Alabau Fatiha
email: alabau@ljll.math.upmc.fr

Boussouira

Piermarco Cannarsa
email: cannarsa@mat.uniroma2.it

Cristina Urbani
email: cristina.urbani@gssi.it

Exact controllability to the ground state solution for evolution equations of parabolic type via bilinear control *

Keywords: bilinear control, evolution equations, exact controllability, parabolic PDEs, ground state 2010 MSC: 35Q93, 93C25, 93C10, 93B05, 35K90

In a separable Hilbert space X, we study the linear evolution equation

where A is an accretive self-adjoint linear operator, B is a bounded linear operator on X, and p ∈ L 2 loc (0, +∞) is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation (p ≡ 0) starting from the ground state of A. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.

Introduction

In a separable Hilbert space X, consider the control system    u (t) + Au(t) + p(t)Bu(t) = 0, t > 0 u(0) = u 0 .

(1)

where A : D(A) ⊂ X → X is a linear self-adjoint maximal accretive operator on X, B belongs to L(X), the space of all bounded linear operators on X, and p(t) is a scalar function representing a bilinear control.

In the recent paper [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF], we have studied the stabilizability of (1) along the ground state solution of the free equation (p ≡ 0). More precisely, denoting by {λ k } k∈N * the eigenvalues of A and by {ϕ k } k∈N * the corresponding eigenfunctions, we call ϕ 1 the ground state of A and ψ 1 (t) = e -λ 1 t ϕ 1 the ground state solution of the equation

u (t) + Au(t) = 0.
In [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF]Theorem 3.4], we have given sufficient conditions on A and B to ensure the superexponential stabilizability of (1) along ψ 1 : for all u 0 in some neighborhood of ϕ 1 there exists a control p ∈ L 2 loc ([0, +∞)) such that the corresponding solution u of (1) satisfies

||u(t) -ψ 1 (t)|| ≤ M e -(e ωt +λ 1 t) , ∀ t ≥ 0 (2)
for some constants ω, M > 0. In the same paper, we have discussed several applications of the above result to parabolic operators. For instance, we have studied the stabilizability of the equation u t -u xx + p(t)µ(x)u = 0 with Dirichlet or Neumann boundary conditions, as well as the equation with variable coefficients u t -((1 + x) 2 u x ) x + p(t)Bu = 0, or n-dimensional problems with radial symmetry. In [START_REF] Cannarsa | Superexponential stabilizability of degenerate parabolic equations via bilinear control[END_REF], we have also shown how to recover superexponential stability for a class of degenerate parabolic operators, still applying the above abstract result.

In this paper, we address the related, more delicate, issue of the exact controllability of (1) to the ground state solution ψ 1 via a bilinear control. Such a property, that is obviously stronger than superexponential stabilizability, holds true in more restrictive settings than those considered in [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF]. Nevertheless, our new results, that we state below, apply to all the aforementioned examples of parabolic problems.

Theorem 1.1. Let A : D(A) ⊂ X → X be a densely defined linear operator such that (a) A is self-adjoint, (b) A is accretive: Ax, x ≥ 0, ∀ x ∈ D(A), (c) ∃ λ > 0, (λI + A) -1 : X → X is compact, [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF] and suppose that there exists a constant α > 0 for which the eigenvalues of A fulfill the gap condition

λ k+1 -λ k ≥ α, ∀ k ∈ N * . (4)
Let B : X → X be a bounded linear operator such that there exist b, q > 0 for which Bϕ 1 , ϕ 1 = 0, and

λ q k | Bϕ 1 , ϕ k | ≥ b ∀ k > 1. (5)
Then, for any T > 0, there exists a constant R T > 0 such that, for any u 0 ∈ B R T (ϕ 1 ), there exists a control p ∈ L 2 (0, T ) for which system (1) is controllable to the ground state solution in time T . Furthermore, the following estimate holds

||p|| L 2 (0,T ) ≤ e -π 2 C K /T f e 2π 2 C K /(3T f ) -1 , (6)
where

T f := min{T, T α }, T α := π 2 6 min 1, 1/α 2 (7)
and C K is a suitable positive constant.

The main idea of the proof consists of applying the stability estimates of [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] on a suitable sequence of time intervals of decreasing length T j , such that ∞ j=1 T j < ∞. Such a sequence, however, has to be suitably chosen in order to fit the error estimates that we take from [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF].

From the above local exact controllability property we deduce two global controllability results. In the first one, Theorem 1.2 below, we prove that all initial states lying in a suitable strip, i.e., satisfying | u 0 , ϕ 1 -1| < r 1 , can be steered to the ground state solution (see Figure 1). Moreover, we give a uniform estimate for the controllability time.

Theorem 1.2. Let A and B satisfy hypotheses (3), (4), and (5). Then there exists a constant r 1 > 0 such that for any R > 0 there exists T R > 0 such that for all u 0 ∈ X that satisfy

| u 0 , ϕ 1 -1| < r 1 , ||u 0 -u 0 , ϕ 1 ϕ 1 || ≤ R, (8)
problem (1) is exactly controllable to the ground state solution ψ

1 (t) = e -λ 1 t ϕ 1 in time T R . ϕ 1 ψ 1 (T R ) R -R r 1 u 0
Figure 1: the colored region represents the set of initial conditions that can be steered to the ground state solution in time T R .

Our second global result, Theorem 1.3 below, ensures the exact controllability of all initial states u 0 ∈ X \ ϕ ⊥ 1 to the evolution of their orthogonal projection along the ground state solution defined by

φ 1 (t) = u 0 , ϕ 1 ψ 1 (t), ∀ t ≥ 0, (9)
where ψ 1 is the ground state solution.

Theorem 1.3. Let A and B satisfy hypotheses (3), ( 4) and (5). Then, for any R > 0 there exists T R > 0 such that for all u 0 ∈ X satisfying

||u 0 -u 0 , ϕ 1 ϕ 1 || ≤ R| u 0 , ϕ 1 | (10)
system (1) is exactly controllable to φ 1 , defined in [START_REF] Cannarsa | Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign[END_REF], in time T R .

Notice that, denoting by θ the angle between the half-lines R + ϕ 1 and R + u 0 , condition [START_REF] Cannarsa | Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign[END_REF]

is equivalent to | tan θ| ≤ R,
which defines a closed cone, say Q R , with vertex at 0 and axis equal to Rϕ 1 (see Figure 2). Therefore, Theorem 1.3 ensures a uniform controllability time for all initial conditions lying in Q R . We observe that, since R is any arbitrary positive constant, all initial conditions u 0 ∈ X \ ϕ ⊥ 1 can be steered to the corresponding projection to the ground state solution. Indeed, for any u 0 ∈ X \ ϕ ⊥ 1 , we define

R 0 := u 0 u 0 , ϕ 1 -ϕ 1 .
Then, for any R ≥ R 0 condition (10) is fulfilled: Finally, we would like to recall part of the huge literature on bilinear control of evolution equations, referring the reader to the references in [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] for more details. A seminal paper in this field is certainly the one by Ball J.M., Marsden J.E., Slemrod M. [START_REF] Ball | Controllability for distributed bilinear systems[END_REF], which establishes that system (1) is not controllable. More precisely, denoting by u(t; p, u 0 ) the unique solution of (1), the attainable set from u 0 defined by

1 | u 0 , ϕ 1 | ||u 0 -u 0 , ϕ 1 ϕ 1 || = R 0 ≤ R. θ θ ϕ 1 ϕ 1 φ 1 (T R ) φ1 (T R ) u 0 û0 Q R
S(u 0 ) = {u(t; p, u 0 ); t ≥ 0, p ∈ L r loc ([0, +∞), R), r > 1}
is shown in [START_REF] Ball | Controllability for distributed bilinear systems[END_REF] to have a dense complement. Among positive results, we would like to mention Beauchard K. [START_REF] Beauchard | Local controllability and non-controllability for a 1d wave equation with bilinear control[END_REF] for bilinear control of the wave equation and Beauchard K., Laurent C. [START_REF] Beauchard | Local controllability of 1d linear and nonlinear Schrödinger equations with bilinear control[END_REF] for the bilinear control of the Schrödinger equation (see also [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF] for a first result on this topic). The local exact controllability results obtained in these papers rely on linearization around the ground state, the use of the inverse mapping Theorem, and a regularizing effect which takes place in both problems. The local exact controllability is proved for any positive time for the Schrödinger equation and for a sufficiently (optimal) large time for the wave equation. Both papers require the condition

Bϕ 1 , ϕ k = 0, ∀ k ≥ 1 ( 11
)
to be satisfied, together with a suitable asymptotic behavior with respect to the eigenvalues. If ( 11) is violated then it has been first shown by Coron J.-M. [START_REF] Coron | On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well[END_REF], for a model describing a particle in a moving box, that there exists a minimal time for local exact controllability to hold. This model couples the Schrödinger equation with two ordinary differential equations modeling the speed and acceleration of the box (see also Beauchard K., Coron J.-M. [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] for local exact controllability for large time). A further paper by Beauchard K., Morancey M. [START_REF] Beauchard | Local controllability of 1d Schrödinger equations with bilinear control and minimal time[END_REF] for the Schrödinger equation extends [START_REF] Beauchard | Local controllability of 1d linear and nonlinear Schrödinger equations with bilinear control[END_REF] to cases for which the above condition is violated, namely there exist integers k for which Bϕ 1 , ϕ k = 0. Under some additional assumptions, the authors prove that the system is locally exactly controllable in a sufficiently large time.

Another example of controllability to trajectories for nonlinear parabolic systems is studied in [START_REF] Fernández-Cara | Local exact controllability of the navier-stokes system[END_REF], where, however, additive controls are considered. In such an example, one can obtain controllability to free trajectories by Carleman estimate and inverse mapping arguments. Such a strategy seems hard to adapt to the current setting.

It is worth noting that the bilinear controls we use in this paper are just scalar functions of time. This fact explains why applications mainly concern problems in low space dimension. A stronger control action could be obtained by letting controls depend on time and space. We refer the reader to [START_REF] Cannarsa | Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign[END_REF][START_REF] Cannarsa | Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign[END_REF] for more on this subject.

This paper is organized as follows. In section 2, we have collected some preliminaries as well as results from [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] that we need for the proof of Theorem 1.1. Section 3 contains such a proof, while section 4 is devoted to examples of applications to parabolic problems.

Preliminaries

In this section we recall some results from [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] that are necessary for the construction of the proof of Theorem 1.1. First, fixed T > 0, consider the following bilinear control problem

   u (t) + Au(t) + p(t)Bu(t) + f (t) = 0, t ∈ [0, T ] u(0) = u 0 . ( 12
)
We introduce the following notation:

||f || 2,0 := ||f || L 2 (0,T ;X) , ∀ f ∈ L 2 (0, T ; X) ||f || ∞,0 := ||f || C([0,T ];X) = sup t∈[0,T ] ||f (t)||, ∀ f ∈ C([0, T ]; X).
The well-posedness of ( 12) is ensured by the following Proposition.

Proposition 2.1. Let T > 0. If u 0 ∈ X, p ∈ L 2 (0, T ) and f ∈ L 2 (0, T ; X)
, then there exists a unique mild solution of (12), i.e. a function u ∈ C([0, T ]; X) such that the following equality holds for every t ∈ [0, T ],

u(t) = e -tA u 0 - t 0 e -(t-s)A [p(s)Bu(s) + f (s)]ds. ( 13
)
Moreover, there exists a constant

C 1 (T ) > 0 such that ||u|| ∞,0 ≤ C 1 (T )(||u 0 || + ||f || 2,0 ). ( 14
)
We refer to [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] for a proof of Proposition 2.1.

Our aim is to show the controllability of the following system

   u (t) + Au(t) + p(t)Bu(t) = 0, t ∈ [0, T ] u(0) = u 0 , (15)
to the ground state solution ψ 1 = e -λ 1 t ϕ 1 , that is the solution of ( 15) when p = 0 and u 0 = ϕ 1 .

We first consider the case λ 1 = 0 and prove the controllability result to the corresponding ground state solution ψ 1 = ϕ 1 . Then, we recover the result also for the case λ 1 > 0.

Set v := u -ϕ 1 , then v is the solution of the following Cauchy problem

   v (t) + Av(t) + p(t)Bv(t) + p(t)Bϕ 1 = 0, t ∈ [0, T ] v(0) = v 0 = u 0 -ϕ 1 . (16)
We observe that the controllability of u to ϕ 1 is equivalent to the null controllabiliy of [START_REF] Komornik | Fourier series in control theory[END_REF].

In order to prove this latter result, we consider the following linearized system

   v(t) + Av(t) + p(t)Bϕ 1 = 0, t ∈ [0, T ] v(0) = v 0 . (17)
and we introduce the following constant

Λ T := k∈N * e -2λ k T e C√ λ k /α | Bϕ 1 , ϕ k | 2 1/2 . ( 18
)
where α is the constant in (4). We observe that, thanks to assumption (5), Λ T converges for any T > 0. The next Proposition guarantees the null controllability of [START_REF] Lebedev | Special functions and their applications[END_REF] and furthermore it yields an estimate of the control p with respect to the initial condition v 0 .

Proposition 2.2. Let T > 0 and let A and B be such that (3), ( 4), (5) hold. Furthermore, assume that λ 1 = 0. Let v 0 ∈ X. Then, defining p ∈ L 2 (0, T ) by

p(t) = k∈N * v 0 , ϕ k Bϕ 1 , ϕ k σ k (t) ( 19
)
where {σ k } k∈N * is the biorthogonal family to {e λ k t } k∈N * given by [12, Theorem 2.4], it holds that v(T ) = 0. Moreover, there exists a constant C α (T ) > 0 such that

||p|| L 2 (0,T ) ≤ C α (T )Λ T ||v 0 || (20)
where Λ T is defined in (18) and α is the constant in (4).

For the proof of Proposition 2.2 we refer to [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF].

Remark 2.3. The behavior of C α (•) with respect to its argument has been studied in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] and is given by

C 2 α (T ) = C •    1 T + 1 T 2 α 2 e C/(T α 2 ) , T ≤ 1 α 2 Cα 2 , T ≥ 1 α 2 , (21)
where C > 0 is a constant independent of T and α.

We now use the control p built in Proposition 2.2 also in the nonlinear system ( 16) and we give an estimate for the corresponding solution v.

Proposition 2.4. Let A and B satisfy hypotheses (3), ( 4), ( 5) and, furthermore, assume λ 1 = 0. Let p ∈ L 2 (0, T ) be defined by (19). Then, the solution v of (16) satisfies

sup t∈[0,T ] ||v(t)|| 2 ≤ e C 3 (T )Λ T ||v 0 ||+C B T (1 + C 4 (T )Λ 2 T )||v 0 || 2 (22)
where

C B ≥ 1 is the norm of the operator B, C 3 (T ) := 2 √ T C B C α (T ), and C 4 (T ) := C B C 2 α (T ).
For the proof of Proposition 2.4 we refer to [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF]. We introduce the function w(t) := v(t) -v(t) that satisfies the following Cauchy problem

   w (t) + Aw(t) + p(t)Bv(t) = 0, t ∈ [0, T ] w(0) = 0. (23)
We define the function K on (0, ∞) by

K 2 (T ) := C B e 2C B √ T +(C B +1)T C 4 (T )Λ 2 T (1 + C 4 (T )Λ 2 T ). ( 24
)
In the following Proposition we estimate how close we are able to steer v to 0 in time T by means of the control p defined in (19).

Proposition 2.5. Let A and B satisfy hypotheses (3), ( 4), [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF], and, furthermore, we assume λ 1 = 0. Let T > 0, p be defined by (19), and let v 0 ∈ X be such that

C α (T )Λ T ||v 0 || ≤ 1. ( 25
)
Then, it holds that ||w(T

)|| = ||v(T )|| ≤ K(T )||v 0 || 2 . ( 26
)
Proof. Observe that w ∈ C([0, T ]; X) is the mild solution of (23). Moreover w ∈ H 1 (0, T ; X)∩ L 2 (0, T ; D(A)) and thus w satisfies the equality

w (t) + Aw(t) + p(t)Bv(t) = 0 (27)
for almost every t ∈ [0, T ].

We multiply equation ( 27) by w(t) and we obtain 1 2

d dt ||w(t)|| 2 ≤ |p(t)|||Bv(t)||||w(t)|| ≤ 1 2 ||w(t)|| 2 + C 2 B 1 2 |p(t)| 2 ||v(t)|| 2 . (28)
Therefore, applying Gronwall's inequality, taking the supremum over [0, T ] and using ( 22) and (20), we get sup

t∈[0,T ] ||w(t)|| 2 ≤ C 2 B e T ||p|| 2 L 2 (0,T ) sup t∈[0,T ] ||v(t)|| 2 ≤ C 2 B e C 3 (T )Λ T ||v 0 ||+C B T +T (1 + C 4 (T )Λ 2 T )||v 0 || 2 ||p|| 2 L 2 (0,T ) ≤ C 2 B C 2 α (T )Λ 2 T e C 3 (T )Λ T ||v 0 ||+(C B +1)T (1 + C 4 (T )Λ 2 T )||v 0 || 4 . (29)
We can suppose, without loss of generality, that C α (T ) ≥ 1. Thus, thanks to (25), we obtain

sup t∈[0,T ] ||w(t)|| 2 ≤ C 2 B C 2 α (T )Λ 2 T e 2C B √ T +(C B +1)T (1 + C 4 (T )Λ 2 T )||v 0 || 4 that is equivalent to sup t∈[0,T ] ||w(t)|| 2 ≤ K(T ) 2 ||v 0 || 4 .
By the last inequality we infer that

||w(T )|| ≤ K(T )||v 0 || 2 . ( 30
)
3. Proof of Theorem 1.1

Fixed 0 < T ≤ min {1, 1/α 2 }, we define the sequence {T j } j∈N * by

T j := T /j 2 , (31)
and the time steps

τ n = n j=1 T j , ∀n ∈ N, (32)
with the convention that 0 j=1 T j = 0. Notice that ∞ j=1 T j = π 2 6 T . The proof of our result relies on the construction of the solution v of ( 16) in consecutive intervals of the form [τ n , τ n+1 ] for which we are able to perform an iterate estimate of (26).

First, through the following Lemma, we study the behavior of the constant K(T ) with respect to T .

We define the function

G M (T ) := M T 2 e M/T ∞ k=1 e -2λ k T +M √ λ k | Bϕ 1 , ϕ k | 2 , 0 < T ≤ 1 ( 33
)
where M is a positive constant.

Lemma 3.1. Let A : D(A) ⊂ X → X be such that (3) and (4) hold and B : X → X be such that (5) holds. Then, there exists a suitable positive constant C M such that

G M (T ) ≤ e C M /T , ∀ 0 < T ≤ 1. ( 34
)
Proof. Thanks to assumption (5), we have that

G M (T ) = M T 2 e M/T ∞ k=1 e -2λ k T +M √ λ k | Bϕ 1 , ϕ k | 2 ≤ M T 2 e M/T e M 2 /(8T ) | Bϕ 1 , ϕ 1 | 2 + 1 b 2 ∞ k=2 λ 2q k e -λ k T e -λ k T +M √ λ k . (35)
For any λ ≥ 0 we set

f (λ) = e -λT +M √ λ . The maximum value of f is attained at λ = M 2T 2 .
So, we can bound G M (T ) as follows

G M (T ) ≤ M T 2 e M/T e M 2 /(8T ) | Bϕ 1 , ϕ 1 | 2 + e M 2 /(4T ) b 2 ∞ k=1 λ 2q k e -λ k T . (36)
Now, for any λ ≥ 0 we define the function g(λ) = λ 2q e -λT . Its derivative is given by

g (λ) = (2q -λT )λ 2q-1 e -λT
and therefore we deduce that

g(λ) is    increasing if 0 ≤ λ < (2q)/T decreasing if λ ≥ (2q)/T
and g has a maximum at λ = (2q)/T . We define the following index:

k 1 := k 1 (T ) = sup k ∈ N * : λ k ≤ 2q T
Note that k 1 (T ) goes to ∞ as T converges to 0. We can rewrite the sum in (36) as follows

∞ k=1 λ 2q k e -λ k T = k≤k 1 -1 λ 2q k e -λ k T + k 1 ≤k≤k 1 +1 λ 2q k e -λ k T + k≥k 1 +2 λ 2q k e -λ k T . ( 37
)
For any k ≤ k 1 -1, we have

λ k+1 λ k λ 2q e -λT dλ ≥ (λ k+1 -λ k )λ 2q k e -λ k T ≥ α( λ 2 + λ 1 )λ 2q k e -λ k T (38)
and for any k ≥ k 1 + 2

λ k λ k-1 λ 2q e -λT dλ ≥ (λ k -λ k-1 )λ 2q k e -λ k T ≥ α( λ 2 + λ 1 )λ 2q k e -λ k T . (39)
So, by using estimates (38) and ( 39), (37) becomes

∞ k=1 λ 2q k e -λ k T = 2 α( √ λ 2 + √ λ 1 ) ∞ 0 λ 2q e -λT dλ + k 1 ≤k≤k 1 +1 λ 2q k e -λ k T . (40)
Furthermore, recalling that g has a maximum for λ = 2q/T , it holds that

k = k 1 , k 1 + 1 ⇒ λ 2q k e -λ k T ≤ (2q/T ) 2q e -2q . (41)
Finally, the integral term of (40) can be rewritten as

∞ 0 λ 2q e -λT dλ = 1 T ∞ 0 s T 2q e -s ds = 1 T 1+2q ∞ 0 s 2q e -s ds = Γ(2q + 1) T 1+2q , (42)
where by Γ(•) we indicate the Euler integral of the second kind. Therefore, we conclude from ( 41) and ( 42) that there exist two constants C q , C q,α > 0 such that

∞ k=1 λ 2q k e -λ k T ≤ C q T 2q + C α,q T 1+2q . ( 43
)
We use this last bound to prove that there exists C M > 0 such that

G M (T ) ≤ M T 2 e M/T e M 2 /(8T ) | Bϕ 1 , ϕ 1 | 2 + e M 2 /(4T ) b 2 C q T 2q + C α,q T 1+2q ≤ e C M /T , 0 < T ≤ 1 as claimed.
Remark 3.2. We recall that K(•) is defined by

K 2 (T ) := C 2 B e 2C B √ T +(C B +1)T C 2 α (T )Λ 2 T (1 + C B C 2 α (T )Λ 2 T ). For any 0 < T ≤ min {1, 1/α 2 }, C 2 α (•) is given by C 2 α (T ) = C 1 T + 1 T 2 α 2 e C/(α 2 T ) .
Thus, we have the following bound for K(•)

K(T ) 2 ≤ C 2 B e 2C B √ T +(C B +1)T G M (T ) (1 + C B G M (T )) , ( 44
)
where G M (•) is defined by (33) and the subscribed M is given by M = C 1 + 1 α 2 . Thanks to Lemma 3.1, we infer that there exists a suitable constant

C K > 0 such that C K > C M and K(T ) ≤ e C K /T , ∀ T ∈ (0, 1]. ( 45
)
In the following Proposition we prove that it is possible to iterate the construction of v in consecutive time intervals of the form [τ n-1 , τ n ]. Proposition 3.3. Let 0 < T ≤ min {1, 1/α 2 }, and consider the sequence (T j ) j∈N * defined by (31). Let v 0 ∈ X for which ||v 0 || < e -6C K /T , let A : D(A) ⊂ X → X be such that (3) and (4) hold and let B : X → X satisfies [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF]. Moreover, we assume that λ 1 = 0. Then, for every n ∈ N * , problem

   v (t) + Av(t) + p(t)Bv(t) + p(t)Bϕ 1 = 0, t ∈ [τ n-1 , τ n ] v (τ n-1 ) = v n-1 , ( 46
)
where v n-1 is determined by induction from the previous steps, p ∈ L 2 (τ n-1 , τ n ) is given by

p(t) = ∞ k=1 v n-1 , ϕ k Bϕ 1 , ϕ k σ k (t -τ n-1 ), ( 47
)
admits a unique mild solution v ∈ C ([τ n-1 , τ n ] , X) that satisfies ||v (τ n )|| ≤ e ( n j=1 2 n-j j 2 -2 n 6)C K /T , ( 48
)
where the time steps {τ n } n∈N are defined by (32).

Proof. To prove the result, we proceed by induction on n. For n = 1, by Proposition 2.5, the hypothesis on v 0 and Remark 3.2, v it satisfies

||v(T )|| ≤ K(T )||v 0 || 2 ≤ e -11C K /T .
Now, suppose the statement is true for all indices k ≤ n -1, we show the validity for index n. Therefore, by inductive hypothesis, the solution v has been constructed in consecutive intervals until [τ n-2 , τ n-1 ] and it satisfies

||v (τ n-1 )|| ≤ e ( n-1 j=1 2 n-1-j j 2 -2 n-1 6)C K /T .
Hence,

C α (T n )Λ Tn ||v (τ n-1 )|| ≤ e C M n 2 /T e ( n-1 j=1 2 n-1-j j 2 -2 n-1 6)C K /T ≤ e (n 2 +(-(n-1) 2 -4(n-1)+2 n-1 6-6-2 n-1 6)C K /T = e -(2n+3)C K /T , (49)
where we have used that C M < C K and the identity

n j=0 j 2 2 j = 2 -n (-n 2 -4n + 6(2 n -1)), n ≥ 0, ( 50
)
which can be easily checked by induction. Consider problem (46) with v n-1 the solution built in the previous interval, evaluated at τ n-1 . By the change of variables s = t -τ n-1 , we shift (46) into the interval [0, T n ]. We introduce the functions ṽ(s) = v (s + τ n-1 ) and p(s) = p (s + τ n-1 ) and we rewrite (46) as

   ṽ (s) + Aṽ(s) + p(s)Bṽ(s) + p(s)Bϕ 1 = 0, s ∈ [0, T n ] ṽ(0) = v n-1 . (51)
Proof of Theorem 1.1. We start the proof by considering the case in which λ 1 = 0. Let T > 0 and let T α and T f be defined by [START_REF] Beauchard | Local controllability of 1d Schrödinger equations with bilinear control and minimal time[END_REF]. We define T = 6 π 2 T f and R T := e -π 2 C K /T f . Observe that 0 < T ≤ 1 and we define the time steps {τ n } n∈N as in (32) with T j := T /j 2 . Fixed v 0 ∈ B R T (0), we apply (55) to obtain

||v (τ n )|| ≤ n j=1 K(T j ) 2 n-j ||v 0 || 2 n ≤ n j=1 e C K j 2 / T 2 n-j ||v 0 || 2 n = e C K 2 n / T n j=1 j 2 /2 j ||v 0 || 2 n ≤ e C K 2 n / T ∞ j=1 j 2 /2 j ||v 0 || 2 n ≤ e 6C K / T ||v 0 || 2 n (59)
where we have used that ∞ j=1 j 2 /2 j = 6. We take the limit as n → ∞ of (59) and we get

u π 2 6 T -ϕ 1 = v π 2 6 T = ||v(T f )|| ≤ 0 (60) since ||v 0 || < e -π 2 C K /T f = e -6C K / T . This means that, we have built a control p ∈ L 2 loc ([0, ∞)), defined by p(t) =    ∞ n=0 p n (t)χ [τn,τ n+1 ] (t), t ∈ (0, T f ] , 0, t ∈ (T f , +∞) (61)
where

p n (t) = ∞ k=1 v (τ n ) , ϕ k Bϕ 1 , ϕ k σ k (t -τ n ) , ∀n ∈ N, (62)
such that the solution u of (1) reaches the ground state solution ϕ 1 in time T , and stays on it forever. Observe that, thanks to (20) and (49), we are able to yield a bound for the L 2 -norm of such a control:

||p|| 2 L 2 (0,T ) = ∞ n=0 ||p n || 2 L 2 (τn,τ n+1 ) ≤ ∞ n=0 C α (T n+1 )Λ T n+1 ||v (τ n )|| 2 ≤ ∞ n=0 e -2(2(n+1)+3)C K / T = e -6C K / T e 4C K / T -1 = e -π 2 C K /T f e 2π 2 C K /(3T f ) -1 (63)
Now we face the case λ 1 > 0. We define the operator

A 1 := A -λ 1 I.
It is possible to check that A 1 satisfies (3) and moreover it has the same eigenfuctions, {ϕ k } k∈N * , of A, while the eigenvalues are given by

µ k = λ k -λ 1 , ∀k ∈ N * . (64)
In particular, µ 1 = 0 and furthermore, {µ k } k∈N * satisfy the same gap condition (4) fulfilled by {λ k } k∈N * . We define the function z(t) = e λ 1 t u(t), where u is the solution of (1). Then, z solves the following problem

   z (t) + A 1 z(t) + p(t)Bz(t) = 0, t > 0, z(0) = u 0 . (65)
For any T > 0, we define T f as in [START_REF] Beauchard | Local controllability of 1d Schrödinger equations with bilinear control and minimal time[END_REF] and the constant R T := e -π 2 C K /T f . We deduce from the previous analysis that, if u 0 ∈ B R T (ϕ 1 ), then there exists a control p ∈ L 2 ([0, +∞)) that steers the solution z to the ground state solution ϕ 1 in time T f ≤ T . This implies the exact controllability of u to the ground state solution ψ 1 (t) = e -λ 1 t ϕ 1 : indeed,

||u (T f ) -ψ 1 (T f )|| = e -λ1T f z (T f ) -e -λ1T f ϕ 1 = e -λ1T f ||z (T f ) -ϕ 1 || = 0.
This concludes the proof of our Theorem.

Remark 3.5. We observe that, from (63), it follows that ||p|| L 2 (0,T f ) → 0 as T f → 0. This fact is not surprising because as T f approaches 0, also the size of the neighborhood where the initial condition can be chosen goes to zero.

Proof of Theorems 1.2 and 1.3

Before proving Theorem 1.2, let us show a preliminary result that demonstrates the statement in the case of a strictly accretive operator. Lemma 4.1. Let A and B satisfy hypotheses (3), ( 4) and [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF]. Furthermore, we assume λ 1 = 0. Then, there exists a constant r 1 > 0 such that for any R > 0 there exists T R > 0 such that for all v 0 ∈ X that satisfy

| v 0 , ϕ 1 | < r 1 , ||v 0 -v 0 , ϕ 1 ϕ 1 || ≤ R, (66) problem (16) is null controllable in time T R .
Proof. First step. We fix T = 1. Thanks to Theorem 1.1, there exists a constant r 1 > 0 such that if ||u 1 (0) -ϕ 1 || < √ 2r 1 then there exists a control p 1 ∈ L 2 (0, 1) for which the solution u 1 of (1) on [0, 1] with p replaced by p 1 , satisfies u 1 (1) = ϕ 1 . We set v 1 = u 1 -ϕ 1 on [0, 1]. We deduce that if ||v 1 (0)|| < √ 2r 1 then there exists a control p 1 ∈ L 2 (0, 1) for which the solution v 1 of ( 16) on [0, 1] with p replaced by p 1 , satisfies v 1 (1) = 0.

Applications

In this section we present some examples of parabolic equations for which Theorem 1.1 can be applied. The hypotheses (3)-( 5) have been verified in [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] and [START_REF] Cannarsa | Superexponential stabilizability of degenerate parabolic equations via bilinear control[END_REF], to which we refer for more details. Furthermore, we observe that also the global results Theorem 1.2 and Theorem 1.3 can be applied to any example.

Diffusion equation with Dirichlet boundary conditions

Let I = (0, 1) and X = L 2 (0, 1). Consider the following problem

           u t (t, x) -u xx (t, x) + p(t)µ(x)u(t, x) = 0 x ∈ I, t > 0 u(t, 0) = 0, u(t, 1) = 0, t > 0 u(0, x) = u 0 (x) x ∈ I. (70)
We denote by A the operator defined by

D(A) = H 2 ∩ H 1 0 (I), Aϕ = - d 2 ϕ dx 2 .
and it can be checked that A satisfies (3). We indicate by {λ k } k∈N * and {ϕ k } k∈N * the families of eigenvalues and eigenfunctions of A, respectively:

λ k = (kπ) 2 , ϕ k (x) = √ 2 sin(kπx), ∀k ∈ N * .
It is easy to see that (4) holds true:

λ k+1 -λ k = π, ∀k ∈ N * . Let B : X → X be the operator Bϕ = µϕ with µ ∈ H 3 (I) such that µ (1) ± µ (0) = 0 and µϕ 1 , ϕ k = 0 ∀k ∈ N * . (71)
Then, there exists b > 0 such that

λ 3/2 k | µϕ 1 , ϕ k | ≥ b, ∀k ∈ N * .
For instance, a suitable function that satisfies (71) is µ(x) = x 2 , for which b = 2π 2 -3 6π 2 . For any T > 0, we define T f as in [START_REF] Beauchard | Local controllability of 1d Schrödinger equations with bilinear control and minimal time[END_REF]. Then, there exists a constant R T f > 0 such that the solution u of (70), with u 0 ∈ B R T f (ϕ 1 ), reaches the ground state solution ψ 1 (t, x) = √ 2 sin(πx)e -π 2 t in time T f and stays on it forever.

Diffusion equation with Neumann boundary conditions

Let I = (0, 1), X = L 2 (I) and consider the Cauchy problem

           u t (t, x) -u xx (t, x) + p(t)µ(x)u(t, x) = 0 x ∈ I, t > 0 u x (t, 0) = 0, u x (t, 1) = 0, t > 0 u(0, x) = u 0 (x). x ∈ I. (72)
The operator A, defined by

D(A) = {ϕ ∈ H 2 (0, 1) : ϕ (0) = 0, ϕ (1) = 0}, Aϕ = - d 2 ϕ dx 2
satisfies (3) and has the following eigenvalues and eigenfunctions

λ 0 = 0, ϕ 0 = 1 λ k = (kπ) 2 , ϕ k (x) = √ 2 cos(kπx), ∀k ≥ 1.
Thus, the gap condition ( 4) is fulfilled with α = π. The ground state solution is just the stationary function ψ 0 (x) = ϕ 0 (x) = 1. We define B : X → X as the multiplication operator by a function µ ∈ H 2 (I), Bϕ = µϕ, such that µ (1) ± µ (0) = 0 and µ, ϕ k = 0 ∀k ∈ N.

It can be proved that, there exists b > 0 such that

λ k | µϕ 0 , ϕ k | ≥ b, ∀k ∈ N * . (74)
For example, µ(x) = x 2 satisfies (74) with b = 2 √ 2. Therefore, equation ( 72) is controllable to the ground state solution ψ 1 = 1 in any time T > 0 as long as u 0 ∈ B R T (1), with R T > 0 a suitable constant.

Variable coefficient parabolic equation

Let I = (0, 1), X = L 2 (I) and consider the problem

           u t (t, x) -((1 + x) 2 u x (t, x)) x + p(t)µ(x)u(t, x) = 0 x ∈ I, t > 0 u(t, 0) = 0, u(t, 1) = 0, t > 0 u(0, x) = u 0 (x) x ∈ I. (75)
We denote by A : D(A) ⊂ X → X the following operator

D(A) = H 2 ∩ H 1 0 (I), Aϕ = -((1 + x) 2 ϕ x ) x .
It can be checked that A satisfies (3) and that the eigenvalues and eigenfunctions have the following expression

λ k = 1 4 + kπ ln 2 2 , ϕ k = 2 ln 2
(1 + x) -1/2 sin kπ ln 2 ln(1 + x) .

Degenerate parabolic equation

In this last section we want to address an example of a control problem for a degenerate evolution equation of the form It is possible to prove that A satisfies (3) (see, for instance [START_REF] Campiti | Degenerate self-adjoint evolution equations on the unit interval[END_REF]) and furthermore, if we denote by {λ k } k∈N * the eigenvalues and by {ϕ k } k∈N * the corresponding eigenfunctions, it turns out that the gap condition (4) is fulfilled with α = 7 16 π (see [START_REF] Komornik | Fourier series in control theory[END_REF], page 135). If γ ∈ [1, 3/2), problem (81) is called strong degenerate and the corresponding weighted Sobolev space are described as follows: given I = (0, 1) and X = L 2 (I), we define In this case the operator A : D(A) ⊂ X → X is defined by

                   u t -(x γ u x
            
∀u ∈ D(A), Au := -(x γ u x ) x , D(A) := u ∈ H 1 γ,0 (I) : x γ u x ∈ H 1 (I) = {u ∈ X : u is absolutely continuous in (0,1] , x γ u ∈ H 1 0 (I), x γ u x ∈ H 1 (I) and (x γ u x )(0) = 0} and it has been proved that (3) holds true (see, for instance [START_REF] Cannarsa | Carleman estimate for a class of degenerate parabolic operators[END_REF]) and that (4) is satisfied for α = π 2 (see [START_REF] Komornik | Fourier series in control theory[END_REF]). For all γ ∈ [0, 3/2), we define the linear operator B : X → X by Bu(t, x) = x 2-γ u(t, x) and in [START_REF] Cannarsa | Superexponential stabilizability of degenerate parabolic equations via bilinear control[END_REF] we have proved that there exists a constant b > 0 such that

λ 3/2 k | Bϕ 1 , ϕ k | ≥ b ∀k ∈ N * .
Finally, by applying Theorem 1.1, we ensure the exact controllability of problem (81) to the ground state solution, for both weakly and strongly degenerate problems.

Figure 2 :

 2 Figure 2: fixed any R > 0, the set of initial conditions exactly controllable in time T R to their projection along the ground state solution is indicated by the colored cone Q R .

  ) x + p(t)x 2-γ u = 0, (t, x) ∈ (0, +∞) × (0, 1)u x ) (t, 0) = 0, if γ ∈ [1, 3/2), u(0, x) = u 0 (x). (81)where γ ∈ [0, 3/2) describes the degeneracy magnitude, for which Theorem 1.1 applies.If γ ∈ [0, 1) problem (81) is called weakly degenerate and the natural spaces for the well-posedness are the following weighted Sobolev spaces. Let I = (0, 1) and X = L 2 (I), we defineH 1 γ (I) = u ∈ X : u is absolutely continuous on [0, 1], x γ/2 u x ∈ X H 1 γ,0 (I) = u ∈ H 1 γ (I) : u(0) = 0, u(1) = 0 H 2 γ (I) = u ∈ H 1 γ (I) : x γ u x ∈ H 1 (I) .We denote by A : D(A) ⊂ X → X the linear degenerate second order operator    ∀u ∈ D(A), Au := -(x γ u x ) x , D(A) := {u ∈ H 1 γ,0 (I), x γ u x ∈ H 1 (I)}.(82)

H 1 γ

 1 (I) = u ∈ X : u is absolutely continuous on (0, 1], x γ/2 u x ∈ X H 1 γ,0 (I) := u ∈ H 1 γ (I) : u(1) = 0 , H 2 γ (I) = u ∈ H 1 γ (I) : x γ u x ∈ H 1 (I) .

Acknowledgments

We are grateful to J. M. Coron and P. Martinez for their precious comments and suggestions.

This paper was partly supported by the INdAM National Group for Mathematical Analysis, Probability and their Applications.

† This author acknowledges support from the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

‡ This author is grateful to University Italo Francese (Vinci Project 2018).

From (49) it follows that C α (T n )Λ Tn ||v(τ n-1 )|| < 1 and thus we can apply Proposition 2.5 to problem (51), obtaining

We shift back the problem into the original interval [τ n-1 , τ n ] and we get

By inductive hypothesis, we can estimate ||v (τ n )|| as follows

Proposition 3.4. Let 0 < T ≤ min {1, 1/α 2 } and consider the sequence (T j ) j∈N * defined by (31). Let v 0 ∈ X be such that ||v 0 || < e -6C K /T , let A : D(A) ⊂ X → X be such that (3) and (4) hold and let B : X → X satisfies (5). Let p ∈ L 2 (τ n-1 , τ n ) be defined by (47). Moreover, we assume that λ 1 = 0. Then, the solution of (46) satisfies

for all n ∈ N * .

Proof. We prove formula (55) by induction on n. The case n = 1 follows from Proposition 2.5, thanks to the assumption ||v 0 || < e -6C K /T . Now, suppose the formula holds for all the indices less than or equal to n -1. We prove it for n as follows. We consider problem (46) and in order to shift it in the interval [0, T n ], we introduce the variable s = t -τ n-1 as before and the functions ṽ(s) = v (s + τ n-1 ) and p(s) = p (s + τ n-1 ). Thus, (46) can be rewritten as

By Proposition 3.3, it holds that C α (T n )Λ Tn ||v n-1 || ≤ 1 and hence, we can apply Proposition 2.5 considering as final time T n (instead of T ), obtaining that

Finally, by inductive hypothesis, we conclude that

that is equivalent to formula (55).

We are now ready to prove our main result.

Second step. Let v 0 ∈ X be the initial condition of ( 16). We decompose v 0 as follows

and in the time interval [0, t R ] we take the control p ≡ 0. Then, for all t ∈ [0, t R ], we have that

In particular, for t = t R , it holds that ||v(t R )|| 2 < 2r 2 1 . Now, we define T R := t R + 1 and set v 1 (0) = v(t R ). Thanks to the first step of the proof, there exists a control p 1 ∈ L 2 (0, 1), such that v 1 (1) = 0, where v 1 is the solution of ( 16) on [0, 1] with p replaced by p 1 .

Then 16) in the time interval (t R , T R ] with the control p 1 (t -t R ) that steers the solution v to 0 at T R .

Proof (of Theorem 1.2). We start with the case λ 1 = 0. Let u 0 ∈ X that satisfies [START_REF] Campiti | Degenerate self-adjoint evolution equations on the unit interval[END_REF]. Set v(t) := u(t) -ϕ 1 , then v satisfies ( 16) and moreover v 0 := v(0) = u 0 -ϕ 0 fulfills (66). Thus, by Lemma 4.1, problem (1) is exactly controllable to the ground state solution ψ 1 ≡ ϕ 1 in time T R . Now, we consider the case λ 1 > 0. As in the proof of Theorem 1.1, we introduce the variable z(t) = e λ 1 t u(t) that solves problem (65). For such a system, since the first eigenvalue of A 1 is equal 0, we have the exact controllability to ϕ 1 in time T R . Namely z(T R ) = ϕ 1 , that is equivalent to the exact controllability of u to ψ 1 :

The proof is thus complete.

The proof of Theorem 1.3 easily follows from Theorem (1.2).

Proof (of Theorem 1.3). Suppose that γ := u 0 , ϕ 1 = 0. We decompose u 0 as

where ζ1 := ζ 1 /γ. We apply Theorem 1.2 to (69) to deduce the existence of T R > 0 such that ũ(T R ) = ψ 1 (T R ). Therefore, the solution of (1) with initial condition u 0 ∈ X that do not vanish along the direction ϕ 1 can be exactly controlled in time T R to the trajectory u 0 , ϕ 1 ψ 1 (•).

Note that if u 0 ∈ X satisfies both u 0 ∈ ϕ ⊥ 1 and ( 10), then we have trivially that u 0 ≡ 0. We then choose p ≡ 0, so that the solution of (1) remains constantly equal to φ 1 ≡ 0. Furthermore, {λ k } k∈N * verifies the gap condition (4) with α = π/ ln 2.

We define the operator B : X → X by Bϕ = µϕ, where µ ∈ H 2 (I) is such that 2µ (1) ± µ (0) = 0, and µϕ 1 , ϕ k = 0 ∀k ∈ N * .

Hence, thanks to (76), ( 5) is fulfilled with q = 3/2. An example of a suitable function µ that satisfies (76) is µ(x) = x, see [START_REF] Alabau-Boussouira | Superexponential stabilizability of evolution equations of parabolic type via bilinear control[END_REF] for the verification. Thus, from Theorem 1.1, we deduce that, for any T > 0, system (75) is controllable to the ground state solution if the initial condition u 0 is close enough to ϕ 1 .

Diffusion equation in a 3D ball with radial data

In this example, we study the controllability of an evolution equation in the three dimensional unit ball B 3 for radial data. The bilinear control problem is the following

where the Laplacian in polar coordinates for radial data is given by the following expression

The function µ is a radial function as well in the space H 3 r (B 3 ), where the spaces H k r (B 3 ) are defined as follows

).

The domain of the Dirichlet Laplacian

). We observe that A satisfies hypothesis (3). We denote by {λ k } k∈N * and {ϕ k } k∈N * the families of eigenvalues and eigenvectors of A, Aϕ k = λ k ϕ k , namely Then, it can be proved that

with b a positive constant. For instance, µ(x) = x 2 verifies (79) and ( 80) with b = 2π 2 -3 6π 2 . Therefore, by applying Theorem 1.1, we conclude that for any T > 0, the exists a suitable constant R T > 0 such that, if u 0 ∈ B R T (ϕ 1 ), problem (77) is exactly controllable to the ground state ψ 1 in time T .


The metaphor of this troubling pachyderm, of course, is well known and widely used (see, for example, [START_REF] Krueger | Tackling the Elephant in the Room[END_REF][START_REF] Thomas | An Elephant in the Consultation Room? Configuring Down Syndrome in British Antenatal Care[END_REF]. The figurative elephant in the room alludes to issues that seem obvious, but are so disturbing that they are actively ignored or put aside to ensure business as usual. In our view, the resulting spectrum of ignorance goes from voluntary blindness to the point where, hidden in plain sight, the obvious and disturbing issues may actually not be known by some of the protagonists.

In this think piece, which accompanies our special section on the problem of classification in ethnography, we suggest that matters of classification, categorization, and typology in all of their shades constitute an 'elephant in the room' of ethnographic field research and writing, both an obvious and invisible part of our anthropological epistemology. We decided to use the metaphor of the elephant in the room to guide us in this text, as we address questions of methodology, positionality, epistemology, and representation in ways we deem heuristic.

The elephants of classification

In their various shades (in terms of ancestry, class differences, diagnostic categories, racial and ethnic labels, and their many intersections), classifications in governmental and medical practice, among others, are extremely powerful, implying material effects for the subject(s) they target. They shape the epistemic space of what is thinkable, conceivable, and targetable. Unavoidably, the resulting categories include as well as exclude, often simultaneously. They can be tools and obstacles at the same time, and have profound impacts on the lives and bodies of both humans and nonhumans. In that sense, they are deeply relational and contingent; their effects vary depending on their specific articulation in practices. Their arbitrariness may play out between different actors as well as within individual subjects, as classificatory violence and social entitlement may go hand in hand; think, for instance, about the boundaries and privileges of citizenship, as explored by [START_REF] Nguyen | The Republic of Therapy: Triage and Sovereignty in West Africa's Time of AIDS[END_REF], [START_REF] Krause | Thinking through Political Subjectivity[END_REF], [START_REF] Heinemann | Biological Citizenship Reconsidered: The Use of DNA Analysis by Immigration Authorities in Germany[END_REF], and others.

In medical anthropology and the anthropology of science, where our own research is situated, we constantly encounter categorizations that help in making up people [START_REF] Hacking | Making up People[END_REF]) and constitute natureculture worlds [START_REF] Subramaniam | Ghost Stories for Darwin: The Science of Variation and the Politics of Diversity[END_REF]. Our interlocutors -among them clinicians, policy makers, epidemiologists, and lab scientists -often base their daily work decisions on standardized classifications. They do so explicitly, as part of their routine epistemic practices. These common categorizations (of disease, risk, or ancestry, for example) are intertwined with attributions of and assumptions about other differencemaking categories: class, race and ethnicity, or religious belonging inform the epistemic practice of classification and have a profound impact on the intended social, scientific, and clinical outputs (see [START_REF] Subramaniam | Ghost Stories for Darwin: The Science of Variation and the Politics of Diversity[END_REF]. We both encountered such intertwinements in our fieldwork: in the context of medical genetics in Oman for instance, clinicians frequently assume that the specific branch of Islam their patients belong to influences their opinions about prenatal diagnosis and medical termination of pregnancy. However, as it is not socially acceptable to inquire about this obedience, practitioners use a patient's name, address, physical appearance, clothing, or accent as a proxy for their religious affiliation. Depending on their resulting impression, they may mention the official ruling of the Saudi clerics (with patients thought to be Ibadi or Sunni) or decide not to raise the religious question at all (with those thought to be Shia). Or, to remain within the field of human genetics, many sampling decisions involved in the creation of biogeographically stratified databases in population genomics operate on assumed correlations among territory, language, culture, and body that link back to the time of colonial race science. So-called ancestral populations are thereby constructed as rather homogenous and place-bound. Despite geneticists' emphasis on the complex relationship between phenotype and genotype, and despite the explicit rejection of racial typology in contemporary genomics, race remains a sticky and troubling absentpresence that is hard to pin down (see M'charek, [START_REF] Amade | Technologies of Belonging: The Absent Presence of Race in Europe[END_REF].

For some of the protagonists of our fieldwork, who perform the daily work of categorization, such (historical) relationships get sidelined in favor of pragmatic decision making. For others, especially those who are subject to the ordering gaze, the resulting categorizations may turn out to be deeply problematic and politically charged. Just like the metaphorical elephant lingering next to the sofa, the presence of difference-making categorizations in clinical and scientific practice is often palpable but not necessarily recognized or addressed by individuals who nevertheless practice them. As anthropologists, we might observe them, yet we are not always prone to discuss them explicitly, especially when considering the multilayered and arbitrary role of categories of difference in practice. The absence of discussion about the existence and performative power of multiple and intersectional categorizations by those who use them tends to simplify and naturalize them: in the local moral worlds where they are valid, they are taken for granted.

The fact that such powerful social phenomena are rarely discussed poses profound methodological, epistemic, and ethical challenges to ethnography. Certainly, we need to ask how categories and standards are forged and employed, and what kinds of work they perform [START_REF] Bowker | Sorting Things Out: Classification and Its Consequences[END_REF][START_REF] Bauer | Contested Categories: Life Sciences in Society[END_REF][START_REF] Lampland | Standards and Their Stories : How Quantifying, Classifying, and Formalizing Practices Shape Everyday Life[END_REF]. However, we also need to acknowledge that our own anthropological practices also include arbitrary classificatory work and may unintentionally reinforce the very categories they intend to unravel. Offering the concept of 'classificatory violence', scholars like [START_REF] Flood | Situated Comparison: A Methodological Response to an Epistemological Dilemma[END_REF], [START_REF] Pollock | Medicating Race: Heart Disease and Durable Preoccupations with Difference[END_REF]Schramm, Krause, andValley (2019) urge anthropologists to attend to inequality and power relations that shape classificatory practices. We ask: what kinds of epistemological dilemmas do anthropologists encounter if they want to both take their interlocutors seriously and provide nuanced accounts of their practices?

In the following sections, we reflect on this issue by considering the elephant of classification as a material-semiotic creature that calls for a careful consideration of positionality and situatedness in our interlocutors' practices as well as our own. We start by zooming out of the first image of this think piece and attending to the surroundings of the pachyderm and their settings. This allows us to focus on the importance of situatedness and ethnographic attentiveness as necessary abilities for anthropologists to acknowledge and write about the elephants of classification in their fieldwork. We also deal with ways of liberating the metaphorical elephant when we propose not to give in to the dichotomous temptation that comes with classification. We stress the importance of historicizing the elephant's presence, shape, and role, as a first step, maybe, to driving it out of the room.

Of concealed, tamed, and painted elephants in anthropology

The image we have chosen to accompany our text offers a felicitous entry point to address these issues. It is titled 'The Elephant in the Room' and is taken from Banksy's 'Barely Legal' show, which took place in Los Angeles in 2006.1 In the position of the audience, we look at the scene from the outside. Spray-painted in red with golden fleurs-de-lis, Tai, a thirty-eightyear-old female Elephas maximus is spectacular, yet camouflaged. The elephant fits into the setting; it takes the center of the stage. In the original Banksy performance, the elephant symbolized 'poverty', that problem of which everybody is aware but willfully ignores. For our purposes, we want to draw attention to the staged character of the scene: neither the living room nor the painted elephant are 'natural'. Likewise, classificatory processes are not natural; they need continuous work in order to be shared, sustained, and performed. This work involves taming: an unruly elephant would destroy the scene, making it impossible to overlook. But Tai, who had performed in numerous popular films, was taught to follow commands and fulfilled her prescribed role. Classifications also imply taming: their very purpose is to organize the world so that it can be grasped, understood, described (and thus discussed), and controlled. This organizing work produces tensions between the general and the specific, the standardized and the personal, as well as the normative and the queer.

Thirty years back, the sociologist Susan Leigh [START_REF] Star | Power, Technology and the Phenomenology of Conventions: On Being Allergic to Onions[END_REF] demonstrated the importance of paying attention to and caring for that which does not fit. In considering 'monstrous' arrangements that do not fit neatly into our binary classifications, she argues that both 1 John Akomfrah's more recent contribution to the 2019 Venice Biennale 'The Elephant in the Room: Four Nocturnes' uses the metaphor to address yet another universe of planetary relations. We have chosen to stay with Banksy's performance because of its staged character and its straightforwardness as an illustration.

violent erasure as well as misrecognition could be potentially deadly. In her seminal piece 'Power, Technology and the Phenomenology of Conventions: On Being Allergic to Onions' she develops a 'theory of multiple membership' in which she questions the self-evident status of standardizations, and asks about the possibilities and limitations of alternative, more accommodating forms of classification (Star 1990, 26). Star points out the cost that is involved in membership: the exclusions, cutting, and covering, in other words, the taming of complexity that is involved in all modes of belonging. She also shows that heterogeneity and multiplicity cannot be resolved by more splinterings or compartmentalizations. Instead, she wants us to acknowledge the tensions, urging us to leave room for more fluid arrangements and to start from what she calls the 'Zero Point', that is, a point 'between dichotomies' that allows to think in new ways about technosocial assemblages (Star 1990, 47, 53).

However, as the think pieces of this special section show, clinical practice is more often than not guided by forms of classification that aim to produce clear-cut boundaries: you receive the test or you don't (see McDowell in this special section); you are entitled to access the health care facility or you aren't (see Brenman in this special section). Better inclusiveness may be attempted by adding or accommodating group labels (see Brenman in this special section), thereby shifting the boundary between 'deserving' and 'undeserving', while keeping the very distinction intact. The work of classification takes place within a complex assemblage of links and relations: making one category fit -including with the help of wellwishing anthropologists who take part in feeding the elephant of classification -may produce new exclusions, new blind spots (see Moyer in this special section). Certainly, the additional presence of the anthropologist in the room already filled by the elephant adds a layer of interesting complexity to the scene.

Moving (around) elephants: Situatedness, positionality, and writing

Star writes (1990, 52): 'Power is about whose metaphor brings worlds together and holds them there. ... Metaphors may heal or create, erase or violate, impose a voice or embody more than one voice'. Returning to Banksy's installation: a journalist tells us that Tai remained calm and played along [START_REF] Oliver | Banksy's Painted Elephant Is Illegal, Say Officials[END_REF]. She was an experienced performer, as her keepers assured the press. But the act of painting her body drew the attention of the animal welfare authorities, who ordered that she be scrubbed clean of the potentially toxic paint.

Local authorities ruled that she had to remain in the staged living room for the time of the performance without her camouflaging paint. Thus cleansed, Tai stood out in the room, losing the purpose the artist had intended for her.

Having shifted positionalities (back and forth from 'art piece' to 'political message' to 'animal welfare symbol'), the example of Tai may help us think through the methodological challenges that we face when dealing with the work of classification. Who indeed has the power to paint or to scrub clean, to display or conceal? With this metaphor of the elephant in the room, we try to maintain an awareness of relationality and power in our consideration of classificatory practices. Here we need to go deeper than merely acknowledging the discomfort and tension that appear when facing the disturbing categorization/elephant. How can we begin to think from Star's point 'between dichotomies' as the space of tension? Facing unfamiliar categorizations, how can we stay in the place of generative epistemic disconcertment, which Helen Verran (2013) marks as the site for 'doing difference together', thereby allowing for new ethnographic alliances? This, we argue, would allow for more inclusive practices of anthropological work that do not pretend to be all encompassing (and thus totalitarian). In our view, the elephant of classification can be tackled methodologically in two ways: a) by paying attention to situatedness and positionality, and b) by cultivating ethnographic attentiveness and forms of writing that do not ignore the elephant, or pretend to, but engage with it.

Questions of positionality have been extensively and critically discussed among anthropologists, entering the mainstream of the discipline with the Writing Culture debate (Clifford and Marcus 1988), if not earlier. These discussions about representation as a powerful and situated practice destabilized the earlier idea that ethnography is equivalent to 'reading over the shoulders' (Geertz 1973, 452) of cultural insiders, and suggested instead that anthropologists have problematically interpreted cultural texts from an unmarked position of presumed superiority. Attempts to stand shoulder-to-shoulder with interlocutors have ranged from collective research and writing [START_REF] Fortun | Experimental Ethnography Online: The Asthma Files[END_REF]Lewis and Russell 2011) to more explicit forms of solidarity (TallBear 2014). These writings emphasize that the researcher's presence always makes a difference, that ethnographers are ourselves protagonists in our stories. As anthropologists have come to realize, the 'God-trick' [START_REF] Haraway | Primate Visions: Gender, Race, and Nature in the World of Modern Science[END_REF]) that would apply a view from nowhere is unfeasible. Ethnography is now widely understood to be all about situatedness, and to draw its strength from the meticulous analysis of concrete settings and relationships (see Gluckman 1940; Clarke, Friese, and Washburn 2018) as well as their historical formation. Ethnographic methods call for an explicit consideration of multiple positionalities and their impacts in our analysis, and of the several blind spots that emerge from our own involvement and standpoints. In fact, acknowledging the situatedness of our knowledge production may allow us to find better ways of recognizing intersectional overlaps in classificatory practices [START_REF] Collins | Learning from the Outsider within: The Sociological Significance of Black Feminist Thought[END_REF][START_REF] Haraway | Primate Visions: Gender, Race, and Nature in the World of Modern Science[END_REF][START_REF] Cho | Toward a Field of Intersectionality Studies: Theory, Applications and Praxis[END_REF].

Positionality makes a difference: you may not realize the elephant's presence when you are inside the room (be it the living room, the triage room, or the project room) because you have already accommodated it; it has become part of the routine scenery. In other words, you could be experiencing 'actors' blindness' [START_REF] Bowker | Sorting Things Out: Classification and Its Consequences[END_REF], 'where the system's description of reality becomes true'. Or you may need to actively ignore it, so that you can keep your focus (like the woman in the opening photograph who is absorbed in her reading). This work of actively ignoring can take different forms, depending on whether you sit on the sofa or lean on one of the walls. During fieldwork, we might bump into the elephant at first, and then follow the example of other individuals who pretend they do not see it as they walk around it. Perhaps we still notice it but then become absorbed in other things, just as some of our interlocutors do. Shifting locations therefore helps us to situate the elephant and to follow its moves. However, reflecting on the ontological weightiness of the elephant and the presumed given-ness of the 'room' as the ethnographic scene does not mean we consider them to be fixed parameters in our ethnographic spaces. Explicit work on positionalities is also necessary to delineate how complicit we might be in setting up the space that holds the elephant, the elephant itself, and their unquestionable materiality. From the outside, we may indeed see different things. The second image shows how the installation is arranged to draw our gaze to the elephant (as well as the artificial space of which it is a part, including here the fence). Familiar with the saying, we may first think of the symbolic meaning of the elephant in the room. We may also consider not only how she interacts with the material environment (the furniture, the tapestry), but also her various relationships with her caregivers, the other actors in the room, or with us, the spectators. Similarly, ethnographers in settings where the work of sorting, typing, and classifying is part of professional routines may choose to shift their position and point of view, scaling up and down, so as to be able to spot the conspicuous elephants as well as the mundane relationships that have stabilized the working arrangement.

At this point, our second methodological consideration -ethnographic attentivenessbecomes relevant, as does the question of how we can nurture such attentiveness in our writing. If we think about writing ethnographically about the fleur-de-lis-covered room, we might imagine ourselves first sitting next to and talking with the woman on the sofa, and then stepping out and talking to the spectators, and then describing these different actors and contexts. However, we should not assume that ethnographers are the only actors on the move, or that the local protagonists in our stories are prisoners of their circumstances (and are therefore bound and fixed to one room). Nina Glick-Schiller's (2010) important critique of methodological nationalism, which remains common in migration studies and other ethnographic work, calls attention to the problem of fixed categories and the epistemological limitations resulting from it.

We are not suggesting that researchers ignore the significance of categorizations and labels. On the contrary, the ethnographer is called upon to describe and analyze how certain categories are brought about, the kind of work they do, and how we can historicize them.

How did the woman come to sit on the sofa? What is she reading? How and when did the elephant enter the room and how is it related to the content of her book? What routines have developed around it, what traces does it set? In genealogical terms, it also involves asking what elephant was there before this one, whether it was bigger or smaller or even standing in another room, and what ghostly presence it might still have. As ethnographers, we cannot afford to ignore the elephant; we must consider it as a living and breathing being, not a prop. As recent scholarship in science and technologies studies has shown, standards, classifications, and indicators are not static, neutral, or dead; they inhabit complex worlds [START_REF] Rottenburg | A World of Indicators: The Making of Governmental Knowledge through Quantification[END_REF].

So, what practices of ethnographic attention and writing allow us to shed light on the actions of classification (without reproducing the unquestioned hierarchies of truths, or falling into the trap of naive relativism)? In their account of 'complexities', John [START_REF] Law | Complexities: Social Studies of Knowledge Practices[END_REF] suggest starting with alternative forms of arranging and ordering ethnographic material. Through listing, mapping, and rearranging, we can focus on multiple relations that constitute, stabilize, or dissolve classificatory objects, rendering them relevant or irrelevant.

We can also draw attention to implicated (and concealed!) actors [START_REF] Clarke | The Social Worlds Framework: A Theory/Methods Package[END_REF] while avoiding the problematic paternalism of 'giving voice' to others. These techniques of writing and analyzing need to achieve the difficult aim of keeping the tension between categories in place, without falling for one dichotomy or another. This is what we call 'liberating the elephant'.

In our view, the edginess, loose ends, and unresolved contradictions of ethnographic writing need not be erased but rather sharply delineated. This involves working through, around, about, and with the elephant, with the very categories of difference and belonging that are part of the field situations we are investigating. Such writing may happen in experimental formats or more conventional ethnographic styles. Against the primacy of visual categorization [START_REF] Haraway | Primate Visions: Gender, Race, and Nature in the World of Modern Science[END_REF], we may acknowledge other sensual aspects in our work, including the epistemic dimension of affective relations [START_REF] Stodulka | Affective Scholarship: Doing Anthropology with Epistemic Affects[END_REF]. Certainly, we sense the elephant's warmth and can smell it, even if we do not see it.

We should also attend to the ghosts that haunt contemporary knowledge production [START_REF] Subramaniam | Ghost Stories for Darwin: The Science of Variation and the Politics of Diversity[END_REF]) in order to arrive at better research practices, both in anthropology and the fields we study. Among these ghostly presences are those racialized, classed, and gendered forms of classification that mark hierarchical differences. They are often inscribed in research designs and analytical practices without explicit mention or recognition. The relevance of ethnographic work, then, is derived from modest interventions in and irritations of such practices, not in providing all-encompassing alternatives (for example, new ad hoc classifications). The goal is not to get rid of the elephant but to engage with it. Such engagement, as we have already outlined, can take different forms. Certainly, the elephant, once recognized, causes disturbance. But it also invites us to share in its tactile sensibility: touching, reaching out, and treading carefully. We might also derive fresh ideas from the surreal arrangement that puts the elephant at center stage of an otherwise well-ordered scheme. The metaphorical elephant might then also signal a move against classification, marking that which cannot be sorted, typed, or classified. In any case, we should not take the categories we encounter, nor their effects and effectiveness for granted.

Some of the elephants of our ethnographic rooms: About the think pieces in this special section

The three think pieces that accompany this introductory essay face their elephants in different ways. They shed light on 'in-between situations', exploring the cracks left open by usual classifications. For example, they discuss circumstances where health care practitioners divert a globalized, standardized logic of categorization in their daily practice [START_REF] Beaudevin | Diversion and Globalization in Biomedical Technologies[END_REF]: basing his treatment decisions on a sense of personal responsibility for the global health funding spent on the testing of his patients, the Indian TB specialist described by McDowell moves away from the clinical categorization framework defined by the program developers. The think pieces also demonstrate how, in other situations, individuals may strive to refine an obviously problematic classification system in order to produce inclusion, thereby creating layers and cumulative changes. As ethnographers, our three colleagues pay attention to the dilemmas and critical moments arising from the tension between generalization and specificity, population and individual that inform their interlocutors' work. They explore the tinkering practices that evolve around triage, and the limitations and capacities of notions of 'deservingness'. Finally, they pay attention to the multiple ways that various actors in their fields relate to boundary-setting elephants, often marked by commonsense notions of race, ethnicity, citizenship, and economic status.

Some of these practices explicitly contest invisibilization, that is, they recognize the discomforting and space-taking presence of the elephant of classification and try to address it. However, as the three think pieces make clear, they cannot fully dissolve it or otherwise make it disappear. In our view, the stickiness2 of categorizations is part of the ethnographic challenge: to acknowledge and demonstrate the power of classificatory practices by paying close attention to how they are brought about as well as to the multifarious and often arbitrary work they perform. Hence, the necessity for anthropologists to accept the discomfort that classifications cause: they are at once tools and obstacles, opening and closing off opportunities. Starting from this location of discomfort allows for a mode of critique that demonstrates that the social, epistemic, and economic relations shaped by classificatory practices are not necessarily static nor fixed, but dynamic and thus open to change.

Natassia Brenman's elephant of classification dwells in a UK mental health care center, where it is tamed and trained by the practitioners. Her piece shows the mental health practitioners and caregivers trying to enrich and broaden a categorization system they deem problematic, with the paradoxical (and expected) impact of increasing inclusiveness. At first disturbed by the 'Black and Minority Ethnic' (BME) category defining the beneficiaries of the center, Brenman could have addressed this trouble by tearing the BME label apart and critically discussing its historical, political, and social stakes without acknowledging the generative and inclusive potential of the categorization. Instead, she decided to do both. Using [START_REF] Haraway | Staying with the Trouble: Making Kin in the Chthulucene[END_REF] evocative metaphor of 'staying with the trouble', Brenman looks at 2 Thanks to Janina Kehr for this relevant image. We use this term in a slightly different sense from Sara [START_REF] Ahmed | The Cultural Politics of Emotions[END_REF] original concept, which emphasizes an embodied dimension of attachment that is not central to our argument here. The elephant of class introduced by Eileen Moyer is, paradoxically, the result of the disappearance of a previous pachyderm: the one of masculinity. In the context of HIV care in eastern and southern Africa, the piece shows the role of global health practitioners as well as anthropologists in shaping risk categories and designing target populations for global health interventions. However, by focusing on gender and sexuality alone (thereby 'making up' specific categories such as adolescent boys, men who have sex with men, or expecting fathers), they produce a huge elephant in the HIV-prevention room, namely class difference. Moyer asks: What might be gained by recognizing and engaging with this particular elephant from the perspective of public health and also from anthropology? What happens to social theories of gender inequality, agency, and risk if we reconfigure middle-class men as 'at risk'?

At the end of the elephant trail, concluding thoughts

We started by exploring how and why we should acknowledge, point to, walk around, write about, liberate, care for, and sometimes help drive away the elephants of classification we face in our ethnographic rooms and field sites. At the end of this trail, we are convinced that a dynamic, relational, and multiscalar practice (in fieldwork, writing, academic policies, and political practice) is the way out of the overcrowded room. Our piece begins to outline such a practice.

We want to acknowledge the ongoing stickiness of categories of difference, as well as their solid, persistent, residual, and/or ghostly presence. These are matters one cannot walk away from in ethnographic practice. The metaphorical incursion into the lives of our communal elephants is driven by a common desire for a better anthropological practice. Our think piece is an account of this incursion, which started with a lively debate in Stockholm in 2018.

Given the scope of the raised issues, we have no conclusive aspirations, but rather aim at launching further discussions with colleagues: what are your own elephant(s), and how do you interact and deal with them?

In posing this question, we acknowledge that the elephant only travels so far. Comparing the practices of sorting, typing, and classifying that we encounter (and sometimes produce) in our research to enormous animals that may be both disruptive and actively ignored is not enough to understand the multiple dimensions of the problems we are dealing with. There is more to the crafting of difference(s) than the simple fact that difference is both huge and rarely discussed, hence our interest in discussing the power dynamics at play in the making and various uses of these categories, their relations to other -seemingly less controversialcategories of differentiation, and their situatedness in wider historical and political fields. These issues indeed point to a complex landscape of epistemological and ethical discomfort that we have to enter and navigate (see [START_REF] Verran | Working with Those Who Think Otherwise[END_REF]. Such a journey invites us to question our own complicity, as authors who have to set themselves in the scenes they are writing about. It also leads us to search for alternative ways of writing, specifically turning discomfort into ways to build new relations across disciplines and epistemic communities as well as in academic and activist spaces. Attending to the elephant in the room, so we hope, may provide a vantage point that opens an unexpected door in the beautifully patterned tapestry of our ethnographic accounts, thus leading us to new pathways in anthropological inquiry.

Sarah Fruendt, Janina Kehr, Kristine Krause, and Charlotte Kühlbrandt for their involvement, as well as to Janina Kehr, Lucile Ruault, and Emilia Sanabria, and the journal's editors for their comments.

About the authors

Figure 2 .

 2 Figure 2. Photo by Bit Boy (2016). Source: https://www.flickr.com/photos/bitboy/246805948.

  how the BME category is crafted, enacted, and adjusted -first to BAME (Black, Asian, and Minority Ethnic) and then to BAMER, with the 'R 'for Refugee -in relation to the shifting institutional needs and clients' claims. Andrew McDowell's paper takes us to a tuberculosis clinic in India, where disease-specific programs in global health are implemented, and where clinicians use a highly technological and informative instant test made freely available by global health philanthropists to better diagnose TB in children. But this test is not used according to the predefined categories of 'deservingness' that come along with it: the physicians add their own category of patient, the 'most needing', which they evaluate on the spot through 'intuitive class assessment'. The elephant in this TB clinic is the relationship of the health care practitioner's morally informed decisions to the patient's healing path. McDowell draws our attention to the limitations of standardizations and categories of deservingness that ignore the richly layered local social fabric in which clinicians work. Their decisions may be impacted by other factors than those shaping the work of the designers of global health programs. McDowell's clinic is a room occupied by several elephants that may stand in each other's way or clot up the clinical practice.

  Katharina Schramm is Chair for Social and Cultural Anthropology at the University of Bayreuth, where she heads a working group on the anthropology of global inequalities. Her research has focused on matters of race, scientific knowledge production, and the formation of political subjectivities. She recently coedited a special issue of Citizenship Studies entitled 'Claiming Citizenship Rights through the Body Multiple' (with Sabine Netz, Sarah Lempp, and Kristine Krause).Claire Beaudevin is a tenured researcher with the French National Centre for Scientific Research (CNRS) at the Cermes3 in Paris. Her current research focuses on the anthropology of genomics and medical genetics. Her main fieldwork is located in the Arabian Peninsula (Sultanate of Oman), where she studies the development of genetics and genomics within the clinic, public health, and research. In France, with Catherine Bourgain and Ashveen Peerbaye, she also investigates the development of routine cancer genomics in state-funded hospitals.

Acknowledgements

This collection of think pieces derives from a joint panel titled 'Sorting, Typing, Classifying: The Elephants in "Our" Rooms' that we convened with Kristine Krause and Janina Kehr at the 2018 biennial conference of the European Association of Social Anthropologists in Stockholm, Sweden. We sought to encourage a conversation on classification as an ethnographic concern between two EASA networks: the Medical Anthropology in Europe (MAE) network (https://www.easaonline.org/networks/medical) and the Anthropology of Race and Ethnicity (ARE) network (https://www.easaonline.org/networks/are). We want to thank all workshop participants, including the audience, for a very engaged and thoughtprovoking discussion. We hope that we have captured the spirit of this lively debate in our reflections. In addition to the contributors to the special section, we offer special thanks to


Arnaud Passalacqua

Exposing urban utopias based on transport systems: a retrospective analysis of the Grand Paris Pavilion for the Architecture and Landscape Biennial in Versailles

This paper proposes to come back on the process that led to the opening of a Grand Paris Pavilion during the Architecture and Landscape Biennial in Versailles (May-July 2019). The installation of this exhibition in an ancient post office temporarily available, in proximity to the Château de Versailles, has become one of the main points of this new event. The choice has been made to focus on utopias, specially those based on transport systems in the Paris area, and on the Grand Paris circle underground project, with the installation of 68 models of the new stations due to be realised during the 2020s. This configuration opens to various questions that are addressed by the paper: the differences of perceptions between car-based utopias and railway-based ones, the way history can be used by the actors of the contemporary project, the differences between the exhibition of authentic mobility systems and the choice, made in Versailles, to propose only reproductions of drawings and planning documents… Another interesting point lies in the interaction between historians, responsible for the choice of utopias and the elaboration of narratives, and architects and designers, who proposed the spatial organization of the exhibition, in a place with its own constraints. Paper 1. Crossing the ephemeral nature of the event and the long time of a transport project

The event that is the subject of this article is the Architecture and Landscape Biennial in Versailles. The first edition of this event took place during the Spring 2019. It is intended to be a regular event that addresses these issues from the perspective of urban ecology, a theme renewed by the importance of climate issues. The event is funded by the Region, whose new president, a conservative, entrusted the organization to a political ally, the Mayor of Versailles, a traditional right-wing city. The choice of this city, which aims to be both one of the jewels of the world heritage and one of the natural poles within the Paris region, is therefore also the result of political proximity. This project therefore wanted to illustrate the ecological will of the head of the Region. The idea of basing this will on an exhibition, in a city that is very touristy because of its palace, was intended to give it a major impact. This event was combined with a project of a completely different origin: the construction of a metro circle line around Paris, known as Grand Paris Express (GPE). This project, which is part of a very long period of time, is supported by the State and has suspected strong tensions with the Region, while it was socialist. But the President of the Region, like the Mayor of Versailles, were part of the presidential team that, around Nicolas Sarkozy, launched this project in 2009. This project should make it possible to significantly improve the mobility of the urban area by allowing direct suburban journeys in a city that suffers from a very strong imbalance between a very dense and well-served centre and a more diffuse periphery with radial links. The lines are expected to open gradually during the 2020s and 2030s, depending on construction hazards and financial availability, since the probable cost of the project is 40 to 50 billion euros.

The state structure behind the project, the Société du Grand Paris (SGP), has chosen to conduct it by developing many forms of territorial consultation, partly to reduce its image of national level imposing its will. Events are very regularly organised, from debates on the choice of the lines to cultural happenings on the works sites. The SGP therefore wants to be present in this territory, which it intend to dynamise by the metro. In the same way that it aims to be a company that conducts its projects in the most ecological spirit possible.

As a result, the crossing of the Biennials in Versailles and this dynamic of the new metro was somehow expected, especially since the metro will have one of its stations in Versailles, which was then a subject of hard discussions. The SGP therefore wished to participate in the event through a pavilion, which became one of the places of this exhibition whose poles were disseminated throughout the city (palace, school of architecture, king's vegetable garden...).

Since I already had relations with the SGP, I had the opportunity to participate in the setting up of the programme of this Grand Paris Pavillon, hosted in a large former post office, before its transformation into a luxury hotel by a real estate transaction.

From the pavilion program to the historical mission

The project's guideline aimed to show that the GPE is part of the logic of urbanization and network deployment in the Paris region, both in terms of time and in terms of the contemporary challenges of building a more ecological city. The latter aspect is essentially driven by the idea that the development of a new public transit offer is a way to reduce car use. This reflection does not address the debate on this issue, that has developed since the beginning of the project.

My mission was to feed the project with long-term elements. For this purpose, I was helped by a very precious assistant, thanks to the SGP. The main guideline was to work on the utopias related to transport on the territory. We had to reflect on the intellectual origins of the GPE project, and then, more broadly, on the origins of the development of mobility in the Paris region. We quickly focused on the idea that the current state of mobility in the urban area is the result of the superposition of multiple projects, some of which have been successful and many of which have not been implemented. We have thus explored the proposals that have marked this history since the beginning of the 19 th century. I already had many classic references in mind, but we looked for other projects, some of which were unknown to me.

As we moved forward, the historical contribution thus took the form of a large timeline presenting various projects that shaped Parisian mobility. We have also included some images to illustrate the configuration of mobility at this or that period, in order to recontextualize the projects presented.

The historical mission then turned into a search for the places where the images envisaged were kept and a legal and financial arrangement of partnerships to expose them, which I did not do myself but which represented the heaviest part of the work.

A loop out of the game

On the theoretical plan, the proposal I initially made was based on a small scheme that seemed quite simple to me. The idea was to question the notion of utopia in the world of transport based on a diagram entitled "from utopia to utopia" made of 4 circular steps:

 1 The formulation of a transport project is a long-term process and involves several opposing projects.  2. The implementation of the project is based on the plans drawn up and evolves according to the hazards of the site, the availability of funding, the phasing of the project over time...  3. The operation of the transport service makes the project concrete, confronts it with everyday problems, the ageing of rolling stock and spaces, the image that can deteriorate…  4. A new urban utopia arises from dissatisfaction with the functioning of the city and its transport network in its current state.

The last stage generates the first, since this utopia ends up taking the form of new transport projects.

This small loop can be read as a cyclical evolution of the purity content of transport projects: while they have an ethereal or even sacred dimension when they are only paper projects, they are gradually confronted with the materiality of the site and then the daily operation, which contributes to a form of disenchantment. The re-enchantment therefore requires the formulation of another utopia, which thus relaunches the circular dynamic.

The whole exhibition project was based on the idea of a loop, since the GPE consists of creating a metro loop around Paris. In particular, a contemporary work of art in the form of a large doublerevolution staircase welcomed visitors to the entrance of the Pavilion. Looking for the loop in history therefore seemed to me a way to echo this dynamic.

However, such a proposal did not appeal to the whole team of curators. It seemed to me to be a simple way of recontextualizing the historical moment we are currently experiencing in the Paris region, which my interlocutors understood well. But they found that this vision was too negative insofar as it locked the project into a form of fatality that would inexorably lead to disillusionment. My proposal was therefore rejected. The theoretical discussions therefore stopped there and it was the visual elements that caught the attention.

The visual empire

As the project progressed, the focus on images increased. This is linked to several dynamics.

First, the exhibition was intended to be accessible to all by offering above all an easily perceptible content. The texts have therefore been reduced to the essentials. Upstairs, the exhibition of 68 models of the GPE's new stations was the highlight of the pavilion. The objective was above all visual: at the top of the stairs, the visitor must have been impressed by the forest of models. Similarly, the ceiling of this level was covered by a series of large maps of the Paris region, of a very different nature and time period. Here too, it was not really a question of giving information carried by a map, but rather of playing on the visual effect of the proliferation of possible representations of the region, and on the colours, scales and graphic choices of the different supports placed on the ceiling. The visitor had to live above all a visual experience.

Secondly, the culture of the team of curators has had a very strong impact on this focus on the visual artefacts. Within this group were at my side architects, a designer and a scenographer, a specialist in contemporary art. Visual and space issues therefore occupied the main focus of our upstream discussions. The urgency of the project also led to the reuse of many existing elements, such as the models scattered in the various architectural firms responsible for building these stations. It was therefore first a question of knowing what objects and images would be available, before reflecting on the substance of the subject.

Thirdly, the orientation of the pavilion on the project's theme has also led to giving priority to graphic representations of the projects, as this is generally the main support that the actors themselves have used to implement their project. This was a form of coherence between the exhibition and the ways of acting of the actors in planning and transport.

The choice of historical images, after a large collection, was organized according to the importance of the projects, but also according to their impressive or aesthetic character. Thus, some images of poorly documented, little-known and marginal projects have played a more important role than others, which have been much more decisive in the region's transport history.

The general organisation of the exhibition was organised in 20-year periods, from 1830 to 2030, from the creation of urban public transport to the inauguration of the GPE. For each period, about ten images were used to illustrate both the state of transport systems at the time and the projects and utopias formulated by the same period. To accompany these images, I was asked to write very short texts, which turned out to be quite difficult to produce. How to give meaning in a few words? The difficulty was all the greater because these texts were taken up by a journalist who was not particularly familiar with the historical issues involved.

The exchanges made it possible to readjust certain elements, but these texts posed two difficulties for me. On the one hand, they were rewritten in a very teleological spirit, aiming to show that the GPE was a hoped-for, logical and successful form of solution to the mobility problems in the Paris region. On the other hand, they have focused very strongly on public transport, at the expense of a system that represents the majority of journeys in the Paris region, the automobile. Yet it is also the source of many utopias and projects.

One of my main contributions was finally to insist that the automobile dimension should not be ignored. I was helped in this by images of automobile projects that were either very aesthetic, such as those of the 1930s, or very striking, with Le Corbusier's Voisin plan and his project for a city of 3 million inhabitants. But I felt the less attractive character of the automobile object, which was generally relegated to smaller images, out of step with the role it plays in our mobility. The texts themselves were initially intended to be very anti-automobiles, as if this unique tension had fuelled the entire history of urban travel, which is in fact much more complex in the games between the different systems.

The last difficulty with the images is due to the less attractiveness of black and white images, those of the 19 th century. The timeline was designed on a reverse pattern, beginning in 2030 and finishing in 1830, following a decreasing size of the 20-year period posters. So the last ones were far less attractive than the first ones and the images were of smaller size. Because most of them were projects of the 19 th century, they were not easily accessible and should have been treated on a more equal basis. But this argument was not able to convince the other curators, seduced by the idea of a sense of progress which was translated into the size of the timeline. Behind this choice, we can read the differences of practices that was a key point of the project.

One team, different practices

If the background was not at the heart of the project, it is mainly because the different actors involved in setting up this exhibition had quite different habits and interests.

Within the SGP itself, various speeches were made. His President wanted to illustrate that transport and urban project take time and evolve during their implementation, in order to better understand the difficulties of the GPE, faced with additional costs, delays and unavoidable route revisions. But the pilots of the operation for the SGP had a vision based on another logic: the project had to be carried out on time and become the support of multiple events, in order to feed a communication policy, towards the inhabitants, visitors and decision-makers. Two different ways of helping their project, not necessarily incompatible, but not always consistent.

For the Mayor, the project had to be part of the Versailles context, which presents many specificities. For him, it was also important that the whole project could be ready on time, so that the Biennial could be inaugurated by the President of the Region as well as by the former President of the Republic, Nicolas Sarkozy.

The scenographer and his team focused on the stakes of costs and deadlines for the material realization of the various supports.

For the architects, participation in this exhibition was only one aspect of the much broader contracts they have for the construction of one or more GPE stations. From then on, it was a matter of highlighting the main concepts that they wanted to place at the heart of their approach to the stations and which essentially revolve around ecological issues. Each of the teams thus remained in the conceptual area they forged themselves to obtain a station market and the exhibition juxtaposed these points of view more than it made them work together. In addition, since the architectural firms involved were large structures, the people attending the meetings were not always the same. The big names in architecture involved in the GPE only became visible when the decision-makers were present.

As a historian, I wanted to propose both an open view of the different transport systems and a problem based on the tension between existing networks and urban utopias. It was also an opportunity to highlight some documents considered original to present well-known projects, but whose iconography could be enriched.

Conclusion

This experience offered me the opportunity to play the role of the historian in a larger team, essentially oriented towards the timely implementation of an exhibition pressed by a timetable that was very narrow. In this sense, it showed me the mechanics of how this type of event works. But I still wonder about the fact that no one in the team has produced a global speech. The operation was conceived as a place of the visual, where visitors had to leave first and foremost by having lived a sensitive experience, the objective of which was to develop an interest for the GPE with the objects presented but also, probably, to impress by the abundance, the dimensions, the stakes... Such is the meaning of the forest of models and the ceiling of maps, which found their meaning above all by the number. This is also the meaning of the accumulation of images of projects that I have proposed.

As a result, no one made a global statement or formulated a problematic line. The ones I proposed were erased by the fact that they offended the actors of the GPE, who often claim to draw historical parallels but have difficulty accepting to conceive themselves as actors of a moment in history. Each architect had its own line and probably did not want this exhibition to be a real place for discussing these choices, which govern the stations that are currently built. The designer was the most likely to unify all this, but he did so on his own issues: organization of space, signage...

Being part of an ongoing project has thus posed many difficulties for me: link to political decisionmakers, interference with other professional logics, link to the temporality of the project... The most striking thing for me was the focus on the visual aspects. The exhibition was extended to include a small leaflet containing the historical timeline that SGP plans to distribute everywhere: it does not give meaning, but offers images.