MSC : 35K35, 93B05, 93B07.
Keywords: Fourth order parabolic equation, global **Carleman** **estimate**, controllability, observability.
1 Introduction
In the present paper, we consider Ω ⊂ R N with (N ≥ 2) a bounded connected open set whose boundary ∂Ω is regular enough. Let ω ⊂ Ω be a (small) nonempty open subset and let T > 0. We will use the notation Q = (0, T ) × Ω and Σ = (0, T ) × ∂Ω and we will denote by ~ n(x) the outward unit normal vector to Ω at the point x ∈ ∂Ω. On the other hand, we will denote by C 0 a generic positive constant which may depend on Ω and ω but not on T .

En savoir plus
axis. Each configuration needs to be addressed separately through a microlocalization procedure. For the Laplace operator at the boundary this was exploited to obtain a **Carleman** **estimate** in [ 35 ] for the purpose of proving a stabilization result for the wave equation.
As in [ 41 ] the method of the present article is based on the study of interior and boundary differential quadratic forms, an approach that originates in the work of [ 17 ] for estimates away from boundaries and in [ 38 , 39 , 37 ] for the treatment of boundaries. In connection with the microlocalizations described above we give a microlocal treatment of those differential quadratic forms. Positivity arguments rely on the Gårding inequality for homogeneous polynomials in connection with the position of the roots of the polynomial p ϕ (x, ξ ′ , τ, ξ n ). In fact the roots are split into three groups: roots with positive imaginary part, roots with

En savoir plus
treatment process, to compute the corresponding state ˆ y h (recall that, actually, this may be avoided
by using directly the optimality condition y = −ρ −2 Lp).
This is in contrast with the dual approach. Indeed, the minimization of J ? by an iterative pro-
cess requires the resolution of wave equations, through a decoupled space and time discretization. As recalled in the introduction, this may lea to numerical pathologies (the occurrence of spurious high frequency solutions) and, therefore, needs some specific numerical approximations and tech- niques. We mention the work [5], where the authors prove, in a close context and within a dual approach, a weaker uniform semi-discrete **Carleman** **estimate** with an additional term in the right hand side, necessary to absorb these possibly spurious high frequencies (see [5], Theorem 2.3).

En savoir plus
treatment process, to compute the corresponding state ˆ y h (recall that, actually, this may be avoided
by using directly the optimality condition y = −ρ −2 Lp).
This is in contrast with the dual approach. Indeed, the minimization of J ? by an iterative pro-
cess requires the resolution of wave equations, through a decoupled space and time discretization. As recalled in the introduction, this may lea to numerical pathologies (the occurrence of spurious high frequency solutions) and, therefore, needs some specific numerical approximations and tech- niques. We mention the work [5], where the authors prove, in a close context and within a dual approach, a weaker uniform semi-discrete **Carleman** **estimate** with an additional term in the right hand side, necessary to absorb these possibly spurious high frequencies (see [5], Theorem 2.3).

En savoir plus
Figure 1: The thin continuous lines represent the level curves of the convex function ρ. Hypothesis (1.10) means that the unitary normal vectors of the interface Γ ∗ are close to the corresponding normal vectors of
those curves.
In order to conclude this introductory section, we highlight that the contribution of this article is a **Carleman** **estimate** for the wave equation that has the noteworthy quality of holding under extended assumptions where the main coefficient is constant by pieces across a possibly non-convex interface. Up to our knowledge, this result is completely new, and paves the way for the usual ap- plications of **Carleman** estimates in controllability results (e.g. [2]) and stability issues for inverse problems (e.g. [4], [5]).

En savoir plus
In preparation to this section, we shall prove here the **Carleman** **estimate** on uniform meshes, for a slightly more general semi-discrete elliptic operator that we define now. For all i ∈ J1, dK, let ξ 1,i ∈ R M and ξ 2,i ∈ R Mi be two positive discrete
functions. We denote by reg(ξ) the following quantity reg(ξ) = max

which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of smoothly varying meshes. As a consequence of the **Carleman** **estimate**, we derive a partial spectral inequality of the form of that proved by G. Lebeau and L. Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the

En savoir plus
1.1. Outline. We start by briefly introducing semi-classical pseudodifferential operators (ψDO) in Sec- tion 2. The Gårding inequality will be one of the important tools we introduce. It will allow us to quickly derive a local **Carleman** **estimate** for an elliptic operator in Section 3. In that section, we present the sub- ellipticity condition that the weight function has to fulfill. We also show the optimality of the powers of the semi-classical parameter h in the **Carleman** estimates. We apply **Carleman** estimates to elliptic equations and inequalities and prove unique continuation results in Section 4. In Section 5, we prove the interpola- tion and spectral inequalities. The latter inequality concerns finite linear combinations of eigenfunctions of the elliptic operator. We show the optimality of the constant e C √µ in this inequality where µ is the largest

En savoir plus
polynomial in ξ n . Of course the configuration of the roots changes as the other parameters (x ′ , ξ ′ , τ ) are
modified. Roots can for instance cross the real axis. Each configuration needs to be addressed separately through a microlocalization procedure. For the Laplace operator at a boundary this was exploited to obtain a **Carleman** **estimate** in [ 38 ] for the purpose of proving a stabilization result for the wave equation. This approach was used for the study of an interface problem in [ 3 , 34 , 33 ] in the case of second-order elliptic operators. The present article provides a generalization of these earlier works, both with respect to the order of the operators and with respect to the generality of the transmission operators used.

En savoir plus
For numerical studies applying **Carleman** estimates to controllability problems, we refer to [ 7 ] for the numerical controllability of the wave equation and [ 12 ] for the numerical controllability of the heat, Stokes and Navier-Stokes equation.
The paper is organized as follows. Sections 2 give some preliminary results. First, in Section 2.1 , we present a **Carleman** **estimate** for the heat operator with Dirichlet boundary conditions. In this **estimate**, we consider two kinds of **Carleman** weights: the classical weights for the heat equation with a double exponential and new weights involving single exponentials which are introduced for numerical purposes. Then, in Section 2.2 , we state a regularity result satisfied by the solution of equation ( 1.1 ). The proofs of the **Carleman** **estimate** and the regularity result are presented in Appendices A and B respectively. At last, in Section 2.3 , we state the stability inequality associated to our inverse problem. Section 3.1 is the core of the paper and presents the numerical reconstruction method of the source term. The latter is an iterative process which requires at each iteration the minimization of a functional based on the **Carleman** **estimate**. This section states the global convergence of the method (Theorem

En savoir plus
For instance, for second-order elliptic operators with real coefficients 3 in the prin- cipal part, Lipschitz continuity of the coefficients suffices for a **Carleman** **estimate** to hold and thus for unique continuation across a C 1 hypersurface. Naturally, pseudo-
differential methods require more derivatives, at least tangentially, i.e., essentially on each level surface of the weight function ϕ. Chapters 17 and 28 in the 1983-85 four-volume book [ 20 ] by L. H¨ ormander contain more references and results.

3. OPÉRATEURS INTÉGRAUX DE **CARLEMAN**
Chaque opérateur correspond à un choix différent de la fonction K, qui s’appelle le noyau de l’opérateur A.
On doit regarder les opérateurs intégraux non pas comme une classe d’opérateurs comme celle des opérateurs différentiels ou celle des opérateurs de substitution, etc., mais plutôt comme un chemin stand arisé de représentation des opérateurs. Dans la littérature ils existent plusieurs approches pour définir les opérateurs intégraux, nous choisirons ici trois des plus répondus.

2 The empirical puzzle
During the end of 80’s and the 90’s, a huge empirical literature has been devoted to the estimation of the public capital rate of return (see Gramlich 1994). If we only consider studies based on times series, two methodological approaches have been used. The di- rect **estimate** of a production function expanded to the stock of public capital, is a …rst empirical way to measure these e¤ects, which has the advantage of a great simplicity. Applied to aggregate data, with a speci…cation in level of the production function, this method generally tends to establish the existence of an important productive contribu- tion of public infrastructures. Indeed, since the seminal article of Aschauer (1989), a lot of empirical studies, based on this methodology, have yielded very high estimated elasticities of output with respect to public capital (see table 1), as well on American as OECD data sets.

En savoir plus
5.3.2 Measurement-Based Probabilistic Timing Analysis
MBPTA is a technique used to reduce the cost of acquiring the knowledge needed for computing trustworthy WCET bounds [24, 43]. MBPTA seeks to determine the WCET estimates for arbitrarily low probabilities of exceedance, namely probabilistic WCET (pWCET). This technique is based on the Extreme Value Theory (EVT) and provides an estimation of the WCET of a task or application running on a hardware platform. In order to defeat the dependence on the execution history, this technique employs randomization. Therefore, MBPTA technique uses the theory of rare events [21]. There are two rare event theories that fit the WCET estimation: theory of extreme values [34] and theory of large deviations [34]. To the best of our knowledge, EVT is the only implemented theory for WCET estimation so far. EVT provides an estimation for the maximum of a sequence of i.i.d random variables [39]. In this work, we have used the MBPTA technique that allows us to find the predicted time of the programs. We have obtained the pWCET estimations that can be applied with high confidence as an upper bound on the execution-time. The execution-times of the programs are observed for 1,000 times (runs). If those execution-times can be modeled with i.i.d random variables, then the pWCET of a program can be obtained by constructing an Empirical Cumulative Distribution Function (ECDF). As we are interested in the EVT and this theory is used to **estimate** the probability of occurrence of extremely large values that are rare events. EVT estimates the distribution function for either the maximal or the minimal values from n set of observations which are formed with the random variables. In short, EVT is used to **estimate** the extremes.

En savoir plus
Workshop on Functional Analysis Valencia 2015 on the occasion of the 60th birthday of José Bonet.. Valencia – June 16, 2015..[r]

Dans l’article [ABDK :05], les auteurs démontrent un résultat de contrôlabilité avec un seul contrôle en utilisant une inégalité de **Carleman** avec une seule observation. L’inégalité de **Carleman** qu’ils obtiennent ne peut être utilisée pour traiter le problème inverse de l’iden- tification de coefficients, car les fonctions poids dans le membre de droite et le membre de gauche de leur estimation sont différentes. Dans les travaux [CGR :06] et [R :07], nous avions obtenu des inégalités de **Carleman** avec une seule observation avec les mêmes poids à gauche et à droite. Dans ce chapitre, on utilise ces nouvelles estimations de **Carleman** pour démontrer à la fois un résultat de stabilité pour un coefficient, deux coefficients et pour les conditions initiales.

En savoir plus
121 En savoir plus

Wklv sdshu hpsor|v wkh rqh0vhfwru Uhdo Exvlqhvv F|foh prgho dv d whvwlqj jurxqg iru irxu glhuhqw surfhgxuhv wr hvwlpdwh G|qdplf Vwrfkdvwlf Jhqhudo Htxloleulxp +GVJH, prghov1Wkh surfhgxu[r]

ε in H per 1 (Y ; L 2 (Ω)), such that the
distance between the unfolded T ε ( ∇ x φ) and ∇ x φ + ∇ y φ b ε is of order of ε in the space [L 2 (Y ; H −1 (Ω))] n .
Theorems 4.1, 4.2 and 4.5 give an **estimate** of the error without any hypothesis on the regularity of the correctors, but with different hypotheses on the boundary of Ω. They require that the right hand side of the homogenized problem be in L 2 (Ω).

ideal of the ring R, because D-stability of a principal ideal means exactly the condition Q|DQ on a generator of the ideal).. Kozlov, On algebraic independence of functions from a certai[r]

Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Ir[r]