Properties of non stationer pseudo vertex with the break of smoothness of the target set boarder curvature in the Dirichlet problem to eikonal type equation

摘要

A number of properties of pseudo-vertices of a boundary value set in the Dirichlet problem to the rst-order PDE of the eikonal type are revealed. Special points of the boundary of the boundary set responsible for the origin of the singularity of the generalized solution of the equation from the corresponding domain the fundamental solution (according to S. N. Kruzhkov) in geometric optics or the minimax solution (according to A. I. Subbotin) in the theory of optimal control, are studied. In this paper, formulas for markers numerical characteristics of pseudovertices are obtained. The formulas are found for the non-stationary case when the smoothness of the curvature of the boundary of the edge set is broken. The necessary conditions for the existence of pseudovertices are also derived in the form of relations generalizing the curvature stationarity conditions. The obtained results are illustrated by the example of building a solution to the velocity control problem.