01167nas a2200109   4500008004100000245010100041210006900142260001300211520077600224100002101000856003601021       2005                        en  d00aRegularity properties of optimal trajectories of single-input control systems in dimension three0 aRegularity properties of optimal trajectories of singleinput con  bSpringer3 aLet q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.1 aSigalotti, Mario  uhttp://hdl.handle.net/1963/4794