Construction of Singular Sets in a Velocity Control Problem with Nonconvex Target

Abstract

Abstract The problem of occurrence of singular sets of solution of the velocity control problem for one class of dynamic problems on a plane with a nonconvex target set is studied. The theoretical apparatus is developed for determining pseudo-vertices of a target set in case when its boundary is a curve with a minimum order of smoothness. Finding pseudo-vertices is a necessary element of the procedure for constructing branches of the singular set of the optimal result function. Necessary conditions for the existence of pseudo-vertices, expressed in terms of one-sided partial limits of differential relations dependent on properties of local diffeomorphisms that determine these singular points, are obtained. Examples of constructing a nonsmooth solution to the velocity control problem are given. The developed procedures for determining the disturbance of smoothness of the solution to the dynamic control problem are also applicable when constructing generalized solutions of Hamilton-Jacobi type equations, as well as when forming a generalized eikonal in geometrical optics.