We first introduce the notion of commutative vector fields and the related infinitesimal object, the Lie bracket. As an instance of application of the interplay between non commutativity and controllability we present a new result for the problem of Global Asymptotic Controllability for control systems (Control systems are O.D.E.s depending on a infinite-dimentional parameter: the control). More specifically, we will consider Control Lyapunov Functions, which in general are not smooth, essentially because of a lack of adequate controllability. To pave the way towards an augmented regularity, we embed the standard dissipative relation in a differential inequality (DI) involving Hamiltonians built from the iterated Lie brackets of the dynamical vector fields. Actually, the solutions of (DI), besides yielding reachability of the target, can likely be expected much more regular than standard Lyapunov functions.
*Texto informado pelo autor.
Franco Rampazzo received his Ph.D. in Mathematics at the International School of Advanced Studies (SISSA) in Trieste. He is currently Full Professor in Mathematical Analysis at the Department of Mathematics of the University of Padova, Italy, where, besides an intense teaching, he has trained several young researchers in the scientific activity. He is the author of about fifty scientific papers, all published in major mathematical journals. He was visiting professor in various scientific institutions, e.g. Rutgers University(NJ), Universitè Paris IX, Université de Brest (France), Imperial College (London), Penn State University (Pa), and engaged many collaborations with their members. His main scientific fields of interest include Analytical and Geometric Control Theory, Hamilton Jacobi equations and Classical Mechanics.