Hugo Parada
email: hugo.parada@univ-grenoble-alpes.fr

Emmanuelle Crépeau
email: emmanuelle.crepeau@univ-grenoble-alpes.fr

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

Hugo Parada \dagger

AND Emmanuelle Cr \' Epeau \dagger

Global well-posedness of the KdV Equation on a star-shaped network and stabilization by saturated controllers

Keywords: Korteweg-de Vries equation, star-shaped network, stabilization, saturating control MSC codes. 93C20, 93D15, 35R02, 35A01, 35Q53

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1. Introduction and presentation of our results. The Korteweg-de Vries (KdV) equation u t + u x + u xxx + uu x = 0 was introduced in [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave[END_REF] to model the propagation of long water waves in a channel. The KdV equation has been very well studied in recent years, in particular, the controllability and stabilization properties; see [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] for a complete introduction to these problems. With respect to the KdV equation on networks, we can mention the work [START_REF] Cavalcante | The Korteweg--de Vries equation on a metric star graph[END_REF] where well-posedness of the KdV equation on a star metric graph was studied. In the works [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF][START_REF] Cerpa | On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network[END_REF], stabilization and controllability problems were studied, for the KdV equation on a star-shaped network, and recently the problem of stabilization using internal delay was addressed in [START_REF] Parada | Delayed stabilization of the Korteweg--de Vries equation on a star-shaped network[END_REF].

In this work, we are interested in the global well-posedness and stability properties of a KdV equation posed on a star-shaped network using internal saturated feedback terms. Let K = \{ k n : 1 \leq n \leq N \} be the set of the N edges of a network \scrT described as the intervals [0, \ell n ] with \ell n > 0 for n = 1, . . . , N , the network \scrT is defined by \scrT = \bigcup N n=1 k n . Specifically, we are going to consider the next evolution problem for the KdV equation, (KdV-N) \left\{ (\partial t u n + \partial x u n + u n \partial x u n + \partial 3 x u n )(t, x) = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0)

\forall n, n \prime = 1, . . . , N,

N \sum n=1 \partial 2 x u n (t, 0) = - \alpha u 1 (t, 0) - N 3 u 2 1 (t, 0), t > 0, u n (t, \ell n ) = \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = u 0 n (x), x \in (0, \ell n ),
where \alpha \geq N 2 . The central node conditions are obtained taking account of the following: If we denote by u n and v n the dimensionless and scaled variables standing, respectively, for the deflection from rest position and the velocity on the branch n of long water waves, then we get from [25, eq. (13.102)] \Biggl\{ \partial t u n + \partial x u n + \partial 3 x u n + u n \partial x u n = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, v n = u n -1 6 u 2 n + 2\partial 2 x u n \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N.

Moreover, at the central node, we can suppose that the elevation of water is the same in all branches and that the sum of the flux is null, which implies \left\{ u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N, N \sum n=1 u n (t, 0)v n (t, 0) = 0, t > 0.

Then we obtain the following problem: \left\{ u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N,

N \sum n=1 \partial 2 x u n (t, 0) = - N 2 u 1 (t, 0) + N 6 u 2 1 (t, 0), t > 0.
We adapt the boundary condition at the central node to have a decreasing energy.

The hypothesis \alpha > N 2 was introduced in [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF] and then in [START_REF] Cerpa | On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network[END_REF] the case \alpha = N 2 was included. (KdV-N) was studied in [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF] by using the following functional setting: Let where the index e is related so that each edge belongs to H 1 r (0, \ell n ). Introduce also the state space

H 1 r (0, \ell n ) = \bigl\{ v \in H 1 (0, \ell n ), v(\ell n ) = 0 \bigr\} ,
\BbbL 2 (\scrT ) = N \prod n=1 L 2 (0, \ell n ) with (u, v) \BbbL 2 (\scrT ) = N \sum n=1 \int \ell n 0 u n v n dx \forall u, v \in \BbbL 2 (\scrT ).
We also define the space \BbbB T = C([0, T ], \BbbL 2 (\scrT )) \cap L 2 (0, T ; \BbbH 1 e (\scrT )) with \| u\| \BbbB T = \| u\| C([0,T ],\BbbL 2 (\scrT )) + \| u\| L 2 (0,T ;\BbbH 1 e (\scrT )) , and \BbbY T be the space of all functions v \in \BbbB T such that \partial \kappa x v n \in L \infty x (0, \ell n ; H .

In [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF][START_REF] Cerpa | On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network[END_REF] the next well-posedness result was proved for small initial condition and for any time horizon.

T > 0. Then there exist \epsilon > 0 and C > 0 such that for all u 0 \in \BbbL 2 (\scrT ) with \| u 0 \| \BbbL 2 (\scrT ) \leq \epsilon , there exists a unique solution of (KdV-N). Moreover, it satisfies \| u\| \BbbB T \leq C\| u 0 \| \BbbL 2 (\scrT ) .

The main problem to get a global well-posedness result is the action of the nonlinear boundary condition on the central node. Similar boundary conditions appear for the first time to our knowledge in the work [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF] where a wave maker control for a single KdV equation was studied and then in the work [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF] where a well-posedness result was given. The system studied in these papers was the next one (1.1) \left\{ \partial t u(t, x) + \partial x u(t, x) + u(t, x)\partial x u(t, x) + \partial 3 x u(t, x) = 0 \forall x \in (0, L), t > 0, \partial 2 x u(t, 0) = - u(t, 0) + 1 6 u 2 (t, 0) + h(t), t > 0, u(t, L) = \partial x u(t, L) = 0, t > 0, u(0, x) = \phi (x),

x \in (0, L), and the following well-posedness result local-in-time for bounded initial data was proven in [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF].

Theorem 1.2 (Theorem 1.1 of [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF]). Let T > 0 and \gamma > 0 be given. There exists T \ast \in (0, T ] such that for any \phi \in L 2 (0, L) and h \in H -1 3 (0, T ) satisfying, \| \phi \| L 2 (0,L) + \| h\|

H -1 3 (0,T )
\leq \gamma . Then the problem (1.1) admits a unique solution u \in C([0, T \ast ]; L 2 (0, L)) \cap L 2 (0, T \ast ; H 1 (0, L)). Moreover, the corresponding solution map is Lipschitz continuous and the solution possesses the hidden regularities (the sharp Kato smoothing properties) \partial \kappa x u \in L \infty x (0, L; H

1 - \kappa 3 (0, T \ast )), \kappa = 0, 1, 2.
The first main result of our work is the following global-in-time well-posedness theorem.

Theorem 1.3. Let (\ell n ) n=1,...,N \in (0, \infty ) N , \alpha \geq N 2 , and T > 0. Then, for all u 0 \in \BbbL 2 (\scrT ), there exists a unique solution u \in \BbbB T of (KdV-N). Moreover, there exist 0 < T \ast \leq T , C > 0 such that u \in \BbbY T \ast and \| u\| \BbbY T \ast \leq C\| u 0 \| \BbbL 2 (\scrT ) .

Note that our result generalized Theorem 1.1 in the sense that the smallness assumption on the initial data is not needed. Our idea is to follow [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF] to obtain a similar sharp Kato smoothing regularity presented in Theorem 1.2 for a linear problem of the KdV equation on a star-shaped network. In order to deal with the nonlinear part, we use a fixed point argument to obtain global well-posedness for small time. Finally, we use an energy estimation to obtain a global well-posedness in time. Similar ideas were applied in the case of a single KdV equation in [START_REF] Perla Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF].

From the point of view of stabilization, we can refer to the work [START_REF] Zhang | Boundary stabilization of the Korteweg-de Vries equation[END_REF] in which the boundary exponential stabilization problem in the bounded spatial domain x \in (0, 1) was studied. It is well known that the length L of the spatial domain plays an important role in the stabilization and controllability properties of the KdV equation.

For example, when L = 2\pi it is possible to find a solution of the linearization around 0 of KdV (u(t, x) = 1 -cos(x)) that has constant energy. More generally, if L \in \scrN , where \scrN is the set of critical lengths defined by

\scrN = \Biggl\{ 2\pi \sqrt{} k 2 + kl + l 2 3 , k, l \in \BbbN \ast \Biggr\} ,
we can find suitable initial data such that the solution of the linear KdV equation has constant energy. For the case of internal stabilization, it is proved in [START_REF] Perla Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF][START_REF] Pazoto | Unique continuation and decay for the Korteweg-de Vries equation with localized damping[END_REF] that for any critical length we achieve local exponential stability for the nonlinear KdV equation by adding a localized damping. In most real-life settings we have to take into account the saturation in the input control due to some (physical, economical, etc.) constraints. With respect to saturated control in infinite-dimensional systems, we can refer to [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] where a wave equation with distributed and boundary saturated feedback law was studied, [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF] where the saturated internal stabilization of a single KdV equation was studied and recently [START_REF] Mironchenko | Local stabilization of an unstable parabolic equation via saturated controls[END_REF] where a saturated feedback control law was derived for a linear reaction-diffusion equation. Our idea closely follows works [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF] and [START_REF] Parada | Delayed stabilization of the Korteweg--de Vries equation on a star-shaped network[END_REF] to prove the stability of the KdV equation in a star-shaped network with saturated internal control. In this work, we consider a saturation map \fraks \fraka \frakt that could be any of the following cases:

\bullet \fraks \fraka \frakt = \fraks \fraka \frakt loc : First consider the following scalar saturation,

sat(f ) = \left\{ - M if f \leq - M, f if -M \leq f \leq M, M if f \geq M,
where M > 0 is given and denotes the saturation level. Then we take the next extension to an infinite-dimensional setting

(1.2) \fraks \fraka \frakt loc (f )(x) = sat(f (x)).
\bullet \fraks \fraka \frakt = \fraks \fraka \frakt 2 : For f \in L 2 (0, L) we define

(1.3) \fraks \fraka \frakt 2 (f )(x) = \left\{ f (x) if \| f \| L 2 (0,L) \leq M, f (x)M \| f \| L 2 (0,L) if \| f \| L 2 (0,L) \geq M.
In what follows, \fraks \fraka \frakt corresponds to either \fraks \fraka \frakt loc or \fraks \fraka \frakt 2 . In order to consider the saturated stabilization problem, we study the next system

(KdV-S) \left\{ (\partial t u n + \partial x u n + u n \partial x u n + \partial 3 x u n )(t, x) + \fraks \fraka \frakt (a n (x)u n (t, x)) = 0, x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N, \sum N n=1 \partial 2 x u n (t, 0) = - \alpha u 1 (t, 0) -N 3 u 2 1 (t, 0), t > 0, u n (t, \ell n ) = \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = u 0 n (x), x \in (0, \ell n ),
where the damping terms (a n ) n=1,...,N \in \prod N n=1 L \infty (0, \ell n ) act locally on all branches, formally written as (1.4) a n \geq c n > 0 in an open nonempty set \omega n of (0, \ell n ), for all n = 1, . . . , N.

In this work, we are going to consider the following energy E(t) of u = (u 1 , . . . , u N ) T \in \BbbL 2 (\scrT ) by

(1.5) E(t) = 1 2 \| u\| 2 \BbbL 2 (\scrT ) .
The second main result of this paper states the semiglobal exponential stability of (KdV-S).

Theorem 1.4. Assume that the damping terms (a n ) n=1,...,N satisfy (1.4). Let (\ell n ) N n=1 \subset (0, \infty ) and R > 0, then there exist C(R) > 0 and \mu (R) > 0 such that for all u 0 \in \BbbL 2 (\scrT ) with \| u 0 \| \BbbL 2 (\scrT ) \leq R, the energy of any solution of (KdV-S) defined by

(1.5) satisfies E(t) \leq C(R)E(0)e - \mu (R)t for all t > 0.
Then, in order to add damped terms only on the critical lengths as in [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF], we neglect the term u n \partial x u n in the KdV equation (KdV-N). Let I c = \{ n \in \{ 1, \cdot \cdot \cdot , N \} ; \ell n \in \scrN \} be the set of critical lengths and I \ast c be the subset of I c where we remove one index.

We consider now the following problem, (LKdV-S)

\left\{ (\partial t u n + \partial x u n + \partial 3 x u n )(t, x) + \fraks \fraka \frakt (a n (x)u n (t, x)) = 0, x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N, \sum N n=1 \partial 2 x u n (t, 0) = - \alpha u 1 (t, 0), t > 0, u n (t, \ell n ) = \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = u 0 n (x), x \in (0, \ell n ),
where the damping (a n ) n=1,...,N \in \prod N n=1 L \infty (0, \ell n ) satisfy

(1.6) \left\{ a n = 0 for n \in \{ 1, . . . , N \} \setminu I \ast c , a n \geq c n in an open nonempty set \omega n of (0, \ell n ), for all n \in I \ast c , and c n > 0 is a constant.

Then we are able to prove the following global stabilization result, which is the last main result.

Theorem 1.5. Assume that the damping terms (a n ) n=1,...,N satisfy (1.6) and let (\ell n ) N n=1 \subset (0, \infty ). Then, there exist C > 0 and \mu > 0 such that for all u 0 \in \BbbL 2 (\scrT ), the energy of any solution of (LKdV-S) defined by (1.5) satisfies E(t) \leq CE(0)e - \mu t for all t > 0.

Remark 1. Note that for the system (LKdV-S) the stabilization result is global, instead of the one for (KdV-S) which is semiglobal. This difference comes from the action of the term u n \partial x u n : The condition \| u 0 \| \BbbL 2 (\scrT ) \leq R is necessary to handle this term. \circ Remark 2. A global stabilization result for (KdV-S) is, to our knowledge, an open problem. \circ 2. Well-posedness. This section is devoted to prove well-posedness results for (KdV-N)-(KdV-S) and (LKdV-S); in particular, we focus on Theorem 1.3. Our scheme will be to consider appropriate linear systems to derive regularity properties. Then, using a fixed point result, we obtain the well-posedness for the nonlinear systems.

2.1. Linear problems. We start by considering the following linear system for the KdV equation on a star-shaped network \scrT :

(LKdV-N) \left\{ \partial t u n + \partial 3 x u n = f n \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N, N \sum n=1 \partial 2 x u n (t, 0) = h(t), t > 0, u n (t, \ell n ) = 0, \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = u 0 n (x) \forall x \in (0, \ell n ), j = 1, . . . , N.
The terms f n and h are internal and boundary functions that are useful for the fixed point approach. First, we deal with the linear system (LKdV-N) with homogeneous initial condition and homogeneous internal source terms (f n = 0):

(2.1)

\left\{ \partial t u n + \partial 3 x u n = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0), t > 0, \forall n, n \prime = 1, . . . , N, N \sum n=1 \partial 2 x u n (t, 0) = h(t), t > 0, u n (t, \ell n ) = 0, \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = 0, \forall x \in (0, \ell n ), n = 1, . . . , N,
The fact that we work with the linear system \partial t u n + \partial 3

x u n = 0 instead of \partial t u n + \partial x u n + \partial 3 x u n = 0 is motivated by [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg--de Vries equation posed on a finite domain[END_REF][START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF]. It is well known, that the term \partial x u n yields problematic behaviors with respect to regularity and controllability properties, as well noted Rosier in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] and then in several works [START_REF] Capistrano-Filho | General boundary value problems of the Korteweg-de Vries equation on a bounded domain[END_REF][START_REF] Zhang | Exact boundary controllability of the Korteweg--de Vries equation[END_REF][START_REF] Caicedo | Neumann boundary controllability of the Korteweg--de Vries equation on a bounded domain[END_REF]. Now, formally we apply the usual Laplace transform with respect to time to the system (2.1) and obtain (2.2)

\left\{ s\û n + \partial 3 x \û n = 0 \forall x \in (0, \ell n ), n = 1, . . . , N, \ u n (s, 0) = \û n \prime (s, 0) \forall n, n \prime = 1, . . . , N, N \sum n=1 \partial 2 x \û n (s, 0) = \ĥ (s), \ u n (s, \ell n ) = 0, \partial x \û n (s, \ell n ) = 0, n = 1, . . . , N, \ u n (0, x) = 0 \forall x \in (0, \ell n ), n = 1, . . . , N,
where

\û n (s, x) = \int \infty 0 e - st u n (t, x)dt, \ĥ (s) = \int \infty 0 e - st h(t)dt \forall x \in (0, \ell n ).
Following [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg--de Vries equation posed on a finite domain[END_REF], we can see that the N component solutions to (2.2) can be written as We write this previous system in its matrix form A N c N = \ĥ e 1 , where e 1 is the first vector of the canonical basis in \BbbR 3N . We can see easily that A N \in M 3N can be decomposed by induction in blocks as (2.5)

A 1 = \left[ (\lambda 1 ) 2 (\lambda 2 ) 2 (\lambda 3 ) 2 e \lambda 1\ell 1
e \lambda 2\ell 1 e \lambda 3\ell 1 \lambda 1 e \lambda 1\ell 1 \lambda 2 e \lambda 2 \ell 1 \lambda 3 e \lambda 3\ell 1 \right] ,

(2.6)

A N = \left[ A N - 1 (\lambda 1 ) 2 (\lambda 2 ) 2 (\lambda 3 ) 2 0 3(N - 1) - 1\times 3 1 1 1 0 0 0 0 3\times 3(N - 2)
0 0 0

D N \right] = \biggl[ A N - 1 B N C N D N \biggr]
for an appropriate choice of B N , C N , and

(2.7)

D N = \left[ - 1 - 1 - 1 e \lambda 1\ell N
e \lambda 2\ell N e \lambda 3 \ell N \lambda 1 e \lambda 1\ell N \lambda 2 e \lambda 2\ell N \lambda 3 e \lambda 3\ell N \right] .

Formally, taking the inverse of the Laplace transform of \û n in (2.3), we get for t \geq 0 and x \in (0, \ell n )

u n (t, x) = 1 2\pi i \int i\infty - i\infty
e st \û n (s, x)ds = \ĥ (s)e \lambda j (s)x ds, we have

(2.8) u n (t, x) = I n (t, x) + J n (t, x).
Now we introduce the notation, super index, +\setminu -which corresponds to taking s = \pm i\rho 3 , \rho > 0, in the characteristic equation. Then the roots of the characteristic equation are given by

\left\{ \lambda + 1 (\rho ) = i\rho , \lambda + 2 (\rho ) = 1 2 \rho ( \surd 3 -i), \lambda + 3 (\rho ) = 1 2 \rho ( - \surd 3 -i), \lambda - j (\rho ) = \lambda + j (\rho ), j = 1, 2, 3.
Let \Delta N,+ (\rho ) be the determinant of A N (i\rho 3 ) and \Delta N,+ 3(n - 1)+j (s) be the determinant of the matrix that is obtained by replacing the column 3(n - 1)+j of the matrix A N (i\rho 3 ) by [1 0 . . . 0] T and \ĥ+ (\rho ) = \ĥ (i\rho 3 ). Assuming that \Delta N,+ (\rho ) \not = 0 (this property will be justified in Proposition 2.1), Cramer's rule implies that c

N,+ 3(n - 1)+j (\rho ) = c N 3(n - 1)+j (i\rho 3 ) is given by (2.9) c N,+ 3(n - 1)+j (\rho ) = \Delta N,+ 3(n - 1)+j (\rho ) \Delta N,+ (\rho ) \ĥ + (\rho ).
Thus, I n and J n can be seen as (2.10)

I n (t, x) = 3 \sum j=1 1 2\pi \int \infty 0 e i\rho 3 t e \lambda + j (\rho )x \Delta N,+ 3(n - 1)+j (\rho ) \Delta N,+ (\rho ) \ĥ + (\rho )3\rho 2 d\rho , (2.11) J n (t, x) = 3 \sum j=1 1 2\pi \int \infty 0 e - i\rho 3 t e \lambda - j (\rho )x \Delta N, - 3(n - 1)+j (\rho ) \Delta N, -(\rho ) \ĥ -(\rho )3\rho 2 d\rho ,
where we use the notation \Delta N, - k (\rho ) = \Delta N,+ k (\rho ), \Delta N, -(\rho ) = \Delta N,+ (\rho ), and \ĥ -(\rho ) = \ĥ+ (\rho ). Our idea now is to obtain estimates for u n ; for that we are going to prove some asymptotic properties for

\Delta N,+ 3(n - 1)+j (\rho )
\Delta N,+ (\rho ) , the following proposition collects these properties.

Proposition 2.1. For all \rho > 0, \Delta N,+ (\rho ) \not = 0. Moreover, the following asymptotic properties hold, for \rho \rightar \infty ,

(2.12)

\Delta N,+ 3(n - 1)+1 \Delta N,+ \sim - \delta N \rho - 2 e -1 2 \rho \surd 3\ell n - i 3 2 \rho \ell n , \Delta N,+ 3(n - 1)+2 \Delta N,+ \sim \delta N \rho - 2 e - \rho \surd 3\ell n +i \pi 3 , \Delta N,+ 3(n - 1)+3 \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 , 3 \sum j=1 \Delta N,+ 3(n - 1)+j \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 , n = 1, . . . , N,
where \delta N > 0 only depends on N and satisfies \delta N = \delta N - 1 \delta N - 1 + 1 .

Proof. The main problem in this proof is to deal with the determinant of the matrix without making explicit computations. Recall that, in the case of N branches, the matrix A N has size 3N \times 3N . Our proof is based on an induction argument over the number N of branches of the network.

\bullet N = 1: in this case, system (2.4) is exactly the system studied in [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF] for

\ell 1 = 1.
By Appendix B, it holds that \Delta 1,+ (\rho ) \not = 0 for all \rho > 0. Moreover, following the explicit calculations given in [START_REF] Caicedo | Well-posedness of a nonlinear boundary value problem for the Korteweg--de Vries equation on a bounded domain[END_REF] we can deduce

\Delta 1,+ 1 \Delta 1,+ \sim - \rho - 2 e -1 2 \rho \surd 3\ell 1 - i 3 2 \rho \ell 1 , \Delta 1,+ 2 \Delta 1,+ \sim \rho - 2 e - \rho \surd 3\ell 1+i \pi 3 , \Delta 1,+ 3 \Delta 1,+ \sim \rho - 2 e - i \pi 3 , 3 \sum j=1 \Delta 1,+ j \Delta 1,+ \sim \rho - 2 e - i \pi 3 .
That gives (2.12) in the case N = 1.

\bullet Suppose now that \Delta N - 1,+ (\rho ) \not = 0 for all \rho > 0 and that the asymptotic property (2.12) is true for any network of N -1 branches. Let us prove that \Delta N,+ (\rho ) \not = 0 for all \rho > 0 and that the asymptotic property (2.12) holds for a network of N branches. As

A N = \biggl[ A N - 1 B N C N D N \biggr] ,
and we have det(A N - 1 ) = \Delta N - 1,+ \not = 0 by hypothesis, we can write

A N = \biggl[ I 3(N - 1) 0 3(N - 1) C N A - 1 N - 1 I 3(N - 1) \biggr] \biggl[ A N - 1 0 3(N - 1) 0 3(N - 1) D N -C N A - 1 N - 1 B N \biggr] \times \biggl[ I 3(N - 1) A - 1 N - 1 B N 0 3(N - 1) I 3(N - 1)
\biggr] ,

which implies directly that

(2.13) \Delta N,+ = det(A N ) = det(A N - 1 ) det(D N -C N A - 1 N - 1 B N ).
The difficulty of the last expression is the role of the matrix A - 1 N - 1 . In fact, to calculate this inverse explicitly is quite complicated. Note now that if

A - 1 N - 1 = \left[ x 1 . . . . . . . . . x 2 . . . . . . . . . x 3 . . . . . . . . . . . . . . . . . . . . . \right]
, then, we have

(2.14) CN A - 1 N - 1 BN = \left[ (\lambda + 1 ) 2 (x1 + x2 + x3) (\lambda + 2 ) 2 (x1 + x2 + x3) (\lambda + 3 ) 2 (x1 + x2 + x3) 0 0 0 0 0 0 \right] ;
from here we can see that it is not necessary to calculate all the entries of the matrix A - 1 N - 1 . Indeed, we only need the 3 first entries of the first column.

Straightforward calculations show that

(2.15)

x 1 = \Delta N - 1,+ 1 \Delta N - 1,+ , x 2 = \Delta N - 1,+ 2 \Delta N - 1,+ , x 3 = \Delta N - 1,+ 3 \Delta N - 1,+ .
Using (2.14) and (2.15) we get

C N A - 1 N - 1 B N = \left[ (\lambda + 1 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ (\lambda + 2 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ (\lambda + 3 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ 0 0 0 0 0 0 \right]
.

(2.16)

WELL-POSED SATURATED STABILIZATION KDV NETWORK

Then with (2.7)

D N -C N A - 1 N - 1 B N (2.17) = \left[ - 1 -(\lambda + 1 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ - 1 -(\lambda + 2 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ - 1 -(\lambda + 3 ) 2 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ e \lambda + 1 \ell N e \lambda + 2 \ell N e \lambda + 3 \ell N \lambda + 1 e \lambda + 1 \ell N \lambda + 2 e \lambda + 2 \ell N \lambda + 3 e \lambda + 3 \ell N \right]
and using the multilinearity of the determinant

det(D N -C N A - 1 N - 1 B N ) = - 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ det(F N ) + det(D N ),
where

(2.18) F N = \left[ (\lambda + 1 ) 2 (\lambda + 2 ) 2 (\lambda + 3 ) 2 e \lambda + 1 \ell N e \lambda + 2 \ell N e \lambda + 3 \ell N \lambda + 1 e \lambda + 1 \ell N \lambda + 2 e \lambda + 2 \ell N \lambda + 3 e \lambda + 3 \ell N \right]
.

Then, it holds that

(2.19) \Delta N,+ = \Delta N - 1,+ \left[ - 3 \sum j=1 \Delta N - 1,+ j \Delta N - 1,+ det(F N ) + det(D N ) \right] . Using (2.
\ell N , (2.21) det(F N ) = \surd 3\rho 3 e - i\rho \ell N + \surd 3\rho 3 e -1 2 \rho ( \surd 3 - i)\ell N + \surd 3\rho 3 e -1 2 \rho ( - \surd 3 - i)\ell N .
Now, to compute \Delta N,+ 3(n - 1)+j , let A n N,j the matrix obtained by replacing the column 3(n - 1)+j of A N by [1 0\cdot \cdot \cdot 0] T , for j=1, 2, 3 and n=1, . . ., N - 1, that is

A n N,j = \left[ (j+3(n - 1) - th) \underbr \underbr 1 0 . . . B N . . . . . . 0 D N \right] = \left\{ \left[ A n N - 1,j B N 0 1 1 0 0 0 0 0 0 0 D N \right] if j = 1, n = 1 \left[ A n N - 1,j B N 1 0 1 0 0 0 0 0 0 0 D N \right] if j = 2, n = 1 \left[ A n N - 1,j B N 1 1 0 0 0 0 0 0 0 0 D N \right] if j = 3, n = 1 \left[ A n N - 1,j B N C N D N \right] if j = 1, 2, 3, n = 2, \cdot \cdot \cdot , N -1.
(2.22)

We claim the following property of \Delta N,+ 3(n - 1)+j .

Lemma 2.2.

(2.23) \Delta N,+ 3(n - 1)+j = \Delta N - 1,+ 3(n - 1)+j det(D N ), n = 1, . . . , N -1, j = 1, 2, 3.

Proof. Using the decomposition given by (2.22), we get

A n N,j = \left[ A n N - 1,j B N C n N,j D N \right]
for an appropriate choice of C n N,j . Thus, with the same idea as (2.13) it holds that

(2.24) \Delta N,+ 3(n - 1)+j = det(A n N,j ) = det(A n N - 1,j ) det(D N -C n N,j (A n N - 1,j ) - 1 B N ).
Similarly, as before, we need to study the product

C n N,j (A n N - 1,j ) - 1 B N , in
particular, the first column of the matrix (A n N - 1,k ) - 1 . To do that, note that \underbr \underbr 1 \cdot \cdot \cdot 0] T which we know coincides with the first column of (A n N - 1,j ) - 1 and, therefore,

A n N - 1,j v = \left[ (j+3(n - 1) - th) \underbr \underbr 1 0 . . . B N - 1 . . . . . . 0 D N \right] v = \left[ 1 0 . . .
C n N,j (A n N - 1,j ) - 1 B N = 0 3\times 3 ; therefore, with (2.24) \Delta N,+ 3(n - 1)+j = \Delta N - 1,+ 3(n - 1)+j det(D N ), n = 1, . . . , N -1, j = 1, 2, 3,
which finishes the proof of Lemma 2.2.

In order to show that \Delta N,+ \not = 0, note that by (2.19) we get \Delta N,+ = -

3 \sum j=1 \Delta N - 1,+ j det(F N ) + \Delta N - 1,+ det(D N ), j = 1, 2, 3.
Using (2.23) recursively, we get

\Delta N - 1,+ j = \Delta 1,+ j N - 1 \prod \ell =2 det(D \ell ).
Noticing that \Delta

\Delta N,+ = \Delta N - 1,+ 3(n - 1)+j \Delta N - 1,+ det(D N ) - 3 \sum l=1 \Delta N - 1,+ l \Delta N - 1,+ det(F N ) + det(D N ) for j = 1, 2, 3, n = 1, . . . ,
\Delta N - 1,+ l \Delta N - 1,+ \sim \delta N - 1 \rho - 2 e - i \pi 3 . Thus \sum 3 l=1 \Delta N - 1,+ l \Delta N - 1,+ det(F N ) \sim \delta N - 1 \surd 3\rho e \rho 2 \surd 3\ell N +i \rho 2 \ell N - i \pi 3 and
then for \rho \rightar \infty

(2.26) det(D N ) - \sum 3 l=1 \Delta N - 1,+ l \Delta N - 1,+ det(F N ) + det(D N ) \sim 1 \delta N - 1 + 1 .

Now by the induction assumption \Delta

N - 1,+ 3(n - 1)+1 \Delta N - 1,+ \sim - \delta N - 1 \rho - 2 e -1 2 \rho \surd 3\ell n - i 3 2 \rho \ell n , \Delta N - 1,+ 3(n - 1)+2 \Delta N - 1,+ \sim \delta N - 1 \rho - 2 e - \rho \surd 3\ell n +i \pi 3 , \Delta N - 1,+ 3(n - 1)+3 \Delta N - 1,+ \sim \delta N - 1 \rho - 2 e - i \pi
\Delta N,+ \sim - \delta N \rho - 2 e -1 2 \rho \surd 3\ell n - i 3 2 \rho \ell n , \Delta N,+ 3(n - 1)+2 \Delta N,+ \sim \delta N \rho - 2 e - \rho \surd 3\ell n +i \pi 3 , \Delta N,+ 3(n - 1)+3 \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 , 3 \sum j=1 \Delta N,+ 3(n - 1)+j \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 , n = 1, . . . , N -1,
where

\delta N = \delta N - 1 \delta N - 1 +1
. It just remains to study the case n = N . Note that using the block decomposition of A N we get

C N \left[ \Delta N,+ 1 \Delta N,+ \Delta N,+ 2 \Delta N,+ \Delta N,+ 3 \Delta N,+ . . . \Delta N,+ 3N - 5 \Delta N,+ \Delta N,+ 3N - 4 \Delta N,+ \Delta N,+ 3N - 3 \Delta N,+ \right] + D N \left[ \Delta N,+ 3N - 2 \Delta N,+ \Delta N,+ 3N - 1 \Delta N,+ \Delta N,+ 3N \Delta N,+ \right] = \left[ 0 0 0 \right] ,
and recalling (2.6) and (2.7) explicit calculations show that (2.28)

\left[ \Delta N,+ 3N - 2 \Delta N,+ \Delta N,+ 3N - 1 \Delta N,+ \Delta N,+ 3N \Delta N,+ \right] = \biggl( - \sum 3 j=1 \Delta N,+ j \Delta N,+ \biggr) det(D N ) \left[ - \rho \surd 3e - i\rho \ell N \Bigl( \rho \surd 3 2 + 3 2 i\rho \Bigr) e \Bigl( -\rho \surd 3 2 + i\rho 2 \Bigr) \ell N \Bigl( \rho \surd 3 2 -3 2 i\rho \Bigr) e \Bigl( \rho \surd 3 2 + i\rho 2 \Bigr) \ell N \right]
,

and using (2.27) we can conclude from (2.28)

\Delta N,+ 3N - 2 \Delta N,+ \sim - \delta N \rho - 2 e -1 2 \rho \surd 3\ell N - i 3 2 \rho \ell N , \Delta N,+ 3N - 1 \Delta N,+ \sim \delta N \rho - 2 e - \rho \surd 3\ell N +i \pi 3 , \Delta N,+ 3N \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 , 3 \sum j=1 \Delta N,+ 3(N - 1)+j \Delta N,+ \sim \delta N \rho - 2 e - i \pi 3 ,
which gives the induction and concludes the proof of Proposition 2.1.

Remark 3. Recently, in [START_REF] Coron | On the Small-Time Local Controllability of a KdV System for Critical Lengths[END_REF], the problem of small-time local controllability of the nonlinear single KdV equation was addressed. To reach the obstruction to smalltime controllability in [START_REF] Coron | On the Small-Time Local Controllability of a KdV System for Critical Lengths[END_REF] new regularity results in the spirit of [START_REF] Bona | A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane[END_REF] were established.

Those results have some connections with the analysis developed in this work. Here, the analysis of the linear problem (2.4) is based on the estimate of the terms I n and J n ((2.10) and (2.11)). These involve two integrals of \rho from 0 to infinity, and Proposition 2.1 shows the integrands are well-defined (\Delta N,+ \not = 0) and deal with their behavior at infinity. However, in [START_REF] Coron | On the Small-Time Local Controllability of a KdV System for Critical Lengths[END_REF] the behavior of the integrands might be infinite for finite \rho . This is the case where L \in \scrN , with 2k + l / \in 3\BbbN \ast [START_REF] Coron | On the Small-Time Local Controllability of a KdV System for Critical Lengths[END_REF]Lemma B1].

The main difference between these two different behaviors is because in [START_REF] Coron | On the Small-Time Local Controllability of a KdV System for Critical Lengths[END_REF] they worked with the linear system including the term, u x which is necessary to study controllability issues. \circ

Now we are going to state the next regularity result for the solution (2.1) using the Laplace representation obtained in (2.8) and Proposition 2.1.

Proposition 2.3. Let T > 0 and h \in H -1 3 (0, T ), then we have a unique solution u \in \BbbY T of (2.1). Moreover, there exists C > 0 such that for all h \in H -1 3 (0, T ),

\| u\| \BbbY T \leq C\| h\|

H -1 3 (0,T )
.

Proof. This proof uses Proposition 2.1 and follows closely [5, Proposition 2.2] and [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg--de Vries equation posed on a finite domain[END_REF], thus it is omitted here.

Note that Proposition 2.3 justifies the formal computations given in (2.8). Let W the operator that corresponds to the integral representation obtained in Proposition 2.3, i.e., given T > 0 and h \in H -1 3 (0, T ), the unique solution u of (2.1) is given by

u = \left( u 1 . . . u N \right) = W h \in \BbbB T .
Our next step is to consider the linear problem including nonhomogeneous initial data and source terms, as follows:

(2.29)

\left\{ \partial t v n (t, x) + \partial 3 x v n (t, x) = f n (t, x) \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, v n (t, 0) = v n \prime (t, 0) \forall n, n \prime = 1, . . . , N, \sum N n=1 \partial 2 x v n (t, 0) = h(t), t > 0, v n (t, \ell n ) = 0, \partial x v n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, v n (0, x) = v 0 n , x \in (0, \ell n ).
We know from [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF] that in the case, h = 0 the solution of (2.29) can be written as

v(t, x) = W 0 (t)v 0 + \int t 0 W 0 (t -\tau )f (\tau )d\tau ,
for any v 0 \in \BbbL 2 (\scrT ) and f \in L 1 (0, T ; \BbbL 2 (\scrT )), where \{ W 0 (t)\} t\geq 0 is the C 0 -semigroup in the space \BbbL 2 (\scrT ) generated by the operator \scrA v = - \partial 3 x v, with domain (2.30)

D(\scrA ) = \Biggl\{ v \in \Biggl( N \prod n=1 H 3 (0, \ell n ) \Biggr) \cap \BbbH 2 e (\scrT ), N \sum n=1 d 2 v n dx 2 (0) = 0, \Biggr\} where H 2 r (0, \ell n ) = \{ v \in H 2 (0, \ell n ), \bigl( d dx \bigr) i - 1 v(\ell n ) = 0,
\left\{ \partial t \psi + \partial 3 x \psi = f, x \in (0, L), t \geq 0, \psi (t, 0) = \psi (t, L) = \partial x \psi (t, L) = 0, t \geq 0, \psi (0, x) = \psi 0 (x),
x \in (0, L), Proposition 2.4 (Lemma 3.3 of [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg--de Vries equation posed on a finite domain[END_REF]). Let T > 0 and L > 0 be given. For any \psi 0 \in L 2 (0, L) and f \in L 1 (0, T ; L 2 (0, L)), the problem (2.30) admits a unique solution \psi \in C([0, T ]; L 2 (0, L)) \cap L 2 (0, T ; H 1 (0, L)), with \partial \kappa x \psi \in L \infty x (0, L; H \leq C \bigl( \| \psi 0 \| L 2 (0,L) + \| f \| L 1 (0,T ;H 1 (0,L)) \bigr) .

Now for any v 0 n \in L 2 (0, \ell n ) and f n \in L 1 (0, T ; L 2 (0, \ell n )), consider

\psi n = \psi n (t, \cdot ) = W n 1 (t)v 0 n (\cdot ) + \int t 0 W n 1 (t -\tau )f n (\tau , \cdot )d\tau ,
where W n 1 (t) is the C 0 -semigroup associated with the boundary-value problem (2.30)

on (0, \ell n ). Let \frakh (t) = \sum N n=1 \partial 2 x \psi n (t, 0) \in H -1 3 (0, T ) by Proposition 2.4. Now take h \in H -1 3 (0, T ), then by Proposition 2.3 the function w = W (t)(h -\frakh ) is well-defined
and is the solution of (2.1) with boundary data h -\frakh . Finally, the solution v of (2.29) can be expressed as

v(t, \cdot ) = W 1 (t)v 0 (\cdot ) + \int t 0 W 1 (t -\tau )f (\tau , \cdot )d\tau + W (t)(h -\frakh )(t).
The next result encapsulates these ideas.

Proposition 2.5. Let T > 0 be given, then, for any v 0 \in \BbbL 2 (\scrT ), h \in H -1 3 (0, T ), and f \in L 1 (0, T ; \BbbL 2 (\scrT )), the problem (2.29) admits a unique solution v \in \BbbY T . Moreover, there exists C > 0 depending only on T and \ell 1 , . . . , \ell n such that \| v\| \BbbY T \leq C \Bigl( \| h\| Proof of Theorem 1.3. Let (u 0 , 0) \in \BbbX T and R, \theta > 0 that will be chosen after.

H -1 3 (0,T ) + \| f \| L 1 (0,T ;\BbbL 2 (\scrT )) + \| v 0 \| \BbbL 2 (\scrT ) \Bigr) .
Consider the closed ball B \BbbY \theta (0, R) := \{ v \in \BbbY \theta , \| v\| \BbbY \theta \leq R\} . Then B \BbbY \theta (0, R) is a complete metric space. Consider the map \Phi : \BbbY \theta \rightar \BbbY \theta defined by \Phi (v) = u, where u is the solution of

(2.31) \left\{ (\partial t u n + \partial 3 x u n )(t, x) = - (\partial x v n + v n \partial x v n )(t, x) \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0)
\forall n, n \prime = 1, . . . , N,

\sum N n=1 \partial 2 x u n (t, 0) = - \alpha v 1 (t, 0) - N 3 (v 1 (t, 0)) 2 , t > 0, u n (t, \ell n ) = \partial x u n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, u n (0, x) = u 0 n , x \in (0, \ell n ).
Clearly, u \in \BbbY \theta is solution of (KdV-N) if u is a fixed point of \Phi . Now we write two lemmas to deal with the source term and boundary conditions.

Lemma 2.6 (Lemma 3.1 of [3]

). There exists a constant C > 0 such that for any

T > 0 and u, v \in Y T \int T 0 \| u(t, \cdot )\partial x v(t, \cdot )\| L 2 (0,L) dt \leq C(T 1/2 + T 1/3 )\| u\| Y T \| v\| Y T ,
where Y T is \BbbY T for N = 1.

Lemma 2.7 (Lemma 3.2 of [START_REF] Jia | Boundary stabilization of the Korteweg-de Vries equation and the Korteweg-de Vries-Burgers equation[END_REF]). There exist of constants C, \beta > 0 such that for any T > 0 and g 1 , g 2 \in H 1 3 (0, T ), it holds that, g 1 g 2 \in H -1 3 (0, T ) and

\| g 1 g 2 \| H -1 3 (0,T ) \leq CT \beta \| g 1 \| H 1 3 (0,T ) \| g 2 \| H 1 3 (0,T )
.

From Proposition 2.5 and Lemmas 2.6 and 2.7 we get for all v \in \BbbY \theta

\| \Phi (v)\| \BbbY \theta =\leq C \Biggl( \| u 0 \| \BbbL 2 (\scrT ) + \bigm\| \bigm\| \bigm\| \bigm\| - \alpha v 1 (t, 0) - N 3 (v 1 (t, 0)) 2 \bigm\| \bigm\| \bigm\| \bigm\| H -1 3 (0,\theta ) + \int \theta 0 \| \partial x v(t, \cdot )\| \BbbL 2 (\scrT ) dt + \int \theta 0 \| v(t, \cdot )\partial x v(t, \cdot )\| \BbbL 2 (\scrT ) dt \Biggr) \leq C \Bigl( \| u 0 \| \BbbL 2 (\scrT ) + \theta \beta (\| v\| \BbbY \theta + \| v\| 2 \BbbY \theta ) + (\theta 1/2 + \theta 1/3 )\| v\| 2 \BbbY \theta + \theta 1/2 \| v\| \BbbY \theta \Bigr) .
We consider \Phi restricted to the closed ball B \BbbY \theta (0, R) and choose \theta , R > 0 such that \left\{ R = 3C\| u 0 \| \BbbL 2 (\scrT ) , C(\theta \beta + \theta 1/2 ) \leq 1 3 , C(\theta \beta + \theta 1/2 + \theta 1/3 )R \leq 1 6 .

(2.32)

WELL-POSED SATURATED STABILIZATION KDV NETWORK

Thus, for u \in B \BbbY \theta (0, R), \Phi maps B \BbbY \theta (0, R) into itself. Take now v and \widetil v \in B \BbbY \theta (0, R), then w = \Phi (v) -\Phi (\widetil v) solves the equation

\left\{ \partial t w n + \partial 3 x w n = - (\partial x v n -\partial x \widetil v n ) -1 2 \partial x ((v n -\widetil v n )(v n + \widetil v n )) \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, w n (t, 0) = w n \prime (t, 0)
\forall n, n \prime = 1, . . . , N,

N \sum n=1 \partial 2 x w n (t, 0) = - \alpha (v 1 (t, 0) -\widetil v 1 (t, 0)), - N 3 ((v 1 (t, 0) -\widetil v 1 (t, 0))(v 1 (t, 0) + \widetil v 1 (t, 0))), t > 0, w n (t, \ell n ) = \partial x w n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, w n (0, x) = 0,
x \in (0, \ell n ).

\| \Phi (v) -\Phi (\widetil v)\| \BbbY \theta \leq \biggl( 1 3 + 1 3 \biggr) \| v -\widetil v\| \BbbY \theta = 2 3 \| v -\widetil v\| \BbbY \theta .
That means that the map \Phi is a contraction on B \BbbY \theta and by the Banach fixed point theorem has a unique fixed point u \in \BbbY \theta . It gives the local-in-time well-posedness for bounded initial data. Now taking T > 0, we can check using integration by parts and the boundary conditions that every solution of (KdV-N) satisfies

(2.33) d dt E(t) = - \biggl( \alpha - N 2
\biggr) | u 1 (t, 0)| 2 - 1 2 N \sum n=1 | \partial x u n (t, 0)| 2 \leq 0
since N \leq 2\alpha . This dissipation law tells us that the energy is a nonincreasing function of the time variable, that means

(2.34) E(t) \leq E(\theta ) \leq E(0) = 1 2 \| u 0 \| \BbbL 2 (\scrT ) \forall t > \theta > 0.
From here, taking the maximum for t \in [0, T ] we can see that

(2.35) \| u\| C([0,T ];\BbbL 2 (\scrT )) \leq \| u 0 \| \BbbL 2 (\scrT ) .
Finally, following [START_REF] Parada | Delayed stabilization of the Korteweg--de Vries equation on a star-shaped network[END_REF][START_REF] Cerpa | On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network[END_REF] we multiply (KdV-N) by q n u n , integrate over (0, T )\times (0, \ell n ), and sum over n = 1, . . . , N to obtain the following equality:

N \sum n=1 \int \ell n 0 q n (t, x)| u n (t, x)| 2 dx \bigm| \bigm| T 0 - N \sum n=1 \int T 0 \int \ell n 0 (\partial t q n + \partial x q n + \partial 3 x q n )| u n | 2 dxdt (2.36) + 3 N \sum n=1 \int T 0 \int \ell n 0 | \partial x u n | 2 \partial x q n dxdt - 2 3 N \sum n=1 \int T 0 \int \ell n 0 | u n | 3 \partial x q n dxdt = N \sum n=1 \int T 0 \bigl[ (q n + \partial 2 x q n )| u n | 2 + 2q n u n \partial 2 x u n - 2\partial x q n u n \partial x u n -q n | \partial x u n | 2 + 2 3 q n | u n | 3 \biggr] (t, 0)dt.
\bullet Taking q n = 1 in (2.36) we can derive (2.37)

N \sum n=1 \int T 0 | \partial x u n (t, 0)| 2 dt \leq \| u 0 \| 2 \BbbL 2 (\scrT ) .
\bullet If we take q n = x(2\ell n - x) \ell n , we can obtain

2N L 2 \| u 1 (\cdot , 0)\| 2 L 2 (0,T ) \leq 2T \ell 2 \| u\| 2 C([0,T ];\BbbL 2 (\scrT )) -2 \int T 0 u 1 (t, 0) N \sum n=1 \partial x u n (t, 0) 2 \ell n dt + \| u 0 \| 2 \BbbL 2 (\scrT ) + 4 3\ell N \sum n=1 \int T 0 \int \ell n 0 u 3 n (t, x)dxdt.
Using (2.35)--(2.37) and Young's inequality we derive

(2.38) \| u 1 (t, 0)\| 2 L 2 (0,T ) \leq C(T + 1)\| u 0 \| 2 \BbbL 2 (\scrT ) + C N \sum n=1 \int T 0 \int \ell n 0 u 3 n (t, x)dxdt.
As

H 1 (0, \ell n ) embeds compactly into C([0, \ell n ]) we get N \sum n=1 \int T 0 \int \ell n 0 | u n | 3 dxdt \leq CT 1/2 \| u 0 \| 2 \BbbL 2 (\scrT ) \| u\| L 2 (0,T ;\BbbH 1 e (\scrT ))
and then with (2.38)

(2.39) \| u 1 (t, 0)\| 2 L 2 (0,T ) \leq C(T + 1)\| u 0 \| 2 \BbbL 2 (\scrT ) + CT 1/2 \| u 0 \| 2 \BbbL 2 (\scrT ) \| u\| L 2 (0,T ;\BbbH 1 e (\scrT )) .

\bullet Finally, considering q j = x and using ( , and T > 0. Then, for all u 0 \in \BbbL 2 (\scrT ), there exists a unique solution u \in \BbbB T of (KdV-S) or (LKdV-S).

Moreover, there exist 0 < T \ast \leq T and C > 0 such that u \in \BbbY T \ast and \| u\| \BbbY T \ast \leq C\| u 0 \| \BbbL 2 (\scrT ) .

Stabilization.

In this section, we are going to prove our stabilization results

inspired by [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF]. The proofs are based on observability inequalities for (KdV-S) and

(LKdV-S), respectively. These inequalities imply the exponential stability. First, note that, given T > 0, we can check that every solution of (KdV-S) and (LKdV-S) has a nonincreasing energy,

(3.1) d dt E(t) = - \biggl( \alpha - N 2
\biggr) | u 1 (t, 0)| 2 - 1 2 N \sum n=1 | \partial x u n (t, 0)| 2 - N \sum n=1 \int \ell n 0
u n \fraks \fraka \frakt (a n u n )dx \leq 0.

Stability of (KdV-S).

We start by studying (KdV-S). First, note that multiplying (KdV-S) by u n and integrating on (0, s) \times (0, \ell n ) we get

N \sum n=1 \int \ell n 0 | u n (s, x)| 2 dx + N \sum n=1 \int s 0 \int \ell n 0 \fraks \fraka \frakt (a n u n )u n dxdt + (2\alpha -N ) \int s 0 | u 1 (t, 0) 2 dt + N \sum n=1 \int s 0 | \partial x u n (t, 0)| 2 dt = \| u 0 \| 2 \BbbL 2 (\scrT ) .
Integrating again this expression with respect to time on (0, T ) we obtain

(3.2) T \| u 0 \| 2 \BbbL 2 (\scrT ) \leq \int T 0 \| u(t, \cdot )\| 2 \BbbL 2 (\scrT ) dt + (2\alpha -N )T \int T 0 | u 1 (t, 0)| 2 dt + T N \sum n=1 \int T 0 | \partial x u n (t, 0)| 2 dt + T N \sum n=1 \int T 0 \int \ell n 0 \fraks \fraka \frakt (a n u n )u n dxdt.
Our goal here is to prove the following observability inequality:

(Obs)

\| u 0 \| 2 \BbbL 2 (\scrT ) \leq C \Biggl( (2\alpha -N ) \int T 0 | u 1 (t, 0)| 2 dt + N \sum n=1 \int T 0 | \partial x u n (t, 0)| 2 dt + N \sum n=1 \int T 0 \int \ell n 0 \fraks \fraka \frakt (a n u n )u n dxdt \Biggr) .
Note that (Obs) is quite similar to (3.2). From (3.2) we can observe that to get (Obs)

it is enough to prove the following inequality:

\int T 0 \| u(t, \cdot )\| 2 \BbbL 2 (\scrT ) dt \leq C \Biggl( (2\alpha -N ) \int T 0 | u 1 (t, 0)| 2 dt + N \sum n=1 \int T 0 | \partial x u n (t, 0)| 2 dt + N \sum n=1 \int T 0 \int \ell n 0 \fraks \fraka \frakt (a n u n )u n dxdt \Biggr) .
Suppose that it is false and take \| u 0 \| \BbbL 2 (\scrT ) \leq R, then we can find (u 0,j ) j\in \BbbN \subset \BbbL 2 (\scrT ) such that \| u 0,j \| \BbbL 2 (\scrT ) \leq R and

lim j\rightar\infty \| u j \| 2 L 2 (0,T ;\BbbL 2 (\scrT )) (2\alpha -N )\| u j 1 (\cdot , 0)\| 2 L 2 (0,T ) + \| \partial xu j (\cdot , 0)\| 2 L 2 (0,T ) + N \sum n=1 \int T 0 \int \ell n 0 \fraks \fraka \frakt (anu j n )u j n dxdt = \infty ,
where u j is the corresponding solution of (KdV-S) with initial data u 0,j . Note now that using (2.33), we deduce (3.3) \| u j (t, \cdot )\| \BbbL 2 (\scrT ) \leq \| u 0,j \| \BbbL 2 (\scrT ) \leq R.

Take \lambda j = \| u j \| L 2 (0,T ;\BbbL 2 (\scrT )) , then \lambda j \leq T 1/2 \| u 0,j \| \BbbL 2 (\scrT ) \leq T 1/2 R. Thus (\lambda j ) j\in \BbbN \subset \BbbR

is bounded. Taking v j n = u j n \lambda j , then v j fulfills (3.4) \left\{ \biggl( \partial t v j n + \partial x v j n + \partial 3 x v j n + \lambda j v j n \partial x v j n + \fraks \fraka \frakt (a n \lambda j v j n ) \lambda j \biggr) (t, x) = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, v j n (t, 0) = v j n \prime (t, 0) \forall n, n \prime = 1, . . . , N, N \sum n=1 \partial 2 x v j n (t, 0) = - \alpha v j 1 (t, 0) -\lambda j N 3 (v j 1 (t, 0)) 2 , t > 0, v j n (t, \ell n ) = \partial x v j n (t, \ell n ) = 0, t > 0, n = 1, . . . , N, \| v j \| L 2 (0,T ;\BbbL 2 (\scrT )) = 1,
and satisfies

(3.5) (2\alpha - N )\| v j 1 (t, 0)\| 2 L 2 (0,T ) +\| \partial x v j (t, 0)\| 2 L 2 (0,T ) + N \sum n=1 \int T 0 \int \ell n 0 1 \lambda j \fraks \fraka \frakt (a n \lambda j v j n )v j n dxdt \rightar 0.
First, note that multiplying (3.4) by v j n and integrating on (0, s) \times (0, \ell n ) we get

(3.6) N \sum n=1 \int \ell n 0 | v j n (s, x)| 2 dx + N \sum n=1 \int s 0 \int \ell n 0 1 \lambda j \fraks \fraka \frakt (a n \lambda j v j n )v j n dxdt + (2\alpha -N ) \int s 0 | v j 1 (t, 0) 2 dt + N \sum n=1 \int s 0 | \partial x v j n (t, 0)| 2 dt = \| v j (0, \cdot )\| 2 \BbbL 2 (\scrT ) ,
which gives us, using that \fraks \fraka \frakt is odd,

(3.7) \| v j \| 2 C([0,T ];\BbbL 2 (\scrT )) \leq \| v j (0, \cdot )\| 2 \BbbL 2 (\scrT ) , \| \partial x v j (t, 0)\| 2 L 2 (0,T ) \leq \| v j (0, \cdot )\| 2 \BbbL 2 (\scrT ) .
Now integrating (3.6) again with respect to time on (0, T ) we obtain (3.8)

T \| v j (0, \cdot )\| 2 \BbbL 2 (\scrT ) \leq \int T 0 \| v j (t, \cdot )\| 2 \BbbL 2 (\scrT ) dt + (2\alpha -N )T \int T 0 | v j 1 (t, 0)| 2 dt + T N \sum n=1 \int T 0 | \partial x v j n (t, 0)| 2 dt+2T N \sum n=1 \int T 0 \int \ell n 0 1 \lambda j \fraks \fraka \frakt (a n \lambda j v j n )v j n dxdt.
This last inequality implies that (v j (0, \cdot )) j\in \BbbN is bounded in \BbbL 2 (\scrT ). Again using that \fraks \fraka \frakt is odd and similar estimates in (2.37)--(2.39)--(2.40) we conclude

(3.9) \| v j \| 2 L 2 (0,T ;\BbbH 1 e (\scrT )) \leq C \Bigl( \| v j (0, \cdot )\| 2 \BbbL 2 (\scrT ) + \| v j (0, \cdot )\| 4 \BbbL 2 (\scrT )
\Bigr) .

Thus (v j ) j\in \BbbN \subset L 2 (0, T ; \BbbH 1 e (\scrT )) is bounded and it holds that

\| v j n \partial x v j n \| L 2 (0,T ;L 1 (0,\ell n )) \leq \| v j \| C([0,T ],\BbbL 2 (\scrT )) \| v j \| L 2 (0,T ;\BbbH 1 e (\scrT )) , which implies that (v j n \partial x v j n ) j\in \BbbN is a subset of L 2 (0, T ; L 1 (0, \ell n )). Using Lemma A.1 we have \bigm\| \bigm\| \bigm\| \bigm\| \fraks \fraka \frakt (a n \lambda j v j n ) \lambda j \bigm\| \bigm\| \bigm\| \bigm\| L 2 (0,T ;L 2 (0,\ell n )) \leq 3\| a n \| L \infty (0,\ell n) \ell 1/2 n \| v j \| L 2 (0,T ;\BbbH 1 e (\scrT )) ,
and then (

\fraks \fraka \frakt (an\lambda j v j n ) \lambda j
) j\in \BbbN is a subset of L 2 (0, T ; L 2 (0, \ell n )). From this, we can see that

\partial t v j n = - (\partial 3 x v j n + \partial x v j n + \lambda j v j n \partial x v j n + \fraks \fraka \frakt (an\lambda j v j n ) \lambda j ) is bounded in L 2 (0, T ; H - 2 (0, \ell n )).
Hence, by the Aubin--Lions lemma ([24, Chapter III, Proposition 1.3]) we can deduce that (v j ) j\in \BbbN is relatively compact in L 2 (0, T ; \BbbL 2 (\scrT )) and we can assume that v j converges strongly at v in L 2 (0, T ; \BbbL 2 (\scrT )) with \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1. Now we are going to study the case \fraks \fraka \frakt = \fraks \fraka \frakt 2 and \fraks \fraka \frakt = \fraks \fraka \frakt loc separately.

3.1.1. Case \bffraks \bffraka \bffrakt = \bffraks \bffraka \bffrakt \bftwo . First, we consider the case \fraks \fraka \frakt = \fraks \fraka \frakt 2 . We know that by (3.3), \| u j (t, \cdot )\| \BbbL 2 (\scrT ) \leq R and then by Lemma A.2 we have that

0 \leq N \sum n=1 \int T 0 \int \ell n 0 a n k n (R)| v j n | 2 dxdt \leq N \sum n=1 \int T 0 \int \ell n 0 1 \lambda j \fraks \fraka \frakt 2 (a n \lambda j v j n )v j n ,
which gives us using (3.5), as j \rightar \infty ,

(2\alpha -N )\| v j 1 (t, 0)\| 2 L 2 (0,T ) + \| \partial x v j (t, 0)\| 2 L 2 (0,T ) + N \sum n=1 \int T 0 \int \ell n 0 a n k n (R)| v j n | 2 dxdt \rightar 0. (3.10)
Furthermore, passing to the limit in (3.10) we get

(2\alpha -N )\| v 1 (t, 0)\| 2 L 2 (0,T ) + \| \partial x v(t, 0)\| 2 L 2 (0,T ) + N \sum n=1 \int T 0 \int \ell n 0 a n k n (R)| v n | 2 dxdt \leq lim inf \Biggl( (2\alpha -N )\| v j 1 (t, 0)\| 2 L 2 (0,T ) + \| \partial x v j (t, 0)\| 2 L 2 (0,T ) + N \sum n=1 \int T 0 \int \ell n 0 a n k n (R)| v j n | 2 dxdt \Biggr) = 0. Thus, v n (t, x) = 0 in (0, T ) \times \omega n and (2\alpha -N )v 1 (t, 0) = \partial x v n (t, 0) = 0 in (0, T )
for all n = 1, . . . , N . Additionally, as (\lambda j ) j\in \BbbN is bounded and nonnegative, we can extract a convergent subsequence such that \lambda j \rightar \lambda \geq 0, consequently v satisfies \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1 and solves the following system:

(3.11) \left\{ \partial t v n + \partial x v n + \partial 3 x v n + \lambda v n \partial x v n = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, v n (t, \ell n ) = \partial x v n (t, \ell n ) = \partial x v n (t, 0) = 0, t \in (0, T ) \forall n = 1, . . . , N, (2\alpha -N )v n (t, 0) = 0, t \in (0, T ), v n (t, x) = 0, (t, x) \in (0, T ) \times \omega n .
1. If \lambda = 0 the system satisfied by v is linear, then we can use Holmgrem's theorem as in [START_REF] Perla Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF] to conclude that v = 0, which contradicts the fact that \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1.

2. If \lambda > 0, we have to prove that v n \in L 2 (0, T ; H 3 (0, \ell n )) in order to apply [START_REF] Saut | Unique continuation for some evolution equations[END_REF]Theorem 4.2]. Consider w n = \partial t v n then

\left\{ \partial t w n + \partial x w n + \partial 3 x w n + \lambda w n \partial x v n + \lambda v n \partial x w n = 0 \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, w n (t, \ell n ) = \partial x w n (t, \ell n ) = \partial x w n (t, 0) = 0 \forall n = 1, . . . , N, (2\alpha -N )w n (t, 0) = 0, t \in (0, T ) \forall j = 1, . . . , N, w n (t, x) = 0, (t, x) \in (0, T ) \times \omega n , w n (0, x) = - v \prime n (0, x) -v \prime \prime \prime n (0, x) -\lambda v n (0, x)v \prime n (0, x) \in H - 3 (0, \ell n ), x \in (0, \ell n ), j = 1, . . . , N.
With [9, Lemma A.2] we can get that w n (0, x) \in L 2 (0, \ell n ) and w n \in C([0, T ],

L 2 (0, \ell n )) \cap L 2 (0, T ; H 1 (0, \ell n )). Thus, \partial 3 x v n = - (\partial t v n -\partial x v n -\lambda v n \partial x v n ) \in L 2 (0, T ; L 2 (0, \ell n )) which implies v n \in L 2 (0, T ; H 3 (0, \ell n )).
Applying [23, Theorem 4.2] we obtain that v n = 0 for all j = 1, . . . , N that contradicts the fact that \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1.

3.1.2. Case \bffraks \bffraka \bffrakt = \bffraks \bffraka \bffrakt loc . Let us consider the case where \fraks \fraka \frakt = \fraks \fraka \frakt loc , by the injection of H 1 (0, \ell n ) into C([0, \ell n ]), we can derive using similar estimate as in (3.9), (3.12)

\int T 0 | u j n (t, x)| 2 dt \leq \ell n \| u j \| 2
L 2 (0,T ;\BbbH 1 e (\scrT )) \leq \ell n \beta for \beta = (R 2 + R 4 ). Now, inspired by [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF], take \Omega n,i \subset [0, T ] defined as follows:

(3.

(2\alpha -N )\| v j 1 (t, 0)\| 2 L 2 (0,T ) + \| \partial x v j (t, 0)\| 2 L 2 (0,T ) + N \sum n=1 \int \Omega c n,i \int \ell n 0 a n k n (R)| v j n | 2 dxdt \rightar 0.
Thus, the limit function v satisfies (2\alpha -N )v 1 (t, 0) = \partial x v n (t, 0) = 0 in (0, T ) for all n = 1, . . . , N and v n (t, x) = 0 in \cup i\in \BbbN \Omega c n,i \times \omega n . Using (3.14), we know that \nu (\cup i\in \BbbN \Omega c n,i ) = T , thus we get that, for almost every t \in [0, T ], v n (t, x) = 0 for x \in \omega n .

Last v is a solution to (3.11) and we conclude as we do in the case \fraks \fraka \frakt = \fraks \fraka \frakt 2 .

Finally, we obtain that (Obs) is valid for a solution (KdV-S) with \| u n n \| \BbbL 2 (\scrT ) \leq R.

Proof of Theorem 1.4. The proof closely follows [START_REF] Parada | Delayed stabilization of the Korteweg--de Vries equation on a star-shaped network[END_REF] (see also [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF]). Note that for u 0 \in \BbbL 2 (\scrT ) such that \| u 0 \| \BbbL 2 (\scrT ) \leq R using that the energy is nonincreasing using Hence we have E(mT ) \leq e - \mu mT E(0), where \mu = 1 T ln( 1 \gamma ) > 0. Let t > 0 then there exists m \in \BbbN \ast such that (m -1)T < t \leq mT , and then again using the nonincreasing property of the energy we get E(t) \leq E((m -1)T ) \leq e - \mu (m - 1)T E(0) \leq 1 \gamma e - \mu t E(0).

This concludes the proof of Theorem 1.4.

Stability (LKdV-S

). Now we study the stabilization of (LKdV-S). For doing that, we follow the approach of section 3.1, and we prove the following observability inequality (Obs2)

\| u 0 \| 2 \BbbL 2 (\scrT ) \leq C \left( (2\alpha -N ) \int T 0 | u 1 (t, 0)| 2 dt + N \sum j=1 \int T 0 | \partial x u n (t, 0)| 2 dt + \sum j\in I \ast c \int T 0 \int \ell n 0 \fraks \fraka \frakt (a n u n )u n dxdt \right)
for any solution u of (LKdV-S). Suppose that it is false, then there exists a sequence (u 0,j ) j\in \BbbN \subset \BbbL 2 (\scrT ) such that \| u 0,j \| \BbbL 2 (\scrT ) = 1 and the corresponding solution of (LKdV-S) satisfies

(2\alpha -N )\| u j 1 (\cdot , 0)\| 2 L 2 (0,T ) + \| \partial x u j (\cdot , 0)\| 2 L 2 (0,T ) + \sum n\in I \ast c \int T 0 \int \ell n 0
\fraks \fraka \frakt (a n u j n )u j n dxdt \rightar 0, as j \rightar \infty . Using the same arguments as in Theorem 1.4 we can find a nontrivial solution v \in \BbbB T of (LKdV-S) such that \left\{ (2\alpha -N ) \| v 1 (\cdot , 0)\| L 2 (0,T ) = 0, \| \partial x v(\cdot , 0)\| L 2 (0,T ) = 0, v n = 0 in (0, T ) \times \omega n , n \in I \ast c , \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1.

We distinguish three cases: \bullet For n \in I \ast c , v n = 0 in (0, T ) \times \omega n . Then, \partial t v n + \partial x v n + \partial 3 x v n = 0 and thanks to Holmgrem's theorem, v n = 0 for all n \in I \ast c . Note that this implies that v n (t, 0) = 0 for all n \in I \ast c and by the continuity condition v n (t, 0) = 0 for all n = 1, . . . , N .

\bullet For n \in \{ 1, . . . , N \} \setminu I c , v n is the solution to

\left\{ \partial t v n + \partial x v n + \partial 3 x v n = 0,
x \in (0, \ell n ), t \in (0, T ), n = 1, . . . , N, v n (t, 0) = 0, t \in (0, T ) \forall j = 1, . . . , N,

\sum N n=1 \partial 2 x v n (t, 0) = 0, t \in (0, T ), v n (t, \ell n ) = \partial x v n (t, \ell n ) = 0, t \in (0, T ), n = 1, . . . , N, v n (0, x) = v 0 n , x \in (0, \ell n ).
Then thanks to [1, Lemma 3.2], v n = 0.

\bullet For n \in I c \setminu I \ast c , v n then satisfies

\left\{ \partial t v n + \partial x v n + \partial 3 x v n = 0, t \in (0, T ) \forall x \in (0, \ell n ), v n (t, 0) = \partial x v n (t, 0) = \partial 2 x v n (t, 0) = 0, t \in (0, T ), v n (t, \ell n ) = \partial x v n (t, \ell n ) = 0, t \in (0, T ), v n (0, x) = v 0 n , x \in (0, \ell n ).
Due to the three null conditions at the central node, we obtain that v n = 0.

Thus v = 0 and we get a contradiction, with \| v\| L 2 (0,T ;\BbbL 2 (\scrT )) = 1 which ends the proof of (Obs2). As we have the observability inequality (Obs2), to derive the exponential decay of the energy of (LKdV-S) given in Theorem 1.5, it is enough to follow the proof of Theorem 1.4. tions (u n (t, \ell n ) = \partial x u n (t, \ell n ) = 0) were considered. These conditions come from the problems studied in [START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF][START_REF] Parada | Delayed stabilization of the Korteweg--de Vries equation on a star-shaped network[END_REF], but in a more general framework the following problem could be studied:

\left\{ (\partial t u n +\partial x u n +u n \partial x u n +\partial 3 x u n )(t, x) = f n (t, x), \forall x \in (0, \ell n ), t > 0, n = 1, . . . , N, u n (t, 0) = u n \prime (t, 0) \forall n, n \prime = 1, . . . , N, N \sum n=1 \partial 2 x u n (t, 0) = - \alpha u 1 (t, 0) - N 3 u 2 1 (t, 0) + h(t), t > 0, u n (t, \ell n ) = g n (t), \partial x u n (t, \ell n ) = p n (t), t > 0, n = 1, . . . , N, u n (0, x) = u 0 n , x \in (0, \ell n ). (4.1)
We expected that adapting the ideas introduced in this paper and in [START_REF] Bona | A nonhomogeneous boundary-value problem for the Korteweg--de Vries equation posed on a finite domain[END_REF], it could be possible to obtain the following result.

\left\{ u 0 n (\ell n ) = g n (0) n = 1, . . . , N if 1 2 < s \leq 3, \partial x u 0 n (\ell n ) = p n (0) n = 1, . . . , N if 3 2 < s \leq 3, \sum N n=1 \partial 2 x u 0 n (0) = h(0) if 5 2 < s \leq 3,
there exists a unique solution u \in \prod

N n=1 C([0, T ]; H s (0, \ell n )) \cap L 2 (0, T \ast ; H s+1 (0, \ell n ))
of (4.1). Moreover \partial \kappa x u n \in L \infty x (0, \ell n ; H s+1 - \kappa 3

(0, T \ast )) for \kappa = 0, 1, 2.

The complications would come from the study of the matrix, which is obtained by replacing the column j + 3(n -1) of A N by [0 1 0 \cdot \cdot \cdot 0] T for the g n case and

[0 0 1 \cdot \cdot \cdot 0] T for the p n case. It is not clear how to derive a result similar to (2.23). Lemma A.1 (Lemma 3.2 of [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF]). For all (f, \widetil f ) \in L 2 (0, L), we have

(A.2) \| \fraks \fraka \frakt (f ) -\fraks \fraka \frakt ( \widetil f )\| L 2 (0,L) \leq 3\| f -\widetil f \| L 2 (0,L) .
Lemma A.2 (Lemma 4.3 of [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF]). Let r be a positive value and a : [0, L] \rightar \BbbR be a function satisfying (A.1) and k(r) defined by

(A.3) k(r) = min \biggl\{ M a \ast r , 1
\biggr\} :

1. Given \fraks \fraka \frakt = \fraks \fraka \frakt 2 and f \in L 2 (0, L) such that \| f \| L 2 (0,L) \leq r, we have

(A.4) (\fraks \fraka \frakt 2 (a(x)f (x)) -k(r)a(x)f (x))f (x) \geq 0 \forall x \in [0, L].
2. Given \fraks \fraka \frakt = \fraks \fraka \frakt loc and f \in L \infty (0, L) such that \forall x \in [0, L], | f (x)| \leq r, we have

(A.5) (\fraks \fraka \frakt loc (a(x)f (x)) -k(r)a(x)f (x))f (x) \geq 0 \forall x \in [0, L].
Lemma A.3 (Proposition 3.4 of [START_REF] Marx | Global stabilization of a Korteweg--de Vries equation with saturating distributed control[END_REF]). Let a : [0, L] \rightar \BbbR satisfy (A.1). If y \in L 2 (0, T ; H 1 (0, L)), then \fraks \fraka \frakt (ay) \in L 1 (0, T ; L 2 (0, L)) is continuous and \forall y, z \in L 2 (0, T ; H 1 (0, L)) we have \| \fraks \fraka \frakt (ay) -\fraks \fraka \frakt (az)\| L 1 (0,T ;L 2 (0,L)) \leq 3L 1/2 T 1/2 a \ast \| y -z\| L 2 (0,T ;H 1 (0,L)) .

Appendix B. For all \bfits \not = 0 with Re(\bfits ) \geq 0, it holds that \Delta \bfone (\bfits ) \not = 0.

This property was stated in [7, Remark 2.5] without proof; here, for the sake of completeness, we give a proof based on [START_REF] Capistrano-Filho | Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictions on the spatial domain[END_REF]. Suppose that \Delta 1 (s) = 0 for some s with Re(s) \geq 0. Then, there exists f \in H 3 (0, \ell 1 ), a nontrivial solution of (B.1) \biggl\{ sf (x) + f \prime \prime \prime (x) = 0, x \in (0, \ell 1 ), f \prime \prime (0) = f \prime (\ell 1 ) = f (\ell 1 ) = 0. Now, consider the conjugate of (B. Appendix C. For all \bfitrho >0 and \bfitj \in \{ 1, . . . , \bfitN \} , it holds that det(\bfitD \bfitj )\not =0.

Let j \in \{ 1, . . . , N \} . Following [START_REF] Capistrano-Filho | Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictions on the spatial domain[END_REF] and Appendix B, suppose that det(D j ) = 0 for some \rho > 0. Then, there exists f \in H 3 (0, \ell j ), a nontrivial solution of (C.1) \biggl\{ i\rho 3 f (x) + f \prime \prime \prime (x) = 0, x \in (0, \ell j ), f (0) = f (\ell j ) = f \prime (\ell j ) = 0. Now, consider the conjugate of (C.1), (C.2)

\biggl\{

- i\rho 3 f (x) + f \prime \prime \prime (x) = 0, x \in (0, \ell j ), f (0) = f (\ell j ) = f \prime (\ell j ) = 0. Then, adding (C.5) and (C.6), we obtain f \prime \equiv 0. Using the boundary conditions of (C.1) we deduce f \equiv 0 which is a contradiction. Hence, det(D j ) \not = 0 for all \rho > 0.

Appendix it can be shown that for all \rho > 0, \ell j > 0 it holds that F (\rho , \ell j ) < 0. Thus,

and \| u\| 2 \BbbH 1 e

 1 where the index r is related to the null right boundary conditions, the space \BbbH 1 e (\scrT ) be the Cartesian product of H 1 r (0, \ell n ) including the continuity condition on the central node (u n (0) = u n \prime (0) \forall n, n \prime = 1, . . . , N ) \BbbH 1 e (\scrT ) = \Biggl\{ u = (u 1 , \cdot \cdot \cdot , u N ) T \in N \prod n=1 H 1 r (0, \ell n ), u n (0) = u n \prime (0) \forall n, n \prime = 1, . . . , N \Biggr\} , (\scrT ) = N \sum n=1 \| u n \| 2 H 1 (0,\ell n) ,

  T )) for \kappa = 0, 1, 2, with the induced norm \| v\| \BbbY T = \| v\| \BbbB T +

2 \lambda j e \lambda j \ell 1 1 )

 211 +j (s)e \lambda j (s)x , where \lambda j (s), j = 1, 2, 3 are the solutions of the characteristic equation s + \lambda 3 = 0 and c N = (c k ) N k=1,...,3N solves the following linear system (+j e \lambda j \ell n = 0, 3 \sum j=1 c N 3(n - 1)+j \lambda j e \lambda j \ell n = 0 \right\} \forall n = 2, . . . , N.

  \ĥ (s)e \lambda j (s)x ds.If we denote, for t \geq 0 and x \in (0, \ell n ), I n (t, x) = \ĥ (s)e \lambda j (s)x ds, J n (t, x) =

,

  by simple inspection; the solution of this problem is v = [0 \cdot \cdot \cdot j+3(n - 1)

1

 1 \leq i \leq 2\} and the space \BbbH 2 e (\scrT ) is the Cartesian product of H 2 r (0, \ell n ) including the continuity condition on the central node (u n (0) = u n \prime (0) \forall n, n \prime = 1, . . . , N ). Using semigroup theory it is possible to show that v \in C([0, T ]; \BbbL 2 (\scrT )) and also using multipliers we can obtain the classical Kato smoothing result v \in L 2 (0, T ; \BbbH 1 e (\scrT )), but it is difficult (if not impossible) to derive the sharp Kato smoothing property established in Proposition 2.3 using energy methods. Now we use the following result obtained in [3] for a single KdV equation posed on a bounded domain.

2 .

 2 Moreover, there exists C > 0 depending only on T and L such that \| \psi \| C([0,T ];L 2 (0,L))\cap L 2 (0,T ;H 1 (0,L))

2. 2 .

 2 Nonlinear problem. With all the tools developed in the last sections we are ready to prove the global well-posedness result established on Theorem 1.3; the main ingredients of this proof are the regularity obtained in the linear cases, energy and multiplier estimates, and a fixed point argument. Let T > 0 and define \BbbX T = \BbbL 2 (\scrT ) \times H -1 3 (0, T ).

\ell 2 n

 2 in (2.36), defining L = max n=1,\cdot \cdot \cdot ,N \ell n and \ell = min n=1,\cdot \cdot \cdot ,N

\ell n 0 1 )v j n dxdt = N \sum n=1 \int \Omega n,i \int \ell n 0 1 \int \ell n 0 a

 110 \lambda j \fraks \fraka \frakt loc (a n \lambda j v j n \lambda j \fraks \fraka \frakt loc (a n \lambda j v j n \fraka \frakt loc (a n \lambda j v j n \fraka \frakt loc (a n \lambda j v j n n k n (R)| v j n | 2 dxdt, which gives us, using (3.5),(3.15)

( 3 . 1 )

 31 and (Obs) we argue the existence of C = C(R) > 0 such that. (3.16) E(T ) \leq \gamma E(0) with \gamma = C 1 + C < 1. Now as the system is invariant by translation in time, we can repeat this argument on [(m -1)T, mT ] for m = 1, 2, . . . to obtain E(mT ) \leq \gamma E((m -1)T ) \leq \cdot \cdot \cdot \leq \gamma m E(0).

4 .

 4 Conclusions and remarks. In this paper, the global well-posedness was studied and the exponential stability of a KdV equation on a star-shaped network with internal saturated feedback terms has been established. The well-posedness was addressed using the Laplace transform of the linearization and obtaining Kato smoothing properties which gave the local-in-time well-posedness, then using multiplier estimates the global-in-time result was deal with. 4.1. Generalization of the well-posedness result. In the work [7] a complete result for general linear boundary conditions for the KdV equation on a bounded domain was derived. In this work, homogeneous Dirichlet and Neumann right condi-

Conjecture 4 . 1 . 3 , 1

 4131 Let (\ell n ) n=1,...,N \in (0, \infty ) N , 0 \leq s \leq 3, and T > 0. There exists 0 < T \ast \leq T such that for allu 0 \in N \prod n=1 H s (0, \ell n ), (h, g, p) \in H (0, T ; L 2 (0, \ell n )),satisfying the compatibility condition,

4. 2 . 4 . 3 .

 243 Exact controllability in the network. In the paper[START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF] the exact controllability of linearization around 0 of (KdV-N) was achieved by acting with N + 1 boundary controls (N controls in the external nodes and one in the central node) if \#\{ \ell n \in \scrN \} \leq 1. Recently, in[START_REF] Cerpa | On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network[END_REF] the authors could reduce the numbers of controls (N controls acting on the external nodes), but the controllability holds for a large time and small lengths. This raises the question of what happens for the boundary control and how many components corresponding to the critical lengths one needs to control in the network case. In particular, we can mention the following open problems: \bullet Is the linearization around 0 of (KdV-N) exactly controllable with N controls acting in the external nodes for T > 0 and \ell n / \in \scrN for all n \in \{ 1, . . . , N \} ? \bullet Is (KdV-N) exactly controllable from the boundary in the case where for some lengths we have \ell n \in \scrN ? A starting point could be, to consider the smallest critical lengths (k = l = 1 or k = l = 2). Generalization of stabilization results. The stabilization results were obtained, proving appropriate observability inequalities working directly on the nonlinear systems. In the work [19] more general feedback laws were considered as cone bounded control laws. Note that Theorems 1.4 and 1.5 hold, replacing \fraks \fraka \frakt by any odd nonlinearity that satisfies the properties given in Lemmas A.1, A.2, and A.3. Appendix A. Useful lemmas. In this section, we present some technical lemmas about the regularity and sector condition of the saturation maps \fraks \fraka \frakt . Let a : [0, L] \rightar \BbbR such that a \ast \geq a \geq a \ast > 0 in an open nonempty set \omega of (0, L). (A.1)

s \int \ell 1 0| f | 2 dx - \int \ell 1 0) s \int \ell 1 0| f | 2 dx + \int \ell 1 0f 1 0 3 \int \ell j 0 x| f | 2 dx + 3 \int \ell j 0 | f \prime | 2 3 \int \ell j 0 x| f | 2 dx + \int \ell j 0 xf

 1111130302300 \biggl\{ sf (x) + f \prime \prime \prime (x) = 0, x \in (0, \ell 1 ), f \prime \prime (0) = f \prime (\ell 1 ) = f (\ell 1 ) = 0. Multiplying (B.1) by f , integrating over (0, \ell 1 ), and performing integration by parts,f f \prime \prime \prime dx + | f \prime (0)| 2 = 0.Similarly, multiplying (B.2) by f and integrating over (0, \ell 1 ), we get (B.4\prime \prime \prime f dx = 0. Then adding (B.3) and (B.4) yields (B.5) 2Re(s)\int \ell | f | 2 dx = - | f \prime (0)| 2 .As f is nontrivial and Re(s) \geq 0, we get f \prime (0) = 0. Then, by (B.5) Re(s) = 0. Thus, we can make the change of variable s = i\rho 3 for \rho \in \BbbR . Multiplying (B.1) by xf , integrating over (0, \ell 1 ), and performing integration by parts, we get(B.6) i\rho dx -\int \ell j 0 xf f \prime \prime \prime dx = 0.Similarly, multiplying (B.2) by xf and integrating over (0, \ell j ), we get (B.7) -i\rho \prime \prime \prime f dx = 0.Then, adding (B.6) and (B.7), we obtain f \prime \equiv 0. Using the boundary conditions of (B.1) we deduce f \equiv 0 which is a contradiction. Finally f \equiv 0 and \Delta 1 (s) \not = 0 for all s \not = 0 with Re(s) \geq 0.

Multiplying (C. 1 ) 3 \int \ell j 0 | f | 2 dx + \int \ell j 0 f 3 \int \ell j 0 x| f | 2 dx + 3 \int \ell j 0 | f \prime | 2 dx - \int \ell j 0 3 \int \ell j 0 x| f | 2 dx + \int \ell j 0 xf

 130030300300 by f , integrating over (0, \ell j ), and performing integration by parts, \prime \prime dx + | f \prime (0)| 2 = 0. Similarly, multiplying (C.2) by f and integrating over (0, \ell j ), we get (C.4) -i\rho \prime \prime \prime f dx = 0. Then, adding (C.3) and (C.4) yields f \prime (0) = 0. Multiplying (C.1) by xf , integrating over (0, \ell j ), and performing integration by parts, we get (C.5) i\rho xf f \prime \prime \prime dx = 0. Similarly, multiplying (C.2) by xf and integrating over (0, \ell j ), we get (C.6) -i\rho \prime \prime \prime f dx = 0.

Remark 4 .

 4 In the case \ell 1 = \cdot \cdot \cdot = \ell N , the proof become easier. In fact,N \sum j=1 det(F j ) det(D j ) = N det(F 1 ) det(D 1 ) \not = 0because, det(F 1 ) = \Delta 1,+ \not = 0 thanks to Appendix B. \circ

  N -1. Then, using (2.21) we get det(F N ) \sim

	\surd	3\rho 3 e	\rho 2	\surd	3\ell N +i \rho 2 \ell N and by the induction assumption	\sum 3 l=1

  + \theta 1/3 )\| v -\widetil v\| \BbbY \theta \| v + \widetil v\| \BbbY \theta +\theta \beta \| v -\widetil v\| \BbbY \theta + \theta \beta \| v -\widetil v\| \BbbY \theta \| v + \widetil v\| \BbbY \theta \bigr) + \theta \beta )\| v -\widetil v\| \BbbY \theta + 1 2(\theta 1/2 + \theta 1/3 + 2\theta \beta )\| v -\widetil v\| \BbbY \theta 2R

	Now from Proposition 2.5 we obtain
	\| \Phi (v) -\Phi (\widetil v)\| \BbbY \theta \leq C (\theta 1/2 \leq C \biggl( \theta 1/2 \| v -\widetil v\| \BbbY \theta + 1 2 \biggl( (\theta 1/2 \biggr)	;
	then with (2.32)

  D. For all \bfitrho > 0, it holds that

																			\bfitN \sum \bfitj =\bfone	det(\bfitF \bfitj ) det(\bfitD \bfitj )	\not = 0.	Letting
	j \in \{ 1, . . . , N \} , we are going to show that Re	\biggl(	det(F j ) det(D j )	\biggr)	< 0. Using (2.20) and
	(2.21) we get
	det(F j ) det(D j )	=	\surd 3\rho	\surd e - i\rho \ell j + \Biggl( \Biggl( 3\rho 3 -1 \Bigl( 2 -e - i\rho \ell j + e -1 2 \rho ( \sqrt{} \surd 3 2 i \Biggr) e \Bigl( -\rho \surd 3 - i)\ell j + e -1 2 \rho ( -\surd 3 2 +i \rho 2 \Bigr) \ell j + \Biggl( -1 2	\surd + 3 - i)\ell j \surd 3 2 i \Biggr) \Bigr)	e	\Bigl(	\rho	\surd 2 +i \rho 3 2	\ell j \Bigr)	\Biggr) .
												\rho 2	\Biggl(	e - i\ell j \rho + 2e	i\ell j \rho 2	cosh	\Biggl( \surd	2 3\ell j \rho	\Biggr) \Biggr)
			=	e - i\ell j -e	i\ell j \rho 2	cosh	\Biggl( \surd	3\ell j \rho 2	\Biggr)	+	\surd	3ie	i\ell j \rho 2	sinh	\Biggl( \surd	2 3\ell j \rho	\Biggr) .
	After some algebraic manipulations and writing the complex numbers in their binomial
	form (Re + iIm), we obtain
	det(F j ) det(D j )	=	cos	\biggl(	\rho 2 3\ell j \rho 2 \biggr) \Biggl( cos -cosh \biggl( 3\ell j \rho 2 \Biggl( \surd \biggr) 3\ell j \rho + 2 cosh 2 \Biggr) + i	\Biggl( \surd \Biggl( \surd 3 sinh 3\ell j \rho 2 \Biggr) \Biggl( \surd -i sin 3\ell j \rho \biggl( 2 \Biggr) 3\ell j \rho 2 -sin \biggr) \Biggr) \biggl(	2 3\ell j \rho	\biggr) \Biggr) .
	Letting \zeta = cos	\Bigl(	3\ell j \rho 2	\Bigr)	-cosh	\Bigl( \surd 3\ell j \rho 2	\Bigr)	+ i	\Bigl( \surd	3 sinh	\Bigl( \surd	3\ell j \rho 2	\Bigr)	-sin	\Bigl(	3\ell j \rho 2	\Bigr) \Bigr)	, and mul-
	tiplying the previous equation by	\zeta \zeta	we get
	Re	\biggl(	det(F j ) det(D j )	\biggr)	=	\rho 2 | \zeta | 2	\Biggl( 1 + cos	\biggl(	3\ell j \rho 2	\biggr)	cosh	\Biggl( \surd	3\ell j \rho 2	\Biggr)	-2 cosh 2	\Biggl( \surd	2 3\ell j \rho	\Biggr)
														-	\surd	3 sin	\biggl(	3\ell j \rho 2	\biggr)	sinh	\Biggl( \surd	3\ell j \rho 2	\Biggr) \Biggr)	.
	By analyzing the function
					F (\rho , \ell j ) = 1 + cos	\biggl(	3\ell j \rho 2	\biggr)	cosh	\Biggl( \surd	3\ell j \rho 2	\Biggr)	-2 cosh 2	\Biggl( \surd	2 3\ell j \rho	\Biggr)
														-	\surd	3 sin	\biggl(	3\ell j \rho 2	\biggr)	sinh	\Biggl( \surd	3\ell j \rho 2	\Biggr)	,

Theorem 1.1 (Theorem 2.7 of[START_REF] Ammari | epeau, Feedback stabilization and boundary controllability of the Korteweg--de Vries equation on a star-shaped network[END_REF]). Let (\ell n ) n=1,...,N \in (0, \infty ) N , \alpha \geq N

Acknowledgment. The authors would like to thank the referees for their valuable comments, which have significantly improved the quality of the article.

The work of the first author is supported by the French National Research Agency in the framework of the ``Investissements d'avenir"" program (ANR-15-IDEX-02). The work of the third author has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003).