Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficientes

The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y| log3/2(1 + |y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.Ministerio de Educación y CienciaFondo Nacional de Desarrollo Científico y Tecnológico (Chile

Controls insensitizing the observation of a quasi-geostrophic ocean model

We consider a linear quasi-geostrophic ocean model with partially known initial conditions. We search for controls that make the observation locally insensitive to the perturbations of the initial data. Their existence is equivalent to the null controllability property for an associated cascade Stokes-like system. Thanks to the presence of the Coriolis term, we are able to prove the existence of such controls. Our strategy is the following. First, we prove a unique continuation property for the adjoint of the state system that leads to approximate controllability; then, under certain assumptions, an observability inequality is established for the adjoint. The proof is inspired by the arguments leading to the unique continuation property. This inequality leads to the desired null controllability result.Ministerio de Educación y CienciaFondo de Financiamiento de Centros de Investigación en Áreas PrioritariasComisión Nacional de Investigación Científica y Tecnológic

Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties.Dirección General de Enseñanza SuperiorFondo de Financiamiento de Centros de Investigación en Áreas Prioritarias (Chile)Comisión Nacional de Investigación Científica y Tecnológica (Chile

Variants of global Carleman weights in one-measurement inverse problems and fluid-structure controllability problems

We review some recent results on variants of global Carleman weights and Carleman inequalities applied to singular controllability and inverse problems partially developed in collaboration with the authors in a series of papers. First of all, we explain how we can modify weights to study one measurement inverse problems for the heat and wave equations with discontinuous coefficients in the principal part, in a case of locally supported boundary observations for recovering coefficients in the wave equation and we mention also some recent results for the Sch¨odinger equation. As another important application, we show how time-variable global Carleman weights are applied to study the null- controllability for a Navier-Stokes-rigid solid problem in moving domains

Application d'une inégalité de Carleman globale avec des poids à direction variable à un problème inverse pour l'équation des ondes

We establish geometrical conditions for the inverse problem of determining a stationary potential in the wave equation with Dirichlet data from a Neumann measurement on a suitable part of the boundary. We present the stability results when we measure on a part of the boundary satisfying a rotated exit condition. The proofs rely on global Carleman estimates with angle type dependence in the weight functions.On établit des conditions geométriques pour le problème inverse consistant à déterminer un potentiel stationnaire dans l'équation des ondes avec une donnée de Dirichlet et à partir d'une mesure de Neumann sur une partie appropriée de la frontière. On présente des résultats de stabilité quand on mesure sur une partie de la frontière satisfaisant une condition sortante à directions variables. Les démonstrations réposent sur des inégalités de Carleman globales avec des fonctions poids qui dépendent d'un angle comme paramètre.Dirección General de Enseñanza SuperiorFondo de Financiamiento de Centros de Investigación en Áreas Prioritarias (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile)Institut National de Recherche en Informatique et en AutomatiqueComisión Nacional de Investigación Científica y Tecnológica (Chile

Potential reconstruction for a class of hyperbolic systems from incomplete measurements

In this article, we study the reconstruction of spatially dependent potentials in n coupled hyperbolic equations in cascade from n - 1 components of the solution of the system. More precisely, we prove local uniqueness and Lipschitz stability for this inverse problem. The main tool is a Carleman estimate for a cascade system with missing observations.FONDECYT 3150089 1151512 CONICYT 2115066

Carleman inequalities and inverse problems for the Schrodinger equation

In this Note, we derive new Carleman inequalities for the evolution Schrodinger equation under a weak pseudoconvexity condition, which allows us to use weights with a linear spatial dependence. As a result, less restrictive boundary or internal observation regions may be used to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrodinger equation from a single boundary or internal measurement, respectively.This work was performed while the third author was visiting the CMM, UMI CNRS 2807.The second author was partially supported by FONDECYT grant 1061263. The first and second authors acknowledge ECOS C04E08 grant

A source reconstruction algorithm for the Stokes system from incomplete velocity measurements

We consider the inverse problem of determining the spatial dependence of a source of the form f (x) sigma (t) in the Stokes system defined in Omega x (0, T), assuming that sigma (t) is known and f (x) is divergence-free. The only available observation is a single internal measurement of the velocity and the acceleration, for which one of its components is missing. Under adequate hypothesis on sigma we prove uniqueness of this inverse problem and we establish an explicit reconstruction formula. This formula provides the spectral coefficients f(k) of the source f in terms of a family of null controls h((tau)) for the corresponding adjoint system indexed by tau is an element of (0, T].DICYT 041633GM CONICYT 2014-21140888 Fondecyt Conicyt 1151512 CONICYT Anillo ACPA ACT1106 MathAmsud 15MATH-02-SOCD

Problèmes inverses et simulations numériques en viscoélasticité 3D

Dans cette thèse, nous abordons plusieurs problèmes mathématiques et numériques relatifs aux équations de la viscoélasticité en trois dimensions. Dans la première partie, nous considérons le système linéaire et nous nous intéressons au problème inverse de récupération d'un coefficient viscoélastique. Pour ce système, nous démontrons une inégalité de Carleman (Chapitre 1) et un résultat de stabilité dans le prolongement unique (Chapitre 2). Nous utilisons ensuite ces résultats pour prouver deux inégalités de stabilité pour le problème inverse, l'une relative à une unique mesure interne et l'autre à une unique mesure sur une partie arbitrairement petite de la frontière (Chapitre 3). Finalement, nous proposons une méthode pour résoudre ce problème numériquement et présentons une application en imagerie médicale (Chapitre 4). Dans la deuxième partie, nous étudions le système de la viscoélasticité non linéaire. Nous présentons des méthodes numériques pour le résoudre et l'implémentation de ces dernières en trois dimensions sur des géométries complexes (Chapitre 5). Une application biomédicale à la simulation des déformations des structures cérébrales est ensuite décrite (Chapitre 6). Enfin, nous abordons une question de modélisation en proposant un modèle couplé viscoélastique/viscoplastique en grandes déformations (Chapitre7).PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Erratum: Cooling in the shade of warped transition disks

This is an Erratum to the paper ‘Cooling in the shade of warped transition disks’ that was published in MNRAS, 486, L58-L62 (2019). The x-axis of Fig. 1 in the original version of the paper is incorrectly labeled, the temperature range of the grid for the Rosseland opacities should start at 10K, not 1K. Here we provide the correct version in Fig. 1 of this Erratum. This is a typographical error that does not impact on the calculations reported in the original paper