Séminaire d'Automatique du plateau de Saclay : Time-extremal trajectories of generic control-affine systems have at most finite-order Fuller singularities

Séminaire le 23 Mars 2017, 10h00 à CentraleSupelec (Gif-sur-Yvette) Salle des séminaires du L2S
Francesco Boarotto (CMAP, Ecole Polytechnique)

Let $M$ be a smooth connected $n$-dimensional manifold, and consider on it the control-affine system $\dot{q}=f_0(q)+uf_1(q),\quad u\in[-1,1].$ Time-extremal trajectories for the time-optimal control problem associated to this system are driven by controls $u$, whose set $\Sigma$ of discontinuities is possibly stratified as follows: $\Sigma_0$ is the set of isolated points in $\Sigma$ (switching times) and, recursively, the $k$-th order Fuller times $\Sigma_k$ are found as the isolated points of $\Sigma\setminus\left(\bigcup_{j=0}^{k-1}\Sigma_j\right)$.

In this talk we show that, in fact, for the generic choice of the pair $(f_0,f_1),$ there exists an integer $N>0$ such that the control $u$ associated to any time-extremal trajectory admits at most Fuller times of order $N$. In particular, $u$ is smooth out of a set of measure zero. This is a joint work with M. Sigalotti (CMAP, Ecole Polytechnique).

Bio: Francesco Boarotto was born in Verona, Italie, in 1988. He received the Master's degree in mathematics from the University of Padou, Italie, in 2012 and the Ph.D degree from SISSA, Trieste, Italie, in 2016. Since then he has been post-doc in CMAP - Ecole Polytechnique. His research interests include geometric control theory and sub-Riemannian geometry.