Complexity of control-affine motion planning
Résumé
In this paper we study the complexity of the motion planning problem for control- affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time- rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities.
As a byproduct, we are able to treat the long time local controllability problem, giving quanti- tative estimates on the cost of stabilizing the system near a non-equilibrium point of the drift.