• GeCoMethods: Geometric Control Methods for the Heat and Schroedinger Equations

Principal investigator: Ugo Boscain (Ecole Polytechnique, Palaiseau) 
Summary of the objectives of the project:
The aim of this project of 5 years is to create a research group on geometric control methods in PDEs with the arrival of the PI at the CNRS Laboratoire CMAP (Centre de Mathematiques Appliquees) of the Ecole Polytechnique in Paris (in January 09). With the ERCStarting Grant, the PI plans to hire 4 postdoc fellows, 2 PhD students and also to organize advanced research schools and workshops. One of the main purpose of this project is to facilitate the collaboration with my research group which is quite spread across France and Italy.
The PI plans to develop a research group studying certain PDEs for which geometric control techniques open new horizons. More precisely the PI plans to exploit the relation between the subRiemannian distance and the properties of the kernel of the corresponding hypoelliptic heat equation and to study controllability properties of the Schroedinger equation.
In the last years the PI has developed a net of international collaborations and, together with his collaborators and PhD students, has obtained a certain number of results via a mixed combination of geometric methods in control (Hamiltonian methods, Lie group techniques, conjugate point theory, singularity theory etc.) and noncommutative Fourier analysis.
This has allowed to solve open problems in the field, e.g., the definition of an intrinsic hypoelliptic Laplacian, the explicit construction of the hypoelliptic heat kernel for the most important 3D Lie groups, and the proof of the controllability of the bilinear Schroedinger equation with discrete spectrum, under some ``generic'' assumptions. Many more related questions are still open and the scope of this project is to tackle them.
All subjects studied in this project have real applications: the problem of controllability of the Schroedinger equation has direct applications in Laser spectroscopy and in Nuclear Magnetic Resonance; the problem of nonisotropic diffusion has applications in cognitive neuroscience (in particular for models of human vision). 
Website: http://www.cmapx.polytechnique.fr/~boscain/gecomethods.html 

 
• CPDENL: Control of partial differential equations and nonlinearity.

Principal investigator: JeanMichel Coron (Université Pierre et Marie Curie, Paris) 
Summary of the objectives of the project:
With the ERC grant, we plan to hire postdoc fellows and PhD students,
to offer 1month positions to confirmed researchers, to organize a
regular seminar and two workshops.
A lot is known on the controllability and stabilization of finite
dimensional control systems and linear control systems modeled by
partial differential equations. Much less is known for nonlinear control
systems modeled by partial differential equations. In particular, in
many important cases, one does not know how to use the classical
iterated Lie brackets which are so useful to deal with nonlinear control
systems in finite dimension. In this project, we plan to develop methods
to deal with the problems of controllability and of stabilization for
nonlinear systems modeled by partial differential equations, in the case
where the nonlinearity plays a crucial role. This is for example the
case where the linearized control system around the equilibrium of
interest is not controllable or not stabilizable. This is also the case
when the nonlinearity is too big at infinity and one looks for global
results. This is also the case if the nonlinearity contains too many
derivatives. Many natural important and challenging problems are still
open. Precise examples, often coming from physics, are given in this
proposal. 
Website: http://www.ann.jussieu.fr/~coron/ 

 
• NUMERIWAVES: New analytical and numerical methods in wave propagation.

Principal investigator: Enrique Zuazua (Basque Center for Applied Mathematics, Bilbao) 
Summary of the objectives of the project:
This project is aimed at performing a systematic analysis, providing a real breakthough, of the combined effect of wave propagation and numerical discretizations, in order to help in the development of efficient numerical methods mimicking the qualitative properties of continuous waves. This is an important issue for its many applications: irrigation channels, flexible multi structures, aeronautic optimal design, acoustic noise reduction, electromagnetism, water waves, nonlinear optics, nanomechanics, etc.
The superposition of the present state of the art in Partial Differential Equations (PDE) and Numerical Analysis is insufficient to understand the spurious high frequency numerical solutions that the interaction of wave propagation and numerical discretizations generates. There are some fundamental questions, as, for instance, dispersive properties, unique continuation, control and inverse problems, which are by now well understood in the context of PDE through the celebrated Strichartz and Carleman inequalities, but which are unsolved and badly understood for numerical approximation schemes.
The aim of this project is to systematically address some of these issues, developing new analytical and numerical tools, which require new significant developments, much beyond the frontiers of classical numerical analysis, to incorporate ideas and tools from Microlocal and Harmonic Analysis.
The research to be developed in this project will provide new analytical tools and numerical schemes. Simultaneously, it will contribute to significant progress in some applied fields in which the issues under consideration play a key role.
In parallel with the analytical and numerical analysis of these problems, a mathematical simulation platform will be set to perform computer simulations and explore and visualize some of the most relevant and complex phenomena.

Website: http://www.bcamath.org/projects/NUMERIWAVES/ 

 