Pierre Lissy
email: lissy@ceremade.dauphine.fr

A non-controllability result for the half-heat equation on the whole line based on the prolate spheroidal wave functions and its application to the Grushin equation

Keywords: Controllability, observability, fractional parabolic equations, prolate spheroidal wave functions MSC 2010 Classification: 93B05, 93B07, 35R11, 33E10

In this article, we revisit a result by A. Koenig concerning the non-controllability of the half-heat equation posed on R, with a control domain that is a measurable set whose exterior contains an interval. The main novelty of the present article is to disprove the corresponding observability inequality by using as an initial condition a family of prolate spheroidal wave function (PSWF) translated in the Fourier space, associated to a parameter c that goes to ∞. The proof is essentially based on the dual nature of the PSWF together with direct computations, showing that the solution "does not spread out" too much during time, with respect to the parameter c. As a consequence, we obtain a new non-controllability result on the Grushin equation posed on R × R.

Introduction

Let T > 0, and let ω be a mesurable subset of R, such that ω = R (which means that R \ ω contains at least a nonempty interval). We are interested in the following control problem, that we will call half-heat equation with distributed control:

∂ t y (t, x) + |∇| y (t, x) = 1 ω v(t, x) in (0, T ) × R, y (0, x) = y 0 (x) in L 2 (R). (1)
The operator |∇| is defined as a Fourier multiplier: for any h ∈ H 1 (R) and any ξ ∈ R, we have

|∇| h(ξ) = |ξ| h(ξ).
Here and in what follows, for h ∈ L 2 (R), h is the Fourier transform of h given by h(ξ) = R e -ixξ f (x)dx.

1

Notice that |∇| can also be defined equivalently (and also in many other ways, see e.g. [START_REF] Kwasnicki | Ten equivalent definitions of the fractional Laplace operator[END_REF]) as a singular integral operator by the following formula: for any h ∈ H 1 (R) and almost any x ∈ R,

|∇| h(x) = 1 π R h(x) -h(y) |x -y| 2 dy.
It is well-known that as soon as v ∈ L 2 ((0, T ) × R) and y 0 ∈ L 2 (R), there exists a unique solution y to (1) verifying moreover y ∈ C 0 ([0, T ], L 2 (R)), and there exists C(T ) > 0 such that for any v ∈ L 2 ((0, T ) × R) and any y 0 ∈ L 2 (R), we have

||y|| C 0 ([0,T ],L 2 (R)) C(T ) ||y 0 || L 2 (R) + ||v|| L 2 ((0,T )×R)
(see e.g. [START_REF] Coron | Control and nonlinearity[END_REF]Theorem 2.37]). Our main result is the following.

Theorem 1.1 System (1) is not null-controllable for no time T > 0, in the following sense: for any T > 0, there exists at least one initial condition y 0 ∈ L 2 (R) such that there exists no v ∈ L 2 ((0, T ) × R) for which the solution y of (1) verifies y(T ) = 0.

Our second result concerns the following Grushin equation on R 2 . We consider Ω = R × ω. We are interested in the following Grushin equation, controlled on Ω:

∂ t f (t, x, y) -∂ 2 xx f (t, x, y) -x 2 ∂ 2 yy f (t, x, y) = 1 Ω (t, x, y)g(t, x, y) in (0, T ) × R 2 , f (0, x, y) = f 0 (x, y) in R 2 . (2)
As for the previous equation, it is easy to prove that for g ∈ L 2 ((0, T ) × R 2 ) and f 0 ∈ L 2 R 2 , there exists a unique solution f to (2) verifying moreover f ∈ C 0 ([0, T ], L 2 (R 2 )), and there exists C(T ) > 0 such that for any g ∈ L 2 ((0, T ), L 2 (R 2 )) and any f 0 ∈ L 2 R 2 , we have

||f || C 0 ([0,T ],L 2 (R)) C(T ) ||f 0 || L 2 (R 2 ) + ||g|| L 2 ((0,T )×R 2 ) .
Our second main result is the following.

Theorem 1.2 System (2) is not null-controllable for no time T > 0.

Comparison with the existing literature

Controllability properties of fractional heat equations have been an increasing subject of interest these last years. The most well-understood case is the fractional heat equation on a bounded interval of R, with a scalar control, for the so-called spectral Laplace operator (i.e. obtained by functional calculus from the decomposition of the Dirichlet-Laplace operator in its basis of eigenfunctions). The framework of a scalar control encompasses most of the 1D situations where we have either a boundary control, or a distributed control that is imposed to be with separated variables, i.e. under the form u(t, x) = v(t)f (x), where f (x) is a fixed profile (with some conditions on its support and the behavior of its Fourier coefficients), and v(t) is a one-dimensional control belonging for example to a L 2 -space. In this context, null-controllability turns out to be essentially equivalent to the construction of appropriate bi-orthogonal functions to a family of exponential and strongly relies on the applications of the generalized Müntz theorem given in [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF] (see also [START_REF] Avdonin | Families of exponentials, the method of moments in controllability problems for distributed parameter systems[END_REF] and [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF]Appendix]). To be more precise, the first positive result for 1D fractional heat equations (of exponent α > 1/2) on a bounded interval with scalar control was provided in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] thanks to the celebrated moment method (see also [START_REF] Biccari | Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects[END_REF] for a close result with a definition of the fractional Laplace operator similar to the one used in the present article). Precise estimates on the cost of controllability in this case have been obtained in [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF][START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF][START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF]. Negative results in the case of exponents α 1/2 have been obtained in [START_REF] Fattorini | Control in finite time of differential equations in Banach space[END_REF] and made more precise in [START_REF] Micu | On the controllability of a fractional order parabolic equation[END_REF]. We also mention [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] for Neumann boundary conditions instead of Dirichlet boundary conditions. See also [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF]Appendix] for some extensions in the case where the control belongs to a finite-dimensional space.

The multi-dimensional setting has been less explored for fractional heat equations, even if the case of the usual heat equation on a bounded domain of R n or on a smooth compact Riemannian manifold with or without boundary has been known since the seminal works [START_REF] Lebeau | Contrôlle exact de l'équation de la chaleur[END_REF] and [START_REF] Fursikov | Controllability of evolution equations[END_REF]. Positive results for distributed control on compact manifolds for exponents α > 1/2 are given in [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF], with precise estimates on the cost of controllability.

The case of unbounded domains is also not well-understood for the moment. In [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF], a positive result is given for exponents α > 1/2 when the control set is the exterior of a compact set. For the usual heat equation, related works are also [START_REF] Micu | On the lack of null-controllability of the heat equation on the half space[END_REF] and [START_REF] Micu | On the lack of null-controllability of the heat equation on the half-line[END_REF], for some counterexamples on the half-line and the half-space, with boundary controls. Let us mention that for the heat equation on R n , the optimal controllability results have been only recently obtained in [START_REF] Wang | Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn[END_REF] and [START_REF] Egidi | Sharp geometric condition for null-controllability of the heat equation on ?d and consistent estimates on the control cost[END_REF], the main tool being the Logvinenko-Sereda uncertainty principle proved in [START_REF] Logvinenko | Equivalent norms in spaces of entire functions of exponential type[END_REF] (see also [START_REF] Havin | The uncertainty principle in harmonic analysis[END_REF]) and the use of the Lebeau-Robbiano strategy (see [START_REF] Lebeau | Contrôlle exact de l'équation de la chaleur[END_REF] and [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF]). The only known negative results for α < 1/2 on R n (n ∈ N * ) are given in [START_REF] Koenig | Lack of null-controllability for the fractional heat equation and related equations[END_REF]. An extension to the half-heat equation ( 1) is provided in [START_REF] Koenig | Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives[END_REF]Chapter 2] (notice that in the two previous references, a little bit more general model is studied here, the "rotated fractional heat equations"). The proof of [START_REF] Koenig | Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives[END_REF]Chapter 2] is based on the study of some inequalities for precise classes of holomorphic functions, but turns out to give a less precise result than ours. Notably, it cannot be applied to the Grushin operator.

Notice that proving negative controllability results for fractional heat equations and distributed control is much more difficult than the case of scalar input controls. For instance, for equation (1), at almost each time t ∈ [0, T ], the control lives in the infinite-dimensional space L 2 (ω), hence, there is no reason that we can restrict to a scalar (or finite-dimensional) input control as one could think for positive controllability results. Notably, the application of the generalized Müntz Theorem is not enough to deduce negative controllability results.

On the other hand, controllability properties of evolution equations involving hypoelliptic operators have recently become an active field of study (see [START_REF] Alphonse | Quadratic differential equations: partial Gelfand-Shilov smoothing effect and null-controllability[END_REF][START_REF] Bernier | Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and nullcontrollability[END_REF][START_REF] Beauchard | Null-controllability of non-autonomous Ornstein-Uhlenbeck equations[END_REF][START_REF] Beauchard | Null controllability of hypoelliptic quadratic equations[END_REF][START_REF] Beauchard | Heat equation on the Heisenberg group: observability and applications[END_REF][START_REF] Beauchard | Null controllability of Kolmogorov-type equations[END_REF][START_REF] Beauchard | Degenerate parabolic operators of Kolmogorov type with a geometric control condition[END_REF][START_REF] Laurent | Tunneling estimates and approximate controllability for hypoelliptic equations[END_REF] for hypoelliptic diffusions different that the Grushin ones, which will be discussed into details later on), due to the specific difficulties arising in this context, namely geometric conditions on the control region together with the appearance of minimal times of controllability. One simple example (but of interest) of such equations is the Grushin equation, that has been widely studied in the literature but is still not completely understood. The example of the Grushin plane can be seen as a first step to study the controllability properties of more general heat equations on singular Riemannian manifolds, which explains the interest of understanding deeply this case. Since the seminal work [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF], some improvements have been obtained in [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF][START_REF] Beauchard | Minimal time issues for the observability of Grushin-type equations[END_REF][START_REF] Koenig | Non null controllability of the Grushin equation in 2d[END_REF][START_REF] Duprez | Control of the Grushin equation: non-rectangular control region and minimal time[END_REF]. Notably, for the Grushin equation posed on (-1, 1) × (0, 1), the situation is the following:

-If the control set is a vertical strip that does not touch the singularity {x = 0}, controllability holds only in large time. The minimal control time can be exactly characterized and depends on the distance between the vertical strip and the singularity. Notably, if the vertical strip touches the singularity, we have controllability in arbitrary small time ( [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF][START_REF] Beauchard | Minimal time issues for the observability of Grushin-type equations[END_REF][START_REF] Allonsius | Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method[END_REF]). -On the contrary, if the control set is the exterior of a horizontal strip, we never have null controllability ( [START_REF] Koenig | Non null controllability of the Grushin equation in 2d[END_REF]). -To finish, if the control region is the neighbourhood of a curve coming from the "bottom" of (-1, 1) × (0, 1) and going to the top of (-1, 1) × (0, 1), we also have a result of nullcontrollability in sufficiently large time ( [START_REF] Duprez | Control of the Grushin equation: non-rectangular control region and minimal time[END_REF]). For some generalizations in higher dimension, see [START_REF] Beauchard | Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type[END_REF]. See also [START_REF] Morancey | Approximate controllability for a 2d Grushin equation with potential having an internal singularity[END_REF][START_REF] Cannarsa | Null controllability in large time for the parabolic Grushin operator with singular potential[END_REF] for a study of the Grushin operator with a singular potential, and [START_REF] Liu | Observability from measurable sets for a parabolic equation involving the Grushin operator and applications[END_REF] for generalizations in the case of measurable sets of controls. Remark that most of the above results concern controllability regions that have a rectangular structure. This is due to the fact that one of the main ingredients of the results described above is the use of Fourier series in one direction in order to reduce the problem to the study of a family of one-dimensional PDEs with a parameter (the Fourier coefficient). Trying to generalize the above results to other geometrical situations (general control domain, global geometry that is not cartesian) seems to be a very difficult challenge. Notice that some of the above results can be extended to the case (-1 -1) × T, R × T or R × (0, 1) (see notably [START_REF] Koenig | Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives[END_REF]Chapter 2]). However, to our knowledge, the case where the variable y belongs to the whole space R has not been investigated so far. This is one of the motivations of the present work.

We also mention [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF][START_REF] Fragnelli | Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations[END_REF][START_REF] Fragnelli | Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients[END_REF][START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for other examples of control problems of parabolic type with interior degeneracy in one space dimension.

Concerning the PSWF, since the seminal works [START_REF] Slepian | Prolate spheroidal wave functions, Fourier analysis and uncertainty, I[END_REF][START_REF] Landau | Prolate spheroidal wavefunctions, Fourier analysis and uncertainty, II[END_REF][START_REF] Landau | Prolate spheroidal wavefunctions, Fourier analysis and uncertainty, III[END_REF][START_REF] Slepian | Prolate spheroidal wavefunctions, Fourier analysis and uncertainty-IV: Extensions to many dimensions; Generalized prolate spheroidal functions[END_REF], they have been widely studied from a theoretical and numerical point of view. They also turned out to have many applications, notably in sampling and signal or image processing. For more explanations on the PSWF and their applications, we refer to the surveys [START_REF] Slepian | Some comments on Fourier analysis, uncertainty and modeling[END_REF][START_REF] Wang | A review of prolate spheroidal wave functions from the perspective of spectral methods[END_REF], the books [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF][START_REF] Hogan | Duration and Bandwidth Limiting[END_REF] and the references therein. However, as far as the author knows, this is the first time that they are used in the context of controllability properties of PDEs.

Heuristic and outline of the paper

Here, our goal is to give a different proof of the results given in [START_REF] Koenig | Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives[END_REF]Chapter 2], relying on an explicit construction of a family of counter-examples given by the PSWF. Another advantage of our technique is that it enables to give a corresponding result on the Grushin equation on R × R, and it is more likely to be extendable in the multi-dimensional case or on more general non-compact manifolds (see our concluding Section 5).

More precisely, we use the first PSWF as initial condition in order to disprove some observability inequality for the free half-heat equation [START_REF] Biccari | Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects[END_REF]. Our approach seems quite natural: as we will see, the first PSWF are bandlimited functions that saturate the Logvinenko-Sereda uncertainty principle outside of the interval (-1, 1). Notably, they strongly concentrate in (-1, 1), and they become more and more concentrated (at an exponential rate) as soon as the band-limit increases. Hence, they are very natural candidates in order to disprove some observability inequality when the observation is some set at least not contains (-1, 1). Of course, because of the dissipativity of the half-heat equation, one cannot expect that the associated solution to the free half-heat equation keeps the same concentration properties. However, it can be proved (and this is the main point of the present study) that during time, the associated solution still remains "quite concentrated". This will enable us to conclude that the observation on (0, T )×ω will be "small" by comparison to the whole L 2 -norm of the associated solution at time T . The extension to the Grushin plane is quite straightforward, using the particular form of our control set Ω and an appropriate Fourier decomposition.

The paper is organized as follows. In Section 2.1, we recall some known properties of the PSWF that we will need. In Section 3.1, we reduce the problem to the disproof of an adequate observability on a dual problem. In Section 3.2, we give some properties of the particular family of solutions of the adjoint problem we will consider. In Section 3.3, we give some useful estimates on the complex PSWF. Section 3.4 is the core of the paper: we give two main Lemmas 3.4 and 3.5 that express the fact that our family of solutions, which is very well concentrated at initial time, does not "spread out" too much during time. We can then conclude to the proof of Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2 (which will mainly be a consequence of the already proved estimates in Section 2). In Section 5, we give some concluding remarks.

2 Proof of Theorem 1.1

Preliminaries on the PSWF

In this section, we regroup some basic facts on the first PSWF. For the sake of clarity, we follow closely the presentation given in [51, Section 2.4] and we refer as often as possible to [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF], even if many results have been in general known for a longer time. Let c > 0, destined to tend to +∞. We introduce the following operator

F c : ϕ ∈ L 2 (-1, 1) → x → 1 -1 e icxξ ϕ (ξ) dξ ∈ L 2 (-1, 1) .
It is easy to prove that F c is a compact and normal operator on L 2 (-1, 1), with distinct eigenvalues (see [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF]Theorem 2.3]). We call λ c the largest (in modulus) eigenvalue of F c . We will call ψ c "the" (up to a normalization and orientation that will be detailed afterwards) first eigenvector of F c . ψ c is called the first (the one of index 0) PSWF with parameter c. By definition, ψ c verifies, for almost every x ∈ (-1, 1),

λ c ψ c (x) = 1 -1 e icxξ ψ c (ξ) dξ.
Hence, ψ c is a bandlimited function with bandlimit c, and its Fourier transform is in L 2 (-c, c) by the Plancherel Theorem and an easy change of variable. From the Paley-Wiener Theorem (see e.g. [53, Theorem 19.3, Page 370]), we deduce that ψ c can be extended on the entire plane C, and ψ c is an entire function of exponential type c which lies in L 2 (R), Moreover, the above expression can be extended in the complex plane as follows: for any z = x + it ∈ C,

ψ c (x + it) = 1 λ c 1 -1 e ic(x+it)ξ ψ c (ξ) dξ. (3)
We have the following properties on ψ c (see [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF]Theorem 2.3]).

Proposition 2.1 ψ c is real and even on R (and hence ψ c is even also on C), and has no roots on (-1, 1).

From (3) and Proposition 2.1, a straightforward computation gives the following property for ψ c on the complex plane C.

Proposition 2.2 For any z = x + it ∈ C,

ψ c (-x + it) = ψ c (x -it) = ψ c (x + it). (4)
Now, we explain how we normalize ψ c . We choose ψ c such that ψ c is normalized in L 2 (R) -norm and ψ c > 0 in (-1, 1) (it has no roots on (-1, 1) by Proposition 2.

1).

In what follows, we will also need to consider the operator

Q c : ϕ ∈ L 2 (-1, 1) → x → 1 π 1 -1 sin (c (x -ξ)) x -ξ ϕ (ξ) dξ ∈ L 2 (-1, 1). Remark that Q c = P c • E
, where E is the extension operator (by 0) from L 2 (-1, 1) to L 2 (R), and

P c : ϕ ∈ L 2 (R) → x → 1 π ∞ -∞ sin (c (x -ξ)) x -ξ ϕ (ξ) dξ ∈ L 2 (R).
P c turns out to be exactly the celebrated projection in L 2 (R) on its closed subspace of bandlimited functions with bandlimit c. By [51, Corollary 2.1], we have

F c F * c = F * c F c = 2π c Q c .
As in [51, (3.48)], we introduce

µ c = c 2π λ 2 c . (5)
Then, ψ c is clearly an eigenvector for Q c associated with the eigenvalue µ c :

µ c ψ c (x) = 1 π 1 -1 sin (c (x -ξ)) x -ξ ψ c (ξ) dξ. (6)
From [51, Theorem 7.6] and our choice of normalization (as explained in [51, Page 236], the authors chose a normalization in L 2 (-1, 1)-norm, which explains the difference of expression with [51, Theorem 7.6]), we have the following equality.

Lemma 2.1 We have 1 -1 ψ c (x) 2 dx = µ c = µ c R ψ c (x) 2 dx . (7)
An important feature of the first PSWF is the following concentration property (see [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF]Theorem 3.53]): any other function f which is c-bandlimited and has L 2 (R)-norm equal to 1 is such that

R\[-1,1] f 2 (x)dx 1 -µ c . (8)
It means that ψ c is the c-bandlimited function which concentrates the most on [-1, 1]. Notice that ( 8) is a version of the Logvinenko-Sereda uncertainty principle for c-bandlimited functions given in [START_REF] Logvinenko | Equivalent norms in spaces of entire functions of exponential type[END_REF], in the very particular case where the observation is made on the thick set R \ [-1, 1]. Hence, (1 -µ c ) -1 is the "best constant" in the Logvinenko-Sereda uncertainty principle for c-bandlimited functions for an observation on the thick set R\[-1, 1], reached only for the linear subspace generated by the first PSWF ψ c . From [26, Theorem 1] applied with a = 2C1c log(c) and n = 0 (see the introduction of [START_REF] Fuchs | On the Eigenvalues of an Integral Equation Arising in the Theory of bandlimited Signals[END_REF]), we also deduce the following estimate.

Lemma 2.2 We have 1 -µ c ∼ 4 √ πc 1 2 e -2c as c → ∞. (9)
Let us now explain the dual nature of the PSWF, that are both eigenfunctions for an integral and a differential operator. Let us call L c the differential operator

L c : ϕ ∈ C 2 (-1, 1) → x → -1 -x 2 ϕ (x) + 2xϕ (x) + c 2 x 2 ϕ (x) ∈ C 0 (-1, 1) .
Then, it can be proved (see [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF]Theorem 2.8]) that F c and L c commute. Hence, ψ c is also an eigenvector of L c , with associated eigenvalue χ c . Moreover, by [51, (3.4)], we have

0 < χ c < c 2 . ( 10
)
We also have the following asymptotic formula (see [51, Theorem 2.6]):

χ c ∼ c as c → ∞. (11)
By definition, ψ c verifies: for any x ∈ (-1, 1),

-1 -x 2 ψ c (x) + 2xψ c (x) + c 2 x 2 ψ c (x) = χ c ψ c (x) .
By analyticity, this expression can be extended on the whole complex plane, so that ψ c verifies: for

any z ∈ C, -1 -z 2 ψ c (z) + 2zψ c (z) + c 2 z 2 ψ c (z) = χ c ψ c (z) . (12)
To conclude, let us now give an equivalent on |ψ c (1)

| 2 as c → ∞. Lemma 2.3 |ψ c (1) | ∼ 2π 1 4 c 3
Proof of Lemma 2.3. We remark that by [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF], ψ c verifies, for any x ∈ (-1, 1),

µ c ψ c (x) = 1 π 1 -1 sin (c (x -ξ)) x -ξ ψ c (ξ) dξ.
Hence, ψ c

• √ c verifies, for any x ∈ (- √ c, √ c), µ c ψ c x √ c = 1 π 1 -1 sin ( √ cx -cξ) x/ √ c -ξ ψ c (ξ) dξ.
Using the change of variable ξ = √ cξ, we obtain

µ c ψ c x √ c = 1 √ cπ √ c - √ c sin ( √ c (x -ξ )) x/ √ c -ξ / √ c ψ c ξ √ c dξ = 1 π √ c - √ c sin ( √ c (x -ξ )) x -ξ ψ c ξ √ c dξ .
Hence, ψ c

• √ c
is exactly the first eigenfunction of the operator

ϕ ∈ L 2 - √ c, √ c → x → √ c - √ c sin ( √ c (x -ξ )) x -ξ ϕ (ξ ) dξ ∈ L 2 (- √ c, √ c),
which is exactly the setting of [START_REF] Fuchs | On the Eigenvalues of an Integral Equation Arising in the Theory of bandlimited Signals[END_REF] with √ c = a and n = 0. Let us set

f (x) = 1 c 1 4 √ µ c ψ c x √ c . (14)
Then, we have by the definition of µ c given in [START_REF] Beauchard | Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type[END_REF] that

√ c - √ c f (x) 2 dx = 1 √ cµ c √ c - √ c ψ c x √ c 2 dx = 1 µ c 1 -1 ψ c (y) 2 dy = 1.
We can now apply [26, III, Page 329] since we are in the correct normalization setting, and we obtain that

|f √ c | ∼ 2π 1 4 c 1 2 e -c µ c as c → ∞.
Going back to ψ c thanks to [START_REF] Beauchard | Degenerate parabolic operators of Kolmogorov type with a geometric control condition[END_REF] and taking into account that µ c → 1 as c → ∞ by ( 9), we deduce that (13) holds.

3 Proof of Theorem 1.1

Reduction to the disproof of an observability inequality

Our proof mainly relies on a classical duality argument: the null-controllability of (1) at time T > 0 is equivalent to the following property: there exists C(T ) > 0 such that for any u 0 ∈ L 2 (R), the solution u of

∂ t u (t, x) + |∇| u (t, x) = 0 in (0, T ) × R, u (0, x) = u 0 (x) (15)
verifies the following inequality, called observability inequality:

R |u(T, x)| 2 dx C(T ) T 0 ω |u(t, x)| 2 dxdt. (16)
For more explanations, see e.g. [19, Theorem 2.44, Page 56]. Hence, in order to prove Theorem 1.1, it is sufficient to exhibit a family of initial conditions u 0 c depending on some parameter c > 0 such that the corresponding family of solutions u c to (15) make the quotient

T 0 ω |u c (t, x)| 2 dxdt R |u c (T, x)| 2 dx go to 0 as c → ∞.
Let us explain some important reduction for what follows. Since ω is some measurable subset of R such that ω = R, R \ ω is an open subset of R which is not empty, so, there exists some ball B (y 0 , ε) in R \ ω with ε > 0 and y 0 ∈ R. Without loss of generality, we can assume that y 0 = 0 (this is just a question of translating in space the initial condition of ( 15)). Moreover, we assume without loss of generality that ω = R \ [-ε, ε]. Indeed, if [START_REF] Biccari | Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects[END_REF] were controllable on ω, it would be also controllable on

R \ [-ε, ε] since ω ⊂ R \ [-ε, ε]. Hence, we will disprove the observability on R \ [-ε, ε].

A particular family of solutions

First of all, let us give the following crucial lemma.

Lemma 3.1 If u 0 (x) = ψ c (x) e icx ,
then the solution u of (15) is given for any t ∈ R and any

x ∈ R by u (t, x) = e ic(x+it) ψ c (x + it) . (17)
Moreover, its Fourier transform in space is given by

u (t, ξ) = 2π cλ c 1 [-c,c] (ξ -c) e -t|ξ| ψ c ξ -c c . (18)
Proof of Lemma 3.1. First of all, we have

u (t, ξ) = e -t|ξ| ψ c (•) e ic• (ξ) = e -t|ξ| ψ c (ξ -c) .
Due to the expression of the PSWF given in (3) for t = 0, an easy change of variable and the fact that ψ c is even on R by Proposition 2.1, we deduce that

ψ c (x) = 1 cλ c c -c e ixξ ψ c ξ c dξ = 1 cλ c R 1 [-c,c] (ξ ) e ixξ ψ c ξ c dξ .
Hence, the Fourier inversion formula of the Fourier transform gives

ψ c (ξ) = 2π cλ c 1 [-c,c] (ξ) ψ c ξ c .
We deduce that ( 18) holds. Using one more time the Fourier inversion formula and the change of variable ξ = ξ-c c , we obtain

u (t, x) = 1 cλ c R e -t|ξ|+ixξ ψ c ξ -c c 1 [-c,c] (ξ -c) dξ = 1 λ c 1 -1 e -ct|ξ +1|+icx(ξ +1) ψ c (ξ ) dξ = 1 λ c 1 -1 e -ct(ξ +1)+icx(ξ +1) ψ c (ξ ) dξ = 1 λ c e -ct+icx 1 -1
e -ctξ +icxξ ψ c (ξ ) dξ .

We conclude by remarking that -ct + icx = ic (x + it) and by applying identity [START_REF] Bernier | Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and nullcontrollability[END_REF].

Remark that if u is a solution of (15) associated with the initial condition u 0 , then, for any β > 0, u(β•, β•) is still a solution of (15) associated with the initial condition u 0 (β•). Hence, we deduce the following result.

Corollary 3.2 Let α > 0. If u 0 c,ε,α (x) = e i αc 2 log(c)ε x ψ c αc log(c)ε
x , then the corresponding solution of (15) is given for any t ∈ R and any x ∈ R by

u c,ε,α (t, x) = e i αc 2 log(c)ε (x+it) ψ c αc log(c)ε (x + it) . (19)

The behaviour of ψ c on the upper half plane for large x

Our proof requires to obtain some fine estimates on ψ c in the complex plane by using growth properties of second order ordinary differential equations with complex coefficients. Let us remind the following general theorem (see [START_REF] Osipov | Prolate spheroidal wave functions of order zero[END_REF]Theorem 2.21] and [START_REF] Lewis | Inequalities for complex linear differential systems of the second order[END_REF]).

w (t) u (t) = 0 β(t) γ(t) 0
w(t) u(t) . ( 20
)
Assume moreover that β and γ do not vanish on

[a, b]. Introduce R(t) = |β(t)| |γ(t)| (21)
and

Q(t) = |w(t)| 2 + R(t)|w (t)| 2 . ( 22
)
Then, for any a t 0 t 1 b, we have

Q(t 0 ) R(t 0 ) 1 4 Q(t 1 ) R(t 1 ) 1 4 exp   t1 t0 R (s) 4R(s) 2 + |β(s)γ(s)| + Re (β(s)γ(s)) 2 ds   . (23)
We can deduce from Proposition 3.1 the following estimate of ψ c on the upper half plane, for large values of x, that might be of independent interest. Proposition 3.2 There exists two constants C, C 1 > 0 such that for any x 2, t 0 and c > 0 large enough (independently of x or t), we have

|ψ c (x + it)| Ce ct ψ c (1) cλ c √ x 2 + t 2 exp C 1 ct x 2 + t 2 . ( 24
)
Proof of Proposition 3.2. In the following proof, C > 0 denotes a numerical constant, that does not depend on x, t or c, and might change from inequality to inequality. For x + it ∈ C, with x 2 and t 0, we introduce

φ c (x + it) = ψ c (x + it) (x + it) 2 -1. (25)
It is well-known that this change of unknown enables to remove the first-order term in [START_REF] Beauchard | Heat equation on the Heisenberg group: observability and applications[END_REF], and that φ c verifies (see e.g. [51, Section 4.2.3])

φ c (x + it) + c 2 (x + it) 2 -χ c (x + it) 2 -1 + 1 ((x + it) 2 -1) 2 φ c (x + it) = 0. (26)
We introduce the following notations, according to (26) and Proposition 3.1 applied with w(x) = φ c (x + it) and u(x) = φ c (x + it) :

β(x) = 1, γ(x) = - c 2 (x + it) 2 -χ c (x + it) 2 -1 + 1 ((x + it) 2 -1) 2 , R(x) = 1 c 2 (x+it) 2 -χc (x+it) 2 -1 + 1 ((x+it) 2 -1) 2 . ( 27
)
Remark that R verifies that for any t 0, any x 2 and any c 1 (thanks to (10)),

1 Cc 2 |R(x)| C c 2 . ( 28
)
We also introduce

Q(x) = |φ c (x + it)| 2 + R(x) |φ c (x + it)| 2 . ( 29
)
From now on, we call y 0 = χ c c 2 ∈ (0, 1) by ( 10). An explicit computation gives

Re (γ(x)) = - c 2 t 4 + t 2 2x 2 + y 2 0 + 1 + x 2 -1 (x -y 0 )(x + y 0 ) 2 t 4 + 2t 2 (x 2 + 1) + (x 2 -1) 2 - t 4 + t 2 2 -6x 2 + x 2 -1 2 t 4 + 2t 2 (x 2 + 1) + (x 2 -1) 2 2
(30) and

Im (γ(x)) = - 2c 2 tx y 2 0 -1 t 4 + 2t 2 (x 2 + 1) + (x 2 -1) 2 - 4tx t 2 -x 2 + 1 t 4 + 2t 2 (x 2 + 1) + (x 2 -1) 2 2 (31)
From ( 30), since for any x 2 and t 0,

t 4 + t 2 2 -6x 2 + x 2 -1 2 t 4 + 2t 2 (x 2 + 1) + (x 2 -1) 2 2
C, and y 0 ∈ (0, 1), it is clear that for any x 2, t 0 and c > 0 large enough (not depending on x or t, which will be assumed implicitly from now on), we have |Re(γ(t))| Cc 2 and Re(γ(t)) 0.

From ( 31), since for any x 2 and t 0, we have

t 4 + 2t 2 (x 2 + 1) + (x 2 -1) (t 2 + x 2 -1) 2 and |t 2 -x 2 + 1| |t 2 + x 2 -1|,
combined with the fact that y 0 ∈ (0, 1), we also deduce that for c 2,

|Im(γ(x))| 2c 2 tx (t 2 + x 2 -1) 2 . ( 33
)
Hence, using the inequality

√ a + b - √ a b/(2 √ a)
, true for any a, b 0, since Re(γ(t)) 0 by [START_REF] Koenig | Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives[END_REF], we obtain that

|β(s)γ(s)| + Re (β(s)γ(s)) 2 = Re(γ(s)) 2 + Im(γ(s)) 2 -|Re (γ(s)) | 2
Im(γ(s)) 2 4|Re(γ(s))| . ( 34
)
From ( 32), ( 33) and ( 34), we deduce that

|β(s)γ(s)| + Re (β(s)γ(s)) 2 C c 2 t 2 x 2 (t 2 + x 2 -1) 4 . ( 35
)
Now, we apply Proposition 3.1. We obtain that for any 2 x x 1 , any t 0 and any c > 0, we have

Q(x) R(x) 1 4 Q(x 1 ) R(x 1 ) 1 4 exp   x1 x R (s) 4R(s) 2 + |β(s)γ(s)| + Re (β(s)γ(s)) 2 ds   .
Using [START_REF] Havin | The uncertainty principle in harmonic analysis[END_REF] and the inequality √ a + b √ a + √ b, true for any a, b 0, we deduce that for c > 0 large enough,

Q(x) C Q(x 1 ) exp x1 x R (s) 4R(s) ds exp x1 x |β(s)γ(s)| + Re (β(s)γ(s)) 2 ds . ( 36
)
Since R is a rational function of x, R /R can change sign only a finite number of times on [2, +∞).

Let us decompose [x, x 1 ] as

[x, x 1 ] = p i=1 [a i-1 , a i ], a i-1 < a i ,
where R /R is of strict constant sign on (a i , a i+1 ). Assume without loss of generality that R /R is positive on (a 0 , a 1 ) and that p is even (if not, it is very easy to adapt the following reasoning). Then,

exp x1 x R (s) 4R(s) ds = exp   (p-2)/2 i=0 a2i+1 a2i R (s) 4R(s) ds - (p-2)/2 i=0 a2i+2 a2i+1 R (s) 4R(s) ds   =   (p-2)/2 i=0 R(a 2i+1 ) R(a 2i ) (p-2)/2 i=0 R(a 2i+1 ) R(a 2i+2 )   1 4
.

Using [START_REF] Havin | The uncertainty principle in harmonic analysis[END_REF] and the fact that p is independent of c on t, x and x 1 , (because the degree in x of the numerator of R does not depend on c or t, so p remains bounded when c , t, x and x 1 vary), the above expression provides

exp x1 x R (s) 4R(s) ds C. (37)
Now, let us remind the following expansions of ψ c and ψ c , coming from the expression provided in [51, Theorems 6.21 and 6.22] and extended in the complex plane:

ψ c (x + it) = 2ψ c (1) sin(c(x + it)) c(x + it)λ c + o 1 x as x → ∞ and
ψ c (x + it) = 2ψ c (1) cos(c(x + it)) (x + it)λ c + o 1 x as x → ∞.
From ( 25), ( 28), [START_REF] Hogan | Duration and Bandwidth Limiting[END_REF] and the above expansions, we deduce that for any t 0 and any c > 0 large enough,

lim sup x1→+∞ Q(x 1 ) Ce 2ct ψ c (1) 2 c 2 λ 2 c .
Combining this estimate with ( 36), ( 37) and ( 35), we deduce that for any x 2, any t 0 and any c > 0 large enough,

Q(x) Ce ct ψ c (1) cλ c exp +∞ x Ccts (s 2 + t 2 -1) 2 ds Ce ct ψ c (1) cλ c exp Cct (x 2 + t 2 -1) . ( 38
)
Let us now investigate

| (x + it) 2 -1|. We have | (x + it) 2 -1| 2 = (x 2 -t 2 -1) 2 + 4x 2 t 2 .
Now, we remark that

(x 2 -t 2 -1) 2 + 4x 2 t 2 = (x 2 -1) 2 + t 4 -2t 2 (x 2 -1) + 4x 2 t 2 (x 2 -1) 2 + t 4 -2t 2 (x 2 -1) + 4t 2 (x 2 -1) (x 2 -1) 2 + t 4 + 2t 2 (x 2 -1) ((x 2 -1) + t 2 ) 2 .
We deduce that for any x 2 and any t 0, we have

| (x + it) 2 -1| 1 4 x 2 + t 2 . (39)
Hence, combining ( 38), ( 29), ( 25) and ( 39) easily leads to [START_REF] Fragnelli | Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations[END_REF].

From now on, we consider

α = 2C 1 , (40)
where C 1 is the constant appearing in [START_REF] Fragnelli | Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations[END_REF]. An important consequence is the following.

Corollary 3.3 There exists C > 0 such that for any c > 0 large enough and any t 0, we have

R\[- 2C 1 c log(c) , 2C 1 c log(c) ] |ψ c (x + it)| 2 dx e -2ct C log(c) 1 -µ c c log(c) + t . (41)
Remark 1 -Estimate (41) roughly says that ψ c (x + it)e ic(x+it) remains "well-concentrated" in space at least on [-2C1c log(c) , 2C1c log(c) ]. Remark that it is important here to consider large enough x. Indeed, if x is too small, the term exp

Cct (x 2 +t 2 )
in [START_REF] Fragnelli | Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations[END_REF] will give an extra exponential term that enters in competition with the exponential decreasing given in [START_REF] Beauchard | Minimal time issues for the observability of Grushin-type equations[END_REF].

-Let us emphasize that if we look at the proof, we would directly obtain that

R\[- 2C 1 c log(c) , 2C 1 c log(c) ] |ψ c (x + it)| 2 dx e -2ct C log(c) |ψ c (1)| 2 c( c log(c) + t)
.

This means that introducing µ c and using estimation (9) is stricto sensu not needed in our proof. However, we think that expression (41) is more eloquent, due to the interpretation in terms of concentration of solutions we gave in the previous point of this remark.

Proof of Corollary 3.3. In the following proof, C > 0 denotes a numerical constant, that does not depend on x, t or c, and might change from inequality to inequality. Using ( 24) and ( 40), for any c > 0 large enough, x 2C1c log(c) and t 0, we have

|ψ c (x + it)| 2 Ce 2ct ψ c (1) 2 c 2 λ 2 c (x 2 + t 2 ) exp   2C 1 ct 4C 2 1 c 2 log(c) 2 + t 2   .
Since the maximum of the function

t 0 → 2C 1 ct 4C 2 1 c 2 log(c) 2 + t 2
is log(c)/2, we deduce that

|ψ c (x + it)| 2 e -2ct C log(c) ψ c (1) 2 c 2 λ 2 c (x 2 + t 2 )
.

Using ( 5), ( 9) and ( 13), we deduce that for any c > 0 large enough, x 2C1c log(c) and t 0, we have

|ψ c (x + it)| 2 e -2ct C log(c) 1 -µ c x 2 + t 2 .
Integrating this expression between 2C1c log(c) and +∞ and using (4) leads to

R\[- 2C 1 c log(c) , 2C 1 c log(c) ] |ψ c (x + it)| 2 dx e -2ct C log(c)(1 -µ c ) Arctan( t log(c) 2C1c ) t .
The easy estimate Arctan(y) y 2 1 + y , y 0, applied to y = t log(c) 2C1c enables to conclude.

Proof of the non-controllability result

Let us remind that T > 0 is fixed. We still consider α chosen in [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF], and we consider the solution u c,ε,α given in [START_REF] Coron | Control and nonlinearity[END_REF] and the quotient

Q(c, ε) = T 0 ||u c,ε,α (t, •) || 2 L 2 (R\[-ε,ε]) dt ||u c,ε,α (T, •) || 2 L 2 (R) . (42)
As already explained in Section 3.1, our goal is to prove that at fixed T , Q(c, ε) → 0 as c → ∞, which will exactly mean that the observability inequality ( 16) cannot be true. We first estimate from above the numerator. This is the purpose of the following Lemma.

Lemma 3.4 There exists C(ε) > 0, depending only on ε, such that for any t 0 and any c large enough (independently on t, but possibly depending on ε),

||u c,ε,α (t, •) || 2 L 2 (R\[-ε,ε]) C(ε) (log(c)) 5 2 (1 -µ c ) c 2 (1 + t) . ( 43
)
Remark 2 The term "(1 + t)" will be particularly useful in the proof of Theorem 1.2.

Proof of Lemma 3.4. From [START_REF] Coron | Control and nonlinearity[END_REF] with α as in [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF] and an easy change of variables, we have

R\[-ε,ε] |u c,ε,α (t, x) | 2 dx = e -4C 1 c 2 log(c)ε t R\[-ε,ε] ψ c 2C 1 c log(c)ε (x + it) 2 = ε log(c) 2C 1 c e -4C 1 c 2 log(c)ε t R\[- 2C 1 c log(c) , 2C 1 c log(c) ] ψ c x + i 2C 1 ct log(c)ε 2 dx.
Using [START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF], we deduce that

R\[-ε,ε] |u c,ε,α (t, x) | 2 dx C log(c) ε log(c) 2C 1 c (1 -µ c ) c log(c) + 2C1ct log(c)ε
.

Our result immediately follows. Now, we estimate the denominator.

Lemma 3.5 There exist K(T, ε) > 0 and C (T, ε) > 0 (depending only on T and ε) such that for any c K(T, ε), we have

R |u c,ε,α (T, x) | 2 dx C (T, ε) |ψ c (1) | 2 log(c) 3 c 3 . ( 44
)
Proof of Lemma 3.5. From [START_REF] Coron | Control and nonlinearity[END_REF] with α given in [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF] and an easy change of variables, we have

R |u c,ε,α (T, x) | 2 dx = e -4C 1 c 2 log(c)ε T log(c)ε 2C 1 c R ψ c x + i 2C 1 c log(c)ε T 2 dx.
Using [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF] and the Plancherel Theorem, we have

R |u c,ε,α (T, x) | 2 dx = log(c)ε 2C 1 c 2πe -4C 1 c 2 log(c)ε T cλ 2 c 1 -1 |ψ c (ξ)| 2 e -4C 1 c 2 log(c)ε T ξ dξ. (45)
We introduce the auxiliary function f defined on (-1, 0) given by

f (ξ) = ψ c (ξ) e -4C 1 c 2 log(c)ε T ξ .
We remark that

f (ξ) = ψ c (ξ) - 4C 1 c 2 log(c)ε T ψ c (ξ) e -4C 1 c 2 log(c)ε T ξ .
Using [START_REF] Beauchard | Heat equation on the Heisenberg group: observability and applications[END_REF] at point x = -1 and t = 0, we know that ψ c (-1) = c 2 -χc 2 ψ c (-1). Hence, we have

f (-1) = c 2 -χ c 2 - 4C 1 c 2 log(c)ε T ψ c (-1) e 4C 1 c 2 log(c)ε T .
Moreover, using [START_REF] Beauchard | Null controllability of hypoelliptic quadratic equations[END_REF] and the fact that for any c > 0, ψ c (-1) = ψ c (1) > 0 by [51, (7.62)], we deduce that for c larger than a constant depending only on T and ε, f (-1) > 0, so that f > 0 on a neighbourhood of -1 (depending on c). If f > 0 on (-1, 0), we call ξ 0 = 0. Otherwise, let us call ξ 0 the smallest point of (-1, 0) where f (ξ) = 0, i.e. the first point ξ 0 verifying

ψ c (ξ 0 ) = 4C 1 c 2 log(c)ε T ψ c (ξ 0 ) . ( 46
)
ξ 0 is well-defined since f is analytic on C, so it has a finite number of zeros on (-1, 0). It is clear that at this point, one must have f (ξ 0 ) 0. Using [START_REF] Micu | On the lack of null-controllability of the heat equation on the half space[END_REF], if we compute f (ξ 0 ), we obtain

f (ξ 0 ) = ψ c (ξ 0 ) - 8C 1 c 2 log(c)ε T ψ c (ξ 0 ) + 16C 2 1 c 4 log(c) 2 ε T 2 ψ c (ξ 0 ) e -4C 1 c 2 log(c)ε T ξ0 = ψ c (ξ 0 ) - 16C 2 1 c 4 log(c) 2 ε T 2 ψ c (ξ 0 ) e -4C 1 c 2 log(c)ε T ξ0 .
Using [START_REF] Beauchard | Heat equation on the Heisenberg group: observability and applications[END_REF] together with [START_REF] Micu | On the lack of null-controllability of the heat equation on the half space[END_REF] in the above expression, we obtain that

1 -ξ 2 0 f (ξ 0 ) e cT ξ0 = c 2 ξ 2 0 -χ c -1 -ξ 2 0 16C 2 1 c 4 log(c) 2 ε T 2 ψ c (ξ 0 ) + 2ξ 0 ψ c (ξ 0 ) = c 2 ξ 2 0 -χ c -1 -ξ 2 0 16C 2 1 c 4 log(c) 2 ε T 2 + 8C 1 c 2 log(c)ε T ξ 0 ψ c (ξ 0 ) .
The smallest root of the polynomial (in the variable ξ 0 ) is given by

ξ 1 (T, ε) = -8c 2 C1T log(c) - 64c 4 C 2 1 T 2 2 log 2 (c) -4 16c 4 C 2 1 T 2 2 log 2 (c) + c 2 - 16c 4 C 2 1 T 2 2 log 2 (c) -χ c 2 16c 4 C 2 1 T 2 2 log 2 (c) + c 2 .
An explicit calculation gives that

ξ 1 (T, ε) ∼ 2 log(c) 2 32c 2 C 2 1 T 2 -1 as c → ∞. (47)
We deduce that for c larger than a constant K(T, ε) depending only on T and ε, we have ξ 1 ∈ (-1, 0). Hence, for c larger than some constant depending on T and ε, since we must have f (ξ 0 ) 0, we are sure that ξ 0 ξ 1 (T, ε). We have finally proved that for any c K(T, ε), f is increasing on (-1, ξ 1 (T, ε)). Going back to [START_REF] Micu | On the controllability of a fractional order parabolic equation[END_REF], we deduce that for c K(T, ε),

R |u c,ε,α (T, x) | 2 dx log(c)ε 2C 1 c 2πe -4C 1 c 2 log(c)ε T cλ 2 c ξ1(T,ε) -1 |ψ c (ξ) | 2 e -4C 1 c 2 log(c)ε T ξ dξ 2π log(c)εe -4C 1 c 2 log(c)ε T 2C 1 c 2 λ 2 c (1 -ξ 1 (T, ε)) |ψ c (-1) | 2 e 4C 1 c 2 log(c)ε T 2π log(c)ε 2C 1 c 2 λ 2 c (1 -ξ 1 (T, ε)) |ψ c (-1) | 2 .
By using ( 5), ( 9), [START_REF] Micu | On the lack of null-controllability of the heat equation on the half-line[END_REF] and the parity of ψ c , the conclusion easily follows.

Proof of Theorem 1.1.

We are now able to conclude. Remind that T > 0 and ε > 0 are fixed. From ( 42), ( 43) and (44), we deduce that for any c K(T, ε), we have

Q(c, ε) C (T, ε) c (1 -µ c ) log(c)|ψ c (1) | 2 ,
for C (T, ε) depending only on T and ε. Hence, using (9) together with (13), we deduce that for c K(T, ε) and some constant C (T, ε) depending only on T and ε,

Q(c, ε) C (T, ε) C (T, ε) log(c) .
This ends the proof of the non-observability result by making c go to ∞.

Remark 3

The family of initial conditions (and then the family of solutions) we use are complexvalued. However, it is quite natural to try to find counter-examples in the case of real initial conditions and real-valued solutions. However, since the half-Laplace operator is a real Fourier multiplier, it can be proved that either Re(u) or Im(u) gives a real counterexample to inequality [START_REF] Boas | Entire Functions[END_REF]. Indeed, assume that we decompose the initial condition u 0 in real and imaginary part as

u 0 (x) = f 0 (x) + ig 0 (x).
Then, since (-∆) 1 2 is a real Fourier multiplier, one easily see that in any time t > 0,

u(t, x) = f (t, x) + ig(t, x),
where f is the solution to (15) with initial condition f 0 and g is the solution to (15) with initial condition g 0 . Hence, the quotient (42) can be rewritten as

Q(c, ε) = T 0 ||f (t, •) || 2 L 2 (R\[-ε,ε]) + ||g (t, •) || 2 L 2 (R\[-ε,ε]) dt ||f (T, •) || 2 L 2 (R) + ||g (T, •) || 2 L 2 (R) . ( 48
)
Our goal is to prove that either

T 0 ||f (t, •) || 2 L 2 (R\[-ε,ε]) dt ||f (T, •) || 2 L 2 (R) or T 0 ||g (t, •) || 2 L 2 (R\[-ε,ε]) dt ||g (T, •) || 2 L 2 (R)
has a subsequence that goes to 0 as c → ∞. By positivity and by passing to the inverse, it is enough to prove that either

InvQ r (c) = ||f (T, •) || 2 L 2 (R) T 0 ||f (t, •) || 2 L 2 (R\[-ε,ε]) + ||g (t, •) || 2 L 2 (R\[-ε,ε]) dt or InvQ i (c) = ||g (T, •) || 2 L 2 (R) T 0 ||f (t, •) || 2 L 2 (R\[-ε,ε]) + ||g (t, •) || 2 L 2 (R\[-ε,ε]
) dt has a subsequence that goes to ∞ as c → ∞. If it was not the case, it would mean that both sequences are bounded and so is InvQ r (c) + InvQ i (c). This is impossible since this quantity is supposed to go to +∞: InvQ r (c) + InvQ i (c) = 1 Q(c,ε) where Q(c, ε) is given in (48) and goes to 0.

Proof of Theorem 1.2

We only give the main ingredients, since the proof is very close to the one of Theorem 1.1. Let T > 0. As for the half-heat equation (1), we can restrict our study to disprove the null-controllability of (2) on the control set ω = R × (R \ [-ε, ε]). As for the half-heat equation, the null-controllability of (2) at time T > 0 is equivalent to the following property: there exists C(T ) > 0 such that for any v 0 ∈ L 2 (R 2 ), the solution v of

∂ t v (t, x, y) -∂ 2 xx v(t, x, y) -x 2 ∂ 2 yy v(t, x, y) = 0 in (0, T ) × R 2 , v (0, x, y) = v 0 (x, y) in R 2 , ( 49
) verifies R R |v (T, •) | 2 dydx C(T ) T 0 R R\[-ε,ε] |v (t, x, y) | 2 dydxdt. (50)
If we consider the Fourier transform in the second space variable y that we still call , we obtain

∂ t v (t, x, ξ) -∂ 2 xx v(t, x, ξ) + x 2 ξ 2 v(t, x, y) = 0 in (0, T ) × R 2 , v (0, x, ξ) = v 0 (x, ξ) in R 2 .
The elliptic operator -∂ 2 xx +x 2 ξ 2 is exactly the harmonic oscillator. Notably, the first eigenfunction is given by the first Hermite function, correctly rescaled, namely x → e -

+ i t + x 2 2 . ( 51
)
Hence v c,ε,α (t, x, y) = u c,ε,α t + x 2 2 , y , where u c,ε,α is the solution of (15) used in Section 2. It is now quite straightforward to conclude. Indeed, according to the inequality (50) that we want to disprove, we introduce the quotient

Q (c, ε) = T 0 R R\[-ε,ε] |v c,ε,α (t, x, y) | 2 dydxdt R R |v c,ε,α (T, x, y) | 2 dydx . (52)
Let us look at the numerator. Using ( 51), [START_REF] Coron | Control and nonlinearity[END_REF] with α given in [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF], and (43), we have Hence, using [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] where T is replaced by T + 1/2, we deduce that there exist K(T, ε) > 0 and C (T, ε) > 0 depending only on T such that for any c K(T, ε), we have Hence, going back to expression (52) and using the same computations as for u c,ε,α in Section 2, we deduce as before that Q (c, ε) → 0 as c → ∞, which gives the desired non-controllability result.

Conclusion

In this article, we explained how we can use the PSWF in order to obtain negative controllability results for the half-heat equation on the whole line R and the Grushin plane on the whole plane R 2 . Since PSWF are functions that are naturally bandlimited and that concentrate very strongly on (-1, 1), they are quite natural candidates as initial conditions in order to disprove observability inequalities. Our hope is that this natural idea can be exploited in order to derive other noncontrollability results for weakly diffusive equations on general geometries. Using the results given in [START_REF] Slepian | Prolate spheroidal wavefunctions, Fourier analysis and uncertainty-IV: Extensions to many dimensions; Generalized prolate spheroidal functions[END_REF], we hope to extend our results in the case of the whole space R n , n 2. A more ambitious goal would be the following. If we consider some smooth Riemannian manifold M (that can be unbounded), can we use some generalized PSWF that saturate some Logvinenko-Sereda uncertainty principle in order to obtain non-null controllability results for the half-heat equation on M ? Now, if we consider a non-compact complete singular manifold for instance with an interior singularity that is an hypersurface, is it possible to use arguments similar to the one developed in Section 1.2 to derive some non-controllability results for the corresponding hypoelliptic diffusion? However, one major limitation is that our study relies strongly on what Slepian calls in [START_REF] Slepian | Some comments on Fourier analysis, uncertainty and modeling[END_REF] "the lucky accident": the integral operator F c commutes with the differential operator L c . Note that this property is crucial in our proof, since it is used both in the proof of Lemmas 3.4 and 3.5. In some particular cases, such a lucky accident still happens (notably in R n for concentration on a ball, see [START_REF] Slepian | Prolate spheroidal wavefunctions, Fourier analysis and uncertainty-IV: Extensions to many dimensions; Generalized prolate spheroidal functions[END_REF]). It seems that in more complex geometries, one cannot expect to exhibit some adequate differential operator L c , making impossible to reproduce the proof given here.

Proposition 3 . 1

 31 Let a < b be real numbers. Consider w, u, β, γ : [a, b] → C that are of class C 1 , verifying the following differential system : for any t ∈ [a, b], we have

|ξ|x 2 2 ,2C 1 c 2 log

 22 associated to the eigenvalue |ξ|. Hence, if we consider the initial condition given byv 0 c,ε,α (x, y) = ei that the corresponding solution of (49) is given byv c,ε,α (t, x, y) = e i 2C 1 c 2 log(c)ε y+i t+ x 2 2 ψ c 2C 1 c log(c)ε y

5 2 ( 1 - 5 2 ( 1 -µ c ) c 2 .

 51512 -ε,ε] |v c,ε,α (t, x, y) | 2 dydxdt C(ε)T (log(c)) Concerning the denominator, let us remark that R R |v c,ε,α (T, x, y) | 2 dydx

  ε,α (T, x, y) | 2 dydx C (T, ε) |ψ c (1) | 2 log(c) 3 c 3 .

e -c as c → ∞.[START_REF] Beauchard | Null controllability of Kolmogorov-type equations[END_REF]

Acknowledgements

The author would also like to thank Boris Andreianov and Luc Molinet for having suggested the strategy of Remark 3. The author would like to deeply thank Morgan Morancey and Dario Prandi for very stimulating and numerous discussions on the topic of this article. The author would also like to thank Sylvain Ervedoza for having pointed out to the author the existence and some properties of the PSWF. The author would also like to thank Luc Miller for its brillant presentation and very interesting discussions during the congress "Control of Distributed Parameter Systems" (Bordeaux, summer 2017), that have attracted the attention of the author to the interest of the use of uncertainty principles in controllability theory and notably to the PSWF. To finish, the author is very grateful to one of the referee in a previous submission, who pointed out a crucial mistake in a previous version of the paper.


Narrative Theory and Discourse Analysis John PIER (Tours and Paris) Narrative Theory and Discourse Analysis

Keywords: discourse, text linguistics, speech genres, enunciation

One important development in French narratology since the decline of structuralism as it was practiced in the 1960s and 70s can be found in French discourse analysis. Emphasizing discourse in social interaction over structure, discourse analysis regards narrative as one type of discourse among others. Three dimensions of discourse that relate to narrative are singled out: the text linguistic categories of text and discourse, the reformulation of langue and parole in the light of the theory of enunciation and the relations between text and context. The role in narrative analysis of enunciation (Benveniste), speech genres (Baxtin) and the scene of enunciation (Maingueneau) are examined.

I would like to set out here a few elements of research in this field which, I believe, deserve to be more broadly known and taken account of by research in narrative theory at large. The field of discourse analysis has developed within several different national and research traditions, and it has both drawn on and been incorporated into a variety of disciplines, mainly in the social sciences. As its name suggests, discourse analysis is not the special reserve of narrative or literary scholarship, but covers all fields of study where discourse plays an influential role. Overall, it can be said that with discourse analysis the emphasis no longer falls on structure, as was the case for the structuralists, but on discourses as they occur within the framework of social interaction, thus favoring the pragmatic dimension of communication in all its forms. On this view, narrative, while it is indisputably pervasive in human societies, is ultimately one form of discourse among others. Now, there are three areas in which French discourse analysis has made contributions that are of consequence for narratologists. The first is a close consideration of text and discourse rather than the story/discourse dichotomy that dominates much narratological research, both classical and postclassical. The second is a rethinking of the Saussurean pair langue and parole. And the third has to do with the interrelations between text and context. Before looking at these questions more closely, however, a few words on the historical origins of French discourse analysis are in order. Research in this field began in the late 1960s, more or less at the same time as the structuralist movement and the rise of narratology. Critical of Saussurean structural linguistics, the so-called pilot science that the structuralists advocated as a model for the social sciences, the French discourse analysts did not participate in the poststructuralist movement spearheaded by Roland Barthes but were focused, at least initially, on a corpus consisting mainly of political discourses. The principal methodological concern was how to conceptualize and model linguistic units beyond the sentence, a dimension of analysis that was opened up by the American school of linguistic distributionalism starting in the 1950s but that was not fully assimilated into French structural linguistics. As it turns out, it was the accomplishment of the early French discourse analysts to have pointed the way to a trio that continues to characterize the field today: a set of concepts and analytical practices coming from pragmatics, linguistic theories of enunciation or utterance and text linguistics. Although some researchers, such as Jean-Michel Adam, began incorporating discourse analytical principles into narratology as early as the mid-1970s (cf. Pier 2011, French discourse analysts were not by and large concerned with narrative theory or literary theory. The changes came starting in the early 1990s with a number of publications by Dominique Maingueneau, who continues to work in the field of literature and discourse analysis today. To be sure, French discourse analysts have drawn lessons from structural narratology, but few if any regard their work as narratology. It is thus important not to confuse Gérard Genette's pioneering contribution to "discourse narratology" and the positions staked out by the French discourse analysts. The latter in fact seldom mention Genette's Narrative Discourse, referring, instead, to his work on transtextuality and paratextuality. This explains, in part, why developments in French discourse analysis are relatively unknown among non-French-speaking narratologists.

It is against this backdrop that the three areas of study I mentioned above can now be looked at more closely.

Why have discourse analysts chosen to give priority to text and discourse rather than to story and discourse? There are three principle reasons for this.

The first is that the term "story" covers only a limited number of varieties of verbal or other communicative activities. Can a logical demonstration, a military command, a commercial contract or instructions on how to install a computer program be construed as stories? Clearly there are many forms of discourse circulating in society which can be qualified as stories only marginally, if at all, even though it is the case that such discourses might in some way be present in narratives without actually being narratives themselves.

A second reason for the choice of text and discourse over story and discourse is of a more conceptual and terminological nature. It was Tzvetan Todorov who, in his important 1966 article, "Les catégories du récit littéraire," proposed the terminology that has become a mainstay of classical and postclassical narratology alike. However, the difficulty with Todorov's proposal is that it superimposes principles taken from the modes of enunciation or utterance in French linguistics (histoire, characterized by the absence of enunciator and enunciatee, and discours, where their presence is marked) onto narrative content and the signifying medium, respectively; this move, in turn, is said to also replace the Russian formalists' principles of fabula and sjužet. Taking things a step farther, in English-language narratology the French and the Russian terms are often presented as equivalent to "story" and "plot." This unhappy mixture of concepts in which histoire, fabula and story, on the one hand, and discours, sjužet and plot, on the other, are presented as an apparently homogeneous system has sometimes been diagnosed and occasionally revised (I myself have also examined the issues from the standpoint of semiotics; cf. [START_REF] Pier | On the Semiotic Parameters of Narrative: A Critique of Story and Discourse[END_REF]. Nevertheless, it remains widespread in narrative research.

The shift from "story" to "text" is not a mere terminological matter, but may serve to shed light on conceptual confusion. By embracing the story/discourse framework, narratologists have sometimes felt the necessity to "export" narratological concepts and principles to disciplines and to forms of expression where they may not yield the most pertinent and fruitful results. The consequence has been the rise of a sort of "pan-narrativism" in which "story" and "discourse" have been called on to do duty in areas lying beyond their conceptual origins or methodological applicability. One author, for example, considers that the story/discourse distinction forms a basis for interdisciplinary research extending up to fields such as feminist studies and international relations [START_REF] Dawson | How Many 'Turns' Does it Take to Change a Discipline? Narratology and the Interdisciplinary Rhetoric of the Narrative Turn[END_REF]. These and other considerations lead me to rally fully to James Phelan's comments on the risks of "narratological imperialism" [START_REF] Phelan | Who's Here? Thoughts on Narratological Imperialism[END_REF].

Taking their distance from this situation, French discourse analysts prefer to speak of text and discourse. This move results in a significant reordering of concepts and categories. For the discourse analyst, text breaks down into a text-structure and a text-product. Text-structure, as studied by textual linguistics, bears on relations beyond the sentence at the level of microlinguistic cohesion (sometimes called 'texture') and at that of macrostructural coherence (also called 'structure'). These dimensions apply to all types of text, whether narrative or not, be they verbal or in other media. As for text-product (to be distinguished from text-archiveroughly speaking, the signifier), this corresponds to the enunciate (énoncé, or utterance in English), in other words, the empirical, observable and describable trace of a discursive activity, whether written, oral or visual: a weather report, a political discourse, a short story, a film, or even an ungrammatical fragment. Moreover, text-product (or utterance) is not something that simply falls out of the sky but is the result of an act of enunciation (énonciation), defined by Émile Benveniste as "the putting into operation of language (langue) by an individual act of use" [START_REF] Benveniste | L'appareil formel de l'énonciation[END_REF][START_REF] Benveniste | L'appareil formel de l'énonciation[END_REF]: 80). In the light of this definition, it is important to remember that an act of enunciation occurs within the context of interpersonal interaction, so that enunciation must actually be thought of as co-enunciation involving the participation of two enunciators or more. In French, as in other European languages, interpersonal interaction in discourse is analyzed through various linguistic markers including the use of personal pronouns and of deictic expressions such as "here," "there," "now," "then." In France, an entire school of enunciative linguistics has grown up around these notions. Now, the problem of enunciation ties in with another feature of language that was paramount in structural narratology and that also continues to exert an influence in present-day research, particularly in its vulgarized form: langue and parole. In structural narratology, as Gerald Prince has pointed out, these concepts are often used to describe "narrative langue" as "the system of rules and norms accounting for the production and understanding of individual narratives," the individual narrative thereby corresponding to "narrative parole" [START_REF] Prince | Dictionary of Narratology[END_REF][START_REF] Prince | Dictionary of Narratology[END_REF]: 70). It is worth remembering, however, that Saussure's formulation of langue and parole is itself problematic and has been the subject of various observations and criticisms.

Langue, the object of linguistic study, is the abstract "system of signs," a code governed by rules and constraints which are external to the individual but are nevertheless lodged in the minds of the members of the speech community. Parole (sometimes called the message) is the material of language and corresponds to an individual and momentary act of use. But by this very token, parole, because it escapes the rules of langue, is not the object of linguistic study: thus, while Saussure spoke of the sentence as a syntagm (a group of words ordered by syntagmatic and paradigmatic relations), he simultaneously excluded it from langue, the system of language, considering the sentence to be a manifestation of parole.

Clearly, then, there is gap between langue and parole, a situation that has drawn numerous proposals and counter-proposals for revisions by linguists and literary theorists. For present purposes, suffice it to say that Roland Barthes, in his famous article "Introduction à l'analyse structurale des récits," postulated a relation of "homology" and even "identity" between language and literature, stating that "structurally, narrative partakes of the sentence, without ever being reduced to a sum of sentences: narrative is a long sentence, just as all constative sentences are, in a way, the outline of a small narrative" (Barthes 1966: 4). This homological relation between narrative and sentence, on the basis of which Barthes called for a "translinguistics" -a linguistics beyond the sentence -pointed in the direction of discourse analysis but fell short of a satisfactory framework for such analysis. Barthes's proposal fails to distinguish between sentence (a grammatical construction) and enunciate, the result of an act of enunciation. Various attempts were undertaken during the 1970s and 80s to remedy this ambivalent situation, notably by enlisting Naom Chomsky's generative-transformational grammar, with its notions of competence and performance in place of langue and parole, in order to create so-called text grammars and even narrative grammars. However, these endeavors ultimately gave way to text linguistics, one of the pillars of French discourse analysis.

The approach to the langue/parole pair adopted by the researchers in question draws on a principle quite different from that taken up by structuralist narratology: Mixail Baxtin's principle of speech genres. Speech genres, present in all spheres of human communication, are defined as "relatively stable thematic, compositional, and stylistic types of utterances" [START_REF] Baxtin | The Problem of Speech Genres[END_REF](Baxtin [1952-53]-53]: 64). Corresponding to neither langue nor parole, speech genres, which are both conventional and normative and thus of a distinctly pragmatic nature, play a crucial role to the extent that they serve as an interface between langue and parole. According to Baxtin, it is thanks to speech genres that verbal communication exists. It must further be noted that speech genres are not limited to the literary genres of epic, dramatic and lyric and their subgenres, since, as sociolinguistic phenomena, they exist in innumerable forms and in all social domains, both in primary or simple forms (the spontaneous exchanges occurring in everyday speech) and in secondary or complex forms (scientific studies, literary works, etc.) as well as in the rich interaction between primary and secondary forms. Within the various social spheres there exist specific functions (economic, scientific, literary, etc.) which, in conjunction with certain conditions peculiar to each sphere, give rise to different genres (stock market analysis, scientific treatise, etc., but also the sonnet, the short story, the essay, etc.). Key to the principle of speech genres is the notion of vyskazyvanie, roughly the equivalent of utterance or enunciate, that is to say, the product of an act of enunciation. The defining feature of Baxtin's utterance as a unit of speech communication, setting it off sharply from grammatical units of language (lexical items, syntactic structures, sentences, etc.), is the change of speakers. The changing of speakers in a sequence of utterances is a significant point for many reasons. Mainly, it can be said that units of language, such as they are presented in the schematic diagrams of general linguistics, communicate nothing: as a grammatical unit, the sentence is constructed by the individual speaker, but it is directed to no one and elicits the response of no one. From the perspective of linguists such as Saussure (who is not alone in this regard), communication consists in the neutral "encoding" of a message by the sender which is then "decoded" by the passive receiver -a system evocative of the transmission of telegrams. Utterances, however, because they are defined by the changes of speaker that occur in the process of communicative exchange, function differently: an utterance, unlike a sentence, elicits an active responsive attitude in the listener, and in doing so it constitutes the initial preparatory stage of the rejoinder which is to come. The clearest example of this process is found in the dialogue, of course, where speaker and listener exchange roles in the ongoing communicative chain. But for Baxtin, the dialogical dimension of language, or dialogism, permeates all aspects of communication; it can be found even in interior monologues and other forms of mental discourse.

So the question now arises: why are Baxtin's speech genres, a reconfiguration of the Saussurean langue/parole distinction taken over by French discourse analysis, important for the narratologist? First of all, speech genres introduce into the study of language a set of principles coming from pragmatics, thereby bringing into focus processes that are closer to actual discourse as a communicative act than, for example, the underlying system of signs that constitutes Saussure's langue. Second, speech genres, which cover a wide spectrum, take form within a sociodiscursive space and include numerous manifestations of discoursenarrative and non-narrative, literary and non-literary and so on. In this way, speech genres tie up with the text/discourse distinction since, contrary to narratological theories that rely on the story/discourse distinction, they are not bound only to stories or narratives: speech genres do enter into the compositional dimension of narratives, of course, but also into that of legal and commercial documents, technical reports and much more, the reason why they play a crucial role in text linguistics (cf. [START_REF] Adam | La linguistique textuelle: Introduction à l'analyse textuelle des discours[END_REF][START_REF] Adam | La linguistique textuelle: Introduction à l'analyse textuelle des discours[END_REF]). Finally, the enunciative properties of discourse, as analyzed by Benveniste, complement the Baxtinian notion of utterance. Thus, formal features such as the use of pronouns and deictic expressions can signal the change of speakers throughout the communicative chain of utterances.

For the final step of my inquiry into the relations between French discourse analysis and narratology, I would now like to look at what Dominique Maingueneau calls scène d'énonciation, or "scene of enunciation" (literally: "theatrical stage of enunciation"). This concept brings together a set of principles that calls for reconsidering the divide between text and context. Maingueneau emphasizes that scène d'énonciation is distinct from "situation of enunciation," which is a linguistic term, and also from "situation of communication," which pertains to sociological conceptions of language. Both approaches view verbal activity "from the outside," one focusing on the physical or social environment, for example, the other on the social status of the participants in an act of communication or on the circumstances of the publication of a book. As for the word scène employed in scène d'énonciation, this is clearly a metaphor: it is chosen in order to stress the idea that the act of enunciation, parole, is a mise en scène, a sort of theatrical production that takes place in the form of a self-constituting process occurring within a defined discursive space. By adopting the theatrical metaphor of scène d'énonciation, Maingueneau endeavors to enter discourse "from the inside." Now, Maingueneau's system breaks down into three scenes, all of them of a pragmatic nature.

The first is the "global scene." This category corresponds more or less to text type or discourse type: recognizing a book as a literary fiction, a collection of historical studies or a treatise on nuclear physics will trigger different kinds of presuppositions in the reader.

However, the global scene, or type, acts in close collaboration with the "generic scene," the latter derived in large part from Baxtin's speech genres. It is at this level that the sociohistorical conditions of enunciation for each genre come into play: for example, the finality of a given genre (the aim of a student research paper in an academic setting is not that of a talk show for late-night television viewers or that of a newspaper editorial intended for readers having a given political persuasion); the linguistic and stylistic resources employed (the lexical choices peculiar to administrative documents as compared to those of popular magazines); the medium or material support (the Internet has given rise to previously unknown genres such as blogs, chats and forums).

Finally, in order to account for the singularity of the individual discourse, Maingueneau introduces the notion of "scenography." This term, normally used to refer to the design of the stage set, refers here to the conditions that render discourse possible and to the product of that discourse; at the same time, it adds to the theatricality of the scène (the spatio-temporal framework in which the discourse occurs) the dimension of -graphie (the process of the discourse's occurrence). Scenography, located both upstream and downstream of the work, is thus not a mere scaffolding for contents, but a true "pivot of enunciation" of discourse (Maingueneau 2004: 192 and 201). "To enunciate," says Maingueneau, "is not only to activate the norms of a pre-existing institution of speech; it is also to construct, on this basis, a singular mise en scène of enunciation: a scenography. […] The notion of scenography is based on the idea that the enunciator, through his enunciation, organizes the situation on the basis of which he is claiming to speak" (Maingueneau 2014: 129). In general -and this is more emphatically the case with literary works -there arises a meaningful tension between the scenography of a discourse and its generic scene.

As one of many possible examples of the scene of enunciation and the relation between generic scene and scenography, we can consider one of the early English novels, Pamela; or, Virtue Rewarded (1740) by Samuel Richardson. Published as a series of letters written by the heroine, a young servant, to her parents for the purpose of illustrating how well-bred working class ladies in eighteenth-century English society should resist the designs of their wicked masters, Pamela is consistent with the epistolary form as a speech genre. At the same time, however, the book's scenography builds up a dramatic situation in which the protagonist's experience is not filtered through the moralizing interventions of a narrative voice, but is witnessed by the reader as though the incidents took place on the stage. Moreover, the avowed didactic purpose of the letters is progressively sidelined as the heroine's moral resistance culminates with the conversion of Mr. B, her social superior, to more noble ways and, in the end, with Pamela's social ascension through marriage. The tensions produced between the epistolary form of Richardson's novel and the scenography of enunciated speech are thus inseparable from the sociohistorical context in which the work appeared. From the discourse analytical perspective, Pamela exemplifies how text as discourse is not a formal structure to which external contextual facts and circumstances must be added, but rather a work in the course of which context is constituted.


Matteo Millan and Alessandro Saluppo (eds), Corporate Policing, Yellow Unionism, and Strikebreaking, 1890-1930: In Defence of Freedom. London: Routledge, 2021, pp. xxv + 271, h/b, £96, ISBN 9780367374129. (The open access version of this book is available at http://www.taylorfrancis.com/books/e/9780429354243, under a Creative Commons Attribution-Non-Commercial-No Derivatives 4.0 License.) Co-edited by two historians from the University of Padua, here is a remarkable book on a neglected topic, crowning years of research by a truly international network on 'The Dark Side of the Belle Époque -Political Violence and Armed Associations in Europe before the First World War'. This European Research Council project, coordinated from beginning (2015) to end (2022) by Matteo Millan, has already led to the publication of a special issue of the European History Quarterly (vol. 49 (4), October 2019), entitled 'The most formidable weapon: Work replacement, armed strikebreaking and private police in Europe's Belle Époque social conflicts'. But the present volume, stemming from a conference held at Oxford on 23-24 October 2018 ('Industrial vigilantism, strikebreaking and patterns of anti-labour violence, 1890s-1930s: A comparative and transnational perspective'), is more ambitious still, covering a longer period and a vaster territory.

Labour historians working on the half century under consideration -a period characterized by the global expansion of capitalism and the connected growth of working-class movements across borders -are familiar with anti-labour repression, its diverse forms and protagonists. Yet, as pointed out by Millan and Saluppo in their excellent introduction, strikebreakers and industrial vigilantes rarely hold centre-stage, in spite of repeated invitations -by Kenneth D. Brown in 1974, by Marcel van der Linden in 2005 -to make the topic more central. As Martin Conway rightly stresses in the concluding chapter, there are archival reasons for that: employers hiring professionals to violently crush workers did not wish to leave traces of such practices. There is also an obvious historiographical reason for that neglect, the deeply ingrained tendency to give prominence to labour activists to the detriment of other actors of industrial conflicts, such as employers, the State or the 'scabs' and 'blacklegs' that this book successfully resuscitates as key participants in the class struggle.

What this volume demonstrates is that in the late nineteenth and early twentieth centuries the use of brutal methods against labour unrest was by no means an American exception, even though the United States did offer some archetypal features. Inevitably, and justifiably, one chapter focuses on the 'the wild west of employer anti-unionism'. But the 'wild east' is also studied here, with chapters on Egypt, India and the Indian Ocean, as is the European continent, from Portugal, Spain, Italy and Greece, to the German, Austro-Hungarian and Russian empires, not to forget Great Britain. The variety of cases examined here testifies to the universal character of the novel problem faced by capitalists around the industrializing world in that era: confronted with the progress of labour solidarity, with movements contesting their power in the workplace, such as strikes, and outside of it, including revolutions, what could be done when attempts at neutralizing or integrating proletarian resistance did not suffice to preserve order? With or without concerted plans, a 'common repertoire' emerged.

The book is structured in three parts ('I: Institutional responses'; 'II: Strikebreaking tactics and practices'; 'III: Civic and industrial vigilantism') that emphasize the diversity of responses elaborated by the defenders of so-called 'free labour', as well as the commonalities. The police and the army could be brought in but, in an age when growing sections of the working class were obtaining the right to vote, politicians could be reluctant to appear too openly as siding with the industrialists. The options left for the bosses were thus either to recruit their own reserve force to replace striking workers, or to sign a contract with independent agents willing to fulfil that task or less respectable ones, from espionage to physical assault. The book confirms that strikebreaking was a tool used above all in settings where workers could easily be replaced because unskilled workers formed the majority, notably in the maritime industry. But the volume broadens that scope, with telling examples from the agricultural and mining sectors, from the steel, iron and cigarette industries.

As almost every chapter illustrates, the borders between public and private coercion, between legal and illegal methods, were seldom clear-cut. Private detectives and civic vigilantes often acted with tacit support from the authorities. But that partnership was not without difficulties, some apostles of free labour having their own political agenda, generally on the far right, which statesmen on occasion saw as too disruptive. One of the merits of this collection is indeed to remind us of how blurred the lines could be. This does not stop Geoff Eley from making a sharp distinction, in his foreword, between pre-1914 forms of anti-labour repression and their post-First World War avatars, marked in Italy and Germany by the transmutation of old style strikebreaking groups into unified fascist movements organized to seize control of the State. The degree of continuity between those two phases, however, remains a contested issue.

Readers of the Labour History Review will probably be particularly interested in the chapters devoted to the British world. Prerna Agarwal's study of 'the murky world of labour politics in Calcutta's docklands', for instance, explores the unlikely collaboration between the colonial State and the Bengal Muslim League, equally preoccupied in the 1930s with the rise of a red-flag union. As for Alessandro Saluppo's thorough analysis of the London-based Volunteer Police Force and its interventions between 1911 and 1914, it comes as a precious complement to this Review's special issue on the Great Labour Unrest (vol. 79 (1), April 2014).

From a methodological point of view, two contributions stand out: Charles Bégué Fawell's chapter on steamship workers, because of his daring attempt to revive the invisibilized struggles taking place at sea; Pierre Eichenberger's essay on the 'transnational mobilisation of industrialists around World War I' and their 'illiberal business internationalism', which the editors did well to place at the very heart of the book. Analysing 'the counter-mobilisation of capital' as a 'a truly global phenomenon', this is a broad-sweeping piece likely to appeal to both lecturers and their students.

In conclusion, the editors and the authors are to be thanked for their efforts to make the comparative and transnational approaches more than catchphrases, and for meticulous studies that fill a gap in the historiography. Though one might regret the absence of chapters on the Frenchspeaking world and on the 'White Dominions', with this edited volume the historians attempting to cover those areas in the future have solid foundations on which to build.

One final comment, inspired by Millan and Saluppo themselves: industrial relations may have become more institutionalized in the Global North between 1945 and the 1970s but the violence encountered in these pages never disappeared from the Global South. This makes the understanding of anti-labour repression in the past all the more important to anyone sympathizing with today's 'workers of the world'.

Yann Béliard, Sorbonne Nouvelle