Nonuniqueness set and the problem of time-optimal motions in a velocity field

We study the problem of time-optimal transfer to the origin for motions described by the law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot x = v(x) + u(t)$$\end{document}, where v: ℝ2 → ℝ2 is a smooth vector field and the control u satisfies the inequality ‖u(t)‖ ≤ u0. We introduce the notion of nonuniqueness set for problems with continuous optimal controls. By using the Pontryagin maximum principle, we single out a family of trajectories that lead to the origin and may be optimal. The nonuniqueness set intersects this family and cuts away the nonoptimal part from each trajectory. We show that the nonuniqueness set for a plane-parallel velocity field with symmetric profile is a ray lying on the symmetry axis. In the case of a nonsymmetric profile, we construct a Cauchy problem whose trajectory is the nonuniqueness set.