Abdelmalek Drici

Boris Haspot
email: haspot@ceremade.dauphine.fr

Remarks on global controllability for the shallow-water system with two control forces

 that the exact global controllability result does not hold for any time T .

Introduction

We consider the viscous shallow water model with friction term. This model is also called by the french community the Saint-Venant equations and is generally used in oceanography. Indeed it allows to model vertically averaged flows in terms of the horizontal mean velocity field u and the depth variation h. In the rotating framework, the model is described by the following system:

           ∂ ∂t h + ∂ x (hu) = 0, (t, x) ∈ Q = (0, T ) × (0, 1)
∂ ∂t (hu) + ∂ x (h|u| 2 ) -∂ x (µh∂ x u) + ∂ x h F r 2 + rhu = 0, (t, x) ∈ Q = (0, T ) × (0, 1) u(t, 0) = v 1 (t), u(t, 1) = v 2 (t), t ∈ (0, T ),

(1.1) F r > 0 denotes the Froude number and we set r = 1 µF r 2 . System (1.1) is supplemented with initial conditions h /t=0 = h 0 , u /t=0 = u 0 .

(1.2)

This model is derived from the compressible Navier-Stokes equations and the turbulent regime (r ≥ 0) is obtained from the friction condition on the bottom, see [START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF]. µ is the viscosity coefficient and verifies µ > 0 .

Here, T > 0 is a given final time and v 1 (t) and v 2 (t) are control functions which are acting over our system at both endpoints of the segment (0, 1). We are now going to 1 assume that the controls v 1 and v 2 belong to the space H 3/4 (0, T ) and the initial data (h 0 , u 0 ) are in H 2 (0, 1)×H 1 (0, 1) with h 0 ≥ c > 0. Furthermore we suppose the following compatibility assumption: v 1 (0) = u 0 (0) and v 2 (0) = u 0 (1). (1.3) Under these assumptions, it is classical to see that there exists a solution (h, u) of system (1.1) which belongs for a T > 0 to the space X T = L ∞ (0, T ; H 2 (0, 1))× L 2 (0, T ; H 2 (0, 1))∩ H 1 (0, T ; L 2 (0, 1)) and a constant C > 0 such that:

(h, u) X T ≤ C( u 0 H 1 (0,1) + h 0 H 2 (0,1) + v 1 H 3/4 (0,T ) + v 2 H 3/4 (0,T ) ), (1.4) (see, for instance, [START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF]). We would like to mention that we can obtain global solution provided that we choose u 0 = -µ∂ x ln h 0 (we shall detail more precisely in the sequel this point which will play a crucial role in your results, we refer also to [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF]).

In the sequel we are interesting in proving some negative results about the global exact controllability of the system (1.1). Before giving some elements of the proof we would like to recall the notion of global exact controllability property for system (1.1).

Definition 1.1 The system (1.1) will be said global exact controllable if for any (h 0 , u 0 ) ∈ (H 2 (0, 1) × H 1 (0, 1)) with h 0 ≥ c > 0, and for any (h 1 , u 1 ) ∈ (H 2 (0, 1) × H 1 (0, 1)) with h 1 ≥ c 1 > 0, for any time T > 0 there exist controls v 1 ∈ H 3/4 (0, T ) and v 2 ∈ H 3/4 (0, T ) such that the corresponding solution of (1.1) satisfies u(T, x) = u 1 (x) in (0, 1) and h(T, x) = h 1 (x) in (0, 1).

Remark 1

In our previous definition, we impose only two controls on the system (1.1) which is natural inasmuch as the result of existence of strong solution for the system (1.1) do not admit boundary condition on the height h.

Let us recall what is known about the controllability of the compressible Navier-Stokes equations, as far as we know there exists very few results. Indeed one of the main difficulty consists in the non controllability of the linear system associated to (1.1) around the stable equilibrium (1, 0). This has been observed indirectly in [START_REF] Rosier | On the exact controllability of a wave equation with structural damping[END_REF] and more recently in [START_REF] Drici | Some remarks on the exact controllability of the compressible Navier-Stokes equations[END_REF]. By developing a moment method in one dimension, we can observe that the operator associated to the linear system of (1.1) admits accumulation point on the eigenvalues, this is due in particular to the behavior of the height which generates a damping effect. However in [START_REF] Drici | Some remarks on the exact controllability of the compressible Navier-Stokes equations[END_REF], we check that the linearized system is approximate controllable. To overcome this difficulty, it would be natural to use the return method developed by Coron (see [START_REF] Coron | Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations[END_REF][START_REF] Coron | Some open problems on the control of nonlinear partial differential equations[END_REF][START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]), however the linearized system around the trajectory of [START_REF] Coron | Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations[END_REF] remains non controllable. It is one of the main reason why the problem of exact controllability for the compressible Navier-Stokes equations is so difficult. However very recently in a very interesting work Ervedoza, Glass, Guerrero and Puel in [START_REF] Ervedoza | Local exact controllability for the 1-D compressible Navier-Stokes equation[END_REF] have obtained the local exact controllability for the 1-D compressible Navier-Stokes equation by using Carleman estimates taking into account the flow associated to the transport equation on the height.

The study of the exact local incompressible Navier-Stokes equations has been investigated by many authors, we would like in particular recall the results of Fernández-Cara et al in [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF][START_REF] Fernández-Cara | Some controllability results for the N dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF]. Global exact controllability in two dimension has been obtained by Coron and Fursikov for a manifold without boundary by combining some result of global approximate control obtained via the return method and some results of local exact controllability via Carleman estimates. The goal of this paper is to show that the compressible shallow water system with friction term in not global exact controllable and not even global approximate controllable. To do this, we are going constructed special solution of the system (1.1) which are not controllable (they are also called quasi solution in [?] when there is no friction term).

Here compared with compressible Navier-Stokes equation we also take into account the friction term rhu physically justified to model the friction condition on the bottom of the ocean ( see [START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF]). Naturally on a mathematical point of view, the term rhu does not add any difficulties for obtaining global weak solution or global strong solution with small initial data, however in our study this term shall turn out to be essential in order to cancel out the coupling between the height h and the velocity u. Initially in [START_REF] Haspot | Existence of strong global solutions for the Korteweg system with large initial data[END_REF] we obtain the existence of global strong solution for Korteweg system with friction term with large initial data (at least for the irrotational part of the initial data) in dimension N ≥ 2. Indeed coupled with the pressure term, the friction term shall introduce a damping effect on a new unknown called effective velocity and introduced in [START_REF] Haspot | New entropy for Korteweg's system, existence of global weak solution and Prodi-Serrin theorem[END_REF]. Roughly speaking this friction term allows to cancel out the coupling between the height h and the velocity u (we can check in particular that (h, -µ∇ ln h) is a particular solution of the system where h verifies a heat equation), it is one of the main difficulty in order to obtain the existence of global strong solution for compressible Navier-Stokes equation (see [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fuids[END_REF], in this case the coupling is between the velocity and the density). Indeed in this last case, it is difficult to obtain a damping effect on the density and the coupling between the velocity and the pressure terms impose a smallness condition on the initial density. This last result was extended for the shallow water system without friction in [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF], in this paper we introduce the notion of quasi-solution (h 1 , -µ∇ ln h 1 ) ( where h 1 verifies a heat equation) which is a particular solution of the shallow-water system when the pressure is considered null. We consider then the pressure term as a perturbation by taking profit of the regularizing effect on h 1 which induces smallness condition on the pressure in high frequencies.

In this paper we are going to verify that (h, -µ∂ x ln h) is a particular solution of (1.1). Via Cole-Hopf formula we verifies that the global controllability problem of (1.1) comes to deal in reality with the viscous Burger equation studied in particular by Guerrero and Imanuvilov in [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF]. They prove that this equation is not global exact controllable by using some comparison argument on the heat equation. We shall follow in our proof the argument of [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF] which shall be an angular stone in our result. Before giving our main result, let us briefly mention some previous works which deal with the exact controllability for the viscous and non viscous Burgers equation in order to understand better this equation.

Exact controllability for the non viscous Burger equations

In [START_REF] Horsin | On the controllability of the Burgers equation[END_REF], Horsin shows that by using one boundary control, the Burgers equation can be driven from the null initial condition to a constant final state M in a time T ≥ 1/M . To do this, he uses is the so-called return method introduced by Coron (see [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] for the case of the global approximate control of the Navier-Stokes equation with slip boundary condition). More general results are obtained in [START_REF] Ancona | On the attainable set for scalar nonlinear conservation laws with boundary control[END_REF] when we consider general scalar non-linear conservation laws.

Exact controllability for the viscous Burger equations

With one boundary control First, in [START_REF] Daz | Obstruction and some approximate controllability results for the Burgers equation and related problems[END_REF], the author proves that the approximate controllability to some target states does not arise for the solutions of the viscous Burgers equation with one boundary control. Indeed he proves the existence of some invariant, more precisely that there exists a constant C > 0 such that, for every T > 0 one has:

u(t, x) ≤ C 0 1 -x ∀(t, x) ∈ (0, T ) × (0, 1).
Next, in [START_REF] Fernández-Cara | On the controllability of Burgers system[END_REF], the authors prove that the global null controllability does not hold with one control force by using in a crucial way the comparison principle.

With two boundary controls

In [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF], Guerrero and Imanuvilov prove that the viscous Burger equation is not exact global controllable even with two boundary controls. To do this they combine the Cole-Hopf transformation and some comparison principle for the heat equation.

In [START_REF] Glass | Uniform controllability of a transport equation in zero diffusion-dispersion limit[END_REF], Glass and Guerrero show the uniform exact controllability for the viscous Burger equation when the viscosity is small enough. In [START_REF] Yu | On global controllability of 2D Burgers equations[END_REF], Imanuvilov and Puel obtain the global exact controllability for a type of Burger equations. To finish we would like to mention the works of Chapouly in [START_REF] Chapouly | Global controllability of nonviscous and viscous Burgers equations[END_REF] who proves the global exact controllability of the viscous Burger equation when we consider three controls: two are the boundary values, one is the right member of the equation and is constant with respect to the space variable. This result is to relate with the paper of Coron [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] inasmuch as the right control replace in some sense the pressure term of the Navier-Stokes equations. In some sense in [START_REF] Chapouly | Global controllability of nonviscous and viscous Burgers equations[END_REF], Chapouly via the source control forces the system to be incompressible as for Navier-Stokes.

In the present paper, we have two main objectives. One concerning the global exact null controllability for small time and the other one concerning the global exact controllability for any time T > 0. Both results are of negative nature.

As long as the first one is concerned, we prove that there exists a final time T and an initial condition u 0 such that the solution of (1.1) is far away from zero. That is to say, the global null controllability for the compressible Navier-Stokes equation with friction and with two control forces does not hold. The precise result is given in the following theorem:

Theorem 1.1 There exists T > 0 and u 0 = -µ∂ x ln h 0 ∈ H 1 (0, 1) with h 0 ≥ c > 0 such that, for any control functions v 1 ∈ H 3/4 (0, T ) and v 2 ∈ H 3/4 (0, T ) satisfying the compatibility conditions (1.3), the associated solution (h, u) ∈ X T to (1.1) satisfies:

u(T, •) H 1 (0,1) ≥ C > 0, (1.5)
for some positive constant depending on T and h 0 .

The second main result is a negative exact controllability result:

Theorem 1.2 For any T > 0, there exists an initial condition u 0 -µ∂ x ln h 0 ∈ H 1 (0, 1) and a target function u 1 = -µ∂ x ln h 1 ∈ H 1 (0, 1) such that, for any v 1 ∈ H 3/4 (0, T ) and v 2 ∈ H 3/4 (0, T ) satisfying (1.3), the associated solution (h, u) ∈ X T to (1.1) satisfies:

u(T, ) -u 1 () H 1 (0,1) ≥ C > 0, (1.6)
for some positive constant C depending on T , h 0 and h 1 .

Remark 2 Let us observe that we have global existence of solution for the system (1.1) via the fact that (h, -µ∂ x ln h) is a particular solution.

Remark 3 Let us remark that in [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF], J-M Coron proves the global approximate controllability of the Navier-Stokes equations with slip boundary condition by using the return method (in [START_REF] Coron | Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary[END_REF], the authors obtain the global exact controllability). In [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF], Guerrero and Imanuvilov prove that the viscous Burger equation in one dimension is not global exactly controllable. We show that it is also the case for the compressible Navier-Stokes equations with friction. This is due in particular to the fact that the flow is compressible, indeed in our case we are working with a specific solution of the form u = -µ∂ x ln h which is naturally not incompressible but irrotational. In particular it gives an idea why in the case of slip boundary condition we can hope global controllability. Indeed by imposing a Neuman condition on the height which verifies an heat equation in the context of the quasi solution, we verify that u = -µ∂ x ln h is such that u • n = 0.

Remark 4

We recall that all global controllability properties obtained up to now for the incompressible NavierStokes system come essentially from the same property for the Euler equation (see the works of Coron [START_REF] Coron | Some open problems on the control of nonlinear partial differential equations[END_REF][START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF]). In the case of the compressible Navier-Stokes equation we use the properties of the viscous Burger equations and not these of the non viscous Burger equations as for incompressible Navier-Stokes equations. On the other side the structure of quasi solution is purely non linear.

Remark 5 In a forthcoming paper we are going to prove the same type of results for the shallow water system without friction by using the same type of argument than in [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] which corresponds to neglect in high frequencies the pressure term.

In order to prove these results, in section 2 we first show the equivalence of the controllability problem for the system (1.1) and some controllability problem for a one-dimensional linear heat equation with positive boundary controls. To do this, we combine a Cole-Hopf transformation and the introduction of particular solution introduced in [START_REF] Haspot | New entropy for Korteweg's system, existence of global weak solution and Prodi-Serrin theorem[END_REF]. Finally, in the section 3 we prove both theorems 1.1 and 1.2 by using a method developed by Guerrero and Imanuvilov in [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF].

Simplification of the problem

In this section, we show that with particular initial data (it means irrotational initial data on the velocity which depends on h 0 ) the exact controllability properties for the system (1.1) (see definition 1.1) can be reduced to some controllability assumptions for the heat equation. In the sequel T > 0 is a fixed final time. Let us rewrite the system (1.1):

               ∂ ∂t h + ∂ x (hu) = 0, (t, x) ∈ Q = (0, T ) × (0, 1) ∂ ∂t u + u∂ x u -µ∂ xx u + ∂ x ln h F r 2 + ru = 0, (t, x) ∈ Q = (0, T ) × (0, 1) u(t, 0) = v 1 (t), u(t, 1) = v 2 (t), t ∈ (0, T ), u(0, x) = u 0 (x), h(0, x) = h 0 (x).
(2.7)

Next we remark that the controllability problem of definition 1.1 implies the controllability problem for a semilinear parabolic system (2.8) with time-dependent controls.We are writing the system (2.8) as follows:

                     ∂ ∂t h -µ∂ xx h = 0, (t, x) ∈ Q = (0, T ) × (0, 1) ∂ ∂t w -µ∂ xx w -µ|∂ x w| 2 = v 5 (t) + v 3 (t), (t, x) ∈ Q = (0, T ) × (0, 1) w(t, 0) = v 5 (t), w(t, 1) = v 4 (t), t ∈ (0, T ), h(t, 0) = e v 5 (t) , h(t, 1) = e v 4 (t) , t ∈ (0, T ), w(0, x) = ln h 0 (x), h(0, x) = h 0 (x).
(2.8)

Lemma 1

The system (1.1) is global exact controllable implies that there exists a solution to the following controllability problem: For any w 0 , w 1 ∈ H 2 (0, 1) and h 0 = e w 0 , h 1 = e w 1 with w 0 (0) = w 1 (0) = 0, there exists control v 3 ∈ L 2 (0, T ) and v 4 , v 5 ∈ H 1 (0, T ) and a solution of (2.8) such that w(T, x) = w 1 (x) in (0, 1) and h(T, x) = h 1 (x) with w ∈ Y T = L 2 (0, T ; H 3 (0, 1)) ∩ H 1 (0, T ; H 1 (0, 1)) and h = e w . system (2.9):

               ∂ ∂t h -µ∂ xx h = 0, (t, x) ∈ Q = (0, T ) × (0, 1) ∂ ∂t h -µ∂ xx h -(v 5 (t) + v 3 (t))h = 0, (t, x) ∈ Q = (0, T ) × (0, 1) h(t, 0) = e v 5 (t) , h(t, 1) = e v 4 (t) , t ∈ (0, T ), h(0, x) = h 0 (x).
(2.9)

Lemma 2

The system (2.9) is global exact controllable implies that there exists a solution to the following controllability problem:

For any h 0 , h 1 ∈ H 2 (0, 1) with 0 < c ≤ h 0 < M , 0 < c 1 ≤ h 1 < M 1 , there exists control v 4 , v 5 ∈ H 1 (0, T ) and a solution of (2.9) such that h(T, x) = h 1 (x) in (0, 1) with h ∈ Y T = L 2 (0, T ; H 3 (0, 1)) ∩ H 1 (0, T ; H 1 (0, 1)).
Proof: By multiplying (2.8) by h, we check that h verifies the second equation of (2.9). In order to ensure the compatibility between the first and the second equation of (2.8), we assume that:

v 5 (t) = - t 0 v 3 (s)ds. (2.10)
To summarize we have proved that if the system (1.1) is global exact controllable, it implies that the following heat equation:

       ∂ ∂t h -µ∂ xx h = 0, (t, x) ∈ Q = (0, T ) × (0, 1)
h(t, 0) = v 6 (t), h(t, 1) = v 7 (t), t ∈ (0, T ), h(0, x) = h 0 (x).

(2.11) is global exact controllable with: h 0 , h 1 ∈ H 2 (0, 1) with 0 < c ≤ h 0 < M , 0 < c 1 ≤ h 1 < M 1 and with v 6 = e v 5 (t) , v 7 = e v 4 (t) ∈ H 1 (0, T ) strictly positive where:

h(t, x) = exp - 1 µ x 0 u(t, y)dy - t 0 v 3 (s)ds , v 3 (t) = - v 2 1 (t) µ + u x (t, 0),
and:

h 1 (x) = K exp - 1 µ x 0 u(t, y)dy , with K = - T 0 v 3 (s)ds.
3 Proofs of theorem 1.1 and 1.2

In this section, we will prove both theorems 1.1 and 1.2. We start with giving technical results for the dual system of the heat equations that we shall use for proving theorem 1.1 and 1.2:

Proposition 3.1 Let 0 < ξ 0 < ξ 1 < ξ 2 < 1.
Then, for each θ > 0 there exists a time T = T (θ) > 0such that the solution of the backwards heat equation:

     -U t -U xx = 0, (t, x) ∈ (0, T * ) × (0, 1), U (t, 0) = U (t, 1) = 0, t ∈ (0, T * ), U (T * , x) = δ ξ 0 -θδ ξ 1 + δ ξ 2 , x ∈ (0, 1) (3.12)
satisfies U x (t, 0) > 0 and U x (t, 1) < 0 ∀t ∈ (0, T * ).

Here δ x is the Dirac measure at x.

The following proposition is a generalization of the previous one which was proved in [START_REF] Belishev | On approximating properties of solutions of the heat equation[END_REF]:

Proposition 3.2 Let T > 0 and 0 < ξ 0 < ξ 1 < ξ 2 < 1.
Then there exists θ > 0 such that the solution of the backwards heat equation:

     -U t -U xx = 0, (t, x) ∈ (0, T * ) × (0, 1), U (t, 0) = U (t, 1) = 0, t ∈ (0, T * ), U (T * , x) = δ ξ 0 -θδ ξ 1 + δ ξ 2 , x ∈ (0 , 1) (3.13)
satisfies U x (t, 0) > 0 and U x (t, 1) < 0 ∀t ∈ (0, T * ).

3.1 Proof of theorem 1.1 via the method of Guerrero et Imanuvilov ( [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF])

We have verified that the global exact controllability of (1.1) implies the exact controllability problem of the heat equation:

       ∂ ∂t h -µ∂ xx h = 0, (t, x) ∈ Q = (0, T ) × (0, 1) z(t, 0) = v 7 (t), z(t, 1) = v 8 (t), t ∈ (0, T ), h(0, x) = h 0 (x). (3.14)
for any h 0 , h 1 ∈ H 2 (0, 1) with h 0 (x), h 1 (x) > 0 in (0, 1) and h 0 (0) = 1 and h 1 (0) = K > 0, and with two controls v 7 (t), v 8 (t) ∈ H 1 (0, T ) which are strictly positive in [0, T ] such that the solution of (2.9) satisfies h(T, x) = h 1 (x) in (0, 1). We are going to show the theorem 1.1 by contradiction. Thus, we assume that for any T > 0 and any h 0 ∈ H 2 (0, 1) with h 0 > 0 and h 0 (0) = 1, there exists a constant K > 0 and two controls 0 < v 1 ∈ H 1 (0, T ) and 0 < v 2 ∈ H 1 (0, T ) such that the solution of (3.14) satisfies h(T, x) = K in (0, 1).

Multiplying the equation of (3.14) by U given by proposition 3.1 (for θ ≥ 2) and integrating over (0, T ) × (0, 1) with T = T * , we obtain that:

T 0 (U x (t, 0) v 1 (t) -U x (t, 1) v 2 (t))dt + K(2 -θ) - 1 0 U (0, x)h 0 (x)dx = 0 (3.15)
for any h 0 ∈ H 2 (0, 1). By using the facts that the normal derivative of U is negative and θ ≥ 2, we observe that the two first terms of (3.15) are non-positive.

Our goal now consists in choosing an initial condition h 0 with h 0 (0) = 1 such that:

- 1 0 U (0, x)h 0 (x)dx < 0,
in order to obtain a contradiction with the equality (3.15) and the fact that the two first terms of this equality are non positive.

Using the fact that the normal derivative of U is negative and U verifies homogeneous Dirichlet boundary conditions, we observe by Taylor formula that there exists δ > 0 such that:

U (0, x) ≥ δx, ∀x ∈ (0, δ) and U (0, x) ≥ δ(1 -x), ∀x ∈ (1 -δ, 1).
On the other hand since U satisfies a heat equation (3.13), we obtain an instantaneously regularizing effect on U in Sobolev norms which depends on some power of 1 T α with α > 1,more precisely there exists a positive constant C * (T * , θ) such that:

U (0, •) L 2 (0,1) ≤ C * .
(3.16)

We are going to follow the choice of h 0 as in [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF], let h 0 = h 0 (x) ∈ (0, 1) for all x ∈ (0, 1) such that

h 0 (x) = 4C * δ 3 ∀x ∈ ( δ 4 , 3δ 4 )U (1 - 3δ 4 , 1 - δ 4 ),
we have by using the fact that h 0 is positive: This achieves the proof.

- 1 0 U (0, x)h 0 (x)dx ≤ -

Proof of theorem 1.2

In this paragraph, we prove the theorem 1.2. The proof follows the same lines than in [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF] and than in the previous section. That why we are just pointing out the differences and explaining how to deal with them. We are interested in showing that for any h 0 , h 1 ∈ H 2 (0, 1) with h 0 (x), h 1 (x) > 0 in (0, 1) and h 0 (0) = 1, h 1 (0) = K, there exists two controls v 7 (t), v 8 (t) ∈ H 1 (0, T ) which are strictly positive in [0, T ] such that the solution of (2.9) satisfies h(T, x) = h 1 (x) in (0, 1). As in the previous section, we multiply the system (3.14) by U checking the proposition 3.2, we have then:

T * 0 (U x (t, 0) v 1 (t) -U x (t, 1) v 2 (t))dt + h 1 (ξ 0 ) -θh 1 (ξ 1 ) + h 1 (ξ 2 ) = 0 (3.17) for all 0 < h 0 ∈ H 2 (0, 1) and 0 < h 1 ∈ H 2 (0, 1). The first term of (3.17) is non positive, we are then using the same argument of contradiction than in the previous proof and in particular showing that:

-1 0 U (0, x)h 0 (x)dx < 0, for a suitable choice on h 0 . We proceed exactly as in the previous section with the same h 0 and furtheremore we take h 1 such that:

h 1 (ξ 0 ) -θh 1 (ξ 1 ) + h 1 (ξ 2 ) < 0.
This is a contradiction with identity (3.17) which concludes the proof.

Aknowledgements: The authors would like to thank Jean-Michel Coron for many helpful and fruitful discussions.

3δ 4 δ 4 4C * δ 3 UU

 43 (0, x)dx -(0, x)h 0 (x)dx ≤ -C * + (1 -2δ)C * = -2δC * < 0.

Proof: It suffices to set:

and with:

From (1.4) and v 1 ∈ H 3 4 (0, T ), we obtain that v 3 ∈ L 2 (0, T ) and v 4 , v 5 ∈ H 1 (0, T ). Let us now prove that the previous controllability result is equivalent to a controllability problem for the system (2.9) with two time-dependent controls, one of bilinear nature (multiplying the state function) and the other one acting at x = 1. We are writing the


Yufan Guo

Anna Korhonen

Thierry Poibeau Lattice

A Weakly-supervised Approach to Argumentative Zoning of Scientific Documents

Argumentative Zoning (AZ) -analysis of the argumentative structure of a scientific paperhas proved useful for a number of information access tasks. Current approaches to AZ rely on supervised machine learning (ML). Requiring large amounts of annotated data, these approaches are expensive to develop and port to different domains and tasks. A potential solution to this problem is to use weaklysupervised ML instead. We investigate the performance of four weakly-supervised classifiers on scientific abstract data annotated for multiple AZ classes. Our best classifier based on the combination of active learning and selftraining outperforms our best supervised classifier, yielding a high accuracy of 81% when using just 10% of the labeled data. This result suggests that weakly-supervised learning could be employed to improve the practical applicability and portability of AZ across different information access tasks.

Introduction

Many practical tasks require accessing specific types of information in scientific literature. For example, a reader of scientific literature may be looking for information about the objective of the study in question, the methods used in the study, the results obtained, or the conclusions drawn by authors. Similarly, many Natural Language Processing (NLP) tasks focus on the extraction of specific types of information in documents only.

To date, a number of approaches have been proposed for sentence-based classification of scien-tific literature according to categories of information structure (or discourse, rhetorical, argumentative or conceptual structure, depending on the framework in question). Some of these classify sentences according to typical section names seen in scientific documents [START_REF] Lin | Generative content models for structural analysis of medical abstracts[END_REF][START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF], while others are based e.g. on argumentative zones [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF][START_REF] Teufel | Towards domain-independent argumentative zoning: Evidence from chemistry and computational linguistics[END_REF], qualitative dimensions [START_REF] Shatkay | Multi-dimensional classification of biomedical text: Toward automated, practical provision of high-utility text to diverse users[END_REF] or conceptual structure [START_REF] Liakata | Corpora for the conceptualisation and zoning of scientific papers[END_REF] of documents.

The best of current approaches have yielded promising results and proved useful for information retrieval, information extraction and summarization tasks [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF][START_REF] Tbahriti | Using argumentation to retrieve articles with similar citations[END_REF][START_REF] Ruch | Using argumentation to extract key sentences from biomedical abstracts[END_REF]. However, relying on fully supervised machine learning (ML) and a large body of annotated data, existing approaches are expensive to develop and port to different scientific domains and tasks.

A potential solution to this bottleneck is to develop techniques based on weakly-supervised ML. Relying on a small amount of labeled data and a large pool of unlabeled data, weakly-supervised techniques (e.g. semi-supervision, active learning, co/tri-training, self-training) aim to keep the advantages of fully supervised approaches. They have been applied to a wide range of NLP tasks, including named-entity recognition, question answering, information extraction, text classification and many others [START_REF] Abney | Semi-supervised learning for computational linguistics[END_REF], yielding performance levels similar or equivalent to those of fully supervised techniques.

To the best of our knowledge, such techniques

have not yet been applied to the analysis of information structure of scientific documents by aforementioned approaches. Recent experiments have demonstrated the usefulness of weakly-supervised learning for classifying discourse relations in scientific texts, e.g. [START_REF] Hernault | Semi-supervised discourse relation classification with structural learning[END_REF]. However, focusing on local (rather than global) structure of documents and being much more fine-grained in nature, this related task differs from ours considerably.

In this paper, we investigate the potential of weakly-supervised learning for Argumentative Zoning (AZ) of scientific abstracts. AZ is an approach to information structure which provides an analysis of the rhetorical progression of the scientific argument in a document [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF]. It has been used to analyze scientific texts in various disciplines -including computational linguistics [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF], law, [START_REF] Hachey | Extractive summarisation of legal texts[END_REF], biology [START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF] and chemistry [START_REF] Teufel | Towards domain-independent argumentative zoning: Evidence from chemistry and computational linguistics[END_REF] -and has proved useful for NLP tasks such as summarization [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF]. Although the basic scheme is said to be disciplineindependent [START_REF] Teufel | Towards domain-independent argumentative zoning: Evidence from chemistry and computational linguistics[END_REF], its application to different domains has resulted in various modifications and laborious annotation exercises. This suggests that a weakly-supervised approach would be more practical than a fully supervised one for the real-world application of AZ.

Taking two supervised classifiers as a comparison point -Support Vector Machines (SVM) and Conditional Random Fields (CRF) -we investigate the performance of four weakly-supervised classifiers on the AZ task: two based on semi-supervised learning (transductive SVM and semi-supervised CRF) and two on active learning (Active SVM alone and in combination with self-training).

The results are promising. Our best weaklysupervised classifier (Active SVM with selftraining) outperforms the best supervised classifier (SVM), yielding high accuracy of 81% when using just 10% of the labeled data. When using just one third of the labeled data, it performs equally well as a fully supervised SVM which uses 100% of the labeled data. Our investigation suggests that weaklysupervised learning could be employed to improve the practical applicability and portability of AZ to different information access tasks.

Data

We used in our experiments the recent dataset of [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF]. [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF] provide a corpus of 1000 biomedical abstracts (consisting of 7985 sentences and 225785 words) annotated according to three schemes of information structure -those based on section names [START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF], AZ [START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF] and Core Scientific Concepts (CoreSC) [START_REF] Liakata | Corpora for the conceptualisation and zoning of scientific papers[END_REF]. We focus here on AZ only, because it subsumes all the categories of the simple section name -based scheme, and according to the inter-annotator agreement and ML experiments reported by [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF] it performs better on this data than the fairly fine-grained CoreSC scheme.

AZ is a scheme which provides an analysis of the rhetorical progression of the scientific argument, following the knowledge claims made by authors. [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF] introduced AZ and applied it first to computational linguistics papers. [START_REF] Hachey | Extractive summarisation of legal texts[END_REF] applied the scheme later to legal texts and [START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF] modified it for biology papers. More recently, [START_REF] Teufel | Towards domain-independent argumentative zoning: Evidence from chemistry and computational linguistics[END_REF] introduced a refined version of AZ and applied it to chemistry papers.

The biomedical dataset of [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF] has been annotated according to the version of AZ developed for biology papers [START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF] (with only minor modifications concerning zone names). Seven categories of this scheme (out of the 10 possible) actually appear in abstracts and in the resulting corpus. These are shown and explained in Table 1. For example, the Method zone (METH) is for sentences which describe a way of doing research, esp. according to a defined and regular plan; a special form of procedure or characteristic set of procedures employed in a field of study as a mode of investigation and inquiry.

An example of a biomedical abstract annotated according to AZ is shown in Figure 1, with different zones highlighted in different colors. For example, the RES zone is highlighted in lemon green.

Table 2 shows the distribution of sentences per scheme category in the corpus: Results (RES) is by far the most frequent zone (accounting for 40% of the corpus), while Background (BKG), Objective (OBJ), Method (METH) and Conclusion (CON) cover The work that needs to be done in the future Figure 1: An example of an annotated abstract Butadiene (BD) metabolism shows gender, species and concentration dependency, making the extrapolation of animal results to humans complex. BD is metabolized mainly by cytochrome P450 2E1 to three epoxides, 1,2-epoxy-3-butene (EB), 1,2;3,4-diepoxybutane (DEB) and 1,2-epoxy-butanediol (EB-diol). For accurate risk assessment it is important to elucidate species differences in the internal formation of the individual epoxides in order to assign the relative risks associated with their different mutagenic potencies. Analysis of N-terminal globin adducts is a common approach for monitoring the internal formation of BD derived epoxides. Our long term strategy is to develop an LC-MS/MS method for simultaneous detection of all three BD hemoglobin adducts. This approach is modeled after the recently reported immunoaffinity LC-MS/MS method for the cyclic N,N-(2,3-dihydroxy-1,4-butadyil)-valine (pyr-Val, derived from DEB). We report herein the analysis of the EB-derived 2-hydroxyl-3-butenyl-valine peptide (HB-Val). The procedure utilizes trypsin hydrolysis of globin and immunoaffinity (IA) purification of alkylated heptapeptides. Quantitation is based on LC-MS/MS monitoring of the transition from the singly charged molecular ion of HB-Val (1-7) to the a(1) fragment. Human HB-Val (1-11) was synthesized and used for antibody production. As internal standard, the labeled rat-[( 13)C(5)( 15)N]-Val (1-11) was prepared through direct alkylation of the corresponding peptide with EB. Standards were characterized and quantified by LC-MS/MS and LC-UV. The method was validated with different amounts of human HB-Val standard. The recovery was >75% and coefficient of variation <25%. The LOQ was set to 100 fmol/injection. For a proof of principal experiment, globin samples from male and female rats exposed to 1000 ppm BD for 90 days were analyzed. The amounts of HB-Val present were 268.2+/-56 and 350+/-70 pmol/g (mean+/-S.D.) for males and females, respectively. No HB-Val was detected in controls. These data are much lower compared to previously reported values measured by GC-MS/MS. The difference may be due higher specificity of the LC-MS/MS method to the N-terminal peptide from the alpha-chain versus derivatization of both alpha-and beta-chain by Edman degradation, and possible instability of HB-Val adducts during long term storage (about 10 years) between the analyses. These differences will be resolved by examining recently collected samples, using the same internal standard for parallel analysis by GC-MS/MS and LC-MS/MS. Based on our experience with pyr-Val adduct assay we anticipate that this assay will be suitable for evaluation of HB-Val in multiple species. [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF] report the inter-annotator agreement between their three annotators: one linguist, one computational linguist and one domain expert. According to Cohen's kappa [START_REF] Chapelle | Semisupervised classification by low density separation[END_REF] the agreement is relatively high: κ = 0.85. 2010) used a variety of features in their fully supervised ML experiments on different schemes of information structure. Since their feature types cover the best performing feature types in earlier works e.g. [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Lin | Generative content models for structural analysis of medical abstracts[END_REF][START_REF] Mullen | A baseline feature set for learning rhetorical zones using full articles in the biomedical domain[END_REF][START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF][START_REF] Merity | Accurate argumentative zoning with maximum entropy models[END_REF] we re-implemented and used them in our experiment1 . However, being aware of the fact that some of these features may not be optimal for weakly-supervised learning (i.e. when learning from smaller data), we evaluate their performance and suitability for the task later in section 4.3.

Background

Objective

Method Result

Conclusion

Related work

Future work

• Location. Zones tend to appear in typical positions in abstracts. Each abstract was there-fore divided into ten parts (1-10, measured by the number of words), and the location was defined by the parts where the sentence begins and ends.

• Word. All the words in the corpus.

• Bi-gram. Any combination of two adjacent words in the corpus.

• Verb. All the verbs in the corpus.

• Verb Class. 60 verb classes appearing in biomedical journal articles.

• Part-of-Speech -POS. The POS tag of each verb in the corpus.

• Grammatical Relation -GR. Subject (ncsubj), direct object (dobj), indirect object (iobj) and second object (obj2) relations in the corpus. e.g. (ncsubj observed 14 difference 5 obj). The value of this feature equals 1 if it occurs in a particular sentence (and 0 if not).

• Subj and Obj. The subjects and objects appearing with any verbs in the corpus (extracted from above GRs).

• Voice. The voice of verbs (active or passive) in the corpus.

These features were extracted from the corpus using a number of tools. A tokenizer was used to detect the boundaries of sentences and to separate punctuation from adjacent words e.g. in complex biomedical terms such as 2-amino-3,8-diethylimidazo[4,5f]quinoxaline. The C&C tools [START_REF] Curran | Linguistically motivated large-scale nlp with c&c and boxer[END_REF] trained on biomedical literature were employed for POS tagging, lemmatization and parsing. The lemma output was used for creating Word, Bi-gram and Verb features. The GR output was used for creating the GR, Subj, Obj and Voice features. The "obj" marker in a subject relation indicates passive voice (e.g. (ncsubj observed 14 difference 5 obj)). The verb classes were acquired automatically from the corpus using the unsupervised spectral clustering method of [START_REF] Sun | Improving verb clustering with automatically acquired selectional preference[END_REF]. To control the number of features we lemmatized the lexical items for all the features, and removed the words and GRs with fewer than 2 occurrences and bi-grams with fewer than 5 occurrences.

Machine learning methods

Support Vector Machines (SVM) and Conditional Random Fields (CRF) have proved the best performing fully supervised methods in most recent works on information structure, e.g. [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Mullen | A baseline feature set for learning rhetorical zones using full articles in the biomedical domain[END_REF][START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF][START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF]. We therefore implemented these methods as well as weakly supervised variations of them: active SVM with and without self-training, transductive SVM and semi-supervised CRF.

Supervised methods

SVM constructs hyperplanes in a multidimensional space to separate data points of different classes. Good separation is achieved by the hyperplane that has the largest distance from the nearest data points of any class. The hyperplane has the form w • xb = 0, where w is its normal vector. We want to maximize the distance from the hyperplane to the data points, or the distance between two parallel hyperplanes each of which separates the data. The parallel hyperplanes can be written as: xb ≤ -1 for x of the other, which can be solved by using the SMO algorithm (Platt, 1999b). We used Weka software [START_REF] Hall | The weka data mining software: an update[END_REF] (employing its linear kernel) for SVM experiments.

w • x -b = 1 and w • x -b = -1,
CRF is an undirected graphical model which defines a probability distribution over the hidden states (e.g. label sequences) given the observations. The probability of a label sequence y given an observation sequence x can be written as:

p(y|x, θ) = 1 Z(x) exp( j θ j F j (y, x))
, where F j (y, x) is a real-valued feature function of the states and the observations; θ j is the weight of F j , and Z(x) is a normalization factor. The θ parameters can be learned using the L-BFGS algorithm [START_REF] Nocedal | Updating Quasi-Newton Matrices with Limited Storage[END_REF]. We used Mallet software [START_REF] Mccallum | Mallet: A machine learning for language toolkit[END_REF] for CRF experiments.

Weakly-supervised methods

Active SVM (ASVM) starts with a small amount of labeled data, and iteratively chooses a proportion of unlabeled data for which SVM has less confidence to be labeled (the labels can be restored from the original corpus) and used in the next round of learning, i.e. active learning. Query strategies based on the structure of SVM are frequently employed [START_REF] Tong | Support vector machine active learning with applications to text classification[END_REF][START_REF] Novak | Text classification with active learning[END_REF]. For example, it is often assumed that the data points close to the separating hyperplane are those that the SVM is uncertain about. Unlike these methods, our learning algorithm compares the posterior probabilities of the best estimate given each unlabeled instance, and queries those with the lowest probabilities for the next round of learning. The probabilities can be obtained by fitting a Sigmoid after the standard SVM (Platt, 1999a), and combined using a pairwise coupling algorithm [START_REF] Hastie | Classification by pairwise coupling[END_REF] in the multi-class case. We used the SVM linear kernel in Weka for classification, and the -M flag in Weka for calculating the posterior probabilities.

Active SVM with self-training (ASSVM) is an extension of ASVM where each round of training has two steps: (i) training on the labeled, and testing on the unlabeled data, and querying; (ii) training on both labeled and unlabeled/machine-labeled data by using the estimates from step (i). The idea of ASSVM is to make the best use of the labeled data, and to make the most use of the unlabeled data.

Transductive SVM (TSVM) is an extension of SVM which takes advantage of both labeled and unlabeled data [START_REF] Vapnik | Statistical learning theory[END_REF]. Similar to SVM, the problem is defined as:

Minimize |w| (in w, b, y (u) ) Subject to

y (l) (w • x (l) -b) ≥ 1, y (u) (w • x (u) -b) ≥ 1 , y (u) ∈ {-1, 1}
, where x (u) is unlabeled data and y (u) the estimate of its label. The problem can be solved by using the CCCP algorithm [START_REF] Collobert | Trading convexity for scalability[END_REF]. We used UniverSVM software [START_REF] Sinz | UniverSVM Support Vector Machine with Large Scale CCCP Functionality[END_REF] for TSVM experiments.

Semi-supervised CRF (SSCRF) can be implemented with entropy regularization (ER). It extends the objective function on Labeled data L log p(y (l) |x (l) , θ) with an additional term U Y p(y|x (u) , θ) log p(y|x (u) , θ) to minimize the conditional entropy of the model's predictions on U nlabeled data [START_REF] Jiao | Semi-supervised conditional random fields for improved sequence segmentation and labeling[END_REF]Mann and Mccal-lum, 2007). We used Mallet software [START_REF] Mccallum | Mallet: A machine learning for language toolkit[END_REF] for SSCRF experiments.

Experimental evaluation

Evaluation methods

We evaluated the ML results in terms of accuracy, precision, recall, and F-measure against manual AZ annotations in the corpus: acc = no. of correctly classif ied sentences total no. of sentences in the corpus p = no. of sentences correctly identif ied as Class i total no. of sentences identif ied as Class i r = no. of sentences correctly identif ied as Class i total no. of sentences in Class i f = 2 * p * r p+r We used 10-fold cross validation for all the methods to avoid the possible bias introduced by relying on any particular split of the data. More specifically, the data was randomly assigned to ten folds of roughly the same size. Each fold was used once as test data and the remaining nine folds as training data. The results were then averaged.

Following [START_REF] Thomas | Approximate statistical tests for comparing supervised classification learning algorithms[END_REF], we used McNemar's test [START_REF] Mcnemar | Note on the Sampling Error of the Difference Between Correlated Proportions or Percentages[END_REF] to measure the statistical significance between the results of different ML methods. The chosen significance level was .05.

Results

Table 3 shows the results for the four weaklysupervised and two supervised methods when 10% of the training data (i.e. ∼700 sentences) has been labeled. We can see that ASSVM is the best performing method with an accuracy of 81% and the macro F-score of .76 (the macro F-score is calculated for the 5 scheme categories which are found by all the methods). ASVM performs nearly as well, with an accuracy of 80% and F-score of .75. Both methods outperform supervised SVM with a statistically significant difference (p < .001).

TSVM is the lowest performing SVM-based method. Yielding an accuracy of 76% and F-score of .73 its performance is lower than that of the supervised SVM. However, it does outperform both CRFbased methods. SSCRF performs better than CRF with 3% higher accuracy and .02 higher F-score. The difference in accuracy is statistically significant (p < .001).

Only one method (ASVM) identifies six out of the seven possible categories. Other methods identify five categories. The 1-2 missing categories are very low in frequency (accounting for 1% of the corpus data each, see table 2). Looking at the results for other categories, they seem to reflect the amount of corpus data available for each category (Table 2), with RES (Results) being the highest and OBJ (Objective) the lowest performing category with most methods. Interestingly, the only method that performs relatively well on OBJ is the supervised SVM.

The best method ASSVM outperforms other methods most clearly on METH (Method) category. Although METH is a high frequency category (accounting for 18% of the corpus data) other methods tend to confuse it with OBJ, presumably because a single sentence may contain elements of both (e.g. scientists may describe some of their method when describing the objective of the study).

Figure 2 shows the learning curve of different methods (in terms of accuracy) when the percentage of the labeled data (in the training set) ranges from 0 to 100%. ASSVM outperforms other methods, reaching its best performance of 88% accuracy when using ∼40% of the labeled data. Indeed when using 33% of the labeled data, it performs already equally well as fully-supervised SVM using 100% of the labeled data. The advantage of ASSVM over ASVM (the second best method) is clear especially when 20-40% of the labeled data is used. SVM and TSVM tend to perform quite similarly with each other when more than 25% of the labeled data is used, but when less data is available, SVM performs better. Looking at the CRF-based methods, SSCRF outperforms CRF in particular when 10-25% of the labeled data is used. However, neither of them reaches the performance level of SVM-based methods.

Figure 3 shows the area under the learning curves (by the trapezoidal rule) at different intervals, which gives a reasonable approximation to the overall performance of different methods. The area under ASSVM is the largest at each of the four intervals, with a value of .08 at (0,10%], .07 at [10%,20%],

.20 at [20%, 40%] and .50 at [40%,100%]. The difference between supervised and weakly-supervised methods is more significant at (0, 20%] than at [20%,100%].

Further analysis of the features

As explained in section 3.1, we employed in our experiments a collection of features which had performed well in previous supervised AZ experiments. We conducted further analysis to investigate which of these features are the most (and the least) useful for weakly-supervised learning. We took our best performing method ASSVM and conducted leaveone-out analysis of the features with 10% of the labeled data. The results are shown in Table 4. Φ: Employing all the features.

We can see that the Location feature is by far the most useful feature for ASSVM. The performance drops 8% in accuracy and .09 in F-score in the absence of this feature. Location is particularly important for BKG (which nearly always appears in the same location: in the beginning of an abstract) and is highly useful for METH and CON as well. Removing POS has almost equally strong effect, in particular on BKG and METH, suggesting that verb tense is particularly useful for distinguishing these categories.

Also Voice, Verb class and GR contribute to general performance, especially to accuracy. Voice is particularly important for CON, which differs from other categories in the sense that it is marked by frequent usage of active voice. Verb class is helpful for METH, RES and CON while GR is helpful for all high frequency categories.

Among the least helpful features are those which suffer from sparse data problems, including e.g. Word, Bi-gram, and Verb. They perform particularly badly when applied to low frequency zones. However, this is not the case when using fully-supervised methods (i.e. 100% of the labeled data), suggesting that a good performance in fully supervised experiments does not necessarily translate into a good performance in weakly-supervised experiments, and that careful feature analysis and selection is important when aiming to optimize the performance when learning from sparse data.

Discussion

In our experiments, the majority of weaklysupervised methods outperformed their corresponding supervised methods when using just 10% of the labeled data. The SVM-based methods performed better than the CRF-based ones (regardless of whether they were weakly or fully supervised). [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF] made a similar discovery when comparing fully supervised versions of SVM and CRF.

Our best performing weakly-supervised methods were those based on active learning. Making a good use of both labeled and unlabeled data, active learning combined with self-training (ASSVM) proved to be the most useful method. Given 10% of the labeled data, ASSVM obtained an accuracy of 81% and F-score of .76, outperforming the best supervised method SVM with a statistically significant difference. It reached its top performance (88% accuracy) when using 40% of the labeled data, and performed equally well as fully supervised SVM (i.e. 100% of the labeled data) when using just one third of the labeled data.

This result is in line with the results of many other text classification works where active learning (alone or in combination with other techniques such as self-training) has proved similarly useful, e.g. [START_REF] Lewis | A sequential algorithm for training text classifiers[END_REF][START_REF] Tong | Support vector machine active learning with applications to text classification[END_REF][START_REF] Brinker | On active learning in multi-label classification[END_REF][START_REF] Novak | Text classification with active learning[END_REF][START_REF] Esuli | Active learning strategies for multi-label text classification[END_REF][START_REF] Yang | Effective multi-label active learning for text classification[END_REF].

While active learning iteratively explores the unknown aspects of the unlabeled data, semisupervised learning attempts to make the best use of what it already knows about the data. In our experiments, semi-supervised methods (TSVM and SS-CRF) did not perform equally well as active learning -TSVM even produced a lower accuracy than SVM with the same amount of labeled data -although these methods have gained success in related works.

We therefore looked into related works using TSVM, e.g. [START_REF] Chapelle | Semisupervised classification by low density separation[END_REF], and discovered that our dataset is much higher in dimensionality than those employed in many other works. High dimensional data is more sensitive, and therefore fine-tuning with unlabeled data may cause a big deviation. We also looked into related works using SSCRF, in particular the work of [START_REF] Jiao | Semi-supervised conditional random fields for improved sequence segmentation and labeling[END_REF] who used the same SSCRF as the one we used in our experiments. [START_REF] Jiao | Semi-supervised conditional random fields for improved sequence segmentation and labeling[END_REF] employed a much larger data set than we did -one including 5448 labeled instances (in 3 classes) and 5210-25145 unlabeled instances. Given more labeled and unlabeled data per class we might be able to obtain better performance using SSCRF also on our task. However, given the high cost of obtaining labeled data methods not needing it are preferable.

Conclusions and future work

Our experiments show that weakly-supervised learning can be used to identify AZ in scientific documents with good accuracy when only a limited amount of labeled data is available. This is helpful thinking of the real-world application and porting of the approach to different tasks and domains. To the best of our knowledge, no previous work has been done on weakly-supervised learning of information structure according to schemes of the type we have focused on [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF][START_REF] Lin | Generative content models for structural analysis of medical abstracts[END_REF][START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF][START_REF] Shatkay | Multi-dimensional classification of biomedical text: Toward automated, practical provision of high-utility text to diverse users[END_REF][START_REF] Liakata | Corpora for the conceptualisation and zoning of scientific papers[END_REF].

Recently, some work has been done on the related task of classification of discourse relations in scientific texts: (Hernault et al., 2011) used structural learning (Ando and[START_REF] Kubota | A framework for learning predictive structures from multiple tasks and unlabeled data[END_REF] for this task. They obtained 30-60% accuracy on the RST Discourse Treebank (including 41 relation types) when using 100-10000 labeled and 100000 unlabeled instances. The accuracy was 20-60% when using the labeled data only. However, although related, the task of discourse relation classification differs substantially from our task in that it focuses on local discourse relations while our task focuses on the global structure of the scientific document.

In the future, we plan to improve and extend this work in several directions. First, the approach to active learning could be improved in various ways. The query strategy we employed (uncertainty sampling) is a relatively straightforward method which only considers the best estimate for each unlabeled instance, disregarding other estimates that may contain useful information. In the future, we plan to experiment with more sophisticated strategies, e.g. the margin sampling algorithm by [START_REF] Scheffer | Active hidden markov models for information extraction[END_REF] and the query-by-committee (QBC) algorithm by [START_REF] Seung | Query by committee[END_REF]. In addition, there are algorithms designed for reducing the redundancy in queries which may be worth investigating [START_REF] Steven | Large-scale text categorization by batch mode active learning[END_REF].

Also, [START_REF] Steven | Large-scale text categorization by batch mode active learning[END_REF] shows that Logistic Regression (LR) outperforms SVM when used with active learning, yielding higher F-score on the Reuters-21578 data set (binary classification, 10,788 documents in total, 100 of them labeled). It would be interesting to explore whether supervised methods other than SVM are optimal for active learning when applied to our task.

Secondly, we plan to investigate other semisupervised methods, for example, the Expectation-Maximization (EM) algorithm. [START_REF] Lanquillon | Learning from labeled and unlabeled documents: A comparative study on semisupervised text classification[END_REF] has shown that EM SVM performs better than supervised and transductive SVM on a text classification task when applied to the dataset of 20 Newsgroups (20 classes, 4000 documents for testing, 10000 unlabeled ones), yielding up to ∼10% higher accuracy when 200-5000 labeled documents are used for training.

In addition, other combinations of weaklysupervised methods might be worth looking into, such as EM+active learning [START_REF] Mccallum | Employing em and pool-based active learning for text classification[END_REF] and co-training+EM+active learning [START_REF] Muslea | Active + semi-supervised learning = robust multi-view learning[END_REF], which have proved promising in related text classification works.

Besides looking for optimal ML strategies, we plan to look for optimal features for the task. Our feature analysis showed that not all the features which had proved promising in fully supervised experiments were equally promising when applied to weakly-supervised learning from smaller data. We plan to look into ways of reducing the sparse data problem in features, e.g. by classifying not only verbs but also other word classes into semanticallymotivated categories.

One the key motivations for developing a weaklysupervised approach is to facilitate easy porting of schemes such as AZ to new tasks and domains. Recent research shows that active learning in a target domain can leverage information from a different but related (source) domain [START_REF] Rai | Domain adaptation meets active learning[END_REF]. Making use of existing annotated datasets in biology, chemistry, computational linguistics and law [START_REF] Teufel | Summarizing scientific articles: Experiments with relevance and rhetorical status[END_REF][START_REF] Mizuta | Zone analysis in biology articles as a basis for information extraction[END_REF][START_REF] Hachey | Extractive summarisation of legal texts[END_REF][START_REF] Teufel | Towards domain-independent argumentative zoning: Evidence from chemistry and computational linguistics[END_REF] we will explore optimal ways of combining weakly-supervised learning with domain-adaptation.

The work presented in this paper has focused on the abstracts annotated according to the AZ scheme. In the future, we plan to investigate the usefulness of weakly-supervised learning for identifying other schemes of information structure, e.g. [START_REF] Lin | Generative content models for structural analysis of medical abstracts[END_REF][START_REF] Hirohata | Identifying sections in scientific abstracts using conditional random fields[END_REF][START_REF] Shatkay | Multi-dimensional classification of biomedical text: Toward automated, practical provision of high-utility text to diverse users[END_REF][START_REF] Liakata | Corpora for the conceptualisation and zoning of scientific papers[END_REF], and not only in scientific abstracts but also in full journal papers which typically exemplify a larger set of scheme categories.

Finally, an important avenue of future research is to evaluate the usefulness of weakly-supervised identification of information structure for NLP tasks such as summarization and information extraction [START_REF] Tbahriti | Using argumentation to retrieve articles with similar citations[END_REF][START_REF] Ruch | Using argumentation to extract key sentences from biomedical abstracts[END_REF], and for practical tasks such as manual review of scientific papers for research purposes [START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF].

3

  Automatic identification of AZ 3.1 Features and feature extraction Guo et al. (

  and the distance between them is 2 |w| . The problem reduces to: Minimize |w| (in w, b) Subject to w • xb ≥ 1 for x of one class, w •

Figure 2 :

 2 Figure 2: Learning curve for different methods when using 0-100% of the labeled data

Table 1 :

 1 Categories of AZ appearing in the corpus of[START_REF] Guo | Identifying the information structure of scientific abstracts: an investigation of three different schemes[END_REF]

	Category	Abbr. Definition
	Background	BKG	The circumstances pertaining to the current work, situation, or its causes, history, etc.
	Objective	OBJ	A thing aimed at or sought, a target or goal
	Method	METH A way of doing research, esp. according to a defined and regular plan; a special form
			of procedure or characteristic set of procedures employed in a field of study as a mode
			of investigation and inquiry
	Result	RES	The effect, consequence, issue or outcome of an experiment; the quantity, formula,
			etc. obtained by calculation
	Conclusion	CON	A judgment or statement arrived at by any reasoning process; an inference, deduction,
			induction; a proposition deduced by reasoning from other propositions; the result of
			a discussion, or examination of a question, final determination, decision, resolution,
			final arrangement or agreement
	Related work	REL	A comparison between the current work and the related work
	Future work	FUT

Table 2 :

 2 Distribution of sentences in the AZ-annotated corpus

		BKG OBJ	METH RES	CON REL FUT
	Word	36828 23493 41544 89538 30752 2456 1174
	Sentence 1429 674 1473 3185 1082 95	47
	Sentence 18% 8%	18% 40% 14% 1%	1%
	8-18% of the corpus each. Two categories are very
	low in frequency, only covering 1% of the corpus
	each: Related work (REL) and Future work (FUT).

Table 3 :

 3 Results when using 10% of the labeled data

		Acc. F-score
		MF BKG OBJ METH RES CON REL FUT
	SVM	.77 .74 .84 .68 .71	.82 .64 -	-
	CRF	.70 .65 .75 .46 .48	.78 .76 -	-
	ASVM	.80 .75 .88 .56 .68	.87 .78 .33
	ASSVM .81 .76 .86 .56 .76	.88 .76 -	-
	TSVM	.76 .73 .84 .61 .71	.79 .71 -	-
	SSCRF .73 .67 .76 .48 .52	.81 .78 -	-

MF: Macro F-score of the five high frequency categories: BKG, OBJ, METH, RES, CON.

Table 4 :

 4 Leaving one feature out results for ASSVM when using 10% of the labeled data

		Acc. F-score
		MF BKG OBJ METH RES CON REL FUT
	Location .73 .67 .67 .55 .62	.85 .65 -	-
	Word	.80 .78 .87 .70 .74	.85 .72 -	-
	Bigram .81 .75 .83 .57 .71	.87 .78 .33 -
	Verb	.81 .79 .84 .77 .73	.87 .75 -	-
	VC	.79 .75 .86 .62 .72	.84 .70 -	-
	POS	.74 .70 .66 .65 .66	.82 .73 -	-
	GR	.79 .75 .83 .67 .69	.84 .72 -	-
	Subj	.80 .76 .87 .65 .73	.85 .72 -	-
	Obj	.80 .78 .84 .75 .70	.85 .75 -	-
	Voice	.78 .75 .88 .70 .71	.83 .62 -	-
	Φ	.81 .76 .86 .56 .76	.88 .76 -	-

MF: Macro F-score of the five high frequency categories: BKG, OBJ, METH, RES, CON.

The only exception is the history feature which was left out because it cannot be applied to all of our methods

Acknowledgments

The work reported in this paper was funded by the Royal Society (UK). YG was funded by the Cambridge International Scholarship.