Bing-Yu Zhang
email: bzhang@math.uc.edu

UNIQUE CONTINUATION PROPERTY AND CONTROL FOR THE BENJAMIN-BONA-MAHONY EQUATION ON THE TORUS

Keywords: Unique Continuation Property, Benjamin-Bona-Mahony equation, Korteweg-de Vries equation, Moving point control, Exact controllability, Stabilization

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2πZ). We prove a Unique Continuation Property (UCP) for small data in H 1 (T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H s (T) for any s ≥ 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H s (T) for any s ≥ 0 and globally exactly controllable in H s (T) for any s ≥ 1.

Introduction

We are concerned here with the Benjamin-Bona-Mahony (BBM) equation

u t -u txx + u x + uu x = 0 (1.1)
that was proposed in [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF] as an alternative to the Korteweg-de Vries (KdV) equation

u t + u xxx + u x + uu x = 0 (1.2)
as a model for the propagation of one-dimensional, unidirectional small amplitude long waves in nonlinear dispersive media. In the context of shallow-water waves, u = u(x, t) represents the displacement of the water surface at location x and time t. In this paper, we shall assume that x ∈ R or x ∈ T = R/(2πZ) (the one-dimensional torus). (1.1) is often obtained from (1.2) in the derivation of the surface equation by noticing that, in the considered regime, u x ∼ -u t , so that u xxx ∼ -u txx . The dispersive term -u txx has a strong smoothing effect, thanks to which the wellposedness theory of (1.1) is dramatically easier than for (1.2) (see [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF][START_REF] Bona | Sharp well-posedness for the BBM equation[END_REF][START_REF] Roumégoux | A symplectic non-squeezing theorem for BBM equation[END_REF] and the references therein). Numerics often involve the BBM equation, or the KdV-BBM equation (see below), because of the regularization provided by the term -u txx . On the other hand, (1.1) is not integrable and it has only three invariants of motion [START_REF] Hamdi | Exact solutions and invariants of motion for general types of regularized long wave equations[END_REF][START_REF] Olver | Euler operators and conservation laws of the BBM equation[END_REF].

In this paper, we investigate the Unique Continuation Property (UCP) of BBM and its applications to the Control Theory for (1.1). We say that the UCP holds in some class X of functions if, given any nonempty open set ω ⊂ T, the only solution u ∈ X of (1.1) fulfilling u(x, t) = 0 for (x, t) ∈ ω × (0, T ), is the trivial one u ≡ 0. Such a property is very important in Control Theory, as it is equivalent to the approximate controllability for linear PDE, and it is involved in the classical uniqueness/compactness approach in the proof of the stability for a PDE with a localized damping.

The UCP is usually proved with the aid of some Carleman estimate (see e.g. [START_REF] Saut | Unique continuation for some evolution equations[END_REF]). The UCP for KdV was established in [START_REF] Zhang | Unique continuation for the Korteweg-de Vries equation[END_REF] by the inverse scattering approach, in [START_REF] Escauriaza | On uniqueness properties of solutions of the k-generalized KdV equations[END_REF][START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF][START_REF] Saut | Unique continuation for some evolution equations[END_REF] by means of Carleman estimates, and in [START_REF] Bourgain | On the compactness of the support of solutions of dispersive equations[END_REF] by a perturbative approach and Fourier analysis. For BBM, the study of the UCP is only at its early age. The main reason is that both x = const and t = const are characteristic lines for (1.1). Thus, the Cauchy problem in the UCP (assuming e.g. that u = 0 for x ≤ 0, and solving BBM for x ≥ 0) is characteristic, which prevents from applying Holmgren's theorem, even for the linearized equation. The Carleman approach for the UCP of BBM was developed in [START_REF] Davila | Unique continuation for the Benjamin-Bona-Mahony and Boussinesq's equations[END_REF] and in [START_REF] Yamamoto | One unique continuation for a linearized Benjamin-Bona-Mahony equation[END_REF]. Unfortunately, Theorems 3.1-3.4 in [START_REF] Davila | Unique continuation for the Benjamin-Bona-Mahony and Boussinesq's equations[END_REF] are not correct without further assumptions, as noticed in [START_REF] Zhang | Unique continuation for the linearized Benjamin-Bona-Mahony equation with spacedependent potential[END_REF]. On the other hand, the UCP in [START_REF] Yamamoto | One unique continuation for a linearized Benjamin-Bona-Mahony equation[END_REF] for the BBM-like equation u xu txx = p(x, t)u x + q(x, t)u, x ∈ (0, 1), t ∈ (0, T ), where p ∈ L ∞ (0, T ; L ∞ (0, 1)) and q ∈ L ∞ (0, T ; L 2 (0, 1)), requires u(1, t) = u x (1, t) = 0 for t ∈ (0, T ) and u(x, 0) = 0 for x ∈ (0, 1).

(1.3) (Note, however, that nothing is required for u(0, t).) Because of (1.3), such a UCP cannot be used for the stabilization problem. More can be said for a linearized BBM equation with potential functions depending only on x. It was proved in [START_REF] Micu | On the controllability of the linearized Benjamin-Bona-Mahony equation[END_REF] that the only solution u ∈ C([0, T ], H 1 (0, 1)) of the linearized BBM equation u tu txx + u x = 0, x ∈ (0, 1), t ∈ (0, T ), (1.4) u(0, t) = u(1, t) = 0, t ∈ (0, T ) (

fulfilling u x (1, t) = 0 for all t ∈ (0, T ) is the trivial one u ≡ 0. It is worth noticing that the proof of that result strongly used the fact that the solutions of (1.4)- (1.5) are analytic in time. On the other hand, several difficult UCP results based on spectral analysis are given in [START_REF] Zhang | Unique continuation for the linearized Benjamin-Bona-Mahony equation with spacedependent potential[END_REF] for the system u tu txx = [α(x)u] x + β(x)u, x ∈ (0, 1), t ∈ (0, T ), (1.6) u(0, t) = u(1, t) = 0, t ∈ (0, T ).

(1.7)

As noticed in [START_REF] Zhang | Unique continuation for the linearized Benjamin-Bona-Mahony equation with spacedependent potential[END_REF], the UCP fails for (1.6)-(1.7) whenever both α and β vanish on some open set ω ⊂ T, so that the UCP depends not only on the regularity of the functions α and β, but also on their zero sets. Bourgain's approach [START_REF] Bourgain | On the compactness of the support of solutions of dispersive equations[END_REF] for the UCP of KdV (or NLS) is based on the fact that the Fourier transform of a compactly supported function extends to an entire function of exponential type. The proof of the UCP in [START_REF] Bourgain | On the compactness of the support of solutions of dispersive equations[END_REF] rests on estimates at high frequencies using the intuitive property that the nonlinear term in Duhamel formula is perturbative. As noticed in [START_REF] Mammeri | Unique continuation property for the KP-BBM-II equation[END_REF], that argument does not seem to be applicable to BBM. Actually, if we follow Bourgain's idea for the linearized BBM equation

u t -u txx + u x = 0 (1.8)
on R, and assume that some solution u vanishes for |x| > L and t ∈ (0, T ), then its Fourier transform in x, denoted by û(ξ, t), is readily found to be û(ξ, t) = exp( -itξ ξ 2 + 1 )û(ξ, 0), ξ ∈ R, t ∈ (0, T ).

The consideration of high frequencies is useless here. By analytic continuation, the above equation still holds for all ξ = ξ 1 + iξ 2 ∈ C \ {±i}. Picking any t > 0, ξ 1 = 0 and letting ξ 2 → 1 -, we readily infer that ∂ n ξ û(i, 0) = 0 for all n ≥ 0, so that û(., 0) ≡ 0 and hence u ≡ 0. Note that

∂ n ξ û(i, t) = ∞ -∞ u(x, t)(-ix) n e x dx, (1.9)
and that it can be shown by induction on n that all the moments M n (t) = ∞ -∞ u(x, t)x n e x dx vanish on (0, T ), so that u ≡ 0. Unfortunately, we cannot modify the above argument to deal with the UCP for the full BBM equation, as the nonlinear term has no reason to be perturbative at the "small" frequencies ξ = ±i. We point out that a moment approach, inspired by [START_REF] Constantin | Finite propagation speed for the Camassa-Holm equation[END_REF], was nevertheless applied in [START_REF] Mammeri | Unique continuation property for the KP-BBM-II equation[END_REF] to prove the UCP for some KP-BBM-II equation.

In this paper, we shall apply the moment approach to prove the UCP for a generalized BBM equation u tu txx + [f (u)] x = 0, where f : R → R is smooth and nonnegative. The choice f (u) = u 2 /2 gives the so-called Morrison-Meiss-Carey (MMC) equation (also called equal width wave equation, see [START_REF] Hamdi | Exact solutions and invariants of motion for general types of regularized long wave equations[END_REF][START_REF] Morrison | Scattering of regularized-long-wave solitary waves[END_REF]). Incorporating a localized damping in the above equation, we obtain the equation

u t -u txx + [f (u)] x + a(x)u = 0, x ∈ T,
whose solutions are proved to tend weakly to 0 in H 1 (T) as t → ∞. Note that similar results were proved in [START_REF] Larkin | Dissipative initial boundary value problem for the BBM-equation[END_REF] with a boundary dissipation. Bourgain's approach, in its complex analytic original form, can be used to derive the UCP for the following BBM-like equation

u t -u txx + u x + (u * u) x = 0
in which the (nonlocal) term (u * u) x is substituted to the classical nonlinear term uu x in BBM.

For the original BBM equation (1.1), we shall derive a UCP for solutions issuing from initial data that are small enough in H 1 (T) and with nonnegative mean values. The proof, which is very reminiscent of La Salle invariance principle, will combine the analyticity in time of solutions of BBM, the existence of three invariants of motion, and the use of some appropriate Lyapunov function.

The second part of this work is concerned with the control of the BBM equation. Consider first the linearized BBM equation with a control force

u t -u txx + u x = a(x)h(x, t), (1.10)
where a is supported in some subset of T and h stands for the control input. It was proved in [START_REF] Micu | On the controllability of the linearized Benjamin-Bona-Mahony equation[END_REF][START_REF] Zhang | Unique continuation for the linearized Benjamin-Bona-Mahony equation with spacedependent potential[END_REF] that (1.10) is approximatively controllable in H 1 (T). It turns out that (1.10) is not exactly controllable in H 1 (T) [START_REF] Micu | On the controllability of the linearized Benjamin-Bona-Mahony equation[END_REF]. This is in sharp contrast with the good control properties of other dispersive equations (on periodic domains, see e.g. [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF][START_REF] Russell | Exact controllability and stabilizability of the Korteweg-deVries equation[END_REF] for KdV, [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF][START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF][START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifold of dimension 3[END_REF][START_REF] Rosier | Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval[END_REF][START_REF] Rosier | Control and stabilization of the nonlinear Schrödinger equation on rectangles[END_REF] for the nonlinear Schrödinger equation, [START_REF] Linares | On the controllability and stabilization of the linearized Benjamin-Ono equation[END_REF][START_REF] Linares | Exact controllability and stabilizability of the Benjamin-Ono equation[END_REF] for the Benjamin-Ono equation, [START_REF] Micu | Control and stabilization of a family of Boussinesq systems[END_REF] for Boussinesq system, and [START_REF] Glass | Controllability and asymptotic stabilization of the Camassa-Holm equation[END_REF] for Camassa-Holm equation). The bad control properties of (1.10) come from the existence of a limit point in the spectrum. Such a phenomenon was noticed in [START_REF] Russell | Mathematical models for the elastic beam and their control-theoretic implications[END_REF] for the beam equation with internal damping, in [START_REF] Leugering | Boundary control of a vibrating plate with internal damping[END_REF] for the plate equation with internal damping, in [START_REF] Micu | On the controllability of the linearized Benjamin-Bona-Mahony equation[END_REF] for the linearized BBM equation, and more recently in [START_REF] Rosier | On the controllability of a wave equation with structural damping[END_REF] for the wave equation with structural damping. It is by now classical that an "intermediate" equation between (1.1) and (1.2) can be derived from (1.1) by working in a moving frame x = -ct with c ∈ R \ {0}. Indeed, letting

v(x, t) = u(x -ct, t) (1.11)
we readily see that (1.1) is transformed into the following KdV-BBM equation

v t + (c + 1)v x -cv xxx -v txx + vv x = 0. (1.12)
It is then reasonable to expect the control properties of (1.12) to be better than those of (1.1), thanks to the KdV term -cv xxx in (1.12). We shall prove that the equation (1.12) with a forcing term a(x)k(x, t) supported in (any given) subdomain is locally exactly controllable in H 1 (T) in time T > (2π)/|c|. Going back to the original variables, it means that the equation

u t + u x -u txx + uu x = a(x + ct)h(x, t) (1.13)
with a moving distributed control is exactly controllable in H 1 (T) in (sufficiently) large time.

Actually, the control time is chosen in such a way that the support of the control, which is moving at the constant velocity c, can visit all the domain T. Using the same idea, it has been proved recently in [START_REF] Martin | Null controllability of the structurally damped wave equation with moving point control[END_REF] that the wave equation with structural damping is null controllable in large time when controlled with a moving distributed control. The concept of moving point control was introduced by J. L. Lions in [START_REF] Lions | Pointwise control for distributed systems[END_REF] for the wave equation. One important motivation for this kind of control is that the exact controllability of the wave equation with a pointwise control and Dirichlet boundary conditions fails if the point is a zero of some eigenfunction of the Dirichlet Laplacian, while it holds when the point is moving under some conditions easy to check (see e.g. [START_REF] Castro | Exact controllability of the 1-d wave equation from a moving interior point[END_REF]). The controllability of the wave equation (resp. of the heat equation) with a moving point control was investigated in [START_REF] Castro | Exact controllability of the 1-d wave equation from a moving interior point[END_REF][START_REF] Khapalov | Controllability of the wave equation with moving point control[END_REF][START_REF] Lions | Pointwise control for distributed systems[END_REF] (resp. in [START_REF] Castro | Unique continuation and control for the heat equation from a lower dimensional manifold[END_REF][START_REF] Khapalov | Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation[END_REF]).

Thus, the appearance of the KdV term -cv xxx in (1.12) results in much better control properties. We shall see that (i) there is no limit point in the spectrum of the linearized KdV-BBM equation, which is of "hyperbolic" type; (ii) a UCP for the full KdV-BBM equation can be derived from Carleman estimates for a system of coupled elliptic-hyperbolic equations. It follows that one can expect a semiglobal exponential stability when applying a localized damping with a moving support. We will see that this is indeed the case. Combining the local exact controllability to the semiglobal exponential stability result, we obtain the following theorem which is the main result of the paper.

Theorem 1.1. Assume given a ∈ C ∞ (T) with a = 0 and c ∈ R \ {0}. Let s ≥ 1 and R > 0 be given. Then there exists a time T = T (s, R) > 2π/|c| such that for any u 0 , u T ∈ H s (T) with

||u 0 || H s ≤ R, ||u T || H s ≤ R, (1.14)
there exists a control h ∈ L 2 (0, T ; H s-2 (T)) such that the solution u ∈ C([0, T ]; H s (T)) of

u t -u txx + u x + uu x = a(x + ct)h(x, t), x ∈ T, t ∈ (0, T ) u(x, 0) = u 0 (x), x ∈ T satisfies u(x, T ) = u T (x), x ∈ T.
The paper is scheduled as follows. In Section 2 we recall some useful facts (global wellposedness, invariants of motion, time analyticity) about BBM. In Section 3 we establish the UCP for BBM. In Section 4 we prove the UCP for other BBM-like equations, including the MMC equation and the BBM equation with a nonlocal term. Section 5 is concerned with the UCP for the KdV-BBM equation. The KdV-BBM equation is first split into a coupled system of an elliptic equation and a transport equation. Next, we prove some Carleman estimates with the same singular weights for both the elliptic and the hyperbolic equations, and we derive the UCP for KdV-BBM by combining these Carleman estimates with a regularization process. Those results are used in Section 6 to prove the exact controllability of KdV-BBM and the semiglobal exponential stability of the same equation with a localized damping term.

Wellposedness, analyticity in time and invariants of motion

Throughout the paper, for any s ≥ 0, H s (T) denotes the Sobolev space

H s (T) = {u : T → R; ||u|| H s := ||(1 -∂ 2 x ) s 2 u|| L 2 (T) < ∞}. Its dual is denoted H -s (T).
Let us consider the initial value problem (IVP)

u t -u txx + u x + uu x = 0, x ∈ T, t ∈ R (2.1) u(x, 0) = u 0 (x).
(2.2)

Let A = -(1 -∂ 2 x ) -1 ∂ x ∈ L(H s (T), H s+1 (T)
) (for any s ∈ R) and W (t) = e tA for t ∈ R. We put (2.1)-(2.2) in its integral form

u(t) = W (t)u 0 + t 0 W (t -s)A(u 2 /2)(s)ds.
(2.3)

For s ≥ 0 and T > 0, let X s T = C([-T, T ]; H s (T)). Note that for u ∈ X s T , u solves (2.1) in D ′ (-T, T ; H s-2 (T)) and (2.2) if, and only if, it fulfills (2.3) for all t ∈ [-T, T ]. The following result will be used thereafter.

Theorem 2.1. ( [START_REF] Bona | Sharp well-posedness for the BBM equation[END_REF][START_REF] Roumégoux | A symplectic non-squeezing theorem for BBM equation[END_REF]) Let s ≥ 0, u 0 ∈ H s (T) and T > 0. Then there exists a unique solution u ∈ X s T of (2.1)-(2.2) (or, alternatively, (2.3)). Furthermore, for any R > 0, the map u 0 → u is real analytic from B R (H s (T)) into X s T . Some additional properties are collected in the following Proposition 2.2. For u 0 ∈ H 1 (T), the solution u(t) of the IVP (2.1)-(2.2) satisfies u ∈ C ω (R; H 1 (T)). Moreover the three integral terms T u dx, T (u 2 + u 2

x )dx and T (u 3 + 3u 2 )dx are invariants of motion (i.e., they remain constant over time).

Proof. Let us begin with the invariants of motion. For u 0 ∈ H 1 (T), u ∈ X 1

T for all T > 0, hence

u t = -(1 -∂ 2 x ) -1 ∂ x (u + u 2 2 ) ∈ X 2 T .
Therefore, all the terms in (2.1) belong to X 0 T . Scaling in (2.1) by 1 (resp. by u) yields after some integrations by parts

d dt T u dx = 0 (resp. d dt T (u 2 + u 2 x )dx = 0.)
For the last invariant of motion, we notice (following [START_REF] Olver | Euler operators and conservation laws of the BBM equation[END_REF]) that

( 1 3 (u + 1) 3 ) t -(u 2 t -u 2 xt + (u + 1) 2 u xt - 1 4 (u + 1) 4 ) x = 0.
Integrating on T yields (d/dt) T (u + 1) 3 dx = 0. Since (d/dt) T (3u + 1)dx = 0, we infer that

d dt T (u 3 + 3u 2 )dx = 0.
Let us now prove that u ∈ C ω (R; H 1 (T)). Since u ∈ C 1 (R; H 1 (T)), it is sufficient to check that for any u 0 ∈ H 1 (T) there are some numbers b > 0, M > 0, and some sequence (u n ) n≥1 in H 1 (T) with

||u n || H 1 ≤ M b n , n ≥ 0, (2.4)
such that

u(t) = n≥0 t n u n , t ∈ (-b, b). (2.5)
Note that the convergence of the series in (2.5) holds in H 1 (T) uniformly on [-rb, rb] for each r < 1. Actually, we prove that u can be extended as an analytic function from D b := {z ∈ C; |z| < b} into the space H 1 C (T) := H 1 (T; C), endowed with the Euclidean norm

|| k∈Z ûk e ikx || H 1 = ( k∈Z (1 + |k| 2 )|û k | 2 ) 1 2 .
We adapt the classical proof of the analyticity of the flow for an ODE with an analytic vector field (see e.g. [START_REF] Hochschild | The structure of Lie groups[END_REF]) to our infinite dimensional framework.

For u ∈ H 1 C (T), let Au = -(1 -∂ 2 x ) -1 ∂ x u and f (u) = A(u + u 2 ). Since |k| ≤ (k 2 + 1)/2 for all k ∈ Z, ||A|| L(H 1 C (T)) ≤ 1/2. Pick a positive constant C 1 such that ||u 2 || H 1 ≤ C 1 ||u|| 2 H 1 for all u ∈ H 1 C (T)
. We define by induction on q a sequence (u q ) of analytic functions from C to H 1 C (T) which will converge uniformly on D T , for T > 0 small enough, to a solution of the integral equation

u(z) = u 0 + [0,z] f (u(ζ))dζ = u 0 + 1 0 f (u(sz))zds. Let u 0 (z) = u 0 , for z ∈ C u q+1 (z) = u 0 + [0,z] f (u q (ζ))dζ, for q ≥ 0, z ∈ C.
Claim 1. u q (z) = n≥0 z n v q n for all z ∈ C and some sequence (v q n ) in H 1 C (T) with

||v q n || H 1 ≤ M (q, b) b n for all q, n ∈ N, b > 0.
The proof of Claim 1 is done by induction on q ≥ 0. The result is clear for q = 0 with

M (0, b) = ||u 0 || H 1 , since v 0 0 = u 0 and v 0 n = 0 for n ≥ 1.
Assume Claim 1 proved for some q ≥ 0. Then, for any r ∈ (0, 1) and any b > 0

||z n v q n || H 1 ≤ M (q, b)r n for |z| ≤ rb,
so that the series n≥0 z n v q n converges absolutely in H 1 C (T) uniformly for z ∈ D rb . The same holds true for the series n≥0 z n ( 0≤l≤n v q l v q n-l ). It follows that

f (u q (ζ)) = A   n≥0 ζ n v q n + n≥0 ζ n ( 0≤l≤n v q l v q n-l )   converges uniformly for ζ ∈ D rb . Thus u q+1 (z) = u 0 + [0,z] n≥0 ζ n A(v q n + 0≤l≤n v q l v q n-l )dζ = n≥0 z n v q+1 n where v q+1 0 = u 0 , v q+1 n = 1 n A(v q n-1 + 0≤l≤n-1 v q l v q n-1-l ) for n ≥ 1.
It follows that for n ≥ 1

||v q+1 n || H 1 ≤ ||A|| n ( M (q, b) b n-1 + nC 1 M 2 (q, b) b n-1 ) ≤ M (q + 1, b) b n with M (q + 1, b) := sup{||u 0 || H 1 , b||A||(M (q, b) + C 1 M 2 (q, b)}. Claim 1 is proved. Claim 2. Let T := (2||A||(1 + 4C 1 ||u 0 || H 1 )) -1 . Then ||u q -u|| L ∞ (D T ;H 1 C (T)) → 0 as q → ∞ for some u ∈ C(D T ; H 1 C (T)). Let Z T = C(D T ; H 1 C (T)) be endowed with the norm |||v||| = sup |z|≤T ||v(z)|| H 1 . Let R > 0, and for v ∈ B R := {v ∈ Z T ; |||v||| ≤ R}, let (Γv)(z) = u 0 + [0,z] f (v(ζ)) dζ. Then |||Γv||| ≤ ||u 0 || H 1 + T ||A||(|||v||| + C 1 |||v||| 2 ) ≤ ||u 0 || H 1 + T ||A||(R + C 1 R 2 ), ||Γv 1 -Γv 2 ||| ≤ T ||A||(|||v 1 -v 2 ||| + |||v 2 1 -v 2 2 |||) ≤ T ||A||(1 + 2C 1 R)|||v 1 -v 2 |||. Pick R = 2||u 0 || H 1 and T = (2||A||(1 + 2C 1 R)) -1 . Then Γ contracts in B R .
The sequence (u q ), which is given by Picard iteration scheme, has a limit u in Z T which fulfills

u(z) = u 0 + [0,z] f (u(ζ))dζ, |z| ≤ T.
In particular, u ∈ C 1 ([-T, T ]; H 1 (T)) (the u q (z) being real-valued for z ∈ R) and it satis-

fies u t = f (u) on [-T, T ] together with u(0) = u 0 ; that is, u solves (2.1)-(2.2) in the class C 1 ([-T, T ]; H 1 (T)) ⊂ X 1 T . Claim 3. u(z) = n≥0 z n v n for |z| < T , where v n = lim q→∞ v q n for each n ≥ 0. From Claim 1, we infer that for all n ≥ 1 v q n = 1 2πi |z|=T z -n-1 u q (z) dz, hence ||v p n -v q n || H 1 ≤ T -n |||u p -u q |||. From Claim 2, we infer that (v q n ) is a Cauchy sequence in H 1 C (T). Let v n denote its limit in H 1 C (T). Note that ||v n -v q n || H 1 ≤ T -n |||u -u q |||,
and hence the series n≥0 z n v n is convergent for |z| < T . Therefore, for |z| ≤ rT with r < 1,

|| n≥0 z n (v n -v q n )|| H 1 ≤ (1 -r) -1 |||u -u q |||,
and hence u q (z) = n≥0 z n v q n → n≥0 z n v n in Z rT as q → ∞. It follows that u(z) = n≥0 z n v n for |z| < T.
The proof of Proposition 2.2 is complete.

Unique Continuation Property for BBM

In this section we prove a UCP for the BBM equation for small solutions with nonnegative mean values.

Theorem 3.1. Let u 0 ∈ H 1 (T) be such that T u 0 (x)dx ≥ 0, (3.1
)

and ||u 0 || L ∞ (T) < 3. (3.2)
Assume that the solution u of (2.1)

-(2.2) satisfies u(x, t) = 0 for all (x, t) ∈ ω × (0, T ), (3.3)
where ω ⊂ T is a nonempty open set and T > 0. Then u 0 = 0, and hence u ≡ 0.

Proof. We identify T to (0, 2π) in such a way that ω ⊃ (0, ε)

∪ (2π -ε, 2π) for some ε > 0. Since u ∈ C ω (R; H 1 (T)) by Proposition 2.2, we have that u(x, .) ∈ C ω (R) for all x ∈ T. (3.3) gives then that u(x, t) = 0 for (x, t) ∈ ω × R. (3.4) Introduce the function v(x, t) = x 0 u(y, t)dy. Then v ∈ C ω (R; H 2 (0, 2π)) and v satisfies v t -v txx + v x + u 2 2 = 0, x ∈ (0, 2π), (3.5)
as it may be seen by integrating (2.1) on (0, x). Let

I(t) = 2π 0 v(x, t)dx.
Note that I ∈ C ω (R). Integrating (3.5) on (0, 2π) gives with (3.1)

I t = - 2π 0 u 0 (x)dx - 1 2 2π 0 |u(x, t)| 2 dx ≤ 0. Since ||u(t)|| H 1 = ||u 0 || H 1 for all t ∈ R, v ∈ L ∞ (R, H 2 (0, 2π)) and I ∈ L ∞ (R).
It follows that the function I has a finite limit as t → ∞, that we denote by l. From the boundedness of ||u(t)|| H 1 (T) for t ∈ R, we infer the existence of a sequence t n ր +∞ such that

u(t n ) ⇀ ũ0 in H 1 (T) (3.6)
for some ũ0 ∈ H 1 (T). Let ũ denote the solution of the IVP for BBM corresponding to the initial data ũ0 ; that is, ũ solves

ũt -ũtxx + ũx + ũũ x = 0, x ∈ T, t ∈ R, ũ(x, 0) = ũ0 (x).
Pick any s ∈ (1/2, 1). As u(t n ) → ũ0 strongly in H s (T), we infer from Theorem 2.1 that

u(t n + •) → ũ in C([0, 1]; H s (T)). (3.7)
It follows from (3.4), (3.7) and the fact that ũ

∈ C ω (R, H 1 (T)) that ũ(x, t) = 0 for (x, t) ∈ ω × R.
On the other hand, 2π 0 ũ0 (x)dx = 2π 0 u 0 (x)dx from (3.6) and the invariance of 2π 0 u(x, t)dx. Let ṽ(x, t) = x 0 ũ(y, t)dy and Ĩ(t) = 2π 0 ṽ(x, t)dx. Then we still have that

Ĩt = - 2π 0 u 0 (x)dx - 1 2 2π 0 |ũ(x, t)| 2 dx ≤ 0. (3.8)
But we infer from (3.7) that

I(t n ) → Ĩ(0), I(t n + 1) → Ĩ(1). Since lim n→∞ I(t n ) = lim n→∞ I(t n + 1) = l, we have that Ĩ(0) = Ĩ(1). Combined to (3.8), this yields ũ(x, t) = 0 (x, t) ∈ T × [0, 1].
In particular, ũ0 = 0. From (3.6), we infer that

2π 0 (u 3 (x, t n ) + 3u 2 (x, t n ))dx → 0 as n → ∞.
As 2π 0 (u 3 + 3u 2 )dx is a conserved quantity, we infer that

2π 0 (3 + u 0 (x)) |u 0 (x)| 2 dx = 0, which, combined to (3.2), yields u 0 = 0. Remark 3.2. Note that Theorem 3.1 is false if the assumptions u 0 ∈ H 1 (T) and (3.1) are removed. Indeed, if u ∈ C(R; L 2 (T)) is defined for x ∈ T ∼ (0, 2π) and t ∈ R by u(x, t) = u 0 (x) = -2 if |x -π| ≤ π 2 , 0 if π 2 < |x -π| < π, then (2.1
) and (2.2) are satisfied, although u ≡ 0.

Unique Continuation Property for BBM-like equations

We shall consider BBM-like equations with different nonlinear terms. We first consider a generalized BBM equation without drift term, and next a BBM-like equation with a nonlocal bilinear term. 4.1. Generalized BBM equation without drift term. We consider the following generalized BBM equation

u t -u txx + [f (u)] x = 0, x ∈ T, t ∈ R (4.1) u(x, 0) = u 0 (x), (4.2)
where f ∈ C 1 (R), f (u) ≥ 0 for all u ∈ R, and the only solution u ∈ (-δ, δ) of f (u) = 0 is u = 0, for some number δ > 0. That class of BBM-like equations includes the Morrison-Meiss-Carey equation u tu txx + uu x = 0 for f (u) = u 2 /2. Note that the global wellposedness of (4.1)-(4.2) in H 1 (T) can easily be derived from the contraction mapping theorem and the conservation of the H 1 -norm. It turns out that the UCP can be derived in a straight way and without any additional assumption on the initial data.

Theorem 4.1. Let f be as above, and let ω be a nonempty open set in T. Let u 0 ∈ H 1 (T) be such that the solution u of (4.1)-(4.2) satisfies u(x, t) = 0 for (x, t) ∈ ω × (0, T ) for some T > 0.

Then u 0 = 0.

Proof. Once again, we can assume without loss of generality that ω = (0, ε) ∪ (2πε, 2π). The prolongation of u by 0 on (R \ (0, 2π)) × (0, T ), still denoted by u, satisfies

u t -u txx + [f (u)] x = 0, x ∈ R, t ∈ (0, T ) (4.3) u(x, t) = 0, x ∈ (ε, 2π -ε), t ∈ (0, T ) (4.4) u ∈ C([0, T ]; H 1 (R)), u t ∈ C([0, T ]; H 2 (R)). (4.5)
Scaling in (4.3) by e x yields for t ∈ (0, T )

∞ -∞ f (u(x, t))e x dx = 0, for ∞ -∞ u txx e x dx = ∞ -∞ u t e x dx by two integrations by parts. Since f is nonnegative, this yields f (u(x, t)) = 0 for (x, t) ∈ R × (0, T ).
Since u is continuous and it vanishes for x ∈ (ε, 2πε), we infer from the assumptions about f that u ≡ 0.

Pick any nonnegative function a ∈ C ∞ (T) with ω := {x ∈ T; a(x) > 0} nonempty. We are interested in the stability properties of the system

u t -u txx + [f (u)] x + a(x)u = 0, x ∈ T, t ≥ 0 (4.6) u(x, 0) = u 0 (x), (4.7)
where f is as above. The following weak stability result holds. for all T > 0. Furthermore, u(t) → 0 weakly in H 1 (T), hence strongly in H s (T) for s < 1, as t → +∞.

Proof. The local wellposedness in H s (T) for any s > 1/2 is derived from the contraction mapping theorem in much the same way as for Theorem 2.1. The global wellposedness in H 1 (T) follows at once from the energy identity

||u(T )|| 2 H 1 -||u 0 || 2 H 1 + 2 T 0 T a(x)|u(x, t)| 2 dxdt = 0. (4.8)
obtained by scaling each term in (4.6) by u. On the other hand, still from the application of the contraction mapping theorem, given any s > 1/2, any ρ > 0 and any u 0 , v 0 ∈ H s (T) with

||u 0 || H s (T) ≤ ρ, ||v 0 || H s (T)
≤ ρ, there is some time T = T (s, ρ) > 0 such that the solutions u and v of (4.6)-(4.7) corresponding to the initial data u 0 and v 0 , respectively, fulfill

||u -v|| C([0,T ];H s (T)) ≤ 2||u 0 -v 0 || H s (T) . ( 4
+ •) → v in C([0, T ]; H s (T)) as n → ∞, (4.11) where v = v(x, t) denotes the solution of v t -v txx + [f (v)] x + a(x)v = 0, x ∈ T, t ≥ 0, v(x, 0) = v 0 (x). Note that v ∈ C([0, T ]; H 1 (T)) for v 0 ∈ H 1 (T)). (4.10) combined to (4.11) yields T 0 T a(x)|v(x, t)| 2 dxdt = 0, so that av ≡ 0. By Theorem 4.1, v 0 = 0 and hence, as t → ∞, u(t) → 0 weakly in H 1 (T), u(t) → 0 strongly in H s (T) for s < 1.

4.2.

A BBM-like equation with a nonlocal bilinear term. Here, we consider a BBM-type equation with the drift term, but with a nonlocal bilinear term given by a convolution, namely

u t -u txx + u x + λ(u * u) x = 0, x ∈ R, (4.12)
where λ ∈ R is a constant and

(u * v)(x) = ∞ -∞ u(x -y)v(y)dy for x ∈ R.
A UCP can be derived without any restriction on the initial data.

Theorem 4.3. Assume that λ = 0. Let u ∈ C 1 ([0, T ]; H 1 (R)
) be a solution of (4.12) such that u(x, t) = 0 for |x| > L, t ∈ (0, T ). (4.13)

Then u ≡ 0.

Proof. Taking the Fourier transform of each term in (4.12) yields

(1 + ξ 2 )û t = -iξ(û + λû 2 ), ξ ∈ R, t ∈ (0, T ). (4.14)
Note that, for each t ∈ (0, T ), û(., t) and ût (., t) may be extended to C as entire functions of exponential type at most L. Furthermore, (4.14) is still true for ξ ∈ C and t ∈ (0, T ) by analytic continuation. To prove that u ≡ 0, it is sufficient to check that Derivating k times with respect to ξ in (4.14) yields

∂ k ξ û(i, t) = 0 ∀k ∈ N, ∀t ∈ (0, T ). ( 4
(1 + ξ 2 )û (k) t + 2kξ û(k-1) t + k(k -1)û (k-2) t = -iξ û(k) + λ k l=0 C l k û(l) û(k-l) -ik û(k-1) + λ k-1 l=0 C l k-1 û(l) û(k-1-l) . ( 4
(ξ, t) = -λ -1 ∀ξ ∈ C,
which contradicts the fact that û(., t) ∈ L 2 (R). Thus (4.16) holds and u ≡ 0.

Unique continuation property for the KdV-BBM equation

In this section we prove some UCP for the following KdV-BBM equation

u t -u txx -cu xxx + qu x = 0, x ∈ T, t ∈ (0, T ), (5.1)
where q ∈ L ∞ (0, T ; L ∞ (T)) is a given potential function and c = 0 is a given real constant.

The UCP obtained here will be used in the next section to obtain a semiglobal exponential stabilization result for BBM with a moving damping.

Theorem 5.1. Let c ∈ R\{0}, T > 2π/|c|, and q ∈ L ∞ (0, T ; L ∞ (T)). Let ω ⊂ T be a nonempty open set. Assume that u ∈ L 2 (0, T ; H 2 (T)) satisfies (5.1) and u(x, t) = 0 for a.e. (x, t) ∈ ω × (0, T ).

(5.2)

Then u ≡ 0.

Proof. Let w = uu xx ∈ L 2 (0, T ; L 2 (T)). Then (u, w) solves the following system

u -u xx = w, (5.3)
w t + cw x = (c -q)u x .
(5.4)

Note that, by (5.2), u = w = 0 a.e. on ω × (0, T ).

(5.5)

Inspired in part by [START_REF] Albano | Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system[END_REF] (which was concerned with a heat-wave system 1 ), we shall establish some Carleman estimates for the elliptic equation ( 5.3) and the transport equation (5.4) with the same singular weight. Introduce a few notations. We identify T with (0, 2π). Without loss of generality, we can assume that c > 0 (the case c < 0 being similar), and that ω = (2π-2η, 2π+η) for some η > 0. Let ω 0 = (2πη, 2π) ⊂ ω. Pick a time T > 2π/c, and some positive numbers δ, ε and ρ < 1 such that

ρT c -2ρδc -2π + η -ε > 0. (5.6)
Pick finally a function g ∈ C ∞ (0, T ) such that

g(t) =    1 t for 0 < t < δ/2, strictly decreasing for 0 < t ≤ δ, 1 for δ ≤ t < T. Let ψ ∈ C ∞ (T × [0, T ]) (i.e. ψ is C ∞ smooth in (x, t) and ψ(., t) is 2π-periodic in x for all t ∈ [0, T ]) with ψ(x, t) = (x + ε) 2 -ρc 2 (t -2δ) 2 for x ∈ [0, 2π -η], t ∈ [0, T ]. Let finally ϕ(x, t) = g(t)(2e ||ψ|| L ∞ -e ψ(x,t) ), (x, t) ∈ T × (0, T ], θ(x, t) = g(t)e ψ(x,t) , (x, t) ∈ T × (0, T ],
where ||ψ|| L ∞ = ||ψ|| L ∞ (T×(0,T )) . The proof of Theorem 5.1 is outlined as follows. In the first step, we prove a Carleman estimate for the elliptic equation (5.3) with the time-varying weight ϕ. In the second step, we prove a Carleman estimate for the transport equation (5.4) with the same weight. In the last step, we combine the two above Carleman estimates into a single one for (5.1) and derive the UCP.

Step 1. Carleman estimate for the elliptic equation.

1 See also [START_REF] Ervedoza | Local exact controllability for the 1-D compressible Navier-Stokes equation[END_REF] for some Carleman estimates for a coupled system of parabolic-hyperbolic equations.

Lemma 5.2. There exist s 0 ≥ 1 and C 0 > 0 such that for all s ≥ s 0 and all u ∈ L 2 (0, T ; H 2 (T)), the following holds

T 0 T [(sθ)|u x | 2 + (sθ) 3 |u| 2 ]e -2sϕ dxdt ≤ C 0 T 0 T |u xx | 2 e -2sϕ dxdt + T 0 ω
(sθ) 3 |u| 2 e -2sϕ dxdt . (5.7)

Remark 5.3. The same Carleman estimate as above with terms integrated over T only is also valid, but with some constants C 0 and s 0 that could a priori depend on t. The above formulation was preferred for the sake of clarity.

Proof of Lemma 5.2: Let v = e -sϕ u and P = ∂ 2

x . Then e -sϕ P u = e -sϕ P (e sϕ v) = P s v + P a v where

P s v = (sϕ x ) 2 v + v xx , (5.8)
P a v = 2sϕ x v x + sϕ xx v (5.9)
denote the (formal) selfadjoint and skeweadjoint parts of e -sϕ P (e sϕ •). It follows that

||e -sϕ P u|| 2 = ||P s v|| 2 + ||P a v|| 2 + 2(P s v, P a v) where (f, g) = T 0 T f gdxdt, ||f || 2 = (f, f ). In the sequel, T 0 T f (x, t)dxdt is denoted f , for the sake of shortness. Then (P s v, P a v) = (sϕ x ) 2 v, 2sϕ x v x + (sϕ x ) 2 v, sϕ xx v + v xx , 2sϕ x v x + (v xx , sϕ xx v) =: I 1 + I 2 + I 3 + I 4 .
After some integrations by parts in x, we obtain that

I 1 = -3 (sϕ x ) 2 sϕ xx v 2 I 3 = - sϕ xx v 2 x I 4 = - v x (sϕ xxx v + sϕ xx v x ) = sϕ xxxx v 2 2 - sϕ xx v 2 x .
Therefore

||e -sϕ P u|| 2 = ||P s v|| 2 + ||P a v|| 2 + [-4(sϕ x ) 2 sϕ xx + sϕ xxxx ]v 2 + (-4sϕ xx )v 2 x .
We notice that

ϕ x = -gψ x e ψ , ϕ xx = -g[(ψ x ) 2 + ψ xx ]e ψ , hence there exist some numbers s 0 ≥ 1, C > 0 and C ′ > 0 such that for all s ≥ s 0 -4(sϕ x ) 2 sϕ xx + sϕ xxxx ≥ C(sg) 3 for (x, t) ∈ (0, 2π -η) × (0, T ), -4sϕ xx ≥ Csg for (x, t) ∈ (0, 2π -η) × (0, T ), while | -4(sϕ x ) 2 sϕ xx + sϕ xxxx | ≤ C ′ (sg) 3 for (x, t) ∈ (2π -η, 2π) × (0, T ), |4sϕ xx | ≤ C ′ sg for (x, t) ∈ (2π -η, 2π) × (0, T ).
We conclude that for s ≥ s 0 and some constant C 0 > 0

||P s v|| 2 + [sg|v x | 2 + (sg) 3 |v| 2 ] ≤ C 0 ||e -sϕ P u|| 2 + T 0 ω 0 [sg|v x | 2 + (sg) 3 |v| 2 ] . (5.10)
Next we show that (sg) -1 |v xx | 2 is also less than the r.h.s. of (5.10). We have

(sg) -1 |v xx | 2 ≤ (sg) -1 |P s v -(sϕ x ) 2 v| 2 ≤ 2 (sg) -1 |P s v| 2 + |sϕ x | 4 |v| 2 ≤ C s -1 ||P s v|| 2 + (sg) 3 |v| 2 .
Combined to (5.10), this gives

{(sg) -1 |v xx | 2 + (sg)|v x | 2 + (sg) 3 |v| 2 } ≤ C ||e -sϕ P u|| 2 + T 0 ω 0 (sg) 3 |v| 2 + T 0 ω 0 sg|v x | 2 (5.11)
where C does not depend on s and v. Finally, we show that we can drop the last term in the r.h.s. of (5.11). Let ξ ∈ C ∞ 0 (ω) with 0 ≤ ξ ≤ 1 and ξ(x) = 1 for x ∈ ω 0 . Then

T 0 ω 0 g|v x | 2 ≤ T 0 ω gξ|v x | 2 ≤ - T 0 ω g(ξ x v x + ξv xx )v ≤ 1 2 T 0 ω gξ xx v 2 - T 0 ω gξv xx v so that 2 T 0 ω 0 sg|v x | 2 ≤ ||ξ xx || L ∞ (T) T 0 ω (sg)|v| 2 + κ T 0 ω (sg) -1 |v xx | 2 + κ -1 T 0 ω (sg) 3 |v| 2 (5.12)
where κ > 0 is a constant that can be chosen as small as desired. Combining (5.11) and (5.12) with κ small enough gives for s ≥ s 0 (with a possibly increased value of s 0 ) and some constant C (that does not depend on s and v)

{(sg) -1 |v xx | 2 + (sg)|v x | 2 + (sg) 3 |v| 2 } ≤ C ||e -sϕ P u|| 2 + T 0 ω (sg) 3 |v| 2 .
(5.13)

Replacing v by e -sϕ u in (5.13) gives at once (5.7). The proof of Lemma 5.2 is complete.

Step 2. Carleman estimate for the transport equation. The functions g, ψ, ϕ and θ are the same as those in Lemma 5.2.

Lemma 5.4. There exist s 1 ≥ s 0 and C 1 > 0 such that for all s ≥ s 1 and all w ∈ H 1 (T×(0, T )), the following holds

T 0 T (sθ)|w| 2 e -2sϕ dxdt ≤ C 1 T 0 T |w t + cw x | 2 e -2sϕ dxdt + T 0 ω
(sθ) 2 |w| 2 e -2sϕ dxdt .

(5.14)

Proof of Lemma 5.4: The proof is divided into two parts corresponding to the estimates for t ∈ [0, δ] and for t ∈ [δ, T ]. The main result in each part is stated in a claim. Let v = e -sϕ w and P = ∂ t + c∂ x . Then e -sϕ P w = e -sϕ P (e sϕ v)

= (sϕ t v + csϕ x v) + (v t + cv x ) =: P s v + P a v. Claim 4. δ 0 T (sθ) 2 |v| 2 dxdt ≤ C δ 0 T |e -sϕ P w| 2 dxdt + T (1 -ξ) 2 (sθ)|v| 2 |t=δ dx + δ 0 ω (sθ) 2 |v| 2 dxdt . (5.15)
To prove the claim, we compute in several ways

I = δ 0 T (e -sϕ P w)(1 -ξ) 2 sθv dxdt.
We split I into

I = δ 0 T (P s v)(1 -ξ) 2 sθv dxdt + δ 0 T (P a v)(1 -ξ) 2 sθv dxdt =: I 1 + I 2 .
Then

I 1 = δ 0 T (ϕ t + cϕ x )(1 -ξ) 2 s 2 θv 2 dxdt = δ 0 T [g ′ (2e ||ψ|| L ∞ -e ψ ) -g(ψ t + cψ x )e ψ ](1 -ξ) 2 s 2 ge ψ v 2 dxdt.
On the other hand

I 2 = δ 0 T (v t + cv x )(1 -ξ) 2 (sge ψ v) dxdt = - δ 0 T s[g ′ e ψ + g(ψ t + cψ x )e ψ ](1 -ξ) 2 v 2 2 dxdt + 1 2 T (1 -ξ) 2 sge ψ |v| 2 |t=δ dx + δ 0 T csξ x (1 -ξ)ge ψ v 2 dxdt
where we used the fact that e -sϕ = O(e -C/t ) as t → 0 + for some constant C > 0. Note that for x ∈ [0, 2πη] and t ∈ (0, δ)

ψ t + cψ x = 2c(x + ε) -2ρc 2 (t -2δ) > 2c(ε + ρcδ) > 0 while g ′ (t) ≤ 0 and g(t) ≥ 1. Thus, for s ≥ s 1 ≥ s 0 , g(ψ t + cψ x )e ψ (s 2 ge ψ + s 2 ) ≥ 2c(ε + ρcδ)(sg) 2 e 2ψ , x ∈ T \ ω, t ∈ (0, δ) -g ′ (t) (2e ψ L ∞ -e ψ )s 2 ge ψ - s 2 e ψ ≥ 0 x ∈ T, t ∈ (0, δ).
It follows that for some positive constants

C, C ′ C δ 0 T (sθ) 2 |v| 2 dxdt ≤ -I + 1 2 T (1 -ξ) 2 sge ψ |v| 2 |t=δ dx + C ′ δ 0 ω (sθ) 2 |v| 2 dxdt. (5.16)
On the other hand, by Cauchy-Schwarz inequality, we have for any κ > 0

|I| ≤ κ -1 δ 0 T |e -sϕ P w| 2 dxdt + κ δ 0 T (sθ) 2 |v| 2 dxdt.
(5.17)

Combining (5.16) with (5.17) gives (5.15) for κ > 0 small enough. Claim 4 is proved. Claim 5.

T δ T (sθ)|v| 2 dxdt + T (1 -ξ) 2 (sθ)|v| 2 |t=δ dx + T (1 -ξ) 2 (sθ)|v| 2 |t=T dx ≤ C T δ T |e -sϕ P w| 2 dxdt + T δ ω (sθ)|v| 2 dxdt . (5.18)
• and (., .) denoting here the Euclidean norm and scalar product in L 2 (T × (δ, T )), we have that ||e -sϕ P w|| 2 ≥ ||P s v + P a v|| 2 ≥ ||(1ξ)(P s v + P a v)|| 2 ≥ 2((1ξ)P s v, (1ξ)P a v).

(5.19)

Next we compute

((1 -ξ)P s v, (1 -ξ)P a v) = T δ T (1 -ξ) 2 s(ϕ t + cϕ x )v(v t + cv x ) dxdt = - s 2 T δ T (1 -ξ) 2 (ϕ tt + 2cϕ xt + c 2 ϕ xx )v 2 dxdt + T (1 -ξ) 2 s(ϕ t + cϕ x ) v 2 2 dx T δ + T δ T cξ x (1 -ξ)s(ϕ t + cϕ x )v 2 dxdt. (5.20)
Recall that ξ ∈ C ∞ 0 (ω) with 0 ≤ ξ ≤ 1 and ξ(x) = 1 for x ∈ ω 0 , and that g

(t) = 1 for δ ≤ t ≤ T , so that ϕ(x, t) = 2e ||ψ|| L ∞ -e ψ(x,t) for x ∈ T, t ∈ [δ, T ]. We have that ϕ t + cϕ x = -(ψ t + cψ x )e ψ = -2c(x + ε -ρc(t -2δ))e ψ for x ∈ [0, 2π -η], t ∈ [δ, T ]. For t = δ -s(ϕ t + cϕ x )(x, δ) > 2c(ε + ρcδ)se ψ > 0 for x ∈ (0, 2π -η),
while for t = T , by (5.6),

s(ϕ t + cϕ x )(x, T ) > 2c(ρT c -2ρδc -2π + η -ε)se ψ > 0 for x ∈ (0, 2π -η). Therefore T (1 -ξ) 2 s(ϕ t + cϕ x ) v 2 2 T δ ≥ C T (1 -ξ) 2 sθ|v| 2 |t=δ dx + T (1 -ξ) 2 sθ|v| 2 |t=T dx .
(5.21)

Next we compute

ϕ tt + 2cϕ xt + c 2 ϕ xx = -{(ψ t + cψ x ) 2 + (ψ tt + c 2 ψ xx )}e ψ ≤ 2(ρ -1)c 2 e ψ for x ∈ (0, 2π -η), which yields - s 2 T δ T (1 -ξ) 2 (ϕ tt + 2cϕ xt + c 2 ϕ xx )|v| 2 dxdt ≥ |1 -ρ|c 2 T δ T (1 -ξ) 2 sθ|v| 2 dxdt. (5.22)
Claim 5 follows from (5.19)- (5.22). We infer from Claim 4 and Claim 5 that for some constants s 1 ≥ s 0 and C 1 > 0 we have for all s ≥ s 1

T 0 T (sθ)|v| 2 dxdt ≤ C 1 T 0 T |e -sϕ P w| 2 dxdt + T 0 ω (sθ) 2 |v| 2 dxdt .
(5.23)

Replacing v by e -sϕ w in (5.23) gives at once (5.14). The proof of Lemma 5.4 is complete.

Step 3. We would like to apply Lemma 5.2 to u and Lemma 5.4 to w = u-u xx ∈ L 2 (0, T ; L 2 (T)), which has not the regularity required. Note, however, that (5.14) is still true when w and f := w t + cw x are in L 2 (0, T ; L 2 (T)). Indeed, in that case w ∈ C([0, T ]; L 2 (T)), and if (w n 0 ) and (f n ) are two sequences in H 1 (T) and L 2 (0, T ; H 1 (T)) respectively, such that

w n 0 → w(0) in L 2 (T), f n → f in L 2 (0, T ; L 2 (T)), then the solution w n ∈ C([0, T ]; H 1 (T)) of w n t + cw n x = f n , w n (0) = w n 0 satisfies w n ∈ H 1 (T × (0, T )) and w n → w in C([0, T ]; L 2 (T)
), so that we can apply (5.14) to w n and next pass to the limit n → ∞ in (5.14).

Here, u ∈ L 2 (0, T ; H 2 (T)), w ∈ L 2 (0, T ; L 2 (T)) and w t + cw x = (cq)u x ∈ L 2 (0, T ; L 2 (T)). Combining (5.3), (5.4), (5.5), (5.7), and (5.14), we obtain for s ≥ s 1 that

T 0 T [(sθ)|u x | 2 + (sθ) 3 |u| 2 + (sθ)|w| 2 ]e -2sϕ dxdt ≤ C T 0 T [|u| 2 + |w| 2 + |(c -q)u x | 2 ]e -2sϕ dxdt. (5.24)
We conclude that u = w = 0 on T × (0, T ) by choosing s ≥ s 1 large enough.

Corollary 5.5. The same conclusion as in Theorem 5.1 holds when u ∈ L 2 (0, T ;

H 2 (T)) is replaced by u ∈ L ∞ (0, T ; H 1 (T)).
Proof. We proceed as in [START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF]. Since u w := uu xx are not regular enough to apply Lemmas 5.2 and 5.4, we smooth them by using some convolution in time. For any function v = v(x, t) and any number h > 0, we set

v [h] (x, t) = 1 h t+h t v(x, s) ds.
Recall that if v ∈ L p (0, T ; V ), where 1 ≤ p ≤ +∞ and V denotes any Banach space, then

v [h] ∈ W 1,p (0, T -h; V ), ||v [h] || L p (0,T -h;V ) ≤ ||v|| L p (0,T ;V )
, and for p < ∞ and

T ′ < T v [h] → v in L p (0, T ′ ; V ) as h → 0.
In the sequel, v

[h] t denotes (v [h] ) t , v [h]
x denotes (v [h] ) x , etc. Pick any T ′ ∈ ( 2π |c| , T ) such that (5.6) still holds with T replaced by T ′ , and define the functions g, ψ, ϕ, and θ as above, but with T replaced by T ′ . Then for any positive number h < h 0 = T -T ′ , u [h] ∈ W 1,∞ (0, T ′ ; H 1 (T)), and it solves

u [h] t -u [h]
txxcu [h] xxx + (qu x ) [h] = 0 in D ′ (0, T ′ ; H -2 (T)), (5.25)

u [h] (x, t) = 0 (x, t) ∈ ω × (0, T ′ ).
(5.26)

From (5.25), we infer that

u [h] xxx = c -1 u [h] t -u [h] txx + (qu x ) [h] ∈ L ∞ (0, T ′ ; H -1 (T)), hence u [h] ∈ L ∞ (0, T ′ ; H 2 (T)). (5.27)
This yields, with (5.3)-(5.4),

w [h] = u [h] -u [h] xx ∈ L ∞ (0, T ′ ; L 2 (T)), (5.28) w [h] t + cw [h] x = (c -q)u x [h] ∈ W 1,∞ (0, T ; L 2 (T)).
(5.29)

From (5.27), (5.28), (5.29) and Lemmas 5.2 and 5.4, we infer that there exist some constants s 1 > 0 and C 1 > 0 such that for all s ≥ s 1 and all h ∈ (0, h 0 ), we have

T ′ 0 T (sθ)|u [h] x | 2 + (sθ) 3 |u [h] | 2 + (sθ)|w [h] | 2 e -2sϕ dxdt ≤ C 1 T ′ 0 T |u [h] | 2 + |w [h] | 2 + |((c -q)u x ) [h] | 2 e -2sϕ dxdt ≤ C 1 T ′ 0 T |u [h] | 2 + |w [h] | 2 + 2|(c -q)u [h] x | 2 + 2|((c -q)u x ) [h] -(c -q)u [h]
x | 2 e -2sϕ dxdt. (5.30)

Comparing the powers of s in (5.30), we obtain that for s ≥ s 2 > s 1 , h ∈ (0, h 0 ) and some constant C 2 > C 1 (that does not depend on s, h)

T ′ 0 T (sθ)|u [h] x | 2 + (sθ) 3 |u [h] | 2 + (sθ)|w [h] | 2 e -2sϕ dxdt ≤ C 2 T ′ 0 T |((c -q)u x ) [h] -(c -q)u [h]
x | 2 e -2sϕ dxdt.

Fix s to the value s 2 , and let h → 0. We claim that

T ′ 0 T |((c -q)u x ) [h] -(c -q)u [h] x | 2 e -2s 2 ϕ → 0 as h → 0.
Indeed, as h → 0,

((c -q)u x ) [h] → (c -q)u x in L 2 (0, T ′ ; L 2 (T)), (c -q)u [h] x → (c -q)u x in L 2 (0, T ′ ; L 2 (T)),
while e -2s 2 ϕ ≤ 1. Therefore,

T ′ 0 T θ 3 |u [h] | 2 e -2s 2 ϕ dxdt → 0 as h → 0.
On the other hand, u [h] → u in L 2 (0, T ′ ; L 2 (T)) and θ 3 e -2s 2 ϕ is bounded on T × (0, T ′ ), so that

T ′ 0 T θ 3 |u [h] | 2 e -2s 2 ϕ dxdt → T ′ 0 T θ 3 |u| 2 e -2s 2 ϕ dxdt
as h → 0. We conclude that u ≡ 0 in T × (0, T ′ ). As T ′ may be taken arbitrarily close to T , we infer that u ≡ 0 in T × (0, T ), as desired.

Control and Stabilization of the KdV-BBM equation

In this section we are concerned with the control properties of the system Theorem 6.1. Let a ∈ C ∞ (T) with a = 0, s ≥ 0 and T > 2π/|c|. Then there exists a δ > 0 such that for any u 0 , u T ∈ H s (T) with

u t -u txx -cu xxx + (c + 1)u x + uu x = a(x)h, x ∈ T, t ≥ 0, (6.1) u(x, 0) = u 0 (x), ( 6
||u 0 || H s + ||u T || H s < δ,
one can find a control input h ∈ L 2 (0, T ; H s-2 (T)) such that the system (6.1)-( 6.2) admits a unique solution

u ∈ C([0, T ], H s (T)) satisfying u(•, T ) = u T .
Proof. The result is first proved for the linearized equation, and next extended to the nonlinear one by a fixed-point argument.

Step 1. Exact controllability of the linearized system We first consider the exact controllability of the linearized system

u t -u txx -cu xxx + (c + 1)u x = a(x)h, (6.4) u(x, 0) = u 0 (x), (6.5)
in H s (T) for any s ∈ R.

Let A = (1 -∂ 2 x ) -1 (c∂ 3 x -(c + 1)∂ x ) with domain D(A) = H s+1 (T) ⊂ H s (T). The operator A generates a group of isometries {W (t)} t∈R in H s (T), with W (t)v = ∞ k=-∞ e -it ck 3 +(c+1)k k 2 +1 vk e ikx for any v = ∞ k=-∞ vk e ikx ∈ H s (T).
The system (6.4)-(6.5) may be cast into the following integral form

u(t) = W (t)u 0 + t 0 W (t -τ )(1 -∂ 2 x ) -1 [a(x)h(τ )]dτ.
We proceed as in [START_REF] Micu | Control and stabilization of a family of Boussinesq systems[END_REF]. Take h(x, t) in (6.4) to have the following form

h(x, t) = a(x) ∞ j=-∞ f j q j (t)e ijx (6.6)
where f j and q j (t) are to be determined later. Then the solution u of the equation (6.4) can be written as

u(x, t) = ∞ k=-∞ ûk (t)e ikx with ûk (t) solves d dt ûk (t) + ikσ(k)û k (t) = 1 1 + k 2 ∞ j=-∞ f j q j (t)m j,k (6.7)
where σ(k) = ck 2 +c+1 1+k 2 , and

m j,k = 1 2π T a 2 (x)e i(j-k)x dx. Thus ûk (T ) -e -ikσ(k)T ûk (0) = 1 1 + k 2 ∞ j=-∞ f j m j,k T 0 e -ikσ(k)(T -τ ) q j (τ )dτ or ûk (T )e ikσ(k)T -ûk (0) = 1 1 + k 2 ∞ j=-∞ f j m j,k T 0 e ikσ(k)τ q j (τ )dτ.
It may occur that the eigenvalues

λ k = ikσ(k), k ∈ Z
are not all different. If we count only the distinct values, we obtain the sequence (λ k ) k∈I , where

I ⊂ Z has the property that λ k 1 = λ k 2 for any k 1 , k 2 ∈ I with k 1 = k 2 . For each k 1 ∈ Z set I(k 1 ) = {k ∈ Z; kσ(k) = k 1 σ(k 1 )} and m(k 1 ) = |I(k 1 )
| (the number of elements in I(k 1 )). Clearly, there exists some integer k * such that k ∈ I if |k| > k * . Thus there are only finite many integers in I, say k j , j = 1, ..., n, such that one can find another integer k = k j with λ k = λ k j . Let

I j = {k ∈ Z; k = k j , λ k = λ k j }, j = 1, 2, ..., n. Then Z = I ∪ I 1 ∪ ... ∪ I n .
Note that I j contains at most two integers, for m(k j ) ≤ 3. We write

I j = {k j,1 , k j,m(k j )-1 } j = 1, 2, ..., n
and rewrite k j as k j,0 . Let p k (t) := e -ikσ(k)t , k = 0, ±1, ±2, ...

Then the set

P := {p k (t); k ∈ I} forms a Riesz basis for its closed span, P T , in L 2 (0, T ) if

T > 2π |c| •
Let L := {q j (t); j ∈ I} be the unique dual Riesz basis for P in P T ; that is, the functions in L are the unique elements of P T such that

T 0 q j (t)p k (t)dt = δ kj , j, k ∈ I.
In addition, we choose

q k = q k j if k ∈ I j .
For such choice of q j (t), we have then, for any k ∈ Z,

ûk (T )e ikσ(k)T -ûk (0) = 1 1 + k 2 f k m k,k if k ∈ I \ {k 1 , ..., k n }; (6.8) ûk j,q (T )e ik j σ(k j )T -û k j,q (0) = 1 1 + k 2 j,q m(k j )-1 l=0 f k j,l m k j,
l ,k j,q if k = k j,q , j = 1, ..., n, q = 0, ..., m(k j )-1. (6.9) It is well known that for any finite set J ⊂ Z, the Gram matrix A J = (m p,q ) p,q∈J is definite positive, hence invertible. It follows that the system (6.8)-(6.9) admits a unique solution f

(..., f -2 , f -1 , f 0 , f 1 , f 2 , ...). Since m k,k = 1 2π T a 2 (x)dx =: µ = 0,
we have that

f k = 1 + k 2 µ ûk (T )e ikσ(k)T -ûk (0) for |k| > k * Note that ||h|| 2 L 2 (0,T ;H s-2 (T)) = T 0 ||a(x) ∞ j=-∞ f j q j (t)e ijx || 2 H s-2 dt ≤ C T 0 ∞ j=-∞ (1 + j 2 ) s-2 |f j q j (t)| 2 ≤ C ∞ j=-∞ (1 + j 2 ) s-2 |f j | 2 ≤ C ||u(0)|| 2 H s + ||u(T )|| 2 H s .
This analysis leads us to the following controllability result for the linear system (6.4)-(6.5). Proposition 6.2. Let s ∈ R and T > 2π |c| be given. For any u 0 , u T ∈ H s (T), there exists a control h ∈ L 2 (0, T ; H s-2 (T)) such that the system (6.4)-(6.5) admits a unique solution u ∈ C([0, T ]; H s (T)) satisfying u(x, T ) = u T (x).

Moreover, there exists a constant C > 0 depending only on s and T such that

h L 2 (0,T ;H s-2 (T)) ≤ C ( u 0 H s + u T H s ) .
Introduce the (bounded) operator Φ :

H s (T) × H s (T) → L 2 (0, T ; H s-2 (T)) defined by Φ(u 0 , u T )(t) = h(t),
where h is given by (6.6) and f is the solution of (6.8)-(6.9) with ( u 0 ) k and ( u T ) k substitued to ûk (0) and ûk (T ), respectively. Then h = Φ(u 0 , u T ) is a control driving the solution u of (6.4)-(6.5) from u 0 at t = 0 to u T at t = T .

Step 2. Local exact controllability of the BBM equation.

We proceed as in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Pick any time T > 2π/|c|, and any u 0 , u T ∈ H s (T) (s ≥ 0) satisfying

u 0 H s ≤ δ, u T H s ≤ δ
with δ to be determined. For any u ∈ C([0, T ]; H s (T)), we set

ω(u) = - T 0 W (T -τ )(1 -∂ 2 x ) -1 (uu x )(τ ) dτ.
Then

||ω(u) -ω(v)|| H s ≤ CT ||u + v|| L ∞ (0,T ;H s (T)) ||u -v|| L ∞ (0,T ;H s (T)) .
Furthermore,

W (t)u 0 + t 0 W (t -τ )(1 -∂ 2 x ) -1 [a(x)Φ(u 0 , u T -ω(u)) -uu x ](τ )dτ = u 0 if t = 0, ω(u) + (u T -ω(u)) = u T if t = T.
We are led to consider the nonlinear map

Γ(u) = W (t)u 0 + t 0 W (t -τ )(1 -∂ 2 x ) -1 [a(x)Φ(u 0 , u T -ω(u)) -uu x ](τ ) dτ.
The proof of Theorem 6.1 will be complete if we can show that the map Γ has a fixed point in some closed ball of the space C([0, T ]; H s (T)). For any R > 0, let

B R = {u ∈ C([0, T ]; H s (T)); ||u|| C([0,T ];H s (T)) ≤ R}.
From the above calculations, we see that there exist two positive constants C

(T)) ≤ C 1 ||u 0 || H s + ||u T || H s + C 2 R 2 , ||Γ(u) -Γ(v)|| C([0,T ];H s (T)) ≤ C 2 R||u -v|| C([0,T ];H s (T)) .
Picking R = (2C 2 ) -1 and δ = (8C 1 C 2 ) -1 , we obtain for u 0 , u T satisfying

u 0 H s ≤ δ, u T H s ≤ δ and u, v ∈ B R that ||Γ(u)|| C([0,T ];H s (T)) ≤ R (6.10) ||Γ(u) -Γ(v)|| C([0,T ];H s (T)) ≤ 1 2 ||u -v|| C([0,T ];H s (T)) . (6.11)
It follows from the contraction mapping theorem that Γ has a unique fixed point u in B R . Then u satisfies (6.1)-(6.2) with h = Φ(u 0 , u Tω(u)) and u(T ) = u T , as desired. The proof of Theorem 6.1 is complete.

6.2. Exponential stabilizability. We are now concerned with the stabilization of (6.1)-(6.2) with a feedback law h = h(u). To guess the expression of h, it is convenient to write the linearized system (6.4)-(6.5) as u t = Au + Bk, (6.12) u(0) = u 0 (6.13) where

k(t) = (1 -∂ 2 x ) -1 h(t) ∈ L 2 (0, T ; H s (T))
is the new control input, and

B = (1 -∂ 2 x ) -1 a(1 -∂ 2 x ) ∈ L(H s (T)). (6.14)
We already noticed that A is skew-adjoint in H s (T), and that (6.12)-(6.13) is exactly controllable in H s (T) (with some control functions k ∈ L 2 (0, T ; H s (T))) for any s ≥ 0. If we choose the simple feedback law k = -B * ,s u, (6.15) the resulting closed-loop system u t = Au -BB * ,s u, (6.16) u(0) = u 0 (6.17) is exponentially stable in H s (T) (see e.g. [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF][START_REF] Rosier | A survey of controllability and stabilization results for partial differential equations[END_REF].) In (6.15), B * ,s denotes the adjoint of B in L(H s (T)). Easy computations show that

B * ,s u = (1 -∂ 2 x ) 1-s a(1 -∂ 2 x ) s-1 u. (6.18)
In particular

B * ,1 u = au. Let à = A -BB * ,1 , where (BB * ,1 )u = (1 -∂ 2 x ) -1 [a(1 -∂ 2 x )(au)]. Since BB * ,1 ∈ L(H s (T)
) and A is skew-adjoint in H s (T), Ã is the infinitesimal generator of a group {W a (t)} t∈R on H s (T) (see e.g. [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 1.1 p. 76]). We first show that the closed-loop system (6.16)-(6.17) is exponentially stable in H s (T) for all s ≥ 1. Lemma 6.3. Let a ∈ C ∞ (T) with a = 0. Then there exists a constant γ > 0 such that for any s ≥ 1, one can find a constant C s > 0 for which the following holds for all u 0 ∈ H s (T) W a (t)u 0 H s ≤ C s e -γt u 0 H s for all t ≥ 0. (6.19)

Proof. (6.19) is well known for s = 1 (see e.g. [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]). Assume that it is true for some s ∈ N * , and pick any u 0 ∈ H s+1 (T). Let v 0 = Ãu 0 ∈ H s (T). Then Plugging the feedback law k = -B * ,1 u = -au in the nonlinear equation gives the following closed-loop system u tu txxcu xxx + (c + 1)u x + uu x = -a(1 -∂ 2

||W a (t)v 0 || H s ≤ C s e -γt ||v 0 || H s . Clearly, W a (t)v 0 = ÃW a (t)u 0 = AW a (t)u 0 -BB * ,1 W a (t)u 0 , hence ||AW a (t)u 0 || H s ≤ ||W a (t)v 0 || H s + ||BB * ,1 || L(H s ) ||W a (t)u 0 || H s ≤ Ce -γt ||u 0 || H s+1 • Therefore ||W a (t)u 0 || H s+1 ≤ C s+1 e -
x )[au], (6.20) u(x, 0) = u 0 (x). (6.21) We first show that the system (6.20)-(6.21) is globally well-posed in the space H s (T) for any s ≥ 0.

Theorem 6.4. Let s ≥ 0 and T > 0 be given. For any u 0 ∈ H s (T), the system (6.20)-(6.21) admits a unique solution u ∈ C([0, T ]; H s (T)).

The following bilinear estimate from [START_REF] Roumégoux | A symplectic non-squeezing theorem for BBM equation[END_REF] will be very helpful. Theorem 6.4: Step 1 : The system is locally well-posed in the space H s (T):

Lemma 6.5. Let w ∈ H r (T) and v ∈ H r ′ (T) with 0 ≤ r ≤ s, 0 ≤ r ′ ≤ s and 0 ≤ 2s-r-r ′ < 1 4 . Then (1 -∂ 2 x ) -1 ∂ x (wv) H s ≤ c r,r ′ ,s w H r v H r ′ . In particular, if w ∈ H r (T) and v ∈ H s (T) with 0 ≤ r ≤ s < r + 1 4 , then (1 -∂ 2 x ) -1 ∂ x (wv) H s ≤ c r,s w H r v H s . Proof of
Let s ≥ 0 and R > 0 be given. There exists a T * depending only on s and R such that for any u 0 ∈ H s (T) with u 0 H s ≤ R, the system (6.20)-(6.21) admits a unique solution u ∈ C([0, T * ]; H s (T)). Moreover, T * → ∞ as R → 0.

Rewrite (6.20)-(6.21) in its integral form

u(t) = W a (t)u 0 - t 0 W a (t -τ )(1 -∂ 2 x ) -1 (uu x )(τ )dτ. (6.22)
For given θ > 0, define a map Γ on C([0, θ]; H s (T)) by

Γ(v) = W a (t)u 0 - t 0 W a (t -τ )(1 -∂ 2 x ) -1 (vv x )(τ )dτ
for any v ∈ C([0, θ]; H s (T)). Note that, according to Lemma 6.3 and Lemma 6.5,

W a (t)u 0 C([0,θ];H s (T)) ≤ C s u 0 H s , and t 0 W a (t -τ )(1 -∂ 2 x ) -1 (vv x )(τ )dτ C([0,θ];H s (T)) ≤ C s θ sup 0≤t≤θ (1 -∂ 2 x ) -1 (vv x )(t) H s ≤ C s c s,s 2 θ v 2 C([0,θ];H s (T)) .
Thus, for given R > 0 and u 0 ∈ H s (T) with u 0 H s ≤ R, one can choose T * = [2c s,s (1+ C s )R] -1 such that Γ is a contraction mapping in the ball

B := {v ∈ C([0, T * ]; H s (T)); v C([0,T * ];H s (T)) ≤ 2C s R}
whose fixed point u is the desired solution.

Step 2: The system is globally well-posed in the space H s (T) for any s ≥ 1.

To this end, it suffices to establish the following global a priori estimate for smooth solutions of the system (6.20)-(6.21):

Let s ≥ 1 and T > 0 be given. There exists a continuous nondecreasing function α s,T : R + → R + such that any smooth solution u of the system (6.20)-(6.21) satisfies

sup 0≤t≤T u(•, t) H s ≤ α s,T ( u 0 H s ). ( 6.23)
Estimate (6.23) holds obviously when s = 1 because of the energy identity

||u(t)|| 2 H 1 -||u 0 || 2 H 1 = -2 t 0 ||au(τ )|| 2 H 1 dτ ∀t ≥ 0. When 1 < s ≤ s 1 := 1 + 1 8
, applying Lemma 6.3 and Lemma 6.5 to (6.22) yields that for any 0

< t ≤ T , u(•, t) H s ≤ C s u 0 H s + C s c 1,s 2 t 0 u(•, τ ) H 1 u(•, τ ) H s dτ ≤ C u 0 H s + Cα 1,T ( u 0 H 1 ) t 0 u(•, τ ) H s dτ.
Estimate (6.23) for 1 < s ≤ s 1 follows by using Gronwall's lemma. Similarly, for s 1 < s ≤ s

2 := 1 + 2 8 , u(•, t) H s ≤ C s u 0 H s + C s c s 1 ,s 2 t 0 u(•, τ ) H s 1 u(•, τ ) H s dτ ≤ C u 0 H s + Cα s 1 ,T ( u 0 H s 1 ) t 0 u(•, τ ) H s dτ.
Estimate (6.23) thus holds for 1 < s ≤ s 2 . Continuing this argument, we can show that the estimate (6.23) holds for 1 < s ≤ s k := 1 + k 8 for any k ≥ 1.

Step 3: The system (6.20)-(6.21) is globally well-posed in the space H s (T) for any 0 ≤ s < 1.

To see it is true, as in [START_REF] Roumégoux | A symplectic non-squeezing theorem for BBM equation[END_REF], we decompose any u 0 ∈ H s (T) as

u 0 = k∈Z ûk e ikx = |k|≤k 0 + |k|>k 0 =: w 0 + v 0 with v 0 ∈ H s (T) satisfying v 0 H s ≤ δ
for some small δ > 0 to be chosen, and w 0 ∈ H 1 (T). Consider the following two initial value problems

v t -v txx -cv xxx + (c + 1)v x + vv x = -a(1 -∂ 2 x )[av], v(x, 0) = v 0 (x) (6.24) and w t -w txx -cw xxx + (c + 1)w x + ww x + (vw) x = -a(1 -∂ 2 x )[aw], w(x, 0) = w 0 (x). (6.25)
By the local well-posedness established in Step 1, for given T > 0, if δ is small enough, then (6.24) admits a unique solution v ∈ C([0, T ]; H s (T)). For (6.25), with v ∈ C([0, T ]; H s (T)), by using Lemma 6.3, the estimate

||(1 -∂ 2 x ) -1 ∂ x (wv)|| H 1 ≤ C||wv|| L 2 ≤ C||w|| H 1 ||v||
H s and the contraction mapping principle, one can show first that it is locally well-posed in the space H 1 (T). Then, for any smooth solution w of (6.25) it holds that 1 2

d dt w(•, t) 2 H 1 - T v(x, t)w(x, t)w x (x, t)dx = -a(•)w(•, t) 2 H 1 , which implies that w(•, t) 2 H 1 ≤ w 0 2 H 1 exp C t 0 v(•, τ ) L 2 dτ
for any t ≥ 0. The above estimate can be extended to any w 0 ∈ H 1 (T) by a density argument. Consequently, for w 0 ∈ H 1 (T) and v ∈ C([0, T ]; H s (T)), (6.25) admits a unique solution w ∈

C([0, T ]; H 1 (T)). Thus u = w + v ∈ C([0, T ]; H s (T))
is the desired solution of system (6.20)-(6.21). The proof of Theorem 6.4 is complete.

Next we show that the system (6.20)-(6.21) is locally exponentially stable in H s (T) for any s ≥ 1. Proposition 6.6. Let s ≥ 1 be given and γ > 0 be as given in Lemma 6.3. Then there exist two numbers δ > 0 and C ′ s depending only on s such that for any u 0 ∈ H s (T) with u 0 H s ≤ δ, the corresponding solution u of the system (6.20)-(6.21) satisfies u(•, t) H s ≤ C ′ s e -γt u 0 H s ∀t ≥ 0. Proof. We proceed as in [START_REF] Pazoto | Stabilization of a Boussinesq system of KdV-KdV type[END_REF]. As in the proof of Theorem 6.4, rewrite the system (6.20)-(6.21) in its integral form

u(t) = W a (t)u 0 - 1 2 t 0 W a (t -τ )(1 -∂ 2 x ) -1 ∂ x (u 2 )(τ )dτ
and consider the map

Γ(v) := W a (t)u 0 - 1 2 t 0 W a (t -τ )(1 -∂ 2 x ) -1 ∂ x (v 2 )(τ )dτ.
For given s ≥ 1, by Lemma 6.3 and Lemma 6.5, there exists a constant C s > 0 such that

Γ(v)(•, t) H s ≤ C s e -γt u 0 H s + C s c s,s 2 t 0 e -γ(t-τ ) v(•, τ ) 2 H s dτ ≤ C s e -γt u 0 H s + C s c s,s 2 sup 0≤τ ≤t e γτ v(•, τ ) 2 H s t 0 e -γ(t+τ ) dτ ≤ C s e -γt u 0 H s + C s c s,s 2γ e -γt (1 -e -γt ) sup 0≤τ ≤t e γτ v(•, τ ) 2 H s
for any t ≥ 0. Let us introduce the Banach space

Y s := {v ∈ C([0, ∞); H s (T)) : ||v|| Ys := sup 0≤t<∞ e γt v(•, t) H s < ∞}. For any v ∈ Y s , Γ(v) Ys ≤ C s u 0 H s + C s c s,s 2γ v 2 Ys . Choose δ = γ 4C 2 s c s,s , R = 2C s δ. Then, if u 0 ≤ δ, for any v ∈ Y s with v Ys ≤ R, Γ(v) Ys ≤ C s δ + C s c s,s 2γ (2C s δ)R ≤ R. Moreover, for any v 1 , v 2 ∈ Y s with v 1 Ys ≤ R and v 2 Ys ≤ R, Γ(v 1 ) -Γ(v 2 ) Ys ≤ 1 2 v 1 -v 2 Ys .
The map Γ is a contraction whose fixed point u ∈ Y s is the desired solution satisfying u(•, t) H s ≤ 2C s e -γt u 0 H s for any t ≥ 0. Now we turn to the issue of the global stability of the system (6.20)-(6.21). First we show that the system (6.20)-(6.21) is globally exponentially stable in the space H 1 (T). Theorem 6.7. Let a ∈ C ∞ (T) with a = 0, and let γ > 0 be as in Lemma 6.3. Then for any R 0 > 0, there exists a constant C * > 0 such that for any u 0 ∈ H 1 (T) with ||u 0 || H 1 ≤ R 0 , the corresponding solution u of (6.20)-( 6 for some positive constants C = C(R 0 ), κ = κ(R 0 ). Finally, we can replace κ by the γ given in Lemma 6.3. Indeed, let t ′ = κ -1 log[1 + CR 0 δ -1 ], where δ is as given in Proposition 6.6. Then for u 0 H 1 ≤ R 0 , u(t ′ ) H 1 < δ, hence for all t ≥ t ′ u(t) H 1 ≤ C ′ 1 u(t ′ ) H 1 e -γ(t-t ′ ) ≤ (C ′ 1 δ/R 0 ) u 0 H 1 e -γ(t-t ′ ) ≤ C * e -γt u 0 H 1 where C * = (C ′ 1 δ/R 0 )e γt ′ . Now we present a proof of Proposition 6.8. Pick for the moment any T > 2π/|c| (its value will be specified later on). We prove the estimate (6.28) by contradiction. If (6.28) is not true, then for any n ≥ 1 (6.20)-( 6 Because of ||v n (0)|| H 1 = 1, (6.35) the sequence (v n ) is bounded in L ∞ (0, T ; H 1 (T)), while (v n,t ) is bounded in L ∞ (0, T ; L 2 (T)). From Aubin-Lions' lemma and a diagonal process, we infer that we can extract a subsequence of (v n ), still denoted (v n ), such that

v n → v
in C([0, T ]; H s (T)) ∀s < 1 (6.36) Thus, v solves v tv txxcv xxx + (c + 1)v x + αvv x = 0 on T × (0, T ), (6.40) v = 0 on ω × (0, T ). (6.41) where ω is given in (6.3). According to Corollary 5.5, v ≡ 0 on T × (0, T ). We claim that (v n ) is linearizable in the sense of [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF]; that is, if (w n ) denotes the sequence of solutions to the linear KdV-BBM equation with the same initial data w n,tw n,txxcw n,xxx + (c + 1)w n,x = -a(1 -∂ (6.35). This completes the proof of Proposition 6.8 and of Theorem 6.7.

v n → v in L ∞ (
Next we show that the system (6.20)-(6.26) is exponentially stable in the space H s (T) for any s ≥ 1. Theorem 6.9. Let a ∈ C ∞ (T) with a = 0 and γ > 0 be as given in Lemma 6.3. For any given s ≥ 1 and R 0 > 0, there exists a constant C > 0 depending only on s and R 0 such that for any u 0 ∈ H s (T) with ||u 0 || H s ≤ R 0 , the corresponding solution u of (6.20)-(6.21) satisfies u(•, t) H s ≤ Ce -γt u 0 H s for all t ≥ 0. (6.48)

Proof. As before, rewrite the system in its integral form

u(t) = W a (t)u 0 - 1 2 t 0 W a (t -τ )(1 -∂ 2 x ) -1 (uu x )(τ )dτ.
For u 0 ∈ H s (T) with u 0 H s ≤ R 0 , applying Lemma 6.3, Lemma 6.5 and Theorem 6.7 yields that, for any 1 ≤ s ≤ 1 +

≤ C s + C(s, m 1 , R 0 )||u 0 || H m 1 γ -1 e -γt u 0 H s .
Thus the estimate (6.48) holds for 1 ≤ s ≤ m 2 := 1 + 2 10 . Repeating this argument yields that the estimate (6.48) holds for 1 ≤ s ≤ m k := 1 + k 10 for k = 1, 2, . . .

Corollary 4 . 2 .

 42 Let u 0 ∈ H 1 (T). Then (4.6)-(4.7) admits a unique solution u ∈ C([0, T ]; H 1 (T))

. 2 ). 3 ) 6 . 1 .

 2361 where c ∈ R \ {0} and a ∈ C ∞ (T) is a given nonnul function. Let ω = {x ∈ T; a(x) = 0} = ∅.(6Exact controllability. The first result is a local controllability result in large time.

|| 2 H 1 ( 6 . 32 )

 21632 .21) admits a solution u n ∈ C([0, T ]; H 1 (T)) satisfying||u n (0)|| H 1 ≤ R 0 where u 0,n = u n (0). Since α n := ||u 0,n || H 1 ≤ R 0 ,one can choose a subsequence of (α n ), still denoted by (α n ), such that lim n→∞ α n = α. Note that α n > 0 for all n, by (6.32). Set v n = u n /α n for all n ≥ 1. Thenv n,tv n,txxcv n,xxx + (c + 1)v n,x + α n v n v n,x = -a(1 -∂ 2 x )[av n ]

  .9) Pick any initial data u 0 ∈ H 1 (T), any s ∈ (1/2, 1), and let ρ = ||u 0 || H 1 (T) and T = T (s, ρ). Note that ||u(t)|| H 1 is nonincreasing by (4.8), hence it has a nonnegative limit l as t → ∞. Let v 0 be in the ω-limit set of (u(t)) t≥0 in H 1 (T) for the weak topology; that is, for some sequence t n → ∞ we have u(t n ) → v 0 weakly in H 1 (T). Extracting a subsequence if needed, we may assume that t n+1t n ≥ T for all n.

		From (4.8) we infer that
	lim n→∞	tn t n+1	T	a(x)|u(x, t)| 2 dxdt = 0.	(4.10)
	Since u(t

n ) → v 0 (strongly) in H s (T), and ||u(t n )|| H s (T) ≤ ||u(t n )|| H 1 (T) ≤ ρ, we have from (4.9) that u(t n

  1 , C 2 (depending on s and T , but not on R, ||u 0 || H s or ||u T || H s ) such that for all u, v ∈ B R

	||Γ(u)|| C([0,T ];H s

  γt ||u 0 || H s+1 , as desired. The estimate (6.19) is thus proved for any s ∈ N * . It may be extended to any s ∈ [1, +∞) by interpolation.

  .21) satisfies ||u(•, t)|| H 1 ≤ C * e -γt ||u 0 || H 1 Proposition 6.8. Let R 0 > 0 be given. Then there exist two positive numbers T and β such that for anyu 0 ∈ H 1 (T) satisfying ||u 0 || H 1 ≤ R 0 ,Indeed, if (6.28) holds, then it follows from the energy identity||u(t)|| 2 H 1 = ||u 0 || 2 H 1 -2 )|| 2 H 1 ≤ (1 -2β -1 )||u 0 || 2 H 1 . Thus ||u(mT )|| 2 H 1 ≤ (1 -2β -1 ) m ||u 0 ||2 H 1 which gives by the semigroup property ||u(t)|| H 1 ≤ Ce -κt ||u 0 || H 1 for all t ≥ 0, (6.30)

		0	t	||au(τ )|| 2 H 1 dτ	∀t ≥ 0	(6.29)
	that
	||u(T
					for all t ≥ 0.	(6.26)
	Theorem 6.7 is a direct consequence of the following observability inequality.
						(6.27)
	the corresponding solution u of (6.20)-(6.21) satisfies
	||u 0 || 2 H 1 ≤ β	0	T	||au(t)|| 2 H 1 dt.	(6.28)

  0, T ; H 1 (T)) weak *(6.37) for some v ∈ L ∞ (0, T ; H 1 (T)) ∩ C([0, T ]; H s (T)) for all s < 1, Note that, by (6.36)-(6.37), we have thatα n v n v n,x → αvv x in L ∞ (0, T ; L 2 (T)) weak * .

	Furthermore, by (6.34),
	0	T	||av|| 2 H 1 dt ≤ lim inf n→∞	0	T	||av n || 2 H 1 dt = 0.	(6.39)
							(6.38)

  w n (t)|| H 1 → 0 as n → ∞. (6.44) Indeed, if d n = v nw n , then d n solves d n,td n,txxcd n,xxx + (c + 1)d n,x = -a(1 -∂ 2 x )[ad n ]α n v n v n,x , d n (0) = 0. Since ||W a (t)|| L(H 1 (T)) ≤ 1, we have from Duhamel formula that for t ∈ [0, T ] ||d n (t)|| H 1 ≤ ∂ 2 x ) -1 (α n v n v n,x )(τ )|| H 1 dτ.Combined to(6.36) and to the fact that v ≡ 0, this gives(6.44). By Lemma 6.3, we have that ||w n (t)|| H 1 ≤ C 1 e -γt ||w n (0)|| H 1 Combined to (6.34) and (6.44), this yields ||v n (0)|| H 1 = ||w n (0)|| H 1 → 0, which contradicts

						2 x )[aw n ],	(6.42)
	w n (x, 0) = v n (x, 0),				(6.43)
	then
	sup
	T
	0	||(1 -for all t ≥ 0.	(6.45)
	From (6.45) and the energy identity for (6.42)-(6.43), namely
	||w n (t)|| 2 H 1 -||w n (0)|| 2 H 1 = -2	0	t	||aw n (τ )|| 2 H 1 dτ,	(6.46)
	we have for Ce -λT < 1
	||w n (0)|| 2 H 1 ≤ 2(1 -C 2 1 e -2γT ) -1	0	T	||aw n (τ )|| 2 H 1 dτ.	(6.47)

0≤t≤T

||v n (t)

  1 10 ,u(•, t) H s ≤ C s e -γt u 0 H s + C s c 1,1,s 2 C s e -γt u 0 H s + C s c 1,1,s (C * ) 2 2 C s c 1,1,s (C * ) 2 2γu 0 H 1 e -γt u 0 H s for any t ≥ 0. Thus the estimate (6.48) holds for 1 ≤ s ≤ m 1 := 1 + 1 10 . Similarly, form 1 ≤ s ≤ m 2 := 1 + 2 10 , we have for ||u 0 || H s ≤ R 0 u(•, t) H s ≤ C s e -γt u 0 H s + C s c m 1 ,m 1 ,s 2 ) 2 H m 1 dτ ≤ C s e -γt u 0 H s + C(s, m 1 , R 0 )

		0	t	e -γ(t-τ ) u(•, τ ) 2 H 1 dτ
	≤ t 0	e -γ(t-τ ) e -2γτ u 0	2 H 1 dτ
	≤	C s +

t 0 e -γ(t-τ ) u(•, τ t 0 e -γ(t-τ ) e -2γτ u 0 2 H m 1 dτ

Acknowledgements

The authors would like to thank E. Zuazua for having brought to their attention the reference [1]. LR was partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02. BZ was partially supported by a grant from the Simons Foundation (#201615 to Bingyu Zhang)


Frédéric Brechenmacher
email: frederic.brechenmacher@euler.univ-artois.fr

Linear groups in Galois fields. A case study of tacit circulation of explicit knowledge

in the proceedings of the Oberwolfach conference "Explicit vs tacit knowledge in mathematics" (January 2012). It presents a case study on some algebraic researches at the turn of the twentieth century that involved mainly French and American authors. By investigating the collective dimensions of these works, this paper sheds light on the tension between the tacit and the explicit in the ways some groups of texts hold together, thereby constituting some shared algebraic cultures.

Although prominent algebraists such as Dickson made extensive references to papers published in France, and despite the roles played by algebra and arithmetic in the development of the American mathematical community, our knowledge of the circulations of knowledge between France and the United States at the beginning of the 20th century is still very limited.

It is my aim to tackle such issues through the case study of a specific collective approach to finite group theory at the turn of the 20th century.

This specific approach can be understood as a shared algebraic culture based on the long run circulation of some specific procedures of decompositions of the analytic forms of substitutions. In this context, the general

linear group was introduced as the maximal group in which an elementary abelian group (i.e., the multiplicative group of a Galois field) is a normal subgroup.

This paper aims at stressing some aspects of my works on the history of algebra in the 19th and 20th century, which are related to the tacit vs the explicit in the ways some groups of texts hold together. These aspects raise issues as to the historian's choices of a corpus of reference and of a scale of analysis. They also address the more general problem of articulating the individual and collective dimensions of mathematics.

Although the category "algebra" points to some collective organizations of knowledge, this category took on changing identities in different times and spaces. Until the 1930s, "algebra" was especially not usually referring to an object-oriented discipline, i.e., as identifying both a corpus of specialized knowledge revolving around some specific objects and the institutionalized practices of transmissions of a group of professional specialists (the "algebraists"). [START_REF] Brechenmacher | On the Identities of Algebra in the 19th Century[END_REF] In France, for instance, algebra was, on the one hand, traditionally considered in the teaching of mathematics as an "elementary," or "intermediary," discipline encompassed by "the higher point of view" of analysis. On the other hand, algebra was also pointing to some procedures that made a "common link" between researches in the various branches of the mathematical sciences. [START_REF] Brechenmacher | On Jordan's measurements[END_REF] What was explicitly identified as "algebraic" therefore often pointed to some implicit circulations between various theories. This situation makes it customary to study carefully the ways texts were referring one to another, thereby constituting some shared algebraic cultures. I shall introduce this paper by making explicit how such issues came up in my research work, before focusing on a case study on "linear groups in Galois fields" at the turn of the 20th century. Although the latter designation may seem to make explicit some collective interests for a theory revolving around a specific object, i.e. Gl n (F p n ) (p a prime number), we shall see that this designation actually supported the implicit reference to a specific algebraic culture that had developed over the course of the 19th century. some tacit knowledge and know-how, such as some conflicting disciplinary ideals on the roles of algebra vs arithmetic, some epistemic values of simplicity vs effectivity, and two opposed internal philosophies of generality. [START_REF] Brechenmacher | La controverse de 1874 entre Camille Jordan et Leopold Kronecker[END_REF] But more importantly for the topic of the present paper, this quarrel also highlights the tacit intertextual relations that lied beneath the expression "theory of bilinear forms."

The controversy started with a public quarrel of priority over two theorems before turning into a private correspondence. As is illustrated by the following letter from Jordan to Kronecker (February 1874), the epistolary communication was mostly devoted to making explicit some tacit relations between some texts of Weierstrass, Christoffel, and Kronecker: I would not like ... that I want to wage war and that I would prefer a controversy / war some public debates to friendly discussions. I was not the one to open fire start the controvery. It is true that I have published (as was my evident right) without consulting you some researches that were completing yours ... But if instead of going abruptly public with this issue, you would have contacted me for exchanging the friendly explanations I would have been legitimately expecting, we would certainly have reached an agreement.

Following your indications, I would have immediately noticed, what I recognized too late, that your method of 1868 reread more carefully your memoir of 1868 and would have noticed, what I did not recognized at first sight, that bilinear forms were implicitly included in your work despite not being cited. 1After Jordan had made himself familiar with the collective of texts Kronecker made explicit, the quarrel eventually went public once again. These successive episodes highlight that even though the "theory of bilinear forms" may have been considered at first sight as the explicit reference to a collective of mathematical methods and notions, it was through some tacit intertextual relations that this theory was making sense. Moreover, the tacit dimension of the theory was recognized by the actors themselves. Both Jordan and Kronecker indeed agreed that it was no more than a "slight fault" ("levis culpa") to fail to grasp the relevant underlying intertextual relations, which had thus to be made explicit through direct communication. Both protagonists also recognized having been already both responsible and victims of such slight faults in the past, as is illustrated by the following letter from Jordan to Kronecker (January 1874):

As I conclude this letter, please allow me Sir to express my regrets that you did not write to me your ask me for some explanations before publishing your criticisms ... I especially regret your decision to involve in this matter the name of M. Christoffel ... I should add that I did not know his memoir at the time which is quite legitimate. I am moreover convinced that [Christoffel] would have agreed with me that one has to be modest about such claims. It is by following this principle that I did not say anything against the publication in the last issue of M. Borchardt's journal of a voluminous paper by M. Sohnke on the symmetry in the plane, in which I was not cited even though I have completely tackled this issue for both the space and the plane five or six years ago in MM. Brioschi's and Cremona's journals. 22 The tacit vs the explicit in building some net-

works of texts

The 1874 controversy highlights the problem of the selection of the corpuses in which a given text is making sense, i.e., the identification of some networks of texts. But such networks cannot be identified as webs of quotations. [START_REF] Goldstein | Sur la question des méthodes quantitatives en histoire des mathématiques : le cas de la théorie des nombres en France (1870-1914)[END_REF] Not only do practices of quotations vary in times and spaces but intertextual relations may also be implicit. My approach to this problem consists in choosing a point of reference from which a first corpus is built by following systematically the explicit traces of intertextual relations. A close reading of the texts of this corpus then gives access to some more implicit forms of references. For instance, in the case of the 1874 controversy, both protagonists were referring to the "equation to the secular inequalities in planetary theory" [START_REF] Brechenmacher | La controverse de 1874 entre Camille Jordan et Leopold Kronecker[END_REF]. Such an explicit reference implicitly pointed to a network of texts that had been published over the course of the 19th century in various theoretical frameworks, e.g., celestial mechanics, analytical geometry, complex analysis, or arithmetic. The 1874 controversy opposed two attempts to turn this traditional algebraic culture into an object-oriented theory, which Jordan aimed to root on group theory while Kronecker laid the emphasis on the theory of quadratic forms.

Linear groups in Galois fields

In the framework of a collective research project,3 a database of intertextual references has been worked out for all the texts published in algebra in France from 1870 to 1914. 4 One of the subgroups of this corpus gives rise to a coherent network of texts which involved mainly French and American authors from 1893 to 1907. 5 A quite representative example is the book " Élements de la théorie des groupes abstraits," published in 1904 by Jean Armand de Séguier, one of the main French authors of the group.

Let us characterize further this network by looking at its main shared references. These were, on the one hand, some French papers published in the 1860s, and, on the other hand, Moore's introduction of the abstract notion of Galois field in 1893 in addition to Dickson's 1901 monograph on Linear groups with an exposition of the Galois field theory. We shall see that the two times and spaces involved here point to a shared algebraic culture that can neither be identified to a discipline nor to any simple national or institutional dimension.

On nations and disciplines

Moore's 1893 "abstract" notion of Galois field has been assumed to highlight the influence of "the abstract point of view then increasingly characteristic of trendsetting German mathematics" on the emergence of both the "Chicago algebraic research school" [START_REF] Parshall | The Emergence of the American Mathematical Research Community[END_REF] and the "American mathematical research community" [START_REF] Goldstein | Hermite's strolls in Galois fields[END_REF]. Here two kinds of categories have been used for making explicit some collective dimensions of mathematics, i.e., on the one hand, some national, or more local, collectives of mathematicians (the U.S.A., Germany, Chicago) and, on the other hand, some mathematical disciplines (abstract algebra). But even though the roles played by German universities in the training of many American mathematicians have been well documented, the influence of this institutional framework on mathematics has been assumed quite implicitly. Here two difficulties arise. First, the role attributed to "abstract algebra" reflects the tacit assumption that the communication of some local tacit knowledge should require direct contact. Because the historiography of algebra has usually emphasized the abstract approaches developed in Germany, and especially in the center of Göttingen, other, more local, abstract approaches, such as in Cambridge or Chicago, have raised issues about the imperfect communication of some tacit knowledge, as exemplified by the late interbreeding in the 1930s of the German Moderne Algebra and the Anglo-American approach to associative algebras. [START_REF] Fenster | A Delicate Collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras in the Early 1930's[END_REF] In this framework, the algebraic developments in France, such as Séguier's, have been either ignored or considered as some isolated attempts modeled on German or Anglo-American approaches. [START_REF] Dubreil | L'algèbre, en France, de 1900 à 1935[END_REF] Second, both disciplines and nations are actors categories, which even though they were much involved in public discourses, cannot usually be directly transposed to the collective dimensions of mathematical developments [START_REF] Brechenmacher | Galois Got his Gun[END_REF]. Public discourses indeed often involved some boundary work that was not only setting delimitations between mathematicians and non mathematicians but was also supporting some hierarchies among the practitioners of mathematics (researchers vs teachers and engineers, pure vs applied, analysts vs algebraists, etc.). 6 . These boundaries often reflect the roles taken on by some authorities as well as the ways some individuals embodied some collective models of mathematical lives or persona. For instance, several authorities of mathematics in France recurrently expressed publicly an opposition between the unifying power of analysis and the poverty of considering algebra as an autonomous discipline, thereby blaming some approaches developed in Germany. Picard was one of the main advocate of such an opposition. But even though he was publicly celebrating Jordan's approach to Galois theory, it was to Kronecker's approach to this theory that he was appealing in his own mathematical work. Recall that Moore's 1893 paper was read at the congress that followed the World Columbian exposition in Chicago. The world fair was dedicated to the discovery of America and was the occasion of much display of national grandeur. [START_REF] Brian | The World's Columbian Exposition[END_REF] In parallel to the elevation of the first great wheel, presented by the Americans as a challenge to the Eiffel tower, or to the viking ship that sailed from Norway as a counterpoint to the replica of Columbus's three caravels, the architectural influence of the French École des Beaux arts was challenged by the German folk village. The latter especially included an exhibit of the German universities in which Klein was delivering a series of lectures which aimed at "passing in review some of the principal phases of the most recent development of mathematical thought in Germany." Klein was also the glorious guest of the congress while Moore was both the host of the congress and one of its main organizers. The latter's concluding lecture was a tribute to Klein's Icosahedron. It indeed aimed at generalizing to a "new doubly infinite system of simple groups," i.e., P SL 2 (F p n ), what was then designated as the three "Galois groups," i.e., P Sl 2 (F p ), p = 5, 7, 11, involved in the modular equations that had been investigated by Galois, Hermite, and Klein among others. [START_REF] Goldstein | Hermite's strolls in Galois fields[END_REF] The generalization consisted in having the analytic form of unimodular binary linear fractional substitutions ax+b cx+d operate on a finite "field" of letters indexed by Galois number theoretic imaginaries.

The nature of the relevant collective dimensions nevertheless change if one shifts the scale of analysis from institutions to texts. As we shall see, even though he had aimed at celebrating the emergence of some abstract researches in the U.S.A. in the framework of the Göttingen tradition, Moore actually collided to the implicit collective dimension that was underlying the use of the analytic representation of substitutions on number theoretic imaginaries [START_REF] Brechenmacher | Self-portraits with Évariste Galois (and the shadow of Camille Jordan)[END_REF]. In 1893, Moore initially appealed to Klein-Fricke's 1890 short presentation of Galois imaginaries but had not yet studied the references to Galois, Serret, Mathieu, Jordan, or Gierster which he cited from Klein-Fricke's texbook. On the one hand, what Moore designated as a Galois field actually corresponded to Cauchy's approach to higher congruences [START_REF] Goldstein | Sur la question des méthodes quantitatives en histoire des mathématiques : le cas de la théorie des nombres en France (1870-1914)[END_REF], as developed later by Serret, and which Moore nevertheless attributed to Galois :

The most familiar instance of such a field ... is the system of p incongruous classes (modulo p) of rational integral numbers. Galois discovered an important generalization of the preceding field ... the system of p n incongruous classes (modulo p, F n (x)).

On the other hand, Moore's "abstract finite field" was actually close to Galois's approach. Moore's remark that "every finite field is in fact abstractly considered a Galois field" thus echoed the connection between two perspectives on number theoretic imaginaries, as it had already been displayed in textbooks such as Serret's in 1866. [START_REF] Brechenmacher | Self-portraits with Évariste Galois (and the shadow of Camille Jordan)[END_REF] But even more dramatically, Moore's system of simple groups had actually already been introduced by Mathieu in 1861. As a result, before the publication of the proceedings of the congress in 1896, Moore and his student Dickson struggled to access the tacit collective dimension of some texts published in France in the 1860s, especially by appealing to Jordan's 1870 Traité des substitutions et des équations algébriques. This appropriation resulted in the publication of a train of papers on "Jordan's linear groups in Galois fields." Moore's 1893 paper thus eventually resulted in the circulation of some works that were foreign to Klein's legacy. This situation highlights the difficult problem of identifying the scales at which various forms of collective dimensions play a relevant role, especially in respect to the articulation of the collective dimensions of texts with the ones of actors, such as disciplines or nations.

The analytic representation of substitutions

Let us now characterize more precisely the collective dimension to which Moore collided to in 1893. Given a substitution S operating on m letters a i , the problem of the analytic representation consists in finding an analytic function φ such that S(a i ) = φ(i). Hermite's 1863 solution to this problem for the cases m = 5, 7 would especially influence Dickson's 1896 thesis. But the analytic representation also requires an indexing of the letters. This issue had already been tackled in Poinsot's 1808 presentation of Gauss's decomposition of cyclotomic equations. [START_REF] Goldstein | Sur la question des méthodes quantitatives en histoire des mathématiques : le cas de la théorie des nombres en France (1870-1914)[END_REF] As had been shown later by Galois in 1830, if m = p n , the indices can be considered as the "imaginary solutions" of the congruence x p n x ≡ 0 (mod p)

that generalized the indexations given by the roots of Gauss's cyclotomic equations. Such analytic forms were nevertheless not limited to representations but were interlaced with some specific procedures. The procedure of reduction of "linear" substitutions (i, ai + b) into combinations of cycles (i, i + 1) and (i, gi) had especially played a key role in Galois's characterization of solvable irreducible equations of prime degree, which roots (x i , x φ(i) ) are permuted by substitutions of "a linear form" (x i , x ai+b ) [START_REF] Brechenmacher | Self-portraits with Évariste Galois (and the shadow of Camille Jordan)[END_REF]. In modern parlance, Galois's theorem and its proofs boil down to showing that the linear group is the maximal group in which an elementary abelian group (the cyclic group F * p in the case n = 1) is a normal subgroup. 7 It was in attempting to generalize this theorem to the analytic forms of roots (x i,i ′ ,...,i (n) , x φ(i),ψ(i ′ ),...,σ(i (n) ) ) of equations of degree p n that Galois introduced the number theoretic imaginaries.

Later on, in the 1860s, Jordan investigated further the group "originating" from the problem of finding the analytic form of the maximal group T in which the group of substitutions (i, i ′ , ..., i (n) ; i + a, i ′ + a ′ , ..., i (n) + a (n) ) (i.e., F * p n ) is a normal subgroup. He followed Poinsot's reformulation of Gauss's decomposition of cyclotomic equations in dividing the letters on which the substitutions are acting into "groups" (i.e., blocks of imprimitivity, which are sets on which the group is acting but that Jordan also called groups), each of a same cardinal p n .

T was then simultaneously partitioned into a "combination of displacements between the groups" [i.e. blocks] and of permutations of the letters within each of the groups [i.e., blocks]." As a result, the substitutions of T were divided into two "species". On the one hand, inside each block, the letters were cyclically substituted by the first specie of substitutions (i, gi) operating by the multiplication of a primitive root g (mod.p). On the other hand, the second specie (i, i+1) substituted cyclically the blocks themselves. Each specie of substitution corresponded to one of the two forms of representation of cycles. Their products generated linear forms (i, ai + b) with i ∈ F p n , i.e., if i = (i, i ′ , i ′′ , ..., i (n) ) :

φ(i) ≡ ai + bi ′ + ci ′′ + ... + vi (n) + d φ(i ′ ) ≡ a ′ i + b ′ i ′ + c ′ i ′′ + ... + v ′ i (n) + d ′ φ(i ′′ ) ≡ a ′′ i + b ′′ i ′ + c ′′ i ′′ + ... + v ′′ i (n) + d ′′ ......

mod(p)

was because they shared this culture that some French and American authors were able to interact with each others, even though most of them did not have any direct contact and did not share any social framework, as is exemplified by such different figures as de Séguier, an aristocrat jesuit abbot, and Schottenfels, one of the first women to graduate in mathematics at Chicago. Communication was nevertheless partial and was actually mostly limited to some shared practices, such as the use of Jordan's canonical form. A telling example is the new formulation that was given repeatedly and indepently to Jordan's "origin" of the linear group as the theorem stating that the group of automorphisms of an elementary abelian group F * p n is Gl(F p n ) (Burnside, Moore, Levavasseur, Miller, Dickson, Séguier).

We have seen that the systematic investigation of explicit traces of intertextual relations also sheds light on some more implicit collective forms of references, such as the one that lied beneath expressions such as "linear groups in Galois fields." This situation highlights the crucial role played by some networks of texts in the shaping of some algebraic cultures at a time when "algebra" was not yet referring to an object-oriented discipline.

Introduction : the "versus"Throughout the whole of 1874, Jordan and Kronecker were quarreling over the organization of the theory of bilinear forms. This controversy made explicit

Je ne voudrais pas ..., que je désire la guerre, et que je préférerais une polémique / guerre des débats publics à des explications amicales. Ce n'est pas moi qui ait ouvert les hostilités commencé la polémique. J'ai publié il est vrai (c'était mon droit évident) sans vous consulter des recherches qui complétaient les vôtres... Si au lieu de jeter brusquement ce débat dans le public, vous vous étiez adressé à moi pour échanger des explications, comme je me voyais en droit de l'espérer, nous nous serions sans doute entendu. Sur votre indication, j'aurais constaté immédiatement, ce que j'ai reconnu trop tard, que votre méthode de 1868 relu plus attentivement votre mémoire de 1868 et constaté, ce que je n?avais pas remarqué à première vue, que les formes bilinéaires non citées dans votre travail, y sont pourtant implicitement comprises.

Permettez moi Monsieur, en terminant cette lettre, de vous exprimer le regret que vous ayez attendu pour m'écrire vos demander des explications d'avoir publié vos critiques ... Je regrette surtout que vous ayez cru devoir introduire dans cette question le nom de M. Christoffel ... J'ajouterai qu'à cette époque, je ne connaissais pas son mémoire ce qui est bien permis. Je suis persuadé d'ailleurs qu'il aurait été d'avis comme moi, qu'il faut être fort réservé dans les réclamations de ce genre. Aussi ai-je C'est d'après ce principe que j'ai laissé passer sans rien dire, dans le dernier numéro du Journal de M. Borchardt, un mémoire volumineux de M. Sohnke sur la symétrique dans le plan où je n'étais pas cité, bien que j'eusse traité complètement cette question pour le plan et pour l'espace dans le journal de MM Brioschi et Cremona, il y a de cela cinq à six ans.

CaaF É : Circulations of algebraic and arithmetic practices and knowledge (1870-1945) : France, Europe, U.S.A ; http://caafe.math.cnrs.fr

The corpus has been selected by using the classification of the Jahrbuch. On Thamous database of intertextual references, see http://thamous.univ-rennes1.fr/presentation.php

On the one hand, Jordan, Borel and Drach, Le Vavasseur, de Séguier, Autonne, etc. On the other hand, Moore, Dickson, Schottenfels, Wedderburn, Bussey, Börger, Miller, Manning, etc.

L.Turner has shown that "research" was a category Mittag Leffler recurrently appealed to in connection to the notions of "contribution" for establishing hierarchies not only between various stratas of practitioners of mathematics, or between a "general public" and a "special public" but also between nations themselves in Mittag Leffler's own construction of an international space[START_REF] Turner | Identities, agendas, and mathematics in an international space[END_REF]

Given (x i , x φ(i) ) the analytic representation of the roots, then φ(i + a) = φ(i) + A because the cyclic group of substitutions (i, i + a) is a normal subgroup. Thus φ(i + 2a) = φ(i) + 2A), ..., φ(i + ma) = φ(i) + mA. Thus φ(m) = Am + B is of a "linear form" (i.e., affine).

Ce travail a bénéficié d'une aide CaaF É (ANR-10-JCJC 0101)

T was therefore a linear group, as Jordan would designate it in the mid 1860. During the following decades, Jordan especially laid the emphasis on the procedures of reductions of the analytic representations of linear substitutions. The statement of Jordan's canonical form theorem between 1868 and 1870 especially gave a generalization to the reduction of the analytic form (i, ai+b) into (i, i+1) and (i, gi) by reducing any linear substitution on F p n to the following form:

For Jordan, such reductions were the very "essence of the question." They were at the core of Jordan, following Poinsot, was designating as the "theory of order," [START_REF] Boucard | Louis Poinsot et la théorie de l'ordre : un chaînon manquant entre Gauss et Galois ?[END_REF] i.e., a theory dealing with the "relations" between classes of objects in contrast with concerns for specific objects. Reductions of analytic representations were indeed supporting links between various branches of mathematics such as number theory (cyclotomy), the theory of substitutions, the "problem of the irrationals" (Galois theory), crystallography (symmetries), mechanics (rotation of a solid body), analysis situs (polyedrons, Riemann surfaces), differential equations, etc. [START_REF] Brechenmacher | On Jordan's measurements[END_REF] The process of reduction of a linear group into chains of subgroups was for instance compared to the unscrewing of a helicoïdal motion into some rotation and translation motions on the model of Poinsot's interpretations of the two forms (i, gi) and (i, i + 1) of analytic representations of cycles as two forms of motions on a circle.

Jordan's 1870 Traité played a key role in the development of a specific algebraic culture based on the reduction of the analytic representation of n-ary linear substitutions. This culture can not only be be traced in France in the works of authors such as Poincaré, Picard, Autonne, Cartan, Séguier, etc., [START_REF] Brechenmacher | Autour de pratiques algébriques de Poincaré[END_REF] but it also circulated in the U.S.A. after Dickson's 1896 thesis [START_REF] Brechenmacher | On Jordan's measurements[END_REF].

Conclusion

The network of texts that revolved around "Jordan's linear groups in Galois field" at the turn of the 20th century had underlying it a specific algebraic culture based on procedures of reductions of the analytic forms of substitutions. It