Swann Marx
email: swann.marx@gipsa-lab.fr.

Eduardo Cerpa
email: eduardo.cerpa@usm.cl.

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr.

Vincent Andrieu
email: vincent.andrieu@gmail.com.

Global stabilization of a Korteweg-de Vries equation with a distributed control saturated in L 2 -norm

Keywords: Partial differential equation, saturation control, nonlinear control system, global stability, Lyapunov function

This article deals with the design of saturated controls in the context of partial differential equations. It is focused on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. The aim of this article is to study the influence of a saturating in L 2 -norm distributed control on the well-posedness and the stability of this equation. The well-posedness is proven applying a Banach fixed point theorem. The proof of the asymptotic stability of the closed-loop system is tackled with a Lyapunov function together with a sector condition describing the saturating input. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.

INTRODUCTION

The Korteweg-de Vries equation (KdV for short) y t + y x + y xxx + yy x = 0, (1) is a mathematical model of waves on shallow water surfaces. Its stabilizability properties have been deeply studied with no constraints on the control, as reviewed in [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF]; [START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation: recent progresses[END_REF]. In this article, we focus on the following controlled KdV equation    y t + y x + y xxx + yy x + f = 0, (t, x) ∈ [0, +∞) × [0, L], y(t, 0) = y(t, L) = y x (t, L) = 0, t ∈ [0, +∞), y(0, x) = y 0 (x), x ∈ [0, L],

(2) where y stands for the state and f for the control. As studied in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], if f = 0 and

L ∈ 2π k 2 + kl + l 2 3 k, l ∈ N * , (3)
then, there exist solutions of the linearized version of (2) written as follows        y t + y x + y xxx = 0, y(t, 0) = y(t, L) = 0, y x (t, L) = 0, y(0, x) = y 0 (x),

This works has been partially supported by Basal Project FB0008 AC3E.

for which the L 2 (0, L)-energy does not decay to zero. For instance, if L = 2π and y 0 = 1 -cos(x) for all x ∈ [0, L], then y(t, x) = 1 -cos(x) is a stationary solution of (2) conserving the energy for any time t.

Note however that, if L = 2π and f = 0, the origin of (2) is locally asymptotically stable as stated in [START_REF] Chu | Asymptotic stability of a nonlinear Korteweg-de Vries equation with a critical length[END_REF]. Then the nonlinear version of the Korteweg-de Vries equation has better results in terms of stability than the linearized one. A similar phenomenon also appears when studying the controllability of the equation with a Neumann control (see e.g. [START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF] or [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF]).

In the literature there are some methods stabilizing the KdV equation ( 2) with boundary control, see [START_REF] Cerpa | Rapid exponential stabilization for a linear Korteweg-de Vries equation[END_REF]; [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]; [START_REF] Marx | Output Feedback Control of the Linear Korteweg-de Vries Equation[END_REF] or distributed controls as studied in [START_REF] Pazoto | Unique continuation and decay for the Korteweg-de Vries equation with localized damping[END_REF]; [START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF].

In this paper, we deal with the case where the control is saturated. Indeed, in most applications, actuators are limited due to some physical constraints and the control input has to be bounded. Neglecting the amplitude actuator limitation can be source of undesirable and catastrophic behaviors for the closed-loop system. Nowadays, numerous techniques are available (see e.g. [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]; [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF]; [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF]) and such systems can be analyzed with an appropriate Lyapunov function and a sector condition of the saturation map, as introduced in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] or [START_REF] Zaccarian | Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation: Control Augmentation for Actuator Saturation[END_REF].

To the best of our knowledge, there are few papers studying this topic in the infinite dimensional case. Among them, there are [START_REF] Lasiecka | Strong stability of elastic control systems with dissipative saturating feedback[END_REF], [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF], where a wave equation equipped with a saturated distributed actuator is considered and [START_REF] Daafouz | Nonlinear control of a coupled pde/ode system modeling a switched power converter with a transmission line[END_REF], where a coupled PDE/ODE system modeling a switched power converter with a transmission line is considered and, due to some restrictions on the system, a saturated feedback has to be designed. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems [START_REF] Logemann | Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities[END_REF], [START_REF] Seidman | A note on stabilization with saturating feedback[END_REF], [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]).

In [START_REF] Marx | Stabilization of a linear Korteweg-de Vries with a saturated internal control[END_REF], in which it is considered a linear Korteweg-de Vries equation with a saturated distributed control, nonlinear semigroup theory is applied. In the case of the present paper, since the term yy x is not globally Lipschitz, such a theory is harder to use. Thus, we aim at studying a particular nonlinear partial differential equation without seeing it as an abstract control system and without using the nonlinear semigroup theory. Moreover, in the case of the present paper, the saturation is borrowed from [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF] and allows us to give explicitely the decay rate for bounded initial conditions, although the saturation used in [START_REF] Marx | Stabilization of a linear Korteweg-de Vries with a saturated internal control[END_REF] is the classical one and no decay rate is given. This article is organized as follows. In Section 2, we present our main results about the well posedness and the stability of (2) in presence of saturating control. Sections 3.1 and 3.2 are devoted to prove these results by using respectively a Banach fixed-point theorem and some Lyapunov techniques together with a sector condition describing the saturating input. In Section 4, we give some simulations of the equation looped by a saturated feedback law. Section 5 collects some concluding remarks and possible further research lines.

Notation: y t (resp. y x ) stands for the partial derivative of the function y with respect to t (resp. x). Given L > 0,

• L 2 (0,L) denotes the norm in L 2 (0, L) and H 1 (0, L) (resp.

H 3 (0, L)) is the set of all functions u ∈ L 2 (0, L) such that u x ∈ L 2 (0, L) (resp. u x , u xx , u xxx ∈ L 2 (0, L)).
A function α is said to be a class K function if it is a nonnegative, an increasing function and if α(0) = 0.

MAIN RESULTS

Setting a control f (t, x) = ay(t, x) with a > 0, we obtain that (2) is stabilized in the L 2 (0, L)-norm topology. Indeed, performing some integrations by parts, we obtain, at least formally 1 2

d dt L 0 |y(t, x)| 2 dx = -|y x (t, 0)| 2 -a L 0 |y(t, x)| 2 dx (5)
and thus y(t, .) L 2 (0,L) ≤ e -at y 0 L 2 (0,L) , ∀t ≥ 0. (6) Let us consider the impact of the constraint on the control. For infinite-dimensional systems, a way to take into account this constraint is to use the following saturation function, that is for all s ∈ L 2 (0, L) and for all x ∈ [0, L],

sat(s)(x) =    s(x) if s L 2 (0,L) ≤ u s , s(x)u s s L 2 (0,L) if s L 2 (0,L) ≥ u s . (7)
Let us consider the KdV equation controlled by a saturated distributed control as follows

       y t + y x + y xxx + yy x + sat(ay) = 0, y(t, 0) = y(t, L) = 0, y x (t, L) = 0, y(0, x) = y 0 (x). ( 8
)
Let us state the main results of this paper. Theorem 1. [Well posedness] For any initial conditions y 0 ∈ L 2 (0, L), there exists a unique mild solution y ∈ C(0, T ;

L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)) to (8).
Theorem 2. [Global asymptotic stability] Given positive values a and u s , there exists a class K function α : R ≥0 → R ≥0 such that for a given y 0 ∈ L 2 (0, L), the mild solution y to (8) satisfies, y(t, .) L 2 (0,L) ≤ α( y 0 L 2 (0,L) )e -at , ∀t ≥ 0. ( 9) Moreover, for bounded initial conditions, we could estimate the decay rate of the solution. In other words, given a positive value r, for any initial condition

y 0 ∈ L 2 (0, L) such that y 0 L 2 (0,L) ≤ r, any mild solution y to (8) satisfies, y(t, .) L 2 (0,L) ≤ y 0 L 2 (0,L) e -µt , ∀t ≥ 0, ( 10
) where µ is defined as follows

µ := min a, u s r . (11)
By a scaling, we may assume, without loss of generality, that either a or u s is 1. However, Theorem 2 shows that saturating a controller insuring a rapid stabilization makes the origin globally asymptotically stable. We cannot select anymore the decay rate of the convergence.

PROOF OF THE MAIN RESULTS

Well-posedness

Linear system. Before proving the well-posedness of ( 8), let us recall some useful results on the linear system (4).

To do that, consider the operator defined by

D(A) = {w ∈ H 3 (0, L), w(0) = w(L) = w (L) = 0}, A : w ∈ D(A) ⊂ L 2 (0, L) -→ (-w -w ) ∈ L 2 (0, L).
It is easy to prove that A generates a strongly continuous semigroup of contractions which we will denote by W (t).

We have the following theorem proven in Rosier (1997) Theorem 3. [Well-posedness of (4), [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]] For any initial conditions y 0 ∈ L 2 (0, L), there exists a unique mild solution y ∈ C(0, T ; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)) to (4). Moreover, there exists C > 0 such that the solution to (4) satisfies

y C(0,T ;L 2 (0,L)) + y L 2 (0,T ;H 1 (0,L)) ≤ C 1 y 0 L 2 (0,L) (12
) and the extra regularity

y x (., 0) L 2 (0,T ) ≤ y 0 L 2 (0,L) . ( 13
)
To ease the reading, let us denote the following Banach space, for all T > 0, B(T

) := C(0, T ; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L))
endowed with the norm

y B(T ) = sup t∈[0,T ] y(t, .) L 2 (0,L) + T 0 y(t, .) 2 H 1 (0,L) dt 1 2 (14)
Before studying the well-posedness of (8), we need a wellposedness result with a right-hand side. Thus, given g ∈ L1 (0, T ; L2 (0, L), let us consider y the unique solution 1 to the following inhomogeneous problem:

       y t + y x + y xxx = g, y(t, 0) = y(t, L) = 0, y x (t, L) = 0, y(0, .) = y 0 . (15)
Note that we need the following property on the saturation function, which will allow us to state that this type of nonlinearity belongs to the space L 1 (0, T ; L 2 (0, L)). Lemma 1. [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF], Theorem 5.1.] For all (s, s)

∈ L 2 (0, L) 2 , we have sat(s) -sat( s) L 2 (0,L) ≤ 3 s -s L 2 (0,L) . ( 16
)
Let us now state the following properties on the nonhomogeneous linearized KdV equation ( 15).

Firstly, we have this proposition borrowed from (Rosier, 1997, Proposition 4.1) Proposition 1. [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]] If y ∈ L 2 (0, T ; H 1 (0, L)), then yy x ∈ L 1 (0, T ; L 2 (0, L)) and the map

ψ 1 : y ∈ L 2 (0, T ; H 1 (0, L)) → yy x ∈ L 1 (0, T ; L 2 (0, L)) is contin- uous.
Secondly, we have the following proposition Proposition 2. If y ∈ L 2 (0, T ; H 1 (0, L)), then sat(ay) ∈ L 1 (0, T ; L 2 (0, L)) and the map ψ 2 : y ∈ L 2 (0, T ; H 1 (0, L)) → sat(ay) ∈ L 1 (0, T ; L 2 (0, L)) is continuous.

Proof. Let y, z ∈ L 2 (0, T ; H 1 (0, L)). We have, using Lemma 1 and Hölder inequality sat(ay)-sat(az) L 1 (0,T ;L 2 (0,L)) ,

≤ 3 T 0 a (y -z) L 2 (0,L) , ≤ 3La √ T (y -z) L 2 (0,T ;H 1 (0,L)) . ( 17
)
Plugging z = 0 in (17) yields sat(ay) ∈ L 1 (0, T ; L 2 (0, L)) and using ( 17) gives the continuity of the map ψ 2 . It concludes the proof of Proposition 2.

•

Finally, we have this result borrowed from (Rosier, 1997, Proposition 4.1) Proposition 3. [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]] For g ∈ L 1 (0, T ; L 2 (0, L)) and y 0 ∈ L 2 (0, L), the mild solution of (15) belongs to B(T ). Moreover the map

ψ 3 : g ∈ L 1 (0, T ; H 1 (0, L)) → y ∈ B(T ) is continuous.
Proof of Theorem 1 We are now in position to prove Theorem 1. Let us begin this section with a technical lemma.

Lemma 2. [ (Chapouly, 2009, Lemma 18)] For any T > 0 and y, z ∈ B(T ),

T 0 (y(t, .)z(t, .)) x L 2 (0,L) dt ≤ 2 √ T y B(T ) z B(T ) (18)
Let us now state our local well-posedness result. Lemma 3. [Local well-posedness of ( 8)] Let T > 0, be given. For any y 0 ∈ L 2 (0, L), there exists T ∈ [0, T ] depending on y 0 such that (8) admits a unique solution y ∈ B(T ).

Proof. We follow the strategy of [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] and [START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF]. From Propositions 1, 2 and 3, we know that, for all z ∈ L 2 (0, T ; H 1 (0, L)), there exists a unique solution y in B(T ) to the following equation

       y t + y x + y xxx = -zz x -sat(az), y(t, 0) = y(t, L) = 0, y x (t, L) = 0, y(0, x) = y 0 (x). ( 19
)
Solution y to ( 19) can be written in its integral form

y(t) =W (t)y 0 - t 0 W (t -τ )(zz x )(τ )dτ - t 0 W (t -τ )sat(az(τ, .))dτ (20)
For given y 0 ∈ L 2 (0, L), let r and T be positive values to be chosen. Let us consider the following set

S T ,r = {z ∈ B(T ), z B(T ) ≤ r}, (21)
which is a closed, convex and bounded subset of B(T ). Consequently, S T ,r is a complete metric space in the topology induced from B(T ). We define the map Γ on S T ,r by, for all z ∈ S T ,r

Γ(z) :=W (t)y 0 - t 0 W (t -τ )(zz x )(τ )dτ - t 0 W (t -τ )sat(az(τ, .))dτ, ∀z ∈ S T ,r . (22)
We aim at proving the existence of a fixed point for this operator.

It follows immediatly from (17), Lemma 2 and the linear estimates written in Theorem 3 that, for every z ∈ S T ,r , there exist positive values C 2 and C 3 such that

Γ(z) B(T ) ≤C 1 y 0 L 2 (0,L) + C 2 √ T z 2 B(T ) + C 3 √ T z B(T ) (23)
We choose r > 0 and T > 0 such that

   r = 2C 1 y 0 L 2 (0,L) C 2 √ T r + C 3 √ T ≤ 1 2 (24)
in order to obtain Γ(z) B(T ) ≤ r, ∀z ∈ S T ,r .

Thus, with such r and T , Γ maps S T ,r to S T ,r . Moreover, one can prove with the same inequalities that

Γ(z 1 ) -Γ(z 2 ) B(T ) ≤ 1 2 z 1 -z 2 B(T ) , ∀(z 1 , z 2 ) ∈ S 2 T ,r (26)
The existence of the solutions to the Cauchy problem (8) follows by using the Banach fixed point theorem (Brezis, 2011, Theorem 5.7).

•

We need the following Lemma inspired by [START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF] and [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] which implies that if there exists a solution for all T > 0 then the solution is unique. Lemma 4. For any T > 0 and a > 0, there exists C 4 (T, L) such that for every y 0 , z 0 ∈ L 2 (0, L) for which there exist mild solutions y and z to

       y t + y x + y xxx + yy x + sat(ay) = 0, y(t, 0) = y(t, L) = 0, y x (t, L) = 0, y(0, x) = y 0 (x), ( 27
)
and        z t + z x + z xxx + zz x + sat(az) = 0, z(t, 0) = z(t, L) = 0, z x (t, L) = 0, z(0, x) = z 0 (x), ( 28
)
one has the following inequalities

T 0 L 0 (z x (t, x) -y x (t, x)) 2 dxdt ≤ e C4(1+ y L 2 (0,T ;H 1 (0,L)) + z L 2 (0,T ;H 1 (0,L)) ) L 0 (z 0 -y 0 ) 2 dx, (29)
T 0 L 0 (z(t, x) -y(t, x)) 2 dxdt
≤ e C4(1+ y L 2 (0,T ;H 1 (0,L)) + z L 2 (0,T ;H 1 (0,L)) )

L 0 (z 0 -y 0 ) 2 dx.

(30)

Proof. Due to space limitation the proof has not been written. The interested reader can however refer to [START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF] and [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] where proofs of very similar results are provided.

•

We aim at removing the smallness condition given by T in Lemma 3. Since we have the local well-posedness, we only need to prove the following a priori estimate for any solution to (8). Lemma 5. For given T > 0, there exists K := K(T ) > 0 such that for any y 0 ∈ L 2 (0, L), for any 0 < T ≤ T and for any mild solution y ∈ B(T ) to ( 8), it holds

y B(T ) ≤ K y 0 L 2 (0,L) . ( 31
)
Proof. Let us fix 0 < T ≤ T . Let us multiply the first line of ( 8) by y and integrate on (0, L). Using the boundary conditions in (8), we get the following estimates

L 0 yy x dx = 0, L 0 yy xxx dx = 1 2 |y x (t, 0)| 2 , L 0 y 2 y x dx = 0.
Using the fact that sat is odd, we get that 1 2

d dt y(t, .) 2 L 2 (0,L) ≤ - 1 2 |y x (t, 0)| 2 - L 0 ysat(ay)dx ≤ 0 (32)
and consequently y L ∞ (0,T ;L 2 (0,L) ≤ y 0 L 2 (0,L) .

(33) It remains to prove a similar inequality for y x L 2 (0,T ;L 2 (0,L)) ,

to achieve the proof. To do that, we multiply by xy (8), integrate on (0, L) and use the following

L 0 xyy x dx = - 1 2 y 2 L 2 (0,L) , L 0 yy xxx dx = 3 2 y x 2 L 2 (0,L) , - L 0 xy 2 y x dx = 1 3 L 0 y 3 (t, x)dx , ≤ 1 3 sup x∈[0,L] |y(t, x)| y 2 L ∞ (0,T ;L 2 (0,L)) , ≤ √ L 3 y x L 2 (0,L) y 2 L ∞ (0,T ;L 2 (0,L)) , ≤ √ Lδ 6 y x L 2 (0,L) + √ L 6δ y 4 L ∞ (0,T ;L 2 (0,L)) , (34)
where δ is chosen such that δ := 3 √ L . In this way, we have 1 2

d dt L 0 |x 1/2 y(t, .)| 2 dx - 1 2 L 0 y 2 dx + 3 2 L 0 |y x | 2 dx - L 0 xy 2 y x dx = - L 0 xsat(ay)ydx. (35)
We get using (34) and the fact that sat is odd

1 2 d dt L 0 |x 1/2 y(t, .)| 2 dx + L 0 |y x | 2 dx ≤ 1 2 y 2 L ∞ (0,T ;L 2 (0,L)) + L 18 y 4 L ∞ (0,T ;L 2 (0,L)) . (36)
Using (33) and a Grönwall inequality, we get the existence of a positive value C 5 = C 5 (L) > 0 such that y x L 2 (0,T ;L 2 (0,L)) ≤ C 5 y 0 L 2 (0,L) , (37) which concludes the proof of Lemma 5.

•

Using Lemmas 3, 4 and 5, for any T > 0, we can conclude that there exists a unique mild solution in B(T ) to (8). Indeed, with Lemma 3, we know that there exists T ∈ (0, T ) such that there exists a unique solution to (8) in B(T ). Lemma 4 states that if there exists a solution to B(T ), then it is unique. Finally, Lemma 5 allows us to state the well-posedness for every T > 0: since the solution y to (8) is bounded by its initial condition for every T > 0 belonging to [0, T ] as stated in (32), we know that there exists an unique solution to (8) in B(T ). This concludes the proof of Theorem 1. 2

Stability

This section is devoted to the proof of Theorem 2. We need then to prove this lemma. Lemma 6. System (8) is semi-globally exponentially stable in L 2 (0, L). In other words, for any r > 0 there exists a positive constant µ given in Theorem 2 such that for any y 0 ∈ L 2 (0, L) satisfying y 0 L 2 (0,L) ≤ r, the mild solution y = y(t, x) to (8) satisfies y(t, .) L 2 (0,L) ≤ y 0 L 2 (0,L) e -µt ∀t ≥ 0.

From this result, we will be able to prove the global asymptotic stability of (8), as done at the end of this section. Note moreover that this result is indeed the second part of Theorem 2.

Technical lemma.

Before starting the proof of the Lemma 6, let us state and prove the following lemma Lemma 7. [Sector condition] Let r be a positive value. Given a positive value a and s ∈ L 2 (0, L) such that s L 2 (0,L) ≤ r, we have β(x) := sat(as)(x) -k(r)as(x) s(x) ≥ 0, ∀x ∈ [0, L],

(39) with k(r) = min u s ar , 1 .

Proof. Consider the two following cases 1. as L 2 (0,L) ≥ u s ; 2. as L 2 (0,L) ≤ u s .

The first case implies that, for all x ∈ [0, L] sat(as)(x) = as(x) as L 2 (0,L) u s .

Thus, for all x ∈ [0, L]

β(x) = as(x) 2 u s as L 2 (0,L) -k(r) . (41)
Since we have

u s as L 2 (0,L) ≥ u s ar ≥ k(r),
then, we obtain β(x) ≥ 0.

The second case implies that, for all x ∈ [0, L] sat(as)(x) = as(x) We obtain that β(x) := (1 -k(r))as(x) 2 ≥ 0.

It concludes the proof of Lemma 7.

•

Semi-global exponential stability. Now we are able to prove Lemma 6. Let r be a positive value such that y 0 L 2 (0,L) ≤ r.

Note that from (32), we get

y L 2 (0,L) ≤ y 0 L 2 (0,L) ≤ r (42)
With the Lemma 7 and (32), we obtain 1 2

d dt L 0 |y(t, x)| 2 dx ≤ - L 0 k(r)a|y(t, x)| 2 dx (43)
Thus, with (40), applying the Grönwall lemma leads to y(t, .) L 2 (0,L) ≤ e -µt y 0 L 2 (0,L) (44) where µ is defined in the statement of Theorem 2. Therefore, it concludes the proof of Lemma 6. 2

Proof of Theorem 2 We are now in position to prove Theorem 2, inspired by [START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF].

Proof. By Lemma 6 and (32), if

y 0 L 2 (0,L) ≤ u s a , (45)
then, by dissipativity, we have y(t, .) L 2 (0,L) ≤ us a for all t ≥ 0. Thus, sat(a y) = a y and we know, from (6), that the corresponding solution y to (8) satisfies y(t, .) L 2 (0,L) ≤ y 0 L 2 (0,L) e -at ∀t ≥ 0.

(46) In addition, for a given r > 0, there exists a positive constant µ, which is given in (11), such that if y 0 L 2 (0,L) ≤ r, then any mild solution y to (8) satisfies y(t, .) L 2 (0,L) ≤ y 0 L 2 (0,L) e -µt ∀t ≥ 0 (47)

Consequently, setting T r := µ -1 ln ar us , we have y 0 L 2 (0,L) ≤ r ⇒ y(T r , .) L 2 (0,L) ≤ e -ln( ar us ) r = u s a Therefore, using (46), we obtain y(t, .) L 2 (0,L) ≤ y(T r , .) L 2 (0,L) e -a(t-Tr) , ∀t ≥ T r ≤ y 0 L 2 (0,L) e aTr e -at , ∀t ≥ 0. Thus it concludes the proof of Theorem 2.

•

SIMULATION

Let us discretize the PDE ( 8) by means of finite difference method (see e.g. [START_REF] Pazoto | Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping[END_REF] for an introduction on the numerical scheme of a generalized Korteweg-de Vries equation). The time and the space steps are chosen such that the stability condition of the numerical scheme is satisfied (see [START_REF] Pazoto | Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping[END_REF] where this stability condition is clearly established).

Given y 0 (x) = 1 -cos(x), L = 2π, a = 1 and T f inal = 6, Figure 1 shows the solution to (8) and with the unsaturated control f = ay. Figure 2 illustrates the simulated solution with the same initial condition and a saturated control f = sat(ay) where u s = 0.5. The evolution of the L 2 -energy of the solution in these two cases is given by Figure 3.

CONCLUSION

In this paper, we have studied the well-posedness and the asymptotic stability of a Korteweg-de Vries equation with a saturated distributed control. The well-posedness issue has been tackled by using the Banach fixed-point theorem and we proved the stability by using a sector condition and Lyapunov theory for infinite dimensional systems. Two questions may arise in this context. In [START_REF] Marx | Stabilization of a linear Korteweg-de Vries with a saturated internal control[END_REF], an other saturation function is used. Is the equation with a control saturated with such a function still globally asymptotically stable? Some boundary controls have been already designed in [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] Tang | Stabilization of Linearized Korteweg-de Vries Systems with Anti-diffusion[END_REF] or [START_REF] Cerpa | Rapid exponential stabilization for a linear Korteweg-de Vries equation[END_REF]. By saturating this controllers, are the corresponding equations still stable?

Fig. 1 .

 1 Fig. 1. Solution y(t, x)with a feedback law f = ay.

Fig. 2 .

 2 Fig. 2. Solution y(t, x)with a feedback f = sat(ay) where u s = 0.5.

Fig. 3 .

 3 Fig. 3. Blue: Time evolution of the energy function y 2 L 2 (0,L) with a saturation u s = 0.5 and a = 1. Red: Time evolution of the theoritical energy y 0 2 L 2 (0,L) e -2µt . Dotted line: Time evolution of the solution without saturation and a = 1.

It follows from the semigroup theory the existence and the unicity of y when g ∈ L 1 (0, T ; L

(0, L) (see(Pazy, 1983, Chapter 4)).


Dr David-Alexandre Trégouët

A Delluc

A Roche

C Derbois

¶ R Olaso

M Germain

† M De Andrade

W Tang

† D I Chasman

‡ A Van Hylckama Vlieg

§ P Reitsma

C Kabrhel

N Smith

P-E Morange

Is there still room for additional common susceptibility alleles for venous thromboembolism ?

Keywords: genome-wide association studies, genetics, polymorphisms, risk factor, venous thromboembolism

Essentials

•

Genetic architecture of venous thromboembolism (VTE) remains to be fully disentangled.

• 11 newly discovered candidate polymorphisms were genotyped in 3,019 VTE cases and 2,605 controls

• None of the 11 polymorphism were significantly associated with VTE risk

• Additional major efforts are needed to identify VTE-associated genetic variants

Introduction

In 2015, the International Network against VENous Thrombosis (INVENT) consortium reported the results of a meta-analysis of twelve Genome-Wide Association studies (GWAS) on venous thromboembolism (VTE) [START_REF] Germain | Meta-analysis of 65,734 individuals identifies TSPAN15 and SLC44A2 as two susceptibility loci for venous thromboembolism[END_REF]. In the discovery phase of this study which was composed of 7,507 VTE cases and 52,632 controls nine loci reached the genome-wide threshold of 5x10 -8 for declaring statistical significance. Three of these loci, SLC44A2, TSPAN15 and ZFPM2, had not previously been reported to associate with VTE risk. The first two replicated in three independent French case-control studies, MARTHA12, EDITH and FARIVE [START_REF] Germain | Meta-analysis of 65,734 individuals identifies TSPAN15 and SLC44A2 as two susceptibility loci for venous thromboembolism[END_REF]. The association of SLC44A2 and TSPAN15 loci with VTE was also confirmed in a recent GWAS analysis of 6,135 self-reported VTE patients belonging to the 23andMe cohorts [2]. A meta-analysis conditioning on the well-established VTE-associated single nucleotide polymorphisms (SNPs) was also performed in the INVENT discovery cohorts but failed to identify any additional genome-wide significant associations [START_REF] Germain | Meta-analysis of 65,734 individuals identifies TSPAN15 and SLC44A2 as two susceptibility loci for venous thromboembolism[END_REF]. However, at a lower statistical threshold of 1x10 -5 , 285 candidate SNP were identified in this conditional, 12cohorts meta-analysis. After excluding associations at known VTE loci, and those at the ZFPM2 locus that did not replicate previously, 125 SNPs mapping to 11 loci remained. We reasoned that there might be true positive VTE-associations among these 11 loci and decided to further genotype the lead SNPs in the three French case-control studies that served as replication for the INVENT meta-analysis.

Material and Methods

Study populations

Three French case-control studies for VTE were used in this work: EDITH, FARIVE and MARTHA12. These are exactly those that had been employed for replicating the main findings of the INVENT meta-GWAS [START_REF] Germain | Meta-analysis of 65,734 individuals identifies TSPAN15 and SLC44A2 as two susceptibility loci for venous thromboembolism[END_REF].

EDITH -

Single nucleotide polymorphisms genotyping and association testing

Eleven SNPs were selected for genotyping in the EDITH, FARIVE and MARTHA12 samples. These SNPs were identified as the lead SNP (or best proxy for it) at the 11 loci that demonstrated significant association with VTE at p < 10 -5 in the INVENT meta-GWAS after adjusting for the known-and replicated-VTE associated SNPs [START_REF] Germain | Meta-analysis of 65,734 individuals identifies TSPAN15 and SLC44A2 as two susceptibility loci for venous thromboembolism[END_REF]. The lead SNPs were USP34 rs10177303, CACNA2D3 rs72624844, FABP6 rs72812220, AKR1C1 rs114155856, WWP2 rs151186597, CPAMD8 rs10421800, KRI1 rs6511703, PPP5C rs3795043, LILRP2 rs8107460, ADRA1D rs78265041 and ICOSLG rs11701174 (Table 1). Genotyping was performed by taqman and sequenom technologies at the Centre National de Génotypage (CNG, Evry, France).

Hardy-Weinberg equilibrium was checked by means of a Chi-square test in case and control subjects from each study separately. Association of genotyped SNPs with VTE risk was assessed in each of the three studies separately by use of a logistic regression model adjusted for age and sex. After having checked for the homogeneity of the associations across these studies using the Cochran's Q statistic, results were then meta-analyzed using a fixed-effects model based on the inverse-variance weighting method. Because the aim of the project is to validate the candidate associations generated in the INVENT cohorts, onesided association tests were conducted.

Results and Discussion

The genotype distributions of all SNPs but one, the KRI1 rs8113381, were compatible with the Hardy-Weinberg equilibirum (HWE). The distribution of the KRI1 rs8113381 that served as a proxy for the rs6511703 significantly (p <10 -3 ) deviating from HWE in all studies, both in cases and controls, this SNP was then excluded from the association analysis. The minimum per-study genotyping success rate over the 10 remaining SNPs was 0.976. Results of the association tests were detailed in Table 2. None of the tested SNPs was statistically associated with VTE in the combined samples, with the smallest p-value being p = 0.027 for the AKR1C rs115641535. Odds ratios for all SNPs were close to unity or in the opposite direction compared to the INVENT findings.

We also investigated whether these associations could be enhanced in specific subgroups of patients, and for this end patients were stratified according to the clinical manifestations of VTE (deep vein thrombosis or pulmonary embolism) or to the presence of strong genetic risk factors (F2 rs1799963 or F5 rs6025 mutations). Also in these analyses we did not observe evidence for preferential association of the tested SNPs in specific strata.

Despite an average power of ~80% to detect in the three case-control populations studied an odds ratio equal to or greater than the INVENT findings (Table 2), we were not able to replicate any of these associations. The lack of validation of the 11 SNPs does not preclude the existence of common variants with weaker effects or with similar effect size but in no linkage disequilibrium with SNPs assessed by imputation techniques, of common variants exercing haplotype or more complex effects, or of rare variants associated with much stronger genetics effects.

Our findings suggest that none of the 11 common SNPs we selected and that demonstrate suggestive statistical evidence at p < 10 -5 (which corresponds to allelic odds ratio greater than 1.12) in the INVENT GWAS tag for VTE-associated SNPs. Additional statistical power and possibly denser genotyping are needed to identify new associations with incident VTE.

  The EDITH study is a case-control study that was designed to test interactions between genetic and environmental risk factors of VTE [3]. Between May 2000 and December 2009, all consecutive unselected in-and outpatients seen at Brest (West of France) University Hospital for symptomatic VTE (deep vein thrombosis (DVT) and/or pulmonary embolism (PE)) were asked to participate in the study. Diagnosis of DVT was confirmed by the absence of full compressibility of a proximal or distal vein of the deep lower limb on compression ultrasonography (CUS). PE Diagnosis was done by a segmental or larger artery filling defect on chest computed tomography, or the combination of high pre-test clinical probability of PE with high probability ventilation-perfusion lung scan according to the PIOPED criteria, or proximal DVT on CUS in a patient with suspected PE. Controls were selected from the roster of patients hospitalized in the same ward in the following 12 months after the case's event date. Controls were not included if they had a past history of VTE or if they were receiving long-term anticoagulant therapy. DNA were available for 1,141 patients and 1,152 controls. FARIVE -The FARIVE study is a multicenter case-control study of 607 patients with a first episode of proximal DVT and/or PE [4]. DVT was diagnosed by venography and CUS while PE was diagnosed by spiral computed tomography,high probability ventilation-perfusion lung scan and pulmonary angiography. Patients younger than 18 years, with previous VT event, that had a diagnosis of active cancer or a history of malignancy less than 5 years previously, or have a short life expectancy because of other causes, were excluded. The control group consists of age-and sex-matched individuals free of venous and arterial thrombotic disease. Potential control subjects with cancer, liver or kidney failure, or a history of venous and/or arterial thrombotic disease are ineligible. DNA were available for 698 cases and 672 controls. MARTHA12 -The MARTHA12 study is composed of an independent sample of 1,245 VTE patients and 801 French healthy individuals. Patients have been recruited between 2010 and 2012 at the Thrombophilia center of La Timone hospital (Marseille, France) as part of the MARTHA study [5]. All patients had a history of a first VTE event documented by venography, Doppler ultrasound, angiography and/or ventilation/perfusion lung scan. They were all free of any chronic conditions and free of any well characterized genetic risk factors including anti-thrombin, protein C or protein S deficiency, homozygosity for FV Leiden or FII 20210A, and lupus anticoagulant. The control group was composed of two subsamples of healthy individuals: one consists of 475 healthy subjects recruited from the Marseille area, the second of 326 healthy heterozygous carriers of the FV Leiden or FII 20210A mutations selected from the national health examination centers of the French Social Security in collaboration with the Hemostasis and Thrombosis Study Group. DNA were available for 1,180 cases and 781 controls.

AddendumA.

  Delluc, A. Roche, M. De Andrade, W. Tang, D. Chasman, A. van Hylckama Vlieg, P.Reitsma, C. Kabrhel, N. Smith, D-A. Trégouët and P-E. Morange participated to data collection and study design. C. Derbois and R. Olaso organized the genotyping analyses. D-A Trégouët and M. Germain performed statistical data analysis. D-A. Trégouët drafted the document that were further reviewed by A. Delluc, M. De Andrade, W. Tang, D.Chasman, P. Reitsma, C. Kabrhel, A van Hylckama Vlieg, N.Smith and P-E Morange.

Acknowledgments: This work was conducted as part of the INVENT consortium whose contributors list is given in the Appendix. This study recieved grant support for NHLBI R01HL116854 and NHLBI 1R01HL116854. This work was partially supported by the GenMed LABEX (ANR-10-LABX-0013), the French Clinical Research Infrastructure Network on Venous Thrombo-Embolism (F-CRIN INNOVTE) and the ICAN Institute for Cardiometabolism and Nutrition (ANR-10-IAHU-05). We thank Cécile Dulary and Fernando Dos Santos for their technical support in genotyping MARTHA12, FARIVE and EDITH samples.

p = 0.34 (1) Allelic Odds Ratio (OR) [95%Confidence Interval] associated with the minor allele.

(2) Association p-value resulting from the meta-analysis of INVENT GWAS conditioning on known VTE associated SNPs (3) p-value of the test for homogeneity of association (Cochran 's Q statistic) across the twelve discovery cohorts of the INVENT consortium.

Table 2 Association of selected candidate SNPs with VTE in the MARTHA12, FARIVE and EDITH replication studies

USP34 rs2694642 (1) CACNA2D3 rs72624844 AKR1C1 rs115641535 (1) FABP6 rs72812220 WWP2 rs80215330 (1) CPAMD8 rs10421800 PPP5C rs3795043 LILRP2 rs7249176 (1) ADRA1D rs78265041 ICOSLG rs4818892 ( (1) rs2694642, rs115641535, rs80215330, rs7249176 and rs4818892 were used as proxies for rs10177303, rs114155856, rs151186597, rs8107460 and rs11701174, respectively. Data for the KRI1 rs8113381 that served as a proxy for the lead rs6511703 were not shown as its distribution did not follow Hardy-Weinberg equilibrium.

(2) Allelic Odds Ratio (OR) [95%Confidence Interval] associated with the minor allele derived from a logistic regression analysis adjusted for age and sex.

(3) Meta-analyzed adjusted allelic OR derived from a standard fixed-effect meta-analysis of the results observed in the three replication studies.

(4) P-value of the combined allelic OR derived from a standard fixed-effect meta-analysis of the results observed in the three replication studies.

(5) P-value of the test for homogeneity of associations (Cochran 's Q statistic) across the 3 studies.

(6) Power of the combined replication studies to replicate, at the 0.05/10 significance level, the allelic effect observed in the INVENT discovery cohorts (one-sided test).