Time-Optimal Trajectories of Generic Control-Affine Systems Have at Worst Iterated Fuller Singularities

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations

Topics in sub-Riemannian geometry

This thesis is concerned with three different problems in sub-Riemannian geometry faced during my PhD. The first one is a problem in differential geometry and is about the local conformal classification of a certain class of sub-Riemannian structures. In the second one we deal with topology, and our main result establish some path-fibration properties for the Endpoint map. In the third and last problem, we begin the development of some variational calculus around critical points of the endpoint map, called abnormal controls, and we estabilish a counterpart of the classical Morse deformation techniques and of the Min-Max variational principle

Third order open mapping theorems and applications to the end-point map

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing

On the smoothness of the value function for affine optimal control problems

We prove that the value function associated with an affine optimal control problem with quadratic cost plus a potential is smooth on an open and dense subset of the interior of its attainable set. The result is obtained by a careful analysis of points of continuity of the value function, without assuming any condition on singular minimizers

: Journées sous-riemanniennes 2018

Let (M, ∆) be a rank-two sub-Riemannian structure on a smooth manifold M, and let x, y be any two points on M. In this talk I will present some recent results concerning the description of the set Ω(y), of all the horizontal curves joining x and y, in the vicinity of a rank-two-nice singular curve γ. This is made possible by the existence of a normal form for the endpoint map F locally around γ, and in turn this result permits to discuss some rather surprising isolation properties of γ among extremal curves. If time permits, we will try to discuss some topological properties of rank-two-nice singular curves, establishing in particular their homotopical visibility. This is a joint work with A. Agrachev and A. Lerario

: Journées sous-riemanniennes 2018

Let (M, ∆) be a rank-two sub-Riemannian structure on a smooth manifold M, and let x, y be any two points on M. In this talk I will present some recent results concerning the description of the set Ω(y), of all the horizontal curves joining x and y, in the vicinity of a rank-two-nice singular curve γ. This is made possible by the existence of a normal form for the endpoint map F locally around γ, and in turn this result permits to discuss some rather surprising isolation properties of γ among extremal curves. If time permits, we will try to discuss some topological properties of rank-two-nice singular curves, establishing in particular their homotopical visibility. This is a joint work with A. Agrachev and A. Lerario

Dwell-time control sets and applications to the stability analysis of linear switched systems

We propose an extension of the theory of control sets to the case of inputs satisfying a dwell-time constraint. Although the class of such inputs is not closed under concatenation, we propose a suitably modified definition of control sets that allows to recover some important properties known in the concatenable case. In particular we apply the control set construction to dwell-time linear switched systems, characterizing their maximal Lyapunov exponent looking only at trajectories whose angular component is periodic. We also use such a construction to characterize supports of invariant measures for random switched systems with dwell-time constraints