Shengquan Xiang
email: shengquan.xiang@ens.fr.

Null controllability of a linearized Korteweg-de Vries equation by backstepping approach

Keywords: Korteweg-de Vries, backstepping, spectral theory, stabilization, null controllability. AMS Subject Classification. 35Q53, 34H05, 35P10

We prove the null controllability of a linearized Korteweg-de Vries equation with a Dirichlet control on the left boundary. Instead of considering classical methods, i.e. Carleman estimates, moment method etc., we use a backstepping approach which is a method usually used to handle stabilization problems.

Introduction

We consider the null controllability of the following linearized KdV control system,      u t (t, x) + u xxx (t, x) + u x (t, x) = 0 in (0, +∞) × (0, L), u(t, L) = u x (t, L) = 0 on (0, +∞), u(t, 0) = κ(t) on (0, +∞), (1.1) where κ(t) ∈ R is a scalar control.

In [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] Rosier introduced the KdV system with a right boundary Neumann control. One surprisingly finds that controllability depends on the length of the interval, which never happens for the linear finite-dimensional system. More precisely, the system is controllable if and only if

L / ∈ N := 2π l 2 + lk + k 2 3 ; l, k ∈ N * . (1.2)
This model has been studied for years, in both controllability [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF][START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF][START_REF] Cerpa | Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain[END_REF][START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF][START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF][START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation: recent progresses[END_REF] and stabilization [START_REF] Chu | Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths[END_REF][START_REF] Coron | Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right[END_REF][START_REF] Coron | Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws[END_REF][START_REF] Alberto | Stabilization of the Korteweg-de Vries Equation with localized damping[END_REF][START_REF] Tang | Local asymptotic stability of a KdV system with a two-dimensional center manifold[END_REF].

Concerning the system studied in this paper, we use the left boundary Dirichlet control. For system (1.1), Rosier (see [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF]) proved that controllability does not depend on the length of the interval. This system was then further studied in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF].

When we study the well-posedness of the control system by using the classical Lions-Magenes method (see [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]), a H 1/3 regularity on the control (with respect to time) is required. Such a problem appears for many boundary control systems, the heat equation and the Burgers equation for example. However, since most control problems are based on evolution models, Sobolev type controls are less preferred than piecewise continuous controls (or even L p type conditions), especially for stabilization problems. In [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF], Coron and Cerpa proved rapid stabilization of the system (1.1) by using the backstepping method. But since they used some stationary feedback laws, this boundary condition problem is avoided. Recently, by using the (piecewise) backstepping approach, Coron and Nguyen proved the null controllability and semi-global finite time stabilization for a class of heat equations (see [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]). They showed how the use of the maximum principle leads to the well-posedness of the closed-loop system. Their method turns out to be a potential way to solve the local (or even semi-global) finite time stabilization problem for systems which can be rapidly stabilized by means of backstepping methods. At the same time, this method provides a visible way to get null controllability directly instead of using observability inequalities and the duality between controllability and observability.

Initially the backstepping is a method to design stabilizing feedback laws in a recursive manner for systems having a triangular structure. See, for example, [START_REF] Coron | Control and nonlinearity[END_REF]Section 12.5]. It was first introduced to deal with finite-dimensional control systems. But it can also be used for control systems modeled by means of partial differential equations (PDEs) as shown first in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF]. For linear partial differential equations, a major innovation is due to Krstic and his collaborators. They observed that, when applied to the classical discretization of these systems, the backstepping leads, at the PDEs level (as the mesh size tends to 0), to the transformation of the initial system into a new target system which can be easily stabilized. This transformation is accomplished by means of a Volterra transform of the second kind. An excellent introduction to this method is presented in [START_REF] Krstic | Boundary control of PDEs, volume 16 of Advances in Design and Control[END_REF]. Krstic's innovation has been shown to be very useful for many PDEs control systems as, in particular, heat equations [START_REF] Liu | Boundary feedback stabilization of an unstable heat equation[END_REF][START_REF] Dejan | Backstepping in infinite dimension for a class of parabolic distributed parameter systems[END_REF][START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF], wave equations [START_REF] Smyshlyaev | Boundary stabilization of a 1-D wave equation with in-domain antidamping[END_REF], hyperbolic systems [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF][START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF][START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF] [2, Chapter 7], Korteweg de Vries equations [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Coron | Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right[END_REF], and Kuramoto-Sivashinsky equations [START_REF] Liu | Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation[END_REF][START_REF] Coron | Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation[END_REF]. It was observed later on that for some PDEs more general transforms than Volterra transforms of the second kind have to be considered: see [START_REF] Coron | Rapid stabilization of 1-D linear Schrödinger equations[END_REF][START_REF] Coron | Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right[END_REF][START_REF] Coron | Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation[END_REF]). Recently, the backstepping method has been adapted to coupled systems, for example the Boussinesq system of KdV-KdV type [START_REF] Roberto | Asymptotic behavior of Boussinesq system of KdV-KdV type[END_REF]. For the case of finite dimensional control system and Krstic's backstepping, see [START_REF] Coron | Stabilization of control systems and nonlinearities[END_REF].

Krstic's backstepping requires solving a kernel equation. In the case of the heat equation, the kernel equation is a wave equation; however, in this paper the kernel equation turns out to be a third-order equation, which generates new difficulties both for the well-posedness of the closed-loop system and for important estimation issues.

In this paper, we prove that the method developed by Coron and Nguyen can be used to get the null controllability of (1.1).

Theorem 1. For any given T > 0, the control system (1.1) is null controllable in time T by using some piecewise continuous controls.

Remark 1. Let us recall that the exact controllability of (1.1) fails, which is proved in [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF].

Remark 2. We study in detail the well-posedness of the system. The approach and tools introduced for this study do not rely on precise structures. In particular the control is not given by a stationary feedback law (compare to [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]) and no maximum principle is used (compare to [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]). Hence, the well-posedness arguments, as well as a priori estimates, and procedure could easily be adapted to many other partial differential equations.

This paper is organized as follows. Section 2 is a preliminary part including the wellposedness of the systems and the rapid stabilization obtained in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]. In Section 3, we design the control and provide some estimates which will lead to the null controllability. In Section 4, we prove the null controllability. We put some further comments in Section 5. It ends with Appendix A (Proposition 1): the proof of the uniqueness of the solution to the kernel equation, which is essential to this paper.

Preliminary

Well-posedness of the control system

We start with the non-homogeneous linear Cauchy problem

           u t + u xxx + u x = h in (T 1 , T 2 ) × (0, L), u x (t, L) = u(t, L) = 0 on (T 1 , T 2 ), u(t, 0) = κ(t) on (T 1 , T 2 ), u(0, x) = u 0 (x) on (0, L), (2.1)
for

-∞ < T 1 < T 2 < +∞, (2.2)
u 0 ∈ L 2 (0, L), (2.3) h ∈ L 1 (T 1 , T 2 ; L 2 (0, L)), (2.4)
κ ∈ L 2 (T 1 , T 2 ). (2.5) Definition 1. A solution to the Cauchy problem (2.1)-(2.5) is a function u ∈ C 0 ([T 1 , T 2 ]; L 2 (0, L))
such that, for every τ ∈ [T 1 , T 2 ] and for every φ

∈ C 3 ([T 1 , τ ] × [0, L]) satisfying φ(t, 0) = φ(t, L) = φ x (t, 0) = 0, ∀t ∈ [T 1 , τ ], (2.6)
one has

- τ T 1 L 0 (φ t + φ x + φ xxx )udxdt - τ T 1 κ(t)φ xx (t, 0)dt - τ T 1 L 0 φ hdxdt + L 0 u(τ, x)φ(τ, x)dx - L 0 u 0 φ(T 1 , x)dx = 0. (2.7)
The uniqueness of the solution to the Cauchy problem (2.1)-(2.5) is straightforward, one can get details from the book by Coron [START_REF] Coron | Control and nonlinearity[END_REF]. For the existence of the solution, in [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain[END_REF], Bona, Sun, and Zhang proved the following result.

Lemma 1. If h ∈ H 1/3 (T 1 , T 2 )
, then the Cauchy problem (2.1) has one and only one solution. This solution is in C 0 ([T 1 , T 2 ]; L 2 (0, L)) ∩ L 2 (T 1 , T 2 ; H 1 (0, L)). There exists a constant c 1 > 0 depending on (T 2 -T 1 ) such that

u C 0 ([T 1 ,T 2 ];L 2 (0,L)) + u L 2 (T 1 ,T 2 ;H 1 (0,L)) + sup x∈[0,L] u x (•, x) L 2 (T 1 ,T 2 ) c 1 u 0 L 2 (0,L) + κ H 1/3 (T 1 ,T 2 ) + h L 1 (T 1 ,T 2 ;L 2 (0,L)) .
(2.8)

Rapid stabilization of (1.1)

We recall some results given in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]. Given a positive parameter λ > 1, we consider the following equations in the triangle T := {(x, y) :

x ∈ [0, L], y ∈ [x, L]},            k xxx + k yyy + k x + k y + λk = 0 in T , k(x, L) = 0 on [0, L], k(x, x) = 0 on [0, L], k x (x, x) = λ 3 (L -x) on [0, L], (2.9)
and

           l xxx + l yyy + l x + l y -λl = 0 in T , l(x, L) = 0 on [0, L], l(x, x) = 0 on [0, L], l x (x, x) = λ 3 (L -x) on [0, L].
(2.10)

In [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF], it is noted that both (2.9) and (2.10) have solutions in C 3 (T ). These solutions are further studied in Section 3, where we provide some estimates on k C 0 (T ) with respect to λ. Actually, the solutions of equation (2.9) and of (2.10) satisfy the following conditions

k xy (x, x) = - λ 3 , (2.11)
l xy (x, x) = - λ 3 , (2.12)
respectively. The Properties (2.11) and (2.12) can be checked as follows: we perform the change of variables,

t = y -x, s = x + y, (2.13)
and define G(s, t) := k(x, y).

(2.14)

Then equation (2.9) of k becomes the following equation of G,

           6G tts + 2G sss + 2G s + λG = 0 in T 0 , G(s, 2L -s) = 0 on [L, 2L], G(s, 0) = 0 on [0, 2L], G t (s, 0) = λ 6 (s -2L) on [0, 2L], (2.15)
where

T 0 := {(s, t); t ∈ [0, L], s ∈ [t, 2L -t]}. From (2.

15), one easily gets

G tts (s, 0) = 0 in [0, 2L]. (2.16)
Hence, G tt (s, 0) = G tt (2L, 0). In order to calculate G tt (2L, 0), one observes from (2.15) that

G tt (2L, 0) = 2G ts (2L, 0) = λ 3 .
(2.17)

Direct calculations show that

k xy (x, x) = -G tt (s, 0) = - λ 3 , (2.18)
which concludes (2.11). The proof of (2.12) is similar. Now, let us define a continuous transformation Π λ :

L 2 (0, L) → L 2 (0, L) by ω(x) = Π λ (u(x)) := u(x) - L x k(x, y)u(y)dy. (2.19)
Moreover, its inverse is given by (let us denote by Π -1 λ )

u(x) = Π -1 λ (ω(x)) := ω(x) + L x l(x, y)ω(y)dy. (2.20)
That is because k(x, y) and l(x, y) are related by the formula

l(x, y) -k(x, y) = y x k(x, η)l(η, y)dη. (2.21)
Actually, one can define l(x, y) := k(x, y)

+ y x k(x, η)l(η, y)dη. (2.22)
Hence one only needs to prove l = l to get (2.21). Direct calculations show that l satisfies

               lxxx + lyyy + lx + ly -λl = 0 in T , l(x, L) = 0 on [0, L], l(x, x) = 0 on [0, L], lx (x, x) = λ 3 (L -x) on [0, L], lxy (x, x) = -λ 3 on [0, L]. (2.23)
Let us define l 0 := l -l. From (2.23), (2.10), and (2.12), one knows that

               (l 0 ) xxx + (l 0 ) yyy + (l 0 ) x + (l 0 ) y = 0 in T , (l 0 )(x, L) = 0 on [0, L], (l 0 )(x, x) = 0 on [0, L], (l 0 ) x (x, x) = 0 on [0, L], (l 0 ) xy (x, x) = 0 on [0, L]. (2.24)
Regarding to the Cauchy problem (2.24), we have the following proposition (hence l = l), whose proof is given in Appendix A.

Proposition 1. The equation (2.24) has a unique solution in C 3 (T ). More precisely, this solution is l 0 = 0.

Remark 3. This proposition is important to this paper. In the following section we construct precisely a solution to equation (2.9)(and of (2.10) respectively), the proof of Theorem 1 relies on some estimates of this solution. Proposition 1 ensures the solution that we construct satisfies (2.21) (hence (2.20)).

We find that by using the transformation Π λ , the solution of (1.1) with control

κ(t) = L 0 k(0, y)u(t, y)dy, (2.25)
is mapped to a solution of the system

     ω t + ω xxx + ω x + λω = 0 in (0, +∞) × (0, L), ω(t, L) = ω x (t, L) = 0 on (0, +∞), ω(t, 0) = 0 on (0, +∞).
(2.26)

For system (2.26), one can easily obtain exponential decay of the solution

ω(t, •) L 2 (0,L) e -λt ω(0, •) L 2 (0,L) . (2.27)
Hence the solution of (1.1) with feedback law (2.25) satisfies

u(t, •) L 2 (0,L) Π -1 λ L 2 (0,L)→L 2 (0,L) ω(t, •) L 2 (0,L) e -λt Π -1 λ L 2 (0,L)→L 2 (0,L) ω(0, •) L 2 (0,L) e -λt Π -1 λ L 2 (0,L)→L 2 (0,L) Π λ L 2 (0,L)→L 2 (0,L) u(0, •) L 2 (0,L) .
(2.28)

From now on, we simply denote Π -1 λ L 2 (0,L)→L 2 (0,L) by Π -1 λ to simplify the notations.

Well-posedness of system (2.26)

For a positive parameter λ > 0, we consider the following linear operator

A λ : D(A λ ) ⊂ L 2 (0, L) → L 2 (0, L) with D(A λ ) := {f ∈ H 3 (0, L); f (0) = f (L) = f x (L) = 0}, (2.29)
A λ := -f x -f xxx -λf, ∀f ∈ D(A λ ). (2.30)
Similarly, for the case where λ = 0 (see [15, page 38-43]), the following properties also hold.

D(A λ ) is dense in L 2 (0, L), (2.31)
A λ is closed, (2.32)
A λ and A * are dissipative.

(2.33)

Hence, A λ generates a strongly continuous semigroup of linear operator {S λ (t)} t 0 on L 2 (0, L). Furthermore, for every initial data ω 0 ∈ D(A λ ), system (2.26) has one and only one solution ω(t, x) ∈ C 0 ([0, +∞); L 2 (0, L)). This solution also satisfies

ω ∈ C 1 ([0, +∞); L 2 (0, L)) ∩ C 0 ([0, +∞); D(A λ )), (2.34
)

ω x (•, 0) L 2 (0,T ) ω 0 L 2 (0,L) , ∀T > 0, (2.35) ω L 2 (0,T ;H 1 (0,L)) C T ω 0 L 2 (0,L)
, where C T only depends on T > 0.

(2.36)

By standard approximation arguments, it follows that when ω 0 ∈ L 2 (0, L) equation (2.26) has a unique solution ω(t, x) ∈ C 0 ([0, +∞); L 2 (0, L)). This solution also satisfies (2.35) and (2.36). For more details on the results and proofs of this subsection, one can refer to Coron's book [15, page 38-43 and page 374-377]. Although the book only describes the case when λ = 0, the general case λ > 0 follows by considering e λt ω.

Control design

Inspired by the work of Coron and Nguyen in [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF], we construct a piecewise control such that on each piece, the solution of (1.1) can be transformed to a solution of (2.26). More precisely, we select {λ n } n∈N , increasing positive numbers that tends to infinity, (

{t n } n∈N , increasing numbers with t 0 = 0 that tends to T as n tends to infinity. Then, for t n < t t n+1 , we successively define

u(t) := Π -1 λn S λn (t -t n )Π λn u(t n ), (3.4)
κ(t) := L 0 k λn (0, y)u(t, y)dy, (3.5)
where S λn is the semigroup given in Section 2.3. One has the following lemma, whose proof is given at the end of this section.

Lemma 2. As defined in (3.3)-(3.4), u(t)| tn t t n+1 is a solution of (2.1) with T 1 = t n , T 2 = t n+1
, h = 0, and κ(t) given by (3.5).

Let us define

s 0 := 0 and

s n := n-1 k=0 λ k (t k+1 -t k ) for n 1, (3.6)
thanks to (3.4) and (2.28), we get

u(t, •) L 2 (0,L) e -sn u 0 L 2 (0,L) n k=0 Π -1 λ k Π λ k , (3.7
)

|κ(t)| e -sn u 0 L 2 (0,L) k λn (0, •) L 2 (0,L) n k=0 Π -1 λ k Π λ k (3.8) for t ∈ [t n , t n+1 ].
Hence, if we have a good estimation on k λ , it will be possible to get u(t) → 0 when t tends to T . Actually, we have the following estimates. Lemma 3. Let λ > 2, then (2.9) has a unique solution k λ ∈ C 3 (T ) (respectively (2.10) has a unique solution l λ ∈ C 3 (T )). Those solutions also satisfy

k λ C 0 (T ) e (1+L) 2 √
λ and l λ C 0 (T ) e (1+L) 2 √ λ .

(3.9)

Proof of Lemma 3. The existence of solution to (2.9) is given in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]. The uniqueness of the solution is proved in Appendix A. Here we focus on the C 0 norm estimate (3.9). Take the following change of variable,

t = y -x, s = x + y (3.10)
and define G(s, t) := k(x, y).

(3.11)

Then we transform (2.9) into an integral equation of G(s, t) (see formula [START_REF] Coron | Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right[END_REF] in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]),

G(s, t) = - λt 6 (2L -t -s) + 1 6 2L-t s t 0 τ 0 (2G sss + 2G s + λG) (η, ξ)dξdτ dη, (3.12)
in

T 0 := {(s, t); t ∈ [0, L], s ∈ [t, 2L -t]}.
We use a successive approximation to give a solution of the equation (3.12). Thanks to Proposition 1, this solution is the unique solution of (3.12). Let us take

G 1 (s, t) := - λt 6 (2L -t -s) (3.13)
and define

G n+1 (s, t) = 1 6 2L-t s t 0 τ 0 (2G n sss + 2G n s + λG n ) (η, ξ)dξdτ dη. (3.14)
For instance,

G 2 (s, t) = 1 108 t 3 λ -λ 2 L + λ 2 t 4 (2L -t -s) + t 3 λ 2 4 [(2L -t) 2 -s 2 ] . (3.15)
But unfortunately, we can not perform such explicit calculation by hand each time. We try to estimate G n (s, t) from another way. Notice that if f (s, t) := g(s)h(t), then

2L-t s t 0 τ 0 ∂ m ∂s m f (η, ξ)dξdτ dη = 2L-t s t 0 τ 0 ∂ m ∂s m g(η)h(ξ)dξdτ dη = 2L-t s ∂ m ∂s m g(η)dη t 0 τ 0 h(ξ)dξdτ. (3.16)
Let P be the space of polynomials of one variable on R. We define operator T by

T : P ⊗ P → P ⊗ P g(s)h(t) → 1 6 2L-t s t 0 τ 0 2 ∂ 3 ∂s 3 + 2 ∂ ∂s + λId (g • h)(η, ξ)dξdτ dη. (3.17)
Equality (3.16) shows that (3.17) is well defined. In fact

T g(s)h(t) = g T (s, t)h T (t), (3.18)
where g T (s, t) and h T (t) are given by g T (s, t) := 1 6

2L-t s 2 ∂ 3 ∂s 3 + 2 ∂ ∂s + λId (g)(η)dη, (3.19)
h T (t) := t 0 τ 0 h(ξ)dξdτ. (3.20)
Observe that, if

|h(t)| t n , ∀ 0 t L, (3.21) then |h T (t)| = | t 0 τ 0 h(ξ)dξdτ | 1 (n + 1)(n + 2) t n+2 , for t ∈ [0, L]. (3.22)
As for g T (s, t), notice that if g(s) = s m with m 3, then

g T (s, t) = 1 6 2m(m -1)(2L -t) m-2 -2m(m -1)s m-2 + 2(2L -t) m -2s m + λ(2L -t) m+1 m + 1 - λs m+1 m + 1 . (3.23)
This inspires us to separate T into the following 6 linear operators {T i } 1 i 6 from P ⊗ P to P ⊗ P.

T 1 : s m h(t) → 1 6 2m(m -1)(2L -t) m-2 h T (t), when m 3, 0, when 0 m 2, (3.24)
T 2 : s m h(t) → -1 6 2m(m -1)s m-2 h T (t), when m 3, 0,
when 0 m 2, (3.25) For any n ∈ N * , for any a = (a(1), a(2), ..., a(n)) ∈ A n , we define the operator

T 3 : s m h(t) → 1 6 2(2L -t) m h T (t), when m 1, 0, when m = 0, (3.26)
T a := T a(n) T a(n-1) ...T a(1) . (3.35)
We define additionally A := {a 0 } and T a 0 := Id (identity operator on P).

Hence for any n ∈ N, we have

G n+1 (s, t) = T n I + J = a∈An (T a I) + a∈An (T a J) . (3.36)
Now we use mathematical induction to conclude the following lemma.

Lemma 4. For every λ > 2, for every n ∈ N and for every a ∈ A n , T a I and T a J are of the form s l h(t). They also satisfy Hence for any a := (a(1), a(2), ..., a(k + 1)) ∈ A k+1 , we have

|h(t)| ( λ 6 ) n+1 t 2n+1 (2n + 1)! (2L + 1) n+1-l 1 l! , t ∈ [0, L]. ( 3
T a = T a(k+1) T (a) . (3.40)
From the assumption, we know that

T (a) I = s l h(t). (3.41)
If T (a) I = 0, then we conclude the proof.

If T (a) I = s l h(t), then we know from (3.37) that

|h(t)| λ 6 k+1 t 2k+1 (2k + 1)! (2L + 1) 1+k-l 1 l! , t ∈ [0, L]. (3.42)
Let us first consider T 1 . We know that T 1 (s l h(t)) = 0 if l 2. Therefore, it suffices to prove the case when l 3. From (3.20)-(3.22), (3.24), and (3.42), we know that

|T 1 (s l h(t))| = 2 6 l(l -1)(2L -t) l-2 h T (t) 2 6 l(l -1)(2L -t) l-2 λ 6 1+k t 2k+3 (2k + 3)! (2L + 1) 1+k-l 1 l! λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) k-1 1 (l -2)! . (3.43)
Notice that T 1 (s l h(t)) can be written as s 0 g(t). Thus, it can be seen from (3.43) that (3.37) is satisfied.

By using the same procedure, we can check that

|T 2 (s l h(t))| λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) 1+k-l s l-2 (l -2)! , (3.44) |T 3 (s l h(t))| λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) 1+k 1 l! , (3.45) |T 4 (s l h(t))| λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) 1+k-l s l l! , (3.46
)

|T 5 (s l h(t))| λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) k+2 1 (l + 1)! , (3.47)
|T 6 (s l h(t))| λ 6 k+2 t 2k+3 (2k + 3)! (2L + 1) 1+k-l s l+1 (l + 1)! . (3.48)
Hence, we complete the proof.

By the same idea of partition and Lemma 4, we can further obtain the following estimate.

Lemma 5. For every λ > 2, for every n ∈ N, and for every a ∈ A n , T a I and T a J are of the form s l h(t). They also satisfy

|∂ t (h(t))| 2 λ 6 n+1 t 2n (2n)! (2L + 1) n+3-l 1 l! , t ∈ [0, L]. (3.49)
Remark 5. One can get similar estimates for C 2 -norm or even C n -norm. However, since in this paper we do not need to use such estimates, this part is omitted. Actually, C 2 -norm estimates can be obtained directly by Lemma 4 (as the way of getting Lemma 5), but the C nnorm (with n ≥ 3) is more complicated. Furthermore, it can be seen from [START_REF] Krstic | Boundary control of PDEs, volume 16 of Advances in Design and Control[END_REF] that for the heat equation the kernel is analytic in the triangle. It is of interest to know if the kernel we obtained in this article is also analytic.

We come back to the estimate (3.36). From Lemma 4, we know that, for every n ∈ N, for every a ∈ A n , for every m ∈ N, and for t ∈ [0, L] we have The same approach shows that the series

∂ m ∂s m T a I (s,
+∞ n=1 ∂ m+1 ∂t∂s m G n (s, t) is uniformly convergent in T 0 . (3.54)
We define

G(s, t) := +∞ n=1 G n (s, t), (3.55)
which is the solution of (3.12) (see [9,

√ (2L+1)λ 3 e (1+L) 2 √ λ , in T 0 . (3.57)
It only remains to prove that G(s, t) ∈ C 3 (T 0 ). Actually, from (3.53) and (3.54) we know that it suffices to prove G tt , G tts , G ttt ∈ C 0 (T 0 ). We know from (2.15) that G tts ∈ C 0 (T 0 ). As for G tt and G ttt , thanks to (3.12), we get

6G t (s, t) = -λ(2L -s -2t) - t 0 τ 0 (2G sss + 2G s + λG) (2L -t, ξ)dξdτ + 2L-t s t 0 (2G sss + 2G s + λG) (η, ξ)dξdη, 6G tt (s, t) = 2λ + t 0 τ 0 (2G ssss + 2G ss + λG s ) (2L -t, ξ)dξdτ - t 0 (2G sss + 2G s + λG) (2L -t, ξ)dξ - t 0 (2G sss + 2G s + λG) (2L -t, ξ)dξ + 2L-t s (2G sss + 2G s + λG) (η, t)dη, 6G ttt (s, t) = - t 0 τ 0 (2G sssss + 2G sss + λG ss ) (2L -t, ξ)dξdτ + t 0 (2G ssss + 2G ss + λG s ) (2L -t, ξ)dξ -2 (2G sss + 2G s + λG) (2L -t, t)dξ + 2 t 0 (2G ssss + 2G ss + λG s ) (2L -t, ξ)dξ + 2L-t s (2G ssst + 2G st + λG t ) (η, t)dη -(2G sss + 2G s + λG) (2L -t, t).
The above formulas together with (3.53)- (3.54) give the continuity of G tt and of G ttt . Hence we complete the proof of Lemma 3.

Remark 6. As we can see from [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF][START_REF] Krstic | Boundary control of PDEs, volume 16 of Advances in Design and Control[END_REF] for the heat equation, the L ∞ -norm of the kernel k λ is of the form e C √ λ . One may naturally ask if the sharp estimate on the L ∞ -norm of kernel k λ is of the form e C 3 √ λ for the KdV case, as KdV is of order 3. However, we do not know how to get such estimates.

At last, it remains to give the proof of Lemma 2.

Proof of Lemma 2. It is equivalent to prove the following statement:

Given u 0 ∈ L 2 (0, L), λ > 0, s > 0, one has (S) κ(t) := L 0 k λ (0, •) Π -1 λ S λ (t)Π λ u 0 (•)dy ∈ L 2 (0, s), and
u(t) := Π -1 λ S λ (t)Π λ u 0 is the solution of (2.1) on [0, s] with h = 0.
We only need to prove the case where u 0 ∈ Π -1 λ D(A λ ), since standard approximation methods then lead to the general case of (S). From Section 2.3, we know that

ω(t) := S λ (t)Π λ u 0 ∈ C 0 ([0, s]; D(A λ )) ∩ C 1 ([0, s]; L 2 (0, L)), (3.58)
which shows that κ(t) ∈ C 0 ([0, s]). Direct calculations, based on (2.10), show that (similar to page 1690 in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF])

u t = ω t + L x (l y + l yyy -λl)(x, z)ω(z)dz -l y (x, x)ω x (x) + l yy (x, x)ω(x), (3.59)
u x = ω x (x) + L x l x (x, z)ω(z)dz, (3.60)
u xx = ω xx (x) -l x (x, x)ω(x) + L x l xx (x, z)ω(z)dz, (3.61)
u xxx = ω xxx -l xx (x, x) + l xy (x, x) ω(x) -l x (x, x)ω x (x) -l xx (x, x)ω(x) + L x l xxx (x, z)ω(x)dz, (3.62)
all these calculations hold on C 0 ([0, T ]; L 2 (0, L)). From (3.59)-(3.62) and (2.10), we know that

u(t, x) ∈ C 1 ([0, s]; L 2 (0, L)) ∩ C 0 ([0, s]; H 3 (0, L)), (3.63)
u t + u x + u xxx = 0, in L 2 (0, L), (3.64)
u(t, L) = u x (t, L) = 0, (3.65) u(t, 0) = κ(t), (3.66)
which show that u satisfies Definition 1.

Remark 7. In fact, by using (2.9) and the hidden inequality (2.35), we can also prove that κ(t) ∈ H 1 (0, s) with its norm controlled by u 0 L 2 (0,L) .

Null controllability

Finally, we are able to prove the null controllability (Theorem 1) by constructing a piecewise continuous bounded control. The construction is explained in Section 3. Thanks to Lemma 2 and (3.1)-(3.8), we only need to find good sequences {λ n } n∈N and {t n } n∈N such that:

lim n→+∞ e -sn n k=0 Π -1 λ k Π λ k = 0, (4.1)
lim n→+∞ e -sn k λn (0, •) L 2 (0,L)

n k=0 Π -1 λ k Π λ k = 0, (4.2)
and that

u(t)| 0 t T is
Π λ L 2 (0,L)→L 2 (0,L) (1 + L k λ C 0 (T ) ) e 2(1+L) 2 √ λ , (4.4)
Π -1 λ L 2 (0,L)→L 2 (0,L) (1 + L l λ C 0 (T ) ) e 2(1+L) 2 √ λ , (4.5)
k λ (0, •) L 2 (0,L) √ L l λ C 0 (T ) e 2(1+L) 2 √ λ , (4.6)
where Lemma 3 is used. Hence it suffices to select {λ n } n∈N and {t n } n∈N such that Remark 8. To deal with the heat equations (by using backstepping approach), one needs to study the wave equation instead, which is already well investigated. In this article, we study the KdV system which has an order of 3. Hence the kernel system (see (A.1)) becomes a third order"wave-like" equation. For this reason, we encountered some difficulties: Lemma 3 for estimation and Proposition 1 for uniqueness. We believe that the Coron-Nguyen method, as well as the techniques introduced in this paper, could be used for other systems whose order is greater than 3. As we know, the backstepping method is well used on the rapid stablization problem of first-order hyperbolic systems (see [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]). Unfortunately, as they are of order one, we are not sure if some good estimates could be obtained for the null controllability or even the finite time stabilization. However, looking for [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF][START_REF] Hu | Boundary exponential stabilization of 1-d inhomogeneous quasilinear hyperbolic systems[END_REF][START_REF] Tsien | Global classical solutions for quasilinear hyperbolic systems[END_REF], this might be possible for quasilinear hyperbolic systems.

Further comments

The above procedure has the following advantages.

• The null controllability is precisely obtained by an explicit piecewise continuous (actually piecewise H 1 ) bounded control instead of some unknown H 1/3 control.

• The well-posedness results and the estimates investigated in this paper should allow smalltime stabilization to be proven (instead of rapid stabilization in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]).

• The backstepping approach as used in [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF] together with the techniques introduced in this paper could be applied to more models, as the backstepping method was widely used in the last 20 years for different models.

We also want to point out a list of open problems which could be further studied.

• The (global or local) controllability and the (global or local) small-time stabilization of the nonlinear KdV equations.

Based on the linear result and the Kato smoothing effect, we may expect the local controllability by standard perturbation. Actually, as it is shown in [START_REF] Roberto | Asymptotic behavior of Boussinesq system of KdV-KdV type[END_REF][START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Coron | Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right[END_REF][START_REF] Marx | Output feedback stabilization of the Korteweg-de Vries equation[END_REF] that the backstepping method can be directly used to treat the nonlinear case. But as the main purpose of this paper is to extend the new method found by Coron and Nguyen to more general models (as we stated in the Introduction), we do not consider nonlinear cases here. However, since there is only one scalar control on the boundary, the global controllability in small time is a real challenge and might be false, as it is the case for the Burgers equations [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF]. If there are more controls there are several global controllability results in small time for the nonlinear KdV equations [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] and for the Burgers equations [START_REF] Chapouly | Global controllability of nonviscous and viscous Burgers-type equations[END_REF][START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF][START_REF] Fernández | Null controllability of the Burgers system with distributed controls[END_REF][START_REF] Gagnon | Lagrangian controllability of the 1-dimensional Korteweg-de Vries equation[END_REF][START_REF] Glass | On the uniform controllability of the Burgers equation[END_REF][START_REF] Marbach | An obstruction to small time local null controllability for a viscous Burgers' equation[END_REF][START_REF] Marbach | Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer[END_REF]. Note that, using the backstepping approach, [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF] allows to recover the global controllability result of [START_REF] Chapouly | Global controllability of nonviscous and viscous Burgers-type equations[END_REF] obtained by means of the return method.

It would be interesting to see if [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF] can be adapted to nonlinear KdV equations.

• In [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Lebeau | Contrôle exacte de l'équation de la chaleur[END_REF], using a "microlocal" approach and Carleman's inequalities, Lebeau and Robbiano proved some e C √ λ type estimates and then deduced from these estimates the controllability of the heat equation. From a "global" point of view, we also obtain the e C √ λ type estimates (Lemma 3), and also get the controllability. An important and interesting question: is there any connection between the backstepping method, Lebeau-Robbiano's strategy, Carleman inequalities, and small-time (local) stabilization?

• Let us also recall open problems raised in Remark 5 about the analytic regularity of the kernel and Remark 6 about sharp kernel estimates. For the sharp L ∞ -norm (or even C nnorm) estimates, considering [START_REF] Chu | Gevrey Class Regularity of a Semigroup Associated with a Nonlinear Korteweg-de Vries Equation[END_REF], perhaps it would be more natural to consider Gevrey class regularity instead of the analytic regularity.

If {ψ n (y)} n is complete in L 2 (0, L), then h(x, 0) is 0. However, we don't know the completeness of the eigenfunctions {ψ n (y)} n . More generally, one could consider eigenfunctionals or even generalized eigenfunctions, following Gel'fand and the coauthors [START_REF] Israel | Generalized functions[END_REF]31]. More precisely, in the non-self-adjoint cases it is not always possible to expand a function as the sum of eigenfunctions. In order to avoid this problem, one uses different generalisations of eigenfunctions. For example, the generalisation introduced by Fokas, augmented eigenfunctions, which is itself a generalisation of Gel'fand's eigenfunctions (allow the appearance of remainder functionals). This generalisation turns out to be a powerful tool to investigate the initial-boundary value problem (IBVP). One can find an almost complete investigation from the papers [START_REF] Locker | Spectral theory of non-self-adjoint two-point differential operators[END_REF][START_REF] Locker | Eigenvalues and completeness for regular and simply irregular two-point differential operators[END_REF][START_REF] Smith | Evolution PDEs and augmented eigenfunctions[END_REF]. In general, let Φ be a function space defined on the closure of a real interval I with sufficient smoothness and decay conditions, L be a linear operator defined on Φ. Let γ be an oriented contour in C, and let E = {E λ : λ ∈ γ} be a family of functionals (imagine as a family of eigenfunction when γ is only defined on a discrete set). Then the corresponding remainder functionals R λ ∈ Φ with respect to eigenvalues λ is R λ (φ) := λ n E λ (φ) -E λ (Lφ), ∀φ ∈ Φ, ∀λ ∈ γ.

(A.13)

One is interested in the cases in which one of the following two conditions is satisfied, As we can see above, the study of augmented eigenfunctions involves complicated asymptotic calculations. In Fokas' work this method is only used to study the evolution equations based on a good transform pair, which does not seem to be a good (easy) option to our problem (A.2). Instead of augmented eigenfunctions, Locker [START_REF] Locker | Eigenvalues and completeness for regular and simply irregular two-point differential operators[END_REF] also considered the generalized eigenspace E G given by N ((λ i I -L) m i ), union for all m i ∈ N, and λ i eigenvalues,

where L denotes the operator, N denotes the kernel. More precisely, to the linearized KdV operator he proved the following. the generalized eigenfunction space E G is complete in L 2 (0, L) space when β = 0.

Remark 9. When β = 0, it does not seem to be known whether generalized eigenfunction space E G is complete in L 2 (0, L). This is one of the reasons that much more complicated augmented eigenfunctions are introduced (the other reasons are about the regularities and some more general boundary conditions). Actually, this case can be regarded as a limit of the cases when the coupling constant β approaches 0.

In fact, in [START_REF] Locker | Eigenvalues and completeness for regular and simply irregular two-point differential operators[END_REF] Locker only considered the operator Lf := f xxx . One can easily verify with the same proof that the same result holds when there is an additional f x in the operator.

Remark 4 .

 4 Inequality (2.35) is a hidden inequality. It was first found by Rosier in[START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Inequality (2.36) is the Kato smoothing effect.

(3. 2 )

 2 First we define u(0) := u 0 and u(T ) := 0. (3.3)

. 37 )

 37 Proof of Lemma 4. When n = 0, one can check that Lemma 4 holds. Let us suppose that Lemma 4 holds when n = k 0. Then we can check in the rest of the proof that Lemma 4 holds when n = k + 1. For any n 1, and for any a := (a(1), a(2), ..., a(n + 1)) ∈ A n+1 , let us define (a) := (a(1), a(2), ..., a(n)). (3.38) For any a ∈ A 1 , let us define (a) := a 0 . (3.39)

7 )

 7 Inspired by the choices given by Coron and Nguyen in [23, Proposition 1], we select t n := T -1/n 2 and λ n := 2n 8 . One easily verifies that (4.7) holds, which completes the proof.

λe

  iλx R λ (φ)dλ = 0, ∀φ ∈ Φ, ∀x ∈ I, (A.14) or λ e iλx λ n R λ (φ)dλ = 0, ∀φ ∈ Φ, ∀x ∈ I, (A.15)where (A.14) (resp. (A.15)) is called the type I (resp. type II) condition of augmented eigenfunctions of L up to the integration along γ.

Theorem 2 .

 2 Let L > 0. For the differential operator Lf := f xxx +af x with boundary conditionsf (0) = f (L) = 0, (A.16) f x (0) + βf x (L) = 0, (A.17)

Acknowledgments. The author would like to thank Jean-Michel Coron for having attracted his attention to this problem, for his constant support, and for fruitful discussions. He also thanks Amaury Hayat, Qi Lü, Peipei Shang, Daniel Wong, Bingyu Zhang, and Christophe Zhang for discussions on this problem. This work was supported by LIASFMA and ANR Project Finite4SoS (ANR 15-CE23-0007).

A Appendix: Proof of Proposition 1

In this part, we give the proof of the uniqueness of the solution to equation (2.24). As the function is defined in the triangle T , we extend l 0 by 0 in the lower triangle [0, L] × [0, L] \ T , and denote by h the extended function. Since on the diagonal x = y, C 3 (T ) function l 0 satisfies l 0x = l 0y = l 0xx = l 0xy = l 0yy = 0, the extended function is H 3 ([0, L] × [0, L]). Moreover, h satisfies

By simple change of variables, x = L -x and ỹ = L -y, it suffices to prove that the solution

2) is similar to the wave equation, it is natural to consider eigenfunctions of the operator (with respect to y variable),

If the eigenfunctions, {ϕ n (y)} n , form a Riesz basis of L 2 (0, L), then the Fourier series decomposition h(x, y)

easily infers the uniqueness required. Unfortunately, this operator is a non-self-adjoint operator and eigenfunctions do not form a Riesz basis, see [START_REF] Vassilis | An example where separation of variables fails[END_REF].

Another idea is to investigate the completeness of eigenfunctions, {ψ(y) n } n , of the adjoint operator A * y ,

Actually, suppose that {ψ n (y)} n is an eigenfunction of the adjoint operator A * y , then from (A.2) as well as the boundary conditions of h and ψ one can deduce that

Combine (A.10) with the fact that

In order to solve our problem, we use another kind of generalized eigenfunctions, which is more general than E G but is less general than augmented eigenfunctions, namely eigenfunctions and associated functions (e.a.f.). The definition of e.a.f., which is defined on equations with λ as parameter, is rather complicated. One can see [51, chapter 1] and [START_REF] Naȋmark | Linear differential operators. Part II: Linear differential operators in Hilbert space[END_REF] for precise description on this subject.

With eigenfunctions and associated functions, Shkalikov in [START_REF] Škalikov | The completeness of the eigen-and associated functions of an ordinary differential operator with nonregular splitting boundary conditions[END_REF] proved the following theorem.

Theorem 3. The eigenfunctions and associated functions of the boundary-value problem generated by an ordinary differential equation with separated boundary conditions l(y) -λ n y = y (n) + p n-2 (x)y (n-2) + ... + p 0 (x)y -λ n y = 0, (A.18)

α jk y (k) (0) = 0, with j = 1, 2, ..., l, (A. [START_REF] Coron | Rapid stabilization of 1-D linear Schrödinger equations[END_REF])

β jk y (k) (L) = 0, with j = 1, 2, ..., n -l, (A.20) form a complete system in the space L 2 [0, L], where p i (x) are arbitrary summable functions, and l > n -l > 0.

Applying Theorem 3 to our case (linearised KdV), we get Corollary 1. For the ordinary differential equation with separated boundary conditions

the eigenfunctions and associated functions form a complete system in the space L 2 [0, L].

Finally, we are able to prove Proposition 1.

Proof of Proposition 1. Let us consider the boundary-value problem (A.21)-(A.24). Let λ 0 be an eigenvalue, and let ϕ 0 (y) = ϕ(y) an eigenfunction for the eigenvalue λ 0 . The associated functions associated with the eigenfunction ϕ(y) are given by the functions ϕ 1 (y), ϕ 2 (y), ..., ϕ k (y).

These functions satisfy (the boundary conditions)

and, for λ = λ 0 , the following relations

Now we prove that for all those functions (e.a.f.), we have

At first, for ϕ 0 , as what we have done in (A.12), clearly

For ϕ 1 , equation (A.26) shows that

Hence from (A.2), (A.25), (A.28), and (A.29) we get

By using the the fact that

Repeating this procedure we get (A.27), which combined with Corollary 1 shows that

Hence the proof of Proposition 1 is completed.

Remark 10. For the y-variable, we only used 3 boundary conditions in the proof to deduce the uniqueness of the solution h: h(x, L) = h y (x, L) = h(x, 0) = 0. This is natural, since once we consider 4 boundary conditions (for a third order differential operator), the eigenfunctions could never become a basis. We may also wonder, if we can get the uniqueness of h by using the other 3 boundary conditions in y-variable: h(x, L) = h y (x, L) = h yy (x, L) = 0? Unfortunately, Theorem 3 can not be applied for these 3 boundary conditions: we observe from (A. [START_REF] Coron | Rapid stabilization of 1-D linear Schrödinger equations[END_REF]) and (A.20) that there should be boundary conditions on both side. Hence, it is difficult to get the uniqueness of h by using the Carleman estimate, see [START_REF] Alabau-Boussouira | Control of partial differential equations[END_REF]Chapter 4], [START_REF] Carleman | Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes[END_REF] and [START_REF] Hörmander | On the uniqueness of the Cauchy problem under partial analyticity assumptions[END_REF], though the Carleman estimate is a standard way to solve the unique continuation problem.


Dr Jean-Luc L Schwartz
email: schwartz@icp.inpg.fr

Frédéric Berthommier

Christophe Savariaux

Seeing to hear better: Evidence for early audio-visual interactions in speech identification

Keywords: Lipreading, audio-visual interactions, speech detection, speech identification, temporal cues

. However, detection is different from comprehension, and it remains to be seen whether improved sensitivity also results in an intelligibility gain in audio-visual speech perception. In this work, we use an original paradigm to show that seeing the speaker's lips enables the listener to hear better and hence to understand better. The audio-visual stimuli used here could not be differentiated by lip reading per se since they contained exactly the same lip gesture matched with different compatible speech sounds. Nevertheless, the noise-masked stimuli were more intelligible in the audio-visual condition than in the audio-only condition due to the contribution of visual information to the extraction of acoustic cues. Replacing the lip gesture by a non-speech visual input with exactly the same time course, providing the same temporal cues for extraction, removed the intelligibility benefit. This early contribution to audio-visual speech identification is discussed in relationships with recent neurophysiological data on audio-visual perception.

Introduction

Lip reading, that is the visual identification of speech gestures from the moving face, improves the intelligibility of speech in noise when audio-visual perception is compared with audio-only perception [START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF][START_REF] Erber | Auditory-visual perception of speech[END_REF]. Another part of our experience as listeners is that speech sounds "louder" when we look at the speaker, as if audio cues were visually enhanced. This has been demonstrated in auditory detection experiments, where the auditory threshold was decreased by 1 to 2 dBs when the sound was accompanied by a lip gesture [START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF]. Analyses of correlations between area of mouth opening and energy fluctuations in different spectral bands showed that for utterances with larger correlation, the visually-driven detection gain was larger: hence, the auditory-visual temporal coherence was basic there [START_REF] Grant | The effect of speechreading on masked detection thresholds for filtered speech[END_REF]Kim & Davis, 2003a).

The question that arises is whether this "speech enhancement" effect due to visual information also improves the intelligibility of speech in noise. In other words, does the gain in detection result in a gain in intelligibility? The answer to this question is not trivial, since these tasks are quite different, and probably involve different mechanisms in the human brain. Searching experimental evidence raises a serious problem. Indeed, it is very difficult in a speech comprehension task to disentangle the contribution of lip reading per se from the potential contribution of an additional mechanism, that is visually-guided sensitivity enhancement of audio cues. The primary objective of this work is to propose and test an original paradigm in order to demonstrate the existence of such an additional component to audio-visual speech intelligibility, different from lip-reading (Experiments 1 and 2). In addition, the "speech-specific" nature of this component is tested in Experiment 3.

The potential contribution of visually-guided sensitivity enhancement of audio cues would be displayed if we could show an intelligibility gain due to visual information using a paradigm that completely eliminated the contribution of lip reading. Our experiments were designed to do just that: we examined the audio-visual identification of speech stimuli with identical lip gestures, embedded in noise. The addition of noise to the stimuli is intended to at least partially simulate real listening conditions-conversations do not typically take place in sound booths and natural settings are often quite noisy. The ability to focus one's attention and understand speech in such conditions has been called the "cocktail party effect" [START_REF] Cherry | Some experiments on the recognition of speech, with one and two ears[END_REF]. In our design, the visual stimulus contains no information about the phonetic content of the sound: it just provides potential cues about when and possibly where (in frequency) the auditory system should expect useful information. We selected the ten French syllables:

[y], [u], [ty], [tu], [ky], [ku], [dy], [du],
[gy], [gu]. All these stimuli are associated with basically the same lip gesture towards a rounded vowel (either [y], the vowel in tu 'you ' or [u], the vowel in tout 'all'). They involve a "mode" contrast between a voiced or unvoiced plosive consonant (e.g., [ty] vs. [dy]) or no plosive consonant at all ([y]); a "consonant place of articulation" contrast between dentals [t d] and velars [k g]; and a "vowel place of articulation" contrast between front [y] and back [u]. If visual information improves intelligibility, this improvement is not likely to be due to visual information per se, since the stimuli are visually similar. Rather, the improvement should be mainly due to visually enhanced detection of acoustic cues.

Experiment 1: Displaying a visual enhancement with visually ambiguous stimuli

Method Experiment 1 tested the potential contribution of visual information using natural stimuli. A French male speaker recorded each of the 10 stimuli [y u ty tu ky ku dy du gy gu] three times in a random order, with variable inter-stimulus intervals (between 1 and 4 s). This variable temporal rhythm ensured that the time of presentation of each syllable was quite unpredictable. A cocktailparty crowd noise was added to the sound signal, with a signal-to-noise ratio of approximately -9dB (measured as the ratio of the mean power of the vocalic portions of the stimuli to the mean noise power). This manipulation severely decreased the audio intelligibility of the stimuli, while preserving the listener's ability to detect the presence of each stimulus syllable because of the high energy in the vowel nucleus. A panel of eight French subjects with no reported hearing problems then completed an identification task in three conditions: audio-only (A), video-only (V) or audio-visual (AV) presentation (Fig. 1a). Half completed the audio-visual (AV) condition, then the audio-only (A) one, half did the reverse order. All subjects completed the visual (V) condition last. They were asked to identify each stimulus and repeat it aloud.

Results

The responses of the subjects were grouped into confusion matrices, from which we computed global A, V or AV identification scores thanks to formula (1) (corrected scores to account for random answers, see [START_REF] Robert-Ribes | Complementarity and synergy in bimodal speech: auditory, visual and audiovisual identification of French oral vowels in noise[END_REF]:

corrected score number correct answers number simuli number categories number categories _ _ _ _ _ _ = - - 1 1 1 (1)
Corrected scores vary between 0 for random responses (or possibly less than 0) to 1 for all correct answers. We also analysed the responses in terms of the identified mode (no plosive vs. voiced plosive vs. unvoiced plosive), plosive place (dental vs. velar) and vowel place (front vs. back).

All the scores are reported in Table 1. It appears that they are all quite low, except for mode in the AV condition. Further analysis of the mode confusion matrices showed that the only distinction that was partly identified was the presence vs. absence of a voiced plosive, that is, [gy gu dy du] vs. [ty tu ky ku y u]. We therefore focussed all further analyses on this distinction. The percentage of identification of the presence vs. absence of a voiced plosive for the three conditions is presented on Fig. 1b. The results show that AV intelligibility was significantly higher than A intelligibility (χ 2 (1)=6.0, p<0.05), while V intelligibility was, not surprisingly, very poor.

Discussion

The likely explanation of this effect is displayed in Fig. 2. The typical acoustic structure of a consonant-vowel syllable is illustrated in Fig. 2a. In French, voiced plosives such as [d] or [g] begin with a low frequency prevoicing bar: before the tongue leaves the palate to produce the consonant, the glottal source has already begun its action, and is audible by the listener as a major voicing cue [START_REF] Lisker | A cross-language study of voicing in initial stops: acoustical measurements[END_REF]. Next comes an acoustic burst, and a spectral transition towards the vowel target (vowel nucleus). In Fig. 2b, we see that the speaker begins to move his lips towards the rounded vowel [y] or [u] about 100 ms before the beginning of the prevoicing bar, if it exists, and 240ms before the vowel nucleus. This provides a temporal cue likely to improve the detection of upcoming acoustic cues, and particularly the prevoicing bar indicating the presence or absence of a voiced plosive.

Experiment 2: Removing any residual lip reading cues

Method

In the previous experiment dealing with natural audio-visual stimuli, we could not yet rule out the possibility that there remained small visual voicing cues that could have provided the increase in intelligibility from the A to the AV condition. The purpose of Experiment 2 was to discard any possibility that vision could enhance intelligibility through direct lip reading, using stimuli in which different sounds were dubbed onto a fixed lip gesture. For this purpose, a new recording of 30 stimuli was made with the same speaker. The speaker's lips were coloured using blue makeup, to allow precise video analyses using a chroma-key process [START_REF] Lallouache | Acquisition et traitement de contours labiaux[END_REF]. We first performed various analyses of the lip profiles and acoustic content of all the stimuli to verify that they were globally compatible with the portrait in Fig. 2. We selected an utterance with a rounding gesture beginning at a lip area value of 1.1 cm 2 typical of the "basis" period, ending at a lip area value of 0.3 cm 2 typical of the rounded target, and changing from the first to the second value in exactly 120 ms. We added a 400 ms rounded plateau and a 240-ms unrounding gesture coming back to the 1.1 cm 2 basis. This fixed 760 ms video stimulus was dubbed onto all 30 acoustic stimuli, in the precise configuration displayed in Fig. 2. We ensured that the lip onset phase occurred at least 100 ms before the acoustic prevoicing phase (if there was one) and roughly 240 ms before the acoustic initiation of the vowel (beginning of the formant transition).

As in Experiment 1 stimuli were merged with high-level cocktail party noise, and a new panel of twelve French subjects with no reported hearing problems completed an identification task in two conditions: audio-only (A) or audio-visual (AV) presentation. Half of the subjects completed the AV condition followed by the A condition, whereas half did the reverse order. The video-only condition was not used here, since it would have involved 30 repetitions of exactly the same video stimulus.

Results

The results are displayed in Fig. 3, still focussing on the [gy gu dy du] vs. [ty tu ky ku y u] contrast. Percentages of correct identification of the presence vs. absence of a voiced plosive computed from the 30 stimuli are displayed. Identification is significantly higher in the AV condition compared to the A condition: χ 2 (1)=3.3, p<0.1. Since the average AV-A difference is rather small, we also compared the number of correct responses per subject, and then evaluated the AV-A difference by a paired t-test. The mean gain is 1.7 correct responses (30 being the total number of responses per subject), which is significantly higher than 0 in a t-test with paired samples (t(11)= 1.95, p<0.04).

Discussion

The significant gain in intelligibility in the AV condition compared to the A condition replicates the result in Experiment 1, but now the interpretation is straightforward. The gain due to visual information cannot be based on lip reading per se; it must be due to the enhanced sensitivity to the voicing cues (basically, the prevoicing bar) contributed by the temporal cue of the lip gesture onset. This fits well with psychoacoustic data showing that audio detection without temporal uncertainty provides a threshold by 2 to 3 dBs lower than detection with temporal uncertainty [START_REF] Egan | Interval of time uncertainty in auditory detection[END_REF]; but it shows for the first time that the audio detection gain may be converted into a speech audio-visual intelligibility gain, providing, independently of lip reading per se, an additional contribution to the visual enhancement of auditory speech intelligibility.

Experiment 3: Testing the speech-specificity of the visual cuing effect

Method

We conducted a third experiment to attempt to test whether the effect displayed in Experiment 2 was speech specific. In Experiment 3, we replaced lip movements by a visual non-speech cue consisting of a red bar appearing and disappearing on a black 720x576 pixel background in synchrony with each stimulus syllable (Fig. 4a). The bar was a rectangle with a width set at 155 pixels and a height equal to 0 in the "basis" period (no bar, just the black background) increasing to 320 pixels by 80-pixel steps in the 120-ms sequence towards the target, stable at 320 pixels during the 400-ms plateau and decreasing back to 0 in 240 ms. The audio tape and the dubbing were the same as in Experiment 2. Hence Experiment 3 literally consisted of replacing a lip rounding-unrounding gesture by a bar increasing and decreasing in height in exactly the same time course. The temporal auditory-visual coherence was therefore preserved, while the nature of the visual input as a speech stimulus was disrupted. Another set of 12 French subjects with no reported hearing problems completed the AV and A conditions in the same way as in Experiment 2.

Results

The percentages of correct identification of the presence vs. absence of a voiced plosive computed from the 30 stimuli are displayed in Fig. 4b. Strikingly, the potential advantage due to visual information disappeared here. Indeed, intelligibility is not significantly higher in the AV condition compared with the A condition: χ 2 (1)=0.4, p>0.5. A paired t-test of the AV-A difference in the number of correct responses per subject is also negative: mean AV-A difference 0.6, t(11)= 0.96, p>0.1. Hence the temporal cueing gain provided by visual information seems to be, at least partly, speech specific.

General discussion

This work shows that seeing the speaker's lips enables the listener to better extract useful acoustic information embedded in cocktail party noise. This results in an additional increase in audiovisual speech intelligibility, different from lip reading per se. In Experiments 1 and 2, while listeners hear all vowel nuclei in the acoustic stimulus, seeing the speaker's lips enables them to extract an additional cue, that is the prevoicing bar, if it exists. The speech understanding system then integrates these features to identify the syllable. These results show that cross-modal interactions can occur early to enhance speech in noise and improve intelligibility.

Visual temporal cueing is likely to provide the explanation of the positive results in Experiments 1 and 2. In this context, the appeal to the concept of "Bimodal Coherence Masking Protection" [START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF] adapted from the audio "Coherence Masking Protection" paradigm [START_REF] Gordon | Coherence masking protection in speech sounds: The role of formant synchrony[END_REF] is logical. It would expand "co-modulation" between frequency bands in the audio spectrum to audio-visual co-modulation reducing the spectro-temporal uncertainty and thus improving audio-visual intelligibility. This mechanism, clearly different from lip-reading, provides an "early stage" for audio-visual interaction in speech identification, which could be part of a multisensorial selective listening process [START_REF] Driver | Enhancement of selective listening by illusory mislocation of speech sounds due to lip-reading[END_REF]. It suggests that auditory scene analysis [START_REF] Bregman | Auditory Scene Analysis: the perceptual organization of sound[END_REF], that is the set of mechanisms that enable the auditory system to separate sounds into streams, could be extended towards audio-visual scene analysis [START_REF] Barker | Is primitive coherence an aid to segment the scene[END_REF].

The negative finding with non-speech visual cues in Experiment 3 is in line with other findings in audio-visual speech detection experiments. [START_REF] Bernstein | Enhanced auditory detection with AV Speech: Perceptual evidence for speech and non-speech mechanisms[END_REF], replacing lip movements with a simple Lissajous curve varying with the audio amplitude, found an audiovisual detection threshold of speech in white noise lower than the audio one, but higher than for audiovisual speech. [START_REF] Kim | Testing the cuing hypothesis for the AV speech detection[END_REF] showed that the AV speech detection advantage was removed by replacing the original video speech stimulus by the synthetic output of a computergenerated talking face. The present results also fit well with a previous negative finding in which replacing lip movements with an audio amplitude driven Lissajous curve lead to no visual benefit in the cocktail party effect [START_REF] Summerfield | Use of visual information for phonetic perception[END_REF]. Notice that the "speech specificity" of the visual cueing effect is not unambiguously demonstrated in the present work. Firstly, the synthetic visual display used in Experiment 3 may have failed to capture critical aspects of the more natural visual display used in Experiments 1 and 2, though the very precise coherence in time makes this assumption a bit unlikely. Second and more importantly, it could well be the case that this display needs some training by the subjects before it can be used efficiently. Checking this assumption in more detail will need further studies.

The neural bases of early audio-visual interaction are of course not completely clear. A number of recent electrophysiological data suggest that audiovisual integration could indeed happen very early in the perceptual processes, both for speech [START_REF] Colin | Mismatch negativity evoked by the McGurk-MacDonald effect: a phonetic representation within short-term memory[END_REF] and non-speech stimuli [START_REF] Giard | Auditory-visual integration during multimodal object recognition in humans: a behavioral and electrophysiological study[END_REF]. Importantly, the fact that the anticipatory lip movement modulates auditory neural processing as early as 50-100 ms after the auditory onset (Lebib, Papo, de Bode & Baudonnière, 2003;[START_REF] Van Wassenhove | Electrophysiology of auditory-visual speech integration[END_REF] suggests that the 100-ms advance of the visual lip onset phase on the acoustic prevoicing phase in Experiment 2 is suitable for audio-visual efficient interaction. These data are compatible with fMRI evidence that the heteromodal cortex could provide a good site for audio-visual binding and signal enhancement [START_REF] Calvert | Evidence from functional magnetic resonance imaging of crossmodal binding in the human heteromodal cortex[END_REF].

Such early interactions in audio-visual speech perception call for the development of a new generation of models, incorporating a low-level processing stage in the general architecture (e.g. [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF]. They provide the basis for new signal processing techniques for joint audiovisual speech analysis, exploiting audio-visual coherence to better process and enhance speech [START_REF] Sodoyer | Separation of audiovisual speech sources: A new approach exploiting the audiovisual coherence of speech stimuli[END_REF] before transmission or recognition in telecommunication systems or hearing aids.

Figure 1 .Figure 2 .

 12 Figure 1. In Experiment 1, French subjects identified natural utterances of the 10 stimuli in three conditions: AV, A and V(a). Percentage of identification of the presence vs. absence of a voiced plosive (b).

FrequencyFigure 3 .

 3 Figure 3. Percentage of identification of the presence vs. absence of a voiced plosive with the dubbed stimuli of Experiment 2.

Figure 4 .

 4 Figure 4. In Experiment 3, the speaker's lips were replaced by a red bar with varying height on a black background (a). Percentages of identification of the presence vs. absence of a voiced plosive computed from the 30 stimuli are displayed (b).

Table 1 .

 1 Corrected global scores and corrected identification scores for individual phonetic features in the three conditions of Experiment 1. All scores are computed according to Eq. (1).

	Corrected scores	Mode	Plosive place	Vowel place	Global score
	A Condition	0.07	-0.03	-0.03	0.02
	AV Condition	0.30	0.06	-0.03	0.06
	V Condition	0.01	0.05	-0.01	0.02

Acknowledgements.

We thank Jon Barker for participation in a preliminary set of experiments, and Pauline Welby for her help in the preparation of the paper. This work was supported by CNRS, INPG and Université Stendhal.

Vitae

Jean-Luc Schwartz is a member of the Centre National de la Recherche Scientifique. He lead the Speech Perception Group at ICP for 10 years. He is now the head of ICP, a laboratory exploring all aspects of speech communication. His main areas of research involve auditory modelling, psychoacoustics, speech perception, auditory front-ends for speech recognition, bimodal integration in speech perception and source separation, perceptuo-motor interactions and speech robotics. He organised with Frédéric Berthommier, Marie-Agnès Cathiard and David Sodoyer the International Workshop "Audio-Visual Speech Processing" AVSP'2003.

Frédéric Berthommier received the M.D. degree from the University of Paris 7 (Lariboisière St-Louis) and his ph.D. in biomedical engineering from the University of Grenoble I in 1992. He his CNRS researcher in the "Institut de la Communication Parlée" in Grenoble since 1993. His research interests include auditory scene analysis, speech perception, audio-visual speech processing and auditory modelling. He now leads the Speech Perception Group at ICP. Christophe Savariaux received his Ph.D. in "Signal, Image and Speech Processing" from the "Institut National Polytechnique of Grenoble" in 1995. He is a CNRS Engineer and Researcher in the "Institut de la Communication Parlée" in Grenoble since 1999. His main research interests are focused on speech production, speech pathology and audio-visual speech processing. He is in charge of data acquisition (acoustic, video, electromagnetic, physiological, etc) at ICP.