Time-optimality of nonsmooth singular trajectories of affine single-input systems in \(\mathbb{R}^3\) (Q5948400)

The paper deals with the issues of existence and optimality of nonsmooth singular trajectories of single input smooth control systems in \(\mathbb{R}^{3}\) that satisfy certain Lie-bracket conditions at a given point, \(q_{0}\). The results are based on normalizations of the system using semicovariants under the feedback pseudogroup. The cases when there exists a nonsmooth singular trajectory through \(q_{0}\) are characterized. The corresponding singular control is discontinuous but bounded. Such trajectories are not strongly locally optimal beyond \(q_{0}\). The issue whether the singular trajectory is still strongly locally optimal at \(q_{0}\) is addressed for the case when the admissible controls are required to be uniformly bounded. In this case, the singular trajectory is time-optimal if the bound is sufficiently small and is not time-optimal if the required bound is large. It is shown that the curvature associated with the Jacobi equation is positive and tends to infinity near \(q_{0}\), though the singular trajectory is weakly locally optimal at \(q_{0}\).