Papers by Judith Vancostenoble
HAL (Le Centre pour la Communication Scientifique Directe), 2016
In this paper, we introduce a new age-structured population model with diffusion and gestation pr... more In this paper, we introduce a new age-structured population model with diffusion and gestation processes and make a complete study of the qualitative properties of its solutions. The model is in the spirit of a model introduced in [13, 15] and studied in [10]. We aim here to correct some weakness of the model that was pointed out in [10].
HAL (Le Centre pour la Communication Scientifique Directe), 2016
In this paper, we are interested in some inverse problem that consists in recovering the so-calle... more In this paper, we are interested in some inverse problem that consists in recovering the so-called insolation function in the 2-D Sellers model on a Riemannian manifold that materializes the Earth's surface. For this nonlinear problem, we obtain a Lipschitz stability result in the spirit of the result by Imanuvilov-Yamamoto in the case of the determination of the source term in the linear heat equation. The paper complements an analogous study by Tort-Vancostenoble in the case of the 1-D Sellers model.
Discrete and Continuous Dynamical Systems - Series S, 2018
A classical and useful way to study controllability problems is the moment method developped by F... more A classical and useful way to study controllability problems is the moment method developped by Fattorini-Russell [12, 13], and based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behavior: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and so to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results of Fattorini-Russell [12, 13] concerning biorthogonal families, using complex analysis techniques developped by Seidman [36], Güichal [20], Tenenbaum-Tucsnak [37] and Lissy [26, 27].
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Oct 1, 2012
We are interested in the climate model introduced by Sellers in 1969 which takes the form of some... more We are interested in the climate model introduced by Sellers in 1969 which takes the form of some nonlinear parabolic equation with a degenerate diffusion coefficient. We investigate here some inverse problem issue that consists in recovering the so-called insolation function. We not only solve the uniqueness question but also provide some strong stability result, more precisely unconditional Lipschitz stability in the spirit of the well-known result by Imanuvilov and Yamamoto (1998) [22]. The main novelties rely in the fact that the considered model is degenerate and above all nonlinear. Indeed we provide here one of the first result of Lipschitz stability in a nonlinear case.
Siam Journal on Control and Optimization, 2009
Abstract. We study the exact controllability of a fluid-structure model. The fluctuations of flui... more Abstract. We study the exact controllability of a fluid-structure model. The fluctuations of fluid velocity and pressure in a domain Ω are described by a potential φ, and the structure is a membrane located in a part Γs of the boundary Γ=∂ Ω of the domain Ω. The potential φ ...
ESAIM: Control, Optimisation and Calculus of Variations, Jun 15, 2004
Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study ... more Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically R+ or R N. Considering an unbounded and disconnected control region of the form ω := ∪nωn, we prove two null controllability results: under some technical assumption on the control parts ωn, we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2 (Ω) can be controlled to zero using control functions in a weighted L 2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.
Advances in Differential Equations, 2005
We prove null controllability results for the degenerate onedimensional heat equation ut − (x α u... more We prove null controllability results for the degenerate onedimensional heat equation ut − (x α ux)x = fχω, x ∈ (0, 1), t ∈ (0, T). As a consequence, we obtain null controllability results for a Croccotype equation that describes the velocity field of a laminar flow on a flat plate.
HAL (Le Centre pour la Communication Scientifique Directe), 2003
... ENS Cachan, Antenne de Bretagne, Campus de Ker Lann, 35 170 Bruz [email protected]... more ... ENS Cachan, Antenne de Bretagne, Campus de Ker Lann, 35 170 Bruz [email protected] Judith Vancostenoble Laboratoire ... Different counterexamples with additional pathological prop-erties were also constructed by Garay [5J, [6J: see also the survey paper [9J ...
ESAIM: Control, Optimisation and Calculus of Variations, 2020
We consider the typical one-dimensional strongly degenerate parabolic operator P u = ut − (x α ux... more We consider the typical one-dimensional strongly degenerate parabolic operator P u = ut − (x α ux)x with 0 < x < and α ∈ (0, 2), controlled either by a boundary control acting at x = , or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as e C/((2−α) 2 T) , when α → 2 − and/or T → 0 +. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).
Evolution Equations and Control Theory, 2019
The goal of this paper is to analyze the cost of boundary null controllability for the 1 − D line... more The goal of this paper is to analyze the cost of boundary null controllability for the 1 − D linear heat equation with the so-called inverse square potential: ut − uxx − µ x 2 u = 0, x ∈ (0, 1), t ∈ (0, T), where µ is a real parameter such that µ ≤ 1/4. Since the works by Baras and Goldstein [4, 5], it is known that such problems are well-posed for any µ ≤ 1/4 (the constant appearing in the Hardy inequality) whereas instantaneous blowup may occur when µ > 1/4. For any µ ≤ 1/4, it has been proved in [52] (via Carleman estimates) that the equation can be controlled (in any time T > 0) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at x = 1. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters T > 0 and µ ∈ (−∞, 1/4]. Our proofs are based on the moment method and very recent results on biorthogonal sequences.
Control and Cybernetics, 2008
Page 1. Control and Cybernetics vol. 37 (2008) No. 4 Generation of analytic semi-groups in L2 for... more Page 1. Control and Cybernetics vol. 37 (2008) No. 4 Generation of analytic semi-groups in L2 for a class of second order degenerate elliptic operators∗ by Piermarco Cannarsa1, Dario Rocchetti1 and Judith Vancostenoble2 ...
Mathematical Control and Related Fields, 2017
We consider the one-dimensional degenerate parabolic equation ut − (x α ux)x = 0 x ∈ (0, 1), t ∈ ... more We consider the one-dimensional degenerate parabolic equation ut − (x α ux)x = 0 x ∈ (0, 1), t ∈ (0, T),
Proceedings in applied mathematics & mechanics, Dec 1, 2007
Motivated by a boundary layer problem, we are interested in the controllability properties of par... more Motivated by a boundary layer problem, we are interested in the controllability properties of parabolic equations degenerating at the boundary of the space domain.We derive new Carleman estimates for a class of degenerate parabolic equation; the proof is based in particular on Hardytype inequalities. Then we deduce observability and null controllability results. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
ESAIM, Apr 1, 2007
We study the exact controllability of a fluid-structure model. The fluctuations of velocity and p... more We study the exact controllability of a fluid-structure model. The fluctuations of velocity and pressure in the fluid are described by a potential, and the structure is a membrane located in a part Γ s of the boundary of the domain Ω. The potential φ and the transverse displacement z satisfy a coupled system of two wave equations, one in the domain Ω × (0, T), the other one in the boundary Γ s × (0, T). Taking two boundary controls, the first one in a boundary condition satisfied by the potential, and the second one in a boundary condition of the structure equation, we identify the space of controllable initial conditions when the geometrical controllability conditions are satisfied. As in the case of the so-called Helmholtz fluid-structure model [10], the difficulty in the treatement of the observability inequalities, in the definition of very weak solutions, and in the proof of controllability result, comes from the coupling terms of the system. To overcome these difficulties, we show that the variant introduced in [10] of the classical Hilbert Uniqueness Method can be adapted to the aeroacoustic model we consider.
Comptes rendus de l'Académie des sciences, Nov 1, 2001
Reçu le 2 juillet 2001, accepté le 13 septembre 2001) Résumé. On étudie la stabilisation de l'équ... more Reçu le 2 juillet 2001, accepté le 13 septembre 2001) Résumé. On étudie la stabilisation de l'équation des ondes en dimension 1 par un feedback frontière ou localement distribué supposé soit positif-négatif, soit intermittent. On donne également des résultats de contrôlabilité en « temps arbitrairement petit » pour l'équation des ondes soumise à un contrôle localement distribué. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS On-off stabilisation and controllability of the wave equation
Discrete and Continuous Dynamical Systems, 2003
We study the exact controllability of the one dimensional semilinear wave equation by a control a... more We study the exact controllability of the one dimensional semilinear wave equation by a control acting on an open subset $(a,b)$ of the domain $(0,1)$. With the aid of d'Alembert's formula and sidewise energy estimates, we obtain sharp conditions in the space-time support of the control, that coincide with the by now well-known geometric control condition. More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0 $:$= 2 max (a , 1-b)$ and fails if $T < T_0$. We weaken strongly their results: given $T>T_0$, we prove that the control can be chosen so that it is supported only on some special time intervals: they are parts of $(0,T)$, in finite number (depending on $a$ and $b$), and their total length can be arbitrarily small. The only condition is that they have to be "close enough" from each other. If this condition holds, we study the observability cost. If it fails, we prove that exact controllability in time $T$ does not hold, but can still be true in time $T'$ large enough.
Mathematical Models and Methods in Applied Sciences, Apr 1, 2005
We study a model of population dynamics describing pregnancy: our model is composed by an equatio... more We study a model of population dynamics describing pregnancy: our model is composed by an equation describing the evolution of the total population, and an equation describing the evolution of pregnant individuals. These equations are of course coupled: one coupling expresses that the total population varies with the number of born people, and another coupling says that the number of fecundated individuals depends on the total population. We study three models of that type: a linear model without diffusion, a nonlinear model without diffusion and a linear model with diffusion. For these three models, we study precisely the qualitative properties and the asymptotic behavior of the solutions.
ESAIM: Control, Optimisation and Calculus of Variations, 2002
Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study t... more Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback a(t)ut. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0, T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: a(t) = a0 > 0 on (0, T), and a(t) = −b0 < 0 on (T, qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual "optimal time condition" in the locally distributed case. These new inequalities provide also new exact controllability results.
Memoirs of the American Mathematical Society, 2016
Cannarsa, Piermarco, 1957-Global Carleman estimates for degenerate parabolic operators with appli... more Cannarsa, Piermarco, 1957-Global Carleman estimates for degenerate parabolic operators with applications / P. Cannarsa, P. Martinez, J. Vancostenoble. pages cm.-(Memoirs of the American Mathematical Society, ISSN 0065-9266 ; volume 239. number 1133) Includes bibliographical references and index. ISBN 978-1-4704-1496-2 (alk. paper) 1. Elliptic operators. 2. Parabolic operators. 3. Carleman theorem.
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Papers by Judith Vancostenoble