Pierre Cornilleau
email: pierre.cornilleau@ens-lyon.org.

Sergio Guerrero
email: guerrero@ann.jussieu.fr.

Controllability and observability of an artificial advection-diffusion problem

may come

Controllability and observability of an artificial advection-diffusion problem

Introduction

Artificial advection-diffusion problem In the present paper we deal with an advection-diffusion problem with small viscosity truncated in one space direction. Our interest for the linear advection diffusion equation comes from the Navier-Stokes equations, but it arises also from other fields as, for example, meteorology. For a given viscosity ε > 0, the incompressible Navier-Stokes equations can be written as

f t + (f.∇)f -ε∆f + ∇p = 0 div(f ) = 0
where f is the velocity vector field, p the pressure, ∇ the gradient and ∆ the usual Laplacian. Considering the flow around a body, we have that f is almost constant far away from the body and equal to a. Then, our system can be approximated by the Oseen equation

f t + (a.∇)f -ε∆f + ∇p = 0 div(f ) = 0
(see, for instance, [9, page 309, (1.2)]). Consequently, we get the following equation for the vorticity u = rot(f ) u t + a.∇u -ε∆u = 0.

In the sequel, we assume for simplicity that a is the nth unit vector of the canonical basis of R n . When one computes the solution of this problem, it can only be solved numerically on a bounded domain. A good way to approximate the solution on the whole space may be given by the use of artificial boundary conditions (see [START_REF] Halpern | Artificial boundary for the linear advection diffusion equation[END_REF][START_REF] Halpern | Artificial boundary conditions for incompressible viscous flows[END_REF]).

In this paper, we will consider an advection-diffusion system in a strip Ω := R n-1 × (-L, 0) (L some positive constant) with particular artificial boundary conditions on both sides of the domain. The following system was considered in [8, section 6]:

       u t + ∂ xn u -ε∆u = 0 ε(u t + ∂ ν u) = 0 ε(u t + ∂ ν u) + u = 0 u(0, .) = u 0 in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω, (1)
where T > 0, Γ 0 := R n-1 × {0}, Γ 1 := R n-1 × {-L} and we have denoted ∂ xn the partial derivative with respect to x n and ∂ ν the normal derivative. In [START_REF] Halpern | Artificial boundary for the linear advection diffusion equation[END_REF]Theorem 3], the author proved that the solution of [START_REF] Coron | Control and nonlinearity[END_REF] converges in some sense to the restriction of the solution of   

u t + ∂ xn u = 0 u = 0 u(0, .) = u 0 in (0, T ) × R n-1 × R -, on (0, T ) × Γ 1 , in R n-1 × R -,
to (0, T ) × Ω (see also section 6 in [START_REF] Halpern | Artificial boundary for the linear advection diffusion equation[END_REF]). Up to our knowledge, there is no controllability result concerning system [START_REF] Coron | Control and nonlinearity[END_REF]. In this paper, we will be interested in the uniform boundary controllability of (1):

   u t + ∂ xn u -ε∆u = 0 ε(u t + ∂ ν u) + u1 Γ1 = v1 Γi u(0, .) = u 0 in (0, T ) × Ω, on (0, T ) × ∂Ω, in Ω, where i = 0 or i = 1.

In the sequel, we shall focus on the one-dimensional problem

(S v )        u t + u x -εu xx = 0 ε(u t + ∂ ν u) = v ε(u t + ∂ ν u) + u = 0
u(0, .) = u 0 in (0, T ) × (-L, 0), on (0, T ) × {0}, on (0, T ) × {-L}, in (-L, 0).

We are interested in the so-called null controllability of this system for given u 0 , find v such that the solution of(S v ) satisfies u(T ) ≡ 0.

This property is classicaly equivalent to an observability inequality for the so-called adjoint system (see Proposition 16 below for a proof), which can be written as

(S ′ )        ϕ t + ϕ x + εϕ xx = 0 ε(ϕ t -∂ ν ϕ) -ϕ = 0 ϕ t -∂ ν ϕ = 0 ϕ(T, .) = ϕ T
in (0, T ) × (-L, 0), on (0, T ) × {0}, on (0, T ) × {-L}, in (-L, 0).

The observability inequality corresponding to the previous controllability property is:

there exists C > 0 such that ϕ(0, .) X ≤ C ϕ(., 0) L 2 (0,T ) ∀ϕ T ∈ X, (2)
where the space X will be defined below. We refer the reader unfamiliar with the links between observability and controllability to the seminal paper [START_REF] Dolecki | A general theory of observation and control[END_REF] (see also [START_REF] Lions | Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués[END_REF] on hyperbolic systems). The phenomenon of degeneration of a parabolic problem to a hyperbolic one has been studied in several papers: see, for instance, [START_REF] Coron | Singular optimal control: a linear 1-D parabolic-hyperbolic example[END_REF] or [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF] (one dimensional heat equation) and [START_REF] Glass | On the uniform controllability of the Burgers equation[END_REF] (Burgers equation). Similar results of interest can also be found in [START_REF] Coron | Control and nonlinearity[END_REF].

Main results

We define X as the closure of D( Ω) := C ∞ ( Ω) for the norm

u X := u 2 L 2 (Ω) + ε u 2 L 2 (∂Ω) 1 2 .
Observe that X is the largest subspace of L 2 (Ω) such that its elements possess a trace in L 2 (∂Ω). In particular, ∀ǫ > 0, H 1/2+ǫ (Ω) X L 2 (Ω).

We will denote by C obs (ε) the cost of the null-control, which is the smallest constant C which fulfills the observability estimate [START_REF] Coron | Singular optimal control: a linear 1-D parabolic-hyperbolic example[END_REF]. Our main result is the following:

Theorem 1 Assume T > 0.
• For any ε > 0, there exists C > 0 such that the solution of problem (S ′ ) satisfies [START_REF] Coron | Singular optimal control: a linear 1-D parabolic-hyperbolic example[END_REF]. Consequently, for every u 0 ∈ X, there exists a control v ∈ L 2 (0, T ) with

v L 2 (0,T ) ≤ C u 0 X
such that the solution of the problem (S v ) satisfies u(T ) ≡ 0.

• Furthermore, if T /L is large enough, the cost of the null-control C obs (ε) tends to zero exponentially as ε → 0:

∃C, k > 0 such that C obs (ε) ≤ Ce -k/ε ∀ε ∈ (0, 1).
Remark 1 One can in fact obtain an observability result for the adjoint system at x = -L, that is

ϕ(0, .) X ≤ C ϕ(., -L) L 2 (0,T ) ,
for any ϕ solution of (S ′ ). This provides some controllability result for the direct system on Γ 1 : a function v can be found depending continuously on u 0 so that the solution of

       u t + u x -εu xx = 0 u t + ∂ ν u = 0 ε(u t + ∂ ν u) + u = v u(0, .) = u 0 in (0, T ) × (-L, 0), on (0, T ) × {0}, on (0, T ) × {-L}, in (-L, 0),
satisfies u(T, .) ≡ 0. Despite the fact that it is more physical to control our system at Γ 1 , we have chosen to present here the result for the system (S v ) since its proof, less obvious, requires more computations.

Remark 2

The fact that the control cost vanishes tells intuitively that the state is almost null for T /L big enough. This is to be connected with the fact that, for ε = 0, the system is purely advective and then that, for T > L, its state is null.

In some context of inverse problems (be able to know the origin of a polluted river for instance), it can be interesting to know if the observation of the solution of the direct problem (S 0 ) n on the boundary part Γ 1 or Γ 0 can allow us to recover the initial data. The corresponding result is presented now and will be proved at the end of the first section.

Proposition 2 Let T > 0 and ε > 0. If the solution of (S 0 ) with initial data u 0 ∈ X satisfies u = 0 on (0, T ) × Γ 1 , then u 0 ≡ 0. However, there is no constant C > 0 such that the following estimate holds

u 0 X ≤ C u(., -L) L 2 (0,T ) ∀u 0 ∈ X. (3)
Remark 3 Using Remark 1, we can also obtain a similar result at x = 0.

The rest of the article is organized as follows: in the first section, we show the well-posedness of the direct and the adjoint problems using some semi-groups approach. In the second section, we adapt a Carleman inequality to the case of our one-dimensional problem. The third section is intended to explain how to get observability in the one-dimensional case and the equivalence between observability and controllability.

Notations:

A B means that, for some universal constant c > 0, A ≤ cB. A ∼ B means that, for some universal constant c > 1, c -1 B ≤ A ≤ cB.

Well-posedness and basic properties of systems

In this section, we work in dimension n.

Homogeneous problems

We will use some semi-group results to show existence and uniqueness of the homogeneous direct problem (that is (S 0 ) n ). This will enable to define solutions of the system (S 0 ) n as a semigroup value. We define H = (X, . X ), V = H 1 (Ω) endowed with the usual norm . and we consider the bilinear form on V defined by

a(u 1 , u 2 ) = ε Ω ∇u 1 .∇u 2 + Ω ∂ xn u 1 u 2 + Γ1 u 1 u 2 . ( 4
)
With the help of this bilinear form, one may now consider the space

D := u 1 ∈ X; sup u2∈D(Ω); ||u2||X ≤1 |a(u 1 , u 2 )| < +∞
equipped with the natural norm

u 1 D = u 1 X + sup u2∈D(Ω); ||u2||X ≤1 |a(u 1 , u 2 )|.
Note that, using an integration by parts, one shows that a(u 1 , u 2 ) is well-defined for u 1 ∈ X and u 2 ∈ C ∞ (Ω) and that the map

u 2 ∈ X → a(u 1 , u 2 ) ∈ R
is well-defined and continuous for any u 1 ∈ D.

Using the Riesz representation theorem, we can define an operator A with domain D(A) = D and such that

∀u 1 ∈ D(A), ∀u 2 ∈ X, < -Au 1 , u 2 > X = a(u 1 , u 2 ).
(S 0 ) n might now be written in the following abstract way u t = Au, u(0, .) = u 0 , since, for u ∈ C(R + , D(A)) ∩ C 1 (R + , X) a classical solution of (S 0 ) n , we directly get that ∀v ∈ X, < u t , v > X = -a(u, v). Proposition 3 Let ε > 0. Then A generates a continuous semi-group (e tA ) t≥0 on X.

Proof Using the Hille-Yoshida theorem, it is sufficient to show that A is a maximal monotone operator.

• First, Green-Riemann formula gives

< Au, u > X = -ε Ω |∇u| 2 - 1 2 ∂Ω |u| 2 ,
so the monotonicity is proved.

• Given v ∈ X, we have to solve (I -A)u = v, that is to find u ∈ D(A) such that ∀u ′ ∈ X, < (I -A)u, u ′ > X =< v, u ′ > X .
The left-hand side term of this equation is a continuous bilinear form B on the space V while the right-hand side term is a continuous linear form L on V . Moreover, using (4), we can easily compute

B(u, u) = u 2 X + ε Ω |∇u| 2 + 1 2 ∂Ω |u| 2 ≥ min{1, ε} u 2 ,
which show that B is coercive on V . Consequently, Lax-Milgram theorem shows that there exists u ∈ V such that B(u, u ′ ) = L(u ′ ), ∀u ′ ∈ V . Using test functions u ′ ∈ D(Ω) additionaly gives u ∈ D(A) and our proof ends.

In particular, for every initial data u 0 ∈ X, we have existence and uniqueness of a solution u ∈ C(R + , X) to (S 0 ) n . We will call these solutions weak solutions opposed to strong solutions i.e. such that u 0 ∈ D(A) and which fulfill u ∈ C(R + , D(A)) ∩ C 1 (R + , X). One can notice that, using the density of D(A) in X, a weak solution can always be approximated by a strong solution.

Let us introduce the adjoint system in dimension n

(S ′ ) n        ϕ t + ∂ xn ϕ + ε∆ϕ = 0 ε(ϕ t -∂ ν ϕ) -ϕ = 0 ϕ t -∂ ν ϕ = 0 ϕ(T, .) = ϕ T in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω.
Defining the adjoint A * of A, we can show in a similar way as before that (S ′ ) n may be written in the following abstract way

ϕ t + A * ϕ = 0, ϕ(T, .) = ϕ T .
The adjoint operator A * is also a maximal monotone operator with domain D(A * ). Thus, for every initial data ϕ T ∈ X, we have existence and uniqueness of a solution ϕ ∈ C(R + , X) to (S ′ ) n given by means of the backward semigroup (e (T -t)A * ) t≥0 . We will also speak of weak solutions or strong solutions in this situation.

Nonhomogeneous direct problems

We consider a slightly more general system

(S f,g0,g1 ) n        u t + ∂ xn u -ε∆u = f ε(u t + ∂ ν u) = g 0 ε(u t + ∂ ν u) + u = g 1 u(0, .) = u 0 in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω, with f ∈ L 2 ((0, T ) × Ω), g 0 ∈ L 2 ((0, T ) × Γ 0 ) and g 1 ∈ L 2 ((0, T ) × Γ 1 )
. We consider a function g on ∂Ω such that g = g i on Γ i (i = 0, 1).

Definition 4 We say that u ∈ C([0, T ] , X) is a transposition solution of (S f,g0,g1 ) n if, for every function ϕ ∈ C([0, T ], D(A * )) ∩ C 1 ([0, T ], X), the following identity holds τ 0 (< u, ϕ t > X + < u, A * ϕ > X + < F, ϕ > X ) = [< u(t), ϕ(t) > X ] t=τ t=0 ∀τ ∈ [0, T ],
where we have defined, using the Riesz representation theorem, F (t) ∈ X such that

< F (t), u ′ > X = Ω f (t)u ′ + ∂Ω g(t)u ′ , ∀u ′ ∈ X. ( 5
)
Proposition 5 Let T > 0, u 0 ∈ X, f ∈ L 2 ((0, T ) × Ω), g 0 ∈ L 2 ((0, T ) × Γ 0 ) and g 1 ∈ L 2 ((0, T ) × Γ 1 ).
Then (S f,g0,g1 ) n possesses a unique solution u.

Proof

Let us first assume that u belongs to

u ∈ C([0, T ], D(A)) ∩ C 1 ([0, T ], X).
Then, it is easy to prove that u is a solution to (S f,g0,g1 ) n if and only if, for any φ ∈ D(A * ),

< u t , φ > X =< u, A * φ > X + < F, φ > X in [0, T ].
Consequently, thanks to Duhamel formula, u is a solution of (S f,g0,g1 ) n is equivalent to

u(t) = e tA u 0 + t 0 e (t-s)A F (s)ds, ∀t ∈ [0, T ], since D(A * ) is dense in X.
In order to prove the existence of a solution u ∈ C([0, T ], X), we approximate (u 0 , f, g) by

(u p 0 , f p , g p 0 , g p 1 ) ∈ D(A) × C([0, T ], L 2 (Ω)) × C([0, T ], L 2 (Γ 0 )) × C([0, T ], L 2 (Γ 1 )) in X × L 2 ((0, T ), L 2 (Ω)) × L 2 ((0, T ), L 2 (Γ 0 )) × L 2 ((0, T ), L 2 (Γ 1 )).
Then, we know that u p given by

u p (t) = e tA u p 0 + t 0 e (t-s)A F p (s)ds, ∀t ∈ [0, T ],
where F p is related to (f p , g p 0 , g p 1 ) by formula [START_REF] Fursikov | Controllability of evolution equations[END_REF], is the solution of (S f p ,g p 0 ,g p 1 ) n . Passing to the limit in this identity, we get that

u p → u in C([0, T ], X) with u(t) := e tA u 0 + t 0 e (t-s)A F (s)ds, ∀t ∈ [0, T ].
Furthermore, passing to the limit in Definition 4 for u p , we obtain that u is a solution of (S f,g0,g1 ) n . Moreover, this argument also proves the uniqueness of the solution.

We now focus on some regularization effect for our general system and for some technical reasons we want to have some explicit dependence of the bounds on ε. For this result, it is essential that ε > 0.

Lemma 6 Let ε ∈ (0, 1), f ∈ L 2 ((0, T ) × Ω), g 0 ∈ L 2 ((0, T ) × Γ 0 ) and g 1 ∈ L 2 ((0, T ) × Γ 1 ). • Let u 0 ∈ X. Then if u is the solution of (S f,g0,g1 ) n , u belongs to L 2 ((0, T ), H 1 (Ω)) ∩ C([0, T ], X)
and

ε 1/2 u L 2 ((0,T ),H 1 (Ω)) + u L ∞ ((0,T ),X) ≤ C( u 0 X + f L 2 ((0,T )×Ω) + g L 2 ((0,T )×∂Ω) ) (6)
for some C > 0 only depending on T .

• Let now u 0 ∈ H 1 (Ω). Then u ∈ L 2 ((0, T ), D(A)) ∩ C([0, T ], H 1 (Ω)), u t ∈ L 2 ((0, T ), X) and ε 1/2 u L ∞ ([0,T ],H 1 (Ω)) + ε( ∆u L 2 ((0,T )×Ω) + ∂ ν u L 2 ((0,T )×∂Ω) ) + u t L 2 ([0,T ],X) ≤ C ε 1/2 ( u 0 H 1 (Ω) + f L 2 ((0,T )×Ω) + g L 2 ((0,T )×∂Ω) ), (7)
where C > 0 only depends on T .

Proof Approximating u by a regular function (in L 2 ((0, T ), D(Ω))), all computations below are justified.

• First, we multiply our main equation by u and we integrate on Ω. An application of Green-Riemann formula with use of the boundary conditions gives the identity 1 2

d dt u 2 X + 1 2 ∂Ω |u| 2 + ε Ω |∇u| 2 = Ω f u + ∂Ω gu,
which immediately yields

d dt u 2 X + ∂Ω |u| 2 + 2ε Ω |∇u| 2 ≤ u 2 X + f 2 L 2 (Ω) + g 2 L 2 (∂Ω) .
Finally, Gronwall's lemma yields [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF].

• Now, we multiply the main equation by u t and we integrate on Ω, which, using Green-Riemann formula and the boundary conditions provide the identity

u t 2 X + Ω ∂ xn uu t + ε 2 d dt Ω |∇u| 2 + 1 2 d dt Γ1 |u| 2 = Ω f u t + ∂Ω gu t , Now - Ω u t ∂ xn u 1 2 Ω |∇u| 2 + 1 2 u t 2 X
and, using again Young's inequality, we have

Ω f u t + ∂Ω gu t 1 4 u t 2 X + C( f 2 L 2 (Ω) + g 2 L 2 (∂Ω) ).
We have thus proven the following inequality

u t 2 X + 2ε d dt Ω |∇u| 2 + 2 d dt Γ1 |u| 2 ≤ 2 Ω |∇u| 2 + C g 2 L 2 (∂Ω) + f 2 L 2 (Ω) . (8)
We integrate here between 0 and t and estimate the first term in the right-hand side using [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit[END_REF] to get

Ω |∇u| 2 C ε 2 T 0 ( f (t) 2 L 2 (Ω) + g(t) 2 L 2 (∂Ω) )dt + u 0 2 H 1 (Ω) ,
for t ∈ (0, T ). We now inject this into (8) to get the second part of the required result

u t 2 L 2 ((0,T );X) C ε T 0 ( f (t) 2 L 2 (Ω) + g(t) 2 L 2 (∂Ω) )dt + u 0 2 H 1 (Ω) .
Writing the problem as

   -∆u = 1 ε (f -u t -∂ xn u) ∂ ν u = g0 ε -u t ∂ ν u = g1 ε -u t -1 ε u in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 ,
we get that u ∈ L 2 ((0, T ), D(A)) and the existence of some constant C > 0 such that

∆u L 2 ((0,T )×Ω) + ∂ ν u L 2 ((0,T )×∂Ω) ≤ C ε ( f L 2 ((0,T )×Ω) + g L 2 ((0,T )×∂Ω) + u L 2 ((0,T ),H 1 (Ω)) + u t L 2 ((0,T ),X) )
which gives the last part of the required result [START_REF] Glass | On the uniform controllability of the Burgers equation[END_REF].

Remark 4 Working on the non-homogeneous adjoint problem and using the backward semigroup e (T -t)A * , one is able to show that the estimates of Lemma 6 hold for the system

(S ′ f,g0,g1 ) n        ϕ t + ∂ xn ϕ + ε∆ϕ = f ε(ϕ t -∂ ν ϕ) -ϕ = g 0 ε(ϕ t -∂ ν ϕ) = g 1 ϕ(T, .) = ϕ T in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω.

Backward uniqueness

We will show here the backward uniqueness of systems (S 0 ) n and (S ′ ) n for ε > 0, thanks to the well-known result of Lions-Malgrange ( [START_REF] Lions | Sur l'unicité rétrograde dans les problèmes mixtes paraboliques[END_REF]) and the regularization effect.

Lemma 7 Let u 0 ∈ X and δ ∈ (0, T ). Then, the solution u of (S 0 ) n satisfies u ∈ C([δ, T ], D(A)).

Proof Our strategy is to show that u ∈ L 2 ((δ, T ); H 2 (Ω)) and u t ∈ L 2 ((δ, T ); H 2 (Ω)). This easily implies that u ∈ C([δ, T ]; H 2 (Ω)).

We first select some regular cut-off function θ 1 such that θ 1 = 1 on (δ/2, T ) and θ 1 = 0 on (0, δ/4). If

u 1 = θ 1 u, we have that        u 1,t + ∂ xn u 1 -ε∆u 1 = θ ′ 1 u ε(u 1,t + ∂ ν u 1 ) = εθ ′ 1 u ε(u 1,t + ∂ ν u 1 ) + u 1 = εθ ′ 1 u u 1 (0, .) = 0 in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω, that is u 1 satisfies (S θ ′ 1 u,εθ ′ 1 u,εθ ′ 1 u )
n . An application of Lemma 6 gives that u 1 ∈ L 2 ((0, T ), D(A)) and u 1,t ∈ L 2 ((0, T ), X). This implies in particular that u ∈ L 2 ((δ, T ), D(A)).

We now focus on u t and we select another cut-off function θ 2 such that θ 2 = 1 on (δ, T ) and θ 2 = 0 on (0, δ/2). We write the system satisfied by θ 2 u and we differentiate it with respect to time. One deduces that u 2 = (θ 2 u) t is the solution of

(S θ ′ 2 ut+θ ′′ 2 u,ε(θ ′ 2 ut+θ ′′ 2 u),ε(θ ′ 2 ut+θ ′′ 2 u) ) n . Using that |θ ′ 2 | |θ 1 |, Lemma 6 gives that u 2 ∈ L 2 ((0, T ), D(A)) that is u t ∈ L 2 ((δ, T ), D(A)).
Proposition 8 Assume that u is a weak solution of (S 0 ) n such that u(T ) = 0. Then u 0 ≡ 0.

Proof If 0 < δ < T , then Lemma 7 shows that u(δ) ∈ D(A). We will now apply Théorème 1.1 of [START_REF] Lions | Sur l'unicité rétrograde dans les problèmes mixtes paraboliques[END_REF] to u as a solution of (S 0 ) n in the time interval (δ, T ) to show that u(δ, .) = 0. The bilinear form a defined in (4) can be split into two bilinear forms on V defined by

a(u 1 , u 2 ) = a 0 (t, u 1 , u 2 ) + a 1 (t, u 1 , u 2 ) with a 0 (t, u 1 , u 2 ) = ε Ω ∇u 1 .∇u 2 , a 1 (t, u 1 , u 2 ) = Ω ∂ xn u 1 u 2 + Γ1 u 1 u 2 .
The first four hypotheses in [START_REF] Lions | Sur l'unicité rétrograde dans les problèmes mixtes paraboliques[END_REF] (see (1.1)-(1.4) in that reference) are satisfied since u ∈ L 2 (δ, T ; V ) ∩ H 1 (δ, T ; H) (from Lemma 6 above), u(t) ∈ D(A) for almost every t ∈ (δ, T ), u t = Au and u(T, .) ≡ 0. On the other hand, it is clear that a 0 and a 1 are continuous bilinear forms on V and that they do not depend on time t. It is also straightforward to see that

∀u ∈ V, a 0 (t, u, u) + u 2 X min{1, ε} u 2
and, for some constant C > 0,

∀u, v ∈ V, |a 1 (t, u 1 , u 2 )| C u 1 u 2 X .
This means that Hypothesis I in that reference is fulfilled. We have shown that u(δ, .) = 0, which finishes the proof since

u 0 = lim δ→0 u(δ) = 0.
Remark 5 The result also holds for the adjoint system: if ϕ is a weak solution of (S ′ ) n such that ϕ(0) ≡ 0 then ϕ T = 0. The proof is very similar to the one above and is left to the reader.

Proof of Proposition 2

The fact that u = 0 on (0, T ) × Γ 1 implies u 0 ≡ 0 is a straightforward consequence of the backward uniqueness of system (S 0 ) n . Indeed, similarly as (2), we can prove the observability inequality u(T, .) X u L 2 ((0,T )×Γ1) ,

(see Remark 6 below), which combined with Proposition 8 gives u 0 ≡ 0.

To show that ( 3) is false, we first need to prove some well-posedness result of (S 0 ) n with u 0 in a less regular space. For s ∈ (0, 1/2), we define X s as the closure of D( Ω) for the norm

u X s := u 2 H s (Ω) + ε u 2 H s (∂Ω) 1 2 ,
where H s (Ω) (resp. H s (∂Ω)) stands for the usual Sobolev space on Ω (resp. ∂Ω). One easily shows that X s is a Hilbert space. We now denote X -s the set of functions u 2 such that the linear form

u 1 ∈ D( Ω) -→< u 1 , u 2 > X
can be extended in a continuous way on X s . If u 1 ∈ X -s , u 2 ∈ X s we denote this extension by < u 1 , u 2 > -s,s .

If u 0 ∈ X -s , we say that u is a solution by transposition of (S 0 ) n if, for every f ∈ L 2 ((0, T ) × Ω)

T 0 Ω uf + < u 0 , ϕ(0, .) > -s,s = 0,
where ϕ is the weak solution (see Remark 4) of

(S ′ f,0,0 ) n        ϕ t + ∂ xn ϕ + ε∆ϕ = f ε(ϕ t -∂ ν ϕ) -ϕ = 0 ϕ t -∂ ν ϕ = 0 ϕ(T, .) = 0 in (0, T ) × Ω, on (0, T ) × Γ 0 , on (0, T ) × Γ 1 , in Ω.
Using the Riesz representation theorem and continuity (see Remark 4) of

f ∈ L 2 ((0, T ) × Ω) -→ ϕ(0, .) ∈ H 1 (Ω),
it is obvious that, for any u 0 ∈ X -s , there exists a solution by transposition of (S 0 ) n , u ∈ L 2 ((0, T ) × Ω), such that u L 2 ((0,T )×Ω) u 0 X -s .

Moreover, if u 0 ∈ X = X 0 , Lemma 6 shows that the solution u satisfies

u L 2 ((0,T ),H 1 (Ω)) u 0 X 0 .
If one uses classical interpolation results (see [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]), we can deduce that for every θ ∈ (0, 1/2), for any

u 0 ∈ X -sθ = [X, X -s ] 1-θ , there exists a solution u ∈ L 2 ((0, T ), H 1-θ (Ω)) such that u L 2 ((0,T ),H 1-θ (Ω)) u 0 X -sθ .
Using classical trace result, we have that

u L 2 ((0,T )×∂Ω) u 0 X -sθ . (9)
On the other hand, estimate (3) implies in particular that

u 0 X u L 2 ((0,T )×Γ1) ∀u 0 ∈ D(Ω). (10)
Finally, ( 9) and ( 10) yields the contradictory inclusion X -sθ ֒→ X.

Carleman inequality in dimension 1

In this paragraph, we will establish a Carleman-type inequality keeping track of the explicit dependence of all the constants with respect to T and ε. As in [START_REF] Fursikov | Controllability of evolution equations[END_REF], we introduce the following weight functions:

∀x ∈ [-L, 0], η(x) := 2L + x, α(t, x) := λ -e η(x) ε 2 t(T -t) , φ(t, x) := e η(x) ε 2 t(T -t) ,
where λ > e 2L . The rest of this paragraph will be dedicated to the proof of the following inequality:

Theorem 9 There exists C > 0 and s 0 > 0 such that for every ε ∈ (0, 1) and every s s 0 (εT + ε 2 T 2 ) the following inequality is satisfied for every ϕ T ∈ X:

s 3 (0,T )×(-L,0) φ 3 e -2sα |ϕ| 2 + s (0,T )×(-L,0) φe -2sα |ϕ x | 2 + s 3 (0,T )×{0,-L} φ 3 e -2sα |ϕ| 2 Cs 7 (0,T )×{0}
e -4sα+2sα(.,-L) φ 7 |ϕ| 2 .

(

) 11
Here, ϕ stands for the solution of (S ′ ) associated to ϕ T .

Remark 6 One can in fact obtain the following Carleman estimate with control term in Γ 1 s 3 (0,T )×(-L,0)

φ 3 e -2sα |ϕ| 2 + s (0,T )×(-L,0) φe -2sα |ϕ x | 2 + s 3 (0,T )×{0,-L} φ 3 e -2sα |ϕ| 2 Cs 7 (0,T )×{-L} e -4sα+2sα(.,0) φ 7 |ϕ| 2 .
simply by choosing the weight function η(x) equal to x → -x + L. The proof is very similar and the sequel will show how to deal with this case too.

In order to perform a Carleman inequality for system (S ′ ), we first do a scaling in time. Introducing T := εT , φ(t, x) := ϕ(t/ε, x) and the weights α(t, x) := α(t/ε, x), φ(t, x) := φ(t/ε, x); we have the following system

       φt + ε -1 φx + φxx = 0 ε 2 ( φt -ε -1 ∂ ν φ) -φ = 0 φt -ε -1 ∂ ν φ = 0 φ|t= T = ϕ T in q, on σ 0 , on σ 1 , in (-L, 0). (12)
if q := (0, T ) × (-L, 0) , σ := (0, T ) × {-L, 0}, σ 0 := (0, T ) × {0} and σ 1 := (0, T ) × {-L}. We will now explain how to get the following result.

Proposition 10 There exists C > 0, such that for every ε ∈ (0, 1) and every s C( T + ε -1 T 2 + ε 1/3 T 2/3 ), the following inequality is satisfied for every solution of (12) associated to φT ∈ X:

s 3 q φ3 e -2s α| φ| 2 + s q φe -2s α| φx | 2 + s 3 σ φ3 e -2s α| φ| 2 Cs 7 σ0 e -4s α+2s α(.,-L) φ7 | φ| 2 . ( 13
)
Observe that Proposition 10 directly implies Theorem 9.

All along the proof we will need several properties of the weight functions:

Lemma 11

• |α t | T φ2 , |α xt | T φ2 , |α tt | T 2 φ3 , • αx = -φ, αxx = -φ.
There are essentially two steps in this proof:

-The first one consists in doing a Carleman estimate similar to that of [START_REF] Fursikov | Controllability of evolution equations[END_REF]. The idea is to compute the L 2 product of some operators applied to e -s α φ, do integrations by parts and make a convenient choice of the parameter s. This will give an estimate where we still have a boundary term on σ 0 in the right-hand side.

-In the second one, we will study the boundary terms appearing in the right hand side due to the boundary conditions. We will perform the proof of this theorem for smooth solutions, so that the general proof follows from a density argument.

The big difference with general parabolic equations is the special boundary condition. Up to our knowledge, no Carleman inequalities have been performed when the partial time derivative appears on the boundary condition. The fact that we still have a Carleman inequality for our system comes from the fact that our system remains somehow parabolic.

We will first estimate the left hand side terms of ( 13) like in the classical Carleman estimate (see [START_REF] Fursikov | Controllability of evolution equations[END_REF]).

We obtain:

Proposition 12 There exists C > 0, such that for every ε ∈ (0, 1) and every s C( T + ε -1 T 2 + ε 1/3 T 2/3 ), the following inequality is satisfied for every solution of ( 12) associated to φT ∈ X:

s 3 q φ3 e -2s α| φ| 2 + s 2 q φe -2s α| φx | 2 + s -1 q φ-1 e -2s α(| φxx | 2 + | φt | 2 ) +s 3 σ1 φ3 e -2s α| φ| 2 s 5 σ0 φ5 e -2s α| φ| 2 + ε 2 s σ0 φe -2s α| φt | 2 . ( 14
)

Proof

Let us introduce ψ := φe -s α. We state the equations satisfied by ψ. In (0, T ) × (-L, 0), we have the identity

P 1 ψ + P 2 ψ = P 3 ψ,
where

P 1 ψ = ψ t + 2sα x ψ x + ε -1 ψ x , (15)
P 2 ψ = ψ xx + s 2 α2 x ψ + sα t ψ + ε -1 sα x ψ, (16)
and

P 3 ψ = -sα xx ψ.
On the other hand, the boundary conditions are:

ε 2 (ψ t + sα t ψ -ε -1 (ψ x + sα x ψ)) -ψ = 0 on x = 0, ( 17
)
ε(ψ t + sα t ψ) + ψ x + sα x ψ = 0 on x = -L. ( 18
)
We take the L 2 norm in both sides of the identity in q:

P 1 ψ 2 L 2 (q) + P 2 ψ 2 L 2 (q) + 2(P 1 ψ, P 2 ψ) L 2 (q) = P 3 ψ 2 L 2 (q) . ( 19
)
Using Lemma 11, we directly obtain

P 3 ψ 2 L 2 (q) s 2 q φ2 |ψ| 2 .
We focus on the expression of the double product (P 1 ψ, P 2 ψ) L 2 (q) . This product contains 12 terms which will be denoted by T ij (ψ) for 1 i 3, 1 j 4. We study them successively.

• An integration by parts in space and then in time shows that, since ψ(0, .) = ψ( T , .) = 0, we have

T 11 (ψ) = q ψ t ψ xx = - 1 2 q (|ψ x | 2 ) t + σ0 ψ x ψ t - σ1 ψ x ψ t = σ0 ψ x ψ t - σ1 ψ x ψ t .
Now, we use the boundary condition (17). We have

σ0 ψ x ψ t ε σ0 |ψ t | 2 -ε T 2 s σ0 φ3 |ψ| 2 -T s σ0 φ2 |ψ| 2 .
Thanks to the fact that ψ(0, .) = ψ( T , .) = 0, the same computations can be done on x = -L using (18) so we obtain the following for this term:

T 11 (ψ) -ε T 2 s σ φ3 |ψ| 2 -T s σ φ2 |ψ| 2 .
• Integrating by parts in time, we find

T 12 (ψ) = s 2 2 q α2 x (|ψ| 2 ) t = -s 2 q αx αxt |ψ| 2 , T 13 (ψ) = s 2 q αt (|ψ| 2 ) t = - s 2 q αtt |ψ| 2 , T 14 (ψ) = ε -1 s 2 q αx (|ψ| 2 ) t = -ε -1 s 2 q αtx |ψ| 2 ,
and using Lemma 11, we get

T 12 (ψ) -T s 2 q φ3 |ψ| 2 , T 13 (ψ) -T 2 s q φ3 |ψ| 2 , T 14 (ψ) -ε -1 T s q φ2 |ψ| 2 .
• Now, integrating by parts in space, we have

T 21 (ψ) = s q αx (|ψ x | 2 ) x = -s q αxx |ψ x | 2 + s σ0 αx |ψ x | 2 -s σ1 αx |ψ x | 2 .
Thanks to Lemma 11 (the choice of η is important here), the last term is positive. Using the boundary condition (17) and Lemma 11, we finally get

T 21 (ψ) s q φ|ψ x | 2 -s 3 σ0 φ3 |ψ| 2 -ε 2 s 3 T 2 σ0 φ5 |ψ| 2 + s σ1 φ|ψ x | 2 -ε 2 s σ0 φ|ψ t | 2 .
• An integration by parts in space provides

T 22 (ψ) = s 3 q α3 x (|ψ| 2 ) x = -3s 3 q α2 x αxx |ψ| 2 + s 3 σ0 α3 x |ψ| 2 -s 3 σ1 α3 x |ψ| 2 .
This readily yields

T 22 (ψ) s 3 q φ3 |ψ| 2 + s 3 σ1 φ3 |ψ| 2 -s 3 σ0 φ3 |ψ| 2 .
• Again an integration by parts in space gives

T 23 (ψ) = s 2 q αt αx (|ψ| 2 ) x = -s 2 q (α t αx ) x |ψ| 2 + s 2 σ0 αt αx |ψ| 2 -s 2 σ1 αt αx |ψ| 2 .
Using Lemma 11, we obtain

T 23 (ψ) -s 2 T q φ3 |ψ| 2 -s 2 T σ0 φ3 |ψ| 2 -s 2 T σ1 φ3 |ψ| 2 .
• The last integral concerning the second term in the expression of P 1 ψ is

T 24 (ψ) = ε -1 s 2 q α2 x (|ψ| 2 ) x .
After an integration by parts in space, we get

T 24 (ψ) ≥ -2ε -1 s 2 q φ2 |ψ| 2 -ε -1 s 2 σ1 φ2 |ψ| 2 .
• We consider now the third term in the expression of P 1 ψ. We have

T 31 (ψ) = ε -1 2 q (|ψ x | 2 ) x ≥ - ε -1 2 σ1 |ψ x | 2 .
• Now, we integrate by parts with respect to x and we have

T 32 (ψ) = ε -1 2 s 2 q α2 x (|ψ| 2 ) x ≥ -ε -1 s 2 q αx αxx |ψ| 2 - ε -1 2 s 2 σ1 α2 x |ψ| 2 ≥ -ε -1 s 2 q φ2 |ψ| 2 - ε -1 2 s 2 σ1 φ2 |ψ| 2 .
• Then, using another integration by parts in x we obtain

T 33 (ψ) = ε -1 2 s q αt (|ψ| 2 ) x -ε -1 s T q φ2 |ψ| 2 -ε -1 T s σ1 φ2 |ψ| 2 -ε -1 T s σ0 φ2 |ψ| 2 .
• Finally, arguing as before, we find

T 34 (ψ) = ε -2 2 s q αx (|ψ| 2 ) x ≥ ε -2 2 s 2 q φ|ψ| 2 -Cε -2 s σ0 φ|ψ| 2 -ε -2 s σ0 φ|ψ| 2 .
Putting together all the terms and combining the resulting inequality with (19), we obtain

P 1 ψ 2 L 2 (q) + P 2 ψ 2 L 2 (q) + I 1 (ψ) + I 2 (ψ x ) J 1 (ψ) + J 2 (ψ t ) + J 3 (ψ x ) + L(ψ) (20)
where the main terms are

I 1 (ψ) = s 3 q φ3 |ψ| 2 + s 3 σ1 φ3 |ψ| 2 , I 2 (ψ x ) = s q φ|ψ x | 2 + s σ1 φ|ψ x | 2 ,
the right hand side terms are

J 1 (ψ) = (ε -1 T s + ε -1 s 2 ) q φ2 |ψ| 2 + (s 2 T + s T 2 ) q φ3 |ψ| 2 +(s T + s 2 ε -1 + ε -1 T s) σ1 φ2 |ψ| 2 + (s 2 T + εs T 2 ) σ1 φ3 |ψ| 2 ,
and

J 2 (ψ t ) = ε 2 s σ0 φ|ψ t | 2 , J 3 (ψ x ) = ε -1 σ1 |ψ x | 2
and the control terms are

L(ψ) = ε -2 s σ0 φ|ψ| 2 + (s T + ε -1 T s) σ0 φ2 |ψ| 2 + (s 3 + s 2 T + ε T 2 s) σ0 φ3 |ψ| 2 + ε 2 s 3 T 2 σ0 φ5 |ψ| 2 .
Let us now see that we can absorb some right hand side terms with the help of the parameter s.

• First, we see that the distributed terms in J 1 (ψ) can be absorbed by the first term in the definition of I 1 (ψ) for a choice of s ε -1/2 T 3/2 + ε -1 T 2 .

• Second, we use the second term of I 1 (ψ) in order to absorb the integrals in the second line of the definition of J 1 (ψ). We find that this can be done as long as s ε -1 T 2 + ε -1/2 T 3/2 since ε < 1.

• Next, we observe that, provided s ε -1 T 2 , the second term in J 2 (ψ x ) absorbs J 3 (ψ x ).

Moreover, we observe that all the control terms can be bounded in the following way:

|L(ψ)| s 5 σ0 φ5 |ψ| 2
as long as s T (1 + ε -1 T ), using that ε < 1.

Next, we use the expression of P 1 ψ and P 2 ψ (see ( 15)-( 16)) in order to obtain some estimates for the terms ψ t and ψ xx respectively:

s -1 q φ-1 |ψ t | 2 s 2 q φ|ψ x | 2 + s -1 ε -2 q φ-1 |ψ x | 2 + s -1 q φ-1 |P 1 ψ| 2 s 2 q φ|ψ x | 2 + q |P 1 ψ| 2
for any s ε -1 T 2 and

s -1 q φ-1 |ψ xx | 2 s 3 T 0 0 -L φ3 |ψ| 2 + s T 2 q φ3 |ψ x | 2 + ε -2 s 2 q φ|ψ| 2 +s -1 q φ-1 |P 2 ψ| 2 s 3 q φ3 |ψ| 2 + s 2 q φ|ψ x | 2 + q |P 2 ψ| 2 , for s T + ε -1 T 2 .
Combining all this with (20), we obtain

s q φ(s 2 φ2 |ψ| 2 + |ψ x | 2 ) + s -1 q φ-1 (|ψ xx | 2 + |ψ t | 2 ) + s 3 σ1 φ3 |ψ| 2 s 5 σ0 φ5 |ψ| 2 + ε 2 s σ0 φ|ψ t | 2 , (21) for s T (1 + ε -1 T ).
Finally, we come back to our variable φ. We first remark that ψ x = e -s α( φx + s φ φ) and so

s q φe -2s α| φx | 2 s q φ|ψ x | 2 + s 3 q φ3 |ψ| 2 .
Then, we have that ψ t = e -s α( φt -sα t φ), hence

s -1 q φ-1 e -2s α| φt | 2 s -1 q φ-1 |ψ t | 2 + s T 2 q φ3 |ψ| 2 s -1 q φ-1 |ψ t | 2 + s 3 q φ3 |ψ| 2
for s T . Analogously, we can prove that

s -1 T 0 q φ-1 e -2s α| φxx | 2 s -1 q φ-1 |ψ xx | 2 + s 3 q φ3 |ψ| 2 + s 2 q φ|ψ x | 2 ,
for s T 2 . We combine this with (21) and we obtain the required result which is true for s T (recall that ε < 1).

s 3 q φ3 e -2s α| φ| 2 + s q φe -2s α| φx | 2 + s -1 q φ-1 e -2s α(| φxx | 2 + | φt | 2 ) + s 3 σ1 φ3 e -2s α| φ| 2
With this result, we will now finish the proof of Proposition 10.

Estimate of the boundary term

In this paragraph we will estimate the boundary term

ε 2 s σ0 φe -2s α| φt | 2 .
After an integration by parts in time, we get

ε 2 s σ0 φe -2s α| φt | 2 = ε 2 2 s σ0 ( φe -2s α) tt | φ| 2 -ε 2 s σ0 φe -2s α φ φtt T 2 ε 2 s 3 σ0 φ5 e -2s α| φ| 2 + ε 2 s σ0 φe -2s α| φ| | φtt |, (23)
for s T + T 2 . In order to estimate the second time derivative at x = 0, we will apply some a priori estimates for the adjoint system. Indeed, let us consider the following function

ζ(t, x) := θ(t) φt := e -s α(t,-L) φ-5/2 (t, -L) φt (t, x).
Then, this function fulfills the following system

       ζ t + ε -1 ζ x + ζ xx = θ t φt ε 2 (ζ t -ε -1 ∂ ν ζ) -ζ = ε 2 θ t φt ζ t -ε -1 ∂ ν ζ = θ t φt ζ |t=T = 0 in (0, T ) × (-L, 0), on (0, T ) × {0}, on (0, T ) × {-L}, in (-L, 0). Using Remark 4 for (t, x) -→ ζ(εt, x) we find in particular ε σ |ζ t | 2 q |θ t | 2 | φt | 2 + ε 2 σ (θ t ) 2 | φt | 2 .
This directly implies that

ε σ θ 2 | φtt | 2 q |θ t | 2 | φt | 2 + ε 2 σ (θ t ) 2 | φt | 2 .
Integrating by parts in time in the last integral, we have

ε σ θ 2 | φtt | 2 q |θ t | 2 | φt | 2 + ε 2 σ ((θ t ) 2 ) tt | φ| 2 + ε 3 σ (θ t ) 4 θ -2 | φ| 2 + ε 2 σ θ 2 | φtt | 2 .
From the definition of θ(t) and multiplying the previous inequality by s -3 , we find that (since ε < 1)

εs -3 σ (e -2s α φ-5 )(., -L)| φtt | 2 T 2 s -1 q φ-1 e -2s α| φt | 2 + ε 2 T 2 s σ ( φ3 e -2s α)(., -L)| φ| 2 T 2 s -1 q φ-1 e -2s α| φt | 2 + s 3 σ ( φ3 e -2s α)(., -L)| φ| 2 (24)
for s ε T . Using Cauchy-Schwarz inequality in the last term of the right hand side of (23), we obtain

ε 2 s σ0 φe -2s α| φ| | φtt | ε T -2 s -3 σ (e -2s α φ-5 )(., -L)| φtt | 2 + ε 3 T 2 s 5 σ0 e -4s α+2s α(.,-L) φ7 | φ| 2
Combining this with (24) and ( 22) yields the desired inequality (13).

Observability and control

In this section, we prove Theorem 1.

Dissipation and observability result

Our first goal, as in [START_REF] Coron | Singular optimal control: a linear 1-D parabolic-hyperbolic example[END_REF], will be to get some dissipation result. Even if we will only use this result in dimension one, we present it in dimension n for the sake of completeness (see also [START_REF] Danchin | Poches de tourbillon visqueuses[END_REF]).

Proposition 13 For every ε ∈ (0, 1), for every time t 1 , t 2 > 0 such that t 2 -t 1 > L and for every weak solution ϕ of (S ′ ) n , the following estimate holds

ϕ(t 1 ) X exp - (t 2 -t 1 -L) 2 4ε(t 2 -t 1 ) ϕ(t 2 ) X .
Proof We first consider a weight function ρ(t, x) = exp( r ε x n ) for some constant r ∈ (0, 1) which will be fixed later. We will first treat the strong solutions case and, using a density argument, we will get the weak solutions case.

We multiply the equation satisfied by ϕ by ρϕ and we integrate on Ω. We get the following identity:

1 2 d dt Ω ρ|ϕ| 2 = - 1 2 Ω ρ∂ xn (|ϕ| 2 ) -ε Ω ρϕ∆ϕ.
We then integrate by parts in space, which due to ∇ρ = r ε ρe n , provides

- 1 2 Ω ρ∂ xn (|ϕ| 2 ) = r 2ε Ω ρ|ϕ| 2 - 1 2 Γ0 ρ|ϕ| 2 - Γ1 ρ|ϕ| 2 ,
and

-ε Ω ρϕ∆ϕ = ε Ω ρ|∇ϕ| 2 + r 2 Ω ρ∂ xn (|ϕ| 2 ) -ε Γ0 ρϕ∂ xn ϕ - Γ1 ρϕ∂ xn ϕ = ε Ω ρ|∇ϕ| 2 - r 2 2ε Ω ρ|ϕ| 2 + r 2 Γ0 ρ|ϕ| 2 - Γ1 ρ|ϕ| 2 -ε Γ0 ρϕ∂ xn ϕ - Γ1 ρϕ∂ xn ϕ .
Using now the boundary conditions for ϕ summing up these identities, we finally get d dt

Proposition 14 If n = 1, T L is sufficiently large and ε sufficiently small, then the observability constant C obs (ε) is bounded by

C exp - k ε
where C, k are some positive constants.

Proof

• We begin with estimating both sides of the Carleman inequality obtained above. We use the same notations as above and we define m = λ -e 2L and M = λ -e L . We first get s 9

T 0 e -4sα(.,0)+2sα(.,-L) (φ 9 |ϕ| 2 )(., 0) s 7 (εT ) -14 exp s(2M -4m) (εT ) 2 T 0 |ϕ| 2 (., 0).

On the other hand, using that φ 1 (εT ) 2 on [ T 4 , 3T 4 , we have the following estimate from below for the left hand-side of the Carleman inequality (13) • We now deduce the result using dissipation estimates. We have just proven We now use the dissipation property with t 1 = 0 and t 2 = t ∈ T 4 , 3T 4 . We easily get, provided T > 4L,

s 3 (εT ) 6 exp - 2sM (εT ) 2

Proof of Theorem 1

To show the controllability result for (S v ), we will adopt some minimization strategy inspired by the classical heat equation.

Proposition 15 A necessary and sufficient condition for the solution of problem (S v ) to satisfy u(T ) = 0 is given by: ∀ϕ T ∈ X, < ϕ(0, .), u 0 > X = T 0 ϕ(., 0)v.

where ϕ is the solution of problem (S ′ ) with final value ϕ T .

Proof We apply the Definition 4 to u against ϕ a strong solution of (S ′ ), which gives T 0 ϕ(., 0)v =< ϕ(0, .), u 0 > X -< ϕ T , u(T, .) > X and we get the desired equivalence by approximation of weak solutions by strong solutions.

Proposition 16

The following properties are equivalent

• ∃C 1 > 0, ∀ϕ T ∈ X; ϕ(0, .) X ≤ C 1 ϕ(., 0) L 2 (0,T ) where ϕ is the solution of problem (S ′ ),

• ∃C 2 > 0, ∀u 0 ∈ X, ∃v ∈ L 2 (0, T ) such that v L 2 (0,T ) C 2 u 0 X and the solution u of problem (S v )satisfies u(T ) = 0.

Moreover, C 1 = C 2 .
Proof (⇒) Let u 0 ∈ X. We define H as the closure of X for the norm defined by ϕ T H = ϕ(., 0) L 2 (0,T ) , where ϕ is the corresponding solution of (S ′ ). Using the observability assumption and backward uniqueness (Proposition 8), one sees that it is indeed a norm on X.

We define a functional J in the following way

J(ϕ T ) = 1 2
T 0 ϕ 2 (., 0)-< ϕ(0, .), u 0 > X .

J is clearly convex and our assumption imply that J is continuous on H. Moreover, thanks to our observability assumption, J is coercive. Indeed, one has:

J(ϕ T ) 1 2 ϕ T 2
According to Proposition 15, we have shown the existence of an admissible control defined by v = φ(., 0). Moreover, choosing ϕ T = ϕ T in (25), we obtain the following estimate v L 2 (0,T ) u 0 X φ(0, .) X .

Using our hypothesis, we are done. (⇐) If v is an admissible control with continuous dependence on u 0 , Proposition 15 gives us, for every ϕ T ∈ X, < ϕ(0, .), u 0 > X = T 0 ϕ(., 0)v.

Choosing now u 0 = ϕ(0) gives us the estimate ϕ(0, .) 2 X C 2 ϕ T H u 0 X that is ϕ(0, .) X C 2 ϕ T H .

s 5 σ0

 5 φ5 e -2s α| φ| 2 + ε 2 s σ0 φe -2s α| φt | 2 , ε -1 T ) + ε 1/3 T 2/3 and using that ε 2 s 3 T 2 σ0 φ5 e -2s α| φ| 2 s 5 σ0 φ5 e -2s α| φ| 2

2 . 2 L 2

 222 Consequently we get that ϕ ((T /4,3T /4);X)s 4 (εT ) -8 exp 4s(M -m) s ∽ (εT ) 2 + (εT ).The above constant C is consequently estimated by ε -4 e c/ε e c ′ /ε for c ′ > c and c well-chosen.

ϕ 2 L 2 (

 22 (T /4,3T /4);X)

  4L) -c ′ > 0 provided that T L > 8 + 32c ′ .

H

  -C ϕ T H for ϕ T ∈ H. Thus J possesses a global minimum ϕ T ∈ H, which gives, writing Euler-Lagrange equations, ∀ϕ T ∈ H, T 0 ϕ(., 0) φ(., 0) =< ϕ(0, .), u 0 > X .

Ω ρ|ϕ| 2 ≥ r(1 -r) ε Ω ρ|ϕ| 2 + (1 -r) Γ ρ|ϕ| 2 -2ε Γ ρϕ t ϕ.On the other hand, it is straightforward that d dt

and, consequently, using that r ∈ (0, 1), we have obtained

Gronwall's lemma combined with exp(-r ε L) ρ(•) 1 successively gives

X

We finally choose

which gives the result.

We will now use this dissipation estimate with our Carleman inequality to get the desired result.

Conclusion and open problems

The question of controllability in higher dimension is open and seems to be hard to obtain using Carleman estimates. Another approach to obtain the observability estimate in the n-dimensional case could be similar to the one of Miller (see [START_REF] Miller | On the null-controllability of the heat equation in unbounded domains[END_REF]Theorem 1.5]) but it seems that our problem is not adapted to that framework (we can check that the link between the one-dimensional problem and the general one is not as simple as it may seem, see Appendix A below).

If n = 1, we can wonder what is the minimal time to get a vanishing control cost when the viscosity goes to zero. The intuitive result would be L, but it seems that Carleman estimates cannot give such a result.

A On Miller's trick

In this paragraph, we detail why we cannot apply Miller's method to deduce observability in dimension n from the one in dimension 1. We refer the reader to the abstract setting of [13, Section 2] (see more specifically Lemma 2.2 in that reference). We denote by Ω 1 or Ω n the domains (resp. X 1 or X n the function spaces, A 1 or A n the evolution operators) in dimension 1 or n. The observability operator in dimension one is defined by

.

where 1 {x=0} is the characteristic function of {x = 0}.

With similar notations, it is obvious that the observability operator on the part of the boundary Γ 0 = R n-1 × {0} is given by

and coincides with the tensorial product I ⊗ O 1 between the identity of L 2 (R n-1 ) and O 1 .

We have now the following result.

Proposition 17 We define the natural "difference" between A n and A 1 by

(∆ ′ denotes the standard Laplacian on R n-1 ) with domain

Then A n does not extend

= Ω 1 ((ε∂ 2 xn -∂ xn )u 1 )ϕ 1 -∂Ω 1 ((∂ ν + 1 {x=-L} )u 1 )ϕ 1 R n-1 wφ

and our claim is proved.


Data formats for phonological corpora

Representing annotated spoken corpora

The annotation of linguistic resources has long-standing traditions (see [START_REF] Cole | Survey of the state of the art in human language technology[END_REF]. The other chapters of this book make clear that the production of annotated resources is a laborious, time-consuming, andexpensive task. In theory, we want to make these resources available in such a way that they can be re-used by as many scholars as possible (see Ide&Romary, 2002). However, a largevariety of annotation formatshave been developed in the previous decades, each one created for a specific research task. Consequently,the resulting resources are frequently only usable by members of the individual research projects.

The goal of the present chapter is to explore the possibility of providing the research and industrial communities that commonly use spoken corpora with a set of well-documented standardised formats that allow a high re-use rate of annotated spoken resources and, as a consequence, better interoperability across tools used to produce or exploit such resources. We hope to identify standards thatcoverall possible aspects of the management workflow of spoken data, from the actual representation of raw recordings and transcriptions to high-level content-related information at a semantic or pragmatic level. Most of the challenges here are similar to those for textual resources, except for, on the one hand, the grounding relation that spoken data has to illocutionary circumstances (time, place, speakers and addressees), and, on the other hand, the specific annotation layers that correspond to speech related information (e.g. prosody), comprising multimodal aspects such as gestures.

We should also not forget, as is well illustrated in this book, the importance of legacy practices in the spoken corpora community, most of them resulting from the existence of specific tools at various representation layers, ranging from basic transcription tools (Transcriber, PRAAT) to generic score-based annotation environments (TASX, Elan, CLAN/CHAT (CHILDES), EMU). By definition, these various tools do not have the same maintenance rate and capacity and it is therefore essential to think about standardised formats as offering the possibility to be embedded with existing practices. This implies that we have two basic scenarios in mind:

 We want to be able to project existing data into a range of standardised representations that bear as little specificity to the original format as possible but as much faithfulness as necessary;

 We want standardised formats to havethe capacity to be used for the development of new technical platforms, thus allowing the integration of new requirements and new features.

These two general requirements both imply standards that can incorporate features and data we have not yet envisioned. To do this, the standards should provide specification or customisation mechanisms that do not hinder their abilityto improve interoperability.

That said, it is clear that such a thorough set of standardscannot be fully describedina single book chapter. Moreover, we acknowledge that there is still some work to be done before we will have a convincing portfolio of standards that cancover all aspects of annotated spoken corpora. For these reasons, we are adopting an intentionally selective (and hence subjective) strategy, with the goal of laying out a foundation that can serve as a basis to complete the standardisation picture step by step.

After a brief introduction1 to existing standardisation activities for language resources in general, we will provide some basic concepts related to the representation of annotated linguistic content. We will present in detail some of the proposals that may be used for the transcription and annotation of spoken data, along with the possibility of defining precise semantics for the corresponding representations.

Standards and standardisation processes

It has become common tospeak of two kinds of standards:de facto standards, whicharisethrough the practices of active communities and are adopted over the years, andde jure standards, which are created "from scratch" and promulgated by official standardisation bodies. Such a dichotomy is misleading, since the actual development of standards is usually accomplished by cooperation from both of these sides. Indeed, we suggest that standardisation isa process with three essential components:

 Consensus building within a technical community, including the involvement of reliable experts and the consideration of existing practices and developments;

 The wide availability of the standard so that any potential user may determine how much he or she is complying to it;  A maintenance process, through which existing defects or necessary improvements maybe implemented in further revisionsof the standard, while taking care of backward compatibility issues.

These processes are the basis for most standardisation bodies, including official national andinternational organisations such as ISO or IETF, or consortium based bodies such as the W3C, OASIS or the TEI. Many standard proposals that do not arise from these processes (usually those initiated within dedicated research and development projects) have failed or suffered due to the lack of communitysupport that could provide for dissemination and maintenance of the standards.

For language resources, we can identify three main organisations thatplay the most important role in standards:

Which standards for linguistic annotation of spoken corpora?

To understand standards for language resources, it is important to understand the various activities that the standardisation organisations mentioned above are pursuing. In the following paragraphs, we suggest a possible overall strategy to achieve the best standard-based approach to the management of linguistic data and justify the biased approach taken up in the rest of the paper.

Various user scenarios -various standards

It is important to consider how standardisation relates to possible organisation levels of spoken corpora. In general, these organisation levels include:

 The first important level of representation in phonological corpora is the transcription, where the source signal, in the form of an audio or video file, as well as any additionalinformation provided by specific sensors (e.g. articulatory) is segmented and classified as a set of symbolic codes.

Such codes may be phonetic or orthographic ones, but may also correspond to any kind of features or patterns that are deemed useful for further analysis of the primary source. Transcription is understood as a process which theoretically should be independent of further annotation steps;

 Anchored to the transcription layers (also referred to as tiers), but also to other prior annotations, a given annotation layer is identified as providing a certain type of interpretation of the primary source, whether this is linguistic (e.g. the identification of syntactic constructs) or of any other possible kind (e.g. identification of pathological features in the speaker's voice). As we shall see, the specification of an annotation level relies on the provision of its internal logic (meta-model) and the corresponding elementary descriptors (data categories);

 Finally, an important aspect of corpus annotation relies on the proper management of the combination of annotation layers (also called tiers in phonological corpora), as well as the corpus of primary sources used within a given transcription and annotation campaign. Tool implementers and project managers are usually those who consider these specific aspects.

The second important aspect to consider is the ecology within which a given corpus creation project will take place and how much this may impact the issue of formats. In general, specific standards for representing a given transcription or annotation layer are chosen based on a wide variety of factors:

 In some cases, the choice will simply be dependant on the formats employed by the software used for the annotation task, and, to a lesser degree, how the tool exports data and files;

 The targeted representation format of an annotated corpus may depend on the kind of treatments that will be further operated upon the data. The capacity, for instance, of a query environment to have a more or less deep understanding of complex annotations or of combinations of various mark-up schemes will increase, or not, the actual requirements on the data formats;

 One has to consider which data structure the final corpus will be recorded in and archived in the long run. Indeed, combining too many heterogeneous formats, which might not all have the same level of stability and documentation, may hinder the further exploitation of the data outside (in time and space) the initial production locus;

 Finally, an important factor is the culture that a given community shares about standards and how difficult it is for community members (and groups of them) to change their practices. This learning curve effect usually explains why communities tend to design their own formats, to be able to progressively add layers of complexity.

Basic components of an annotation schema

As explainedin the various contributions tothis book, each annotation tool tends to come with its own annotation schema and, in turn, each annotation schemais defined according to its own technical principles, mostly resulting from both legacy practices in the corresponding research environment and the actual preferences of the implementer. As a whole, it is seldom the case that an annotation schema results from a clear conceptual analysis where, in particular, the modelling (e.g. based on a UML specification) and representation (in the form of an XML schema for instance) levels are clearly differentiated (cf. [START_REF] Zipser | A model oriented approach to the mapping of annotation formats using standards, Workshop on Language Resource and Language Technology Standards[END_REF]). If we want, in this context, to move toward better interoperability across the existing initiatives within the spoken corpora community, it is necessary for us to introduce some basic elements that will act as references for comparing existing schemas and above all for mapping them onto common principles and standards.

The first stage for us is to define what is meant by an annotation and identify its various components. As illustrated in Figure 1, we consider that an annotation is a combination of three components, a source, a range and a qualifier, that have the following characteristics:

 The source3 is the information upon which some additional statement is made in the context of the annotation. It is considered as a fixed object from the point of view of the annotation (i.e., changing the source invalidates the annotation);

 The range identification characterises a portion of the source (a markable) that is being qualified by the annotation, either as one or several already identified parts of the source, or by reference to a certain identification scale (e.g. a temporal or spatial reference) that maps onto the source;

 The qualification expresses a constraint on the actual portion of the source as elicited by the range. This constraint is made of an elementary piece of information, mostly expressible as a feature-value pair.  Direct reference to a (generally temporal) scale that transitively relates to the source. This is the basic mechanism provided by simple models such as annotation graph [START_REF] Bird | A formal framework for linguistic annotation[END_REF]. In such situations, there is no possibility to express an explicit co-occurrence relation between two annotations, except for identifying that one temporal reference is, for instance, the same. This is typically the strategy implemented in tools such as ANVIL or Praat (see [START_REF] Schmidt | A TEI-based Approach to Standardising Spoken Language Transcription[END_REF] ;

 Reference to reified objects on a scale. In the case of a temporal scale, this corresponds to the identification of events to which more than one qualifier may refer (i.e. a timeline, as in EXMARaLDA and ELAN, see [START_REF] Schmidt | A TEI-based Approach to Standardising Spoken Language Transcription[END_REF];

 Reference to explicit components of the source, allowing one to skip in some sense the actual ranging mechanism, or at least to make it boil down to a simple pointer or group of pointers. The important partof this last possibility is that it allows annotations to be about any kind of entity, including annotations themselves. This is usually the case for all annotation tools adopting a pure stand-off strategy such as MMAX (Müller andStrube, 2001).

An annotation is defined at a very low level of granularity, so thateach elementary statement upon a source (e.g. an elementary provision of some part of speech information about a word) is potentially embedded within a single annotation. Naturally, this does not preventspecific implementations fromproviding explicit factorisations that may facilitate the reduction of redundant information across annotations. For instance, a morpho-syntactic annotation schema may want to combine all information relevant toa given word by conflating all descriptors associated to a single range as a tagset label.

Because there are so many types of linguistic annotation, annotations are often grouped according to different criteria. The reason for the grouping is technical and/or conceptual. To distinguish between these two different groupings, a distinction between annotation layer and annotation levelhas been introduced

(see for exampleGoecke et al. 2010).A short description of this distinction can also be found in [START_REF] Witt | Multiple Hierarchies: New Aspects of an Old Solution[END_REF]:

To avoid confusion when talking about multiply structured text and text ideally organized by multiple hierarchies, the terms "level" or "level of description" is used when referring to a logical unit, e. g. visual document structure or logical text structure. When referring to a structure organizing the text technically in a hierarchically ordered way the terms "layer" or "tier" are used. A level can be expressed by means of one or more layers and a layer can include markup information on one or more levels.

Furthermore, it is possible to conceptualise the underlying coherence that is required when optimizing an annotation schema by defining the notion of annotation level as a coherent set of annotation types sharingthe following characteristics:

 Same underlying source, or set of sources (in the case of a corpus);

 Same ranging mechanism, by which we mean not only the same referring mechanisms (component or scale), but also a coherent description of ranges from the point of view of their linearity, possible overlapping or alternation;

 Precisely defined and comprehensive data category selection that is applicable for qualifiers. We predict a general notion of tagset as such a selection.

With this general analysis in mind, we can now take a more precise look at the current state of standardisation processes forlanguage resources.

Providing a reference semantics for linguistic annotation

One important aspect ofrepresenting anykind of annotation is the capacity to provide a clear and reliable semantics for the various descriptors that are being used, either in the form of features and featurevalues, or directly as objects in a representation expressed, for instance, in XML. In order to be shared across various annotation schemas and encoding applications, such a semantic should be implemented as a centralised registry of concepts, which we will henceforth refer to as data categories. As such, data categories should bear the following constraints:

 From a technical point of view, they must provide unique and stable references (implemented as persistent identifiers) such that the designer of a specific encoding schema can refer to them in his or her specification.

By doing so, two annotations will be considered as equivalent when they are actually defined in relation to the same data categories (as feature and feature-value);

 From a descriptive point of view, each unique semantic reference should be associated with precise documentation combining a full text elicitation of the meaning of the descriptor with the expression of specific constraints that bear upon the category.

In recent years, ISO has developed a general framework for representing and maintaining such a registry of data categories, encompassing all domains of language resources. This work, carried out in the context of ISO project 12620

[ISO 12620], has led to the implementation of an online environment providing access to all data categories which have been standardised in the context of the various language resource-related activities within ISO, or specifically as part of the maintenance of the data category registry. It also provides access to the various data categories that individual language technology practitioners have defined in the course of their own work and decided to share with the community.

The ISO data category registry, as available through the ISOCat implementation, is meant to be a "flat" marketplace of semantic objects, providing only a limited set of ontological constraints. {/masculine/ and /feminine/}.

In this section, we have tried to delineate a comprehensive view on annotations that, as it were, encompasses all types of representations within a multi-tier annotated corpus. Indeed, any kind of information added to a bare primary source (like an audio recording), from low-level segmentation markers to highlevel discourse relation identification, can be seen as an annotation in the sense presented here.

Language resource management -an ISO perspective

Specific ISO models and formats for linguistic annotation ISO committee TC 37/SC 4, launched in 2002, focuses on the definition of models and formats for the representation of annotated language resources. To this end, ISO/TC 37/SC 4 has generalised the modelling strategy initiated by its sister committee SC 3 for the representation of terminological data [START_REF] Romary | An abstract model for the representation of multilingual terminological data: TMF -Terminological Markup Framework[END_REF],

and through which linguistic data models are seen as the combination of a generic data pattern (a meta-model), which is further refined through a selection of data categories, which provide the descriptors for this specific annotation level. Such models are defined more or less independently from any specific formats (not even bound to an XML framework), and ensure that an implementer has the necessary tool to design and compare formats, with regard to their degrees of interoperability. In the rest of this section, we will go through several projects 5 from ISO/TC 37/SC 4 that are important for phonological corpora.

One of the early proposals of ISO/TC 37/SC 4 has been to outline a possible standard for morpho-syntactic (also referred to as part-of-speech) annotation.

Such an annotation level corresponds more or less to the first linguistic abstraction level for a corpus and, depending on the language to be annotated and the actual characteristics of the tool that is being used, can vary enormously in structure and complexity. In order to deal with the complex issues of ambiguity and determinism in morpho-syntactic annotation, ISO 24611/MAF makes a clear distinction between the two levels of tokens (representing the surface segmentation of the source) and word forms (identifying lexical abstractions associated to groups of tokens). These two levels have the specificities that, on the one hand, they can be represented as simple sequences as well as local graphs (e.g. multiple segmentations, ambiguous compounds, etc.), and, on the other hand, any n to n combination can stand between word forms and tokens6 . Associated to this meta-model, MAF provides a default XML syntax, but as we shall see later in this chapter, it is also possible to contemplate a TEIbased implementation for it.

For syntactic annotation, however, ISO committee TC 37/SC 4 did not reach an early consensus on a possible XML syntax that would cover the variety of possible syntactic frameworks (constituency or dependency based, theory specific) that can be observed either within existing treebanks (Abeillé, 2003) or as export formats of syntactic parsers (Ide&Romary, 2003). The published standard (SynAF, ISO 24615) is thus centred on a comprehensive meta-model informing the whole spectrum of syntactic representation practices, coupled with an extensive list of data categories that are now available within ISOCat (see section (Broeder et al., 2008). The standard can presently be used to specify new formats or make interoperability checks7 , and a reference serialisation of SynAF that would cover the kind of features now available in such formats as Tiger2 Finally it offers a default XML serialisation that fully implements the features of the intended model10 .

As the preceding examples make clear, the focus on modelling and interoperability issues facilitates the design of a given corpus as the combination of basic standardisation building blocks, which can then be adapted by projects to handle legacy data or tools. It also allows one to anticipate possible transitions to make existing data more and more compliant to international standards when they are adopted within a scholarly community.

Genericity made a principle: LAF -GRAF

In cases where no standardisation activity for a specific annotation level exists, or, as is usually the case, when a variety of annotation levels have to be merged within one single information pool in order to carry out cross-level queries or visualisation, there is a need for a high level representation that basically unifies all types of specific annotation structures. Various proposals have been suggested to addressthis situation, including projects such as ATLAS [START_REF] Steven | ATLAS: A flexible and extensible architecture for linguistic annotation[END_REF], Mate (McKelvie et al., 2001) or more recently the American National Corpus (ANC; [START_REF] Ide | The American National Corpus: A Standardized Resource of American English. Corpus Linguistics[END_REF]. The American National Corpus project was an opportunity to experiment and finalise the principles enunciated in the ISO LAF project, on the basis of a generic graph representation where nodes represent the reification of linguistic annotation components and edges relations between them. Based on the ISO-TEI feature structure standard for the further qualification of nodes and edges, LAF offers a default format (called GraF) for the serialisation of any type of linguistic structure.LAF wascreatedto provide easy mapping with similar past and present initiatives such as annotation graphs, or PAULA. It is also an important step in contemplating generic query mechanisms and perhaps a standardised query language for language resources.

Linguistic annotation with the TEI

In many respects, the TEI appears to be a most appropriate method to a) describe primary transcription of phonological corpora and b) implement the models provided by ISO standards [START_REF] Romary | Questions & Answers for TEI Newcomers[END_REF]. Indeed, the Text Encoding

Initiative can be a good entry point for anyone looking for a generalpurpose XML vocabulary, which in turn may be connected to -and thus made interoperable with -many other corpora and encoding initiatives.

In the rest of the paper, we show how the TEI guidelines already offer a variety of constructs and mechanisms to cope with many issues relevant to spoken corpora and their annotations. When applicable, we will make the necessary links with ongoing ISO/TC 37 activities so that some clues are given as to how a possible transition to more elaborate annotation schemas, or possibly a mapping from basic TEI representations to other annotation schemas, could be implemented.

The TEI framework for transcribing spoken corpora

The 1994), the TEI also addresses the topic of annotating transcribed speech. After a revision of the Guidelines in 2002 that mainly switched from an SGML-to a fully XML-compliant syntax of the annotation, the most recent version of the TEIannotation scheme was published as TEI P5 in 2009 as a "living document" that is continuously updated. This section describes TEI's approach totranscribing spoken language according to P5 (TEI 2011). However, as the TEI consortium has been very careful with their updates and changes -especially the chapter on the transcription of spoken languages, which has only seen a few minor changes over the years -older TEI-based annotations are still usable without much effort.

The general structure of the TEI encoding framework is highly modularised.

About 30 specialised TEI modules exist, for instance for dictionaries, verse text, dramas, linguistic analysis, and speech transcriptions. Moreover, it is also possible to define freely specialised tag sets for all purposes not addressed by existing TEI tags.

Independent from the type of the annotated document, i.e. regardless of the used TEI modules, all TEI documents are subdivided into two major parts: the TEI-Header containing the metadata of the annotated resource, for instance information on the time and place a dialogue took place; and the annotated resource itself, for instance the transcription of the spoken dialogue. (see listing)

<TEI> <teiHeader> <!--Comment: --> <!--the metadata of the annotated resource are included here --> </teiHeader> <text> <!--Comment: --> <!--the annotated resource itself is included here --> </text> </TEI>

The following sections describe the TEI-metadata and TEI-annotations with a strong focus on options to deal with spoken language. This entails the omissionof many aspects of TEI. The complete guidelines with its some 1300 pages are available on the TEI website (http://www.tei-c.org).

The TEI Header

The header of the TEI document contains all the metadata associated with a spoken text. This information is subdivided into four different major classes: (1) the file description, (2) the encoding description, (3) the profile description, and

(4) the revision description. While the revision description does not contain information specifically relevant tophonological resources, the other three do.

Apart from the file description, all other parts of the header can be omitted.

<teiHeader> <fileDesc> <!--... --> </fileDesc> <encodingDesc> <!--... --> </encodingDesc> <profileDesc> <!--... --> </profileDesc> <revisionDesc> <!--... --> </revisionDesc> </teiHeader>

Information about the file

There are only three necessaryparts to a TEI Header. All of them must be included as children of the file description, annotated as <fileDesc>. These necessaryelements are used to provide information about the title (<titleStmt>), a publication statement (<publicationStmt>), and a description of the source of the annotated text (<sourceDesc>). In some respects, the file description contains information usuallyregarded as metadata. In case of annotated speech resources, this class also allows the representation of information about the source of the transcription, almost always a recording. Technical data of a speech recording can be included in the information contained in<sourceDesc>. Such data include file format information (e.g. uncompressed wav, compressed mp3

or ogg, the sampling frequency), specifications of the audio equipment (e.g. the number and the type(s) of microphone(s)), the source of the recording (e.g.

original recording, broadcast transmission), etc. For this kind of information the <recordingStmt> (recording statement) with its sub-element <recording> (recording event) are available in the header of a TEI document that contains the transcription of speech. The type of recording could also be 'video'. In addition tothe description of the <equipment> used to prepare the <recording>, the element <broadcast> could be used if the source wererecorded from radio or TV. Of course, since the broadcast speech was also recorded before transmission, it is possible to include the element <recording> in <broadcast>, as well. This exemplifies how rich the TEI's metadata description can be whenneeded.

Information about the encoding

The encoding declaration "documents the relationship between an electronic text and the source or sources from which it was derived" (TEI P5). Besides other information the element <encodingDesc> allows a tagging declaration to provide detailed information about the tagset used in the document, the feature system declaration <fsdDecl> that could be used when applying feature structures, and the element <geoDecl> for the declaration of the geographic coordinates.

Because a lot of transcriptions of spoken language are prepared (semi-)automatically, for instance with the tools described in this volume, one might want to mention which tools have been used for this task in the metadata. The element <appInfo> allows the specification of a list of applications used for preparing the transcription. <appInfo> <application version="1.4.4" ident="EXMARaLDA"> <label>EXMARaLDAPartitur-Editor</label> <ptr target="#dialog2"/><ptr target="#dialog132"/> </application> </appInfo> This example defines the application EXMARaLDAPartitur-Editor 1.4.4 and specifies two dialogues that have been transcribed with this tool.

Information about the profile

A comprehensive description of the languages used by the speakers, information about the situation in which the speech recording took place and other nonbibliographic metadata can be specified in a profile description.

One important component for the transcription of speech, especially when elicited in an experiment, is the <settingDesc>. By means of this element it is possible to provide information about the place, date, activities etc. of the speech interaction. It could also be used to refer to controlled settings as e.g. in Maptask- [START_REF] Anderson | The HCRC Map Task Corpus[END_REF] and Tinkertoy [START_REF] Senft | Spatial reference in Kilivila: The Tinkertoy Matching Games -A case study[END_REF] experiments.

It is possible to provide very fine-grained metadata with very detailed specifications of a participant in a dialogue. Within the <profileDesc> the element <partDesc> can be used to include information about participants in a conversation by means of a list of <person>-elements. This element enablespersonal data for a person to be included, for instance: <person sex="2" age="infant"> <birth when="2010"> <date>12 Jan 2010</date> <name type="place">Berlin, Germany</name> </birth> <langKnowledge tags="de "> <langKnown level="first" tag="de">German</langKnown> </langKnowledge> </person>

TEI-based transcription

In this section, we discusshow the TEI can be used for spoken data, using a dialogue from Thomas Schmidt's article in this volume as an example. In this example the persons communicate verbally in French as well as through gestures. A translation into English and additional information are also provided.

Furthermore, the alignment of the characters and the timeline indicate the sequence and the overlap of information. The <body> embeds one or more 'utterances' (<u>). Within an element <u> an orthographic or a phonetic transcription is included. Since this element may contain text, it is possible to include annotations in non-XML-based conventions.

The following example uses the convention GAT (see Schmidt, this volume) to mark a non-linguistic event.

<u>Alorsçadépend ((cough))un petit peu.</u> Such an approach allows researchers to continue to use conventions they are used to. At least, they can do so to a certain extent, as long as the annotation conventions do not contradict constraints pertaining to text data in XML documents. This means, in particular, that characters like '<' or '&' cannot be included directly, instead they have to be represented as so-called XML entities.

As a result of this restriction, the widely used CHAT conventions (REF) cannot be included here directly.For example, the event in the samplesentence above would be annotated as &=coughs in CHAT syntax. But even in cases that do not lead to such difficulties, it is not recommended to mix syntactic variants. The TEI tagset for transcriptions of spoken language defines several elements for the integration of annotation. The TEI-conformant representation of the utterance given above would be: <u>Alorsçadépend<vocal><desc>cough</desc></vocal> un petit peu.</u>

In general, the element <vocal>should be used for non-or semi-lexicalised sounds. Other elements, like <kinesic> and <incident> could be used to mark gestures, environmental noise, etc.:

<kinesic><desc>right hand raised</desc></kinesic>
Because XML documents are, technically, nothing but a sequence of characters, indentation and visual alignment are not usable in order to indicate relations like the synchronicity or overlap of utterances, gestures, occurrences in the environment etc. Instead of visual alignment, XML enforces the use of special mark up in order to make such relations explicit. This can be done according to varying degrees of detail. On the one hand, we can mark up information corresponding to simple statements like "a speaker started an utterance before the other speaker finished her utterance". On the other hand, we may have something like an explicit reference to a timeline. The following example shows an approach whose grade of granularity ranges between these two extremes: <body> <u who="#SPK1">Okay. Trèsbien, <anchorxml:id="tp1u"/>trèsbien.<anchor xml:id="tp2u"/></u> <u who="#SPK2"><anchor synch="#tp1u"/>Alorsçadépend <vocal><desc>cough</desc></vocal> <anchorxml:id="tp2u"/>un petit peu.</u> <kinesic who="SPK1" type="nv" start="#tp1u"> <desc>right hand raised</desc></kinesic><anchor synch="#tp2u"/> <u who="#SPK1">Ah oui?.</u> </body>

In this example, the overlapping speech of the two speakers is indicated by the inclusion of an anchor within the first utterance at the point where the second speaker starts his or her first utterance. At this very point the first speaker starts a gesture that ends when the second speaker begins the phrase "un petit peu".

Besides this explicit information about the temporal relations of the different utterances and gestures, implicit temporal information is also included in the XMLfile, simply due to the serialisation of the XML document. If there is no explicit information about overlaps, then it is implied that the communication events (speech, gestures etc.) have been produced sequentially one after the other. In the example above, this means that the last utterance 'Ah oui?.' starts after the completion of its previous speech turn "Alorsça depend ((cough)) un petit peu." The most precise approach to keep the temporal information is referencing each event to relative or absolute time points. This can be done by including the TEI element <timeline>, the definition of relevant time points and linking from utterances etc. to them. In the annex of this chapter a complete example that makes use of this technique is given.

One of the most interesting benefits when using a TEI-based approach to annotate speech corpora is the possibility of including elements from all other TEI modules. One of these modules is described in the TEI guidelines in chapter 17,"Linking, Segmentation, and Alignment". It not only provides elements for a highly sophisticated addressing and linking mechanism, but also an element <seg> that allows the grouping of text fragments as long as the XML constraints are met. So, naturally, it is not possible to split elements in a way that results in overlapping markup. The element <seg> might be used with the attribute 'xml:id' to provide unique identifiers. This allows, whenever needed, a direct referencing to arbitrary text segments. Another example of the use of the <seg>element given in the TEI Guidelines (TEI 2011, pp. 464f.) is reproduced below: <seg type="sentence" subtype="declarative"> <seg type="phrase" subtype="noun"> <seg type="word" subtype="adjective">Literate</seg> <seg type="word" subtype="conjunction">and</seg> <seg type="word" subtype="adjective">illiterate</seg> <seg type="word" subtype="noun">speech</seg> </seg> <seg type="phrase" subtype="preposition"> <seg type="word" subtype="preposition">in</seg> <seg type="word" subtype="article">a</seg> <seg type="word" subtype="noun">language</seg> <seg type="word" subtype="preposition">like</seg> <seg type="word" subtype="noun">English</seg> </seg> <seg type="phrase" subtype="verb"> <seg type="word" subtype="verb">are</seg> <seg type="word" subtype="adverb">plainly</seg> <seg type="word" subtype="adjective">different</seg> </seg> <seg type="punct">.</seg> </seg>

In this example the element <seg> is used to segment a sentence into phrases and words and to associate more detailed information like the phrase type or the part of speech with the segments. However, the guidelines also make clear that a more appropriate annotation of linguistic information is available in the module "Simple Analytic Mechanisms", because this module defines not only specialised elements for sentences (<s>), phrases (<phr>) and words (<w>) but also for morphemes (<m>) and syllables (<syll>).

Annotating corpora with the core mechanisms of the TEI Using feature structures within an annotation scheme

In this section we address the implementation of what we named the'qualification level' by means of feature structures and compare it with the general model for elementary annotations describedabove. Feature structures [START_REF] Pollard | Information-Based Syntax and Semantics[END_REF] are formal structures which combine a basic representation mechanism by means of a possibly recursive combination of feature-value pairs, where values can in turn be feature structures and associated operations in order to access, filter or unify such structures. Feature structures have been used as the reference mechanism for various unificationbased formalisms andalso as a descriptive tool in order to attach basic properties to a linguistic segment (e.g. for phonetic descriptions, [START_REF] Bird | Phonological analysis in typed feature systems[END_REF]).

Complementing this well-established scientific background, an XML-based representation for feature structures has been developed since the early days of the Text Encoding Initiative by Terry Langendoen and Gary Simons (1995), and has been further improved and stabilised in the context of a joint TEI-ISO activity (ISO 24610-1, henceforth ISO-TEI-FSR).

The representation of feature structures in ISO-TEI-FSR is based upon two central elements:

 <f> which contains a single feature-value pair  <fs> which groups together one or several feature-value pairs A simple feature-value pair is described by means of the name of the feature (attribute @name) and its value, expressed as the content of the <f> element. In the canonical ISO-TEI-FSR this value is systematically typed by means of an embedded element, which can either be <binary> (with attribute @value=true/false), <symbol>, <numeric> or <string>.

For instance, the expression of a part of speech value for a noun would typically look like this: <f name="partOfSpeech"> <symbol>noun</symbol> </f> When combined, several feature-value pairs should be embedded within a feature structure, which can optionally be further typed, for instance to provide direct access to all feature structures associated withthe same annotation level.

For instance, a basic morphosyntactic qualification block could be represented as:

<fs type="morphosyntacticAnnotation"> <f name="partOfSpeech"> <symbol>noun</symbol> </f> <f name="grammaticalGender"> <symbol>masculine</symbol> </f> <f name="grammaticalNumber"> <symbol>plural</symbol> </f> </fs> As an illustration of the way feature structures can be used to describe the basic components of an annotation scheme, let us show how atagset can be covered with this framework.

Creating tagsetsthrough feature structure libraries

Rationale

The main issue regardingtagsets11 as reference descriptions for morphosyntactic annotations is that they can be shared across corpora and annotation tools. In particular, a tagset articulates the relation between a concrete syntactic representation within a set of annotationsanda reference semantics that may allow one to interpret the annotation further when exploring the annotated data.

To this end, the ISO-TEI standard provides mechanisms for declaring feature and feature-value libraries that perfectly match the objective stated here.

In the following section we will briefly outline a possible method for declaring tagsets in the feature structure framework, to show that such a method could be used as reference to actually document, record and compare the various tagsets used within the linguistic and computational linguistic communities.

Description of an elementary tag

The first step in the process of declaring a tagset is the ability to describe elementary features. This can easily be achievedwiththe ISO-TEI standard by combining elementary feature statements such as those seen above within a feature library (fLib), together with a systematic identification of each feature (by means of an xml:id attribute).

In the following example, the three elementary features corresponding to the grammatical gender possibilities in German are described accordingly.

<fLib n="grammatical gender" > <f name="grammaticalGender" xml:id="fem"> <symbol value="feminine"/> </f> <f name="grammaticalGender" xml:id="mas"> <symbol value="masculine"/> </f> <f name="grammaticalGender" xml:id="neu"> <symbol value="neuter"/> </f> </fLib> It can be noted here that if desired, one may fragment the various types of features (grammatical category, gender, number etc.) within separate <fLib> constructs or just group them all together within a single one. For instance, and in order to have all the illustrative material at hand, we could have the following series of declarations for grammatical categories: <fLib n="grammatical category"> <f name="partOfSPeech"> <symbol value="commonNoun" xml:id="#NC"/> </f> <!--further grammatical categories here --> </fLib> as well as for grammatical number: <fLib n="grammatical number"> <f name="grammaticalNumber"> <symbol value="singular" xml:id="sing"/> </f> <!--further values for grammatical number here --> </fLib>

Description of a complete tagset

Once all the elementary declarations are made, the ISO-TEI framework allows one to combine them to declare feature-value libraries (fvLib), within which a feature structure combining elementary morpho-syntactic features corresponds to a tag in the tagset in a one-to-one manner. In the following (simplified) example, for instance, the tag for a masculine singular common noun is declared and provides the appropriate identifier for further reference: <fvLib> <fsxml:id="Ncms__" feats="#NC #mas #sing"/> <!--further tags declared here --> </fvLib> Once such a full tagset is described, the various entries maybe reused in many different ways. In a proprietary format, it may simply be referred to in the documentation in order to provide a formal reference to the corresponding annotation scheme, or, when available, it can be referred to within the declaration section of an annotation file. In the case of a fully TEI-based representation, a possible mechanism is to see a tag as an analysis of a linguistic segment and point to the declaration by means of the @ana attribute, as in the following example 12 : <p><w>Le</w><w>petit</w><w ana="#Ncms__">chat</w><w>est</w><w>mort</w><pc>.</pc></p>

Towards maintainable and sustainable specifications

The standard-based description of tagsets outlined above only makes sense if the actual specifications can actually be re-used as a reliable reference across various annotation projects. Even in basic cases, where such feature-structure libraries can be imbedded within the document containing the data itself (like in the <back> component of the TEI document), this is obviously not a good strategy if one wants to maintain and disseminate a tagset specification in a sustainable way. It is thus a recommended best practice to integrate tagset specifications within their own TEI document, which in turn lets one document and record origin and versioning information in the corresponding TEI header.

Once one has a stable tagset specification at hand, it is probably time to consider a dissemination and standardisation strategy. First, we recommend storing the specification in a stable registry, with version control mechanisms (such as SourceForge). This can be a way to involve a wider community in using and reacting to the proposed tagset. The second stage is to build a real standardisation strategy, either by making the tagset a recommendation of an institution or a research infrastructure (such as CLARINor DARIAH), or by actually making this a contribution to ISO/TC 37/SC 4 (as a technical report, for instance). It should be noted here that any such move toward a wider publication of a specification will result in requests for evolution.

A final word on the issues of publication, dissemination and above all standardisation: we recognise the need for several reference tagsets for a given language. Depending on the use case, or the expected granularity of description, tagsets may vary in the way they use and combine morpho-syntactic features.

Still, the proper publication -in a standardised format, as suggested here -of the tagset specification, as well as its systematic anchoring tothe data categories in ISOCat, will improve our capacity to provide better comparisons between them.

Range identification in the TEI framework

To complement the use of feature structures that we presented above, the TEI provides mechanisms for the annotation of ranges and their linking to qualifications (as described in Figure 1). In the following section we will briefly describe this mechanism in order to provide a comprehensive package for linguistic annotation.

The central element for range identification in the TEI guidelines is <span>, which specifies, by means of a @from and a @to attribute, a sequence within a document to which ones want to make an annotation. In its simplest form, <span> allows one to make a plain text comment in the element content. In the case of formal annotations, <span> bear the @ana attribute that we have already seen to point to a structured qualification such as a feature structure. Furthermore, the TEI provides a <spanGrp> element to put together all span descriptions that correspond, for instance, to the same annotation level.

To illustrate the possible use of the <span> element in a concrete annotation case, let us consider the morpho-syntactic annotation of a linguistic sequence in conformance with the ISO MAF proposal. By construction, the MAF meta-model makes a clear (and essential) distinction between a token level and a word form level. The token level corresponds to the identification of elementary segments on the linguistic surface, whereas word forms are abstract lexical items identified across spans of one or several tokens. This model can be implemented within a full TEI-based representation by means of <span> as follows: the transcription is initially tokenised by means of the <w> element, as presented before. We take here a simple sequence ("pomme de terre", potato) corresponding to a compound lexical item:

<uwho="#speakerA"> ... <w xml:id="t1">pomme</w> <w xml:id="t2">de</w> <w xml:id="t3">terre</w> ...

</u>

The word form level is then implemented by means of <span>s that can be set together within a single <spanGrp>: <spanGrp type="wordForm"> ... <span from="#t1" to="#t3" ana="#pomme_de_terre_sing"/> ... </spanGrp> Each <span> is actually pointing to a reference lexical entry and more precisely to the corresponding inflected form. Such a lexical entry can be implemented, in compliance with ISO LMF, as a TEI <entry> element, as follows (excerpt): <entry> <form type="inflected" xml:id="pomme_de_terre_sing"> <orth>pomme de terre</orth> <gramGrp> <number>singular</number> </gramGrp> </form> <form> ... </form> </entry>

Conclusion -further standard developments in the domain of spoken corpora

In roughly the last 25 years, pioneering work has laid the groundwork for a wide-coverage standardisation framework, which, combining the existing background from both the TEI and ISO, offers a wide range of possibilities to deal with both primary transcriptions and higher level annotations. In this paper we hope we have conveyed the message that, within what could appear as an intricate jungle of standards, it is possible to identify some baseline formats allowing one to start putting together a corpus project within some stable normative environments such as the TEI. By doing so, we also want to suggest that the phonological corpora community should become less and less dependent upon proprietary formats created for specific projects and design its own standardisation roadmap to both improve existing proposals and fill in the gaps that still exist in this domain (e.g. representation of multiple tiers within an annotated corpus). This should be accompanied by a stronger involvement by the spoken corpus community in standardisation bodies such as ISO or the TEI, as well as more effortstoward the identification and dissemination of an optimal combination of standardswhich could be delivered as guidelines of best practices to the community.

Annex

The following XML excerpt represents a fully compliant example of a TEI-based representation for a transcription of speech. It illustrates in particular the use of a timeline mechanism to anchor the transcription toreference temporal points in the source audio file.

<TEI xmlns="http://www.tei-c.org/ns/1.0"> <teiHeader> <fileDesc> <titleStmt> <title>Title</title> </titleStmt> <publicationStmt> <p>Publication Information</p> </publicationStmt> <sourceDesc> <p>Information about the source</p> </sourceDesc> </fileDesc> <profileDesc> <particDesc> <personxml:id="SPK0"> <persName> <abbr>Peter Black</abbr> </persName> </person> <personxml:id="SPK1"> <persName> <abbr>Judith White</abbr> </persName> </person> </particDesc> </profileDesc> </teiHeader> <text> <timeline unit="ms"> <whenxml:id="T1"/> <whenxml:id="T2"/> <whenxml:id="T3"/> <whenxml:id="T4"/> <whenxml:id="T4bar"/> <whenxml:id="T5"/> <whenxml:id="T6"/> <whenxml:id="T7"/> </timeline> <body> <u who="#SPK0"><anchor synch="#T1"/>Okay. <anchor synch="#T2"/>Très bien, <anchor synch="#T3"/>très bien.<anchor synch="#T4"/></u> <u who="#SPK1"><anchor synch="#T3"/>Alors ça<anchor synch="#T4"/>depend <anchor synch="#T4bar"/><kinesic type="cough"/><anchor synch="#T5"/>un petit peu. <anchor synch="#T6"/></u> <incident who="SPK0" type="nv" start="T3" end="T5"> <desc>right hand raised</desc> </incident> <u who="#SPK0"><anchor synch="#T6"/>Ah oui?. <anchor synch="#T7"/></u> </body> </text> </TEI>

Figure 1 :

 1 Figure 1: generic structure of an annotation "Range" is defined hereabstractlybecauseexisting annotation schemas implement ranging mechanisms in different ways. These can be classified along the following lines:

(

  Romary et al., preprint) 8 is planned.Work carried out within ISO project 24617-2 provided a comprehensive framework for the annotation of dialogue acts[Bunt et al., 2010], applicable to any kind of multimodal interaction. ISO/DIS 24617-2 (Dialogue acts) can be seen at various levels of abstraction. It first provides a well-defined theoretical framework where the basic concepts of dialogue act, semantic content and communicative function are defined. Building upon the numerous initiatives and projects 9 that have taken place in the last twenty years, it defines a domainindependent meta-model providing a multidimensional description of dialogue act phenomena, coupled with data categories registered in the ISOCat registry.

  Text Encoding Initiative (TEI)beganin the late 1980s to propose approaches to annotate different types of textually represented resources. Beginning with the 3rd major edition of the TEI Guidelines (Sperberg-McQueen &Burnard,

Figure 2 :

 2 Figure 2: Example of a transcription using EXMARalDa Whereas the metadata of a speech transcription are embedded in the <teiHeader>, the actual transcriptions are part of the <body> of a TEI document.

  The objective there is to facilitate the maintenance of a comprehensive descriptive environment where new categories are easily inserted and reused without requiring any strong consistency check with the registry at large. Indeed, the following basic constraints are actually part of the data category model, as defined in ISO 12620:

 Simple generic-specific relations, when these are useful for the proper identification of interoperability descriptors between data categories. For instance, the fact that /properNoun/ is a sub-category of /noun/ allows one to compare morphosyntactic annotations which are based on different descriptive levels of granularity;  Description of conceptual domains, in the sense of ISO 11179 ([ISO 11179]), to identify, when known or applicable, the possible value of socalled complex data categories 4 . For instance, this can be used to record that possible values of /grammaticalGender/ (limited to a small group of languages, see

[START_REF] Romary | Stabilizing knowledge through standards -A perspective for the humanities[END_REF]

) could be a subset of {/masculine/, /feminine/ and /neutral/};  Language-specific constraints, either in the form of specific application notes or as explicit restrictions bearing upon the conceptual domains of complex data categories. For instance, one could express explicitly that /grammaticalGender/ in French can only take the two values:

For a precise presentation of background activities which lead to the current standardization picture, see(Ide and Romary

2007)

In the remaining text of this paper, we assume that the reader has some basic understanding of XML technologies, and in particular have no difficulty in reading through the XML samples we introduce. See also(Bray et alii 1998)

One may want to distinguish between a primary source, which is not anchored on any previous information layers and secondary source, when this can be seen as being derived from or built upon another source.

Complex data categories will typically be implemented as place-holders (or features), whereas simple data categories, will be implemented as values.

One token can correspond to several word forms, and vice versa.

Usually to assess the conformity of a data set with an expected input of a tool, and design a possible filter accordingly.

See the recent proposals by Głowińska and Przepiórkowski (2010) and Erjavec et al. (2010) for the encoding of SynAF compliant annotations by means of the TEI framework.

Cf. annotation schemes defined in such projects as TRAINS, HCRC Map Task, Verbmobil, DIT, SPAAC, C-Star, MUMIN, MRDA, AMI, and more recent attempts towards domain-independence, interoperability and standardization in DAMSL, MATE, DIT++ or the EU project LIRICS.

Even if space prevents us from providing further details on this, this serialization is inspired from the annotation framework provided by the TEI guidelines.

See for instance(Monachini and Calzolari, 1994) for the corresponding work carried out within the Multext project.