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1. Project scope
   We want to study, from a theoretical viewpoint, some control and inverse properties for PDE. We are mainly concerned with properties such that controllability, stabilization and identification of both coefficients and source terms. We will study systems with different behavior as dispersive equations (Korteweg-de Vries and Schrödinger equations), fluid-structure systems, transport equations, etc.
   
   
2. Abstract
   Controllability and inverse problems for PDEs are strongly related topics. The principal objective of this project is to connect research groups in inverse and controllability theory working in France, Brazil and Chile. The main idea is to share know-how in both disciplines to deal with the study of two representative models: Schrödinger and Korteweg-de Vries (KdV) equations. More specifically, in this project, our aim is to study: 1. Control and inverse problems for the Korteweg-de Vries equation, including: existence of a minimal time for controllability, stabilization of both linear and nonlinear systems, bilinear control systems, source identification inverse problems, and numerical ill posed problems. 2. Control and inverse problems for the Schrödinger equation, including: sharp estimates of the minimal time for controllability, 2-D or 3-D moving potential wells, feedback approaches to generate new trajectories, systems of coupled Schrödinger equations, and inverse problems with partial boundary measurements. 3. Other controllability and inverse problems: fluid-structure problems, Lagrangian controllability, and transport dominated problems.
   
   
3. Main Goals
   More specifically, our aim is to study: 1. Control and inverse problems for the Korteweg-de Vries (KdV) equation: a very interesting open problem in controllability of KdV equations is the existence of a minimal time for controllability in the nonlinear case. Other open problem is the study of the stabilization of both linear and nonlinear KdV systems. Both problems appear also for bilinear KdV systems. From the point of view of inverse problems, there is the problem of source identification, where, in the simplest case, the source can depend on the spatial variable only and the design of numerical methods for source recovering. 2. Control and inverse problems for the Schrödinger equation: in the case of moving potential wells modeled by bilinear Schrödinger systems, it is well known that there exists a minimal controllability time, but sharp estimates of the minimal control time are unknown. In the same system, and for seek of robustness, one can try to generate interesting trajectories using a feedback approach. Regarding inverse problems for Schrödinger equations, we are interested in some source recovering problems from partial boundary measurements and the extension to systems of coupled Schrödinger equations. 3. Other controllability and inverse problems: we are also interested in other controllability and inverse problems in fluid-structure interaction, Lagrangian controllability, and transport dominated problems.