On the structure of the singularity of the solution to the time-optimal control problem in the case of a discontinuity in the curvature of the boundary of the target set

Abstract

A time-optimal control problem in two-dimensional space with a spherical velocity vectogram and a non-convex target set with a boundary having a finite number of points of discontinuity in the curvature is considered. Identification and construction of scattering curves that form a singular set of the optimal result function are studied for the case when the points of discontinuity in the curvature have one-sided curvatures of different sign. It is shown that these points belong to pseudo-vertices that are characteristic points of the target set boundary responsible for generation of singular set branches. The structure of scattering curves and optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed. It is proved that one pseudo-vertex can generate two different singular set branches. The results obtained are illustrated by an example of solving the control problem for the target set as a one-dimensional manifold.