Assia Benabdallah

Yves Dermenjian

Jérôme Le Rousseau

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications

Keywords: Carleman estimate, non-smooth coefficients, parabolic equations, control, inverse problem Inégalité de Carleman, coefficients non réguliers, équations paraboliques, contrôle, problème inverse

published or not. The documents may come

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications

Assia Benabdallah, Yves Dermenjian, Jérôme Le

Introduction

The question of controllability of partial differential systems with discontinuous coefficients and its dual counterpart, observability, are not fully solved yet. Recently, a result of controllability for a semilinear Email addresses: assia@cmi.univ-mrs.fr (Assia Benabdallah), dermenji@cmi.univ-mrs.fr (Yves Dermenjian), jlerous@cmi.univ-mrs.fr (Jérôme Le Rousseau). 1 Corresponding author.

February 9, 2008 heat equation with a discontinuous coefficient was proven in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] by means of a Carleman observability estimate. Roughly speaking, as in the case of hyperbolic systems (see e.g. [2, page 357]), the authors of [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] proved their controllability result in the case where the control is supported in the region where the diffusion coefficient is the 'lowest'. In both cases, however, the approximate controllability holds without any restriction on the 'monotonicity' of the coefficients. It is then natural to question whether or not an observability estimate holds in the case of non-smooth coefficients and arbitrary observation location.

In the one-dimensional case, the controllability result for linear parabolic equations was proven for coefficients with bounded variations in [START_REF] Fernández-Cara | On the null controllability of the one-dimensional heat equation with BV coefficients[END_REF]. The proof relies on Russel's method [START_REF] Russel | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF]. However, the question of the existence of a Carleman-type observability estimate remains open. The present paper provides a positive answer in the case of piecewise C 1 coefficients.

Carleman estimates for parabolic equations with smooth coefficients were proven in [START_REF] Fursikov | Controllability of evolution equations[END_REF]. The proof is based on the construction of suitable weight functions β whose gradient is non-zero in the complement of the observation region. In the non-smooth case, in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF], to obtain the observability, the authors have to add the assumptions on the 'monotonicity' of the coefficients mentioned above. In both cases, the weight function β was chosen in the domain of the operator ∇ • (c∇). Here, we do not impose this constraint, which, with the jump of the derivative of β as a new parameter, enables us to control the interface terms in the derivation of the Carleman estimate and therefore allows us to relax the 'monotonicity' condition on the coefficient. Note however that the results of [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] are for the multidimensional heat equation. The relaxation of the 'monotonicity' condition in the n-dimensional case, n ≥ 2, remains, to our knowledge, open.

With such a Carleman estimate at hand, we treat the problem of the null controllability for classes of semilinear parabolic equations of the form

     ∂ t y -∂ x (c∂ x y) + G (y) = 1 ω v in Q, y(t, x) = 0 on Σ, y(0, x) = y 0 (x) in Ω, (1)
where T > 0, Ω = (0, 1), Q = (0, T ) × Ω, Σ = (0, T ) × {0, 1} and where G : R → R is locally Lipschitz and G (0) = 0 (further assumptions on the nonlinear function G or on the initial condition will be introduced below). We also provide a stability result for the inverse problem of the identification of the diffusion coefficient.

Global Carleman estimates

We consider a piecewise C 1 diffusion coefficient with a finite number of singularities. We shall thus here assume that 0 = a

0 < a 1 < a 2 < • • • < a n = 1 and c | [a i ,a i+1 ] ∈ C 1 ([a i , a i+1 ]
), i = 0, . . . , n -1. Let j ∈ {0, . . . , n -1} be fixed in the sequel and ω 0 ⋐ ω ⋐ (a j , a j+1 ) be a non-empty open set. Let T > 0. We shall use the following notations

S = {a 1 , . . . , a n-1 }, Ω ′ = Ω \ {a 1 , . . . , a n-1 }, Q = (0, T ) × Ω, and Q ′ = (0, T ) × Ω ′ .
We first introduce a particular type of weight functions, which are constructed using the following lemma. Lemma 2.1 There exists a function

β ∈ C (Ω) such that β | [a i ,a i+1 ] ∈ C 2 ([a i , a i+1 ]), i = 0, . . . , n -1, satisfying β > 0 in Ω, β = 0 on {0, 1}, ( β | [a j ,a j+1 ] ) ′ = 0 in [a j , a j+1 ]\ω 0 , ( β | [a i ,a i+1 ] ) ′ = 0, i ∈ {0, . . . , n-1}, i = j,
and the function β satisfies the following trace properties: for some α > 0, (A i u, u) ≥ α|u| 2 , u ∈ R 2 , with the matrices A i , defined by

A i =   [ β ′ ] ai β ′ (a + i )[c β ′ ] ai β ′ (a + i )[c β ′ ] ai β ′ (a + i )[c β ′ ] 2 ai + [c 2 ( β ′ ) 3 ] ai   , i = 1, . . . , n -1.
where

[ρ] x = ρ(x + ) -ρ(x -) for x ∈ (0, 1).
Choosing a function β, as in the previous lemma, we introduce β = β + K with K = m β ∞ and m > 1. For λ > 0 and t ∈ (0, T ), we define the following weight functions

ϕ(t, x) = e λβ(x) t(T -t) , η(t, x) = e λβ -e λβ(x) t(T -t) , (2)
with β = 2m β ∞ (see [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]). We introduce

ℵ = q ∈ C (Q, R); q | [0,T ]×[a i ,a i+1 ] ∈ C 2 ([0, T ] × [a i , a i+1 ]), i = 0, . . . , n -1,
q |Σ = 0, and q satisfies (TC), for all t ∈ (0, T ) , with

q(a - i ) = q(a + i ), c(a - i )∂ x q(a - i ) = c(a + i )∂ x q(a + i ), i = 1, . . . , n -1. (TC) Theorem 2.2 There exist λ 1 = λ 1 (Ω, ω) > 0, s 1 = (T + T 2 ) s 1 > 0 and a positive constant C = C(Ω, ω)
so that the following estimate holds

s -1 Q e -2sη ϕ -1 (|∂ t q| 2 +|∂ x (c∂ x q)| 2 ) dxdt+sλ 2 Q e -2sη ϕ |∂ x q| 2 dxdt+s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dxdt + sλ n-1 i=1 T 0 ϕ(t, a i )e -2sη(t,ai) |∂ x q(t, a - i )| 2 dt + s 3 λ 3 n-1 i=1 T 0 ϕ 3 (t, a i )e -2sη(t,ai) |q(t, a i )| 2 dt ≤ C s 3 λ 4 (0,T )×ω e -2sη ϕ 3 |q| 2 dxdt + Q e -2sη |∂ t q ± ∂ x (c∂ x q)| 2 dxdt , (3)
for s ≥ s 1 , λ ≥ λ 1 and for all q ∈ ℵ.

Proof. Arguing as in [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF], we set ψ = e -sη q. We obtain, in the derivation of the Carleman estimate, integral terms over Q ′ and some time integrals over (0, T ) with trace terms at a i , i = 1, . . . , n -1. In fact, the leading order terms for these time integrals at a i (w.r.t. to the parameters s and λ) are given by

µ i := sλ T 0 ϕ(t, a i ) [β ′ |c∂ x ψ| 2 (t, .)] ai dt + s 3 λ 3 [c 2 (β ′ ) 3 ] ai T 0 ϕ 3 (t, a i ) |ψ(t, a i )| 2 dt, i = 1, . . . , n -1.
We obtain

µ i = sλ T 0 ϕ(t, a i ) A i u(t, a i ), u(t, a i ) dt with u(t, a i ) = (c(a - i )∂ x ψ(t, a - i ), sλϕ(t, a i )ψ(t, a i ))
t , with the 2 × 2 matrix A i as in Lemma 2.1. This term is thus positive and can 'absorb' the remaining time integrals at a i if we choose the parameters s and λ to be sufficiently large. The rest of the proof can be adapted from [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF].

Remark 1 In a similar fashion, we can obtain Carleman estimates with a 'side observation' for the operators ∂ t ± ∂ x (c∂ x ), for instance with the term sλ T 0 ϕ(t, 0)e -2sη(t,0) |∂ x q| 2 (t, 0) dt replacing the volume integral over (0, T ) × ω on the r.h.s. of (3). To this purpose, we need to introduce a weight function β according to the following lemma.

Lemma 2.3 There exists a function

β ∈ C (Ω) such that β | [a i ,a i+1 ] ∈ C 2 ([a i , a i+1 ]), i = 0, . . . , n -1, satisfying β > 0 in Ω, β(1) = 0, ( β | [a i ,a i+1 ] ) ′ ≤ ν < 0, i ∈ {0, .
. . , n -1}, and the function β satisfies the following trace properties: for some α > 0, (A i u, u) ≥ α|u| 2 , u ∈ R 2 with the matrices A i , defined as in Lemma 2.1.

Remark 2 Note that an inequality, of the form of (3), with these pointwise terms on the l.h.s of the Carleman estimates can still be obtained in the case of a smooth coefficient by simply choosing the weight function β to have a jump condition for its derivative and satisfying the properties given by Lemma 2.1.

Controllability results

The Carleman estimate (3) allows to give observability estimates that yield results of controllability to the trajectories for classes of semilinear heat equations. Considering a piecewise C 1 diffusion coefficient with n -1 points of discontinuities, a 1 , . . . , a n-1 , with 0 = a 0 < a 1 < • • • < a n-1 < 1 = a n . We let ω ⋐ (a j , a j+1 ) be an non-empty open set for some j ∈ {0, . . . , n -1}. We first state an observability result with an L 2 observation. We let a be in L ∞ (Q) and q T ∈ L 2 (Ω). From Carleman estimate (3) we obtain the following proposition.

Proposition 3.1 The solution q to      -∂ t q -∂ x (c∂ x q) + aq = 0 in Q, q = 0 on Σ, q(T ) = q T in Ω, (4)
satisfies q(0) 2 L 2 (Ω) ≤ e CK(T, a ∞ ) (0,T )×ω |q| 2 dxdt, where K(T, a ∞ ) = 1 + 1 T + T a ∞ + a 2/3 ∞ .
The proof of this proposition can be found in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]. Such an observability estimate yields the null controllability of the following semilinear parabolic equation [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] where : R → R is locally Lipschitz and G (0) = 0. This implies that G (s) = sg(s), with g in L ∞ loc (R). Theorem 3.1 Let c be a piecewise C 1 diffusion coefficient with n -1 points of discontinuities, 0 < a 1 < • • • < a n-1 < 1. We let ω ⋐ (a j , a j+1 ) be an non-empty open set and we assume that G is locally Lipschitz. Let T > 0 :

(i) Local null controllability: There exists ε > 0 such that for all y 0 in L 2 (Ω) with y 0 L 2 (Ω) ≤ ε, there exists a control v ∈ L 2 ((0, T ) × ω) such that the corresponding solution to system (1) satisfies y(T ) = 0. (ii) Global null controllability: Let G satisfy in addition lim |s|→∞ |G (s)| |s| ln 3/2 (1+|s|) = 0. Then for all y 0 in L 2 (Ω), there exists v ∈ L 2 ((0, T ) × ω) such that the solution to system (1) satisfies y(T ) = 0.

The proof is based on a fixed point argument and is along the same lines as that in [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] and originates from [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF].

Remark 3 A similar result holds for a boundary control, i.e., by imposing y = v at either 0 or 1, with v ∈ L 2 (0, T ). Note that as usual, y(T ) = y * (T ) can replace y(T ) = 0 in the previous statements, where y * is any trajectory defined in [0, T ] of system (1), corresponding to some initial data y * 0 ∈ L 2 (Ω) and any v * in L 2 ((0, T ) × ω) (L 2 (0, T ) in the case of a boundary control). For the local controllability result, one has to assume y 0y * 0 L 2 (Ω) ≤ ε, with ε sufficiently small.

Stability for a discontinuous diffusion coefficient

In [START_REF] Benabdallah | An inverse problem for the heat equation with discontinuous diffusion coefficients[END_REF], the authors establish a uniqueness result for the discontinuous diffusion coefficient c as well as a stability inequality. This inequality estimates the discrepancy in the coefficients c and c of two materials (with the same geometry) with an upper bound given by some Sobolev norms of the difference between the solutions y and y to

     ∂ t y -∂ x ( c∂ x y) = 0 in Q, y(t, x) = h(t, x) on Σ, y(0, x) = y 0 (x) in Ω,      ∂ t y -∂ x (c∂ x y) = 0 in Q, y(t, x) = h(t, x) on Σ, y(0, x) = y 0 (x)
in Ω.

(

) 5
Set u = yy and q = ∂ t u. Then q is solution to the following problem

     ∂ t q -∂ x (c∂ x q) = ∂ x ((c -c)∂ x ∂ t y) in Q ′ , q = 0 on Σ, transmission conditions (TC g ) on S × [0, T ],
with q(x -) = q(x + ), (c∂ x q)(x -) = (c∂ x q)(x + ) + g(x, t), (TC g ) where x ∈ S = {a 1 , . . . , a n-1 }, the set of singularities for both c and c, and

g(x, t) = ((c -c)∂ x ∂ t y)(x + ) -((c -c)∂ x ∂ t y)(x -).
If the solutions y and y to (5) satisfy some (regularity) conditions (that can be achieved with some choices of boundary conditions h and initial conditions y 0 and y 0 in L 2 (Ω) -see [START_REF] Benabdallah | An inverse problem for the heat equation with discontinuous diffusion coefficients[END_REF] for details) we have the following stability result. Theorem 4.1 We assume that the diffusion coefficients c and c are piecewise constant with the same singularity locations. Then there exists a constant C such that

|c -c| 2 L ∞ (Ω) ≤ C|∂ x (∂ t y -∂ t y)(., 0)| 2 L 2 (0,T ) + C|∆y(T ′ , .) -∆ y(T ′ , .)| 2 L 2 (Ω ′ ) , (6)
where Ω ′ is the open set Ω with the singularities of c removed.

A Carleman estimate is the key ingredient in the proof of such a stability estimate. In [START_REF] Benabdallah | An inverse problem for the heat equation with discontinuous diffusion coefficients[END_REF], this Carleman estimate was proven in any dimension but with an additional 'monotonicity' assumption on the discontinuous diffusion coefficient. In the present case, we can establish such a Carleman estimate for a piecewise C 1 diffusion coefficient. Choosing the weight function as in Lemma 2.3, we have the following estimate. Theorem 4.2 Let t 0 > 0, in (0, T ) and g(., a i ) ∈ H 1 (t 0 , T ), i = 1, . . . , n -1. There exist λ 1 > 1, s 1 = s 1 (λ 1 ) > 0 and a positive constant C so that the following estimate holds for s ≥ s 1 , λ ≥ λ 1 and for all q ∈ ℵ g , with M 1 and M 2 given by

|M 1 (e -sη q)| 2 L 2 (Q ′ ) + |M 2 (e -sη q)| 2 L 2 (Q ′ ) + sλ 2 Q e -
M 1 ψ = ∂ x (c∂ x ψ) + s 2 λ 2 ϕ 2 (β ′ ) 2 cψ + s(∂ t η)ψ, M 2 ψ = ∂ t ψ -2sλϕcβ ′ ∂ x ψ -2sλ 2 ϕc(β ′ ) 2 ψ,
and ℵ g is given by ℵ g = q ∈ H 1 (t 0 , T, H 1 0 (Ω)); q | (t 0 ,T )×(a i ,a i+1 ) ∈ L 2 (t 0 , T, H 2 ((a i , a i+1 )), i = 0, . . . , n -1, q |Σ = 0 and q satisfies (TC g ) a.e. w.r.t. t . (8)

Remark 4 Observe that in Theorem 4.1 and Theorem 4.2, we need not assume that jumps for c are greater than some positive constants ∆ at its points of discontinuities, as is done in [START_REF] Benabdallah | An inverse problem for the heat equation with discontinuous diffusion coefficients[END_REF]. This is due to the choice made on the weight function β in Lemma 2.3. This remark is to be connected to the proof of Theorem 1.3 in [8, estimate (1.16) and following arguments].

2sη ϕ|∂ x q| 2 dxdt + s 3 λ 4 Qee -2sη ϕ 4 |g| 2 dσdt +s - 2 T

 42 -2sη ϕ 3 |q| 2 dxdt ≤ C sλ T t0 e -2sη ϕ |∂ x q| 2 (t, 0) dt + Q e -2sη |∂ t q ± ∂ x (c∂ x q)| 2 dxdt + sλ t0 Se -2sη |∂ t g| 2 dσdt ,[START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]


Jérôme Le Rousseau

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic operators ∂ t ± ∂ x (c∂ x ) with a coefficient c with bounded variations. These estimates are obtained by approximating c by piecewise regular coefficients, c ε , and passing to the limit in the Carleman estimates associated to the operators defined with c ε . Such estimates yield results of controllability to the trajectories for a classe of semilinear parabolic equations.

Résumé

Inégalités de Carleman et résultats de contrôle pour une équation de la chaleur à coefficient BV en dimension un. On obtient des inégalités de Carleman pour des opérateurs paraboliques linéaires ∂ t ± ∂ x (c∂ x ) avec un coefficient c à variations bornées. Ces inégalités sont obtenues par approximations de c par des coefficients réguliers par morceaux, c ε , et en passant à la limite dans les inégalités de Carleman associées aux opérateurs définis par c ε . De telles inégalités donnent des résultats de contrôlabilité aux trajectoires pour une classe d'équations semilinéaires paraboliques. Pour citer cet article : J. Le Rousseau, C. R. Acad. Sci. Paris, Ser. I - (-).

Version franc ¸aise abrégée

On considère les opérateurs ∂ t ± A dans Q = (0, T ) × Ω, avec T > 0 et Ω = (0, 1). L'opérateur A est défini par -∂ x (c∂ x ) sur L 2 (Ω), où c est à variations bornées (BV) et vérifie 0 < c min ≤ c ≤ c max . Le domaine de A est D(A) = {u ∈ H 1 0 (Ω); c∂ x u ∈ H 1 (Ω)}. On prouve l'inégalité de Carleman globale suivante. Théorème 0.1 Soit c ∈ BV(Ω), avec 0 < c min ≤ c ≤ c max . Soit ℵ l'ensemble des solutions (faibles) de ∂ t q ± ∂ x (c∂ x q) = f dans Q, q = 0 sur (0, T ) × {0, 1}, q(T, x) = q 0 (x) (resp. q(0, x) = q 0 (x)) dans Ω, avec q 0 ∈ L 2 (Ω) et f ∈ L 2 (Q).

(i) ∃λ 1 > 0, s 1 > 0 et C > 0 tels que l'inégalité de Carleman (4) soit valable pour s ≥ s 1 , λ ≥ λ 1 et pour tout q ∈ ℵ.

Email address: jlerous@cmi.univ-mrs.fr (Jérôme Le Rousseau).

(ii) Si c de est de classe C 1 sur O ⋐ Ω, avec O un ouvert non vide de Ω, alors on a une inégalité de Carleman avec une 'observation' sur (0, T ) × O.

On donne une idée de la preuve du Théorème 0.1 dans le cas (i), i.e. une 'observation' frontière. Si c est C 1 par morceaux, une inégalité de Carleman (4) est obtenue [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF][START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF] en construisant des fonctions poids données dans (3), à partir d'une fonction β, régulière par morceaux, qui à chaque point de discontinuité de c satisfait la condition [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF].

Dans le cas où c ∈ BV(Ω), pour ε > 0 on choisit c ε , constant par morceaux, de manière à avoir c -

c ε L ∞ (Ω) ≤ ε, V 1 0 (c ε ) ≤ V 1 0 (c), où V b a (ρ)
est la variation totale de ρ sur (a, b) [START_REF] Kolmogorov | Eléments de la théorie des fonctions et de l'analyse fonctionnelle[END_REF]. On construit des fonctions β ε adaptées à c ε : posant

[ρ] x := ρ(x + ) -ρ(x -), on exhibe une fonction f telle que i) si [ β ′ ε ] a = f ([c ε ] a ) en chaque point de discontinuité a de c ε , alors β ε satisfait la condition (2) ; ii) de plus V 1 0 ( β ′ ε )
est majorée uniformément par rapport à ε. À une sous-suite près, ces fonctions β ′ ε convergent partout quand ε tend vers 0, par le théorème de Helly. On déduit, par passage à la limite et intégration, une fonction β.

À l'aide de β ε et β, on construit des fonctions poids selon [START_REF] Kolmogorov | Eléments de la théorie des fonctions et de l'analyse fonctionnelle[END_REF]. Celles associées à c ε permettent d'obtenir des inégalités de Carleman pour ∂ t ± ∂ x (c ε ∂ x ). Une inspection détaillée de la preuve des inégalités de Carleman dans le cas C 1 par morceaux permet de prouver l'existence d'une borne uniforme, par rapport à ε, pour la constante C à droite de (4) pour les opérateurs ∂ t ± ∂ x (c ε ∂ x ). De même, les constantes s 1 et λ 1 dans le Théorème 2.2 peuvent être choisies uniformes par rapport à ε. Ces résultats permettent un passage à la limite dans les inégalités de Carleman associées à

∂ t ± ∂ x (c ε ∂ x ) : les solutions de ∂ t q ε ± ∂ x (c ε ∂ x )q ε = f sont telles que ∂ t q ε , q ε , ∂ x q ε , ∂ x (c∂ x q ε )
convergent vers les fonctions associées pour la solution q de ∂ t q ± ∂ x (c∂ x )q = f , pour f régulière et les conditions finales (resp. initiales) bien choisies ; les fonctions poids convergent et les constantes restent bornées. On relaxe alors la régularité sur f et les hypothèses sur les conditions finales (resp. initiales).

Le Théorème 0.1 permet d'obtenir des résultats de contrôlabilité aux trajectoires pour une classe d'équations semilinéaires paraboliques. Si G (x, y) = x g(x, y) + y G(x, y), avec g et G dans L ∞ loc (R). Théorème 0.2 Soit T > 0. Sous les hypothèses du Théorème 0.1, on a : (i) Contrôlabilité à zéro locale : Il existe ε > 0 tel que pour tout y 0 dans L 2 (Ω) avec y 0 L 2 (Ω) ≤ ε, il existe un contrôle v ∈ L ∞ (0, T ) tel que la solution de [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF] satisfasse y(T ) = 0.

(ii) Contrôlabilité à zéro globale : Si G satisfait de plus [START_REF] Fernández-Cara | On the null controllability of the one-dimensional heat equation with BV coefficients[END_REF], alors pour tout y 0 ∈ L 2 (Ω), il existe v ∈ L ∞ (0, T ) tel que la solution de [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF] satisfasse y(T ) = 0.

On obtient des résultats similaires pour le système [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] avec un ouvert non vide ω ⋐ Ω, et v ∈ L ∞ ((0, T ) × ω), en supposant une régularité C 1 du coefficient c sur une partie ouverte non vide de ω.

Le preuve du Théorème 0.2 repose sur un argument de point fixe [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF][START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF].

Introduction

We consider the elliptic operator A formally defined by -∂ x (c∂ x ) on L 2 (Ω) in the one-dimensional bounded domain Ω = (0, 1), with 0 < c min ≤ c ≤ c max , and c is assumed to be of bounded variations (BV). The

domain of A is D(A) = {u ∈ H 1 0 (Ω); c∂ x u ∈ H 1 (Ω)}. Let T > 0. We shall use the following notations Q = (0, T ) × Ω, Γ = {0, 1}, and Σ = (0, T ) × Γ.
Here, we show that we can achieve global Carleman estimates for the operators ∂ t ± A in Q, with a boundary 'observation' or an interior 'observation'. With such Carleman estimates at hand, we treat the problem of the controllability to the trajectories for semilinear parabolic systems of the form

             ∂ t y -∂ x (c∂ x y) + G (y, ∂ x y) = 0 in Q, y(t, 0) = v(t), y(t, 1) = 0, y(0, x) = y 0 (x) in Ω, (1)
where v is the control, G : R 2 → R is locally Lipschitz and G (0, 0) = 0 (further assumptions on the nonlinear function G or on the initial condition will be introduced below). A null controllability result for a linear parabolic equation with BV coefficients was proven in [START_REF] Fernández-Cara | On the null controllability of the one-dimensional heat equation with BV coefficients[END_REF]. The proof relies on Russell's method [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF]. However, the question of the existence of a Carleman-type observability estimate was open. In a recent work [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF], a result of controllability for the one-dimensional heat equation with general bounded coefficients was proved. The method of [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF] is based on that of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. However, with such a method, the knowledge of the observability constant is not sufficient to address the controllability of non-linear heat equations. Carleman estimates, of the form proved here, provide a more precise knowledge of this constant with respect to lower-order terms in the partial differential operator, which allows to treat the case of a class of semilinear equations following the (fixed-point) method of [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] (generalized in [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF]). Carleman estimates also have other applications in analysis, e.g., uniqueness results, inverse problems.

The Carleman estimates derived here for the operators ∂ t ± ∂ x (c∂ x ) are obtained through a limiting process from the Carleman estimates associated for ∂ t ± ∂ x (c ε ∂ x ), for c ε piecewise regular converging to c in L ∞ (Ω). The main issue in this limiting process is to keep both the weight functions and constants in the Carleman estimate under control. Controllability results of this type can be motivated by problems from biology for example. Note that the approximation of the BV coefficient c by some piecewise regular coefficient c ε is closely related to numerical methods.

Carleman estimates in the case of a BV coefficient

We first consider a piecewise C 1 diffusion coefficient c with a finite number of singularities. Here, we shall assume that 0

= a 0 < a 1 < • • • < a n = 1 and c | [a i ,a i+1 ] ∈ C 1 ([a i , a i+1 ]), i = 0, . . . , n -1. We shall use the following notations S = {a 1 , . . . , a n-1 }, Ω ′ = Ω \ S , and Q ′ = (0, T ) × Ω ′ . Lemma 2.1 ([2]) There exists a function β ∈ C (Ω) such that β | [a i ,a i+1 ] ∈ C 2 ([a i , a i+1 ]), i = 0, . . . , n -1, satisfying β > 0 in Ω, β(1) = 0, ( β | [a i ,a i+1 ] ) ′ < 0, i ∈ {0, . . . , n -1}
, and the function β satisfies the following trace properties, for some α > 0,

(A i u, u) ≥ α|u| 2 , with A i =           [ β ′ ] a i β ′ (a + i )[c β ′ ] a i β ′ (a + i )[c β ′ ] a i β ′ (a + i )[c β ′ ] 2 a i + [c 2 ( β ′ ) 3 ] a i           , u ∈ R 2 , ( 2
)
for i = 1, . . . , n -1, and where [ρ] x = ρ(x + )ρ(x -) for x ∈ (0, 1).

Choosing a function β, as in the previous lemma, we introduce β = β + K with K = m β ∞ and m > 1. For λ > 0 and t ∈ (0, T ), we define the following weight functions,

ϕ(t, x) = e λβ(x) t(T -t) , η(t, x) = e λβ -e λβ(x) t(T -t) , (3)
with β = 2m β ∞ (see [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]). We introduce

ℵ = q ∈ C (Q, R); q | [0,T ]×[a i ,a i+1 ] ∈ C 2 ([0, T ] × [a i , a i+1 ]), i = 0, . . . , n -1, q | Σ = 0,
and q satisfies, q(t, a

- i ) = q(t, a + i ), c(a - i )∂ x q(t, a - i ) = c(a + i )∂ x q(t, a + i ), i = 1, . . . , n -1, for all t ∈ (0, T ) .
With the function β satisfying the properties of Lemma 2.1, we obtain the following Carleman estimate.

Theorem 2.2 ([2]

) There exist λ 1 > 0, s 1 = (T + T 2 ) s 1 > 0 and C > 0 so that the following estimate holds, for s ≥ s 1 , λ ≥ λ 1 and for all q ∈ ℵ,

s -1 Q e -2sη ϕ -1 (|∂ t q| 2 + |∂ x (c∂ x q)| 2 ) dxdt + sλ 2 Q e -2sη ϕ |∂ x q| 2 dxdt + s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dxdt ≤ C sλ T 0 ϕ(t, 0)e -2sη(t,0) |∂ x q| 2 (t, 0) dt + Q e -2sη |∂ t q ± ∂ x (c∂ x q)| 2 dxdt . (4)
We now consider a diffusion coefficient c ∈ BV(Ω), with 0 < c min ≤ c ≤ c max . We denote the total variation of c on (0, 1) by ϑ := V 1 0 (c). Let ε > 0. There exists a piecewise-constant [START_REF] Bressan | Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem[END_REF]). We denote by a 1 , . . . , a k the points of discontinuity of c ε . We then have k i=1 |c ε (a

function c ε such that c -c ε L ∞ (Ω) ≤ ε, V 1 0 (c ε ) ≤ ϑ (see e.g.
+ i ) -c ε (a - i )| ≤ ϑ. Let Y i = c ε (a + i )/c ε (a - i ) and X i , i = 1, . . . , k, be defined by X i = f (Y i ), with f (s) = s if s ≥ 1 and f (s) = 2 -s if s < 1.
We define the piecewise-constant function γ ε as γ ε (x) := γ(1) x<a j X j , for x {a 1 , . . . , a k }, for some fixed γ(1) < 0. We set the function β ε (x) := x 1 γ ε (y) dy, which satisfies the properties listed in Lemma 2.1 by the jump conditions imposed on γ ε = β ′ ε at a 1 , . . . , a k . Concerning the total variation of the functions γ ε we have the following lemma. Lemma 2.3 There exist K > 0 and ε 0 > 0 that solely depend on the diffusion coefficient c ∈ BV(Ω) such that, for all

0 < ε ≤ ε 0 , V 1 0 (γ ε ) ≤ K |γ(1)
|. By Helly's theorem [START_REF] Kolmogorov | Eléments de la théorie des fonctions et de l'analyse fonctionnelle[END_REF][START_REF] Bressan | Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem[END_REF], up to a subsequence, the functions γ ε converge everywhere to a function γ as ε goes to 0. Moreover, this function satisfies V 1 0 (γ) ≤ K|γ(1)|. By dominated convergence, the associated functions β ε converge everywhere to the continuous function β(x) := x 1 γ(y)dy. Let a be any point of discontinuity of c ε . As above, we set Y = c ε (a + ) c ε (a -) and define the matrix A as in Lemma 2.1. Then we have the following crucial lemma. Lemma 2.4 The eigenvalues ν 1 , ν 2 of the matrix A satisfy ν i ≥ C|Y -1|, i = 1, 2, with C uniform w.r.t. ε and the considered singularity of c ε .

With β ε and β, we define weight functions according to [START_REF] Kolmogorov | Eléments de la théorie des fonctions et de l'analyse fonctionnelle[END_REF]. With a detailed inspection of the proof of the Carleman estimate in the case of a piecewise C 1 coefficient [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF], the previous lemma yields Proposition 2.5 The constant C on the r.h.s. of the Carleman estimate ( 4) for ∂ t ± ∂ x (c ε ∂ x ) and the constants s 1 and λ 1 can be chosen uniformly w.r.t. ε for 0 < ε ≤ ε 0 , with ε 0 sufficiently small. The previous results show that both the weight functions and the constants in the Carleman estimate remain 'well behaved', as ε goes to zero. We consider q and q ε (weak) solutions to

             ∂ t q ± ∂ x (c∂ x q) = f in Q, q = 0 on Σ, q(T, x) = q 0 (x) (resp. q(0, x) = q 0 (x)) in Ω.              ∂ t q ε ± ∂ x (c ε ∂ x q ε ) = f in Q, q ε = 0 on Σ, q ε (T, x) = q 0,ε (x) (resp. q ε (0, x) = q 0,ε (x)) in Ω,
with the initial data q 0 and q 0,ε given by ∂ x (c∂ x q 0 ) = µ and ∂ x (c ε ∂ x q 0,ε ) = µ, with the same µ, where µ and f are sufficiently regular. This choice of initial data, yields the convergence of q 0,ε to q 0 in H 1 0 (Ω) as ε goes to zero. Since ∂ t q ε , q ε , ∂ x q ε and ∂ x (c∂ x q ε ) converge to ∂ t q, q, ∂ x q and ∂ x (c∂ x q) in L 2 norm, the previous results show that the Carleman estimate for q ε and the operators ∂ t ± ∂ x (c ε ∂ x ) yield a similar estimate for q and the operators ∂ t ± ∂ x (c∂ x ) with the same constants, as ε goes to zero. We can now relax the assumptions on µ and f . We have thus obtained the following theorem. Theorem 2.6 Let c ∈ BV(Ω) with 0 < c min ≤ c ≤ c max . There exists λ 1 > 0, s 1 = (T + T 2 ) s 1 > 0 and C > 0 so that the Carleman estimate (4) holds for s ≥ s 1 , λ ≥ λ 1 and for all q (weak) solution to ∂ t q ± ∂ x (c∂ x q) = f in Q, q = 0 on Σ, q(T, x) = q 0 (x) (resp. q(0, x) = q 0 (x)) in Ω, with q 0 ∈ L 2 (Ω) and f ∈ L 2 (Q).

In the case where O ⋐ Ω, with O a non-empty open set and c is of class C 1 in O, we obtain Carleman estimates, for the operators ∂ t ± ∂ x (c∂ x ), with an interior 'observation' on (0, T ) × O i.e. with the term s 3 λ 4 (0,T )×O e -2sη ϕ 3 |q| 2 dxdt replacing the time integral on the r.h.s. of (4).

In the case of an interior 'observation', the proof is similar to the previous one. The function β ε is constructed on both sides of O, through its derivative, and the regularity of c in O allows to smoothly connect the two parts of the function, with some convergence in C 2 (O) as ε goes to zero. As above, we then pass to the limit in a Carleman estimate for ∂ t ± ∂ x (c ε ∂ x ), with c ε piecewise C 1 , with an interior 'observation', proven in [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF].

3. A Carleman estimate for the heat equation with a r.h.s. in L 2 (0, T, H -1 (Ω))

Following [START_REF] Yu | Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications[END_REF], we obtain a Carleman estimate for O,c) so that the following estimate holds, for s ≥ s 2 , λ ≥ λ 2 and for all q ∈ ℵ ± ,

∂ t q ± ∂ x (c∂ x q) = f if f ∈ H -1 . We set ℵ ± = q ∈ C ([0, T ], H 1 0 (Ω)); q(t) ∈ D(A) ∀ t ∈ [0, T ], ∂ t q ± ∂ x (c∂ x q) = F 0 + ∂ x F 1 , F 0 , F 1 ∈ L 2 (Q) .
sλ 2 Q e -2sη ϕ |∂ x q| 2 dxdt + s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dxdt ≤ C         s 3 λ 4 (0,T )×O e -2sη ϕ 3 |q| 2 dxdt + Q e -2sη |F 0 | 2 dxdt +s 2 λ 2 Q e -2sη ϕ 2 |F 1 | 2 dxdt         . (5)

Controllability results

The Carleman estimate (5) proven in the previous section allows to give observability estimates that yield results of controllability to the trajectories for classes of semilinear heat equations. We let ω ⋐ Ω be a non-empty open set and c ∈ BV(Ω) with 0 < c min ≤ c ≤ c max and c of class C 1 in some non-empty open subset of ω. We let a and b be in L ∞ (Q) and q T ∈ L 2 (Ω). From Carleman estimate (5) we obtain the following lemma. Lemma 4.1 The solution q to -∂ t q -∂ x (c∂ x q) + aq -∂ x (bq) = 0 in Q, q = 0 on Σ, and q(T ) = q T satisfies q

(0) 2 L 2 (Ω) ≤ e CH (0,T )×ω |q| dxdt 2 , where H = H(T, a ∞ , b ∞ ) = 1+ 1 T +T +(T +T 1/2 ) a ∞ + a 2/3 ∞ +(1+T ) b 2 ∞ .
Since the coefficient c is C 1 in some open subset of ω, the proof of [6, Theorem 2.5, Lemma 2.5] can be adapted. See also [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]Proposition 4.2,Lemma 4.3]. Such an observability estimate yields the null controllability of the semilinear parabolic system (1), as well as the following system

             ∂ t y -∂ x (c∂ x y) + G (y, ∂ x y) = 1 ω v in Q, y(t, .) = 0 onΣ, y(0, x) = y 0 (x) in Ω, (6)
where G : R 2 → R is locally Lipschitz and G (0, 0) = 0, which implies that G (x, y) = x g(x, y) + y G(x, y), with g and G in L ∞ loc (R). For all y 0 ∈ L 2 (Ω), there exists v ∈ L ∞ (0, T ) such that the solution to (1) satisfies y(T ) = 0. We have similar results for system [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] with a non-empty open subset ω ⋐ Ω, and v ∈ L ∞ ((0, T ) × ω), assuming that c of class C 1 in some non-empty open subset of ω.

The proof is based on a fixed-point argument and is along the same lines as that in [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] and originates from [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. Note that as usual, y(T ) = y * (T ) can replace y(T ) = 0 in the previous statements, where y * is any trajectory defined in [0, T ] of system [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications[END_REF], corresponding to some initial data y * 0 ∈ L 2 (Ω) and any v * in L ∞ (0, T ) (L ∞ ((0, T ) × ω) in the case of an interior control). For the local controllability result, one has to assume y 0y * 0 L 2 (Ω) ≤ ε, with ε sufficiently small.

Theorem 3 . 1

 31 Let O ⋐ Ω be a non-empty open set and c ∈ BV(Ω) with 0 < c min ≤ c ≤ c max and c of class C 1 in O. There exists λ 2 = λ 2 (O, c) > 0, s 2 = (T + T 2 )s 2 (O, c) > 0 and a positive constant C = C(

Theorem 4 . 2

 42 Let T > 0. Let c ∈ BV(Ω) with 0 < c min ≤ c ≤ c max . (i) Local null controllability: There exists ε > 0 such that for all y 0 in L 2 (Ω) with y 0 L 2 (Ω) ≤ ε, there exists a control v ∈ L ∞ (0, T ) such that the solution to (1) satisfies y(T ) = 0. (ii) Global null controllability: Let G satisfy in addition lim |(x,y)|→∞ |g(x, y)| ln 3/2 (1 + |x| + |y|) = 0 and lim |(x,y)|→∞ |G(x, y)| ln 1/2 (1 + |x| + |y|)

Acknowledgements

The author wishes to thank A. Benabdallah and Y. Dermenjian for numerous discussions on the subject of this article.


Aymen Laadhari

Pierre Saramito

Chaouqi Misbah

Improving the mass conservation of the level set method in a finite element context

Keywords: level set method, mass conservation, finite element, Lagrange-Galerkin method

In this paper, a new algorithm is proposed for improving the mass conservation of the level set method in the finite element context. Two kinds of Lagrange multipliers are introduced, associated respectively to the redistancing and advection equations. The first one, is located at the vicinity of the interface, while the second one is associated to a correction that is global to the domain. The performances of the proposed method are tested on the Zalesak test case, and the convergence rate versus the element mesh size are founded to be improved. Amélioration de la conservation de la masse pour la méthode des fonctions de niveaux en éléments finis Résumé -Dans cet article, nous proposons un nouvel algorithme pour améliorer la conservation de la masse dans la méthode des fonctions de niveau dans un cadre éléments finis. Deux types de multiplicateurs de Lagrange sont introduits, associés respectivement à l'équation de redistanciation et à celle d'advection. Le premier est localisé au voisinage de l'interface, tandis que le second est associé à une correction globale au domaine de calcul. Les performances de la méthode proposée sont testées avec le cas test du disque de Zalesak, et nous observons que le taux de convergence par rapport au la taille des éléments du maillage est amélioré.

Version française résumée -Les méthodes numériques pour résoudre des problèmes à surface libres et des problèmes d'interfaces ont pris ces dernières années une importance croissante en physique et réalité virtuelle. Les applications recouvrent des domaines aussi variés que les interfaces liquide-vapeur, les mousses et les émulsions, les tsunamis ou le comportement d'un globule rouge dans une artère. La méthode des fonctions de niveau (voir par exemple [START_REF] Osher | The Level Set Method and Dynamic Implicit Surfaces[END_REF][START_REF] Sethian | Level Set Methods and Fast Marching Methods[END_REF]), s'appuyant sur un maillage fixe du milieu continu tridimensionnel, est devenue de plus en plus courante pour ce type de problème, car elle s'avère particulièrement souple lorsque les interfaces se déforment fortement ou présentent des changements topologiques (coalescence, etc). Une autre approche très répandue est la méthode de suivi lagrangien de l'interface par un maillage surfacique variable : moins apte aux changements topologiques, elle a cependant l'avantage de conserver mieux la masse, c'est à dire le volume du fluide, particulièrement pour des interfaces à courbure discontinue ou très minces. Des tentatives pour améliorer la conservation de la masse dans méthode des fonctions de niveau a a conduit à de nombreuses approches, telles que augmenter l'ordre des schémas [START_REF] Jiang | Weighted eno schemes for hamilton-jacobi equations[END_REF]. Enright et al. [START_REF] Enright | A hybrid partical level set method for improved interface capturing[END_REF], dans le cadre d'une méthode particulaire, a proposé d'utiliser des marqueurs lagrangiens pour reconstruire la fonction de niveau dans les régions sous-résolues. Une autre approche, proposée par Sussman et Fatemi [START_REF] Sussman | An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow[END_REF] dans le cadre de la méthode des différences finies, consiste à ajouter une contrainte à l'équation de redistanciation discrète afin de mieux conserver la masse.

Dans ce papier, dans le cadre de la méthode des éléments fini, nous proposons deux améliorations indépendantes de la conservation de la masse discrète. La première amélioration concerne l'équation d'advection et introduit un multiplicateur de Lagrange scalaire, associé à la conservation globale de la masse. Ce multiplicateur peut être calculé explicitement et conduit à un coût négligeable en temps de calcul. La seconde amélioration est une extension aux éléments finis de la méthode proposée par Sussman et Fatemi : il s'agit d'introduire un multiplicateur de Lagrange local. Ce multiplicateur peut-être également explicité et est nul hors d'un voisinage de l'interface. La correction est ici également très peu coûteuse. Les performances de la méthode proposé sont testées avec le cas test du disque de Zalesak, et nous observons que le taux de convergence par rapport à la taille des éléments du maillage est amélioré. Soit Λ ⊂ R N un domaine borné, avec N = 2 ou 3, et T > 0. Pour tout t ∈]0, T [, la surface fermée Γ(t) ⊂ Λ s'écrit comme : Γ(t) = {(t, x) ∈]0, T [×Λ; φ(t, x) = 0}, c'est-à-dire comme le zéro d'une fonction de niveau φ(t, .). Notons Ω(t) ⊂ Λ la région où φ(t, .) est négatif et Γ(t) = ∂Ω(t). Soit u la vitesse de Γ(t) et

Dφ Dt = ∂φ ∂t + u.∇φ = 0, (1)
où Dφ/Dt désigne la dérivé matérielle. Le champs de vitesse u est supposé à divergence nulle. Cette équation de transport est complétée par une condition initiale φ(t = 0) = φ 0 , où, pour tout x ∈ Λ, la fonction φ 0 (x) est une distance signée entre x et la surface Γ(0) :

φ 0 (x) = inf {|y -x|; y ∈ Γ(0)} si x / ∈ Ω(0), inf {-|y -x|; y ∈ Γ(0)} sinon.
Cependant, le transport déforme la forme initiale de la fonction de niveau, qui n'est plus une distance signée pour t > 0. Pour palier à cela, nous effectuons une ré-initilisation à une distance signée, appelée redistanciation, en calculant, à tout instant t ∈]0, T [, la solution stationnaire du problème suivant, utilisant le pseudo-temps τ :

∂d ∂τ (τ, x; t) + sgn(φ) (|∇d| -1) = 0 p.p. (τ, x) ∈]0, +∞[×Λ, d(0, x; t) = φ(t, x) p.p. x ∈ Λ. ( 2
)
où sgn(φ) est la fonction signe qui prend respectivement les valeurs 0, -1, +1 sur l'interface Γ(t), dans Γ(t) et hors de Γ(t). La solution stationnaire vérifie |∇d| = 1 presque partout dans Λ, c'est-à-dire que d(∞, .; t) est une distance signée, elle est prise comme la nouvelle fonction de niveau φ(t, .) à l'instant t.

Remarquons que la solution d du problème de redistanciation (2) conserve la position de Γ(t): pour tout τ > 0, l'ensemble de niveau zéro de d(τ, .; t) est égal à l'ensemble de niveau zéro de φ(t, .). Il en résulte que le volume meas(Ω(t)) est également conservé, ce qui est de première importance pour de nombreuses applications. Cependant, après discrétisation par différences finies ou éléments finis, cette propriété n'est satisfaite que de façon approchée.

Nous proposons un nouvel algorithme pour améliorer la conservation de la masse dans la méthode des fonctions de niveau dans un cadre éléments finis. Deux types de multiplicateurs de Lagrange sont introduits, associés respectivement à l'équation d'advection et à celle de redistanciation. Le premier est localisé au voisinage de l'interface, tandis que le second est associé à une correction globale au domaine de calcul. Les performances de la méthode proposée sont ensuite testées avec le cas test du disque de Zalesak, et nous observons que le taux de convergence par rapport à la taille des éléments du maillage est amélioré.

------------

1.
Maintaining the volume constraint. -Let us introduce the velocity field v = sgn(φ) ∇d |∇d| .

Then, problem (2) expresses equivalently as: The constraint (4) develops as:

∂d ∂τ (τ, x; t) + v.∇d = sgn (φ (t, x)) a.e. (τ, x) ∈]0, +∞[×Λ, d(0, x; t) = φ(x, t) a.e. x ∈ Λ. ( 3
d dt Ω(t) dx = d dt Λ H(φ) dx = Λ δ(φ) ∂φ ∂t dx = 0 (6) March 12, 2010
Then, combining ( 5) and ( 6) leads to an explicit expression for ζ(t):

ζ(t) = -Λ u.∇φ δ(φ) dx Λ δ(φ) dx (7)
Let us check that that since u is divergence free, then ζ is equal to zero and that the global volume correction will act at the discrete level only. For any integrable function f on Γ(t), let f denotes an extension of f in Λ. Then

Γ(t) f ds = Λ f δ(φ) |∇φ| dx (8)
Let n = ∇φ |∇φ| denotes the unit outward normal to Γ(t). By using [START_REF] Zalesak | Fully multidimensional flux-corrected trasport algorithms for fluids[END_REF] with f = u.n and then the Green formula:

Λ u.∇φ δ(φ) dx = Γ(t) u.n ds = Λ div u dx 1.2.
The volume constraint for the redistancing problem. -Let us consider the redistancing problem at any fixed time t > 0, while the pseudo-time τ is varying. Sussman and Fatemi [START_REF] Sussman | An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow[END_REF] introduce the constraints that the volume remains constant in any V ⊂ Ω(t) during the redistancing process:

∂ ∂τ V H(d) dx = 0. ( 9
)
These authors introduces also a Lagrange multiplier λ(τ, x; t) that enforces this constraint locally at x ∈ Λ. The evolution equation in (3) becomes:

∂d ∂τ (τ, x; t) + v.∇d = sgn (φ (t, x)) + λ(τ, x : t) g(d) a.e. (τ, x) ∈]0, +∞[×Λ (10)
In order to ensure the volume conservation while not perturbing the redistancing process, the authors propose g(d) = δ(d)|∇d|. Notice that (9) leads to

∂ ∂τ V H(d) dx = V δ(d) ∂d ∂τ dx = 0 (11)
Combining ( 10) and (11), Sussman and Fatemi [START_REF] Sussman | An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow[END_REF] proposed to exploit numerically an explicit average value λ V over an arbitrary finite volume V ⊂ Λ:

λ V (τ ; t) =        V δ(d) (v.∇d -sgn (φ)) dx V δ(d) g(d) dx when V ∩ Γ(t) = ∅ 0 otherwise (12)
In the next section, based on the regularization procedure, a new local explicit expression of λ(τ, x; t) can be exhibited and that λ can be formally eliminated from the redistancing equation.

2.

Regularization. -Three sharp functions have been introduced: the Heaviside function H(φ), that acts as the indicator of Λ -Ω, the Dirac function δ(φ) that localizes the interface Γ, and the sign function sgn(φ). In order to avoid the discretization of the interface Γ, a transition region with a width of 2ε is introduced and the Heaviside, the Dirac and the sign functions are replaced respectively by H ε (φ), δ ε (φ) and sgn ε (φ) as:

H ε (φ) =          0, if φ < -ε, 1 2 1 + φ ε + sin( πφ ε ) π , if |φ| ≤ ε, 1, otherwise, (13)
δ ε (φ) = dH ε dφ (φ) = 1 2ε 1 + cos( πφ ε ) , if |φ| ≤ ε 0 , otherwise (14)
sgn ε (φ) = 2H ε (φ) -1 (15) March 12, 2010
The regularization parameter is chosen as proportional to the the mesh size h.

After regularization the evolution equation (10) becomes:

∂d ∂τ (τ, x; t) + v ε .∇d = sgn ε (φ (t, x)) + λ(τ, x; t) f ε (d) a.e. (τ, x) ∈]0, +∞[×Λ (16)
where

v ε = sgn ε (φ) ∇d |∇d| and f ε (d) = δ ε (d) |∇d|.
Conversely, the regularized version of the mass conservation (11) writes:

V δ ε (d) ∂d ∂τ dx = 0, ∀V ⊂ Λ ⇐⇒ ∂d ∂τ = 0, when |d| < ε (17)
Combining the two previous equations leads to a local version of the explicit expression for λ:

λ =      sgn ε (φ) (|∇d| -1) δ ε (d) |∇d| when |d| < ε 0 otherwise (18)
Finally, the previous expression of λ can be replaced in ( 16) and the evolution equation becomes, for any (τ, x) ∈]0, +∞[×Λ:

∂d ∂τ (τ, x; t) = 0 when |φ| < ε -v ε .∇d + sgn ε (φ) otherwise (19)
Notice that the set Γ ε (t) = {x ∈ Λ; |φ(t, x)| < ε} represents a thin layer around the boundary Γ(t). Then, with the mass correction, the redistancing process acts only around a thin layer outside the boundary.

3. Time discretization. -This section presents the numerical approximation of the advection problem (5) together with the redistancing problem (19), where the Lagrange multipliers are expressed by [START_REF] Sussman | An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow[END_REF] and (12). Let t n = n∆t, n ≥ 0, where ∆t > 0 is the time step. Conversely, let τ m = m∆τ , m ≥ 0, where ∆τ > 0 is the pseudo time step. Let φ n , ζ n be approximations of φ(t), ζ(t), respectively, at time t n and d m , v m be approximations of d(τ ), v(τ ) respectively at τ m . The time discretization is performed by using the method of characteristics: for any t > 0 and x ∈ Λ, the characteristic curve X(., x, ; t) passing at time t through x is defined by the following ordinary differential equation:

∂X ∂t (s, x; t) = u (X(s, x; t), t) , s > 0 X(t, x; t) = x. (20)
For any function f (t, x), the total derivative Df /Dt expresses:

Df Dt (t, x) = ∂f ∂t + u.∇f (t, x) = ∂ ∂t f X(t, x; t n+1 ), t n+1 | τ =t (21)
Following Pironneau [START_REF] Pironneau | On the transport-diffusion algorithm and its applications to the Navier-Stokes equations[END_REF], this derivative is approximated by a first-order backward Euler scheme:

Df Dt (t n+1 , x) ≈ f (t n+1 , x) -f (t n , X n (x)) ∆t ( 22
)
where X n (x) = x -∆t u n (x) denotes the first-order forward Euler approximation of X(t n , x; t n+1 ). The n-th step of the discrete scheme for the transport problem (5) becomes an explicit computation of φ n+1 , where φ n is known:

φ n+1 (x) = φ n • X n u + ∆t ζ n (23)
Conversely, the redistancing problem ( 19) is solved also explicitly:

d m+1 = d m when |φ n | < ε d m • X m vε + ∆τ sgn ε (φ n ) otherwise (24)
March 12, 2010

Here, the characteristic have subscripts u and v ε in order to avoid confusion. These algorithms are implemented in the C++ finite element library Rheolef [START_REF] Saramito | Rheolef: A finite element environment, i.e. some C++ classes and unix commands[END_REF], where the localization of X n (x) in an unstructured mesh is performed by using a fast searching algorithm based on a quad-tree data structure.

4. Space discretization. -Let T h be the finite element triangulation of Λ, and V h the space of continuous and piecewise linear functions on T h , where h > 0 denotes as usual the mesh size parameter. Conversely, let W h be the space of piecewise constant functions on T h . At any step n ≥ 0, the discrete version of the algorithm leads to a succession of explicit computations:

φn+1 h = π h (φ n h • X n u ) ζ n+1 h = -Λ u.∇ φn+1 h δ ε ( φn+1 h ) dx Λ δ ε ( φn+1 h ) dx φn+1 h = φn+1 h + ∆t
g m h .ψ h dx = Λ ∇d m h .ψ h dx, ∀ψ h ∈ V N h
A mass lumping procedure is used for this linear system: the integrals involved in the computation of the coefficients of the matrix associated to the L 2 scalar product are evaluated by using the trapeze quadrature formulae. By this way, the matrix of the linear system is replaced by a diagonal one, and the computation of g m h becomes explicit. Then, let

v m ε,h = π h sgn ε φn+1 h g m h |g m h |
The discrete version of the redistancing algorithm writes also explicitly:

d m+1 h = d m h when |φ n h | < ε π h d m h • X m vε + ∆τ sgn ε (φ n h )(1 -|∇d m h |) otherwise (25)
5. Numerical experiments. -The improvement of the accuracy with the mass conservation schemes is demonstrated for a classical test case: the rigid body rotation of Zalesak's disk in a constant rotating velocity field [START_REF] Zalesak | Fully multidimensional flux-corrected trasport algorithms for fluids[END_REF]: u(x, y) = (1/2-y, x-1/2) in the domain Λ = ]0, 1[ 2 . The initial data φ 0 is associated to a slotted circle centered at (1/2, 7/10) with a radius r = 1/5, the slope depth is 3/10 and the width is equal to 1/10. The triangular mesh is based on a N h ×N h grid, where each square is subdivided in two triangles. The mesh size is h = √ 2/N h and ∆t = 2π/N h is chosen. Fig. 1.b plots the error H ε (φ) -H ε (φ h ) 0,2,Λ after N h time steps, at t = 2π: notice that the disk has performed a full revolution. Observe that the error decreases versus the mesh size. The error is plotted for the following variants of the method: (a) without mass correction, (b) with local mass correction (c) with local and global mass correction. The rate of convergence, i.e. the slope on the log-log graph, is respectively 0.30, 0.46 and 0.50. For N h = 400, the mass lost is about 10% without any correction. It is improved to 4.9 % the local correction and to 1.8 % with both local and global corrections.

) 1 . 1 .

 11 The volume constraint for the advection problem. -A new modification is introduced in the advection equation (1) related to the divergence free velocity field u. At any time t ∈]0, T [, the global volume conservation writesmultiplier ζ(t) ∈ R is introduced and the advection equation (1) becomes ∂φ ∂t + u.∇φ + ζ(t) = 0 (5)

Fig. 1 .H 2 1. 5 × 3 7 × 10 - 1 10 - 1 7

 125311 a shows the solution for N h = 400 with and without corrections: the corners become less rounded and the the disk is more conserved. ǫ (φ) -H ǫ (φ h ) 0,2,Λ h √ 10 -2 1.5 × 10 -× 10 -2

Fig. 1 .

 1 Fig. 1. The Zalesak disk: (a) initial position (black and dotted) and after one rotation: without correction (blue and dotted line), with global correction (magenta and dashed), with local correction (green and dashed), with local and global corrections (red and solid line) ; (b) convergence properties in log-log scale. Fig. 1. Le disque de Zalesak : (a) position initiale (trait noir et pointillé) et apès une rotation: sans correction (bleu et pointillé), avec correction globale (violet et pointillé), avec correction locale (vert et pointillé), avec corrections locale et globale (rouge et continu) ; (b) propriétés de convergence en échelle log-log.

  ζ n h where π h denotes the Lagrange interpolation in V h . Then, φ n+1

	h h = φn+1 discrete version of the redistancing algorithm. Let d 0 h algorithm, suppose d m h ∈ V h being known, and let g m h ∈ V N h be the approximation of ∇d m is obtained from φn+1 by the following h . At any step m ≥ 0 of the redistancing h defined h ∈ W N by the following linear system:

Λ


Huu Nghia Nguyen

Jean-Pierre Jacquot

A tool for checking CSP||B specifications ⋆

Introduction

This paper reports about our experience with building a simple tool to assist us in the verification of CSP||B specifications.

Our primary research goal is methodological: we aim at understanding and modeling how people develop formal specifications. Thus, a language and its supporting environment is simply a tool that specifiers use for a specific purpose. Tools are always designed as a trade-off between range of applicability and efficiency. The most efficient tools often have a narrow purpose. Like carpenters or plumbers, specifiers must carry heavy toolboxes rather than a single do-it-all tool.

An important difference between the tools used for hardware and for software is that, in the latter case, they have very strong and subtle interactions. Actually, they must often be used together at the same time. One specification may require the use of B for managing some complex data structure, of CSP for maintaining some liveness properties, and of UML for validation by the stakeholders. "Gluing tools" are needed.

In a recent specification, in a context of multi agents system, we used CSP||B to model the bevahiors of agents and their communications. While most of the work could be done within language specific tools (FDR2 1 for the CSP part and AtelierB 2 for B part), the verification of the coherence between B and CSP prompted us to design and implement a small assistant. We had at least three reasons:

-Practical use of formal methods entails a bit of intelligent work and a huge amount of boring stuff which requires an attention to all details. Computers, with the proper tools, are good at the second type of work while we human are bad at it; of course, the opposite is true for the first type of work. -The issue of the coherence proofs between CSP and B is quite subtle and needs a precise understanding. Working on an implementation is a good way to make the fundamental questions pop out. -We believe that even incomplete tools can be useful: they can leave out some of the intelligent part so long as they take care of all the tedious part. Prototyping is the best way to discover whether there is a real gain.

2 Requirements for the tool

CSP||B and verification

CSP||B is a multi-formalisms specification technique developed in the last 10 years. It focuses on the modeling of component-based systems. A component is an entity which manages some internal state and which communicates with other components. B is used to describe the state management part of the component. CSP is used to describe the communication part of the system. A component is modeled by two texts: a B machine and a CSP controler. To each operation of the B machine is associated a "machine channel" in the CSP controler. The semantics is that when a process synchronizes on a machine channel, this corresponds to a " call" of the machine operation.

The theory of the verification of a CSP||B specification has been developed in [START_REF] Treharne | Combining Control Executives and Software Specifications[END_REF]. It is based on a process with three independent steps:

the B machines are verified by the usual B technology: generation of proof obligations and proofs with B provers, the CSP controlers are verified by standard CSP techniques to assess the absence of deadlocks or divergences, the protocol constraints expressed in CSP processes guarantee that the precondition of an operation holds when it is called through a machine channel synchronization.

This last step is a consequence of two implicit assumptions in the preceding steps: B machine invariants are maintained only if operations are called when their precondition holds and CSP synchronizations always happen when they are possible, which is always the case for "calling" a machine operation through a machine channel synchronization. Hence, CSP||B mandates that operations in machines are coded with preconditions instead of guards since the B semantics states that the former are always executable but may lead to any result if the preconditions do not hold.

Expected functions

[1] proposes a verification technique where CSP controlers are translated into B machines. The calling order imposed by the protocol is modeled by a counter. The translation involves two kinds of works. The first is the application of a set of transformation rules, it is the easy but tedious part. The second is the introduction of a Control Loop Invariant (CLI), which is the intelligent part. CLI is such that

CLI = (cb = 1 =⇒ PRE(op 1 )) ∧ . . . ∧ (cb = n =⇒ PRE(op n ))
where PRE(op i ) is the precondition of the operation called when cb = i. CLI can be simplified by removing some conjuncts after analysis of the structure of the operations. The first and most important requirement is to apply the transformation rules and generate files that can be used by AtelierB. The second requirement is to assist specifiers with invention of the CLI. Several strategies have been studied, from making CLI a parameter of the translation program to generating it. We ended up with the idea to generate the form of the CLI needed by the specification.

Realization

Architecture and design

We see our tool as a protoytpe and, thus, followed a process based on short and fast cycles. Architecture and design are, in a sense, results of the development.

An important decision was the structure of the output of the translation. For each CSP controler it consists of two controler machines: an abstract machine which introduces the counter and a refinement which introduces the calls to the operations of the controled B machine. When needed, CLI are introduced in the refinements.

The current translation process is a sequence of six semantic steps which follow the parsing of the files containing the CSP controler and the B machine:

1. generation of the abstract machine, as a text file, 2. removal of non pertinent variables. CSP syntax requires to introduce names when values are passed through a channel. With respect to the verification of coherence between B and CSP, only the names associated with machine channel are pertinent and must be translated, all others can be safely ignored, 3. removing name conflict. A pertinent variable can lead to a conflict on three cases: it is one letter long, it is identical to a keyword of B, it is identical to an existing B variable visible through the include clause, 4. type inference. B, contrary to CSP, requires all variables to be typed, 5. computation of the CLI form, 6. generation of the refinement, as a text file.

Implementation and test

As an implementation language, we used OCaml3 . Three reasons motivated this choice: OCaml is a powerful language for symbolic manipulations, it has strong parsing libraries, and the project Brillant 4 provides us with several tools, notably a B parser.

The implementation consists in two packages: one for B analysis, around the Brillant parser, and one for CSP analysis. A parser for CSP, accepting the syntax used by FDR2, has been implemented.

The tool needs two input files: C.mch and C.csp which contain the texts for the B part and the CSP controler. Included files in CSP texts are read when needed.

A required CLI is indicated by a comment containing the skeleton of the invariant.

The program has been tested on three published specifications [START_REF] Schneider | Communicating B machines[END_REF][START_REF] Treharne | Combining specification with composition[END_REF][START_REF] Colin | Towards validating a platoon of cristal vehicles using CSP B[END_REF]. It was used to verify the extension of [START_REF] Colin | Towards validating a platoon of cristal vehicles using CSP B[END_REF] which motivated this development.

Observations

Picking OCaml for developing the tool has been an excellent choice. Beside the Brillant library, the major advantage is the clarity of the code. Ideas could be implemented very quickly and the intuitive correctness of the algorithms is easy to assess. In the three months of programming, we were able to test several translation strategies.

Starting without a firm design is another advantage of fast programming. In fact, we "discovered" the technical problems (typing, name clashes, etc.) as we progressed and processed new exemples. Of course, production tools must use much more structured development processes. However, such processes rely on that, from the outset, the shape of the solution is well known. Our work only aims at drawing the shape! A problem that we encountered is the paucity of exemples of specifications. This limited the territory which our exploratory approach could chart. While we have a reasonable confidence in the correctness of our prototype, we do not know how many difficulties, either technical or theoretical, remain to discover. For instance, we know that the generated CLI is pessimistic when operations contain a conditional substitution. Which other traits of the languages could cause a similar phenomenon should be studied. A library of exemples would be a real bonus here.

Another problem is the communication between the tools which should enter the toolbox of a specifier. In our case, we used the minimum common denominator: text files. Inefficiency is not only technical, repeatedly parsing files, it is also conceptual. Parsers are complicated to build and to use, and thus consumme a lot of developer's time, yet the interesting points are elsewhere. The problems cited above could be better solved if we had libraries with "semantic" functions such as the set of variables modified by an operation, or read only, etc. Such libraries are much needed.

Conclusion

A question remains about the potential utility of a tool such as ours: do we need it in the toolbox? Should we invest in the development of a "certified" and stronger version, knowing that it does an incomplete job? We think so. Even for a formal developer, not all tools need be fully formal and complete. Time is short, boring tasks lead to errors. For the practical developer, there is room in the box for tools which take care of mundaneness and leave out the smart bits.

Our prototype is available to brave users through the Brillant project.

⋆ This work is supported by project TACOS ANR-06-SETI-017 and by project CRISTAL 1 http://www.fsel.com/ 2 http://www.clearsy.com

http://caml.inria.fr

https://gna.org/projects/brillant