Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

We discuss the problem of local attainability for finite-dimensional nonlinear control systems with quite general assumption on the target set. Special emphasis is given to control-affine systems with a possibly nontrivial drift term. To this end, we provide some sufficient conditions ensuring local attainability, which involve geometric properties both of the target itself (such as a notion of generalized curvature), and of the Lie algebra associated to the control system. The main technique used is a convenient representation formula for the power expansion of the distance function along the trajectories, made at points sufficiently near to the target set.