Some open problems

We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry. It is getting harder to prove theorems and easier to force other people to prove them when you are sixty. Some colleagues asked me to describe interesting open problems in geometric control and sub-Riemannian geometry. Here I list few really challenging problems; some of them are open for a long time and were publicly or privately stated by well-known experts: J.-M. Coron, I. Kupka, R. Montgomery, B. Shapiro, H. Sussmann, and others. I. Singularities of time-optimal trajectories. Let f, g be a pair of smooth (i. e. C∞) vector fields on a n-dimensional manifold M . We study time-optimal trajectories for the system q̇ = f(q) + ug(q), |u| ≤ 1, q ∈M, with fixed endpoint. Admissible controls are just measurable functions and admissible trajectories are Lipschitz curves in M . We can expect more regularity from time-optimal trajectories imposing reasonable conditions on the pair of vector fields. A. (f, g) is a generic pair of vector fields. Optimal trajectories cannot be all smooth; are they piecewise smooth? This is true for n = 2. More precisely, if dimM = 2, then any point of M has a neighborhood such that any contained in the neighborhood time-optimal trajectory is piecewise smooth with atmost 1 switching point (see [32, 11]). According to the control ∗SISSA, Trieste & Steklov Math. Inst., Moscow