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

Some controllability results for linear viscoelastic fluids

We analyze the controllability properties of systems which provide a description, at first approximation, of a kind of viscoelastic fluid. We consider linear Maxwell fluids. First, we establish the large time approximate-finite dimensional controllability of the system, with distributed or boundary controls supported by arbitrary small sets. Then, we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets. The proofs of these results rely on classical arguments. In particular, the approximate controllability result is implied by appropriate unique continuation properties, while exact controllability is a consequence of observability (inverse) inequalities. We also discuss questions concerning the controllability of viscoelastic fluids and some related open problems.Ministerio de Ciencia e Innovació

A geometric inverse problem for the Boussinesq system

In this work we present some results for the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Boussinesq equations. First, we establish a uniqueness result. Then, we show the way the observation depends on perturbations of the rigid body and we deduce some consequences. Finally, we present a new method for the partial identification of the body assuming that it can be deformed only through fields that, in some sense, are finite dimensional. In the proofs, we use various techniques, related to Carleman estimates, differentiation with respect to domains, data assimilation and controllability of PDEs.Dirección General de Enseñanza SuperiorComisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile

Control de EDPs orientado a la terapia de un tumor cerebral

Consideramos un sistema de ecuaciones en derivadas parciales que modela los efectos de una terapia sobre un tumor cerebral (glioblastoma). La densidad de células tumorales verifica una EDP parabólica semilineal, que está acoplada con otra EDP similar para un agente citotóxico. El control es distribuido, su soporte es un pequeño subdominio del abierto que representa el cerebro y actúa a través del segundo miembro de la ecuación para la densidad de anticuerpos. Presentamos resultados de control para este modelo, así como algunas experiencias numéricas

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

Análisis y control de algunas EDP no lineales con origen en mecánica

"En esta Memoria, analizaremos diversos problemas de controlabilidad relacionados con ecuaciones en derivadas parciales (EDP) de evolución de tipo parabólico o hiperbólico, motivadas por problemas con origen en Mecánica"

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