive interaction parameter that exists between an electronegative atom and a hydrogen atom bonded to another electronegative atom. Since water has two hydrogen atoms and one oxygen atom, it can form an H-bond between the Oxygen and the H atoms. Each water molecule can form up to four hydrogen bonds at the same time (two through its two lone pairs, and two through its two hydrogen atoms). The average number of H-bonds per molecule calculated in bulk water varies approximately from 2.3 to 3.8 according to the water model and the way used to define the H-bond. [START_REF] Nobakht | Heat flow diversion in supported graphene nanomesh[END_REF][START_REF] Jorgensen | Optimized Intermolecular Potential Functions for liquid Hydrocarbons[END_REF] In this study the number of hydrogen atoms between water molecules and water-methanol molecules were calculated. The results, shown in Figs. 8 and9, indicate that the number of hydrogen bonds between water molecules is from 2 to 3 but it decreases for water and methanol molecules down to 1. The hydrogen bond distribution is also plotted in the box, where the most probable ones form near the center, Fig 10.

Conclusion

In this paper molecular dynamic theoretical method is presented to calculate density, shear viscosity, hydrogen bonding and diffusion coefficient for the mixture of methanol-water. Gromacs software was used for this purpose. The simulation was performed up to 200 ps. The results in Table 1 and Fig. 3 clearly show that the calculated diffusion coefficient by MD method indicate a good agreement and an almost similar trend with those obtained from the Wilke-Chang equation, both of which are approved by experimental data. These values, however increase linearly with temperature (D~T). Density and viscosity similar to that of diffusion coefficient are affected by temperature. The density decreases with temperature with a decreasing deviation between MD and experimental data. This shows that MD is much more reliable at higher temperatures. In the case of viscosity, it also decreases with temperature with a decreasing deviation. After that, it decreases but with a lower deviation of MD with those of the experimental data. The reference temperature for MD calculation in the input files of the simulation is 300 K, so some of the calculated properties such as viscosity and density at the points near this temperature have the minimum deviation from the experimental data. The MD calculation for pure water in this study estimated H-bond numbering between 2 to 3 which is in a good agreement with the previous studies [START_REF] Nobakht | Heat flow diversion in supported graphene nanomesh[END_REF][START_REF] Jorgensen | Optimized Intermolecular Potential Functions for liquid Hydrocarbons[END_REF][START_REF] Zielkiewicz | Structural properties of water: Comparison of the SPC, SPCE, TIP4P, and TIP5P models of water[END_REF][START_REF] Nobakht | Anisotropic control of thermal transport in graphene/Si heterostructures[END_REF][18]. This number decreases considerably for the case of water-methanol molecules. This is due to the lower polarity of methanol molecules, compared to z axis (nm) Probability Probability that of water molecules. Hydrogen bond formation has the most distribution probability at the center of the box and the temperature has no effect on distribution. In the center of the box, there are water and methanol molecules in all the directions, so the hydrogen bond has the maximum distribution in this part of the system, but in the points near the sidewall of the system the number of molecules decreases so the hydrogen bonding has the minimum distribution in these points.

Overall, the results show the reliability of the MD method to calculate the physical properties of the systems. One can conclude that, this method may be used to determine the macroscopic properties of the systems, instead of practical methods in the laboratory.

Nomenclature

Figure 1 .

 1 Figure 1. Computer generated image of a methanol-water box

Figure 2 .Figure 3 .

 23 Figure 2. Root Mean square displacement of methanol molecules at different temperatures

Figure 4 .Figure 5 .

 45 Figure 4. The effect of temperature on density for methanol-water mixture

Figure 6 .

 6 Figure 6. Viscosity of the methanol-water mixture at different temperatures

Figure 8 .Figure 9 .Figure 10 .

 8910 Figure 8. Hydrogen bond of water molecules in the mixture



  Viscosity (kg/m.s)VAsolute molar volume at boiling point (m 3 /kmol)

Table 1 .

 1 The partial charge and Lenard-Jones parameters for methanol molecule[START_REF] Gunsteren | Computer Simulation of molecular dynamics methodology, applied and perspectives in chemistry[END_REF]

	Atom type	Charge	Lenard Jones parameters
			C6	C12
	CH3	0.29	8.8758e-3	17.8426e-6
	O	-0.69	2.6169e-3	2.5231e-6
	H	0.4	0	0

Table 2 .

 2 Calculated Diffusion Coefficient of Methanol in Water by MD method

Table 3 .

 3 Calculated and experimental density in different temperatures

	Temperature (K)	Cal. Density (kg/m^3)	Exp. Density (kg/m^3) [11]	Error %
	273	936.10	928.70	0.8
	283	927.27	922.10	0.6
	288	918.15	918.50	0.04
	293	915.54	915.60	0.006

Box (nm )

Table 4 .

 4 Average calculated and experimental viscosity of the methanol-water mixture

	Temperature	273 K	283 K	288 K	293 K	300 K
	MD Calculated value	1.27	1.07	0.95	0.71	0.51
	Experimental value [12]	1.06	0.9	0.85	0.74	0.6
	Error %	19.8	18.0	11.7	4.1	7