Title
Lie brackets with low regularity: from commutativity to local controllability
Abstract
Commutativity of the integral trajectories of two vector fields f, g is a trivial property for constant vector fields. However, it is well-known that such a property is verified also for non-constant vector fields as soon as their Lie bracket [f,g](x):= Dg(x)f(x) – Df(x)g(x) vanishes identically. On the other hand, Local Controllability for a non-linear control system (linear in the controls) consists in the fact that, for every (sufficiently small) t>0, the points reached at some time s£t by the trajectories of the system starting from a point x* form a full neighborhood of x*. Rashevskii-Chow’s theorem states that the so-called rank-condition, namely the fact that iterated Lie brackets of vector fields of the system generate the whole tangent space, is sufficient for local controllability. These results are classically obtained under strong hypotheses of regularity. In this talk, it will be illustrated how these theorems (and other similar ones) can be extended to a non-smooth setting by means of a generalized notion of Lie bracket. Besides being desirable for technological applications, such extensions might constitute a starting point for a non-smooth version of sub-Riemannian geometry.
Bio
Franco Rampazzo obtained his PhD at SISSA (Trieste, Italy) in 1989 under the supervision of Prof. A. Bressan and Prof. A. Cellina. He is currently Full Professor at the University of Padova. His early scientific work was devoted to Control Theory with unbounded and impulsive controls, together with analytical and geometrical aspects of applications to Classical Mechanics. Later on, he addressed topics in Hamilton-Jacobi theory for both state-constrained control systems and differential games. Further researches concerned interactions between non-smooth analysis and a differential geometric approach to system theory. He has also exploited a set-separation approach to highlight the geometric contents of infimum gap issues. He has also worked on higher order controllability and stabilizability of contol -affine systems. Furthermore, he has investigated problems with state constraints, in particular high order conditions for the continuity of the value functions. He has supervised several Master theses and some post-doc positions, and he authored several scientific papers, most of which published on highly ranked journals. He has been Visiting Scholar in several foreign scientific institutions in the USA, UK, Germany, and France.