- Project scope
This project is devoted to the analysis of several physical process and phenomena modeled by Partial Differential Equations (PDE) and more precisely coupled systems of PDE. Coupled PDE often arise in the modeling of complex phenomena where each component of the system possesses different behaviors, which are, in turn, modeled by a different PDE. The interaction between each component of the system is reflected mathematically by the coupling term of PDE. Complex systems modeled by coupled PDE include applications in biology, engineering, fluid mechanics, aeronautics, chemistry and medicine. Hence, knowing whether it is possible to control or not this type of systems, and how to build controls, is crucial for a better understanding of the world we live in.
From a mathematical point of view, we will study several qualitative properties for these problems, such as controllability, observability, unique continuation, asymptotic behavior, uniqueness, stability, etc. The techniques for uncoupled equation are now quite standard but their transference and application to coupled systems is not straightforward and have shown several unexpected behaviors and so the development of new tools to treat such problems are also of great interest from the point of view of applications.
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Abstract
This project is devoted to the analysis of several physical process and phenomena modeled by Partial Differential Equations. In particular, the problems to be addressed are of great interest in biology, engineering, fluid mechanics, aeronautics, chemistry and medicine. From a mathematical point of view, we will study several qualitative properties for these problems, such as controllability, observability, unique continuation, asymptotic behavior, uniqueness, stability, etc.
Also, this project contributes to create and consolidate strong research groups at an international level in the fields of Control and Inverse problems in France, Brazil and Chile.
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Project Goals
The main goal of this project is the creation and consolidation of a high-level network of collaboration in the field of Partial Differential Equation focused on Control and Inverse Problems between young researchers from Brazil, Chile and France.
From a mathematical point of view, the aim is to obtain a better understanding of the behavior of several phenomena modeled by Partial Differential Equations, with focus on Controllability and Inverse problems for such models. In particular, we aim to develop new tools and methods to treat the problems proposed, with special attention to those coming from real applications.
In summary, our main scientific goals are:
º To analyze controllability properties for fluid-structure interaction systems that couple hyperbolic and a parabolic systems with a free boundary.
º Studying the controllability of cardiac models and in particular the bidomain model that involves coupling terms with time derivatives.
º To use a switching strategy in order to improve the controllability properties of parabolic systems.
º To develop numerical studies for the inverse obstacle problem and in particular detecting structures immersed into a fluid.
º To study the controllability limits in some elasticity models when one of the speed of propagation goes to infinity.
º To investigate the optimal shape and location of sensors for thermoelastic models