Aim of the Conference
Differential
geometry has been the source of an original viewpoint to investigate typical
questions arising in the control framework: controllability, observability, stabilization, optimal control... Such
viewpoint, and the powerful tools developed in its framework, is known as
geometric control theory [1,2]. Geometric control theory, in turn, has shown
its strengths as an approach to study geometric problems, and in particular
sub-Riemannian geometry [3]. The goal of this workshop is to present recent advances
sharing this approach, both theoretical and applicative (with a particular
emphasis to quantum control).
In the conference,
three main lines of research will be presented. They share many common points,
especially for what concerns the techniques used to tackle them.
1. Sub-Riemannian geometry from the
control viewpoint and applications. We will present recent results
concerning sub-Riemannian structures and other generalized Riemannian
structures both from the geometric and the analytic viewpoint, focusing on
diffusive phenomena
and applications.
2. Structural properties in geometric control.
We will present recent contributions to a fundamental problem in geometric
control theory, namely, the characterization of invariant quantities and
equivalence between control systems. In particular, we will include
presentations about the inverse optimal control approach to neurophysiology.
3. Control of quantum dynamics. The
goal of quantum control is to design efficient population transfers between
quantum states. This task is crucial in atomic and molecular physics, with applications
ranging from photochemistry to quantum information, and has attracted
increasing attention among quantum physicists, chemists, computer scientists
and control theorists alike [4]. Depending on the physical system and on the
properties under consideration, a cornucopia of models have been proposed (from
low-dimensional conservative ODEs to dissipative systems of PDEs). The conference
will include geometric control contributions to the theory of quantum control,
both of theoretical, algorithmic, and experimental nature.
References
[1] A. Agrachev, Y. Sachkov. Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences, 87. Control Theory
and Optimization, II. Springer-Verlag, Berlin, 2004.
[2] V. Jurdjevic. Optimal control and geometry: integrable systems. Cambridge Studies in Advanced
Mathematics, 154. Cambridge University Press, Cambridge, 2016. xx+415 pp.
[3] A. Agrachev, D. Barilari, and U. Boscain. A
Comprehensive Introduction to sub-Riemannian Geometry, volume 181 of Cambridge
Studies in Advanced Mathematics. Cambridge University
Press, Cambridge,
2020. xviii+746 pp.
[4] D. D'Alessandro. Introduction to quantum control and dynamics.
Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series.
Chapman & Hall/CRC, Boca Raton, FL, 2008.
xiv+343 pp.