## Section: New Results

### Analysis of singularities in minimum time control problems

Participants : Jean-Baptiste Caillau, Jacques Féjoz [Université Paris-Dauphine & Observatoire de Paris] , Michaël Orieux [SISSA] , Robert Roussarie [Université de Bourgogne-Franche Comté] .

An important class of problems is affine control problems with control on the disk (or
the Euclidean ball, in higher dimensions). Such problems show up for instance in space
mechanics and have been quite extensively studied from the mathematical (geometric)
and numerical point of view. Still, even for the simplest cost, namely time
minimization, the analysis of singularities occuring was more or less open. Building
on previous results of the team and on recent studies of Agrachev and his
collaborators, we give a detailed account of the behaviour of minimum time extremals
crossing the so-called singular locus (typically a switching surface). The result is
twofold. First, we show that there the set of initial conditions of the Hamiltonian
flow can be stratified, and that the flow is smooth on each stratum, one of them being
the codimension stratum leading to the singular locus. This generalizes in higher
codimension the known case of switching conditions of codimension one encountered, for
instance, in ${\mathrm{L}}^{1}$-minimization (consumption minimization, in aerospace
applications). We give a clear geometric interpretation of this first result in terms
of normally hyperbolic invariant manifold.
Secondly, we provide a model for the singularity on the flow when strata are crossed,
proving that it is of logarithmic type. This paves the way for *ad hoc* numerical
methods to treat this kind of extremal flow. The crucial tool for the analysis is a
combination of blow-up and normal form techniques for dynamical systems.