# Geometric control methods for heat and Schroedinger equations

## Final Report Summary - GECOMETHODS (Geometric control methods for heat and Schroedinger equations)

GECOMETHODS is a project in which techniques of geometric control theory have been applied to study problems of geometric analysis and of control of quantum mechanical systems.

In the first part of the project we studied the heat diffusion on sub-Riemannian manifolds. The birth of the subject goes back to Carateodory’s 1909 seminal paper on the foundations of Carnot thermodynamics, followed by E. Cartan’s 1928 address at the International Congress of Mathematicians in Bologna. Sub-Riemannian manifolds model media with a constrained dynamics: motion at any point is only allowed along a limited set of directions (called horizontal ones), which are prescribed by the physical problem at hand. When the set of horizontal directions coincides with the whole tangent space, we obtain Riemannian manifolds. In applications, sub-Riemannian geometry appears in the study of many mechanical problems (robotics, vehicles with trailers, etc.) and recently in new fields of research such as mathematical models of human behaviour, quantum control or motion of self-propelled microscopic organisms.

On such a structure we have studied how to define intrinsically the heat diffusion and we have studied the relation between geometric properties (the sub-Riemannian distance, the volume, the curvature) and the heat diffusion. Such sub-Riemannian diffusions are today used to model the flow of information in the visual cortex of mammals and provide an exceptional tool for algorithms of image reconstruction.

In the second part of the project, we have studied problems of control of quantum mechanical systems. In particular we studied how it is possible to induce a jump in the energy level of a molecule or on a spin system by means of some external fields. Such problems are for instance encountered in medical imaging (in particular in MRI) and in photochemistry (to induce chemical reactions using electromagnetic fields). We developed a technique to prove, under suitable conditions, that the Schroedinger equation is approximately controllable, i.e. that for every pair of states there exists a control (an external field) that makes the system evolving from one state arbitrarily close to the other one. Also we proved that the conditions under which this result holds are "generic" in the sense that they are satisfied by "most" of the systems. We applied these results to the problem of controlling the rotational degrees of freedom of a molecule and to the problem of controlling a two-level system in interaction with a distinguished mode of a quantized bosonic field.

The project permitted to finance PhD students and post doc and to finance travels for collaborations, schools and conferences. The project was very successful. We published more than 40 articles in peer reviewed journals, more than 20 proceedings of international conference and three books. Two of them contain the lectures notes of the courses given at the trimester "Geometry, Analysis and Dynamics on sub-Riemannian manifolds" held at the Institut Henri Poincare in Paris in 2014, that was partially financed by GECOMETHODS.
One of the most important success of GECOMETHODS is that 6 people involved in the project as post-docs or PhD students got a permanent academic position.