A bound on the iteration order of Fuller singularities for generic control-affine time-optimal systems

It is known that a time-optimal control-affine system can have any kind of singularity. It is also known that the Fuller phenomenon (i.e., the accumulation of switching times) occurs generically for extremal trajectories of control-affine time-optimal systems if the dimension of the manifold is large enough and the control takes values in a polytope. In this talk we consider the case where the control takes values in a m-dimensional ball. We present recent results stating that, for generic systems with m=1 or m even, the control u associated with an extremal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on m and the dimension of the ambient manifold, 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. The talk is based on joint works with F. Boarotto and Y. Chitour.