Mathematics

Given a $n$-dimensional manifold $M$, we study the time-optimal problem for the following system: $$\dot{q}=f_0(q)+\sum^k_{i=1}u_if_i(q)$$ where $q\in M$, $k < n$, $f_0,f_1,...,f_k$ are smooth vector fields, and $u=(u_1,...,u_k)$ is an admissible control taking values in a $k$-dimensional closed ball. We analyse local regularity of time-optimal controls and trajectories for such a n-dimensional affine control system. In the case of $k =n-1$, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities. This is a joint work with A. Agrachev.